EVALUATION OF GEOSAT DATA

AND APPLICATION TO VARIABLITY OF

THE NORTHEAST PACIFIC OCEAN

Jeffrey William Campbell

B.S., Oceanography

United States Naval Academy

1982

SUBMnTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

and the

WOODS HOLE OCEANOGRAPHIC INSTITUTION

September, 1988

This thesis is not subject to U.S. copyright.

0

EVALUATION OF GEOS AT DATA

AND APPLICATION TO VARIABLITY OF

THE NORTHEAST PACIFIC OCEAN

by

Jeffrey William Campbell

Submitted to the Massachusetts Institute of Technology -

Woods Hole Oceanographic Institution Joint Program

in September, 1988 in partial fulfillment of the

requirements for the Degree of Master of Science

ABSTRACT

A portion of the northeast Pacific ocean was chosen within which to evaluate and
use altimetric data from the U.S. Navy Geodetic Satellite GEOSAT. The zero-order
accuracy of the major GEOSAT geophysical data record (GDR) channels was verified, and
occasional gaps in the altimeter coverage were noted. GEOSAT'S 17-day repeat orbit
allowed use of collinear-track processing to create profiles of the difference between the sea
surface height along a given satellite repeat, and the mean sea surface height along that
repeat's groundtrack. Detrending of sea surface bias and tilt on each repeat reduced orbit
and other long wavelength errors in the difference profiles.

The corrections provided on the GEOSAT GDR were examined for their effects on
the difference profiles of three test arcs. It was found that only the ocean tide,
electromagnetic bias, and inverted barometer corrections varied enough over the arc lengths
(-4400 km) to have any noticeable effect on the difference profiles. Only the ocean tide
correction was accurate enough to warrant using it to adjust the sea surface heights. The
recommended processing of GEOSAT data for the area included making the ocean tide
correction, three-point block averaging successive sea surface heights, and forming the
mean height profiles from 1 8 repeat cycles (to reduce aliasing of the M2 tidal component).
A set of difference profiles for one GEOSAT arc indicated that a reasonable estimate of
GEOSAT's system precision was -4.5 cm (RMS). The mid wavelength range (100-500
km) of these profiles was found to be the only range in which oceanic mesoscale features
could be separated from altimeter errors.

Mean alongtrack wavenumber spectra of oceanic variability for two GEOSAT arcs
were compared with a SEASAT-derived regional spectrum of Fu (1983). Agreement was
good, with GEOSAT showing less system noise at short wavelengths, and greater oceanic
variability at long wavelengths. The GEOSAT spectra were fit well by a k"1-5 slope at
wavelengths from 100 to 1000 km.

Sea surface temporal variability as a function of location was compared with the
SEASAT results of Cheney et al. (1983). Qualitative agreement was excellent and
quantitative differences were largely accounted for. GEOSAT picked up the variability of
the major current systems of the northeast Pacific, including the Alaskan, Califomian, and
North Equatorial currents. Error bounds on GEOSAT-derived oceanic variability showed
that the effects of uncorrected electromagnetic bias, inverted barometer, and wet
troposphere were significant. Further work in the areas of error modelling, orbit
determination, and geoid calculation were called for.

Thesis Supervisor: Dr. Carl Wunsch

Title: Cecil and Ida Green Professor of Physical Oceanography
Secretary of the Navy Research Professor

TABLE OF CONTENTS

ABSTRACT 2

CHAPTER 1: BACKGROUND 4

1.1 INTRODUCTION TO SATELLITE ALTIMETRY 5

1.2 OCEANOGRAPHY FROM ALTIMETRY 15

1.3 GEOGRAPHY AND TERMINOLOGY 21

1.4 THE GEOPHYSICAL DATA RECORD 22

1.5 SUMMARY 36

CHAPTER 1 FIGURES 38

CHAPTER 1 TABLES 51

CHAPTER 2: EVALUATING THE DATA 53

2.1 COLLINEAR PROCESSING SCHEMES 54

2.2 A COLLINEAR ALGORITHM 55

2.3 THE TEST ARCS 58

2.4 ARC 82 59

2.5 ARC 209 67

2.6 ARC 293 73

2.7 SUCCESSIVE POINT AVERAGING 83

CHAPTER 2 FIGURES 84

CHAPTER 2 TABLES 110

CHAPTER 3: USING THE DATA 114

3.1 OVERVIEW 115

3.2 VARIABILITY ON ARC 293 116

3.3 DATA COMPARISON 126

3.4 ERROR BOUNDS ON GEOSAT VARIABILITY CALCULATIONS ... 135

3.5 SUMMARY AND CONCLUSIONS 137

CHAPTER 3 FIGURES 142

APPENDIX A 156

REFERENCES 158

ACKNOWLEDGEMENTS 163

CHAPTER ONE - BACKGROUND

1.1 INTRODUCTION TO SATELLITE ALTIMETRY

The purpose of this thesis is to evaluate data from the U.S. Navy altimetric satellite
GEOS AT (Geodetic Satellite), and use that data to investigate oceanic variability in the
northeast Pacific Ocean. In a broader sense, however, this thesis also acquaints the reader
with the history, fundamentals, and limitations of satellite altimetry, as well as
oceanographic uses to which altimetric data may be put. Sections 1 . 1 and 1 .2 are directed
to these background topics, while Sections 1.3 and beyond are devoted primarily to the
evaluation and use of the GEOS AT data set.

The primary function of a satellite radar altimeter is to measure the distance between
itself and the surface beneath it. This measurement, when combined with a determination
of the satellite's position, allows the topography of that surface to be mapped. Radar beam
scattering constraints limit the use of altimetric measurements to relatively smooth surfaces
such as flatlands and ice fields. Most commonly, however, satellite altimetry is used to
determine the changing topography of the sea surface, and that is the application to which
this thesis is directed.

To lowest order, the sea surface lies along a gravitational equipotential surface
known as the geoid. This equipotential may be thought of as that surface to which sea level
would correspond if the oceans were at rest with respect to the earth and not subject to any
wind or other forcing. The geoid varies by some ±100 m worldwide, as a result of the
earth's mass distribution; with its short wavelength undulations being primarily a function
of bathymetry, and features of increasing horizontal scale reflecting conditions deeper in the
earth (Stacey, 1977). The departure of the sea surface from the geoid is known as the sea
surface topography and is usually less than 1-2 m in amplitude. Causes of such
topography include the tides*, winds, atmospheric pressure changes, and current systems
with their associated mesoscale features such as eddies (Stewart, 1985). With increased

The tides also modify the geoid itself, as is discussed in the section on
altimeter errors.

accuracy from the altimeter system, increasingly useful information about these features
may be determined. For example, overall system accuracies of 1-2 m allow for a
reasonable estimate of the geoid, while accuracies of several decimeters enable strong
current systems such as the Gulf Stream and Kuroshio (with typical topography of >100
cm) to be tracked (Stewart, 1985). Repeated altimeter measurements of the same area, with
instrument precisions of a few centimeters, may be processed so as to minimize the
uncertainties due to orbit and geoid, and allow for observations of the variable surface
geostrophic currents. These are typically associated with eddies and are manifested in
topography with magnitudes of ten centimeters or more. Chapter 14 of Stewart (1985)
contains a useful description of phenomena which influence the sea surface topography and
is also highly recommended as an introduction to the field of altimetry. His excellent
review of that discipline served as the basis for Section 1 . 1 of this work and uncited
descriptions of altimetric methods found therein are credited to him.

The basic idea of an altimeter is straightforward. The altimeter transmits sharp
radar pulses straight down to the sea surface, and the height of the satellite relative to the
surface is determined from the travel time there and back. If the satellite's position relative
to the center of the earth is also known, the absolute sea surface height with respect to the
earth may be calculated. In addition, the return pulses' amplitude and shape give
information about waves and winds at the sea surface. The former phenomenon is
explored extensively in later passages.

The construction of workable altimeters requires several modifications of the basic
theory, however. For example, while a sharp pulse is necessary to ensure good
measurement precision, a long pulse can carry more radio energy and ensure a higher
signal to noise ratio at the satellite receiver. To satisfy both criteria, altimeters use the
technique of pulse compression (Rihaczek, 1969): The sharp pulse generated by the
transmitter is lengthened by a dispersive filter prior to leaving the satellite, and is inverted to
the original form upon its return.

Determining the arrival time of the return pulse with high accuracy requires an
additional modification. From Fourier transform theory (see Bracewell [19861, Chapter 8),
it can be shown that high time resolution requires a large pulse bandwidth. Unfortunately,
the resolution needed to ensure height precisions of a few cm necessitates a bandwidth
which is unacceptably large (Stewart [1985] links a measurement precision of 1 cm to a
bandwidth of 30 GHz). This problem may be solved with the technique of pulse
averaging, which maintains high measurement precision while allowing the use of
narrowband pulses. The individual returns created by the reflection of each narrowband
pulse off the wavecrests and troughs on the sea surface are plotted for received power as a
function of time, within the tracking circuitry onboard the spacecraft. For a number of
consecutive pulses averaged together, the set of returns will lie generally along a smooth
waveform curve such as those shown in figure 1.1a. To reduce the noise in this curve, a
large number of pulses must be averaged (according to the GEOS AT specifications in
MacArthur et al. [1987], GEOSAT averages together around a hundred pulses for each
height measurement). Travel time is reckoned from the half-power point on the leading
edge of the averaged waveform, and is converted to height using the speed of the pulses.
These calculations are computed onboard the satellite and stored with the spacecraft clock
count.

The final modification of theory which is inherent in satellite altimeter design
involves the extent of the sea surface illuminated by the radar pulse. For the altimeter to
have sufficient horizontal resolution, this footprint size should be kept fairly small, and
antenna theory gives two options. The first is the beam-limited situation whereby a narrow
raaar beam is used. A narrow beam requires a large and expensive antenna, as well as very
good pointing acuracy. The second involves pulse-limiting whereby a fairly wide beam is
used, but the pulses are kept short. With this technique, the returned power from a flat sea
increases linearly with time until the trailing edge of the pulse reaches the water. Then, the
radar footprint becomes a spreading annulus of constant area until the outer rim of the

annulus exceeds the beamwidth. The latter technique is the one used in altimetry, and is
depicted in figure 1.1b.

Altimeter errors

Following Stewart (1985), altimeter errors may be divided into two categories:
those which influence the actual altimeter measurement, and those which affect the
interpretation of those measurements. Within the first category, instrument noise has been
alluded to already. As has been mentioned, the limitation on pulse bandwidth requires the
altimeter to determine the pulse travel time from the sloping front edge of a broad averaged
waveform. This determination introduces error, as the shape of the waveform is uncertain
due to the variability of the individual reflections from which it is determined. The
variability is the main source of the instrument noise, and it increases with increasing
significant wave height.* Instrument noise (often called measurement precision) is random
in nature, of short wavelength (<10's of km), and is at the few cm level (RMS) in current
generation altimeters. Correcting for it is not possible, though averaging successive sea
surface height measurements together is an option for reducing it. This possibility is
explored in the next chapter.

Of the remaining errors which influence the actual altimeter measurement, the most
severe is the orbit error. Orbit error is a result of the calculated satellite position being
different from the true satellite position. While various ground tracking stations fix the
position of the altimeter at isolated spots, the vast majority of orbit determination is from
ephemeris calculations. Sources of ephemeris error are discussed in Section 1.4, but their
effects on the altimeter-derived sea surface topography are easily understood. For
example, if the satellite is actually higher than calculated, the measured distance to the ocean
will be larger than it should be for the alleged position, and the sea surface will be
calculated to be lower than it really is. Alongtrack errors, whether produced by ephemeris

* Significant wave height (S WH) is defined as the average height of the highest third of the
waves at a point, and is about 4 times the RMS wave height (Fedor and Brown, 1982).

or clock error, also result in height error, as the satellite possesses a vertical velocity
component with respect to the ocean surface (mainly due to the ellipticity of the orbit).
Altogether, orbit error accounts for an RMS error in the current GEOSAT sea surface
heights of about 3 m (Born, 1988). Fortunately, the alongtrack error wavelength is very
large; tending to be once per orbit (see Douglas and Goad [1978]), so the orbit error over
short distances on the ocean surface may be greatly reduced by methods discussed later in
this chapter.

Satellite attitude is another factor influencing the altimeter measurement.
Essentially, off-nadir altimeter pointing results in the distance to the surface being
calculated as too large. Current generation altimeters such as GEOSAT are able to correct
for this through waveform analysis, however, because off-nadir pointing causes the slope
of the trailing portion of the return waveform to decrease. This change in slope is
calculated and used to correct the altimeter measurement (MacArthur et al., 1987).

Surface waves influence the altimeter measurement in a number of ways. As
alluded to earlier, the shape of a return pulse is a function of its reflection history off the
peaks and valleys of the wave-modified sea surface. As the significant wave height
increases, the individual reflections will be more spread out in time and the leading edge of
the final waveform will be less steep, making it more difficult to find the half-power point.
In addition, the irregularity of the sea surface caused by waves is a major source of the
instrument noise discussed earlier. Finally, high significant wave height will cause the
spherically expanding radar pulses to intersect the sea surface at more distant radii from the
beam center than would occur with a flat sea. The radar footprint size is thus increased,
and horizontal resolution is decreased. None of these errors are correctable. There are,
however, certain wave-induced biases which may be corrected for, and these will be
investigated in subsequent sections.

The remaining errors which affect the actual altimeter measurement are
environmental in nature and are often termed the delay corrections. This is so because

10

these phenomena slow the radar pulses as they travel to and from the sea surface, delaying
the travel time and thus depressing the calculated topography.* These effects are due to the
gases in the atmosphere (including water vapor), and free electrons in the ionosphere.
While they are described in more detail later in the chapter, the point to note here is that they
may have to be corrected for in order to determine the real sea surface topography. The
word "may" is a key one, because in a number of circumstances, such correction is
unnecessary or even impossible. Chapter 2 is largely devoted to investigating these
circumstances and to determining whether or not the various error corrections should be
applied to the GEOSAT sea surface height data.

The second category of altimeter errors consists of those phenomena which
influence how one interprets the altimeter-derived sea surface topography. These
phenomena are the tides - both solid and ocean. The solid tide essentially biases the surface
topography up or down on long wavelength scales as the earth adjusts its shape to the tidal
potentials induced primarily by the sun and moon. The ocean tide involves similar height
biases, as well as tidal currents and long waves. The height biases are essentially the result
of the ocean's attempt to follow the changes in the earth's geoid height induced by the
apparent motion of the sun and moon. The additional dynamic nature of the ocean tide
makes this a more complicated phenomenon than the solid tide, however (see Section 1.4).

Often, oceanographers treat tidal effects as noise when studying oceanic flow;
modelled tides are removed from the sea surface topography. But, altimeters are also
important tools in improving the tidal models themselves - as evidenced by Cartwright and
Alcock (1981), who used SEAS AT data to map the M2 tide in the north Atlantic; Brown
and Hutchinson (1981), who mapped the M2 tide in the northeast Pacific, and Mazzega

The delay corrections slow the pulses relative to a vacuum. In practice, the mean value of
a delay phenomenon along a satellite track causes the apparent sea surface to be biased
downwards all along the track. As the phenomenon fluctuates alongtrack about its mean
value, so too will the calculated sea surface. Thus the sign of a delay correction is always
the same, but the apparent effect on the observed sea surface can be positive or negative.

1 1

(1983), who determined tides in the Indian Ocean from altimeter data. This thesis seeks,
among other things, to determine the effectiveness of modelled ocean and solid tide
corrections in removing unwanted features from the surface topography, for purposes of
studying the circulation. In this regard, the earth's geoid is considered invariant over all
GEOS AT repeats, as its tide-induced fluctuations are assumed to be separable (and
removable).

An additional source of altimeter system error, though not germaine to this work,
nevertheless bears mention. This involves the coordinate systems used in satellite
altimetry. There are a number of such systems (see Mueller [1981]), including the satellite
tracking coordinates, earth shape and geoid coordinates, and inertial reference coordinates,
among others. These systems do not coincide and thus lead to errors when attempting to
define an "absolute" sea surface height. This is not a problem to lowest order, however,
when one seeks the variability of the sea surface, as variability is relative in nature and not
dependent on the coordinate system in use. Since this thesis will be concentrating on the
latter approach, further discussion of coordinate systems is unnecessary.

Satellite Altimeter Systems

Prior to GEOSAT, three other satellite altimeters were flown: on SKYLAB (1973)
GEOS-3 (1975), and SEASAT (1978). The SKYLAB altimeter was essentially a proof of
concept mission, and with a range resolution of only about 1 m it produced few
oceanographic results.

GEOS-3 was the first dedicated altimetric mission, and its altimeter operated at 13.9
GHz with an 80 MHz bandwidth. Pulse compression was a factor of 30, and pulsewidth
was 12.5 ns (or 3.8 m). The rated range resolution (consisting of all errors except orbital)
was 0.5 m after 0.2 s of data averaging. Significant wave height, calculated from the slope
of the leading edge of the waveform, was rated at ±25% of the actual SWH, for SWH
values from 4-10 m (altimeter specifications from Stewart [1985]).

12

GEOS-3 spawned a number of oceanographic and geophysical advances, as well as
various algorithms and data handling techniques whose refined versions are still in use.
Huang et al. (1978), for example, studied the temporal and spatial changes of the Gulf
Stream front and its associated eddies by using the crossover method on GEOS-3 data (this
method is discussed in Section 1.2). Mather et al. (1980) investigated variable features
south of the Gulf Stream via the collinear method discussed in Douglas and Gaborski
(1979), and expressed confidence in locating eddies larger than 50 cm in magnitude at
scales of 30-100 km with the GEOS-3 data. Marsh et al. (1980) computed the mean sea
surface in a variety of locales using GEOS-3 outputs, while Diamante and Nee (1981)
determined tidal constituents from GEOS-3 derived sea surfaces. Cheney and Marsh
(1981b) mapped mesoscale variability measured by GEOS-3, and others used the data to
improve geoid models (e.g., Rapp [19791, Lerch et al. [1979]).

GEOS-3 was followed by SEASAT; which, along with four other microwave
sensors, flew an improved altimeter similar to GEOSAT's. The SEASAT altimeter
operated at 13.5 GHz with a 320 MHz bandwidth and pulse compression factor of 1000.
Pulsewidth was 3.6 ns (1.1 m); shorter than that of GEOS-3 and thus allowing more
accurate SWH determination. The latter output was rated at ±.5 m or 10% of the actual
SWH (whichever was greater), for SWH values from 1-20 m. Range resolution was
comparable to the current generation GEOSAT, being 0.1 m after 1 s of data averaging
(Stewart, 1985).

SEASAT was launched on 28 June 1978 and failed due to a massive short circuit
on 10 October 1978. Additional down time in the altimeter further reduced the amount of
sea surface height measurements - the final data set consisted only of some 70 days of
altimeter readings. Nevertheless, SEASAT continued to advance the field of satellite
altimetry and increase knowledge of the oceans. Two excellent sources on the evaluation
and results of SEASAT are the SEASAT Special Issues I and II reprinted from the Journal
of Geophysical Research in 1982 and 1983 (Vols. 87 [C5] and 88 [C3], respectively).

13

Some notable results from the SEASAT altimeter include the following: Cheney and Marsh
(1981) observed the Gulf Stream position and movement of a cyclonic ring, and Cheney et
al. (1983) mapped global mesoscale variability from collinear tracks of SEASAT data.
Wavenumber spectra of the oceanic mesoscale were computed by Fu (1983), while Marsh
and Martin (1982) produced global contour maps of the mean sea surface topography. Tai
and Wunsch (1983) mapped the subtropical gyre in the north Pacific; the first direct
measurement of such a feature not dependent on conventional hydrography. Fu and
Chelton (1984) used SEASAT data to provide the first direct evidence for zonal coherence
in the temporal variablity of the Antarctic Circumpolar Current, and Douglas et al. (1984)
observed global oceanic circulation with the SEASAT data set. Geoid model improvement
continued with work such as Rapp (1983).

Eight years elapsed from the untimely demise of SEASAT to the launch of
GEOSAT, which is described in the next subsection. Future plans for satellite altimeters
include the European Space Agency's ERS-1 in late 1990, and the joint French-American
mission of TOPEX/POSEIDON in mid 1991.

GEOSAT

On 30 September 1986, the U.S. Navy altimetric satellite GEOSAT completed its
classified primary mission of marine geoid mapping, after some 18 months of operation.
On 1 October, a series of maneuvers was initiated to alter the geodetic orbit to produce
groundtracks within a few km of the previously released SEASAT tracks, allowing the data
to be unclassified. The new orbit was designed to repeat witihin a 1 km band after 244
revolutions (17.05 days), and yields an equatorial groundtrack spacing of 164 km. This
configuration, termed the Exact Repeat Mission (ERM), became operational on 8
November 1986, and was optimized for the study of oceanic variability (Cheney et al.,
1987).

GEOSAT carries an improved SEASAT class altimeter operating at 13.5 GHz with
a 320 MHz bandwidth. Pulse compression is over 32,000 and the (compressed)

14

pulsewidth is 3.125 ns - somewhat shorter than for SEAS AT. The specified measurement
precisions are: Sea surface heights to 3.5 cm (RMS) for 2 m SWH and less, after 1 s of
data averaging; and significant wave height to 10% of the actual SWH or 0.5 m, whichever
is greater (specifications from MacArthur et al. [1987]).

It is important to note the distinction between measurement precision and range
resolution. The former is inherent to the altimeter itself and may be thought of as the ability
of the altimeter to repeat the distance measurement to a surface, all other factors constant.
As an example, the 3.5 cm precision figure indicates that a motionless GEOSAT taking
repeated measurements to an unchanging sea surface would still have a 3.5 cm RMS
variability in those measurements, just due to the altimeter noise level. Range resolution,
on the other hand, includes all of the altimeter errors mentioned earlier (except the orbit
error), and reflects the accuracy with which the altimeter makes a "real world"
determination of the distance to the sea surface, after corrections for the errors are
implemented. GEOSAT's rated range resolution is similar to SEASAT's - about 0.1 m
(refer to Summary Section at the end of this chapter).

This thesis focuses on the evaluation and use of the GEOSAT data with regards to
the study of mesoscale variability in the ocean. This is an important work for a number of
reasons. First, the GEOSAT data set is a relatively new and untried product. As with any
such product, initial care must be taken to ensure that the data appears reasonable to the
knowledgeable eye. This is the purpose of Section 1.4. The second contribution of this
thesis is an analysis of the corrections provided with the GEOSAT sea surface height data,
which purport to reduce the aforementioned altimeter errors. Many of these corrections are
new; and some, like the wet troposphere models, are actually a step back from SEASAT
(which could compute its own correction on board). Rather than indiscriminately applying
the corrections and focusing on some area of the ocean with obvious mesoscale features,
this thesis takes a subtler approach. To this end, Chapter 2 uses the GEOSAT data in an
essentially quiet part of the ocean (the northeastern Pacific), where any oceanographic

15

signals may be easily obscured by altimeter errors. Spectral and statistical methods are
employed to observe the actual effects the corrections have on the sea surface heights. Do
the corrections make it easier or harder to spot the oceanography, or do they make little
difference at all? If one of the latter two turn out to be the case, then the correction should
not be made. Chapter 2 in a sense passes judgement on the GEOS AT corrections and also
explores the possibility of averaging successive sea surface heights together to decrease
noise and computer storage space.

The final chapter in the thesis uses GEOS AT data edited according to Chapter 2, to
investigate oceanic variability in the northeastern Pacific. GEOS AT-derived results are
compared with independent studies and new insights into the oceanography of the region
are explored. Suggestions for further work are also presented.
1.2 OCEANOGRAPHY FROM ALTTMETRY

Prior to discussing the specifics of GEOS AT, it is both useful and motivational to
examine what sorts of oceanographic knowledge may be gained from satellite altimeter data
and why that data is often preferable to more conventional measurements. Basically, an
altimeter is a useful oceanographic tool because of the high spatial and temporal density of
its measurements. This density allows for synoptic scale coverage of the oceans while
avoiding the expense, difficulty, and limited scope of traditional shipboard measurements.

Given synoptic scale sea surface height measurements, what may be learned about
the oceans? Perhaps the most fundamental product is the determination of the surface
geostrophic currents. Additional knowledge of the three-dimensional density distribution
in the ocean allows the geostrophic currents to be fixed at all depths (Wunsch and
Gaposchkin, 1980). Knowing where the water goes is absolutely central to physical
oceanography, as this knowledge enables quantities such as mass, heat, and chemical
transports to be calculated. These in turn are crucial variables in such diverse fields as
climatology, marine biology and pollution control.

1 6

To a good approximation, surface currents are in geostrophic equilibrium and
hydrostatic balance (Wunsch and Gaposchkin, 1980). The former states that the pressure-
induced force caused by a sloping sea surface is largely balanced by the Coriolis force
acting on the moving water parcel. Thus water tends to flow clockwise around oceanic
"hills", instead of down them. Pure geostrophy allows no evolution of the flow with time,
however (see eq. 3), and the actual flow is really only quasigeostrophic. In non-
dimensional terms, the deviation of the ocean currents from geostrophy is of the order of
the Rossby number, defined by Rq = U/fL, where U is a characteristic velocity, L is a
characteristic length scale, and f = 2Qsin(latitude) is the Coriolis parameter (Q being the
rotation rate of the earth). Over most of the oceans the Rossby number is very small
(typically <10~2), indicating that the geostrophic approximation is a good one. However,
geostrophy breaks down at the oceanic boundaries, the equator (where f approaches zero),
and in the cross-stream velocity components of major currents such as the Gulf Stream
(Wunsch and Gaposchkin [1980], Stommel [1965]). In these regions, one may still
calculate the surface velocity fields from the sea surface slopes, but more complex
relationships than the geostrophic one (soon to be presented) must be used.

The approximation of hydrostatic balance simply states that pressure changes in the
water column are due to the weight of the water above and are independent of water
motion. In a local Cartesian coordinate system with x horizontal and z upward, the
hydrostatic balance implies

1 dp

g =

p dz

where g is the local vertical component of gravity, p is the water density, and p is the
pressure. The aforementioned geostrophic balance says that

(1)

17

p dx (2)

where f is again the Coriolis parameter and v is the velocity normal to the x,z plane. The
connection between the oceanic density structure and the geostrophic currents is found by
eliminating pressure between (1) and (2), and integrating vertically from some reference
level zr, to obtain

v=j_rz^p^„ (3)

J*zaP
— dz + vr
zr dx

where vr is the velocity at the arbitrary reference level and v is the velocity at z. Thus, if
the density structure between two levels is known (from hydrography, for example), the
integral may be evaluated, determining the relative velocity between them. The desired
result is the calculation of absolute velocities, however, and for that, the reference level
velocity vr must be fixed. Traditionally, two approaches have been used. The first has
been to go out and measure the current at some level, which is expensive, time-consuming,
and inherently limited in scope. The second was to assume that little was going on in the
deep oceans, and so choose some deep zr and assign it a reference velocity of zero. Recent
work indicates that this is not an accurate assumption (Wunsch and Gaposchkin, 1980),
which is where the real usefulness of altimetry becomes apparent.

Altimeters allow the geostrophic velocity at the surface to be fixed, because that
velocity is related to the surface topography through the hydrostatic and geostrophic
equations by

g c^l
vs=f— (4)

dx

where vs is the surface geostrophic velocity and r| is the displacement of the sea surface

from the geoid, which may be obtained from an altimeter. Thus, (3) and (4) allow for the

calculation of the geostrophic velocity at any depth from

18

v= -%-f — dz + vs

pf 9x (5)

Even with highly accurate altimetry, the practical apphcation of (5) has its
limitations. First, only the geostrophic component of the total velocity may be determined
from the sea surface slope. Near the surface, other components such as the wind-driven
Ekman flow may be of much larger magnitude than the geostrophic current.* Second, lack
of knowledge of the oceanic density field limits the usefulness of altimeter data alone in
defining the oceanic circulation. Acoustic tomography may eventually replace conventional
hydrography and ease this restriction, but that time still lies in the future. An additional
method of some promise is the technique of assimilating altimeter-derived sea surface
heights into global and regional circulation models. Here, the in situ density field can be
solved for from the surface boundary conditions as the model propagates the surface
velocities downward through the water column (e.g., Hurlburt [19861). Such models are
currendy far from perfect, however, and it may be a number of years before they may be
confidently used with only sea surface heights for boundary conditions.

The third practical limitation on the use of (5) is the lack of sufficiently accurate
geoids. The sea surface must be referenced to the local geoid to determine its true slope
and hence the surface geostrophic current. The use of mean altimeter-derived sea surfaces
to approximate the geoid is prohibited because they already contain the effects of the mean
circulation (See Section 2.1). Thus independent gravity-derived geoids are required, which
are costly and have been determined only in a few areas. The Marsh and Chang (1978)
gravimetric geoid for the Northwest Atlantic is a noteworthy example, which has been used
in several altimetric studies to draw inferences about circulation in the Gulf Stream region.

There are ways to gain valuable information about the oceans while circumventing
the need for an accurate geoid, however. These involve looking at the variability of the sea

* Knowledge of the geostrophic velocity is still in itself a very useful thing, as this
component of the total velocity is important throughout the water column.

19

surface height with time, which allows for the computation of the variable geostrophic
currents. Essentially, these techniques use the altimeter data to create a mean sea surface
over all the satellite repeat cycles. They then calculate the difference between the surface
topography during any one repeat and the mean surface topography. Since the geoid (and
the mean circulation) are expressed by non-varying surface topography, the differencing
procedure causes them to fall out of the problem and highlights the variable features. This
technique forms the basis of the so-called crossover and collinear methods of satellite
altimetry, which have the additional advantage of reducing orbital and other long
wavelength errors.

The crossover technique uses sea-surface height data obtained at points where the
ascending and descending ground tracks cross each other. Excluding the small
contribution of the variable surface topography, these separate heights should agree with
each other. They rarely do, however, mainly because of the different orbit error between
the two tracks. As mentioned before, this orbit error is of very long wavelength, and so
over arc segments it may be approximated by a linear (or quadratic) polynomial (see Tai
and Fu [19861). The crossover technique seeks to minimize the crossover differences in a
least squares sense by adjusting the sea surface heights along the tracks with the desired
polynomial. The result is a separate sea surface height map for each satellite repeat cycle
(approximately every 17 days for GEOSAT). These maps may then be used to form the
mean and differenced surfaces described above. Examples of the use of the crossover
technique may be found in Huang et al. (1978), Cheney and Marsh (1981b), and Tai and
Wunsch (1983), among others. An additional advantage of the crossover technique is that
determining the satellite orbit at various times with precision tracking or ephemerides of
high accuracy can essentially fix the crossover mesh with respect to the earth and allow for
absolute sea surface height mapping. This was performed on GEOS-3 data by Marsh et al.
(1980), and on SEAS AT data by Marsh and Martin (1982), for example. A recent
variation of the crossover technique is to represent the orbital error by a Fourier series

20

whose coefficients are then adjusted to minimize the crossover differences (Douglas et al.,
1984).

The crossover technique is not without its drawbacks, and three of these in
particular prevented its use in this work. First, it is time and storage intensive on a
computer, and difficult to master in a limited period of time. Second, effective error
estimates are difficult to define as errors of all wavelengths (from altimeter noise to orbit
error) contribute to the calculated RMS crossover differences. Most importantly, though,
the large gaps between the crossover points (see figure 1.3) preclude use of the short
spatial scales in investigating the effectiveness of the altimeter corrections or the smaller
mesoscale features. Since these pursuits are central to this work, the collinear approach
was adopted instead. This technique is presented more fully in Chapter 2, but a few points
are salient at this time.

Sea surface variability can be obtained from the collinear method when the satellite
groundtracks repeat after some interval. A mean sea surface height profile for a segment of
a selected groundtrack is calculated by averaging together the profiles of individual passes
along it. Then, the difference between any given repeat and the mean may be determined,
again taking the geoid out of the problem and highlighting variable features. Orbit error is
reduced as in the crossover method by removing a low order polynomial from each sea
surface height profile. The length of the groundtracks used in this work (some 4400 km)
allows a simple bias and tilt removal to suffice. Note that this linear trend removal will also
reduce long wavelength errors in any applied corrections, as well as eliminate the need to
correct for any long wavelength processes in the first place. Bias and tilt removal also
eliminates any oceanographic signals of wavelengths comparable to the groundtrack length,
but this loss is acceptable if mesoscale effects are the focus; as they are in the present case.
Examples of the collinear technique may be seen in the work of Douglas and Gaborski
(1979), Mather et al. (1980), Cheney et al. (1983), and Fu (1983).

21

1.3 GEOGRAPHY AND TERMINOLOGY

The area under investigation in this work is a rectangular portion of the northeastern
Pacific, bounded by 20°N, 55°N, 180°W and 1 15°W (see figure 1.2). It is primarily deep
ocean, though it also contains the western coast of the United States, the Hawaiian islands,
and portions of the Aleutian island chain. This region was chosen because it is fairly well
known in an oceanographic sense and independent sources exist which may be used for
data comparison. In addition, much of it is relatively quiet in mesoscale variability, and so
this area may also be used to find a lower noise threshold on GEOS AT-derived sea
surfaces, and more rigorously determine the effects of corrections.

The geometry of the area, along with GEOS AT's orbit, gives rise to a pattern of
satellite sub-tracks which may be described as follows (this terminology will be adhered to
throughout this work): In a given 24-hr period, GEOS AT overflies the area about 6 times;
each overflight is called a pass. The six overflights appear in 2 clusters of 3 passes each.
One cluster contains all ascending passes, and the other, only descending. Clusters are
separated by from about four and a half to seven and a half hours when no passes are made
over the area. Passes within a cluster are separated in their starting times by about 101
minutes, and in perpendicular distance by about 2200 km. Figure 1.2 shows the area and
satellite sub-tracks for day one of the ERM.

During any 17-day period, each pass will be along a distinct arc within the area, and
the set of passes during a 17-day repeat cycle will fill out a skewed checkerboard pattern
(see figure 1.3). After 17 days however, the passes start to repeat themselves, tracing over
the same arcs as before. Thus, each arc is a great circle passing through the area, and may
be considered as an ensemble of those passes separated by 17 days which are traced along
it. The spatial and temporal sampling which result from this scheme are fairly close to the
20-day repeat and 140 km equatorial groundtrack spacing suggested by Mitchell (1983) as
being nearly optimal for adequately resolving the oceanic mesoscale (which has typical
spatial scales of 100 km and time scales of 30+ days).

22

Note from figure 1.3 the significant amount of missing data in the area. These data
dropouts occur mainly on descending passes and result from the satellite's waveform
tracker being unable to lock onto the return pulses after passing over land. Extensive
dropouts present obvious problems for the collinear technique, and repeat passes which
contain them are largely avoided in this work.
1.4 THE GEOPHYSICAL DATA RECORD

The GEOSAT Geophysical Data Record (GDR) is prepared from Sensor Data
Records and provides the user with 34 channels of data approximately every second.
These 1/sec values include time, latitude and longitude of the satellite subpoint, orbit
height, sea surface height, geoid height, several diagnostics, and corrections which were
already applied to the sea surface heights. In addition, corrections for the errors mentioned
in the preceding sections are given, which may be applied as the user sees fit. The Sensor
Data Records (SDR's) are described in detail in the GEOSAT Interface Control Document
(Cole, 1985). The main reference on the GDR is the GDR User Handbook (Cheney et al.,
1987).

The following subsections decribe the initial forays into the GDR. Quite simply,
these were attempts to see if what was actually on tape made sense compared to
observational and/or theoretical knowledge of what was supposed to be there.
Accordingly, discussions of applicable physical processes are presented, with descriptions
of their corresponding GDR channels. In addition, the literature search necessitated by the
above revealed numerous (and sometimes conflicting) error estimates for the various
GEOSAT measurements and corrections. I attempt to reconcile these estimates and provide
as meaningful an error budget as is currendy achievable.

The "zero-order" look at the GDR data was at a long arc passing completely
through the area and comprised of six gap-free passes. For identification purposes, it was
arc 82; a descending track with approximate limits of 55°N, 138°W and 20°N, 161°W

23

(identified on figure 1.3). During the first few repeat cycles, this arc consisted of passes
on days 3, 21, 38, 55, 72 and 88 of the ERM.

Time

GEOSAT's raw measurements are recorded on board the satellite with the
spacecraft clock count, and then dumped about twice a day when the satellite is in view of
the Ground Station. The station keeps its own time as well, which is used to calibrate the
spacecraft clock during periodic real-time downlinks. The calibration is used to generate a
corrected "time tag" for each processed altimetry data frame (see Jones et al. [1987] and
Cole [1985] for further discussion). The accuracy of the final time tagging of the data is
rated at <100 jisec (Jones et al., 1987), and studies by Cheney et al. (1987b) of crossover
differences have revealed no time tag biases thus far. As will be seen shortly, GEOSAT's
time accuracy is not a limiting factor in its error budget.

The GDR has two 4 byte (integer) channels devoted to time. They indicate seconds
and microseconds of UTC (Universal Time Coordinated, or Greenwich Mean Time)
referenced to 1 January 1985. A printout of these channels shows that successive points
on a pass are separated by the correct time of about .98 seconds, and the repeat passes on
the arc are separated by the published value of 17.05 days (cited times are from Cheney et
al., 1987).

Latitude and Longitude

The GDR has two 4 byte (integer) channels for location information - one each for
latitude and longitude of the satellite subpoint, in microdegrees. Printouts show that
successive subpoints on a pass are separated by approximately 6.7 km, which is
appropriate for the groundspeed of a satellite with a nodal period of 6037.55 sec (Born et
al, 1987). They also indicate that the groundtracks of the repeats on arc 82 are separated
by a crosstrack distance of less than 1 km, as desired. The subpoints are a product of the
ephemeris calculated by the Navy Astronautics Group (NAG), and their horizontal
accuracy is about 3 to 5 times the radial orbit accuracy (Born, 1988); or about 15 m (RMS).

24

Horizontal positioning uncertainties of this size should not significantly affect the
accuracy of the measured sea surface topography, for two reasons. First, the deep ocean
surface height simply doesn't change by much on that horizontal scale, assuming waves are
averaged out (which they are for GEOS AT). Second, GEOSAT's radar footprint is itself
of order 4 to 5 km, depending on significant wave height (Shuhy et al., 1987), so errors of
a few meters in its horizontal position are negligible.

Orbit

GEOSAT is currently in a 17 day exact repeat and frozen orbit. Exact repeat refers
to the fact that groundtracks should repeat to within ±1 km for each repeat cycle. Frozen
means that the mean argument of the perigee (w ) and the mean eccentricity (e) have no
long term variation, being chosen such that perturbing forces in the earth's gravity field will
cancel each other out (see Born et al. [1987]). In the hypothetical case of no other orbit
perturbations, a frozen orbit will result in a near invariant satellite altitude profile from
repeat to repeat on any arc (in actuality, even in a frozen orbit, w and e will vary
cyclically). The mean argument of the perigee and mean eccentricity for GEOSAT are 90°
and .0008 respectively (orbit parameters from Born et al. [1987]). The perigee argument
places apogee for GEOSAT over the equator and perigee nearest the poles. The low
eccentricity indicates that GEOSAT is in a near circular orbit, which is desireable to keep
small the radial component of the satellite's velocity (less than about 50 m/s in GEOSAT'S
case [Cheney et al., 1987b]). Small radial speeds are necessary to maintain the rate of
satellite-sea surface distance variation within the limits of the altimeter tracking circuitry,
and to minimize altimeter measurement error due to timing inaccuracies. At the cited radial
speeds, GEOSAT's timing uncertainty of 100 |isec gives rise to only a .5 cm error in the
sea surface heights, which is negligible compared to the other uncertainties, as will be seen
shortly. Two final orbital parameters of interest are GEOSAT's mean semimajor axis a,
and its mean inclination i. These are 7167.4 km, giving an actual altitude of about 800 km;
and about 108°, putting GEOSAT in a retrograde orbit. Chapter 15 of Stewart (1985)

25

contains an excellent description of orbital terminology, and is recommended for further
reading.

Any satellite's orbit is perturbed by gravitational forces from such sources as the
earth's geopotential, the sun and moon, solid and earth tides, as well as by nongravitational
forces. The latter include atmospheric drag and the radiation pressure due to the solar flux
(see Parke et al. [19871). Incomplete knowledge of these forces naturally affect the
accuracy of the orbit calculation. GEOS AT's ephemeris is computed from doppler tracking
data in conjunction with a calculated trajectory using the NASA GEM (Goddard Earth
Model) 10B geopotential model (Lerch et al., 1981). This model does not include
gravitational effects other than the earth's fixed geopotential, or any non-gravitational
effects. Cheney et al. (1987) cite the overall radial accuracy of the ephemeris as
approximately 4 meters (RMS), which will direcdy propagate into uncertainty in the sea
surface heights.* Fortunately, as mentioned earlier, most of the orbit error may be
removed by simple detrending of the sea surface heights on each pass.

Each orbital value on the GDR is a 4 byte integer and gives the satellite height in
mm above the earth's reference ellipsoid defined by a = 6378.137 km and f =
1/298.257223563. Printouts show the orbit height close to the known range near 800 km
and the difference between repeat passes on arc 82 to be on the order of hundreds of
meters.

Sea Surface Height

The following discussion is based upon MacArthur et al. (1987), and Cole (1985).
Repeated citation is avoided in the interest of readability.

The GDR sea surface height channel (H) is a 2 byte integer (as are all the
subsequent channels discussed in the GDR section), and is the one-second average sea
surface height in cm above the reference ellipsoid. H is computed at the GDR rate of

Recent work puts the uncertainty at about 3 m RMS (Born, 1988).

26

~l/second from a linear fit with iterative outlier rejection of the 10 surrounding height
values HI through H10 which are at the full GEOSAT data rate of 10/sec (Cheney et al.,
1987). Each full-rate height is calculated by H(n) = ORBIT/10 - ALTIMETER, where
ALTIMETER is the measured height in cm between the satellite and the sea surface and
ORBIT is the satellite height in mm above the reference ellipsoid. The height is determined
by the travel time taken from pulse transmission to the half power point on the leading edge
of the averaged return waveform.

Prior to the computation of H(n), the altimeter measurement is corrected for a
number of effects. The GEOSAT waveform processing module takes 8 return waveform
samples at the full rate and averages them every second. Corrections are made to the
waveforms based on prior calibration, and the corrected waveforms are used to produce a
voltage term proportional to satellite attitude (VATT) at the 1/second rate. VATT also
depends on significant wave height (SWH), however, and its accuracy is degraded at high
values of SWH. The VATT values are corrected for SWH based on adjustable coefficients
initially determined by prelaunch tests.

The corrected 1/second VATT values are edited to delete outliers, as well as any
values lying outside the range corresponding to the expected off-nadir excursions of
GEOSAT. The remaining values are sent to a first order fit routine that assigns a VATT
value at the 1/second GDR rate, which is the average of the edited VATT values in a
window of up to ±2 minutes. The window span is determined by SWH, as the increased
noise at high SWH requires a longer averaging period. The fitted VATT values, in
conjunction with the SWH, are used to compute a 1/second height correction to the
ALTIMETER measurement. In essence, this height correction compensates for the
apparent depression of the sea surface caused by an off-nadir radar beam. Cole (1985) lists
the correction uncertainty as ±2 cm. It appears in the GDR channel labelled DH (for "Delta
H") and has already been applied to the sea surface heights.

27

Another 1/second correction applied to the ALTIMETER measurement is due to the
range/doppler crosstalk of linear FM (Frequency Modulated) waveforms like those of the
GEOS AT radar. In essence, the doppler shift caused by a vertically travelling GEOS AT is
seen as a height bias in the ALTIMETER measurement (This is another reason why the
orbital eccentricity was kept small). Cole (1985) writes that it is only necessary to know
the vertical speed to a precision of 1 m/s to determine the FM correction to better than .5
cm. This precision is easily achieved by averaging ten of the raw satellite altitude
measurements before determining the vertical speed. The FM crosstalk correction is given
in the GDR as FM/DH, and like DH, has already been applied to sea surface heights in the
GDR.

Three other corrections are made to the sea surface heights before being placed in
the GDR. The first accounts for the location and movement of the spacecraft center of
gravity with respect to the altimeter and was determined prior to launch (this corrects for the
ORBIT values being computed to GEOS AT's center of gravity while altimeter
measurements are actually made from the radar antenna). The second is a bias periodically
computed from the on-board calibrate mode of the satellite, and the third corrects for
inherent system effects such as antenna focal length and internal signal delays.

The reader will note that all corrections are at the GDR rate of 1/second, while the
H(n) which are averaged to calculate H occur at a rate of 10/sec. This is not a discrepancy.
The ten H(n) each have the same correction applied, which is the one appropriate to their
central time value (Cole, 1985).

The GEOSAT altimeter is designed to measure the satellite-sea surface distance with
a precision of 3.5 cm for 2 m significant wave height (MacArthur et al., 1987). Ground
tests have verified that the altimeter noise level is near 2 cm at low SWH after 1 second
averaging. In flight, the performance was evaluated by computing the standard deviation
about a linear fit to a set of 10 heights at the full 10/sec rate, repeating this for 10
consecutive sets, and averaging the resultant standard deviations. MacArthur et al. (1987)

28

found the agreement between the ground test data and this in-flight evaluation to be quite
good and so assigned an in-orbit RMS precision of 3.5 cm for 2 m SWH to H values
averaged over 1 second (as on the GDR). Being an in-orbit test, the 3.5 cm figure should
include the aforementioned corrections of attitude, crosstalk, etc. One must remember,
though, that the in-orbit evaluation used standard deviations calculated from short spans of
time (-10 sec for the final averages), over which effects such as attitude excursions and
bias drift would be small. For the longer times appropriate to the arcs under study in this
work, the system noise level might therefore be greater.

As previously mentioned, the GDR H channel is determined not only from the
ALTIMETER measurement, with its nominal 3.5 cm precision, but also from the ORBIT
calculation, with its 3 m accuracy (RMS). In this regard, the GDR looks good to zero
order. At a given time on a pass, the HI through H10 values (whose relative magnitudes
over their span of 1 second are negligibly affected by the long wavelength orbit error),
fluctuate on the order of a couple cm, which is appropriate to the altimeter precision. (The
actual standard deviation for each set of HI -HI 0 values is included as a separate GDR
channel denoted as SIG H, for "sigma H". Printouts show that this too is in the few cm
range for the passes on arc 82). The H values at a given geographic location on the arc
differ by up to -300 cm between passes, which is primarily a reflection of the orbit error.
Figure 1.4 is a plot of the values of H along arc 82 during the fourth repeat cycle. By
comparison with figure 1.5, note how, to first order, the sea surface merely follows the
geoid, which is now discussed.

Geoid

The GDR Geoid channel contains geoid height in cm above the reference ellipsoid,
computed with bilinear interpolation of 1 degree geoid height estimates computed from
Rapp (1978). The Rapp geoid was originally determined to degree and order 12, which
represents features in the geoid down to a wavelength of 2rcR/12 3300 km (where R is
the radius of the earth). While this resolution is far too coarse for oceanographic

29

applications, the Rapp geoid is nevertheless included in the GDR as a way of reducing the
variability of the sea surface heights. Since the sea surface topography follows the geoid to
lowest order; subtracting out the geoid reduces the sea surface height variation from the
±100 m range to the ±5 m range. This makes data plotting and visual inspection much
easier and is used in this work.

GDR printouts look good with respect to the known characteristics of the Rapp
geoid. Geoid heights on successive passes for the same geographic area differ by a few
cm, which is appropriate, since cross and alongtrack track distances can differ by a few
kilometers for the different repeats. A plot of the geoid channel along the fourth repeat on
arc 82 (see figure 1.5), agrees quite well with the sea surface height profile of the same
repeat (figure 1.4); though one can clearly see that the geoid model does not fully resolve
Hawaii at ~22°N. Orbit error is also evident in the offset between the profiles. Figure 1.6
is a plot of figure 1.4 minus figure 1.5, or H minus geoid for repeat 4 on arc 82, and
indicates the efficacy of a geoid correction in reducing variability. The major features of
this profile are due to uncertainty in the modelled geoid.

Significant Wave Height

The GDR Significant Wave Height (SWH) channel is listed once per second, and
similar to the H channel, is the one second average of the surrounding full-rate SWH data.
The averaging process is identical with that used for H; a linear least squares fit with
iterative outlier rejection (the maximum number of rejections allowed for both SWH and H
is four. After that, flag values of 32767 cm are set in the GDR). The 10/second SWH are
determined onboard GEOS AT by measuring the leading edge slope of the reflected pulse.
Since this measurement varies with attitude and is also a function of SWH itself, a
correction is made for these effects, as well as a small calibration adjustment (Cole, 1985).

The design precision of the GEOS AT SWH after averaging is 10% of SWH or 0.5
meters (whichever is greater). Based on comparisons with buoy data it appears that this
goal has been met (Dobson et al., 1987). As noted in the sea surface height section, SWH

30

is a necessary input in calculating the attitude/SWH correction to H. Since the verification
of the H precision was performed inflight with these corrections applied, though, SWH
uncertainties in this regard should be included in the 3.5 cm precision value.

Significant wave height is also a measure of other effects that have an impact on the
sea surface height measurement. These effects have, singly or in concert, been termed
electromagnetic (EM) bias. Wave peaks tend to disperse radar energy, while troughs tend
to focus it in the direction of the spacecraft. Thus the satellite receives more return energy
from the troughs than the peaks and so sees the sea surface as depressed from the actual
mean level (Tapley et al., 1982). An additional factor is the wave skewness, or asymmetry
(i.e. narrow high peaks and broad shallow troughs) that tends to further bias the altimetry
measurement towards the troughs. Lipa and Barrick (1981), extending the theory of
Jackson (1979) determined that the EM bias correction at the SEAS AT altimeter frequency,
which is the same as GEOS AT's, should be about 2% of the SWH. This was supported
by SEASAT data as calculated by Bom et al. (1982). For these reasons, the present
recommendation for GEOS AT is that the sea surface heights should be increased by 2% of
the significant wave height (Cheney et al., 1987). A recent GEOSAT error budget
(Lybanon and Crout, 1987) ascribes an RMS uncertainty of 2.2 cm to this correction,
assuming a SWH of 2 m and a wave skewness of 0.1. The relatively short wavelength
(200-1000 km) that they ascribe to this error indicates that bias and tilt removal would not
reduce it by much when the correction is applied to sea surface heights in the area.

Printouts of the GDR SWH channel for early repeats along arc 82 show quite a bit
of variability, both along and between passes. Differences between the repeats are on the
order of 7 meters for the northern part of the arc; and there are a respectable number of
SWH values of greater than 2 meters (meteorological causes of SWH behavior are
examined in a later section). This variability would equate to EM bias correction variability
on the order of 10+ cm, but more importantly perhaps, some questions must be raised as to
the applicability of a 3.5 cm instrument precision in regions of high SWH. This, too, will

31

be addressed later. Figure 1.7 shows a plot of SWH along arc 82 during the fourth repeat
cycle.

Solid Tide

The GDR solid tide (STIDE)* correction, like the remaining correction channels, is
given in mm and is to be subtracted from the H values as the user sees fit. This correction
is based on the solid earth response to the tide generating potentials of Cartwright and
Taylor (1971) with subsequent modification by Cartwright and Edden (1973). The solid
tide height is extrapolated at the GDR 1/second interval from calculations of the tide
generating potential and gradient at 30 second intervals along the ground track (Cheney et
al., 1987).

The amplitude of this phenomena is about 20 cm (Tapley et al., 1982), with
printout values on arc 82 varying smoothly within this range. Of more interest, however,
is the low error of this correction, given by Cheney et al. (1987) as 1 cm (RMS).
Additionally, as can be seen by its plot for the fourth repeat on arc 82 (figure 1.8), the
Solid Tide correction is of very long wavelength, and so any errors in it should be further
reduced via bias and tilt removal, as was discussed earlier.

Ocean Tide

This channel gives the surface ocean tide (OTTDE) correction to H and is based on
the work of Schwiderski (1980). The correction is interpolated along the ground track at
the GDR 1/second interval from a 1 degree global grid of 1 1 tidal components. The ocean
tide reaches amplitudes over 100 cm in the open ocean and 200 cm in coastal areas
(Schwiderski, 1980). Printouts of the GDR OTIDE channel show the data to be consistent
with these bounds. Unfortunately, the accuracy of this correction is much worse than that
of the solid tide correction. This is due not only to the larger amplitude of the ocean tide,
but to the more complex nature of the oceanic response to tidal forcing. As opposed to the

The various GDR corrections will often be referred to by acronym in this work. The
appropriate acronyms are defined at the beginning of each correction's subsection.

32

solid tide, which is largely static, the ocean tide also includes dynamic response to tidal
forcing, in the form of long waves which are affected by the Coriolis force and the inherent
resonances of the ocean basins (see Hendershott [1981]). Uncertainties in modelling these
waves leads to the rather large figure of 10 cm (RMS) error in an OTIDE correction to sea
surface heights (Schwiderski, 1980).

Figure 1.9 profiles the OTIDE correction for repeat 4 on arc 82. It is apparent that
this correction has a richer spectrum at short wavelengths than the STIDE correction,
indicating that the error introduced by applying an OTIDE correction might not be
significantly reduced by bias and tilt removal.

As will be seen later, the magnitude and variability of the ocean tide necessitates a
correction be made to the sea surface heights before investigating mesoscale features. This
result emphasizes the importance of testing the zero-order accuracy of the GDR product
While this check may be done by statistical or spectral methods (and will be in Chapter 2),
a quick comparison with actual tidal station predictions is a useful exercise. To this end,
three arcs were chosen that passed the closest to the tidal reference stations at Honolulu,
San Diego, and Sweeper Cove, Alaska. For each arc, GDR OTIDE values for the closest
point of approach to the tide station were obtained for 3 different repeat cycles. The GDR
tidal values were then compared with tabulated predictions from the U.S. Dept. of
Commerce Tide Tables for the stations at the appropriate times. The results may be seen in
table 1.1 and indicate that the GDR channel is appropriate in sign and amplitude. Of note is
the column entitled R, which is the ratio of the GDR tidal amplitude to the station
amplitude. This turns out in nearly all cases to be <1.0, which is proper, since the GDR
data is for open ocean, whereas the station values are taken in more enclosed waters, where
the tidal amplitude normally will be greater.

Wet Troposphere

The remaining three corrections account for the lengthening of the radar travel time
caused by delay of the signal as it propagates through the troposphere and ionosphere.

33

Since any delay in travel time will depress the measured sea surface below its actual state,
these corrections should wind up raising that surface. As previously mentioned, the GDR
corrections are to be subtracted from the H channel, so their negative values as seen on arc
82 printouts do in fact increase the sea surface heights as desired.*

The wet troposphere correction channels compensate for the travel time delay
caused by water vapor in the troposphere. One GDR channel is termed wet FNOC
(WFNOC) and is computed as follows: Surface values of air temperature and water vapor
pressure are interpolated along the satellite track from the Fleet Numerical Oceanographic
Center (FNOC) model output, which appears in a 2.5 degree grid at 12 hr intervals. These
values are then used to compute the wet troposphere correction based on the Saastamoinen
(1972) model. The other channel is termed wet SMMR and is based on Prabhakara et al.
(1985) and Tapley et al. (1982b). Values for the vertically integrated atmospheric water
vapor are interpolated along the ground track at the GDR interval from a climatic monthly
mean data set from the NIMBUS 7 SMMR prepared by C. Prabhakara. This data appears
in a 3° latitude by 5° longitude grid of monthly averages between 1979-1981 (Cheney et
al., 1987). The water vapor values are then converted to a height correction using a simple
formula due to Tapley et al. (1982b). It may be noted that these modelled corrections are
necessary because GEOSAT, unlike SEASAT, does not possess the means of determining
its own wet troposphere correction (i.e., a Scanning Multichannel Microwave Radiometer,
or SMMR, which SEASAT carried).

Tapley et al. (1982b) indicate the amplitude of the wet tropospheric correction as
being up to 50 cm, and GDR printouts are consistent with this range. Unfortunately for the
determination of mesoscale phenomena, however, this correction is spatially highly
variable. Figures 1.10 and 1.11 show typical profiles of the wet correction - one the

* The GDR delay corrections are referenced to a vacuum, and so their sign is always
negative (see figures 1.10 to 1.13). Note that bias and tilt removal will eliminate the mean
effect of the delay corrections, however, allowing their fluctuations to both raise and lower
the apparent sea surface.

34

FNOC channel and the other the SMMR. Qualitatively, the two look similar, but
differences on the order of several centimeters are evident and Cheney et al. (1987) report
discrepancies of 10 cm as not uncommon. The obvious question arises as to which (if any)
of these corrections should be applied.

During SEASAT's geophysical evaluation, Tapley et al. (1982b) compared FNOC-
derived wet troposphere corrections of the same sort as GEOSAT'S, with corrections
obtained from coincident radiosonde data. The RMS difference was found to be 5.73 cm.
Cheney et al. (1987) estimate the accuracy of the GDR WSMMR correction as about 5 cm
(RMS) over the observation period of the NIMBUS-7 SMMR. The very variability of the
wet tropospheric phenomenon argues against the use of this climatic correction, however,
so the FNOC correction would seem to be the one of choice. A quantitative comparison
between the GEOS AT WSMMR and WFNOC corrections is undertaken in the next
chapter.

Lybanon and Crout (1987) present an error wavelength of only 200 km for the wet
tropospheric correction, which would preclude much error reduction from bias and tilt
removal.

Dry Troposphere

The dry troposphere (DFNOC) correction compensates for the radar travel time
delay caused by air molecules in the troposphere, and like the preferred wet tropospheric
correction, is derived from FNOC data. Values of surface atmospheric pressure are
interpolated at the GDR rate from the FNOC model output and are used to calculate the
height correction, again based on Saastamoinen (1972). This correction is given by the
simple expression DFNOC = -.2277 P (1 + (0.0026 x cos(2 x LATITUDE))) where p =
surface atmospheric pressure in mbar and the correction is in cm.

The dry troposphere correction is much better known for GEOS AT than the wet
correction, because it is much less variable than the latter. Figure 1.12 displays the
DFNOC correction for arc 82 on the fourth repeat cycle and agrees with the known fact that

35

this correction is always about 2.3 m with an additional long wavelength variation of about
10 cm. This was also observed on printouts of given areas for different repeats on arc 82.

Assuming a 3 mbar surface pressure uncertainty, one can estimate the uncertainty in
the Dry Tropospheric correction to be .2277 x 3 = .7 cm. Lybanon and Crout (1987) list
the wavelength of this very small error as 1000 km.

An additional phenomenon that affects the sea surface heights is related to the
DFNOC correction; and while it is not explicitly included on the GDR, it may be calculated
from that correction. This phenomenon is the inverse barometer (IB) effect, where sea
level rises and falls with changes in the atmospheric pressure. In essence, one may think
of HIGH pressure zones pushing the water away so that it piles up under areas of LOW
pressure. A standard rule of thumb is that there is a 1.01 cm rise (fall) in sea level for
every 1 mb drop (rise) in pressure below (above) the standard value of 1013.3 mb. This
can result in a significant effect of up to -50 cm.*

Wunsch (1972) has shown that the Rossby radius of deformation is the upper
wavelength limit of an expected static IB response (at longer wavelengths, the pressure
gradients may be supported by geostrophic currents). What is less clear, however, is the
time scale within which the ocean responds to pressure changes. Wunsch (1972), for
example, determined that an IB effect exists at mid-ocean islands at periods of greater than
1 day. The inability of subsequent studies to determine more rigorous bounds on the IB
effect was the reason for its non-inclusion in the GDR. This interesting problem is
explored in the next chapter.

Ionosphere

The ionospheric (IONO) correction to H found in the GDR accounts for the
altimeter signal travel time delay caused by free electrons in the ionosphere. Since

* It is important to note that the IB and DFNOC corrections arise from the same source - the
atmospheric pressure. This indicates probable correlation between the two, so one must be
careful when determining how they affect the sea surface heights. This topic will be treated
in more detail in Chapter 2.

36

GEOSAT does not have a dual frequency altimeter, it cannot measure the delay directly.
Instead, the correction is based on the Global Positioning System (GPS) climatic
ionosphere model, which is driven by daily values of the solar flux (Cheney et al., 1987).

The amplitude of this correction could be on the order of several decimeters (Lorell
et al., 1982), but effects of this magnitude would only be seen during periods of peak solar
activity. The next such period will be around 1991, so GEOSAT is gradually experiencing
ionospheric error of increasing size. In early 1987, Cheney et al. (1987) reported that the
GEOSAT GDR ionospheric correction rarely exceeds 5 cm, which agrees with printouts of
arc 82 at that time. They put its accuracy at the 50% level, which is slightly lower than
Lybanon and Crout's (1987) estimate of a 4 cm (RMS) uncertainty. An RMS uncertainty
of 3 cm is a reasonable choice, and Lybanon and Crout's (1987) error wavelength estimate
is >1000 km. Figure 1.13 indicates the long wavelength nature of the ionospheric
correction. This behavior is naturally a result of the GPS model containing only long
wavelength effects, but these are in fact the dominant ones (Cheney et al., 1987).
1.5 SUMMARY

A zero-order look at the GDR for arc 82 revealed no obvious discrepancies - what
is actually seen on GDR printouts and plots agrees with what one expects to find. This
simple, but vital step in the verification of the GEOSAT data set has also been useful in
providing some insight into the actual physical processes which the GDR attempts to
measure and/or model. A result of this insight is an error budget for GEOSAT, which was
presented piecemeal in the preceding subsections. The values cited were obtained from a
number of sources. When possible, correction errors and their wavelengths were obtained
from the original papers of the models used by GEOSAT. In addition, three previously
published satellite error budgets provided useful information: Lybanon and Crout's (1987)
GEOSAT error budget, Tapley et al.'s (1982) SEASAT error budget, and the budgets of
the TOPEX SCIENCE WORKING GROUP (1981). The Lybanon and Crout budget was
most applicable, though it did not include tidal errors (other than long wavelength tidal

37

perturbations of the orbit), and it cited orbit errors appropriate to a different orbital
processing scheme than the one currently used in producing the GDR. A summary of the
present error budget appears in table 1.2. Note that some of the sources (such as altimeter
and orbit) are not broken down into separate components. Such detail is unnecessary for
current purposes.

The range resolution of GEOSAT may be determined from table 1 .2 by taking the
square root of the sum of the squares of the individual correction errors (excepting the orbit
error). This results in an overall range resolution of about 13 cm - promising some hope in
detecting oceanographic features.

38

1.94 M SWH

Fig. 1.1a Altimeter return waveform curves. Formed by averaging together many
consecutive radar pulses (from Stewart [1985]).

Antenna Beamwidth

Fig. 1.1b Pulse-limiting in an altimeter. The received power rises linearly with time until
the trailing edge of the pulse reaches the surface (from Stewart [1985]).

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53

CHAPTER TWO - EVALUATING THE DATA

54

2.1 COLLINEAR PROCESSING SCHEMES

The purpose of this chapter is to determine which of the GDR corrections should be
made to the GEOSAT sea surface heights and if any successive-point averaging is possible.
To this end, I used an algorithm designed to investigate oceanic variability while
minimizing the effects of unknowns. Various collinear schemes fit these criteria, and their
methodology is as follows (equations 1 through 3 are due to Caiman [1987]).

For each repeat pass, one seeks the time varying component of the dynamic ocean
topography T(x,t), where the total dynamic topography TJ(x,t) is the sum of T|'(x,t) and the
mean dynamic topography r\(x). But the total dynamic topography for a repeat is also
given by

7i(x,t) = h(x,t)-G(x) (1)

where h(x,t) is the measured sea surface height profile for the repeat (i.e., the H channel in
the GDR), and G(x) is the geoid height. By the collinear scheme, one averages together an
ensemble of repeat passes on a given arc to form a mean height profile h(x), where

h~(x)=:n(x)+G(x) (2)

That is, the mean profile is the combination of the mean dynamic topography experienced
by all repeats, and the geoid - which is the same for all repeat passes on a given arc (not
quite, but that will be seen shortly). Adding (1) and (2) and simplifying, one finds that

ri'(x,t) = h(x,t) - h~(x) (3)

This result shows that for a given pass, the dynamic topography (also known as a
difference profile) may be found by subtracting the mean sea surface height profile for the
arc from the sea surface height profile for the pass. Note that no explicit knowledge of the
geoid is needed with this method.

There are three drawbacks to collinear schemes. The first is that knowledge of the
mean circulation is lost because the mean dynamic topography has been used to form the
mean sea surface. I accept this limitation in view of the aims of this work. The second is
more troublesome and is residual geoid contamination. This effect is due to the fact that the

55

data points for the successive repeat passes do not exactly coincide, either along or cross
track (the cross-track spread is less than ±1 km and the alongtrack spread can be limited to
±A/2, where A is the distance between successive data points). Wherever there is a
significant geoid gradient, the resultant will be that the ensemble of passes will not all see
the same geoid, and the mean profile h(x) will be contaminated by an extra geoid term
G'(x). When difference profiles are created by subtracting out the mean, this
contamination will remain and may be mistaken for oceanographic signals. Fortunately,
residual geoid contamination is usually easy to spot as it habitually occurs in those areas
where the geoid is changing rapidly. Unfortunately, though, these same regions often
contain ocean currents of interest which may not be visible over the contamination. These
areas may be found by looking at arc mean profiles, as will be seen presently. The third
drawback to collinear schemes is the possibility of temporal aliasing. Since these
techniques rely on repeated sampling of the same arcs, any phenomena having periods at or
near integral multiples of the repeat period will appear invariant. Similarly, those processes
which change very slowly over the length of time needed to form a mean profile may add
unwanted energy to that mean and substantially alter the difference profiles. The latter
occurrence is explored later in this chapter.
2.2 A COLLINEAR ALGORITHM

The algorithm which created the difference profiles used to determine GEOSAT
correction applicability is as follows:

1) Delete data points not over water, or flagged bad in the H channel

2) Remove the low order Rapp geoid from the H values on a pass

3) Apply a chosen GDR correction (if any) to the resultant values

4) Remove any spikes from the corrected profile

5) Detrend the profile (i.e. - remove bias and tilt)

56

6) Average a collinear ensemble of such profiles together to form a mean sea
surface height profile for that arc. Figures 2.1-2.3 show such a profile for arcs 82, 209,
and 293.

7) Subtract this mean profile from each pass in the ensemble (where each pass is
always edited with steps 1) thru 5)) to create the set of difference profiles (i.e. - T|'(x,t)).
Such a set for arc 82 may be seen in figures 2.4 and 2.5.

From these difference profiles, quantities such as RMS alongtrack variability and
wavenumber spectra may be calculated, in order to evaluate the efficacy of the chosen
correction. Prior to passing such judgement, however, a number of checks on the
algorithm were performed. These are now summarized as some of them illuminate the
finer points of the subsequent analysis:

Subtraction of the Rapp geoid was done to make plotting and inspection of the
height profiles easier. Since this geoid is of low order, it is nearly unchanged from repeat
to repeat (even with cross and along-track offset), and so removing it from the sea surface
heights has negligible effect on the difference profiles.

The proper application of the correction channels was verified by comparing non-
detrended profiles before and after correction with plots of the correction channels
themselves. That is, if a plot of the OTIDE correction for a repeat pass showed that the sea
surface height should be decreased by 50 cm at 40°N, for example, profiles of that repeat
before and after OTIDE correcting were inspected to ensure exactly that happened. No
problems were found in this regard.

The spike editor is based on a simple first difference scheme of each point on a
corrected height profile with its predecessor. If the difference exceeds a set value (30 cm
was found to be a good criterion), then the spike value is replaced by the average of its
neighbors to either side. This technique was found to work well on isolated spikes, but not
in areas with sustained large first differences (such as regions of high geoid variability).
This limitation was not a problem in practice as such areas were deleted anyway to avoid

57

geoid contamination. The editor was checked visually on a number of spikes and ensured
that any spikes which did appear in the final difference profiles were less than 40 cm in
magnitude. As shown in Appendix 1, such residual spikes have only a small effect on
variability calculations, and the amount may be quickly estimated, if necessary.

The detrender was checked numerically to ensure it removed bias and tilt from the
corrected height profiles; and also visually, to ensure it did not change the "look" of the
profiles. An additional benefit of this process was a printout of the bias removed from each
pass, where the bias is mostly the orbit error. The RMS bias over 10 passes on arc 82 was
around 1.5 m, which is comparable with the known orbit error.

The averaging method used is a point-to-point match up of corresponding data
points on the various repeat passes along the chosen arc. Alongtrack offsets are minimized
by sliding the passes so that all starting latitudes are as closely packed as possible around
some reference latitude. The sliding results in the data points of the various passes being
within a range of ±A/2 where A is the interval between successive data points (-6.7 km).
As mentioned in the last section, geoid contamination is a problem in regions of large geoid
variability, which are identified and excluded from the analysis. To check the sensitivity of
the difference profiles and variability calculations to the sliding scheme, the algorithm was
performed several times on a collinear ensemble, with slight perturbations in the sliding. It
was found that variability calculations and the difference profiles were affected only very
slightly by such changes except in areas of large geoid gradient.

The difference profiles were compared with variability calculations and
wavenumber spectra to ensure that all were corroborative. For example, a difference
profile which showed a significant long wavelength variation would be found to have a
large RMS alongtrack variability and high power density at long wavelengths. As a
quantitative check, the area under several power density spectra was determined for
comparison with the calculated alongtrack variance (RMS variability squared). No
discrepancies were noted.

58

2.3 THE TEST ARCS

Once the proper operation of the collinear algorithm was assured, three arcs within
the area were chosen to investigate the efficacy of the correction channels when applied to
the sea surface heights. These were arcs 82, 209, and 293. They were chosen for length,
good coverage through a number of repeat cycles, and to sample different portions of the
area. They are specially marked in figure 1.3.

Arc 82 is a descending arc, and its mean profile was formed from 10 repeat passes
(numbers 1-6,8-10,13). Eighteen repeat cycles were available at the time, and the eight not
used were rejected due to numerous data dropouts. Arc 209 is an ascending arc whose
passes during a given repeat cycle occur some 4 1/2 days later than those along arc 82.
This arc experienced fewer dropouts, enabling 14 repeat passes to be used in creating the
arc 209 mean (numbers 1-6,8-14,17). Arc 293 is an ascending arc, whose passes are
about 3 days later than those of arc 209. Arc 293 experienced no major dropouts
throughout the 18 repeat cycles and so all passes were used in forming its mean. The mean
sea surface height profiles along these three arcs, with no correction channels applied, are
shown in figures 2.1 through 2.3.

For each arc, the effects of making the various corrections were investigated in the
following way: First, the mean profile was examined to spot those areas where the mean
sea surface changed rapidly. This examination was done even though the Rapp geoid was
already removed, because the features on the mean were still largely a result of the real
geoid (because the Rapp model is of too low an order to accurately reflect shorter
wavelength geoidal variations). Thus, those areas where the mean sea surface was highly
variable with horizontal distance were the same areas where possible geoid contamination
of the difference profiles could be expected. A simple look at the mean profile often
sufficed here, though a first difference plot of the profile was also useful. Those areas
where geoid contamination was suspected were then examined on a set of non-corrected
difference profiles. If strong signals habitually appeared there, that particular latitude band

59

was eliminated from the arc (the test arcs were chosen to minimize this possibility). This
elimination was done in the southern region of arc 82 (Hawaiian islands) and the
northernmost part of arc 209 (Aleutian chain). While some oceanographic features may
have been lost in the process, that was considered acceptable since the corrections, and not
the oceanography, were the focus at this point.

With geoidal contamination largely removed, the algorithm was first used to create a
set of difference profiles (one for each pass that helped form the arc's mean profile), with
no correction applied in step 3. For each difference profile, the RMS alongtrack variability
(a) was calculated. The set of a were then averaged together to determine the mean
alongtrack variability of the difference profiles on the arc (o). This mean variability was
used in evaluating the effect a particular GDR correction had on the ensemble of difference
profiles for the test arc. An additional statistic computed was the sample standard deviation
of the individual a about CJ (SSD). The SSD was essentially a measure of how different
the alongtrack variablities of the individual difference profiles were from each other. This
statistic was of some use in evaluating random corrections, but was most helpful when
used in conjunction with o>, in cases where the mean profile was aliased. The statistics for
the no-correction case comprised the standard against which analogous statistics were
compared after rerunning the algorithm separately for each possible correction. The results
of these runs for the three test arcs are listed in tables 2. 1 through 2.3, and are discussed
individually.

Note that this work uses RMS variability (or standard deviation) instead of
variance. This choice was made because much of the GEOSAT literature does likewise,
and the altimeter errors under investigation have uncertainties typically given as RMS
values.
2.4 ARC 82

As previously mentioned, the mean profile for arc 82 is shown in figure 2.1. The
difference profiles for the 10 passes used to make the mean are shown in figures 2.4 and

60

2.5, with no GDR corrections applied. Note how the region of high geoid variability to the
south has been deleted. I refrain from looking for mesoscale features at this point and
focus on the GDR corrections.

Table 2.1 lists the statistical results for the various corrections applied to the sea
surface heights of the repeat passes along arc 82. As an example of how to use this table,
note the row corresponding to repeat 4. With no corrections applied to any passes, the
difference profile for this repeat had an alongtrack RMS variability of 9.58 cm. When the
EM BIAS correction was applied throughout, the RMS variability for the repeat 4
difference profile increased to 1 1.43 cm. Now observe the last two rows denoting all
repeats. With no corrections applied, the mean of the ten difference pass variabilities was
7.28 cm with an SSD of 1.91 cm. When the EM BIAS correction was applied, o rose to
7.34 cm, and the SSD rose to 2.21 cm. The significance of these numbers will be
explained shortly.

For a random phenomenon, alongtrack variability should almost always decrease
upon the application of a well-modelled correction, as the correction will reduce an apparent
perturbation to the sea surface. Accurate corrections are desirable in that they can remove
non-oceanographic features from the difference profiles and so make it easier to spot the
oceanographic ones. If the phenomenon is poorly modelled, variability will tend to
increase upon correcting, and oceanographic signatures may be obscured. The behavior of
the variability can best be stated in terms of the coherent and incoherent power as follows:

Assume that the sea surface observed by the altimeter (o), is the desired surface (d),

plus some phenomenon (x), which may be modelled. We wish to remove x from o, in

order to study d. Thus

d = o - x

Fourier transforming and taking the complex conjugate of both sides gives

*

d d = (o-x)(o-x)
Expanding the right hand side and taking expected values throughout produces

61

* * * * *

<dd > = <oo > - <ox > - <o x> + <xx >
The terms within the brackets may be replaced by the appropriate spectra, giving

<*>dd = $00 -2Re[O0x] + ®xx (4)

where O^ is the power spectrum of the desired sea surface profile, O0o is the power
spectrum of the observed (difference) profile, Oxx is the power spectrum of the modelled
correction for the particular profile, and 00x is the cross power spectrum between the
observed profile and the modelled correction. It is assumed that the observed profile is
incoherent with any modelling error. If the model of the phenomenon accurately reflects its
true effect on the sea surface, then Oox^rOxx and (4) becomes

Odd^Ooo -3>xx (5)

Thus, the variance (and hence the alongtrack variability) of the corrected profile (d) will
tend, on the average, to be less than that of the uncorrected profile (o) when the correction
is very accurate. On the other hand, if the model of the phenomenon is a poor one, its
profile will bear little resemblance to the observed sea surface profile, meaning that Oox-0.
In that case, (4) becomes

^dd-^oo + ^xx (6)

Alongtrack variability in general increases when the model of the phenomenon is
inaccurate.

This discussion has implicitly assumed that the mean surface from which the
observed difference profiles are created has not had the phenomenon aliased into itself.
When this occurs, additional terms appear on the right hand side of (4). The short time
scale and aperiodic nature of all the non-tidal corrections suggests that they do not cause
this difficulty, however; and aliasing of the tides is treated separately in this chapter.

The results of (5) and (6) indicate how the statistics mentioned earlier may be used
to judge the effectiveness of a particular GDR correction. If the correction accurately
models the phenomenon in question on a particular repeat, then that repeat's difference
profile should have a lower a than without the correction. If the correction does well over

62

the ensemble of repeats, then a will be lower than for the no-correction case. In addition,
if one uses the a priori knowledge that the test arcs are in a fairly quiet region of the Pacific,
then removing non-oceanographic signals with an accurate correction will make the
difference passes look more alike. This will result in similar variabilities between the
repeats, and a smaller SSD after correcting. Of course, if the phenomenon has very small
variability to start with, then O^ will be very small, and correcting for it will have little
effect on the difference profiles. In that case, there is no need to bother making the
correction.

Referring back to the statistics on EM BIAS mentioned earlier, one can conclude
that this correction was not well modelled for the repeats on arc 82.

The first step taken in examining the correction channels as they affected arc 82 was
to identify those which made no difference. To this end, a simple criterion was used: If the
effect of a correction on the variability of a difference profile did not exceed an assumed
instrument noise of 3 cm (RMS) the correction was deemed to have made no difference. 3
cm was used instead of the published 3.5 cm in order to reject fewer values. To illustrate
this, again consider repeat 4 in table 2.1, with an RMS variability of 9.58 cm. Adding and
subtracting noise of 3 cm (RMS) produces upper and lower bounds of 10.04 cm and 9.10
cm respectively (combining RMS values entails taking the square root of the sum or
difference of the squares of the values). The only corrections which caused the difference
profile variability of repeat 4 to exceed these bounds were EM BIAS and WFNOC, which
were thereby judged to be 'significant' corrections.

By the stated criteria, the STIDE, WSMMR, and IONO corrections made no
difference for arc 82. This judgement, based on the statistics of table 2.1, was also
corroborated with difference profile printouts. In no instance was there any discernable
difference between the profiles before and after making the indicated corrections. Figures
2.6a and 2.6b compare spectra of a typical difference profile (rep 9 in this case), with the
spectra of its STTDE and WSMMR correction channels (the IONO channel variability was

63

very slight and produced a power spectrum orders of magnitude lower than those of the
STIDE and WSMMR channels). Note how, even at long wavelengths, the variance of the
correction channels is some two orders of magnitude less than the variance of the difference
profile. The low power of the corrections indicates that they do not vary much over the
arc, and as mentioned, this is why they had no significant effect when applied. As an
aside, note the red spectrum for the difference profile, which is characteristic of oceanic
processes.

Using table 2.1, the remaining corrections were examined by finding those repeats
where the corrections had the most significant effect on variability, in comparison to the
noise bounds. For each correction, four repeats were chosen - the two with the most
significant variability reduction upon application of the correction, and the two with the
most significant variability increase. The appropriate repeats for each correction appear in
figures 2.7 through 2.10. On each figure, the corrections for the individual repeats are
plotted with the correction causing the most significant variability reduction on the top,
down to the correction causing the most significant variability increase on the bottom.
These figures were used to try and determine what factors are involved in whether or not a
given correction will cause a variability increase or decrease.

Electromagnetic Bias

Note that figure 2.7 is a plot of significant wave height for the listed repeats, which
is the source of the EM BIAS correction (the latter being 2% of SWH). Two observations
may be made from these plots: First, the EM BIAS correction appears to do very poorly
when the SWH is large. Repeat 4 has much higher SWH than the other 3 repeats pictured
in figure 2.7, and also had the most significant variability increase upon application of the
EM BIAS correction. Figure 2. 1 la is a spectral plot comparing repeat 4's difference
profile before and after correcting for electromagnetic bias, and shows that the variability
increase occurred mainly in the long wavelengths (>400 km), though the confidence limits
prevent one from stating this with conviction. A better indicator of which difference profile

64

wavelengths would be most affected by the EM BIAS correction may be seen in figure
2.1 lb, which compares the spectra of repeat 4's difference profile and its EM BIAS
correction. Here it is very evident that only at wavelengths around 400 km and longer,
does the EM BIAS variability approach that of the difference profile. Before proceeding, it
must be pointed out that the converse of the first observation on electromagnetic bias is not
true. That is, a small SWH does not guarantee variability reduction upon correcting for EM
BIAS. See repeat 13 for an example of this. A plausible reason for the variar- "ity increase
when SWH is large is presented in the section on arc 209.

The second observation made from figure 2.7 is that when SWH is large, the short
wavelength noise of the difference profile is increased. From figure 2.7, compare repeat
13, having small SWH alongtrack (~2 m); with repeat 4, that has peak SWH values
around 10 m. From figures 2.4 and 2.5, note how much quieter repeat 13's difference
profile is at the shortest wavelengths compared to repeat 4's. This effect can be seen more
quantitatively in figure 2.12, where, through most of the short wavelengths, repeat 13's
power density is around an order of magnitude less than repeat 4's. As a final example,
one may even note that the latitude where repeat 4 has its largest SWH (~33°-46°N) is the
area with the largest spikes on its difference profile. The most obvious reason for this
phenomenon is that high SWH worsens the point-to-point precision of the altimeter itself.
Refer back to figure 2.12; if one considers the generally white portion of both spectra
(X<40 km) as system noise, one can graphically integrate under each spectrum from X =
40 km to 14 km (the Nyquist limit) and obtain the following estimated noise-induced
variances: for repeat 13, 4.6 cm^, and for repeat 4, 19.5 cm^. These variances
correspond to altimeter precisions of 2.1 cm and 4.4 cm (RMS), respectively. Repeat 13
does better than the rated noise floor of 3.5 cm and repeat 4 does worse, due to the high
significant wave height. A useful pursuit in future studies would be to empirically find the
altimeter noise floor as a function of SWH, especially for the larger wave heights.

65

Electromagnetic Bias is a rich enough subject to warrant further discussion in this
work, though based on the ensemble statistics thus far, it is not recommended for
application to the sea surface heights.

Ocean Tide

Figure 2.8 shows the repeat cycles whose OTIDE corrections had the most
significant effect on variability. It is plain that this effect is due to the long wavelength
curvature of the ocean tide profiles along arc 82. Of all the correction channels, OTIDE
caused the largest changes in difference profile variability upon application. Figure 2.13a
shows the spectrum of the repeat 10 difference profile before and after OTIDE correcting,
and indicates that the wavelengths affected are the longer ones (>200 km). As with EM
BIAS, the confidence limits make the results in the 200-400 km wavelength band
somewhat ambiguous, and so figure 2.13b compares the difference profile spectrum with
the OTIDE channel spectrum. Here, the confidence limits are less of a problem, and one
sees that the OTIDE power density is much too small compared to the difference profile
power density at 200 km wavelength to have any significant effect. The actual wavelengths
affected by the OTIDE correction are thus more likely 300-400 km and longer, as one
would expect due to the long wavelength nature of this phenomenon. The OTIDE profiles
in figure 2.8 indicate that the tide is probably being time aliased. This can be seen in the
similarity of the tidal profiles between repeats near to each other in time.

Examining the ensemble statistics in table 2.1, the OTIDE results at first appear
confusing. On one hand, o rises when the OTIDE correction is made, which is not
desireable. In contrast, however, the SSD falls after the data is corrected for ocean tide,
indicating that the repeats tend to look more alike in terms of variability. As mentioned
before, if one accepts that this is a quiet portion of the Pacific, then this similarity is
desireable. A clear example of how the OTIDE correction makes the difference profiles
more alike may be seen in figure 2.14. Figure 2.14a shows spectra of the two difference
profiles with the most difference in variability before OTIDE correcting (repeats 3 and 10).

66

Figure 2.14b shows them after OTIDE correcting. Note how the long wavelength
variability of repeat 10 has been reduced to the point where the two repeats, within the
confidence limits, have essentially the same power spectrum.

As will be shown later, the ambiguous ensemble statistics are a characteristic of
aliasing, which will be examined in more detail in the arc 293 section.

Wet FNOC

Table 2. 1 shows that, while some repeats had a significant change in variability
upon application of the WFNOC correction, the change was not large, and was just as
likely to be a variability increase as a decrease. Figure 2.9 does not indicate any special
criterion for determining whether the WFNOC correction will increase or decrease the
variability, and the ensemble statistics indicate that the correction really did not make much
difference anyway. This result is supported by figure 2.15, which indicates that the power
density of the WFNOC correction is small compared to even a relatively quiet difference
profile (repeat 9). For these reasons, the GDR WFNOC correction is deemed unnecessary,
pending investigation of the remaining test arcs.

Of note at this point is the distinction between the modelled WFNOC correction and
the true wet troposphere delay correction. The former had little effect on the difference
profiles, while the latter is often much more robust. That is, true wet tropospheric effects
of short wavelength and large amplitude may easily be mistaken for oceanographic signals.
Until coincident measurements of this phenomena are available from radiometers, it appears
wisest to avoid making the inherendy smooth model-derived corrections.

Dry FNOC

As opposed to WFNOC, where variability increased or decreased with near equal
regularity, table 2.1 indicates that the DFNOC correction yielded a consistent, albeit small,
variability decrease. Figures 2.16a and 2.16b indicate the affected wavelengths as being
around 400 km and greater, which is to be expected from the gentle undulations observed
in the repeats plotted in figure 2.10. With the exception of repeat 10 however, none of the

67

variability decreases even approach the noise level, and difference profiles made before and
after DFNOC correction showed only the most minor differences - none of which could be
mistaken for oceanographic signals. For this reason, unless the subsequent test arcs show
otherwise, making the DFNOC correction is deemed not to be worth the effort.

Arc 82 Summary

Based upon statistical and spectral methods, the Solid Tide, Wet FNOC, Wet
SMMR, Dry FNOC, and Ionospheric GDR corrections were determined to be unnecessary
over the length of the arc. Over a longer arc, such might not be the case, although the
methodology presented herein would still be applicable. These judgements were fully
corroborated by the difference profiles which showed that the phenomena (at least as
modelled for the GDR) could not be mistaken for oceanographic signals.

The Ocean Tide correction had the greatest effect on difference profile variabilities,
though the results were not always favorable. Aliasing was suspected and will be treated in
more detail later.

The Electromagnetic Bias correction also showed some significant effects on the
variability, though the ensemble results indicated the danger of indiscriminandy correcting
the data. I examine this in some detail as the next test arc is discussed.
2.5 ARC 209

The mean sea surface height profile along arc 209 is shown in figure 2.2.
Representative difference profiles of 5 of the 14 passes used to make this mean are shown
in figure 2.17. Again, no corrections (other than spike editing and detrending), were
performed to the sea surface heights of these profiles, and a region of high geoid variability
was deleted. In passing, note the short-to mid-wavelength features in the southern region
of profiles 5, 8, 9, and 10. This region is that of the Musician Seamounts north of Hawaii,
and the features were not eradicated by any corrections. Residual geoid contamination may
be ruled out because the features are seen to move some 100 km between repeats 8 and 10,

68

leaving oceanographic signals as a promising source. I refrain from further discussion of
oceanography for the present.

Table 2.2 shows the effects of the corrections on the difference profile variablities
for arc 209, and like table 2.1, indicates that the STIDE, WSMMR, and IONO corrections
make no real difference (repeat 17 experienced a variability decrease just outside the noise
bound after being WSMMR corrected, but this was the only instance of this, and the
ensemble statistics are nearly identical with those of no correction). As before, the
WFNOC and DFNOC corrections occasionally cause a significant change in variability on a
difference profile, but their ensemble statistics continue to differ only slightly from those of
the no correction case. Note that the 7.78 cm mean variability after WFNOC correcting is
only about a 3% decrease from the no correction figure of 8.04 cm, while the DFNOC
correction decreases c by less than 1%. Spectral plots and difference profiles concur that
none of the above corrections need to be made.

The two corrections which bear further study are OTIDE and EM BIAS. As was
seen for arc 82, the former correction has the most significant effect on individual
difference profiles; either by increasing or decreasing variability. Spectral plots for arc 209
concur with earlier results showing the OTIDE correction as affecting wavelengths around
300 km and longer. In contrast to earlier, however, observe from table 2.2 how the mean
variability is decreased upon OTIDE correcting, now that more repeats are used to form the
mean profile. This behavior continues to indicate that the ocean tide is aliased by the 17
day sampling period, which is no surprise. (As a further check, an arc 209 mean was
formed by only using 10 repeat cycles. In this case, the mean variability increased by 4%
upon OTTDE correcting). These results will be explained in detail in the arc 293 section.

Electromagnetic Bias

Figure 2.18 shows SWH for five representative repeats along arc 209. Repeats 9
and 5 both experienced a significant increase in variability when the EM BIAS correction
was made, whereas repeats 10, 3, and 2 had their variability reduced. This figure supports

69

the same two observations made previously - repeats that have the greatest SWH fare the
worst after the EM bias correction, and the areas where SWH is largest also show the
greatest noise on the difference profiles. The wavelengths affected most by the EM BIAS
correction continue to be around 400 km and longer.

Since the EM BIAS correction can have a significant effect on the alongtrack
variability for a given repeat, a better understanding of it is essential. I begin by looking
more closely at those repeats where correcting for EM BIAS increased the variablity. Note
on the difference profiles of repeats 5 and 9 (figure 2.17) that the sea surface height is
higher than the mean profile in those areas where SWH is largest. If EM BIAS was the
major effect here, one would expect just the opposite. That is, large waves should depress
the non-corrected difference profiles below the mean surface. I suspect that the EM BIAS
correction failed because the sea surface was actually responding to the larger inverted
barometer (IB) effect. To test this assertion, a closer look was taken at the situation in the
area at the time of repeat 5, since its SWH plot shows such a prominent feature.

Figure 2.19 is the meteorological surface analysis in the area at 12Z, 22 January
1987; 3 hours prior to repeat 5 on arc 209. The dominant feature is the strong LOW
pressure system centered at 50°N, 166°W.* The small triangles indicate its position and
strength at 12 and 18 hrs prior to 12Z. The large values of SWH on repeat 5 from around
38°-49°N seen on figure 2.18 may be corroborated by the isobaric spacing along the arc.
Observe how the isobars are closest together in the same latitude range, indicating the
highest winds which would give rise to the largest waves. If the SWH profile for repeat 5
thus looks appropriate, then why didn't the EM BIAS correction work? The answer lies in
the pressure field. As one approaches the center of the LOW while travelling on the arc,
the sea surface will rise according to the IB effect. From the surface analysis, this effect

The LOW can also be seen in a plot (not pictured) of the DFNOC correction, which
reduced the RMS variability of the repeat 5 difference profile by 7.4%, when applied (see
table 2.2).

70

should be a maximum at around 49°N, which is what the difference profile indicates. If
one considered only EM BIAS, one would expect the uncorrected sea surface to be at its
lowest where SWH is highest - near 47°N in this case. The fact that this is not seen is
because the IB effect is of opposite sign and greater magnitude than EM bias. Thus the sea
surface, to lowest order, follows the IB effect and EM BIAS is not appropriate. (In this
case, the maximum IB effect is 1013 mb - 972 mb = 41 cm to be subtracted, and the
maximum EM BIAS is 1200 cm x .02 = 24 cm to be added. I do not recommend making
the two corrections jointly, as will be discussed shortly).

A point by point inverted barometer correction was calculated by first recovering
sea level pressure from the GDR DFNOC correction and then using the 1.01 cm/mbar
relation mentioned earlier. The results of this correction, as applied to arc 209, appear in
the rightmost column of table 2.2. Prior to examining the results in this column, it is
important to recall a point made in Chapter 1 ; namely, that care must be taken in
investigating the IB effect due to its close connection with the Dry Troposphere correction.
This can be seen in the following way: if the Dry Troposphere correction is made to the sea
surface heights before making the IB correction, and the Dry Troposphere correction is not
perfect; it will introduce a false pressure signal into the sea surface heights. Since this same
pressure signal is used to compute the IB correction, there will be a false coherence
between the observed profile and the modelled IB correction - manifested by a change in
the Oox term in equation (4). This may invalidate the results of (5) and (6), making
determination of the success of the IB correction very difficult. To avoid this problem here
and throughout, all corrections were investigated individually.

In the case of repeat 5, the IB correction significandy decreased variability, by some
26%. One can see from the plot of the repeat 5 difference profile after correcting for IB
(top of figure 2.20), that this variability decrease was indeed a result of taking out the
gentle rise in the sea surface due to the LOW. Such a simple correction does not always
bring success, however. In the case of repeat 10, variability was increased by 40% after

7 1

IB correcting. From the repeat 10 difference profile after correcting for IB (middle of
figure 2.20), note that the IB correction added a convex signature centered around 36°N.
Figure 2.21, a diagram of surface analysis 3 hours prior to overflight, shows this to be a
consequence of the HIGH pressure system located there. The problem is that the
uncorrected difference profile for repeat 10 shown in figure 2.17 is basically flat in this
region - indicating that the sea surface did not respond in an IB sense to the HIGH, and that
a correction is therefore inappropriate. Why the difference between repeats 5 and 10?

As alluded to earlier, the IB effect is a function of the time and space scales of the
forcing from the surface pressure field. Looking at the spatial scales of the LOW and
HIGH, it is apparent that the HIGH has a much larger spatial scale than the LOW. I assert
that the LOW is small enough for long gravity waves to quickly communicate its presence
throughout the affected ocean area, resulting in a largely static IB response. This
communication is not possible over the much larger HIGH, and so oceanic response is
more dynamic in nature in this case. This prevents the simple IB correction from working
well for repeat 10. This can be quantified using the result of Wunsch (1972) that the
Rossby radius of deformation is the upper spatial limit of an expected static IB response.
The Rossby radius of deformation is defined by the ratio of the wave speed to the Coriolis
parameter, and may be thought of in this context as the shortest spatial scale at which the
geostrophic balance may occur. Long deep water waves travel at an approximate speed of
(gh)1/2, where g is the gravitational acceleration and h is the water depth (about 5500 m
under arc 209). The Coriolis parameter at mid latitudes is about 10"4 s_1, giving an
approximate barotropic Rossby radius of 2300 km for the region in question. Spatial
scales for the LOW and HIGH may be estimated from the diameter of the largest closed
(and reasonably symmetric) isobar, giving 1 100 km and 2500 km respectively. The LOW
forces the sea surface on a scale less than the Rossby radius, and so an inverted barometer
response is possible. The forcing scale of the HIGH exceeds the Rossby radius and so the

72

pressure gradients tend to be supported by geostrophic currents - preventing a static IB
response.

Another scenario which may be envisioned relates to the time scale of the IB
response. Suppose that a pressure system with a spatial scale otherwise conducive to a
static IB response rapidly approaches a groundtrack in the hours/days preceding overflight,
being quite close to that arc at the time of overflight. The pressure field at overflight from
which the IB correction is calculated will thus result in a large calculated correction,
whereas the true IB response will be small, since the ocean hasn't yet had time to respond
to the changing surface pressure. A similar outcome may result from a system which
rapidly strengthens prior to overflight. I believe a combination of these effects accounts for
the variability increase upon IB correcting repeat 8. As can be seen in figure 2.22, a
strengthening LOW approached from the Southwest prior to the repeat. The IB correction
assumed that the sea surface had statically responded to the alongtrack pressure profile at
the time of overflight and thus depressed the sea surface in the northern region to
compensate for the LOW (compare figures 2.17 and 2.20). In reality, though, figure 2.17
shows that the sea surface had not risen in response to the LOW, and thus the IB correction
was inappropriate. (Note that the surface analysis for repeat 5 in figure 2. 19 was for 3
hours prior to overflight, giving the ocean more time to respond in that instance and
resulting in a smaller calculated IB correction, since the LOW's center would have been
further from the arc. These factors, in addition to the relatively constant strength of the
LOW in that case could explain why the IB correction worked well for repeat 5 and not for
repeat 8).

Overall, the application of the pressure-derived IB correction to arc 209 resulted in a
mean variability increase of about 4%. As just alluded to, the inefficacy of this correction
over the ensemble of passes is due to the fact that the ocean's response to surface forcing is
a complicated function of the forcing scales and not just a simple reflection of the
alongtrack pressure profile at overflight. For this reason, I recommend not making the

73

inverted barometer correction, at least in its simplest form, to the GEOS AT sea surface
heights. A better approach would be to determine a frequency/wavenumber dependent
correction.

Returning to EM BIAS, note that its "failure" in regions where SWH was greatest
has been shown, in the instances studied, to have been the result of being overpowered by
the IB effect (other cases like repeat 9, and repeat 4 of arc 82, were also investigated, with
similar results). Because the latter effect is so difficult to predict in a simple fashion,
neither can one say, a priori, when the EM BIAS correction will be desireable. Since the
indiscriminate application of this correction, as showed by table 2.2, can result in a mean
variability increase, it is also recommended that the EM BIAS correction not be made.

Arc 209 Summary

Based on the first two test arcs, all corrections except OTIDE are deemed
unnecessary or undesireable. The OTIDE correction is examined in detail in the following
section, wherein final judgement on all corrections is also made.
2.6 ARC 293

The mean profile for arc 293 is shown in figure 2.3. The first five difference
profiles of the 18 repeats used to form that mean are shown in figure 2.23. Geoid
contamination was not enough of a problem to warrant the deletion of any areas.

Table 2.3 presents the results of applying the various corrections to arc 293. It is
immediately apparent that the IONO correction is again insignificant, both from individual
repeats and ensemble results. Four other corrections are also seen to be unnecessary,
based on ensemble results. These are STIDE, WFNOC, WSMMR, and DFNOC. While
these corrections did affect some individual difference profiles beyond the noise bounds; it
was never by much, was often harmful to the variability, and always involved repeats that
had a large variability before correcting. Such repeats have narrower noise bounds than
those which have small variability before correcting, meaning that more correction results
will be tagged as being significant purely as a matter of course. The EM BIAS correction is

74

not listed because it was compared to a mean profile formed from only 14 repeats (due to
data problems in the SWH channel). It continued to behave as before, however, having
significant effect on several difference profiles; but doing poorly over the ensemble. Such
was also the case with the IB correction, as shown by table 2.3.

Thus far, the results of arc 293 completely support the earlier assertions that all
corrections except OTIDE are either unnecessary or undesireable. Though not included as
figures, numerous spectra and difference profiles before and after correcting corroborate
these findings.

Ocean Tide

Table 2.3 shows that the OTIDE correction significantly reduced the ensemble mean
variability and standard deviation, as well as having a good deal of effect on the individual
difference profiles. Note from figure 2.23 how the first five difference profiles look before
OTIDE correcting. Significant long wavelength variations are evident in repeats 1, 2, and
3, which account for the high variability seen in table 2.3. Now observe figure 2.24 and
note how these variations largely match the ocean tide signals alongtrack. After the OTTDE
correction was made, figure 2.25 shows the long wavelength signals were greatly reduced,
as was the variability in table 2.3. Spectral analysis concurred with earlier estimates of
wavelengths >300 km being affected by the correction.

From the results of this arc, the OTIDE correction looks like a desireable one.
Before making a final judgement though, I seek an understanding of why the OTIDE
correction increases variablity in some repeats, and why the ensemble results for arcs 82
and 209 were not nearly so positive. The answer is due to the aliasing of the ocean tide
introduced by the way GEOSAT samples the ocean.

GEOSAT passes over a given arc every 17.05 days, so employing the collinear
method to produce an arc mean profile entails using snapshots of the ocean surface at
widely spaced time intervals. If some non-random signal affects the surface and does not
change appreciably from one repeat to the next, it will be aliased into the mean profile. To

75

understand exactly how this aliasing affects the ocean tide, it is necessary to take a closer
look at GEOSATs orbit.

Following Parke et al. (1987), the earth's equatorial bulge causes GEOSAT's
orbital plane to precess at a relatively slow rate Q, given by

n=~lJ2(3)1/2[_ R!2 ]2cosi (sl) (7)

a a ( 1 -e )

where i is satellite inclination, a is the mean semimajor orbit axis, e is the mean orbital
eccentricity, |1e is the gravitational constant times the earth's mass, J2 is the second zonal
harmonic of the earth's gravity field, and Re is the earth's equatorial radius. The sign
convention is that Q is positive for a retrograde orbit such as GEOSAT's. As mentioned in
the previous chapter, e for GEOS AT is only 0.0008, allowing (7) to be simplified as

-7/2
7 - - 1

Q = - 4.17x10 a cosi (s ) (g)

where a is in km. For GEOSAT, with i ~ 108.04 (varying periodically), and a = 7167.4
km, (8) gives Q = 4.144 x 10~7 s~l as the precession rate of the GEOSAT ascending node
(GEOSAT orbital values from Born et al., 1987).

GEOSAT completes 244 revolutions in 17 days, where in this context, a day is the
time it takes the earth to rotate 360° with respect to the GEOSAT ascending node. The
conversion between this nodal time and actual clock time is given by

2k

tclock= tnodal (9)

co-O

where co is the earth's sidereal rotation rate of 7.2921 15 x 10"^ s~*. Thus, the actual repeat

period P is slightly longer than 17 days by the conversion factor, giving P = 1,473,162.2

sec (= 17.05 days or 409.211722 hrs).

76

The principal alias period of any tidal component may be obtained from the change
of phase of that component over the repeat period. The phase change of a tidal component
of period T over the repeat period is just

A$ = ^- (-JC to 71)

T (10)

and the resulting principal alias period X is

x = abs(^-) (11)

A0 v }

again, following Parke et al. (1987).

A sample calculation for the M2 tidal component's principal alias period will

illustrate the use of (10) and (11): The M2 tide has a period of T = 12.420601 hrs, and

using P = 409.21 1722 hrs, equation (10) gives A<|> = 207.0071392. This A0 actually

includes a number of complete cycles, so to find the -tx to k phase change, one subtracts

from it the nearest integral multiple of 2tc (in this case, 2k x 33 = 207.3451 152). This

gives A(j)]vj2 = -0.337976 and indicates that between any two "snapshots" of an arc, taken

17.05 days apart; the M2 tide will have cycled nearly 33 times. In fact, it will have cycled
so close to 33 times exactly, that its apparent phase change A(j)^j2 W1^ De extremely small.
Plugging the value of A<{)M2 into (11) gives an alias period of Tj^2 = 7607.5 hrs or 316.98
days. Thus, the small apparent change in the M2 tide over a repeat period means that the
satellite must sample the arc more than 18 such periods to see the M2 tide complete one
apparent cycle. This is a serious alias, especially since M2 is by far the largest tidal
constituent. Table 2.4 lists the results of alias calculations for M2 and the next three largest
constituents.

The aliasing of the ocean tide is readily seen in figure 2.24, as the OTIDE profiles
vary slowly from one repeat to the next. Aliasing is not a problem for arc 293 however, as
its mean profile was formed over a 306 day span ( 1 8 repeats), and so sampled nearly all
possible tidal profiles. Contrast this with the arc 82 and arc 209 means - formed from 10

77

and 14 repeats, respectively. I would expect arc 82 to be the most seriously aliased,
followed by arc 209 with minor aliasing. This assertion is supported by the results shown
previously. The ensemble mean variability suffered the most upon OTTDE correction on
arc 82, rising some 5%. It fared better on arc 209, falling by 1%; and did best on the non-
aliased arc 293, falling by 32%. A discussion of the mechanisms by which this happened
follows.

Equation (4) showed the relationship in the Fourier domain between the observed
difference profile, a modelled phenomenon, and the desired profile after the phenomenon
was removed. It is written again here for reference:

3>dd = 3>oo - 2Re[O0x] + Oxx (4)

Now the observed profile is actually computed from each observed repeat profile r,
minus the mean profile m. Thus o=r-m, and the first two terms on the right hand side
expand to

and

<D00=<(r-m)(r-m)*>
= Oir-2Rc[0>J + Oinm

Oox=<(r-m)x >
= 0 . 0

rx mx

(12)

(13)

allowing (4) to be rewritten as

<&„= 0 +0 +0 -2RerO +0 -0 1

(14)

where the subscripts d,r,m, and x respectively denote the desired difference profile after
correcting for the phenomenon, the observed repeat profile, the mean profile, and the
modelled correction for the repeat. The 0 with repeated subscripts are power spectra and
0 with different subscripts are cross power spectra.

If the phenomenon being modelled is essentially random over the length of time
used to form the mean profile, then that mean will be uncorrelated with the phenomenon on

78

any given repeat, meaning that Omx~ 0. Similarly, the mean will also be uncorrelated with
each repeat profile, giving Cv^O.* From (12) and (13), one sees in this case that
^n+ ^mm-^oo and ^rx-^ox- Thus, when the phenomenon is random, the full
expression (14) reduces back to (4), which was the special case derived earlier.

The full form of (14) is the key to understanding the statistical behavior of the ocean
tide correction over the three test arcs. Arc 293's mean was formed from repeats which
represented a full set of ocean tide phases (summing to 2k), and so would be uncorrelated
with any repeat profile or its modelled tide. Thus Orm^O and Omx-0, allowing this arc
to be described by the random phenomenon equation (4). As derived before, the accuracy
of the phenomenon model may be easily evaluated in this case by observing the effect that
the correction has on the RMS variability of each repeat's difference profile. Since the
mean variability o~ fell substantially after applying the OTTDE correction to the passes on arc
293, it can be concluded that the tidal model is a good one, at least in the vicinity of that
arc.

The other two arcs had means formed from partial sets of possible tidal phases,
with the most common phase thus aliasing into the mean profile. From (14), it is apparent
that the RMS variability calcuations are now a function of the aliasing as well as the model
accuracy, and several options are possible. The simplest is the utilization of a near-perfect
tidal model. In this case, the model completely removes the tide from each repeat used to
form the mean profile, and so the mean will be uncorrelated with any observed repeat
profile or its modelled tide. That is, Onn-0 and Omx:^0. The observed repeat profiles
will be highly correlated with the modelled phenomenon,and so Orx:^Oxx. Using these
results, and the relations in (12) and (13), equation (14) reduces to Odd-^oo-^xxi the

Actually, the mean is always correlated with each repeat to some extent, due to the mean
sea surface topography. This natural correlation is ignored here, since it is independent of
the phenomenon being investigated.

79

random case with excellent modelling, derived before. This result indicates that, if the tide
model is good enough, aliasing is not a factor.

The statistical results from arc 293 indicate that, while the ocean tide model is a
good one, it is not nearly perfect (as seen in the variablity increases of some repeats after
correcting for OTIDE). Considering (14) with this knowledge, two limiting cases apply in
the study of the statistical behavior of arcs 82 and 209. The first is for a repeat having a
tidal phase similar to the one aliased into the mean. In this case, the repeat profile and its
modelled tide will both be positively correlated with the mean; Orm>0 and Omx>0. Since
the model is a good one, once again O^^O^. Thus (14) may be rewritten as

^dd^Orr+Omm^RetOrmJ-Oxx + 2Re[Omx] (15)

where all O are positive. The power spectrum of the observed (i.e., non-corrected)
difference profile, as shown in (12),. is just the first three terms on the right hand side of
(15), where the amount of aliasing determines Omv If the repeat has a tide very similar to
the one aliased into the mean, Ormwill approach Omm, giving Ooo^On-Omm and the
uncorrected difference profile will have low power and variability. This explains what
happened with repeats 1-3 on arc 82, for example, as that arc's mean was formed from a
number of early repeats with tidal phases similar to theirs. Note in table 2.1 the low RMS
variability of these repeats before correcting for ocean tide, as expected. This behavior is
more difficult to spot in the arc 209 repeats, as that mean was aliased to a lesser extent than
arc 82's. Nevertheless table 2.2 indicates that repeats 8, 10, and 13 on arc 209 may have
had their tides aliased into the mean.

The effect of the modelled correction in the first limiting case is described by the last
two terms in (15). The familiar Oxx term indicates the usual tendency of a well-modelled
correction to reduce the variance. The new <£>mx term, however, indicates that even a well
modelled correction will increase the variance of the difference profile to which it is
applied, if it has been aliased into the mean profile. The balance between Oxx and
2Re[Omx] determines whether there will be a variability increase or decrease upon making

80

the correction. If the repeat tide highly resembles the one aliased into the mean, then Omx
will approach Oxx, and the variability will increase. The highly aliased first three repeats
on arc 82 again illustrate this result, as may be seen in table 2.1. When the OTIDE
correction is made to them, their RMS variability increases. Repeats 8, 10, and 13 on arc
209 also show this behavior, as indicated in table 2.2.

The second limiting case to (14) is for a repeat having a tidal phase very dissimilar
to the one aliased into the mean. In this case (assuming the model is good), the repeat
profile and its modelled tide will be highly correlated with each other but not with the mean;
or Orx~Oxx, Orm~0, and Omx~0. Then (14) becomes

Odd-^rr + Omm.cDxx (16)

where from (12) the first two terms on the right hand side describe the power spectrum of
the non-corrected difference profile ( Oqq), and the last term indicates the effect of making
the correction. As opposed to the case where the repeat tide was highly aliased and
Ooo-^rr^mm; here the power of the uncorrected difference profile is given by
Ooo^Orr+Omm. Thus the RMS variability of a non-aliased repeat will tend to be larger
than that of an aliased repeat before correcting for ocean tide. Note also from (16) how, in
this case, there is no aliasing term involving the correction. This indicates that for a repeat
whose tidal phase has not been aliased into the mean, variability will tend to decrease upon
making the correction.

These results are clearly demonstrated by repeat 10 on arc 82. Note from figure 2.8
how the tidal phase of this repeat is clearly different from the phase of the early repeats
which was aliased into the arc 82 mean. One would thus expect from (16) that this repeat
should have high RMS variability before making the OTIDE correction, which will be
reduced upon making the correction. The results listed in table 2.1 verify this. Such

81

behavior is also seen in repeat 17 on arc 209, which has the unique tidal phase of all the
repeats used to form the arc 209 mean.

When the mean profile of an arc is highly aliased; by definition a majority of its
repeats will have tidal phases similar to the one aliased into it. Thus, most repeats will also
experience a variability increase upon making the OTIDE correction, and the mean
variability over the ensemble of repeats (CJ) will rise. This explains the behavior of cr for
arc 82. The mean of arc 293 was not aliased, and so the accuracy of the tidal model caused
G to drop substantially when the OTIDE correction was performed. Arc 209 falls in
between the two extremes, as does the behavior of its o.

The behavior of the sample standard deviation (SSD) can also be explained from the
results of this subsection. In the case of a non-aliased mean, the OTIDE correction is
merely reducing a major source of oceanic variability. The difference profiles with the tide
removed will thus look more alike, and so the standard deviation of the individual repeat
variabilities about (J will be less. With the aliased means, SSD will still tend to drop for a
well modelled correction. This drop occurs because the aliased repeats with initially low
variability will experience a variability rise upon correcting for OTIDE; and the non-aliased
repeats with initially high variability will tend to experience a variability decrease upon
OTTDE correction. The spread in variabilities between repeats is thus lessened, and so is
the SSD. The data for arcs 82 and 293 behave in just this way, as seen in tables 2.1 and
2.3. Arc 209 experiences a slight rise in SSD after making the OTTDE correction, which
may indicate that the tidal model is not quite as good along arc 209 as it is for the other

There is a simple way of estimating which repeats will be the most (and least) aliased into
the mean: Draw a circle and divide it into 18 equal slices. The slices represent the full set of
tidal phases sampled over the M2 alias period. Then, for each repeat used to form the mean
profile, color in its appropriate slice. From the resultant picture; those colored repeats
which appear in bunches are the most likely to be aliased into the mean. Those colored
slices which are isolated on the phase circle indicate repeats having tidal phases which are
probably different from the aliased phase. Estimates of the behavior of the repeats on the
aliased test arcs using this method correspond fairly well with the actual results.

82

arcs. This result is not surprising as arc 209 passes through the Aleutian Island chain;
which is a difficult area to model.

Test Arc Summary

The results of applying the GEOSAT GDR corrections to the three test arcs may be
summarized as follows: Of all the phenomena which are modelled in the GDR, only the
ocean tide, electromagnetic bias and inverted barometer corrections vary enough over the
length of the arcs to have a significant effect on the difference profiles when applied (actual
effects such as the true wet troposphere correction may also be significant, but are not
currently modelled well enough to reflect it). Out of these three effects, only the ocean tide
is modelled accurately enough to allow consistent reduction in the variability which it
produces in the sea surface. The ocean tide, however, may be significantly aliased into the
mean profiles formed by the collinear method; resulting in erroneous variability in the
difference profiles.* The following recommendations are therefore made for editing
GEOSAT data in the test region:

1 ) Apply only the GDR OTIDE correction to the sea surface heights.

2) For each arc, use an ensemble of OTTDE-corrected repeats that will prevent the
ocean tide from aliasing into the mean sea surface height profile..

When only a limited number of repeats are available (due to data dropouts, for
example), the latter recommendation may be achieved by ensuring that the repeats used are
evenly spaced around the phase circle discussed earlier. If data dropouts are not an issue,
the best mean profiles will be formed from repeat cycles spanning some integral multiple of
the M2 alias period (-18 repeat cycles); though as more repeats are used, aliasing will
become less of a problem as a matter of course.

* There is nothing to prevent the other corrections from aliasing into the means either,
though their random nature would result in more broadband contamination of the mean
profiles. In any case, the generally low variability of most of the non-OTIDE corrections
would preclude their possible aliasing from being much of a problem.

83

2.7 SUCCESSIVE POINT AVERAGING

The final evaluation of the GDR concerned the possibility of successive - point
averaging the sea surface heights to reduce noise and computer storage space. The benefits
of the latter are obvious, and the former was deemed desireable due to the short wavelength
jitter seen in most of the difference profiles. The algorithm of choice was a three point
block (e.g. - not running) average of the sea surface heights on each pass. This algorithm
was chosen mainly for ease of computation, as the average sea surface height of three
consecutive data points could be assigned in time and space to the central point.

Before including the block average in the collinear algorithm (between spike
removal and detrending), spectral methods were employed to determine its effect on the
difference profiles. The power spectra of the eighteen OTIDE corrected difference profiles
on arc 293 were averaged together to form an arc spectrum with 144 degrees of freedom.
This spectrum is shown in figure 2.26. Aside from the peak at a wavelength of 130 km,
note how the power falls off between k'1 and k"2 (where k is the wavenumber) down to
wavelengths of about 100 km, and then levels off. Now, the three point block average
results in an increased sampling interval of 3A or 20 km; meaning that signals of ?i<40 km
will be unresolvable and possibly aliased. Remembering that the power is plotted on a
logarithmic scale, however, figure 2.26 shows that there is minimal variance in the
wavelengths shorter than 40 km. Thus, little information about ocean variability is missed
at these scales, and the averaging is acceptable. Additionally, the mesoscale features of
interest have typical spatial scales of 100+ km, which the block average will not obscure.
The spectral peak at 130 km is due to geoid contamination, as will be shown in Section
3.2.

84

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DEGREES NORTH LATITUDE

Fig. 2.4 Difference profiles along arc 82. No corrections applied. The repeat numbers
appear at the right of each profile.

38

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20 22 24 26 26 30 32 34 36 38 40 42 44 46 48 50 52 54 56

DEGREES NORTH LATITUDE

Fig. 2.5 Difference profiles along arc 82. No corrections applied.

89

WAVELENGTH (KM]

10'

10'

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Fig. 2.6a Power spectra of the difference profile (solid line) and solid tide correction

(dashed line) for repeat 9 on arc 82. The 95% confidence limits appear at the
upper right.

10'

10'

WAVELENGTH (KM)
10'

10'

z 101

10''

10''
10

10 '

WflVENUMSEfl 1CTCLE5/KM,

Fig. 2.6b Power spectra of the difference profile (solid line) and wet SMMR correction
(dashed line) for repeat 9 on arc 82.

90

CJ

1400-1
1200-
1000-
800-
600-
400-
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20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56

DEGREES NORTH LATITUDE

Fig. 2.7 Significant wave height along arc 82. Repeats whose difference profiles

experienced a significant variability decrease upon making the SWH-derived EM
BIAS correction appear at the top. Repeats which experienced a significant
variability increase appear at the bottom.

91

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DEGREES NORTH LATITUDE

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56

Fig. 2.8 Ocean tide correction along arc 82.

92

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20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56

DEGREES NORTH LATITUDE

Fig. 2.9 Wet FNOC correction along arc 82.

93

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DEGREES NORTH LATITUDE

Fig. 2.10 Dry FNOC correction along arc 82.

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Fig. 2. 1 la Power spectra of the difference profile for repeat 4 on arc 82 before (solid line)
and after (dashed line) correcting for electromagnetic bias.

WAVELENGTH

[KM)

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(dashed line) for repeat 4 on arc 82.

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Fig. 2. 12 Power spectra of the difference profiles for repeat 4 (solid line) and repeat 13
(dashed line) on arc 82. No corrections applied.

96

WAVELENGTH (KM)
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WAVENUMBER (CTCLE5/KM)

Fig. 2.13a Power spectra of the difference profile for repeat 10 on arc 82 before (solid line)
and after (dashed line) correcting for ocean tide.

10'

10'

WAVELENGTH (KM)
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Fig. 2. 13b Power spectra of the difference profile (solid line) and ocean tide correction
(dashed line) for repeat 10 on arc 82.

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WAVELENGTH (KM I
10'

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Fig. 2. 14a Power spectra of the difference profiles for repeat 3 (solid line) and repeat 10
(dashed line) on arc 82, before correcting for ocean tide.

10'

WAVELENGTH (KM)
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10''

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Fig. 2.14b Power spectra of the difference profiles for repeat 3 (solid line) and repeat 10
(dashed line) on arc 82, after correcting for ocean tide.

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Fig. 2.15 Power spectra of the difference profile (solid line) and wet FNOC correction
(dashed line) for repeat 9 on arc 82.

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and after (dashed line) correcting for dry FNOC.

WAVELENGTH (KM)
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20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56

DEGREES NORTH LATITUDE

Fig. 2.17 Difference profiles along arc 209. No corrections applied.

101

1400-1

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z: 800-

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20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54

DEGREES NORTH LATITUDE

56

Fig. 2.18 Significant wave height along arc 209.

102

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20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56

DEGREES NORTH LATITUDE

Fig. 2.20 Difference profiles along are 209. Simple inverted barometer correction applied

104

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20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56

DEGREES NORTH LATITUDE

Fig. 2.23 Difference profiles along arc 293. No corrections applied.

107

150-

100-

50-

0-

-50-

-100-

O

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108

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109

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1 14

CHAPTER THREE - USING THE DATA

115

3.1 OVERVIEW

The purpose of this chapter is to describe the use of GEOS AT data to investigate
oceanic variability in the northeast Pacific, after editing it according to the lessons learned in
Chapter 2. The collinear algorithm presented previously was slightly modified to handle
data dropouts while maintaining the alongtrack matchup of data points between the repeat
passes. Its steps are as follows:

1) Purge data points over land or flagged bad.

2) Subtract the Rapp geoid and Ocean Tide from the H values of each pass on the
arc.

3) Remove spikes from the corrected profiles.

4) Detrend the profiles.

5) Identify a pass with no data droputs; and starting with the second data point,
define the latitude of every third point as a reference latitude.

6) For each pass, identify the set of data points which lie within ±A/2 of the
reference latitudes, and make them the center points of the three point block averages.
Assign an average sea surface height value to each of the center points unless a data
dropout prevents the use of three points at the normal interval A. In that case, assign a
latitude and height of zero to the center point involved.

7) Average a collinear ensemble of these edited profiles together to form a mean
profile for the arc. If an edited profile is missing a center point for a particular reference
latitude (i.e., zeroes assigned), interpolate for the sea surface height from the neighboring
values.

8) Subtract the mean profile from each edited pass profile in the ensemble to create
the set of difference profiles.

This algorithm was applied to an ensemble of 18 repeat passes on arc 293. The
resultant difference profiles are shown in figures 3.1 through 3.4. The following section
shows what knowledge of the oceanic mesoscale may be inferred from such data.

116

3.2 VARIABILITY ON ARC 293

Many observations may be made from the full set of arc 293 difference profiles.
Perhaps the best place to start is by noting how generally quiet the Pacific ocean appears to
be in them. Arc 293 runs roughly parallel to the California coastline, some 1000 km
offshore; and except for its northern section near Alaska, it is largely removed from
significant current systems and their associated mesoscale features. The ensemble average
of the RMS alongtrack variabilities computed from the difference profiles (o) is only 5.6
cm, and the quietest repeats such as 8 and 9 have RMS values down near 4.5 cm.* If the
ocean itself is fairly quiescent at these times, the latter figure may be thought of as
representing a lower bound on the noise of the GEOS AT system. This limit is a
combination of the instrument noise, positioning errors (including orbit and horizontal
offsets), ocean tide modelling errors, and the environmental effects not corrected for. It is
well within the worst case limits presented in the table 1.2 Error Budget, and certainly low
enough to justify post facto the use of linear detrending rather than some higher order
means of reducing the orbit error.

The difference profiles are by no means featureless, however, and one's eye is
quickly drawn to a number of things. For example, what are the short wavelength
oscillations seen on repeats 1 and 3 (among others), and what are the periodic undulations
found in the northern regions of repeats 10 and 13? Is the U-shaped feature at ~30°N, seen
in various stages of growth and decay on repeats 6 through 9, a cold eddy or some
environmental effect not corrected for? At longer wavelengths; what is the dip in sea
surface height in repeat 2 centered on 50°N, or the gentle rise in the northern region of
repeat 13? Are these real oceanographic features or something else? The signatures seen
on the difference profiles will be investigated in order of increasing wavelength within the
following subsections.

* These variabilities are smaller than those seen in table 2.3 in the OTIDE column, due to
the 3 point averaging.

1 17

Short Wavelength Features

"Short wavelength" refers to the features of smallest horizontal extent seen on all
the difference profiles - the spikes. Each spike is the result of one sea surface height being
significantly (-5-10 cm) different from its neighbors. A single outlying height results in a
simple spike with a horizontal extent of some 40 km (the distance spanned by the outlier
and its neighbors to either side). Two successive outlying points, one high and one low,
create alternating spikes with a combined "wavelength" of 60 km. A reasonable choice as
the upper limit for short wavelength features provides some extra margin, and is here
defined to be 100 km.

The assertion is that these signatures on the difference profiles are not real
oceanographic features. Rather, they are a manifestation of the high wavenumber altimeter
noise seen previously - aliased down by the increased sampling interval. Aside from
spectral methods, this assertion is supported by two observations. First, as mentioned in
Section 1.4, there is simply no independent data which suggests that the sea surface height
fluctuates some 10 cm or more within such a short distance. Second, the short wavelength
features appear to be highly correlated with regions of large significant wave height. Such
regions were seen in Chapter 2 to be associated with a worsening of the altimeter precision
and a corresponding rise in instrument noise - not with mesoscale features. Comparison of
figure 3.5 (which plots SWH along five of the arc 293 repeat passes) and the difference
profiles demonstrates this association.

Spectral methods give further support to the idea of the spikes being non-
oceanographic in origin. If one accepts that the spikes are random noise, then an increase
in their number and/or amplitude should serve only to raise the general power level at high
wavenumbers. This is demonstrated in figure 3.6. Repeats 4 and 5 have very similar
difference profiles, except that repeat 5 has larger and more numerous spikes (although not
pictured, repeat 4 had an average SWH alongtrack of <3 m). Their respective power
density plots show that the high wavenumber power has risen rather evenly from repeat 4

118

to 5, indicating that the spikes are probably random in nature. Further indication that the
short wavelength spikes are system noise may be seen in coherence plots between
successive difference profiles. As an example, figure 3.7 indicates the coherence phase
and amplitude between repeats 4 and 5. Note how the amplitude drops rapidly below the
95% level of no significance as wavelengths get smaller than -100 km; indicating that the
short wavelength features are essentially uncorrected between the repeat passes. This is
characteristic of random noise.*

It must be remarked that the above behavior is by no means isolated. The power
spectra of the profiles with the most spikes all have elevated power densities at wavelengths
£ 100 km; and coherence plots between successive passes show no significant coherence
in the same wavelength regime. The conclusion is as proposed - the short wavelength
features are altimeter noise. As such, even more successive point averaging could be
applied to the sea surface heights, with five points instead of three being a reasonable
choice.

In order to complete the discussion of short wavelength features, mention must be
made of the effects of clouds and rain upon the altimeter measurement. Goldhirsh and
Rowland (1982) have shown that any delaying of the radar pulses due to the presence of
rain and clouds results in a height error which is small enough (<2 cm) to be ignored.
Clouds and rain may create a significant height error, however, if they attenuate the radar
pulse in a spatially variable manner over the radar footprint area. Monaldo et al. (1986)
showed that this variable attenuation distorts the return waveform and so causes the half-
power point to be mistracked. The tracking error results in erroneous pulse travel time
calculations; giving RMS sea surface height errors (after 3s of averaging, as performed

* Non-sinusoidal features such as those on the difference profiles cannot be linked with one
particular wavenumber on a power spectrum. Thus the 100 km cutoff in the power spectra
should not be confused with the horizontal size of the short wavelength features. The
wavenumber regime is used more in a qualitative sense here, to indicate the noise-like
behavior of the shortest horizontal scales in the difference profiles.

1 19

here) of up to 0.6 cm and 14.9 cm respectively, for idealized single-cloud and rain-cell
models encountered by a SEAS AT class altimeter (with a 2 m SWH). The cloud induced
error is therefore negligible, and Goldhirsh (1983) has estimated that the spatial scales of
rain cells are very short - about 3 km on the average for cells with a rain rate >10 mm/hr.
Thus, rain-induced altimeter error occurs on much too short a scale to be mistaken for
oceanographic phenomena, and may just be considered as increased altimeter noise.

Mid Wavelength Features

Mid wavelength is here defined as those features having a horizontal extent from
100 to 500 km on the difference profiles. This range includes such features as the U-
shapes seen around 30°N in repeats 6 through 9, and the periodic variations in sea surface
height found in the northern regions of repeats such as 10 and 13. The mid wavelength
regime is a good one in which to find oceanography - for two reasons. First, it contains
most mesoscale features (Pond and Pickard, 1983). Second, it lies above the upper
wavelength limit of the altimeter noise and below the horizontal scales of many corrections.
Thus it is less likely than either the short or long wavelength regimes to be contaminated by
these non-oceanographic effects.

There are three possible sources for the mid wavelength features seen in the arc 293
difference profiles. These are geoid contamination, environmental effects, and mesoscale
oceanography. Geoid contamination is examined first.

Geoid contamination was seen in Chapter 2 to be a result of collinear processing in
those areas where the geoid is changing rapidly with horizontal distance. Contamination
was the reason why regions of test arcs 82 and 209 were excluded from the analysis
conducted on the corrections; but was not deemed to be enough of a problem to warrant
any deletions from arc 293. Nevertheless, the mean sea surface height profile and its
alongtrack gradient for arc 293, shown in figure 3.8, indicate that geoid contamination may
not be completely ignored. Note especially the large gradients in the mean sea surface
(which is mainly a reflection of the geoid) at 40°N and north of ~44°N. Any repeat passes

120

which are not coincident with the mean profile, either along or across track, will incur false
signatures in these areas on their difference profiles. The horizontal scale of these
signatures will be the same as the horizontal scale of the geoid variation (-140 km
wavelength for the somewhat sinusoidal variations seen north of 44°), and their magnitude
may be estimated from the alongtrack gradient. This latter value is around 3-4 cm/km in
the most troublesome regions. Since the sliding scheme discussed earlier allows any pass
to be as much as -3.4 km (A/2) away from the reference latitudes of the mean profile;
features of around 10-15 cm in magnitude may appear on the difference profiles, due to
geoid contamination. (Note that crosstrack offset between the repeat passes also gives rise
to geoid contamination. In general, this is not a problem, as crosstrack offsets are limited
by the orbit to ±1 km. In the case of arc 293, there is even less difficulty, because the arc
is roughly perpendicular to the strike of the Aleutian island chain - which means that the
geoid variations seen in that region are of greatest magnitude along the arc).

The regions of large alongtrack sea surface gradient in the mean profile are the ones
which must be scrutinized most carefully for geoid contamination on the difference
profiles. As remarked, these are at 40°N and north of 44°N for arc 293. A particularly
fortunate occurrence is the periodic nature of the geoid variation in the northern region of
the arc; making contamination easy to spot in this case. Using this periodicity as a
guideline, one is immediately drawn to the regular features north of 44° which appear to
some extent in many of the difference profiles (figures 3.1 through 3.4); but are most
noticeable in repeats 10, 13, and 16. The scale and amplitude of these signatures closely
match those just predicted as being geoid-induced; and output data from the collinear
sliding scheme indicate these passes as being the furthest south of all 18 passes in the
ensemble. Thus they are the most likely ones to be contaminated by the geoid, as expected.
Further proof of the artificial nature of the periodic features may be seen by sliding passes
10, 13, and 16 even more to the south, which should exacerbate any geoid contamination.
This is exactly what happens, as seen in figure 3.9. Note how features longer than the mid

121

wavelength ones are relatively unchanged by the sliding, but mid wavelength ones grow all
along the arcs, especially north of 44° and at 40°. In general, all the periodic features north
of 44°, as well as signatures at 40°N seen on the arc 293 difference profiles, behaved in
this way and so were classified as non-oceanographic in origin.

Two other aspects of geoid contamination bear discussion before finishing with the
topic. The first regards the effect which geoid contamination has on the power spectra of
the arc 293 difference profiles, and is a result of the periodic geoid fluctuations seen in that
arc's mean profile. These regular features result in increased energy in the power spectrum
of the mean profile at those wavelengths surrounding the periodic one. This increase can
be seen in figure 3.10, which shows a definite peak in the power density of the mean
profile at a wavelength of around 130 km. When a repeat pass is shifted in an alongtrack
sense from the mean, that mid wavelength energy will bleed into the difference profile, as
shown in figure 3.1 la. Note the very large coherence at 130 km between the arc 293 mean
profile and the highly shifted repeat 10. (The 90° phase difference is a direct result of the
shift). This bleedover results in a peak in the wavenumber spectrum of the contaminated
difference profile (figure 3.1 lb). Since a number of the difference profiles for arc 293
contained some sort of geoid contamination, a mean spectrum formed from their individual
power spectra would also show increased energy at wavelengths around 130 km. This
explains the significant peak at this wavelength in the arc 293 average power spectrum,
shown at the end of Chapter 2 (figure 2.26).*

The final aspect of geoid contamination which should be covered concerns the
feasibility of investigating oceanographic features in regions of large geoid gradient.
Although there are no cut and dried answers to this question, a few general comments may
be made. First, areas of large geoid gradient often correspond with regions of high

* The narrowness of the peak seen in Chapter 2 is a result of the increased wavenumber
resolution afforded by averaging fewer Fourier coefficients together. This could be done
without degree of freedom limitations, as 18 spectra were always available for averaging.

122

oceanographic activity, so these places are ignored at the investigator's peril. If the activity
is vigorous enough, or the offsets happen to be small enough, then oceanographic features
may still be detected with no special data processing. If these cases are not in effect, more
care must be taken in the collinear algorithm to minimize alongtrack offsets between the
repeat passes. This may be done by interpolating between the original data points with a
cubic spline, for example, and then resampling the resultant continuous profiles at the exact
reference latitudes desired. Alongtrack offsets may be almost completely eliminated in this
way, although at the expense of increased computer time and storage space.
Unfortunately, if the crosstrack geoid gradient is large enough, the difference profiles will
remain contaminated even after alongtrack offsets are removed. The interpolation
technique is not used here, as a number of difference profiles such as repeats 2 and 6 are
still able to pick up interesting features without it.

The second possible source of mid wavelength features on the difference profiles is
the altimeter error caused by environmental effects. Within the mid wavelength range, table
1.2 indicated that only the EM BIAS and wet troposphere effects should have any
significant affect on the difference profiles. Chapter 2's investigation of all the GDR
corrections confirmed that these two phenomena could be quite noticeable at times
(especially the true wet troposphere effect; which often exceeds the modelled WFNOC
correction). The lack of consistent variability reduction when corrections were made for
these effects prevented their use in the data set used in this chapter, however. Thus, the
difference profiles for arc 293 (figures 3.1 through 3.4) may well show mid wavelength
features which are due to EM BIAS or the wet troposphere effects, rather than
oceanography.

There are ways of picking out the environmental signatures from the difference
profiles, however. First, Chapter 1 related how increases in the EM BIAS and wet
troposphere effects depress the apparent sea surface when not corrected for. Thus, isolated
increases in sea surface height are probably not due to these phenomena, as intuition and

123

experience suggests that highly localized decreases in wave height and water vapor are
uncommon. This observation is not of much use on the arc 293 difference profiles; but it
does, for example, disqualify EM BIAS or the wet troposphere effects as having caused the
peak in sea surface height seen in the repeat 13 difference profile on arc 82 (figure 2.5).

The second way of spotting the environmental signatures is based upon knowledge
of the time scale of these phenomena. Both EM BIAS and water vapor evolve with the
time scale of weather - which is only a few days. Therefore, features which persist
through successive difference profiles are very likely not environmental but oceanographic
in origin (assuming geoid contamination has already been ruled out).

Given a set of difference profiles and the known geoid and environmental effects,
the identification of mid wavelength oceanographic features may be summarized as follows:
First, line up the difference profiles vertically in time order (as done in figures 3.1 through
3.4). Then, look for features which persist in the same general region over two or more
repeat cycles. This is the time scale over which most mesoscale activity occurs, and it also
rules out false signatures from the EM BIAS or wet troposphere effects. If the features
move noticeably between the repeats, geoid contamination may be ruled out, as this effect's
movement is limited to ±A/2 (i.e., ±3.4 km) by the sliding technique. If the features
appear stationary, the mean sea surface height profile and its gradient may be checked in
that area to determine the magnitude of any geoid contamination. The product of A/2 and
the maximum local gradient give a good estimate of the contamination magnitude. If the
observed features exceed it, they are likely oceanographic in nature.

Use of the above technique highlighted some interesting mid wavelength features
on the arc 293 difference profiles as being probably oceanographic in origin. The most
striking of these is the U-shaped dip in the sea surface which first appears in repeat 6 at
~30°N, grows through repeats 7 and 8, and decays by repeat 9. The dip is located
approximately at 29.5°N, 128°W, putting it about 1000 km southwest of Los Angeles and
750 km from the California shelf break (see figure 1.3 for the arc location). It has a

124

magnitude of at most -15 cm, a horizontal extent of around 240 km (using repeat 8), and
persists through at least 3 repeat cycles (50+ days). These are reasonable values for a cold
eddy in this region (e.g., McNally[1981], Pickard and Emery [1982]). A likely source for
such an eddy would be the southward flowing California current, which regularly sheds
cold mesoscale features (the eddy is classified as cold because the observed difference
profile dip is an indication of the increased density of cold water). There is also some
indication that the eddy could be moving to the southwest (perpendicular to the arc), as it
does not appear to change position with respect to the arc, but does change in horizontal
size. Thus, the small, weak signature seen in repeat 6 could signify the first intersection of
the arc with an eddy moving from the northeast. As the eddy passes through the arc, its
cross section increases through repeat 8, and then decreases as it moves to the southwest of
the arc. A rough estimate of the eddy speed may be made from its maximum diameter
(-240 km in repeat 8) and time of passage (estimate 60 days), as being 4 km/day. This
figure is reasonable (e.g., Pickard and Emery [1982], Chap. 7).

Mid wavelength features which appear less simple than the cold eddy, but
nevertheless seem to be oceanographic in origin, may be seen from 28° to 32°N on
difference profiles 1 1 through 14 (figure 3.3). A suggestion for future work would be to
compare these signatures with other data sources such as infrared imagery or tomographic
data. Work of this kind could eventually catalogue typical difference profile expressions of
mesoscale features and so allow the altimeter data to stand alone as a primary source of
descriptive oceanography. Future goals notwithstanding, the encouraging result of this
subsection is that the GEOS AT data picks up probable oceanographic features of relatively
small amplitude, even with rather unsophisticated processing.

Long Wavelength Features

Long wavelength features are defined as those having a horizontal extent greater
than 500 km on the difference profiles. It is difficult to find oceanographic signatures in
this wavelength range, because it is here that the altimeter errors have their greatest

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expression. In addition, oceanographic features with wavelengths approaching the arc
length will not appear in the difference profiles anyway, as seen in Section 1.2.

The prevalence of altimeter errors in the long wavelength features seen in figures
3.1 through 3.4 was verified by comparing the difference profiles with the most energetic
GDR corrections identified in Chapter 2, and with meteorological data. For example,
meteorological surface anlaysis charts indicated that the long wavelength features seen in
repeats 1 (32°-54°N), 3 (34°-48°N), 6(50°N+), and 13(44°N+) could all be attributed to
the inverse barometer effect. Plots of the WFNOC correction and satellite maps of
integrated atmospheric water vapor accounted for the long wavelength variations all along
repeats 4 and 17, as well as the significant dip in sea surface height from 46° to 54°N on
repeat 2. Electromagnetic bias explained the smooth curve in the sea surface beginning at
28°N on repeat 5.

The reader should remember that these three significant environmental errors remain
in some of the difference profiles because they were not modelled well enough overall to
allow correcting for them. Two things would largely alleviate these deficiencies, however.
First, a water vapor measurement coincident with the altimeter reading would pick up many
of the fluctuations in this phenomenon which are not adequately depicted in the FNOC
model. The upcoming TOPEX/POSEIDON mission should provide this capability with an
onboard SMMR. Second, an accurate frequency/wavenumber dependent inverted
barometer correction would not only remove this phenomenon from the sea surface
heights, but allow the EM BIAS correction to be more effective as well (see Chapter 2 for
discussion on the interplay between EM BIAS and the inverted barometer effect).

In addition to the long wavelength features that are caused by uncorrected altimeter
errors; any errors in applied corrections will also result in false signatures on the distance
profiles. For example, non-linear orbit error will not be removed with the bias and tilt
correction, and will result in very long wavelength curvature in the difference profiles (see
repeat 15 in figure 3.3). Also, the large error (10 cm RMS) in the applied ocean tide model

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may produce unwanted difference profile features in the 500-1000 km wavelength range.
This can be especially serious due to the previously mentioned aliasing of the ocean tide;
which may result in correlated model errors over several repeat cycles.

Upon examining the full set of arc 293 difference profiles for long wavelength
features, no signatures were found that could be considered as probably oceanographic in
origin. This is not unexpected, and will continue to be a problem with altimeter data until
better orbits and corrections are calculated.
3.3 DATA COMPARISON

Section 3.2 indicated that mesoscale variability may be observed in GEOSAT
difference profiles if one knows how and where to look. With this basic capability thus
established, the current section compares GEOSAT - derived variability results for the
northeast Pacific with two other studies of the region. The comparisons not only aid in
determining the general accuracy of the GEOSAT product, but also lend new insights into
the dynamics of the test area.

Waven umber Spectra

Fu (1983) used SEAS AT data from repeat orbits during the last 24 days of its
operation to calculate wavenumber spectra along the satellite tracks. The spectra were
determined from difference profiles using the collinear technique, and so were actually
spectra of the oceanic variability. Fu calculated average wavenumber spectra for two types
of oceanic regions - those of high and low energy (i.e., variability). The regions were
chosen according to the eddy-energy maps of Wyrtki et al. (1976), compiled from ship-
drift data; and of Cheney et al. (1983), who utilized SEAS AT data.

One of Fu's low energy regions (his area 3) closely corresponds with the test
region of this work. Within it, Fu chose 6 SEASAT arcs; each having approximately 8
repeats along it (the SEASAT repeat period was 3 days). Alongtrack wavenumber spectra
for each of the 50 odd difference profiles in the region were calculated, and then averaged
together. From the resultant regional spectrum, Fu found that SEASAT system noise

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extended from the shortest wavelengths out to 100 km, and that the regional spectrum
followed a kr1 slope between wavelengths of 100 and 1000 km.

Arcs 209 and 293 were chosen to compare GEOS AT data with the results of Fu.
For each arc, a wavenumber spectrum for each difference profile was calculated, and the
resultant set of spectra was averaged together. These averaged spectra, one each for arcs
209 and 293, appear in figure 3.12. For comparison, the power density of the Fu regional
spectrum is plotted as circles at wavelengths of 100, 1000, and -1 100 km, within their
95% confidence intervals. The confidence interval in the upper right of the figure is
appropriate to the arc 209 spectrum and is slightly larger than the one (not shown) for arc
293, which had more spectra available for averaging.

A number of important observations may be obtained from figure 3.12. First, note
how closely the power levels of the three spectra correspond at 100 km. This similarity is
to be expected, as power at 100 km consists primarily of altimeter system noise (see
Section 3.2), which is comparable between SEAS AT and GEOS AT (-0.1 m, as shown in
Chapter 1). While Fu's spectrum went fully white (i.e., horizontal) at wavelengths shorter
than 100 km, however, observe how the GEOSAT spectra still continue to drop, albeit not
as steeply as at longer wavelengths. This would indicate that GEOSAT is actually slightly
more accurate than SEASAT, which is not surprising, as the GEOSAT noise floor was
estimated in this region as ~5 cm RMS (see Section 3.2), while SEASAT is usually rated at
-13 cm RMS.

At the longest wavelengths, note how Fu's power level decreases at wavelengths
longer than 1000 km. Fu attributed this decrease to his use of a quadratic trend removal,
rather than the linear one used herein. Even without this power drop, the SEASAT
regional spectrum is still somewhat lower in the mid to long wavelengths compared to
those of GEOSAT. Some of this difference may be attributed to the fact that Fu's data had
all corrections applied, while only ocean tide was removed from the GEOSAT data. Most
of the difference in power, however, is due to Fu's use of only 24 days of SEASAT repeat

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data; which is all that was available to him. Over such a short period of time, much of the
long wavelength mesoscale variability simply cannot be seen, and so power levels at those
long wavelengths will be lower than when measurements cover a longer time span. The
lower powers at long wavelengths led Fu to fit a k_1 slope to the ocean variability spectrum
in the northeast Pacific from 1000 to 100 km. Using the GEOSAT data available over
some 10 months, however, one sees that the spectrum is actually closer to behaving as k"
15 in this wavelength range. As more GEOSAT repeats become available, oceanographers
may be able to find some length of time beyond which the ocean variability spectrum does
not change with increased time. This discovery would aid in defining the "mean" state of
the ocean - a problem which has heretofore eluded solution.

As was shown in Section 3.2, the peak at around 130 km seen on the GEOSAT arc
293 wavenumber spectrum is due to geoid contamination. Similar analysis confirms that
this contamination is also the cause of the peak at a slightly shorter wavelength found on
the arc 209 spectrum. Fu's regional spectrum appeared to have a small peak at -130 km as
well, though it is difficult to tell, as that spectrum was plotted amidst 4 others. In any case,
there is no indication in Fu's results of a peak at 130 km of the magnitude seen in the
GEOSAT spectra. This is not unexpected, however, as Fu interpolated for sea surface
heights alongtrack using a cubic spline, and so did not incur alongtrack geoid
contamination (the lesser crosstrack errors caused by the ±2.5 km SEAS AT repeat tracks
would explain any small peaks at wavelengths near 130 km).

Additional information about both the ocean and the altimeter data may be gained
from comparing the two GEOSAT spectra. Note how, at wavelengths longer than
-200 km, the arc 209 power is noticeably greater than the arc 293 power. There are three
possible reasons for this. First, the environmental errors which are not corrected for may
be more severe along arc 209. This seems unlikely, as one would expect, if anything,
greater environmental variability near the California and Alaska coastal regions than in the
middle of the Pacific. Second, arc 209 may be an area of greater mesoscale variability than

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arc 293. Figure 3.13 shows these two arcs plotted on the mesoscale variability map of
Cheney et al. (1983), which was compiled from SEASAT repeat track data. Even after
truncating the northern part of arc 209 (due to geoid contamination), it still appears that this
arc covers a region of slightly higher oceanic activity than arc 293. This is especially the
case northeast of Hawaii, where arc 209 passes quite close to the higher variability region
surrounding the Hawaiian islands. An example of this variability may be seen near 25°N
on a number of the arc 209 difference profiles (figure 2.17).

The third possible reason for the difference in power between arcs 209 and 293 is
due to the ocean tide and its correction. As shown in Section 2.6, the aliasing of this
phenomenon into the arc 209 mean caused increased variability in its difference profiles.
Additionally, as the behavior of the sample standard deviation indicated before and after
making the GDR Ocean Tide correction to arc 209; it is quite possible that the ocean tide is
not as well modelled along this arc as it is for arc 293. Section 2.6 indicated how poor
modelling of a correction would increase the difference profile variabilities, which would in
turn raise the power density in the applicable wavelengths (the mid to long ones, in this
case).

Overall, the GEOSAT-derived wavenumber spectra agree very well with the
regional spectrum of Fu. As just seen, spectra of this sort are valuable not only in
indicating the presence and level of mesoscale activity; but also in spotting areas for
improvement in the altimeter product. It should be noted that oceanic wavenumber spectra
of the scope and accuracy shown here are nearly unobtainable with conventional
hydrography, which is better suited to taking repeated measurements in a limited area.
Continued oceanic coverage by GEOS AT and follow on altimeters will provide additional
information on oceanic activity in the wavenumber domain.

Variability as a Function of Position

The other domain within which GEOSAT provides valuable information on oceanic
mesoscale variability is the frequency domain. That is, rather than investigating how the

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sea surface height changes alongtrack (as was done in the prior subsection), one may ask
how it varies with time at a given location. Such a calculation was performed by Cheney et
al. (1983), using the same SEASAT repeat-track data as Fu.

Cheney et al. used a collinear processing scheme on SEASAT arcs divided
worldwide into 2000 km long segments. Most geophysical phenomena were corrected for,
including the delay of the radar pulse due to the ionosphere and troposphere (both wet and
dry components), tidal effects, and electromagnetic bias. The data was smoothed by fitting
a quadratic curve to every eight successive 1/sec data points, with iterative oudier rejection.
The resultant curves, centered on each data point, also allowed for alongtrack interpolation
of sea surface height. All repeats on each arc segment were processed in this way, and a
mean profile was formed on each segment from their averages. Difference profiles were
then calculated, with linear detrending being used to reduce the orbit error. Due to the
alongtrack interpolation of sea surface height, the set of ~8 difference profiles for each arc
segment had their 1/sec values aligned to be as close as possible to one another - greatly
reducing geoid contamination. RMS variability at each 1/sec aligned point could then be
determined by calculating the standard deviation of the corresponding sea surface heights
on the 8 difference profiles. The resultant set of variabilities, determined approximately
every 7 km on the SEASAT groundtracks, were then gridded using a bilinear modelling
function. Cheney et al.'s subsequent map of mesoscale sea height variability is reproduced
in figure 3.13. The overall map is remarkably consistent with what is independendy
known of the oceans, and so is valuable for comparison with the GEOS AT data.

Five GEOS AT arcs were chosen for comparison with the results of Cheney et al.
for the northeast Pacific. They were arcs 321, 293, 265, 237, and 209, spaced at an
approximate 700 km interval from the California coast west to Hawaii. These arcs are also
indicated on figure 3.13. On each arc, RMS variability as a function of latitude was
determined by calculating the standard deviation of the 1 8 sea surface heights available
there from the difference profles (only 14 profiles were used for arc 209). To reduce short

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wavelength jitter in the variability plots, three consecutive sea surface heights were
averaged together around each chosen latitude before computing the standard deviation.
The results of the GEOSAT variability calculations appear in figure 3.14. The arcs are
arranged from top to bottom going westwards through the test area. As a reminder, the
GEOSAT data used to create these plots was corrected only for ocean tide and was not
interpolated alongtrack, although the 3 point block average was applied to reduce noise.

Comparing figures 3.13 and 3.14, one sees that the GEOSAT data agrees very well
with the results of Cheney et al. in mapping the qualitative picture of mesoscale variability
in the northest Pacific. The GEOSAT variabilities do tend to be higher, though, for a
number of reasons: First, the SEAS AT arc segments were less than half as long as those
used in this thesis, resulting in smaller orbit error in its difference profiles (as well as more
limited wavelengths within which to spot oceanic variability). Second, Cheney et al.'s
alongtrack data interpolation significantly reduced their geoid contamination (although
SEAS AT,with its ±2.5 km repeat, would have a larger crosstrack component of this effect
than GEOSAT). Third, SEASAT's SMMR allowed for computation of a coincident wet
tropospheric correction to its sea surface heights. Fourth, the SEASAT results were
determined from only 24 days of data, over which time oceanic variability would be less
than for the 307 days used in the GEOSAT calculations. Fifth, Cheney et al.'s gridding
scheme resulted in reduction of variability for narrow current systems, such as the Alaskan
and Californian ones.

Quantitative estimates of the effects on variability calculations caused by these
differences may be made as follows: Tai (1988) derived analytically the average error
commited in representing the orbit error along a groundtrack segment by a bias and tilt.
For a known RMS bias difference between the various repeats on an arc, and a given arc
length (expressed in degrees relative to a 360° earth circumference), his table 1 gives the
RMS error of the final sea surface height variability calculations. Using GEOSAT arc 209,
with an arc length of 3600 km (remember that arc 209 was truncated due to geoid

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contamination), and an RMS bias difference over 14 repeats of 101 cm, Tai's table gives a
1.18 cm RMS error in the arc 209 variability estimates. If Cheney et al.'s 2000 km arc
length was used instead, the RMS error would be only 0.38 cm. Each of these errors
reflects the tendency of the difference profiles to overestimate oceanic variability when orbit
error remains in them - the longer arcs used in the GEOS AT calculations thus overestimate
the oceanic variability by (1 . 182-0.382)1^2 = 1.18 cm more than shorter arcs would have
(this corresponds to a variance difference of 1.25 cm2). Now refer again to figures 3.13
and 3.14. Integrating the location-dependent variabilities along arc 209, the GEOSAT-
determined variablity is 7.0 cm, while the Cheney et al. variability appears to be about 3.5
cm. The difference in variance between the two estimates is thus (7.02-3.52) = 36.8 cm2.
The portion of this difference which may be attributed to the residual orbit error difference
just calculated is only about 3%.

The contribution of alongtrack geoid contamination to the difference in variance
between the GEOSAT and SEASAT results may be estimated in the following way: The
spectral peak seen at wavelengths of about 130 km on the mean spectra of the various
GEOSAT arcs was shown in Section 3.2 to be largely due to geoidal effects. Using the arc
209 mean spectrum in figure 3.12, the variance of this peak is calculated to be 0.72 cm2;
which accounts for some 2% of the variance difference between the GEOSAT results and
those of Cheney et al.* Since crosstrack offsets are less than the alongtrack ones, the
crosstrack geoid contamination contribution to the variance calculations is negligible in this
case (remember also that the geoid is changing most rapidly with location in the alongtrack
direction for arc 209).

* Note that the geoid variance estimate comes from alongtrack calculations of sea surface
height, while the arc 209 variance estimates are temporal in nature. Meaningful
comparisons of spatial and temporal variance may be made under the assumption that the
ocean is essentially homogeneous in space and time (if the ocean is completely
homogeneous, spatial and temporal variance are equivalent). This is a fairly good
assumption for the test area, which is largely free of major current systems and their
accompanying inhomogeneity.

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Assuming that the SEASAT SMMR provided an accurate wet tropospheric
correction for Cheney et al., a significant portion of the higher variance of the GEOS AT
results may be attributed to not making any such correction for the latter data set. This
portion may be roughly estimated by calculating the variance of the wet FNOC correction
along the GEOS AT repeats on arc 209, from table 2.2 (see Section 3.4 for the
methodology). This calculation gives a wet-tropospheric variance of 21.14 cm2, which
accounts for about 57% of the GEOSAT-SEASAT variance difference.

The fact that the GEOS AT data spans a longer time than the SEASAT data allows
GEOS AT to see more of the low-frequency oceanic variance than SEASAT. Using the
tide-free power spectrum of Honolulu sea level from 1938 to 1957, calculated by Munk
and Cartwright (1966), the increased sea surface height variance seen over 307 days (as
opposed to 24 days) is graphically estimated as 1 3 cm2. This extra oceanic variance seen
by GEOS AT explains some 35% of its difference with the SEASAT variances calculated by
Cheney et al.

The reduction of variance in strong current regions caused by Cheney et al.'s
gridding technique is difficult to estimate; but when the variance is integrated alongtrack,
the effect should be minimal. Overall, the other four calculated effects account for some
97% of the difference in integrated alongtrack variance between GEOSAT arc 209, and the
corresponding results of Cheney et al. This is outstanding agreement.

Returning to qualitative results, it is apparent that the two altimeters provide a very
consistent view of the northeast Pacific. For example, all five GEOSAT arcs pick up the
variability of the Alaskan current (seen in figure 3.13) at their northern ends, although arc
293 almost misses it as the current veers into the Gulf of Alaska. GEOSAT arcs 237 and
209 also show the increasingly southward location of the Alaskan current as one looks
further west along the Aleutian island chain. Similarly, the GEOSAT arcs all see slightly
higher variabilities in the south; at the northernmost reaches of the North Equatorial Current

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system. In addition, arc 209's southern segment (up to ~38°N) indicates the additional
region of high oceanic variability near the Hawaiian islands depicted by Cheney et al.*

The latitudes covered by the middle portions of the GEOS AT arcs provide more
information about variability in the northeast Pacific. Refer to figure 3.14 and note the
region of high variability around 38°N on arc 321, corresponding to the most energetic
portion of the California current. In addition, observe the ability of GEOS AT to pick up
Cheney et al.'s isolated region of slightly elevated variability (>3 cm RMS), corresponding
to the central segment of arc 265. The GEOS AT arcs to either side of arc 265 do not show
this elevated signature, which is consistent with Cheney et al.

Each of the GEOS AT variability plots of figure 3. 14 may be integrated alongtrack
to determine its temporal RMS variability (as was done for arc 209). The following
variabilities are determined: Arc 321-7.53 cm, arc 293-5.27 cm, arc 265-6.76 cm, arc 237-
6.76 cm, and arc 209-7.00 cm (the figure for arc 237 is slightly high, due to the residual
spikes in its difference profiles). GEOS AT data thus describes a northeastern Pacific with
high temporal variability at its eastern California boundary, a relatively quiet central
portion, and increased variability again in the west. This qualitative picture is consistent
with Cheney et al. and the work of others using non-altimetric sources (see Wyrtki et al.
[1976]).

It may be noted that oceanic quantities of interest such as the eddy kinetic energy
can also be calculated with the collinear scheme. Here, the sea surface slope variability is
computed as a function of position, and then converted to surface current variability (from
which eddy kinetic energy arises) through the geostrophic equations. Menard (1983), for

* The GEOSAT difference profiles from which the plots in figure 3.14 originated were
examined to ensure that regions of high variability could not have been due to geoid
contamination, orbit error, etc. The only regions of falsely high variability which were

noted, appear as peaks at 31°N and 45°N on arc 237, and are due to data spikes not fully
eradicated in the editing process.

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example, performed these calculations for the Gulf Stream and Kuroshio Current regions

using the SEAS AT repeat- track data.

3.4 ERROR BOUNDS ON GEOSAT VARIABILITY CALCULATIONS

The integrated temporal variabilities for the five GEOSAT arcs were shown to agree
in a qualitative manner with a priori knowleldge of the northeastern Pacific. An important
question to answer, however, is what are the error bounds on these variabilities? That is,
do the non-modelled and poorly-modelled effects discussed in earlier sections cause
enough uncertainty in the final GEOSAT results to warrant concern? At the two extremes
of the wavenumber range, the answer is no. Instrument noise is largely averaged out, and
residual orbit error for a 4400 km long arc is only some 1.80 cm RMS, after bias and tilt
removal (using Tai's [19881 table with a typical RMS bias difference between repeats of 1
m). Within the mid wavelengths, the effects of geoid contamination on the final variances
of a typical arc were estimated as 0.72 cm2, giving an RMS error of only 0.85 cm. These
errors are not significant compared to the typical temporal variability estimates of 5+ cm
(RMS).*

The remaining causes of error in the arc variability estimates stem from the
corrections studied in Chapter 2. Of these corrections, the ocean tide was usually the most
energetic, and was applied in the final editing scheme. Its error may be roughly determined
in the following way (using arc 209 as the example): Observe from table 1.2 that the ocean
tide correction error has a magnitude 1/10 of the correction itself. From table 2.2, the effect
of the ocean tide correction upon the actual difference profiles can be seen in their change in
variance after making that correction. Using only the 7 repeats marked with an asterisk
(denoting those repeats where the OTIDE correction had a significant effect on the variance
of the difference profiles), the average change in difference profile variance caused by the

* Recall that Tai's calculations were for the average residual orbit error. The maximum
error seen on a particular difference profile might be greater.

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OTIDE correction is 31.87 cm2.* Assuming the error is 1/10 of the correction, then the
GEOSAT OTIDE error variance is 3.19 cm2, This corresponds to an ocean tide-induced
error of 1.79 cm (RMS) in the arc 209 variability estimate, and is not significant.

Three other corrections were seen in Chapter 2 to be energetic enough to be a
problem in any GEOSAT results. These involved the electromagnetic bias, inverted
barometer, and wet troposphere phenomena. Since these effects were not corrected for in
the editing scheme used here, their total effect on the difference profiles must be considered
- not the residual error after correcting for them (as in the ocean tide). Using the method
described in the previous paragraph, the average change in difference profile variance
caused by these phenomena for arc 209 are as follows: electromagnetic bias - 34.42 cm2,
inverted barometer - 30.80 cm2, and wet FNOC - 21.14 cm2. These correspond to errors
in the arc 209 variability estimate of 5.87 cm, 5.55 cm, and 4.60 cm (RMS), respectively.
Recall from Chapter 2, however, that the simple inverted barometer correction tends to
overcorrect the sea surface heights, while the wet FNOC correction tends to undercorrect
them. Thus, the error in the arc 209 variability estimate due to no IB correction being made
is likely smaller than 5.55 cm, while the error due to not making a wet troposphere
correction is probably larger than 4.60 cm. In any case, these three effects can be a
significant source of error in any GEOSAT oceanic variability estimates.

The reader should remember that the electromagnetic bias, inverted barometer, and
wet troposphere error estimates given here are rough, and are biased towards the worst
case. In practice, the phenomena themselves can usually be separated from oceanographic
signals in the difference profiles, with knowledge of their spatial and temporal
characteristics - as demonstrated in Section 3.2. Nevertheless, the manual separation of
non-oceanographic signals from GEOSAT difference profiles is a tedious process at best,

* No attempt is made to distinguish tidal phases, or whether the OTIDE correction increased
or decreased the variance - this is just a zero-order estimation of the magnitude of the
correction. It is assumed that any model error is uncorrelated with the difference profiles.

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and rather defeats the goal of near real-time sampling of the oceans. Therefore, further
work is called for in correcting these three major sources of error, as already seen in
Section 3.2.
3.5 SUMMARY AND CONCLUSIONS

A region of the northeast Pacific, from 20° to 55°N and 1 15° to 180°W, was
chosen within which to evaluate and use altimetric data from the U.S. Navy Geodetic
Satellite GEOSAT. A 4400 km satellite track (arc 82) spanning the region was used to
verify the zero order accuracy of the major GEOSAT geophysical data record (GDR)
channels; including time, position, orbit height, sea surface height, geoid height, significant
wave height, tidal values, and environmental corrections. No discrepancies in the data
were noted, and the information gained in the accuracy check allowed for the presentation
of a concise GEOSAT error budget, listed in table 1.2. While the available GEOSAT data
looked good, a number of gaps in altimeter coverage were noted, especially in the
descending passes. These gaps limited the choice of arcs for later processing.

The effects of applying various geophysical corrections to the GEOSAT sea surface
heights were investigated using a collinear processing scheme: For a given repeat pass, bad
data points were edited/smoothed, the low order Rapp geoid was removed, and the
resultant alongtrack sea surface height profile was detrended of bias and tilt (to reduce orbit
and other long wavelength errors). All repeat passes on a chosen arc were processed in
this way and then averaged together to form a mean sea surface height profile for the arc.
No interpolation for height between alongtrack data points was done, although the various
repeats were slid to be within ±A/2 of gridded reference latitudes (where A = 6.7 km).
This was performed to reduce geoid contamination.* The mean profile was then subtracted
from each repeat profile on the arc to create a set of difference profiles, which highlighted
the variable features of the ocean.

* Regions of very high spatial geoid variability were too much for the sliding scheme to
handle on portions of arcs 82 and 209, so these regions were not processed.

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The various GEOS AT corrections were examined with the collinear scheme by
individually applying them to the repeat profiles on three test arcs (#'s 82,209,293), and
observing their effect on the alongtrack RMS variability of the resultant difference profiles.
Derivations using the theory of coherent and incoherent power showed that a well-modelled
correction would tend to decrease the alongtrack variability when applied, while a poorly
modelled one would increase it. Corrections which varied only slightly over the length of
the test arcs (-4400 km) would have very little effect on the difference profile variabilities.

The results from applying the GEOS AT GDR corrections to the three test arcs
indicated that only the ocean tide, electromagnetic bias, and inverted barometer corrections
varied enough over the arc length to warrant using them to adjust the sea surface heights.
Referral to satellite maps of integrated atmospheric water vapor, in conjunction with
difference profiles, indicated that the true wet tropospheric correction could also be
significant; but the GEOS AT wet FNOC and wet SMMR corrections did not reflect it. Of
the three significant GDR corrections, the electromagnetic bias and inverted barometer
effects were shown to be (negatively) correlated with one another, through the atmospheric
pressure. The dependence of a static inverted barometer response upon the temporal and
spatial scales of the atmospheric forcing was demonstrated, which explained the failure of a
simple model to correct for it. Therefore, no correction for the inverted barometer effect
was made, and the resultant contamination of the difference profiles by the atmospheric
pressure prevented the GDR EM BIAS correction from working well, either. It, too, was
therefore not recommended for application. The remaining ocean tide correction did work
well enough to warrant its use, although the long alias period (-317 days) of its M2
component led to the recommendation of using multiples of 18 repeat cycles to form the arc
mean profiles.

Spectral analysis indicated that a three-point block average of ocean tide-corrected
sea surface heights would result in the loss of very little oceanographic information, and so
was recommended to reduce short wavelength instrument noise and lessen computer

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storage space requirements. The GEOS AT data used in the rest of the thesis had this
averaging performed, was corrected for ocean tide, and used 18 repeat cycles worth of
data.

A set of 1 8 difference profiles on arc 293 was investigated for mesoscale
variability. The overall alongtrack variability on this arc was quite low, with the quietest
difference profiles having an RMS variability of -4.5 cm. This figure was suggested as a
reasonable estimate of GEOS AT's working precision.

Oceanographic features could be found on the arc 293 difference profiles, but only
after sifting through a variety of other signals. At scales of less than 100 km, the difference
profiles were seen to be dominated by instrument noise. At scales from 100 to 500 km,
geoid contamination was seen to be a problem, although the alongtrack gradient of the arc
mean profile could be used to identify it. The electromagnetic bias and wet troposphere
effects also significantly affected these length scales, but knowledge of their temporal
variation allowed their signatures to be catalogued on the difference profiles. Scales longer
than 500 km were seen to be dominated by altimeter errors, and no oceanographic features
were noted within them. Oceanic mesoscale signals were seen in the 100 to 500 km range,
however, and included a probable cold eddy lasting for 50+ days, with a horizontal extent
of some 200 km and a surface dynamic expression of 10-15 cm.

Edited GEOSAT data from the area was then compared with other investigations of
the region - mostly from SEASAT data. Mean alongtrack wavenumber spectra of oceanic
variability for arcs 209 and 293 were compared with a regional variability spectrum of Fu
(1983), with good overall agreement. Differences included slightly less system noise for
GEOSAT, and higher power densities at long wavelengths in the GEOSAT spectra (mainly
due to the longer time span covered by the GEOSAT data). These differences in power led
to a k*1-5 fit at wavelengths from 100 to 1000 km in the GEOSAT spectra; slighdy steeper
than the k_1 slope determined by Fu.

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Sea surface temporal variability as a function of location was determined along 5
GEOSAT arcs for comparison with similar calculations by Cheney et al. (1983), who used
SEAS AT data. Qualitative agreement was excellent, with GEOSAT picking up the
variability of the Alaskan, Californian, and North Equatorial currents, as well as the region
of elevated variability near the Hawaiian islands. Absolute variability levels were higher
for the GEOSAT results, due mainly to lack of a wet tropospheric correction and a greater
span of covered time. These two factors accounted for some 92% of the difference in
variance between GEOSAT and SEASAT calculations for one arc. The temporal variability
as a function of latitude along each of the 5 GEOSAT arcs was then integrated alongtrack to
estimate an overall temporal arc variability. These estimates showed elevated variabilities at
the eastern and western boundaries of the test area, surrounding a quiet central region;
which was consistent with independent knowledge of the northeast Pacific. Error bounds
on the arc variabilities were estimated for arc 209; and the uncertainties in oceanic
variability caused by the uncorrected electromagnetic bias, inverted barometer, and wet
tropospheric effects were seen to be significant.

A number of areas for improvement in altimetric methods were noted through the
course of the thesis: First, more accurate orbit determination is necessary to monitoring the
ocean over longer length scales - currently, monitoring is limited to the relatively short arc
segments over which simple detrending can be used. These long scales are the ones in
which the "planetary" oceanic waves (Rossby and Kelvin) may be studied. Second,
independent gravimetric geoids must be determined if there is to be much open in using
satellite altimeters to study the mean oceanic circulation. Third, collinear processing should
include interpolating for sea surface height between the 1/sec data points. A fair amount of
GEOSAT data was unusable in this work due to the simple sliding scheme being unable to
sufficiently reduce the alongtrack geoid error. In addition, geoid error had to be manually
identified on a number of difference profiles, in order that it not be mistaken for
oceanographic features. Alongtrack interpolation would alleviate much of these problems.

141

Finally, a call is made for continued improvement in error modelling and correction.
A good frequency/wavenumber inverted barometer model would be a great asset to future
altimetric studies. Radiometer measurements of integrated atmospheric water vapor,
coincident with the altimeter measurement, are further desired for correcting the sea surface
height data. Perhaps the bottom line to the necessity of having good corrections is that it
allows data analysis to be more automated, as one then need not continually sift through
difference profiles to remove unwanted features. The real attraction of satellite altimetry is
its potential of providing accurate, near real-time sampling of the oceans, and anything
which helps to realize that goal is to be pursued.

142

40 -i
30-
20-

— 10-

O

0-

— -10
-20-
-30-
-40-

t r

40-i
30-
20-
10-

S o-

— -10-
-20-
-30-
-40-

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 r

40
30
20

— 10

E o

— -10
-20
-30
-40

40
30

20

— 10

O

0-

— -10
-20
-30
-40

i 1 r

i 1 1 1 1 1 1 1 r

T 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

40-]

30-

20-

~ 10-

s °-

— -10-
-20-
-30-
-40--
20

—I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56

DEGREES NORTH LATITUDE

Fig. 3.1 Difference profiles along arc 293. Three-point block average and ocean tide
correction applied

143

40

30

20

z:

CJ

10

0

-10

-20

-30

-40

40

30

20

10

0

-10

-20

-30

-40

40

30

20

z:

10

0

-10

-20

-30

-40

40

30

20

z:

CJ

10

0

-10

-20

-30

-40

40

30

20

CJ

10

0

-10

-20

-30

-40

-I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

"'"VW^

i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

-i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

10

— I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56

DEGREES NORTH LATITUDE

Fig. 3.2 Difference profiles along arc 293. Three-point block average and ocean tide
correction applied.

144

40-|

30-
20-

— 10-

E °-

— -10-'
-20-
-30-

-40 H 1 1 r

40 -.
30-
20-
10-

S °-

— -10-
-20-
-30-
-40-

i r

11

T 1 1 1 1 1 1 1 1 1 1

12

i 1 1 1 1 1 1 1 1 1 1 1 1 1 r

T 1

13

14

15

-i 1 1 1 1 1 1 1 1 1 1 1 1 i 1 1 r

20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56

DEGREES NORTH LATITUDE

Fig. 3.3 Difference profiles along arc 293. Three-point block average and ocean tide
correction applied.

40 -|
30-
20-

— 10-

E °-

— -10-
-20-
-30-
-40-

O

40-i
30-
20-
10-

5 o-

— -10-

40 -j

30-

20-

10-

0-

-10-

-20-

-30-

-40-

145

16

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

17

-20-
-30-
-40-1 1 1 1 r

~i 1 1 1 1 1 1 1 1 1 1 1 i

18

— i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56

DEGREES NORTH LATITUDE

Fig. 3.4 Difference profiles along axe 293. Three-point block average and ocean tide
correction applied.

146

1200-1

1000-

_ 800-

§ 600-

400-

200-

0

T 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1200-j

1000-

_ 800-

g 600-

400-

200-

% ^^*J*^0^^* » **H^»

18

— i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56

DEGREES NORTH LATITUDE

Fig. 3.5 Significant wave height along arc 293.

147

io:

•XL

in ' u

C_)

l_J

- 103

en

z

LU

a

102

10'

103

WAVELENGTH
io2

:km

10'

"i — i — i 1 r

-i — i — i 1 r

10'

\Q'2

WRVENUMBER (CTCLE5/KM

ict

Fig. 3.6 Power spectra of the difference profiles for repeat 4 (solid line) and repeat 5
(dashed line) on arc 293. The cutoff at 40 km is due to the increased sampling
interval caused by the three-point block average.

148

ISO'

d

r

Q_

LU

<-) 0°

en

a _Qn°

-901

•180

io-s

10"J

WflVENUMBER (CYCLES/KM

10'

1. 00

WflVENUMBER 1CTCLES/KM)

Fig. 3.7 Coherence phase and amplitude between the difference profiles of repeats 4 and 5
on arc 293. Note the lack of significant coherence at wavelengths shorter than 100
km.

149

MEAN PROFILE

o

z:

flLONGTRRCK GRADIENT

"i 1 1 1 1 1 1 1 r

20 22 24 26 28 30 32 34 36 38

r
40

1 1 1 1 1 1 1 1

42 44 46 48 50 52 54 56

DEGREES NORTH LATITUDE

Fig. 3.8 Mean sea surface height profile and alongtrack gradient for arc 293. Three-point
block average and ocean tide correction applied.

150

40

30

20

10

0

-10

-20

-30

-40

10

i 1 1 1 1 1 1 1 1 1 r

1 1 1 1 r

o

o

13

16

T 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56

DEGREES NORTH LATITUDE

Fig. 3.9 Difference profiles along arc 293. Three-point block average and ocean tide
correction applied. The passes from which the profiles originated were shifted
south alongtrack by 6.7 km.

151

io3

106 i i i i

WAVELENGTH (KM
ioJ

-i r

10'

z: io5
en

LU
_l
LJ
>—
U
\
PnJ 10'

*
*

C_)

CO IC"

Z.

LU

o

a

Q_

10'

10'

10"

WflVENUMBER (CTCLE5/KM

10"

Fig. 3. 10 Power spectrum of the mean sea surface height profile along arc 293. The peak
at 130 km is due to geoid contamination.

152

l/l 90°

u 0"

z
I

I 80

10

10''

WflvCNUMBER ICTCLE5/KM)

HflVENUMBER ICTCLE5/KM)

Fig. 3. 1 la Coherence phase and amplitude between the arc 293 mean profile and its

difference profile for repeat 10. The high coherence at 130 km indicates that the
geoid contamination seen in the mean profile has worked its way into the
difference profile.

10'

HOVElENCTh (KM)

10'

10'

WflVENUMBER ICTClES/KM)

Fig. 3. 1 lb Power spectrum of the difference profile for repeat 10 on arc 293. Note the
significant geoid contamination at 130 km.

153

io5

10'

>—
<_>

x.

CM

*
«

lO

LU

a

LU

10'

10'

,04

WAVELENGTH [KM

io3 10"

— \-r-i — i — i — i 1 1 1 1 i i i — i— i 1 r

10'

io-«

10-' 10''

HP1VENUMBER ICTCLE5/KMJ

10'

Fig. 3.12 Mean power spectra of the difference profiles of arcs 209 (solid line) and 293
(dashed line). The circles indicate values on a corresponding regional spectrum
calculated by Fu (1983), using SEAS AT data.

154

8*8

<o

C/3 0
< >
W rT
C/3 '

g-ao-

° « 2

jggf

1,18 8

_£oo

82 6

o©".S

8 u 3

•§ 2 E
-2«n 5

0~ °

m

CO

155

o

o

20

18

16

14

12

10

8

6

4

2

0

20

18

16

14

12

10

8

6

4

2

0

20

18

16

14

12

10

8

6

4

2

0

321

T 1 1 1 1 1 1 1 r~ —I r

T 1 1

293

T 1 1 1 1 1 1— I 1 1 I

I I 1- 1 1- 1

265

t 1 1 1 1 r— i 1 1 1 r— i r

t 1 1 1

zz
o

CJ

20-1

18-

16-

14-

12-

10-

8-

6-

4-

2-

0-

20

18

16

14

12

10

8

6

4

2

0

237

209

— i 1 1 1 1 1 1 1 1 1 r— t 1 1 1 1 1 1

20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56

DEGREES NORTH LATITUDE

Fig. 3.14 Sea surface height variability (RMS) as a function of latitude, for five GEOSAT
arcs.

156

APPENDIX A

We seek a simple measure of the effect a single spike in a difference
profile has on the RMS variability of that profile. For a large number of points N
making up the profile, the variability a is calculated from

' (i)

where the Vj are the squares of the individual sea surface heights on the difference
profile. Now suppose that all the Vi but one represent "good" data such that the
true variability C7 is the square root of their average, v. The remaining "bad" data
point is denoted as v. Separating (1) into good and bad constituents gives the
calculated variability Gc as

N-l

(2)

N-l _

where for large N, ^ ' • Thus, (2) may be rewritten as

i

or

- y 1/2

osod +-U)

NV (3)

The difference plots have shown that the spike editor keeps individual
spikes smaller than 40 cm, corresponding to a v' of 1600 cm2. For a

conservative N = 540 and v = 40 cm2 (from o = 6.3 cm) this still makes

V
— = « 1 allowing use of the binomial theorem on (3) to give

NV

157

ac^o(l+-^)

2Nv

or

~ v '
oc-o+ =

2Na (4)

Using equation (4), a difference profile consisting of 540 points with a
true variability of 6.0 cm and a spike of 40 cm, will have a calculated variability
of 6.25 cm. This variability is not significant compared to the noise bounds

discussed in Chapter 2.

V

The perturbation term in (4) may be approximated by 2Na to estimate the

true variability from the calculated variability. This was done occasionally during
Chapter 2 when investigating the correction channels and had no bearing on the
outcome; mainly because the corrections themselves had no effect on the spikes.

158

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163

ACKNOWLEDGEMENTS

I would like to thank the following persons and organizations for their contributions
to this thesis:

My advisor, Dr. Carl Wunsch - for guidance, advice, knowledge, helpful criticism,
and being a pleasure to work with.

The United States Navy - for its commitment to graduate education, and funding
my stay at M.I.T. and the Woods Hole Oceanographic Institution through the Naval
Postgraduate School Civilian Institutions program.

Barbara Grant, Linda Meinke, and especially Charmaine King - for programming
assistance, and computing beyond the call of duty. Barbara and Charmaine are supported
in part by ONR grant #N00014-85-J-1241, and Linda is supported in part by ONR grant
#M00014-86-K-0751.

Dorothy Frank - for rapid, accurate word processing, and the patience to put up
with numerous revisions.

Anne Gettelman - for editorial comments, support, and lots of inspiration.

My parents, Dr. Donn V. and Mrs. Jeanne K. Campbell - for instilling a love of
learning in their children, and their support throughout my scholastic years.

Thesis
C19365
c.l

Thesis
C19365
c.l

Campbell

Evaluation of GEOSAT
data and application to
variability of the north-
west Pacific ocean.

Campbell

Evaluation of GEOSAT
data and application to
variability of the north-
west Pacific ocean.

ff

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