:a 9;
14 10
108
ii 1 1 n n *
Monterey, California
ii
.-B
THESIS
SPATIAL STRUCTURES OF OPTICAL PARAMETERS IN
THE
CALIFORNIA CURRENT AS MEASURED WITH THE
NIMBUS-7 COASTAL ZONE COLOR SCANNER
by
John T. McMurtrie, Jr.
March 1984
Thesis Advisor: J. L.
Mueller
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Spatial Structures of Optical Parameters in the
California Current, As Measured with the
Nimbus-7 Coastal Zone Color Scanner
5. TYPE OF REPORT 4 PERIOD COVEREO
Master's Thesis
March 1984
6. PERFORMING ORG. REPORT NUMBER
7. AUTHORS
John T. McMurtrie, Jr.
8. CONTRACT OR GRANT NUMBER^*)
9 PERFORMING ORGANIZATION NAME ANO AOORESS
Naval Postgraduate School
Monterey, California 93943
10. PROGRAM ELEMENT, PROJECT, TASK
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March 1984
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The research reported here was supported by The Office of Naval Research
(Code 425 OA) Work request N0001484 WR24001.
19. KEY WORDS (Continue on ravaree aide II neceeeary and Idantity by block numbar)
Ocean Optical Depth Variability, Remote Sensing, Ocean Color, Coastal
Zone Color Scanner(CZCS) , California Current System, Empirical Orthogonal
Functions.
20. ABSTRACT (Continue on ravaraa alda II nacaeaary and Identity by block number)
Optical variability across the continental slope and shelf off Central
California was studied using Nimbus-7 Coastal Zone Color Scanner (CZCS) data.
CZCS estimates of k(490) , the irradiance attenuation coefficient at 490 nm,
were expressed as optical depth l/k(490). A modified atmospheric correction
algorithm was used to account for water radiance at 670 nm. Time sequences
of l/k(490) were assembled and partitioned into four zonal transects, at
different latitudes, spanning May through November in 1979, 1980 and 1982.
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Empirical Orthogonal Functions(EOFs) were calculated for each partition. The
first EOFs are dominated by scales of order 180 km, with in all cases, a band
of low optical depth water in the first 100 km adjacent to the coast. Scales
decrease in successive EOFs, to about 40 km in the fifth EOF. The feasibility
of joining EOFs from different partitions was demonstrated as a precursor for
future applications to piecewise analysis of oceanic satellite data.
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Spatial Structures cf Optical Parameters in the California
Curren t,
As Measured with the Nimbus-7 Coastal Zone Color Scanner
by
John T. McMurtrie, Jr.
Lieutenant, United States Navy
B.S., University of South Carolina, 1977
Submitted in partial fulfillment of the
requirements for the degree Df
MASTER OF SCIENCE IN METEOROLOGY AND OCEANOGRAPHY
from the
NAVAL POSTGRADUATE SCHOOL
March 1984
ABSTRACT
Optical variability across the continental slope ana
shelf off Central California was studied using Nimbus-7
Coasxal Zone Color Scanner (CZCS) data. CZCS estimates of
It (490), the irradiance attenuation coefficient at 490 nmr
were expressed as optical depth 1/k(490). k modified atmos-
pheric correction algorithm was used tD account for water
radiance at 670 nm. Time sequences of 1 / Ic (4 90) were assem-
bled and partitioned into four zonal transects, at different
latitudes, spanning May through November in 1979, 1980 and
1982. Empirical Orthogonal Functions (EOFs) were calculated
for each partition. The first EOFs are dominated by scales
of order 180 km, with in all cases, a band of low optical
depth water in the first 100 km adjacent to the coast.
Scales decrease in successive EOFs, to about 40 km in the
fifth EOF. The feasibility of joining EOFs from different
partitions was demonstrated as a precursor for future
applications to piecewise analysis of oceanic satellite
da ta .
TABLE OF CONTENTS
I. INTRODUCTION 14
II. OCEANOGRAPHY OF THE CENTRAL CALIFORNIA COAST ... 17
A. THE STUDY DOMAIN 17
1. Coverage 17
a- Area Domain 17
b. Time Domain 19
2. Gecmatry 19
a. Coastal 19
b. Bathymetry 20
3. Descriptive Oceanography 20
a. Coastal Upwelling 20
b. Currents 24
c. tfater Masses 30
III. CZCS OCEAN COLOR IMAGES AND UPPER OCEAN OPTICAL
PROPERTIES 34
A. INTRODUCTION 34
B. SYSTEM DESCRIPTION 35
1. The Nimbus-7 Coastal Zone Color Scanner
(CZCS) 35
2. Measured Signal 37
C. CZCS GEOPHYSICAL ALGORITHMS 38
1. Atmospheric Corrections 38
6
2. Clear Water P.adiance .41
3. Bio-optic Parameters 43
a. Chlorophyl Concentrations 43
fc. Diffuse Attenuation Coefficient .... 45
D. SIGNAL FACTORS 46
IV. EMPIRICAL CRTHOGONAL FUNCTION ANALYSIS METHODS . . 43
A. INTRODUCTION 48
B. EOF EQUATIONS 52
1. Raw Data Conversion 52
2. Principal Direction of Scatter 53
3. Principal Component, Eigenvalue and
Eigenvector Representaton 55
C. PARTITIONED EOF ANALYSIS 57
1. Purpose 57
2. Rules and Methods 58
3. Equation Development 61
D. INTERPRETATION 64
V. RESULTS 65
A. INTRODUCTION 65
B. CORRECTIONS FOR NON-ZERO L (670) IN COASTAL
w
WATERS 65
C. DATA STRUCTURE 70
1. Partition 1 (Zonal Transect at 35 53N) . . 71
2. Partition 2 (Zonal Transect at 35 40N) . . 73
3. Partition 3 (Zonal Transect at 35 22N) . . 75
4. Partition 4 (Zonal Transect at 35 00N) . . 76
D. EOF ANALYSIS 79
1- Eigenvalues and Degrees of Freedom .... 79
2. Data Reconstruction Using Eigenvectors
and Principal Components 81
3. Mean Structure 88
4. Structural Content of Eigenvectors and
Principal Components 90
5. The Joining Cf Two Partitions 112
VI. DISCUSSION AND CONCLUSIONS 116
APPENDIX A. SATELLITE DATA PROCESSING METHODS .... 122
A. INTRODUCTION 122
B. LEVEL-I PROCESSING 123
C. LEVEL-II PROCESSING 128
D. LEVEL-III PROCESSING 130
APPENDIX B. DATA CONDITIONING 132
APPENDIX C. EOF PROCESSING 140
LIST OF REFERENCES 144
INITIAL DISTRIBUTION LIST 149
LIST 0? FIGURES
Figure 1. Ocean 3athymetry Off the California CDasr. . . 13
Figure 2. Ocean Bathymetry Off the California Coast . . 21
Figure 3. Graph Showing T-S Curves Defining
Subarctic Water ....33
Figure 4. Plot Showing the Difference 3etween
Minimization of Distances 49
Figure 5. Trackline plots 60
Figure 6. Comparison Plots For 1/K(490> Between
Track 4 and Selected 69
Figure 7. The Optical Depth Parameter, 1/K(490),
Across Partition 1 (35 53N) 72
Figure 8. The Optical Depth Parameter, 1/K(490),
Across Partition 2 (35 40N) 74
Figure 9. The Optical Depth Parameter, 1/K(49Q),
Across Partition 3 (35 22N) 77
Figure 10. The Optical Depth Pa rameter, 1/K (490) ,
Across Partition 4 (35 00N) 80
Figure 11. Eigenvalues for Partition One 83
Figure 12. Eigenvalues for Partition Two 84
Figure 13. Eigenvalues for Partition Three 85
Figure 14. Eigenvalues for Partition Four 86
Figure 15. Reconstruction of ODtical Depth Transect
of 3 June 1980 . .* 89
Figure 16. Mean and Eigenvectors 1 to 5 for
Partition One 93
Figure 17. Mean and Eigenvectors 1 to 5 for
Partition Twc 94
Figure 18. Mean and Eigenvectors 1 to 5 for
Partition Three 95
Figure 19. Hear, and Eiaenvectcrs 1 to 5 for
Partition Four 96
Figure 20. Principal Components 1 to 5 for Partition
One 97
Figure 21. Principal Components 1 to 5 for Partition
Two 98
Figure 22. Principal Components 1 to 5 for Partition
Three 99
Figure 23. Principal Components 1 to 5 for Partition
Four 100
Figure 24. Mean and Eigenvectors 6 to 10 for
Partition One 104
Figure 25. Mean and Eigenvectors 6 to 10 for
Partition Two 105
Figure 26. Mean and Eigenvectors 6 to 10 for
Partition Three 106
Figure 27. Mean and Eigenvectors 6 to 10 for
Partition Four 107
Figure 28. Principal Components 6 to 10 for
Partition One 108
Figure 29. Principal Components 6 to 10 for
Partition Two 109
Figure 30. Principal Components 6 to 10 for
Partition Three 110
Figure 31. Principal Components 6 to 10 for
Partition Four 111
Figure 32. Level-I Processing Schematic Diagram . . . 124
Figure 33. Level-II Processing Schematic Diagram . . . 129
Figure 34. Level-Ill Processing schematic Diagram . . 131
Figure 35. Data Conditioning Schematic Diagram .... 133
Fiaure 36. Partitioning Scheme for Track One (35 53
N) 136
10
Fiaure 3 7. Partitioning Scheme fcr Track: Two (35 uo
N) . . . 137
Figure 3 8- Partitioning Scheme fcr Track Three (35
22 N) 138
Figure 39. Partitioning Scheme for Track Four (35 53
N) 139
11
LIST OF T ABLZS
TABLE I. Characteristics of the CZCS 36
TABLE II. Eigenvalue Data for Partitions 1 through
4 82
TABLE III. Eigenvalue Data for Joining Process ... 115
TABLE IV. Satellite Data Tapes 125
TABLE V. Partition dimensions 134
12
ACKNOWLEDGEMENT
The tremendous effort of Ms. Melissa Ciarir3, BDH Servi-
ces Company, in processing the unending chain of program
changes and updates deserves special recognition. Also, her
presence served as an organizational factor to keep me on
track for the completion of this thesis.
This thesis presented many problems that were resolved
by the expertise of the thesis advisor. Dr. James Mueller,
Adjunct Professor of Oceanography. These problems have left
me with a keen awareness of the scope and breadth involved
in the processing of satellite data. Support in the hij -he-
ma tical development came from my second reader, Dr. A. J.
Willmott. Finally, a special -hanks to my wife who accepted
my long hours away from heme with no complaints.
13
I- U^iODUCTION
Satellite remote sensing systems offer fast, economical
means of determining the horizontal structure of the oceans
on a global basis. The objective of this thesis is to con-
tribute to the development of empirical methods for using
satellite images of optical parameters and sea surface temp-
erature (SST) to infer th€ upper ocean's vertical structure,
through interpolation and extrapolation of relatively lim-
ited in situ data.
The objective is being approached through regional case
studies of correlations between optical parameters and phys-
ical water mass properties in the upper ocean in different
regions of the world. Mere specifically, this thesis is a
preliminary case study of the California Current region.
The ultimate goal is to relate statistically the horizontal
structure of optical properties observed with the Coastal
Zone Color Scanner (CZCS) to the underlying vertical struc-
tures of temperature and salinity, as well as bio-optical
parameters, for a given region and season.
The study domain encompasses the continental slope and
shelf off the coast of California between Point sur and
Point Arauello. This area was selected to investigate an
14
ocean up welling front which is known to persist throughout
the upwslling season (Traganza,et al., 197 9) . The northern
and southern portions of the study domain are typifisd by
complex sddy structure associated with irregular features in
the bathymetry, such as off Point Sur. Between Point Sur
and Point Arguellc, on the other hand, isolines of S3T and
optical parameters tend to be aligned roughly parallel to
the underlying isobaths.
An ensemble of data acquired with the Nimbus 7 CZCS dur-
ing the summer and fall seasons of 1979, 1980 and 1982 is
analyzed in this study. The horizontal structure in oio-cp-
tical parameters dstermired from cloud-fuss portions of CZCS
imagery are investigated using a Partitioned Empirical
Orthcgonal Function (PEOF) decomposition. The spatial par-
titions examined here consist of four zonal transects cross-
ing the shelf/slope region at different latitudes. The
specific goals of this analysis are:
1. To characterize the meridional and- zonal spatial
correlation structures of ocean color parameters
(specifically optical depth 1/K(490) in meters).
2. Tc compare the spatial scales and structures of
optical variability highlighted by the PEOF
15
decompositions, and to relate zhsse to the historical
descriptive oceanography of the study region, and
3. To develop preliminary statistics related to the
feasibility of joining data from different: spatial par~
titicns en the basis of partial subsampies, and to thus
provide an optimal interpolation of satellite image
data into cloud coversd areas.
16
II. OCEANOGRAPHY CF THE CENTRAL CALIFORNIA
A. THE STUDY DOMAIN
1 . Coverage
a. Area Domain
The region investigated in. -his project is
located between 32 and 40N, and from the coast of California
offshore to approximately 126W, Fig. 1. This area was
selected because it contains water mass structures, includ-
ing fronts, which strongly influence phyzoplankton concen-
trations, and therefore the optical properties of the ocean
water. Furthermore, an adequate sample of data was availa-
ble fcr this area.
A subarea of this region is labelled Insert A in
Figure 1 and presented in greater detail in Figure 2 Insert
A is bounded by 34 to 38 N, and by 126 zo 12 0w. It is the
primary study domain of this thesis. The background hydrog-
raphy and dynamics of this region are described in subsec-
tions 2, 3, and 4 of this chapter.
17
CALIFORNIA COAST BATHYMETRY
40 N ~ .....
4400"
2800 / j
'Cape Mendocino
3200
;4000
38 N —
36 N
4800
36Q0
Francisco
34 Ni
4400
INSERT A
4400
32 N -
- ■
30N|_
130 W
128W
3200 i ^lonterey
400
4000
3600
4000
126 W
124 W
I22 W
120 W
Figure 1. Ocean Bathymetry Off rhe California
Coast (Synthetic Bathy metric Profiling
System (SYNBAPS) Data Contoured at 40u
Intervals) .
18
b. Time Domain
In the time domain, the available CZCS data
include scenes from summer through early fall seasons in
1979, 1980 and 1982. Originally, a single season ensemble
of CZCS data (May through September 1980) was sought. How-
ever, the time span had to be expanded to three years to
assemble a reasonably large sample siza of cloud-free sub-
scenes. The sample analyzed contains three scenes in 1979,
eleven scenes in 1980 and eight scenes in 1982. Detailed
characteristics of these CZCS images are presented in
Appendix E.
2. Geometry
a. Coastal
The California coast bounding the area of study
is characterized by a steep, mountainous coastal range run-
ning roughly parallel to the coastline. The coastline
stretching frcm San Francisco to Point Arguello is oriented
roughly northwest to southeast, but is interrupted by Monte-
ray Bay at 36 45'N and by smaller bays in the vicinity of
Morro Bay at 35 20^. No major rivers drain into this
coastline, although many local rain-generated drainage
creeks empty here.
19
b. Bathyrae-ry
The prsdo minairt orientation of the bathymetry is
roughly northwest :o southeast (parallel to the coast) , Fig.
2. Interruptions of this orientation are evident in the
vicinity of the Monterey Canyon, Point Sur and the Sur Can-
yon, the Davidson Seamount, the Taney Seamount and the Santa
Lucia Banks and Escarpment off Point Arguelio. A very
abrupt shelf creak is evident all along this section of the
California coast. Isobaths tend to diverge south of Monte-
ray, due to a broadening of the continental shelf and slope
with distance south of Monterey.
3« De §c rigtive Oceanography
a. Coastal Upwelling
Coastal upwelling is an oceanic phenomenon which
has a pronounced impact upon many physical and biological
processes. Predominantly southward winds during spring and
summer off the central California coast, yield offshore sur-
face Ekman transports, which forces compensation water to
rise from depths of the order of 200 to 300 m (Smith, 1968).
The upwelling season off the coast of California
is generally confined to the late spring through early fall.
The onset of the seasonal upwelling commences in more
20
INSERT A
38 N
^7
4200
4000
36 N
3600
4400
34 N
4600
4200
126 W
124 W
122 W
120 W
Figure 2. Ocean Bathymetry Off the California Coast
(SYNBAPS Data, Contoured at 230 m Intervals)
21
southern waters off the California coast and prcgrsssss
northward as the season unfolds (Yoshida and Mao, 19 57;
Wooster and Rsid, 1963; Pavlova, 1966; Hickey, 1979). The
CZ CS data set spans the upwelling season and includes images
from beyond this season into early winter.
Upwelling has a marked effect on the sea surface
temperature, causing it to be much lower than would other-
wise be normal for the latitude and season (Smith, 1968).
The relatively lowar temperatures are evident in IR images
of the region (Johnson, 1980; Nestor, 1979). Accompanying
this decrease in temperature is an incraase in surface
salinity, an upwelling property unique to the regime off the
west coast of North America (Smith, 1963) .
The ocsanographic properties of upwelling have
been documented in many areas of the world, but nowhere with
the thoroughness of the work off California and Oregon.
Ship and satellite observations have allowed us to identify
seasons, centers, and the extent of the upwelling event
along the west coast of North America. Traganza, et ai.
(1979) used combined satellite and shipboard observations to
infer nutrient upwelling distributions off the coast of Cal-
ifornia. Frontal structures and mesoscale eddies that can
22
result from the upwelling phenomenon have been examined with
relevance to Anti-Submarine Warfare (ASW) by Traganza
(1979). The use of infrared (IR) imagery in the detection
and description of upwelling was examined by both Johnson
(1980) and Nestor (1979).
The introduction of nutrient rich waters to the
nearshore euphotic zone greatly enhances the development of
the in situ phytcplankton population. This enhancement in
turn causes the upwelled water mass moving offshore to have
distinctly different optical properties than adjoining off-
shore waters. The boundary (frontal region) between the
upwelled water mass and the normal surface water mass is
thus readily detectable and of great interest.
Nutrient enrichment off the California coastal
zone is observed in the regions of upwelling events. These
nutrients, which are classified as "biochemically new" on
the basis of nitrate-to-phosphate ratios which approach
15:1, are brought to the surface from depths up to 300 m.
By way of contrast, nutrients also present in the open ocean
surface water approach 5:1 (Nestor, 1979). The added
nitrates are a primary factor in the increase in phytopiank-
ton concentrations during the upwelling season. Coastal
23
nifty times sore productive than
_ ,. f ^ f j.
waters, on the whole,
open ocean waters and this difference can be increased dur-
ing periods of upwelling (Sverdrup, et al., 1942). The phy-
toplankton concentrations, with their associated
chlorophyll-like pigments, have a profound effect or. the
upwelled radiances measured by the CZCS, as discussed in
Chapter III.
Another aspect of upwelling, and its relation to
satellite data, is its effect on regional climare. The rel-
atively cold sea surface temperature in upwelling zones
cools the air above and thus increases its relative
humidity. As a result, low stratus and fog commonly occur
here in a shallow (marine) layer with warm air aloft. The
frequent occurrence of low stratus and fog, seriously limits
infrared and visible satellite coverage during the upwelling
season. The cool sea water also contributes to a diurnal
sea breeze by increasing the onshore-offshore pressure gra-
dient. Onshore winds bring cool, moist air as far as 50
miles inland (Smith, 1968) .
b. Currents
The California Current System may be discussed
and studied in terms of four large scale currents:
24
the California Current, the California Undercurrent, the
Davidson Inshore Current, and the Southern California Car-
rent (Hickey, 1979) . The first thres of these currents
directly influence the study domain. Masoscale currents
associated with seasonal upwelling are also important here.
O) California Current. Th = California Current
is a broad wind-driven equatorward current which exhibits
significant seasonal variations proportional to the changes
in the wind field (Brown, 1974) . off Point Conception the
mean annual location of the current axis is located 270 km
offshore while the shoreward boundary extends to 200 km off-
shore. The current is of the order 700 km wide and flows
south at 10 to 30 cm per second (Hickey, 1979).
The California Current is a continuation of
the West wind Drift in th€ North Pacific and flows southward
along the California coast between 48 and 23N. It turns
westward between 20 and 30N where it becomes part of the
North Equatorial Current. This flow regime comprises the
eastern extent of the anticyclonic NE Pacific Subtropical
Gyre, which is centered near the Hawaiian Islands (Sverdrup,
et al., 1942; Chelton and Davis, 1982).
25
(2) California Undercurrent. The California
Undercurrent, also, referred tc as the California Counter-
current, is the poleward subsurface flow over the continen-
tal slope. Maximum poleward flew occurs during the summer
and fail seasons at depths of 200 to 250 m (Pavlova, 1966
and Hickey, 1979). The flew can be described as a broad
current with a central jet. It is this jet structure that
is most often measured and referred to when applying
specific values to the Undercurrent. The broad poleward
flow has a geostrophic component alongshore near the shelf
break of approximately 15 cm/sec (Ccddington, 1979).
The flow appears to have a jet-like struc-
ture, both vertically and horizontally, and to extend to the
bottom over the slope. The existence of a high speed jet
core of the order of 20 to 70 km in width, was first sug-
gested by Beid (1962, 1963). Subsequent, direct measurements
of these jets have produced values as high as 40 cm/sec off
Northern Eaja and values of 16 cm/sec off Washington
(Booster and Jones, 1970) . The depth of the high-speed cere
varies seasonally. It rises from depths of 200 to 300 m to
the surface during the late fall and winter north of Point
Conception. Here it is referred as the Davidson Inshore
26
Current by many authors (Hickey, 1979; Pavlova, 1966; Ingra-
ham, 1967). 2vent-scale fluctuations (of the order of 100
km and 10 days) in the flew appear to be correlated with the
alongshore component of wind stress (Nelson, 1977).
The extent and time scale of continuous
alongshore flow, and the width of the region of northward
flow below 500 m, are important topics yet to be answered
about the California Undercurrent (Hickey, 1979).
(3) Davidson l5£k£?£ Current. North of Point
Conception, the poleward surface flow in the nearshore
regions off the West Coast is known as the Davidson Inshore
Current. It is associated with winter weather circulation
patterns. As the southward winds weaken and tend toward a
northwestward flow, the Davidson Inshore Current becomes
established (Hickey, 1979). The current flows near the
coast, usually within 100 km, well inshore of the California
Current and is confined to the continental shelf and slope.
Pavlova (1966) reported that north of Point Conception, the
Davidson Inshore Current reaches its maximum development at
depth (200 to 250 m) in the summer and autumn. In August
the Davidson Inshore Current is scarcely noticeable at
the surface despite active development a depth. Maximum
27
surface development is reached frcra October through April,
i.e., late autumn to early spring. In December, the cere of
maximum velocity emerges at the surface ana in the Late
spring it almost completely disappears (Raid, 1960; Reid,
et al.f 1958; Pavlova, 1966). Poleward velocities of up to
25 cm/sec were recorded (Beid and Swartziose, 1962) within
80 Jem of csntral California in January.
The Davidson Inshore Current and the Cali-
fornia Undercurrent are often discussed as though they were
separate currents. Both currents transport Equatorial-type
water northward at least as far as Cape Mendicino (Pavlova,
1966). Also, no subsurface maximum has been found in the
flow of the Davidson Inshore Current. These characteristics
support a view that the Davidson Current is simply the sur-
face expression of the California Undercurrent, rather than
a separate current superimposed en it.
(4) Other Currents^ The presence of eddies
throughout the the California Current System has been docu-
mented for many years (Bernstein, et al. , 1977). The time
scales for these eddies, as well as the processes responsi-
ble for their generation and subsequent dissipation, is an
area of increasing study.
28
Between San Francisco and a point about
half the distance to Point Conception, chere is a permanent
counterclockwise eddy that produces northward flow curing
all months except April (Erown, 1974). A second eddy just
north of Point Conception forms during the summer months and
makes northward flow continuous from Point Conception to San
Francisco (Hickey, 1979) . Willmott (1983) has shown -hat
these features may be produced by flow separation of the
California current in the vicinity of major coastal capes.
Reid, at al. (1963) made direct measurements of
an eddy (90 km in diameter) off the northern coast cf Baja
California. Hypotheses for eddy formation discussed in
their paper are as follows:
(1) The process of upwelling and the offshore movement of
the cclder, mere saline waters might degenerate into eddies.
The lateral shear between the upwelling flow away from the
coast and the California Current and Undercurrent (bare-
tropic instability) could produce eddy structures. Tempera-
ture and salinity differences set up strong baroclinic zones
along the upwelling boundaries which could result in eddy
formation. (Sverdrup and Fleming, 19U1)
29
(2) The offshore surface flow during upwelling should pro-
duce a counter current (Hunk, 1953). If there is
substantial north-south variation in the intensity of the
winds, then seperate countercur rents of different strengths
might occur along the coast.
(3) A second hypothesis proposed by Reid, et al. (1963), is
that the deeper count ar current may transfer momentum upward
to the surface layers, at times when, or in regions where, a
surface current does not prevail. This could cause spot
intrusions of colder circulating waters that form eddies
where neither surface countercur rents nor coastal upwelling
produce them.
Additionally, the effects of bottom steering by coastal
topography, and the associated trapped motions must be con-
sidered when discussing eddy formation . (Hurlburt, 197U;
Johnson, 1982; Willmott, 1983)
c. Water Masses
Descriptions of the water masses that contribute
to the California Current System are given in Tibby (1941),
Sverdrup, et al., (1942) and Reid, et al., (1958). Four
major sources are discussed by the authors:
(1) Subarctic Water Mass - from tha north.
(2) Central Water Mass - from the west and northwest.
30
(3) Equatorial Water Mass - from the south.
(4) Water derived from upwelling sources.
These sources were simplified in Tibby (1941) and Sverdrup,
et al., (1942) into two extreme sources named "Subarctic
North Pacific" and "Equatorial Pacific".
The percent of each water mass comprising a sam-
ple can te defined by entering Figure 3 with a T-S pair.
However, the determination of percentage composition by this
means cannot te used for water above depths of about 100 m.
This restriction is due to vertical mixing in the nearsur-
face layer related to the effects of wind and local changes
due to heat and mass fluxes across the air-sea interface.
Any mixing along surfaces of constant a would be severly
masked in these shallower depths by the effects of turbulent
vertical mixing. Also, below 1000 m tha differences in the
T-S relationships of the two extreme water masses are negli-
gible. For intermediate depths, as might be expected, the
percentage of equatorial water decreases in the direction of
northward flow. The Undercurrent is characteristically
warmer and more saline than the California Current, and it
has a salinity maximum en the a = 26.54 surface. Cff
31
Monterey and below 8 00 m, the water is greater than 603
Equatorial Water and this percentage increases both with
depth and movement towards lower latitudes (Brown, 1974).
32
O -oo
T(t)8-
Figure 3. Graph Showing T-S Curves Defining
and Equatorial Eacific Water, ana
Various Percentages of Equatorial
Assuming Mixing Along Surfaces of
(Brown, 1974) .
Subarctic Wate:
Curves for
Pacific Water
Equal a
33
k. INTRODUCTION
The physical processes of absorption and scattering
relate the upweliing radiance just beneath the sea surface
to the constituents of the water (Gordon, 1976) . Except for
coastal waters and waters influenced by river discharge,
biological constituents play a dominant role in these pro-
cesses (Smith and Baker, 1978; Jerlov, 1976). Optically, the
most important biological constituent is phytoplankton ,
microscopic plant organisms that photosynthesize and make up
the first link of the oceanic food web (Steele, 1970).
Ch lorophy 11-a' is the dominant photosynthe tic pigment, and
absorbs light strongly in the blue and red regions of the
visible spectrum (U00 to 700 nm) (Hovis, et al., 1980).
Therefore, as the concentration of phytoplankton increases,
the color of the water is shifted toward green hues from the
dsep blue of its pure state. By measuring upwelled radiance
(backscattered daylignt) in specific spectral bands, we can
determine the concentrations of phytoplankton pigments in
the ocean (Gordon, et al. , 1980; Gordon, e_ al. , 1983).
34
This chanter first describes the CZCS sensor and
ts
capabilities, and then the measured signal is discussed.
Algorithms that are currnetly applied to this signal to cor-
rect for atmospheric effects are discussed. Finally, the
algorithms designed to convert the corrected radiance values
to phytoplanktcn concentrations, C, and irradiance attenua-
tion coefficient, k, are presented.
B. SYSTEM DESCRIPTION
I- IJSJ Nimbus- 7 coastal Zone Color Scanner (CZCS)
The CZCS was built by the Ball Brothers Research
Corporation to NASA's specifications. The instrument is a
spatially imaging multispectral scanner. Six spectral bands
are precisely coregistered and internally calibrated. The
swath width of the CZCS is slightly more than 1600 km.
Characteristics of its five visible (443, 520, 550, 670, 750
nm) and one thermal IR (10.5 to 12.5 ym) channels are summa-
rized in Table I. The CZCS has an active scan of 78 degrees
centered en nadir and a field of view of 0.0485 degrees,
yielding a geometric instantaneous field of view of 825 m
(at nadir) from a spacecraft altitude of 955 km. It can
tilt the scan plane 20 degrees from nadir in 2 degree incre-
ments along the satellite track to minimize the influence of
35
direct sa:. glint. The Nimbus- 7 spacecraft is in 2 sun'
synchronous orbit with ascending node near local neon.
3and
Number
TABLE I
Characteristics of the CZCS
(Hovis, et al., 1980)
Wavelength
(nm)
Saturation
Gain Radiance
(mW/cm2sr ^m
1
433 to 453
3
2
1
0
5.41
7.64
9.23
11 .46
2
510 to 530
3
2
1
0
3.50
5.10
6.20
7.64
3
540 to 560
3
2
1
0
2.86
4.14
5.10
6.21
4
660 to 680
3
2
1
0
1.34
1 .91
2.32
2.88
5
6
700 to 800
10,500 to 12,500
23.90
Measured
signal/noise
158/1
200/1
176/1
118-/1
350/1
0.22 K*
* Noise equivalent temperature difference at 270 K.
36
2 • Measured Signal
The designed purpose of the CZCS experiment was to
provide estimates of the nearsurface concentrations of phy-
toplankton pigments (defined to be chiorophyll-a and its
associated degradation products, called "phaeop igments") by
measuring the spectral radiance backscatter ed out of the
ocean (Gordon and Clark, 1881) . The radiance scattered out
of the ocean that reaches the sensor is a very small portion
of the total radiance received. Consider the physical set-
ting where solar irradiance F (A) at a wavelength X is inci-
dent on the top of the atmosphere at a zenith angle 8 and
azimuth 0 and the scanner is' detecting total radiance L f.\)
o t
at a nadir angle 8 and azimuth angle 0 . L ( A) consists of
radiance which has been scattered by tha atmosphere and sea
surface, radiance generated by Fresnel reflection of the
direct (unscattered) solar irradiance from the rough ocean
surface (sun glint), and solar irradiance scattered from
beneath the sea surface t<A)L (A), where t ( A) is the diffuse
d W d
tr ansmitt ance.
Observations (Gordon, et al . , 1983; Gordon, et al. ,
1980) produce values cf L (A) in the blue that are ten times
greater than L (A) . These effects are principally due to
37
scattering by the air (Rayleigh scattering) and by micro-
scopic particles suspended in the air (aerosol scattering),
both of which increase the radiance detected at the sensor.
Fresnel reflection (sun glint) can be ignored as the tilting
capability of the CZCS minimizes its effect. However the
scattering effects, both Rayleigh and aerosol, must be
removed from I (X) to give usable values for the upwslled
radiance L (A).
w
C. CZCS GEOPHYSICAL ALGORITHMS
1. Atmospheric Corrections
From the signal description in section A of this
chapter, we can construct the following formula
(1)
L (X) = Lr(X) + La(X) + td(X)Lw(X)
where
L = Total radiance
Lr= Radiance due to Rayleigh scattering
La= Eadiance due to aerosol scattering
L , = Ocwelled radiance from beneath the sea surface
w *
t , = Diffuse transmi ttancs of the atmosphere
X = Wavelength
38
As previously mentioned L is the total radianca
measured cy the CZCS. The Rayleigh scattering term car. be
expressed as
F (A) t (A) r -,
L (A) = — Pr('^_) + Cp(8) + p(9o)}Pr(a + ) j T (X) ,
4tt cos 6 L J 3
(2)
Where
F = The instantaneous extr aterr estial solar
o
irradiance.
xr = The Rayleigh optical thickness of the atmosphere.
Fr = The Rayleigh scattering phase function.
a_ = The scattering angle through which photons are
backscatter ed frcm the atmosphere to the sensor
without interacting with the sea surface.
a+ = The forward scattering angle of those photons
which are scattered in the atmosphere toward the sea
surface (sky radiance) and then specularly reflected
frcm the surface into the field of view of the sen-
sor (p(8 ) term) as well as photons which are first
o
specularly reflected from the sea surface and then
scattered by the atmosphere into the field of view
of the sensor (p(6) term).
P = The Fresnel reflectance of the air-sea interface.
39
T = The tvic-way ozone transmittar.cs of the atmosphere.
03 J
9 = The sensor zsntith angle at the observed point on
the sea surface.
9 = The solar zenith angle at the observed point on
0
the sea surface.
The aerosol scattering term is found using the 670
nm channel, where there is only a negligibly small contribu-
tion by the L term. (This is referred to as the "black
* w
ocean" assumption.) We calculate aerosol radiance at X =
670 as
L (670) = L (670) - L (670)
a t r
(3)
The key assumption in this algorithm is that the ratio of
aerosol wavelengths is constant over a scene, and is given
as
e(X,670) =
L (X)
a
L (670)
a
(4)
e is calculated using either simultaneous direct radiance
measurement from a ship, or upwelled radiance values mod-
elled at a clear water pixel (Gordon and Clark, 1981) . The
latter method is discussed in section 2 of this chapter.
40
Returning to equation (1), t (x) is the diffuse
d
tr an s mitt a nee of the atmosphere and sea surface, which may
be approximated as
[1 - p(e) ]
td(x) =
exp
m
(X)
+ To,(x)
/ cos
(5)
where all tens have teen previously defined except m, which
is the index of rafracticn of water relative to air and is
assumed to be 4/3 for the wavelengths (400 - 700 nm) .
We have now developed the basics for extracting the
upwelled radiance values, 1^, from the CZCS detected signal.
Lt-
2. Clear Water Radiance
The scene constant, e , given in equation (4) is cal-
culated as
e(X,X ) = (X/X )
F (A) exp[-xn (X)(sec9 + sec8 )]
n __o 0_3 o
F (a ) exp[-Tn (X )(sec3 + sec6 )]
oo 0 3 o o
(6)
where n is called the Angstrom coefficient. Equation (6)
can be rewritten as
e(X,X ) = (X/X )
O 0
F (X) T (X)
0 0 3
F (X ) Tn (X )
oo 0 3 o
(7)
41
where T ( x) = exp
the CZCS X = 670 nm .
- t ( a ) (sec e + sec 6
U 3 J
and where for
Gordon and Clark (1981) developed the concept of
clear water radiance for atmospheric correction of CZCS
imagery. The strategy employed in this study was to find an
area cf the image that cculd be assumed to have a chloro-
3
phyll concentration less than 0.25 mg/m . At this low con-
centration, L at 520 and 550 nm are assumed to be
essentially constant for a given solar elevation. Then,
given these "clear water values" cf L ( A) at one position,
L (a) is calculated using equation (1). e(A,670) are found
3.
from equations (4) using the computed L A A) and L-(670)
value. Finally, rearranging equation (7) we find that
n(A) =
In e(X ,X ) /
o
F (X) T_ (X)
o 03
F (X ) T (A )
o o 0 3 o _
(8)
ln(A/A )
o
Values for n at 520 and 550 nm (n(520) and n(550)) are com-
puted, then averaged to estimate n(443) . The Angstrom coef-
ficient at 443 nm cannot be directly measured in this way
42
because L (443) is highly sensitive to even atinute fluctua-
w
-ions in chlorophyll concentrations at low concentration.
An important aspect of this algorithm is that nei-
ther surface measurements of L (A) , nor any properties of
the aerosol are required to implement the atmospheric
correction.
3 • B ic^o^tic Parameters
a. Chlorophyl Concentrations
Determination of chlorophyll concentrations
C from ratios of L (A) relates the surface value of C to the
w
ratio of the upwelled radiance at two different wavelengths
(Morel and Prieur, 1977; Gordon and Clark, 1981). The basis
for this is that to a first approximation L is proportional
to the ratio of the volume backscattering coefficient,
B(A)b(A) , and the volume absorption coefficient, a (A ) , of
the water plus its constituents (Gordon, et al. , 1983) . The
contributions from the individual constituents can be summed
to fiovide a total value for each optical coefficient.
Moreover the contributions to B(A), b (A) , and a (A ) arising
from phytoplankton and their pigments are assumed to be pro-
portional tc chlorophyll concentration C. Taking a ratio of
Lw at two different wavelengths and applying the assumption
L (A) a 3 (A )b (A) /a (A) , we obtain
w
43
L ( A i ) B(X1)b(X1)a(X2)
(9)
Lw(\2) B(x2)b(>,^^ a(;Vl)
Secause of the non-linearities involved in the individual
constituent contributions to B(A), b (A ) , and a (A ) , we appeal
to a heuristic observation that
Lw(Xi) = R(C,K,...)
Lw(A2)
(10)
i.e., the ratio of two upwelled radiances is a function R
of the chlorophyll concentration, C, and the diffuse attenu-
ation coefficient, k, as well as ether optically important
constituents of seawater. It was then assumed that. R is
related to C through a log linear mcdel of the form
Log C = Log A + Ax Log R(Ai,X2) , (11)
which was empirically fit to observations to determine coef-
ficients A and Ai . Thus, pigment concentrations are com-
puted from CZCS data using the eguation
Al
C = A R (12)
o
The empirical coefficients presented by Gordon,
et al. (1983) have been adopted by NASA and are:
Case I: C 1.5 for R (443,550)
A = 1.1297959
o
A, = -1.71
44
Case II: C > 1.5 for P(443,55G), bat
C < 1. 5 for B (520,550)
A = 1.1297950
o
Ai = -1.71
Case III: C > 1.5 for R (520, 550)
A = 3.3265955
Ai = -2.44
3
whare C is ip. mg/m .
b. Diffuse Attenuation Coefficient
A similar development of tha algorithm for the
determination of the diffuse attenuation coefficient, k (A ) ,
is given by Austin (1981) . Like the chlorophyll concentra-
tion algorithm, this algorithm derives a value based on tha
ratio of L at two wavelenaths. k can be defined as
w
k(A) =
-1
dF(A,z)
F( A, z) dz
(13)
Equation (13) can be solved for irradianoe F(a,z) to obtain
F(A,z2) = F(\,Zl) e
xp T-k(A) (z2-Zl)l
(14)
Hence
k(A) =
z2"zl
In
F(A,z2)
F(A,Zl)
(15)
45
Empirically derived coefficients frccn spectral data vie
— 1 .-+91
id
*L (44 3)
— i
(16)
k(490) = 0.0 8 33 — +0.02 2
(Austin, 1981).
D. SIGNAL FACTORS
Many factors have been accounted for with these algor-
ithms by either mathematical and empirical models or heuris-
tic assumptions. The determination of the total radiance
values in the first four channels of the CZCS allows us to
apply the corrections to determine upwelled radiance. The
constituents of the water which affect ins absorption and
scattering properties are then empirically derived.
The distribution of phytcpla nkt en is controlled by many
local, mesoscale and global factors, including solar radia-
tion, global weather patterns, and ocean circulation pat-
terns. The mesoscale events of upwelling or eddy
circulation can have important regional effects. These fac-
tors are too numerous and varied tc be modelled on a theo-
retical basis. However, empirical modelling can produce
relatively accurate and consistent results.
The measurement of these bio-optioal parameters from
space allows us to remotely determine their relationships to
46
physical events in the regions under study. Tim? scales,
spatial scales and specific features can. be discerned using
the known (or hypothesized) relationships between inherent
optical properties of the ocean water constituents and the
forcing involved in their distribution.
47
IV. EMPIRIC A I iTHOGONAL FUNCTION ANALYSIS
A. INTRODUCTION
The concept of principal component analysis has been
presented and utilized in different forms over the past
eighty years. Fitting a line tc a data set was usually
accomplished using a least squares method. Distances to
this line from each point were measured parallel to an arbi-
trarily set axis. From the early work of Pearson ( 190 1) ,
this method was adapted sc that the perpendicular distances
from each point to the best fit line were measured. Figure
H illustrates this difference and shows that the first
method is tied to a coordinate system while the Pearson
approach is independent of coordinate systems. This new
method laid the foundation for the development of principal
component decomposition techniques. These techniques have
since been utilized in many forms and referred to by similar
names in a number of disciplines. Applications in psychol-
ogy ty Eckert and Young ( 1936 , 1 939) , although somewhat dif-
ferent in their development, contain the essential elements
of data analysis and principal component decomposition as
used in geophysical disciplines tcday.
48
Figure H. Plot, showing the Difference Between Minimization
of Distances to a Line Parallal to an Axis (d
Values) and Perpendicular to the Line (p Values)
(Preisenaorf er , et al . , 1980)
49
Meteorological applications by Lorenz f 1955) , Kutz-
bach(1967) , and Rinne, et al. (1979) demonstrated the convie-
nence of representing a large clinatological field with a
smaller set of values. These areas include:
(1) Non-linear statistical prediction (Lorenz, 1956),
(2) Ncn-linear dynamical prediction (Lcrer.z, 1956) ,
(3) 500 mb height field representation (Rinne, et al. ,
1979) , and
(U) Sea level pressure, surface temperature, and pre-
cipitation pattern representations (Kutzbach, 1967)
Other uses of 20F analysis techniques in oceanography
include the representation of ocean color spectra (Mueller,
1976) and of wave spectra (Aranu vachupun and Thorton, 1983).
The principal difficulties encountered in principal com-
ponent analysis problems relates to the selection of the
•meaningful' subset of components and to their physical
interpretation. Methods of selection of the principal com-
ponents are also widely varied. Preisendorf er , et al. (1981)
discussed two methods which together involve seventeen dif-
ferent testing rules. Empirical selection of a cutoff value
for variance or forcing factors can also be utilized.
50
Visual inspection of the data which leads to a clear cut
(albeit subjective) choice is also an option.
The enormous data volume inherent in satellite data sets
begs application of the techniques of principal component
analysis. Frincipal component analysis techniques often
allow the efficient representation of a large data set by
its first few principal components with a negligible loss of
information- The advantage gained is reduction in the num-
ber of variables needed tc represent the data. Reducing a
data set to its principal components can also aid in the
interpretation of the data by separating noise from the sig-
nal. Principal component analysis theory can be applied to
preliminary explorations within a relatively unstructured
domain of knowledge, one in which the fundamental laws gov-
erning the processes under study are still being
defined. (Preisendcr f er, et al., 1981)
A brief review of the EOF analysis follows to provide
background for the later analyses. The reader is referred
to Priesendorf er, et al. (1981) for a more complete devel-
opment and history. The following matrix algebra notation
is adopted throughout this thesis.
1. No underscore denotes a scalar X
2. A straight line underscore denotes a vector.... JC
51
3. A curved line underscore denotes a matrix.,.
u. A s-raiaht line cverbar denotes a mean value
5. The use of a superscript "T" denotes a matrix
trans cose.
6. HOF EQUATIONS
1- la* Oat a Conversion
Following Preisen dorf er , et al. (1931) let F' be the
raw (uncentered) data matrix,
f ' (1,1) f ' (1,2)
f ' (2,1) f ' (2,2)
. f (1,p)
. f* (2,P)
(17)
f « (n,l) f (n,2) ... f ' (n,p)
where f (i,j) is the measurement in tha i ' th time point and
j* th spatial position. In the present investigation, each
member of F* will correspond to an optical parameter meas-
ured by the CZCS at a particular time and spatial position.
To convert the raw data matrix, F*, to a centered data
matrix, F, the temporal means are computed and subtracted
frcm^F1. The temporal mean vector f(x) is calculated as
f (x)
l n
= n *-"
f ' (t,x)
(18)
t = l
The centered data matrix, F, is then defined as
52
F =
f ' ( 1 , 1) - 1(1) f (1,2) - f (2). ..f ' (l,p) - f(p)
f (2,1) - f(l) f (2,2) - f(2) ...f '(2,p) - f(p)
(19)
f'(n,l) - f(l) f (n,2) - f (2) . . .f (n,p) - f(p)
Each element f (trx) of J consists of a raw data measurement
with the temporal mean removed. The centered data matrix ,
F, can be written as
F = F' - F
(20)
where £ is the matrix containing as rows the transpose of
the mean vector f = f(1),£(2),...,f(p) .
2- Principal Direction of Scatter
To find the direction, e.i, (in the space domain)
along which the scatter (or variance) of the data set is
greatest, consider the projection of the data vectors f^(t)
along an arbitrary direction e_i
D( t ,ei) = f (t)e
(21)
53
Squaring this length and summing over all n observations,
gives a measure cf the scaiter cf the data along the direc-
tion, e i , namely
D (ei) = 2 f (t)^i
t = 1 .
The righthand side cf (22a) can be expanded -co yield
(22a)
(ei) = E eTf(t)fT(t)e 1
t = iL "1_ ~ ~l]
E £(t)fT(t) e
T
(22b)
(22c)
The next stap is to define the "Scatter Matrix", S,
s = F F
(23)
with elements
(i,j) = E f(t)fT(t)
t = i
(24)
Expanding the abcve equation produces for each member of S
n
T
(i,j) = E (f * (t,D-f (D) (f ■ (t,j)-f (j))
t = i
(25)
If the matrix is normalized by dividing by (p - 1), then
when i / i, the members of S are covarianca values, and when
i = j (the trace of the matrix) the members are variance
values (i.e. each element is the variance of f at a single
spatial grid pcint) . The scatter matrix, S, is symmetric.
54
Therefor e, it generally has p non-negative eigenvalues 1. (j
= 1, -.., p) and associated eigenvectors e ^ (j = 1, ..., p) ,
provided that the rank of S is equal to p.
3. Principal Component, Eigenvalue and Eigenvector
Represent alicr.
The first principal component of an observation vec-
tor f is defined to be the linear combination
a i = eufi + eiofo + ...+ e i f = e f
1 " el 1 r 1
12 J- 2
(26)
P P
whose sample variance
P P
1 £ei
(27)
Sa = E E eiieiJSi3 = ^
1 i = 1 j = 1
is a maximum for all possible vectors e, subject to the con-
straint that
T 1
eiei = 1
(28)
Introducing the Lagrange multiplier \\, the maximum variance
must satisfy
T.
s r 2 t "I 3 r
i s +X1(l-ete1) =- e
Se, + X i ( 1-
l$ll T Al
= 2(S -XlI)e1 = 0
Si£iH
(29)
For non-trivial solutions r A lr must be chosen such that
S - Ail
»/>
»/i
- 0 ,
(30)
55
and, therefore, Xi , is an eigenvalue of S, and Sj is its
associated eigenvector. Furthermore
\ £1 = Ai ®i
^
and since eiei = 1
T
S e = \ = s
1
<\j -
al
(31)
(32)
i.e., the first eigenvalue of S is interpr atable as the sam-
ple variance of S. If we expand this development to the
other eigenvalues and eigenvectors of S, we obtain
E =
[*e 1 ; j = 1 , . . .p
the eigenvector matrix and
(33)
L =
-DM
; J = i,
(34)
the diagonal eigenvalue matrix.
Thus ,
S E = E L
(35)
T T
In terms of E the constraint e e = 1 becomes E E = I. Where
I is the identity matrix. Therefore, if we multiply both
T
sides of equation (35) by E ' we obtain
S = E L E
t a. ^ %
(36)
56
Now, using the definition provided for f and equation (20) ,
the principal component matrix can be defined as
A = F E (37)
This is the desired principal decomposition of F where
AT A = (F E)T F E (3 8a)
(38b)
(38c)
(33d)
(38e)
=
ET
F
T ?
E
=
ET
S
E
=
ET
E
a.
L
=
L
.
C. PARTITIONED EOF ANALYSIS
1 . P ur£Ose
The EOF analysis method outlined above wcrks very
well for a large continuous data set. However, geophysical
data sets are rarely continuous. In the case of satellite
data, cloud cover results in many gaps. Sometimes these
gaps can be bridged by linear interpolation, e.g., when they
are small and surrounded by good data. Often this is not
the case and so a scheme of utilizing non-continuous data is
necessary. Here, the purposes of partitioning are:
57
(1) To maximize the sample size in the presence of
cloud cover, thus allowing statistical computations for
subregions ;
(2) To highlight spatial structures of variance fea-
tures locally, before absorbing them into the modes of
the overall domain; and
(3) To achieve computational convience.
Briefly, partitioning permits EOF analysis using
small subsets of the overall data set. These subsets are
partitioned to yield continuous data in each subdomain. An
EOF analysis is completed on each individual subset, and an
eigenvalue matrix, an eigenvector matrix and a principal
component matrix are obtained. The next step is to perform
an EOF analysis to join the principal components of the sub-
sets. This second EOF analysis produces 'joining functions'
which relate twc non-overlapping subsets.
2« £u=i§ §2^ M e t ho d s
When performing the partition of any data set cer-
tain rules must be observed to maintain the statistical
reliability cf the computations. Two obvious and basic
rules are:
58
(1) The minimum partition size (number of pixels) must
re greater than or equal to the sample size (i.e., if
there are 25 sample days each partition must have 25 or
more pixels). In practice, the spatial dimension will
fce required to be significantly greater than the sample
size .
(2) The partition size should net be so small that the
spatial structure is dominated by noise (e.g., a parti-
tion boundary will not be placed in a major feature,
such as a front or eddy of length scale much less than
the partition size).
The methods involved in the partitioning are subject
to the above principles, together with a general understand-
ing cf the physical processes occurring in the study domain.
Four tracklines at 35N, 35 22«N, 35 UO'N and 35 53' N, were
used to aid in this initial trial cf partitioning (Fig. 5) .
The radiance values and computed optical parameters along
each trackline were plotted versus distance from the coast.
These plots were aligned to pictcrialiy represent the data
and its gaps (due to clouds). The partitioning scheme was
59
INSERT A
SO M
^7
rSarv Francisco
4200
36 N
4000
V
!\
: /
3600
4400
\pnr)
4600
34 N
126 W
200
>
4200
124 W
122 W
120 W
Figure 5. Tracklir.e plots.
60
then applied to try and produce subsets that were as com-
plete (continuous) as possible over the time domain. The
total data matrix is thus partitioned into ? subdomains
F = F ; p = 1 ,
(39)
where the subscript t denotes the total data set and sub-
script p denotes the partitions of the data ss+. Each F is
r * 6 v> p
the data matrix for grid points falling within grid parti-
tion p, and contains all time points for which complete data
were acquired in that subdcmain.
3- i2i3^ij:211 Dey elqpment
The EOF decomposition discussed in B, is applied to
each partition separately, such that for each partition, p,
the scatter matrix is given by
8
T
= F F
p ^p ^p
(40)
and from (36)
S = E L E
^p 'vp p ^p
(41)
where E
^ P
= Eigenvector matrix of the spatial partition p,
and
L = Eigenvalue matrix for partition p.
61
The matrix of principal components for each partition is
given by
A = F E
"up 'Vl p % p
(42)
Equations (38) require that
L = A A
%p ^p ^p
(43)
and so (4 1) can te written as
T T
S = E A A E
o-p 'Vp ^p ^p 'Xp
(44
Relating this to a global scatter matrix, S ^
W
>G =
C 1 ?
(SYM)
^ip %^p o-p
(45)
Where C12 represents the natrix of covariances between grid
points in domains 1 and 2, and so forth.
Now the joining process is developed. For any num-
ber cf partitions (two are used in this development) ,
<\,
T
FT Pi
= Ei Ai Ai Ei
(46)
and
S2
T
F o F o
T T
E 9 A 9 A 9 E 9
o.z a,'1 ^ ^
(47)
62
where the subscripts denote partition number.
For this combined set the scatter matrix is given by
^
1 2
li
Fl
?2
F2
Qsing equations (46) and (47) ,
(U8)
"«1
8"
s12 =
<K, l *■
0
^
■ T T
Ai Ai Ao A i
T T
Ai A? A? Ao
eT 0
T
0 Eo
(49)
The joining functions, Jf are defined as the eigenvectors of
the central matrix given in (49) . Finally using (4 1 ) , (43)
and (44) , (49) becomes
9 1 ?
El
0
'v
J 1 2 L. 1 2 aJ 1 2
% A *- a.
a.
"El
'I. 1
0
a.
(50)
The joining functions, J, relate the separate subscenes to
each ether across an overall study domain. The interpreta-
tion of these functions should allow examination of the var-
iations that occur throughout the domain and localized
effects or. the individual partitions of the domain.
63
D. INTERPRETATION
The coordinate system defined by the eigenvectors gives
the domain fcr the principal components. In the present
study, as well as most geophysical applications, the princi-
pal components can be thought of as temporal amplitudes and
the eigenvectors as their spacial modulators (Preisendorf er,
et al., 1981) .
The i-th principal component is that linear combination
of the data field which explains the i-th largest portion of
the total field variance. Essentially -che eigenvectors
define a direction of variance, while the principal compo-
nents give the amplitude of the variance in the direction of
the associated eigenvector.
Once a data sat has been reduced to a set of eigenvec-
tors and associated amplitudes, the guestion of signal ver-
sus noise arises. A decision as to which components of the
data field have significance, and which components of the
data field have no physical meaning must be made. Seme sort
of a selection process must be defined and applied. Bases
for these selection processes should have their roots in the
physical processes being studied.
64
V . 5 Z S J 1 1 5
A. INTRODUCTION
The major fccus of this thesis was to achieve a first
step towards the analysis of the obtained data set. Much of
the energy in producing these results was directed toward
the processing of the data to a usable form for EOF analy-
sis. Appendices A and B give a detailed discussion of the
processing techniques utilized and an accounting of all
adjustments applied to the data.
The results obtained in this thesis encompass three dis-
tinct areas. The first result emerged from the data
processing and a discovery of the breakdown in the black
ocean assumption. The remaining areas are interrelated as
one deals with the data S€t prior tc the EOF analysis, while
the ether attempts to relate this to a statistical meaning
using EOF analysis methods.
B. CORRECTIONS FOR NON-ZERO L (670) IN COASTAL WATERS
w
Preliminary examination of this data set showed that,
near the California coast, the assumption that L (670) = 0
breaks down (Chaptsr III, Section C) . This finding pre-
sented a need for an adjustment algorithm.
65
At pixels ahare the upwelled radiance at 670 r.m ,
L (6 70), is significantly greater than zero, calculated val-
w
ues cf L (443) are often < 0.01 (mv/(cra -sr-ij) (approxi-
w
mately 1 digital count in CZ CS channel 1). This is
unreasonable even in moderately turbid ooean waters. Smith
and Wilson (1981) observed that in coastal waters off Cali-
fornia, where pigments and/or sediment concentration are
relatively high, it is not uncommon for the subsurface
upwelling radiance Lw(670) to be non-zero. They developed
an iterative procedure to account for this, which is similar
to that developed independently and used in the present
processing.
The procedure involves two major sreps.
In the first step, which is invoked when L (443) < 0.01:
1. Set I (443) = 0.01, a minimal value for daylight
W
tackscatter and slightly less than one digital count in
CZCS channel 1 .
2. Decrease L a(670) and increase Lw(670) to be consis-
tent Kith the new value of L (443) (using equations (4)
w
and (1)).
3. Recalculate L (520) and L (550).
w wv
4. Recalculate C1# C2, K(4 90), K(520).
66
The second step is based on the assumption that the
C2 algorithm (equation (12), case III) is robust and insen-
sitive to moderate errors in L (670) . This assumption was
a
supported ty sensitivity calculations which showed C2 values
to vary by less than 30 percent for wile variations in e (X
,670). If Ci - C2 > 0.5 and C 1 > 2, the correction is pre-
sumed unreasonable due to L (670) being too large (and cor-
resocndingly , L (670) being too small). Next, values of
w
K(490) and K(520) are estimated, which are consistent
with C 2:
1. Estimate a ratio L (443) /L (550) consistent with
C 2 by inverting the C 1 algorithm (Equation ( 1 2) with case
I coefficients) using the C 2 values.
2. Increase L (670) and decrease L (670) to be consis-
w a
tent with the new values of Lw(443).
3. Recalculate L (443), L (520), and L (550)
WW w
4. Recalculate C lf C 2, K(490), K(520).
5. Iterate this procedure until C1 and C2 agree.
Data acquired aboard the R/V Acania durinq the Optical
Dynamics Experiment (ODEX) provide a tentative basis for
assessing the validity and performance of the above adjust-
ment algorithm. In Figure 6 values of 1/K(490) calculated
67
from CZCS data, acquired en 16 October 1982, are compared
with preliminary calculations of VK(490) from selected CDEX
stations. The transect shewn is along 35N (partition 4) .
Stations 24 and 25 were occupied 1.5 hoars before the Nim-
bus-7 CZCS observation, and 2 hours after it, respectively.
Station 2 1 was occupied 9 hours , and stations 19 and 13 one
and two days, respectively, prior to the satellite pass.
The 1/K(U90) values at these stations were calculated from
the graphical displays of raw irradiance profiles (at a
wavelength of 490 n ii) presented in the preliminary E/V
ACANIA ODEX CRUISE REPORT (Mueller, Zaneveld and Smith
1982) .
Panel 6a compares the CZCS and in situ 1/K (490) values
before the above adjustment was applied, and figure 6b com-
pares them after the correction. Agreement in both cases is
excellent in the transparent waters at stations 2 1, 24 and
25: no adjustment for L (670) was required in this region.
w
In the inshore portion of the transect, however, agreement
is obviously poor before the L (670) adjustment, and much
w
improved afterwards. This result is preliminary, and sub-
ject to possible revision by cognizant ODEX investigators
when their data have been brought to publishable form.
68
Nevertheless, the L (670) adjustment alisr i*:hm so overwhela-
w
ingly improves the CZCS estimates of K(490) that its use in
this thesis project is fully justified and essential.
U3
ID
CM
£E
u
c
K
O
CD
a.
13
T
e
Ayy
25
T
iWww
i
24
i i i
-as
(\l
en
H
CO
<
o
U13
II
19
T
0
\h^h^4m
-tM,— t 1— — T"
I I I
500
400 300 200
DISTANCE (KM)
100
Figure 6. Comparison Plots For 1/K (490) Between Track 4 and
Selected ODEX Stations. Panels (a) and (b)
Respectively are Before and After Adjustments for
Non-Zero Values of L
w
(670) .
69
C. DATA STRUCTURE
Figures 7 through 10 shew the optical depth parameter,
1/K(490) = 290{U90) along each track for the available data
scenes. (Gordon and acCluney (1975) showed that Z90(a) is
the depth over which 90 percent of L (\) is backscattered. )
The plots are oriented sc that the ccast is on the right-
hand side (positive x) , while time of the data sc2?.^ gees
from earliest to latest in the positive direction along the
ordina-e cf each figure. The scale of 1/K (490) is in
meters.
Chapter II and III give background into the oceanography
of the region and how that can be related to ocean optical
parameters. The structures depicted in figures 7 through 10
will be discussed in terms cf ocean eddy and front visuali-
zations which result from these relationships. Relatively
high values of 1/K(u90) indicate water with lower concentra-
tions of chlorophyll and sediment. In g=neral these concen-
trations may be expected to decrease with distance offshore.
Abrupt changes in 1/K(U90) are usually associated with ocean
frontal structure and eddies.
70
1- £llii:ii25 J (Zonal Transect at 35 53 N)
The only data available from 197 9 (2 3 Nov) is from
winter and shows relatively little structure, (Fig- 7) .
This image was obtained after the end of the upweliing sea-
son and the surface waters were homogeneous to at least 225
km offshore.
The 1980 data series shows mere structure. Begin-
ning en 17 May 80, an eddy of approximately 40 km diameter
was centered approximately 180 km offshore. Sixteen days
later the entire track shows several eddy-like features
ranging in size from 4 km to 20 km. Three days later, en 6
June 80, the track has lost much cf this structure, although
1/K(U90) generally increases in the offshore direction.
This trend persisted and strengthened slightly through June
1980. By 1 August 1980, a distinctive pattern had developed
with rearly uniform turbid waters adjacent to the coast, and
an abrupt (15 km) frontal transition to much more transpar-
ent waters at a distance approximately 95 km offshore. This
pattern is suggestive of the zonal scale of bio-optical
response to coastal upweliing over a single season.
71
o
CM J.
<y
<1)
e
o
30 Sep 82
1 Aug 80
,? 1 Nnv, 7 9
211
111 Land
Distance Offshore km
F-i
igure 7.
The 0
ParJ
Optical Depth Parameter, 1/K(490), Across
tition 1 (35 53N)
72
The 1982 scenes are all in the late fall to early
winter. The transect from 30 September 1982 shews uniformly
turbid water (5 to 10 m optical depth) with little struc-
ture, except for a weak, clear -water eddy signature 200 km
offshore. Between 30 September and 16 October, mors trans-
parent (20 to 25 m optical depth) water intruded from off-
shore to within 190 km of the coast, with a frontal
transition region of scale approximately 15 km. Over the
ensuing month, this clear intrusion appeared to evolve into
a field of less organized, eddy-like anomalies with scales
ranging frcm 10 to 50 km. The 2-dimensional character of
this pattern evolution should be studied in a future analy-
sis of this data set. Such an analysis, which is beyond the
scope of the present thesis, may indicate whether these
changes are test interpreted as breakdown of a spatially
continuous intrusion of offshore water, or as simply due to
advection transporting entirely different water mass
features into view at this location.
2- Partition 2 (Zonal Transect at 35 00 N)
Track two (Fig. 8) is located along latitude 35
40' N (south of the first track) .
73
Date
14 Nov 8 2
3 Nov 82
1 Nov 82
28 Oct 82
27 Oct 82
16 Oct 82
30 Sep 82
1 Aug 80
24 Jun 80
23 Jun 80
12 Jun 80
7 Jun 8 0
6 Jun 8 0
3 jun 80
17 May 80
6 May 8 0
33 Nov 79
261
201 101
Distance Offshore km
Land
Figure 8. The Optical Decth Parameter, 1/K(U90), Across
Partition 2 (3! 40N)
7U
Again, the 1979 data set is limited tc one day late
in the year and is virtually featureless.
The 1980 series starts in late spring (6 Say 19 80}
and progresses through late summer (1 August 1980) . The
same general features found in the northern track are also
evident in track 2. However, the general appearance of the
frontal boundary is broken by additional eddy structures.
In addition, organized ocean frontal structures, which are
apparently related to similar features in Partition 1 data,
are here displaced approximately 15 km further offshore.
The 1982 scenes from partition 2 show the same
front/eddy development presented in the data from the north-
ern transect (partition 1). An offshore frontal boundary
formed between 30 September 198 2 and 16 October 1982, and
evolved to a less organized pattern of eddy signatures.
3- Partition 3 (Zonal. Transect at 35 22 N)
Along track 3 (Fig. 9) the features already dis-
cussed for the previous partitions occurred. An additional
feature here is a strong clear-water eddy signature approxi-
mately 220 km offshore, evident on the third of June 1980.
The data suggest an eddy-like intrusion of transparent off-
shore water, approximately 40 km in diameter. (A feature of
75
similar diameter, but centered approximately 40 Jem closer
inshore appears simultaneously in partition i (see below)).
This feature is short-lived , however, as no evidar.ee of it
remains three days later. This strange phenomenon is
another candidate for future investigation in a
2-dimensioral analysis.
In general, frontal boundaries along partition three
are farther offshore in both the 1980 and 1982 series than
apparently related frontal signatures in partitions one and
two. Again, this may suggest the importance of bottom
steering of the mean flow ever diverging isobaths.
4. Partition a (Zonal Transect at 35 00N)
The southernmost track at 35 00 N (Fig. 10) has sim-
ilar patterns, including the offshore displacement of fron-
tal boundaries. The location of the frontal boundary is at
its farthest offshore position in this track.
The transient eddy feature of 3 June 1930, as dis-
cussed in partition 3, appears here to have a latitudinal
extent large enough to span partitions 3 and 4 which are
roughly 45 km apart. This is consistent with the zonal
scale of the feature (approximately 40 km) .
76
268
208 108
Distance Offshore km
Date
14 Nov 82
3 Nov 82
30 Sep 82
■— 2 3 Nov 7 9
La'nd
Fiqure 9. The Optical Depth Parameter, 1/K(U90) , Across
Partition 3 (35 22N)
77
There also is a longitudinal displacement of approximately
40 km which suggests an sddy of slightly oblong shape that
roughly parallels the coast. Questions of whether such fea-
tures are coherent and continuous from one transect to
another can te easily resolved in future 2-dimensional
analyses.
It is reasonable to expect a band of surface water
with low optical depth to lie adjacent to the coast, due to
enhanced nearshore biological productivity during the
upwelling season and due tc northward transport of more tur-
bid water masses by the Davidson Inshore Current in winter.
In contrast, offshore waters tend to be far more transpar-
ent, at least during the seasons covered by this data set.
It may be anticipated therefore, that an obvious optical
front will persistently delineate the boundary between what
may be classified as nearshore and offshore bio-optical
regimes, and that intrusions of eddy- like surface water fea-
tures from ere regime to the other will be illuminated by
optical contrast. Additionally, horizontal gradient struc-
ture in bio-optical processes will, within each regime,
often (but not always) accompany the physical structure
associated with ccean frcnts and eddies and produce optical
78
depth structure of similar scales. The scales of previously
observed structures in tie California Current region may
thus be expected to be present in the horizontal structure
of optical depth. It is clear from the above discussion and
the review cf Chapter II that this is in fact the case. In
the next section these structures will be discussed from a
statistical viewpoint using EOF analysis.
D. EOF ANALYSIS
1 ■ Eigenvalues and Decrees of Freedom
Table II is a listing cf the eigenvalues for the
spatial covariance matrix calculated from the data in each
partition analyzed. The cumulative percentages cf the total
variance are included. The first ten eigenvalues are listed
here. In each case, they account for roughly 98 percent of
the variance. The eigenvalues are presented graphically in
figures 11 through 1 4, which include both the eigenvalues
listed in Table II, and the additional eigenvalues that con-
tain the nciser-appearing, higher spatial freguencies of
variability (and together account for lees than 2 percent of
the total sample variance). We have assumed that 98 percent
of the variance is an adequate cutoff for calculations.
79
286
206 106
Distance Offshore km
23 Nov 79
Land
Figure 10. The Optical Depth Parameter , 1 /K (490) , Across
Partition 4 (35 00N)
80
The major features seen in the eigenvalues ars the
differences in the first value between parti-ions 1, 2, and
3, and partition 4. The variance in partition u is rccre
evenly distributed over the first three eigenvalues. All
four partitions have reached roughly 90 percent of the vari-
ance by the sixth eignevalue. The difference in the struc-
ture of the eigenvalues suggests that partition four is
either affected by additional factors not found in the
northern partitions, or that some factors which influence
the northern partitions are absent here.
2. Data He const ruction Usin q Eigenvectors and Principal
Components " ~ "" -—.—— —
Before the eigenvectors and principal components are
interpreted, how they are combined to reconstruct a particu-
lar observation is explained. Recall that each eigenvector
defines a direction of spatial varibility, and that its
associated principal components represent the amplitude of
variations in that direction at certain time points. In the
present context a "direction" takes the form of 1/K(U90)
variations that are coupled at all grid points of the
domain, and "direction" in this sense may be best
81
1
2
3
U
5
6
7
8
9
10
TABLE II
Eigenvalue Data fcr Partitions 1 through 4
Partition 1
Order Eigenvalue Cumulative
2
(m )- Percentage
Partition 2
Eigenvalue Cumulative
2
(m ) Percentage
1
271-40
41. 60
36 5.90
43.70
2
174.90
68. 41
217.40
69.67
3
72. 17
79. 47
99.03
81 .49
4
37.72
85.25
45. oO
db .94
5
26.90
89.37
28.85
90.39
6
17.69
92.08
20.39
92.83
7
14.80
94. 35
15.12
94.64
8
10.59
95. 97
13.63
96.27
9
7.02
97. C5
7.90
97.21
10
5.08
97. 83
5.98
97.92
Partition 3
Orier Eigenvalue Cumulative
2
(i ) ♦
557.40
10
97
58,
32,
60
76
08
93
20.72
13.86
9.48
8.29
5.74
P ercentage
54. 18
74. 65
84. 15
89. 79
92. SS
95.00
96. 35
97
98
98
27
06
64
Partition 4
Eigenvalue Cumulative
2
(m ) Percentage
273.70
133.60
119.80
44. 14
40.87
35.96
24.42
13.33
8.22
7.00
35
59
74
80
85
52
35
90
63
93
90.60
93.77
95.51
96.58
97.49
illustrated either as a curve (for one-dimensional
transects) or contour plots (fcr two-dimensional domains).
Given a temporal mean value for each grid point of a domain,
the eigenvector multiplied by the principal component fcr a
specific time yields a modifier tc the mean signal.
For example, to view the contributions of the first
five eigenvectors to the observed signal for 3 June 1980
82
EIGENVALUES
»■
s-
<c«-
SsS-
8-
».0 8.0 10. 0
EIGENVALUE
'igure 11. Eigenvalues for Partition Ona
83
EIGENVALUES
£
W c
4.0 6. a 9.0 10.0 12.0 11.0
EIGCNVftULC
Figure 12. Eigenvalues for Partition Two
8a
EIGENVALUES
«-
I
e
u
o
8
40 4.0 9.0 ICO 12.0 11.0
eiGCNVRLUt
Figure 13. Eigenvalues for Partition Three
85
EIGENVALUES
1.0 1.0 10.9
Figure 14. Eigenvalues for Partition Four
86
. ...v eigenvector by its principal
JitlOB 3), We BUltipU each e-9-r
Tn, stepwise reconstruction of the ob = .r
T"' 1980 a. (Partition 3, fro. if P— i
:n 3 June 198U aa^a vr
4-^^c is illustrated (Fig. 15).
linwts and e.g«nvcc
■^^noe^nr primarily
, r- that the fir£t eigenvector y
It is clear tnax ta
ffclinre This aspect of ttie
,,Litudes only beyond 80 k. offshore.
• a in eiaenvectors will be dis-
.L-informetion contained in eigen
a 4 „ i9*er sections.
Figur€ 15b illustrates the previously a==uaula.-d
4-h- flotted curve
•a-n^ now tne u - - i —
„ith a solid curve, and
• .aaition =f the second eigenvector as »odi-
tresents tne addition
„„ + This mode accounts
ld ty the second principal component. Th-
, the en-ire transact with scales of
:r variations across the entir
«c aunvs -he contribution or the
i -?-nnre 15c ShOWo ua«
::der 100 km- r.gurs
-* Thi= mode has
„ n.jnriDai component. m —
nird eigenvector and principal
• -i fas will be subsequently dis-
4. «« effect on the signal (as w_ix
Lmost no eii«<-<- *->»,
M. thl. happens to be an anxious case, . «- -tri-
,ations0ft»e fourth and fifth eigenvectors and Principal
:a.pcnent are illustrated in rigs. «. - - -paring
87
from partition thrae of VIS113 on 25 Jun 1983 and the recon-
structed curve agree with the use of five eigenvalues, and
that a fair reprssen tation can be reconstructed using only
the first twc eigenvectors.
3- M§lfi Structure
The mean optical depth (1/k(490) = Z90 ) transect
profiles for partitions 1 through 4 are illustrated in Figs.
16 through 19 and are repeated for ease of comparison in
Figs. 24 through 27. The mean vector in each track
represents the tendency of the signal, while the eigenvec-
tors scaled by the principal components give the perturba-
tions of the mean. In all four partitions, the mean value
of optical depth tends to increase with distance offshore.
This tendency is expected since the coastal waters should
contain higher concentrations of sediment and phytoplankton,
especially during the upwelling season (Traganza, et al.,
1979). There is a general lack of significant eddy-like
structure in the mean vectors from all four transects
(although very lew amplitude perturbations of scale five-to-
ten km are apparent in the means).
88
en
u
0)
jj
E
o
200 100
Distance Offshore km
Land
Figure 15.
Reconstruction of Optical Depth Transect of 3
June 1980 (Partition 3) from Mean (Solid Curve,
Panel a) and Successive Contributions of
Eigenvectors 1 to 5 (Dashed Curves) in Panel a
to e Respectively.
89
4. Structural Content of Eigenvectors and Principal
Compcnen t s~"
The eigenvector discussion involves many intercom-
parisons of the partitions. Each partition's first ten
eigenvectors are plotted, (Figs. 16 through 19 and 24 through
27). To organize the discussion, the first eigenvector will
be discussed for all four partitions before proceeding to
discuss the second, and so forth. The associated principal
components are also illustrated, (Figs. 20 through 23 and 28
through 3 1) .
The structure in the first eigenvector of each of
the partitions is characterized by a band of low variability
adjacent to the coast, and the structure offshore of that
band is dominated by a scale extending from there to the
offshore end of the domain. The "node" marking the onshore
limit of significant variation in this mode is progressively
farther offshore, proceeding from the south through the par-
titions. The "node" of partition 1 begins at approximately
45 km offshore, and by partition 4, the "node" is 100 km
offshore. There is a tendency for variance to decrease in
the amplitude of the first eigenvector as the offshore
boundary is approached. This may be an artifact of the
90
outer boundary and should be investigated farther over a
larger doirain to better estimate the dominant scale. Parti-
tion U, which has the larcest spatial extent, shows mors and
larger offshore structure to beyond 180 km.
The associated principal components, which modify
the eigenvectors before they are applied to the mean, show
the time variations. Across the four tracks, the first
eigenvectors/principal components vary in phase with each
other. In all four transects there is a large difference
between the first principal component of the only 1979 image
(early winter) and those from 1980 (early spring) . This
marked difference is certainly a manifestation of seasonal
variations in the California Current system (Pavlova, 1966;
Hickey; 1979) . Most of the first principal components vari-
ability in all cases is observed in the 1980 series (upwell-
ing season) , and the record contains relatively little
variability in the 1982 series (Davidson Current season).
Coherency of the variations differs from partition-to- parti-
tion with no apparent pattern. The first eigenvector and
principal component appear to have their foundations in the
offshore seasonal variation and large scale eddy structure
that occurs during the upwelling season. In the first mode,
91
the inshore zone influenced by upwelling tends to Terrain
turbid throughout the year, whereas the dominant variations
in optical depth occur offshore of the upwelling zona.
The shapes of second eigenvectors from the four par-
titions are similar, but vary in an oscillatory fashion from
the northern partition to the southern partition. The first
partition (northern) shows negative values beyond approxi-
mately 180 km offshore, and then small amplitude positive
values from there to the coast . Partition two and three
depict a mirror image pattern to that of the first parti-
tion. Partition four shews much the same pattern as parti-
tion one. The phase relation in the principal components
shows no pattern between partitions one and two, but the
series for partitions three and four both suggest phase
reversal from the first partition. This negative-to-posi-
tive-back-to-negative pattern cf behavior weakly suggests a
wave-like meridional oscillatory structure, with an offshore
peak (180 km offshore) in the vicinity of partitions two and
three. Resolution of this meridional characteristic feature
will require a 2-dimensional analysis. The distance of sep-
aration of the partitions suggests a wavelength of the order
of 120 km. Again, the majority cf the variability occurs
92
■"1
o
Ci 01
rr s-i
*" <° -
« 4J 3-
«CHN HK3 I - •> LIW.NVH.IUK">
i i I
Til'
I I I I '
I I 111
211 111
Distance Offshore km
Land
Figure 16. Mean and Eigenvectors 1 to 5 for Partition One,
93
o
X 4S 3.
h a 2.
rtCPW «C 1 - S C1CCNVCCT0RS
1 1 1 1 1 1
1 1 1
?J
a.,^^V
1 1 1
261 201 101
Distance Offshore km
Land
Figure 17. Mean and Eigenvectors 1 to 5 for Partition Two.
9H
«e«w hoi -s ciia-NvixTURs
o ei
& en u
^ u ■
w a) ~
^ 4-> •
\ OJ *
r-t S °.
t ■ i r*
-T-- I I
268 208 108
Distance Offshore km
Land
Figure 18. Mean and Eigenvectors 1 to 5 for Parti-ion
Thr€€.
95
81
o =
*T U 2
— a)
h e s
1i^ WI-5 CISENVCCTORS
i ■ »
i i i i ii
— 1
7—3
y*^v ■ — ' w\W" V^'
?j
?j
^V
'^■nu''V
^v ;»■
i i i i i i
i i i
286
206 106
Distance Offshore km
Land
Figure 19. Mean and Eigenvectors 1 to 5 for Partition Four,
96
CKlNCIPflL COMPOTCNTS 1 " S
-x^-
*—
1 I I
«J
"T
»J
I »■— T-
10
Time Point
18
Figure 20. Principal Components 1 to 5 for Partition One,
97
PRINCIPflL COMPONENTS I - 5
I / I '\s
ZX
n 1
\y
■ i i
sj
10
18
Time Point
Figure 21. Principal Components 1 to 5 for Partition Two
98
PRINCIPAL OWONCNTS t - 5
■ 1
1 ■ -»■ *l ^ ■ I lJ 1 ^V. '
^»— »- I t I I I I
, /x
10
15
Time Point
Figure 22. Principal Components 1 to 5 for Partition Three
99
HI
PUINCIP*. COTPOtCHTS 1 - 5
, 2
i * m »o
- 3
+. ^X
■^
10
18
Time Point
Figure 23. Principal Components 1 to 5 for Parxition Fou:
100
daring the 1980 series. More nearshore structures are
apparent in -his eigenvector, as is an increase in nearshore
variability, as compared tc the first eigenvector.
The ^hird eigenvector has a similar behavior for the
first two partitions. The perturbations are of roughly the
same spatial scale (45 km) and appear to be in phase. How-
ever, the third and fourth partitions show an opposite
behavior in the far offshore region (beyond 180 km). Numer-
ous smaller scale features (of the order of 10 km or less)
are apparent in this eigenvector. In general, higher spa-
tial frequencies become increasingly important in higher
order eigenvectors. The principal components show an
increase in the variability of the 1982 series with the wide
range of variability still present in the 1980 series. This
eigenvector shows the largest nearshore amplitudes of all
the eigenvectors, which suggests it may be closely linked to
the nearshore structure of upwelling. There is little sug-
gestion of a temporal relation evident in the third princi-
pal components of the four tracks.
The fourth eigenvector shows an increase in fre-
quency (decrease in wavelength) of the represented variabil-
ity scales. Features range in size from 18 - 45 km, with
101
numerous smaller scale perturbations. The average
wavelength and range of variability is approximately the
same for all four partiticns. The principal components for
partitions one , three and four qualitatively suggest coher-
ency. However, eigenvector behavicr is opposite in parti-
tion three and similar in partition four when compared to
partition one. A prominent feature approximately 180 km
offshcre in the fourth eigenvector of partition one, seems
to be shifted outward to 210 km offshors in partition four.
This time/space relationship between the structures of par-
tition one and four again suggests a meridional oscillation
worthy of future investigation through a 2-dimsnsional
analysis.
The fifth eigenvector shows features of scale that
range from 5-42 km. The much s nailer features (less than 5
km) are not dealt with as they are essentially part of the
background noise expected in any natural system. Little can
be said about the correspondence of the four partitions with
just a visual inspection. However, nearshore structure
appears in the eigenvectors with more variability than in
previcus eigenvectors of the 198 2 series. The 1980 series
still demonstrates the largest overall variability, and the
102
large interannual differences between the one scene in 1979
and the 1980 scenes is still evident. The principle compo-
nents of partition one and fear show excellent, agreement in
amplitude and phase for the first five time points, but as
structural variability decreases with time, so does corre-
spondence. The quantitative correspondence of these varia-
tions is beyond the scope cf this thesis, but it should be
investigated in future analyses of this data set.
The sixth through the tenth eigenvectors are charac-
terized by variation cf such high frequency that little can
be said of the relationships bet ween the partitions. Scales
of structural features in these eigenvectors range from 1 to
35 km, with no suggesticn of a temporal relation between
partitions. The principal components show that the 1980
data again dominates the variance, but it these higher fre-
quencies the increased contribution cf the 1982 data to the
total variance is very apparent. Eecause of this disorgan-
ized structure, detailed interpretation is not attempted for
eigenvectors cf order greater than 5.
103
MCAN W4J 6 - !0 ElGCNVCCTtWS
O ?
h a :
I I I I I
li*vA'AW tf=*>^r*=f ■ 9
^v^aa^^v^
10
i i i
211
111
Distance Offshore km
Land
Figure 24. Sean and Eigenvectors 6 tc 13 for Partition One,
104
o 2
rH e a
KfH »<3 6 • 1C E1GCNVCCT0R5
i i 1 1 i ' i >
/^~\
fA.-Ov^v
. 6
v'v/^T^^ 7
I \ ) I «J*I
T7
17 V t I "
8
V
^
^X^y
rAj^ ^^^yA^^v
10
261 201 101
Distance Offshore km
Land
Figure 25. Mean and Eigenvectors 6 tc 10 for Partition T
wo
105
no* ANO 6 - 10 E1CCNVCCWS
\^y £r- \*f*\
t >«oli\ r
mjv
"V
9-1
^
^-, 8
9-i
r\ t I r
■^Aa r^'
I I I I
268 208 108
Distance Offshore km
10
Land
Figure 26 .
Mean and Eigenvectors 6 tc 10 for Partition
Three.
106
o
<r> w »
"^ u '
«— » <o »
X. V *
N. » 3
.H S •
HSN ftflS-IO CISCNVCCTORS
'i i i i i i i i i i i i i i i
^vv^A^V^
8
JU
-rv\^/v^ V »'"1A^
«*n
V^V'W^^TV^'
I »' I
10
II
286
206 106
Distance Offshore km
Land
Figure 27. Mean and Eigenvectors 6 tc 10 for Partition
Four.
107
PRINCIPfH. CCMP0NCNT5 6-10
m
aj
i i
8
Time Point
Figure 28. Principal Components 6 to 10 for Partition One,
108
PRINCIfflL COMPOfCNTS 5 - 10
• i
10
10
18
Time Point
Figure 29. Principal Components 6 to 10 for Partition Two,
109
PRIUCIFflL COnPOMCNrb s - to
*J
- 7
T 1 I O
10
Time Point
gure 30.
Principal Components 6 tc 10 for Partition
Three.
110
A
/ ■ V-v/'V ■
fRIMTIPflL CDTPWCNTS 6-10
<^L
->7
8 J
» — ' n>" i i o
-^
10
"s^"
r v^*-t— -■■ f^ ■ ■¥■
10
18
Time Point
Figure 31. Principal Comfcnents 6 tc 10 for Partition Four
1 11
5- The Joining Of Two Parti tic r.s
Only a cursory treatment of the joining function
results is presented. Partition one and two were examined
and analyzed as per the joining function development given
in Chaptsr IV. The analysis was based on performing the EOF
analysis on partitions one and two combined, and then com-
paring this result to these obtained from the seperate F.OF
analyses that were joined. It was found that the joining
function principle components were within .0001 m cf these
computed using the two partitions as one data set. Further-
more, using ten degrees of freedom, it w=s found that
-6
j J - I
(within 2.5 X 10 ) (51)
and
E E = I
-6
(within 8.0 X 10 ) (52)
This result is cased on the orthogonality of the eigenvec-
tors as J is the matrix cf eigenvectors of the covariance
matrix of principal components for the two partitions com-
bined.
Additionally, a comparison of the principal compo-
nents yielded (For a joint sample size of 17)
W
A A = L
-3
(within 1.4 X 10 ) (53)
112
Representing the principal components of the joining
tions with "i , it follows that
sj*
(*
T "^
Y Y = L (within 5.0 X 10 ) { 54)
Again the small difference demonstrates the utility of the
joining process.
The eigenvalues obtained by thus joining the eigen-
vectors and principal components frcm partitions 1 and 2 are
given in Table III, together with fractions of total sample
variance. At face value, the first ten eigenvalues account
for 98.55* of total sample variance. Rscall however, -hat
the input data were represented in truncated form, using
only the first ten principal components from each of the
partitions. This original approximation retained only
97.89% of the total variance computed from the original
data, hence, it is necessary to adjust the apparent trunca-
tion of the joined result accordingly. The results of this
adjustment are given in the third column of Table III and
show that assuming that ocly the first ten eigenvectors are
significant actually leads to a truncation to 96.50% of the
total sample variance. While 3.5% precision is an accepta-
ble level of approximation for most problems in geophysical
1 13
>a . xv u
da ta interpretation, the affects of successive trunc«
must be giver, careful attention when applying the parti-
tioned method to EOF analysis.
This example indicates strongly that the partition-
ing approach to EOF analysis can provide computationally
acceptable results when applied tc satellite image data.
The example also emphasizes that proper care must be given
to controlling successive truncation in the partition join-
ing process. It is left to future projects to investigate
questions such as joining partitions on the basis of par-
tially intersecting samples to provide optimal functions
for interpolating satellite data into cloudy regions, and to
interpretation of partition joining functions to illuminate
spatial correlations between locally important structures
(e.g., topographically generated mesoscale eddies) and the
dominant structure of the overall domain (e.g., that associ-
ated with the evolution of the synoptic scale upwelling
front over the continental slope and shelf over the course
of the upwelling season) . It is questions of this kind that
address the ultimate utility of partitioned EOF analysis.
The present effort is limited tc preliminary work to estab-
lish foundations of feasibility and procedural constraints.
1 14
TABLE III
Eigenvalue Data for Joining Process
Order Eigenvalue Percentage Cumulative
2
(m ) of Variance Percentage
1
594.00
40.74
40.74
2
U01.60
27.54
68.28
3
171.70
11.78
80.06
a
67.35
4.62
9U.63
5
6 3.86
4.38
89.06
6
U7.57
3.26
92.32
7
34.34
2.36
94.68
8
23.17
1.59
96.27
9
20.52
1.41
97.68
0
12.70
0.87
98.55
1 15
VI- 2I2CUSSION AND CONCLUSIONS
Zonal transects of optical depth (1/k(490) m) ueasursd
with the Nimbus-7 C2C5 have been analysed to investigate
bio-optical structure ovsr the continental shelf and slope
off central California. Samples of cloud free data were
selected and processed for latitudes 35-53N, 35-40N, 35-22N
and 35-OON. The data were observed in 1979, 1980, and 1982
during the months May through November. The zonal structure
in these samples was analysed using EOF's computed sepa-
rately for each section. Meridional variance structure was
analysed only qualitatively through inspection of similari-
ties in features contained in EOF's of the different tran-
sects and in the temporal sequences of associated principal
components. Finally, the computational feasibility of
applying partiticned EOF analysis methods to this type of
data was investigated by joining the EOF's of the two north-
ernmost transects to form estimates of the EOF's of the com-
bined spatial domain.
The first eigenvectors for four zonal transects of opti-
cal depth 1/(kU90) each contained dominant scales of order
116
20 0 km or greater, and accounted for between 35 and 5 4 per-
cent of ths total variance. They are each also charac- pr-
ized by a hand of low variability in optical depth in the
inshore region influenced by upwelling and the Davidson
Inshore Currant. This band is confined within 45 km of the
coast at 35 53' N, and icnoton ically broadens to approxi-
mately 100 km at 35N latitude. This behavior is possibly
related to the broadening of the continental shelf and slope
with longshore distance scuth of Monterey. Hurlburt (1979)
showed that the topographic beta effect plays a fundamental
role in the dynamics associated with mssoscale (order 100
km) longshore variations in topography by affecting the
strength of the longshore flow. Also, the influence of
topography can produce barctropic flew beyond its immediate
vicinity. For mssoscale variations in coastline geometry,
the coastal currents and the patterns of vertical motion
tend to follow the coastline, but net with uniform strength.
Coastal current widths tend to be narrower than the scale of
coastline variability. In these terms, the meridional vari-
ation in scales present in the first EOF* s are gualitatively
consistent with the longshore variations in bathymetry of
the study domain.
117
The second eigenvectors account for zonal structure with
dominant scale cf order 120 km, and with nearly uniform
amplitude frcm the coast tc a node approximately 150 km off-
shore in partitions 1, 2f and 4. The second eigenvector for
partition 3 (35 20N) is anomalous in that it is dominated by
a zonal waveform with nodes spaced at approximately 80 km,
or roughly half the dominant scale of its counterparts. The
reason for this behavior should be investigated.
The third eigenvectors are dominated by scales ranging
from approximately 60 to 100 km (between nodes) . The shapes
and scales vary more strongly from partition-to-partition
than was the case with the first two eigenvectors.
Across each transect, zonal features with wavelengths
100 km and greater appear. The suggestion of an oscillatory
behavior in the meridional direction needs to be studied
further. Resolving such a feature requires a more detailed
study involving a 2- dimensional analysis cf the study
domain.
The large eddy field associated with the shoreward
boundary of the California Current was observed in the data
set. The scales of this eddy field were of the same
magnitude as the spatial scales employed in the partitions.
1 18
This necessitated placing the partition boundary within this
eddy field anc cutting away seme of the features. The sea-
sonal development of a synoptic scale iipwelling front off
the California coast is strongly suggested in the data and
its eigenvectors. The s sailer eddies associated with this
pattern ranged from 5 to 100 km in scale.
The convergence of the eigenvalues to roughly 98 percent
of the variance after the tenth value was of particular
interest. This was true for all four partitions and
although this is not an overwhelming reduction in the
degrees of freedom of the initial system, it is significant.
Satellite images, and ether fields of oceanic and atmos-
pheric variaDles, provide massive data sets. Large amounts
of computer time must often be expended for processing these
data sets at even relatively primitive levels. Analyses and
interpretations are, morever, made difficult by the sheer
volume of data. EOF analysis provides a viable method for
mathematically representing satellite data fields in a com-
pact and easily manipulated form. Data transformed using
EOF's illuminates, and facilitates analysis of, the time and
space scales associated with a given variable over the
domain; the present study has exercised this attribute of
119
EOF's on a descriptive level. In addition, the ccapact
principal component representation cf satellite images pro-
vides an efficient form for analysing the response cf spa-
tial structure in, for example, optical depth to forcing by
wind stress and currents, acting through a bio-cpticai
model; this is a logical avenue fcr future research to build
on the present results.
Considering purely ccmputat ional aspects of SOFs, the
well-known symmetry of eigenvector solutions in the time and
space domain can be used tc great advantage in the analysis
of satellite image data. The number cf spatial grid points
in even the single trackline partitions of the present
study yield large, but computationally tractable, scatter
matrices. The larger arrays associated with 2-dimensicnal
area partitions, each with several hundred grid points, will
clearly exceed sizes admitting direct computation of spatial
EOFs. The linear algebra and scalicgs involved in using the
smaller time domain scatter matrix for computation cf space
domain EOF*s is reviewed in Appendix C.
The partitioned method of EOF analysis illuminates cor-
relations between variability in spatially separate sub-re-
gions. The present results demonstrate the computational
120
faasiblity of this piecewise approach when applied to CZCS
optical depth data. There is every reason to believe -hat
the method may be equally well applied to other CZCS parame-
ters and tc infrared imagery of SSI. Further work in this
area should aim to determine whether the joining functions
linking EOF's from separate domains are sufficiently sta-
tionary tc provide a basis for optimally interpolating sat-
ellite image cata of these types over cloud-covered areas of
a particular day^s image. Other applications to be explored
include determination of the extent to which correlations
between 3-dimensional in situ data and 2-dimensicnal satel-
lite data in small sub-regions may be extended to other
parts of the larger domain covered by satellite data alone.
121
APPENDIX A
SATELLITE DATA PROCESSING METHODS
A. INTRODUCTION
Data processing was divided into three major levels.
Level-I processing includes all steps required to take the
original data tape to a Level-I tape. Level-II processing
includes all stsps between a Level-I tape and a Level-II
tape. Level-Ill processing includes the steps involved to
take the Level-II output to a usable form. The following
sections briefly describe the steps involved in the three
levels of processing.
Computer hardware utilized was that resident at the
Naval Postgraduate School, Monterey, California. The main
frame computer used was the IBM 3033AP while the mini-com-
puter used was the Apple^II. Computer software referred to
in this section is either a system utility resident to the
IBM system or a locally generated program. Documentation of
the locally generated programs can be obtained from:
Dr. J. L. Mueller (Code 68My)
Department of Oceanography
Naval Postgraduate School
Monterey, California 939U3
122
scftware invclv
i.._-.. u.
any system- da pendent features,
well as features inserted for convienence. Users of these
programs en other systems are cautioned to review the docu-
mentation carefully prior to attempting to transfer the
software .
B. LEVEL-I FBOCESSING
figure 32 is a schematic diagram illustrating the
processing steps for Level-I and should be referred to
throughout this discussion. The master tape (raw satellite
data) was obtained from the Scripps Institution of Oceanog-
raphy, San Diego, California. Table IV gives a summary of
the master tapes utilized in this study. The data were in
the fcrm of a standard magnetic tape in a binary format with
6250 bits per inch (BPI) . The tapes were originally created
using a Hewlett Packard (HE) - 3 000 which has a characteris-
tic high crder, low order bit arrangement opposite to the
IBM system. Therefore, tefore using this raw data in the
I3M 3033AP, it had to undergo a byte swap routine. This
byte swap was accomplished when the unformatted working bac-
kup tape was made using local program VISBKV. After the
unformatted backup tape is made a variable blocked spanned
(VBS) format tape is produced using the system utility
IEBGENER.
123
JER tape
INFORMATTED
WORKING
BACKUP
V8S
WORKING
BACKUP
— S"\
[navigation!
'PARAMETERS'
i
DATA AND
CONTROL FLOW
CONTROL FLOW
INTERNAL DATA
EXTERNAL DATA
ZIPSIO
GRAYSCALE
IMAGE
ZIPSIO
EFILE
GENERATION
ZIPPIC
PICPRT
MAPS
'NAVIGATE
AND
\ ADJUST ,
BACKWARD
(FLIP
IMAGE)
i L
EFILE
(COPY TO
TAPE)
CZCSNAV
NAVDUMP
NAV MATRIX
».
DISK
FILE
L FILE
E FILE
G FILE
+n
TEMPORARY
TAPE
LEVEL-I
TAPE
FILE
VBS
LEVEL-I PROCESSING
SCHEMATIC DIAGRAM
PROCESSING
VENUE
* IBM 3023AP
** APPLE-II
Figure 32. Level-I Processing Schematic Diagram
124
TABLE
IV
Sat€
tllite Da
ta Tapes
!
Tape
Designation
Source
I
)a~s
V I S 0 1 7
Nimbus
7 i
[CZCS)
16
OCT
1979
VI SO 32
Nimbus
7 |
'CZCS)
12
NOV
1979
VI SO 4 0
Nimbus
7
[CZCS)
23
NOV
1979
VIS094
Nimbus
7
[CZCS)
6
MAY
1980
VIS095
Nimbus
7 |
'CZCS)
5
MAY
1980
VIS097
Nimbus
7
[CZCS)
17
MAY
1980
VIS104
Nimbus
7 I
CZCS)
3
JUN
1980
VI S 1 0 5
Nimbus
7 |
;czcs)
6
JUN
1980
VIS106
Nimbus
7
[CZCS)
7
JON
1980
VIS1 17
Nimbus
7 {
'CZCS)
12
JUN
1980
VI S 1 1 1
Nimbus
7
[CZCS)
23
JUN
1980
VIS1 12
Nimbus
7
[CZCS)
24
JUN
1980
VIS1 13
Nimbus
7 <
'CZCS)
25
JUN
1 980
VIS126
Nimbus
7
[CZCS)
1
AUG
1980
AR0000
Nimbus
7 |
[CZCS)
30
SEP
1982
AR2642
Nimbus
7 |
[CZCS)
5
OCT
1982
AR2668
Nimbus
7
(CZCS)
16
OCT
1982
AR2685
Nimbus
7 i
[CZCS)
27
OCT
1982
AR2686
Nimbus
7
[CZCS)
28
OCT
1982
AR26 91
Nimbus
7
[CZCS)
1
NOV
1982
AR2693
Nimbus
7
[CZCS)
3
NOV
1982
AR2704
Nimbus
7
[CZCS)
14
NOV
1982
ioi
Vdij.aJ_'
no
no
no
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
no
no
no
no
no
no
no
no
125
This format is used in conjunction with unformatted read
statements to minimize computer time. These two copied
tapes serve as the working tapes for the remainder of the
Level-I processing, and the master tape is archived.
Using the VBS formatted tape, a Versatec plotter grays-
cale is produced using local program ZIPSIO. This program
also unpacks the 2vent file (hereafter referred to as the
E-file) and writes it to a storage disk. The grayscale
depicts the satellite pass in picture form for hand analysis
of landmarks. Line numbers and pixel numbers are taken off
the grayscale for clear, cloud-free landmarks. These values
are entered into local program ZIPPIC to generate a
'PICPEINT'. This is a matrix of radiance value centered on
the individual landmarks line and pixel number. These
PICPRINTS are then contoured by hand (using a threshold
value of 18 counts for land or clouds) to determine an exact
time and pixel number for the landmark. The landmarks
latitude, longitude, line number and pixel number with addi-
tional housekeeping data are entered into local program
CZCSNAV on the Apple II. This program is interactive and
prompts for necessary inputs. Additionally this program
adjusts roll, pitch and yaw to reduce the root mean squared
126
distance error in the navigation problem. The mean ras
value obtained for all the adjusted, utilized data was
approximately 1.09 n.mi.. The final product of this step
generates a set of navigation parameters that are used to
generate a navigation matrix. This step is accomplished
using local program CZCSNAV2 to generate the navigation
matrix and NAVDUMP to write the navigation matrix (here-
after referred to as the G-file) to a temporary formatted
tape.
The E-file is copied from its temporary disk storage to
the temporary tape as the G-file. Additionally the Data
file (hereafter referred to as the L-file) is first reversed
from its fccttcm-to-to? orientation to a top-to- bottom orien-
tation using local program BACKWARD. This program also puts
the L-file to the previously mentioned temporary storage
tape. Finally, these files on the temporary tape are copied
to a Level-I tape using the system utility IEBGENER. The
only difference between the temporary tape and the final
Level-I tape is that the L-file is copied into an unformat-
ted file which will aid in the speed of further processing.
127
C. LEVEL-II FBOCESSIBG
Figure 33 is a schematic diagram illustrating the
processing steps for Level-II. The Lavel-I tape generated
by the steps discussed in the previous section is the input
tape for this processing. Only the L-file is affected by
the Level-II processing as the E-file and G-file are copied
straight to the Level-II tape using the system utility
IE3GENER. The L-file is used to generate output for calcu-
lating the proper values cf the Angstrom coefficient for
each scene. This is done using local program CZPARMS2 and
an assumed value for the Angstrom coefficient. Chapter III
Section C.2. discusses the importance and method of finding
these values. Next, the computed Angstrom coefficients with
the L-file are rerun through CZPARMS2 to regenerate the
L-file. This regeneration involves talcing the raw counts of
each channel and applying the bio-optic algorithms discussed
in Chapter III to produce values for chlorophyll and K. At
this point the adjustment algorithm discussed in Chapter V
has not been applied. Analyses cf the initial Level-II cut-
put precipitated the nesd for the corrective algorithm,
which was then applied during Level-Ill processing.
128
LEVEL-I
Or
FILE
CZPARMS2
L2A
i
! ANGSTROM i
f COEFF. '
E-FILE
G-FILE
CZPARMS2
L2B
L-FILE
IEBGENER
(COPY
FILE)
E-FILE
IEBGENER
(COPY NAV
FILE)
G-FILE
LEVEL-II
-a
DATA AND
CONTROL FLOW
:ONTROL FLOW
INTERNAL DATA " "
EXTERNAL DATA *
LEVEL-II PROCESSING
SCHEMATIC DIAGRAM
PROCESSING
VENUE
* IBM 3033AP
** APPLE-II
Figure 33. Level-II Processing Schematic Diagram
129
D. LEVEL-III PROCESSING
Figure 3U illustrates the Level-Ill processing
The Level-Ill processing basically takes the data obtained
in Level-II and marries it to the navigation matrix gener-
ated during Level-I processing. Using the four designated
tracks (Fig. 5 ), the coastal starting points from each
track were entered into local program TEDDOHP to provide the
navigation block of the track origin. The G-file contains
data (in latitude and longitude values) every sixteenth
pixel and sixteenth line. Once the origin block is estab-
lished the exact line and pixel was interpolated using local
program FINDPIX on the Apple II. With this starting point
local program DATA4 was entered to generate every 1 km along
each track an associated line and pixel number which was
then converted into the appropriate data values. This cut-
put was written to storage for later processing. It was
here that the adjustment algorithm was applied, producing
the final version of the data in a navigated form.
130
LEVEL II
■y
T^DDUMP
*
J G-FILE
i
'
DATA4
•
L-FILE
■
'
•
LEVEL
III
)CEANSAT/
ADJUSTMNT
ALGORITHM
FINAL
LEVEL
III DAT;
•i PULL i
i ORIGIN |
I SLOCKS J
t.
| DETERMINE l
i INITIAL !
^TRAC_K_PTSj
**
DATA AND
CONTROL FLOW
CONTROL FLOW
INTERNAL DATA
EXTERNAL DATA
LEVEL-III
PROCESSING
SCHEMATIC DIAGRAM
PROCESSING
VENUE
* IBM 3033AP
** APPLE-II
Figure 34. Level-Ill Processing Schematic Diagram
131
APPENDIX 3
DATA CONDITIONING
Tc apply the Level-Ill data tc the analytic techniques
certain conditioning steps were necessary prior to begin-
ning. Much of the conditioning applied to the data was
dependent on the data itself as to its completeness and
behavior. This discussion focusses on the steps necessary
prior to using the EOF analysis techniques.
Figure 35 depicts the steps involved in this discussion
and should be referred to as a guide. First the Level-Ill
data for the four tracks and twenty-two scenes were
extracted using local program PACKJOB. Twenty-two files
each contained the data for the four tracks for each partic-
ular scene. These data were plotted using local program
PARPLOT and the DISSPLA utilities resident on the
IBM-3033AP. The format of the plot was chosen to give an
indication of either good data or bad data with no struc-
ture. This plot was used to decide on the partitioning
scheme. Four partitions.were selected and their details are
listed in Table V.
132
/LEVEL- 1 iK
< OCEANSAT ,
PACKJOB
PAROAT
load partTtton
data to 1 file
PEOF1
Eigenvalues
Eigenvectors
Principal Comp,
Plots
DATA TO
DISK
JE0F2
! LEVFI.-III
*l SEPARATF
! FILES
•i
PARPLOT
| FAKFLUI
I I
L.„.r„.J
! PARTITI0N~1
DATA !
i
PARPLT
Raw data
Discussion
Plots
Joining Functions
DATA AND
CONTROL FLOW
DONTROL FLOW *
INTERNAL DATA — •"
XTERNAL DATA — •"
PROCESSING
VENUE
* IBM 3033AP
+* APPLE-II
Figure 35. Data Conditio ring Schematic Diagram
133
Track
No.
1
2
3
U
Partition
No.
TABLE V
Partition dimensions
Sin Grid Max Grid Ntime Nspace
1 ec
165
200
2CC
410
425
467
485
17
17
16
17
231
261
268
2 86
Figures 36 through 39 show the plots generated by par-
plot and the partitioning given in Table IV. The objective
of the partitioning was tc find the most complete data over
time and space possible given ten samples.
Cata from these four partitions was then entered into
local program PARDAT. This program applied most of the con-
ditioning to the data set. Only the K(490) data was uti-
lized from this point on although this program could be
easily altered to focus on another optical parameter. The
data wera searchad to find good points and bad points and a
control arrangement for later use was made. The raw data
were scaled and inverted to produce 1/K(490) values in
meters. Next the data were averaged by every fourth point
to smooth out noise features. At this point data strings
with gaps existed for each applicable scene. Next a linear
interpolating routine was applied to obtain continuous data
at each time point. Finally, the data for all scsnes and
134
tracks were combined and written to diss in a single data
file. To this pcint the conditioning applied has consisted
of partitioning the data into four partitions, rejecting
incomplete scenes, scaling the K (490) values, averaging the
data by every four values, and applying linear interpolation
to fill in the remaining gaps.
The conditioned data was then plotted using local pro-
gram PARPLT and the DISSEIA system utilities. The plots
generated are figures 7 through 10 and were used in Chapter
V section B to discuss the data and its relationship to the
regional oceanography.
The final st<=ps of the data conditioning involved appli-
cations of the EOF analysis techniques. Local program PEOF1
produced eigenvalues, eigenvectors, and principle components
for each partition and plotted the output. Figures 11
through 14 and 16 through 31 are the plots produced. The
eigenvalues, eigenvectors and principle components were all
written to disk for later use. The local program JEOF2 was
designed to produce the joining function that related parti-
tion cne tc partition two.
135
o
CM
o
*A».» >*v. «, ^V^o^s^s-Wl I
. Partition One
wW(H^'-W^^_
,.*A» -< -
...wt1' \ PWV.^_
V
\»Wi i^ d^#»<1l T*M»»«, ■*«,»■ ,
lL..'.
>**« — «i
lv-v«nvwy\Uv— vA^W^^^,
Au
■ A- ■ * - ■ , -A *
\
f/rli 1
Date
14 Nov 82
3 Nov 8 2
1 Nov 82
28 Oct 82
27 Oct 82
16 Oct 82
5 Oct 82
30 Sep 82
1 Aug 80
25 Jun 80
24 Jun 80
23 Jun 80
12 Jun 80
7 Jun 8 0
6 Jun 80
3 Jun 80
17 May 80
6 May 80
5 May 80
23 Nov 79
12 Nov 79
16 Oct 79
410
310
210
110
Land
Distance Offshore km
Figure 36. Partitioning Scheme for Track One (35 53 N)
136
o
u
£
o
"9"
"\
^fl/*fo* r^+s/Kx^
"**v—v
Partition Two
«^VV^« i
"XH v ■ ^^tn^-Vr-, Ni
Date
14 Nov 82
3 Nov 8 2
1 Nov 82
28 Oct 82
27 Oct 82
16 Oct 82
5 Oct 82
30 Sep 82
1 Aug 8 0
25 Jun 80
24 Jun 80
23 Jun 80
12 Jun 80
7 Jun 80
6 Jun 80
3 Jun 8 0
17 May 80
6 May 80
5 May 80
23 Nov 79
12 Nov 79
16 Oct 79
400 300 200 100
Distance Offshore km
Land
Figure 37. Partitioning Schsme for Track Two (35 40 N)
137
I
w
u
E
O
'•I-.I III
\
Partition Three
\
\
fSfWi/^*
^wwv.
'^i.'O/^i^Mii^^^
\*V* V*
"v^v^yvAi-^M^
"^
Wy-^ /^VA->-nj^Ma^vjrW^
V^^nW^A
Dp. te
14 Nov 82
3 Nov 82
1 Nov 82
28 Oct 82
27 Oct 82
16 Oct 82
5 Oct 82
30 Sep 82
1 Aug 80
25 Jun 80
24 Jun 80
23 Jun 80
12 Jun 80
7 Jun 80
6 Jun 80
3 Jun 80
17 May 80
6 May 80
5 May 80
23 Nov 79
12 Nov 79
16 Oct 79
417 317 217 117
Distance Offshore km
Land
Figure 38. Partitioning Scheme for Track Three (35 22 N)
138
o
u
o
E
O
. i 'i.vi-
Partition Four
MM* M< ■ #WA
W^« ^ V V £-
\ v^.v V -^\
v »
V
*\AUw»» fr^mW^* 1
r^****""** *
,w"-v.l>.i/^' ■',"■
i1 V-
^
Ml^M f~<^ V*Wr iiiO < ., r ■ irl .lln ift )» t,
■ ■, . I . rt •■»
T^VVUUaN h'
a*.*-*-"4
J>^^i^ yM*Afs ■ y
^,
410
310
210
110
Date
14 Nov 32
3 Nov 82
1 Nov 82
28 Get 82
27 Oct 82
16 Oct 82
5 Oct 82
30 Sep 82
1 Aug 80
25 Jun 80
24 Jun 80
23 Jun 80
12 Jun 80
7 Jun 8 0
6 Jun 80
3 Jun 80
17 May 80
6 May 80
5 May 80
23 Nov 79
12 Nov 79
16 Oct 79
Land
Distance Offshore km
Figure 39. Partitioning Scheme fcr Track Four (35 53 N)
139
APPENDIX C
EOF PROCESSING
The desired EOF ' s are those in the space dcmair., which
for satellite data, is dimensioned much larger than the time
dimension. It is possible to significantly expedite compu-
tations by computing eigenvalues, eigenvectors and principal
components using the smaller covariance matrix of the time
domain, and to then scale these results :o obtain the eigen-
vectors and principal components in the space domain. The
algebraic basis for this approach is reviewed in this
appendix .
Consider the following convention for dimension nota-
tions
Space . . . . n = 1,...,N
Time . . . . m = 1,...,M
EOF (order) . . . . k = 1,...,K
where K < min(M,N). As before the raw data matrix is given
by
F» - If (55)
a.
|_ mn J
of H rows by N columns. The sample mean in the space domain
is given by
140
M
£ - M
E
1
mn
(5b)
of dimension 1 X N. The centered data matrix in the spaca
domain is given by
F =
f - f
-n
(57)
From Chapter IV -he sample space ccvariance matrix (of size
N X N) is
1
S = M-l F F
(53)
and the sample time covariance matrix (cantered in space and
of size H X M) is now defined as
1
T = M-l F F
(59)
If A, and e, are an associated eiganvalue and eigenvector
k -k
respectively cf S, then
S e. = \ e. f 0
t -k k -k
(60)
since
X f Q and e, j* 0
(61)
Then
F e. ^ 0
% -k
(62)
and therefore
141
1 T
— — - F F F e. = A. F eu /gq»
(noting the commutative property of ^k and F ). Thus , Xfc is
an eigenvalue of T with associated eigenvector (in the time
domain)
-k ^ -k
where a , is the k'th principal component (in the space
mk.
domain) at time point m.
We now consider the eicenvalues a and eigenvectors %.
k -K
of T, normalized such that
M
E5 . S . - «4 • (65)
"mi *mj lj , v '
m - 1
with time domain principal components
Sk = f Sk S k - 1, ...,K , (66)
where the 'significance order K is selected either to retain
an arbitrary fraction of the total sample variance, or
according to one of the more objective statistical selection
rules discussed by Preisendorfer , et al. (1981) .
The principal components in the time and space domain
have the property that
N M
(67)
a a (.M-JJ A . o . .
mi mj i ij
n = 1 m = 1
ZZ, 5 = V^ a . a = (M-l) X. 6 . .
ni nj / j mi mj i ij
142
a A r*.r\ -P
Recall that F was centered on I in trie space domain
, and
that both S and I were therefore normaiizad by the factor
(1/M-1) .
Ey analogy to equation (63)
T
1 T T
M-l l i i H 'k i -k k ^k
(63)
where c, is now seen to be an eigenvec
length
.or of s ,
but of
(69)
rather than being an orthcnormal eigenvector of length 1.
The two are thus related by
-k • ( M - l ) x .
(70)
(principal components in the time domain scale to orthonor-
mal eigenvectors in the space domain),
and by similar arguments
ak = r/(M - l)Xk ] Ck
(71)
(orthonormal eigenvectors in the time domain scale to prin-
cipal components in the space domain).
143
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148
INITIAL DISTRIBUTION LIST
1. Defense Technical Information Center
Cameron Station
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2. Library, Code 0 142
Naval Postgraduate School
Monterey, CA 93943
3. Professor Robert J. Bsnard, Code 63Rd
Department of Meteorclogy
Naval Postgraduate School
Monterey, CA 93943
Copies
4. Professor Christopher N. K.
Department of OceanograDhv
Naval Postgraduate School"
Monterey, CA 93943
Mooers, Code 68Mr
5. Adjunct Professor James L.
Department of Oceanography
Naval Postgraduate School
Monterey, CA 93943
Mueller, Code 68My
6. Assistant Professor Andrew J. fcillaott,
Code 68Wt
Department
Naval Post'
Monterey, i
of Oceanography
rraduate School
:A 93943
7. Lt. John T.
Route 7 Box
aiks n , s . c .
McMurtrie
801
8. Director
Naval Oceanography Division
Naval Observatory
34th and Massachusetts Ave. NW
Washington, D.C. 20390
9. Commander
Naval Oceanography Command
NSTL Station
Eay St. Louis, MS 39522
10. Commanding Officer
Naval Oceanographic Office
NSTL Station
Bay St. Louis, MS 39522
149
Thesis
M2571+
c.l
20751
McMurtrie
Spatial structures of
optical parameters in
the California current ,
as measured with the
Nimbus-7 Coastal Zone
Color Scanner.
207513
Thesis
M25T^
c. 1
McMurtrie
Spatial structures of
optical parameters in
the California current,
as measured with the
Nimbus-7 Coastal Zone
Color Scanner.
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