\ 93943

1216158

NAVAL POSTGRADUATE SCHOOL

Monterey, California

THESIS

AN ERROR ANALYSIS OF
RANGE-AZIMUTH POSITIONING

by
David A. Waltz
September 1983

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AN ERROR ANALYSIS OF RANGE -AZIMUTH POSITIONING

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Master's Thesis
September, 1983

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7. AUTHORS

David A. Waltz

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Naval Postgraduate School
Monterey, California 93943

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Naval Postgraduate School
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September, 1983

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1*. KEY WOROS (Continue on roetto aid* It noeotety and Identity by block number)

Hydrography, surveying, range-azimuth, theodolites, hydrographic surveying

20. ABSTRACT 'Condnua on rovmrmo tide II neceeeery and identity by block number)

Pointing error standard deviations for two theodolites, the Wild T-2 and
Odom Aztrac, were determined under conditions closely approximating those of
range-azimuth or azimuth-azimuth hydrographic surveys. Pointing errors found
for both instruments were about 1.3 meters, and were independent of distance.
No statistical difference between the errors of the two instruments was found.
The accuracy of the interpolation methods used by the National Ocean Service
(NOS) for range-azimuth positioning were investigated, and an average inverse

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distance of about 2.5 meters was observed between interpolated positions and
corresponding observed positions. The overall range-azimuth position errors
of the two theodolites were then compared to positioning standards of NOS and
the International Hydrographic Organization, using assumed ranging standard
deviations of 1.0 and 3.0 meters. Both instruments met all standards except
the NOS range-azimuth standard for 1:5,000 scale surveys. Interpolated
positions may fail to meet more of the standards because of additional
inherent error.

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An Error Analysis
of
Range-Aziaath Positioning

by

David A. Waltz
Lieutenant, National Oceanic and'' Atmospheric Administration
B.S., University of Alabama, 1971

Submitted in partial fulfillment of the
requirements for the degree of

MASTER OF SCIENCE IN OCEANOGRAPHY (HYDROGRAPHY)

from the

NAVAL POSTGRADUATE SCHOOL
September, 1983

ABSTRACT

Feinting error standard deviations for two theodolites,
the Wild T-2 and Odom Aztrac, were determined under condi-
tions closely approximating those of range-azimuth or
azimuth-azimuth hydrographic surveys. Pointing errors found
for both instruments were about 1.3 meters, and were inde-
pendent of distance. No statistical difference between the
errors of the two instruments was found. The accuracy of
the interpolation methods used by the National Ocean Service
(NOS) for range-azimuth positioning were investigated, and
an average inverse distance of about 2.5 meters was observed
between interpolated positions and corresponding observed
positions. The overall range-azimuth position errors of the
two theodolites were then compared to positioning standards
of NOS and the International Hydrographic Organization,
using assumed ranging standard deviations of 1.0 and 3.0
meters. Both instruments met ail standards except the NOS
range-azimuth standard for 1:5,000 scale surveys.
Interpolated positions may fail zc meet more of the stan-
dards because of additional inherent error.

TABLE OP CONTENTS

I. INTRODUCTION 9

A. THE RANGE-AZIMUTH POSITIONING METHOD 9

B. HYDROGRAPHIC POSITION ERROR STANDARDS .... 12

C. OBJECTIVES 14

II. ERROR INDICES AND RANGE-AZIMUTH GEOMETRY 20

A. DEFINITIONS 20

1. E landers 2 2

2. Systematic Errors 23

3. Random Errors 24

3. TWO-DIMENSIONAL ERROR FI3URES 26

1. Concentric and Eccsntric G=cm=try .... 27

2. The Error Ellipse 29

3. Root Bean Square Distance 33

4. Circular Standard Error 35

C. THE ERROR OF AN INTERPOLATED FIX 37

1. Interpolation Algorithms 37

2. Error Propagation 39

III. EXPERIMENT DESIGN AND IMPLEMENTATION 43

A. FIELD WORK 43

3. THEODOLITE POINTING ERROR 47

C. INTERPOLATION ALGORITHM EVALUATION 49

D. CHOICE OF EXPERIMENTAL CONDITIONS 51

IV. RESULTS AND DATA ANALYSIS 56

A. DATA PROCESSING SYSTEM 56

3. POINTING ERROR DETERMINATION 56

C. INrERPOLATION EVALUATION 64

D. ANALYSIS OF FACTORS AFFECTING THE RESULTS . . 66

S. APPLICATION TO POSITION ERROP STANDARDS ... 72

V. CONCLUSIONS AND R ECO MMENDATIO N S 79

APPENDIX A: SEANS AND STANDARD DEVIATIONS OF

ACQUIRED DATA SETS 85

APPENDIX B: 3E0DETIC POSITION OF HORIZONTAL CONTROL

STATIONS 91

APPENDIX C: EMPIRICAL PROBABILITY DENSITY FUNCTION

PLOTS 92

BIBLIOGRAPHY 98

INITIAL DISTRIBUTION LIST 101

LISP OF TABLES

I. Circular Error Formulae 36

II. Data Acquisition Sequence of Events H5

III. Pointing Error Standard Deviation (pooled
estimates) 58

17. Summary of ANOVA Results at 95% confidence . ... 63

V. Results of Interpolation Evaluation 65

VI. Corrected Aztrac Standard Deviation 70

VII. Experiment Precision and Theololite Error .... 72

VIII. Position Standards Comparison 78

IX. 3000 Meter Arc 35

X. 1500 Meter Arc 86

XI. 1000 Meter Arc 37

XII. 700 Meter Arc 88

XIII. 500 Meter Arc 89

XIV. 300 Meter Arc 90

LIST OF FIGURES

1.1
1.2
1.3
2. 1

2.2
2.3
2.4
2.5
2.6
2.7
2.8
3.1
3.2
3.3
3.4
3.5
4. 1
4.2
4.3
4.4
4.5
C. 1
C.2
C.3
C.4
C.5
C.6

Illustration of Range- Aziauth Positioning ... 10

Illustration of Aztrac Shoes Station 17

The Aztrac Transmitting Unit 19

The Normal Probability Curve 25

Eccentric Range-Azimuth Geometry 27

Concentric Range-Azimuth Geometry 28

Eccentric Error Compensation 29

Error Ellipse and d^* 31

A Range-Azimuth Position 32

Variation of dr^s Probability 34

Example of Angular Interpolation 38

Sketch of the Survey Area 44

Lines of Position Observed to the Vessel .... 45

Uncertainty of an Observed Error 48

Interpolation of Angle Only 50

Interpolation of Angle and Distance 50

One-Dimensional Difference 3etween Means .... 66

Two-Dimensicnal Difference Between Means .... 6"^

Probability Density vs. Error 68

Results Compared to d^s 74

Results Compared to 90% Probability 76

Probability Density Plot: 300 m arc 92

Probability Density Plot: 500 m arc 93

Probability Density Plot: 700 m arc 94

Probability Density Plot: 1000 m arc 95

Probability Density Plot: 1500 m arc 96

Probability Density Plot: 3000 m arc 97

I- 5NIS0DUCTIDN

A. THE RANGE-AZIHUTH POSITIONING METHOD

The fundamental purpose of a hydrographic survey is
defined by Ingham (197U) as being to "depict the relief of
the seabed, including all features, natural and manmade, and
to indicate the nature of the seabed in a manner similar to
the topographic map of land areas." He goes on to describe
two factors defining a single point on the seabed:

(i) "The position of the point in the
horizontal plane in, for example, latitude
and longitude, grid co-ordinates or anales
and distances from known control points.

(ii) The depth of the point below the sea
surface, corrected for the vertical
distance between the point of measurement
and water level and for the height of the
tide above the datum or reference level to
which depths are to be related. "

Thus the hydrographer must answer the two primary ques-
tions of "hew deep" and "where" for each of the thousands of
soundings acquired on every survey. Because every area to
be surveyed has different geophysical characteristics and
levels of use, the hydrographer must possess a suite of
tools and techniques to accomplish each survey. A survey of
a large metropolitan harbor requires different equipment and
measurement precision than one for a deep ocean area.

Only the first of Ingham's two factors cited above is
considered, and it is further narrowed in scope to techni-
ques used in the most precise surveys. Such a survey might
be one of a winding, narrow river carrying deep draft
vessels, or perhaps a very large scale survey of an inner
harbor. Eoth areas require the highest positioning accuracy
and a minimum of shore control stations.

Any method of positioning employs the intersection of
lines of position (LOP's) to construct a fix. Although
advanced methods may use multiple LDP's, traditional hydrog-
raphy uses the simple intersection of two lines to fix the
vessel's position. The vessel is located somewhere along
each of two lines of position, and the only point satisfying
these conditions is the intersection of the lines.

The error associated with one of the simplest posi-
tioning methods, that of the two LDP range-azimuth fix,
which is illustrated in Figure 1.1, is analyzed.

r

range
and azimuth

Figure 1. 1 Illustration of Range- Azimuth Positioning.

Also called the rho/theta method, range-azimuth positioning
consists of the observation of a distance and an azimuth to
a vessel from either one or two known locations [Ombach,
1976 ]. An example of this method is the use of radar aboard
ship. A relative position for a radar contact is determined
by observing a radar range and azimuth, or a radar range and
visual azimuth, to a contact. The two lines of position

10

always intersect at right angles because the observation is
made from a single point, and this concentric geometry
provides the strongest fix possible. Mariners also know
that the fix obtained via a visual azimuth is stronger than
the one using a radar azimuth, because the visual bearing is
more accurate.

This example shows the advantages that make the range-
azimuth method popular for hydrography. It provides the
geometrically strongest possible fix, and only one location
on shore need be occupied to control the survey. Such a
positioning method is ideal in harbors or rivers where
maximum accuracy is needed but where obstructions make ether
types of fix geometry impractical. In 1982 the U.S.
National Ocean Service (NOS) obtained twenty thousand linear
nautical miles of launch hydrography, and sixty percent of
this was controlled by the range-azimuth method [Wallace,
1983 ].

There are limitations associated with this method just
as with any fix geometry. It is labor intensive and
requires more radio communication (to establish fix timing)
than mest other methods. In totally nonautoraated situ-
ations, distances to the survey vessel are recorded manually
aboard the vessel, and azimuths are recorded ashore by the
theodolite observer at prescribed intervals. These fix data
are later put into computer compatible digital form via a
process called logging. Manual recording of these fix data
are generally toe slow to position every sounding.
Therefore, individual sounding positions must be
interpolated from the observed fixes.

Systems have been designed that have an intermediate
level of automation. The NOS Hydroplot System is an example
of this type [ Wallace, 1967], When used in the range-
azimuth mode, the vessel is usually steered along arcs of
constant range from the theodolite station, and the

11

Hydroplct System autcmatica lly records a distance measure-
ment for each sounding. Azimuths are not telemetered to the
vessel but are relayed ovsr voice radio and are manually
entered into the computer system. Since the maximum data
rate is atout two angles per minute, the interpolation of
angles for sounding positions between fixes is necessary.

Recently a digital theodolite, the Odom Aztrac, has been
developed which can record and telemeter angles with great
speed -- up to ten angles per second [Odom Offshore Surveys,
Inc., 1982]. A computer system aboard the survey vessel can
thus record and plot an observed position for each sounding.
This rapid position fixing, combined with a computer's
ability to provide cross-track errsr indications to the
helmsman, enables the hydrographer to systematically cover a
survey area with maximum efficiency by running straight and
parallel sounding lines.

The Aztrac system is still considered a semiautcmated
system because an observer is required to manually track the
vessel with the theodolite. Two fully automated range-
azimuth systems which feature fully automatic tracking have
been devolcped. One is the Polarfix system developed by
Krupp-Atlas Elektronik in Germany [Smith, 1983], and the
ether is the Artemis system developed by Christiaan
Huygenslaboratcr ium in the Netherlands [Newell, 1981],

B. HYDROGRAPHIC POSITION ERROR STANDARDS

Historically, most national hydrographic organizations,
as well as the International Hydrographic Organization
(IKO) , have used linear plotting error at the scale of the
survey to be the standard for sounding position accuracy.
Prior to 1982, the standards recommended by IHO [IHO, 1968]
were:

"The indicated repeatability of a fix
(accuracy of location referred to shore

12

control) in the operating area, whether
observed by visual or electronic methods,
combined with plotting error- shall seldom
exceed 1.5 mm 70.05 in) at the scale of
the survey. "

The IHO recently published new recommendations for error
standards [IHO, 1982] which are:

"... any probable error, measured relative
tc shore control, shall seldom exceed
twice the minimum plottable error at the
scale of the survey (normally 1.0 mm on
paper) . "

Neither of the IHO standards make any reference as to
what probability level they apply. Munson (1977) inter-
preted the words "shall seldom exceed" in the above state-
ments to mean "less than 10% of the time", which seems
reasonable. The 1982 IHO standard is somewhat confusing due
to its use of both the terms "seldoi exceed" and "probable
error". The latter term is associated with a 50% prob-
ability by most statisticians including Green wait (1971).
However, the author of these standards. Commodore A. H.
Cooper, HAN (retd) , has stated that he intended no statis-
tical significance tc the term "probable error" [Wallace,
1983 ].

The NOS has not yet incorporated the latest IHO stan-
dards, but such action is being considered in some form
[Wallace, 1983]. Current NOS standards have been developed
to ensure that "accuracies attained for all hydrographic
surveys conducted by NOS shall equal or exceed the specifi-
cations" of the 1968 IHO standards [Ombach, 1976]. Unlike
the international standards, the NOS standards for all elec-
tronic positioning systems use the concept of root mean
square error (drr*s or rmse) , which has a somewhat variable
probability of between 68.3 and 63.2 percent. The NOS stan-
dards for fully visual and for hybrid (combination elec-
tronic and visual) positioning have no explicit reference to
probability.

13

Specific operational standards for range- azimuth posi-
tioning have been neglected by many hydrographic organiza-
tions. However, NOS [Umbach, 1976] requires the following
observational procedures bs follows! for all range-azimuth
positions.

"Objects sighted on should be at least 500
m from the theodolite... the azimuth
check should not exceed one minute of
arc... observed azimuths or directions to
the sounding vessel for a position fix
shall be read to the nearest 1 min of arc
or better if necessarv to produce a posi-
tional accuracy of 0.5 mm at the scale of
the survey. "

Since the range-azimuth method is classified as a hybrid
positioning system, it is not referenced to any particular
probability, but a reasonable assumption may be made that
the drfW<; concept also applies in this case.

The U.S. Naval Oceanographic Office (NAVOCEANO) also
requires that its surveys meet the standards of the NOS
Hydrographic Manual. The Army Corps of Engineers presently
have no formal positioning requirements that must be met by
all districts, although draft specifications are being
written at this time [Hart, 1983]. The range-azimuth tech-
nique and its applicability to Corps of Engineers surveys is
discussed in Hart (1977) . No specific requirements for
range-azimuth positioning could be found for either the
Canadian Hydrographic Service or the British Hydrographic
Service. Palikaris (1983) also reports no published stan-
dards for these organizations.

C. OBJECTIVES

All position error standards using an explicit prob-
ability are based on the idea that an observation is a
normally distributed random variable with zero mean and
standard deviation <T . These standards require a value for

14

the standard deviation of the component lines of position
that make up the fix. The standard deviation is a value
such that there is a 68.27% probability of an observation
falling within ±1 <T cf the mean. It is unfortunate that
of ~en the 1 <T values of a hydrographic measurement are
simply not known, or known only for ideal conditions and
provided by manufacturers who have a vested interest in the
measuring instrument.

This paper and experiment, then, has as its primary
purpose the determination of a pointing error standard devi-
ation for two theodolites used to measure azimuths under
hydrographic conditions. One instrument (the Wild T-2) is
the standard used by NOS field unizs. The other (the Odom
Aztrac) is a new digital telemetering theodolite, which may
prove useful in automating the presently tedious manual
methods. No estimation of standard deviation has ever been
made on these devices under typical range-azimuth
conditions.

The T-2 is the standard instrument used by NOS surveying
parties for land surveying and, to a lesser extent, hydrog-
raphy. It is a very precise instrument used in third-order
horizontal control and provides an angular resolution of one
second of arc. However, it dees possess features that are
less than ideal for range-azimuth work. The horizontal
tangent screw of this instrument is awkward for range-
azimuth surveying because it is not an infinite gearing
device. The observer often encounters the end of the drive
mechanism, stopping the instrument's movement while tracking
the vessel, k solution to this problem for many observers
is to track the vessel with the tangent screw undamped,
then clamp the screw only seconds before the fix occurs.

The inverted image feature of the T-2's in use by NOS
simplifies the optical system and reduces optical error.
This is satisfactory for land survey work but creates some

15

confusion when tracking a fast moving vessel. This is
because the observer sees an image of the survey boat upside
down and coving apparently in the opposite direction from
its actual movement. Another disadvantage is that the mech-
anism for reading the horizontal circle of a T-2 is more
complicated than desired for the rapid observations neces-
sary in hydrographic survey work. The operator is required
to step tracking the vessel, remove his eye from the tele-
scope, and use an auxiliary eyepiece to read the angle.
There is a practical limit of about 30 seconds to the speed
with which successive angles can be observed and read. The
particular unit tested was serial namber 30504.

The Odom Aztrac theodolite is a semi-automated,
line-of-sight angle measuring system. The Aztrac system
consists of a Wild T-16 theodolite (serial number 2534880
was tested) which was modified for infinite tangent drive
and to provide angular information in a digital format. The
shore unit decodes the observed angle, determines the direc-
tion of rotation of the instrument and displays the angle on
its front panel. The angle is then converted to binary
coded decimal {BCD) format and used to frequency shift key
(FSK) an FM transmitter to link the data with the survey
vessel. The data is transmitted at the rate of 10 angles
per second. On board the vessel the Aztrac receiver
converts the received data to parallel form and displays it
on the front panel for manual recording. For automated
recording or processing by onboard computer a serial data
ouout is provided. The Aztrac has an angular resolution of
0.01 degree (36 arc seconds) (Odom, 1983].

With the notable exception of angular resolution, this
theodolite is more appropriate for range-azimuth hydrography
than the Wild T-2. It has an erect image and infinite
tangent screw which allow the vessel to be constantly
tracked. Its digital output requires no action on the

16

Figure 1.2 Illustration of Aztrac Shore Station.

17

operator's part to record observed angles. An additional
positive feature of the digital output is the ability to
rapidly zero the instrument on an initial pointing. This is
done by pressing a manual reset button on the instrument
control panel. Figures 1.2 and 1.3 show photographs of the
Aztrac equipment. Figure 1.2 illustrates a typical shore
station. The observer is adjusting the Aztrac instrument,
and the transmitter unit is on the ground to ths right of
the Aztrac tripod. The distance measuring equipment is
mounted on a tripod behind the observer, and the Aztrac
transmitting antenna is at the top of a pole on the extreme
right of the photograph. Figure 1.3 shows the Aztrac trans-
mitter unit, with its digital angular display in hundredths
of a degree. Both photographs were provided by Odom
Offshore Surveys, Inc.

A seccnd objective of this paper is to evaluate the
interpolation methods used by the NOS for range-azimuth
work. The availability of the digital theodolite, with its
direct measurement of all positions, enabled a comparison of
observed and interpolated fixes to be made. This made
possible an estimate of whether interpolated positions meet
required accuracy standards , and whether direct measurement
of all positions is needed. The estimate presented here was
not made statistically rigorous so that the thesis could be
kept to a manageable size. More theoretical statistical
work is needed to fully reduce the interpolation data.

The final objective of this investigation is to compare
the position errors of these two instruments with the
various position error standards discussed in section 3 of
this chapter. The conclusions resulting from this objective
will assist the hydrographer to select equipment and oper-
ating conditions that meet required position error
standards.

18

im -ml

t ■ ■■■>,■' "<^

\

Figure 1.3 The Aztrac Transmitting Unit,

19

II. ERROR INDICES AND RA^GE-iZIMOTH GEOMETRY

The method of range-azimuth positioning is usually
selected for large scale surveys because of its simplicity
and accuracy. This is the result of both angle and distance
measurement devices being co-located, with the intersection
of the two lines of position always being ninety degrees.
In practice, however, the co-location of both instruments is
often not achieved. The result is an eccentric geometry for
the position fix. This section will analyze the geometry of
both eccentric and concentric fixes. Position error indices
in common use will be reviewed and analyzed for the special
cases of range-azimuth methods, and the error of an
interpolated fix will be derived.

A. DEFINITIONS

Although a complete and general treatment of error
theory will not be presented, some basic definitions are
necessary to understand the data analysis presented here.
The ideas in this section are included in many basic statis-
tics textbooks, and were specifically drawn from Wonnacott
(1977), Bowditch (1977), Heinzen (1977), Kaplan (1980), and
Davis (1981).

Error may be defined as "the difference between a
specific value and the correct or standard value" [Bowditch,
1977], or as "the difference between a given measurement and
the "true" or "exact" value of the measured quantity"
[Davis, 1981]. Mathematically it can be defined as:

e = x.- T (2.1)

A.

20

where e is the error, X; is an observation, and T is the
"correct" or "true" value. The word error implies that
there is a known true value for a quantity, with which a
measurement ma y be compared to find the "error" associated
with that measurement. Since the true value of a measured
quantity is rarely kncwn, the term "error" is not precisely
correct. Davis (1981) states that it is more appropriate to
speak of the theory of observations rather than the theory
of errors, but it can be shown that the difference between
the two is largely one of semantics.

A single aeasurement of a particular quantity may be
considered sufficient for many purposes, even if it is known
that additional measurements will probably be slightly
different than the first. If the quantity to be measured is
of sufficient importance, then multiple measurements are
made and the sample mean, Y, is used. Each of these
multiple measurements can be a considered numerical value
for a random variable. A random variable is one that takes
on a range of possible values, each associated with a
particular probablility.

The sample mean may be expressed mathematically by equa-
tion 2.2 [Wcnnacott, 1977]:

i *

X * - IX; (2.2)

where n is the sample size. If the sample size were
increased without limit (n — > <*>) , aquation 2.2 would give
the population mean^< . The sample mean is always an esti-
mate of the population mean, which is never directly
computed. This leads to the concept of the residual, v,
which is the difference between the estimate 7 of the popu-
lation mean and the observation x^ . This is shown in
equation 2.3.

21

v = X - x^ (2.3)

The residual is computationally the negative of the
error. Nevertheless, equation 2.3 is more appropriate
because it uses an estimate, X~, of the unknowable population
mean^>< . The presence of X in equations 2.3 implies that
multiple measurements have been made, and allows e partic-
ular confidence to be assigned to the estimate of ^<
depending on the number of such measurements. Because the
word error is still used in much of the hydrographic profes-
sion, it will be used interchangeably in this paper with the
term residual. It is important, however, to understand that
the concept of the residual, whatever its name may be, is
fundamental to any measurement operation.

Errors are classically divided into three groups: blun-
ders, systematic errcr, and random errors [Greenwalt, 1962].
Eowditch (1977) and Davis (1981) do not classify errors as
including blunders, but like the term error itself, the
distinction is largely a semantic one. Ideally blunders and
systematic errors are completely eliminated from the data.
The most precise measurements reduce random error as much as
possible, but it can never be completely eliminated.

1 . Blunders

Blunders are mistakes that are "usually gross in
magnitude compared to the other two types of errors" [Davis,
1981], and are most often caused by carelessness on the part
of the observer, or by grossly malfunctioning observing
equipment. They are usually detected and eliminated by
procedural checks during the data aoquistion process. The
recognition of a blunder is not always easy, since a blunder
"may have any magnitude, and may be positive or negative"
[Bowditch, 1977].

22

2 • Systematic Errors

Systematic errors are defined by Davis (1981) as
those that occur "according to a system which, if known, can
always be expressed by mathematical formulation." This
mathematical model results in correctors that are applied to
all measurements obtained, thus eliminating the systematic
errors from the observations. The model may be as simple as
a constant corrector subtracted from lengths obtained with a
steel tape, or it may be as complicated as modelling the
effects of atmospheric refraction on electronic distance
measuring equipment.

If the systematic error is such that it cannot be
modelled, it is then estimated by a process known as cali-
bration. Kaplan (1980) defines calibration as the process
of comparing the measuring instrument against a "known"
standard. The word "known" is usually operationally defined
as a measurement operation or instrument that is much more
accurate than the one being calibrated. The difference
between the observed and standard value is used as an esti-
mate of the total effect of all systematic errors present.
This process is very close to the classical concept of
"errors" presented above, and is entirely proper for use in
the correction of systematic errors [Davis, 1981]. Of
course, one must be careful to apply the corrector only to
those measurments made under the same conditions as the
calibration.

A systematic error found in theodolite or sextant
observations is known as the personal error of the observer
[Mueller, 1969], [Bowditch, 1977]. This type of error is
rarely quantified for hydrogaphic applications, but never-
theless it does exist. The observer must rely on the senses
of hearing and vision to make measurements, which vary
between individuals as well as with time in one individual.

23

Some personal errors are constant and some are erratic
[Davis, 1981]. These errors are minimized by training and
standardizing observational procedures. The best way to
eliminate personal error is by the use of completely
automated observation equipment.

3 . Random Errors

"Random errors are chance errors, unpredictable in
magnitude or sign", and are "governed by the laws of prob-
ability" [Bowditch, 1977], If one assumes that all blurders
and systematic errors have been removed from the observa-
tions, the remaining values can be regarded as sample values
for a random variable. As noted earlier, a random variable
can take on a range of values, each associated with a
particular probability. A random arror has high probability
of being close to the population mean,^ , and a low prob-
ability of being very much different than^a [Greenwalt,
1962 ].

A probability density function expresses the rela-
tion between a value for a random variable and the prob-
ability of its occurrence. Hydrographic survey measurements
often use the normal or Gaussian probability density func-
tion. A concise explanation of this function is given in
Greenwalt (1962) and Kaplan (1980). The function itself is
given as equation 2.4, where p (v) is the probability of the
occurrence of a particular residual v, and <T is the popula-
tion variance which is approximated by the sample variance,
St given by equation 2.5.

a

p(v) =

I

exp

2*x

(2.4)

sx =

\r\-

&-*)'

(2.5)

^-i

24

To find the probability of a residual falling
between two residuals v, and vx , equation 2.4 must be inte-
grated over the interval v, to vz . This corresponds to the
area under the gaussian curve between -chose two points, as
is shewn in Figure 2.1. If p(v) were integrated from -1<rto
♦ 1cT, the area under the curve would be 68.27% of the total
area. This means that there is a 63.27% probability of a
particular residual falling between plus or minus one stan-
dard deviation of the mean, where the standard deviation, C,
is defined as the square root of the variance given in
equation 2. 5.

Figure 2.1 The Normal Probability Curve.

The combined effect of blunders, systematic errors
and randem errors can now be seen in overview. If it is
assumed that blunders and systematic errors have been
completely eliminated from a set of observations, there
remain only random errors. If the sample size is large
enough, then the sample mean and variance are good

25

approximations of the population mean and variance. That
is, there is a 68.27% probability that any future measure-
ments made under the same conditions will not fall farther
than plus or minus one standard deviation from the mean.

B. TWO-DIHENSIONAL EEHOR FIGOBES

The two-dimensional error of a position, as applied to
the special case of range-azimuth fixes, must next be exam-
ined. Two figures, the ellipse and the circle, are used to
characterize two-dimensional error. The "error diamond" is
also sometimes used but has no statistical significance
[Thomson, 1977]. The error ellipse is discussed first since
it is the most general index of error. Another is root mean
square distance (drrnS ) , also known as root mean square error
[Bowditch, 1977 ], which is the radius of a circular figure
commonly used in hydrography. It is the error index used
for NOS positioning standards [Umbaoh, 1976]. A second
circular figure, known as circular standard error, is also
examined briefly because of the ease of converting it to
circular figures which have different probabilities.

For clarification of the issues involved in this
section, the following assumptions are made.

(i) Only random errors are considered.

(ii) Errors associated with each LOP are normally
distributed.

(iii) Errors are independent.

(iv) Errors are limited to the two-dimensional case.

(v) LOP's are straight lines at their point

of intersection.

26

These are the same assumptions made by Kaplan (1980),
except that Kaplan allows systematic error to be considered
in assumption (i). This appears to be an oversight since
the remainder of his discussion, from which this section
draws heavily, considers only random error.

1 . Concentric and Eccentric Geometry,

Before proceeding further into a discussion of error
figures, it is necessary to examine the two special cases of
range-azimuth positioning, that of eccentric and concentric
geometry. Each is illustrated in Figures 2.2 and 2.3.

vessel position

distance device

theodolite

Figure 2.2 Eccentric Range-Azimuth Geoaetry.

In actual practice the geomatry used is often eccen-
tric, but the concentric assumption is made. This is
because it is usually difficult to co-locate both theodolite
and ranging equipment. Hence, they are offset one or two
meters from each other. It should be noted that this
assumption will introduce a systematic error in all

27

/

/

/

vessel position

theodolite and
distance device

Figure 2.3 Concentric Range-Aziauth Geoaetry.

positions, which must be eliminated before an analysis of
random errors can be made. The systematic error is often
ignored (by use of the concentric assumption) if the total
uncompensated error is within the tolerance of the standards
being used. An algorithm for eliminating this error is
shown in Figure 2.4 and given by equation 2.6. This algo-
rithm assumes thai: the size of d in Figure 2.4 is small
compared to r.

c = d cos <J>

(2.6)

where:

c = the corrector to be applied to r
r = observed range to the vessel

d = distance between theodolite and ranging device
$ - the angle between the visual LOP and the
line connecting the two stations

28

range
arc

distance
device

^V

vessel position

\
\

^ ^theodolite

Figure 2.4 Eccentric Error Compensation.

2. The Error Ellipse

Detailed discussions of the development of the error
ellipse can bs found in many references, especially in
Greenwalt (1962) and in Burt (1966). This paper will only
present enough background to apply the error ellipse concept
to two LOP range-azimuth positioning. The error ellipse
formed when multiple LOP observations are made is not
considered here.

A range-azimuth position is formed by the intersec-
tion of two LOP's, each having an associated standard devia-
tion. By applying the two-dimensional normal distribution
to the errors, elliptical contours of egual probability
density are formed. The contours center on the intersection
point of the lines of position. This is illustrated in
Figure 2.5, and shown mathematically by

P(v<,,vb) -

2i> <Ta <rb

e*i

(2.7)

29

wh e r e :

\ = residual (error) in the direction of <£
vb = residual (error) in the direction of <J£

<£ = Isngth of the semi-major axis
<Tb = Isngth of the semi-minor axis

It can be shown "that

Vn

= K

(2.8)

wh e r e

:"■ = -XJU

p(v«,vb) <Ta(Tb 21T

(2.9)

"For values of p (v , v ) from 0 to oo , a family of equal
probability density ellipses are fornei with axes K <& and
K(Tt" [Greenwalt, 1962], The probability density function in
equation 2.7, when integrated over a particular area,
becomes the probability distribution function. This yields
the probability that the residuals va and vto will occur
simultaneously within that region. This probability distri-
bution function of an ellipse is given by

-K/2

?(va ,vb) = 1 - e

(2. 10)

The solution of equation 2. 8 for different values of K
yields different probabilities. For example, for 39.35^
probability, the axes of the ellipse are 1.000 (^ and 1.000
cTb [Greenwalt, 1962]. In other words, a one-sigma error
ellipse around a measured position indicates a 39.35% prob-
ability that the position is actually within that ellipse.

It is seen in Figure 2.5 that the standard devia-
tions <T( and <T7 of the measured LOP's are not the same as the
standard deviations d^ and c£ of the error ellipse. A

30

Figure 2.5 Error Ellipse and dr„s .

coordinate -transformation is required to obtain them. This
transformation is found in Heinzen (1977) and will not be
discussed here. Results of the traQ sf ormazion are presented
in equations 2.11 and 2.12, from Bowditch (1977).

<r* ■

2 sirf/3

<j. + <r,

z +-W(<r,%cr^"4sin/30"'^

(2.11)

^" 2sin\3

<£ + <£- V ((j;l* °0 ~4si n/S cr* *;

(2. 12)

31

where :

fla = length of the semi-major axis
tf~b = length of the semi-minor axis
o*« = standard deviation of the range LOP
<SX ■ standard deviation of the angle LOP
when converted to distance units

/3 = angle of intersection of LOP's

For range-azimuth positioning, <S[ and (fx are not
equal. (Tt is the error in distance measurement, and it is
dependent on the equipment used for ranging. A diagram of a
range-azimuth position is shown in Figure 2.6.

— ■*-

""}

1

'

L

— ^

V

4,

I

<r,

<4

*,

~>f

I
I

->■

Figure 2.6 & Range-Azimuth Position,

32

The error in the visual LOP, (Tx , is a function of the
angular error of the theodolite and the distance to the
vessel, where r is the distance and C@ is the angular stan-
dard deviation of the theodolite, in units of degrees. This
is given by equation 2.13, which is a modification from
Heinzen [1977], Heinzen uses the tarm angular resolution in
place of the more correct <Te .

^ r C&

<L =

1 f7. 2<?6»

(2. 13)

The error ellipse concept can now be applied to the
eccentric geometry by using equation 2.11, 2.12, and 2.13
directly. In the concentric case, since the angle of inter-
section i ft * is always ninety degrees, equations 2.11 and
2.12 can be simplified to equations 2.14 and 2.15.

*-T

<rN <c +

<r, + °l - 4 <rt tr%

(2. 14)

<-T

cr, * <rx +

<r,+ <rz

- 4 d, cr,

(2. 15)

3 • Root Mean Square Distance

Root mean square distance (drrv<s) is presented here
because of its common use in hydrography. It is not
commonly known among hydrographers that unlike the error
ellipse, drmShas a variable probability depending on the
eccentricity of its associated error ellipse, and ranges
from 68.3% to 63.2%, as shown in Figure 2.7 from
[Burt, 1977]. Eccentricity is defined as the ratio of the
semi-miner to the semi-major axes of the error ellipse.
Root mean square distance is also called Mean Square
Positional Error (MPSE) by Greenwalt (1962), who recommends

33

0 0.1 0.2 0.3 0.4 OS 0« 0.7 0.8 0.9 1.0

eccentricity

Figure 2.7 Variation of d^s Probability,

that this index not be used because of its variation in
probability.

Root mean square distance is defined in Bovditch
(1977) as equation 2.16. An alternative form is given by
Heinzen (1977) as equation 2.17. If the errors are assumed
independent, the correlation coefficient o , is zero and
equation 2.17 is reduced to equation 2.18. An alternate
method of arriving at equation 2.18 is to substitute equa-
tions 2.11 and 2.12 directly into equation 2.16.

<Vm< =

d

crb

(2. 16)

drmS =

5/aJ

fi

-\/<J\+ <£ + 2^<T,<TX Cos/2

(2. 17)

34

If concentric geometry is assumed, the angle & is equal to
ninety degrees and equation 2.18 is reduced to equation
2. 19.

&r«« s\/^ + ^ (2- 19)

Substituting aquation 2.13 into equation 2.19, the final
form for range-azimuth d^m$ is obtained.

««.. - "\/<r. + i^4 <2-20.

** • Circular Standard Error

Circular standard error has experienced little use
in hydrography but is valuable because it allows easy
conversion between circles of different probability. It is
derived in Greenwalt (1962) and given by equation 2.21. It
should be noted that this equation is only an approximation,
although a very good one.

CI = 0.50 00 ( C~^ + <Tb ) (2.21)

Circular standard error has a probability of 39.35% for a
completely circular error ellipse. It is preferred over
drmS because it can be converted to other circular error
indices cf different probability by a constant conversion
factor, as long as the ratio of ^b/<Ta is between 0.2 and
1.0. The equation for circular standard error, and for
other circular error figures, is given in Table I, which is
taken directly from Greenwalt (1962) . This table gives all
error indices in terms of either (Tc or the error ellipse

35

TABLE I
Circular Error Formulae

Percentage
Precision Index Probability

Formula

Circular
Standard Error

39-35%

ac = 0.5000 (ax + cry)

when Qmln/°max> °-2

Circular
Probable Error

50%

CPE = l.llJh 0c

CPE = 0.5887 (ax + oy)

when °miiAniax> °"2
CFE~ (0.21*1 amin ♦ 0.6621 0mflV)

rain

max'

when 0.1 < omin/amax<0.21
CPE— (0.0900 a . + 0.67*5 a )

mm

max

when 0.0<o . /a < 0.1*
— mm' max ^

Circular Map 90^
Accuracy Standard

CMAS = 2.1*60 a
CMAS

1.0730 (ox + <ry)

when <j ■ /a > 0.2
nun' max —

Circular Near-
Certainty Error
(Three -five sigma)

99.78^

3-5000 a

semi-major and semi-minor axes d"<* and <jj . Applying the
assumptions of the previous section on drr*s to this case, 0~c
for the concentric range-azimuth case is given by equation

2.22.

o! - 0.5000 («*5£^6

(2.22)

36

C. THE EBEOB OF AN INTERPOLATED FIX

The error in an interpolated position for both methods
of interpolation used in range-azimuth positioning will now
be derived. Fundamental to the derivation is an under-
standing of the concept of error propagation, which is
explained in detail in Greenwalt (1962) and Davis (1981).
Error propagation is summarized here for the special case of
range-azimuth positioning. It must be noted that any inter-
polation discussed in this chapter is strictly due to errors
in the observed positions between which the interpolation is
made. Error due to the vessel not being at its interpolated
position (due to steering or wind and sea conditions) will
be considered in later chapters.

1 . Interpolation Algor ithms

The present NOS methods of interpolating range-
azimuth fixes are of two types. One interpolates both the
range and angle between two observed positions. The second
is used when actual range information is acquired on each
sounding. In this case only the angle is interpolated, and
is used with the observed range to compute a position. In
each case a linear interpolation is used [Ehrhardt, 1979].
Algorithms for the interpolated value of the range, r^ , and
the angle 9; , are given in equations 2.23 and 2.24.

r^ = r, ♦ (j/K*1) (r^-r,) (2.23)

6^=6, ♦ (j/K+1) (ei-9, ) (2.24)

where :

Subscript i denotes interpolated.

Subscript 1 denotes observed position number 1.

37

Subscript 2 denotes observed position number 2.
K denotes the number cf intsrpolations.
j steps from one to K.

The following example of ths interpolation process
is also sketched in figure 2.8.

a

/

/
--X

/
i

i

Figure 2.8 Example of Angular Interpolation.

Let:

e, = 251
ez = 244*

r, = 1501m j ■ 1 to 2

rz = 1485m K = 2

The values of the interpolated angles and ranges are
computed by equations 2.24 and 2.23.

6., = 251 ♦ (1/3) (244-251) = 251 - 2.3 = 248.7° (2.25)

e:Z = 251 + (2/3) (244-251) = 251 - 4.6 = 246.4 (2.26)

38

r., = 1501 ♦ (1/3) (1485-1501) = 1501-5.3 = 1495. 7m (2.27)

^U " 1501 + (2/3) (1485-1501) = 1501-10.7 = 1490.3^(2.28)

2 • Error Propagation

"Error propagation is better termed the propagation
of variances and covariances" [Davis, 1981]. The following
paragraphs are a general derivation of error propagation,
and will be applied below. In reading this section, the
terms x and y are general, yet tha reader should remember
that they will be applied specifically to range-azimuth
interpolation. Let y be a set of quantities each of which
is a function of another set of random variables x. The
random variables in cur application are the observed x
given in equation 2. 1, and y is the interpolated e^ or ru
given in equations 2.24 or 2.23.

The covariance matrix l_>yy ^-s 9^ven bY the matrix
equation 2.29, where Jyx is called the Jacobian matrix and
is given in equation 2.30.

1—tyy - Jyx^-iJCX^y

T
%

(2. 29)

yx

ax.

3P-

^xN

(2. 30)

i>rxt

39

Jyx is the transpose of Jy* , and ^^ is another covariance
matrix given by :

I

I

I

t.

or,

<r* <S

a*

Ml

(2.31)

The covariancG matrix in equation 2.3 1 is written
for the general case of correlated random variables. If y
is a single quantity rather than a sst of quantities, and if
the variables are assumed uncorrelated, equation 2.29
reduces to:

*>x.

<.0
0"'v

<r„

3*2

(2.32)

By carrying out the multiplication in equation 2.32, the
expanded form is given in equation 2.33.

y

(2.33)

If y is a linear function of random variables, the
partial derivative terms in equation 2.33 become constants.
Thus the matrix equation 2.29 becomes equation 2.34, where
a,b,c, and d are constants.

a. i

= a <r, +

lJ1 ^ * *

D <TZ ■+■ c cr. + - - -

cl <r„

(2.34)

40

Now let us apply equation 2.34 to the range-azimuth
case. The assumptions made above are:

(i) y is a single quantity

(ii) The random variables x are assumed
un correlated.

(iii) y is a linear function of x

If these are applied to the two interpolation algorithms,
equations 2.24 and 2.23, it can be seen that:

(i) 9^ and r^ are single quantities.

(ii) The random variables 6, , 92 and
r, , rz are assumed uncorrelated.

(iii) The two interpolation algorithms are linear.

From the foregoing general discussion the interpo-
lated variances of equations 2.23 and 2.24 can be given as:

4. ■ <£« + (lo7~) (f01 ' ^7 i2m36)

but,

<£i ■ 4a. " $r (2.37)

41

and:

<£i = 4z ■ <Te (2- 3Q)

By substituting equations 2.37 and 2.38 into equations 2.35
and 2.36, we have:

<Jrx ■ <Tr (2.39)

and

Gel = cre <2.a0)

This error propagation applies only to vessels
moving in an arc. For a vessel moving in a straight line
the angular interpolation algorithm is not linear and the
partial derivative terms analagous to those in equation 2.33
are not constants. The error ellipse for an interpolated
position can now be formed. The ellipse is seen to be the
same as fcr an observed position (figure 2.5) since equa-
tions 2.39 and 2.40 show that errors in interpolated
distance and angle are the same as for an observed distance
and angle.

42

III. EXPERIMENT DESIGN AND IHPLEHENT ATION

The experimental work performed in this investigation
can be conceptually divided into two parts, although both
parts were accomplished simultaneously. Part one involved
pointing error and variance determination for the two theo-
dolites, and part two investigated the accuracy of
interpolated positions.

A. FIELD WOBK

The actual field work was reprasentati ve of a typical
range^azimuth survey. Two full days <8 and 15 April, 1983)
were required to obtain 443 position fixes and over 2500
Aztrac angles. The experiment took place in southern
Monterey Bay near the Monterey Harbor Coast Guard Pier, as
shown in Figure 3.1. The vessel usad was a chartered
36-foct Uniflite with a fiberglass hull and twin engines,
and its operator had about two months' hydrographic survey
experience, including steering range arcs.

For each position of the vessel, six lines of position
were observsd. Three wild T-2 theodolites located at
stations MUSSEL, SOFAE, and USE MON were used to obtain the
"reference" or best estimate positions of the vessel. Two
additional test theodolites, a Wild T-2 and an Odom Aztrac,
made observations from stations T2 and AZTRAC. Finally, a
Del Norte Trisponder (model R04) provided a distance LOP to
the vessel from station GEOCEIVER. The reference positions
were obtained at one minute intervals, and five Aztrac
angles were observed between each of these positions for
later use in evaluating interpolation methods. Figure 3.2
shows the sources of the various LOP's to the vessel. It

43

Monterey Bay

reir .
7HG.oooi.ires

ScfAd

USE. **oisl

Figure 3.1 Sketch of the Survey Area.

represents an enlargement of the dashed circle area shown in
Figure 3.1.

Geodetic control for the reference positions consisted
entirely of monumented third-order stations. Station
GEOCEIVER, although not a published station, was located to
third order specifications by Mr. William Anderson of
NAVOCEANO in 1982, and the other two stations were located
as eccentrics of this station. Each was less than one meter
from GEOCEIVER. GEOCEIVER is a monumented station, while
Aztrac and T2 are marked by masonry nails driven into the
concrete pier. Positions for Aztrac and T2 were computed on
an HP-9815 computer using the NGS geodetic direct program.
The initial pointing by both instruments was to station USE

44

mussel

error ellipse

sofar

\ 3000 m
\ range arc

use mon

Figure 3.2 Lines of Position Observed to the Vessel.

MON. Initial pointings at all stations were made to third-
order stations at least 500 meters distant. Geodetic posi-
tions of all control stations is given in Appendix 3.

Underway operations were very similar to a nonautomated
range-azimuth survey, except that five theodolites, rather
than one, were trained on the boat for each fix. The survey
boat steered along the appropriate range arc, and fix marks
were given over voice radio to the observers on shore. All
T-2 angles were obtained once each minute, and Aztrac angles
were recorded every ten seconds. The sequence of events for
each reference position is given in Table II.

This process continued throughout the two-day field
operation. Breaks in data collection occurred at the ends
of each range arc, and also when theodolite observers were
rotated. The monotony of the events shown in Table II was
sufficent to approximate the monotony (with its associated
observer difficulties) of an actual survey.

45

TABLE II

Data Acquisition Sequence of Events

tii
to

20

ce prior J
fix j action

sec

depress Aztrac manual fix button

15

sec

"15 second standby" over voice radio I

10

sec

depress Aztrao manual fix button |

5

sec

"standby" over voice radio |

0

sac

"mark" over voice radio, {
depress Aztra^ manual fix button,
T-2 operators make observations

j

Observers at stations MUSSEL and USE MON were required
to record angles as well as to observe. T-2 observers at
stations T2 and SOFAR were provided with separate recorders,
because at times the high angular speeds of the vessel at
these stations (due to closeness of the boat) made observing
and recording difficult for one person.

The Trisponder was calibrated using the standard NOS
method over a geodetic baseline (GEOCEIVER to USE MON) and
no systematic errors were observed. The geodetic baseline
was determined by computing an inv=rse distance between
stations USE MON and GEOCEIVER using the HP98 15 computer and
NGS geodetic software. The Trisponder was reported by Odom
Offshore Surveys to have a standard deviation of its ranging
error of 1.0 aeters. The master unit and antenna were
mounted at the highest part of the boat, about one meter
above and aft of a radar antenna enclosure, which was about
one meter in diameter and 0.4 meter high. The master unit
was a cube about 0.3 meter on a side and was covered with

46

green signal cloth. This a nit was the target to which
angles were observed. The Trisponiar Distance Measuring
Unit (DMU) was mounted next to the steering station inside
the boat.

Data logging for both Trispondsr and Aztrac was accom-
plished by an Odom Navtrace computer system. Both the
Trisponder DMU and Aztrac receiver were interfaced to this
unit. The Navtrace computer is programmed to automatically
log fix data not on intervals of equal time, but on equal
distance intervals from a reference line. For this reason,
all fixes were legged by using the manual fix feature of the
computer. This simply caused the computer to log distance
and angle each time a button was manually pressed. Timing
for fixes was provided by an NOS standard sounding clock.
This is a mechanical clock with a buzzer set to ring every
ten seconds.

The weather during both days of field work was good to
fair for survey work. Visibility was good at all times, and
winds were calm on each morning of operations. Afternoon
northwest winds were a maximum of about 15 knots on both
days, which produced two to three foot seas in the offshore
part cf the operating area.

B. THEODOLITE POINTING ERROR

The pointing error, x^ , is defined as the difference
between an observed and computed angle given by

x.= e0- 9C (3.1)

and the mean pointing error, T, is therefore

* = -£-£*- <3-2)

47

where n is the number of observations, 9 is the observed
azimuth and 6C is the computed azimuth from the theodolite
position to the reference position of the vessel. The stan-
dard deviation, <f& , cf a set of pointing errors can be
expressed mathematically as

CL =

r»-

n*-*T

(3.3)

This is the usual definition in most statistics texts and
agrees with the discussion in section II. A. of this paper.

The computed azimuth, 9C , was determined by computing a
geodetic forward azimuth from either test theodolite posi-
tion to the reference position of the vessel. There is some
potential error in 9C due to an uncertainty in the reference
position of the vessel. The reference position of the
vessel was determined by a least squares adjustment cf the
three LOP's from the three Wild T-2 theodolites. A
by-product of this adjustment is an error ellipse for each
position. The angle subtending this error ellipse from each
test theodolite is the error in 9C . This is illustrated in
Figure 3.3.

error
ellipse

Figure 3.3 Uncertainty of an Observed Error,

48

The least squares method was developed in 1794 by Gauss.
It is based on the ccncept that as soon as redundant obser-
vations are present there appear discrepancies among the
observations. In order to eliminate these discrepancies a
residual, v, must be added to each observation. In the
method of least squares the residuals are determined such
that the sum of their squares becomes a minimum, provided
all observations have the same accuracy [Mueller, 1979]. It
is assumed that the three LOP«s used in this adjustment do
in fact have the same accuracy. Further information on the
method of least squares as applied to hydrography is given
by Kaplan (1990) .

Reference positions were calculated by the least-squares
FORTRAN program AZLSQ2, written by the author, which is a
modification of program SILVA1 [Silva, 1979]. The program
generated plane and geodetic coordinates for each position
fix, as well as the lengths of the semi-major and semi-minor
axes of its error ellipse. The angle made by the semi-major
axis and the x-axis was also computed.

C. INTERPOLATION ALGORITHM EVALUATION

The present NOS method of observing azimuths to the
sounding vessel is totally nonautoma ted , as explained in
section A. This precludes recording an azimuth for each
sounding because of the speed with which soundings are
taken. Consequently interpolation algorithms are employed
to plct soundings between observed positions. Two algo-
rithms are used: one interpolates only the azimuth and uses
an observed distance to the vessel, while the other interpo-
lates both azimuth and distance. Figures 3.4 and 3.5
illustrate the situation.

49

actual
track

aztrac positions
used for interpolation

interpolated aztrac

position position

Figure 3.U Interpolation of Angle Only

aztrac positions
used for interpolation

T

aztrac
positions

interpolated
positions

Figure 3.5 Interpolation of Angle and Distance.

The method used to evaluate thass interpolation methods
is simple. Actual distances and angles to the vessel were

50

recorded every ten seconds, using the telemetering theo-
dolite and associated computer logging system. A ncrauto-
mated system was then simulated by interpolating between
positions obtained every minute. Thus two data sets were
produced, an interpolated set and a corresponding set of
actually observed positions. These two sets of positions
were then compared and their differences examined. Both NOS
interpolation algorithms were evaluated in this manner. An
analysis cf the experimental results is given in Chapter IV.

D. CHOICE OF EXPERIMENTAL CONDITIONS

Any experiment must be performed under conditions
similar to those under which -he results will be applied.
This section will describe the specific error sources to be
examined under range-azimuth conditions. By carefully
choosing the experimental conditions, these error sources
can be brought into the foreground for examination, while
all ether sources of error can be leapt in the background.

It is necessary to introduce some standard statistical
terminology to help understand the experiment- design. The
classical experimental method studies the effect of only one
variable. That is, it holds all effects but one to be
constant, and varies that single one systematically. In the
case of this thesis, one factor (a theodolite) was varied
systematically by introducing two levels (Aztrac and T-2) of
the factor. It is desired to find the effect of that factor
(the theodolite) on the error of a range-azimuth position.
It is known, however, that the observer also has a consider-
able effect on the error of a position. Thus another factor
(the observer) is introduced into the experiment. Four
levels (four different individual observers) of this factor
were selected.

51

The following four observers, who are attached to the
Naval Postgraduate School, constitute the four levels of the
observer factor.

LCDR Gerald E. Mills, NOAA. Instructor in
Hydrography. Ten years experience in
geodetic and hydrographic surveying.

LT Maureen Kenny, NOAA. Student in
hydrography. 2 years field experience
atoard the NOAA Ship Davidson, primarily
in Alaskan and West Coast waters.

LT Mary C. Schomaker, NOAA. Student in
hydrography. 4 years field experience
aboard NOAA Ship Davidson, and on NGS
horizontal control and leveling parties.

Mr. James R. Cherry, Supervisory
Geodesist, Naval Postgraduate School. 23
years field experience in hydrography and
geodesy, with the Naval Oceanographic
Office.

Although additional observers would have been desired, none
were available who had any experience with theodolites.

Aside from the factors to be investigated, it was recog-
nized that there were background conditions that affected
the results of the experiment [Crow, 1955]. Some of these
were taken into account explicitly in the design. The
influence of all others was minimized by scheduling the
experiment such that each combination of instrument and
observer was evaluated in random order, thereby randomizing
the effects of these other conditions.

52

Blunders and some systematic errors were explicitly
accounted for in the design. Blundsrs were identified visu-
ally by their large magnitude and were simply deleted from
the data set using the editor on ths NPS computer. The
Aztrac instrument essentially eliminates all blunders due to
its automatic data legging feature.

Systematic errors integral to the theodolites, such as
collimation and eccentricity, were not accounted for, since
they were very small compared to the size of the errors
under investigation. One systematic error that could be
quite large is the initial pointing error, which is due
largely to the observer. The NOS Hydrographic Manual
[Umbach, 1976] requires the initial pointing to be "accurate
to within ±30 seconds of arc". That is, the difference
between beginning and ending pointings is not to exceed one
minute. For this experiment, the mean of the beginning and
ending pointings to the initial azimuth was algebraically
subtracted from all observed angles for that set. It could
be argued that a more accurate way of removing this system-
atic error would be to prorate the difference in initial
readings between the beginning and ending pointings.
However, the former method is often used by NOS hydrographic
survey units.

One background condition, the angular speed of the
vessel, was explicitly taken into account in the design by
dividing the experiment into groups. Six separate subsets
of the experiment were created by measuring pointing error
at six different angular speeds. These speeds were chosen
to be closely representative of speeds found under actual
surveying conditions. To this end, integral values of
angular speed were not used, but rather integral distances
from the vessel to the theodolite station. This corresponds
to actual practice in most nonautomated or semiautomated
surveys. The vessel maintains constant engine speed, and is

53

navigated in a circle such that a constant distance value is
displayed in the ranging equipment. For this experiment,
distances of 300, 500, 700, 1000, 1500, 3000 meters were
used. Using the approximate vessel speed of 2 meters per
second (4 knots) , the angular speeds in minutes per second
corresponding to these distances were 22.9, 13.8, 9.8, 6.9,
4.6, and 2.3, respectively.

Another background condition that was explicitly
accounted for in the design was the experience of the
observers. For a completely general investigation, one
would desire to evaluate the error using observers of widely
varying experience. Only experienced observers were used in
order to conserve resources of time and money, although ncne
of them were experienced in using the new Aztrac instrument.
All observers except LCDR Mills had acted as observer for
range-azimuth hydrography within tha past year using the
Wild T-2.

There were other background conditions that affected the
results of the experiment, including weather and lighting
conditions, and observer fatigue. Other subtle factors may
have also contributed to the error. In nonautomated
systems, there might have been a time lag by the radio oper-
ator in the vessel as the fix mark was relayed to the theo-
dolite operator ashore. The design of the experiment could
not explicitly account for all these conditions, so several
steps were taken to randomize the order of observations for
each combination of instrument and observer.

It was assumed that the theodolites used were randomly
drawn from the entire population of T-2 and Aztrac instru-
ments. This is a fairly good assumption for the T-2, since
the particular unit tested is one of ssveral maintained by
the Naval Postgraduate School, and has seen several years of
service. The assumption for the Aztrac theodolite is not as
good, since it is one of only five in existence at this
time. It was provided by the manufacturer for testing.

54

Theodolite observers ware required to be paired, since
the experimental theodolites observed simultaneous angles to
the vessel, A random selection of observer pairs was accom-
plished by consulting a random number table in Wonnacott
(1977). The order of observation for the six different
range arcs was important and required randomization.
Failure to do so could allow the error associated with each
to be influenced by time varying conditions. Examples of
these are the changing effect of sun glare on the instru-
ment, observer fatigue, the effect of repetition acting to
decrease error, or increasing afternoon winds disturbing the
theodolite. Randomization was again accomplished with the
aid of a random number table. A random number was assigned
to each range arc, and the order of observation was estab-
lished by selecting the arcs from the lowest to highest
number.

55

IV. RESULTS AND DATA ANALYSIS

A. DATA PROCESSING SYSTEM

All data acquired for this experiment were manually
entered into the NPS computer facility. A data processing
system was designed and implemented to log the raw data and
perform the necessary computations to arrive at the finished
form given in this chapter. This system was divided into
numerous subsystems to enter data, determine the pointing
error, compute interpolated positions, compute means and
variances, test for randomness, and compute analysis of
variance (ANOVA) statistics. ANOVA is a standard statis-
tical technique used in testing for a difference between the
effects of two or more factors.

The data processing system was designed and programmed
by the author. As mentioned previously, the NGS geodetic
inverse and direct subroutines [Pfeifer, 1975], as well as
the least-squares adjustment program [Silva, 1979], were
adapted for use here. The ANOVA computations were performed
by the Statistical Package for Social Sciences (SPSS) on the
Naval Postgraduate School mainframe computer [Hull, 1981].
The pointing error and interpolation subsystems were
designed as described in Chapter III.

B. POINTING ERROR DETERMINATION

The pointing error standard deviation for each theo-
dolite needs to be quantified, and i determination must be
made as to whether there is a statistically significant
difference between the two instruments. Pointing error for
each combination of instrument and observer was determined
by the methods discussed in Chapter III. Standard deviation

56

of the pointing error was computed using equation 3.3f for
each combination of instrument and observer, at each range
arc, and are given in Appendix A. This entire procedure was
performed first for the T-2 angles observed to the nearest
second of arc, which is the maximum resolution for this
instrument, and second, for T-2 angles rounded to the
nearest minute of arc, which is spacifically allowed by the
NOS Hydrographic Manual [Urabach, 1976].

Since eight different instrument-observer combinations
exist, the sample means and standard deviations for each
combination were slightly different because each sample is
only an estimate of the mean, , and standard deviation, ,
for the entire population. The poolsd standard deviation is
the best estimate of for this case of multiple samples
because it takes the differences among sample means into
account. If an overall standard deviation is computed using
equation 3.3, the population standard deviation will be
overestimated because of the differences among sample means.
The pooled standard deviation is mathematically expressed by
equation a. 1 * crow, 1955], [Box, 1978].

/("■-')*» ("»-0£ + +K-')<

". + ni * - ■ +nk-k

where:

k = total number of observer-instrument
combinations

n = number of observations for each observer-
instrument combination

s = sample variance for each observer-
instrument combination

Thus a pooled standard deviation was computed for the eight
samples available at each different range arc. These are

57

given in Table III for the Aztrac and both the rounded and
unrounded T-2 data. This table gives angular speeds for the
vessel in units of minutes of arc per seconds of time. Note
that the pointing error of the Aztrac is twice as large as
that of the T-2 and that the rounded T-2 pointing error is
only slightly greater than the unrounded.

r

TABLE

III

Pointing Error

Standard Deviation |

[pooled

estimates)

Ranae
Arc

(m)

Approx.
Angular

Speed
(min/sec)

Aztrac
m (sec)

T-2

(unrounded)

i (sec)

T-2
(rounded)

m (sec)

I

I 22.9
1

I <71 Cs

I

I
i

1. 26

(868)

I 1.31

<r&

300

I 3.33 (2290)

(902)

500

| 13.8
I j

| 3.56 (1470)

!

1
i

1.72

(712)

j 1.73

(715)

700

| 9.8

I

| 2.10 (619)

1

1
i

1.30

(382)

| 1. 31
j

(336)

1000

I 5.9

2.97 (6 13)

1

1
i

1.08

(225)

1 .09

(226)

1500

4. 6
j j

4.25 (584)

1

i

0.92

(126)

I 0.93
i

(128)

3000

2. 3
I I

2.06 (142)

1
1
1

1.41

( 97)

| 1 .43

I

( 99)

The technique of analysis of variance (ANOVA) is used to
determine whether a statistically significant difference
exists between the two theodolites. The ANOVA technique
allows the partitioning of overall pointing error variance
into portions caused by each factor (observer and theo-
dolite) , by interaction, and by experimental error.
Interaction exists if the variance for a particular combina-
tion of instrument and observer is greater than the variance
for any other such combination. The experimental error is

58

primarily due to the conditions described in Chapter III,
concerning the reference positions for the vessel. This
error is a measure of the precision of the experiment, where
precision is defined as the closeness with which repeated
measurements made under similar conditions are grouped
together [Greenwalt, 197 1]. Experimental error is assumed
to have a population mean of zero. Further details of the
ANOVA method can be found in many statistics textbooks, such
as Wo im a cot t (1977), Box (1978), Crow (1955), and Walpole
(1978). This discussion was taken primarily from Crow
(1955).

The following assumptions must be made when using the
analysis of variance technique.

(i) Observations are random.

(ii) Means and variances are additive, as given
in the mathematical model below.

(iii) Experimental errors are independent.

(iv) Variances of the experimental errors
are equal.

(v) Distribution of the axperimental errors
is normal.
A mathematical model can now be given, using these
assumptions, which specifies the total effects of the vari-
ances acting on a particular observation, x^.+ .

where:

^<- overall mean for all observations

59

a^ = effect of the instrument factor at level i

hi - effect of the observer factor at level j

fc; = effect of the interaction of instrument and
observer at level i and j, respectively .

e--f = random effect caused by the variance of the
experimental error.

i * 1,2

j ■ 1,2,3,4

t = 1,2,3, ■ • ■ ,n

n = the number of observations for a
particular observer-instrument
combination (usually 15)

It should be emphasized that the a, b, fr , and e of equa-
tion 4.2 are not actual variances, but a realization of the
effect of those variances on a particular observation x-f .
The variances associated with observer, instrument, and
interaction are computed by the ANQVA procedure, and these
form the basis of the test for differences. This test is
called the F-test in honor of Sir Ranald A. Fisher
[Wcnnacott, 1977], and is based on a ratio of variances. To
test for a difference between instruments, we form a ratio
with the variance among instruments in the numerator, and a
denominator composed of an estimate of the variance of the
experimental error. In terms of our mathematical model, a
(the numerator) is being compared with e>c;f (the denomi-
nator). More simply stated, it is a comparison of the
precision of the instrument (the numerator) , with the preci-
sion of the experiment (the denominator) . This ratio, F, is
then compared to a ratio FM , which is computed for for a
particular confidence level from the F-distribution function
given as equation 4.3 [Crow, 1955].

60

p(fi = / ^)jW)i fi fi F (f,t*F) dF <a'3'

where: F = the ratio discussed above

f, = the number of degrees of freedom
in the numerator of F

fx = the number of degrees of freedom
in the denominator of F

A precise hypothesis must now be stated that can be
tes-ced by the F-test. Walpole (1974) states that "a statis-
tical hypothesis is an assumption or statement, which may or
may not be true, concerning one or mora populations."
Experiments are designed to test hypotheses, and "the rejec-
tion of an hypothesis is to conclude that it is false, while
acceptance merely implies that we have no reason to believe
otherwise" [Walpole, 1978]. A null hypothesis is an
"initial hypothesis, or one we hope to reject" [Crow, 1955],
and is usually stated in terms of an assumption of no
difference between the effects to be investigated by the
experiment. Phis experiment uses three null hypotheses,

which are:

(i) There is no difference between observers,
that is, the ratio F, for observers, is
small compared to F^ .

(ii) There is no difference between instruments,
that is, the ratio F, for instruments, is
small compared to F^ .

(iii) There is no interaction, that is, the ratio
F for interaction, is small compared to F^ .

61

These three null hypotheses can be accepted or rejected
on the basis of the F-test and the ANOVA procedure. If any
are to be rejected, the value of tha ratio F must exceed
that of the critical F^ for that confidence level (95%) . A
summary of ANOVA results is given in Table IV. For each
range arc, the computed values of F and the corresponding
critical value needed to reject the null hypothesis are
given. The rightmost column of the areas labelled as
rounded or unrounded data indicate acceptance or rejection
of the null hypothesis. It can be seen from this table that
in no case could the null hypothesis be rejected for either
instrument or observer. In other words, it may be said with
95% confidence that there is no reason to believe there is a
difference between the four observers or between the two
instruments. It should be noted that data for the 1500
meter arc indicate a rejection of the null hypothesis for
interaction. According to Crow (1955), a significant inter-
action usually occurs because unrandomized background condi-
tions are present. There is little apparent reason why this
interaction should occur, other than that there were some
unidentified, time varying conditions affecting the measure-
ments. The 1500 and the 1000 meter range arc were only
observed en April 8, while all the other range arcs were
investigated on both days of the experiment.

A final assumption of the ANOVA technique is that the
observations within the eight combinations of instrument and
observer are randomly drawn from their populations. Several
tests are available to determine if a particular sample is
random. The test used here was the Run Test, as given in
Crow (1955) , and a confidence level of 95% was used. This
test showed all eight samples were random at the 95%
confidence level. '

62

—

TABLE

IV

-T

Suamary

of ANO?A Results

at 95% Confidence

Rounded

Data

Unrounded Data

I

F*

I

F

I
I

null
hyp.

F

null
hyp.

300 b arc

!

instrument
observer

3.97

2.70

j

0.10
t 1 .69

I

I
l

accept
accept

0.09

1.69

l

accept
accept

interaction

2.70

1 .36

1
1

accept

1. 35

accept

500 m arc I

instrument

3.92

0.01

1

accept

0.01

accept |

observer

2.68

0.13

■

1
accept

0. 14

accept i
accept 1

interaction

2.68

0.25

1
1
1

accept

0. 25

700 m arc 1

instrument

3.84

I 0.05

I
i

accept

0.05
1.86

accept |

observer

2.60 I

1 .86
0.28

■
1

1

I
accept

accept |

interaction

2.60

accept

0. 29

accept |

1000 b arc I

instrument i

3.92 |

0.01

1
i

accept

0.01 |

accept |

observer

2.68

0.45

1
1

accept

0.45

1
accept |

interaction

2.68

1 .74

1

accept

1.79

1

accept |

1500 b arc I

instrument

3. 8U

0.78

i

i

accept

I 0.73

accept l

observer

interaction

2.68
2.68

1 .44
5 .01

1
1
1
1

1
accept

reject

1.41
5.02

i

accept |

reject |

3000 b arc I

instruBent

3.95

0.10

1

accept

0. 12

accept

observer

2.70

1 .69

1

I

accept

,.7«

accept

interaction

2.70

| 1 .36

1

1

1

1
accept

| 1.37

accept

1

i

63

C. INTERPOLATION EVALUATION

The two methods of interpolation used by NOS were next
evaluated. As explained in section III.C. , three sets of
positions were computed. These were a set of actually
observed positions using Aztrac, and two sets of corre-
sponding positions computed by the two different, interpola-
tion algorithms discussed in section II. C. These two
algorithms were evaluated by computing the distance between
each observed position and each corresponding interpolated
position. A FORTRAN program written by the author performed
the computations. The NGS geodetic direct subroutine
[Pfeifer, 1975] was used to compute positions from observed
distances and Aztrac directions to the vessel. The interpo-
lated positions were computed using equations 2.23 and 2.24,
which are the same algorithms used in the NOS interpolation
subroutine TCARC [Ehrhardt, 1979]. Distances between the
two corresponding positions were computed on a plane, rather
than using a geodetic computation. There is negligible
difference between plane and geodetic methods at the
distances (about ten meters) under consideration here.
Results of these computations are given in Table V, which
shows for each range arc the average distance in meters
separating the interpolated and the corresponding observed
positions. An indication of the variability of these values
is shown by the percentage of interpolated positions falling
farther than 1.0 meter away from the actual position.

Little can be inferred from the results in Table V
because this table is really an intermediate step towards a
rigorous evaluation of the raw data. This table should be
viewed as only a general indication of the effectiveness of
interpolation. Since it is a direct result of the ability
of the boar operator to steer the vessel in an arc, wind and
sea conditions have a tremendous effect on interpolation

64

Range

m

3000

1500

1000

700

500
300

TABLE V
Results of Interpolation Evaluation

Angle Only

I Angle and Distance

% > 1.0 m I —

from mean j X

% > 1.0 m

1.15

47%

2.09

85%

1.63

72%

2.0 4

75%

1.86

79%

2.59

79%

2.41

4.98
4.07
3.01
4. 15
5.32

rrom me

an

55S

9 1%

78%

80%

83%

34%

effectiveness. Although this experiment was carried out in
representative survey conditions, Table V should not be
viewed as being applicable to all situations. The table
does indicate that, whenever possible, automatic recording
of range data should be used.

Full analysis of the interpolation algorithms should be
the two-dimensional equivalent of tasting for the difference
between means. This is because both the interpolated and
observed positions are not "true" positions, but have some
error. A one-dimensional test of differences between means
is well established, and is discussal in several references,
including Wcnnacott (1977) . The null hypothesis for such a
test is

(j = yMi ' y^:

(4.4)

65

where^, an & ^c*^ are the means of tha two populations, and d
is some arbitrary distance selected by the experimenter.
The two-dimensional problem has tha same null hypothesis but
the mathematics of the test have not been established. This
problem is illustrated in Figures 4.1 and 4.2.

Figure 4.1 One-Dinensional Difference Between Means.

A proper analysis of the data would inquire for each
interpolated-obsarved pair of positions, whether the
distance between the two positions was greater than d for a
particular confidence. More work than could be incorporated
into this thesis is required to fully evaluate the data.

D. AHALYSIS OF FACTORS AFFECTING THE RESULTS

The results of this experiment were presented in Tables
III, IV, and V. An attempt will now be made to analyze the
experiment for errors in logic and technique, in order to
better understand these findings.

66

interpolated

observed

Figure 4.2 Two- Dimensional Difference Between Means.

The most important results from this thesis are the
estimates of (T for theodolite pointing error. It is obvious
from Table III that standard deviations of both the rounded
and unrounded T-2 error values are about one-half that of
the Aztrac, for all angular speeds considered. The ANOVA
technique, however, shows no statistical difference between
the instruments at the 95% confidence level. An analysis of
the data used to obtain the results yields a potential
explanation for this apparent contradiction.

The original data (pointing errors in seconds of arc)
were made the subject of empirical probability density plots
using the subroutine HISTG [Robinson, 1974] on the NPS
computer. These plots show probability density versus
error, as well as mean and standard deviation. An example
of these plots, for the 500 meter range arc, is given in
Figure 4.3. The remaining plots are found in Appendix C.
Plots are shown for both Aztrac and T-2 (unrounded) , for
each range arc. A striking feature of the Aztrac curves is

67

r " ■

500 METER ARC

•*•

•

T-2: SOLID LINE
flZTRflCJ OflSHEO LINE

to

•

f\f s

O"

'"* / \ *

^■^

/ \ / '\ «

>

* x / ' \ \

w

M * 1 i \

O-oi

7 \ \

d"

»*

1 \\

•

o"

* / v *

o

./ V^ — \N»

^w.. _^*

• -

4000.0 -2000.0 0.0 2000.0

- 1
4000.0

ERROR IN SECONDS

Figure 4.3 Probability Density vs. Error.

their bimcdal shape, as compared to a single peak for the
T-2 curves. It can be easily seen from the curves how the
spread of this bimcdal distribution would increase the
computed standard deviation for the Aztrac data.

The method of data acquistion for this thesis was
semiautcmated in that all T-2 angles were manually recorded,
while the Aztrac angles were recorded by pressing a button
aboard the vessel. It is probable that a time lag existed
between all the T-2 observations and the Aztrac observa-
tions, despite the best efforts of the observers, because
the observation procedure was not totally automated. If
this were true, there would be little difference between the
observed angls 0 for the T-2 and the computed angle 9C ,
because 6C is associated with a reference position also
derived from T-2 observations. The observed angle 0O for
the Aztrac would, however, be consistently different from 9C
because of this time lag and because the vessel was moving
to the left or right with respect zo the Aztrac observer.

68

Since approximately equal numbers of observations were made
with the boat moving to the left or right along the range
arc, the distribution of Aztrac pointing errors would take
on a bimodal shape.

It is understood that the data analyzed by these curves
come from not one but four different samples, and that the
curves should not be expected to ba perfectly peaked. The
T-2 curves, however, also come from four samples and do not
have multiple peaks. This analysis is further supported by
finding the distance between one peak of the Aztrac carve
and the single peak of the T-2 curve in figure 4.3. The
distance in arc seconds, when converted to meters, is
roughly the distance the vessel traveled in one second. One
second of time is certainly a reasonable figure for the time
lag discussed above. A manual check of the raw data
recorded in the field also suggests such a time lag. The
original data were sorted into two sets of "left'' and
"right" observations, which were analyzed for mean and stan-
dard deviation. Results of the analysis are shown in Table
VI. This table gives the mean and standard deviation for
the "left" and "right" data sets, and shows that "the mean of
both sets was about two meters to the left or right cf the
reference position of the vessel. This two meter difference
corresponds closely to a nominal vessel speed of two meters
per second (four knots) for the boat used, and a time lag of
one second. Means for the 1500 and 3000 meter ranges are
somewhat unequal because sea conditions at these offshore
ranges caused the boat to travel slower in one direction.
The rightmost column in Table VI gives the pooled standard
deviation of each "left" and "right" data set, which is the
best estimate of the population standard deviation , <f0 , for
the Aztrac.

69

TABLE 71
Corrected Aztrac Standard Deviation

Arc

left/
right

Mean , X
m (sec)

s
in (sec)

Aztrac Pooled
Standard
Deviation

I L \ 2.95 ( 2035) | 1.17 ( 806) |
300 | || 1.07 (741)
| R -2.67 (-1836) | 0.99 (682) |

| L | 2.40 ( 993) | 1.50 (660) |
500 1 j || 1.44 (598)
R j -2.69 (-11 12) | 1.31 (539) j

700

L | 1.61 ( 476)
R I -2.37 ( -700)

0.90 (266) |

J 1.05 (312)
1.28 (379)

I L | 2.49 ( 514) | 1.55 (321) |
1000 || I 1-»3 (296)
1 R | -2.92 (-604) 1.30 (268) |

1500

L | 1.93 (266) | 1.56 (228) |

I I I 1.45 (199)
R -3.22 (-444) | 1.36 (147) |

3000

L

R

0,48 (33) |
-3.97 (-273)

1.42 (98) |
1.22 (84)

I 1.3 0 (90)

The systematic error caused by a time lag as the boat
moved left or right in the observer's field of view was the
result of fauity design of the expariment. The proper way
to correct this problem would be to duplicate the experiment
using better synchronization of all observations. An alter-
native would be to model the systematic error and apply
corrections to the existing data. Such a model should
include an estimate of the boat speed and its left or right
direction with respect to the observer.

Different ANOVA results might ba obtained using the data
corrected for systematic error, but this would not explain
the ANOVA results in Table IV. The conditions affecting
data acquistion must again be • considered, as well as an

70

understanding of the ANOVA process, when offering an
explanation.

The denominator of the F-ratio discussed in section 3 is
essentially the experimental error of the measurement
process. This error is primarily a function of the error
ellipse for the "reference" position shown in Figure 3.3.
Table VII gives an estimation of this experimental error, by
comparing the size of the error in 30 and 9C . Uncertainty
in ec is given as the mean major axis of all error ellipses
for a given range arc. Uncertainty in 0O is the pooled
standard deviation , <TB , of each test theodolite. The
values for Ce in the Aztrac column are from Table VI, and the
T-2 values are from Table III. An sxamination of Figure 3.3
shows that use of the major axis of ths ellipse is a worst
case estimate of the uncertainty, since the ellipse could
have any orientation in the x-y plane. Thus it can be seen
from Table VII that the uncertainty in 9fc is smaller than
that of the observed azimuth 80 . This comparison is an
indicator of the precision of the sxperiment.

If this error in the computed angla, ec , could be
reduced by decreasing the size of the arror ellipse, the
denominator of the F-ratio would be smaller and the ratio
itself would be larger. Thus a more precise experiment
could produce a rejection of the nail hypothesis, although
this is net indicated in light of the values of (Te for the
Aztrac and T-2 shown in Table VII. Reducing experimental
error any further than this study would be difficult under
typical hydrographic conditions, because the three LOP
theodolite intersection position, adjusted by the least-
squares method, is one of the most precise positioning means
available today. The size of the reference position error
ellipse could possibly have been reduced if better intersec-
tion angles were available, but this was not possible with
the particular geographic shape of Honterey Bay.

71

TABLE VII
Experiment Precision and Theodolite Error

Range, | Mean major
meters j axis

neters (sec)

Aztrac pooled | T-2 pooled
std. deviation j std. dev.
meters (sec) meters (sec)

1.07 (741)

1.44 (598)
1.05 (312)
1.43 (296)

1.45 (199)
1.30 (90)

T

5*

1.26
1.72
1.30
1.08
0.9 2
1.41

(868)
(712)
(382)
(225)
(126)
(97)

In summary, the analysis of variance performed on these
data do, in fact, shew that there is no statistical differ-
ence between Aztrac and T-2 at 95% confidence. This proce-
dure is a strictly numerical one and must be viewed in light
of the original research conditions. When these conditions
are carefully analyzed to remove as much systematic error as
possible, the data strongly suggest that the &N3VA results
are indeed correct.

E. APPLICATION TO POSITION ERROR STANDARDS

From Chapter I it can be seen that there is some confu-
sion in the hydrographic community as to the application of
probability to positioning standards. With this in mind,
there appear to be four possibilities for consideration as
standards with which to compare the results of this thesis,
as given below.

(i) The 1968 I HO standard* of 1.5 mm at the

72

survey seals, with tha 90% probability
suggested by Munson (1977).

(ii) The 1982 IHO standard of 1.0 mm at the

scale of the survey, with 90% probability.

(iii) The current range-azimuth standard of NOS
(0.5 mm at the survey scale) assuming a
probability associated with d^s •

(iv) The drwS standard of microwave range-
range positioning found in Umbach (1976).
This requires that d^^s values at the
survey scale not exessd 0.5 mm for 1:20,000
scale surveys and smaller, 1.0 mm for
1:10,000 scale surveys, and 1.5 mm for
surveys of 1:5,000 scale and larger.

These four possibilities may now be compared to the
position errors of the Aztrac and 1-2 by using the pointing
error results given in Table VII. Since the precision of a
range-azimuth position is also a function of the standard
deviation of ranging error, <J[ , assumed values of 1.0 and
3.0 meters are used in this analysis. Some manufacturers of
microwave ranging equipment used in range-azimuth posi-
tioning report a 1.0 meter value foe <T, , but 3.0 meters is
most often used by NOS personnel [Wallace, 1983] and has
some supporting experimental evidence by unbiased
experimenters [Munson, 1977].

Using these assumed values for C, and the observed
values for <TZ given in Table VII, the error ellipse axes may
be computed using equations 2.11 and 2.12. However, the
error ellipse must be converted to a circular error figure
in order to be compared with the standards above. The
assumed standards (i) and ( ii) must use a-90% probability
circle, which can be computed using Table I, and standards

73

-5

-4

E

a

Aztrac (uncorrected)

a

-^

=*— *

©

T-2

L

Aztrac (corrected)

CO

L.
<D

E

-1

10

15

20

J I 1 l I I L__J I L

J L

I ' I 1_

J L.

J L.

angular speed (arc min/sec)

Figure 4.4 Results Compared to drwS .

(iii) and (iv) must use the drwS formula given by equation
2.16. Results of these computations are shown graphically
in Figure 4.5 for 90% probability, and in Figure 4.4 for
drms . Each figure has angular speed of the theodolite along
the abscissa, and an ordinate consisting of a distance
scale, in metars, indicating the radius of the error circle.

74

The error circle for Figure 4.5 is associated with the 907o
probability level, while the dr^s circle for Figure 4.4 is
of somewhat variable probability, as discussed in Chapter
II. The points plotted in Figure 4.5 have eccentricity
values ( ^/(j; ) ranging from 0.3 to 3.6, which indicate a
probability range of from about 65)5 to 67.5%.

These figures also clearly show an improvement in the
estimate of Gq for the Aztrac as a result of the time lag
correction discussed earlier. The position error values for
the uncorrected Aztrac error is shown by triangles in both
figures, while position error computed using the corrected
(f9 values are shown by solid dots. For both Aztrac cases,
two solid linear regression lines are drawn. Both figures
show that the uncorrected Aztrac values have a much greater
variability than the corrected values, and that the
corrected values are almost the same as the T-2 position
error values indicated by open dots and dashed linear
regression lines. If the corrected Aztrac values are taken
to be the best estimate of position error for this instru-
ment, then a relatively constant error is indicated for the
entire range of angular speeds considered here. This is
about 3.3 meters d^s and 4.6 meters (90% probability) for
both instruments using a <T, value of 3.0 meters, and 1.6
meters drmi and 2.5 meters (90%) for a CT, value of 1.0
meters. Plots are not shown for computations using a range
error of 1.0 meter.

The four possible assumptions for position error stan-
dards are compared to Aztrac and T-2 position errors in
Table VIII. The roman numerals heading the columns of this
table refer to the position standards associated with the
same numerals at the beginning of this section. The reader
should use the table by selecting one of these columns and
inspecting it from the top to the bottom of the table. The
first row of Table VIII indicates the probability associated

75

8

to

<D
♦->

E

u
L.

U

-7

-6

Aztrac (uncorrected)

Aztrac (corrected)

5 10 15 20

i i i i i i i i i t i i t i i i i i i i i i i.

angular speed (arc min/sec)

Figure 4.5 Results Compared to 90% Probability.

with each positioning standard. The second row lists tha
maximum position error allowed by that standard, at the
scale of the survey. Rows three and four show the errors
allowed in row two, when converted to actual distances for
two representative survey scales of 1:5,000 and 1:10,000.
Rows five and six shew the radius of the associated

76

probability circle fcr both Aztrac and T-2, for ranging
error values of 3.0 and 1.0 meters, respectively. The dual
probability percentages in columns (iii) and (iv) indicate
the variable probability of dr^s . Dual values in column
(iii) for maximum error at the survey scale result from the
NOS standard for range-range positioning. The remainder of
the table presents conclusions as to whether the T-2 and
Aztrac meet the various standards. For example, in column
(iv) the observed 3.3 meter drrr,5 value in row five is less
than the maxiium allowable error of 5.0 meters shown in row
four. Therefore, the T-2 and Aztrao do meet the NOS range-
azimuth standards of 0.5 mm at the scale of the survey for
1:10,000 seals surveys.

77

TABLE VIII

Position Standards Comparison

(i) I (ii) I (iii)
IHO | IHO | NOS
(1968) | (1982) | (r/r)

(iv)

NOS

(r/a)

Assumed
Probability

I
I

90%

I
I

90%

I

68%-
-63%

| 6 8%-

| -6 3%

Allowable
Max Error
at Scale

I

I

1 . 5 mm

I

I

1 .0 mm

I
I
I

1.5,

1.0 mm

| 0.5 mm

Allowable
Max Error
at 1:5,000

I

I

7.5 m

I
I
I

5.0 m

I
I
I

7.5 m

| 2.5 m

Allowable
Max Error
at 1: 10,000

Aztrac & T-2j
Error
(a, = 3.0m) |

15.0m 10.0 m ( 10.0 m 5.0 m

4.6 m

Aztrac & T-2|

Error |

( cr, = 1 . 0m) |

2. 5 m |

4.6 m | 3.3 m

2.5m | 1.6m
I

3.3 m

I 1 .6 m

Conclusion As To Meeting Standards: OT, = 3.0 m

I
I

(i)

I
I

(ii)

i (iii)

(iv)

1:5,000
scale

I
I

yes

I
I

yes

i yes

no

1: 10.000
scale

I

yes

I

yes

yes

yes

Conclusion As To Meeting Standards: 0", = 1 . 0 m

I

(i)

I (ii)

I
I

(iii)

(iv)

1:5,000
scale

I

yes

l yes

I

yes !

yes

1:10.000
scale

I
i

yes

I yes

I
I

yes I

yes

78

7. CONCLUSIONS AND RECOMMENDATIONS

Of the original objectives for this thesis discussed in
Chapter I, ths first and most basic was the determination of
pointing error standard deviation for the Aztrac and T-2
theodolites. No investigation of this type had ever been
done for conditions typical of a range- azimuth survey. An
experiment was carefully designed to determine this pointing
error and to determine if there was a statistically signifi-
cant difference between the instruments. The initial esti-
mates of pointing error were given in Table III, which shows
the Aztrac to have an error, when converted to distance, cf
about 3.0 meters, while the estimate was about 1.3 meters
for the T-2.

An uncompensated systematic error in the data, due to
the time lag discussed in Chapter III, was discovered when
empirical probability density function plots were made of
the entire data set for each instrument. This led to a
revised estimate of the pointing error for the Aztrac,
because the bimodal distribution caused by this time lag
adversely affected the original estimate for <f& . This
revised estimate is about 1.3 meters, as shown in Table VII,
and is almost the same as the value of <SQ for the T-2
instrument. »hen these estimates for CQ are viewed in light
of the precision of the experiment, as shewn in the table,
it is seen that the actual values of tf9 could be smaller
than indicated, because smaller values would be masked by
the relative imprecision of the experiment. It is to be
concluded, however, that the actual values of the pointing
error of each instrument are no larger than those given
here.

79

The question of a statistically significant difference
between the instruments was then considered using the ANOVA
technique, which can be said to compare a variance component
due to the instruments with a variance component due to the
precision of the experiment. This precision was not very
much greater than the variance of the instruments, but was
based on the most precise positioning method generally
available for hydrography -- an intersection position using
three theodolites. The ANOVA procedure indicated no signif-
icant difference between instruments, but if the precision
of the experiment had been increased, a significant differ-
ence could possibly have been detected. In light of the
subsequent discovery and elimination of the systematic error
due to a time lag, this conclusion of no difference between
theodolites appears to be well justified.

An evaluation of interpolation methods was the second
objective of this thesis, and although the analysis was not
as rigorous as it could be, it can be concluded that there
is a measurable distance between an interpolated position
and a corresponding observed position. This has never been
done for the oase of range-azimuth positioning because the
rapid position fixing available with Aztrac has not been
available. It has been shown, through an error propagation
analysis of the interpolation algorithms, that the interpo-
lated error is not inherently due to the algorithms them-
selves. The error is therefore due to the inability of the
vessel tc follow the range arc, whioh is caused primarily
because of environmental conditions and the vessel opera-
tor's track keeping capability. The distance between the
interpolated and corresponding observed positions may be as
much as two to four meters, as indicated in Table V, and is
roughly twice as great for a position that is computed using
both distance and angular interpolation, as for a position
using angle interpolation alone. It is therefore

80

recommended that, whenever possible, automatic recording of
range data should be used.

The third and most important objective of the thesis is
a comparison of the total position error using these instru-
ments with the required error standards of the major hydro-
graphic survey organizations. The lower half of Table VIII
gives these results, which are that all the standards
considered are indeed met, except the NOS range-azimuth
standard at 1:5,000 scale using a range error of 3.0 meters.
This conclusion requires a very important qualification
regarding the T-2 instrument and the errors encountered
while pursuing the first two objectives. These are errors
due to interpolation, and to the time lag discussed in
Chapter IV.

The approximately one second time lag discovered with
the Aztrac data set is not actually associated with the
Aztrac at all but is associated with the T-2. It appeared
to be a systematic error of the Aztrac in this experiment
only because the reference postions were obtained using T-2
instruments similar to the test T-2. It must be concluded
from the data acquired in this project that there exists,
for any angle measured with a T-2, a time lag of about one
second between angle observations and any measurement made
aboard the vessel, including both automatic and manually
recorded depth and range data. There is then an associated
position error for these measurements, the magnitude of
which depends upon vessel speed, which was about two meters
for the four knot speed used in this experiment. The
conclusions and position accuracies for observed T-2 posi-
tions in Table VIII do not take this additional error into
account. When the error contributions from both the time
lag and interpolation are considered, it can be concluded
that positions interpolated between observed T-2 positions
have an additional error of about two to four meters. Thus

81

tha total actual position error for T-2 positions and posi-
tions interpolated between T-2 positions might fail to meet
more of the standards than are indicated in Table VIII.

Having reached conclusions related to the objectives of
the thesis, another set of conclusions and recommendations
can be made far the Odom Aztrac theodolite, regarding its
ease of use and suitability for range-azimuth hydrography.
The Aztrac instrument was expressly designed for range-
azimuth or azimuth-azimuth positioning, and has features
which are advantageous to the operator of the instrument in
the field. Such advantages are discussed in Chapter I, and
include ease of tracking, because of an upright telescope
image and an infinitely geared tangent screw.

The Aztrac theodolite possesses advantages much more
important than ease cf use in the field, and these addi-
tional advantages are derived primarily from its ability to
be interfaced with a computer aboard the survey vessel.
Sesides eliminating systematic error due to a time lag, a
computer based survey system offers the additional advantage
of being able to measure and record a position every few
seconds. This allows three important advantages over a
system that can only measure positions once per minute.
First, since each position is individually measured, no
interpolation is required and thus better accuracy is
obtained than with a nonautomated system. Second, no manual
data logging is required, which reduces blunders and greatly
increases the speed with which a survey may be processed.
Third, an automated system allows the surveyor to run
straight sounding lines rather than curved arcs, because an
automated system can provide an almost real-time cross track
error indication to the helmsman. Running straight sounding
lines increases survey efficiency by orienting the lines
more normal to the depth contours, and by requiring fewer
total linear miles of hydrography for each survey. A vessel

82

may of course steer straight lines with a nonautomated
system, but it is extremely difficult to maintain the strict
line spacing requirements for hydrographic surveys without
an indication of cross-track error, so curved range arcs are
usually followed.

Disadvantages encountered during this experiment include
additional operator fatigue caused by the requirement to
constantly track the vessel. This is necessary if the
substantially increased data rate available with this
instrument is to be utilized. The extra effort to track the
vessel is more than offset, however, by the elimination of
manual data lagging. Care is required by the operator when
rotating the instrument through an arc, because if the
instrument is moved too rapidly the maximum telemetry data
rate is exceeded and erroneous angle data will be trans-
mitted. This can only be detected by checking the original
initial pointing. Although this problem was observed during
a manufacturers demonstration, it did not occur during the
experimental field work. Finally, the transmitter range of
5 km is rather short for the distances used by NOS, which
can be up to about 10 km. The manufacturer has stated that
the system range can be easily extended by increasing the
transmitter power [ Apsey, 1983]. Although this thesis meas-
ured Aztrac error at a maximum distance of 3 km from the
shore station, the conclusions stated here should not be
blindly extrapolated to increased ranges. Still, if the
Aztrac pointing error standard deviation is reduced to its
angular resolution (0.01 degree) at the very slow angular
speed of the vessel at long ranges, the error appears to
remain acceptable. For example, 0.01 degree pointing error
at a range of 10 km results in an error of only 1.74 meters.

The advantages of the Aztrac clearly outweigh its disad-
vantages, so it is therefore recommended that the Odom
Aztrac system be incorporated into the computer based equip-

83

ment used by NOS. It should be used only in situaxions
where it meets required positioning standards based on the
value for (Te found in this investigation. It is also recom-
mended that the systematic error induced by the time lag
discussed here should be accounted for in operational stan-
dards for range-azimuth positioning using the Wild T-2.
This could be done fcr semi -automated systems by providing
an automatic radio signal to the observer on shore that
precedes the actual depth measurement by one second. A
simpler method of reducing this systematic error could be a
limit on the speed of the vessel, depending on the accuracy
standard required for the particular survey.

A final recommendation must be made regarding the posi-
tioning standards of NOS. At present there are conflicting
standards for a given survey scale, depending on whether
electronic, hybrid (including range-azimuth) , or visual
methods are used. For example, positions for a 1:5,000
scale survey may be required to have an accuracy of either
7.5 meters drrn$ , or 2.5 meters at some unspecified prob-
ability, depending on whether microwave range-range or
range-azimuth methods are employed. The NOS is certainly
the most progressive hydrographic organization with regard
to position error specifications, but it is recommended that
the concept of probability be applied to all positioning
methods and not only electronic ones. Further, if the
meaning of "seldom exceed" in the IHO standards is to be
interpreted as a 90% probability circle, then all the NOS
standards should be changed to reflect that standard.

84

APPENDIX A
MEANS AND STANDARD DEVIATIONS OP ACQUIRED DATA SETS

All values for means and standard deviations are in
units of seconds of arc.

TABLE IX
3000 Meter Arc

I Mills
I

Kenny | Schoraaker I Cherry

X ==> 53

Aztrac s ==> | 257

overall mean = -

306
93

106

66 | -242
76 63

pooled s = 142

T-2

un-
rounded

X ==> j -34
s ==> | 67

overall mean

29
59
23

113

164

| -16
I 60

i

pooled s = 97

T-2
rounded

X ==> | -39
s ==> 75

overall mean

26

62 163

89 pooled s «

-10
57

85

TABLE X
1500 Meter Arc

Mills I Kenny | Schomaker | Cherry
I I

Aztrac j

X ==>

5 ==>

o veral

-124
392

mean

| 476 37
990 | 431

88 pooled s

| -3,

476

584

T-2

un-

ounded

X ==>
s ==>

overall

27
95

mean

-116 | 47
145 j 164

8 pooled s

74

. »l

| 107

T-2

rounded

X ==> 32
s ==> j 102

overall mean

-114 48
145 | 167

11 pooled s

. il

75
106

86

TABLE XI
1000 Meter Arc

I Mills | Kenny
I I

Schomaker | Cherry

Aztrac

X ==> 115
s ==> 482

overall mean

| -116
| 534

-120

768

= 20 pooled s =613

199
711

T-2

un-
rounded

X ==>

5 ==>

overal

-45
286

mean

17
111

125

269

= 11 poolsd s =

-53
196
225

T-2
rounded

X ==>
s ==>

o veral

-49
282

mean

13
| 114

133
274

■ 11 pooled s =

-54
| 199
226

87

TABLE XII
700 Meter Arc

Mills | Kenny | Schomaker | Cherry

X ==>

I

Aztrac j 5 ==>
Dveral!

123
585

mean

-114 | 36

777 | 508
63 pcolad s

208
542

= 619

T-2

X ==>

un- s ==>

rounded j

I overal

-53 | -55
338 | 210

mean = 9 pooled s

= 382

156
681

T-2 |X ==>

rounded I s ==>
| o v e ra 1

-14 j -50 j -60

311 j 342 | 205

mean = 9 pooled s

160
690

= 386

88

Aztrac

TABLE XIII
5 00 Meter Arc

I Mills J Kenny | Schomaker | Cherry
III I

X ==>
s ==>

o v e ra 1

284
1543

mean

<* I

98
1957

I 1423 j

65 pooled s

.,.1

-127
930

0

T-2

un-
rounded

X ==>

3 ==>

overall.

56
710

mean

9

452
57 pooxsd s

. ,i

160
618

T-2 | X ==> | 62

rounded | s ==> | 710

I

overall mean

0 I 15

1041 | 4 55

- 60 poolsa s =

163
623

*15

89

Aztrac

TABLE XIV
3 00 Meter Arc

I Mills | Kenny | Schoraaker I Cherry
ill I

X ==> | 425
s ==> I 2274

overall mean

-200
1605

699
2104

-1219
3516

= -74 pooled s = 2290

T-2

un-
rounded

X ==> | -210 | 320 | 276
■

s ==> | 652 | 933 | 1290

overall mean = 124 pooled s

109
682

868

T-2 X ==>

oundsd s ==>

I overal

-215 | 313
654 j 929

mean = 10 4

I 274
I 1289

pooled s =

43

t

843
902

90

APPENDIX B
GEODETIC POSITION OP HORIZONTAL CONTROL STATIONS

Station Nam*

SOFAR (1947)

USE MON (1978)

MUSSEL (1932)

AZTRAC

T2

GSOCEIVER

Latitude

3

Longitude

36

36'

32". 117

121

53«

24 ".0 04

36

36'

04". 685

121

52

' 35". 9 00

36

37'

18". 151

121

54

1 11 ".6 28

36

36'

32". 530

121

53'

• 25". 310

36

36'

32". 493

121

5 3

• 25". 254

36

36'

3 2 " . 5 1 2

121

53

1 25". 2 86

91

APPENDIX C
EHPIBICAL PROBABILITY DENSITY FUNCTION PLOTS

o

to

o

300 METER ARC

T-2: SOLID LINE
flZTRRC: DASHED LINE

■4000.0 -2000.0 0.0 2000.0

ERROR IN SECONDS

4000

Figure C. 1 Probability Density Plot: 300 a arc.

92

■ —

in

.. 1

o~

500 METER ARC

■^

•

T-2: SOLID LINE
flZTRPIC: DASHED LINE

to

■

l\ t ^

o~

'**> / \ v

/— V

' x / / \ \

>

' \ 1 ' \ \

«-^

! > // \

a. oj

7 \ \

d"

«^

t i \ \

•

o

•S Nw-X^^N. V>-

o_

^ ^

-

1 II

4000.0 -2000.0 0.0 2000.0

1

4000

ERROR IN SECONDS

_ j

Figure C. 2 Probability Density Plot: 500 ■ arc.

93

ir>

o~

700 METER ARC

"**

•

T-2: SOLID LINE
flZTRflC: DASHED LINE

to

•

' \

<-^

>

Q_. ct

\ 1

t

d~

\ J
\ 1

A /

\ 1

\

' v /

l\

•

' x /

1
1

\ / M

o

/ A

1

\ / \ ■* N

/ f \

, ftt /

j J +—'

* /

1 4

/ /

^

1 »

y /

1 \

o

1 \

o

1

1 I

1 1 1

-2000.0 -1000.0 0.0

ERROR IN SECONDS

1000.0

2000

Figure C. 3 Probability Density Plot: 700 m arc.

94

o

to
o

CL cvj

■

O

o

o

1000 METER ARC

T-2: SOLID LINE
RZTRflC: DASHED LINE

.\

-1500.0-1000.0 -500.0 0.0 500.0 1000.0

ERROR IN SECONDS

1500

Figure C. 4 Probability Density Plor: 1000 m arc.

95

LO
O

to
o

o

o
o

1500 METER ARC

T-2: SOLID LINE
AZTRflC: DASHED LINE

-1500.0-1000.0

-500.0 0.0 500.0

ERROR IN SECONDS

looo. n i5on

Figure C. 5 Probability Density Plot: 1500 m arc.

96

O'

CO

o

o

o
o

3000 METER ARC

T-2: SOLID LINE
flZTRFiC: DASHED LINE

t 1 1 1 r 1 1

-SOChffllO. 0-300. 0-200.0- 100. 0 0.0 100.0 200.0 300.0 400.0 500

ERROR IN SECONDS

Figure C. 6 Probability Density Plot: 3000 m arc.

97

BIBLIOGRAPHY

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Inc., Baton Rouge, LA, 1983.

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100

INITIAL DISTRIBUTION LIST

1.
2.
3.

4.

5.

6.

7.
8.

9.

10.
11.
12.

Defense Technical Information Canter
Cameron Station
Alexandria, VA 22314

Librarv, Code 0 142

Naval Postgraduate School

Monterey, CA 93943

Chairman (Code 63Mr)
Department of Oceanography
Naval Postgraduate School
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LCDR Gerald B. Mills, NOAA (Coda 58mi)
Department of Oceanography
Naval Postgraduate School
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Fleet Numerical Oceanography Canter

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Director Naval Oceanograohy Division
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Naval Oceanography Command

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Naval Ocaan Rasearch and DeveloDment Activity

NSTL Station

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Chairman, Oceanography Departaant
U. S. Naval Academy
Annapolis, MD 2 1402

Chief of Naval Research
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Director (Code PPH)

Defense Mapping Aaency

Bldg. 56, U. S. Naval Observatory

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13. Director (Coda HO)

Defense Mapping Agenoy Hydrographic

Topographic Center
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15. Director, Charting and Geodetio Services (N/CG)
national Oceanic and Atmospheric Administration
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and Trainma (NC2)
National Oceanic and Atmospheric Administration
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NOAA Ship FAIR WEATHER
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NOAA Shio DAVIDSON
Pacific Marine Center, NOAA
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NOAA Ship WHITING
Atlantic Marine Centar, NOAA
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102

26. Commanding Officer
NOAA Ship PEIRCE

Atlantic Marine Center, NOAA
439 West York Street
Norfolk, VA 23510

27. Chief. H? drographic Field Parties section
MOA 233

Atlantic Marine Center, NOAA
439 West York Street
Norfolk, VA 23510

28. Maureen R. Kenny, LT, NOAA
N/MOP 21x2

7600 Sand Point Way N2
Bin C15700 Blda. 3
Seattle, WA 98115-0070

29. LT David A. Waltz, NOAA
147 5 Soring Meadow Lane
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30. LT Christine W. Schomaker, NOAA
NOAA/NGDC E/GC3

325 Broadway
Boulder, CO 80303

31. IHO/FTG International Advisory Board
International Hydrograohic Bureau
Avenue President J. F. Kennedy

Monte Carlo, Monaco

32. LT Athanasios E. Palikaris
72 Timotheou Street -
Athens Tr512

Greece

33. Mr. Dunny Green
SMC 2790 NP5
Monterey, CA 93943

34. LCDR Lima Williams, Vsnezuelan Navy
SMC 1070 NPS

Monterey, CA 93943

35. Ing. Walter Herrera

Instituto Nacional de Canalizaciones
Zona Industrial de la Ferrominsra
Puerto Ordaz
Estado Bolivar
Venezuela

103

352

Thesis
W22528
c.l

Waltz

An error analysis of
range-azimuth position-
ing.

r .

"Tiesis
W22528
c.l

Waltz

An error analysis of
range-azimuth position-
ing.