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A GEOMETRIC APPROACH TO DETERMINATION OF SATELLITEEPHEMERIS OVER A LIMITED AREAby

Keith R. Thackrey

Thesis submitted to the Faculty of theVirginia Polytechnic Institute and State University

in partial fulfillment cf the requirements for the degree of

MASTER OF SCIENCEin

Civil Engineering

APPRDVED: grzäé v} 2 ES. D. Johnscn, Co—ChairmaWOP.

J. -ell, Co-Chairman T. Sc er

December, 1988

Blacksburg, Virginia

A GEOMETRIC APPROACH TO DETERMINATIONOF SATELLITE EPHEMERIS OVER A LIMITED AREA

byKeith Richards Thackrey

Committee Co-Chairmen: Steven D. Johnson, Patrick J. FellCivil Engineering

(ABSTRACT)

Range and interferometric observations have been

examined for their potential application in a geometric

approach to determination of satellite ephemeris. The

approach differs from the normal (dynamic) approach in that

(_ each satellite position is treated as an independent statety variable or benchmark.

F%Ä Programs have been developed that simulate and formatR’ the input data for the least squares estimation routines,

{Siand perform statistical analyses of those results. Random

errors, tropospheric refraction errors, and atomic clock

errors have been considered, and the range observation

adjustment program directed to solve for clock errors.

Tests have been conducted, using different error

sources, and varying the quality of the initial estimates of

the satellite positions, to examine the sensitivity of the

programs to various parameters.

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To test the accuracy of the solution, the results were

compared to the true satellite positions, which were used

to generate the input data. Interpolations to determine

positions at intermediate times have also been performed,

and compared with true values.

The results support the use of the geometric approach

for satellite positioning.

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TABLE OF CONTENTS

ÄBSTRAÜT•••••••••••••••••••••••••••••••••••••••••••••••ACm¤WLEDGEMENTS•••••••••••••••••••••••••••••••••••••••

FIGURES••••••••••••••••••••••••••••••••••••••••

TABLES•••••••••••••••••••••••••••••••••••••••••1•

INTR¤DUCTION••••••••••••••••••••••••••••••••••••••• 1

1.1 Satellite Ephemeris Determination............ 1l•2 Objectiv€•••••••••••••••••••••••••••••••••••• 31•3 Sc¤pe•••••••••••••••••••••••••••••••••••••••• 3

1•42•

BÄ€KGROUND••••••••••••••••••••••••••••••••••••••••• 6

2.1 Global Positioning System.................... 6

2.1.1 The Satellite System.................. 62.1.2 Observation Types..................... 7

2.2 Previous Studies............................. 9

3• PRÜCEDURE••••••••••••••••••••••••••••••••••••••••••3•1

Data•••••••••••••••••••••••••••••••••••••3.2

The Software Programs........................ 17

3•2•l3•2•2RNGB••••••••••••••••••••••••••••••••••3•2•3

PHASE•••••••••••••••••••••••••••••••••3•2•4

PÜSDAT••••••••••••••••••••••••••••••••3.3

Observation Equations........................ 33

3.3.1 Range Observations.................... 353.3.2 Phase Observations.................... 39

3.3.2.1 Undifferenced Phase

3.3.2.2 Single Difference Phase

3.3.3 Observation Errors.................... 47

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IPI

3.4 The Normal Equations......................... 524• ANALYSIS•••••••••••••••••••••••••••••••••••••••••••4•l

R€SultS•••••••••••••••••••••••••••••••••4•2

R€SUltS••••••••••••••••••••••••••••••••5•

• • •••• • •• • •• • •• ••••5.1

Conclusions.................................. 745 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

REFERENCES•••••••••••••••••••••••••••••••••••••

A•••••••••••••••••••••••••••••••••••••••••••A.1

Source Code Listings......................... 79A.2 Sample Run and Data.......................... 161

A• 2 • 1 • • • • • • • • • • • • • • • • • • • • • • • • • •

•A.2.2Program Runs.......................... 168A• 2 • • • • • • • • • • • • • • • • • • • • • • • • • • • •

2VITA•••••••••••••••••••••••••••••••••••••••••••••••••••vi

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LIST OF FIGURES

3.1 GPS Ground Track................................. 15

3.2 Structure of the Normal Matrices................. 56

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LIST GFTABLES3.1

GPS Orbit Parameters............................. 13

3.2 Ground Stations.................................. 16

4.1 Run Descriptions................................. 64

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CHAPTER 1INTRODUCTION

1.1 Satellite Ephemeris Determination

The classical approach to satellite orbit estimation

employs a dynamic model of the satellite orbit to find its

position and velocity at any given point in time. To

produce the necessary accuracies for most applications, the

force model needs to consider, among other things, the

gravitational model of the Earth, yielding an extremely

complex mathematical solution requiring a great deal of data

and numerical calculation.

The geometric approach yields a much simpler solution

to the problem of ephemeris determination. Ephemerides can

be derived with comparable accuracies, in some cases

considerably better than those of the dynamic approach.

There are, however, certain constraints, which prevent

universal application of the geometric approach. In those

cases where it can be used, it offers some advantages.

A dynamic model is a time varying force function of the

orbital elements, with a gravity model which normally is

described by a set of spherical harmonic coefficients. The

degree and order of the harmonic function will determine the

sufficiency of the model. As a result, the complexity of

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the problem increases significantly when a higher degree of

accuracy is required.

The geometric approach treats each satellite position

as an independent benchmark in space. By using a network of

precisely known ground locations, each satellite position

may be determined independent of any other, presumably with

a high degree of accuracy. In this case, the accuracy is

less a function of the complexity of the model, and more a

function of the knowledge of the ground positions and the

accuracy and density of the observations. The equations are

not necessarily time varying, nor do they require knowledge

of the gravity field. If intermediate positions are needed,

they can be generated using some type of interpolation

scheme.

There are some restrictions to the geometric approach.

It requires a network of accurately known ground station

locations, or at least accurately known baselines between

the stations. Since many of the stations must be visible to

the satellite simultaneously, this usually restricts the use

of the geometric approach to a limited geographic area. Of

course smaller areas may be tied together, but it is

extremely difficult to tie continents together across the

oceans using the geometric approach, especially using

A satellites of low altitude.

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For low orbit satellites, the ground network can only

cover a small area and still provide visibility. To perform

simultaneous positioning over larger areas, a higher orbit

satellite must be used, such as those of the Global

Positioning System.

1.2 Objective

In order to determine the utility of the geometrio

approach, the model will be tested against a known system of

ground stations, satellite positions, and observations with

their associated errors. Using the geometrio approach in

simulation, with individual observations, the accuracy of

derived satellite positions will be compared with the "true"

positions. The sensitivities of the models to such factors

as the quality of the initial estimates, a priori weights

(including correlation of variables), addition of error

sources, and adding additional unknowns (parameters) to the

model, will also be determined. The result will be a

demonstration of the hypothesis that the geometric approach

actually can replace the dynamic approach for certain

applications.

1.3 Sggpg

To study the characteristics of the geometrio approach,

a network of precisely known ground stations and simulated

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satellite positions was considered over North America.

Observations were generated, varying from perfect

observations, to observations containing such major error

sources as random error, tropospheric refraction, and clock

error. Initial estimates were varied to determine

sensitivities, giving a general idea of the strength and

flexibility of the geometric solution. All observation

types were not considered, nor were all error sources.

Alternative models might prove more effective for certain

applications, but were not examined.

1.4 Organization

Chapter 2 outlines some useful background information

for satellite positioning with an overview of the Global

Positioning System (GPS) of satellites presented in Section

2.1.1, and the observation types supported by GPS presented

in Section 2.1.2. Some literature related to satellite

positioning is reviewed in Section 2.2.

Chapter 3 describes the procedures employed, including

a description of the simulated data, a discussion of the

programs used, and finally an explanation of the algorithms

and the mathematical model employed by the geometric

solution.

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An analysis of the results is included in Chapter 4.

Also in that Chapter is a discussion of some of the problems

encountered during the data processing.

A list of the conclusions is given in Chapter 5, along

with recommendations for implementation and possible further

studies.

All source code for the programs used, is contained in

the appendix.

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CHAPTER 2BACKGROUND

2.1 Global Positioning System

2.1.1 The Satellite System

The Global Positioning System (GPS) network of

satellites, according to Fell [1986], will be a system of 18

satellites in six orbital planes, designed to provide at

. least four satellites in view world—wide, at any instant in

time. Stansell [1978] stated that employing range and

range-rate measurements, GPS will provide navigational

accuracy and availability well in excess of that provided by

the Navy Navigation Satellite System, or Transit System,

which it is intended to replace.

The six orbital planes of the GPS are inclined at 55

degrees, and three satellites will be equally spaced within

each plane, according to Fell [1986]. Currently, a reduced

constellation is in place which provides four satellites in

view at various times during the day.

The satellites fly in a high orbit with a semi-major

axis in excess of 25,000 km. This reduces the effect of

local variations in the gravity field on the orbit, and

increases the time that a given satellite is in view, as

well as the area on the ground which can view the satellite

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*at a given time. Satellite 4 on day 17, 1983, was in viewover much of North America for more than four and one—half

hours. The satellite contains two high precision clocks, a

cesium clock and a rubidium clock, designed to reduce the

synchronization error between the satellite and a uniform

time scale.

Fell [1980] described the broadcast frequencies of the

GPS. The satellite broadcasts on two L band frequencies.

The frequencies are 1575.4 MHz and 1227.6 MHz, known as L1and L2 respectively. L1 has a pseudo random noise sequence,

known as the P code, which has a repetition rate of 38

weeks. L2 also has a pseudo random noise sequence, known as

the C/A code, but with lower frequency and a repetition

cycle of 1 millisecond.

2.1.2 Observation Types

The GPS yields three types of observations. The first

is a range observation, produced using the P code to

calculate the time for a particular pulse to travel from the

satellite to the ground. This observation type is highly

dependent on the clock synchronization error between the

satellite and the ground station.

The other two observation types are more in the nature

of range difference observations. Doppler observations are

based on the shift in frequency (known as the Doppler shift)

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due to the motion of the satellite relative to the observer.

The shift is the difference between the observed frequency,

with errors removed, and the broadcast frequency, and is

caused by the fact that the distance between the satellite

and the receiver is changing. Doppler observations are

based on the frequency shift over a time interval, and since

they are dependent on the range, a range difference equation

over the interval can be developed.

The final observation type is the phase, or

interferometric observation, and can take two forms. One is

the measurement of a single pulse at two geographically

separated receivers. The difference between the times of

receipt of the pulse at the two stations, after other errors

are removed, is due to the difference in the ranges between

each station and the satellite. This requires some type of

interference to make a single pulse recognizable. The

second form results from the measurement of the phase of a

continuous signal from a single satellite and takes the form

of a biased range measurement rather than a range

difference. In this case there are two considerations at

any given time, the phase of the wave at that point in time,

and a whole cycle count from some initial time. Using these

two modeled quantities, an integer wave count for the

initial observation can be determined. Multiplying the

number of waves by the wavelength and combining this with

_ 9

the measured phase will give the range at each observation

time.

2.2 Previous Studies

Numerous articles and papers have been written on the

subject of satellite positioning, and in particular, using

GPS. Fell [1980] was one of the earliest investigations of

the geodetic positioning applications of GPS. His

dissertation presented three models for positioning, using

range, integrated Doppler, and interferometric observations

with dynamic models to perform ground station positioning

and baseline determination. The study also looked at the

relative strengths of the different models, rank

deficiencies and the sensitivities due to different error

sources.

Each of the observation types was found to have

advantages for certain applications. In particular, the

range solution showed the greatest geometric strength, and

the range and phase solutions were most suited to shorter

tracking intervals, while the Doppler observations were

suited to longer tracking intervals. In general, depending

on the errors in the known satellite position, ground

locations could be determined to an accuracy of less than 2

meters and baselines in the neighborhood of 10 centimeters.

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A number of papers were presented on the subject of

satellite positioning at the Fourth International Geodetic

Symposium on Satellite Positioning held in Austin, Texas in

1986. This was in fact, the primary purpose of the

symposium. Wu, et al. [1986] presented a consolidation of

two earlier studies on orbit determination of GPS and of

LANDSAT—5. The first study, conducted in March-April 1985,

was presented by Davidson, et al. [1985] on the fiducial

network concept, using ten sites in the continental United

States. The second experiment was conducted in November

1985 and was also presented at the Fourth International

Geodetic Symposium, by Davidson, et al. [1986]. The results

of the two studies indicated that orbits for both satellites

can be obtained with true errors under 2 meters.

Several other studies were presented on orbit

determination. Abbot, et al. [1986] performed a comparative

study using March 1985 data, as did Abusali, et al. [1986].

Both studies varied some of the parameters, but obtained

comparable results.

Beutler, et al. [1986] presented a paper employing

March 1985 test data to perform orbit determination using

double differenced phase observations. Eren and Leick

[1986] looked at triple differenced phase observations and a

1983 test data set over the same area. Qifeng [1986]

studied single differenced phase observations for orbit

11

[ determination, using simulated data over China. All of the

studies attempted to make use of the more precise

interferometric observations, in spite of the lack of

geometric strength of the solution, and all were able to

demonstrate a high degree of positioning accuracy under

their controlled circumstances.

Two other studies looked at alternate ground networks

with similar success to that of the previously described

papers. The topic presented by Thornton, et al. [1986]

described networks in Mexico and the Caribbean. Landau and

Hein [1986] presented their results of a feasibility studyfor a European GPS tracking network.

A variety of other studies have been performed, and

presented at this and other conferences. The papers of the

Fourth International Geodetic Symposium on Satellite

Positioning represent a good cross—section of the available

literature.

All of the studies deal with some type of correlation

between satellite positions. While there is a fair amount

of literature on the geometric approach, the literature

directly addressing the geometric approach applied to the

GPS is somewhere between scant and non—existent.

CHAPTER 3PROCEDURE

Two approaches were examined, one using range

observations and the other using phase observations. Whilethe basic problem to be solved is the same, there are many

peculiarities to programming the phase observation

equations, so they will be treated separately. In addition

to the two reduction programs, there are two dataprocessors, a pre-processor which builds the simulated data

sets, and a post—processor which performs some statistical

analyses on the output data.

Chapter 3 is an examination of the approach used to

study the problem. Section 3.1 will look at the setting for

the simulation study. The programs and their subroutines

will be described in detail in Section 3.2. Section 3.3will contain a description of the observation equations

employed, as well as a brief discussion of some alternative

observation equations. An explanation of the algorithms

used to carry out the least squares adjustment is contained

in Section 3.4.

3.1 The Data

The simulation data is derived from an orbit generation

routine using actual GPS orbit parameters. Table 3.1 lists

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Table 3.1GPS Orbital Parameters

Satellite 4, Day 17, 1983

Parameter yglgeSemi-major Axis 26,561.983 km.

Eccentricity .003187

Inclination 630 13' 45.46"

Argument of Perigee -1520 39' 39.68"

Right Ascension of 1720 25' 18.99"the Ascending Node

Mean Anomaly 1550 50' 47.56"

14 '

the orbital parameters for the satellite, which represent

satellite 4 on day 17 of 1983. A listing of the ephemerides

at five and fifteen minute intervals was generated, and

using a display routine on an HP9845, Figure 3.1 was

generated. This figure shows graphically at which time the

satellite was first and last in view (assuming an elevation

of 100 above the horizon) at the five ground stations listed

in Table 3.2, simultaneously. The programs used to generate

this data were developed in support of other work, and thus,

they will not be discussed in detail.

The five ground stations are actual satellite tracking

stations in North America. Two of these stations, the ones

located at Richmond, Florida and Westford, Massachusetts,

are a part of the Cignet network, a worldwide network which

provides continuous tracking of GPS satellites. The station

at Ft. Davis, Texas was originally a part of the GPS

monitoring system, but is no longer used for that purpose.i The station at Owens Valley, California was used in the

early testing stages of the GPS system, and together with

the stations at Ft. Davis and Westford, formed what was

known as the Iron Triangle. The final station, located in

Vancouver, British Columbia, contains a satellite receiver,

although it is not a part of the GPS network. The five

stations were chosen because they form a strong network of

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Table 3.2Ground Stations

Q

Station x(km) y(km) z(km)

Richmond, Fla. 961.374 -5673.945 2740.832

Ft. Davis, Tex. -1324.339 -5331.448 3232.978

Westford, Mass. 1492.211 -4458.116 4296.000

Owens Valley, Cal. -2409.585 -4478.332 3838.587

Vancouver, B.C. -2287.436 -3493.350 4805.126

All coordinates are geocentric coordinates referenced

to the WGS-84 coordinate system.

17

control stations, giving good geometric coverage of North

America.The locations of the stations, given in Table 3.2, are good

to an accuracy of 1 m. in each component, with the exception

of Vancouver. The coordinates used for that station were

rounded to the nearest minute on both latitude and

longitude, but will be treated as if known to the same

degree of accuracy as for the other stations. The baselines

between the stations are known to about 10 cm. For

satellite positioning purposes, the ground stations are

given a 10 cm. accuracy, as their relative positions are

used to determine the locations of the satellite relative to

the network. The position of the whole system will then

have about a 1 m. bias.

3.2 The Software Programs

There are four programs for processing the data, a pre-

processor called BUILD, two least squares processors called

RNGE and PHASE, and a post—processor called POSDAT. Each of

these programs will be described separately. The source

code listings for all of the programs and their associated

subroutines are contained in Appendix A.1, and the

mathematical models behind the programs are described in

Sections 3.3 and 3.4.

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g 3.2.1 DQTLQ

BUILD is a pre-processor which takes the ground station

positions and satellite positions, simulates and thenreformats the data to be used by both of the least squares

processors. The program computes both types of

observations, can scale the satellite positions to put some

error on the initial estimates, and can generate three types

of error and place them on the observations.

Random error can be generated, with a gaussian

distribution and specified standard deviation. A 1 m. errorwas placed on the range observations, consistent with what

is generally obtained in practice. An error of about one-

half of a pulse, or 10 cm., was placed on the phaseobservations, again consistent with actual observations.

Atmospheric refraction error is generated using the

Hopfield model for tropospheric refraction. Anderle [1974]

presents a good description of the Hopfield refraction

model. Default values of 15° C for local temperature, 980

mbars local atmospheric pressure, and 50% relative humidity

were used. On the assumption that most of the refraction

error can be modeled and removed from the observation in

actual practice, only 5% of the computed error is actually

added into the observations.

The program generates random correlated clock error

known as flicker noise, using a specified Allan variance,

19‘

for either a cesium or a rubidium atomic clock. A cesiumclock is generally used for both the satellite and the

ground stations, treating the clock at the first station,

that in Richmond, Florida, as truth. The clock error is

then computed by converting the flicker noise to a range

error, and subtracting the satellite clock error from the

ground station clock error, for each station and each

satellite position. Any bias and drift can be removed from

the computed error.

The driver program is titled BUILD, and like all of the

routines, was written in the FORTRAN 77 version FORTVS2,

which is resident on the IBM 360/370 mainframe at Virginia

Polytechnic Institute and State University. The driver

calls the subroutines GAUSS and RANDOM, which are general

subroutines, accessed by several routines, and the

subroutine OBSBLD, which is specific to BUILD. GAUSS is

designed to generate random errors with a gaussian

distribution and RANDOM is a random number generator.

OBSBLD is the heart of the pre-processor. Its purpose

is to build the observations for the range and phase

processing routines. Using the known ground station

locations and the "true" satellite positions, the routine

computes the ranges for each station, at each discrete time

of interest. Upon request, the routine will add random

error, atmospheric refraction error, and/or atomic clock

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error to the true ranges. Using these ranges the program

will then compute the phase observations, and the number of

whole waves counted since the first observation, at time to.Finally, OBSBLD will compute a mean and standard deviation

of the different errors, and of the total errors, and print

the results.

OBSBLD calls the subroutines LSTBLD, REF, and GAUSS.

It receives as input, the station locations and satellite

positions, along with support data such as the frequency of

the satellite broadcast, the time between observations, the

standard deviations for the different observation types,

which are used in computing the random error, the seed for

the random number generator, and the time since the last

calibration of the ground station and satellite clocks,

which is used in computing the clock error. The subroutine

also receives instructions as to whether to compute the

different errors.

On output, the routine gives the observations and the

total means and standard deviations of the errors.

Additionally, the random number seed, which has been

updated, is returned to the main program.

REF was developed at the Naval Surface Warfare Center

(NSWC) on a CDC computer and converted for use by BUILD on

the IBM. REF generates atmospheric refraction using the

Hopfield refraction method, which is discussed more fully in

21

Anderle [1974]. It takes as input, the distance from the· center of the earth to the ground station and the zenith

angle of the satellite, and outputs the amount of refraction

as a range error.

LSTBLD was created to extract clock error when called,

and upon request, to filter any bias and drift from the

generated error. The subroutine computes a linear least

squares solution to determine the bias and drift of the

clock errors, then removes each from the computed clock

errors. This process assumes that, in actuality, the bias

and drift of the clock errors could be modeled, knowing the

type of clock, and the time since its last calibration,

leaving only the random and higher order error components.

LSTBLD calls MAINV, MAMULT, and FLICKE, all of which were

adapted from existing programs.

MAINV and MAMULT are modified programs for matrix

manipulation, which are used often in the programs. MAINV

computes the inverse of a matrix. MAMULT computes the

product of two matrices.

Like REF, FLICKE was developed at the NSWC and adapted

for use by BUILD on the IBM. It generates clock noise using

an Allan variance. FLICKE computes both a fractional

frequency error and a range error, determined for either a

cesium or a rubidium clock. The errors can also be computed

for either ground station or satellite clocks. BUILD

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assumes that the first ground station is the master clock,

and calls FLICKE twice, once to compute a set of errors for

the other four stations, and a second time to compute a

separate set of errors for the satellite clock.

FLICKE receives the number of ground stations and

satellite positions for which the error is to be determined,the time between observations, the time since the lastcalibration of the clocks, and a flag which says whether the

satellite clock is a cesium or rubidium clock. On output

are two arrays, one containing the flicker noise for each

observation, and the other, the range error.FLICKE calls a package of subroutines, including STM,

MATSM, COVMAT, with its entry point COVSET, PACK, MAMSNG,

and RANDOM, which has already been described. With the

exception of RANDOM and MAMSNG, all of these were created

for FLICKE at the NSWC, and will not be described further.

A good description of the subroutines can be found in Cadzow

[1970]. MAMSNG is a single precision version of MAMULT.

3.2.2 ggg;

RNGE is the main thrust of the work. The program is

designed to take range observations and known ground control

and solve for satellite positions. Currently, the ground

station location uncertainties are 10 cm. The program can

r23 g

also, upon request, solve for a clock bias and drift. The

weights for those variables are taken as input.

RNGE can either accept satellite position estimates, or

by performing a smaller least squares adjustment, can

generate initial estimates for the satellite positions. Theobservation weights can be input as a constant, or as a

matrix containing a different weight for each observation.The program can solve for a maximum of five ground

stations, with three positional and two clock variables, for

a maximum of twenty-five parameters. It can also solve for

a maximum of thirty satellite positions, with three

variables per position, yielding a maximum of ninety

satellite variables, for a total of 115 variables. The

program employs a partitioning scheme which preventsinversion of any matrix larger than 25x25. All of the

normal equations, and the partitioning scheme are described

more fully in Section 3.4.The driver is titled RNGE. It calls the subroutines

RGETDT, CONSTR, RPARTL, TILDA, MATEQ, and LEAST.

RGETDT collects the input data for RNGE. The routine

reads the number of ground stations and satellite positions,

and tests to be sure there are at least four ground stations

in view while not more than five, and at least four

satellite positions with not more than thirty. It then

24

reads the ground station coordinates and the range

observations.

The subroutine also reads several records

conditionally, depending on the value of flags contained in

the input file. If instructed to solve for the clock

parameters, RGETDT will read the sigmas for the clock bias

and drift. Next the observation weights are read, as either

a single value if the weights are constant, or as an array

_ of standard deviations if they vary. The program will not

read any off-diagonal elements. RGETDT also converts all of

the sigmas to weights.

Two flags are read, concerning the satellite positions.

The first determines whether to generate covariance matrices

for the satellite positions. The second instructs the

program whether to generate initial position estimates, or

to read them. If so instructed, RGETDT reads the satellite

positions. Finally, it reads the time between observations.

All times are in seconds and all distances are in

kilometers.

RGETDT calls the subroutine POSITS, while getting no

input data from the calling routine. Through the call line,

RGETDT returns the delta time, the range observations, the

weights, and the flags which are needed by the rest of the

program. The ground station data is passed in one common

block, and the satellite data is passed in another.

25

POSITS is called by RGETDT when the operator has

instructed the program to compute initial estimates for the

satellite positions. The subroutine uses three of the

ground stations, and three iterations to perform a small

least squares solution on each satellite position

independently. This gives rough estimates for the

positions, which improve over time, because the final

estimate of the previous position is used as the initial

estimate for the current solution.l

POSITS calls MAINV and MAMULT, and receives the

observations and ground station information as input, along

with the number of satellite positions. The output data

contains the estimates of the satellite positions.

CONSTR is called by RNGE to add the ground station

constraints to the normal equations. If solving for a clock

bias and drift, those weights will also be added by CONSTR.

The subroutine calls MATEQ, and it calls NORMAL, with

its entry points NITIAL and FILL. It receives as input, the

number of ground stations, and satellite positions, the

number of variables per ground station, and the bias and

drift weights. It returns N-dot, T—dot, and N-tilda, the

matrices for the normal equations.

MATEQ is called to move matrices from one array

location to another. It receives as input, the two arrays,

the starting row and column for the matrix within each

II

26

array, the row dimension of each array, and the row and

column order of the matrix.

NORMAL is called to add observations to the normal

equations, which are described in Sections 3.3 and 3.4. The

subroutine was modified from one in another program. It

receives the normal matrices, the observation partials and

the weights, and the locations within the matrices of the

non—zero elements. Normal returns the updated normal

equations.

NITIAL is an entry point of NORMAL. Its purpose is to

clear the normal matrices before processing begins. It

receives the matrices to be cleared and their rank, and

returns the cleared matrices.

FILL is the other entry point for NORMAL. Its purpose

is to fill in the lower triangular portion of an upper

triangular matrix, forming a symmetric normal matrix. The

subroutine receives the upper triangular matrix, and its

rank, and returns the symmetric matrix.

RPARTL is a workhorse of RNGE. It is called once for

each satellite position and computes the partial derivatives

for the observation equations, which are described in

Section 3.3. The subroutine then loads the partials into

the normal matrices, using NORMAL along with NITIAL and

FILL. Most of the work is performed in temporary locations,

and transferred to the universal arrays using MATEQ.Ith

'lmu

Se.-----................................;...........................-.-.----

27

also computes the B and F matrices which are used to

calculate the residuals. RPARTL also calls MAMULT, which is

discussed in Section 3.2.1.The input data for RPARTL includes the time and number

for each satellite position, and its associatedobservations. The routine also receives the number of

ground stations and their locations, and the same

information for the satellite positions. The number of

ground station variables, the observation weights, and the

bias and drift values, as well as a flag indicating whether

to use the clock variables, are also included among the

input data. The output data includes the B and F matrices,

and the different normal matrices, as described in Section

3.3.

TILDA is employed to build the N-tilda and T—tilda

matrices. It is called for each satellite position,

immediately after RPARTL. The subroutine calls MATEQ, as

well as MAINV and MAMULT, which are explained in Section

3.2.1.

The input data includes the satellite position number,

the number of ground variables, and the normal matrices

built by RPARTL. It outputs the tilda matrices of

Section 3.3.

The other workhorse of RNGE is LEAST. The purpose of

this routine is to perform the least squares adjustments and

28

update the unknowns. The least squares adjustment for the

ground stations is actually computed in MSOLVE, and for the

satellites, in SATSLV. LEAST also calls RESID to compute

the residuals.

Among the input data for LEAST is the number of ground

variables. The normal matrices, the delta matrices, the

number of variables per ground station, the observation

weights, and a flag indicating whether to compute the

satellite position covariances, are also passed to LEAST.The ground station and satellite position information isobtained through common blocks. The output data includes

the updated positions and delta matrices, and the weighted

variance of the solution.

MSOLVE was created to solve the equation:

N * A = U (3.1)

for A. It is employed by RNGE to solve for the ground

station variables, due to the partitioning used by the two

reduction programs. MSOLVE calls MAINV and MAMULT.

The input data consists of the rank of the N matrix,

and the N and U matrices, which are the matrices of the

normal equations from LEAST. The output contains the

updates to the ground station unknowns.

29

SATSLV is also required due to the partitioning, andsolves for the satellite position unknowns. The subroutine

calls MATEQ and MAMULT. The inverse of the N matrices are

passed to SATSLV, as are the rest of the normal equations,

the number of satellite positions, the number of ground

station variables, and the ground station updates. The

updates to the satellite position unknowns are returned to

LEAST.

RESID is the last subroutine called by LEAST. It is

used to compute the observation residuals and the weighted

variance of the solution, and upon request will compute the

satellite position covariances. It calls MATEQ, MAMULT,

MAINV, and HDNG.

RESID receives the number of ground variables,

satellite positions, ground stations, and variables perground station. It also receives the observation weights, aflag directing the program whether to compute the satellite

covariances, and the B, F, and delta matrices used in

computing the residuals. Finally, it receives the different

N matrices. The subroutine returns the weighted variance,

and prints out the residuals and the covariances.

3.2.3 Qgggg

Work on the phase reduction program was halted at a

relatively early stage, when it became apparent that the

I

30 Iapproach was flawed. Consequently, the data file created by

BUILD does not match what is expected for the program. The

possible reasons for the failure, and recommendations for

future work are contained in Chapters 4 and 5.

The program is called PHASE, and uses undifferenced

phase observations to compute satellite positions. The

different type of observation equations which support

interferometry, including the undifferenced observations,

are described in Section 3.3.

The program can solve a maximum of five ground

stations, with three positional unknowns, and an unknown

integer number of whole waves between the first satellite

position and each ground station. This yields four unknowns

per station for a maximum of twenty ground variables. The

biases and drifts of the clocks were never added to the

program. The program can also solve for up to thirty

satellite positions, with three unknowns per position for a

maximum of ninety satellite unknowns, and a total of 110

unknowns.

The driver is titled PHASE. It calls GETDAT, COMPUT,

CONSTR, PARTIL, TILDA, and LEAST. The processing performed

by CONSTR, TILDA, and LEAST is explained in Section 3.2.2.

GETDAT performs much the same function for PHASE as

RGETDT does for RNGE. The observations are phase Iobservations, which should not exceed 2N, and a number of

I

II

31 {

I

whole waves counted since an initial observation. The other

item which is unique to PHASE is the frequency of the pulse.

From that frequency, GETDAT also calculates a wavelength.

GETDAT will not compute initial estimates as was the case in

RGETDT.

Like RGETDT, GETDAT receives no data from the main

driver, and the common blocks are the same. The output data

on the call line include the observations, the frequency,

the time between observations, the wavelength, the a priori

sigmas and weights, and a flag directing whether to compute

the satellite covariances.

COMPUT is used to compute the estimated ranges using

the ground station information, along with the estimated

satellite positions. The subroutine also computes the

difference between the number of pulses transmitted by the

satellite and the number of pulses received on the ground,

since the receiver began receiving.

COMPUT receives the frequency and time between

observations, along with the observed number of waves, and

the ground and satellite positional information. It returns

the computed ranges, and the computed pulse deltas, which

are defined in Section 3.3.2.1 and by Equation 3.28.

PARTIL is the sister subroutine of RPARTL. While the

partial derivatives are slightly different, and there are

some different variables involved, the processing is

32

essentially the same. PARTIL calls NORMAL and its entry

points NITIAL and FILL, which are discussed in Section

3.2.2. It also calls MAMULT, which is first addressed in

Section 3.2.1, and MATEQ, which is first described in

Section 3.2.2.

The input for the routine includes the positional

information, the information calculated by COMPUT, the

observed phases, the observation weights, and the

wavelength. PARTIL also receives the number of whole pulses

computed from the ranges of the first satellite position

estimate. The other input data are the number of ground

variables and the number of variables per ground station.

The output data contains the normal equations and the

observation equations.

3.2.4 POSDAT

The post—processor for RNGE is called POSDAT. Its

purpose is to take the satellite positions generated by RNGE

and compare them to truth values. If there are intermediate

truth positions available, the program will also compute

intermediate positions, using a seven point formula of a

Lagrangian interpolation. The formula is

6 lk(x)f(x) ¤ L6 = käo qäifk (3.2)

33

where

l0(x) = (x—xl)(x—x2) ... (x-x6)

lk(x) = (x—x0) ... (x—xk_l)(x-xk+l) ... (x-x6)

l6(x) = (x-xO)(x-xl) ... (x-x5)

POSDAT takes the computed values from RNGE and computes

a root mean square (RMS) of the x, y, and z component

differences from the truth values. It also computes an RMS

for the differences between the interpolated positions and

the truth values.

The driver is called POSDAT. The input data includes

the computed satellite positions, and the true satellite

positions. The program also gets the number of positions

and the time between positions for both the computed and the

true satellite positions.

The computed positions are output first. This is

followed by the true positions and any interpolated

positions. Finally, the RMS values are written to the

output file.

3.3 Observation Eggations

Section 3.3 contains a discussion of the observation

equations which can be employed to perform orbit

determination for the GPS. In addition to the basic

34

observations, the partial derivatives will be examined, and

the rank deficiencies for the equations actually used by

RNGE and PHASE.

All observation equations used, employ the special case

of the adjustment with conditions only, whose matrix form is

defined by Mikhail [1976] as follows:

Av + BA,= f (3.3)

with

f:d··A1 (3.4)

In the programs, A represents the partials of the functions

with respect to the observations, v is the residual vector,

B represents the partials of the functions with respect to

the unknowns, A is the difference between the latest

estimate of the unknown and its updated value, d is a column

vector of constants, and 1 represents the observations.

Since the equations each contain only one observation, the A

matrix is the identity, yielding what Mikhail [1976] refers

to as an adjustment of indirect observations. The equation

is solved for A, which is then added to the last estimates

for the unknowns.

35

Doppler observations yield stronger solutions when

continuous observations are taken over a long period of

time. Conseguently, they were deemed less amenable to the

geometric approach and there was no attempt to include them.

Range observations are discussed in Section 3.3.1, with a

look at the interferometric equations in Section 3.3.2. An

examination of the error sources, and their inclusion in the

observation equations, takes place in Section 3.3.3.

3.3.1 Range Observations

The range observation is the simplest form of all of

the observations. In some manner, each of the other

observation equations will contain the range equation. The

basic form, without error sources considered, is

Rij = :2 + 2 + 21l/2 (3-5)

where Rij is the distance between the ith ground station and

the jth satellite position, Xj, Yj, and Zj are therectangular coordinates of the jth satellite position, and

xi, yi, and zi are the rectangular coordinates of the ith

ground station. This is a simple application of the

distance formula from the pythagorean theorem.

The linearized observation equation used to create the

B matrix is

(

36

Bfii Bf;. Bfi. Bfi. _Y1; ay. AY1 + az. M1 °' ax. AX;1 1 1 ;8f.. Bf,. (3.6)+——’*-·lAY.+-$·‘—Az. =-11..+s..BY. ; BZ. ; 1; 01;J J 0

where

fij = •[(Xj • Xi)2 + • yi)2 + • Zi)2]1/2 (3.7)

and Rij is the observed range from the ith ground station to

the jth satellite position, foij is the value of fij usingthe latest estimates of Xj, Yj, Zj, xi, yi, and zi. Xj, Yj,

Zj, xi, yi, and zi, are as in Equation 3.5. The zero

subscript on the parentheses in Equation 3.6 implies that

the partial derivatives are evaluated using the latest

estimates of the unknowns.

In Equation 3.6, the partial derivatives are defined as

follows:Bfi. X. · xiBx. = r., (3°8)1 13Bfi. Y. · yi—‘lay = ·‘];* (3.9)i ij

Bfi. Z. - zi· —-·‘— = -·‘——- (3.10)Bz. r,.1 1JBf,. —(X. - x.)..&l =...;L...l. (3.11)BX. r,.J 1JBf,. - Y.- .1; =

(; Y1) (3.12)BYj rij

37

asi. -(zi - zi)az. ‘ 1~.. (3-33)

J 1J

where

Iij = · Xi)2 + · yi)2 +• Zi)2]l/2 (3.14)

The preceding equations describe the basic processing for

the range observation equations.

There is a singularity in the system at this point,

which prevents direct processing. As all of the ground

stations and satellite positions are undefined, there is no

frame of reference for the coordinate system. A minimum of

six constants are required to establish the reference frame,

and as the five ground stations are known to a relative

accuracy of 10 cm., they can be entered as constraints to

the system. The coordinates are entered as weighted

observations using the following equations:

Axi = 0 (3.15)

Ayi = 0 (3.16)

Azi = 0 (3.17)

with a standard deviation of 10 cm. This is identical to

processing using the following constraint equation:

38

CA= 9 (3.18)

where C represents the constraint equations and g = 0.

There are ngr equations generated for each satellite

position, where ngr is the number of ground stations, and

there are nsat satellite positions. As a result, the numberof observation equations, neq, ignoring the constraints, is

neq = ngr * nsat (3.19)

There are three coordinate constraints per ground station,

so the total number of equations, n, is

n = 3*ngr + nSat*ngr (3.20)

Each satellite position and each ground station have

three unknown coordinates, so the total number of unknowns,

u, is

u = 3*ngr + 3*nSat (3.21)

which is the rank of the B matrix. The redundancy, r, or

degrees of freedom, is found by the equation

r = n - u (3.22)

39 :or

r = ngr*nSat — 3*nSat = (ngr - 3)*nSat (3.23)

To be able to solve the normal equations, r must be greater

than zero, which means that there must be at least fourground stations. As a result of the constraints on the

ground stations, each satellite position is essentially

solved independently, and thus, the number of satellite

positions is immaterial.The complete range observation equations have now been

presented, provided no solution is desired for error

sources. The errors are discussed in Section 3.3.3.

3.3.2 Phase Observations

3.3.2.1 Undifferenced Phase Observations. The phase

observation equations can take several forms. The first,

and the one used by PHASE, is the undifferenced phase

observation, which is basically a range equation.

The observation itself, 9, is a measure of the position

within the sine wave, between zero and 2w, received at thestation at a discrete time. Assuming that the transmitter

is at zero at the time of the measurement, the equation for

the observation is

I40 I

Öij = 21T/Ä(Rij • ANij) (3.24)

where Gij is the phase observation from the jth satellite

position to the ith ground station, A is the wavelength of

the pulse, Rij is the range, composed of six unknowns, as

defined in Equation 3.5, and Nij is an integer unknown which

represents the number of whole waves which makes up the

signal from the satellite to the ground.

The assumption is made here that A is the same value,

when sent by the transmitter and when received at the

ground. The fact that the wavelength at the receiver will

change is ignored. The change is due to the Doppler shift

and is a function of the changing distance between the

satellite and the receiver. The change is of a fairly small

order, but could be considered in a real life situation.

As the equation stands, there will be one new integer

unknown, Nij, for each equation, in addition to the

positional unknowns, and consequently the system could never

be solved.

For each phase observation, there is also an

accompanying whole wave count, Ncbs, which measures thenumber of whole pulses received since the first observation.

At any given time tl, the number of waves transmitted, Nt,

by the satellite, since the initial time to, is defined by

Q

41

Nt = fAt (3.25)

where f is the frequency of the transmission and

At = tl - to (3.26)

The difference, ANij, between the number transmitted

and number received can be added to the wave count for that

ground station, from the first satellite position, yielding

Nij = Nil + ANij (3.27)

and the observation equation becomes

= 2TT/Ä[Rij ' Ä(Nil + (3.28)

In this way, ANij can be computed, and the only additional

unknowns in the equation are the integer counts from the

first satellite position to each ground station.

The total number of unknowns is the same as in Equation

3.21, with the addition of one unknown per ground station,

or

u = 4*ngr + 3*nSat (3.29)

u

42

The rank of the B matrix is identical to that in Equation

3.20, given as

n = 3*ngr + nSat*ngr (3.30)

The degrees of freedom are then

r = (ngr - 3)*nSat - ngr (3.31)

and for r to be greater than zero, ngr must be greater than

three, and nsat should be greater than ngr.The partial derivatives of the constraints for the

undifferenced phase observations are identical to those

defined in Equations 3.15 through 3.17. The B matrix uses

the following equations:

öfi. öfi. Ef,. Bf,.v.. - ——·lAx. + ——·lAy. + —¥·lAz. + —i‘l./AX.1J 3x. 1 Sy. 1 öz. 1 BX. J1 1 1 J

öfi. öfi. öfi.1 61. MJ az. MJ + 611. ANi1 = 'R1;5 J' foij (3*32)J J 11 o

where

= ·2TT/Ä[Rij • Ä(Nil + (3.33)

43i

and fOij is the value of fij using the initial values of xi,yi, zi, Xj, Yj, Zj, and Nil. All other values are as

previously explained, and the zero subscript on the

parentheses in Equation 3.32 implies that the partials are

evaluated using the initial estimates of the unknowns.

In Equation 3.32, the partial derivatives are defined

as follows: Sfii -2n SR1.Gx. =

—T—3x.1

1

öfia -2n SR1.--t•L - —..—„ —

—·‘—(3.36)öyi A öyi

Sfi. -2w SR_.TM = T TM. @36)1 1

Sfi. -2n 3RiiSX. = T V. J JS51. -2n SR1.S1'. “ T SY. (3·38)J JSfi. -2n SRi.SZ. = T az. >J JSfi.g$J'= ZV (3.40)‘11

The partials of Rij are as outlined in Equations 3.8 through

3.14, with SL1bStitI1t€d fOI° fij.There is a peculiarity which makes this system

difficult to solve. One of the unknowns, Nil, is an integer

value, so there is a problem in assuring that the final

optimum solution contains integer values for Nil.

44

0ne answer to this problem, and the one employed by

PHASE, is to use a two-step solution process. The first

solution allows the integers to vary and converge. The

integers are then locked down at the nearest whole number

and removed from the solution as unknowns.

Technically, to assure the optimum solution, all

integer values within a set number of standard deviations

about the solution should be tested, but if that amounts to

as few as three values each, over five ground stations, that

would mean 243 solutions would need to be computed and

compared, so only the closest integer value is used.

3.3.2.2 Single Difference Phase Observations. Another

potential solution to the problem of integer unknowns is to

use a single difference phase observation. Given that the

unknown for a particular ground station, over multiple

satellite positions, is the same, taking the difference of

two observations Gij and Gik will yield

Aöijk = Gik - Gij = (zw/k)Rik - 2¤Ni1 — 2¤ANik· (ZV/)~)Rij + 2TTNi1 + 2TTANij (3.41)

or

Aöijk = (2'TT/Ä) (Rik• • 2TT(ANik • ANij) (3.42)

45 l I

The integer unknown is now removed, and the observation

becomes a range difference equation, rather than a range

observation.In this case, the equation which defines the B matrix

is Sfi.k Sfi.k Sfi.k Sfi.ky.. - ——·L—Ax. + %·‘L—Ay. + -)-112. + --1-Ax.1J SX. 1 Sy. 1 Sz. 1 SX. 3

( 1 1 l J

SY. J SZ. 3 SX k S1 K:1 1 k k

+ - + fazk "k " ' 1311 oijk (3-43)O

where

fijk = -2TF/Ä(Rik - + 2TT(ANik " ANi.j) (3.44)

and the other variables are as previously detailed. The

zero subscript indicates that the values are determined

using the latest estimates of the unknowns.

In Equation 3.43 the partials are represented as

follows: 61*.. -2y 612. 612._ .112 _ ..1Sx. — A Sx. Sx. (3-45)1 1 1)6s.. -2n 612. 612.. _ ..112 _ .1.1ay. 1 ay. ay. (3.).6)1 1 1)

46

(3.47)32.

—A 32. 32.l ( l l

Bf,. 2n SR., (3°48)_l.1Ä=...;..laxj A 3Xjaf,. 2¤6R.. (3'49)...£.1Ä=.....iJ.3Yj A 8YjBf,. &röR.. (3°5O)..?LZ&=....i.lözj A 3Zjar., -2¤ aa, (3'Sl)..e..1§=.__&3Xk A 6Xk

(3.52)Bf., -2n 3R..;.1.12:....A3Yk A 3Yk„ (3.53)of,. -2TI BR.l]k_____ ik3Zk A 3Zk

The partials of Rij and Rik are as described in Equations3.8 through 3.14, with Rij or Rik substituted for fij.

The number of unknowns is the same as in Eguation 3.21.

As a result of the differencing, there is one less

observation equation per ground station than in Eguation

3.20, resulting in

n = 3*ngr + (nsat — 1)*ngr (3.54)

which makes the degrees of freedom

r = ngr*(nSat - 1) - 3*ngr= (ngr — 3)*nSat — ngr (3.55)

1

471

As was the case with the undifferenced equations, the number

of ground stations must exceed three, and the number of

satellite positions should exceed the number of ground

stations.

Double and triple differencing have been used to reduce

the effect of errors, and therefore are described in Section

3.3.3.

3.3.3 Observation Errors

Four error sources were considered in the program RNGE.

Several other error sources were not included, but will

briefly be touched upon.

The first error source is random, uncorrelated error

drawn from a normal distribution. Random error is included

in the residual error, which is solved for by any least

squares solution. In fact the purpose of an optimizing

solution is to make sure that the residual error is as small

as possible, ideally representing only the random

measurement error.

Atmospheric refraction is caused by the increasing

density of the troposphere bending the wave by increasing

amounts. It can be removed in several manners. The method

used by RNGE and BUILD is to model the error, using known

atmospheric conditions, and remove it from the observation,

prior to the least squares solution. The Hopfield

I48

refraction method, as described by Anderle [1974], removes

all but about 5% of the error, which is added to the

residual error.

A second possible method is to solve for the refraction

as an unknown in the normal equations. This adds a great

deal of geometric complexity to the system, and as the bulk

of the refraction is easily removed, it is a less desirable

alternative.

The third error source treated by RNGE is clock error.

As with the tropospheric refraction, clock error can be

modeled and the majority removed prior to a least squares

solution. Given the type of atomic clock, cesium or

rubidium, and the time since last calibration, an Allan

variance model can be used to simulate clock error affecting

the observations. There is a small random component to the

clock error as well as a small ageing term, both of which

increase over time, and will affect the results; but if the

atomic clocks are calibrated regularly, these components

should remain small.

The majority of the clock error can be modeled as a

bias and drift. The equation representing the error, gc, is

Sc = b + dAt + rc (3.56)

49

where b is the clock bias, d is the drift rate, At is the

time since calibration, and rc is the random component of

the clock error, which will include any unmodeled systematic

and random error.Each clock in the system will have its own error

characteristic. In the range equation, this can be

represented by

Rij = r(xj - xi)2 + (Yj — yi>2 + (zj - zi)211/2+ (Bi + DiAt) - (BS + DSAt) (3.57)

where Bi and Di are the bias and drift of the clock at theith ground station, and BS and DS are the bias and drift of

the clock on the satellite. The other variables are as

previously defined.

The partial derivatives for the bias and drift terms in

Equation 3.57 areöfi.3B_ = -1 (3.66)

' 1

Bfi.616. = ‘^*° (M9)

1

Bfi.SB = 1 (3.60)Söfi.

s

I

50

In addition, one of the clocks, in the case of RNGE and

BUILD, the first ground station clock, can be the master

clock against which the others are measured, so the bias and

drift for that clock are defined to be zero. Therefore, if

i = 1, the Bi and Di terms in Equation 3.57 are notincluded.

Under these assumptions, two additional unknowns would

be added for each clock, including the satellite clock, but

excluding the master clock, for a total addition of

Au = 2*ngr (3.62)

This is not a large addition, and consequently the clock

errors can be solved, without an excessive impact to the

system.

There are other considerations related to observational

error or treatment of observations in terms of modeling.

Instrument error was not modeled within the programs

RNGE or PHASE, but was accounted for by the weights placed

on the observations. The total error contribution,

including random error, to the range observations is one

meter, and the contribution to the interferometric

observations is ten centimeters.

51‘

l

Ionospheric refraction is a second form of atmospheric

refraction, and was not considered in the solutions. It is

caused by the effect of the charged particles in the

ionosphere on the electromagnetic signal. The amount of

error is a function of frequency and altitude along the

signal path. According to Fell [1980] the upper bound on

the error, given the GPS frequency of 1575 MHz, is 4.8

millimeters, which is small compared with the error sources

that were included.

An observation effect which was not considered, but

could be applied to the phase observations, is the Doppler

shift. The wavelength is actually a function of the

velocity of the transmitter, relative to the receiver. To

get a good estimate of this value, a priori knowledge of the

satellite velocity vector over time is required. This

complicates the model, and means that the satellite

observations would not truly be independent.

Another error results from the fact that the PHASE

program made the assumption that the transmitter began at a

phase of zero, which is not necessarily true. As a result,

there is an initial phase offset, 59j, at each satellite

position. The observation equation then becomes

l52

Using the single difference formula at two ground stations,

i and l, yields

+ ' (ZTI/Ä)Rlj + ZTTNI1 + 2TI'ANlj "°2nNi1 + 2nANik - öök + (2n/A)Rij

• 211*Nil • 21rAN-ij + ßßj (3.64)

or

^@115k (Rlk ‘ R15 " R1k + R15)-2n(ANlk - ANlj — ANik + ANij) (3.65)

thus removing the initial offset. Equation 3.65 is the

double difference formula.

The triple difference formula takes the difference for

two satellites, at the same two ground stations, at the same

two times. The formula can be used to eliminate errors at

the ground station which are a function of time, such as the

drift terms for the ground clocks.

3.4 The Normal Eguations

The normal equations lent themselves to some unusual

partitioning schemes. The basic observation equation is

BA = f (3.66)

53 I

I

and can be partitioned according to ground station variables

and satellite variables, yielding

(B IB)[A} = f (3.67)Ü)

where the single dot refers to ground station variables,

including all clock bias and drift variables when they are

employed, and the double dot refers to satellite position

variables.

Multiplying both sides of Equation 3.66 by BTW gives

the normal equation

NA. = t (3.68)

where

N = BTWB (3.69)t = ¤Tw£ (3.70)

In the case of RNGE, with clock error unknowns

included, there could be as many as five ground stations,

and thirty satellite positions. Inserting these values in

Equations 3.21 and 3.62 gives a maximum of 115 unknowns,

which would be the rank of the N matrix. Solving for A

V54

requires inverting the N matrix, which would be extremely

time cohsuming on the computer used.This equation also can be partitioned, using Equation

3.67, into

= (3.71)LET N1 [tl

where

11 = 6Tw13 (3.72)E = 6Tw°é‘ (3.73)ä = ESTWE (3.74)t= 6Tw: (3.75)t = ÄTW: (3.76)

Solving the two equations simultaneously results in the

standard partitioning solution of

5 = ü'1E (3.77)

where

(3.78)t= - §°1i'l°£ (3.79)

I

55 I

As a result, the largest matrix which needs to be inverted

is N. N has a maximum rank of 90, and the inversion, while

faster than N, is still extremely slow.

As it turns out, N is a sparse matrix, as shown in

Figure 3.2. In Figure 3.2, the j subscript implies that the

matrix is a submatrix which is only the portion of the

complete matrix that applies to observations from satellite

position j.

This structure leads to additional partitioning, which

results in the equations

NA + = t (3.80)-•·T• |•‘O••=

OO.NJ A + NBA] t] (3.81)

where j ranges from 1 to nsat. Solving this system for

results in the following:

Ä = Ä'1; (3.82)

where

~=

•

¢NN ZNJNJ N] (3.83)“ - ° - —.".-1".t - t NJN] tl (3.84)

56

•• •• ‘

N = N1 0 ... 0o gz o

0 0...ii

= (El E2 iim)

•• ••]

E2

where

m = nsatOO' =.OO·T ·•O•NJ B] WJB]-—•

'Z

.•T••••N] B] WJBJ

= °°.T . .tl BJ Wjfn

Figure 3.2Structure of the Normal Matrices

!57

Additionally

N = 2Nj (3.85)

and

t = Ztj (3.86)

which results in the following values for N and t

“= °. - ’.°°.·1“.TN Z(N] NJN] NJ ) (3.87)” - °. - ".°°.—l°°.t — Z(tJ N]N] tj) (3.88)

and allows the matrices to be built one satellite position

at a time.

The largest matrix which is inverted in these equations

is N which has a rank of ngrv, or the number of ground

variables. That number has a maximum value of 25, while all

of the Nj matrices have a rank of three. This considerably

speeds the process, to the point that RNGE can be run in

real time in a matter of seconds, rather than in batch mode,

requiring many minutes, or even hours.

There are problems with this type of partitioning. One

problem is that it makes the process of coding and debugging

extremely difficult, and increases the lines of code

581

I

significantly. A second problem is that the covariancel

terms relating a satellite position to the ground, and

relating different satellite positions to each other, are

lost.

The §°1 matrix yields the covariance for the ground

variables. The satellite position covariances, relating the

different elements of a single position, can be generated

through a more complex process, by solving the system of

Equations in 3.80 and 3.81 for·ZB rather than Ä. This

results in the following normal equation:

Kj = ;1j°1€j (3.89)

where

= + + l-lüj (3.90)Ej = °£j - + §jiz3°°l'1ijT)‘1(€ + iji¢5‘l¥j) (3.91)

The covariance can then be generated from §j”l, and tj doesnot need to be computed.

In both programs, the observation equations are

linearized, thus requiring one or more iterations to achieve

a solution. There are four criteria, any one of which will

cause the iteration process to halt. The weighted variance

is determined for each iteration by the equation

59

soz = vTwv/r (3.92)

where v is the residual vector defined by the equation

v = f - BA (3.93)

and W and r are as previously defined. If soz is

sufficiently small, less than 10-6 for RNGE and PHASE, the

solution has converged, and the iterations are halted

successfully.

Two other criteria relate the weighted variance of the

current iteration to that of the previous iteration. If the

two are not sufficiently different, less than 10% for RNGE

and less than 10-7 for PHASE, the solution is again

considered to have converged, and the iterations are

successfully halted. If the variance of the current

iteration is greater than that of the previous iteration,

then the solution is diverging, and the program is halted in

error. The variance of the previous iteration is

initialized to 1030 to ensure that neither of the last two

conditions will be met on the first iteration.

When the solution for RNGE has converged, it has always

done so within five iterations. Therefore, an iteration

limit of five,has been imposed on both programs. If the

60 *l

solution does not meet any of the other criteria by the end

of the fifth iteration, the program is again halted in

error, as the solution is not converging as rapidly as it

should.

Whenever the programs are halted during the iteration

process, whether after successful completion, or in error,

the output files are created.

CHAPTER 4ANALYSIS

Simulations were performed using both programs,

resulting in varying degrees of success. Processing and

development were halted at an early stage on PHASE, as the

6 redesign required was extensive. A large number of runs,

with a variety of input data, were performed on RNGE. Those

results are addressed in Section 4.1. The results of the

few PHASE simulations, along with a discussion of the

probable causes of its failure, are contained is Section

4.2.

4.1 RNGE Results

All runs involving the RNGE program used five ground

stations. In addition, nineteen satellite positions at

fifteen minute intervals were used. This represents the

complete 4.5 hours of coverage during which the satellite

was visible from all five stations, simultaneously.

The first runs performed with RNGE were for test and

debug purposes, and consequently, used perfect data. There

were no errors on the observations, and the initial

estimates for the satellite positions were the truth values.

As expected, when the code was functioning properly, the

solution converged on the first iteration, and the weighted

variance was negligible.

61

I

62

POSDAT determines the difference between the computed

values from RNGE and the truth values, and prints them out

to six places. The smallest unit displayed is one

millimeter. The program also performs a Lagrangian

interpolation to determine intermediate positions, as

described in Section 3.2.4, and compares those values with

truth values, printing the results to six places, as was the

case with the computed values from RNGE.

In all cases, for the run with perfect data, these

results were displayed as zero, meaning the values were less

than .5 millimeters. The primary purpose of the run was to

test the basic algorithm, and make sure it would perform the

desired processing. As such, the run was a success.

The next test runs were similar tests, which involved

using perfect observations, with no errors from any source.

The difference in these runs being that the satellite

„ positions were scaled to create initial estimates which

deviated from the true values.

In the first of these, the scale factor was one part in

105, creating initial errors on the order of 100 meters.

Again, the solution converged in one iteration, with a

weighted variance of 10°7. The RMS of the differences of

the computed values and the true values were all under 4

centimeters. The interpolations differed from the true

IIIII

63 I

Ppositions by a value of less than 35 centimeters in all

cases.

The same observation data was used, allowing RNGE to

generate the initial estimates, as described in Section

3.2.2. This scenario seemed to tax the system more than any

other, as the solution required five iterations to converge.

The weighted variance was just less than the cutoff value of10°6. The output RMS for the computed variables was less

than 20 centimeters and for the interpolated positions, less

than 40 centimeters, in each component.

A final run using the perfect observations was

attempted. The initial estimates had a scaling factor of

1.0. When entered into RNGE, however, some estimates

exceeded the allowable range, and the program halted in

error.

Table 4.1 gives a description of thirteen simulation

runs which completed successfully. The table describes each

run, and states whether the initial estimates for the

satellite positions were estimated by BUILD from scaled

values of the true positions, or were generated by RNGE.

For the positions which were estimated, the table gives the

scale factor. The final two columns show the means and

istandard deviations, in meters, of the errors placed on the

observations.

Go N M

65

Table 4.2 shows the results for the simulations listed

in Table 4.1. The table shows the number of iterations

required for each, which is useful in determining the

sensitivity of the program to different input data. The

next column gives the weighted variance from the final

iteration. All elements of the table are given in meters,

with the exception of the weighted variance, which is

unitless.

The output errors show the root mean square of the

errors in the results of RNGE. The error listed in those

columns reflects the difference between the computed

position and the true satellite position for the same time.

The final three columns list the differences between the

interpolated values for the intermediate times and the

actual satellite positions. In each case, the error is

given for the X, Y, and Z components individually. The mean

and standard deviations of the total input errors which were

added to each range in order to compute the observation,

were computed by BUILD.

A variety of input data was used to test the program's

sensitivities to different errors. Comparing the

magnitudes, it is apparent that the majority of the input

error is random. In fact, by chance, the direction of the

clock and refraction errors is, on average, opposite to that

l

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67 l

lof the random error, so the net effect of the clock and

refraction errors is to reduce the total error.

The first five runs in Table 4.2 had only random errors

added to the observations. The program used a standard

deviation of one meter to generate the errors. The results

of BUILD show a mean of -.016 meters and a standard

deviation of 1.084 meters. The difference between the runs

was in the estimated satellite positions.

Because most of the error in any of the runs was

random, these runs were used as a test of the ability of the

program to complete the adjustments even though the initial

estimates were bad. Run number four scaled the initial

estimates by 10-5, or about 100 meters. Run three varied

the initial estimates by a 10-4 scale factor, or about one

kilometer. Run two was scaled by 10-3 and run one was

scaled by 10-2. The fifth run allowed RNGE to generate the

initial estimates. Another run was performed using a

scaling factor of 0.1, but the initial estimates exceeded

the allowable range, and the program was terminated in

error.

The five runs concluded successfully, with nearly

identical variances. The satellite positions were also

nearly identical, for both the computed and the interpolated

positions. In each component, the error is on the order of

ten meters, which is good for most ephemeris applications.

68

Given the input error, and the ground and observation

weights, ten meters might seem large, but this is probably

an indication that the program, and particularly the

application of the weights, require some fine tuning. For

both the Y and Z components, the interpolated positions were

closer to truth than the computed positions. This is not

statistically significant, however, and is probably due to

coincidence.

The primary difference between the five runs is

indicated by the number of iterations required for

convergence. Runs two, three, and four, each needed only

two iterations for a solution. Run one, with initial

estimates in error by about 1000 kilometers, took three

iterations to complete, and run five required the maximum of

five iterations, still yielding approximately the same

results. This indicates that the solution is largely

insensitive to the error in the initial estimates,

presumably because of the geometric strength of the range

equations.

The next two runs were conducted using both the random

errors, described in the previous runs, and tropospheric

refraction error. The refraction error is the 5% residual

error detailed in Section 3.3.3. The total mean refraction

error, as computed in BUILD was 3.6 meters, meaning that

about 18 centimeters of error was added to the observations.

69 I

Run six used the initial estimates which were scaled by

a factor of 1O”5. The input errors had a mean of 16.6

centimeters, but the standard deviation was close to that of

the runs with only the random errors applied. The run

required two iterations and finished with a weighted

variance which was almost identical to that in the previous

five runs. The output values were also almost identical,

between 7 and 13 meters for the computed positions, and

between 8 and 11 meters for the interpolated errors.

The seventh run used the same observations, but the

initial estimates were generated in RNGE. As was the case

in the other types of runs, the results were nearly

identical to run six, but run seven required five iterations

to converge.

Runs eight through thirteen all had clock errors, as

well as the refraction and random errors, added to the

observations. Runs eight through eleven employed a simple

linear least squares model in BUILD to filter the bias and

drift of the clock error before it was added to the

observations, as described in Section 3.3.3. The clock bias

and drift terms were not solved during RNGE, so that the

only variables determined were the ground and satellite

positional variables.

Runs eight and nine assumed that all clocks had been i

calibrated at the time the first observation was taken, so TIIIII

I70

that to was zero. After removing the bias and drift, the

mean clock error added to the observations was zero, as

expected, and the standard deviation was 2.9 centimeters.

The errors of the output data showed slight improvement

over those of the previous seven runs in all but the X

component of the least squares positions. The weighted

variances also showed some slight improvement. The change

was small in all cases, and not statistically significant.

As was the case with the previous run groupings, the

difference between runs eight and nine was in the estimated

satellite positions. Those of run eight were scaled by

lo'5, while those of run nine were generated by RNGE.

Again, the results were almost identical, except that run

eight required two iterations, and run nine required five,

for convergence.

Runs ten and eleven duplicated runs eight and nine

respectively, for both input and output. The only

difference between the sets was that runs ten and eleven

assumed that the ground clocks were last calibrated 2.5

hours before the observations began, and the satellite clock

was checked 250 hours prior to the start of the

observations. After filtering, the input data looked almost

identical to that of runs eight and nine, as did the results

of RNGE.

71 ,The final two successful runs used the same input

assumptions as runs eight and nine. The difference for

twelve and thirteen was that the clock errors were not

filtered, with the total error passed to the observations.

RNGE was run without solving for the bias and drift, just

solving for the positional variables. For the computed

positions, the results fell between the results of runs

eight through eleven and those of runs one through seven,

for all components. The results were very close to runs

eight through eleven for the interpolated positions. The

weighted variances also fell in the middle of the values for

the thirteen runs.

Four additional runs were attempted using the clock

errors generated for runs eight through eleven, but without

filtering the bias and drift. In these runs, RNGE attempted

to solve for the bias and drift, but the solutions diverged

with differences from the true positions in excess of 10

kilometers. The reasons for the failure have not been fully

analyzed, but are believed to be in the coding or design,

not in the basic concept.

4.2 PHASE Results

In contrast with RNGE, only two runs were performed

using the PHASE program. Both runs used a reduced data set

from that used by the RNGE program, with five ground

72|

stations and five satellite positions, the latter at fifteen

minute intervals.

The first simulation was the same as the first for

RNGE, using perfect observations and perfect estimates.

Again, this data set was used for test and debug purposes.

The run was brought to a successful conclusion in both

parts. The first adjustment, using the integer unknowns,

converged in one iteration with a weighted variance on the

order of 10'26. The second adjustment, which held the

integers constant, and only allowed the positional variables

to change, also concluded in a single iteration, with a

variance on the order of 1O”27. The differences between the

computed components and the true values were not observable.

The second run used the same perfect observations, but

with scaled errors of 10°7 on the estimates, or on the order

of a meter. Even with such good initial estimates, the

solution diverged on the second iteration. After

preliminary analysis, it was determined that extensive

rewriting of the program was required to correct the

problems.

There are several constraints for which the program did

not account. The