Clyde C. GOAD, Benjamin W. RE MONDI National Geodetic Survey

Charting and Geodetic Services National Ocean Service

National Oceanic and Atmospheric Administration Rockville, Maryland 20852, U.S.A.

I N I T I A L R E L A T I V E P O S I T I O N I N G R E S U L T S U S I N G

T H E G L O B A L P O S I T I O N I N G S Y S T E M

Abstract

The National Geodetic Survey (NGS) has become increasingly involved with research and development for relative positioning using the Global Positioning System

(GPS). The NGS has procured two Macrometers T M and is also a participant in the development o f the Texas Instruments ' TI4100.

The Macrometers were delivered in March 1983 and 3 months o f testing has now been completed. These data have been processed using a variety o f newly developed processing techniques, and numerical intercompaHsons o f several base line solutions are given. A byproduct o f one technique is the estimation o f the relative variations of the ground clocks to the subnanosecond level

1. Introduction

Over the past three years the National Geodetic Survey (NGS) has become increasingly involved in relative positioning using the Global Positioning System (GPS). At first our activities were restricted to theoretical or numerical investigations (B ossler et aL, 1980 ; Goad, 1981). In 1981 NGS joined with the U.S. Geological Survey and the Department of Defense to fund the development of the TI4100, a single channel, multiplexing receiver which uses the p-code (Ward, 1982; Remondi, 1983). B y 1982 this development was in progress. By the Spring of 1982 Professor Counselman had demonstrated centimeter-level relative positioning with the Macrometer T M (C ounsel- man and Steinbrecher, 1982) a GPS receiver which does not use the p-code. In January 1983 many base lines were measured with the Macrometer system for the Federal Geodetic Control Committee (F GC C) test established for that purpose (Hothem and Fronczek, 1983). In March 1983 the NGS took delivery of two Macrometers and have had them in the field ever sinCe.

During 1982 software was developed to process simulated phase data. In 1983 this development continued, culminating in several different approaches to processing phase data. Using the data from the F GC C test along with approximately three months of our own field measurements, we were able to exercise our various processing software

T M Macrometer is a registered trademark of Macrometrics, Inc., Woburn, Massachusetts, U.S.A. Presented at the International Union of Geodesy and Geophysics, XVIII General Assembly, Hamburg, August 15--27, 1983.

Bull. GEod. 58 (1984) pp. 193-210. 193

C.C. GOAD, B.W. RE MONDI

as well as software provided with the Macrometer. The results of these various processing approaches are the essence of this report.

Section 2 presents a description of the raw observable and other observables derived from them. Section 3 discusses alternative techniques for processing these observables. Sections 4 , 5 , and 6 document actual numerical results.

Section ? discusses the quality of receiver clock difference estimates obtained from single difference processing. T he results imply that subnanosecond clock variations can be resolved. Finally, Section 8 discusses some of the more operational aspects of relative positioning using GPS.

2. T he GPS Phase Observable

Much has been written about the GPS satellites and the various signals and data transmitted from them. As in Spilker's (1978) excellent paper, the preponderence of literature discusses the p-code and/or C/A code pseudo range observable. The observable upon which this research and development have been based is the measured phase of the L1 carrier (1575.42 HHz) .

The GPS geodetic receivers typically have rubidium or high quality crystal oscillators. The GPS satellites have high quality rubidium and cesium oscillators (Bartholomew, 1980). Whereas the satellite oscillators have proportional stabilities in the range 10"* -11 to 10"* -13 over 1 second, the crystal oscillators in the subject geodetic receivers have proportional stabilities in the range 1 0 " * - ] 0 to 10 " * -12 over ] second. A proportional error in frequency of 10"*-11 will cause phase instability over 1 second, at the L I frequency, of about 5 degrees or 3 mm in range. Root mean square (rms) residuals as small as 1.5 mm have actually been encountered over very short lines which implies that the Macrometer's oscillators are no worse than 5 parts in 10" '12 over ] second.

T he results reported here are based on the Macrometer's single difference phase observable, but apply as well to the phase observable of the T14100 (both described below).

Let us symbolize the phase of the L1 signal at an instant, t , b y ~ ( t ) . B ecause the oscillators involved are quite stable over short periods of time, the following linear approximation is valid "

q~(t + A t ) ---" ~ ( t ) + f . At, (1)

where f is the oscillator frequency.

In modeling the phase observable one realizes that the true receipt time must be determined (as contrasted with the indicated receipt time or time tag), and one must determine the true transmit time (including signal propagation time and propagation delays). T he instantaneous phase difference between the satellite signal, q~s ' and that of

the i - t h ground receiver clock, ~R i , can be rigorously modeled as

(~s (tt i) -- ~R i (tR i ) (2)

where t t i and t Ri are transmit and receipt time respectively. Neglecting the transmission

medium delays, etc., expression (2) can be rewritten as follows :

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INIT IAL RE LAT IVE POSIT IONING RE SULTS .....

~) ~ s ( t R i - c - - ~ R i ( t R i ) (3)

where Pi is the slant range from the satellite to the i - t h receiver at transmit time and

c is the velocity of light. Using equation (1), expression (3) can now be approximated by

f sP i ~s (tR i ) -- C t~Ri (tR i ) + N i (4)

where the integer N i simply compensates for the fact that the first expression is a

quantity less than 1 cycle whereas fs " Pi / c represents many cycles.

How these geodetic receivers actually accomplish these measurements is outside the scope of this paper. T he instantaneous phase observable at a single station, described above, is what the TI4100 provides. Although the Macrometer also measures this instantaneous phase, only phase differences between ground receivers were available to us. The Series observable (MacDoran et al., 1982) is formed by combining phase measurements of GPS signals at many frequencies (and difference frequencies). Thus the Series starts with a rough a priori satellite position and then deduces near-instantaneous slant range from these many phase measurements.

Based on the above discussion, the observable provided by the Macrometer can be approximately modeled as

( ) fs c " - ~ R 2 ~ ~R1 Jr N12 �9

T his is what we call the single difference observable, tt is the difference of the quantity in (4) which is measured at each end of a base line. The first term is just the difference of slant ranges ; the second term represents the phase difference of the station clocks (which varies from one observation epoch to the nex t - see the clock drift discussion below) ; the third term is the difference of two integers.

Because the Macrometer maintains lock on the L1 signal, from epoch to epoch, the change in the integer from the first observation epoch is also provided. T hus only the integer number associated with the first observation epoch is unknown. In summary, the single difference phase observation of satellite j at epoch i can be approximately modeled as

where we have chosen to define slant range as a function of receipt time.

Notice that at a given epoch i , ~bR2 - ~b R 1 will be the same for all satellites.

The ambiguity function approach exploits this fact (C ounselman and Gourevitch, 1982).

Only for the correct base line vector -~ is it true (for M i satellites observed at the i - t h epoch) that

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C.C. GOAD, B.W. REMONDI

I Mi ] f ( - ~ ) = 2~ e i "* j = l ( O j - c j ( b ) ) = M i (6)

at all epochs and in absence of noise, where the 0 i are the preprocessed fractional phase

observations and the cj (~') are the fractional phase portions of the fs" (p i - P ~ ) / c .

The ambiguity function technique requires the use of a search algorithm for

finding the b which maximizes f .

The double difference observable (Bossier et aL, 1980) is a single difference using one satellite minus a single difference using another satellite, where the ground stations are the same for both single differences. Thus the station clock misalignment cancels since it is a common contributor to both single differences. A triple difference is simply the change in a double difference from one epoch to the next. In their basic form these created observables are dependent ; however, one can uncorrelate them if desired. In fact, the Macrometer processing system generates uncorrelated double differences.

Why are there so many processing approaches to the same data ? T he answer is that they each have advantages and disadvantages. Our triple difference algorithm is fast and automatically rejects data with cycle slips. It is somewhat less accurate than the other methods since some single difference data is not used. The results of generating double differences is the cancellation of local clock errors and common jumps in the single differences. Thus in double difference processing a cycle slip might not show up whereas with single difference processing it will. Also, to be competitive accuracy--wise with the single differences, the data dependence must be modeled, but this is not difficult.

We believe that, with little extra work, single difference processing results in slightly more accurate base line solutions than does double difference processing. Additionally, the station clock behavior can be monitored as a check on the processing and as a check on field clock logs.

In contrast to single, double, and triple differences, the ambiguity function approach requires only the fractional phase and is therefore completely insensitive to cycle slips or integer ambiguities. A disadvantage is that it requires considerable computation. One should start with agood initial base line vector estimate to minimize the computations. T he triple difference method can provide this.

In Macrometer processing we find that uncorrelated double differences do a rather nice job ; however, a considerable amount of interaction is required. T he fact that the Macrometer uses a limited analytic orbit model rather than precise orbit tables has also been a factor in our decision to process the data on other machines.

On our Hewlett Packard model 1000 computer we find the single difference technique to be the most powerful. The results are excellent, but still interaction is required. The triple difference is useful in cases where a good a priori base line vector is unavailable. Also a good a priori vector is valuable in cases where there are data problems such as losses of lock. Combining the triple difference and ambiguity function techniques is also very useful since neither requires interactive processing. This autonomy could prove to be invaluable.

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INIT IAL RE L AT IVE POSIT IONING RESUL TS .....

3. NGS Processing T echniques

To date, the only GPS phase data available to us at NGS (Remondi, 1983) has been the phase difference data from the Macrometer. The NGS took delivery of two Macrometer receivers in March of 1983, installed them in vans, and, since April, has been making field measurements. These measurements, coupled with earlier measurements made during the 1983 Federal Geodetic Control C ommittee (F GC C) test (Hothem and F ronczek, 1983) have provided sufficient data for analysis.

Once a base line has been selected and scheduled, a magnetic cassette which contains orbital data (almanacs) for the satellites (up to six) to be observed, is prepared for each field unit. This data cassette is site-dependent and provides expected Doppler signals to isolate the phase history of each satellite from wide band reception.

During the observing session the satellite signals are tracked and, at preestablished epochs, data are recorded. It is typical to record observations every one to three minutes. After all observations have been taken, the data are mailed or telephoned to the home office for processing.

After receipt, the raw data are entered into the Macrometer processing system. Next the recorded signal data from the site are combined resulting in a fi le of phase difference data. These data are now ready to be reduced using the programs provided with the Macrometer (V-J000) system processor. Macrometrics, Inc., also provided a program which allows access to these phase observations. This has permitted the transfer of the data to a Hewlett Packard 1000 series computer, a more powerful machine. Qn this machine resides our own processing software. These programs use, through interagency agreement, precise GPS orbital ephemerides from the Naval Surface Weapons Center (NSWC), Dahlgren, Virginia.

Besides the raw phase difference data, and Earth centered fixed satellite ephemerides, a fi le of a priori site coordinates and ancillary data, such as the relative position of the antenna to the survey mark and weather data , are prepared. After preprocessing the data to account for effects of troposphere, any of the four NGS- developed programs can be used to process the data.

Briefly, the authors have developed code to process the data using least-squares techniques in single difference, correlated double difference, or correlated triple difference mode. The ambiguity function approach (C ounselman and Gourevitch, 1981) has also been developed. Additional code to uncorrelate the double or triple difference observations has not been generated since these programs are used only to obtain good starting conditions for the single difference and ambiguity function techniques. We have found the triple difference technique to be ideally suited to this supportive role.

T he triple difference processing is straight forward, simple, and automatic. 8 y not uncorrelating the observations, the process has been made faster and less complex. B y automatic, it is meant that it is not necessary to f ix cycle slips. No case has been encountered so far where triple difference processing has failed to achieve a solution within a few centimeters. This is its uti l i ty. In fact, to determine the base line vector from [C00 to IC 22 in the New Mexico test, the coordinates of Hamburg, Federal Republic of Germany were used for a priori coordinates for the second station (amounting to an initial error of about 10000km) . This did not present a convergence problem. The solution thus obtained was then passed on to single difference and ambiguity function software for final processing.

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C.C. GOAD, B .W. RE MONDI

4. Numerical Comparisons

As stated earlier, NGS repeated observations on many of the base lines that were observed during the F GCC Macrometer test. Unfortunately, one of the satellites, NAVSTAR 1, experienced attitude control problems during this repeat session. Thus only four of the five satellites were available, and therefore a comparison of these results with the F GC C results would be inappropriate. However, this situation did raise questions concerning the quality of results when fewer satellites are used, and this will be discussed subsequently.

In late June and early July 1983, NGS collected data at a very accurately measured base line at Holloman Air Force Base, New Mexico. The line runs almost perfectly south to north and is known to one part per million (ppm) or better.

Results from these two survey projects are presented here. The New Mexico sites are named [COO, [C06, I C ] ] , and [C22, and their relative positions are shown in Figure 1. The F GC C Washington survey consisted of sites Athey (AT HY), Gaithersburg Observatory (ORM1), and Goddard Space F light Center (GORF) which are all part of the U.S. Transcontinental Traverse. Two sites, only ?50 meters apart, were also occupied at the National Bureau of Standards in Gaithersburg, Maryland. Except for the two sites at the National Bureau of Standards, Figure 2 gives a sketch of these base lines.

15.4 k m ~ ! IC22

ICU

7.5 km ---I- IC06

3.61 kn ICO0

I I I

12' 10' 08' 106 ~

I

06'

33002 '

33000 '

32058 '

32~ '

32~ '

32o52 '

Fig. 1 - Base Lines Observed in New Mexico

Table 1 compares all base line length results with terrestrial reference values. No adjustment of the satellite results for scale or orientation has been made. The estimated uncertainty in the terrestrial values is 1 ppm (one standard deviation). All four NGS- generated programs along with the independent double difference program of the Macrometer are compared. All results presented here were obtained by NGS personnel.

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INIT IAL RE L AT IVE POSIT IONING RE SULTS .....

Gai thersburg Observatory N

ORMI Athey _ ~ . 3 ~ ~

ATHY L~ ~// " "~r k, God,.rd Space "=~ S ~ F l ight Center

42. ! GORF

OPTK Optrack Fig. 2 - B ase Lines Observed Near Washington, D.C.

The base line lengths generally agree with the terrestrial reference values to about ! ppm. Two exceptions are the O R M 1 - A T H Y and OPT K-GORF base lines. These base line terrestrial distance values are estimated to be 2.2 and |.9 ppm, respectively (Hothem and F ronczek, 1983). There is excellent agreement between the terrestrial reference and satellite solutions for the Holloman base lines.

Tables 2 and 3 give the azimuthal and vertical angle comparisons, respectively. These comparisons show that all methods agree with each other to the 0.5 second of arc level except for the ambiguity function technique. One possible explanation for this difference is the lack of an editing capability in the ambiguity function software. All the other techniques, automatically or by hand editi ng, routinely require rejection of outliers. Generally speaking, most data sets required rejection of a few measurements (usually less than 2 ~o ) .

Even though the angle results were in agreement on a given base line, variations at the 2 to 3 seconds of arc level are noted among base lines - mostly in vertical angle comparisons. However this is possibly due to effects of the ionosphere and troposphere or unknown local variations in deflections of the vertical which are required for a comparison to the terrestrial reference.

5. Vector Closures

A powerful check on precision is to measure how well the vectorial addition of sides of a polygon agree in closure. Table 4 gives the closure discrepancies for the five available cases in this study. The rms of all component discrepancies is 3.9 cm. The average line length is 20 krn. This yields an average closure error of 2.0 ppm, well within the expected accuracy of the single frequency system.

Notice that since no closure constraint was used during any of the reductions, even solutions with sides determined with data collected during the same tracking session yield nonzero closure.

199

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LEG I LEG 2 LEG 3 DISTANCES SURVEY DATES Ax fly Az (km) (Day of Year, 1983) (ca) (ca) (cm)

GORF-ORMI ORMI-OPTK GORF-OPTK 35-18-42 20-19-20 -1.5 -3.9 -0.8

OPTK-ORM1 ORMI-AIHY OPTK-ATHY 18- 9-12 19-19-19 4.0 -1.2 8.6

OPTK-ORMI ORMI-ATHY OPTK-ATHY 18- 9-12 20-19-19 6.2 1.8 -0.8

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ICO0-1CI1 ICII-IC22 IC00-IC22 8- 8 - 1 5 183-181-179 2.3 -2.5 2.6

6. Suboptimal Solutions

In Apri l and May 1983 NAVSTAR 1 developed an attitude control prol:lem, and its signal could not be acquired. Since NAVSTAR 1 is visible for the duration of most of our observing sessions, this seemed to be quite a loss in terms of geometry. The four remaining satellites were simultaneously in view for only ! hour of our normal 3 -hou r observing span. Actual ly it is not necessary to observe for 3 hours, but in our quest for high accuracy we usually have done so.

Because three or less satellites were visible for much of the observing period, it was natural to investigate the dependency of the solution on the number and geometry of satellites. In these investigations the single difference technique was employed.

All of the lines that were processed using five satellites and presented in Tables I, 2 , and 3 , except the l km line, were reprocessed using four satellites. In all cases but one, NAVST AR 1 was excluded ; NAVST AR 3 was excluded in that other case.

It is apparent from Table 5 that no serious degradation of accuracy resulted from the loss of an important satellite. The variation in the length estimate was always smaller than ] ppm.

Note particularly the New Mexico stations where the terrestrial line lengths are known to I ppm or better. For these stations, there are larger length differences between the terrestrial values and the five-satell i te values (see Table 1) than between the five-satell i te values and the four-satel l i te values (Table 5).

In fact, the results were so good it was natural to also compare three-and t w o - satellite solutions with a five-satell i te solution. These results are presented in Table 6. Notice that with two orbital planes (having nodes of 46 and 167 degrees) the three- satellite solutions were still quite good. It should be clear from the number of observations used that these three satellites were not visible for the entire 3 hour span (for that would imply approximately 180 observations).

7. Clock Drift E stimation

Referring again to expression (5), it is seen that the clock difference between the two ground receivers is an unavoidable contribution to the single difference observation modeling. E yen if the ground receivers were to use highly accurate oscillators with a proportional error 1 0 " * - 1 4 , the contribution to the observation would be of the order

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of ] /5 of a wavelength. Although available, such equipment is far too expensive and requires too much maintenance for use in the types of applications under discussion. One way to obviate this is to combine measurements taken at common epochs in the form of double differences as was suggested in Bossier et al., (1980).

F or the same reason that this clock dr i f t term can be eliminated when two or more observations are differenced at the same time, one should be able to estimate the clock dr i f t if the measurements were not differenced. Although the single difference base line solutions are sensitive to a constant relative offset in the clocks at the ]0 microsecond level, they are sensitive to the variation or dr i f t from epoch to epoch at the subnanosecond level.

Notice that with this technique the noise contributions to the measurements are independent. No special software is required, as is the case for double or t~iple differences, to obtain independent observations.

Also note that clock variations are estimated for each epoch. The mathematical model does not require any degree of smoothness in the individual estimates. However it is pleasing to see smooth, almost linear, clock difference histories. A typical example of clock dr i f t estimates is given in Figure3. This particular figure gives the clock drif t estimates for the base line from [C06 to [C22 of Holloman Air Force Base, New Mexico. This estimate is in very good agreement with the clock differences measured with a time interval counter 30 minutes before and 30 minutes after the 3 - h o u r survey session.

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Fig. 3 - C lock Difference

2 0 6

INITIAL RELATIVE POSITIONINGRESULTS .....

Sometimes, though infrequently, blockages of the satellite signals result in only one of the satellite's signals being received. It can be shown that if only one single difference observation is available at an epoch, this observation can be used for clock estimation ; but then no information remains for base line determination.

Another benefit to using this measurement model is its ease in searching the data for cycle slips. We have found it just as easy to analyze the single difference history as the double difference history. This often has to be done when searching for the integer jump that occurs during loss of lock of the signal. When a loss of lock is analyzed in double difference mode, at least two sets of double difference histories are required to isolate the satellite whose signal was temporarily lost. This is seen immediately in the single difference history.

It is natural to question the quality of these clock dri f t estimates. Figure 4 shows what remains from the estimated clock differences given in Figure3 when an eighth degree least-squares polynomial is removed. Notice that the total variation is 2 nanoseconds. It is seen from Figure4 that these residuals still have a systematic character at or below the 0.] nanosecond level.

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Fig. 4 Clock Difference After Removal of 8-.th Degree Polynomial

Typical ly the rms of the single difference residuals is 1 cm or lower. Considering the GPS transmission frequency (1575 t-~Hz), the speed of light, and the fact that the Macrometer actually tracks half cycles, a clock difference resolution of 0.03 nanoseconds should be possible. This is not the case here, however. Such contributions

207

c.c. GOAO, B.W. REMONDI

as ionospheric delay, tropospheric refraction errors, orbit error, etc. which have a common contribution to all single difference measurements at an epoch, wil l go directly into the clock difference estimates. It is interesting to note that for the two-satell i te solutions discussed in section 6, the estimated clock variations still agreed with the five--satellite solution to within [ and 2 nanoseconds.

But it is a tantalizing thought to consider just how close to this potential clock difference resolution one could get with the use of dual frequency measurements along with microwave radiometers and very accurate orbits.

8. Asides : Theory versus practice

Since this paper is primarily one of results, we shall interpret "results" broadly and include not only what is achievable but also what has been learned. So far what is achievable has been emphasized.

We have learned that setting the clocks to UTC, occasionally, and accidentally, results in a full second offset. All of the programs discussed above can deal with this offset, however. Usually one clock is set to within 1 ms of UTC (using time transferred by the National Bureau of Standards via the Geostationary Orbiting Earth Satellite system) and the second received clock is synchronized to within a fraction of a microsecond of the first prior to the observing session. Therefore common clock error has not been a factor.

We have also realized that although the data can be processed when the relative clock drift is high (e.g., ] 0 - 2 0 / ~ s / h r ) , it is best to keep this drift rate low (e.g., ] / l s / h r ) . Keeping the drift low allows one to model the clock differences during the survey session as a constant in all but the single difference technique. As discussed earlier, the dri f t is an inherent part of the single difference model. Otherwise it becomes more diff icult to interpolate for either the clock difference at the first epoch (for the single difference technique) or the midpoint (which is required for the other techniques) when the drift rate is large.

Another problem encountered (and expected) is integer cycle slips. Occasionally satellite visibil ity is obstructed by some object (e.g., a building). The full cycles encountered between epochs are provided as part of the observation. But during the period of obstruction the integer cycle count is lost and only the fractional phase at the next observation epoch is reliable. Normally this is a surmountable problem, and the lost integer count can be supplied during preprocessing. This is occasionally diff icult, however. In such cases the triple difference method or the ambiguity function approach becomes particularly attractive. B esides-signal blockages, cycle slips are also occasionally the result of temporary power outages.

F or high stability the receiver clocks need to be keptwarm. Occasionally, through loss of power, an oscillator gets cold. When the power is reapplied the clock's frequency can vary wildly during warmup. It can happen that a clock is found to be cold just before the observing period ; this can lead to extreme drift rates making the time logs essential.

During April and May 1983 NAVST AR 1 was inoperable so that there were only four GPS satellites- available. As discussed above this degraded the solutions somewhat, especially since NAVST AR 1 provided the strongest geometry, and because during much of the observing session only three satellites were in view. In the future this is not expected to be a problem.

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INIT IAL RE L AT IVE POSIT IONING RESULTS .....

A problem not discussed is propagation delay. F or many applications this is not a problem. The NGS Macrometers are single frequency receivers and therefore cannot supply information to correct for the ionospheric delay. The results are nevertheless excellent. In fact, for all but the very high accuracy requirements of geodynamics and long base line geodesy, the single frequency Macrometer appears to be sufficient. However, for the ultimate in accuracy, dual frequency receivers will be required and more attention will have to be given to tropospheric propagation delay.

Data transfer from the home office to and from the field via commercial telephone service has been rather unsatisfactory. At first it was thought that the mailing of magnetic cassettes could be avoided using the telephone service. We have made this procedure work, but far more time has been required than was originally expected. The main problem encountered has been to establish a good communication link. Now we use an overnight delivery service, and we find this satisfactory. The capability to use the telephonic transmission of data is still used ,occasionally when timely data delivery cannot be achieved.

9. C onclusions

The theoretical modeling of the GPS phase observable has been given along with several possible algorithms for processing it. We favor the triple difference technique for determining a very good initial estimate of the base line vector. We employ a single difference technique for final vector determination. A byproduct of single difference processing is the determination of receiver clock variation histories to the subnanosecond level. The results obtained have given us at NGS the confidence to proceed routinely with these single frequency receivers for use in control surveying over base lines up to 50 km.

We look forward to the time when dual frequency receivers will be available. Perhaps then, longer base lines can be surveyed routinely. The use of microwave radiometers may also be needed for very precise measurements. However, the radiometers are not, at this time, required for control surveying purposes.

We believe that NGS made the proper decision to get involved with GPS in the early stages of receiver development. This technology is now paying dividends for us ; we are confident that the surveying and geophysical communities will benefit soon.

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R E F E R E N C E S

C.A. B A R T H O L O M E W : Satellite frequency standards. Global Positioning Systems. The Institute of Navigation, Washington, D.C., pp. 21-28, 1980.

J.D. B OSSL E R, C .C. GOAD, and P.L. B E NDE R : Using the Global Positioning System (GPS) for geodetic surveying. Bull. Geod., 54, pp. 553-563, 1980.

C . C . C O U N S E L M A N , and S.A. G O U R E V I T C H : Miniature interferometer terminals for Earth surveying : ambiguity and multipath with Global Positioning System. I E E E Transaction on Geoscience and Remote Sensing. Vol. GE -19, No. 4, Oct. 1981.

C.C.COUNSELMAN, and D.H. STEINBRECHER : The Macrometer T M : a compact radio interferometry terminal for geodesy with GPS. Proceedings of the Third/nternational Geodetic Symposium on Satellite Doppler Positioning. pp. 1165-1172, Feb. 8--12, 1982.

No produc t e n d o r s e m e n t is i n t e n d e d or impl ied, by NOAA, in this article.

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C.C. GOAD : Positioning with the Global Positioning System. Reference Coordinate Systems for Earth Dynamics. Gaposchkin and Kolaczek (eds.), pp. 191-194, Reidel, 1981.

L. D. HOT HE M, and C. J. F RONCZ E K : Report on test and demonstration of MacrometerTM model V-1000 interferometric surveyor, Federal Geodetic Control Committee, Report F GC C - - IS-83-2 , 1983, NGS, NOAA, Rockville, MD.

P. G. MAC DORAN, D. J. S P I T Z M E SSE R, and L.A. B UE NNAGE L : Series : Satellite emission range inferred Earth surveying. Proceedings of the Third/nternationa/Geodetic Symposium on Satellite Doppler Positioning, pp. 1143-- 1164, Feb. 8-12, 1982.

8.W. RE MONOI : GPS geodetic receivers - a status update report. Proceedings of the ACSM/ASP Convention. Washington, D.C., pp. 441--448, March 13--18, 1983.

J.J. SPILKE R, Jr. : GPS signal structure and performance characteristics. Navigation. Vol. 25, pp. 121-146, 1978.

P. WARD : An advanced NAVSTAR GPS geodetic receiver. Proceedings of the Third International Geodetic Symposium on Satellite Doppler Positioning, pp. 1123--1142, Feb. 8--12, 1982.

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