^3-B00a NAVAL POSTGRADUATE SCHOOL Monterey, California "£7317 OPTIMIZED OBSERVATION PERIODS REQUIRED TO ACHIEVE GEODETIC ACCURACIES USING THE GLOBAL POSITIONING SYSTEM by Richard II. Bouchard March 1988 Co-Advisor Co-Advisor Stevens P. Tucker Narendra K. Saxena Approved for public release; distribution is unlimited. T238737 Unclassified Security classification of this caze REPORT DOCUMENTATION PAGE la Report Secuntv Classification Unclassified lb Restrictive Markings 2a Secunt) Classification Authority 2b Declassification Downgrading Schedule 3 Distribution Availability of Report Approved lor public release: distribution is unlimited -i Performing Organization Report Number(s) 5 Monitoring Organization Report Number(s) 6a Name of Performing Organization Naval Postgraduate School 6b Of nee Symbol (if applicable) 35 7a Name of Monitoring Organization Naval Postgraduate School nc Address i rir, stale, and ZIP code) Monterev. CA 93943-5000 7b Address i city, surf, and ZIP code) Monterev. CA 93943-5000 Sa Name of Funding Sponsoring Organization 8b Office Symbol i if applicable) 9 Procurement Instrument Identification Number Sc Address < cicv. state, and ZIP code; 10 Source of Funding Numbers- Program Element No Project No Task No Work Unit \ccession ii Tale (include securitv classification) OPTIMIZED OBSERVATION PERIODS REQUIRED TO ACHIEVE GEODETIC ACCURACIES USING THE GLOBAL POSITIONING SYSTEM 12 Personal Author(s) Richard H. Bouchard 13a T>pe of Report Master's Thesis 13b Time Covered From To 14 Date of Report (vear, month, day) March 1938 I 5 Page Count 74 16 Supplementary Notation The views expressed in this thesis are those of the author and do not reflect the official policy or po- sition of the Department of Defense or the U.S. Government, Cosau Codes Field Group Subgroup 18 Subject Terms I continue on reverse if necessary and identify by block number) Global Positioning System, relative geodesy, Trimble 4000SX, long baseline determination, triple difference, GPS, PDOP, optimized observing periods, geodetic accuracy. 19 Abstract i continue on reverse if necessary and identify by block number) Measurements of a 1230-km baseline were made during an eight-week period in the fall of 1987 using Trimble 400USX single-frequency, five channel Global Positioning System (GPS) receivers. Twenty-eight days of carrier phase data were processed using correlated triple differences with fixed satellite orbits, the broadcast ephemerides, a modified Hopfield tropospheric model, and without ionospheric correction to determine the accuracies and precisions of the slope distance and baseline components. The data were processed in ever increasing observing sessions to detennine the optimized observation periods required to achieve various orders of geodetic accuracies. The accuracies of the slope distances were better than 1.0 ppm for any observing period. The day-to-day repeatabilities oi the slope distance measurements were better than 1.0 ppm (2a) alter 20 minutes of observations. Accuracies and repeat- abilities {2a) of the baseline components were better than 10.0 ppm after 20 minutes of observations. The correlated triple difference results were on the order of previous GPS surveys that used higher resolution differencing or external timing aids. Discussions include the effects of ephemeris, tropospheric and ionospheric errors, and dilution of precision. Observation periods and mean slope distance errors were reduced when observations started close to and included the infinite peak of the Position Dilution of Precision (PDOP). The smallest variances were associated with observations about the infinite PDOP peak. 20 Distribution Availability of Abstract '3 unclassified unlimited D same as report D DTIC users 21 Abstract Security Classification Unclassified 22a Name of Responsible Individual Stevens P. Tucker 22b Telephone {include Area code) (408)-646-3269 22c Office Svmbol 68Tx DD FORM 1473.84 MAR 83 APR edition may be used until exhausted All other editions are obsolete security classification of this page Unclassified Approved for public release; distribution is unlimited. Optimized Observation 1'eriods Required to Achieve Geodetic Accuracies Using the Global Positioning System by Richard II. Bouchard Lieutenant, United States Navy B.S., Lyndon State College, 1979 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN METEOROLOGY AND OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL March 1988 ABSTRACT Measurements of a 1230-km baseline were made during an eight-week period in the fall of 1987 using Trimble 4000SX single-frequency, five channel Global Positioning System (GPS) receivers. Twenty-eight days of carrier phase data were processed using correlated triple differences with fixed satellite orbits, the broadcast ephemendes, a modified Hopfield tropospheric model, and without ionospheric correction to determine the accuracies and precisions of the slope distance and baseline components. The data were processed in ever increasing observing sessions to determine the optimized obser- vation periods required to achieve various orders of geodetic accuracies. The accuracies of the slope distances were better than 1.0 ppm for any observing period. The day-to-day repeatabilities of the slope distance measurements were better than 1.0 ppm (2a) after 20 minutes of observations. Accuracies and repeatabilities (2a) of the baseline components were better than 10.0 ppm after 20 minutes of observations. The correlated triple difference results were on the order of previous GPS surveys that used higher resolution differencing or external timing aids. Discussions include the ef- fects of ephemeris, tropospheric and ionospheric errors, and dilution of precision. Observation periods and mean slope distance errors were reduced when observations started close to and included the infinite peak of the Position Dilution of Precision (PDOP). The smallest variances were associated with observations about the infinite PDOP peak. in ■1$ TABLE OF CONTENTS I. INTRODUCTION 1 II. BASELINE DETERMINATION USING GPS 3 A. INTRODUCTION 3 B. THE MONTEREY-SAND POINT BASELINE 3 1. General 3 2. Monterey Coordinates 5 3. Sand Point Coordinates 7 4. Monterey-Sand Point Baseline Components 8 C. THE ONE WAY CARRIER BEAT PHASE 8 D. DIFFERENCING THE ONE-WAY CARRIER BEAT PHASE 9 1. Single Difference 9 2. Double Difference 10 3. Triple Difference 10 E. ERROR EFFECTS 11 III. DATA COLLECTION, PROCESSING, AND ANALYSIS 14 A. EQUIPMENT CONFIGURATION 14 1. Hardware 14 2. Software 14 B. SATELLITE OBSERVATION PLAN 15 1. Satellite Selection 15 2. Position Dilution of Precision (PDOP) 15 C. METEOROLOGICAL PARAMETERS 16 D. PROCESSING SOFTWARE 18 E. PROCESSING PROCEDURES 18 1. General 18 2. Batch Processing 19 F. ANALYSIS PARAMETERS 20 G. DATA AVAILABILITY 21 IV IV. RESULTS AND DISCUSSION 24 A. GENERAI 2-4 B. ACCURACY 24 1. Slope Distance 24 2. Baseline Components ', 27 C. REPEATABILITY 30 1. Slope Distance 30 2. Baseline Components 31 3. Standard Deviation of the Mean 33 D. ERROR EFFECTS 33 1. 7-Day Means 33 2. Ephemeris Errors 34 3. Ionospheric Errors 35 4. Tropospheric Errors 37 5. 7-Day Repeatability 39 E. EFFECTS OF THE C/A CODE 40 F. COMPARISON WITH PREVIOUS STUDIES 42 G. MISSING EPOCHS 43 H. DILUTION OF PRECISION AND RANGE ERRORS 46 V. CONCLUSIONS AND RECOMMENDATIONS 50 A. CONCLUSIONS 50 B. RECOMMENDATIONS 51 APPENDIX . BATBLD.BAS LISTING 54 REFERENCES 57 INITIAL DISTRIBUTION LIST 61 LIST OF TABLES Table 1. MONTEREY ANTENNA LOCATION SURVEYS 5 Table 2. RESULTS OF MONTEREY ANTENNA LOCATION SURVEYS 6 Table 3. RESULTS OF SAND POINT ANTENNA LOCATION SURVEY 7 Table 4. STANDARD BASELINE DISTANCES AND 2-SIGMA VALUES. ... 8 Table 5. DAYS USED IN THE DATA ANALYSIS 22 Table 6. OBSERVATION DAYS NOT USED IN HIE ANALYSIS 22 Table 7. MEAN SLOPE DISTANCE ERROR FOR CASES 1, 2, 4, AND 5 ... 26 Table 8. CASE 3 MEAN ERRORS 26 Table 9. MEAN AX ERROR FOR CASES 1, 2, 4, AND 5 27 Table 10. MEAN AY ERROR FOR CASES 1, 2, 4, AND 5 28 Table 11. MEAN AZ ERROR FOR CASES 1, 2, 4, AND 5 28 Tabic 12. SLOPE DISTANCE 2-SIGMA VALUES FOR CASES 1, 2, 4, AND 5 30 Table 13. CASE 3: 2-SIGMA VALUES 31 Table 14. AX 2-SIGMA VALUES FOR CASES 1, 2, 4, AND 5 32 Table 15. AY 2-SIGMA VALUES FOR CASES 1, 2, 4, AND 5 32 Table 16. AZ 2-SIGMA VALUES FOR CASES 1, 2, 4, AND 5 33 Table 17. CASE 1 ERROR: 7-DAY MEANS 34 Table 18. GROUP 2-SIGMA VALUES FOR THE ERROR SOURCES 39 Table 19. CASE 1: 2-SIGMA VALUES, 7-DAY GROUPINGS 40 Table 20. BEST C/A CODE RESULTS AND ERRORS 41 Table 21. CASE 1: DATA SEGMENTS BEFORE AND AFTER REPROCESS- ING 41 Table 22. ELEVATION ANGLES 49 VI LIST OF FIGURES Figure 1. Monterey-Sand Point Baseline and environs A Figure 2. Satellite availability 15 Figure 3. Poor and good PDOP 17 Figure 4. PDOP versus time 17 Figure 5. Sky Plots of satellite tracks for Monterey 29 Figure 6. Mean age of data (ephemeris) 35 Figure 7. DilFerence between observation end time and 0600 PST 36 Figure 8. Flectron fluence 36 Figure 9. DilFerence in refractivity between Monterey and Sand Point 38 Figure 10. Mean Monterey-Sand Point refractivity 39 Figure 1 1. Results of station 2 offset 43 Figure 12. Fraction of available triple difference observations 44 Figure 13. Missing epochs while tracking SV 6 45 Figure 14. Continuous tracking without SV 6 45 Figure 15. Relative positioning geometry 47 vu LIST OF SYMBOLS b Baseline distance (slope distance) bM Measured baseline distance bT True baseline distance c Speed of light d Day e Elevation angle f Frequency h Highest satellite i Observation epoch identifier 1 Integer number of epochs n Refractivity n, Number of epochs ns Number of satellites r Receiver identifier s Satellite identifier v Angle between the vector tangent to the ellipsoid and the slope distance vector A(r,s,i) Initial integer ambiguity C Component or coordinate DD(h,s,i) Double difference E Error E Mean error MTD Maximum number of triple difference observations N Refractive index SD(s,i) Single difference TD(i) Triple difference X X coordinate Y Y coordinate Z Z coordinate o(i) Difference between receiver clock times VUl y Angle between slant range vectors from a satellite to two ground stations 4> Phase of the GPS carrier signal p Satellite-receiver slant range p Time rate of change of p a Standard deviation t(r,i) Tropospheric delay 6 Angle between the slope distance vector and the elevation angle to a satel- lite £(/') Average olTset of receiver clock time C(r,i) Common receiver clock errors A Difference AX Baseline component in the X direction AY Baseline component in the Y direction AZ Baseline component in the Z direction IX LIST OF ABBREVIATIONS AND ACRONYMS ppm parts per million AODE Age of Data (Ephemeris) C A Coarse/Acquisition Code DMA Defense Mapping Agency DMAHTC DMA Hydrographic Topographic Center DOD Department of Defense EF Electron Fluence GDOP Geometric Dilution of Precision GPS Global Positioning System MRY Monterey GPS antenna location NAD83 North American Datum 1983 XGS National Geodetic Survey NOAA National Oceanic and Atmospheric Administration NPS Naval Postgraduate School Ob Observing or observation PDOP Position Dilution of Precision PST Pacific Standard Time ( + 8 hours UTC ) SEA-TAC Seattle-Tacoma International Airport SPt Sand Point GPS antenna location SV Satellite Vehicle TDOP Time Dilution of Precision TEC Total Electron Content TOW Time Of the Week UTC Universal Coordinated Time VLBI Very Long Baseline Interferometry WGS84 World Geodetic System 1984 WSO National Weather Service Office I. INTRODUCTION A priori knowledge of the observation periods required to achieve specified orders of geodetic accuracies is important in planning efficient and productive geodetic surveys. While terrestrial surveys have specified field and processing procedures and standards to categorize the geodetic accuracies of surveys [Federal Geodetic Control Committee, 1984], only recently have standards been proposed for surveys conducted with the Global Positioning System (GPS) [Federal Geodetic Control Committee, 1986]. Among the proposed requirements are standards for the length of observing periods and satellite geometry. Field studies by Remondi [1984] and numerical simulations by Fell [1980], Langley et al. [1984], and Landau and Eissfeller [1986] studied optimized observation periods, but for baselines less than 100 km. Cannon et al. [1985], Bock et al. [1984], Goad et al. [1985], Mader and Abell [1985], and Bertiger and Lichten [1987] conducted long baseline surveys, but did not study optimized observation periods. One of the objectives of this thesis is to fill the gap between the above studies, i.e., examine the optimized observation periods for a long baseline. The optimized times will be examined using the correlated triple dilference carrier beat phase observable because of its insensitivity to integer ambiguities and loss of lock of the GPS carrier by the receiver. Another objective of this thesis is to add to the body of triple difference accuracy testing following a recommendation by Remondi [1984, p. 259]: "More testing is required to establish the full accuracy potential of the triple dif- ference method. " GPS carrier phase and pseudorange measurements were made during an eight-week period in the fall of 1987 with Trimble 4000SX single-frequency, 5 channel receivers. The long baseline is approximately 1230 km in length between the National Oceanic and Atmospheric Administration's (NOAA) Western Regional Center located at Sand Point in Seattle, Washington, and the Naval Postgraduate School (NTS), Monterey, California. The baseline was determined by locating the positions of its ends by con- necting them by independent short baseline surveys from nearby Very Long Baseline Interferometry (VLB I) horizontal control points. The results of those surveys form the reference to which accuracy will be determined. Additionally, studies for repeatability were conducted following another recommen- dation by Remondi [1984, p. 263] to enhance the capabilities of GPS measurements. The recommendation was to perform extensive repeatability studies on non-varying baselines for verifying and improving the GPS modelling. II. BASELINE DETERMINATION USING GPS A. INTRODUCTION The Department of Defense's (DOD) Global Positioning System is intended to provide accurate positioning and precise timing for navigation purposes by broadcasting codes superimposed on two radio carrier frequencies from satellites. The satellites are placed in a constellation so that at least four satellites are visible globally. The Precise Code (P code) will be limited to authorized DOD users. The Coarse Acquisition (C/A) code provides real-time accuracies to about 100 m [Baker, 1986] and is available to anyone. The codes provide their transmit times, satellite orbit and clock information, and information to enable any receiver to lock onto other GPS satellites. The orbital infor- mation (ephemeris) provides the position of the satellite. The receiver measures the time delay between the receipt of the C/A code and its transmission time. The time delay can be transformed into an apparent slant range from the satellite's known position to de- termine the location of the receiver. Since it includes delays due to receiver clock errors and the effects of atmospheric refraction, the apparent slant range is referred to as the pseudorange. A minimum of four satellites are required to solve the system of range equations for the receiver's coordinates and clock errors. While the C/A code provides real-time location, it does not meet the accuracy re- quired for precise geodetic work. Nevertheless, GPS makes possible a higher resolution via the carrier signal. Though the carrier itself does not contain the orbital and timing information, which would have to be supplied by some other means, it does offer a higher resolution because of its 19-cm wavelength. B. THE MONTEREY-SAND POINT BASELINE 1. General The length of the Monterey-Sand Point baseline (Figure 1) was computed by differencing the World Geodetic System 1984 (WGS84) [Defense Mapping Agency, 1987] Cartesian coordinates determined for Monterey and Sand Point by short baseline GPS surveys from known horizontal control points. The precision and agreement with terrestrial survey results of short baseline GPS measurements using single frequency, double difference solutions are well documented {e.g., [Remondi, 1984], [Goad and Remondi, 1984], [Bock et ai, 1984]). 45° n -j-y -f- Or 35°N~{- 125° W 120° W Figure 1. Monterey-Sand Point Baseline and environs: Insets not drawn to scale. At Sand Point, on-site meteorological measurements were made near the middle of the observing session. For the Monterey antenna determination meteorological measurements were made every half hour and the mean of all the measurements was used in the processing. The Trim640 solutions were obtained using uncorrected double differences and estimating initial integer ambiguities. The ambiguities were fixed to the integer values that produced the smallest residuals. A tropospheric factor was estimated along with the integer ambiguities and the baseline components in the least-squares processing. The horizontal control points used for the reference stations in determining the coordinates of the antennas were mobile Very Long Baseline Jnterferometry (VLBI) sites. The NAD83 Cartesian coordinates for the VLBI sites were provided by the the Gravity, Astronomy and Space Geodesy Branch of the National Geodetic Survey (NGS) [Abell, 1987]. The NAD83 coordinates were determined in August 19S7 from a global adjustment of VLBI surveys. Carter et al. [1985] described the NGS VLBI program. The Defense Mapping Agency Hydrographic.'Topographic Office (DMAHTC) validated the direct transformation of the VLBI Cartesian coordinates to WGS84 Cartesian coor- dinates [Kumar, 1988]. 2. Monterey Coordinates The coordinates for the Monterey antenna location were determined by aver- aging two surveys conducted on separate days from the VLBI site FT ORD NCMN 19S1. Table 1 lists the observing sessions used to determine the WGS84 coordinates of the Monterey antenna. Table 2 lists the results of the GPS survevs. Table 1. MONTEREY ANTENNA LOCATION SURVEYS: NCMN 1981. PST = Pacific Standard Time. From FT ORD Date Start Time (PST) End Time (PST) Number of Double Difference Observations Percent Rejected Slope Distance (m) ab 09 16 87 09 18 87 1018 0914 1210 1120 996 1136 7 4 12139.8125 12139. SI 19 0.025 0.025 The Monterey coordinates were computed from: Cmry~ Cord + Ac where: CMBV Monterey coordinate Ft Ord coordinate Variance-weighted mean baseline component The uncertainties of the Monterey coordinates, oc^RY , were computed from: MRY (-ORD 5. Table 2. RESULTS OF MONTEREY ANTENNA SURVEYS: WGS84 Cartesian Coordinates (meters). LOCATION DATE FT ORD X aORD X AX °.\x MRYX ® MRY X 09/16 87 09 1887 -2697026.493 -2697026.493 0.007 0.007 -10313.601 -10313.602 0.032 0.037 -2707340.093 -2707340.095 0.032 0.037 Mean: -2707340.094 0.036 DATE FT ORD Y aORD Y AY °1Y MRY Y G \(9Y Y 09 16,87 09 T 8 87 -4354393.309 -4354393.309 0.010 0.010 917.693 917.689 0.048 0.049 -4353475.617 -4353475.620 0.049 0.050 Mean: -4353475.618 0.05 1 DATE FT ORD Z aORDZ AZ <7*Z MRYZ 0 MRYZ 09 16,87 09 18 87 3788077.778 3788077.778 0.009 0.009 -6337.391 -6337.389 0.041 0.044 3781740.387 37S1740.389 0.042 0.045 Mean: 3781740.388 0.044 i / 2 2 LMRY — \ LORD \ where ornnn is the uncertainty of the Ft Ord coordinate. LORD One month prior to the surveys originating from Ft Ord, two other surveys were conducted from the satellite Doppler horizontal control point NAVAL POST GRAD 31965,'DOPPLER. The Doppler station is approximately 300 m north of the Monterey antenna location (Figure 1). The double differenced GPS carrier phase solutions of the two Doppler-originating surveys yielded the following Monterey coordinates: X -2707339.725 m Y -4353475.654 m Z 3781740.264 m The baseline components had an uncertainty of ±0.002 m, but the Doppler station has an uncertainty of ±2 m in each coordinate before transformation to WGS84. The results of the Doppler surveys were not used in determining the Monterey coordinates because of the large uncertainty in the Doppler station location. The three-dimensional positions of the Monterey antenna from the Doppler and the VLBI originating surveys agree to better than 0.5 m. The two days of pseudorange observations at the Monterey receiver were each subjected to a least-squares estimation and then averaged together to yield the Monterey coordinates: X -2707334.248 ±1.3 m Y -4353466.S07 ±0.6 m Z 3781741.330 ±2.2 m where the two-day standard deviations are given. The deviation of the pseudorange from the differenced carrier phase derived Monterey coordinates is expected because of the coarser resolution of the C/A code and because the pseudoranges are corrected for neither tropospheric nor ionospheric delays. 3. Sand Point Coordinates The Sand Point antenna coordinates were determined by one 90-minute GPS survey (Table 3) from the mobile VLBI site Aviation 2 which is 530 m distant (Figure 1). Table 3. RESULTS OF SAND POINT ANTENNA LOCATION SURVEY: From double difference carrier phase solutions. Distances and WGS84 Cartesian Coordinates are in meters. Aviation 2 Coordinate °A\1 A Sand Point Coordinate GSFt X= -2295347.760 Y= -363S029.429 Z= 4693408.964 0.017 0.028 0.032 -408.361 330.200 73.813 0.002 0.002 0.003 -2295756.121 -3637699.22S 4693482.777 0.017 0.0 28 0.032 The Sand Point antenna coordinates computed from the pseudorange data were: X -2295749.623 m Y -3637694.279 m Z 4693488.472 m and standard deviations are not given because this is a single observing session and the program does not provide solution standard deviations. 4. Monterey-Sand Point Baseline Components First the Monterey-Sand Point baseline components were computed by sub- tracting the Sand Point Cartesian coordinate from the respective Monterey Cartesian coordinate. The uncertainties in the baseline components were computed as the square root of the sum of the squares of the uncertainty of the Monterey and Sand Point co- ordinates. The slope distance was computed as the square root of the sum of the squares of the baseline components. The slope distance uncertainty was computed as: /, AX .2 , , AT .2,, AZ ,2 where b is the slope distance. Using the information from Tables 2 and 3 gives Table 4, which is used as a standard to estimate accuracy. Table 4. STANDARD BASELINE DISTANCES AND 2-SIGMA VALUES. AX -411583.973 + 0.080 m (0.19 ppm) AY -715776.390 + 0.116 m (0.16 ppm) AZ -911742.388 ± 0.109 m (0.12 ppm) Slope Distance 1230045.280 ± 0.109 m (0.09 ppm) C. THE ONE WAY CARRIER BEAT PHASE The development of the carrier beat phase technique and a model for its application are given by Remondi [1984]. The phase measurement is done by beating the received carrier with a local oscillator internal to the GPS receiver. The slant range from a GPS receiver to a GPS satellite can be modelled in terms of the time it takes the signal to travel or the number of cycles that occur between the satellite and the receiver. The range in cycles will consist of an integer and fractional number of cycles. When a GPS receiver locks onto the carrier signal, it can immediately measure the fractional part and begin counting subsequent integer cycles, but it cannot measure or account for the initial integer number of cycles that preceded the initial fractional part. These missing cycles bias subsequent measurements and are called the initial integer ambiguity biases. The signal does not take a direct path to the receiver as it is refracted by the ionosphere and troposphere. Additional errors are caused by the satellite deviating from its predicted orbit, errors in the satellite clock, and errors in the receiver clock. Anti- cipating that the observables will be used in relative positioning, that a single frequency receiver will be used and ignoring other error sources, such as multipath, the one-way carrier beat phase, 4>b(r,s.i), observed at receiver, r, from satellite, s, at observation epoch, i, can be modelled to first-order as: fs b(r,stt) = s{i) - (f)r{r,i) - — p{r,s,i) - x{r,i) + A{r,s,i) where: / Epoch identifier r Receiver identifier 5 Satellite identifier 0.(0 Phase of received carrier signal 4>r(r,i) Phase of receiver generated carrier signal fs Transmitted frequency of carrier signal p{r,s,i) Satellite-receiver slant range p{r,sj) Time rate of change of slant range c Speed of light fr Receiver generated carrier frequency r(r,i) Tropospheric delay A{r,s,l) Initial integer ambiguity S(i) 2 S(i) C(i,0 = ^(0- C(2,o = «o + c{i) Mean clock offset for both receivers ()(/') Clock, drift between both receivers and parentheses do not indicate factors or functions, but simply enclose identifiers. Brackets indicate factors. D. DIFFERENCING THE ONE-WAY CARRIER BEAT PHASE 1. Single Difference The single difference (SD) is formed by differencing carrier beat phase observa- bles from two receivers at the same observation epochs. Following Remondi [1984], taking the difference, expanding the £(r,i) terms of Equation (2.1), and expressing dif- ferences in the between-satellite and between-receiver phases as f.S(i) gives: SD(s,i) = b(2,s,?)-(t>b{l,s,i) = 4-Cp(2,v')-p(U0] + 4-Cp(2.v)-p(l,5./)]^(/) (2.2j -tfi-filtW+fAQ + -±L-lp(2,s,i) + p{l,s,i)ld(i) +y;[r(2,/) - t(1,/)] + ^(2,5,1) - ^(1,5,1) where the terms are the same as Equation (2.1). Single differencing reduces or eliminates satellite orbital and clock errors because they are common to both receivers. 2. Double Difference A double difference {DD) is formed by differencing single differences between a reference satellite, h, and another satellite at the same epoch: DD{h,s,i) = SD{h,i) - SD(s,i) Because the differences at the same epoch are taken with the same reference satellite, the double differences for each epoch are correlated. The advantage of the double differences is that the clock dependent terms - fsS(i) and [f2 — fx ]£(/) - are elim- inated. The significance of the removal of those terms is to reduce from nanoseconds to microseconds the timing accuracy required to achieve one cycle accuracy. The Trimble 4000SX achieves sub-microsecond accuracy by using the C/A code timing in- formation [Ashjaee, 1985]. 3. Triple Difference A triple difference (TD) is formed by differencing the double differences for the same satellite pair at some integer number of succeeding epochs, / : TD{i) = DD(h,s,i + /) - DD(h, sf?) The advantage of the triple difference is that it eliminates all the time inde- pendent terms, namely the initial integer ambiguities, A(r,s,\), and becomes insensitive to the initial ambiguities and any cycle slips when the receiver loses lock. The disadvantages of the triple difference are: another level of correlation, loss of resolution, and reduced number of observations. The triple differences are already correlated with respect to satellite because of the underlying double differences, and are 10 further correlated with respect to time because consecutive triple differenced observa- tions will have the DD{h,s,i + I) term in common. For short baselines, integer ambiguities can easily be resolved because unmod- elled errors are highly correlated between the two antenna sites and are mostly elimi- nated by the differencing. Algorithms can take advantage of the integer nature of the initial ambiguities and solve for them. At longer baselines the unmodelled errors are not as highly correlated and not eliminated by the differencing. These errors fold into the initial ambiguities, so that the ambiguities are no longer integers. In some cases the ambiguities cannot be resolved (e.g., Henson and Collier, 1986, Tables 1 and 2). Because of the advantages the triple difference offers for long baselines, I use the triple difference scheme. While the triple difference can be decorrelated by forming a weight matrix [Remondi, 1984], only the correlated triple difference software was avail- able to me. E. ERROR EFFECTS For long baseline GPS surveys, the primary errors are satellite orbit errors, ionospheric and tropospheric delays [Remondi, 1984, and Beutler et ai, 1986]. Orbit (ephemeris) errors are the result of the departure of the satellite from the broadcast ephemeris orbit. The ephemeris is a predicted orbit for the satellite. Orbit errors prop- agate directly into the baseline measurements when the GPS orbit coordinates are fixed in the processing [Hothem and Williams, 19S5]. Orbit errors can be the dominant error source affecting the repeatability of long baseline measurements [Lichten and Border, 1987]. The magnitudes of baseline errors increase with increasing baseline length because of the increasing projection of the ephemeris error onto the baseline component [Fell, 19S0J. Estimates of the broadcast ephemeris error range from 25 m [Beutler et ai, 1986] to 100 m [Wells et al, 1986]. The magnitude of the effect of the ephemeris error on baseline accuracy has been traditionally approximated [Beutler et al., 19S4, Equation (2.1)]: A.ie. (2.3) b P K } where: b baseline length p slant range (receiver to satellite) 11 Eb error in baseline length Ep error in slant range The slant range to a GPS satellite is about 20,000 km, which translates to a baseline er- ror ranging from 1 ppm to 4 ppm using Equation (2.3). It is expected that the ephemeris error for the Monterey-Sand Point baseline will be towards the lower end of the range because the ephemerides are uploaded prior to the satellites entering the Yuma Proving Ground [Russell and Schaibly, 1980] near the California border. The ephemeris linearization error specification is to 1 m per day [Weils et al., 1986]. The ionosphere disperses the code (the group velocity) from the carrier phase (phase velocity) because the C/A code has a frequency of 1 MHz while the carrier signal upon which it is superimposed has a frequency of 1575 MHz. The effect is to increase the pseudoranges, but decrease the carrier phase derived ranges [Smith, 1987]. Field exper- iments [Beutler et al., 1986] showed that ionospheric dispersion shortens baselines on the order of a few tenths to perhaps 2 ppm. Ionospheric error is proportional to the Total Electron Content (TEC) along the signal path and the cosecant of the elevation angle of the satellite [Smith, 1987]. Thus the error is greatest for low elevation angles and least at the zenith. Wells et al. [1986] estimated that the range errors due to the ionosphere are from 150 m at the horizon to 50 m at the zenith. Ionospheric activity is a function of latitude, longitude, time of day, season, and sunspot activity. Ionospheric activity increases towards the equator and towards the sunlit portions of the earth. Diurnally, it has a minimum near 0600 local time with a maximum around 1600 local. Ionospheric activity increases with the peak in the sunspot cycle. The minimum in the current 11-year sunspot cycle occurred in 1986. Upon these systematic characteristics sporadic ionospheric disturbances are superimposed. [Henson and Collier, 1986] The tropospheric error is proportional to the refractivity along the satellite-receiver path and proportional to the cosecant of the elevation angle [Martin, 1980]. The index of refraction, n, is the ratio of the speed of light in a vacuum to the speed of light in a particular medium, in this case the troposphere. Because the refractive index is a small fraction greater than 1.0, a more convenient unit to work with is refractivity, N, where N = {n - 1) x 106 . The magnitude of the tropospheric biases range from 20 m for 10° elevation angles to 2.3 m at the zenith [Wells et al., 1986]. 12 Henson and Collier [1986] have shown that triple difference measurements are un- affected by path-dependent ionospheric bias errors, but by path-dependent gradient ionospheric errors, and Martin [1980] has estimated that the combined ionospheric and troposphenc gradient errors are on the order of meters per hour and are proportional to the cosecant of the elevation angle. 13 III. DATA COLLECTION, PROCESSING, AND ANALYSIS A. EQUIPMENT CONFIGURATION 1. Hardware A complete description of the Trimble 4000SX receiver is given by Trimble Navigation [1937a]. NPS operates three Trimble 4000SX GPS Surveyor receivers, of which two were used in this study. The 4000SX is capable of observing the C A code, integrated Doppler and carrier beat phases of up to five satellites simultaneously. Its ability to use the C/A code allows the receiver to be used as a stand alone navigation system which determines position using Doppler-smoothed pseudoranges and velocities [Ashjaee, 1985]. For precise relative positioning the 4000SX can transmit its data through an RS232 port to a microcomputer for storage on floppy disk for post-processing. The 4000SX's ability to use the C/A code allows it to decode the GPS navigation messages so that it can track satellites automatically and determine. Most importantly, it uses the C/A code in a time transfer mode to determine any offset and drift of its own clock and thus provide accurate time tags for the observations without an external atomic clock or synchronization with the receiver at the other end of the baseline. The receivers were left on continuously to allow unattended data collection. Multipath-resistant Trimble microstrip antennas were installed at both the Monterey and Sand Point locations. 2. Software For relative positioning, the receiver is controlled from the microcomputer by Version D of Trimble's Datalogger program. The reference position (the geodetic coor- dinates of the antenna) and the particular options chosen must be entered into the re- ceiver via the receiver keypad. The receivers were set to use the reference position height for point positioning when less than four satellites were available. Each observation session was initialized to log data when a minimum of four satellites were 15° above the antenna's horizon. Five satellites were designated for each observing session. The software logs the observables and receiver clock parameters to a floppy disk every 15 seconds and the C/A code-determined antenna position and Po- sition Dilution of Precision (PDOP) every Five minutes. The GPS navigation message is logged to a separate file at the beginning of the session. 14 B. SATELLITE OBSERVATION PLAN 1. Satellite Selection The same five satellite (SV) (6, 9, 11, 12, and 13) were used for the entire eight- week observation period. These five satellites were visible at both stations for over 100 minutes, and of these five satellites four were visible for three hours (Figure 2). START DATE/TIME: 1987/10/17 STOP DATE/TIME: 1987/10/17 DATA AVAILABLE: STATION: SAND POINT SV 6 | 15:21:15. 18:16: 1. DAY OF YEAR 290 DAY OF YEAR 290 TON TOW 573675. 584160. 1 1 SV 9 I SV 11 | | SV 12 | | SV 13 1 1 STATION: MONTEREY SV 6 I 1 1 SV 9 | SV 11 | I SV 12 | I SV 13 | I Figure 2. Satellite availability: Each dot represents 10 observations; each col- umn, 10 epochs. From Trim640 output. 2. Position Dilution of Precision (PDOP) Trimble Navigation [1987b] recommends that observations include the time that the PDOP goes to infinity. The Federal Geodetic Control Committee [1986] notes that initial results from investigations indicate that best results may be achieved when the Geometric Dilution Of Precision (GDOP) is changing value during the observing ses- sion, and proposes that observing sessions start with a high GDOP and stop with a low GDOP. Position Dilution of Precision is a component of the GDOP. GDOP is a measure of how satellite geometry degrades point position accuracy [Jorgensen, 1984]. For computational ease in the navigation solution the GDOP is defined as the square root of the trace of the co variance matrix of the errors imposition and time with the range errors set to one [Milhken and Zoller, 1980]. The role and definition of GDOP and PDOP in GPS point positioning were applied to GPS relative positioning, i.e., good 15 PDOP would provide better accuracy than poor PDOP [King et al. 1985]. Good and poor PDOP are shown in Figure 3. Landau and Eissfeller [1986], using numerical simulations in which they assumed a full 18 satellite constellation and a receiver that could track those satellites that mini- mized GDOP, found that better accuracy for triple difference solution for a 68 km baseline generally corresponded to high GDOP. They used a more complete GDOP that included consideration of ionospheric, tropospheric and satellite position errors which are neglected in the conventional GDOP. The 4000SX does not record GDOP, but it does record PDOP every five min- utes. PDOP relates to GDOP as: GDOP2 = PDOP2 + TDOP2, where TDOP is the Time Dilution Of Precision, the error in the user clock bias multiplied by the velocity of light. The expected uncertainty in a GPS point positioning solution is a product of the PDOP and the expected slant range error. The difference between PDOP's at Monterey and Sand Point remained less than 1.0 for the entire eight week observing period. The PDOP peaks at two times in an observing session (Figure 4) - 60 minutes and 150 minutes. The PDOP peaks occur near when the satellites lie in a common plane causing the sol- ution of the linearized range equations to diverge and the PDOP becomes infinite [Jorgensen, 1984]. C. METEOROLOGICAL PARAMETERS Meteorological parameters are needed for the tropospheric correction model used in the processing software. On-site meteorological observations were not available for Sand Point. Instead observations from the Weather Service Office (WSO) of the Seattle-Tacoma International Airport (SEA-TAC) were used for Sand Point. SEA-TAC is approximately 27 km from Sand Point (Figure 1). The NPS Department of Meteor- ology routinely collects real-time hourly observations of sea-level pressure, temperature and relative humidity from the National Weather Service's data network. Observations that pertained to GPS observing sessions were entered into a file that is accessed by the batch file building program. While the Federal Geodetic Control Committee [1986] has proposed the use of on- site meteorological parameters, researchers have had success using standard atmosphere parameters for satellite geodesy [Fell, 1976] or extrapolated meteorological data [Rathacher et al., 1986]. 16 POOR PO°p GOOD PDOP Figure 3. Poor and good PDOP: From King et at, 11985, Fig. 3.2]. Figure 4. PDOP versus time 17 Hourly observations for Monterey were obtained from the NTS Department of Meteorology. The instruments were located approximately 300 m south of the Monterey antenna location (Figure 1). D. PROCESSING SOFTWARE A complete description of the Trimble supplied Trimvec software can be found in Trimble Navigation [1987b]. The data was processed using the Trimvec Trim640 pro- gram, Revision AB. Trim640 limits processing to 700 epochs, so the first 700 epochs for each observing session are used. Sand Point was used as the reference station and its coordinates kept fixed in the least-squares processing. Sand Point was chosen as the reference station because four satellite availability occurred later at Sand Point than at Monterey. This avoided having to load the not-in-common epochs from Monterey at each processing. Trim640 uses the C/A code derived positions obtained at the lowest PDOP for the initial estimates of the baseline components. Trim640 culls the best C, A code position during the data loading. No ionospheric correction is provided by the software, and only the broadcast ephemeris can be used to compute fixed orbit satellite positions. In the triple difference processing, the only parameters estimated by the least-squares process- ing are the baseline components, AX, AY and AZ. A modified Hopfield tropospheric model [Goad and Goodman, 1974] is used to correct the carrier phase delay caused by the troposphere. The correction is a function of the atmospheric refractivity computed from surface meteorological values of pressure, temperature, and humidity, and the elevation angle of the satellite. Larger corrections are required for low elevation angles, as the signal travels a longer path through the troposphere. The model corrects for at least 90% of the tropospheric delay [Remondi, 198--1]. Tritn640 allows only one pressure, temperature and humidity entry for each site per session. E. PROCESSING PROCEDURES 1. General To study optimized times, the data from each observing session were segmented. Each successive segment contained 10 minutes more data than its predecessor. For ex- ample, for an entire observing session that started when four satellites were available and stopped when less than four were available provides 175 minutes of observations. The first segment will use the first 10 minutes of data , the sixth segment will use the first 60 minutes, while the eighteenth will use all 175 minutes. For each segment, the entire 18 processing was restarted from the data loading. Reloading the data for each segment takes considerably longer than using the Trim640 option to flag data for processing, but reloading was done so that the processing does not use a best C/A code position from later in the observing session. Convergence of the least-squares solution was achieved by doing five iterations using every tenth triple difference formed from every tenth double difference, followed by five iterations decreasing the triple and double difference increments to five, and finally five iterations using all triple differences formed from all double differences. Trim640 rejects those observations whose residuals exceed a multiple oi" the mean residual. The multiple of the mean residual is known as the edit multiplier. Trim640 uses 3.5 as the default value for the edit multiplier. I used the default value for the initial processing. Any segment that had more than ten percent of its observations rejected or whose solution slope distance standard deviation (<7S) was greater than 10 m was re- processed. The reprocessing was identical to the initial processing except that before invoking the triple difference process the pseudoranges for both stations were subjected to separate least-squares adjustment. The pseudorange processing improves the C/A code derived initial estimates for the baseline components and corrects the carrier beat phase time tags. The carrier beat phase time tags are computed from the C/A code times, and are earlier than the C A code times. If the pseudorange processing failed to lower the rejections to ten percent, the edit multiplier was increased until the rejections reached ten percent. A ten percent re- jection level was observed for a few sessions and always occurred within the first thirty minutes of observations. The data were transferred to the Naval Postgraduate School's IBM 3033 computer for analysis. The data were analysed and graphics produced using the ^PL-based GRAFSTAT program. To study the effects of reducing observation time, five case studies were under- taken in which the observation start times were changed for processing. Each case study followed the processing procedures outlined above. 2. Batch Processing Processing is performed in a batch mode. A batch file passes parameters to a template. Trimble supplies command files that tell the Trim640 to use the template pa- rameters in processing. 19 The batch file is built using the program Batbld (Appendix A). Baibld builds a batch file by providing the appropriate file names and start and stop times. Baibld computes the appropriate meteorological parameters for each segment by locating the applicable weather observations from the weather observation file, interpolating values at the start time, computing running means from each hourly weather observation, then interpolating the running means to the stop time of each segment. Two millibars were subtracted from the SEA-TAC sea-level pressure to compensate for the 20-m elevation above sea-level for the Sand Point antenna. Initially, processing was done on an IBM XT with a math coprocessor and a hard disk. Processing the 18 segments of an observing session took ten hours of com- puting time. Later, processing was performed on an 80286 based microcomputer run- ning at 10 MHz with an 80287-8 math coprocessor that reduced the processing time to three hours. Two minor problems with Trim640 were discovered during the processing. First, large values in range differences were found when using the range differences rather than the pseudoranges to improve the C/A code positions. The data were for- warded to Trimble Navigation for evaluation and an error was found in their software. The error had no apparent effect upon carrier phase difference processing. Second, Trim64Q is incompatible with one or more of the AST Research, Inc. device drivers supplied with the 80286 microcomputer: ram disk, print spooler, and extended memory. Removing the drivers allowed Trim640 to execute normally. At the conclusion of the batch processing, the slope distance, the baseline components, their standard deviations (er,), the number of observations, the number of observations rejected, and the RMS cycle fits were extracted from the Trim640 output file and collected into files that held the data for a particular segment for each case study. F. ANALYSIS PARAMETERS The statistical parameters that will be used to evaluate the results are the error, the sample mean and the standard deviation defined as [Davis et al ., 1981]: Ed=Cd-Cs (5.1) r-_iVr L ~ m Lid 20 o2= l-r- > {E,-E)2 in - 1 where: Ed Error for day d Q Measured component for day d Cs Expected values E Mean error or bias a2 Variance a Standard deviation of £rf 07 Standard deviation of £ 5.2 Accuracy describes the closeness between the measurements and the expected values [Davis et al., 1981]. The degree of accuracy is determined to the magnitude of the mean error (£). The repeatability of the measurements will be expressed in terms of 2a because it approximates the 95% confidence level for single-dimension measurements [Federal Geodetic Control Committee, 1986]. The slope distance and individual baseline com- ponents are one-dimensional measurements. G. DATA AVAILABILITY Observations were made simultaneously at Monterey and Sand Point for an eight- week, period beginning 29 September 1987. Observations were made Tuesday through Saturday except the days after federal holidays. Forty observing sessions were con- ducted of which 28 were used in the analysis and are listed in Table 5. The remaining 12 days of observations were not used in the analysis for various reasons, which are listed in Table 6. For brevity the observing days will be referred to by their Julian day. Times in Table 5 are given in Pacific Standard Time (PST) rather than Universal Coordinated Time (LTC) for ease in the later discussions on diurnal effects. 21 Table 5. DAYS USED IN THE DATA ANALYSIS Date Julian Day Start Time (PST) End Time (PST) Number of Triple Difference Observations 09 29 87 272 0833 1129 1599 09 30 S7 273 OS 30 1125 1485 10 01 87 274 0825 1121 1501 10 02/87 275 0822 HIS 1479 10 03 87 276 0817 1113 1518 10 06 87 279 0S05 1100 1563 10 07 87 280 0801 1056 1490 10 OS 87 281 0757 1053 1508 10.09 87 282 0753 1049 1500 10, 10 87 283 0749 1044 1522 10/14/87 287 0732 1028 1524 10/17/87 290 0721 1016 1504 10 20/87 293 0707 1002 1540 10 21/87 294 0703 0958 1562 10/22/87 295 0658 0953 1551 10/24/87 297 0651 0946 1493 10/27/87 300 0638 0934 1567 10/28/87 301 0634 0930 1516 10/29/87 302 0631 0926 1681 11,03 87 307 0613 0908 1S48 11/04/87 308 0605 0901 1553 11/05/87 309 0601 0S57 1695 11/10/87 314 0541 0837 1535 11/11/87 315 0536 0832 1927 11/13/87 317 0528 0824 1923 11/14/87 318 0524 0819 1550 11/21/87 325 0455 0750 1595 11/25/87 329 0438 0734 1854 Table 6. OBSERVATION DAYS NOT USED IN THE ANALYSIS. Date Reason for not analyzing 10/15/87 No SEA-TAC weather observations 10/16 87 No satellites at Monterev for first 10 minutes 10/23/87 Slope distance a, > 10 m for first 10 minutes 10/30/87 Disk error 10/31/87 No SEA-TAC weather observations 11/06/87 Unhealthv satellites 11/07/87 Unhealthv satellites 11/17/87 Unhealthv satellites 11/18/87 Unhealthy satellites 11 19/87 Unhealthv satellites 11/20/87 Unhealthv satellites 11/24/87 No satellites at Monterey for first 20 minutes 22 Day 283 was processed with Monterey as the fixed reference station because the data set would not partition into 10-minute segments when Sand Point was used as the reference station. The days that were missing one or two segments were excluded from the analysis, so that changes in the sample variances would not be due to unequal sample populations between the segments. Reprocessing the first segment for 23 October failed to reduce the slope distance sigma to less than 10 m because that segment had only nine triple difference observa- tions because the receiver frequently lost lock on the satellites. That day was not used in the analysis, even though its other segments had slope distance sigma's less than 10 m without reprocessing. 23 IV. RESULTS AND DISCUSSION A. GENERAL To study optimized times, five cases are studied: 1 Process all data when four satellites at least 15° above horizon 2 Begin processing when five satellites at least 15° above the horizon 3 Begin processing as in 2, but delete fifth satellite 4 Begin processing data 40 minutes later than in 1 5 Begin processing data 70 minutes later than in 1 Case 1 is essentially the processing of the full data set. Each of the other cases is a subset of Case 1. Trim640 allows the user to designate at what time within the full ob- serving period that the data loading should begin. For Case 3, Trim640's ability to flag data was used to exclude the fifth satellite (SV 12) from the processing. Each day of Table 5 was processed for each of the five cases. During the course of the discussions it will be necessary to distinguish between ob- servation periods and the time of the observations fixed with respect to satellite geom- etry. As the satellite geometry (PDOP) begins with the availability of four satellites, the time of observations can be defined in terms of the Case 1 start time. Observation pe- riods are determined from the start time for each case. Times of observations will given as equivalent Case 1 times and is obtained by adding the case observation length to the case's time offset from Case 1 (Case 2 and 3's offsets are 20 minutes; Case 4, 40 minutes and Case 5, 70 minutes). B. ACCURACY 1. Slope Distance The slope distance errors for Cases 1, 2, 4, and 5 are presented in Table 7. Then results show that accuracy to better than 1.0 ppm is achieved for any observation period, but that there are differences among the cases and with changing observation periods within each case. Cases 1, 2, and 4 exhibit similar behavior as the observation period increases - they become less positive (or more negative) as the observation period increases until they reach a minimum, then they reverse their trends and become less negative. Positive error indicates that the measured baseline is longer than the standard values, so that the 24 measured baselines are exhibiting an accordion effect as they shorten then lengthen as more observations are included in the solutions. While the minimum occurs after dif- ferent observing periods (140, 110, 90 minutes for Cases 1, 2, and 4 respectively), they occur at about the same absolute time with respect to the Case 1 start time (140, 130, and 130 minutes for Cases 1,2, and 4 respectively). As the start times occur later with each case, the errors for shorter observing periods become less positive and the ranges of the errors for each case decrease (the range of Case 1 errors is + 36 to -18.4 cm; + 7.8 to -18.1 cm. for Case 2; and -10.3 to -15.1 cm, for Case 4). The error after the entire observation session decreases with later observing start times ( -14.8 cm, -9.1 cm, and -0.4 cm for Cases 1, 2, and 4 respectively). Case 5 behaves similarly to the previous cases except that its minimum occurs after only 10 minutes of observations and adding more observations causes the error to become positive. The error is largest after the entire observing session (38.4 cm). Cases 1, 2, and 4 start their observations prior to the first PDOP peak that occurs at 60 min- utes (Figure 4) while Case 5 starts after the PDOP peak, so that starting the observa- tions close to the larger PDOP peak and including the PDOP peak observations can reduce the error and the required observation period. The effect of the PDOP peak upon the mean slope distance error is readily apparent in comparing Cases 4 and 5. Case 4 remains negative without the early positive error, and Case 5 remains positive as it lacks most of the observations from about the first PDOP peak. The first PDOP peak at 60 minutes differs from the second PDOP peak in that it has a higher value, is symmetric, and occurs farther from the four satellite observation start time than the second peak is from the four satellite observation stop time, i.e., the second peak occurs with lower elevation angles than the first which implies larger tropospheric and ionospheric errors. The results for Case 3 (excluding the fifth satellite) slope distance errors are presented in Table 8. Case 3 was studied because the Case 2 results showed a decrease in the initial slope distance errors, and Case 3 was to study the effects of observing five satellites. The results show that using only four satellites makes a difference of only a few centimeters from the results using all available satellites. It should be noted that Case 3 uses only three satellites once SV 6 sets after 100 minutes of observations. The Case 3 minimum, -11.5 cm, was less negative than the Case 2 minimum of -18.1 cm. 25 Table 7. MEAN SLOPE DISTANCE ERROR FOR CASES 1, 2. - 4. AND * I Observation Period Case 1 Case 2 Case 4 Case 5 [min] cm ppm cm ppm cm ppm cm ppm 10 36.0 0.29 7.8 0.06 -10.3 -0.08 -2.3 -0.02 20 30.4 0.25 2.5 0.20 -10.9 -0.09 5.7 0.05 30 21.7 0.18 -0.4 -0.00 -12.2 -0.10 19.2 0.16 40 15.1 0.12 -3.3 -0.03 -13.0 -0.11 19.7 0.16 50 9.8 0.08 -6.0 -0.05 -12.9 -0. 1 1 17.7 0.14 60 5.3 0.04 -8.6 -0.07 -10.9 -0.09 17.9 0.15 70 0.8 0.01 -10.4 -o.os -13.2 -0.11 20.6 0.17 SO -3.0 -0.02 -10.7 -0.09 -15.0 -0.12 23.2 0.19 90 -6.0 -0.05 -14.2 -0.12 -15.1 -0.12 27.2 0.22 100 -7.8 -0.06 -16.8 -0.14 -13.4 -0.11 31.7 0.26 110 -12.1 -0.10 -18.1 -0.14 -11.2 -0.09 38.4 0.31 120 -15.9 -0.13 -17.2 -0.14 -8.0 -0.07 - - 130 -18.1 -0.15 -16.0 -0.13 -4.8 -0.04 - - 140 -18.4 -0.15 -14.0 -0.11 -0.4 -0.00 - - 150 -18.0 -0.15 -11.7 -0.10 - - - - 160 -17.0 -0.14 -9.1 -0.07 - - - - 170 -15.6 -0.13 - - - - - - 175 -14.8 -0.12 - - - - - - Table 8. CASE 3 MEAN ERRORS Observation Sic pe AX AY A7 Period [min] Dist .ince cm ppm cm ppm cm ppm cm ppm 10 8.8 0.07 -287.0 -6.97 177.6 2.48 -21.7 -0.24 20 3.7 0.03 -247.0 -6.00 155.5 2.17 -15.6 -0.17 30 -0.4 -0.00 -214.3 -5.21 126.3 1.76 -1.9 -0.02 40 -4.6 -0.04 -199.4 -4.84 110.4 1.54 9.5 0.10 50 -8.5 -0.07 -170.6 -4.15 87.6 1.22 19.7 0.22 60 -10.9 -0.09 -142.2 -3.44 64.9 0.91 27.9 0.30 70 -11.5 -0.10 -115.7 -2.82 44.8 0.63 32.6 0.36 80 -10.1 -0.08 -90.4 -2.19 24.5 0.34 35.1 0.38 90 -10.3 -0.08 -58.7 -1.43 7.2 0.10 34.7 0.38 100 -10.0 -O.OS -38.5 -0.95 9.2 0.13 23.7 0.26 110 -10.3 -0.08 -15.8 -0.39 3.9 0.05 18.0 0.20 120 -10.2 -0.08 5.9 0.15 -1.0 -0.01 11.8 0.13 130 -9.7 -0.08 28.7 0.70 -7.3 -0.10 5.9 0.06 140 -8.8 -0.07 52.9 1.29 -18.6 -0.26 2.5 0.03 150 -7.5 -0.06 73.7 1.S0 -29.8 -0.42 0.3 0.00 160 -5.2 -0.04 97.2 2.36 -45.0 -0.63 -1.5 -0.02 26 2. Baseline Components The AX and AY accuracies are better than 10.0 ppm for all observing periods while AZ accuracy is better than 1.0 ppm for all observing periods. The accuracy of the baseline components is expected to be less than the accuracy of the baseline because the baseline errors are mostly perpendicular to the baseline itself [Remondi, 1984], The baseline component results (Tables 9, 10, and 11) show that the AX and AY errors are greater than the baseline components while the AZ errors are about the same order of magnitude as the slope distance errors. The AX and AY errors are negatively correlated which is the result of the correlations of the triple differences and the semi-circular tracks of the satellites (Figure 5). Case 3 (Table 8) shows little difference from Case 2 in the AX error, and a more negative AY error is offset by a less negative AZ error. The smallest mean errors for the baseline components are found at various ob- serving periods. Zero mean error for all the baseline components is achieved with fewer observations as the observing start times occur closer to and before the larger PDOP peak. Table 9. MEAN AX ERROR FOR CASES 1, 2, < 4, AND ? Observation Period Case 1 Case 2 Case 4 Case 5 [min] cm ppm cm ppm cm ppm cm ppm 10 -252.4 -6.13 -296.3 -7.19 -136.3 -3.30 29.6 0.73 20 -255.4 -6.21 -256.0 -6.22 -136.4 -3.30 83.9 2.04 30 -255.5 -6.21 -212.1 -5.15 -107.6 -2.62 132.9 3.23 40 -238.0 -5.78 -194.8 -4.74 -80.1 -1.94 183.9 4.47 50 -211.2 -5.12 -167.8 -4.08 -52.7 -1.29 191.7 4.66 60 -197.0 -4.79 -137.4 -3.33 -24.8 -0.61 192.9 4.69 70 -171.8 -4.17 -107.3 -2.60 21.9 0.53 193.4 4.70 80 -145.1 -3.52 -76.8 -1.S7 48.4 1.17 195.4 4.75 90 -116.9 -2.84 -31.8 -0.78 67.7 1.65 201.7 4.91 100 -87.3 -2.11 3.1 0.08 82.3 1.99 208.7 5.08 110 -45.2 -1.09 28.4 0.70 96.2 2.33 221.0 5.37 120 -10.7 -0.27 48.6 1.18 112.0 2.72 - - 130 15.2 0.36 65.7 1.60 124.8 3.04 - - 140 35.6 0.87 83.3 2.02 139.9 3.40 - - 150 53.6 1.30 97.7 2.37 - - - - 160 70.7 1.72 110.1 2.67 - . - - 170 84.3 2.04 - - - - . - 175 91.7 2.24 - - - - - - 27 Table 10. MEAN AY ERROR FOR CASES 1, 2, 4, AND 5 Observation Period Case 1 Case 2 Case 4 Case 5 [min] cm ppm cm ppm cm ppm cm ppm 10 188.0 2.63 199.2 2.78 110.0 1.54 9.1 0.13 20 181.2 2.53 181.4 2.53 106.3 1.49 -28.0 -0.39 30 179.3 2.50 152.4 2.13 93.6 1.31 -68.6 -0.96 40 171.2 2.39 139.3 1.95 79.8 1.11 -95.2 -1. jj 50 153.0 2.14 125.3 1.75 66.6 0.93 -92.6 -1.29 60 141.0 1.97 109.5 1.53 52.2 0.73 -91.9 -1.28 70 127.7 1.78 94.3 1.32 29.3 0.41 -89.7 -1.25 80 114.0 1.59 79.3 1.11 21.6 0.30 -89.3 -1.25 90 100.5 1.40 58.4 0.82 11.1 0.16 -95.6 -1.33 100 86.4 1.21 46.4 0.65 4.0 0.06 -105.4 -1.47 110 66.7 0.93 34.0 0.47 -3.3 -0.05 -121.0 -1.69 120 55.8 0.78 23.5 0.33 -15.8 -0.22 - - 130 43.3 0.60 14.8 0.21 -27.7 -0.39 - - 140 33.9 0.47 2.9 0.04 -42.5 -0.59 - - 150 25.5 0.36 -9.0 -0.13 - - - - 160 14.4 0.20 -17.0 -0.24 - - - - 170 4.0 0.06 - - - - - - 175 -1.8 -0.03 - - - - - - Table 11. MEAN \Z ERROR FOR CASES 1, 2, 4, AND 5 Observation Period Case 1 Case 2 Case 4 Case 5 [min] cm ppm cm ppm cm ppm cm ppm 10 -82.2 -0.90 -33.2 -03.6 -11.0 -0.12 -17.4 -0.19 20 -68.0 -0.75 -30.2 -0.33 -7.3 -0.08 -23.6 -0.26 30 -54.8 -0.60 -23.4 -0.36 -8.5 -0.09 -32.0 -0.36 40 -47.3 -0.52 -17.0 -0.19 -8.9 -0.10 -34.8 -0.38 50 -38.1 -0.42 -14.6 -0.16 -11.1 -0.12 -37.7 -0.41 60 -28.9 -0.32 -12.3 -0.13 -15.1 -0.17 -39.1 -0.42 70 -23.8 -0.26 -11.6 -0.13 -15.1 -0.17 -44.7 -0.49 80 -20.0 -0.22 -13.1 -0.14 -18.5 -0.20 -49.5 -0.54 90 -18.1 -0.20 -12.4 -0.14 -18.9 -0.20 -52.7 -0.58 100 -17.9 -0.20 -15.1 -0.17 -22.2 -0.24 -54.2 -0.59 110 -15.6 -0.17 -15.1 -0.17 -25.7 -0.28 -56.5 -0.62 120 -17.5 -0.19 -17.2 -0.19 -27.3 -0.30 - . 130 -16.4 -0.18 -19.8 -0.22 -28.0 -0.31 - - 140 -17.9 -0.20 -21.0 -0.23 -29.3 -0.32 . - 150 -19.9 -0.22 -21.3 -0.23 . . . . 160 -20.3 -0.22 -21.9 -0.24 - . - . 170 -20.1 -0.22 - - - - . - 175 -20.1 -0.22 - - - - - - 28 NORTH Figure 5. Sky Plots of satellite tracks for Monterey: Elevation angles are dotted concentric circles. Zenith is at the center. 29 C. REPEATABILITY 1. Slope Distance The day-to-day repeatabilities, represented as the 2a level, for Cases 1 , 2,4, and 5 are presented in Table 12, and for Case 3, in Table 13. All the cases achieve 1.0 ppm repeatability for any observing period except Case 1 which requires 20 minutes of ob- servations. Repeatability eventually reaches better than 0.5 ppm after 60 minutes of observations for any case. The minimum 2a levels for all cases are reached at the 80 to 90 minute time of observation, which is about 30 minutes after the larger PDOP peak. A slight increase in the 2a level is centered about the 120 to 130 minute time of observation for all cases which is near the PDOP minimum, after which the 2a level decreases slightly as obser- vations from the second PDOP peak are included in the solutions. Case 4 had the narrowest range of 2a values, and Case 1 had the widest range of 2a values. Table 12. SLOPE DISTANCE 2-SIGMA VALUES FOR CASES 1, 2, 4, \ND 5 Observation Period [min] Case 1 Case 2 Case 4 Case 5 cm ppm cm ppm cm ppm cm ppm 10 176.8 1.44 64.8 0.53 57.8 0.47 52.0 0.42 20 118.2 0.96 63.2 0.51 56.8 0.46 53.4 0.43 30 84.1 0.68 58.6 0.4S 59.2 0.48 60.8 0.49 40 69.8 0.57 58.4 0.47 53.8 0.44 57.0 0.46 50 64.0 0.52 58.6 0.48 54.4 0.44 55.4 0.45 60 61.8 0.50 55.2 0.45 57.6 0.47 55.0 0.45 70 60.4 0.49 55.0 0.44 59.6 0.48 53.6 0.44 80 57.4 0.47 56.8 0.46 60.4 0.49 53.6 0.44 90 56.2 0.46 59.0 0.48 60.8 0.49 57.6 0.46 100 57.2 0.47 60.0 0.49 59.6 0.48 62.6 0.51 110 59.2 0.48 60.4 0.49 57.8 0.47 67.8 0.55 120 60.0 0.49 59.6 0.48 56.2 0.46 - - 130 61.0 0.50 59.0 0.48 56.0 0.46 . - 140 60.6 0.49 58.2 0.47 55.4 0.45 . . 150 60.0 0.49 57.8 - - - . . 160 59.0 0.48 58.4 - . . . . 170 58.6 0.48 - _ _ . . 175 58.6 0.48 - - - - - 30 Table 13. CASE 3: 2-SIGMA VALUES Observation Sic >pe AX AY AZ Period [min] Distance cm ppm cm ppm cm ppm cm ppm 10 62.4 0.51 326.0 7.92 140.8 1.97 76.6 0.84 20 59.4 0.49 304.2 7.39 135.8 1.90 80.8 0.89 30 57.8 0.47 318.8 7.75 169.0 2.36 76.6 0.84 40 58.0 0.47 301.0 7.31 173.4 2.42 82.2 0.90 50 58.8 0.48 287.4 6.98 195.8 2.60 95.4 1.04 60 54.0 0.44 272.2 6.61 197.0 2.75 97.0 1.06 70 53.0 0.43 251.8 6.12 199.8 2.79 95.8 1.05 80 53.6 0.44 244.6 5.94 199.4 2.79 S9.0 0.98 90 56.4 0.46 235.6 5.72 187.0 2.61 83.4 0.91 100 58.0 0.47 219.8 5.34 165.8 2.32 76.2 0.84 110 59.2 0.48 199.2 4.84 138.8 1.94 71.8 0.79 120 59.0 0.4S 18S.8 4.59 127.4 1.78 6S.2 0.75 130 59.0 0.48 184.2 4.48 122.6 1.71 65.0 0.71 140 58.8 0.48 1S8.4 4.58 127.2 1.78 65.4 0.72 150 58.8 0.48 190.4 4.62 128.4 1.79 64.8 0.71 160 57.8 0.47 203.0 4.93 137.0 1.91 63.8 0.70 2. Baseline Components All the baseline components have repeatabilities better than 10.0 ppm for any observing period except for the Case 1 AX, which required 20 minutes of observations (Tables 13, 14, 15, and 16). It is interesting to note that while the AX 2a values for the first segment of Cases 2, 3, 4, and 5 are less than Case l's first segment, the final Case 1 2a value is less than the final segment of any other Case. The final 2a values for Cases 2, 3, 4. and 5 are greater than the Case 1 2a values after an equivalent number of observations. The minimum 2a levels for AX and AY occur at or near the end of the observing sessions for Cases 1,2, 4 and 5. For Case 3 the minimum 2a levels occur after 130 min- utes of observations. The minimum AZ 2a level occurs after various observation peri- ods, but generally in the vicinity of 130 to 150 minute observation time, which is between the PDOP minimum and the second PDOP peak. Case 3 behaves in an opposite fashion from the other cases in that its minimum AX and AY 2a levels occur at the 150-minute time of observation while its AZ minimum occurs at the end of the observation period. 31 Table 14. AX 2-SIGMA VALUES FOR CASES 1, 2, 4, AND 5 Observation Period Case 1 Case 2 Case 4 Case 5 [min] cm ppm cm ppm cm ppm cm ppm 10 646.4 15.69 347.2 8.45 422.6 10.27 395.2 9.00 20 531.4 12.93 311.4 7.57 343.4 8.34 375.4 9.12 30 345.0 8.41 312.2 7.58 312.2 7.32 375.0 8.87 40 248.2 6.03 311.0 7.31 2S6.8 6.97 336.2 8.17 50 239.6 5.83 286.0 6.95 251.0 6.10 315.8 7.43 60 230.4 5.59 269.2 6.54 240.8 6.00 272.8 6.63 70 221.2 5.39 246.6 5.99 233.4 5.67 252.8 6.14 80 209.2 5.10 243.2 5.91 212.8 5.17 231.0 5.61 90 195.0 4.74 221.4 5.38 188.4 4.43 224.4 5.45 100 202.0 4.91 194.8 4.73 16S.8 4.10 206.6 5.02 110 180.8 4.39 168.8 4.10 159.2 3.87 203.4 4.94 120 162.4 3.94 155.2 3.77 155.6 3.78 - - 130 141.4 3.44 146.4 3.56 141.4 3.43 - - 140 134.6 3.26 144.4 3.51 139.8 3.40 - - 150 129.0 3.11 133.0 3.23 - - - - 160 129.4 3.14 134.4 3.27 - - - - 170 121.8 2.96 - - - - - - 175 121.8 2.96 - - - - - - Table 15. AY 2-SIGMA VALUES FOR CASES 1, 2, 4, AND 5 Observation Period Case 1 Case 2 Case 4 Case 5 [min] cm ppm cm ppm cm ppm cm ppm 10 110.8 1.55 204.0 2.85 244.2 3.41 325.0 4.54 20 130.4 1.82 191.2 2.67 224.2 3.13 311.4 4.34 30 140.8 1.97 190.6 2.67 220.4 3.08 298.4 4.17 40 130.8 1.83 193.2 2.70 221.0 3.09 264.8 3.70 50 135.4 1.89 192.2 2.69 212.0 2.96 237.0 3.31 60 142.8 1.99 188.6 2.63 203.8 3.00 200.0 2.79 70 144.4 2.02 185.2 2.59 214.6 2.85 181.4 2.53 80 144.6 2.02 185.0 2.58 181.6 2.54 170.2 2.38 90 145.4 2.03 173.4 2.42 147.4 2.06 177.2 2.48 100 145.0 2.10 152.4 2.13 133.8 1.87 174.6 2.44 110 139.0 1.94 127.6 1.78 124.6 1.74 182.2 2.55 120 126.8 1.77 115.2 1.61 124.8 1.74 - - 130 108.0 1.51 110.6 1.55 119.0 1.66 - - 140 102.6 1.43 112.6 1.57 120.4 1.68 - - 150 100.0 1.40 109.0 1.52 . . . . 160 101.8 1.42 118.0 1.65 . . . . 170 100.0 1.40 - - . . . . 175 101.8 1.42 - - - - - - 32 Table 16. ^Z 2-SIGMA VALUES FOR CASES 1, 2, 4, AND 5 Observation Case 1 Case 2 Case 4 Case 5 Period [min] cm ppm cm ppm cm ppm cm ppm 10 73.6 0.81 65.2 0.76 92.8 1.02 88.0 0.97 20 7S.0 0.86 73.4 0.81 87.4 0.96 74.6 0.82 30 53.8 0.59 69.8 0.77 82.8 0.91 65.2 0.72 40 54.6 0.60 71.6 0.79 78.4 0.86 57.8 0.63 50 55.4 0.61 71.6 0.79 72.0 0J9 49.2 0.53 60 60.2 0.66 68.4 0.75 68.4 0.75 47. S 0.52 70 62.6 0.69 63.8 0.70 65.0 0.71 52.2 0.57 SO 61.0 0.67 59.8 0.70 59.8 0.66 53.2 0.58 90 59.2 0.65 58.2 0.64 55.2 0.61 56.2 0.62 100 5S.0 0.64 55.4 0.61 53.2 0.58 5S.2 0.64 110 56.6 0.62 53.4 0.59 52.2 0.57 59.4 0.65 120 55.8 0.61 52.8 0.58 53.6 0.59 . . 130 54.4 0.60 52.4 0.58 54.0 0.59 - . 140 53.8 0.59 54.2 0.59 53.6 0.59 . . 150 53.6 0.59 54.4 0.60 . . . . 160 56.0 0.61 53.4 0.59 . . . . 170 56.6 0.62 - - - . . . 175 56.0 0.61 - - - - - - 3. Standard Deviation of the Mean The repeatability values can be used to estimate the standard deviations of the mean errors given in Tables 7 through 11 by using Equation (5.2). The values of Tables 12 through 16 should be divided by J28 (where 28 is the sample population) to compute the standard deviations of the means (at the 2a level). Generally, the repeatabilities were about five times the magnitudes of the mean errors; therefore, the uncertainties of the mean errors are on the order of the mean errors themselves. Allowing for the 0.1 ppm uncertainty in the baseline and the baseline components and for the possible standard deviation of the means, accuracies to better than 1.0 ppm for the slope distances and 10.0 ppm for the baseline components remain valid. D. ERROR EFFECTS 1. 7-Day Means Because of the observations were made over a long period of time, Case 1 was subdivided into four groups comprised of seven consecutive observation days to study trends in the slope distance error to identify the contribution of various error sources to 33 optimizing observing times. The results for the slope distance error are presented in Table 17. Table 17. CASE I ERROR: 7-DAY MEANS Observation Period Gro Lip 1 Group 2 Gro ip 3 Group 4 [min] cm ppm cm ppm cm ppm cm ppm 10 80.5 0.65 -19.0 -0.15 53.5 0.43 29.0 0.24 20 62.6 0.51 21.1 0.17 26.9 0.22 10.9 0.09 30 42.8 0.35 1.4 0.01 26.3 0.21 16.4 0.13 40 30.6 0.25 -8.0 -0.07 27.3 0.22 10.4 0.08 50 24.3 0.20 -12.9 -0.10 22.0 0.18 6.1 0.05 60 21.5 0.17 -18.2 -0.15 16.1 0.13 1.8 0.01 70 17.8 0.14 -24.6 -0.20 10.7 0.09 -0.8 -0.01 80 13.2 0.11 -25.4 -0.21 6.1 0.05 -5.9 -0.05 90 10.2 0.08 -27.6 -0.22 2.6 0.02 -9.2 -0.07 100 10.7 0.09 -30.9 -0.25 0.8 0.01 -12.0 -0.10 110 6.4 0.05 -36.4 -0.30 -2.4 -0.02 -16.2 -0.13 120 2.0 0.02 -40.7 -0.33 -6.1 -0.05 -19.0 -0.15 130 -0.6 -0.00 -43.6 -0.35 -6.6 -0.05 -21.6 -0.18 140 -2.1 -0.02 -43.5 -0.35 -6.3 -0.05 -21.8 -0.18 150 -2.3 -0.02 -42.5 -0.35 -5.6 -0.05 -21.8 -0.18 160 -1.6 -0.01 -40.6 -0.33 -4.8 -0.04 -21.0 -0.17 170 -0.2 -0.00 -38.3 -0.31 -3.4 -0.03 -20.5 -0.17 175 0.7 0.01 -37.4 -0.30 -2.3 -0.02 -20.1 -0.16 The seven-day mean slope distance errors remain below 1.0 ppm for all groups for any observation period. The differences between the groups are mainly in the predominance of negative errors in Groups 2 and 4 while Groups 1 and 4 have pre- dominantly positive errors. To account for the change in the characters of the groups, the change in the major sources of error are examined. 2. Ephemeris Errors Examining the AODE (Age of Data Ephemeris) for all satellites during the eight-week observing period, showed that the oldest AODE for a single satellite was nine hours, but most were less than five hours. Averaging the AODE for all satellites for each observing day showed a range of seven hours during the eight weeks of observa- tions (Figure 6). Assuming 12 hours is the largest AODE, the maximum ephemeris linearization error would be 50 cm. Using Equation (2.3) results in 0.025 ppm baseline relative error. 34 1 1 1 1 1 Mill i i 1 1 1 i mi i i i ii 1"! i i ••••■ ■ MEAN AODE 1^ - 7-DAY MEAN »■. ■ 0 • • '■■•»< 1 O X in . | 9 ■ i y r -.. 1 ' ! * : f • ■ » i '\ ■ 1 . t : « - * i • :' Mill 1 1 1 1 i i i i i i i 1 1 1 1 1 1 ii ii i i : !72 279 287 29* 301 JOB 315 325 J29 DAY Figure 6. Mean age of data (ephemeris) The changes in the seven-day mean AODE's do not correspond well to the changes in the groupings, especially as the largest change in AODE from group 3 to 4 does not correspond to a similar change in the group 3 to 4 mean slope distance errors. This is not surprising because while the change is relatively large, the magnitude of the orbital errors is expected to be small. The ephemeris error will appear as a bias during any one observing session. By averaging over eight weeks some of the errors will cancel and appear as variance. Be- cause of the short observing sessions, little variation in the ephemeris error is expected during any single sessions 3. Ionospheric Errors All the observing sessions completed well before 1600 local standard time and approached the normally ionospheric activity minimum of 0600 (Figure 7). The near north-south orientation of the Monterey-Sand Point baseline places both stations in the same time zone, so that ionospheric errors between the two stations will be correlated to some degree and reduced in the differencing. 35 s fl o o to o ' 8 UJ N a i- □ O *" ill li 1 1 1 1 l i III 1 1 1 1 III MM I L 7-DAY MEAN AT AT : ' 1- ♦ '■» - • \ k 8 1 1 1 1 1 1 1 1 1 i i mi i i i i i i i ii i i i 'i „ 272 279 2B7 294 301 JOB 315 325 329 DAY Figure 7. Difference between observation end time and 0600 PST tttp — rrm i — i — rm — rrr w K _ u ° — Z ^ - : UJ - : 3 -: d ;j z o a UJ 3 1 1 1 1 7-DAY MEAN EF EF i\ i i i-LLLI LLLLI I ' U-L-t I I I rl *■■-'" i j i i J I 272 279 2B7 294 301 DAY 30B 315 325 329 Figure 8. Electron fluence: Units electrons — cm~2 —day1 —sr~\ 36 The weekly Preliminary Report and Forecast of Solar Geophysical Data (Space Environment Services Center, 1987] for the eight weeks of the observations described the solar and geomagnetic activity as generally quiet or low. Though the observing period occurred during a general lull in ionospheric ac- tivity, there are two ionospheric effects that can be easily examined: the diurnal effect as the observing periods occur closer to the diurnal ionospheric activity minimum and sporadic effects from electron fluxes. The change in ionospheric range error for the eight-week observing period was computed from Henson and Collier [19S6. Equation (3)] using the difference in Total Electron Content (TEC) determined from Spilker [1980, Figure 1-11] and Henson and Collier [1986, Figure 2] for the mean observation times for the First and last groups. The maximum change in range error, because of diurnal ionospheric changes from the First to the last week, is approximately 1.5 m which by- Equation (2.3) is equivalent to a change in the relative baseline error of only 0.075 ppm. While the 0.075 ppm trend is not discernible because of the 0.1 ppm uncertainty in the baseline and the larger uncertainties in the measurements, the mean number of observations (Table 5) increased with each later group. The mean number of triple difFerence observations available were : 1519 ±45, 1522 ±22, 1601 ±124, and 1726 ±174 for the First through the last groups respectively. The increased number of observations could be attributed to stronger signal-to-noise ratio with the reduced ionospheric dispersion as the observations approach the diurnal ionospheric activity minimum. The group to group changes did not show an apparent trend, but alternated between predominantly negative and positive mean slope distance errors indicating a sporadic effect, or complex interaction between the error sources. To examine sporadic ionospheric effects, daily Electron Fluence (EF) from the weekly Preliminary Report and Forecast of Solar Geophysical Data [Space Environmental Services Center, 1987], and seven-day means of the EF were computed (Figure 8). Electron fluence ( electrons — cm~2 —day1 —sr'1 ) is the daily average of electron flux with energies greater than 2 Mev as measured by the GOES-7 satellite. High values of the seven-day mean EF corresponded to predominantly positive seven-day mean slope distance errors (Groups 1 and 3), and low mean EF corresponded to negative mean slope distance error. 4. Tropospheric Errors Refractivity, N, was computed using the meteorological parameters from the time closest to the middle of each observation session and Remondi [1984, equations 2.30 and 2.31]. Possible indicators of the influence of the troposphere upon the baseline 37 solution are the difference in refractivity, AA', between the baseline stations (Figure 9) and the mean N of both stations (Figure 10). The difference can indicate the level of correlation of the tropospheric errors. Correlated tropospheric errors are reduced in differencing. The mean N (N) will indicate the magnitude of the refraction of the carrier signal to both sites. The changes in the difference in the seven-day mean N (Figure 9) did not have an apparent correspondence to the changes in the seven-day group mean slope distance errors. Instead N shows a pattern similar to the group slope distance errors (Figure 10). High mean refractivity corresponded to positive mean slope distance errors (Groups 1 and 3), and low mean refractivity corresponded with negative mean slope distance errors. The difference between the high and the low mean N are small and do not readily account for the magnitude of the changes. 8- Z O < — TTTTT Hill i m i rn m n rr 7- DAY MEAN AN AN ur^Tt J-LLU I I I I I J III I J-i. Li ■ ■ 272 279 287 204 301 DAY 303 315 325 329 Figure 9. Difference in refractivity between Monterey and Sand Point 38 1 — rrm 1 — i — rm — ttt ti — n r h 1 1 1 1 1 1 1 1 1 272 279 237 7-DAY MEAN N N ♦ 1 1 1 1 III 111 284 301 30S DAY _L 315 325 329 Figure 10. Mean Monterey-Sand Point refractivity 5. 7-Day Repeatability The Group repeatabilities could not be given a general characterization as the Group means were because the was no consistent pattern to the Group-to-Group changes in their repeatabilities. For completeness, the la values for the various error sources are presented in Table 18, and the Group la values are presented in Table 19. Table 18. GROUP 2-SIGMA VALUES FOR THE ERROR SOURCES Error Source Group 1 Group 2 Group 3 Group 4 AODE [hours] 1.6 2.0 1.6 2.4 Observation End Time - 0600 PST [minutes] 24.8 45.0 37.6 56.8 Electron Fluence [electrons — cm'2 —day-1 — sr~l] 610.0 59.6 294.8 21.4 AX 27.4 21.2 27.8 23.8 .V 15.2 10.6 9.0 17.8 39 Table 19. CASE 1: 2-SIGMA VALUES, 7-DAY GROUPINGS Observation Period Gro up 1 Group 2 Group 3 Group 4 [min] cm ppm cm ppm cm ppm cm ppm 10 129.8 1.06 81.2 0.66 155.4 1.26 232.0 1.89 20 S9.4 0.73 110.2 0.90 141.8 1.15 124.8 1.01 30 61.8 0.50 64.0 0.52 108.2 0.88 89.6 0.73 40 60.8 0.49 56.2 0.46 80.6 0.70 64.S 0.53 50 58.0 0.47 53.6 0.44 72.6 0.59 52.4 0.43 60 59.8 0.49 53.4 0.43 65.4 0.53 45.6 0.37 70 62.0 0.50 46.2 0.38 60.2 0.49 44.8 0.36 80 57.0 0.46 49.0 0.40 53.4 0.43 48.2 0.39 90 57.0 0.46 47.4 0.39 51.4 0.42 47.6 0.39 100 55.6 0.45 46.0 0.37 50.4 0.41 50.0 0.41 110 57.4 0.47 46.8 0.38 50.4 0.41 53.2 0.43 120 61.8 0.50 46.8 0.38 48.2 0.39 55.4 0.45 130 64.6 0.53 44.4 0.36 45.4 0.37 58.0 0.47 140 65.0 0.53 43.0 0.35 43.4 0.35 60.2 0.49 150 64.4 0.53 43.4 0.35 42.2 0.34 60.2 0.49 160 63.4 0.52 42.8 0.35 42.6 0.35 59.2 0.48 170 62.8 0.51 42.6 0.35 43.8 0.36 58.6 0.48 175 63.0 0.51 41.6 0.34 45.0 0.37 57.6 o.47 E. EFFECTS OF THE C/A CODE The best C/A code position for Monterey was extracted from each observing session for comparison with the carrier phase results. The coordinates of Sand Point were sub- tracted from the 28-day mean of the Monterey C,'A code coordinates to form the baseline components. The slope distances were computed from the square root of the sum of the squares of the baseline components. The mean errors of the C/A code de- rived baseline were on the order of the carrier phase mean errors, but the 2a levels are several times greater than the carrier phase results (Table 20). These results are for one instantaneous position determination and would improve with time averaging within the observing period or Doppler-smoothing. The C/A code has its most significant impact upon the differenced carrier phase solutions when there are few carrier phase observations. Table 21 shows that reproc- essing the carrier phase observations was required for several segments because either more than ten percent of the observations were rejected or the slope distance solution sigma was greater than 10 m, and that all the reprocessed segments were confined to the first two segments. 40 Table 20. BEST C/A CODE RESULTS AND ERRORS Component AX AY AZ Slope Distance Distance (m) -411587.334 -715776.997 -911739.358 1230044.511 Relative Error (ppmi -8.2 +30.0 . 0.8 +28.5 -3.3 +17.7 -0.6 -1- 7.2 Table 21. CASE 1: ING DATA SEGMENTS BEFORE AND AFTER REPROCESS- Day Segment Total Obs Before After Rejected Obs Slope Distance (m) Rejected Obs Slope Distance (m) Edit Multiplier Case 1 272 275 276 280 2 SO 287 294 308 1 1 1 1 2 1 2 1 62 42 51 61 110 48 124 45 14 17 5 41 44 6 23 * -> 1230050.192 1230268. 9S8 1230047.276 1230227.492 1230093.548 1230048.636 1230044.230 1230042.920 0 0 0 0 0 0 12 0 1230045.846 1230045.294 1230047.020 1230045.717 1230045.915 1230045.326 1230046.146 1230045.972 99 5 99 99 5 99 5 99 Case 2 314 1 78 56 1230074.209 0 1230045.282 3.5 Case 3 314 1 59 37 1230074.209 0 1230045.373 3.5 Case 4 2S1 281 282 1 2 1 78 169 62 22 47 20 1230034.198 1230041.504 1230039.213 0 11 0 1230045.082 1230045.031 1230045.024 3.5 3.5 3.5 Case 5 287 308 314 2 1 1 159 S6 52 43 24 * 1 1230043.418 1230043.047 1230051.620 3 0 0 1230045.177 1230045.472 123004S.0OO 3.5 3.5 99.0 The carrier phase solutions were improved by the least-squares processing of the pseudoranges to obtain better initial antenna position estimates and also to correct the 41 carrier phase time-tags. All the Case 1 reprocessed segments were improved by forcing the processing program to accept more carrier phase observations by increasing the edit multiplier. Because of the success of improving the solutions by the pseudorange pre- processing, all segments for three days (276, 287, and 294) were reprocessed to see if their already acceptable solutions would improve. The solutions either did not change or became slightly less accurate. F. COMPARISON WITH PREVIOUS STUDIES Remondi [1984] concluded that 100 ppm accuracy was achievable in 30 minutes and 10 ppm, in one hour - regardless of baseline length - based upon single-frequency, single difference solutions using precise ephemerides over baselines less than 100 km. To study the required observation times for various accuracies, he partitioned his data into 15-minute observation spans. While decorrelated triple differences would be capable of performing relative geodesy at 1 ppm level, correlated triple differences may provide 5-10 ppm or better [Remondi, 1985]. Some long baseline surveys include: Bock et al. [1984] achieved 2 ppm accuracy in closure of a transcontinental net aided by external atomic clocks using 10 hours of single frequency, single difference obser- vations. Cannon et al. [1985] achieved 0.1 to 0.7 ppm repeatability using single frequency non-differenced network solutions without ionospheric correction for 1700-km baselines between two California stations and Calgary, Alberta, Canada over a four- day period. The range in repeatability was attributed to satellite clock errors. They also found that the largest errors occurred in the AX component. Mader and Abell [1985] found that single frequency long baseline GPS results using single differenced carrier phase observables agreed to one ppm with VLB I measured baseline lengths. Their 2-day repeatability was 0.24 ppm. Goad et al [1985] measured a 1302-km baseline between California and Texas to better than 1.0 ppm compared to the VLB I measured distance, and their two-day repeat- ability was better than 0.5 ppm. Bertiger and Lichten [1987] achieved 6 parts per billion repeatability over a 1314-km east-west baseline over separate four and seven-day periods using dual-frequency re- ceivers with a fiducial network for orbit determination, and some water vapor radiometers. Except for the Bertiger and Lichten [1987] results, the Case 1 results are competitive with the above studies despite the inherent low resolution and the added burden of the correlations of the correlated triple ditferences. A possibility for the optimum results of Case 1 is that a good solution is insured by setting both receivers to their known WGS84 42 coordinates. For real-time surveys, one end of the baseline will usually not be known to one meter accuracy. During the eight-week observation period, the third 4000SX was delivered to NTS. The antenna was installed on the same rooftop, but approximately 35 m to the north of the Monterey antenna. Two days of data using Monterey as- the reference station were used to fix the location of the new antenna. On Day 294, the new receiver's coor- dinates were offset 37 m, equivalent to 30 ppm baseline error. The dilference between the offset antenna and Monterey solutions for the baseline to Sand Point is 0.7 ppm for the first segment (Figure 1 1 ). > A£p. 47 Similar arguments can be extended to point positioning using two satellites at one epoch or two epochs of one satellite by inverting the geometry' of Figure 15, but the reduction in range errors would apply to point positioning as well. The effects of y have been ignored because y is small. From the law of sines : b sin 8 ,, ,v sin y = — (6.4) where 0 = e + v. e is the elevation angle to the satellite and v is the vertical angle be- tween the slope distance and the local horizon. Expanding sin 8 as sin e cos v + cos e sin v , and ( sin e cos v) > > ( cos e sin v) because cos v^l ( v is -5° for the 1230 km baseline ), then sin 8 can be approximated by sin e. Approximating b by 1230 km and p by 20000 km. then, by the law of sines, y is approximately 1° when e is 15°; by the law of cosines, y is approximately 2.5° when the satellite is at the baseline midpoint. With cos 1°~0.9998 and cos 2.5°^0.999l, a very insignificant change, most of the deviation of the measured baseline from the true baseline must come from the error dif- ference factor, AEP. Most importantly having satellites at high elevation angles with re- spect to the termini of the baseline is not detrimental to relative positioning as high elevation angles are for point positioning. A second area of investigation is the path dependent errors. The ionospheric and tropospheric errors, approximated by cosecant functions of the elevation angle, are minimized at high elevation angles. Simultaneous high elevation angles for both stations is another by-product of high PDOP. Finally, the effects of differencing must be explored. Qualitatively, when the satel- lites are bunched close together at high PDOP (Figure 3), some of the correlation in errors that was lost in lengthening the baseline is restored. The change in tropospheric and ionospheric errors with time will also be minimized because: d esc e — cos e d e ,, e\ (6.5) dt sin2e dt and high elevation angles minimize Equation (6.5). Such time derivative minimization of errors, will greatly affect the triple difference because the triple difference is itself a time dillerence. The sum of the elevation angles for all satellites at both stations (Table 22) reaches a maximum after 70 minutes of observations, which is ten minutes later than the PDOP 48 peak. The minimum variance is achieved in the 80 to 90 minute Case 1 time. The run- ning mean of the sum of the elevation angle peaks at the 100 minute observing time while the running mean elevation angle readies a local maximum in the 80 to 100 minute range. The running means provide a better estimator of the behavior of the accuracy and precision because of the accumulation of observations. Table 22. ELEVATION ANGLES Observation Time Number of SV's Sum of Elevation Angles Mean Elevation Angle Running Mean oC Sum Running Mean Elevation Angle 0 S 404 50.5 4(»4 50.5 10 8 446 55.8 425 53.1 20 10 509 50.9 453 52.3 30 10 525 52.5 471 52.3 40 10 547 54.7 486 52.8 50 10 559 55.9 498 53.4 60 10 569 56.9 508 53.9 70 10 562 56.2 515 54.2 80 10 557 55.7 520 54.4 90 10 54S 54.8 523 54.4 100 10 535 53.5 524 54.3 110 10 517 51.7 523 54.1 120 8 505 50.5 519 54.7 130 8 464 5S.0 516 54.9 140 8 452 56.5 511 55.0 150 8 441 55.1 507 55.1 160 8 418 52.3 502 55.0 170 8 396 59.4 496 54.8 49 V. CONCLUSIONS AND RECOMMENDATIONS A. CONCLUSIONS Following a recommendation by Remondi [1984] that more testing of the accuracies of triple difference carrier phase measurements was needed, I studied the optimized GPS observation times required to achieve geodetic accuracies by partitioning observation periods for a 1230-km baseline into ten-minute segments and changing the length and starting time of observations. Accuracy was determined by comparing measured baseline values with reference values obtained by locating the ends of the baselines from very precise VLB I horizontal control points using double difference GPS carrier phase measurements over short baselines. The 1230-km baseline was directly measured for 28 days over a period of eight weeks using Trimble 4000SX receivers. The Trimble supplied Trimvec software was used to process the carrier phase measurements. Using broadcast ephemerides in a fixed orbit, triple difference carrier phase solution with no ionospheric corrections and a tropospheric delay model that used only surface meteorological data, 1.0 part per million (ppm) accuracy in the slope distance was achieved for any observing period with a day- to-day repeatability better than 1.0 ppm (2a). The AX, AY, and AZ (the components of the baseline parallel to the WGS84 axes) achieved accuracies better than 10.0 ppm for any observing period. AY and AZ repeatabilities were better than 10.0 ppm for any observing period, while AX required 30 minutes of observations to reach 10.0 ppm. I had not expected to achieve 1.0 ppm accuracy in the slope distance measurements because Remondi [1984] had suggested that 1.0 ppm required dual frequency measure- ments using highly accurate orbit information and water vapor measurements, and my solutions are further burdened by the correlations of the triple differences. The 1.0 ppm accuracies were on the order of the uncorrelated single frequency results of previous long baseline GPS surveys: Cannon et al. [1985], Bock et al. [1984], Goad et al. [1985], and Mader and Abell [1985]. The unexpected 1.0 ppm accuracy can be attributed to low ionospheric activity be- cause of the orientation of the baseline, the time of year, and the minimum in the sunspot cycle and solar activity during the observing period. Ephemerides errors are considered low because of the low age of the ephemerides. Those ameliorating factors must be considered before applying the results of these case studies to planning surveys 50 on different baselines at different times of day and year. Ionospheric errors were on the order of the tropospheric errors. The superposition of the opposing ionospheric (short- ening) and tropospheric (lengthening) errors reduced mean slope distance errors. Slope distance errors and observation periods were reduced when GPS observations started near the infinite, symmetric PDOP peak. The reduction of errors about the PDOP peak can be attributed to the simultaneous minimization of: range errors, the projection of range errors onto a baseline, satellite and epoch separation. A high sum of the satellite elevation angles for a single station can also be used to determine favor- able observing times, as simultaneous high elevation angles correspond to high PDOP values. My results reconfirm Trimble Navigation's [1987b] recommendation to include the infinite PDOP peak in the observing session; however, I could not confirm the Federal Geodetic Control Committee's [1986] proposal to stop at a GDOP of 5.0. I could not identify any consistent observing stop point that improved accuracy other than to stop when four satellites were no longer available. When less than 30 minutes of data are used, the goodness of the C A code has a great effect upon the carrier phase solution because C/A code positions are used to es- timate the carrier phase solution and to compute the time tags of the earner phase measurements. The number of rejected triple dilference measurements provided the best indicator of the quality of the C/A code estimates and the carrier phase solutions. In- cidences of poor accuracy of the carrier phase solution were found to be caused by poor C A code estimates. The carrier phase solutions could then be improved by correcting the C.'A code solutions, and in turn the carrier phase time tags, by pre-processing the C A code measurements or accepting more carrier phase measurements. Pre-processing the C/A code measurements did not improve the accuracy of the carrier phase solutions when the number of rejected triple differences was less than 10% or when the solution standard deviation of the slope distance was less than 10 m. The long-term average of the best pseudorange solutions produced results compa- rable to the mean carrier phase solution, except that the pseudorange solutions had variances several times the magnitude of the carrier phase solutions. B. RECOMMENDATIONS The results of these case studies should be tested on surveys conducted with baselines of varying orientation and length and with different satellite configurations. 51 Currently, TRIMVEC does not support the use of precise ephemerides or the com- putation of uncorrected triple differences. Should those capabilities be implemented in the future, or provided by third-party software, the data should be reprocessed and an- alyzed to isolate the tropospheric and ionospheric error contributions and to improve the baseline component results. Both of those capabilities would allow better insight into the effects of geometry and the tropospheric and ionospheric errors. The usual role of the correlated triple difference is to aid in fixing cycle slips when the receiver loses lock and to provide the initial estimates for the double difference least-squares processing. The current data should be reprocessed in the double differ- ence mode to determine whether the correlated triple difference solution provides suffi- cient accuracy to fix the cycle slips and estimate the initial integer ambiguities in long baseline surveys. The standard values used to estimate accuracy could be improved by a more rigor- ous fixing of the antenna locations. Both sites should be subjected to a network ad- justment, either locally or simultaneously (possibly in conjunction with the NGS VLBI crustal motion studies). The extra baselines required for a network solution could be measured using the third Naval Postgraduate School receiver. Studies should be conducted to determine the effects of using meteorological obser- vations far removed from the Sand Point antenna site. A temporary meteorological station could be set up near the antenna site (possibly in cooperation with the Weather Service Forecast Office (WSFO) located on the grounds of the Western Regional Center or with the nearby University of Washington). Future Naval Postgraduate School GPS surveys would be aided by portable meteorological instruments, preferably that made digital records of the temperature, humidity, and pressure. Because of the importance of elevation angles to accurate results, studies should be conducted to determine the performance of the TRIMVEC supplied tropospheric mod- els( modified Hopfield and Marini ) at various elevation angles. While this study has shown the utility of the single-station PDOP, a more complete DOP may provide a better satellite selection aid. Such a complete DOP ( or Dilution Of Relative Position (DORP)) should incorporate covariances for baseline components, satellite orbital errors, receiver timing errors, ionospheric and tropospheric delays, un- certainties in the reference station coordinates, and cross-covariances. The DORP would also be formulated with respect to the type of differencing to be used. 52 Lastly, the amount of data processed for these case studies was limited by the speed of the microcomputer and the necessity of transferring the results to the Naval Post- graduate School mainframe for analysis. As more GPS surveys will be conducted in the future, the demands for processing power will increase. It will become imperative that the processing software be ported to the mainframe or to a networked mini-computer. Those computers will allow multiple access to the software as well as speeding the processing. As an interim measure, the RAM of the current ensemble of microcomput- ers should be increased to several megabytes. This increase in RAM will allow the use of RAM -disks which will speed up the data loading - which is the most time consuming of the processing procedures. 53 APPENDIX . BATDLD.BAS LISTING 10 REM PROGRAM BATBLD. BAS BY R. BOUCHARD NOV87 20 REM PROGRAM BATBLD: BUILDS THE BATCH FILE FOR TRIMVEC PROCESSING. 30 REM READS WX OBS FROM SEA. MET FILE 40 REM DAY 283 REQUIRES ITS OWN BATBLD PROGRAM. 50 INPUT "batch file name";0FL$ 60 SP$=" " 70 Sl$=" command /c tbf nodd. tern " 80 OPEN OFL$ FOR APPEND AS #2 90 DIM A$(18),JD$(29),MM$(29),DD$(29),PH(29),PM(29) 95 REM initialize Julian Day array JD$(2)="273": JD$( 3)="274": JD$(4)="275" JD$(7)="280": JD$(8)="28l": JD$(9)="282" JD$(12)="293": JD$( 13)="294": JD$( 14)="295" 100 JD$(n="272": : JD$(5)=''276" 110 JD$(6)="279": : JD$(10)="287" 120 JD$(m="290" : JD$(15)="297" 130 JD$(16)="300": JD$( 17 )="30l" 140 JD$(20)="308": JD$( 21)="309" 145 JD$(23)="315": JD$( 24)="317" 147 JD$(26)="325": JD$( 27)="329" 150 MM$(1)="09": MM$(2)="09" 160 FOR 1=3 TO 18: MN$(I)="l0": NEXT I 170 FOR 1=19 TO 28: MM$(I)="ll": NEXT I 180 DD$m="29": DD$( 2)="30": DD$(3)="0l": : DD$(6)= 06": DD$( 7)="07": DD$(8)="08": DD$(9)="09": DD$( 10)="l4" DD$(12)="20" JD$(18)="302": JD$(22)="314" JD$(25)="318" JD$(28)="321" JD$(19)="307" DD$(4)="02": DD$(5)="03" DD$(11)="17" DD$(14)="22" DD$(19)="03" DD$(24)="13" DD$(27)="25" A$(2)="a02": 190 DD$(13)="21" 200 DD$(18)=U29" 205 DD$(23)="ll" 207 DD$(26)="21" 210 A$(n='aOl": : A$(6)=r,a06" 220 A$(7)="a07": : A$(12)="al2" 230 A$(13)="al3": A$(14)="al4M : A$(17)="al7": A$(18)="al8" 240 FOR 1=1 TO 7: PH(I)=19: NEXT I FOR 1=8 TO 13: PH(I)=18: NEXT I DD$(15)="24": DD$( 16)="27": DD$(17)="28" DD$(20)="04": DD$( 21)="05": DD$(22)="lO" DD$(25)="l4" DD$(28)="17" A$(3)="a03": A$(4)="a04": A$(5)="a05" A$(8)="a08": A$(9)="a09": A$( 10)="al0": A$( ll)="all" A$(15)="al5n: A$( 16)="al6" 250 260 270 275 280 PH(I)=17: NEXT I PH(I)=16: NEXT I PH(I)=15: NEXT I PM(2)=31: PM(3)=26: PM(4)=23: PM(5) = 18: PM(6)=5: PM(7) = 1 PM(9)=54: PM(10)=33: PM(11)=22: PM(12)=8: PM(13)=3 FOR 1=14 TO 21 FOR 1=22 TO 25 FOR 1=26 TO 28 PM(1)=34: 290 PM(8)=58: : PM(14)=59 300 PM(15)=51: PM(16)=39: PM(17)=35: PM(18)=31: PM(19) = 14 310 PM(20)=6: PM(21)=2: PM(22)=42 315 PM(23)=37: PM(24)=29: PM(25)=24: PM(26)=56: PM(27)=39: 317 REM begin building the batch file for each day. 320 FOR IK=1 TO 28 330 OPEN "i",l,"sea. met" PM(28)=8 54 340 INF$="sa"+JD$(IK)+" ma"+JD$(IK) 350 S2$=S1$+INF$ 360 PRINT "FOR JD: ";JD$(IK) 370 INPUT "start hour ; SH 380 INPUT "start minute"; SM 390 PRINT JD$,PH(IK),PM(IK) 400 SH$=STR$(SH) 410 SM$=STR$(SM) 420 PC=SM/60 430 1=1 435 REM Find the WX OB 440 INPUT#1,ID$,IH$,P1(I),T1(I),R1(I),P2(I),T2(I),R2(I) 450 IF((ID$=JD$(IK))AND(SH--VAL(IH$))) THEN 470 460 GOTO 440 470 Pl(I)=Pl(I)-2 : T2(I)=(T2(I)-32)*5/9 480 1=1+1 490 INPUT,J/l,ID$,IH$)Pl(n,Tl(I)>Rl(I)JP2(I),T2(I),R2(I) 500 IF(ID$<>JD$(IK)) THEN 540 505 REM Subtract 2 rab from SEA ob 510 Pl(I)=Pl(I)-2! 515 REM Convert NPS Temp to Celsius 520 T2(I)=(T2(I)-32)*5/9 530 GOTO 480 540 IE=I-1 550 REM interpolate WX OBS TO GPS START TIME 560 PR1(1)= P1(1)+(P1(2)-P1(1))*PC 570 PR2(1)=P2(1)+(P2(2)-P2(1))*PC 580 TR1(1)=T1(1)+(T1(2)-T1(1))*PC 590 TR2(1)=T2(1)+(T1(2)-T1(1))*PC 600 RR1(1)=R1(1)+(R1(2)-R1(1))*PC 610 RR2(1)=R2(1)+(R2(2)-R2(1))*PC 615 REM Compute running mean of first hour 620 PR1(2)= (PRl(l)+Pl(2))/2 630 PR2(2)=(PR2(l)+P2(2))/2 640 TRl(2)=(TRl(l)+Tl(2))/2 650 TR2(2)=(TR2(l)+T2(2))/2 660 RRl(2)=(RRl(l)+Rl(2))/2 670 RR2(2)=(RR2(l)+R2(2))/2 680 WT=1-PC 685 REM Compute remaining running means 690 FOR 1= 3 TO IE 700 IA=I-1 710 IWT=1+WT 720 PR1(I)=(PR1(IA)*WT + ((Pl(IA)+Pl(I))/2))/(IWT) 730 PR2(I)=(PR2(IA)*WT + ( (P2( IA)+P2( I) )/2) )/( IWT) 740 TR1(I)=(TR1(IA)*WT + ( (Tl( IA)+T1( I) )/2) )/( IWT) 750 TR2(I)=(TR2(IA)*WT + ( (Tl( IA)+T2( I) )/2) )/( IWT) 760 RR1(I)=(RR1(IA)*WT + ( (Rl( IA)+R1( I) )/2) )/( IWT) 770 RR2f I)=(RR2(IA)*WT + ( (R2( IA)+R2( I) )/2) )/( IWT) 780 WT=IWT 790 NEXT I 800 M1=SM 810 H=SH 820 C=0 830 1=2 840 DHR=PH(IK)-SH: DMIN=PM( IK) -SM 55 850 IF(DMIN<0) THEN DHR=DHR-1: DMIN=DMIN+60 860 NP=DHR*6+(DMIN/10) 865 REM Interpolate hourly running means to 10 minute intervals 870 FOR IL=1 TO NP 880 M1=M1+10 890 IF(M1>60) THEN Ml=Ml-60: H=H+1: 1=1+1 900 WT=Ml/60 910 IF(M1=60) THEN Ml=Ml-60: H=H+1: 1=1+1 920 MP=M1 930 HP=H 940 IWT=1-WT 950 IA=I-1 960 PW1=PR1(IA)*(IWT)+PR1(I)*WT 9 70 PW1=INT((10*PW1)+. 5)/ 10 980 PW2=PR2( IA)*( IWT)+PR2( I)*WT 990 PW2=INT((10*PW2)+. 5)/10 1000 TW1=TR1( IA)*( IWT)+TR1( I )*WT 1010 TW1=INT((10*TW1)+. 5)/10 1020 TW2=TR2(IA)*(IWT)+TR2(I)*WT 1030 TW2=INT((10*TW2)+.5)/10 1040 RW1=RR1( IA)*( IWT)+RR1( I)*WT 1050 RW1=INT((10*RW1)+. 5)/10 1060 RW2=RR2(IA)*(IWT)+RR2(I)*WT 1070 RW2=INT((10*RW2)+. 5)/10 1080 REM convert to strings 1090 PS1$=LEFT$(STR$(PW1),7) 1100 PS2$=LEFT$(STR$(PW2),7) 1110 TS1$=LEFT$(STR$(TW1),5) 1120 TS2$=LEFT$(STR$(TW2),5) 1130 RS1$=LEFT$(STR$(RW1),5) 1140 RS2$=LEFT$(STR$(RW2),5) 1150 HS$=STR$(HP) 1160 S3$=S2$+SP$+A$(IL)+". "+JD$(IK) 1170 MS$=STR$(MP) 1180 PRINT S3$+PS1$+TS1$+RS1$+PS2$+TS2$+RS2$+SP$+MM$(IK)+SP$ +DD$( IK)+HS$+MS$+SH$+SM$ 1190 PRINT#2, S3$+PS1$+TS1$+RS1$+PS2$+TS2$+RS2$+SP$+MM$(IK) +SP$+DD$( IK)+HS$+MS$+SH$+SM$ 1200 LPRINT S3$+PS1$+TS1$+RS1$+PS2$+TS2$+RS2$+SP$+MM$(IK)+SP$ +DD$( IK)+HS$+MS$+SH$+SM$ 1210 NEXT IL 1220 CLOSE #1 1230 NEXT IK 1240 CLOSE #2 1250 END 56 REFERENCES Abell, M.. National Geodetic Survey, private communication. 1987. Ashjaee, J., New results on the accuracy of the C/A code GPS receiver, in Proc. First International Symposium on Precise Positioning with the Global Positioning System, 1, 207-214. U.S. Department of Commerce, Rockville, MD, Apr. 15 to 19. 1985. Baker. P. J., Global Positioning System (GPS) policy, in Proc. of the Fourth International Geodetic Symposium on Satellite Positioning, I, 51-64. Sponsored by the Defense Mapping Agency and the National Geodetic Survey, at Austin, TX, Apr. 28 to May 2, 1986. Bertiger, W., and S. M. 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Collins, Surveying with GPS, 128 pp., School of Surveying, The University of New South Wales, Kensington, N.S.W, Australia, 1985. 58 Kumar, M., Defense Mapping Agency Hydrographic, Topographic Center, letter to the author, dated Feb. 10, 1988. Landau, IT., and B. Eissfeller, Optimization of GPS satellite selection for high precision differential positioning, in GPS Research at the Institute oj Astronomical and Physical Geodesy, edited by H. Landau, B. Eissfeller, and G. \V. Hem, 65-105, University FAF, Munich, FRG. 1985. Langley, R. B., G. Beutler, D. Delikaraoglou, B. Nickerson, R. Santerre. P. Vanicek, and D. E. Wells, Studies in the application of the Global Positioning System to differ- ential positioning. Department of Surveying Engineering Tech. Rep. 108 , 266 pp., University of New Brunswick, Fredericton, Canada, 1985. Lichten. S. M., and J. S. 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Sponsored by the Defense Mapping Agency and the National Geodetic Survey, at Austin, TX, Apr. 28 to May 2, 1986. Remondi, B. W., Using the Global Positioning System (GPS) phase observable for rel- ative geodesy: modelling, processing, and results, Ph.D. Dissertation, 361 pp.. The University of Texas at Austin, May 1984. Remondi, B. W., Global Positioning System carrier phase: description and use, Bulletin Geodesique, 59, 361-377, 1985. Russell, S. S., and J. H. Schaibly, Control segment and user performance, in Global Po- sitioning System, I, 74-102. Institute of Navigation, Washington, DC, 1980. 59 Smith, C. A., Ionospheric total electron content estimation for single-frequency Global Positioning System receivers, Ph.D. Dissertation, 111 pp., University of California, Los Angeles, 1987. Space Environment Services Center, Preliminary Report and Forecast of Solar Geophysical Data, SESC PRF's 631 through 640. Boulder, CO, 06 Oct. to 08 Dec. 1987. Spilker, J. J., GPS signal structure and performance characteristics, in Global Positioning System, I, 29-54. Institute of Navigation, Washington, DC, 1980. Trimble Navigation, Trimble Model 4000SX GPS Surveyor - PRELIMINARY - Instal- lation and Operation Manual Rev. 8; I ,'87, 96 pp., Sunnyvale, CA, 1987a. Trimble Navigation, TRIMVEC GPS Survey Software Preliminary User's Manual, Rev. B, 62 pp., Sunnyvale, CA, 1987b. Wells, D. E. (Ed.), Guide to GPS Positioning, Canadian GPS Associates, Fredericton, New Brunswick, Canada, 1986. 60 INITIAL DISTRIBUTION LIST No. Copies 1. Defense Technical Information Center 2 Cameron Station Alexandria, VA 22304-6145 2. Library, Code 0142 2 Naval Postgraduate School Monterey, CA 93943-5002 3. Chairman (Code 68Co) 1 Department of Oceanography Naval Postgraduate School Monterey, CA 93943 4. Prof. Stevens P. Tucker 3 Department of Oceanography (Code 68Tx) Naval Postgraduate School Monterey, CA 93943 5. Prof. Narendra K. Saxena 3 Department of Civil Engineering University of Hawaii at Manoa 2540 Dole Street, Holmes 383 Honolulu, HI 96822 6. LT Richard H. Bouchard, USN 3 NAVOCEANCOMCEN/JTWC COMNAVMARIANAS Box 12 FPO San Francisco 96630-2926 7. Director Naval Oceanography Division 1 Naval Observatory 34th and Massachusetts Avenue NW Washington, DC 20390 8. Commander 1 Naval Oceanography Command NSTL Station Bay St. Louis, MS 39522 9. Commanding Officer 1 Naval Oceanographic Office NSTL Station Bay St. Louis, MS 39522 61 10. Commanding Officer Naval Ocean Research and Development Activity NSTL Station Bay St. Louis, MS 39522 11. Chairman, Oceanography Department U.S. Naval Academv Annapolis. MD 21402 12. Chief of Naval Research (Code 420) Naval Ocean Research and Development Activity S00 N. Quincv Street Arlington, VA 22127 13. Director (Code PPH) Defense Mapping Agency Bldg. 56, U.S. Naval Observatory Washington, DC 20305 14. Director (Code HO) Defense Mapping Agency Hydrographic/Topographic Center 6500 Brookes Lane Washington, DC 20315 15. Director (Code PSD-MC) Defense Mapping School Ft. Belvoir, VA 22060 16. Director, Charting and Geodetic Services (N/CG) National Ocean and Atmospheric Administration Rockville, MD 20852 17. Chief, Program Planning, Liaison and Training (NC2) National Oceanic and Atmospheric Administration Rockville, MD 20852 18. LTJG James Waddell, NOAA SMC 2590 Naval Postgraduate School Monterey, CA 93943 19. Mr. Gary Fredrick NOAA, NOS Code N7 MOP 222 Bin CI 5700 7600 Sand Point Wav, NE Seattle, WA 98115-0070 20. Mr. Paul Perrault Trimble Navigation, Ltd. 585 North Mary Avenue Sunnyvale, CA 94088-3642 62 21. Ma. Wei-Ming SMC 2229 Naval Postgraduate School Monterev, CA 93943 ■>*) Dr. Stephen M. Lichten Jet Propulsion Laboratory California Institute of Technology Mail Stop 238-640 4S00 Oak Grove Drive Pasadena, CA 91109 63 <37- m Thes B731 c.l Bouchard Optimized observation periods required to achieve geodetic acuura- cies using the Global Positioning System. Thesis E7317 c.l Bouchard Optimised observation periods required to achieve geodetic acuura- cies using the Global Positioning System.