Accuracy Assessment for Processing GPS Short Baselines using Ionosphere-Free Linear Combination

Transcription

1 Australian Journal o Basic and Applied Sciences, 5(9): , 0 ISSN Accuracy Assessment or Processing GPS Short Baselines using Ionosphere-Free Linear Combination Khaled Mohamed Abdel Mageed Dept. o Civil Eng., Higher Technological Institute, 0 th o Ramadan City, Egypt. Abstract: The biases that aecting the GPS measurements all into three categories, which are: satellite biases, station biases, and signal propagation biases. Accordingly, processing the GPS data using L & L data is the most proper way to minimize, but not to eliminate, all errors. The ionosphere reraction, which constitutes a major part in the biases, can be eliminated using linear combination o L and L data called Ionosphere-Free linear combination, which can aect the accuracy o the processing or GPS short baselines up to 0 km. This paper investigates the accuracy o GPS short baselines (up to 0km), in case o using Iono-Free linear combination in processing the GPS data, compared to processing data using L & L without combination. The results supported with statistical analysis showed that the dierence between processing the GPS data using L & L, and using the Iono-Free model gives discrepancies o mean values 3.6mm, 3.mm, and.4mm in X, Y, and Z coordinates, respectively. In addition, the positional discrepancy between the two solutions has a mean value o 5.4mm. These indings are considered to be insigniicant in the daily work o cadastral survey; but it should be taken into consideration in case o the monitoring the deormation o structures or ancient antiquities; where a sub-millimeter accuracy is considered in the measurement Key words: GPS; GPS Errors; GPS Observables; L & L GPS data; Ionosphere-Free Combination, INTRODUCTION Positioning with GPS can be classiied into Single, Dierential, and Relative positioning. Single positioning is meant by the process o inding out the 3-d coordinates o a certain point, while the Dierential and Relative positioning are concerned with the determination o the dierences in coordinates (vector) between two dierent points (baseline) (Homann et al, 00). There are two types o GPS observables, namely the code and phase observables. In general, the code observations are used or coarse navigation, whereas the phase observations are used in high-precision surveying applications. That is due to the act that the accuracy o the carrier phase observations is much higher than the accuracy o code observations (Langley, 993). The code pseudorange is a measure o the distance between the satellite and the receiver. The P-code, C/A-code can be used to determine the code pseudorange. These ranges can be determined by multiplying the speed o light by the time shit required to match the code generated in the receiver with the code received rom the satellite (Erickson, 99). The phase observable is the dierence between the phase o generated signal in the receiver and the carrier signal o the satellite, measured at the receiver. The phase measurement is made at instant time, so, the number o ull cycles between the satellite and the receiver cannot be measured, and this is what so called initial phase ambiguity. This integer number o cycles is constant or the same receiver with same satellite, until a loss o lock happens, or the receiver is switched o (Leick, 995). Both GPS observables types are aected by many systematic biases, dierent in their source, nature, value and the suitable method o treatment. GPS biases can be classiied into three groups, which are the satellite errors group; the receiver errors group; and the signal propagation group (Grant et al, 990). In addition to these 3 groups, the accuracy o the computed GPS position is also aected by the geometric locations o the GPS satellites as can be detected by the receiver. The more spread out the satellites are in the sky, the better the obtained accuracy o the GPS derived 3-d coordinates. There are our methods to eliminate or at least reduce the GPS biases. The irst method is by applying a mathematical model to correct some errors. The second method, is using the GPS dierence modes. The third method is based on making linear combinations between the GPS observables (Abdel Mageed, 006). The ourth method is the using the GPS precise products like IGS products (IGS, 00). This paper investigates the accuracy o the discrepancies in Cartesian coordinates X, Y, and Z and the spatial position P, in case o using Ionosphere Free linear combination model compared to using the original GPS dual requency data L, and L or processing the GPS baselines up to 0 km. in this context, the GSP observations and errors will be presented. The dierent types o the Relative GPS technique will be discussed. Corresponding Author: Khaled Mohamed Abdel Mageed, Dept. o Civil Eng., Higher Technological Institute, 0 th o Ramadan City, Egypt. 793

2 Aust. J. Basic & Appl. Sci., 5(9): , 0 The methodology o investigation and the description o the ield test will be presented. Finally, the analysis o the obtained data supported with the statistical analysis will be shown, rom which the important conclusions and recommendations will be extracted. GPS Observations and Errors: The GPS observables are ranges which are deduced rom measured time or phase dierences based on a comparison between received signals and generated signals. Unlike the terrestrial distance measurements, GPS uses the so-called one-way concept, where, two clocks are used, namely one in the satellite, and the other in the receiver. Thus, the ranges are aected by satellite and receiver clocks errors and, consequently, they are denoted as pseudoranges. Mainly, there are two types o GPS observables, namely the code pseudoranges and carrier phase observables. In general, the pseudorange observations are used or coarse navigation, whereas the carrier phase observations are used in high-precision surveying applications. That is due to the act that the accuracy o the carrier phase observations is much more higher than the accuracy o code observations, (Rizos, 997). Beside the two GPS observables, the GPS satellite transmits a navigation message. The navigation message is a data stream added to both L and L carriers as binary biphase modulation at a low rate o 50 Kbps. It consists o 5 rames o 500 bits each, or bits in total. This means that, the transmition o the complete navigation message takes 750 seconds. The navigation message contains, along with other inormation, the coordinates o the GPS satellites as a unction o time, the satellite health status, the satellite clock correction, the satellite almanac, and atmospheric data. Each satellite transmits its own navigation message with inormation on the other satellites, such as the approximate location and healthh status (Seeber, 993). Code Pseudoranges Observations: The code pseudorange is a measure o the distance between the satellite and the receiver. The P-code, C/A- code can be used to determine the code pseudorange. These ranges can be determined by multiplying the speed o light by the time shit required to match the code generated in the receiver with the code received rom the satellite (Figure ). Fig. : Pseudorange observables Analogously, the delays o the clocks with respect to GPS system time rame will lead to timing error. The tropospheric and ionospheric delays aect the measured code pseudorange (Leick, 995). The general orm o code pseudorange observation equation is: P = + c( dt dt ) + d ion + d trop + d orb + p () Where: P is the observed pseudorange, is the unknown geometric satellite to receiver range, c is speed o light which is approximately equal to 300,000 km/s, dt and dt are satellite and receiver clock errors respectively, d ion, d trop, are the error due to ionospheric, tropospheric reraction respectively, d orb is the orbital error and p is the code measurement noise. The precision o a pseudorange derived rom code measurement has been about % o the chip length. Consequently, a precision o about 3m, 0.3m is achieved with C/A-code and P-code pseudoranges respectively. However, recent development indicates that a precision o about 0.% o the chip length may be obtained (El-Rabbany, 00). The bias term d ion can be determined, with high percentage, in case o using dual requency receivers, since the ionospheric eect is requency dependant, and its estimation parameters are usually transmitted within the satellite message. Concerning the tropospheric eect term d trop, it can be evaluated to more than 95%, using an adopted model, which is a unction o measured meteorological quantities, such as humidity, pressure, and temperature, o the atmosphere surrounding the receiver position. Concerning the orbital bias term d orb, it can be estimated rom satellite orbital dynamics continuous analysis, at the master control station, and included in the satellite-transmitted message also. The satellite and receiver clocks biases term (dt-dt), is usually treated as one unknown parameter. Hence, equation () o observed pseudorange will be let out with only our unknowns 794

3 Aust. J. Basic & Appl. Sci., 5(9): , 0 parameters, which are the 3-d geocentric cartesian coordinates (X, Y, Z) o the receiver antenna position, in addition to the clock bias term. O course, in order to solve such an equation, or the our unknown parameters, one needs to have our o these observation equations. Phase Observations: The range between the receiver and satellite can be obtained through the carrier phase. The range would simply be the sum o the total number o ull carrier cycles plus ractional cycle at the receiver and the satellite, multiplied by the carrier wave length (Figure ). The ranges determined with the carriers are more accurate than those obtained by the codes (Leick, 995). This is due to the act that, the wavelength o the carrier signal (9cm in case o L) is smaller than the codes. However, there is a problem that the carriers are just pure sinusoidal waves, which means thatt all cycles look the same. Thereore, the GPS receiver has no means to dierentiate one cycle rom the other. In other words, the receiver cannot determine the total number o the complete cycles between the satellite and receiver when switched on. The receiver can only measure a raction o a cycle accurately, while the initial number o complete cycles remains unknown, or ambiguous (Kaplan, 996). This initial cycle ambiguity remains unchanged over time as long as no signal loss or cycle slip occurs. Fig. : Phase Observables. The observation equation o the phase pseudorange is: = + c( dt dt) + N d ion + d trop + d orb + () Where, the measured phase is indicated in meters by, is the carrier wavelength, N is the phase ambiguity, and is the combined receiver and multipath noise, and the other remaining symbols are the same as deinedd in equation (). The same analysis o the bias terms and unknownn parameters, as given in the previous subsection, holds true here also or the case o carrier phase observation equation. The only dierence here is the ambiguity term N, which can be solved or, using a certain adopted technique. This means, again, that at least our satellites should be in view at the time, which can be simultaneously tracked rom the same ground receiver. On the other hand, or both cases o code pseudorange and carrier phase pseudorange, most o the bias terms can be eliminated, or minimized by ollowing a certain technique or collecting GPS measurements, such as single, double, and triple dierences; and/ /or using mathematical model; and/or using linear combination. GPS Errors: GPS measurementss are subjected to some errors, which will aect the accuracy o the inal results. There are two basic types o errors, which are the systematic errors or biases, and the random errors. Generally, the biases aected the GPS measurements all into threee categories which are: satellite biases, receiver biases, and signal propagation biases (Grant et al, 990). Satellite biases consist o biases in satellite ephemeris, satellite clock, and the eect o selective availability SA. The later was internationally terminated by the U..S. Government in May, 000 (Divis, 000). Satellite biases are aecting both code and phase pseudorange measurements. Receiver biases usually consist o receiver clock bias, receiver noise and antenna phase center variation. The signal propagation biases appear due to tropospheric reraction, ionospheric reraction, and multipath (Klobuchar, 99). To give an idea about GPS errors and their values, table () shows the absolute navigationn error budget contained in GPS observables. Beside the eect o these biases, the accuracy o the computed GPS position is also aected by the geometric locations o the GPS satellites as seen by the receiver. Generally, the more spread out the satellites are in the sky, the better the obtained accuracy, which is denoted as dilution o precision DOP. 795

4 Aust. J. Basic & Appl. Sci., 5(9): , 0 Table : GPS absolute error values Source Error Absolute value Satellite clock error 3.0m Space segment Ephemeris error.7m Selective availability (not active) 7.0m Atmospheric eect Ionosphere eect 8.m Troposphere eect.8m User segment Multipath 0.6m Receiver noise 0.3m In order to achieve the expected high accuracy o GPS, the above mentioned errors must be taken into consideration in the data processing stage, then modeled and minimized as possible. The minimization o these errors can be done through our approaches (El-Maghaby et al, 005). The irst is by modeling these errors mathematically and counts or them in the adopted observation equation. The second approach is based on a dierential solution to cancel out, or at least greatly reduce many o these errors. The third approach is concentrated on using linear combination between the GPS observables. The ourth approach is depending on using precise products such as precise satellite ephemeris and satellite clock osets, through multinational GPS agencies such as the International GPS Service IGS (IGS, 00). In addition, the GPS measurements include some observational random errors, moreover, the un-modeled small systematic errors inherent on the system due to multipath and imaging, antenna phase center movement, and residual biases, are usually treated in practice as contributing part o the resulting random errors. The main sources or the random errors in the GPS system can be stated as (Seeber, 993):. Instrumental source, which causes multipath, satellite clock error, and receiver clock error.. Atmospheric source causing ionospheric and tropospheric reraction un-modeled eects. 3. Satellite orbital source, which includes shortage in dynamic models used to determine the relative motion o GPS satellites and the observing stations. Relative GPS Observation Technique: The GPS observation techniques include: Single Point Positioning SPP; Dierential Positioning DGPS; and Relative GPS positioning. GPS Single Point Positioning employs one GPS receiver, while DGPS and Relative GPS positioning employ two or more GPS receivers, simultaneously tracking the same satellites. Surveying works with GPS have conventionally been carried out in the Relative and Dierential positioning techniques. This is mainly due to the higher positioning accuracy obtained rom the relative and dierential techniques, compared to that o the GPS Single Point Positioning. A major disadvantage o GPS Relative and Dierential techniques, however, is the dependency on the measurements or corrections rom the reerence receiver (Rizos, 997). There are dierent GPS Relative positioning techniques. These techniques are Static, Stop & Go, Post Processing Kinematic PPK, and Real Time Kinematic RTK. Static GPS technique, is an accurate and reliable technique, however, it is relatively slow in production. On the other hand, each one o other remaining techniques, is represented a ast solution to the problem o obtaining high productivity, such as measuring many baselines in a short period o time, or the ability to obtain results even while the receiver in motion, that is real time solution, however, with a relatively less accuracy than the static case (Abdel Mageed, 006). Static relative positioning by carrier phase is the most requently used method by surveyors, as it is more accurate as compared to the code pseudorange measurements (Kaplan, 996). The principle o static relative positioning is based on determining the vector between two stationary receivers, using code and phase data. The range o accuracy or static survey, is normally 3mm + 0.5ppm. The static surveying is usually applied in high accuracy surveying projects, such as establishing new geodetic networks, densiication o existing irst order control networks or lower order network, crustal movements, and structural deormation. The intention o the Stop & Go and PPK techniques, is to determine the position o the antenna while it is in motion. The main dierence between Stop & Go and PPK techniques, is that in PPK technique, the coordinates o roving receiver are calculated at all points separated by pre-speciied time or distance interval, along the survey trajectory, whereas, in Stop & Go technique, the coordinates o the roving receiver are calculated at selected points. In many other respects, the PPK technique is similar to Stop & Go technique, that is the ambiguity must be resolved beore starting the survey, and the ambiguity must be reinitialized i a cycle slip occurs during the survey. Providing that the ambiguities are resolved, in the initialization part o the chain, and lock to the satellites is maintained while moving, positional accuracy o about 0 mm + ppm can be achieved. Thus, the GPS Stop & Go, and PPK observing techniques, is well suited when many points close together, have to be surveyed. It may be also used in detailed engineering surveying. However, the main constraint is that, no obstacles are allowed between the satellite position and the roving receiver locations (Homann et al, 00). 796

5 Aust. J. Basic & Appl. Sci., 5(9): , 0 Real Time Kinematic RTK technique is used to determine the coordinates in real time. In this method, the base receiver remains stationary over the known point and is attached to a radio transmitter. The rover receiver is normally carried out and attached to a radio receiver. This method is similar to DGPS, except that in case o DGPS corrections are transmitted, however, in case o RTK the known coordinates o the base along with the receiver measurements are transmitted to the rover receiver through the communication link, using a data rate o Hz, which means one sample per second. The built-in sotware in the rover receiver combines and processes the GPS measurements collected at both the base and the rover receivers, to obtain the rover coordinates. The initial ambiguity parameters are determined instantaneously using a technique called on-the-ly OTF ambiguity resolution. Once the ambiguity parameters are ixed to integer values, the receiver will display the rover coordinates right in the ield. That is, no post processing is required. The expected positioning accuracy is o the order o 5mm+ppm (Shaw et al, 000). Characteristics o Ionosphere-Free Linear Combination: GPS observables are obtained rom the code inormation or the carrier phase in the broadcast satellite signal. Recall that the P-code is modulated on both carriers LI and L, whereas the C/A-code is modulated on LI only; consequently, one could measure the code ranges P L, P L, the carrier phase Φ L, Φ L, and the Doppler shits D L, D L or a single epoch (Langley, 993). GPS observables can be gathered or the L carrier wave only, when single requency receivers are used, or it may be collected or both carrier waves L and L, using dual requency receivers. In the second case, when both L and L observables are available, it is possible to construct, mathematically, a new kind o observables, using dierent ratios o both L and L observables. These observables are naturally linear combinations. Hence, the main objective o GPS phase linear combinations is to eliminate, or at least greatly reduce, the dierent GPS biases. The general ormulation o the resulting GPS phase linear combination can be given as, (Homann et. al, 00): a, b a. b. (3) a, b a. b. (4) c a, b (5) a, b a. b. ion ion d a, b.. d (6) a. b. Where: a,b two arbitrary numbers, whose values are assigned by the considered type o the linear combination. Φ a,b the resulted GPS phase linear combination observable, is the measured L phase, is the measured L phase. a,b, l a,b the corresponding requency and wavelength o the considered phase linear combination., the requencies o L and L signals which are MHz and 7.60 MHz respectively ion the ionospheric delay on the linear combination as a actor o the ionospheric delay on d a, ion b L signal d Considering a certain noise level or the phases, the noise level is increased or the linear combination. Basically, the noise level is about % o the wave length. So, the noise o the p-code is about 30 cm, while the noise o the phase-carrier is about mm., which indicates that the noise level o code observations is greater than that o phase observations by about 50 times (Homann et. al, 00). Applying the error propagation law and assuming the same noise or both phases, the noise o the linear combination is calculated rom (Comery et. al, 989): a, b. a b. a. b. Where: L =9 cm, L =4.4 cm, and e is the noise on the L carrier Regarding the ionosphere ree combination, the ionospheric reraction bias can be eliminated by deining the two coeicients a, and b as ollows (Yang, 995): (7) 797

6 Aust. J. Basic & Appl. Sci., 5(9): , 0 a.54 and b. 54 (8) Accordingly, the phase observation equation o the ionosphere-ree is: (9) trop orb c( dt dt ). N d d The resulting wavelength o the ionosphere-ree linear combination is =4. cm. Also, the noise level is about. times that aecting L carrier. The phase ambiguity o the ionosphere-ree can be calculated rom the relation: N N N L.54N.54N (0) P P Similarly, the ionosphere-ree linear combination or the code measurements is ormed as [Rizos, 997]:. P. P.54P. 54 P trop orb c( dt dt ) d d The advantage o the ionosphere-ree linear combination is the totally removal o the ionosphere eect. The drawback o this linear combination is that due to the non-integer nature o the coeicients, the ambiguities have lost their integer characteristics, which make the process o ixing the phase ambiguity an impossible task. Methodology o Investigation: The objective o this paper is based on comparing the coordinates o GPS baselines processed using dual requency L, L data, and using the linear combined Ionosphere-Free model. The methodology o our investigation herein, will be based on the statistical analysis o the behavior o the discrepancies in the 3-D cartesian coordinates o 8 baselines with approximate distances rom.5 km to 0 km. The ield test was done at New Cairo City on Aug, 0 and Aug 4, 0. The ield procedure o the test was started by setting up a dual requency GPS receiver o Topcon GR3 at a reerence control point. A second dual requency receiver o the same type o Topcon GR3 was set up on 8 control points at approximate distances rom.5 km to 0 km, rom the reerence receiver. The observational operating parameters were the same or the two receivers, which are: static mode, elevation angle 5 0, and 0 seconds rate o observations. The observational duration o each baseline was as ollows: 5 minutes or the baselines o approximate distance.5 km, and 5 km; 45 minutes or the baselines o approximate distances 7.5 km, 0 km, and.5 km; 65 minutes or the baselines o approximate distances 5 km, 7.5 km, and 0 km. Ater completing the GPS ield campaign, the raw data were downloaded and transerred to RINEX ormat using TOPCON LINK sotware. The processing o the Rinex data was done using Leica Geo Oice sotware, on two steps. The irst step was the processing o the 8 baselines using L, and L data. The second step was the processing o the same 8 baselines using the Ionosphere-Free. In each step the 3-D Cartesian coordinates were archived or the statistical analysis. Analysis o Results: The analysis o the results will be based on comparing the discrepancies in X, Y, and Z coordinates between processing 0 GPS baselines using L, L data, and using Iono-ree model. The discrepancies are: X X X DUAL, Y Y YDUAL, Z Z Z () DUAL Where: X Dual is the X coordinate rom processing the data using L, L data. X is the X coordinates rom processing the data using Iono-Free model. The same abbreviations are valid or Y and Z coordinates. Also, the positional discrepancy and the standard deviation can be calculated rom [Comery et. al, 989]: (3) P ( X ) ( Y ) ( Z) P () 798

7 Aust. J. Basic & Appl. Sci., 5(9): , 0 p (4) x y z The discrepancies in X, Y, Z and position P, as well as the approximate length o the 8 baselines are shown in Table. Table : The discrepancies in X, Y, Z, and position P Baseline No. Approx. Length (km) X (mm) Y (mm) Z (mm) P (mm) Figures (3) shows the X, Y, and Z coordinate discrepancies or 8 baselines between the processing o data using L, L and using the Iono-Free linear combination model. In addition, Figure (4) shows the positional discrepancies P or the same 8 baselines. Fig. 3: Variation o the X, Y, and Z coordinate discrepancies Fig. 4: Variation o the Positional discrepancies The previous igures are supported by descriptive statistics to measure the quality o the obtained results. Table (3) shows these descriptive statistics. For instance, the X-coordinate discrepancies are ranging between.7mm and -.5mm, with mean value 3.6mm and SD 8.mm or single determination. The Y-coordinate discrepancies are luctuating between 07mm and -4.3mm, with mean value o 3.mm and SD or single observation o 5.8mm. The Z-coordinate discrepancies are varying between 6.4mm and -3.6mm, with mean value o..4mm and SD or single determination o 3.3mmm. Finally, the positional discrepancies between the dual data and Iono-Free combination are diering rom 4.4mm to.5mm, with most probable value o 5.4mm and SD 5.mm respectively. Table 3: Descriptive statistics o the discrepancies (mm) Discrep. ΔX ΔY ΔZ ΔP Max Min Range Mean S.D single S.D mean Conclusions: The present study investigates an accuracy study or the discrepancies in Cartesian coordinates X, Y, Z in case o using Ionosphere Free linear combination model compared to using the original GPS dual requency 799

8 Aust. J. Basic & Appl. Sci., 5(9): , 0 data L, and L. To achieve such an objective a ield test was done to observe 8 GPS baselines varying rom.5km to 0km. The GPS data were processed one time using the original dual requency data L, L; and the second time the data were processed using the Iono-Free linear combination. The results supported with statistical analysis showed that the dierence between processing the GPS data using L, L, and using the Iono-Free model gives discrepancies o mean values 3.6mm, 3.mm, and.4mm in X, Y, and Z coordinates, respectively. In addition, the positional discrepancy between the two solution has a mean value o 5.4mm. The above indings are considered to be insigniicant in the daily work o cadastral survey; but it should be taken into consideration in case o the monitoring the deormation o structures or ancient antiquities; where a sub-millimeter accuracy is considered in the measurement. Accordingly, in case o using GPS in the monitoring o deormation, it is highly recommended to process the all data using one technique either L, L; or Iono-Free combination; to sustain a sub-millimeter level between the initial observation and the repeated observations. REFERENCES Abdel Mageed, Kh. M, 006. Towards Improving the Accuracy o GPS single Point Positioning. Ph.D Thesis, Faculty o Engineering, Department o Public Works, Ain Shams University, Cairo, Egypt. Comery A., P. Bott and H. Lee 989. Elementary Statistics: A Problem-Solving Approach. WM. C. Brown Publishers, Caliornia, USA. Divis, D.A., 000. SA: Going the way o Dinosaur. GPS World, : 9. El-Maghraby, M.F., M.M. Rabah and Kh. Abdel Mageed, 005. Developing and Assessment o Epoch by Epoch Linear Model or GPS Single Point Positioning. The Scientiic Engineering, Ain Shams University, 40:. El-Rabbany, A., 00. Introduction to the Global Positioning System GPS. Artech House Mobile Communications Series, Boston, London. Grant, D.B., C. Rizos and A. Stolz 990. Dealing with GPS Biases; some Theoretical and Sotware Considerations. School o Surveying, University o New South Wales, Australia. Homann-Wellenho B., H. Lichtenegger and J. Collins, 00. Global Positioning System - Theory and Practice. 5 th Revised Edition, Springer-Verlag, New York. IGS, service., 00. IGS annual report. Kaplan, E., 996. Understanding GPS: Principles and Applications. Norwood, MA: Artech House. Klobuchar, A., 99. Ionospheric Eects on GPS. GPS World, : 4. Langley, R. B., 993. The GPS Observables. GPS World, 4: 4. Leick, A., 995. GPS Satellite Surveying. A Willey Inter-Science Publications, John Willey & Sons, New York. Rizos, C., 997. Principles and Practice o GPS Surveying. Monograph 7, School o Geomantics Engineering, the University o New south Wales. Seeber, G., 993. Satellite Geodesy: Foundations, Methods and Applications. Walter de Gruyter, Berlin. Shaw, M., K. Sandhoo and D. Turner, 000. Modernization o the Global Positioning System. GPS World, :