Datum Definition in the Long Range Instantaneous RTK GPS Network Solution

Transcription

1 Journal of Global Positioning Systes (23) Vol. 2, No. 2: -8 Datu Definition in the Long Range Instantaneous RK GPS Network Solution Israel Kashani,2, Pawel Wielgosz,3, Dorota Grejner-Brzezinska he Ohio State University, CEEGS, 47 Hitchcock Hall, 27 Neil Avenue, Colubus, OH e-ail: tel.: , fax: echnion - Israel Institute of echnology 3 University of Waria and Mazury in Olsztyn, Poland Received: 7 October 23 / 6 Deceber 23 Abstract. his paper presents the ultiple reference station approach to wide area (and regional) instantaneous RK GPS, ipleented in the MPGPS M (Multi Purpose GPS Processing Software) software, developed at the Ohio State University. A weighted freenet approach (WFN) was applied in the instantaneous RK software odule, which enabled optial estiation of the rover coordinates, properly reflecting the accuracies of the observations and the coordinates of the CORS stations. he effect of using the distancedependent weighting schee in the WFN approach on the final rover solution is analyzed. he influence of the different weights was studied by introducing the distance-dependent weights as a function of the CORS station separation L to the rover (/L 2 ). he results show that alost % of the differences between the coputed horizontal rover coordinates and the known reference coordinates are below a decieter, and 95% of the differences in the vertical coordinate are below 2 c, when using the suggested approach. In addition, the accuracy analysis of two other solutions, with different datu definition (iniu constraint and overconstrained), is presented. his analysis verifies the suitability of the stochastic odels used in the RK odule and the rigorous approach, taking into account inter-baseline correlation as well as the correlation inbetween each baseline coponent. Key words: RK, GPS, Datu Definition Introduction. RK GPS Positioning he ost challenging ode aong all the kineatic GPS techniques is long range instantaneous RK, while no data accuulation is ade throughout the adjustent process. With longer distances to the reference stations, the RK technology faces great challenges, since any distance-dependent biases (atospheric, orbital errors, etc.) need to be precisely odeled. If not properly accoodated, these effects ay seriously corrupt the positioning and tie transfer results. he success of precise GPS positioning over long baselines depends on the ability to resolve the integer phase abiguities when short observation tie spans are required, which is especially relevant to RK applications (IAG SSG.79, 2-23). Instantaneous abiguity resolution has several advantages over the on-the-fly (OF) ethod; it is resistant to negative effects of cycle slips and can provide iediate centieter level positioning accuracy without any delay needed for initialization as required by the OF technique (Pratt et al., 998). However, the abiguity resolution can only be achieved if the atospheric, orbital and tiing errors are properly accounted for. Over the past few years, the use of a GPS reference station network approach has shown great proise in extending the inter-receiver separation. he ipleentation of ultiple reference stations in a peranent array offers several advantages over the standard single-baseline approach. It iproves the accuracy of the obile receiver and akes the results less sensitive to the length of the rover-reference baselines (Chen et al., 2; Dai et al., 2; Cannon et al., 2). Consequently, the precision of the baseline coponents, Soe results presented here were reported at ION GNSS 23, Portland, OR

2 Kashani et al.: Datu Definition in the Long Range Instantaneous RK GPS Network Solution in ters of their standard deviation is iproved. he ost iportant contribution of the network solution, as copared to the baseline approach, is not, however, the iproveent in precision, but rather the iproveent in the reliability and availability, as well as the new possibilities for odeling atospheric errors (Marel, 998). Moreover, it is still possible to get an acceptable solution even if there are gaps, outliers or cycle slips in soe of the reference station data. In order to evaluate the quality and effectiveness of the WFN approach, two ore ethods using different, datu definition were tested; these are the iniu constraints (MC) and the over-constrained (OC) ethods, and the coparison between the three odels was carried out. he results show noticeable iproveents with respect to the ean and standard deviation (SD) in the estiated rover coordinates while using the suggested WFN approach..2 Geodetic Control and Datu Definition Observation equations relating to any geodetic network are norally rank-deficient due to the lack of datu reference. here is, therefore, a need to define the datu in order to solve for the positioning paraeters. Datu definition can be accoplished by a careful selection of soe constraints added to the observation equation odel. his could be done while fixing one or ore reference stations during the adjustent procedure. When only one reference station is held fixed we establish a datu referred to as a iniu constraint solution; if ore than one station is fixed, the datu will be referred to as an over-constrained solution. For a free network approach, it is often desirable that the property of the iniu su of the variances and iniu su of squares of the correction to the coordinates is restricted to certain points (Koch, 999). hese points are undoubtedly those that correspond to the reference points in the geodetic network. When dealing with GPS ultiple reference station processing, the CORS stations should be considered a high order reference network. Free net adjustent techniques play a ajor role in the analysis of geodetic networks. Free networks are very popular in geodesy, surveying and apping, ainly due to their unique property of being independent of errors in external data. he results of the free net adjustent and their superior quality serve as an authentic reflection (undistorted irror iage) of the easureent quality (Papo, 999). here are several unique properties of the free networks, as follows: he estiated accuracies of the coordinates are optial (Meissl, 969). hey depend solely on the actual accuracies of the easureents. One of the ost iportant properties of a free network adjustent is that of generating the iniu trace in the dispersion atrix of the unknowns (Fritsch and Schafrin, 982). hey are an unbiased representation of the adjusted easureents. he coordinates of the network points define the coordinate syste. A ore flexible approach ay be adopted by applying weights to these coordinates (WFN)..3 he MPGPS M Software he MPGPS M software, currently under developent at OSU, includes different processing odules. he priary goal of the developent of the instantaneous RK odule is to achieve reliable instantaneous real-tie solutions for distances over k fro the reference stations. In order to accoplish this goal, a ultireference station approach is ipleented. his paper presents the test results fro the current software ipleentation, based on the actual field data. he results analyzed here were obtained using instantaneous processing and the abiguity resolution technique in the post-processing ode. he ethod is, however, suitable for real tie. At the current stage of the software developent, we assue a unidirectional radio link fro the reference network stations (CORS) to the user receiver (rover). he ipleented algorith is based on the doubledifferences (DD) ode, which requires receiving observations fro one or ore reference stations (i.e. baseline or network solution). It should be noted that all the processing is carried out at the rover. he applicability of the unidirectional radio link was tested by the Federal Highway Adinistration (FHWA) over long distance, and real-tie kineatic application was deonstrated at 25 k (Reondi, 22). he ongoing developents of the MPGPS M software include transitting the DD atospheric corrections derived fro the reference network stations. hese corrections ay further increase the syste capabilities, allowing for increased rover-network distances..4 he VCV Matrix he reliability of the variance-covariance (VCV) atrix of the estiated paraeters is of great iportance, especially in a ulti-station processing ode (i.e. network-based solution), when several baselines are processed together (Kashani, 22). he estiation of a realistic VCV atrix by GPS processing software is a function of the easureent weights (e.g. phase and code) and the stochastic odels taking into account both atheatical and physical correlation between the

3 2 Journal of Global Positioning Systes easureents. While the atheatical correlation can be easily odeled, the physical correlation, unfortunately, is ore coplicated to odel. Consequently, if siplified stochastic odels are used, overly optiistic statistics ight result even in ore coprehensive software, which processes the easureents in a ulti-station ode, and produce a fully populated VCV (i.e. considering the between baselines atheatical correlation). VCV atrices derived fro a nuber of the existing GPS software ight not be realistic, especially in the case of coercial packages (Howind et al., 999). he practical iplication is that the produced accuracies are too good, while the derived coordinates are in fact less accurate. herefore, a wrong decision could be ade, based on the software-derived accuracies, that the solution the software presents is reliable and acceptable. he ajority of the GPS processing software packages consider the VCV atrix of the observations to be either diagonal - i.e., no correlation is considered, or block diagonal - i.e., only the atheatical correlation is considered (El-Rabbany et al., 23). All the odules in the MPGPS M software include rigorous stochastic odels. he stochastic odel used in the network-based RK odule, which is based on DD easureents, includes the inter-baseline atheatical correlation (i.e., fully populated atrix). Even though, the physical correlations are not taken into account, the presented accuracy analysis indicates that the VCV atrix sees realistic, rendering convincing coordinate standard deviations. 2 Methodology In this section, the review of the paraeter estiation procedure based on the odel of DD GPS observations is given, with a particular ephasis on different network adjustent strategies. In general, the instantaneous RK software Module consists of three ajor steps as presented by Kashani et al. (23):. Float solution - based on network adjustent. 2. Integer abiguity estiation - using the classical wide-lane and 6/77 search algorith. 3. Fixed solution - network adjustent introducing the fixed abiguities. he paraeter estiation is carried out by the leastsquares principle. An elevation-depended stochastic odel (weighting schee) and atheatical correlation including inter-baseline and baseline-coponent correlations were taken into account and applied in the algorith. In addition, a fast Cholesky decoposition was applied during the adjustent process. he network solution was perfored including all the independent baselines in the network (i.e. fro all the reference stations to the rover). 2. Instantaneous RK 2.. Float Network Solution he instantaneous RK algorith requires the use of both the pseudo-range and phase observations due to the low redundancy. Hence, the DD observation equations for the GPS easureents are as follows: Φ = ρ + I + λ N Φ = ρ + ( f / f ) I + λ N,, 2 2 2, 2 2 2, P = ρ + + I, P = ρ + + ( f / f ) I 2 2 2, 2 Where: Φ DD phase observation on frequency n (n=,2) n, P DD code observation on frequency n n, ρ DD geoetric distance DD tropospheric delay I DD ionospheric delay f, f 2 first and second GPS frequencies while i,j denote the two receivers and k,l the two satellites. he Modified Hopfield odel was used to calculate the tropospheric DD corrections ( ), and the DD ionospheric delay ( I () ) paraeters were estiated in the adjustent procedure. hus, the adjustent odel becoes a siple one including only three types of unknowns: rover coordinates (and soe reference station coordinates in the MC and the WFN), DD abiguities, and the DD ionospheric delay paraeters. he syste of observation equations, which after linearization with respect to the unknown paraeters gives the linear syste of equations, is presented in equation (2): V + L = AX (2) Where: L V Measured inus coputed easureent Measureent corrections

4 Kashani et al.: Datu Definition in the Long Range Instantaneous RK GPS Network Solution 3 A X Design atrix relating easureents and coordinates Corrections to the approxiate unknowns he unknown paraeters X and the corresponding design atrix A could be subdivided with respect to the unknown abiguities and the reaining paraeters (i.e., the baseline coponents and the ionospheric delay paraeters), as follows: V + L = Ax + A2x2 (3) Where x and the corresponding design atrix A refer to the abiguities; x 2 and the corresponding design atrix A 2 refer to the reaining paraeters. As a result of the float network solution, real valued estiates for both the abiguities and the baseline coponents are obtained, together with their corresponding VCV atrices. Following eunissen (998), the resulting float network adjustent solution and its VCV atrix will be: K = round( Nˆ Nˆ ), 2, ˆ ˆ K2 = round(6n, 77 N2, ) his approach deals with the DD code and phase easureents fro one pair of satellites and receivers at a tie that we call one at a tie approach. hus, at this stage the network was decoposed to individual DD pairs in order to fix the abiguities. Although only a particular cobination yields the correct integer solution for the two abiguity sets, the K and K 2 values ay not be free of errors. Consequently, not all of the abiguities can be recovered to their correct integer values, especially for longer baselines. However, in the test data analyzed here, alost all the abiguities (96%) were successfully fixed to their integer values. Since the priary objective of this paper is the deonstration and perforance of the WFN approach, we do not concentrate on the abiguity search and validation techniques here, but rather on the practically achievable accuracy within the current software/algorithic configuration. xˆ Q Q ; xˆ Q Q xˆ xˆxˆ2 2 xx ˆ2ˆ xˆ2 he solution of the integer least-squares proble is denoted as x and x 2 ; the corresponding float solution is denoted as ˆx and ˆx 2 ; Q ˆx is the VCV atrix of ˆx ; Q ˆx 2 is the VCV atrix of ˆx Wide-Lane Abiguity Search At this stage of the algorith, the real-valued abiguities ˆx, denoted as and N, fro the float network Nˆ ˆ, 2, solution are fixed to their integer values, x, designated as N, and N 2,. his step follows the classical widelane and 6/77 search ethod with the integer approxiation on baseline-by-baseline basis (Yang et al., 994): N = (6 K K ) /7 2, 2 N = N + K, 2, Where: N and N are the integer abiguities fro the center, 2, of the search space. (4) (5) 2..3 Fixed Solution Once the fixed solution x has been obtained, as described in the previous section, the abiguity residuals ( ˆx - x ) are used to correct the float solution x 2 in order to produce the final positioning solution for the rover, x = xˆ ( x ), shown in equation (6) (eunissen 998): 2 2 x = xˆ ( x ) = xˆ Q Q ( xˆ x ) xx ˆˆ 2 xˆ his equation refers to the relationship between the float and the fixed solutions. It shows how the deference between these two rover coordinate estiates depends on the difference between the real-valued least squares abiguity estiates, ˆx, and the integer least-squares abiguity estiate, x. In order to get the VCV atrix of x 2 one should apply the law of error propagation to (6) as follows: x2 xˆ2 xˆxˆ2 xˆ xˆ2ˆ (6) Q = Q Q Q Q x (7) 2.2 Free-Net Adjustent Approach he atheatical odel of n easureents, which produces a set of n observation equations linearized at the approxiated values of the coordinates, is given in equation (2). he basis of the null space of A (Papo,

5 4 Journal of Global Positioning Systes 987), known also as the Helert transforation atrix E, has the following iportant property: AE = (8) Datu based on selected points is defined by subjecting the vector X in (2) to the following linear constraints (Papo and Pereluter, 98; Papo, 986; Papo, 989): E P X = = E X (9) where: E = P E and P = P Equation () is derived fro seeking a iniu of the following quadratic for: X P X = in. () First, we iniize the V PV quadratic for in order to create the set of noral equations: NX = U () he above set is singular due to the need for datu definition. It can be solved only if we apply at least d independent linear constraints, as the order of the datu defect to X. If, for exaple, we apply (9) to (), we will obtain a particular solution denoted as X. here is an infinite nuber of sets of d constraints, which, when applied to (), will produce different solution vectors X, all of which satisfy the noral equation (). Let us consider one particular branch of such solutions, which is characterized by the contents of the P weight atrix. Out of the ultitude of P atrices (and respective datus), there is one particular solution, which is unique, since the P atrix is equal to the identity atrix. We will denote that unique solution as X k. In this case, =k when each of the k points in the network contributes to the datu definition on an equal basis, as P k =I. Meissl (969) and Koch (999) have shown that such a solution, as copared to the rest of the existing solutions, is optial and unique in the sense that the trace of the VCV atrix is iniu. In addition, it can be shown (Koch, 999) that Q k, the un-scaled (σ 2 ) VCV atrix of X, is the pseudo-inverse of N, known also as the Moore- Penrose inverse of N (Q k = N + ). According to Meissl (969), the above optial solution is the only one that faithfully reflects the accuracy of the easureents, where all the other (external) sources of error have been effectively filtered out. According to Wolf (977), any two X vectors, which pertain to two different datus - for exaple, X and X k, - are related through the following siple equation: X = X + E t (2) k k where t k is a d vector containing transforation paraeters fro the k - to the - datu. he solution for t k is derived fro (2) under the iniu condition (), as follows (Wolf, 977 and Papo, 987): t = ( E E) E X = H X (3) k k k When using a network-based approach with CORS stations in RK ode, only the CORS station coordinates should be used to establish a datu, and the solution should be the -datu one, as presented in equation (9). he P diagonal atrix entries include zeros or ones, and define the datu points by setting ones in the correct places. his solution is known as a Weighted Free Net (WFN) adjustent. However, the P atrix could be populated by different weights for the datu-defining points, such as distance-dependent weights, as presented here. Moreover, any a priori inforation, such as VCV atrix of the CORS station coordinates, could also replace the respective entries in the P atrix (Kashani, 22). 3 Experient 3. Data Source he results shown herein were generated fro a static network of the Ohio CORS ( In addition to the three reference stations: SIDN, COLB and LEBA fro the Ohio CORS, a City of Dayton CORS static station served as a rover. Figure shows the test network with the analyzed baselines arked. All the stations were equipped with RIMBLE 57 high quality geodetic receivers and antennas. he reference stations were equipped with choke-ring GPS antennas, and the Dayton station was connected to a geodetic antenna. his reference-rover configuration reflects a possible scenario coon in surveying practice. Precise dual frequency data collected on August 3, 23 at a sapling rate of 3-seconds for 3 hours between :- 3: U were used. A total of 588 observation epochs with the phase-soothed code (Springer, 999) and carrier phase easureents were processed and analyzed. he distances between the reference stations varied fro 98 to 2 k, and the distances to the rover ranged fro 4 to k. he GPS data were collected under oderate geoagnetic and ionospheric conditions with the axiu Kp value reaching 3+.

6 Kashani et al.: Datu Definition in the Long Range Instantaneous RK GPS Network Solution 5 6k 4k 4 Analysis of the Results k Fig. est area ap. In order to analyze the perforance of the WFN approach, two other odels, MC and OC, were introduced and tested against the WFN odel. he influence of a weighting schee was studied by introducing distance-dependent weights as a function of the CORS station separation L fro the rover (/L 2 ). he final rover solution is analyzed in ters of the deviation of the estiated coordinates fro the reference truth, and the reliability of their estiated uncertainties. hree odels were tested as follows: a) MC - only the closest to the rover CORS station was fixed. b) OC - all the CORS stations were fixed. c) WFN - distance-dependent weighting schee, where the reference stations serve as datu-defining points. he ean and the standard deviations based on the processed tie series of the rover coordinate coponents (n,e,u) were calculated to copare the different odels. he results obtained for the three cases (a, b and c) are shown in ables and 2; graphical presentation is shown in Figure 2. In addition, Figure 3 presents a scatter plot of the n,e coponents derived using the WFN approach. What is iediately noticeable in Figure 2 is that the WFN produces the ost stable and the least noisy Miniu Constraints Over Constrained dn, de [] dn de dn de du [].4 du du Weighted Free-Net dn de du Fig. 2 Coparison of n,e (left side) and u (right side) coponents derived using different constraints in epoch by epoch ode.

7 6 Journal of Global Positioning Systes solution both in the horizontal and vertical coponents, as copared to the other two odels. he absolute eans of the calculated rover position residuals (the differences fro the known position) equal to 3. c and 2.3 c for east and north coponents, respectively. his is about 5-2 percent iproveent as copared to the OC ethod (the second best) and about 3-4 percent better than the MC. he SD tie series is also the lowest for the WFN in all three coponents. As one could expect, the height coponent is the less accurate (~ 2-3 ties) than the horizontal one. [] WFN - n,e scatter plot [] Fig. 3 Scatter plot of the n,e coponents derived using the WFN approach in epoch by epoch ode. In order to investigate the capabilities of the instantaneous RK algorith presented here, a success rate analysis was perfored. able 3 presents suary statistics of the residual tie series, represented as the success rate of the true coordinate recovery by the different odels. he horizontal threshold was set to c and the vertical one to 2 c. he results presented in able 3 confir that the WFN approach perfors the best with about 98- percent success rate in the horizontal coponents and 95 percent in the vertical one. It should be entioned that in the WFN approach the observation equation syste becoes undeterined when only four satellites are visible. In this test, there were only four visible satellites in epochs 28-9, and during that tie there was no solution provided. able. Absolute ean of n,e,u coponents derived using ethods with different constraint schees. ie Series abs. Mean [] Coponent MC OC WFN n e u able 2. Standard deviations of n,e,u coponents derived using ethods with different constraint schees. ie series SD [] Coponent MC OC WFN n e u able 3. Suary statistics of the coordinate residuals; threshold level of c and 2 c was applied to the horizontal and the height coponents, respectively. Percentage Coponent MC OC WFN n ( c) e ( c) u (2 c) he obtained accuracy analyses are presented in Figure 4, which shows the derived n,e,u residuals, obtained using all three odels, versus ±3σ liit. he plots in Figure 4 show high correlation between the residuals and the estiated SDs, and also confir that the WFN approach provided optial accuracies, as copared to the MC and the OC ethods. he results prove that the rigorous stochastic odel used in the instantaneous RK odule is correct. 5 Suary and Conclusions he results presented here were based on the analyses of 3 hours of dual-frequency GPS data collected at the Ohio CORS and the Dayton CORS stations. he data were processed with the beta version of the MPGPS M software; the ulti-base instantaneous RK odule was used. It should be noted that no data accuulation was applied during the processing, thus a one-epoch solution was carried out for all 588 observed epochs (i.e., every epoch is considered independent fro the previous epoch solution). he distances between the reference stations varied fro 98 to 2 k, and distances to the rover ranged fro 4 to k. hree different datu definition odels were applied and copared, MC, OC and WFN with distance-dependent weighting schee. he WFN provided the best results in ters of SD and the ean of the tie series of the fit residuals fro the known reference solution. he WFN approach provides the ost accurate results as copared to the MC and the OC odels. he absolute ean of the calculated rover position residuals (the differences fro the known position) equals to 3. c and 2.2 c for east and north coponents respectively in the WFN odel (while e.g., MC offers 5. c and 3.3 c, respectively).

8 Kashani et al.: Datu Definition in the Long Range Instantaneous RK GPS Network Solution 7 Miniu Constraints Over Constrained Weighted Free-Net dn [] de [] du [] Fig. 4 he derived n,e,u versus ±3σ in all the three odels. In addition to the ean and SD analyses of the tie series, a suary analysis of the fit of the WFN solution to the true rover coordinates was perfored, with a threshold of c for the horizontal coponents and 2 c for the vertical one. he results show 98- and 95 percent of success rate for the horizontal and the vertical coponents respectively, indicating rather high level of perforance. In the accuracy analysis the fit residuals were tested against ±3. σ (estiated standard deviation), in order to check the reliability of the coordinate uncertainties. he results show a very good correlation between the residuals and the estiated SDs. As expected, the WFN approach provided optial accuracies, as copared to the MC and the OC ethods. According to the results, these correlations and the fact that alost percent of the residuals lies in-between the tolerance interval reflect the effects of the rigorous stochastic odel used in the instantaneous RK odule. he algorith takes into account correlation between observations within each baseline as well as inter-baselines correlation, which results in a realistic stochastic odel of observations, rendering the optial solution with a realistic VCV atrix of the paraeters. It was confired here that the WFN approach iproved the solution and ade it ore stable and less noisy, as copared to the MC and OC odels. he perforance level verified with the test data indicates that the current ulti-base instantaneous RK odule of the MPGPS M software, based on WFN approach, ight be suitable for navigation as well as for engineering/geodetic applications. More testing, especially using real kineatic data is needed. Acknowledgeents his project is supported by NOAA, National Geodetic Survey, N/NGS (project DG33C-2-SE-759), and SOI - Survey of Israel (project 22-). Dr. Pawel Wielgosz is supported by the Foundation for Polish Science. he authors would like to thank Patrick M. Ernst, P.S., fro the Ohio Division of Civil Engineering, City of Dayton CORS, for providing Dayton station data, and David Conner, the Ohio Geodetic Advisor, National Geodetic Survey, NOAA, for his help in obtaining the data.

9 8 Journal of Global Positioning Systes References Cannon, M.E., G. Lachapelle, P. Alves, L.P. Fortes and B. ownsend (2) GPS RK Positioning Using a Regional Reference Network: heory and Results, Proceedings of 5th GNSS International Syposiu, Seville, May 8-. Chen, H.Y. (2) An instantaneous abiguity resolution procedure suitable for ediu-scale GPS reference station networks, 3th Int. ech. Meeting of the Satellite Division of the U.S. Inst. of Navigation, Salt Lake City, Utah, 9-22 Septeber, pp Dai, L., J. Wang, C. Rizos and S. Han (2) Real-tie carrier phase abiguity resolution for GPS/GLONASS reference station networks, Int. Syp. on Kineatic Systes in Geodesy, Geoatics & Navigation (KIS2), Banff, Canada, 5-8 June, p El-Rabbany, A.E. (23) Effect of eporal Physical Correlation on Accuracy Estiation in GPS Relative Positioning, Journal of Surveying Engineering, Vol. 29, No., pp Fritsch, D. and B. Schafrin (982) he Choice of Norn Proble for the Free Net Adjustent with Orientation Paraeters, Boll. Geod. Sci. Aff., Vol. 4, pp Howind, J., H. Kutter and B. Heck (999) Ipact of teporal correlation on GPS-derived relative point positions, Journal of Geodesy, 73, pp IAG SSG.79, Wide area odeling for precise satellite positioning, ers of Reference. last accessed in 23. Kashani, I., (22) Geodetic control networks in the era of peranent GPS stations, Ph.D thesis, echnion - Israel Institute of echnology, 65p. Kashani, I., P. Wielgosz and D. Grejner-Brzezinska (23) Free Net Adjustent in Multi Reference Stations Approach for Instantaneous RK, Proceedings, ION GPS/GNSS, Sept 9-2, Portland, Oregon, pp Koch, K.R. (999) Paraeter Estiation and Hypothesis esting in Linear Models, Springer Verlag, Berlin Heidelberg New York, 333p. Marel, H. van der (998) Virtual GPS reference stations in he Netherlands, Proc. of the th International echnical Meeting, Algoriths and Methods ION-GPS '98, Nashville, USA, 5-8 Septeber 998, part I: pp Meissl, P. (969) Inner heory of Errors of a Cluster of Points. Annex F in Systeatic Investigations of Geodetic Networks in Space, (ed) K. Rinner. Papo, H. and A. Pereluter (98) Datu Definition by Free Net Adjustent, Bulletin Geodesique, Vol.55, No.3. Papo, H. (986) Extended Free Net Adjustent Constraint, NOAA ech. Rep., NOS 9 NGS 37, Rockville MD. Papo, H. (987) Bases of Null Space in Analytical Photograetry, Photograetria, Vol. 4. Papo, H. (989) Modeling in Non-Cartesian Reference Fraes, IAG General Meeting, Edinburgh, Scotland, UK. Papo, H. (999) Datu Accuracy and its Dependence on Network Geoetry, International Scientific and echnical Conference, dedicated to the 22 th anniversary of Moscow State University of Geodesy and Cartography, May 24-29, 999 Pratt, M., B. Burke and P. Misra (998) Single epoch abiguity resolution with GPS-GLONASS L-L2 data, Proc. of the th International echnical Meeting, Algoriths and Methods ION-GPS '98, Nashville, USA, pp Reondi, B. W. (22) Support of the Syste est and Analysis Progra for the Nationwide Differential Global Positioning Syste Modernization Progra Phase I High Accuracy-Nationwide Differential Global Positioning Syste Report, FHWA-RD-2- (July 22). Springer,.A. (999) Modeling and Validating Orbits and Clocks Using the Global Positioning Syste, Ph.D. dissertation, Astronoical Institute, University of Berne, Berne, Switzerland, 55p. eunissen, P.J.G and A. Kleusberg (Eds.), (998) GPS for Geodesy, Springer - Verlag Berlin Heidelberg New York, 65p. Yang, M., C. Goad and B. Schaffrin (994) Real-tie on-thefly abiguity resolution over short baseline in the presence of anti-spoofing, in Proceedings of ION GPS-94, Salt Lake City, pp