|Year : 2009 | Volume
| Issue : 1 | Page : 28-34
Comparative Analysis of the Techniques for Estimation of GPS DOP over Indian Region
Quddusa Sultana1, Dhiraj Sunehra2, AD Sarma2, PVD Somasekhar Rao3
1 Deccan College of Engineering and Technology, Hyderabad, A.P, India
2 R & T Unit for Navigational Electronics, Osmania University, Hyderabad - 500 007, A.P, India
3 Department of E. C. E., J. N. T. U., Hyderabad, A.P, India
Deccan College of Engineering and Technology, Hyderabad, A.P
Estimation of Dilution of Precision (DOP) plays an important role in aviation, terrestrial and marine navigation applications. In this paper results due to four techniques to estimate DOP namely, Kihara's method, 'all-in-view' SVs, 'best-5' SVs and 'best-4' SVs are investigated. Experiments are performed using Novatel (DL-4plus) GPS receiver and the collected data is used for the analysis. Bancroft algorithm is used to find preliminary position of the receiver. Using this position of the receiver and satellite coordinates, DOP is estimated. Comparative analysis of the specified techniques is performed. For most of the time, DOP values due to 'all-in-view' SVs are found as the best.
Keywords: DOP, GPS, navigation
|How to cite this article:|
Sultana Q, Sunehra D, Sarma A D, Somasekhar Rao P. Comparative Analysis of the Techniques for Estimation of GPS DOP over Indian Region. IETE J Res 2009;55:28-34
|How to cite this URL:|
Sultana Q, Sunehra D, Sarma A D, Somasekhar Rao P. Comparative Analysis of the Techniques for Estimation of GPS DOP over Indian Region. IETE J Res [serial online] 2009 [cited 2013 Dec 12];55:28-34. Available from: http://jr.ietejournals.org/text.asp?2009/55/1/28/51324
| 1 Introduction|| |
Global Positioning System (GPS) provides three dimensional position, navigation and time information to the equipped user. Dilution of Precision (DOP) plays an important role in judging the accuracy performance of a navigation system. DOP indicates the influence of satellite geometry on positioning accuracy. The better the satellite geometry the lesser will be the DOP and better will be the accuracy.
For the ship to sail in oceans or the vehicle to move in terrestrial environment information of horizontal position accuracy is adequate. For estimating the horizontal accuracy the measure of Horizontal DOP (HDOP) is needed. Therefore, HDOP plays an important role in position fixing and navigation for terrestrial and marine applications. However, the knowledge of user altitude or height is an important factor in vertical movements. Especially in civil aviation, the information of user height is most essential while aircrafts land or take off. Timely and accurate measurement of Vertical DOP (VDOP) and user position can save the user from facing hazardous situations.
There are various techniques and algorithms used in literature for estimating various DOPs and user position accuracy ,,, . In this paper, four prominent techniques namely 'all-in-view' SVs  , 'best-5' SVs , , 'best-4' SVs  and Kihara's method  are used for the estimation of DOP for a low latitude Indian station. Also comparative analysis of the techniques is performed.
| 2 Significance of DOP|| |
DOP is one of the important aspects that define the performance of a navigation system  . DOP is a number signifying the effect of satellite geometry on computed position  . It describes how good the satellite geometry is for estimation of user position. The satellite geometry is assumed to be better if user-satellite unit vectors are more spread [Figure 1].
A lesser value of DOP gives better accuracy in the user position estimate and vice versa. A high DOP (say >10) defines a situation where the position solution is unreliable. The main factor affecting the DOP is the satellites' position with respect to user. The more the satellites are distributed with respect to the user the better is the satellite geometry. The better the satellite geometry is the lesser will be the DOP  [Figure 2]. Lot of valuable work has been done on DOP for geo-location systems predating GPS  .
DOP can be envisaged based on the predicted locations of the satellites relative to the user. As the satellite constellation geometry continuously vary over a period of 24 hours, the number and position of satellites for the user at any point on the earth will seldom be ideal. Therefore most of the times the maximum obtainable precision will be diluted in practice  .
DOP also varies due to the factors such as elevation cutoff angle, number of satellites used by the receiver to find position fix etc. Some factors such as obstructions blocking the satellite signals and unexpected problems during satellites' operation may also change the actual value of DOP at the time of observation. Further, DOP varies with time of the day and geographic location. However, the pattern of DOP at a particular location repeats itself every day as the GPS satellite constellation repeats. Due to this reason DOP is highly predictable.
The product of Position DOP (PDOP) and User Range Error (URE) yields the error in user position. The combined effect of measurement errors on user range is referred to as URE (σURE) or User Equivalent Range Error (UERE). The URE is defined as the root-sum-square (rss) of all the components of the measurement errors, all expressed in units of length  .
where σRE/CSis the error due to satellite clock and ephemeris parameters,σRE/p denotes the atmospheric propagation modeling error and σRE/RNM is the receiver noise and multipath error.
| 3 DOP Calculations|| |
If four satellites are considered for position solution, then the linearized equations for their pseudoranges can be written in the form of geometry matrix as  ,
Let H be a covariance matrix given by,
Mathematically, DOPs are defined as , ,
While defining position error it is more meaningful to a user to think in terms of horizontal and vertical components of the error (HDOP and VDOP) defined relative to the local ENU (East, North, UP) coordinate frame  . Hence,
where σ2x and σ2y denote the variances of the x and y component of the position error, respectively. σ2e, σ2n and σ2u are the variances of the east, north and height components. All these parameters are obtained from the diagonal elements of the co-factor matrix of the least squares position solution H (All elements have been divided by the variance of unit weight). The corresponding co-factor matrix for the local geographic components (e, n, u) is written as  .
where σ2z is the variance of the z-component of the position error, σ2b is the variance of the clock bias estimate and 'R' is the transformation matrix given by  ,
where φ, and λ are the latitude and longitude respectively.
From 'H' and 'R' matrices only (3*3) matrices i.e., first 3 rows and first 3 columns of each matrix shall be extracted to estimate HDOP and VDOP.
Furthermore, several researchers have made efforts towards the simplification of the GDOP calculations , . Zhu  has presented a closed-form formula for fast evaluation of GDOP which cannot be applied for three satellite case  . Leva  has shown the relationship between vertical error, VDOP and pseudorange error.
| 4 DOP Estimation Techniques|| |
There are several algorithms available in literature to select the GPS satellites to estimate DOP. For instance, Kalman filter is a recursive algorithm which provides optimum estimates of user position, velocity and time (PVT)  . The filter is first initialized with approximate values (of position, velocity, receiver clock offset and clock drift) for each user state. These user values are given as input to a dynamical model consisting of system equations. The predicted values of the model can be updated by observations containing information on some components of the user state vector  . If there are n (>4) number of visible satellites, then Kalman filter calculates the DOP values for all satellite-combinations taking four at a time and selects those four satellites which gives the best DOP.
Another example of DOP estimation technique is RAIM (Receiver Autonomous Integrity monitoring) algorithm. In aviation the user travels at high speed and can quickly deviate from the right path. Therefore, integrity problem is crucial. Integrity refers to the ability of the system to provide timely warnings to the users when the system should not be used for navigation  . RAIM is a technique which provides integrity for airborne applications of GPS  . RAIM requires redundant satellite range measurements to detect faulty signals and alert the pilot. For fault detection the algorithm requires a minimum of five satellites and for Fault Detection and Exclusion (FDE) six visible satellites are needed. A variety of RAIM schemes have been proposed and all are based upon a self consistency check of the measurements , .
As this paper focuses on the effect of satellite geometry on DOP, the quality of the signals is not paid attention to estimate DOP. Since the quality of the signals is not taken care, the estimated DOP value may not reflect the true DOP if their quality is below certain threshold level  . Further, to save time in the estimation of DOP, most of the existing receivers prefer rejecting some of the weak signals before initiating to estimate DOP. The DOP is then estimated based on the availability of the satellites. The trade-off between signal quality and satellite geometry is not practiced.
Four prominent techniques which are investigated in this paper are presented in the following section.
4.1 DOPs Due to 'Best-4' SVs
For GPS positioning solution, ranging signals from a minimum of four satellites (SVs) are sufficient. These SVs can be selected from a group of visible SVs based on geometry which gives least DOP. However, for several epochs, number of possible combinations can be in hundreds. Selecting one of the combinations for best DOP is cumbersome and time consuming process. For instance, if there are 9 SVs visible over the horizon at a particular epoch then selecting 4 SVs is possible in 15,120 ways. Selecting one best combination to get the best DOP out of these many combinations is highly unproductive. For this reason, most of the GPS receivers adopt 'all-in-view' technique  .
4.2 DOPs Due to 'Best-5' SVs
To fulfil crucial requirements of aviation and to have better integrity more than four SVs are taken into account to estimate user position. The RAIM algorithm selects either 5 or 6 SVs based on the trade-off between the geometry and the quality of signal. However, in this method 5 SVs are selected based on the geometry only. The number of combinations formed out of the available SVs taking 5 SVs at a time will be less than that of four SVs. For example, if 9 SVs are visible then selecting combinations of 5 SVs is possible in 3024 ways. Hence the time requirement of this method is lesser than that of 'best-4' SVs.
4.3 DOPs Due to 'All-in view' SVs
When 'all-in-view' or more than four SVs are considered for estimation of DOPs the solution is said to be over determined. To find the solution 'Least square' technique is used  . Since no special significance is given to any SV and all visible SVs are given equal importance, less number of computations are required. As no combinations of SVs are to be formed and all are to be utilized for position solution, the time consumption for this technique is very less compared to other techniques  .
4.4 Kihara's Method
In most of the applications there will be necessity of either VDOP or HDOP information. Selecting four SVs to get good PDOP may not always give best HDOP or best VDOP. Kihara  has suggested selection of two sets of SVs which can give best HDOP and VDOP respectively. Though the estimated horizontal and vertical position error will be lesser, the computation time will be more. For instance, selecting best combination of 4 SVs from 9 visible SVs to estimate PDOP, HDOP, and VDOP individually can be done in 15,120 ways. Hence three such iterations will be done in 45,360 ways. Thus increases the DOP estimation time to thrice that of 'best-4' SVs.
| 5 Results and Discussions|| |
Data is collected from DL-4 plus GPS receiver for a typical day and converted to RINEX format for the analysis. Bayes filter is used to filter out tropospheric error and receiver clock bias error. It also filters out the error in SVs' coordinates due to earth rotation  . Bancroft algorithm  is used for estimating initial user position. The information on user position and the coordinates of SVs are required as inputs for the programs developed to find DOPs.
5.1 Estimation of user position
[Figure 3] shows the variations in user position estimate with respect to local time. Variations in latitude are negligible [Figure 3a]. Variations in longitude are minimal [Figure 3b]. Variations in height are relatively large [Figure 3c]. From [Figure 3c] it can be observed that the algorithm gives unstable results for the beginning hour therefore standard deviation in height is 53.34m. However, over a period of 23 hours (1:00-24:00 hrs) standard deviation in height is 20.88m only.
Maximum, mean and standard deviation values of latitude, longitude and height are shown in [Table 1]. Standard deviations of latitude and longitude are minimal. Standard deviation of height is significant (53.34).
5.2 Estimation of DOPs
Taking the user position [Table 1] into consideration and the satellites visible over Hyderabad horizon on 19-01-2008, GDOP, PDOP, VDOP, HDOP and TDOP are estimated.
5.2.1 Estimation of GDOP
Variations in GDOP are plotted with respect to local time using four specified techniques [Figure 4]. GDOP due to 'all-in-view' SVs is found to be the least for most of the times. 'Best- 5' SVs method has given GDOP values more than that of 'all-in-view' method but lesser than that of 'best-4' and Kihara's methods.
Comparing 'best-4' method with Kihara's method it can be observed that GDOP curves due to these methods are coinciding, indicating equal performance. It is due to the same combination of 4 SVs used by both the methods to estimate GDOP.
5.2.2 Estimation of PDOP
For almost all the epochs, PDOP due to 'all-in-view' method is found to be the best [Figure 5]. PDOP values due to 'best-5' method are more than that of 'all-in-view' method and lesser than that of other two methods. PDOP due to best-4' and Kihara methods are found to be the same.
PDOP is a function of GDOP and TDOP (Eqs. 4-6). Further, TDOP values due to all the methods are almost same for most of the epochs. Hence, the curves of PDOP are similar to that of GDOP due to all the methods.
5.2.3 Estimation of VDOP
VDOP depends on the vertical spread of the satellites. The more the SVs are spread vertically the less will be the VDOP. There is more possibility of having more vertical spread when all-in-view SVs are taken into consideration. Hence, 'all-in-view' method provides the best VDOP [Figure 6]. For similar reasons, VDOP due to 'best-5' method is better than the remaining two methods.
In case of 'best-4' method, the combination of 4 SVs used to estimate GDOP is also used to estimate VDOP, even though it may not yield the best VDOP. Where as, in case of Kihara's method, combination of 4 SVs is selected exclusively which could give best VDOP, neglecting the SVs combination which had given the best GDOP. Hence, Kihara's method proves better than 'best-4' method.
5.2.4 Estimation of HDOP
HDOP depends on the horizontal spread of satellites. It is more possible to have maximum horizontal spread of SVs when 'all-in-view' SVs are taken into consideration. Hence 'all-in-view' method gives the best HDOP values [Figure 7]. For similar reasons, HDOP due to 'best-5' method is better than the remaining two methods.
In case of 'best-4' method, the combination of 4 SVs used to estimate GDOP, is also used to estimate HDOP even though it may not yield the best HDOP. Where as, in case of Kihara's method, combination of 4 SVs is selected exclusively which could give best HDOP, neglecting the SVs combination which had given the best GDOP. Hence, Kihara's method proves better than 'best-4' method in case of HDOP.
5.2.5 Estimation of TDOP
TDOP values due to all the methods are almost same most of the time [Figure 8]. For some of the epochs 'all-in-view' method proves to be better and for some of the epochs 'best-4' method proves to be the worst.
Similarity in graphs due to all methods, indicate that the number of satellites (n ≥ 4) does not have much effect on TDOP at a particular epoch (Eq. 6).
[Table 2] shows the minimum, maximum and standard deviation of DOP values due to four methods for a period of 24 hours. 'Minimum value' of GDOP, PDOP, HDOP, VDOP and TDOP respectively is least (best) due to 'all-in-view' method.
Comparing all the methods for the 'maximum value' of DOPs, it can be observed that the least among these is due to 'best-5' method.
Evaluating all the methods for standard deviation (Std), it is found that Kihara's method proves to be the most stable (least Std) for the estimation of HDOP. Standard deviation is least due to 'best-5' method in case of VDOP, PDOP, TDOP and GDOP. Hence, it can be concluded that the method which could give the best value of DOP does not necessarily give the least standard deviation.
| 6 Conclusions|| |
Significance of DOP is explained and DOP calculations are briefly described. User position is computed using Bancroft algorithm. Latitude (17.408 deg) and longitude (78.518 deg) remained almost constant for the whole day. However, height varied to a large extent (170m-521m). DOPs are estimated using 'all-in-view' SVs, 'best-5' SVs, 'best-4' SVs and Kihara's method. Most of the time, 'all-in view' method is found as the best method for all the DOPs except for TDOP. TDOP is found to be almost the same by all the techniques. 'Best-5' method has proved to be the next best. 'best-4' method and Kihara's method performed equally in case of GDOP as both methods use the same combination of SVs. However, in case of HDOP and VDOP, Kihara's method has proved to be better than 'best-4' method. This is due to the selection of different combinations of SVs for the estimation of HDOP and VDOP by Kihara's method, compared to that of 'best-4' method. Further, it can be concluded that the method which could give the best value of DOP does not necessarily give the least standard deviation.
| 7. Acknowledgements|| |
The work presented in this paper is carried out under the project sponsored by Dept. of Science and Technology (DST), New Delhi. Vide sanction order No: SR/S4/AS-230/03, dated:21-03-2005.
Quddusa Sultana graduated in Electronics and Communication Engineering from Osmania University in 1994 and M Tech in Digital Science and Computer Engineering from JNTU in 2001. Presently she is Associate Professor in DCET, Hyderabad. She has 14 years of teaching experience. She has several publications to her credit. Her present areas of interests include GPS, GAGAN, IIRNSS and Pseudolites.
Dhiraj Sunehra graduated in Electronics & Communication Engineering from CBIT, Hyderabad in 1997, and M. Tech. in Digital Systems & Computer Electronics from JNTU, Hyderabad in 2001. He has 7 years of teaching experience in MGIT/CBIT, Hyderabad. Later he joined NERTU, Hyderabad as a Research Scientist. His areas of interest includes GPS, SBAS, Ionospheric delay, and Instrumental bias estimation. He has several publications to his credit.
A D Sarma is Ph D. in Electronic and Electrical Engineering from London University (UK) in 1986. He was associated with DEAL, Dehradun, Electrical Engineering Dept. of IIT, New Delhi, DLRL, Hyderabad, Telecommunications Division, Electrical Engineering Dept of. Eindhoven University of Technology, The Netherlands Wave propagation Laboratories, NOAA, Boulder, U.S.A. He has about 25 years of experience in the areas of GPS, Radio Wave propagation and Mobile Communication. He has been guiding several Ph.D students in these areas. He has more than 160 papers to his credit. Presently he is the Director for the Research and Training Unit for Navigational Electronics (NERTU), Osmania University, Hyderabad.
P. V. D. Somasekhar Rao received his M.Tech. and Ph.D. degrees from IIT, Kharagpur in 1979 and 1990 respectively. He worked as Senior Research Asst. in Radio Centre of IIT, Kharagpur and served as Electronics Engineer in Radio Astronomy Centre Group, Tata Inst. of Fundamental Research, Ootcamund. He joined the Faculty of ECE Dept. at JNTU in 1981 and became the Professor and Head of ECE Dept. He is presently working as Professor of ECE at JNTU, Hyderabad. He has more than 25 years of Teaching/Research experience and has published more than 20 Research papers in IEEE, IEE, JEWA, IETE journals and International/National conferences. He has been on the visiting Faculty of at the School of Engineering, Assumption University, Bangkok, during 1997-99. He is a Fellow of IETE (India), Life Member of ISTE (India) and a Senior Member of IEEE.
| References|| |
|1.||Phatak M. S., "Recursive Method for Optimum GPS Satellite Selection", Aerospace and Electronic Systems, IEEE Transactions , Vol. 37 (2), pp: 751-4, Published: April, 2001. |
|2.||Sairo H., Akopian D., and Takala J., "Weighted Dilution of Precision as Quality Measure in Satellite Positioning", Radar, Sonar and Navigation, IEEE Proceedings , Vol. 150 (6), pp: 430-6, 2003. |
|3.||Kovach K., Maquet H., and Davis D., "PPS RAIM algorithms and their Performance", Navigation , Vol. 42, (3), pp: 515-29, 1995. |
|4.||Park C., Ilsun K., Lee J. G., and Jee G., "A Satellite Selection Criterion Incorporating the Effect of Elevation Angle in GPS Positioning", Control Eng. Pract ., 4, (12), pp: 1741-6, 1996. |
|5.||Youg Y, and Lingjuan M., "GDOP Results in All-in-view Positioning and in Four Optimum Satellites Positioning with GPS PRN Codes Ranging", Position Location and Navigation Symposium , pp: 723-7, 2004. |
|6.||Lee Y. C., "Analysis of Range and Position Comparison Methods as a Means to Provide GPS Integrity in the User Receiver", Proceedings of the Annual Meeting of the Institute of Navigation, Seattle WA , 24-6 June, 1986. |
|7.||Brown R. G., and Hwang P.Y.C., "GPS Failure Detection by Autonomous Means Within the Cockpit", Proceedings of the Annual Meeting of the Institute of Navigation, Seattle, WA, pp:5-12, 24-6 June, 1986. |
|8.||Parkinson W. B., and J. J. Spilker, " Global Positioning System: Theory and Applications - vol. I and II", Institute of Aeronautics and Astronautics, Inc, Washington , 1996. |
|9.||Kihara M., and Okada T., "A Satellite Selection Method and Accuracy for the Global Positioning System", Navigation: The Journal of the Institute of Navigation , Vol. 31, (1) pp: 8-20, 1994. |
|10.||Langley R.B., "Dilution of Precision", GPS World , 10 (5), pp: 52-9, 1999. |
|11.||Chaffee J., and Abel J., "GDOP and the Cramer-Rao Bound", Position Location and Navigation Symposium, IEEE , pp: 663-8, 1994. |
|12.||El-Rabbany A., "Introduction to GPS", Artech House, 2002. |
|13.||Lee H. B., "Accuracy Limitations of Hyperbolic Multilateration Systems", IEEE Transactions, Aerospace Electronic Systems , AES-11, (1), pp: 16-29, 1975. |
|15.||Misra P., and Per Enge, "Global Positioning System-Signals, Measurements and Performance", Ganga-Jamuna Press, Lincoln, Massachusetts, USA, 2001. |
|16.||Kai Borre, and Gilbert Strang, "Linear Algebra Geodesy and GPS", Wellesley-Cambridge Press, USA, 1997. |
|17.||Hofmann Wellenhof B., Lichtenegger H., and Collins J., "Global Positioning System-Theory and Practice", Springer Wien, New York, fifth edition, 2001. |
|19.||Massatt P., Rhodus W., and Rudnick K., "2-D and 3-D Characterization of GPS Navigation Service" Proceedings of IEEE PLANS′86, Position Location and Navigation Symposium, Las Vegas, NV, pp: 481-7, 4-7 Nov., 1996. |
|20.||Milliken R. J., and Zoller C. J., "Principle of Operation of NAVSTAR and System Characteristics", Navigation: The Journal of the Institute of Navigation , 25, (2), pp: 95-106, Summer 1978. |
|21.||Zhu J., "Calculation of Geometric Dilution of Precision", Aerospace and Electronic Systems, IEEE Transactions , Vol.28 (3), pp: 293-895, July, 1992. |
|22.||Stein B. A., "Satellite Selection Criteria During Altimeter Aiding of GPS", Navigation: The Journal of the Institute of Navigation , 32, (2), pp: 149-57, Summer 1985. |
|23.||Leva J. L., "Relationship between Navigation Vertical Error, VDOP, and Pseudo-range Error in GPS", Aerospace and Electronic Systems, IEEE Transactions , Vol. 30 (4), pp: 1138-42, Published Oct., 1994. |
|24.||Kaplan E. D., "Understanding GPS -Principles and Applications", Artech House, INC ., 1996. |
|25.||SimGEN User Manual (DGP00686AAA), Issue 1-06, Spirent Communications Ltd, USA, Oct., 2003. |
|26.||Bancroft S., "An Algebraic Solution of the GPS Equations", IEEE Trans. Aerosp and Elec. Systems , AES-21, pp:56-59, 1985. |
[Figure 1], [Figure 2], [Figure 3a], [Figure 3b], [Figure 3c], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7], [Figure 8]
[Table 1], [Table 2]