Indian Journal of Radio & Space Physics Vol. 38, February 2009, pp. 57-61 Comparative analysis of the effect of ionospheric delay on user position accuracy using single and dual frequency GPS receivers over Indian region Ashish K Shukla #,*, Priya Shinghal, M R Sivaraman & K Bandyopadhyay SATCOM & IT Applications Area, Space Applications Centre, ISRO, Ahmedabad 380015 India # E-mail: ashishs@sac.isro.gov.in Received 12 February 2008; revised 3 October 2008; accepted 6 October 2008 The ionosphere acts as a prominent source of range errors for users of Global Positioning System (GPS) satellite signals requiring accurate position determination. Various models and mathematical formulations have been devised to calculate the absolute range error caused due to ionospheric delay. The present study aims at comparing two methods for calculating delay due to ionosphere: (i) using grid based model at L1 carrier frequency with bilinear interpolation technique; and (ii) using pseudo-range measurements at both L1 and L2 carrier frequency. For analyzing the effect of ionospheric delay on the seasonal behaviour of positional accuracy, a quantitative analysis has been done for all quiet days (Ap index < 50) in 2005 using GPS data for International GNSS Service (IGS) Bangalore (IISc) receiver in January, March and June. Various corrections such as satellite clock bias, transit time, ionospheric delay corrections, etc. are applied to pseudo-ranges to calculate the user coordinates. For single frequency (L1) receivers, ionospheric delay corrections have been applied using IGS total electron content data derived from grid based ionospheric model; and for the dual frequency receivers, pseudorange measurements at L1 and L2 carrier frequencies have been used. It has been observed that there is an improvement of 1-4 m in the standard deviation of position errors when the ionospheric delay correction is applied using pseudo-range measurements at L1 and L2 frequencies (dual frequency receiver) as compared to L1 frequency only. It has also been observed that some residual errors still remain in the estimated user position even after using dual frequency receivers. Keywords: Ionospheric delay, Global Positioning System, Pseudo-range measurement, Single frequency receiver, Dual frequency receiver, Position accuracy PACS No.: 94.20.Vv; 94.20.cf 1 Introduction The signals from GPS satellites experience a delay when passing through the ionosphere. The delay due to ionosphere results in range errors that can vary from few meters to tens of meters. The ionosphere is a dispersive medium, i.e. its refractive index is a function of the operating frequency 1-3. Hence, suitable methods can be adopted for determining the magnitude of delay due to ionosphere using code observables at L1 (1.575 GHz) or both L1 (1.575 GHz) and L2 (1.227 GHz) GPS frequencies. When working with single frequency GPS receivers, a grid based ionospheric model can be used to determine the delay due to ionosphere. Since the magnitude of delay depends linearly upon the number of free electrons passed when traveling through the ionosphere, the model determines the slant ionospheric delay at user s position in terms of total electron content (TEC). The details of the model used in the present study are mentioned in section. Instead of modeling the delay, another approach using GPS code observables at both L1 and L2 frequencies can also be adopted. From the two observables, one at each frequency, a so-called ionosphere free range observable is formed, the mathematical formulation of which is described in later section. In the present study, both the above-mentioned techniques are used to calculate the delay (in meters) due to ionosphere that is separately applied as corrections to the pseudo-range measurements. The measured pseudo-range (R) is written as: R = ρ - c ( b- b ) + c ( T + I ) + ε (1) it ut where, ρ it, is the geometric range between satellite and user; b, is the satellite clock bias; b ut, is the receiver clock bias; T, is the tropospheric delay error; I, is the ionospheric delay error; ε, is the error term consisting of multipath, receiver measurement noise and due to other factors, respectively. A comparative analysis of the standard deviations in positional errors is done using both the methods that shows that accuracy in estimation of user position is more when ionospheric delay corrections are
58 INDIAN J RADIO & SPACE PHYS, FEBRUARY 2009 applied using both the L1 and L2 frequency code observables. Thus, this study proves to be helpful in applications where centimeter level accuracy in position computation is required. A detailed description of the two methods used for calculating ionospheric delay is illustrated. 2 Ionospheric corrections using Grid-based model GPS signal delay caused by the ionosphere is directly proportional to the number of free electrons along the ray path of GPS received signal. The TEC along the ray path of the signal from satellite to receiver may be written as the path integral of the electron density along the line of sight. R TEC = Ne( l) dl (2) S where, the subscripts S, and R, identify the satellite and receiver in question, respectively and Ne, is the electron density. This integral can be rewritten as an integral over altitude h: R TEC(E,A,θ,φ) = Ne( h, Ψ( E, A, θ, ϕ)). M ( E, h) dh...(3) where, S M (E,h) [1 (R cos E / (R h)) ] 2 1/2 = e. e+...(4) Here, M (E, h) is known as mapping function or obliquity factor; R e, is the radius of the Earth; and ψ (h, E, A, θ, Ф) defines the ray path; our parameters mentioned are required to specify the ray path uniquely at the ionospheric pierce point. Here, E, is the elevation angle; A, is the azimuth angle; θ, is ionospheric pierce point (IPP) latitude; and Ф, is IPP longitude, respectively. Thus, solving for slant delay is an inherently threedimensional problem. The single-shell model reduces this three-dimensional problem to two-dimensions by introducing the simplifying assumption that the whole ionosphere is compressed only in a neighbourhood of a specified reference height, taken as 350 km 4-5. Due to the presence of equatorial ionospheric anomaly (EIA) over Indian region 6, altitude of maximum electron density varies 7. For the year 2005, a low solar activity period, it is expected to vary from 350-400 km. Here, for the simplicity it has been taken as 350 km. Mathematically, it is equivalent to setting h as a constant, which allows M (E, h) to be pulled outside the integral. Hence, TECslant = M (E,h). TECVertical (5) Above relation allows estimating the vertical delay at the IPP to be inferred from a measurement of slant delay 3,8. 3 Ionospheric delay estimation at ionospheric grid point (IGP) For the present study, the vertical total electron content (VTEC) at the four IGPs covering IGS Bangalore station has been obtained from the International GNSS Service (IGS) TEC data files downloaded from the site http://www.sec.noaa.gov/ustec/. VTEC values have been calculated at every 5 5 grid spacing in latitude and longitude using global ionospheric maps. Maximum electron density altitude has been taken as 350 km as described in the last section. 4 Slant delay estimation at IPP in grid-based model After getting the VTEC values at IGPs, user needs to interpolate from the broadcast IGP delays to that at its computed IPP locations. For this, bilinear interpolation technique is adopted 9,10. This technique has been explained in the following algorithm that computes the delays at user position using the delays at four IGPs. Geometry of the interpolation is illustrated in Fig. 1. For four point interpolation, the mathematical formulation for interpolated vertical IPP delay τ vpp (Φ PP, λ PP ) as a function of IPP latitude Φ PP and longitude λ PP, is: Fig. 1 Four-point interpolation at user s end
SHUKLA et al.: COMPARATIVE ANALYSIS OF IONOSPHERIC DELAY EFFECT ON USER POSITION ACCURACY 59 4 vpp ( pp, pp) = Wi ( xpp, ypp) vi (6) i= 1 τ ϕ λ τ where, the general equation for the weighing function is: f (x,y) = xy (7) here, τ vi, are the broadcast grid point vertical delay values at four corners of the IGP grid; and τ vpp, is the desired output value at IPP; and x pp, and y pp, are dimensionless quantities required to calculate the weights at the four grid points 9,10. Weights are calculated in an unbiased manner such that total sum of all the weights at four grid points is equal to unity. Their formulation is given as below: W = x y 1 pp pp W = (1 x ) y 2 pp pp W = (1 x )(1 y ) 3 pp pp W = x (1 y ) 4 pp pp λpp = λpp λ1 ϕpp = ϕpp ϕ1 and x y pp pp λpp = λ λ 2 1 ϕpp = ϕ ϕ 2 1 where, λ 1, is longitude of IGP west of IPP; λ 2, is longitude of IGP east of IPP;, Φ 1, is latitude of IGP south of IPP; and Φ 2, is latitude of IGP north of IPP. The slant delay at user s position can be estimated by multiplying the vertical delay, calculated as mentioned above, with a mapping function, as described earlier. 5 Ionospheric corrections using dual frequency receiver Instead of using grid based ionospheric model for estimation of ionospheric delay at user s location, another method can be adopted. This method uses GPS pseudo-range measurements at both L1 and L2 frequencies. The total electron content (TEC) is computed and converted into ionospheric delay in meters using a conversion factor. Following relation has been used to get total ionospheric delay (including receiver bias and P1-P2 bias): TEC = 9.483( R R ) TEC TEC (8) L2 L1 RC P1-P2 where, R L1, is pseudorange at L1 frequency; R L2, is pseudorange at L2 frequency; TEC RC, is receiver bias error/0.351; and TEC P1_P2, is P1_P2 bias error/0.351, respectively. Therefore, the total ionospheric delay in meters is given as: I = 0.163.TEC (9) 6 Method of analysis In order to estimate user position accurately using GPS code observable, certain correction terms are applied to remove the bias from pseudo range to get the correct user to satellite range. Since delay due to ionosphere is one of the most important sources of error, in our analysis this delay has been estimated using GPS code observables and method using TEC values. Ionospheric correction terms from both the methods are applied to the corresponding pseudoranges and user position is estimated. Difference in the actual (known) and calculated user position is found which gives the error estimates in x, y and z components. Standard deviation of these errors terms are estimated for both the methods used. A comparative analysis is done using the results obtained from both the methods. 7 Results and discussion Analysis of standard deviations in the position errors with ionospheric correction is applied using single and dual frequency GPS code measurements. SD(x), is the standard deviation of error in x; SD(y), is the standard deviation of error in y; SD(z), is the standard deviation of error in z; and RMSE, is the root mean square error in x, y and z coordinates, respectively at single frequency (L1) and dual frequency (L1, L2) measurements. A comprehensive analysis and comparison of the effect of ionospheric delay using single and dual frequency measurements on the accuracy of
60 INDIAN J RADIO & SPACE PHYS, FEBRUARY 2009 estimation of user position has been presented. Variation of root mean square error and the standard deviation of errors in x, y & z coordinates of user position are shown for January, March and June 2005 representing the three different seasons namely, winter, equinox and summer, respectively. The results in Figs 2-4 provide the variation in root mean square error (RMSE) and standard deviation of error estimates for single and dual frequency GPS receivers. A comparison of the results at single and dual frequency measurements indicates that the RMSE estimate is considerably less (1-6 m) when Fig. 2 Variation in x, y and z coordinates of user position when ionospheric delay is calculated from L1 and L1-L2 GPS frequency measurements for four quiet days of January 2005: (a) root mean square of errors; (b) standard deviation of errors Fig. 3 Variation in x, y and z coordinates of user position when ionospheric delay is calculated from L1 and L1-L2 GPS frequency measurements for nine quiet days of March 2005: (a) root mean square of errors; (b) standard deviation of errors Fig. 4 Variation in x, y and z coordinates of user position when ionospheric delay is calculated from L1 and L1-L2 GPS frequency measurements for eight quiet days of June 2005: (a) root mean square of errors; (b) standard deviation of errors
SHUKLA et al.: COMPARATIVE ANALYSIS OF IONOSPHERIC DELAY EFFECT ON USER POSITION ACCURACY 61 Fig. 5 Variation of errors in x, y and z coordinates of user position when ionospheric delay is calculated from L1 and L1-L2 GPS frequency measurements for 27 January 2005 ionospheric correction is applied using dual frequency code measurements as compared to single frequency. Similar observations can also be seen for standard deviation plots that provide a statistical view of the observations. Since the vertical total electron content values at the IGP have been taken from IGS data, it is quite possible that the estimation of VTEC at these points may not be precise as there are only few IGS stations around Indian region. A denser network of IGS receivers over Indian region may help to provide more accurate VTEC values at the grid points over Indian region. Another reason for inferior performance from single frequency receiver may be due to the estimation inaccuracy of VTEC values at IGPs from the model used. Figure 5 shows a comparison between the absolute errors in x, y and z coordinates of user position when delay due to ionosphere has been applied using the two above-mentioned methods. Decrease in absolute errors is evident when the correction is applied using dual frequency measurements, but at some instances it increases, which may be due to some instrumental bias still present or other unknown error sources. Further, it has been observed from the results [Figs 4(a) and (b)] that RMSE and standard deviation in errors reaches a maximum in the month of June around the summer solstice, i.e. 21 June. It has also been observed that some residual errors still remain in the estimated user position even after using dual frequency receivers. Acknowledgements Authors are thankful to Dr K S Dasgupta, Deputy Director, SITAA, and Dr Deval Mehta, for their constructive suggestions. Authors are also thankful to the anonymous reviewers for their suggestions towards the improvement of the manuscript. Authors express their sincere thanks to scientists/engineers from IGS responsible for providing precise GPS satellite ephemeredes, clock offset files and grid based IGS TEC data used for carrying out this study and continuous support from all the team members from ISRO. References 1. Davies K, Ionospheric radio propagation, (Dover Publications Inc., New York), 1966, 69. 2. Kaplan E D, Understanding GPS: Principles and applications, (Artech House), 1996, 247. 3. Misra P & Enge P, Global Positioning System signals: Measurements and performance, (G J Press, Massachusetts), 2001, 137. 4. Walter T, Hansen A, Blanch J, & Enge P, Robust detection of ionospheric irregularities, Proceedings of Institute of Navigation Global Positioning System, (Salt Lake City, USA), 2000. 5. Mannucci A J, Wilson B D, Yuan D N, Ho C H, Lindqwister U J & Runge T F, A Global mapping technique for GPSderived ionospheric total electron content measurements, Radio Sci (USA), 33 (1998) 565. 6. Rama Rao P V S, Jayachandran P T & Sri Ram P, Ionospferic irregularities: The role of the equatorial ionization anomaly, Radio Sci (USA), 32 (1997) 1551. 7. Rama Rao P V S., Niranjan K, Prasad D S V V D, Krishna S G & Uma G, On the validity of the ionospheric pierce point (IPP) altitude of 350 km in the Indian equatorial and lowlatitude sector, Ann Geophys (France), 24 (2006) 2159. 8. Tsui J B Y, Fundamentals of global positioning system receivers: a software approach, (John Wiley & Sons), 2000, 69. 9. RTCA Special Committee 159, Minimum operational performance standards for airborne equipment using Global Positioning System/Wide Area Augmented System, RTCA/DO 229 C, November 2001. 10. Sivaraman M R, Grid based ionospheric model, Internal Technical Note TN-336, (SAC-ISRO, India), 2001, 8.