Ionospheric Time Delay Modelling using Anisotropic IDW

Satya Srinivas Vemuri, Dattatreya Sarma Achanta and Krishna Reddy Desi Reddy Chaitanya Bharathi Institute of Technology (CBIT), Gandipet, Hyderabad (A.P), India

Abstract— Precise estimation of ionospheric time delay is

essential for achieving better positional accuracy. The trans-ionospheric signals are affected due to the existence of anisotropic plasma in the ionosphere. For the first time, a novel model based on anisotropic Inverse Distance Weighting (IDW) with ‘Jackknife’ technique is proposed and is applied in the context of ionospheric time delay modelling for Global Positioning System (GPS). The performance of the proposed anisotropic IDW model is evaluated by comparing with the prominent grid based models (Kriging, Spline and Modified Planar Fit (MPF)).

I. INTRIDUCTION A dual frequency GPS receiver can be used to interpret

the ionospheric time delay effects on the signals. But for most of the applications including aircraft navigation only a single frequency GPS receivers are used normally. Therefore, inorder to account for the time delay, modelling is essential. Global, regional and local models are being developed for supporting Global Navigation Satellite Systems (GNSS) worldwide. The Indian Satellite Based Augmentation Systems (SBAS), namely GPS Aided Geo Augmented Navigation (GAGAN) system is expected to cater civil aviation over Indian region and is likely to be operational in the near future. As a part of the project, about 20 TEC stations are established all over India to support ionospheric radio wave propagation studies with a scope of development of new ionospheric models. In view of this, several models are being developed and investigated [1]. The ionospheric time delay model plays crucial role in improving the performance of SBAS system.

II. IONOSPHERIC TIME DELAY MODELLING The ionospheric time delay modelling includes

approaches that may combine regression analyses, distance based weighted averages etc. The key difference among the techniques is, the criteria used to ‘weight values’ in relation to distance. The criteria may include simple distance relations (e.g., inverse distance methods), minimization of variance (e.g., Kriging and co-Kriging), minimization of curvature, and enforcement of smoothness criteria (Splining), planar fit etc. The Inverse Distance Weighted (IDW) interpolation is one of the techniques where the weight being assigned depends on several factors such as, distance, data redundancy, anisotropy, variability of data etc. There are several other forms of IDW algorithm such as Anisotropic IDW (AIDW), Angular distance weighting, De-trended IDW etc., [2].

The ionospheric models used in SBAS rely on the thin-shell approximation of ionosphere with peak electron density assumed at an altitude of 350kms [1]. The intersection of line of sight from satellite to receiver at this altitude is known as

Ionospheric Pierce Point (IPP). Under the assumption that the presence of peak ionospheric irregularities at a height of 350 km, in a layer of 50 km thickness, and the anisotropic irregularity axis ratio is 5 [3]. In view of this, one of the forms of IDW technique namely anisotropic IDW’ with Jackknife technique is considered and is applied in the context of ionospheric time delay modelling for GNSS.

A. Anisotropic IDW and Jackknife The ionospheric time delay is anisotropic and depends on

observation direction. Therefore, anisotropic IDW is considered to model the effect of ionosphere on GPS signals. using anisotropy IDW for known vertical delays ( )iw due to

‘n’ (i=1,2…n) visible satellites at IPP latitudes ( )iφ and

longitudes ( )i

λ , the estimated delay ( )k

w at user location

( ),k k

φ λ can be written as [2],

( )( )

( ),

1 1,

1( )k k k

n ni i i

i iik ik

wwd d ββφ λ

φ λδ δ= =

=+ +

∑ ∑ (1)

where, β : distance weighted power, δ : smoothing parameter, and ‘ ikd ’ is the effective distance. ikd is in the cartesian coordinates (x,y), is given as,

( ) ( )2 2. . . .ikd A x B x y C y= Δ + Δ Δ + Δ (2)

where,

( )2 2 , 2. . .xx yy xx xy yx yyA R R B R R R R= + = + and 2 2

yy yxC R R= +

The terms, xxR , xyR , yxR and y yR are obtained from anisotropy angle ( θ : elevation), and anisotropic ratio ( ρ : ratio of earth’s semi-major axis to semi-minor axis as per WGS-84 standard) and are given as,

( ) ( )( ) ( )

cos ;

sin cos

sin ;xx xy

yx yy

R R

R and R

θ ρ

θ θ

θ ρ= =

= − = (3)

β and δ are assumed as ‘1’ and ‘0’ respectively. Since the IDW technique does not quantify the uncertainty of estimation, a new technique called ‘Jackknife’ is incorporated for model refinement. In this, the algorithm is repeated ‘n’ times for estimating ionospheric delay at user location by removing each time one IPP delay data. The ‘n’ estimates thus obtained are processed to obtain ‘n’ pseudo estimates ( iw∗ ) as,

( ) ( )( )( 1). 1 .i k i nw n w n w∗= −= − − (4)

267978-1-4799-3540-6/14/$31.00 ©2014 IEEE AP-S 2014

The mean of Eq.(4) gives precise estimation of ionopsheric delay ( kw∗ ) at user location. The standard error (σ) is given as,

( ) ( )2

* *

1

1 . 1n

i ki

w wn nσ=

= −− ∑ (5)

The theoretical background with relevant mathematical expressions to estimate ionospheric time delay using Kriging, Spline and MPF is available in open literature [4].

III. RESULTS AND DISCUSSION Multi-station data of GAGAN network of several days spanning over different seasons is considered for evaluating the proposed model. To highlight the significance of this model the analysis corresponding to a typical day (15th April 2006, 4≤Kp≤5) is presented. Fig.1 shows the total number of SVs visible at every epoch. It can be observed that the satellite visibility dropped to two SVs at certain epochs.

0 2 4 6 8 10 12 14 16 18 20 22 240

2

4

6

8

10

12

Local Time (Hrs)

No

of S

Vs

Station: HyderabadDate: 15 April 2006

Fig.1 Number of satellites visible at Hyderabad (15th Apr.06)

Due to ‘n’ visible satellite at an epoch, there are ‘n’ IPP

vertical delays available. Among them whichever IPP location is nearest to the station/user coordinates, the vertical delay of only that IPP is considered as a reference and all other models including the proposed one are compared. In this way it is possible to compare the results due to models with experimental data. Fig.2 shows the time delay estimates due to the models and delays observed at nearest IPPs. The rms error of delay estimates due to proposed models, anisotropic IDW with Jackknife, kriging, spline and modified planar fit are 0.21 m, 0.23 m, 0.24 m and 0.34 m respectively. During 5.5 Hrs to 6.1 Hrs only two SVs are visible (Fig.1). The anisotropic IDW with Jackknife technique estimated reasonably well even with two IPP delays (Fig.2a).

0 3 6 9 12 15 18 21 240

2

4

6

8

10

Local Time (Hrs)

Ver

tical

Del

ay (m

)

Nearest IPPAIDW Jkf

Station: HyderabadDate: 15 April 2006

0 3 6 9 12 15 18 21 24-4

-2

0

2

4

6

8

10

Local Time (Hrs)

Ver

tical

Del

ay (m

)

Nearest IPPKriging

Station: HyderabadDate: 15 April 2006

(a) (b)

0 3 6 9 12 15 18 21 240

2

4

6

8

10

Local Time (Hrs)

Ver

tical

Del

ay (m

)

Nearest IPPSpline

Station: HyderabadDate: 15 April 2006

0 3 6 9 12 15 18 21 24-4

-2

0

2

4

6

8

10

Local Time (Hrs)

Ver

tical

Del

ay (m

)

Nearest IPPM odified Planarfit

Station: HyderabadDate: 15 April 2006

(c) (d)

Particularly, at 5.33 Hrs local time (LT), the experimental vertical delay value is 0.73 m and the estimated delay value due to AIDW with JackKnife is 0.71 m. But, when only two SVs are available the delay estimates due to kriging and modified planar fit are abnormal (Fig.2b and 2d). Further, the Spline technique fails to interpolate with two IPPs data (Fig.2c). This is because QR factorization or decomposition is used to attain least square approximation solution in Spline techniques. Therefore, a minimum of 3 data points are required to perform QR factorization.

IV. CONCLUSIONS To model the ionospheric delay variations a new approach known as ‘anisotropic IDW with Jackknife’ is proposed. For the first time the parameters, anisotropic ratio and anisotropy angle that determine the anisotropic instability of ionosphere electron density are considered in the model for better time delay estimation. Even with less number of data points, the proposed method is giving reliable results. The model is capable of reliably producing diurnal variations of ionospheric time delay.

ACKNOWLEDGEMENT The above work has been carried out under the project entitled “Data Processing Techniques to Improve the Real Time Performance of GNSS Systems” sponsored by DST, New Delhi, India sponsored project, vide sanction letter No. SR/S4/AS-89/2012, dated: 25th September 2012.

REFERENCES [1] Sarma, A. D., N. Prasad, and T. Madhu, “Investigation of Suitability of

Grid‐based Ionospheric Models for GAGAN, Electron. Letters, Vol.42, Issue 8, pp.478–479, 2006.

[2] Tomczak, M., “Spatial Interpolation and its Uncertainty using Automated Anisotropic Inverse Distance Weighing (IDW) - Cross-Validation/Jackknife approach”, Journal of Geographic Information and Decision Analysis, Vol.2, No.2, pp.18-30, 2003.

[3] Pi X., S. Nandi, D.A. Stowers, M.R. Marcin, U.J. Lindqwister, M.J. Reyes, A.W. Moore, D.N. Fort, and J.A. Klobuchar, “Development of Ionospheric Scintillation Monitoring System using Receivers of the IGS Global GPS Network”, Proc. of the 55th Annual ION meeting, Cambridge, Massachusetts, pp.395-402, June 1999.

[4] Ratnam, D.V., A. D. Sarma, V. Satya Srinivas and P. Sreelatha, “Performance Evaluation of Selected Ionospheric Delay Models during Geomagnetic Storm Conditions in Low Latitude Region”, Radio Science Journal, March, 2011.

Fig.2 Comparison of nearest IPP delays with the delay estimates due to the models (a) Anisotropic IDW with Jackknife (b) Kriging (c) Spline and (d) Modified Planar fit

268