CURRENT IONOSPHERIC MODELS AND ANALYSIS

Chapter 4: Ionospheric D elay and Modelling

Many ionospheric models have been developed for the estimation of the ionosphere delays. An ionospheric estimation about 50 percent accuracy based on the broadcast model can be conveniently obtained from the coefficients of these polynomials in navigation message files (Klobuchar, 1987, Feess, 1987, and Leick, 1995). This model may be suitable for general GPS users but nevertheless does not satisfy with the need of high precision GPS users. A model based on P-code was tried by using Kalman filtering technique to smooth the P4 observations (Stewart, 1997 and Webster, 1993). The advantage of this modelling is that no ambiguity is involved but nevertheless the instrument biases have to be previously calibrated and the code multipath is too noisy to be exactly modelled. For a high precision ionospheric modelling the precise phase data is used optimally. These models (Lanyi, 1988, Georgiadou and Kleusberg, 1987, Wild et. al., 1989, Mannucci et. al., 1993, Sardon et. al., 1994a, Schaer et. al., 1995, Chao et. al., 1995, Schaer et. al., 1996, and Feltens et. al., 1996) are summarized and analyzed as below.

1. For these models using L4 observation, model (C), (E), and (F) keep the constant (ambiguity and instrument bias) as a parameter of the observation equations and Model (B) and (D) take double differencing strategy with pre-resolution of ambiguities to eliminate the constant. For these models using Lc observation. Model (G) and (H) keep the instrument bias as a parameter of the observation equations, and Model (A) and (I) use the preprocessing of hardware calibration (at least one station the biases have to be fixed for Model (A)).

2. All current models need to remove the cycle slips for a time series of data.

3 Data below the elevation angle of 20 degrees are not used in most of current models. 4 The most popular equation of the mapping function is shown as in (4.12).

5 The equations used to describe the vertical TEC can basically be divided into polynomials, Taylor expansion, spherical harmonic functions, and triangular interpolation.

6 The height of the shell adopted by most models is 350 km.

7 The sun-earth reference system is adopted by most models to deal with the problem of time variation of the ionosphere.

8 Basically the parameter estimation strategy of these models is the least squares adjustment or Kalman filtering technique.

Table 4.2 A summary of ionospheric models

Item \ Model (A) (B) (C) (D) (E) (F) (G) (H) (I)

l.OBS Lc L4 L4 L4 L4 L4 Lc Lc Lc

2.DIFF UD DD UD DD U/SD UD UD UD UD

3.PREl-slip Yes Yes Yes yes Yes Yes yes Yes yes

4.PRE2-bias Yes no No no No No no No yes

5.PRE3-amb No Yes No yes No No no No no

6.MAP 4.12 4.12 — 4.12 4.11 4.11 4.12 4.15 4.12

7.VTEC 4.16 4.17 4.17 4.17 4.20 4.19 4.19 4.19 4.18

8.SHELL 350 400 — 350 350 400 350 — 350

9.SOLAR Yes Yes Yes yes Yes No yes No yes

a.ESTM K-B L-N L-C L-N L-C L-C L-B K-B —

b.Cut-off 20 20 — 20 — — — 20 20

c.VERF A-a B-p B-p B-n B-p B-1 A-a No A-a

d.RMS (unit: cm) 5 tecu 17.6/ 6.2 82/ 37 85%/ 75% 1.2/ 1.0 0.07 ppm 3 ns No 10 tecu e.Area/Term G G/R G G L L L L G

(A) JPL, A J. Mannucci et. a l, 1993 (C) Stanford, Y.Chao et. a l, 1995 (E) Berne, U. Wild et. al., 1989 (G) G.E. Lanyi, 1988

(I ) ESOC, J. Feltens et. al., 1996

(B) Berne, S. Schaer et. a l, 1995 (D) CODE, S. Schaer et. al., 1996

(F) Y. Georgiadou and A. Kleusberg, 1988 (H) E.Sardon et. al., 1994a

l.OBS: 2.DIFF: 3.PREl-slip: 4.PRE2-bias: 5.PRE3-amb 6.MAP: 7.VTEC: 8.SHELL: 9.S0LAR: a.ESTM: b.Cut-off: c.VERF: d.RMS e.Area ionospheric observations

UD-undifferenced, SD-single differenced, DD-double differenced preprocessing o f cycle-slip detection and repair

precalibration o f instrument biases preresolution o f ambiguities the mapping function (equation no.) the vertical TEC (equation no.) the height of spherical shell (unit: km) sun-earth reference system

parameter estimation strategy,

L-least squares adjustment, K-kalman filtering

B-with instrument parameters,N- with ambiguity parameters C-with ambiguity and instrument parameters

elevation angle cutoff (unit: degree) verification o f the modelling,

A-differences with comparison of another method, B-comparison without/with the model, (default: cm)

p-positioning accuracy a-absolute vertical TEC (0.9cm=3ns=8.5tecu), n-ambiguity resolution(%), 1-baseline length (ppm)

accuracy o f the result: t-TECU, s-nanosecond, p-ppm G-global, R-regional, L-local

no comments

9 The resultant comparison with other models such as Model (A), (G), and (I), or the result of the comparison with and without the model such as Model (B), (C), (D),

Chapter 4: Ionospheric D elay and Modelling

can be the absolute vertical TEC (e.g. cm, ns, or TECU), the positioning accuracy (cm), the percentage of ambiguity resolution (%), and the ratio of the error and baseline length (ppm). By using the former verification method, the accuracy of vertical TEC of Model (A), (G), and (I) can be achieved at about 5 to 10 TECU (~ 52 cm to ~ 104 cm, see Feltens J. et. ah, 1996). By using the latter verification method, the improvement of GPS positioning accuracy or ambiguity resolution has been made by Model (B) to (F). However the evaluation of these models seems impossible to be accomplished without using the same data set and under the same processing conditions.