Improving the performance of time varying spherical radial basis functions in regional VTEC modeling with sparse data

Abstract

Least-squares parameterization of retrieved Vertical Total Electron Content (VTEC) from permanent dual-frequency GPS stations in terms of Time Varying Spherical Radial Basis Functions (TV-SRBFs) with optimal shape parameter leads to proper regional VTEC reconstruction at Ionospheric Pierce Points (IPPs). However, in the area where there is no observation, the modeling result is not satisfactory. This feature is especially tangible when the number of IPPs in the study area is low with poor geometric distribution, which leads to a rank deficient problem. Whereas, if the global ionospheric maps (GIMs) are used as a background in estimating the optimal shape parameter of TV-SRBFs and coefficients of the expansion, the stated obstacle to produce accurate regional ionospheric maps will be removed. For this purpose, an objective function is defined and optimized. The objective function consists of three terms. First, L-2 norm of the differences between the reproduced VTECs and values obtained from the receivers’ observations. Second, L-2 norm of the differences between the reproduced values on a regular grid with the values obtained from the spatial interpolation of GIM at a specific moment in the modeling interval. Third, the discrete smoothing norm of the coefficients of TV-SRBFs. Since the obtained VTECs from GPS receivers are more accurate than the resulted VTECs from interpolation of GIMs and also a regularization parameter is needed due to the ill-posedness, the three objectives cannot be considered with the same weights in the multi-objective optimization. Hence the weighted sum method with the preference-based solution is applied to obtain the optimal relative weight of GIMs, the regularization parameter and the coefficients. Moreover, utilizing the least-squares collocation (LSC) can also improve the modeling accuracy to some extent. The proposed method is applied for hourly regional VTEC modeling over Iran from 3 May to 9 May 2016, which had both low and very high geomagnetic activities. A portion of the retrieved VTECs which is excluded from the modeling dataset, is considered as test data. The Root Mean Square (RMS) of the differences between obtained VTECs from regional constructed model and test data is 0.94 TECU in the study period. RMS of the differences between obtained VTECs from the regional model and VTECs of two control stations which are not involved in the modeling are 1.12 TECU and 1.24 TECU. Also, the regional model performs well in ionospheric delay estimation and the accuracy of single-frequency positioning at control stations in the study period is better than 0.7 m.