GNSS-Based Non-Negative Absolute Ionosphere Total Electron Content, its Spatial Gradients, Time Derivatives and Differential Code Biases: Bounded-Variable Least-Squares and Taylor Series

Global navigation satellite systems (GNSS) allow estimating total electron content (TEC). However, it is still a problem to calculate absolute ionosphere parameters from GNSS data: negative TEC values could appear, and most of existing algorithms does not enable to estimate TEC spatial gradients and TEC time derivatives. We developed an algorithm to recover the absolute non-negative vertical and slant TEC, its derivatives and its gradients, as well as the GNSS equipment differential code biases (DCBs) by using the Taylor series expansion and bounded-variable least-squares. We termed this algorithm TuRBOTEC. Bounded-variable least-squares fitting ensures non-negative values of both slant TEC and vertical TEC. The second order Taylor series expansion could provide a relevant TEC spatial gradients and TEC time derivatives. The technique validation was performed by using independent experimental data over 2014 and the IRI-2012 and IRI-plas models. As a TEC source we used Madrigal maps, CODE (the Center for Orbit Determination in Europe) global ionosphere maps (GIM), the IONOLAB software, and the SEEMALA-TEC software developed by Dr. Seemala. For the Asian mid-latitudes TuRBOTEC results agree with the GIM and IONOLAB data (root-mean-square was < 3 TECU), but they disagree with the SEEMALA-TEC and Madrigal data (root-mean-square was >10 TECU). About 9% of vertical TECs from the TuRBOTEC estimates exceed (by more than 1 TECU) those from the same algorithm but without constraints. The analysis of TEC spatial gradients showed that as far as 10–15° on latitude, TEC estimation error exceeds 10 TECU. Longitudinal gradients produce smaller error for the same distance. Experimental GLObal Navigation Satellite System (GLONASS) DCB from TuRBOTEC and CODE peaked 15 TECU difference, while GPS DCB agrees. Slant TEC series indicate that the TuRBOTEC data for GLONASS are physically more plausible.