A Two-Step Estimation of GPS Differential Code Biases and Local Ionospheric TEC Based on Orthogonal Transformation

We have developed a fast and efficient algorithm for estimating the global positioning system (GPS) differential code biases (DCBs) and ionospheric vertical total electron content (VTEC) using a local GPS network. The algorithm does not simultaneously estimate the DCBs and VTEC parameters, it instead determines the parameters in a two-step estimation. At the first estimation step, the algorithm determines the DCBs parameters and, at the second estimation step, the algorithm determines the VTEC parameters. While the DCB estimations (the first step) are carried out by recursively accumulating the normal equation for each observation batch (or for each epoch), the VTEC estimations (the second step) are recursively executed every batch. By using the algorithm, the users who only need the DCBs estimates may stop when the first estimation step finishes, making the computational time efficient. In constructing the algorithm, we applied the orthogonal transformation to ensure that the algorithm is numerically reliable. We then fully utilized the structure of the problem to enable recursive calculations for each estimation step, making the algorithm fast and efficient. We found that the algorithm produces identical results to those calculated using the standard method, but the calculation time is tremendously reduced. Furthermore, the algorithm enables epoch-wise solutions of the VTEC estimates to be rapidly executed using an ordinary laptop computer. In addition, the algorithm is applicable to determine the DCBs and VTEC estimates both in a network mode and a single receiver mode.