<p>The normal equations are widely used to combine elementary least squares solutions, to solve very large problems which are not possible to handle directly. The principle is to reduce each problem to a minimal set of parameters present in the global problem, without removing the corresponding information, and connect them. For instance, one important application is the combination over years of daily network solutions, as performed for the ITRF (Altamimi et al., 2016) [1].</p><p>The approach can also be used in orbit determination to connect arcs solutions in order to construct the solution of a global arc. This was applied for example for GPS constellation solutions as in the article written by Beutler et al. (1996) [2]. Due to the size of the problems, it is interesting to divide for example a three days solution into three one day solutions. Another advantage is that the one day solutions are usually efficiently processed by the orbit determination software. For rapid or ultra-rapid GNSS products this is also very interesting, as the solutions are needed very often for small shifts of the global arc (for example 24 hours arcs, shifted every 6 hours in the case of ultra-rapid products). A further extension is to construct recursive solutions from these elementary arcs, leading to a filter similar to a Kalman filter.</p><p>We propose a unified methodology, associated with an efficient implementation compatible with our least squares software GINS, allowing us to solve the various problems ranging from arc connection to sequential filtering. The final objective is to construct efficient GNSS ultra-rapid products.</p><p>The application on a simple problem consisting in connecting different SLR arcs is shown, as a test case to develop and implement the methodology. In this case, the global solution can also be directly constructed for validation purposes. This study includes the construction of the solution at the end points of the elementary arcs, and also the recovery of the global solution state vectors at every epoch.</p><p>The next step will be to implement more complex parameterizations (including measurement parameters, which are not present in the SLR test case), and to apply this for GNSS constellation solutions.</p>
Helmert Variance Component Estimation for Multi-GNSS Relative Positioning