The truncated singular value decomposition (TSVD) regularization applied in ill-posed problem is studied. Through mathematical analysis, a new method for truncated parameter selection which is applied in TSVD regularization is proposed. In the new method, all the local optimal truncated parameters are selected first by taking into account the interval estimation of the observation noises; then the optimal truncated parameter is selected from the local optimal ones. While comparing the new method with the traditional generalized cross-validation (GCV) andLcurve methods, a random ill-posed matrices simulation approach is developed in order to make the comparison as statistically meaningful as possible. Simulation experiments have shown that the solutions applied with the new method have the smallest mean square errors, and the computational cost of the new algorithm is the least.

Integer ambiguity resolution plays a key role in applications related to GNSS precise positioning. This contribution focuses on three commonly used integer estimators (IEs), i.e., integer rounding (IR), integer bootstrapping (IB) and integer least-squares (ILS). Contributions are mainly of four aspects. First, the objective function of IR and IB is given, respectively. Second, an upper bound for ILS is proposed. Third, a sorting technique is introduced to tighten the upper bound of IR/ILS after decorrelation. Fourth, the success-rate approximation for IR and ILS with bounded error is developed, respectively. Finally, real-collected data in PPP validate the following arguments. (i) Applying sorting technique after decorrelation can improve the tightness of IR upper bound a lot, but can only slightly tighten the ILS upper bound. (ii) The proposed ILS upper bound is almost as the same tight as the one in PS-LAMBDA software, much tighter than other known upper bounds. Unlike the PS-LAMBDA upper bound, the proposed ILS upper bound has the advantage of time efficiency for real-time applications. (iii) The proposed approximations produce more accuracy approximated success-rate than the frequently-used ambiguity dilution of precision (ADOP) based approximation.