Sparse Hardy function model of regional velocity field from GNSS data

Classical Hardy function interpolation method is often necessary in establishment of regional velocity field. In this work, a regularization method, namely the L1-norm regularization, also called the least absolute shrinkage selection operator (Lasso), is employed to improve the traditional method. With the new method, a sparse model can be obtained with many zero elements in the parameter vector. Compared to L2-norm regularization, which is also called the Tikhonov regularization, the L1-norm regularization will select the best model automatically by making the coefficient of the unnecessary kernels zero, through solving a convex optimization problem. In this paper, the velocity field dataset derived from global navigation satellite system data is used to establish models in different directions. By comparing the interpolation accuracy of velocity at the same unknown points of Hardy function improved by Lasso and Tikhonov regularizations respectively, the feasibility of the former is verified. The results show that L1-norm regularization method has a slightly worse interpolation accuracy than Tikhonov regularization method in North and Up directions, but in East direction, the interpolation effect is much better and all directions can get great sparse performance. More specifically, the prediction accuracy of the proposed method in the East direction is improved by 17.3%, but in North and Up directions is decreased by 5.4% and 6.3% respectively in comparison with that of Tikhonov regularization method, though the sparse rates of them are over 60%. In addition, besides selecting the root mean square error as the evaluation standard for model selection, we can also select the sparse rate as the evaluation criteria to make the model much sparser if necessary. A sparse model would be beneficial in terms of better interpretability and improved variable assigning efficiency. To summarize, L1-norm regularization can be viewed as a potential alternative in velocity field modelling.

Automated summary

In this study, the traditional Hardy function interpolation method was improved by using L1-norm regularization, which automatically selects the best model and achieves sparse performance. The results showed that in the East direction, the interpolation effect of the L1-norm regularization method was significantly better than that of the Tikhonov regularization method, while there was a slight decrease in the North and Up directions.