Combination of the Levenberg–Marquardt and differential evolution algorithms for the fitting of postseismic GPS time series

Abstract

Postseismic global positioning system (GPS) time series are of fundamental importance for investigating the physical mechanisms of postseismic deformations, as well as the construction and maintenance of terrestrial reference frames. Particularly, methods for constructing accurate fitting models for such time series are critical. Based on the physical features of postseismic deformation models, we propose a new algorithm that combines the strengths of the Levenberg–Marquardt (LM) and differential evolution (DE) algorithms, that is, the LM + DE algorithm. In this algorithm, the parameters are initialised by the constrained DE algorithm; the final parameters of the postseismic model are then solved by the LM algorithm. To validate the proposed method, DE, LM, and LM + DE were compared using synthetic and observational data from the 2011 Tohoku Earthquake. For all tests based on synthetic data, the LM + DE algorithm consistently converged to the global solution and the residual is small, regardless of how the independent parameter was varied. In the 2011 Tohoku earthquake, the parameters calculated by the LM + DE algorithm matched consistently for the global solution with a 100% passing rate after constraints were provided for the ratios of the initial relaxation time parameters. In contrast, the LM and DE algorithms individually achieved passing rates of only 22% and 1%, respectively. These results demonstrate that the proposed LM + DE algorithm effectively solves the initial estimate problem in the fitting of nonlinear postseismic models, and also ensures that the fits are mathematically optimal and consistent with physical reality.