An Iterative Least Squares Method Based on the F-J Inversion for Linear-Non-Linear Models

Fukuda and Johnson (2010) proposed an inversion method associated with the Bayesian theory (hereinafter termed the F-J method) for linear-non-linear geophysical/geodetic problems. Since then this method has been widely applied to estimating globally optimal parameters and their precisions for linearnon-linear geophysical/geodetic models. Yet for the application of the F-J method the choice of Markov chain Monte Carlo (MCMC) sampling parameters somewhat affects the invered results. Additionally, searching the most appropriate sampling parameters is time-consuming. On the other hand, an iterative least squares method can also be applied to the inversion for linear-non-linear models if their non-linear parameters are linearized beforehand. However, this method depends on optimum initial values. To make full use of the advantages of the both methods, here we propose a hybrid method termed as the iterative least squares with initial values constrained by using the F-J method. The proposed method refers to the calculation of linear-nonlinear parameters using the F-J method just one time and to refining those parameters using the iterative least squares method a few times. Thus, it reduces the computation time relevant to the F-J method on the one hand and gains a global solution after a few times of employing the iterative least squares method on the other hand. To verify the efficacy of the hybrid method, we make comparisons using synthetic and real data sets. We employ the F-J method, iterative least squares method with random initial values and iterative least squares method with initial values provided by the F-J method, respectively. Our results show that:(1) the choice of sampling parameters indeed affects the results by using the F-J method; (2) based on the iterative least squares method with random initial values, inverted results generally diverge and sometimes converge to wrong results for some synthetic tests, and (3) using the iterative least squares method with initial values provided by the F-J method produces converged results without the dependence on MCMC sampling parameters and initial values, as expected thanks to absorbing the merits of the global optimality of the F-J method and efficiency of the iterative least squares method.