Iterative Least Squares Method For Inversion Of Linear-non-linear Models

Fukuda and Johnson(2010)proposed an inversion method based on the Bayesian theory(hereinafter termed the F-J method)for linear-non-linear problems,which has been widely used to estimate the globally-optimal parameters and their precisions of linear-non-linear geophysical models.However,the choice of Markov Chain Monte Carlo(MCMC)sampling parameters affects the results based on the F-J method and finding appropriate sampling parameters is time-consuming.On the other hand,the iterative least squares can be applied to the inversion of linear-non-linear models after linearizing the non-linear parameters,but the results of this method depends on the choice of initial values.For the inversion of linear-non-linear problems,this paper introduces the basics of the Bayesian theory,the fully Bayesian inversion method,the F-J method,MCMC sampling algorithms and iterative least squares,respectively.Based on this,a hybrid method termed as the iterative least squares with initial values constrained by using the F-J method is proposed.Afterwards,using the methods mentioned,the inversion on synthetic data of the inter-seismic model and the inversion on Mogi model of synthetic and real data sets are carried on:different MCMC sampling methods are employed to investigate the influence of sampling algorithms on the inversion results,then the F-J method,iterative least squares with random initial values and iterative least squares with initial values provided by the F-J method,respectively,are used in the inversion of linear-non-linear problems.The results show that(1)the choice of sampling parameters affects the results of the F-J method,(2)the sampling algorithms based on Gibbs sampling and simulated annealing are better than the Metropolis algorithm in some degree,(3)the inversion results based on the iterative least squares with random initial generally diverge and even worse,converge to wrong results,and(4)the iterative least squares with initial values provided by the F-J method can result in convergence,retaining the advantages of the global optimality of the F-J method and efficiency of the iterative least squares.