Joint inversion of multiple data types with the use of multiobjective optimization: problem formulation and application to the seismic anisotropy investigations

Abstract

S U M M A R YIn geophysical studies the problem of joint inversion of multiple experimental data sets obtained by different methods is conventionally considered as a scalar one. Namely, a solution is found by minimization of linear combination of functions describing the fit of the values predicted from the model to each set of data. In the present paper we demonstrate that this standard approach is not always justified and propose to consider a joint inversion problem as a multiobjective optimization problem (MOP), for which the misfit function is a vector. The method is based on analysis of two types of solutions to MOP considered in the space of misfit functions (objective space). The first one is a set of complete optimal solutions that minimize all the components of a vector misfit function simultaneously. The second one is a set of Pareto optimal solutions, or trade-off solutions, for which it is not possible to decrease any component of the vector misfit function without increasing at least one other. We investigate connection between the standard formulation of a joint inversion problem and the multiobjective formulation and demonstrate that the standard formulation is a particular case of scalarization of a multiobjective problem using a weighted sum of component misfit functions (objectives). We illustrate the multiobjective approach with a non-linear problem of the joint inversion of shear wave splitting parameters and longitudinal wave residuals. Using synthetic data and real data from three passive seismic experiments, we demonstrate that random noise in the data and inexact model parametrization destroy the complete optimal solution, which degenerates into a fairly large Pareto set. As a result, non-uniqueness of the problem of joint inversion increases. If the random noise in the data is the only source of uncertainty, the Pareto set expands around the true solution in the objective space. In this case the 'ideal point' method of scalarization of multiobjective problems can be used. If the uncertainty is due to inexact model parametrization, the Pareto set in the objective space deviates strongly from the true solution. In this case all scalarization methods fail to find the solution close to the true one and a change of model parametrization is necessary.

Proterozoic–Archean boundary in the mantle lithosphere of eastern Fennoscandia as seen by seismic anisotropy

“…Therefore, at present stage we are hunting for more shear-wave splitting parameters in surprisingly rather noisy SVEKALAPKO dataset . All anisotropic parameters evaluated in 3D, both the shearwave splitting and the P-spheres, will input into joint inversion of multi-method geophysical data with the use of multi-objective optimisation (Kozlovskaya, 2001;Kozlovskaya et al, 2005).…”

Integrating Electromagnetic Data with Other Geophysical Observations for Enhanced Imaging of the Earth: A Tutorial and Review

“…Thus, plotting combinations of misfits for the different data sets and the constraints reveals if they are compatible with each other or minimizing one comes at the cost of deteriorating the match for the other. In practical inversion, applications such plots can be constructed by varying the weight for the different terms of the objective function (Thompson et al 2016) or using specially designed multi-objective minimization algorithms (Kozlovskaya et al 2007;Moorkamp et al 2007Moorkamp et al , 2010Roux et al 2011;Schnaidt and Heinson 2015;Niri and Lumley 2016).…”

A multi-objective stochastic optimization approach for estimation of subsurface geomodels

“…Conventionally, there are two broad approaches to handle the multi-purpose subsurface geomodeling process (Kozlovskaya et al, 2007):…”