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A matrix-free reformulation of the multi-parameter descent and conjugate-gradient method for isotropic elastic iterative reverse-time migration

Multi-parameter inversion of linear systems appears in many problems. The focus here is on isotropic elastic iterative reverse-time migration for three position-dependent subsurface model parameters, which amounts to data fitting of processed seismic data with synthetics from the Born approximation of the elastic wave equation. In that case, the matrix of the linear system is the hessian. As it is impractical to form, a matrix-free formulation is needed, which is readily derived for the gradient descent method. For single-parameter inversion, the conjugate-gradient (CG) method is generally more efficient than simple descent. However, the multiple-parameter CG method has a significantly higher cost than the descent method. Here, first a matrix-free data-domain reformulation is derived. Then, its performance is compared to the simple descent method to see of its faster convergence justifies the higher cost. A comparison on a marine 2-D toy problem with a salt body and sea-bottom receivers shows that the multiple-parameter descent method wins in terms of efficiency if the number of iterations is limited and that the single-parameter CG method is even faster.