# Eigenvector Model Descriptors for Solving an Inverse Problem of Helmholtz Equation: Extended Materials

## TL;DR

This paper introduces an eigenvector-based regularization method for seismic inverse problems, improving subsurface property recovery by reducing ill-posedness and effectively reconstructing complex media like salt-domes.

## Contribution

The study proposes a novel eigenvector model descriptor approach for seismic inversion, demonstrating its effectiveness in 2D and 3D reconstructions, especially for challenging salt-dome media.

## Key findings

- Eigenvector representation enhances seismic imaging accuracy.
- The method compensates for low-frequency data limitations.
- Effective in 2D and 3D seismic inverse problems.

## Abstract

We study the seismic inverse problem for the recovery of subsurface properties in acoustic media. In order to reduce the ill-posedness of the problem, the heterogeneous wave speed parameter to be recovered is represented using a limited number of coefficients associated with a basis of eigenvectors of a diffusion equation, following the regularization by discretization approach. We compare several choices for the diffusion coefficient in the partial differential equations, which are extracted from the field of image processing. We first investigate their efficiency for image decomposition (accuracy of the representation with respect to the number of variables and denoising). Next, we implement the method in the quantitative reconstruction procedure for seismic imaging, following the Full Waveform Inversion method, where the difficulty resides in that the basis is defined from an initial model where none of the actual structures is known. In particular, we demonstrate that the method is efficient for the challenging reconstruction of media with salt-domes. We employ the method in two and three-dimensional experiments and show that the eigenvector representation compensates for the lack of low frequency information, it eventually serves us to extract guidelines for the implementation of the method.