# Gradient-based deterministic inversion of geophysical data with Generative Adversarial Networks: is it feasible?

## TL;DR

This paper investigates the feasibility of gradient-based deterministic inversion methods within the latent space of GANs for geophysical data, revealing significant limitations due to nonlinearity and emphasizing the robustness of probabilistic approaches.

## Contribution

It demonstrates the challenges of applying gradient-based deterministic inversion in GAN latent spaces for geophysical problems, highlighting the importance of inversion approach and initial conditions.

## Key findings

- Gradient-based methods often fail to find suitable solutions within limited iterations.
- Gauss-Newton method was unable to recover solutions with correct data misfit.
- Probabilistic global optimization reliably finds appropriate solutions.

## Abstract

Global probabilistic inversion within the latent space learned by a Generative Adversarial Network (GAN) has been recently demonstrated. Compared to inversion on the original model space, using the latent space of a trained GAN can offer the following benefits: (1) the generated model proposals are geostatistically consistent with the prescribed prior training image (TI), and (2) the parameter space is reduced by orders of magnitude compared to the original model space. Nevertheless, exploring the learned latent space by state-of-the-art Markov chain Monte Carlo (MCMC) methods may still require a large computational effort. As an alternative, parameters in this latent space could possibly be optimized with much less computationally expensive gradient-based methods. We show that due to the typically highly nonlinear relationship between the latent space and the associated output space of a GAN, gradient-based deterministic inversion may fail even when considering a linear forward physical model. We tested two deterministic inversion approaches: a quasi-Newton gradient descent using the Adam algorithm and a Gauss-Newton (GN) method that makes use of the Jacobian matrix calculated by finite-differencing. For a channelized binary TI and a synthetic linear crosshole ground penetrating radar (GPR) tomography problem involving 576 measurements with low noise, we observe that when allowing for a total of 10,000 iterations only 13% of the gradient descent trials locate a solution that has the required data misfit. The tested GN inversion was unable to recover a solution with the appropriate data misfit. Our results suggest that deterministic inversion performance strongly depends on the inversion approach, starting model, true reference model, number of iterations and noise realization. In contrast, computationally-expensive probabilistic global optimization always finds an appropriate solution.