[Paper Review] Transdimensional 2D Full-Waveform Inversion and Uncertainty Estimation

Reetam Biswas, Mrinal K. Sen|arXiv (Cornell University)|Jan 23, 2022

Seismic Imaging and Inversion Techniques6 citations

TL;DR

This paper introduces a transdimensional 2D full-waveform inversion (FWI) framework using Reversible Jump Hamiltonian Monte Carlo (RJHMC) with Voronoi-based velocity parametrization to simultaneously estimate model parameters and the optimal number of nuclei. The method enables efficient, gradient-guided sampling of high-dimensional model space, achieving accurate reconstruction of the Marmousi model with ~15,500 nuclei and quantifying uncertainty via posterior predictive distributions.

ABSTRACT

Full-Waveform Inversion (FWI) has now become a widely accepted tool to obtain high-resolution velocity models from seismic data. Typically, the velocity model in its discrete form is represented on a rectangular grid, and we solve for the elastic properties at these grid points. FWI is mostly solved employing a local optimization method, where one obtains a velocity update by minimizing the misfit between the observed and the calculated seismograms. Note also that FWI is a highly non-linear problem which is known to be prone to non-uniqueness. The convergence to a globally optimum solution is not guaranteed; it depends on the choice of the starting model. Thus, a Bayesian formulation of the inverse problem with subsequent sampling of the posterior distribution is a preferred choice, since it enables uncertainty quantification. However, with the increase in the dimension of a model, sampling search space becomes computationally expensive. We employ a recently developed trans-dimensional sampling method called Reversible Jump Hamiltonian Monte Carlo (RJHMC), to the 2D full waveform inversion problem. We represent our velocity model using Voronoi cells, determined from the distribution of certain nuclei points in the model space. This method offers two advantages. First, it solves for a variable dimensional velocity updates by using a trans-dimensional reversible jump Markov Chain Monte Carlo (RJMCMC) step and thus tries to achieve an optimum number of nuclei to represent the model and minimize the misfit. A smaller number of parameters helps in an efficient sampling of the model search space. Second, it applies the gradient-based Hamiltonian Monte Carlo (HMC) step, which further improves the sampling by allowing the algorithm to take a large step guided by the gradient. This two-step algorithm proves to be a useful tool for model exploration and uncertainty quantification in FWI.

Motivation & Objective

• To address the non-uniqueness and high-dimensionality of full-waveform inversion (FWI) by enabling automatic model parametrization selection.
• To improve model space exploration and uncertainty quantification in 2D FWI using a Bayesian transdimensional framework.
• To overcome limitations of fixed-dimensional parametrization in FWI, which can lead to overfitting or underfitting due to suboptimal number of parameters.
• To implement a computationally efficient sampling strategy that balances model complexity and data fit through parsimonious Bayesian inference.
• To demonstrate the feasibility of using Voronoi cells with RJHMC for realistic 2D FWI with uncertainty quantification.

Proposed method

• The velocity model is represented using Voronoi cells generated from a variable number of nuclei points in the 2D domain, enabling adaptive spatial resolution.
• A transdimensional reversible jump Markov Chain Monte Carlo (RJMCMC) step dynamically adjusts the number of nuclei during sampling to find the optimal model complexity.
• The Hamiltonian Monte Carlo (HMC) step uses gradient information from the misfit function to enable large, efficient jumps in model space, improving mixing and convergence.
• The algorithm combines RJMCMC and HMC in a two-step process: first, it proposes changes in model dimension (number of nuclei), and second, it performs gradient-guided updates to nuclei locations and velocities.
• A two-level checkpointing strategy is employed to manage high memory usage from storing forward and adjoint wavefields, enabling efficient GPU computation with NVMe storage.
• The posterior distribution is sampled using a Bayesian framework, where models with fewer parameters are naturally favored, promoting parsimony and reducing overfitting.

Experimental results

Research questions

• RQ1 Can transdimensional FWI with adaptive parametrization improve model resolution and uncertainty quantification compared to fixed-grid FWI?
• RQ2 How does the combination of RJHMC and Voronoi-based parametrization affect sampling efficiency and convergence in 2D FWI?
• RQ3 What is the optimal number of nuclei required to accurately reconstruct a complex velocity model like Marmousi, as dictated by the data?
• RQ4 To what extent does the Bayesian framework with parsimony reduce overfitting in high-dimensional FWI problems?
• RQ5 How does the use of gradient-based HMC steps enhance exploration of complex, non-linear model spaces in FWI?

Key findings

• The RJHMC algorithm successfully reconstructed the Marmousi velocity model using only approximately 15,500 nuclei, significantly fewer than the maximum possible 192,517 grid points.
• The posterior distribution peaked at around 16,000 nuclei, indicating that the data dictated an optimal, parsimonious model complexity, consistent with Bayesian parsimony.
• The method achieved accurate velocity model recovery with minimal over-parameterization, as evidenced by the concentration of posterior samples around the optimal number of nuclei.
• The use of gradient-based HMC steps enabled large, efficient jumps in model space, significantly improving sampling efficiency compared to standard MCMC methods.
• The two-level checkpointing technique effectively managed high memory demands, allowing sustained GPU computation with background data transfers.
• Multiple models sampled from the posterior predictive distribution enabled robust uncertainty quantification of P-wave velocity estimates in 2D FWI.