insight - Geophysics seismic inversion - # Elastic full-waveform inversion

Robust Elastic Full-Waveform Inversion Using an Alternating Direction Method of Multipliers with Reconstructed Wavefields

Q: How can the proposed ADMM-based EFWI algorithm be extended to handle more complex geological structures, such as anisotropic media or heterogeneous density distributions

The proposed ADMM-based EFWI algorithm can be extended to handle more complex geological structures by incorporating anisotropic media and heterogeneous density distributions. For anisotropic media, the elastic wave equation can be modified to account for the directional dependence of wave propagation. This can involve introducing additional parameters to characterize the anisotropy, such as Thomsen parameters or azimuthal anisotropy parameters. The algorithm can then be adapted to update these parameters along with the traditional elastic parameters. In the case of heterogeneous density distributions, the physical constraints in the algorithm can be expanded to include constraints on density variations. This can involve incorporating prior information about the expected density variations in the subsurface, such as from well logs or geological studies. By constraining the density variations within certain bounds, the algorithm can better capture the complex density structures in the subsurface. By integrating these enhancements, the ADMM-based EFWI algorithm can effectively handle the challenges posed by anisotropic media and heterogeneous density distributions, leading to more accurate subsurface property estimations.

Q: What are the potential limitations of the current physical constraint formulation, and how could it be further improved to better capture the underlying geological relationships

The current physical constraint formulation in the EFWI algorithm may have limitations in capturing the full complexity of geological relationships in the subsurface. One potential limitation is the static nature of the constraints, which may not adapt well to dynamic geological environments. To address this, the physical constraints can be made more adaptive by incorporating machine learning techniques to update the constraints based on the inversion results. Another limitation could be the oversimplification of the constraints, leading to under-constrained or over-constrained inversion problems. To improve this, the constraints can be refined by incorporating additional geological information, such as structural trends, lithological boundaries, or seismic attributes, to better capture the underlying geological relationships. Furthermore, the current formulation may not fully account for uncertainties in the constraints. By introducing probabilistic constraints or incorporating uncertainty quantification methods, the algorithm can provide more robust estimations that consider the uncertainty in the geological relationships. Overall, by addressing these limitations and refining the physical constraint formulation, the EFWI algorithm can better capture the complex geological relationships in the subsurface.

Q: Given the computational efficiency gains from the source sketching method, are there other randomized techniques that could be explored to further accelerate the EFWI process without compromising accuracy

While source sketching has shown promising results in improving the computational efficiency of the EFWI algorithm, there are other randomized techniques that could be explored to further accelerate the process without compromising accuracy. One such technique is random projection, where the model and data are projected onto a lower-dimensional space using random matrices. This reduces the computational complexity of the inversion while preserving the essential information for accurate estimation. Another technique is stochastic optimization, where subsets of the data or model parameters are randomly selected for each iteration of the optimization process. This introduces randomness into the optimization procedure, leading to faster convergence and reduced computational burden. Additionally, techniques such as randomized sampling and Monte Carlo methods can be employed to explore the solution space more efficiently and identify optimal solutions in a shorter time frame. By integrating these randomized techniques into the EFWI algorithm, further computational efficiency gains can be achieved, making the inversion process faster and more scalable.

Core Concepts

The core message of this article is to present a new elastic full-waveform inversion (EFWI) algorithm that leverages the Alternating Direction Method of Multipliers (ADMM) and reconstructed wavefields to accurately estimate subsurface properties. The proposed approach addresses key challenges in EFWI, such as incorporating physical constraints, mitigating interparameter cross-talk, and demonstrating robust convergence even from inaccurate initial models.

Abstract

The article presents a new EFWI algorithm that utilizes the ADMM framework to solve the constrained optimization problem. Key highlights and insights include:

The algorithm formulates EFWI as a constrained optimization problem, where the constraints include the partial differential equation (PDE) governing elastic wave propagation and physical model constraints.
The ADMM method is used to solve the problem, resulting in an iterative algorithm with well-conditioned subproblems. The wavefield reconstruction is the most challenging part, which can be addressed using sparse linear algebra techniques and frequency domain formulations.
Different parameterizations, including Lamé parameters and velocities, are explored. The Gauss-Newton Hessian matrix exhibits a block-diagonal sparse structure, enabling efficient model updates.
Physical constraints are incorporated into the inversion process to promote physically plausible models, which is efficiently addressed using the ADMM method.
Numerical examples using benchmark models demonstrate the performance of the proposed EFWI algorithm, including its ability to handle noise, free surface effects, and the effectiveness of source sketching for computational efficiency.
The algorithm outperforms traditional reduced-space FWI and the Wavefield Reconstruction Inversion (WRI) approach in terms of stability and convergence, particularly when starting from inaccurate initial models.

Stats

The article presents the following key figures and statistics:
"The Hessian matrix is blocky with diagonal blocks, making model updates fast."
"Various numerical examples are used to investigate algorithmic components, including model parameterizations, physical model constraints, the role of the Hessian matrix in suppressing interparameter cross-talk, computational efficiency with the source sketching method, and the effect of noise and near-surface effects."

Quotes

"Elastic full-waveform inversion (EFWI) is a process used to estimate subsurface properties by fitting seismic data while satisfying wave propagation physics."
"The alternating direction method of multipliers is used to solve the problem, resulting in an iterative algorithm with well-conditioned subproblems."
"Gradient ascent is used to update Lagrange multipliers by summing PDE violations."

Key Insights Distilled From

Robust elastic full-waveform inversion using an alternating direction method of multipliers with reconstructed wavefields

by Kamal Aghaza... at arxiv.org 04-12-2024

https://arxiv.org/pdf/2404.07927.pdf

Robust elastic full-waveform inversion using an alternating direction method of multipliers with reconstructed wavefields

Deeper Inquiries

How can the proposed ADMM-based EFWI algorithm be extended to handle more complex geological structures, such as anisotropic media or heterogeneous density distributions

The proposed ADMM-based EFWI algorithm can be extended to handle more complex geological structures by incorporating anisotropic media and heterogeneous density distributions. For anisotropic media, the elastic wave equation can be modified to account for the directional dependence of wave propagation. This can involve introducing additional parameters to characterize the anisotropy, such as Thomsen parameters or azimuthal anisotropy parameters. The algorithm can then be adapted to update these parameters along with the traditional elastic parameters.
In the case of heterogeneous density distributions, the physical constraints in the algorithm can be expanded to include constraints on density variations. This can involve incorporating prior information about the expected density variations in the subsurface, such as from well logs or geological studies. By constraining the density variations within certain bounds, the algorithm can better capture the complex density structures in the subsurface.
By integrating these enhancements, the ADMM-based EFWI algorithm can effectively handle the challenges posed by anisotropic media and heterogeneous density distributions, leading to more accurate subsurface property estimations.

What are the potential limitations of the current physical constraint formulation, and how could it be further improved to better capture the underlying geological relationships

The current physical constraint formulation in the EFWI algorithm may have limitations in capturing the full complexity of geological relationships in the subsurface. One potential limitation is the static nature of the constraints, which may not adapt well to dynamic geological environments. To address this, the physical constraints can be made more adaptive by incorporating machine learning techniques to update the constraints based on the inversion results.
Another limitation could be the oversimplification of the constraints, leading to under-constrained or over-constrained inversion problems. To improve this, the constraints can be refined by incorporating additional geological information, such as structural trends, lithological boundaries, or seismic attributes, to better capture the underlying geological relationships.
Furthermore, the current formulation may not fully account for uncertainties in the constraints. By introducing probabilistic constraints or incorporating uncertainty quantification methods, the algorithm can provide more robust estimations that consider the uncertainty in the geological relationships.
Overall, by addressing these limitations and refining the physical constraint formulation, the EFWI algorithm can better capture the complex geological relationships in the subsurface.

Given the computational efficiency gains from the source sketching method, are there other randomized techniques that could be explored to further accelerate the EFWI process without compromising accuracy

While source sketching has shown promising results in improving the computational efficiency of the EFWI algorithm, there are other randomized techniques that could be explored to further accelerate the process without compromising accuracy. One such technique is random projection, where the model and data are projected onto a lower-dimensional space using random matrices. This reduces the computational complexity of the inversion while preserving the essential information for accurate estimation.
Another technique is stochastic optimization, where subsets of the data or model parameters are randomly selected for each iteration of the optimization process. This introduces randomness into the optimization procedure, leading to faster convergence and reduced computational burden.
Additionally, techniques such as randomized sampling and Monte Carlo methods can be employed to explore the solution space more efficiently and identify optimal solutions in a shorter time frame. By integrating these randomized techniques into the EFWI algorithm, further computational efficiency gains can be achieved, making the inversion process faster and more scalable.

Robust Elastic Full-Waveform Inversion Using an Alternating Direction Method of Multipliers with Reconstructed Wavefields