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Improved impedance inversion by the iterated graph Laplacian

Davide Bianchi, Florian Bossmann, Wenlong Wang, Mingming Liu

TL;DR

Impedance inversion is inherently ill-posed, especially under noise. The authors propose an iterative graph Laplacian regularization (it-graphLaΨ) that builds a data-adaptive graph from an initial impedance estimate and refines the solution via a Tikhonov-like variational problem, updating the graph at each step and solving with MMGKS. The framework is validated on synthetic Marmousi2 and SEAM models and real Volve field data, showing faster convergence, better edge preservation, and improved robustness across four initializers (two neural networks and two classical methods). The approach effectively blends data-driven priors with model-based regularization, yielding higher fidelity impedance reconstructions while maintaining stability under noise. The work highlights a flexible path to combine diverse initialization schemes with graph-based priors, with potential extensions to non-linear forward models and reduced data requirements.

Abstract

We introduce a data-adaptive inversion method that integrates classical or deep learning-based approaches with iterative graph Laplacian regularization, specifically targeting acoustic impedance inversion - a critical task in seismic exploration. Our method initiates from an impedance estimate derived using either traditional inversion techniques or neural network-based methods. This initial estimate guides the construction of a graph Laplacian operator, effectively capturing structural characteristics of the impedance profile. Utilizing a Tikhonov-inspired variational framework with this graph-informed prior, our approach iteratively updates and refines the impedance estimate while continuously recalibrating the graph Laplacian. This iterative refinement shows rapid convergence, increased accuracy, and enhanced robustness to noise compared to initial reconstructions alone. Extensive validation performed on synthetic and real seismic datasets across varying noise levels confirms the effectiveness of our method. Performance evaluations include four initial inversion methods: two classical techniques and two neural networks - previously established in the literature.

Improved impedance inversion by the iterated graph Laplacian

TL;DR

Impedance inversion is inherently ill-posed, especially under noise. The authors propose an iterative graph Laplacian regularization (it-graphLaΨ) that builds a data-adaptive graph from an initial impedance estimate and refines the solution via a Tikhonov-like variational problem, updating the graph at each step and solving with MMGKS. The framework is validated on synthetic Marmousi2 and SEAM models and real Volve field data, showing faster convergence, better edge preservation, and improved robustness across four initializers (two neural networks and two classical methods). The approach effectively blends data-driven priors with model-based regularization, yielding higher fidelity impedance reconstructions while maintaining stability under noise. The work highlights a flexible path to combine diverse initialization schemes with graph-based priors, with potential extensions to non-linear forward models and reduced data requirements.

Abstract

We introduce a data-adaptive inversion method that integrates classical or deep learning-based approaches with iterative graph Laplacian regularization, specifically targeting acoustic impedance inversion - a critical task in seismic exploration. Our method initiates from an impedance estimate derived using either traditional inversion techniques or neural network-based methods. This initial estimate guides the construction of a graph Laplacian operator, effectively capturing structural characteristics of the impedance profile. Utilizing a Tikhonov-inspired variational framework with this graph-informed prior, our approach iteratively updates and refines the impedance estimate while continuously recalibrating the graph Laplacian. This iterative refinement shows rapid convergence, increased accuracy, and enhanced robustness to noise compared to initial reconstructions alone. Extensive validation performed on synthetic and real seismic datasets across varying noise levels confirms the effectiveness of our method. Performance evaluations include four initial inversion methods: two classical techniques and two neural networks - previously established in the literature.
Paper Structure (14 sections, 1 theorem, 22 equations, 14 figures, 3 tables, 1 algorithm)

This paper contains 14 sections, 1 theorem, 22 equations, 14 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The graph Laplacian and the graphLa$\Psi$ method
  3. Images and graphs
  4. Induced graph
  5. The standard graphLa$\Psi$ method: convergence and stability results
  6. The iterated graphLa$\Psi$ method: it-graphLa$\Psi$
  7. Numerical Experiments
  8. Error measures
  9. Initialization methods
  10. Neural network approaches
  11. Classic approaches
  12. Model data experiments
  13. Volve field data
  14. Conclusion

Key Result

Theorem 2

Fix a sequence $\{\delta_k\}$ and $\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+$ be such that Then every sequence $\{{\boldsymbol{x}}_k\}$ of elements that minimize the functional graphModel, with $\delta_k$ and $\Theta(\delta_k, {\boldsymbol{y}}^{\delta_k})$, has a convergent subsequence. The limit ${\boldsymbol{x}}$ of the convergent subsequence $\{{\boldsymbol{x}}_{k'}\}$is a graph-minimi

Figures (14)

  • Figure 1: Pipeline of the iterative method \ref{['eq:iterative_graphLaNet']}.
  • Figure 2: Simple illustration that demonstrates a step-by-step process for converting an image ${\boldsymbol{x}}$ into a graph. Starting from the left, a $7\times7$ pixel image—composed of red, orange, and blue squares—is shown, where each pixel’s color intensity is determined by evaluating the function ${\boldsymbol{x}}$. Moving right, each pixel is mapped to a graph node (depicted as a black circle) with its position corresponding to an ordered pair in $\mathbb{Z}^2$ based on its location in the grid. Next, nodes are connected by an edge whenever their $\ell^1$-distance is one, with each connection initially assigned a weight of one. Finally, the edge weight is adjusted by the function $g_{\boldsymbol{x}}(p,q)$, which is visually represented by the edge thickness: thicker lines indicate that adjacent pixels have very similar intensities, while thinner lines indicate a larger difference.
  • Figure 3: Left: The original image ${\boldsymbol{x}}$ in grayscale. Right: $|\Delta_{\boldsymbol{x}}{\boldsymbol{x}}|$, where $\Delta_{\boldsymbol{x}}$ is computed using equation \ref{['eq:edge-weight-function:applications']}. As it can be seen, $|\Delta_{\boldsymbol{x}}{\boldsymbol{x}}|\approx \boldsymbol{0}$ and all the edges are well identified.
  • Figure 4: data from the Marmousi2 (left) and SEAM model (right): impedance profile (a,b) and seismic data without noise (c,d), medium noise PSNR $\approx33$ (e,f), high noise PSNR $\approx27$ (g,h). The colormap of all seismic images is the same and cuts of extreme values for a better contrast.
  • Figure 5: Linear forward operator $K$ learned for Marmousi2 (a) and SEAM (b).
  • ...and 9 more figures

Theorems & Definitions (2)

  • Definition 1
  • Theorem 2