PaperlayerView exact revision on arXiv

Source-linked AI summary

Data-driven Seismic Waveform Inversion: A Study on the Robustness and Generalization

Zhongping Zhang, Youzuo Lin

arXiv:1809.10262v8eess.SP

TL;DR

Seismic waveform inversion is a nonlinear, ill-posed, and costly problem, and data-driven approaches face robustness and generalization challenges. The paper develops VelocityGAN, a conditional adversarial encoder-decoder with data-learned regularization and transfer learning. Experiments report stronger reconstruction than physics-driven methods and selected deep-learning baselines, with transfer learning improving generalization.

  • Problem

    Seismic waveform inversion is ill-posed, vulnerable to cycle skipping and local minima, computationally expensive, and difficult for data-driven models to generalize beyond their training data.

  • Method

    VelocityGAN maps raw seismic waveforms to velocity images with a conditional adversarial generator-discriminator framework that learns regularization from data, combined with transfer learning.

  • Results

    VelocityGAN outperforms physics-driven methods and selected deep-learning baselines in quantitative and qualitative experiments, while transfer learning improves its generalization results.

  • Takeaways & Limitations

    The results support VelocityGAN as a promising data-driven tool for subsurface velocity estimation with basic generalization ability that can be improved through transfer learning.

Abstract

from arXiv · show

Acoustic- and elastic-waveform inversion is an important and widely used method to reconstruct subsurface velocity image. Waveform inversion is a typical non-linear and ill-posed inverse problem. Existing physics-driven computational methods for solving waveform inversion suffer from the cycle skipping and local minima issues, and not to mention solving waveform inversion is computationally expensive. In recent years, data-driven methods become a promising way to solve the waveform inversion problem. However, most deep learning frameworks suffer from generalization and over-fitting issue. In this paper, we developed a real-time data-driven technique and we call it VelocityGAN, to accurately reconstruct subsurface velocities. Our VelocityGAN is built on a generative adversarial network (GAN) and trained end-to-end to learn a mapping function from the raw seismic waveform data to the velocity image. Different from other encoder-decoder based data-driven seismic waveform inversion approaches, our VelocityGAN learns regularization from data and further impose the regularization to the generator so that inversion accuracy is improved. We further develop a transfer learning strategy based on VelocityGAN to alleviate the generalization issue. A series of experiments are conducted on the synthetic seismic reflection data to evaluate the effectiveness, efficiency, and generalization of VelocityGAN. We not only compare it with existing physics-driven approaches and data-driven frameworks but also conduct several transfer learning experiments. The experiment results show that VelocityGAN achieves state-of-the-art performance among the baselines and can improve the generalization results to some extent.

I. INTRODUCTION

Seismic waveform inversion reconstructs subsurface velocity from seismic observations, but its nonlinear, ill-posed, computationally expensive physics-driven formulations face cycle skipping and local-minimum challenges. The paper introduces VelocityGAN, a data-driven conditional GAN with learned regularization and transfer learning to improve reconstruction and generalization.

  • Seismic full-waveform inversion infers a 2D/3D seismic-velocity map from observations to characterize subsurface geology and material properties.
  • AEWI is ill-posed and under-constrained because data coverage is limited, while nonlinear sensitivity to initialization causes cycle skipping and local minima.Low-frequency components are important for alleviating cycle skipping.
  • Physics-driven regularizers such as Tikhonov-like, total-variation, high-order, and prior-based methods are largely hand crafted and loosely related to the physical problem.These approaches remain within computationally expensive physics-driven AEWI frameworks.
  • VelocityGAN maps raw seismic waveform data to velocity images with a generator and discriminator, learning a regularization term from data and imposing it on the generator.The discriminator distinguishes ground-truth velocity maps from generated maps.
  • The paper addresses generalization by combining the data-driven method with network-based transfer learning and fine-tuning all model parameters on the new dataset.
  • The reported contributions include conditional adversarial AEWI, improved velocity-image accuracy, GAN-based regularization, robustness experiments, and transfer learning to alleviate generalization issues.

II. RELATED WORK

Prior work on learning-based inverse problems includes end-to-end regression, representation learning, iterative refinement, and hybrid analytical-denoising methods. In seismic inversion, this paper is positioned within end-to-end data-to-velocity mapping and network-based deep transfer learning.

  • A. Data-driven Inverse Problems: Deep-learning inverse-problem methods include end-to-end regression, higher-level representation learning, gradual inversion refinement, and hybrid analytical-denoising approaches.
  • A. Data-driven Inverse Problems: End-to-end seismic inversion directly learns a mapping from seismic data to velocity models, whereas low-wave-number learning precedes traditional full-waveform inversion iterations.
  • C. Deep Transfer Learning: Deep transfer learning approaches include instance-based, mapping-based, network-based, and adversarial-based methods.
  • C. Deep Transfer Learning: Network-based transfer learning uses either feature extraction, which updates selected layers, or fine tuning, which updates all parameters for the new task.
  • The paper organizes its method around governing physics, VelocityGAN architecture, loss functions, and connections to inverse and regularization theory.

1) Governing Physics - Wave Equation:

The governing physics models acoustic or elastic wave propagation from subsurface parameters and source terms to wavefields, while inversion estimates unknown subsurface properties from limited surface measurements. Because this inverse problem is under-constrained and ill-posed, regularization stabilizes solutions by balancing data fit against model complexity.

  • 1) Governing Physics - Wave Equation:: The acoustic and elastic wave equations describe wave propagation using density, material tensors, source terms, pressure or displacement wavefields, and time.
  • 1) Governing Physics - Wave Equation:: The forward model is written as P = f(m), where f maps the velocity-model parameter vector m to pressure or displacement wavefields P.The parameter vector includes density and compressional- and shear-wave velocities.
  • 1) Governing Physics - Wave Equation:: Time-domain staggered-grid finite differences are used to solve the acoustic- or elastic-wave equation.
  • 1) Governing Physics - Wave Equation:: Limited measurements taken at or near the surface make seismic full-waveform inversion extremely under-constrained and severely ill-posed.
  • 1) Governing Physics - Wave Equation:: Regularized physics-driven inversion combines waveform data misfit with a regularization term weighted by λ to favor simple models consistent with the data.The regularization term measures model complexity; L1- and L2-norm regularizers do not fully express expert prior knowledge.

2) Data-driven Approach:

End-to-end data-driven waveform inversion reframes seismic inversion as a mapping from seismic data to velocity models. VelocityGAN implements this mapping with an image-to-image generator and CNN discriminator.

  • End-to-end methods transfer physics-driven minimization into a direct mapping from seismic data to velocity models.
  • The inverse operator g is learned from paired velocity maps and corresponding seismic data.
  • Existing end-to-end AEWI methods commonly use encoder-encoder structures for learning the mapping function.
  • VelocityGAN uses an image-to-image generator and a convolutional neural network discriminator.

1) Generator:

The generator maps multi-channel seismic shot-gather data to velocity images, while adversarial training distinguishes generated from real velocity maps. Its content loss combines MAE and MSE to represent complementary geological features.

  • 1) Generator:: The generator uses convolutional encoding followed by a decoder with five upconvolution blocks to produce velocity maps.
  • 1) Generator:: The discriminator uses convolutional blocks and PatchGAN classification to capture local style statistics in velocity maps.
  • 1) Generator:: Wasserstein loss with gradient penalty distinguishes the ground-truth velocity-map distribution from the generator's predicted distribution.
  • 1) Generator:: The generator loss combines adversarial loss with content loss based on mean absolute error and mean square error.
  • 1) Generator:: MSE captures geological faults, whereas MAE better reveals geological interfaces.

D. Connection to Regularization Theory

VelocityGAN interprets adversarial training as data-learned regularization for seismic inversion. The adversarial term constrains generated maps to resemble ground-truth velocity-map distributions, while MAE and MSE enforce pixel consistency.

  • D. Connection to Regularization Theory: The GAN formulation connects the learned generator with the inversion operator in inverse-problem regularization.
  • D. Connection to Regularization Theory: The adversarial loss acts as regularization learned from training data, penalizing velocity models that do not resemble ground-truth distributions.
  • D. Connection to Regularization Theory: MAE and MSE content losses promote consistency between predicted and ground-truth velocity maps.
  • D. Connection to Regularization Theory: Data-learned regularization is presented as more customized to the physical problem than generic L1- or L2-norm penalties.

IV. EXPERIMENTS

The experiments use synthetic waveform–velocity pairs to evaluate VelocityGAN and test generalization across geological fault configurations. The main dataset is large and structurally varied, while two smaller datasets alter fault counts.

  • IV. EXPERIMENTS: The experiments compare reconstruction quality and generalization across datasets and methods.
  • IV. EXPERIMENTS: Synthetic velocity images and corresponding seismic waveforms are generated to evaluate VelocityGAN because large real paired datasets can be expensive to obtain.
  • IV. EXPERIMENTS: CurvedData contains 50,000 velocity models of dimension 150 by 100 with corresponding seismic waveforms and complicated curved geological layers.
  • IV. EXPERIMENTS: The CurvedData input size is 32 × 1000 × 6, with 32 receivers, 1000 time samples, and 6 channels.
  • IV. EXPERIMENTS: Two smaller datasets test generalization by changing the number of geological faults to two or zero.

2) Training Details:

The experiments use train/validation/test splits, compare physics-driven and data-driven baselines, and evaluate velocity reconstruction with error and accuracy metrics.

  • Each dataset is split into 20% testing, 10% validation, and the remaining data for training.
  • The baselines include AEWI-Pre and AEWI-MTV as physics-driven methods, plus FCN, U-Net, and modifiedFCN as data-driven methods.
  • Reconstruction accuracy is evaluated using relative mean absolute error, mean log10 error, and accuracy thresholds at t=1.01, 1.02, 1.05, and 1.10.
  • Qualitative evaluation uses velocity-image samples and vertical velocity profiles, with particular attention to geological fault imaging.

C. CurvedData

CurvedData is designed to test reconstruction of irregular geological layers and faults, motivating a combined MAE and MSE objective.

  • CurvedData targets curved geological-layer interfaces, where MAE can suppress faults while MSE can blur layer boundaries.
  • Combining MAE and MSE is intended to balance geological-layer interfaces and fault reconstruction in velocity images.
  • The ablation study finds that the combined-loss VelocityGAN achieves a better overall tradeoff across the reported metrics than either single-loss variant.
  • The CurvedData experiments indicate that the generator structure, adversarial training, and combined losses each improve reconstructed accuracy.

2) Qualitative Results:

Qualitative and generalization experiments compare VelocityGAN with physics-driven and data-driven alternatives on layer, fault, and cross-dataset reconstruction. The combined loss and transfer learning improve the reported reconstruction behavior, while dataset size affects generalization results.

  • 2) Qualitative Results:: The combined MAE-MSE model balances sharp geological-layer interfaces with visible faults better than either single-loss model.
  • 2) Qualitative Results:: VelocityGAN with combined MAE and MSE reveals fault zones in low- and high-velocity regions where single-loss variants can miss them.
  • 2) Qualitative Results:: VelocityGAN-L1 reveals a high-velocity geological fault more effectively than Generator-L1, which the authors attribute to adversarial training.
  • 2) Qualitative Results:: VelocityGAN produces more accurate velocity reconstructions of both global and local geological features than the compared physics-driven methods.
  • D. Generalization Experiments: On 2-Fault CurvedData, VelocityGAN-TF achieves the best reported quantitative results, whereas AEWI-MTV performs better than it on 0-Fault CurvedData.
  • D. Generalization Experiments: The generalization experiments associate better data-driven performance with the larger training set available for 2-Fault CurvedData.
  • D. Generalization Experiments: Visual comparisons show VelocityGAN-TF slightly outperforming VelocityGAN-org on 2-Fault CurvedData and both generally exceeding the physics-driven baselines.
  • D. Generalization Experiments: Transfer learning improves generalization from one-fault training data to zero- and two-fault velocity images, reaching performance competitive with physics-driven methods.

V. CONCLUSION

VelocityGAN combines a modified encoder-decoder with conditional adversarial training and an improved loss for seismic velocity reconstruction. Experiments report stronger reconstruction performance than physics-driven and selected deep-learning baselines, with transfer learning improving generalization to some extent.

  • VelocityGAN uses a modified encoder-decoder with conditional adversarial training and an improved loss for velocity-image reconstruction.
  • VelocityGAN achieves the most accurate reconstructed velocity images among the compared physics-driven and data-driven methods on CurvedData.
  • The combined MAE and MSE loss offers a better tradeoff between geological-layer interfaces and geological faults than either loss alone.
  • VelocityGAN outperforms both physics-driven methods and selected deep-learning baselines in quantitative and qualitative experiments.
  • Transfer learning improves VelocityGAN's generalization results to some extent.
Loading 1809.10262v8…