Seismic Traveltime Tomography with Label-free Learning

TL;DR

This work addresses label scarcity in seismic traveltime tomography by combining LSQR with dictionary learning and a shallow neural network in a label-free framework. The method uses a MAP formulation and derives a tractable LSQR solution for perturbations s_g, then applies a two-step procedure—warming up dictionary learning from LSQR patches and NN-driven dictionary optimization—to obtain a high-resolution VM that blends a reference slowness, LSQR estimate, and the optimized dictionary. It demonstrates strong performance on synthetic smooth-discontinuous and Marmousi models as well as field traveltime data, achieving lower traveltime misfit and substantially smaller RMSE than LSQR or dictionary learning alone, while maintaining robustness to noise and varying atom counts. By delivering an interpretable, low-cost alternative to end-to-end deep learning, the approach provides a practical label-free pathway to reliable initial models for imaging workflows such as FWI and DL-based geophysical inversions. The key idea is to let a neural network refine a learned dictionary rather than directly predicting velocities, yielding improved resolution with greater controllability and transferability.

Abstract

Deep learning techniques have been used to build velocity models (VMs) for seismic traveltime tomography and have shown encouraging performance in recent years. However, they need to generate labeled samples (i.e., pairs of input and label) to train the deep neural network (NN) with end-to-end learning, and the real labels for field data inversion are usually missing or very expensive. Some traditional tomographic methods can be implemented quickly, but their effectiveness is often limited by prior assumptions. To avoid generating and/or collecting labeled samples, we propose a novel method by integrating deep learning and dictionary learning to enhance the VMs with low resolution by using the traditional tomography-least square method (LSQR). We first design a type of shallow and simple NN to reduce computational cost followed by proposing a two-step strategy to enhance the VMs with low resolution: (1) Warming up. An initial dictionary is trained from the estimation by LSQR through dictionary learning method; (2) Dictionary optimization. The initial dictionary obtained in the warming-up step will be optimized by the NN, and then it will be used to reconstruct high-resolution VMs with the reference slowness and the estimation by LSQR. Furthermore, we design a loss function to minimize traveltime misfit to ensure that NN training is label-free, and the optimized dictionary can be obtained after each epoch of NN training. We demonstrate the effectiveness of the proposed method through the numerical tests on both synthetic and field data.

Figures (20)

• Figure 1: 2-D slowness image divided into pixels (dashed boxes) according to bianco_travel_2018. $W_{1}$ and $W_{2}$ represent the number of pixels in vertical and horizontal directions, respectively. The green triangles are receivers and the orange lines represent the rays between them. The gray region represents the i-th patch containing $n$ pixels.
• Figure 2: Schematic diagram of our method. The symbol $\bigotimes$ is matrix multiplication. The vectors vector $\mathbf{t}$ and $\mathbf{s}_{\mathrm{g}}$ denote the traveltime and desired perturbations, respectively. $\mathbf{s}^{\ast}_{\mathrm{g}}$ is the perturbations inverted by LSQR, $\mathbf{D}$ is the initial dictionary trained by dictionary learning, while $\mathbf{X}$ is the initial code obtained by sparse coding. $\mathbf{D}^{\dagger}$ is the dictionary optimized by NN, and $\mathbf{X}^{\dagger}$ is the code for $\mathbf{D}^{\dagger}$ by sparse coding.
• Figure 3: Illustration of our neural network. The input and output represent the initial and optimized dictionaries, respectively. Conv2D is a 2-D convolution layer, BatchNorm refers to batch normalization, and LeakyReLU is a nonlinear activation function. $n$ denotes the number of hidden layers.
• Figure 4: (a) Synthetic slowness map with dimensions of $W_1=W_2=100$ pixels (1 km/pixel). (b) Ray sampling with 64 receivers (red crosses).
• Figure 5: Comparison of 2-D slowness, 1-D slowness (from the black lines in 2-D slowness) against true slowness (Fig. \ref{['model']}(a)), and slowness errors by LSQR, dictionary learning, and the proposed method ($\sigma=0.02$). (a)-(d) Results by LSQR. (e)-(h) Results by dictionary learning with 150 atoms. (i)-(l) Results by our method with 150 atoms. RMSE values are printed on these slowness maps.