Markov-random-field modeling for linear seismic tomography

We apply the Markov-random-field model to linear seismic tomography and propose a method to estimate the hyperparameters for the smoothness and the magnitude of the noise. Optimal hyperparameters can be determined analytically by minimizing the free energy function, which is defined by marginalizing the evaluation function. In synthetic inversion tests under various settings, the assumed velocity structures are successfully reconstructed, which shows the effectiveness and robustness of the proposed method. The proposed mathematical framework can be applied to inversion problems in various fields in the natural sciences.

Figure 1

Schematic illustration of ray paths through a cell slowness model. Triangles indicate the receiver points and stars indicate the seismic source. In this study, we use a linear approximation for the tomographic model and this imposes the condition that all ray paths are straight lines (indicated by dashed lines).

Figure 2

Graphical sketch of the MRF model assumed in this study.

Figure 3

Results of the simulations presented in Sec. V: (a) assumed true seismic velocity structures $1/{\mathit{x}}_{\mathbf{t}}$ for the 20 $\times$ 20 cells and (b) ray paths. In the numerical simulations, the number of observation travel times $m$ was 4000. The triangles indicate the receivers.

Figure 4

(a) Dynamic behavior of the estimated hyperparameters $\widehat{\alpha }$ and $\widehat{\beta }$ and (b) the RMSE of the target physical parameter $\mathit{x}$ in the recursion formulas (27) and (28). The red square indicates the true value of the hyperparameters ${\alpha }_t$ and ${\beta }_t$. Black lines indicate the results of the MRF method. For comparison, the regularization parameters ${\lambda }_L$ estimated by the $L$-curve method are shown as a green dashed line. The trial number indicates the number of iterations. The green dashed lines indicate the RMSE for the $L$-curve method.

Figure 5

Comparison of the generalized inverse method, $L$-curve method, and MRF model. (a) Estimated results for the reconstructions of the velocity structure $1/\mathit{x}$ and (b) the residuals of the difference between the estimated and true model parameters.

Figure 6

Influence of the observation number $m$ on the restoration results for ${\beta }_t=100$: (a) RMSE, (b) estimated regularization parameter $\lambda$, (c) estimated smoothness hyperparameter $\widehat{\alpha }$, and (d) estimated precision $\text{hyperparameter}\widehat{\beta }$. Black lines indicate the results of the MRF method. Green dashed lines indicate the results of the $L$-curve method. Red dotted lines indicate the assumed true hyperparameters in (b)–(d). Error bars indicate one standard deviation $\sigma$.

Figure 7

Influence of the assumed precision hyperparameter ${\beta }_t$ on the restoration results in when $m=4000$: (a) RMSE, (b) estimated regularization parameter $\lambda$, (c) estimated smoothness hyperparameter $\widehat{\alpha }$, and (d) estimated precision hyperparameter $\widehat{\beta }$. Black lines indicate the result of the MRF method. Green dashed lines indicate the results of the $L$-curve method. Red dotted lines indicate the assumed true hyperparameters in (b)–(d). Error bars indicate one standard deviation $\sigma$.