Parametric elastic full waveform inversion with convolutional neural network

Abstract

Elastic full waveform inversion (EFWI) is a powerful tool for estimating elastic models by reducing the misfit between multi-component seismic records and simulated data. However, when multiple parameters are updated simultaneously, the gradients of the loss function with respect to these parameters will be coupled together, the effect exacerbate the nonlinear problem. We propose a parametric EFWI method based on convolutional neural networks (CNN-EFWI). The parameters that need to be updated are the weights in the neural network rather than the elastic models. The convolutional kernel in the network can increase spatial correlations of elastic models, which can be regard as a regularization strategy to mitigate local minima issue. Furthermore, the representation also can mitigate the cross-talk between parameters due to the reconstruction of Frechét derivatives by neural networks. Both forward and backward processes are implemented using a time-domain finite-difference solver for elastic wave equation. Numerical examples on overthrust models, fluid saturated models and 2004 BP salt body models demonstrate that CNN-EFWI can partially mitigate the local minima problem and reduce the dependence of inversion on the initial models. Mini-batch configuration is used to speed up the update and achieve fast convergence. In addition, the inversion of noisy data further verifies the robustness of CNN-EFWI.

Appendix I

In this appendix, we derive the gradient of cost function with respect to Lamé parameters ($\lambda$ and $\mu$) and density ($\rho$). For simplicity, we ignore the non-reflecting PML regions. The sum of squared errors is used as the cost function:

$$\min J = \sum\limits_{i = 1}^{{N_{s} }} {J_{i} = \sum\limits_{i = 1}^{{N_{bs} }} {\sum\limits_{t = 1}^{{N_{t} }} {\left\| {d_{i}^{obs} - d_{i} } \right\|_{2}^{2} } } },$$

(15)

where $N_{t}$ is the number of time points. In the adjoint state method algorithms, the gradient for elastic model parameters $\partial J/\partial \lambda$, $\partial J/\partial \mu$ and $\partial J/\partial \rho$ depend on the adjoint wavefields, which is calculated by chain rule. We obtain the adjoint equations:

$$\begin{aligned} \tilde{\sigma }_{xx} (r,t - {\Delta }t) = & \tilde{\sigma }_{xx} (r,t) + \frac{{\partial \tilde{v}_{x} (r,t)}}{{\partial_{ - } x}}{\Delta }t\frac{1}{\rho } + S_{x} (t) \\ \tilde{\sigma }_{zz} (r,t - {\Delta }t) = & \tilde{\sigma }_{zz} (r,t) + \frac{{\partial \tilde{v}_{z} (r,t)}}{{\partial_{ - } z}}{\Delta }t\frac{1}{\rho } + S_{z} (t) \\ \tilde{\sigma }_{xz} (r,t - {\Delta }t) = & \tilde{\sigma }_{xz} (r,t) + \left( {\frac{{\partial \tilde{v}_{x} (r,t)}}{{\partial_{ + } z}} + \frac{{\partial \tilde{v}_{z} (r,t)}}{{\partial_{ + } x}}} \right){\Delta }t\frac{1}{\rho } \\ \tilde{v}_{x} (r,t - {\Delta }t) = & \tilde{v}_{x} (r,t) + \left[ {\frac{{\partial \tilde{\sigma }_{xx} (r,t)}}{{\partial_{ + } x}}(\lambda + 2\mu ) + \frac{{\partial \tilde{\sigma }_{zz} (r,t)}}{{\partial_{ + } x}}\lambda + \frac{{\partial \tilde{\sigma }_{xz} (r,t)}}{{\partial_{ - } z}}\mu } \right]{\Delta }t \\ \tilde{v}_{z} (r,t - {\Delta }t) = & \tilde{v}_{z} (r,t) + \left[ {\frac{{\partial \tilde{\sigma }_{zz} (r,t)}}{{\partial_{ + } z}}(\lambda + 2\mu ) + \frac{{\partial \tilde{\sigma }_{xx} (r,t)}}{{\partial_{ + } z}}\lambda + \frac{{\partial \tilde{\sigma }_{xz} (r,t)}}{{\partial_{ - } x}}\mu } \right]{\Delta }t, \\ \end{aligned}$$

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where the wavefields with a tilde denote the adjoint wavefield. In other words, they are the partial derivatives of $J$ with respect to corresponding wavefields. For example, $\tilde{v}_{z}$ indicates $\partial J/\partial v_{z}$. ${\Delta }t$ is discrete time interval. $S_{x} (t)$ and $S_{z} (t)$ are adjoint source.

As the adjoint wave fields propagate backward in time, the derivatives of the objective function with respect to the elastic parameters are calculated as:

$$\begin{aligned} \frac{{\partial \sigma_{xx} }}{\partial \lambda } &= \frac{{\partial \sigma_{zz} }}{\partial \lambda } = \left( {\frac{{\partial v_{x} }}{\partial x} + \frac{{\partial v_{z} }}{\partial z}} \right)\Delta t \\ \frac{{\partial v_{x} }}{\partial \rho } &= - \rho^{ - 2} \left( {\frac{{\partial \sigma_{xx} }}{\partial x} + \frac{{\partial \sigma_{xz} }}{\partial z}} \right)\Delta t \\ \frac{{\partial v_{z} }}{\partial \rho } &= - \rho^{ - 2} \left( {\frac{{\partial \sigma_{zz} }}{\partial z} + \frac{{\partial \sigma_{xz} }}{\partial x}} \right)\Delta t \\ \frac{{\partial \sigma_{xx} }}{\partial \mu } &= 2\frac{{\partial v_{x} }}{\partial x}\Delta t,\frac{{\partial \sigma_{zz} }}{\partial \mu } = 2\frac{{\partial v_{z} }}{\partial z}\Delta t \\ \frac{{\partial \sigma_{xz} }}{\partial \mu } &= \left( {\frac{{\partial v_{x} }}{\partial z} + \frac{{\partial v_{z} }}{\partial x}} \right)\Delta t. \\ \end{aligned}$$

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Note that the system is solved backwards in time. Connect Eqs. (3) and (2), the gradients of the objective function $J$ with respect to elastic models $\lambda$, $\mu$ and $\rho$ are:

$$\begin{aligned} \frac{\partial J}{{\partial \lambda }} = & - \mathop \sum \limits_{t = 1}^{{N_{t} }} \left( {\tilde{\sigma }_{zz}^{t + 1} + \tilde{\sigma }_{xx}^{t + 1} } \right)\left( {\frac{{\partial V_{z}^{t} }}{\partial z} + \frac{{\partial V_{x}^{t} }}{\partial x}} \right){\Delta }t \\ \frac{\partial J}{{\partial \mu }} = & - \mathop \sum \limits_{t = 1}^{{N_{t} }} \left[ {\tilde{\sigma }_{xz}^{t + 1} \left( {\frac{{\partial V_{z}^{t} }}{\partial x} + \frac{{\partial V_{x}^{t} }}{\partial z}} \right) + 2\tilde{\sigma }_{zz}^{t + 1} \frac{{\partial V_{z}^{t} }}{\partial z} + 2\tilde{\sigma }_{xx}^{t + 1} \frac{{\partial V_{x}^{t} }}{\partial x}} \right]\Delta t \\ \frac{\partial J}{{\partial \rho }} = & \mathop \sum \limits_{t = 1}^{{N_{t} }} \rho^{ - 2} \left[ {\tilde{v}_{x}^{t + 1} \left( {\frac{{\partial \sigma_{xz}^{t} }}{\partial z} + \frac{{\partial \sigma_{xx}^{t} }}{\partial x}} \right) + \tilde{v}_{z}^{t + 1} \left( {\frac{{\partial \sigma_{xz}^{t} }}{\partial x} + \frac{{\partial \sigma_{zz}^{t} }}{\partial z}} \right)} \right]\Delta t. \\ \end{aligned}$$

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