Physically Guided Deep Unsupervised Inversion for 1D Magnetotelluric Models

TL;DR

The paper tackles 1D magnetotelluric inversion where supervised deep learning is data-hungry and sensitive to training distributions. It introduces a differentiable forward operator within TensorFlow and a neural inverse operator to perform unsupervised inversion guided by physics. The objective combines data misfit and model regularization, enabling learning from observed impedance data without labeled training pairs. Across synthetic and field tests, the method achieves faster convergence and more accurate resistivity models than the compared methods, demonstrating practical potential for reliable subsurface imaging with reduced tuning.

Abstract

The global demand for unconventional energy sources such as geothermal energy and white hydrogen requires new exploration techniques for precise subsurface structure characterization and potential reservoir identification. The Magnetotelluric (MT) method is crucial for these tasks, providing critical information on the distribution of subsurface electrical resistivity at depths ranging from hundreds to thousands of meters. However, traditional iterative algorithm-based inversion methods require the adjustment of multiple parameters, demanding time-consuming and exhaustive tuning processes to achieve proper cost function minimization. Recent advances have incorporated deep learning algorithms for MT inversion, primarily based on supervised learning, and large labeled datasets are needed for training. This work utilizes TensorFlow operations to create a differentiable forward MT operator, leveraging its automatic differentiation capability. Moreover, instead of solving for the subsurface model directly, as classical algorithms perform, this paper presents a new deep unsupervised inversion algorithm guided by physics to estimate 1D MT models. Instead of using datasets with the observed data and their respective model as labels during training, our method employs a differentiable modeling operator that physically guides the cost function minimization, making the proposed method solely dependent on observed data. Therefore, the optimization algorithm updates the network weights to minimize the data misfit. We test the proposed method with field and synthetic data at different acquisition frequencies, demonstrating that the resistivity models obtained are more accurate than those calculated using other techniques.

Figures (6)

• Figure 1: The neural network $\mathcal{I}_{\mathbf{\Theta}}$ has inputs given by $\mathbf{d}=[d_1, d_2, d_3, \cdots, d_J]$ and outputs with the conductivities of the model $\pmb{\sigma}=[\sigma_1, \sigma_2, \sigma_3, \cdots, \sigma_N]$. The number of neurons is constant within the $L$ hidden layers.
• Figure 2: Data misfit curves over 500 iterations for the proposed method (left) from Eq. \ref{['eq:datamisfit']} in black line and the solution based solely on the $\ell_2$--norm with Tikhonov regularization (right) from Eq. \ref{['eq:tikhonov']} in blue dashed line.
• Figure 3: Visual comparison of inversion results with the proposed method and the $\ell_2$-norm: a) in the impedance response and b) in the 1D resistivity model.
• Figure 4: a) Impedance response where both methods fit the measurements with high accuracy. b) The resistivity model shows that the proposed method performs better than SimPEG in delineating the first four layers, especially in areas with sharp resistivity changes. In contrast, SimPEG exhibits smooth resistivity transitions between consecutive layers.
• Figure 5: a) 1D magnetotelluric inverted models with the proposed method compared with SimPEG Kang2017 and ZondMT1D Patel2020 inversions. b) Zoom between 0-10 km, highlighting the shallow resistivity anomalies R1 and R2.