for Blind Inversion

Zachary E. Ross

^{1}Caltech Computing and Mathematical Sciences
^{2}Caltech Seismo Lab

(a) A simulated cross section of the Earthâ€™s interior (velocity structure), along with the locations of receivers on the surface (red triangles) that collect measurements. (b) The time it takes for a wave traveling from a source below the surface to reach the specified receiver is visualized for each location in the region of interest. The overlaid dots represent the true locations of simulated earthquakes and indicate the measured travel times that constrain optimization. (c) The subsurface velocity reconstruction obtained using a baseline approach optimized with the help of a seismologist. Note that the bright anomaly is missing from this reconstruction. Overlaid dots represent the inferred earthquake locations. (d) DeepGEM reconstructed subsurface velocity and inferred earthquake locations. Note that DeepGEM is able to accurately recover the gradient of the velocity field as well as partially recover the central anomaly.

Typically, inversion algorithms assume that a forward model, which relates a source to its resulting measurements, is known and fixed. Using collected indirect measurements and the forward model, the goal becomes to recover the source. When the forward model is unknown, or imperfect, artifacts due to model mismatch occur in the recovery of the source. In this paper, we study the problem of blind inversion: solving an inverse problem with unknown or imperfect knowledge of the forward model parameters. We propose DeepGEM, a variational Expectation-Maximization (EM) framework that can be used to solve for the unknown parameters of the forward model in an unsupervised manner. DeepGEM makes use of a normalizing flow generative network to efficiently capture complex posterior distributions, which leads to more accurate evaluation of the source's posterior distribution used in EM. We showcase the effectiveness of our DeepGEM approach by achieving strong performance on the challenging problem of blind seismic tomography, where we significantly outperform the standard method used in seismology. We also demonstrate the generality of DeepGEM by applying it to a simple case of blind deconvolution.

**paper [pdf]** **code [Github]**
**supplement [pdf]**

Angela F. Gao, Jorge C. Castellanos, Yisong Yue, Zachary E. Ross, and Katherine L. Bouman (2021). "DeepGEM: Generalized Expectation-Maximization for Blind Inversion." *Neural Information Processing Systems, 2021.*