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Project Abstract

In this publication we introduce a scheme for probabilistic hypocenter inversion with Stein variational inference. Our approach uses a differentiable forward model in the form of a physics informed neural network (PINN), which we train to solve the Eikonal equation (see EikoNet project for more info). This allows for rapid approximation of the posterior by iteratively optimizing a collection of particles against a kernelized Stein discrepancy. We show that the method is well-equipped to handle highly multimodal posterior distributions, which are common in hypocentral inverse problems. A suite of experiments is performed to examine the influence of the various hyperparameters. Once trained, the method is valid for any seismic network geometry within the study area without the need to build travel time tables. We show that the computational demands scale efficiently with the number of differential times, making it ideal for large-N sensing technologies like Distributed Acoustic Sensing. The techniques outlined in this manuscript have considerable implications beyond just ray-tracing procedures, with the work flow applicable to other fields with computationally expensive inversion procedures such as full waveform inversion.

HypoSVI: Bio

The software package is available at with interactive Google Colab notebook.

The accepted manuscript can be found at the link below.

HypoSVI: Text

In this work we use a Stein Variational Inference method to update a series of particle locations, selected initially from a uniform prior, till the density of the particle locations best approximate a posterior distribution which we wish to learn. Unlike other Variational inference approaches this does not assume any prior knowledge of the shape of the posterior distribution (e.g. Gaussian).

The figure below show examples distributions, each composed of a user imposed mixture of Gaussians. Stage 1 demonstrates the imposed true posterior field. Stage 2 represents the initial randomised particle locations selectedfrom a uniform prior. Stage 3 involves updating the particle locations to minimise the kernelized Stein discrepancy. By Stage 4, the optimised particle locations with locations have stabilized. Stage 5 represents the particle kernel density as contours, demonstrating the similarity with the imposed posterior colormap.

HypoSVI: Image

Outlined above is the use case of Stein Variational Inference approach when the posterior is known.

For our problem the likelihood function encompasses uncertainty in Eearthquake signal arrivals at a series of instrument locations on the Earths surface, and uncertainty in the knowledge of the subsurface velocity structure. We derive a forward model for the travel-time by solving the Eikonal partial differential equation, the fastest travel-time between through the velocity; using a Physics Informed Neural network between any two points within the user defined velocity structure. Due to the quick recovery of the travel-time from the neural network, and the analytical differentiability, we can quickly determine the gradients required for the Stein Variational Inference particle updates. 

HypoSVI: Text

Shown in the figure below is the overview of the inversion procedure. Panel 1 represents the optimisation block that is applied to the particles to minimize the kernelized Stein discrepancy. (a) initial particle locations are supplied, where their kernel density approximates the initial posterior distribution. (b) predicted travel times are determine from all particle locations to observation locations using the physics informed neural network for the Eikonal equation. (c) particle locations are then updated by a step in the direction that minimises the kernelized Stein discrepancy. (d) updated particle locations are returned. Panel 2 demonstrates the optimisation scheme applying these optimisation blocks to update all the particle locations as a single batch between epochs. Red triangles represent observation locations, the seismic stations. Black points represent particle locations. Contours represent the particle kernel density. White star represents the median location of the particles representing the optimal hypocentral location.

HypoSVI: Image

The attached video shows the updating of the particle locations until convergence when the posterior earthquake location is optimised.

HypoSVI: Video

Using this method we can determine earthquake locations independent of meshes or grids, reducing artefacts that you typically return from grid/meshed based inversion procedures.

Comparison in earthquake location for 10,000 micro earthquakes from 2019, in a very active region of Southern-California. (a)-(b) are the locations determined from our HypoSVI inversion scheme harnessing the EikoNet forward model. (c)-(d) are the locations determined from the industry standard a McMC inversion procedure using finite-difference forward models (NonLinLoc) . Notice the returned gridding artefacts in depth within the NonLinLoc locations

HypoSVI: Image
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