An Adjoint-based Method for Inverting for Heterogeneous Material Properties and Fault Slip From Earthquake Surface Deformation Data

Analysis of geodetic and seismological data helps constrain earthquake dynamics and the physics of lithospheric deformation. Here, we discuss a new modeling approach based on an open-source finite-element framework to invert surface deformation data for constitutive laws and their parameters, such as the Poisson’s ratio or shear modulus in the crust and mantle wedge.

These inversions can be realized by using adjoint-based optimization methods which efficiently reduce the misfit between the calculated and observed displacements. To quantify the associated model uncertainties, we extend the inverse approach to a Bayesian inference problem. Since the data are usually informative only in a few directions in parameter space, we use a low-rank Laplace approximation of the posterior distribution to make the inverse problem computationally tractable. The mean and the posterior covariance are approximated by the solution of the inverse problem (MAP point) and the inverse of the Hessian of the negative log posterior evaluated at the MAP point, respectively. We show how smoothly varying parameter fields can be reconstructed satisfactorily from noisy data.

To improve the spatial resolution of the inverse solution we solve a Bayesian optimal experimental design problem to find the best station configuration by maximizing the expected information gain, defined as the Kullback-Leibler divergence between posterior and prior distributions. We show how and why the optimal network improves the material property inference more than evenly spaced stations. Based on our previous work on inverting for fault slip without Green’s function computations, we combine the two inversion schemes to jointly infer both model parameters, the coseismic slip, and material properties distribution. Lastly, we test this numerical forward/inverse framework with an application, the 2011 Tohoku-oki M9 earthquake. Both continuous land-based and six offshore acoustic GNSS stations located around the earthquake epicenter are inverted to jointly estimate the shear modulus and the fault slip during the megathrust event.

Our results demonstrates the potential of our computational framework and the general approach for inferring constitutive laws to evaluate sensitivity to parameters, and define strategies to improve our understanding of relevant parameters for earthquake dynamics.