Neuro-Fuzzy Kinematic Finite-Fault Inversion: 1. Methodology
Abstract
Kinematic finite-fault source inversions aim at resolving the
spatio-temporal evolution of slip on a fault given ground motion
recorded on the Earth’s surface. This type of inverse problem is
inherently ill-posed due to two main factors. First, the number of model
parameters is typically greater than the number of independent observed
data. Second, small singular-values are generated by the discretization
of the physical rupture process and amplify the effect of noise in the
inversion. As a result, one can find different slip distributions that
fit the data equally well. This ill-posedness can be mitigated by
decreasing the number of model parameters, hence improving their
relationship to the observed data. In this article, we propose a fuzzy
function approximation approach to describe the spatial slip function.
In particular, we use an Adaptive Network-based Fuzzy Inference System
(ANFIS) to find the most adequate discretization for the spatial
variation of slip on the fault. The fuzzy basis functions and their
respective amplitudes are optimized through hybrid learning. We solve
this earthquake source problem in the frequency domain, searching for
independent optimal spatial slip distributions for each frequency. The
approximated frequency-dependent spatial slip functions are then used to
compute the forward relationship between slip on the fault and ground
motion. The method is constrained through Tikhonov regularization,
requiring a smooth spatial slip variation. We discuss how the number of
model parameters can be decreased while keeping the inversion stable and
achieving an adequate resolution. The proposed inversion method is
tested using the SIV1-benchmark exercise.