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.

We validate the fuzzy kinematic finite-fault inversion method by studying the rupture process of the $M_w 6.2$, 24/Aug/2016, Amatrice, central Italy, earthquake. We jointly invert three different datasets to infer the spatio-temporal slip distribution, including static and high-rate GNSS data ($<=0.06$ Hz) and strong-motion data ($>0.06-0.5$ Hz). Each dataset is used to constrain a different frequency range of the source model, depending on the sensitivity of each dataset. Our slip solution confirms the main rupture features revealed by previous studies, including a slow nucleation phase at shallow depths, between 3–4 km, followed by a bilateral rupture that forms two asperities, one to the NW (Norcia) and another to the SE (Amatrice) of the hypocenter. To select an adequate number of fuzzy basis functions, we propose two alternative procedures, that yield the same general slip features concerning the amplitude, distribution, and velocity of slip. The first approach consists of ensuring the inverse problem is formally over-determined and uses a constant number of basis functions at all frequencies. The second is based on a maximum likelihood analysis of the model misfit and selects a different number of basis functions for each frequency. The maximum likelihood approach allows for more basis functions at high frequencies, where more detail in the spatial slip distribution obtained. The solution obtained with the maximum likelihood approach provides a more physically-plausible source time function, which shows less back-slip artifacts. The accurate prediction of high-rate GNSS traces not used in the inversion testifies the robustness of the inversion