An inequality-constrained regularized inversion of slip distributions on multiple faults with applications to the 2016 Kumamoto Mw 7.0 earthquake

Abstract

SUMMARY Although slip inversion has been almost always reported to be inherently non-unique, we prove that Fredholm integral equations of the first kind for slip inversion are mathematically of a unique solution, theoretically assuring that earthquake rupture can be properly reconstructed if there exist a sufficiently large number of measurements. The statement about non-uniqueness of slip inversion can be misleading and misunderstanding, since it is theoretically not true but simply caused by a practical issue of lack of measurements. We propose an inequality-constrained regularized inversion of slip distributions on multiple faults. The method implements a physically more general inequality constraint to accommodate more complex dislocation models and allows reconstructing a complex earthquake mechanism on multiple faults from measurements. The corresponding inequality constraints are a natural extension of positivity constraints proposed by Olson & Apsel and Hartzell & Heaton, or the equivalent inequality constraints that only allow slips to take place between −45° and 45° around the main rupture direction. The regularization parameter is chosen by minimizing the mean squared errors of inverted slip solutions. The proposed method is applied to the 2016 Kumamoto Mw 7.0 earthquake with GNSS measurements. Our slip inversion results show that the 2016 Kumamoto earthquake is only of magnitude Mw 6.7 and ruptures shallowly up to 10 km under the surface. More precisely, about 94 and 88 per cent of the energy released from Hinagu and Futagawa faults take place up to this depth, with the maximum slips of 4.81 and 7.89 m on each fault, respectively.