On the inverse problem for earthquake rupture: The Haskell-type source model

In order to gain insight into how to invert seismograms correctly to estimate the details of the earthquake rupturing process, we perform numerical experiments using artificial data, generated for an idealized faulting model with a very simple rupture and moment release history, and solve the inverse problem using standard widely used inversion methods. We construct synthetic accelerograms in the vicinity of an earthquake for a discrete analog of the Haskell-type rupture model with a prescribed rupture velocity in a layered medium. A constant level of moment is released as the rupture front passes by. We show that using physically based constraints, such as not permitting back slip on the fault, allow us to reproduce many aspects of the solution correctly, whereas the minimum norm solution or the solution with the smallest first differences of moment rates in space and time do not reproduce many aspects for the cases studied here. With the positivity of moment rate constraint, as long as the rupturing area is allowed to be larger than that in the forward problem, it is correctly found for the simple faulting model considered in this paper, provided that the rupture velocity and the Earth structure are known. If, however, the rupture front is constrained either to propagate more slowly or the rupturing area is taken smaller than that in the forward problem, we find that we are unable even to fit the accelerograms well. Use of incorrect crustal structure in the source region also leads to poor fitting of the data. In this case, the proper rupture front is not obtained, but instead a ''ghost front'' is found behind the correct rupture front and demonstrates how the incorrect crustal structure is transformed into an artifact in the solution. The positions of the centroids of the moment release in time and space are generally correctly obtained.