Application of an inhomogeneous stress (patch) model to complex subduction zone earthquakes: A discrete interaction matrix approach

In recent years it has been recognized that the level of shear and normal stress along a fault can vary; thus the stress is spatially and temporally inhomogeneous. Moreover, it has also been suspected that faults might interact in some way, with the result that a variety of earthquake magnitudes might be produced along a given length of fault at varying times. In order to explore these ideas we have developed a quantitative formalism, which we call the interaction matrix method, to express the influence of one fault upon another. This matrix is calculated by use of the energy change for a system of interacting cracks or faults and therefore gives energy-consistent results. Specifically, the interaction matrix relates the area-averaged stress on the fault segment to the area-averaged slip state on all the other fault segments in the system. Since any fault can be subdivided into an arbitrary number of fault segments, the interaction matrix can have arbitrary dimension; in fact, the continuum limit is recovered as the dimension of the matrix approaches infinity. We combine this matrix method with a segmentation, or “patch,” model for earthquakes, in which each discrete segment of a fault has the same coseismic stress change (defined as the difference between the driving stress at which healing occurs minus the driving stress at which sliding starts) each time it slips. We show that slip on a patch during an earthquake can vary substantially, depending on how it interacts with other nearby patches. In this model it is quite possible for the spatial distribution of stress on the fault following an event to be again in a spatially inhomogeneous state, rather than in a uniform state, as is often assumed. Hence the seismic moment produced by an earthquake on a given set of patches can vary substantially, depending on the sequence of sliding and healing on the different patches. To apply these ideas, we devised a means to calculate the interaction matrix elements and used them to quantitatively examine earthquake sequences off the Colombia-Ecuador coast and in the Nankai Trough near Japan