Two Combinatorial Optimization Methods that Determine On-Fault Earthquake Magnitude Distributions

Two combinatorial optimization methods have been developed to determine the spatial distribution of earthquakes along a single fault or among faults within a fault system. For a synthetic earthquake catalog that spans millennia, the objective is to determine the optimal arrangement of earthquakes that minimizes misfit to target fault slip rates. The first method is a spatial gap-filling method that uses a greedy-sequential algorithm in which earthquakes are sequentially placed in a locally optimal sense. The second method is integer programming (IP) that globally optimizes earthquake placement subject to a number of constraints, including slip-rate uncertainty (minimum and maximum limits). The binary decision vector in the IP model is composed of every possible location that each earthquake in the synthetic catalog can occur. General mixed-integer programming (MIP) software is used that consists of a number of different algorithms, including the simplex algorithm for the initially relaxed binary constraints and the branch-and-cut algorithm to recursively search a tree of binary solutions. We find that the greedy-sequential algorithm efficiently scales to long catalog durations and complex fault systems. Global optimization, in contrast, is likely a nondeterministic polynomial-time (i.e., NP-Hard) problem, similar to the knapsack and bin-packing problems well known in operations research. MIP software has greatly advanced in recent years, making fault-scale problems tractable, although high-performance computing platforms are required for the largest problems. We apply both of these methods to both a single fault (Nankai subduction megathrust) and a complex fault system (California transform system) and compare the results. We demonstrate that, in general, combinatorial optimization methods provide a valuable and independent way to determine on-fault earthquake magnitude distributions.