Combinatorial Optimization of Earthquake Spatial Distributions under Minimum Cumulative Stress Constraints

We determine optimal on-fault earthquake spatial distributions using a combinatorial method that minimizes the long-term cumulative stress resolved on the fault. An inte-ger-programming framework was previously developed to determine the optimal arrangement of a millennia-scale earthquake sample that minimizes the misfit to a target slip rate determined from geodetic data. The resulting cumulative stress from just slip-rate optimization, however, can greatly exceed fault strength estimates. Therefore, we add an objective function that minimizes cumulative stress and broad stress constraints to limit the solution space. We find that there is a trade-off in the two objectives: minimizing the cumulative stress on a fault within fault strength limits concentrates earthquakes in spe-cific areas of the fault and results in excursions from the target slip rate. Both slip-rate and stress objectives can be combined in either a weighted or lexicographic (hierarchical) method. Using a combination of objectives, we demonstrate that a Gutenberg-Richter sample of earthquakes can be arranged on a constant slip-rate finite fault with minimal stress and slip-rate residuals. We apply this method to determine the optimal arrangement of earthquakes on the variable slip-rate Nankai megathrust over 5000 yr. The sharp decrease in slip rate at the Tokai section of the fault results in surplus cumulative stress under all scenarios. Using stress optimization alone restricts this stress surplus to the north-east end of the fault at the expense of decreasing the slip rate away from the target slip rate at the southwest end of the fault. A combination of both slip-rate and stress objectives provides an adequate fit to the data, although alternate model formulations for the fault are needed at the Tokai section to explain persistent excess cumulative stress. In general, incorporating stress objectives and constraints into the integer-programming framework adds an important aspect of fault physics to the resulting earthquake rupture forecasts.