Multi-Resolution Grids in Earthquake Forecasting: The Quadtree Approach

The Collaboratory for the Study of Earthquake Predictability (CSEP) is an international effort to evaluate probabilistic earthquake forecasting models. CSEP provides the cyberinfrastructure and testing methods needed to evaluate earthquake forecasts. The most common way to represent a probabilistic earthquake forecast involves specifying the average rate of earthquakes within discrete spatial cells, subdivided into magnitude bins. Typically, the spatial component uses a single-resolution Cartesian grid with spatial cell dimensions of 0.1° × 0.1° in latitude and longitude, leading to 6.48 million spatial cells for the global testing region. However, the quantity of data (e.g., number of earthquakes) available to generate and test a forecast model is usually several orders of magnitude less than the millions of spatial cells, leading to a huge disparity in the number of earthquakes and the number of cells in the grid. In this study, we propose the Quadtree to create multi-resolution grid, locally adjusted mirroring the available data for forecast generation and testing, thus providing a data-driven resolution of forecasts. The Quadtree is a hierarchical tree-based data structure used in combination with the Mercator projection to generate spatial grids. It is easy to implement and has numerous scientific and technological applications. To facilitate its application to end users, we integrated codes handling Quadtrees into pyCSEP, an open-source Python package containing tools for evaluating earthquake forecasts. Using a sample model, we demonstrate how forecast model generation can be improved significantly in terms of information gain if constrained on a multi-resolution grid instead of a high-resolution uniform grid. In addition, we demonstrate that multi-resolution Quadtree grids lead to reduced computational costs. Thus, we anticipate that Quadtree grids will be useful for developing and evaluating earthquake forecasts.