Towards improving the spatial testability of aftershock forecast models

Aftershock forecast models are usually provided on a uniform spatial grid, and the receiver operating characteristic (ROC) curve is often employed for evaluation, drawing a binary comparison of earthquake occurrences or non-occurrence for each grid cell. However, synthetic tests show flaws in using the ROC for aftershock forecast ranking. We suggest a twofold improvement in the testing strategy. First, we propose to replace ROC with the Matthews correlation coefficient (MCC) and the inline-formulaF₁ curve. We also suggest using a multi-resolution test grid adapted to the earthquake density. We conduct a synthetic experiment where we analyse aftershock distributions stemming from a Coulomb failure (inline-formulaΔCFS) model, including stress activation and shadow regions. Using these aftershock distributions, we test the true inline-formulaΔCFS model as well as a simple distance-based forecast (R), only predicting activation. The standard test cannot clearly distinguish between both forecasts, particularly in the case of some outliers. However, using both MCC-inline-formulaF₁ instead of ROC curves and a simple radial multi-resolution grid improves the test capabilities significantly. The novel findings of this study suggest that we should have at least 8 % and 5 % cells with observed earthquakes to differentiate between a near-perfect forecast model and an informationless forecast using ROC and MCC-inline-formulaF₁, respectively. While we cannot change the observed data, we can adjust the spatial grid using a data-driven approach to reduce the disparity between the number of earthquakes and the total number of cells. Using the recently introduced Quadtree approach to generate multi-resolution grids, we test real aftershock forecast models for Chi-Chi and Landers aftershocks following the suggested guideline. Despite the improved tests, we find that the simple R model still outperforms the inline-formulaΔCFS model in both cases, indicating that the latter should not be applied without further model adjustments.