Adaptive Spatiotemporal Smoothing of Seismicity for Long-Term Earthquake Forecasts in California

We present new methods for time-independent earthquake forecasting that employ space-time kernels to smooth seismicity. The major advantage of the methods is that they do not require prior declustering of the catalog, circumventing the relatively subjective choice of a declustering algorithm. Past earthquakes are smoothed in space and time using adaptive Gaussian kernels. The bandwidths in space and time associated with each event are a decreasing function of the seismicity rate at the time and location of each earthquake. This yields a better resolution in space-time volumes of intense seismicity and a smoother density in volumes of sparse seismicity. The long-term rate in each spatial cell is then defined as the median value of the temporal history of the smoothed seismicity rate in this cell. To calibrate the model, the earthquake catalog is divided into two parts: the early part (the learning catalog) is used to estimate the model, and the latter one (the target catalog) is used to compute the likelihood of the model's forecast. We optimize the model's parameters by maximizing the likelihood of the target catalog. To estimate the kernel bandwidths in space and time, we compared two approaches: a coupled near-neighbor method and an iterative method based on a pilot density. We applied these methods to Californian seismicity and compared the resulting forecasts with our previous method based on spatially smoothing a declustered catalog (Werner et al., 2011). All models use small M >= 2 earthquakes to forecast the rate of larger earthquakes and use the same learning catalog. Our new preferred model slightly outperforms our previous forecast, providing a probability gain per earthquake of about 5 relative to a spatially uniform forecast.