Predicting the intensity function of point processesbeyond observation areas.

Seismic networks provide data that are used a basis both for public safety decisions and for scientific research. Theirconfigurationaffects the data completeness, which in turn, critically affects several seismological scientific targets (e.g., earthquake prediction, seismic hazard…). How to map earthquakes density in seismogenic areas that are not covered by the network? We propose to predict the spatial distribution of earthquakes from the knowledge of presence locations and geological relationships, taking into account any interactions between records.Namely, in a more general setting, we aim to estimate the intensity function of a point process in windows where it has not been observed, conditional to its realization in observed windows, as in geostatistics for continuous processes. We define a predictor as the best linear unbiased combination of the observed point pattern. We show that the weight function associated to the predictor is the solution of a Fredholm equation of second kind. Both the kernel and the source term of the Fredholm equation are related to the second order characteristics of the point process through the pair correlation function. Results are presented and illustrated on simulated nonstationary processes, using continuous covariates or the realization of additional point processes, and real data for mapping Greek Hellenic seismicity in a region with unreliable and incomplete records.;;