Spatial and Spectral Interpolation of Ground-Motion Intensity Measure Observations

Following a significant earthquake, ground-motion observations are available for a limited set of locations and intensity measures (IMs). Typically, however, it is desirable to know the ground motions for additional IMs and at locations where observations are unavailable. Various interpolation methods are available, but because IMs or their logarithms are normally distributed, spatially correlated, and correlated with each other at a given location, it is possible to apply the conditional multivariate normal (MVN) distribution to the problem of estimating unobserved IMs. In this article, we review the MVN and its application to general estimation problems, and then apply the MVN to the specific problem of ground-motion IM interpolation. In particular, we present (1) a formulation of the MVN for the simultaneous interpolation of IMs across space and IM type (most commonly, spectral response at different oscillator periods) and (2) the inclusion of uncertain observation data in the MVN formulation. These techniques, in combination with modern empirical ground-motion models and correlation functions, provide a flexible framework for estimating a variety of IMs at arbitrary locations. Electronic Supplement: Demonstration Python script for the evaluation of the multivariate normal (MVN) with additional uncertainty.