Functional multiple-point simulation

We present a new paradigm, called functional multiple-point simulation, in which multiple-point geostatistical simulation can be performed when functions or curves are observed at each location of a random field. Multiple-point simulation is a non-parametric method used for conditional geostatistical simulation of complex spatial patterns by inferring multiple-point statistics from a training image, rather than from a two-point variogram or covariance model. When the observable at each spatial location is a functional random variable, such multiple-point simulation must take into account not only the spatial correlation among locations but also the similarity of functions or curves observed at each location. The data events to be compared in this case are now functional, in the sense that they consist of spatial arrangements of functions. Consequently, we propose four distances, inspired by the functional data analysis literature, for measuring similarities between functional data events and use these to extend the direct sampling method to perform multiple-function geostatistical simulation with functional fields. We coin the new method Functional Direct Sampling and carry out extensive qualitative and quantitative performance comparison between the four proposed distances using simulation techniques on two well-known applications of multiple-point simulation: simulating copies of a functional random field and gap-filling of locations in a functional random field. We apply the proposed method to a gap-filling task of simulated wind profiles spatial functions over the Arabian Peninsula.