Ii. Introduction to Space/time Random Field Modelling in the Light of Uncertain Physical Knowledge

Most natural variables assume values at points within a spatiotemporal continuum. Each point in this continuum is represented by its space and time coordinates p = (s,t). Randomness of a natural variable manifests itself as an ensemble of possible realizations over the space/time continuum. This leads to the concept of the spatiotemporal random field (S/TRF) X(p), which is a collection of realizations (possibilities) of the spatiotemporal distribution of the natural variable. The value taken by the random field X(p) at a given space/time point p i is called a random variable and is noted as x i = X(p i). Herein, we will use capital English letter, e.g. X(p), to denote random fields, small English letters x 1 , x 2 , etc. to denote random variables, and small Greek letters χ 1 , χ 2 , etc. to denote their realizations. For convenience we will let the vector of random variables x = [x 1 ... x m ] T denote the random field X(p) at points p i (i=1,...m), and the vector χ χ = [χ 1 ...χ m ] T denote its realization. In spatiotemporal mapping we are interested in estimating the values of X(p) at point p k given physical data at space/time points p i knowledge is available, is noted as x data = [x 1 ,..., x m ] T. Having stated the framework in which STRF will be used, we will need some foundation for their mathematical interpretation. In this chapter we will give some short mathematical preliminaries on probability space, random variables and random fields. We will then introduce the concept of uncertain physical knowledge, and we will revisit traditional STRF modelling in the light of uncertain physical knowledge. The formal description of probability space and random variable are based on set-theoretic notions, and some useful mathematical preliminaries are provided in Appendix 2.1. In short a random variable x may acquire any one value from a distribution of values. The distribution of values χ that the random variable x may take is described for a given probability space (Ω , F , P) by the cumulative distribution function (cdf) F x (χ)= P[x ≤ χ] The derivative of the cdf F x (χ) with respect to χ , when it exists, is called the probability density function (pdf) of the random variable x , namely f x (χ) = ∂F x (χ) …