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 pi is called a random variable and is noted as xi X( pi ). Herein, we will use capital English letter, e.g. X( p), to denote random fields, small χ χ English letters x1, x2 , etc. to denote random variables, and small Greek letters 1, 2 , etc. to denote their realizations. For convenience we will let the vector of random variables = T x [x1 ...xm ] denote the random field X( p) at points pi (i=1,...m), and the vector χχ= χ χ T [ 1 ... m ] denote its realization. In spatiotemporal mapping we are interested in estimating the values of X( p) at point pk given physical data at space/time points pi = T = (i=1,...m). We let xmap [x1,...,xm, xk ] denote X( p) at points pi (i 1,2,...,m,k), and = the subsets of these values at points pi (i 1,2,...,m ), where (possibly uncertain) physical = T knowledge is available, is noted as xdata [x1,...,xm ] . 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. 5 2.1. Introduction to Space/Time Random Fields Modelling 2.1.1. The Space/Time Random Field 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 Ω probability space ( , F , P) by the cumulative distribution function (cdf) χ ≤ χ Fx ( )= P[x ] χ χ The derivative of the cdf Fx ( ) with respect to , when it exists, is called the probability density function (pdf) of the random variable x, namely ∂F (χ) f (χ) = x (2.1) x ∂χ As explained in more details in Appendix 2.1, the probability measure P on a measurable Ω Ω σ Ω space ( , F ), where is the sample space and F a -algebra of subsets of , is a → function P: F [0,1] satisfying Kolmogorov's axioms, i.e. (a) P(Ω) = 1 (b) 0 ≤ P(A ) ≤ 1 for all sets A ∈ i i F ∞ ∞ ∩ =∅ ≠ = (c) if Ai Aj (i j), then P( U Ai ) ∑ P(Ai ) i=1 i=1 Any function possessing these properties is a valid cdf, and a random variable is said to be completely characterized if its cdf or pdf is known. 6 A random field describes a natural process which takes values at points within a space/time continuum. Space/time is viewed as a continuous spatial arrangement combined with a temporal order of events. This union of space and time is defined in terms of their = Cartesian product. Spatial coordinates s (s1,...,sn ) are elements of the Euclidean space ℜn , i.e. s ∈S ⊂ℜn , and coordinate t is along the time axis T ⊂ℜ1. The space/time coordinates p = (s,t) are defined on the Cartesian product S × T , i.e. p = (s,t) ∈S × T . A formal description of random field arises from set-theoretic notions, and the readers is referred to Christakos; 1992 for a detailed treatment. Conceptually a spatiotemporal random field X( p) may be defined as a parameterized collection of random variables X( p) = {X } (2.2) p p∈ℜ n ×[0,∞) where the random variables Xp are defined on a probability space, and the space/time point p takes values in the space/time continuum S × T . For mapping purposes it is sufficient to consider the random field X( p) as being the family of random variables {x1, x2 ,...} at some specified points p1, p2 ,... (usually the data and estimation points). In this latter view the random field X( p) is completely described by the join (or multivariate) cdf χ χ χ = ≤ χ ≤ χ ≤ χ Fx ( 1, 2 , ... , m ) P[x1 1, x2 2 , ... ,xm m ] (2.3) In order to be valid, the multivariate cdf must satisfy Kolmogorov's axioms, and the χχ= χ χ χ T derivative of the cdf with respect to [ 1, 2 , ... , m ] , when it exists, is called the = T multivariate pdf of the random vector x [x1, x2 ,...,xm ] , namely ∂ nF (χ ,χ , ... ,χ ) f (χ ,χ , ... ,χ ) = x 1 2 m (2.4) x 1 2 m ∂χ ∂χ ∂χ 1 2 , ... m 7 A random field would be completely characterized by its multivariate cdf or pdf if known, however in most cases this probability law is not known a priori. In this case the random field may be incompletely characterized in terms of ensemble functions of the general form Φ = χ χ Φ χ χ χ χ [X( p1),...X( pm )] ∫ d 1 ...∫ d m [ 1,..., m ] f x ( 1,..., m; p1,..., pm ) (2.5) χ χ where f x ( 1,..., m; p1,..., pm ) is the multivariate pdf for the S/TRF X( p) taken at points Φ ⋅ p1,..., pm ., [...] denotes the operation of stochastic expectation and [ ] is a function of χ χ Φ ⋅ 1,..., m . Statistical moments of order p are obtained by setting the functional [ ] to a χ χ polynomial in 1,..., m of order p. Certain moments can often be estimated from the data or physical models, and they provide information regarding the properties of the pdf. Commonly used moments are the moments of order one and two, namely the mean and covariance. The mean is given by = = χ χ χ mx ( p) X( p) ∫ f x ( ; p) d (2.6) χ where f x ( ; p) represents the (univariate) pdf for the random field at point p. The (centered) space-time covariance function is given by = − − cx ( p, p') (X( p) mx ( p))(X( p') mx ( p' )) (2.7) = χ − χ − χ χ χ χ ∫ ∫ ( mx ( p))( ' mx ( p')) f x ( , ';p, p') d d ' χ χ where f x ( , ';p, p') represents the bivariate pdf of the random field taken between points p and p' . The covariance function provides information about the spatial distribution of the variable and the fluctuation correlations, but not its strength. The strength of the fluctuations of the variable at a given point, or local variance of the fluctuations, is given by the value of the covariance at zero lag (i.e. when p' = p) as 8 σ 2 = x ( p) cx ( p, p) (2.8) Then the correlation function gives information about the strength of the fluctuation correlations by normalizing the covariance with the local variances as follows ρ = σ σ x ( p, p') cx ( p, p') x ( p) x ( p'). (2.9) We now consider briefly some symmetry properties of S/TRFs. If all multivariate pdf's are invariant for all translations in space and time the S/TRF is called statistically homogeneous-stationary in the strict sense. Hence, strict homogeneity-stationary requires that χ χ = χ + + τ χ + + τ FX [ 1(s1,t1),..., N (sN ,tN )] FX [ 1(s1 r,t1 ),..., N (sN r,tN )] (2.10) = τ for all N 1,2,..., and for all spatial lags r and temporal lags , where FX denotes the joint probability distribution. On the other hand, if the first two moments are invariant for all translations in space and time, the S/TRF is called statistically homogeneous-stationary in the wide sense (also, second order or weakly homogeneous). While strict homogeneity-stationary implies weak homogeneity-stationary, the converse is not always true. Thus, weak homogeneity-stationary requires only that the S/TRF X(s,t) have a constant mean, = mX ( p) mX ; (2.11) and that the covariance be a function only of the spatial lag vector r = s − s′ and temporal lag τ = t − t′ between the two points, i.e., 9 ′ = − ′ − ′ = τ Cx ( p; p ) Cx (s s ;t t ) Cx (r, ), (2.12) If a S/TRF is homogeneous-stationary in the wide sense, it can be further classified as isotropic (in the wide sense) if the covariance function depends only on the spatial lag distance; e.g., ′ = − ′ = CX ( p; p ) CX (r;t t ), r |r|. (2.13) A S/TRF that is not isotropic will display anisotropic spatial dependence. If the second order ensemble moments are equal to the second order sample moments then the S/TRF is called statistically ergodic in the wide sense. While the ensemble moments are calculated in terms of the corresponding pdf's, the sample moments are calculated on the basis of a single realization which is usually available in practice. This will become important later, when we will discuss the subject of covariance estimation of a S/TRF from experimental data. A S/TRF will be homogeneous-stationary only in the absence of any spatial or temporal trend.