Page 79 - Modern Spatiotemporal Geostatistics

60 Modern Spatiotemporal Geostatistics — Chapter 2
DEFINITION 2.10: An S/TRF X(p) is a collection of complementary
field realizations x associated with the values of a natural variable at
points p = (s, t) of a spatiotemporal continuum S x T. Mathematically,
the S/TRF X(p) is the mapping

where J7 is the sample space that includes all possible field realizations,
F is a family of realizations, P(x) € [0,1] is a probability associated
with each realization, and L Q(fl,F,P), q>l denotes the norm on the
probability space ($7, F, P).
I/2-norms are usually considered in geostatistics. By "complementary field
realizations" are meant all physically possible realizations of the natural vari-
able. An S/TRF X(p) is, therefore, a collection of realizations for the distri-
bution of the natural variable in space/time. The S/TRF can be viewed also
as a collection of correlated random variables, say, x map = (0:1,. . ,x m,Xk) at
-
the space/time points p map = (PI, • • • ,p m,p k). A realization of the S/TRF at
these points is denoted by the vector x map = Xi, • • • >Xm» Xfc)- We assume
(
that the S/TRF takes values in the space of real numbers, since this assump-
tion represents the majority of applications in the natural sciences. A detailed
discussion of recent developments in the mathematical S/TRF theory may be
found in Christakos and Hristopulos (1998).

COMMENT 2.10: I t must b e clear to th e reader that here we use th e symbol
X map because our goal later in the book will be to obtain spatiotemporal
maps displaying estimates at points p k of the unknown values Xk of the

natural variable from it s observed values xi > • • • , Xm • In th e same context,

the elements o f th e vector p data ar e the data points p t ( i = 1,... , m) an d

the vector p map ha s as elements th e points p i ( i = 1,..., m, k).

The complete characterization of an S/TRF is provided by the multivariate
probability density function (pdf) f K defined as
where the subscript 9£ denotes the physical knowledge used to derive the pdf
(physical knowledge bases are studied in Chapter 3). Notice that the notation
(Xmop! Pmap) 'n tne rest °f tne book nas been kept only when necessary
(e.g., when differentiation with respect to space and/or time takes place); in all
other situations, p map has been dropped and the simpler notation f^(x. map) is
used. A generally incomplete—yet in many practical applications satisfactory—
characterization of the S/TRF is provided by a limited set of statistical mo-
ments (also called, simply, space/time statistics), which are defined as follows,