f.-f. Royer and A. Shtuka : Stochastic Imaging of Environmental Data
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3.2 Estimation of a Distribution
The cumulative frequency distribution (cdf) of a RV, denoted F(zd, is defined as
the proportion of values below the thresholds Zc. By convention, the cumulative
frequency below the minimum value is zero and the frequency below the
maximum value is one:
F(Zmin) = 0
F(zmax) = 1
The local expression of the cumulative distribution function will be written below
as F(x;z). The relation between the global and the local cumulative distribution
function is:
1
F(z) = IIDII J F(x; z) dx
(1)
XED
where D is the domain of definition of Z(x).
3.3 The Variogram
The variability of a given regionalized variable is characterized by the variogram
function 2y(x,h), which is defined as the expectation of the random variable
[Z(x) - Z(x+h)]2, i.e.,
2y(x, h) = E ( [Z(x) - Z(x+h)]2}
(2)
In general, the variogram 2y(x,h) is a function of both the point x and the vector h,
but in geostatistical applications it is assumed that the variogram depends only on
the distance h between data point x and x+h and not on location x (intrinsic
hypothesis). It is then possible to estimate the variogram 2y(x,h) from sample
points as the arithmetic mean of the squared differences between two experimental
measures [Z(Xi), Z(Xi+h)] at any two points separated by the lag h.
1 N(h)
2y*(x, h) = NCh) L [Z(Xi)-Z(xj+h)]2
(3)
1=1
where N(h) is the number of experimental pairs [z(Xj)-Z(Xi+h)] of data separated
by the vector h .
3.4 Indicator Function
The indicator of a random function Z(x) is the binary function lex; zc> defined
by:
lex; zd = 1 if Z(x) ~ Zc else 0
where Zc is a threshold value defined in [Zmin, zmax]. The stationary mean of the
binary indicator is the cumulative distribution function of the random function
Z(x) itself; indeed: