104
Water In Ecosystems: A Non Renewable Resource
Other works treat such simulation problems using the Indicator kriging. The initial
RV Z(x) is coded by a vector of indicators [Joumel, 1986], then using a MonteCarlo procedure the Zs is simulated at each unknown point together with the
estimation of the indicator function.
Such a procedure implies that the cdf of the simulated values Zs is the same as the
cdf of the experimental Zj. However, it does not guarantee the variogram
constraints (i.e. the variogram computed on Zs is the same as that computed on
Zj). This last constraint is generally assumed by a postponed treatment using
simulated annealing (Deutch and Joumel, 1992). This approach involves a considerable amount of computer time and memory.
3
SOME BASIC GEOSTATISTICAL CONCEPTS
Etymologically, the term geostatistics designates the statistical study of the Earth.
The first to use this term was G. Matheron (1962), and his definition was:
"Geostatistics is the application of the formalism of random functions to the
reconnaissance and estimation of natural phenomena". Several basic notions will
be briefly presented below. The reader will find an exhaustive text book
presentation in Matheron (1968), Joumel and Huijbregts (1978), Deutch and
Joumel (1992).
3.1 Regionalized Variables
A natural phenomenon can often be characterized by the distribution in space of
one or more variables, called regionalized variables. The distribution of
concentration in space, the level of pesticides, the porosity or the permeability of a
soil can be considered as regionalized variables. The regionalized variables have
two apparently contradictory characteristics:
• a local, random, erratic behaviour which resembles that of a random
variable;
• a general (or average) structured aspect on a larger scale which can be
represented as a function;
Geostatistical theory is based on the observation that the variability of regionalized
variables has a particular structure. The values z(x) and z(x+h) of a given
regionalized variable Z(x) at points x and x+h are auto-correlated. This autocorrelation depends both on the vector h and on location x.
The direct study of the mathematical function z(x) is excluded because its spatial
variability is usually extremely erratic, with all kinds of discontinuities and
anisotropies. The random and structured aspects of the regionalized variable are
expressed by the probabilistic language of random functions.
A random function Z(x) can be seen as a set of random variables Z(Xj) defined at
each point Xj of the domain D: Z(x) = { Z(Xj), V Xj ED}. The Xj represent the
space coordinates { x, y, z } of a given point i.
Water In Ecosystems: A Non Renewable Resource
Other works treat such simulation problems using the Indicator kriging. The initial
RV Z(x) is coded by a vector of indicators [Joumel, 1986], then using a MonteCarlo procedure the Zs is simulated at each unknown point together with the
estimation of the indicator function.
Such a procedure implies that the cdf of the simulated values Zs is the same as the
cdf of the experimental Zj. However, it does not guarantee the variogram
constraints (i.e. the variogram computed on Zs is the same as that computed on
Zj). This last constraint is generally assumed by a postponed treatment using
simulated annealing (Deutch and Joumel, 1992). This approach involves a considerable amount of computer time and memory.
3
SOME BASIC GEOSTATISTICAL CONCEPTS
Etymologically, the term geostatistics designates the statistical study of the Earth.
The first to use this term was G. Matheron (1962), and his definition was:
"Geostatistics is the application of the formalism of random functions to the
reconnaissance and estimation of natural phenomena". Several basic notions will
be briefly presented below. The reader will find an exhaustive text book
presentation in Matheron (1968), Joumel and Huijbregts (1978), Deutch and
Joumel (1992).
3.1 Regionalized Variables
A natural phenomenon can often be characterized by the distribution in space of
one or more variables, called regionalized variables. The distribution of
concentration in space, the level of pesticides, the porosity or the permeability of a
soil can be considered as regionalized variables. The regionalized variables have
two apparently contradictory characteristics:
• a local, random, erratic behaviour which resembles that of a random
variable;
• a general (or average) structured aspect on a larger scale which can be
represented as a function;
Geostatistical theory is based on the observation that the variability of regionalized
variables has a particular structure. The values z(x) and z(x+h) of a given
regionalized variable Z(x) at points x and x+h are auto-correlated. This autocorrelation depends both on the vector h and on location x.
The direct study of the mathematical function z(x) is excluded because its spatial
variability is usually extremely erratic, with all kinds of discontinuities and
anisotropies. The random and structured aspects of the regionalized variable are
expressed by the probabilistic language of random functions.
A random function Z(x) can be seen as a set of random variables Z(Xj) defined at
each point Xj of the domain D: Z(x) = { Z(Xj), V Xj ED}. The Xj represent the
space coordinates { x, y, z } of a given point i.