f.-f. Royer and A. Shtuka : Stochastic Imaging of Environmental Data
103
Stochastic simulations of random fields have been suggested as possible non linear
techniques to produce a "Stochastic image" or a realistic simulation map of the
unknown parameters. Such a simulation method is of fundamental importance to a
number of other technological applications (risk analysis, environmental risk
assessment) including hydrogeology (fracturation or ground water flow modeling)
and the petroleum industry (reservoir modeling).
This work presents a stochastic simulation method for continuous random
variables using an indicator coding technique. It is based on a DSI type of
algorithm under a set of constraints which guarantees that the cumulative
distribution function (cdf), the variogram or the drift of the estimates are
respectively the same as those of the initial data. The technique can be applied to
estimate a set of risk assessment maps in environmental problems.
2
OVERVIEW OF GEOSTATISTIC OR STOCHASTIC
SIMULATION TECHNIQUES
The stochastic simulation technique has received increasing attention in
geostatistics since the early theoretical work of Matheron (1968). Because of its
practical importance, particular interest arose during the last decades in methods
based on Indicator Functions (Journel, 1989), especially in risk analysis. Much of
the work on this topic has been concerned with two kinds of problems:
the simulation of a Random Field imposing a model of spatial
variability using the covariance or variogram function;
the simulation imposing a statistical property of the estimated
parameters such as the distribution function (Monte Carlo methods).
Most of the time, these two constraints are processed independently according to
the following steps:
• estimation of the mean Random Fields Zm using a classical estimation
procedure such as least squares, spline, ARMA or kriging methods;
Monte-Carlo simulation of the random component Zr imposing the
distribution (Gaussian anamorphosis) function and the variogram
(turning band methods, LU decomposition);
• the resulting simulation field Zs is then the sum of Zm and Zr ;
Such a two-step procedure does not guarantee that the simulated field Zs has the
same cumulative distribution function (cdf) as the initial sampled value Zj, except
when Z(x) has a Gaussian distribution function. To tackle this problem a
considerable amount of work has been done in the last decade (Sequential
Gaussian Methods, Orthogonal Polynomials, Indicator kriging ... ) [Deutch and
J ournel, 1992].
103
Stochastic simulations of random fields have been suggested as possible non linear
techniques to produce a "Stochastic image" or a realistic simulation map of the
unknown parameters. Such a simulation method is of fundamental importance to a
number of other technological applications (risk analysis, environmental risk
assessment) including hydrogeology (fracturation or ground water flow modeling)
and the petroleum industry (reservoir modeling).
This work presents a stochastic simulation method for continuous random
variables using an indicator coding technique. It is based on a DSI type of
algorithm under a set of constraints which guarantees that the cumulative
distribution function (cdf), the variogram or the drift of the estimates are
respectively the same as those of the initial data. The technique can be applied to
estimate a set of risk assessment maps in environmental problems.
2
OVERVIEW OF GEOSTATISTIC OR STOCHASTIC
SIMULATION TECHNIQUES
The stochastic simulation technique has received increasing attention in
geostatistics since the early theoretical work of Matheron (1968). Because of its
practical importance, particular interest arose during the last decades in methods
based on Indicator Functions (Journel, 1989), especially in risk analysis. Much of
the work on this topic has been concerned with two kinds of problems:
the simulation of a Random Field imposing a model of spatial
variability using the covariance or variogram function;
the simulation imposing a statistical property of the estimated
parameters such as the distribution function (Monte Carlo methods).
Most of the time, these two constraints are processed independently according to
the following steps:
• estimation of the mean Random Fields Zm using a classical estimation
procedure such as least squares, spline, ARMA or kriging methods;
Monte-Carlo simulation of the random component Zr imposing the
distribution (Gaussian anamorphosis) function and the variogram
(turning band methods, LU decomposition);
• the resulting simulation field Zs is then the sum of Zm and Zr ;
Such a two-step procedure does not guarantee that the simulated field Zs has the
same cumulative distribution function (cdf) as the initial sampled value Zj, except
when Z(x) has a Gaussian distribution function. To tackle this problem a
considerable amount of work has been done in the last decade (Sequential
Gaussian Methods, Orthogonal Polynomials, Indicator kriging ... ) [Deutch and
J ournel, 1992].