I-I Royer and A. Shtuka : Stochastic Imaging of Environmental Data
109
This technique is very fast and provides a set of realizations Zs(x) with the
following properties:
Zs(x) honors the observed data being equal to Z(x) at each location Xj E
K where data are available (conditioned simulation);
The cdf (histogram) of Zs(x) is equal to the imposed cdf F(z). The F(z)
can be the experimental cdf or it can also be an arbitrary given cdf
imposed by external information.
One of the disadvantages of the simple simulation is that the variogram
computed on the simulated value Zs(x) is not equal to the a priori
variogram computed on the initial data.
Moreover, the estimated indicators are smoothed because of the
roughness criteria used in DSI.
To overcome these difficulties, especially the smoothness of the simulated values, a
sequential simulation method was investigated below.
5.2 Sequential simulation
The sequential simulation consists in conditioning the simulated value by the
original data and by the previous simulated values computed at each step. The
algorithm is built as follows:
define a random path to visit each grid point at which the simulation is
performed;
for a given grid point Xi, interpolate the indicator cdf I(Xi;Z) = F(x;z)
using the original or previously simulated data points located in the
neighbourhood;
simulate a value z(Xj,) using the cdf F(x;z) by a simple Monte-Carlo
method and save it in the simulated data set;
transform the estimated cdf I(Xi;Z) into a step function computed on
Z(Xj), and put it in the initial data set;
repeat the last three procedures for all grid points.
The sequential simulation provides less smooth simulation results, but it implies a
reprocessing for each simulation. The computing time is then longer than in the
simple simulation case. However, in the sequential simulation, the indicator vector
can be coded in binary format (one bit per threshold), implying less memory. The
sequential simulation does not take into account the variogram constraint.
However, using a post-processing technique at the DSI iteration level, it is possible
to constrain the simulation by the local variogram.
5.3 Accounting for the variogram
In order to take account of the variogram, we suggest applying the Monte Carlo
simulation technique to a restricted interval of values in such a way that the local
variogram is satisfied. More precisely, let a be the location at which the simulation
is performed and A(a) the neighbourhood of location a. Two cases can be found:
109
This technique is very fast and provides a set of realizations Zs(x) with the
following properties:
Zs(x) honors the observed data being equal to Z(x) at each location Xj E
K where data are available (conditioned simulation);
The cdf (histogram) of Zs(x) is equal to the imposed cdf F(z). The F(z)
can be the experimental cdf or it can also be an arbitrary given cdf
imposed by external information.
One of the disadvantages of the simple simulation is that the variogram
computed on the simulated value Zs(x) is not equal to the a priori
variogram computed on the initial data.
Moreover, the estimated indicators are smoothed because of the
roughness criteria used in DSI.
To overcome these difficulties, especially the smoothness of the simulated values, a
sequential simulation method was investigated below.
5.2 Sequential simulation
The sequential simulation consists in conditioning the simulated value by the
original data and by the previous simulated values computed at each step. The
algorithm is built as follows:
define a random path to visit each grid point at which the simulation is
performed;
for a given grid point Xi, interpolate the indicator cdf I(Xi;Z) = F(x;z)
using the original or previously simulated data points located in the
neighbourhood;
simulate a value z(Xj,) using the cdf F(x;z) by a simple Monte-Carlo
method and save it in the simulated data set;
transform the estimated cdf I(Xi;Z) into a step function computed on
Z(Xj), and put it in the initial data set;
repeat the last three procedures for all grid points.
The sequential simulation provides less smooth simulation results, but it implies a
reprocessing for each simulation. The computing time is then longer than in the
simple simulation case. However, in the sequential simulation, the indicator vector
can be coded in binary format (one bit per threshold), implying less memory. The
sequential simulation does not take into account the variogram constraint.
However, using a post-processing technique at the DSI iteration level, it is possible
to constrain the simulation by the local variogram.
5.3 Accounting for the variogram
In order to take account of the variogram, we suggest applying the Monte Carlo
simulation technique to a restricted interval of values in such a way that the local
variogram is satisfied. More precisely, let a be the location at which the simulation
is performed and A(a) the neighbourhood of location a. Two cases can be found: