Simulation - Geostatistics

This page presents the main concepts related to the simulation functionality by indication available in the SPRING Geostatistics module.


CONDITIONAL STOCHASTIC SIMULATION

An stochastic simulation is a process to get realizations, equally likely, from RV probability distributions. Given a RV Z(u), each realization l of the RV is denoted by z(l)(u). A conditional simulation is conditioned to a set with n samples. In this case the realizations honor the attribute values at the data sample positions, that is, z(l)(ua ) = z (ua ), " l.

This text presents concepts related to conditional stochastic simulations which will be described in the following sections.


INDICATION SEQUENTIAL SIMULATIONS (IS)

The geostatistics models attribute values, inside a region A in the earth surface, as a random function. For each position u Î A the attribute value of an spatial data is modeled as a random variable (RV) Z(u). This means that, at position u, a RV Z(u) can assume different attribute values, each value with a given probability associated to it. In the n sampled positions, ua, a =1,2,...,n, the z(ua ) values are considered deterministic, or even, can be considered RV's where the measured values has an occurrence probability of 100%. A distribution function of Z(u) conditioned to the sampled data, F(u;|(n)), which is defined by:

F(u; z|(n)) = Prob{Z(u) £ z|(n)} when the attribute is numerical

                     and

F(u; z|(n)) = Prob{Z(u) = z|(n)} when the attribute is thematic

A F(u; z|(n)) models the uncertainty over the z(u) values at positions u not sampled, considering the n samples. This function can be inferred from an inference procedure called Kriging by Indication. It is an statistical non linear inference technique because it is applied over the attribute values transformed by a non linear mapping, the codification by indication.

A codification by RV indication Z(u=ua), in a threshold value z = zk, generating a RV value I(u=ua ; zk) using the following non linear mapping function:

for numerical attributes and,

for thematic attributes

The threshold values, zk, k=1,2...,K, are defined as samples number function. It is required an enough number of samples with a 1 value to define, successfully, a variographic model for each threshold value (Journel, 1983).

The RV expected conditional by indication I(u; zk) is computed by:

The equation above represents a very important result with respect to a distribution probability inference of a random variable: "The conditional expectation of I(u; zk) gives, for the threshold value z =zk, an estimate of the conditioned distribution function, cdf, value, of Z(u) in the thematic attributes case and an estimate of the accumulated conditioned distribution function, acdf, for numerical attributes".

Deutsch, 1998, presents a method called indication sequential simulation which uses local acdf's approximations to get RV realizations Z(u). This simulation can be used to create raster representations of continuous and categorical attributes. For this, an univariated acdf is defined for each visited node in a random sequence. To assure the covariance model reproduction, each univariated acdf is conditioned to samples and also to the grid nodes previously simulated (Goovaerts, 1997).

The realizations are obtained from the probability values of an uniform distribution model, which are mapped to values z given by the acdf of the RV which represents the considered attribute. Figure 1 shows this process.

          This process is used for the random fields generation, represented as regular grids, which represents the realizations of the numerical attribute surface in the region of interest.

 

Figure 1: Shows the process to get realizations from an acdf of an RV

Parameters estimate of the acdf from the RV realizations

The realization set, obtained for a given node of the random fields, can be used to determine statistical parameters of the local acdf at the node position.

The average m of the acdf is computed from the simple average of all realizations obtained for the node. The variance and the standard deviation s of the acdf are easily obtained from their average m and the realized values.

The median q.5 is determined from the realized sorted values z(u) and posterior definition of an attribute value that divides the ordered set into two sub-sets with the same cardinality. Values that are approximately the same as the average and the median indicating similarity of the RV probability distribution function. The median is a more robust estimator for the asymmetric probability distribution (Isaaks, 1989). The ordered realization set is also used to determine other quantis, associated to the RV distribution.

The average and median are typically used with numerical estimated attribute values represented as random variables.


See how to execute:

SPRING Procedures Sequence

Exploratory Analysis
Spatial Analysis by Semivariogram
Semivariogram Adjustment
Model Validation
Kriging, Kriging by Indication or Simulation by Indication

 See also:
Regionalized Variables
Variogram
SPRING Spatial Analysis
Digital Terrain Model
Digitalization of Maps