Kriging - Geostatistics

This page presents the main concepts related to kriging estimation methods that are available at the Geostatistics module of SPRING.


KRIGING

The term Kriging is derived from the name Daniel G. Krige, who was the pioneer to introduce the use of moving averages to avoid systematic super estimation of mining reserves (Delfiner e Delhomme, 1975).

Initially, the kriging method was developed to solve geological mapping problems, but its use was expanded with success for soil mapping (Burgess e Webster, 1980a, b), hydrologic mapping (Kitanidis e Vomvoris, 1983), atmospheric mapping (Lajaunie, 1984) and other correlated fields.

The difference between kriging and other interpolation methods is the way the weights are assigned to the different samples. In the case of simple linear interpolation, for example, the weights are equal to 1/N (N = number of samples); in the inverse square distance interpolation method, the weights are defined as the reciprocal of the square distance from the point to be interpolated to the observed points. In kriging, the procedure is similar to the weighted moving average interpolation, except that here the weights are determined from a spatial analysis, based on the experimental semivariogram. Besides, kriging estimators have no bias and minimum variance.

Following Oliver e Webster (1990), linear kriging includes a set of estimation methods, such as: simple kriging, ordinary kriging, universal kriging, co-kriging, disjunctive kriging, etc. There exists also non-linear kriging, from which the indicator kriging should be mentioned.

In the following, concepts related to simple kriging, ordinary kriging and indicator kriging are presented.


SIMPLE KRIGING (SK)

Consider a surface over which a soil property, Z, is observed, in different points, with coordinates represented by the vector x. Then, let {z(xi), i=1, ..., n} be a set of values, where xi identifies a two dimensional position represented by the pair of coordinates (xi, yi). Suppose that the objective is to estimate the value of Z at the point x0. The unknown value Z(x0) can be estimated from a linear combination of the n observed values plus a parameter l 0 (Journel, 1988):

 

. (27)

We would like to have an unbiased estimator, that is:

E [ - ] = 0 . (28)

This relation presumes that the two means are equal, that is:

E [] = E[] . (29)

But, (30)

The parameter l 0 is obtained substituting Equation (30) in (29), therefore:

. (31)

Substituting the value of l 0 in Equation (27), we have the estimator:

. (32)

The simple kriging method assumes that the mean (m) is a constant, known a priori, and therefore:

= m . (33)


Substituting Equation (33) in (32), the simple kriging estimator is:

. (34)

Journel (1988) shows that, minimizing the error variance (), the weights l i are obtained from the following system of equations, denominated simple kriging system:

for i = 1, ..., n (n equations) (35)

where,

  • C(xi, xj) refers to the covariance function corresponding to a vector, h, with origin xi and extremity in xj.
  • C(xi, x0) refers to the covariance function corresponding to a vector, h, with origin xi and extremity in the point to be estimated x0.

For example, for n = 2, the simple kriging system has 2 equations and 2 unknowns (l 1, l 2):


The corresponding minimized error variance, also called simple kriging variance (), is given by (Journel, 1988):

. (36)

In matrix notation, the simple kriging system is written as:

K . l = k Þ l = . k, with (37)


, l and k

where K e k are the covariance matrices and l is the vector containing the weights.

The simple kriging variance is given by (Journel, 1988):

(38)


ORDINARY KRIGING (OK)

As in the simple kriging, the unknown value Z(x0) can be estimated by a linear combination of the n observed values plus a parameter, l 0 (Journel, 1988):

. (39)

We would like to have an unbiased estimator, that is,


E [- ] = 0 . (40)

The above relation implies that the two means are equal; therefore, applying Equation (39) in (40), we get:

. (41)

Differently from simple kriging, the ordinary kriging does not require the previous knowledge of the mean m. In this case, for Equation (41) to be satisfied is necessary that:

.

Therefore, the ordinary kriging estimator is:


, with . (42)

Journel (1988) shows that, by minimizing the error variance, () under the condition that , the weights l i are obtained from the following system of equations, denominated ordinary kriging system:

(43)

where,

  • C(xi, xj) and C(xi, x0) are as previously defined; and
  • a is the Lagrange parameter necessary for minimizing the error variance.



The corresponding minimized error variance, denominated ordinary kriging variance (), is given by the following expression (Journel, 1988):

. (44)

The ordinary kriging system (43) can be written in matrix notation as:


K . l = k Þ l = . k (45)

where,

K and k are covariance matrices and l the vector containing the weights.


K =, l = e k =

The ordinary kriging variance is given by (Journel, 1988):

(46)

Practical example of KRIGING:

Consider the sample space shown in the Figure below. Suppose that the objective is to estimate the value of the variable Z in the point xo, from z(x1), z(x2), z(x3) and z(x4). Consider also that the experimental variogram was adjusted by a spherical model, with a contribution C1 = 20, nugget effect C0 = 2 and range a = 200.

Fig. - Sample points grid.

Applying Equation (45), we have:

=

The matrices elements are computed in the following way:

Cij = C1 + C0 - g (h), where h is the distance vector between the points xi and xj. Then, for the given example, we have:

C12 = C21 = C04 = C1 + C0 - g (50)

= 20 + 2 - = 9.84

C13 = C31 = C1 + C0 - g = 1.23

C14 = C41 = C02 = C1 + C0 - g = 4.98

C23 = C32 = C1 + C0 - g = 2.33

C24 = C42 = C1 + C0 - g = 0.29

C34 = C43 = C1 + C0 - g = 0

C01 = C1 + C0 - g (50) = 12.66

C03 = C1 + C0 - g (150) = 1.72

C11 = C22 = C33 = C44 = C1 + C0 - g (0) = 22

Substituting the values of Cij in the matrices, the following weights are found: l 1 = 0.518 , l 2 = 0.022 , l 3 = 0.089 and l 4 = 0.371. Finally, the estimator of is given by:

= 0.518 z(x1) + 0.022 z(x2)+ 0.089 z(x3) + 0.371 z(x4)

Comments:

Although the samples Z2 and Z3 have few influence in the final estimate of Z0, their relative influences are not linear in relation to their distance from Z0. The sample Z3 is more distant than Z2; however, it has more influence (8.9%) than Z2 (2.2%). This occurs because Z0 is directly under the influence of Z3, while Z2 is very close to Z1. When the covariance are introduced for computing the weights, improper weights to clusters of samples are avoided; this does not occur with other methods based only on distances.


INDICATOR KRIGING (IK)

Geostatistics models the values of an attribute, inside a region A of the earth surface, as a random function. For each position u Î A the attribute value of a spatial data is modeled as random variable (RV) Z(u). This means that, in the position u, the RV Z(u) can assume different values of this attribute, each value having an associated probability of occurrence. In the n sampled position, ua , a =1,2,...,n, the values z(ua ) are considered deterministic, that is, they can be considered RV’s with probability 100% of occurrence. The distribution function of Z(u) conditioned to the sample data , F(u; z|(n)), is defined by:

F(u; z|(n)) = Prob{Z(u) £ z|(n)} for a numerical attribute,

and

F(u; z|(n)) = Prob{Z(u) = z|(n)} for a thematic attribute.

The function F(u; z|(n)) models the uncertainty about the values of z(u), in non sampled positions u, considering the n samples. This function can be inferred from the inference procedure denominated indicator kriging. It is a non-linear statistical inference technique, since it is applied to the attribute values transformed by a non-linear mapping, the indicator codification.

The indicator codification of the RV Z(u=ua ), for a cut-off value z = zk, generates the RV I(u=ua ; zk) using the following non-linear mapping function:

for a numerical attribute, and

for a thematic attribute.

The cut-off values, zk, k=1,2...,K, are defined depending on the number of samples. It is necessary that the amount of codified samples with value 1 are enough for defining, with success, a variography model for each cut-off value (Journel, 1983).

The conditional expectation of the indicator RV I(u; zk) is computed by:

The above equation represents a very important result with respect to the inference of the distribution of probability of a random variable: "The conditional expectation of I(u; zk) gives, for the cut-off value z = zk , an estimation of the value of the conditional distribution function, cdf, of Z(u) for the case of thematic attributes and an estimation of the conditional cumulative distribution function, ccdf, for numerical attributes".

The simple indicator kriging is a simple linear kriging procedure applied to the sample set by the indicator codification in z = zk, that is:

where FS*(zk) is the mean of the random function of the stationary region and the weights l Sa (u; zk) are determined with the objective of minimizing the error variance.

Considering that the sum of the weights are equal to 1, one can get a more simplified version of the simple indicator kriging, the ordinary indicator kriging, for which the estimation expression is:

 

The weights l Oa (u; zk) are obtained by solving the following ordinary indicator kriging system of equations:

where m (u; zk) is a Lagrange parameter, ha b is the vector defined by the positions ua and ub , ha is the vector defined by the positions ua and u, CI(ha b ; zk) is the autocovariance defined by ha b and CI(ha ; zk) is the covariance defined by ha . The autocovariance are determined by the theoretical variography model defined by the set I when z = zk.

The indicator kriging, simple or ordinary, gives, for each cut-off value k , an estimator which is also the better least square estimator of the conditional expectation of the RV I(u; zk). Using this property it is possible to compute estimates of the cdf values of Z(u) for several values of zk, belonging to the Z(u) domain. The set of the estimated values of the cdf’s of Z(u), for the cut-off values, is considered a discrete approximation of the real cdf of Z(u). The greater the number of cut-off values, the better the approximation.

The indicator kriging is non-parametric. It does not consider any type of a prior distribution for the random variable. Instead, it permits the construction of a discrete approximation of the cdf of Z(u). The values of the discreted probabilities can be directly used to estimate the values that characterize the distribution, such as: mean, variance, mode, quantiles, etc.


Check how to execute:

SPRING Procedures Sequence
Exploratory Analysis - Geostatistics
Semivariogram Generation - Unidirectional Analysis
Semivariogram - Modeling or Adjustment
Model Validation
Kriging, Kriging by Indication or Simulation by Indication

 See also :
Regionalized Variables
Variogram - Geostatistics
Spatial Analysis
Digital Terrain Modeling
Digitalization of Maps