Kriging is a geostatistical technique for estimating attribute values at a point, over an area, or within a volume. It is often used to interpolate grid node values in mapping and contouring applications. In theory, no other interpolation process can produce better estimates (being unbiased, with minimum error); though the effectiveness of the technique actually depends on accurately modeling the variogram. The accuracy of kriging estimates is driven by the use of variogram models to express autocorrelation relationships between control points in the data set. Kriging also produces a variance estimate for its interpolation values.
The technique was first used for the estimation of gold ore grade and reserves in South Africa (hence the origin of the term Nugget Effect), and it is named in honor of a South African mining engineer, Danny Krige. The mathematical validity and foundation was developed by Georges Matheron, who later founded the Centre de Geostatistiques, as part of the Ecole’ des Mines in Paris, France.
(Henley, 1981; Hohn, 1988; Journel, 1989; Isaaks and Srivastava, 1989; Deutsch and Journel, 1992; Wackernagel, 1995).
KRIGING FEATURES
Kriging is a highly accurate estimation process which:
minimizes estimation error (the difference between measured value - the re-estimated value)
honors “hard” data
does not introduce an estimation bias
does not reproduce inter-well variability
produces a “smoothed” result; like all interpolators
is a univariate estimator; requiring only one covariance model
weighs control points according to a spatial model (variogram)
tends to the mean value when control data are sparse
uses a spatial correlation model to determine the weights ()
assigns negative or null weights to control points outside the correlation range of the spatial model
indicates the global relative reliability of the estimate through RMS error (kriging variance), as a by-product of kriging
has a general and easily reformulated kriging matrix, making it a very flexible technique to use more than one variable
declusters data before the estimation Types Of Kriging
There are a number of kriging algorithms, and each is distinguished by how the mean value is determined and used during the estimation process. The four most commonly used methods are:
Simple Kriging: The global mean is known (or supplied by the user), and is held constant over the entire area of interpolation.
Ordinary Kriging: The local mean varies, and is re-estimated based on the control points in the current search neighborhood ellipse.
Kriging with an External Drift: Although this method uses two variables, only one covariance model is required, and the shape of the map is related to a 2-D attribute which guides the interpolation of the primary attribute known only at discrete locations. A typical application is time-to-depth conversion, where the primary attribute (such as depth at the wells) acquires its shape from the secondary attribute, referred to as external drift (such as two-way travel time known on a 2-D grid).
Indicator Kriging: estimates the probability of an attribute at each grid node (e.g., lithology, productivity). The technique requires the following parameters:
Coding of the attribute in binary form, as 0 or 1.
Prior Probabilities of both classes.
Spatial covariance model of the indicator variable.
The Kriging Process
We will illustrate the estimation process with an example problem, as shown in Figure 1: Arrangement of three data points.
Figure 1
Given samples located at (Z), where = 1, 2, 3 Find the most likely value of the variable Z at the target point (grid node: Z0*
, Figure 1). In this graphic, we see the geometrical arrangement of three data points Z, the location of the point whose value we wish to estimate Z0*, and the unknown weights, .
Consider Z0*
as a linear combination of the data Z
Z0* = 0 + Z
Where: = 1 and 0 = mz -
Determine so that:
Z0*
is unbiased: E [Z0*
-Z] = 0
Z0*
has minimum mean square error (MSE)
E [Z0*
-Z]2 is minimum Recall that the unknown value Z0*
is estimated by a linear combination of n data points plus a shift parameter 0:
Z0* = 0 + Z (1)
By transforming the above equation into a set of linear normal equations, we solve the following to obtain the weights . The set of linear equations takes the following form:
j C (x, xj) - = c (x, x0) for all j = 1,n (2) or in matrix shorthand notation:
C = c (3)
All three terms are matrices where:
C (x, xj) represents a covariance between sample points x and xj
c (x, x0) represents a covariance between a sample located at x and the target point x0; the estimated point
are the unknown weights, j
is a Lagrange multiplier that converts a constrained minimization problem into an unconstrained minimization.
Determine the matrix of unknown weights by solving the matrix equation for as follows:
C = c (4)
Where
= C-1 c (5)
Note that equation 3 is written in terms of covariance values, however we either modeled a variogram or correlogram, not the covariance. We use the covariance values because it is computationally more efficient.
The covariance equals the sill minus the variogram (Figure 2 : Relationship between a spherical variogram and its covariance equivalent):
Figure 2
C(h) = 2 (sill) -(h) (6)
Kriging Variance
In addition to estimating the value of a variable at an unsampled location, the kriging technique also provides an estimation of the likely error (in the form of error variance) at every grid node. These error estimates can be mapped to give a direct assessment of the reliability of the contoured surface. Because the kriging variances are determined independent of the data values, the kriging error is not a measure of local reliability (Deutsch and Journel, 1992). Do not attempt to use the kriging standard deviation like the true classical standard deviation statistic.
The kriging variance equation is:
2k = C(x0, x0) i(x, x0) - (7) Search Neighborhood Criterion
All interpolation algorithms require good data selection criteria, specified by a search neighborhood. The model variogram plays a role in controlling extent of the neighborhood. The variogram range defines the maximum size of the neighborhood from which control points should be selected for estimating a grid node, in order to take advantage of the statistical correlation among the
observations.
A typical geostatistical routine might interpolate values for a specific location using nearest neighbor values weighted by distance and the degree of autocorrelation present for that distance (as defined by the variogram model).
The neighborhood searches would be limited to a specified number of nearest neighbors, and might also be restricted to a particular direction.
Search neighborhood parameters include:
Search radius
Neighborhood shape (isotropic or anisotropic: Figure 3a and 3b
3b
)
Number of sectors ( 4 or 8 are common)
Figure 3a
Number of data points per sector
Unique Neighborhoods use all data points (practical limit is 100)
Moving Neighborhoods use a limited number of points per sector (e.g., 4)
Azimuth of major axis of anisotropy
In this graphic, note the elliptical shape of the anisotropic search neighborhood and the circular shape for the isotropic neighborhood. Both neighborhoods are divided into octants, with a maximum of two data points per sector.
This graphic shows the radii for the anisotropic neighborhood are: minor axis = 1000 meters and major axis = 4000 meters, aligned at N15E. The isotropic model has a 1500-meter radius length. The center of the neighborhood is the target grid node for estimation. There are 55-sample points (x) within the study area.
Weights are shown for data control points used for the estimation at the target point.
Practical Considerations in Designing the Search Neighborhood
Align the search axis with the direction of maximum anisotropy.
Search radii (if anisotropic) should be to the correlation ranges.
Each quadrant should have enough points ( 4) to avoid directional sampling bias.
CPU time and memory requirements grow rapidly as a function of the number of data points in neighborhood.
In theory, more data in the kriging system reduces the mean square error.
In practice, the covariance is poorly known for distances exceeding about
½ to 2/3 the size of the field. Including points that are more distant may actually increase the error.
The kriging estimator is built from data within the search neighborhood centered at the target location Zo*
.
Practical Considerations: Unique versus Moving Search Neighborhoods
In a moving search neighborhood, a new simple kriging (SK) or ordinary kriging (OK) system of equations is solved at each grid node.
A unique search neighborhood uses all the data, so the left side of the kriging matrix, C, is solved only once and used at each grid node.
If sufficient wells are available for ordinary kriging, then a moving neighborhood is preferable to the unique neighborhood.
Unique neighborhoods tend to prevent artifacts from abrupt changes in the number and values of the data points.
Unique neighborhoods smooth the data more than moving neighborhoods.
Practical Considerations: Ordinary (OK) versus Simple Kriging (SK)
Simple kriging does not adapt to local trends; rather it relies on a constant, global mean.
Ordinary kriging uses a local mean (mz), which amounts to re-estimating mz at each grid node from the data within the search neighborhood.
When all data points are used (unique neighborhood), ordinary kriging and simple kriging yield similar results.
If only a few data points are available in the local search neighborhood, ordinary kriging may produce spurious weights because of the constraint that the weights must sum to 1.
If the wells are known to provide a biased sampling, it may be better to impose your own mz with simple kriging rather than use ordinary kriging.
Effects of Variogram Parameters on Kriging
Kriging applies weighting functions according to a mathematical model of the variogram. The resulting kriging estimates are best linear unbiased estimates of the surface, provided that the surface is stationary and the correct form of the variogram has been determined. As the shape of the model variogram changes, so do the kriging results.
Rescaling the variogram or correlogram (to create a larger or smaller sill):
Has no affect on kriging estimate
Changes the kriging variance
Increasing the nugget component:
Acts as a smoothing term during kriging (makes weights more similar)
Increasing the range tends to increase the influence of more distance data points and leads to smoother maps.
The shape of the variogram or correlogram near the origin influences the continuity of the interpolation process (e.g., the gentler the slope, the smoother the interpolation). See Figure 4a, 4b,
Figure 4a
4c,
4b
4d,
4c
4e,
4d
4f,
4e
4g,
4f
4h,
4g
and 4i:
4i
Kriging results from a common data set, based on different variogram models.
4h
In this graphic, Frames A-H use the isotropic neighborhood design shown in Figure 3b. The nested model (Frame F) used two spherical variograms, with a short range = 1000 meters and a long range of 10,000 meters. Nested models are additive. The anisotropic model (Frame I) used the anisotropic neighborhood design shown in Figure 3. The minor axes of the variogram model = 1000 meters, with a major axis = 5000 meters (5:1 anisotropy ratio), rotated to N15E. The color scale is equivalent for all figures. Purple is 5% porosity and red is 13%. All these illustrations were created using the same input data set.
Advantages Of Kriging
Kriging is an exact interpolator (if the control point coincides with a grid node).
Kriging variance:
Relative index of the reliability of estimation in different regions.
Good indicator of data geometry.
Smaller nugget (or sill) gives a smaller kriging variance.
Minimizes the Mean Square Error.
Can use a spatial model to control the interpolation process.
A robust technique (i.e., small changes in kriging parameters equals small changes in the results).
Disadvantages Of Kriging
Kriging tends to produce smooth images of reality (like all interpolation techniques). In doing so, short scale variability is poorly reproduced, while it underestimates extremes (high or low values). It also requires the specification of a spatial covariance model, which may be difficult to infer from sparse data.
Kriging consumes much more computing time than conventional gridding techniques, requiring numerous simultaneous equations to be solved for each grid node estimated. The preliminary processes of generating variograms and
designing search neighborhoods in support of the kriging effort also require much effort. Therefore, kriging probably is not normally performed on a routine basis;
rather it is best used on projects that can justify the need for the highest quality estimate of a structural surface (or other reservoir attribute), and which are supported by plenty of good data.