Chapter Contents |
Previous |
Next |

The KRIGE2D Procedure |

In all the theoretical models considered previously,
the lag distance *h* entered as a scalar value.
This implies that the correlation between the spatial
process at two point pairs
*P _{1}*,

However, real spatial phenomena often show directional effects. Particularly in geologic applications, measurements along a particular direction may be highly correlated, while the perpendicular direction shows little or no correlation. Such processes are called anisotropic. Refer to Journel and Huijbregts (1978, section III.B.4) for more details.

There are two types of
anisotropy. The
simplest type occurs when the same covariance
*form* and scale parameter *c _{0}* is
present in all directions
but the range

This type of anisotropy is called "geometric" and is discussed in the following section.

As you can see from the figure, the
SE -NW curve gets "close" to the sill
at approximately *h*=2, while the
NE -SW curve does so
at *h*=6. The ratio of the shorter to longer
distance is (2/6) = (1/3).
This is the value to use in the RATIO= parameter
in the MODEL statement in PROC KRIGE2D.
Since the longer, or major, axis is in the NE -SW direction,
the ANGLE= parameter in the MODEL statement in PROC KRIGE2D
should be 45^{o} (all angles are measured clockwise from north).

The terminology associated with geometric anisotropy is that of ellipses. To see how this comes about, consider the following hypothetical set of calculations.

Let {} be a geometrically anisotropic process, and assume that there are sufficient data points to calculate an experimental semivariogram at a large number of angle classes }. At each of these angles , the experimental semivariogram is plotted and the effective range is recorded. A diagram, in polar coordinates, of yields an ellipse, with the major axis in the direction of the largest and the minor axis perpendicular. Denote the largest by , the smallest by , and their ratio by

By a rotation, a new set of axes are aligned along the major and minor axis. Then, a rescaling elongates the minor axis so its length equals that of the major axis of the ellipse.

First, the angle of
the major axis of the ellipse (measured clockwise from north)
is transformed to standard
Cartesian orientation or counter-clockwise from the x-axis
(east). Let denote the transformed angle.
The matrix to transform the distance *h* is in terms of
and the ratio *R* and it is given by

For a given point pair *P _{1}*

The transformed interpair distance is then

The original semivariogram, a function of *both*
*h* and , is then transformed to a function
only of *h*':

This single semivariogram is then used for kriging purposes.

The interpretation of the major/minor axis in the case of geometric anisotropy is that the direction of the major axis is the direction in which the spatial process is most highly correlated; the process is least correlated in the perpendicular direction.

In some cases, these directions are known a priori. This can occur in mining applications where the geology of a region is known in advance. In most cases, however, nothing is known about possible anisotropy. Depending on the amount of data available, using four to six directions is usually sufficient to determine the presence of anisotropy and find the approximate major/minor axis directions.

The most convenient way of performing this is to use the NDIR= option in the COMPUTE statement in PROC VARIOGRAM to obtain a separate experimental semivariogram for each direction. After determining the direction of the major axis, use a DIRECTIONS statement on a subsequent run of PROC VARIOGRAM with this direction and its perpendicular direction. For example, if the initial run of PROC VARIOGRAM with NDIR=6 in the COMPUTE statement indicates that is the major axis (has the largest ), then rerun PROC VARIOGRAM with

DIRECTIONS 45,135;

Then, determine the ratio of for
the minor and major axis for the RATIO= parameter
in the COMPUTE statement of
PROC KRIGE2D. This ratio is for
modeling geometric anisotropy.
In the other type of anisotropy,
*zonal* anisotropy, the
RATIO= parameter is set to a large number
for reasons explained in the following section.

Instead, nesting and geometric anisotropy are used together
to approximate zonal anisotropy. For example,
suppose the spatial process has a correlation structure in
the N -S direction described by , a spherical model with
sill at *c _{0}*=6 and range

You can approximate this structure in PROC KRIGE2D by specifying two nested models with large RATIO= values. In particular, the appropriate MODEL statement is

MODEL FORM=(S,S) ANGLE=(0,90) SCALE=(6,3) RANGE=(2,1) RATIO=(1E8,1E8);

The large values of the RATIO= parameter for each nested structure have the effect of an "infinite" range parameter in the direction of the minor axis. Hence, there is no variation in in the E -W direction and no variation in in the N -S direction.

Chapter Contents |
Previous |
Next |
Top |

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.