KRIGING Option of GRID Command

WARNING:  A user should not attempt to run the kriging option without some
	understanding of the theory of geostatistics.  The discussion below 
	is intended to be a user's guide to the prompts and parameters.  It 
	is not intended to be an introduction to geostatistical theory.  For
	such an introduction, read any of the materials listed in the biblio-
	graphy.

The options available if user chooses kriging are summarized as follows:
(variance map option)* (ordinary/universal kriging)* (nugget value)*
(number of variograms)* (variogram types)* (variogram parameters)*
(maximum distance)* (maximum number of points)* (re-enter option)*

(variance map option)* is answered "Y" if a map of kriging variances
     is wanted.  If the response is "Y", user is then prompted for the
     new map name.  A new cell map is then created, of the same size as
     the kriged cell map, with each cell value assigned the value of the
     kriging variance for that cell. "N" is the default for this option.

(ordinary/universal kriging)* allows user to specify whether ordinary
     or universal kriging is wanted.  If universal kriging is selected,
     the user is then prompted for exponents of the drift terms. See
     the section entitled "How to Find and Enter Drift Term Exponents" ,
     below.  There is no default for this option.

(maximum distance)* defines the search radius for data points to be used
     in kriging a cell.  If a data point is farther from the center of
     the cell than this distance,it is not used in kriging to that cell.
     If no points are found within the search radius, cell is assigned a
     default value of zero and the cell of the variance map, if any, is
     assigned a value of -1.  This parameter is analogous to the 'roving
     window' parameter of the WEIGHT and QUAD options.

(maximum number of points)* is the maximum number of points to be used
     to be used in kriging to any cell.  If the number of points found
     within the search radius exceeds the maximum number of points, the
     closest points are used and the rest ignored.
        The processing speed for kriging each cell varies approximately
     in inverse proportion to the square of the number of points.  In
     other words, kriging using the 20 nearest points takes about four
     times as long as kriging using the 10 nearest points.
        The largest number that can be entered in response to this
     parameter is '50'.  Adequate accuracy can usually be obtained with
     a value of about 11 or 12.

(nugget value)* allows user to specify the nugget value or random var-
     iation observed in the data.  See the bibliography.

(number of variograms)* the nugget value is considered to be the basic
     variogram.  User is then prompted for "other" variograms in the
     model.  Between zero and five "other" variograms may be specified.
     However, if user has entered a nugget value of 0, there must be
     at least one other variogram.
          Normally, a user would have only one variogram.  In case of
     anisotropy or an exceptionally complicated distribution, several
     superimposed variograms can be used.

(variogram type)*  the choices are spherical,exponential,linear,gaussian
     and cubic.  Usually a variogram is spherical or linear.  For an
     explanation of these model types, see Journel and Huijbregts, pp.
     163 - 171.

(variogram parameters)*  for a linear model, you must enter a sill
     and a slope.  For all other models, you must enter a sill and
     a range of influence. See the section entitled "How to Determine
     Variogram Parameters".

(anisotropy)* allows user to specify that a variogram is isotropic
     or anisotropic.  If "anisotropic" is selected, anisotropy factors
     must then be entered.  See the section entitled "How to Enter
     Anisotropy Factors".

(re-enter option)* after all variogram parameters have been entered,
     parameters are displayed on screen and user is asked if he wants
     to re-enter.  If the response is 'Y', all prompts are repeated
     beginning with 'Do you want ordinary or universal kriging?'.
     The default for this option is 'N'.
Limitations of Kriging

     1) On a 16-bit system, no more than 1,000 data points may be
        kriged.  On a 32-bit system, no more than 15,000 points may
        be kriged.
     2) Data set must not have any duplicate points.
     3) At present, kriging cannot be run in batch mode.


HOW TO FIND AND ENTER DRIFT TERM EXPONENTS
__________________________________________

When universal kriging is selected, the user must decide on the
kind of polynomial in x and y which best represents the trend:

         A  +  B*(x**i)*(y**j)  +  C*(x**k)*(y**l)  +  . . .

The individual terms of this polynomial are known as drift terms.
The exponents (i,j,k,l,etc.) are a measure of the kind of trend in
a given area: flat(exponent=0), linear(exponent=1), parabolic
(exponent=2), cubic(exponent=3).  Larger exponents are possible
but are seldom used.  This program does not permit negative or
fractional drift term exponents.  A maximum of ten drift terms
are permitted.  There is no need to know the values of the coef-
ficients A,B,C, etc.; the program will estimate these according
to a least squares fit.

The user may know enough about a region to determine the kind of
trend without analysis.  For example, in the Great Plains of eastern
Colorado, where the terrain slopes upward to the west, the drift
in elevation might be represented as linear in the x-direction and
flat in the y-direction.

Directional variograms may also be used to determine a drift.  For
more information about drift terms and universal kriging, see the
bibliography.

EXAMPLE.  Suppose that you have decided that the polynomial representing
the drift in a given area is of the form:

       A  +  B*(x)  +  C*(y)  +  D*(x**2)*(y)

This expression is a polynomial consisting of four terms, or monomials .
The first term is a constant; thus,its x- and y- exponents are both 0.
The second term has an x-exponent of 1 and a y-exponent of 0.  The
third term has an x-exponent of 0 and a y-exponent of 1.  The fourth
term has an x-exponent of 2 and a y-exponent of 1.  Thus, you would
make the following entries in response to the computer prompts:


ENTER DRIFT TERM EXPONENTS

     ENTER NUMBER OF MONOMIALS (MAX = 10):  4
     REMEMBER THAT THE X- AND Y- EXPONENTS OF THE FIRST (CONSTANT) TERM ARE
     BOTH ENTERED AS ZEROES

     ENTER X-EXPONENT OF TERM 1: 0
     ENTER Y-EXPONENT OF TERM 1: 0

     ENTER X-EXPONENT OF TERM 2: 1
     ENTER Y-EXPONENT OF TERM 2: 0

     ENTER X-EXPONENT OF TERM 3: 0
     ENTER Y-EXPONENT OF TERM 3: 1

     ENTER X-EXPONENT OF TERM 4: 2
     ENTER Y-EXPONENT OF TERM 4: 1


HOW TO DETERMINE VARIOGRAM PARAMETERS

Linear Variogram Model

STEP ONE:    Use the VARIOGRAM command to get a plot of the variogram.
STEP TWO:    Determin the range of influence.  For linear variograms,
             this parameter is arbitrary, but should not be larger than
             the maximum distance parameter you entered above.
STEP THREE:  Draw a vertical line through the range of influence.  The
             point on the graph where this line intersects the variogram
             is the sill.
STEP FOUR:   The slope is the sill divided by the range of influence.
             Alternatively, pick any convenient part of the variogram.
             The "rise" divided by the "run" on the variogram is the slope.


All Other Models (spherical,exponential,gaussian,cubic)

STEP ONE:    Use the VARIOGRAM command to get a plot of the variogram.
STEP TWO:    Determine the point at which the graph intersects the
             vertical axis.  This is the nugget value.
STEP THREE:  Determine the height at which the graph levels off.
             This is the "sill".  (If the graph doesn't seem to
             level off, you might consider using a linear model.)
STEP FOUR:   For spherical and cubic models, the range of influence is
             approximately the projection on the horizontal axis of the 
             point where the graph levels off.  For exponential and
             gaussian models, it is the projection on the horizontal
             axis of the point where the graph has reached about two
             thirds of the sill.  For more exact computation of the
             range of influence, see Journel and Huijbregts, p. 164.


HOW TO FIND AND ENTER ANISOTROPY ANGLES

An "isotropic" variogram is one which is the same in all directions;
an "anisotropic" variogram is one in which the variation is different
in different directions.

If you tell the program that the variogram is anisotropic, the program
will respond with:

IS ANISOTROPY OF VARIOGRAM NUMBER 1 (1) GEOMETRIC OR (2) ZONAL?
          ANSWER 1 OR 2:

If you respond "1" (Geometric anisotropy) you will be asked for an
anisotropy angle and an anisotropy factor.  An explanation of these
terms can be found in Journel and Huijbregts, pp. 175-183.

The less experienced user will probably prefer to respond "2" (zonal
anisotropy).  To prepare for this question, you will have examined
several directional variograms and determined that the model exhibits
a markedly higher sill in one direction than in others.  This direction
is known as the "anisotropy angle"  and is measured in degrees, starting
from north at 0.0 degrees and increasing clockwise.  Thus, east would be
90.0 degrees and southwest, 225.0 degrees.  The anisotropy angle is
entered as either a positive or negative number.
 

The program will ask:

ENTER ANISOTROPY ANGLE (DEC DEG, NORTH=0):

And you will enter your pre-determined anisotropy angle.




BIBLIOGRAPHY

Clark, Isobel, 1979.  Practical Geostatistics.  Applied Science
               Publishers, London.  150 pp.
Journel, A.G., and Huijbregts, Ch.J., 1978.  Mining Geostatistics.
               Academic Press, New York, 1978.  600 pp.
Knudsen, H.P., and Kim, Y.C.  A Short Course on Geostatistical Ore
               Reserve Estimation.  College of Mines, University of
               Arizona, Tucson, Arizona.  202 pp.
