Kriging with Cov. or Var. function?

[Chris Skelly]

I have always been a little unsure as to why some researchers
insist on using a covariance function for kriging estimation
rather than the variogram.  Having picked up Issaks and
Srivastava (1989), I am still confused...if YOU are less
confused than I please read on...I could use the help...

Lam (1983) and others suggest that one should use a variogram
over covariogram because,

	Within the system of simple Kriging, two different ass-
	umptions may...be distinguished and these relate to two 
        approaches for estimating the autocovariance matrix. In
	the first approach, the covariogram function...is used.
	It is expected that the covariogram is a decreasing function
	of distance; however, in actual applications the 
	covariograms will diverge from this theoretical behavior.
	This approach to simple Kriging is based on the stationarity
	assumption...However, natural phenomena with this stationarity
	charateristic seldom exist. Hence, interpolation may be based
	upon the...quasi-stationarity assumption...Instead of the 
	covariogram, the variogram...is now used (Lam, 1983, 132).

So, when reading through my newly purchased copy of Issaks and Srivastava
(1989), It seems to have a major inconsistancy with respect to the
use of either the variogram or covariogram for estimation purposes.  They
develop most of the theory on spatial structure using the variogram not
the covariogram.  They even go as far as to make the covariogram look like
a variogram for consistancy and then they NEVER use the variogram for
estimation purposes.  A couple citations from their book, which I hope
are fair extractions of what they are trying to get across to their
readers are as follows,

	...we suggested three methods for describing spatial continuity:
	the correlation function, the covariance function, and the var-
	iogram.  As descriptive tools, any one of these three serves as
	well as the other.  For the purpose of estimation, however, these 
	three functions are not equivalent...we will see that the classical
	theory of estimation places most relevance on the covariance 
	function.  For this reason we will use the covariance function
	to describe our exhaustive data set (Issaks and Srivastava, 1989,93).

somewhat of an explanation is later attempted as

	The common practice in geostatistics is to calculate modeled var-
	iogram values then, for reasons of computational efficiencey, to 
	subtract them from some constant...The net result is that although
	geostatisticiancs eventually resort to solving the ordinary
	kriging equations in terms of covariances, most of the intitial 
	calculations are done in terms of variograms (Issaks and 
	Srivastava, 1989, 290).

Hence my confusion...do you use covariance because it is theoretically
more satisfying (contrary to what Lam and others have suggested...if I
read them correctly) or do you use covariance because it computationally
more effecient?  Issaks and Srivastava seem to argue it both ways!

Is this worth a discussion out there?  Or is this old hat, trivial stuff
to which someone out there has THE answer?

Lam, N.S-N. 1983. "Spatial Interpolation Methods:  A Review",
The American Cartographer, vol 10, no 2, pp 129-149.

Issaks, E.H. and R.M. Srivastava. 1989. AN INTRODUCTION TO APPLIED
GEOSTATISTICS, Oxford University Press, New York, pp 561.


cheers,
chris			<-- @mqatmos.cic.mq.edu.au