Re: What is kriging and co-kriging?
You can have a look here for an explanation: static-modeling-f39/kriging-and-the-nugget-t205.htmlFor a more scientific explanation see below:
Any unsampled value, a porosity for example, can be estimated by generalized regression from surrounding measurements of the same value once the statistical relationship between the unknown being estimated and the available sample values has been defined. This is exactly what the correlogram (or variogram) provides: a prior model of the statistical similarity between data values. This generalized regression can also include nearby measurements of some different variable, like seismic travel time, or a facies code. When using such secondary information, one also needs a prior model of the cross-correlogram, which provides information on the statistical similarity between different variables at different locations. Kriging only uses a correlogram while co-kriging takes a secondary variable into acount using the cross-correlogram.
They generalize traditional regression as applied in well log analysis in two senses:
A) The data used (the “independent variables” in traditional statistical jargon) need not be independent one from another. This allows redundancy to be taken into account, an important factor when using several related well logs simultaneously.
B) The sample correlation is modeled before being applied to the regression analysis. This allows for filtering certain aspects (frequencies) of the sample correlation. For example, white noise and more generally the high frequency components of the data can be filtered out, leaving an interpolated map that reflects only the large scale trends of these data’s. Conversely, a particular trend known from indirect information can be built into the map in such a way that it honors as closely as possible the available data.
However, multivariate regression, or “cokriging” as a geostatistician would call it, is usually not a convenient framework for the integration of too largely different types of data such as qualitative geological information, which is usually only indicative in nature, and direct laboratory measurements. At a particular location where one does not yet have a porosity measurement, a consideration of the litho-facies information might provide a reasonable interval within which the unknown value should fall. If we are certain that we are in a particular type of sandstone, we might know that the porosity must be somewhere in the interval from 15% to 25%. If, in addition, we also have enough core plug measurements within that particular sandstone to build a histogram, we could use that histogram to provide a probability distribution that might, for example, tell us that the unknown porosity is more likely to be on the low end of our 15% to 25%
range than on the high end.
The indicator framework of geostatistics and the soft kriging algorithm allow an updating of such prior distribution using nearby data that may be either “soft” or “hard”. To continue with our previous example, we could locally update the previous probability distribution obtained from lithofacies considerations with more specific local information. This could be soft information, such as the fact that the location in question is in the upper half of a fining-upwards sequence, or hard information, such as the fact that a full core measurement taken in a well only 50 feet away had a porosity of 19,6%. The result of this updating is a “posterior” probability distribution that provides the probability for the unsampled value to belong within any given class of values, say, between 15% and 20% porosity. From such a distribution, any “optimal” estimate for the unsampled value can be derived once an optimality criterion has been specified. The optimality criterion preferred by most statisticians is the least square error criterion; for this definition of optimality, the mean of the posterior probability distribution provides the best estimate. This is why in kriging outside any data control (defined by the range of the variogram) the local average (ordinary kriging) or the global average (simple kriging) is used.