In Arcgis,the cross-covariance models are estimated for cokriging. 

I would like to confirm that in ArcGIS Cokriging is developed. It is possible to calculate variance and covariance models up to 4 variables.

dowload from file exchange of matlab

https://www.mathworks.com/matlabcentral/fileexchange/29025-ordinary-kriging

this produce variogram during computations. Software will be matlab

Arcgis had developed approximately a complete package for geostatistics and Radial Basis Function (RBF). a complete variogram, cross-variogram is presented for all spatial models. all neighbor parameters are adjustable with a preset properties.  

Whether you are using covariance functions or variograms, for cokriiging it is essential to have a matrix valued function (covariance or variogram) that satisfies the positive definiteness or conditional negative definiteness condition. It is also important to note the distinction between computing an experimental cross covariance ffunction or experimental cross variogram and fitting a matrix valued model to data. In the univariate case a number of models (e.g. spherical, exponential, gaussian) are known  and which are known to satisfy the relevant positive definiteness or conditonal negative definiteness condition.

In the multivariate case (cokriging) the only known model is the "linear coregionalization model".. A cross variogram or cross covariance function model must be related to a specific pair of variograms/covariance functions hence there is no list of "valid" cross variogram models or cross covariance models (other than the LCM). Note also that cross variograms might be symmetric or not, similarly for cross covariance function models. The LCM forces the  cross variograms to be symmetric.

There is also a practical problem, in many applications there may not be data for all variables at all data locations (the so-called undersampled problem).  In fact it is even possible that there are no data locations with data for both variables hence computing an experimental cross variogram or experimental cross covariance function is not possible. Cross variograms do not have to look like variograms nor do cross covariance functions have to look like covariance functions.

You need to look carefully at the actual algorithm in the software to see what they are doing, it is most likely they are using LCM if they claim to do automatic fitting. Even in the univariate case automatic fitting of a variogram model is not perfect.

See the books by M.E. Hohn, H. Wackernagel and the following

1982, Myers,D.E., Matrix Formulation of Cokriging Mathematical Geology 14, 249-257

1988, Myers,D.E.,Multivariate Geostatistics for Environmental Monitoring Sciences de la Terre 27, 411-427

1991, Myers,D.E., Pseudo-Cross Variograms, Positive Definiteness and Cokriging Mathematical Geology 23, 805-816

Finally ArcGis doesn't recognize that interpolation using RBF's is equivalent to universal kriging  (see references to "dual kriging" and also

1992, Myers,D.E.,Cokriging, Radial Basis Functions and the role of Positive Definiteness Computers Math. Applications 24, 139-148

1994, Myers,D.E., Statistical Methods for Interpolation of Spatial Data Inter. J. Applied Science and Computations 1, 283- 318

Thanks for all, 

Farhad Alizadeh, do you know or have the  version of ArcGIS that has this ability, cross variogram.? 

Hello Ahmed,

Version 10 of Arc GIS has the cross variogram option on the Geostatistical module.

Ahmed Assad

You might want to talk to Andy Long in the Mathematics Dept at Northern Kentucky University. Any automatic or semi-automatic algorithm for fitting cross-variograms has to use a "Linear Coregionalization Model', you will find that the gstat package in R has a good explanation (and is open source software). Unlike variograms where there is known list of valid models, i.e. functions satisfying the necessary statistical conditions, there is no list of known cross-variograms. See the book by M.E. Hohn on Multivariate Geostatistics or the book by Hans Wackernagel

on cokriging on arcgis 10.5 it is possible to introduce more than three covariates or a maximum of three !

Excuse me, have you solved the problem yet？

If you use the term "calculate" with respect to variograms, covariance functions, cross variograms or cross covariance functions then either you are confusing experimental with theoretical or you are assuming that you know the multivariate probability density function.

If you do a search on Google for "ArcGIS 10 cokriging" you will find a lot of incorrect information. Don't rely on it