- Let's study how to apply, how to use, variogram in Kriging. Let's make a summary. Taking a variogram model, for this example we have an isotropic variogram model in 2D. Semi-major axis equal to 25, it is the correlation range with the maximum continuity. Semi-minor axis equal to 15. It is the correlation range perpendicular to the major axis which is minimum continuity. Azimuth equal to 30 degrees, it is the clockwise rotation angle of major axis from node. Spherical model with C zero equal to one, A zero will be defined, wherever use it, U is the location. We are figure out the value for that. There are three data in its neighbors, U one, U two, and U three. Since this is a simple Kriging equation for three known data, for details, refer to the lecture for derivation of this equation. W's are weights, which we already obtained by solving this equations. C's on left side, on right side, are covariances, which will be obtained from variogram model. These three covariances are data variances which can be computed from data side. Left side yields no more lines of variogram model, so they are one. Assume covariances are symmetric, so covariance U two, U one, is equal to covariance U one, U two. Covariance U three, U one is equal to covariance U one, U three. Covariance U three, U two, is equal to covariance U two, U three. So, we only need to compute six covariances. They are covariance U one, U two, covariance U one, U three, covariance U two, U three, covariance U, U one, covariance U, U two, covariance U, U three. Take U and U one as example to compute variogram. The black arrow is the vector from U to U one, OK. To take the two variogram ellipse model. Figure out the correlation integer A zero for this vector, the length of this blue line is A zero. For example, it is 18. Use it to update spherical model A zero equal to 18. Given an edge equal to the length of vector U and U one, the length vector, we get gamma value. For example, it is zero point eight. Variogram is defined as the first equation. Covariance is defined as the second equation. Expand as a variogram equation, and then plug the covariance equation in, we get the third equation. This is the relationship between variogram and the covalence. It normalize this C zero equal to one. So C equal to one minus gamma, that is one minus zero point eight, equal to zero point two. Replace, say U, U one, is zero point two. Follow this procedure we can get the other five covariances. So, this matrix equation three ways can be obtained. Let's make a summary. Variogram model can be created from the data side. Covariances can be converted from variograms. Weights in Kriging Equation can be obtained from covariances. Weights are used to get Kriging estimator and the estimation variance.