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  How to Compute a Variogram?

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- This is to address how to compute the imperical variogram starting with regularly spaced, 2D data set, then proceed to handle irregularly spaced data set. A vector is defined by a certain distance in a particular direction. For example,and h are all vectors. Addition and subtraction operations apply to vectors. h is the difference of two location vectors: u+h and u. It will be used in variogram computation. The magnitude of h is a scalar. It is equal to the length of h. It is called lag distance. The direction of h is defined by azimuth angle. It is the degree clockwise from North. The range is zero to 360 degrees. H-Scatterplot shows all possible pairs of data values whose locations are separated by a vector. On left side, there are regularly spaced data. Let's pair the data using the vector h1. We get 16 pairs. See 16 red arrows. Put 16 heads into the column, Head. See: 0.6, 1.1, 0.7. And put 16 pairs into the column, Tail. See: 0.2, 0.7, 0.3, et cetera. Let's pull out heads versus tails. H-scatterplot has been completed. Variogram 2 gammafor lag h is defined as the average squared difference of values separated approximately by h. 2 gammais variogram. Gammais semivariogram Nis the number of pairs for lag h. ziis the ith Z value at u. zis the ith Z value at u+h. We have already paired data in H-scatterplot. Head data are in column B. Tail data are in column C. The scalar difference of B and C are in column D. The average of D is equal to 0.16. It is the variogram value for lag h1. This is to compute the second variogram value for h2. First, let's pair the data for h2. You see nine red arrows paired as: 1.1, 0.3; 1.6, 0.8; 1.2, 0.4; et cetera. The average squared difference of paired values are computed and plotted in the right-side graph. Please see the point in the red circle. When data are irregularly spaced, the azimuth tolerance, lag tolerance, and bandwidth are recurred to obtain the second datum in the pair. The bandwidth is used to limit the maximum derivation. There are 10 irregularly spaced porosity data. Data in the green outlined area, 0.1, 0.2, can be paired with the first data, 0.21, for variogram computation. This is the lag considerations. Lag separation distance should coincide with data spacing. Variogram is only valid for one half of the field size. This is the under of theof variogram computation.