- Experimental variogram provides information on the spatial autocorrelation of data sets. However, they don't provide information for all possible directions and distances. For this reason, and to ensure that kriging predictions have positive kriging variances, it is necessary to fit a model. In other words, a continuous function or curve to the experimental variogram. Variogram modeling is the foundation for geostatistical analysis. The variogram model is a key component in kriging. Kriging is a powerful interpolation method that can provide predicted values, errors associated with the predictions, and information about the distribution of possible values for each location in the study area. Once you have created the experimental variogram, you can fit a model to the points forming the experimental variogram. The modeling of a variogram is similar to fitting a least squares line in regression analysis. Select a function or curve to serve as your model for fitting to these points. This function is an Exponential model. The goal is to calculate the parameters of the curve to minimize the deviations from the points according to some criterion. There are many different variogram models to choose from. Nugget model: Theoretically, it is zero at H equal to zero, and it is the variance of data, or one for normal scores, at H is greater than zero. Spherical model: it is commonly applied variogram model, which increases in a linear fashion and then curves to the sill. Exponential Model: It is similar to spherical model, that is, linear near the origin and curving toward the sill. It differs from spherical model in it rises more steeply than spherical model and reaches the sill asymptotically. Gaussian Model: H powered to the second in the exponential results this variogram model to have a parabolic shape at short distances. Hole effect model: It is periodic function. The ranges can be set to infinity. Power model: The power model does not reach a sill, and so labels the second parameter "b". Any power model between zero and two may be used to construct a valid power variogram model. When b equals to one, it is the linear variogram model. Both the power and linear variogram models are appropriate if there is long-range correlation or if samples were not collected at a sufficiently large distance to reach the point where pairs of points are uncorrelated. If the experimental variogram does not seem to follow any of the standard structures, it is possible to combine structures to obtain a variogram with the characteristics of more than one of the standard structures. A linear combination of valid variogram structures is also a valid variogram model and is called a nested variogram structure. Each model has its own partial sill, or contribution. here is an example with three models. The nugget model and its contribution in green. A short-range exponential model and its contribution in red. A longer-range spherical model and its contribution in blue. The purple is the total model. An ellipse is used to demonstrate anisotrophic rose diagram. The ellipse has two axes and one angle. Major axis: it is the correlation range with the maximum continuity. Minor axis: it is the correlation range perpendicular to the major axis, with minimum continuity. Azimuth: it is the clockwise rotation angle of major axis from North. The range is zero to 360 degrees. A variogram is anisotropic if it changes in some way with respect to direction. if value of variogram depends not only on length of vector H, but depends also on direction of vector then we have anisotropic variogram. Geometric anisotropy means that the correlation is stronger in one direction than it is in the other directions. The range, but not the sill, of the variogram changes in different directions. Mathematically, if one plots the directional ranges, in two dimensions they would fall on the edge of an ellipse. See right side of figure. Please see the blue line on the diagram, it's length is the correlation range for this direction. Where major and minor axes of ellipse correspond to the largest and shortest ranges of directional variograms. Dip angle is the second angle of rotation, it is rotating the vertical ellipse, and it represents a downward rotation of the major axis from the horizontal plane. The angle is measured in negative degrees down from horizontal. The range is zero to minus 90 degrees. When 3D anisotropy exists, that requires to compute a number of directional variograms in 3D. An ellipsoid is utilized to demonstrate this anisotropy. the magnitude of vector defined from the ellipsoid center to one point at the ellipsoid skin is the correlation range for the 1D variogram at this specific direction. Please see the blue line on the graph, it's length is the correlation range for this direction. The ellipsoid has three axes and two angles. Major axis, minor axis, and azimuth angle have been defined above. Vertical axis: it is the correlation range perpendicular to horizontal plane. Dip has been defined above. Theoretically, geometric anisotrophy means that the correlation is stronger in one direction than it is in the other directions. Mathematically, if one plots the directional ranges, in three-dimensional case they would fall on the surface of an ellipsoid, see figure, where major and minor axes of ellipsoid correspond to the largest and shortest ranges of horizontal variograms. The ellipsoid equation is illustrated here. If value of variogram depends only on length of vector H, that is, Major axis equal to minor axis, and equal to vertical axis, the ellipsoid becomes a sphere. It is an isotrophic variogram. Geostatistical property modeling utilizes variogram model which includes model type, sill, nugget, and ellipsoid parameters as shown above, major axis, minor axis, vertical axis, azimuth angle, and dip angle. If it is a nested variogram model, more than one mathematical model types are provided, each of them has its partial sill or contribution. The total model is a linear combination of two or more variogram models. After the variogram models have been created, what are we going to do? We are very familiar with this diagram. In statistics, when given measured data X and Y, for example, X is porosity, Y is permeability, the model y equal to K X plus B can be created and used to predict the new Y using new X. When predict the new Y from the new X, just use the mathematical model. In geostatistics, when given the measured data with spatial locations, for example, porosity located at X and Y, the model variogram can be created and used to predict the new value when given the new spatial location. Let us take a look at this diagram. The variogram is created based on five data, and used to predicate the value at x six and y six. The key is we are not only interested in data values, we are also interested in data locations. When predict the new datum at the new location, both variogram model and data with locations used for computing variogram are required, in sequential Gaussian simulation, the simulated data are also included. Let us review what we covered in variogram modeling. It covers: six mathematical models. How to use the mathematical model to model experimental data points, eclipse/ellipsoid model brief introduction of applying models in geostatistics, This is the end of the presentation variogram modeling.