Ordinary Kriging Process in ArcGIS The target is center of a cell. The value of Zi will be computed for the cell based on the center point (like a lattice). Compute distance between cell and the selected sample . Estimate the semivariance for each sample point and for the target point (i.e. the nugget) Find the weights to apply to each sample.

Theoretical semivariogram models for kriging are based on isotropic models, so correction for any anisotropies are necessary to use kriging methodology * Experimentally determine the directions corresponding to minimum and maximum range * Develop a model that describes change in direction as well as distance – or models based on direction * Combine directional models into a single common model that is consistent in all directions – range standardized to 1

Estimate unknown point Zp based on three samples (Z1, Z2 & z3) at known locations.

In Matrix form

Solution Zp = W1*Z1 + W2*Z2 + W3*Z3 Est Variance = W1*SV(h1p) + W2*SV(h2p) + W3*SV(h3p) + L

Types Kriging produces the best linear unbiased estimate of an attribute at an unmeasured site, once the semivariogram has been modeled. Ordinary Kriging: produces interpolation values by assuming a constant but unknown mean value, allowing local influences due to nearby neighboring values. Because the mean is unknown, there are few assumptions. This makes ordinary kriging particularly flexible, but perhaps less powerful than other methods. Used when there is no drift (i.e. trend) in the data. Simple Kriging: produces interpolation values by assuming a constant but known mean value, allowing local influences due to nearby neighboring values. Because the mean is known, it is slightly more powerful than ordinary kriging, but in many situations the selection of a mean value is not obvious.

Types Universal Kriging: produces interpolation values by assuming a trend surface with unknown coefficients, but allowing local influences due to nearby neighboring values. It is possible to overfit the trend surface, which does not leave enough variation in the random errors to properly reflect uncertainty in the model. When used properly, universal kriging is more powerful than ordinary kriging because it explains much of the variation in the data through the nonrandom trend surface. In ArcGIS trend is modeled by a constant, linear, second or third order equation.

Types Addressing trend. Trend can be a simple constant; that is, μ(s) = m for all locations s, and if μ is unknown, then this is the model on which Ordinary Kriging is based. It can also be composed of a linear function of the spatial coordinates themselves; for example, μ(s) = β0 + β1x + β2y + β3x2 + β4y2 + β5xy, where this is a second-order polynomial trend surface and is just linear regression on the spatial x- and y-coordinates. Trends that vary, and where the regression coefficients are unknown, form models for Universal Kriging. Whenever the trend is completely known (i.e., all parameters and covariates known), whether constant or not, it forms the model for Simple Kriging (In ArcGIS you can only make μ(s) constant).

Types Indicator Kriging: uses thresholds to create binary data (0 or 1 values, also called indicator values), and then uses ordinary kriging for this indicator data. Predictions using indicator kriging are interpreted as the probability of exceeding (or, depending on how the binary variables are defined, not exceeding) a threshold. Additional Cutoffs can compensate for the loss of information caused by coding data with indicator functions, but it requires fitting cross-covariances which requires more modeling decisions and parameter estimation. Indicator kriging is not recommended for data having a trend.  

Types Probability Kriging is considered an improvement over indicator kriging by using the original, continuous data for ordinary cokriging of the indicator data. Probability kriging uses more information than indicator kriging so it can be more powerful, but it requires fitting cross-covariances which involves more modeling decisions and parameter estimation. Probability kriging is not recommended for data having a trend.   Disjunctive Kriging is a nonlinear method that is more general than ordinary kriging and indicator kriging. Disjunctive kriging tries to do more than ordinary kriging and indicator kriging by considering functions of the data, rather than using only the data. As usual, to get greater rewards requires stronger assumptions. Disjunctive kriging assumes all data pairs come from a bivariate normal distribution. This assumption can be examined in the Geostatistical Wizard.

Types Cokriging uses information on several variable types. The main variable of interest is Z1, and both autocorrelation for Z1 and cross-correlations between Z1 and all other variable types are used to make better predictions. It is appealing to use information from other variables to help make predictions, but it comes at a price. Cokriging requires much more estimation, which includes estimating the autocorrelation for each variable as well as all cross-correlations. Theoretically, you can do no worse than ordinary kriging because if there is no cross-correlation, you can fall back on just autocorrelation for Z1. But, each time you estimate unknown autocorrelation parameters, you introduce more variability, so the gains in precision of the predictions may not be worth the extra effort.

Co-Kriging If y and x are highly correlated we can use the information about x to improve the prediction of y. If the primary variable of interest is y, the x values that were sampled can be used to improve the y predictions at any point in the region We first need a formal method for estimating and modeling the correlation Do this by extension of the covariogram and variogram for single variables to cross-covariogram and cross-variogram

Co-Kriging ( for two variables) A random process relating to the primary variable: A random process relating to the secondary variable If both processes are assumed stationary, then the cross-covariogram is defined as: and the cross-variogram is defined as:

Sample estimator for cross-variogram is given by: This sample cross-variogram is then fit with a smooth model.

Prediction of y is performed through a weighted average of nearby y and x values Where i = 1, …, n measurements of Y and j= 1,..,m measurements of X The weights are a function of distance modified by the variogram and cross-variogram.