The spline, polygonal, and thiessens statistical interpolation methods available in ArcView 3.1 were not investigated further because of their overall poor performance as determined by Flores-Garnica (2001). 

Standard statistical interpolation methods can be classified as Global or Local. Inverse Distance Weighted (IDW) interpolation is a local method, meaning that the unknown value, u(x), is determined based upon points in a designated local neighborhood; (x) is the unknown point, (xk) is the known point, d is the distance from (x) to (xk), and p is the power parameter usually set equal to 2.  The interpolated point is calculated using the following equation (Table 3):

In the ArcGIS Geostatistical Analyst Wizard it is possible to set the minimum and maximum number of neighbors used in the interpolation.  For IDW the assumption is made that the closer neighbors are a better estimator of the unknown value than the farthest neighbors, and therefore are weighted according to this inverse distance relationship (ESRI 2003).

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Kriging: Geostatistical interpolation methods (Kriging) have their origins in mining exploration for the prediction of ore trends (Armstrong 1998).  Kriging techniques create surfaces that incorporate the statistical properties (e.g. Ore concentrations) of the measured data (ESRI 2001).  There are two categories of Kriging: linear and non-linear.  Linear methods include Simple (SK), Ordinary (OK) and Universal (UK) Kriging; Non-linear methods include Indicator (IK), Probablility (PK) and Disjunctive (DK) Kriging (Olea 1999).  In addition, Ordinary Kriging can be performed on point variables, with Point Kriging usually regarded as synonymous with Ordinary Kriging; or across a regionalized variable as in Block Kriging (Armstrong 1998). CoKriging is a multivariate adaptation of Ordinary Kriging (Olea 1999), and is applicable when the parameter of interest has a strong co-linear relationship with another parameter. Since previous applications of geostatistics to forest fuel estimation have demonstrated strong linear and co-linear relationships (Flores-Garnica 2001), the non-linear methods will not be investigated here.

Simple Kriging requires that the mean of the data must be known, whereas Ordinary Kriging substitutes a local mean (the actual mean is an unknown constant), and Universal Kriging eliminates assumptions regarding the mean altogether (Olea 1999).  The Ordinary Kriging equation is given in Table 4.  For further details regarding the method refer to Armstrong (1998), Olea (1999), and Schabenburger & Gotway (2005). 

For Kriging to be applied the data must meet three criteria (Armstrong 1998):

1.    A normal data distribution

2.    The sample data represent a random function Z(x), where x is the spatial location

3.    The data are second order stationary (Intrinsic Stationarity)

    A normal (Gaussian) or near normal distribution for the data has been demonstrated in the preliminary statistical analysis above. A random function in geostatistics is the realization that the data set represents only an incomplete set of observations that can be used to predict the unknown data (Schabenburger & Gotway 2005).  Stationarity requires that four statistical moments (i.e.  mean, variance or covariance, skewness, and kurtosis) must be constant.   Intrinsic stationarity requires that only the first two moments (the mean and covariance) must be constant, since it is difficult to prove that skewness (symmetry of the distribution) and kurtosis (flatness or “peakedness” of the distribution curve) are constant (Armstrong 1998; Olea 1999; Schabenburger & Gotway 2005). 

Regardless of the method applied, the Kriging process is divided into two tasks (Armstrong 1998):

1.Quantifying the spatial structure of the data (the variogram)

2.Producing a prediction surface (a matrix solution)

To quantifying the spatial structure of the data a variogram is calculated using the equation from Table 5. The resulting values are plotted as the Experimental Variogram, and the user fits a spatial-dependence model to the data.  Formulas for the most common variogram structures are given in Table 6 (Armstrong 1998).  Kriging then uses the fitted variogram model, values derived from the variogram (sill, range, and nugget; Table 5 & Figure 6), and the measured values of the surrounding data points to predict the unknown value (ESRI 2001).

 

Figure 6.  Theoretical Variogram.  The Sill (h) = 2; Range = distance a; The Nugget Effect occurs when the y-intercept of the variogram is not equal to zero.  This example shows no Nugget Effect. (Reproduced from Armstrong 1998).

    An important consideration when determining the best fit variogram equation is the number of lags (or bins) and the lag distance (the minimum value of h) used in the calculation of experimental variogram equation (Table 6).  The lag size is dependant upon the minimum and maximum distances between sample points within the field area.  For Mesa del Huracan the minimum sampling distance is 40 meters and the maximum is 1000 meters.  Both should be tested along with intermediate values to determine the appropriate lag size.  However, as a rule of thumb the optimal lag distance times the number of lags (bins) should be less than one half the maximum sampling distance (ESRI 2003).  For the study area this value should be less than 500m.  Based upon this estimation, the minimum lag distance corresponds to the minimum sampling distance: 40m times 12 lags = 480m, which is less than 500m.  Flores-Garnica (2001) used the minimum 40m for the lag, but had not explained how that number was determined.  The optimal number of neighbors sampled was chosen at 25 following the model refinements of Flores-Garnica and Omi (2003).  The Geostatistical Analyst Wizard default lag distance was consistently around 660m using 12 lags, with a default number of 5 neighbors, demonstrating that the default mode cannot be relied upon exclusively when applying Kriging methods.

    The ArcGIS 9.3 Geostatistical Analyst Wizard includes all of the Kriging methods mentioned in this section, both linear and non-linear (ESRI 2001).  A step-by-step example of the Ordinary Kriging Process is presented in Appendix A.

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