1. Spatial Analysis & Geostatistics Methods of Interpolation Linear interpolation using an equation to compute z at any point on a triangle

2. Linear Interpolation

3. Clough-Tocher Interpolation – Using cubic polynomial defined by 12 parameters

4. Inverse Distance Weighted Interpolation – nodal function constants

5. Inverse Distance Weighted Interpolation –gradient method

6. Inverse Distance Weighted Interpolation –cubic method

7. Inverse Distance Weighted Interpolation –cubic method - truncated

8. Natural Neighbor Interpolation

9. Kriging

10. Differences between kriging and nn

11. Kriging is based on the assumption that the parameter being interpolated can be treated as a regionalized variable. A regionalized variable is intermediate between a truly random variable and a completely deterministic variable in that it varies in a continuous manner from one location to the next and therefore points that are near each other have a certain degree of spatial correlation, but points that are widely separated are statistically independent (Davis, 1986). Kriging is a set of linear regression routines which minimize estimation variance from a predefined covariance model. Once the experimental variogram is computed, the next step is to define a model variogram. A model variogram is a simple mathematical function that models the trend in the experimental variogram More formal definition of kriging

12. The first step in ordinary kriging is to construct a variogram from the scatter point set to be interpolated. A variogram consists of two parts: an experimental variogram and a model variogram. Suppose that the value to be interpolated is referred to as f. The experimental variogram is found by calculating the variance (g) of each point in the set with respect to each of the other points and plotting the variances versus distance (h) between the points. Several formulas can be used to compute the variance, but it is typically computed as one half the difference in f squared.

14. Construction of a ccdf Using the Pu Isotopic Ratios. Normal Score Transform of the Pu Isotopic Ratio into Y value. Variogram Modeling of the Normal Score Values Y Using Trial & Error Approach to Fit a Model. Bivariate Normality Test If Normal  Proceed with sGs. If Normal Assumption Violated Select Other Simulation Routines Define Random Path that visits Each Grid Node Once. Specified Number of Neighboring Conditioning Data Including Original Data and Previously Simulated Values. Use SK with the Above Variogram Model to Determine the Parameters of the ccdf of the Y Variable at Location (u). Draw a Simulated Value y(l)(u) from that ccdf and Add this Value to the Data Set. Proceed to the Next Node and Loop Until all Nodes Are Simulated. Backtransformed the Simulated Normal Values {y (l) (u), u  A} into Simulated Values of the Original Variable z. Where l represents realization, u location, and A the study area. One Hundred Realizations Were Conducted. Setting A Different Random Path For Each Realization. Stochastic Simulation Techniques

16. Probability of Exceedance

17. The level of variation present in all scales can be described by a single parameter, the fractal dimension D, defined by Mandelbrat, (1982): where N is the number of steps used to measure a pattern unit length and r is the scale ratio. In practice, we estimated the D values from the following relationships: where h is the sampling interval and  (h) is the spatial structure. By plotting log  (h) versus log(h), the slope of the line is equal to 4 – 2D

18. Elevation501.2 Mean Snow Depth 501.4 Organic C351.7 pH351.7 Bulk density501.8 Soil Moisture Content 501.7 L*351.6 a*351.7 b*351.7 L*101.8 a*101.8 AttributeLag (m) D b*101.8