1. Deterministic Solutions Geostatistical Solutions
Spatial Structure
The relationship between a value measured at a point in one place, versus a value from another point measured a certain distance away.
Describing spatial structure is useful for:
Indicating intensity of pattern and the scale at which that pattern is exposed
Interpolating to predict values at unmeasured points across the domain (e.g. kriging)
Assessing independence of variables before applying parametric tests of significance
Spatial Structure
Deterministic Solutions
Geostatistical Solutions

2. Geostatistical Techniques
Kriging
Kriging has
Two tasks:
Quantify Spatial Structure, and
Predict an unknown value
Recall:
Spatial Data may be:
Continuous; any real number ( , )
Integer; interval (-2, -1 , 0 , 1 ….)
Ordinal; an “ordered categorical code” ( worst, medium, best)
Categorical; unordered code (forest, urban)
Binary; nominal, (0 or 1)
IF:
Data are spatially continuous, AND continuous in value with a normal distribution, AND you know the autocorrelation of the distribution--
Then:
Kriging is an OPTIMAL predictor
Different forms of Kriging can accommodate all types of data (see above) and is an “approximate” method that works well in practice
However:

3. Recall our earlier example:
Kriging Models
Most fundamentally; kriging depends upon mathematical and statistical models, of which the statistical model of probability distinguishes kriging methods from the deterministic methods of spatial interpolation.
Correlation (autocorrelation) as a function of distance is a defining feature of geostatistics
Recall our earlier example:
is the variable of interest at location (s)
is the mean, and
Is spatial dependent of an intrinsic stationary process

4. Anistotropy Stationarity
Assumptions about :
Expected to be 0 on average and the autocorrelation between
and Does not depend upon actual location of s
All of the random errors are second-order stationarity – zero mean and the covariance between any two random errors depends only on the distance and direction that separates them, not their exact locations
Anistotropy
Stationarity
spatial structure of the variable is consistent in all directions.
Spatial structure of the variable is consistent over the entire domain of the dataset.

5. Ordinary Kriging unknown Assumes that is an unknown constants
One spatial dimension (x-Coordinate) representative of perhaps elevation.
There is no way to decide , based upon the data alone, whether the observed pattern is the result of autocorrelation alone or a trend ( changing with s)

6. Simple Kriging Known Assumes that us an known constants
Because you assume to know then you also know
This assumption (knowing )is often unrealistic, unless working with physical based models.

7. Universal Kriging Deterministic function
Assumes that is some deterministic function
The mean of all is 0
Universal kriging is basically a polynomial regression with the spatial coordinates as the explanatory variables which (instead of being modeled as independent), they are modeled to be autocorrelated.

8. Indicator Kriging Binary variable Assumes that is unknown constant
is assumed to be autocorrelated
Interpolations will be between 0 and 1, and can be interpreted as probabilities of the variable being a 1 or of being in the class indicated by a if binary class is a “threshold” – imterpolation shows the probabilities of exceeding the threshold.

9. cokriging Assumes that and are unknown constants
The variable of interest is , and both autocorrelation for and cross-correlations between are used.

10. Dissimilarity Similarity
Anatomy of a typical semivariogram
Anatomy of a typical covariance function
Where:
var is the variance
Where:
cov is the covariance
Then:

11. Variogram Modeling Suggestions
Check for enough number of pairs at each lag distance (from 30 to 50).
Removal of outliers
Truncate at half the maximum lag distance to ensure enough pairs
Use a larger lag tolerance to get more pairs and a smoother variogram
Start with an omnidirectional variogram before trying directional variograms
Use other variogram measures to take into account lag means and variances
(e.g., inverted covariance, correlogram, or relative variograms)
Use transforms of the data for skewed distributions (e.g. logarithmic transforms).