Geostatistics

[Name of the Author]

Geostatistics

What is Geostatistics?

"Geostatistics: study of phenomena that vary in space and/or time" (Deutsch, 2002). "Geostatistics can be regarded as a collection of numerical techniques that deal with the characterization of spatial attributes, employing primarily random models in a manner similar to the way in which time series analysis characterizes temporal data." (Olea, 1999). "Geostatistics offers a way of describing the spatial continuity of natural phenomena and provides adaptations of classical regression techniques to take advantage of this continuity." (Isaaks and Srivastava, 1989)

Exploratory Analysis of Data

From the main menu click Grid ( Data…, then select the main file of data.

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Our data consist of vertically averaged porosity values, in percent. Porosity values are available from 85 wells distributed throughout the field, which is approximately 20 km in east- west extent and 16 km north-south. The porosities range from 12% to 17%. Here are the data values posted at the well locations:

For analyzing the data we have utilized the software of SURFER. Geostatistical methods are optimal when data are - normally distributed and - stationary (mean and variance do not vary significantly in space). Significant deviations from normality and stationarity can cause problems, so it is always best to begin by looking at a histogram or similar plot to check for normality and a posting of the data values in space to check for significant trends (Olea, 1999). The posting above shows some hint of a SW-NE trend. Looking at the histogram (with a normal density superimposed) and a normal quantile-quantile plot shows that the porosity distribution does not deviate too severely from normality:

Spatial Covariance, Correlation and Semivariance

We have already learned that covariance and correlation are measures of the similarity between two different variables. To extend these to measures of spatial similarity, consider a scatterplot where the data pairs represent measurements of the same variable made some distance apart from each other. The separation distance is usually referred to as "lag", as used in time series analysis. We'll refer to the values plotted on the vertical axis as the lagged variable, although the decision as to which axis represents the lagged values is somewhat arbitrary. Here is a scatterplot of porosity values at wells separated by a nominal lag of 1000 m:

Because of the irregular distribution of wells, we cannot expect to find many pairs of data values separated by exactly 1000 m, if we find any at all. Here we have introduced a "lag tolerance" of 500 m, pooling the data pairs with separation distances between 500 and 1500 m in order to get a reasonable number of pairs for computing statistics. The actual lags for the data pairs shown in the crossplot range from 566 m to 1456 m, with a mean lag of 1129 m.

The three statistics shown on the crossplot are the covariance, correlation, and semivariance between the porosity values on the horizontal axis and the lagged porosity values on the vertical axis. To formalize the definition ...