12.747 Lecture 5: Section 2:

Objective Mapping and Kriging

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5.2 Moving Averages

Sometimes it is possible to put your data onto a regular grid through various averaging schemes. One of the most common is the moving average. These averaging schemes are an outgrowth of a school of thought largely credited to mining operations in France and South Africa and are precursors to kriging, the main topic of this segment. Each averaging scheme applies some variant of the following mathematical equation:

where the grid estimate (Y_ij) is the sum of a weighting scheme (W_k) times the actual observations (Y_k). The nature of W_k varies as we have seen in the first part of this segment (n-nearest neighbors, inverse of the distance, inverse of the square of the distance, etc.).

5.2.1 Block Means

The first and simplest of these averaging techniques is the block mean. This technique involves dividing your field area (containing somewhat randomly located samples) into equal area/volume "blocks". Consider a two-dimensional field divided into nine sub-areas, or blocks, of equal area.

An estimator for the center of this "design" is then given by equation 5.2.2 and each sub-area can be estimated by making it the center of its own 3-by-3 block.

Here the are the weights applied to the block means. These weights are determined by a number of methods, some of which are outlined in Section 5.1.3 or from field data that allows the inversion of the system of equations in equation 5.2.2.

One drawback to this approach is that although the mean of the block is relatively independent of the size of the block (once the block is above a certain, data dependent, size), the variance of the block estimate tends to decreases with increasing block size. It is quite possible that the variance of the block estimate may be too large to make the estimate of much use in your investigation.

5.2.2 Moving Average Design

To produce estimates with lower variance and increase the reliability of the estimate we can use a variation of the above block mean called a moving average. This moving average is a variation upon the design of the block averaging. Once the study area has been divided into blocks, these same blocks can be re-divided to give you more 's in equation 5.2.2. Consider Fig. 5.2.2:

It is left to the readers imagination as to how other geometries could be used to divide and re-divide the study area into blocks for estimating the blue cross. Keep in mind that only one blue cross was shown for demonstration purposes, but that each block has its own blue cross that is estimated in a fashion similar to the one we just discussed.

The averages from the blocks, the 's, can also be weighted, or windowed. The results, in histogram form, are shown in Fig. 5.2.3.

Now in equation 5.2.2 the 's are not equal over the entire block, but rather are a function of how distant the block centroid is from the point to be estimated.

5.2.3 Trend Surface Analysis

There are two aspects of trend surface analysis that are important for gridding your data and kriging (which may sound redundant, but there are subtle differences between the two). In the first case, by fitting a trend surface to your data you can use the fit function to re-sample your data field on a regular grid. This reflects an interest in the trend surface itself. In the second case, you may want to remove a trend surface from your data before proceeding with the kriging operation.

Sometimes it is desirable or just convenient to have a function that represents your data in terms of the coordinate system of your study area (e.g. in terms of the longitude and latitude). In these cases it is possible to make your study variable a function of your coordinate system. You are, in fact, fitting a trend surface to your data in terms of the coordinates that you use to locate your samples. It can be a trend surface of any order and rank; meaning that the trend can be 1^st order (a straight line, a flat plane) or 2^nd order (quadratic curve, surface or hyper-surface). The order refers to the highest power any independent variable is raised to, the rank refers to the dimensionality. You can set up the equations and solved them with either normal equations or the design matrix, in certain advanced cases you may need to apply the non-linear fitting technique of Levenberg-Marquardt. Or, in most cases, you can use a handy little m-file Bill Jenkins wrote up called surfit.m. It uses the repetitious nature of higher and higher order polynomials and the SVD solution to the normal equations to fit surfaces to your data of the form:

Most grid generation schemes work best when , in order to accomplish this it is important to remove any trend surface from your data first. At the very least you should remove a first order, n-dimensional surface (n refers to the rank of your coordinate system) from your data before proceeding to run you grid generation routine. You can always add it back in to your grid estimation points because you now have an analytical equation that relates your property to the coordinate system of your study area. Higher order, n-dimensional, surfaces can also be fitted. The higher order you go, the better your fit will be regardless of what you use as a goodness-of-fit parameter. But keep in mind the "better fit" may not be statistically significant and you can use ANOVA to test for this (see also Davis, 1986, pp 419-425).

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