I. Moving average
The southwide map for black cherry depicts the range of the species as recorded by FIA. Estimates show the average cubic foot volume per acre for trees 1.0 inches and larger diameter at breast height for specific locations. A single colored "blotch" represents about 181,000 ha (447,000 ac), and an average of data from 1 to 49 1-ac sample plots with the species.

I converted point values to an indicator probability surface with a moving average interpolation procedure, with ArcView/Spatial Analyst. I used a radius of 24 km [15 km]--a size with resolution suitable for multi-county decisions, e.g., multi-agency, federal, and regional planning. The surface was interpolated to a 2.4 km [1.5 mi] grid. I estimated cubic foot averages from sample plots where the species was present within a 24 km [15 mi] radius. In other words, the grain size may be too coarse for local management purposes, but may provide the regional manager with  the "bigger picture."

A National Forest example shows the probability of an attribute occurring in forest land from forest grid cells within a 9.6 km radius  (encompassing up to 13 sample plots).  This illustration includes National Forest boundaries for Arkansas. I used beverage containers because it is a human intrusion commonly associated with problem behavior (teen drinking) and could negatively impact "primitive" or wilderness recreational experiences. Grain size is smaller to accomodate local management needs.

(Tutorial: Moving average interpolation is a relatively quick, intuitive way to obtain a "surface" of values, with all points (plots) included within a specified radius. Choice of the radius will yield different results, with small radii yielding fine-grained details but also include much random noise--these are "diamonds in the rough." Using large radii yield coarse-grained maps with minimal random noise--these are "big picture" results.)

II. Kriging
The map for east Texas depicts land use, as recorded by FIA for the 1992 survey. I converted point values to an indicator probability surface interpolated with kriging, using ArcView/Spatial Analyst with GS+ software. I interpolated and depicted the surface with a 2.4 km grid, and overlain by ounty-level ecological subregion boundaries. The map is in Lambert equal-area azimuthal projection (central meridian -100 degrees, reference latitude 45 degrees).

(Tutorial: Unlike straightforward "averaging" technique listed above, kriging is a weighting scheme that makes an assumption about the point's context among other sample points. Kriging takes advantage of the relationship between a point's correlation with nearby points, and provides estimates that weight a point's value according to its distance from other values. The weights may be exponential (4-16-256), spherical, linear (1-to-1), gaussian, etc. In the linear case,  a point value will correspond directly (1-to-1) with the next adjacent piont value. In the case of land use, linear kriging yielded the highest r-square autocorrelation (>0.65) and lowest residual sums-of-squares. Unlike simple moving average interpolation, changing the radius will usually yield only small differences in resulting patterns. Kriging commonly uses only the nearest points (16 to 20) into account when determining the weighting scheme to be used. Often the radius specified is large enough to accomodate the selection of 16 to 20 points for a sparsely populated sample region.)

III. Miscellaneous examples for National Forest land.

IV. Estimates of Certainty/Conditional Simulation
Interpolation of point data yield results with questions about certainty that the patterns are real. A straightforward appraisal of certainty is to list the number of points used to estimate the value of each grid cell. We have greater certainty in regions with more samples. Regularly-spaced samples are best with multipurpose resource objectives, but compromises must always be made. As a rule of thumb, both kriging and moving average interpolation yield results with similar confidence when samples are regularly-spaced. Greater confidence in regions with sparse, or irregularly-spaced samples are highest with kriging interpolation.

With moving average interpolation, a sparsely-sampled region will have a grid cell represented by only a few samples. Classic (nonspatial) statistics, like t-tests, indicate that confidence in data based on only a few samples is quite low, which is not very comforting to those making high-cost decisions. In these cases, one can say simply that we have more confidence in regions with many samples, and less confidence in regions with few samples. With FIA, for example, our confidence in mapped forest attributes will be low in sparsely forested regions, e.g.,  the Delta region of Mississippi. Confidence in mapped forest attributes will be high in densely forested regions, such as the Boston Mountains of Arkansas.

With kriging interpolation. each grid cell's estimate is dependent on a range of samples weighted by their distance from the grid cell. Calculation of confidence becomes more difficult. So we are left with variable confidence, or certainty, that the patterns depicted are real. Our confidence is greater that with moving average interpolation, as the estimates use the nearest 16 to 20 forest samples--regardless of the sparseness of the forests. Conditional simulation is a way to estimate the variation.in our confidence (remainder is under construction) .

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