Disregarding for a moment problems with delineating the precise location of mounds and the propagation of the margin of error, is there some way we can determine whether TM distances (or any particular distances) occur across a site more commonly than we should expect by chance?  This is, after all, the basis for deriving the TM at the Toltec Mounds site, for the 48 m module Pluckhahn (2004) tentatively suggests for Kolomoki, and for the 1.666 and 86.63 m units Clark (2004) proposes for numerous Archaic sites.  Multiples of these particular distances seem to come up quite frequently when researchers look for them, and the distances are therefore interpreted as significant.  Deriving a statistical background is necessary for any test of significance, but can be quite difficult in the case of spatial entities.  Kvamme (1999:169-170) offers several ways GIS can be employed to test for statistical significance in spatial contexts even where populations may be nonparametric and overall statistical parameters are difficult to determine.  It is possible to derive the statistical parameters for the TM at the Toltec Mounds site by using GIS to measure all possible occurrences of this distance, to and from the target points deemed significant by Sherrod and Rolingson.

            Idrisi version 132.2 was used for all GIS operations in this analysis.  Mound locations were digitized on-screen from a scanned 1:1000 scale paper map provided by Dr. Martha Rolingson, Arkansas Archeological Survey Station Archaeologist at the Toltec Mounds site.  The mound edges are defined by the solid polygons in this raster, and center points were determined through a weighted-mean method using the outer edges of each mound (Figure 4). 

Figure 4.  GIS model of the Toltec Mounds site showing mound and mound center point locations. 

            The first step in the analysis was the creation of distance surfaces from the edge of the embankment and edges and centers of the mounds (no center point was generated for the linear embankment).  The distance surfaces were then reclassified to keep only those values corresponding to the TM and its multiples.  For this analysis, a conservative +/- 2 m on either side of TM distances was used as a margin of error.  Figure 5 illustrates TM distances derived from the edge of Mound A.  The area within the embankment is shaded.  All points on the ground that are covered by the concentric TM distances in this image would be considered a TM multiple – that is, it is possible to measure from the edge of Mound A to all points covered by these lines, and come within 2 m of a multiple of 47.5 m.  The 'hits' from the edge of Mound A derived this way cover about 8% of the core site area.  This is the statistical background of TM distances from the edge of Mound A with a margin of error of +/- 2 m: every point on the ground within the enclosure has an 8% chance of being considered an occurrence of the TM distance, simply because of the size of the module in question, the margin of error, and the overall size of the site. 

Figure 5.  TM increments (+/- 2 m) as measured from the edge of Mound A.  Of the area within the embankment (shaded area), 8% is covered by a TM multiple.

            Sherrod and Rolingson employ numerous target points in their analysis, primarily the edges of all mounds and the embankment, and the centers of all mounds.  In order to derive the statistical background taking into account all of these potential targets, it is necessary to sum the TM occurrences from all of them.  Figure 6 is the result of this summation: the overlap of all rasters of TM distances (+/- 2 m) from the edges and centers of all mounds and from the edge of the embankment.  Each cell in this raster is coded with a value corresponding to the number of target points from which it is possible to measure within 2 m of a TM distance.  The values in the figure are collapsed into three categories for display.  Only 6% of the area is not covered by at least one occurrence of a TM, 41% is covered by one to two occurrences, and a full 53% is covered by three or more.  Table 2 shows the percent of area covered by each overlap increment.  From this analysis it is clear that every point within the enclosure has a 94% chance of being within 2 m of a TM distance from a target point simply by chance.  Note that this analysis does not even include the full suite of targets used by Sherrod and Rolingson, particularly the five gaps in the embankment, the surrounding ditch, and mound 'midslopes'.  Note also that this analysis employs a very modest margin of error of +/- 2 m.  A realistic margin of error at most mound sites may be significantly larger.

Figure 6.  Overlap of all TM increments (within the embankment only) from the edges and centers of all mounds and the edge of the embankment.  Overlaps have been reduced to three categories for display. 

Table 2.  TM distance overlaps by percent of area covered, derived from the raster shown in Figure 6. 

            Any similarly short measurement would cover a large percentage of the site and occur quite frequently simply by chance as well.  The smaller the distance tested in this way, the more commonly it will occur, and the larger the distance the less commonly it will occur.  This is an inherent property of any such site, regardless of the actual layout of the mounds.  The percentage of area covered is a function of the size of the site, the number of targets from which to measure, the length of the unit being tested, and the margin of error employed. 

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