Modified Method:  Conceptually, this modified process can be pictured as taking each polygon separately, using it like a cookie cutter to clip out only those grid cells that lie within the polygon, then running 'ZonalStatsTable' on only that polygon and only those clipped grid cells.  Technically, the modified version takes each polygon in the theme and converts it into a grid with the same cell size as the Input Grid, producing a grid of "1" values in the shape of the polygon.  It then multiplies that grid by the Input Grid, thereby converting all cells outside of the polygon into "No Data" cells while retaining the original cell values inside the polygon.  Finally, this extension uses the 'ZonalStatsTable' request on the modified input grid to generate statistics for that single polygon. This process eliminates the possibility that the 'ZonalStatsTable' request will select cells outside of the polygon boundary by converting all cells outside of the polygon boundary into "No Data" values.

     Polygon radii are in meters and grids had a cell size of approximately 92.6 meters.  The modified process causes the circular polygons to have identical numbers of cells for each radius size while the standard process can have variable cell counts in each radius size.  The means, standard deviations and ranges reflect the ratios of (GRID Surface Area) divided by (TIN Surface Area), or the ratio of (GRID Surface Ratio) divided by (TIN Surface Ratio).

     In all cases, Surface Ratio values using the Standard method were very close between the TIN-based and GRID-based calculations.  Accuracy and precision increased with higher cell counts, but even in the worst case the GRID-based calculation was only 4% different than the TIN-based calculation.

     At low cell counts, this Standard method produced highly variable results with low correlations between TIN-based and GRID-based surface areas.  Circular polygons can have variable cell counts depending on their placement on the grid and these different cell counts can have significant effects on the internal areas.  The scatterplot for the 100m circles comparing Surface Areas clearly shows 4 groupings of points, and the groups are explained by the cell counts.  There are very distinct separations between those circles with 2, 3, 4 or 5 cells.  These distinct groupings are still visible in the 200m and 400m circles, but they start to blend together at higher cell counts.

     At high cell counts this Standard method generally produced very accurate and precise surface areas, with the exception of the 2,000m circles.  Something strange happened there, and surface areas for 19 of these circles apparently included grid cells that were outside the circle.  The scatterplot for the 2,000m surface areas shows two distinct groupings of points and I suspect that one of these groups was somehow calculated incorrectly.  This problem with the 2,000m points led me to develop the Modified method of generating statistics.

Surface Area/Ratio Comparison using Modified Method:

     In all cases, Surface Ratio values using the Modified method were very close between the TIN-based and GRID-based calculations.  Accuracy and precision increased with higher cell counts, but even in the worst case the GRID-based calculation was only 1% different than the TIN-based calculation.

     Surface Area values were always highly correlated, but GRID-based calculations tended to be higher than TIN-based calculations at low cell counts.  Surface area values for the 100m circles, with 4 grid cells each, tended to be over 9% higher when calculated from the Grids.  GRID-based surface areas for circles with up to 60 cells tended to be about 2.5% higher than their corresponding TIN-based surface areas.

     Surface areas for circles with cell counts >= 234 tended to be almost identical between the GRID- and TIN-based calculations, and correlations were almost perfect.

DISCUSSION:

RECOMMENDATIONS:

     I feel that this extension does produce reliable surface area and surface ratio values when the area of interest includes a large number of cells (>= 264).  I leave it to the user to decide whether the Standard or Modified method of generating statistics is preferable.  This extension offers the choice to generate statistics using either method.  The Standard method is much faster, but did produce some mysterious results in this data set.  The modified method appears to consistently overestimate the surface area at low cell counts, but appears highly accurate at high cell counts.

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