Explanation:

Angle Calculations

## Elevation Analysis using the National Elevation Dataset

### Elevation Analysis of 7.5' USGS Quadrangles

What follows is an analysis of USGS quads covering various US states and mountainous areas, displayed by category. Simple categories include: Quad Highpoint, Quad Lowpoint, Elevation Range, and Average Elevation. More complex calculations are Tile Variation Average, which measures the overall elevation change within a given quad, Canopy Average, which measures the overall tree cover in a quad, and Surface Area Ratio which determines the surface area of a quad in comparison to its geometric area.

More states and mountainous regions within the NED coverage area will be added as they become available.

Digital data used is 1 arc second NED. All algorithms to develop the lists are authored by myself.

Quads Ranked by Surface Area Ratio:

### Tile Variation Average Explained:

At 1 arc second NED resolution, each 7.5' USGS quad consists of ~202500 30m elevation tiles. The differences in elevation between a single tile and all its adjacent tiles are determined and then averaged. 8 adjacent tiles exist for each interior tile, 5 for edges, 3 for corners. This average figure for a tile is its Tile Variation Average(TVA). The TVAs for all tiles in a quad are averaged to come up with a TVA figure for the entire quad, in feet. TVA values generally range from 0 to 30, with the highest reading so far calculated being the Temple of Sinawava quad(UT) at 34.215. What is being stated is, using that quad as an example, that the average difference between any given tile and all of its adjacent tiles is 34.215 feet.

### Canopy Average Explained:

This figure makes use of the "National Land Cover Database 2001 Tree Canopy" dataset available from the USGS. A RGB value indicating amount of tree cover for each of the ~202500 30m tiles in a quad is obtained and converted to a single value on a 0 to 100 scale based on tree cover density, with 0 being open land and 100 equivalent to maximum cover. All tiles on a given quad are averaged to come up with a Canopy Average(CA) for a given quad. The greatest CA calculated so far is the Bear Mountain Quad in Colorado, at 76.24.

### Surface Area Ratio Explained:

The surface area of a mountainous tract of land may be significantly greater than its geometric area, due to the presence of sloped surfaces. The more sloped surfaces within a quad, and the greater the slope values, the greater the surface area will be, which makes it a useful measure to describe the overall elevation change in a given quad. The Surface Area Ratio(SAR) figure shown in the lists is not true surface area, rather it is a ratio of the surface area compared to the geometric area.

#### Calculation:

Given that digital data is comprised of discrete area squares, the method below calculates the surface area in a non-continuous manner. Starting at the west edge of a quad, the elevation reading of a given DEM square is compared to the elevation reading of its adjacent square to the east. From these two readings, a hypotenuse linking the two is calculated. Surface area for a given square is then determined using the hypotenous as the x component, with the y kept at 30.92 meters. Hypotenuses are then calculated for the next two adjacent squares moving eastward, continuing on for all 449 sets of DEM tiles comprising a row of squares within a quad, moving from west to east. Then, the next row of 449 tiles is similarly calculated, until all 450 rows which make up a quad have been processed. What results are 202050 hypotenuse-based surface area figures which are averaged to come up with single surface area figure representative of the entire quad in the E-W direction. The raw hypotenuse figure(in meters) is compared to the x component (30.92m) to come up with a Surface Area Ratio figure, usually between 1 and 1.2. A perfectly flat plane with no slope would have a SAR of 1.

Next, the process is repeated in the N-S direction, starting with the first column and running through all 450 columns, east to west. As before, the 202050 N-S hypotenuse-based, per square surface area figures are averaged to come up with single surface area figure, with this raw amount again converted to a SAR figure which summarizes the ratio of surface area to geometric area in the N-S direction.

Finally the E-W, N-S SAR figures are averaged which results in a SAR figure for the entire quad. So, given a SAR figure of, say 1.056, what is being stated is that the surface area of the given quad is 1.056 times the geographic area of the quad. To calculate the actual surface area, multiply SAR with the geometric area of a quad.