Elevation Interpolation

Terrain Elevation Interpolation

Q: Which interpolation method should I use?

A: The topographic elevation data base includes elevation values on a grid of latitude and longitude (e.g., a 3-second grid for the 3-second data base). For a given path under study, it is likely that few if any of the data base points will be exactly on the path. Therefore, each desired point on the path is determined by interpolation of the surrounding data base points

TAP provides four methods of data interpolation:

• The FCC specified method using four data points for the interpolation. This method should be used for pertinent terrain averaging calculations to be submitted to the FCC. This method us typically used in most instances

• A weighted interpolation method using twelve data points for the interpolation. This method may provide some improvement in elevation accuracy, but will also increase the execution time for the program since more topographic data must be processed.

• A maximum elevation method using the elevation of the highest elevation of the four points around the radial point. This method is often used to test for "worst case" conditions, such as a microwave line of sight check.

• A nearest point method which selects the elevation of the nearest point in the data base.

FCC interpolation

The FCC specified the interpolation procedure to be used with the point elevation data base in a public notice (84-705)  dated July 13, 1984:

The prescribed method is a simple linear interpolation of the four data base "corner" points (A,B,C,D in the diagram) surrounding a desired point (G) as shown.

Following the FCC Public Notice, the interpolation is first performed between points AB to determine the elevation of point E, and between points CD to determine the elevation of point F. Finally, a linear interpolation is performed between points EF to predict the elevation of the desired point G.

This method is intended for data retrieval for the purpose of terrain averaging, and the averaged results for typical radials compare very favorably with the averages obtained using elevations taken manually directly from U.S.G.S. 7-1/2 minute topographic maps.

However, the FCC interpolation method includes an inherent limitation of the accuracy for individual data points. For example, if point "G" is a mountain peak, the four surrounding corner points will all be lower, and a simple linear interpolation of the four corners will result in a value lower than the actual elevation of the peak. Conversely, if point "G" is the bottom of a valley, the corner points will be higher, and the interpolated result will be higher than the actual elevation. Thus the FCC linear interpolation tends to "flatten" the terrain elevation values, lowering the peaks and raising the valleys.

While this effect is substantially overcome in the normal averaging process of fifty or more points on each radial, the use of the FCC linear interpolation for the retrieval of the elevation of a particular individual point may not provide sufficient accuracy. (The FCC accepts the use of the interpolated data for path elevation averaging, but requires the exact elevation of the end point, usually a transmitter site, to be taken from a topographic map.)

Weighted interpolation

The second interpolation method available attempts to reduce the impact of this "flattening" effect by using additional surrounding data points to determine the elevation of the point on the path under study. Twelve points, instead of four, are used, and the interpolation process uses data base points outside of the four immediate corner points to determine the slope of the terrain in different directions around the point under consideration.

Figure 4:  Weighted interpolation - Twelve Points

The elevations are interpolated based on weighting factors computed from the slope of the terrain in the direction of each corner.

For example, using the point marked "H" in Figure 4, the weighed contribution is computed as follows:

The bearing from point H to the target point G is computed.

Using this bearing, the location of the crossing point I is determined.

A linear interpolation between points AC determines the elevation of point I.

The vertical slope of line HI is computed based on the horizontal distance between the points and the difference in their elevations.

Based on the slope of line HI, the extrapolated elevation of point G is computed.

This process is repeated for each of the outer points, for a total of eight predicted values of the elevation of point G. These values are averaged together to obtain a weighted value. Finally, the weighted value is averaged with the FCC value determined as described above.

Maximum Elevation

The maximum elevation method selects the highest elevation from the four closest points and uses that value. This method provides a "worst case" profile using the highest elevations within approximately 300 feet (in the 3-second data base) of the path.

Figure 5:  Maximum elevation method

Regardless of the interpolation method used, the software cannot represent topographic features that are not in the data base. Even in the case of topographic features that are in the data base, the elevation in the data base may differ from the actual elevation shown on topographic maps or determined by surveys or other means. (See the Technical Reference section Elevation Data Accuracy for more information on data base accuracy.) For this reason, all elevations that could critically affect the results of studies based on the topographic data should be determined from topographic maps or surveys or other means.

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