Correlation coefficients and their use in the prediction of mean anomalies
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Date
1962-04
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Ohio State University. Division of Geodetic Science
Abstract
The correlation of point anomalies with the separation between them has been investigated by means of the correlation coefficient ro,s. Correlation curves for free air anomalies have been established in Ohio and Finland, and for Bouguer anomalies in Ohio alone. The individual curves showed a wide variation, but the curve obtained by meaning the values of a particular ro,s from the individual curves gave a reasonably smooth curve that turned negative at 107 km in Ohio and 147 in Finland. Various equations were applied to these mean curves in order to find an analytical representation. Several equations given by Hirvonen were applied with limited success, primarily because of the inability of the equations to fit the actual curves in its negative portions. A satisfactory representation was found from a third order polynomial representation: ro,s = 1+a1 s+a2 s2+a3 s3. If we assume that Δgs= Δgo° ro,s where Δgs is a mean anomaly on a circular ring of radius s about Δgo, equations for the prediction of a mean anomaly from the point anomaly at the center of the figure can be derived. The simplest case to consider is that of a circle. It is shown, however, that once equations have been established for a circle, a simple conversion between the radius of the circle and the side of the square, enable the mean anomaly in a square to be derived from a mean anomaly in a circle. Tests of the prediction formulas for free air anomalies were carried out using circles of 20 km and 40 km in radius. An accuracy analysis was made by comparing the predicted mean anomaly with a "true" mean anomaly. Based on 14 points in Ohio a mean error of one prediction was ±4 mgals and on 15 points in Finland was ±3 mgals (for R=40 km). In both cases this was an increase in the accuracy of mean anomaly above that if the mean anomaly was simply taken as the point anomaly at the center of the circle. The Ohio data was applied to predicting in Finland to find that there was only a slight decrease in the accuracy of the prediction. Thus it seems valid to assume that data from one area can be applied to another area as long as the areas are similar (i.e., in this case, relatively flat). More work needs to be done in areas that have appreciable elevation differences. It is possible that also in areas of rugged topography the data of an area A can be applied in another area B, but then the elevation correlation (of the free air anomalies) has to be considered. [Some mathematical expressions are not fully represented in the metadata. Full text of abstract available in document.]
Description
Prepared for Geophysics Research Directorate, Air Force Cambridge Research Laboratories, Office of Aerospace Research, United States Air Force, Bedford, Massachusetts: Contract No. AF 19(604)-6201, Project No. 7600, Task No. 76002 (OSURF Project 1058)