National Geodetic Survey (NGS) Gravity Prediction Methodology

This process is designed to predict surface gravity at specified geographic position and topographic height.

Rudolf J. Fury M.S.,M.Eng.

04/02/1999

I. METHOD

The data are modeled with polynomial equations as described by Dr. Rolland Hardy's (professor at Iowa State University - retired) multiquadric biharmonic method. This involves representing the gravity data as a sum of individual cones, each adding to form the model.

The equation is of the form:

  gi  =  j ( a[i,j]  .  Cj )      j=1,2,...,n



where

       a[i,j]  =  [(x(i) - x(j))2 + (y(i) - y(j))2 + D2]1/2

a[i,j] is an n x n symmetric matrix of distances from observed data points (g_i for i=1,2,...n) to all data points (j=1, 2,...n) having zero diagonal elements, and D (delta) is a smoothing factor. C_j's represent coefficients to be determined from the data. Delta has been omitted because a value of zero was found to produce the best values. Once the coefficients are known, the predicted value becomes

  g =  i { j ( a[i,j]  .  Cj ) }       j=1,2,...n    i= 1,2,...n



 
References:


 Hardy, R.L., 'Multiquadric equations of topographic and other irregular surfaces',
             Journal of Geophysical Research, 76, 1905-1915, 1971.  
 Hardy, R.L., and Gopfert, W.M., 'Least squares prediction of gravity anomalies,
              geoidal undulations, and deflections of the vertical with multiquadric
              harmonic functions', Geophysical Research Letters, 2, no. 10, 423-427, 1975.
 Hardy, R.L., and Nelson, S.A., 'A multiquadric-biharmonic representation and approximation of
              disturbing potential', Geophysical Research Letters, vol.13, no. 1, 18-21, 1986.
 Jekeli, C., 'Hardy's Multiquadric-Biharmonic Method for Gravity Field Predictions'
             Computers Mathematical Applications, Vol. 28, No. 7, pp 43-46, 1994.

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II. DATA SELECTION

Surface gravity, Bouguer anomaly, and topographic elevation data are retrieved from the Integrated Data Base (IDB) of NGS;

Data retrieval from the Integrated Data Base :

1. Retrieval area boundaries are set to 0.1 degree from the prediction point

2. When the number of points in the area is less than the minimum requirement of 150 observed data, the area is extended iteratively by 0.1 deg. four times (to a maximum of 0.5 deg.) to get more observations, up to a maximum of 300 data points; after that, whatever number of data were available are accepted

3. All available data (i.e., proprietary and non-proprietary), edited and unedited are retrieved (the latter to increase data distribution density; there are few unedited data points in the IDB, only some recently acquired data).

Data Selection for Modeling

1. Data spacing: some data points may be dropped when they are closer than 0.0001 arc minutes

2. When the average distance from prediction to data points is less than 10 arc minutes, the needed data points are cut back to 100 points

3. When the shortest distance from prediction to data points is less than 0.1 arc minutes, the needed data points are cut back to 50

4. The data are centered with respect to a best fitting (least-squares) plane;

5. It was found that while predicting at the coast, the volume of ship data 'inundated' the areally retrieved data set, thus squeezing out valuable land data; due to the very close spacing of ship data, the selection of every second data point was sufficient.

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III. SURFACE GRAVITY PREDICTION

Multiquadric equations of surface gravity, Bouguer anomaly and topographic elevation are formed and solved; all three values are predicted from the data;

Surface gravity is predicted by two methods:

1. Interpolated Bouguer anomaly is transformed into surface gravity values using

     a. the user-specified mean sea level height (when provided),  and also 
     b. the predicted height 

2. Surface gravity is also interpolated directly.

Comparison of surface gravity values obtained by the two methods provide gross error detection.

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IV. ESTIMATION OF THE ERROR OF PREDICTION

1. A data set is assembled from the ten closest data points in the vicinity of prediction point

2. One data point is removed from the set in turn; then the remaining nine points are used to predict the value of removed point

3. The difference between the observed and predicted value is calculated

4. The RMS of these ten differences of the set represents the estimate of the error of prediction.