The rms (root mean square) error of the deflection of the vertical computed from gravity anomalies by the Vening Meinesz formula was calculated for two sets of hypotheses: (1) perfect knowledge of gravity within given radii, and none beyond; (2) ideal distribution of given numbers of gravity stations out to given radii, and no stations, or a uniform distribution, beyond.
The calculation for lack of gravity data beyond given radii was based on rms anomalies G8, as found by Hirvonen, and on correlation of free‐air anomalies determined from profiles in central USA.
The general formula employed to compute the rms error in the deflection is
where
the effect of a one milligal anomaly for an arbitrary elemental area; rii is the coefficient of correlation between the mean anomalies of two elemental areas; and (αj−αi) is the angle between two elemental areas subtended at the deflection point.
Assuming the rms point anomaly Go = ±28 mgal, the rms modulus of the deflection was found to be 6.0″. The rms effects of anomalies beyond given radii were found as follows: 3000 km, 1.2″; 2000 km, 1.7″; 1000 km, 2.2″; 500 km, 3.3″.
The rms geoid elevation computed in a similar manner was found to be ±21.3 m. The rms effects of anomalies beyond given radii were found as follows: 150°: ±5.1 m; 90°: ±7.6 m; 60°: ±11.0 m; 45°: ±11.5 m; 3000 km: ±11.9 m; 2000 km: ±13.4 m; 1000 km: ±17.2 m.
The discrepancy between the results obtained by these methods and those obtained by Cook in 1950 and 1951 using spherical harmonics is roughly proportional to the difference between the rms anomalies used.
Six ideal distributions, for which each gravity station makes an equal contribution to the deflection error, were computed out to 35°, and graphs constructed therefrom. The effects due to the errors of representation of these distributions were combined with those due to lack of gravity data beyond given radii to give a total uncertainty. For example, 200 stations out to 1000 km will give an rms error of 2.7″; to 2000 km, 2.0″; to 3000 km, 1.6″. For 50 stations, the corresponding figures are: 1000 km, 3.0″; 2000 km, 2.6″; 3000 km, 2.5″. A uniform distribution of one station to every 10° block will improve the rms error with 200 stations out to 1000 km to 1.2″; one to every 5° block will improve it to 0.8″.