On the basis of the gravity material available, the author studies two statistical functions: G2 = the rms (root mean square) anomaly in a square with side s, and E2 = the rms deviation of one actual point anomaly from the actual mean anomaly in a square with side s.
The function E2, is called the error of representation, for if inside a square there is only one observed anomaly and this anomaly is accepted to represent the mean anomaly of the entire square, a standard mean error E can be used for the estimation of accuracy. On the other hand, if there are no observations inside the square and the mean anomaly of the square is assumed to be zero, G can be used as the standard mean error.
For points or for very small squares, E is zero and G has a maximum value G0. For a hemisphere, G is zero and E has a maximum value G0. There is a critical size at about s = 3°, where E = G. When s is greater, it is not advisable to use the observed anomaly at a single station, as the representative of the mean anomaly of the square, because for zero the error to be expected is smaller. The weighted mean of zero and the observed anomaly is recommended.
Because the regions without any observations are still large, it is necessary to estimate the size of the smallest squares, where the mean anomaly can be assumed to be independent of the mean anomalies of the adjacent squares. On the basis of the present gravity data, an estimated value of s = 30° seems to be acceptable.
Using the functions E and G and the accepted values s = 3° and s = 30°, the precision obtainable for the gravimetric determination of the elevations N of the geoid (Stokes' formula) and of the deflections δ of the vertical (Vening Meinesz' formula) has been estimated. In the most favorable cases (Central Europe and the central parts of the United States) the standard mean error of N is ±10 meters and that of δ ± 0.″85. The former figure is almost entirely due to the great unexplored areas of the Earth'; the latter depends half on these unexplored areas and half on the small gaps within a distance of 50° from the point where δ is computed.
