On the precision of the gravimetric determination of the geoid

Abstract

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.

Accuracy of gravimetrically computed deflections of the vertical

“…These uncertainties should be comparable to those computed from the world wide gravity field for actual points. Hirvonen [1956] made such a computation for two points: one in North America, with dense gravity data out to 12°, and scattered data beyond, such as is represented by the Jeffreys' formula; and one in Europe, with dense data out to about 14°, and scattered data beyond. Com paring the uncertainties found by Hirvonen with values interpolated from those derived in the preceding paragraph gives the following ratios for fa -12°: 1.94'70.88" = 2.2 and 18.3/10.4 = 1.8; for fa « 14°: 1.72 ,/ /0.82 // -2.1 and 17.2/ 10.4 = 1.7; all again very close to the ratio 2.0 of the rms anomalies used.…”

“…Gravity anomaly correlation- Hirvonen [1956] computed the covariances of the mean isostatic anomalies of 1°, 5°, and 10° squares with centers a distance d ~ hs apart C M = (AgiKAgi+h) (0…”

The gravimetric geoid of Great Britain and Ireland

“…In view of the massive computation involved in the numerical evaluation of Stokes's integral, a purely theoretical determination of the errors involved is difficult, and has never been achieved. The few investigators in the field - de Graaff Hunter (1935), Hirvonen (1956) and…”

Reconciliation of Stokes' function and astro-geodetic geoid determinations

“…The variance of Ag, defined here as the mean anomaly for an arbitrary area, is a function of the size of the area and the distribution of observations within it, as described by Hirvonen [1956] and Kaula [1957]. In cases where we have areas empty of observations, we must allow for covariance as well.…”