A statistical method for estimation of block gravity means.
The block gravity means are necessary for the computations of geoid, deflection of the vertical and other quantities gravimetrically. The block means can be estimated by a linear combination of point observations, e.g., gravities and topographical data. The coefficients of the linear combination are calculated by the least-squares method to minimize the estimation error. The topographical data are feasible only when a relation between gravity and topography is available. As gravity and topography.are often correlated linearly (Fig. 1), it may be effective to utilize the topographical data in estimation of point gravities and block gravity means. In order to apply the least-squares method, various co-variance functions are needed, e.g., gravity-gravity, gravity-topography, topography-topography, and point-point, point-block, block-block co-variances. When a co-variance function of point gravity anomaly and a relation between gravity anomaly and topography are known, most of the co-variance functions are calculated from the co-variance function of point gravity anomaly (as shown in eqs. (12), (14), (16),（17) and (38)), adding the error co-variance functions (N in eq. (12) and U in eq. (15)). Therefore, the co-variance function of point gravity anomaly is basically important. The estimation error of block mean of free-air anomaly is the sum of those of Bouguer anomaly and topographical height. As Bouguer anomaly is usually less scattered than free-air anomaly, the estimation error of block mean of Bouguer anomaly is expected to be smaller than that of free-air anomaly. Therefore,when accurate block means of topographical height are available, it is effective to estimate the block mean of free-air anomaly through Bouguer anomaly (eq. (24)). In this case, the estimation error of block mean of free-air anomaly becomes the same as that of Bouguer anomaly (eq. (25)). The representation error is defined as the error of estimation of the block mean when a point value in the block is taken as the mean value of the block. Concerning the representation error, the situation of error quantities existing among the representation errors of free-air anomaly （m2F）, of Bouguer anomaly （M2B) and of topographical height (mH) is the same as the case of the estimation error mentioned above (see eq. (32)). A co-variance function of the local gravity anomaly in Japan （eq. (33)) obtained from the deflection of the vertical is used to estimate the representation error of free-air anomaly (by eq. (29)). The estimated representation errors for various block sizes are compared with the actual data obtained by Ono (1976) from gravity distribution in certain areas (Table 1: the 3rd row (estimation) and the 4th row (actual), and Fig. 3). The coincidence between the estimated and actual values are quite satisfactory for larger blocks of 10 km square, but for smaller blocks, the opposite is true. The co-variance function (eq. (33)) should be modified for the part near the origin because it has not a zero first-derivative at the origin. The zero first-derivative is a reasonable requirement from the physical characteristic of the anomaly field. In the ocean areas,since the density of gravity measurements is small, the utilization of depth data may also be effective to estimate better block gravity means there. The depth data are converted to gravity anomalies through conversion function (such as Fig. 6 by McKenzie and Bowin, 1976). Although this kind of conversion function has not been tested widely, we may expect that the depth data can be used effectively in estimating the block gravity means in the ocean areas. Key words: block gravity mean.
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| Format: | Journal Contribution biblioteca |
| Language: | Japanese |
| Published: |
1978
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| Online Access: | http://hdl.handle.net/1834/16300 |
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