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1.
A solution to the downward continuation effect on the geoid determined by Stokes' formula 总被引:2,自引:1,他引:2
L.E. Sjöberg 《Journal of Geodesy》2003,77(1-2):94-100
The analytical continuation of the surface gravity anomaly to sea level is a necessary correction in the application of Stokes'
formula for geoid estimation. This process is frequently performed by the inversion of Poisson's integral formula for a sphere.
Unfortunately, this integral equation corresponds to an improperly posed problem, and the solution is both numerically unstable,
unless it is well smoothed, and tedious to compute. A solution that avoids the intermediate step of downward continuation
of the gravity anomaly is presented. Instead the effect on the geoid as provided by Stokes' formula is studied directly. The
practical solution is partly presented in terms of a truncated Taylor series and partly as a truncated series of spherical
harmonics. Some simple numerical estimates show that the solution mostly meets the requests of a 1-cm geoid model, but the
truncation error of the far zone must be studied more precisely for high altitudes of the computation point. In addition,
it should be emphasized that the derived solution is more computer efficient than the detour by Poisson's integral.
Received: 6 February 2002 / Accepted: 18 November 2002
Acknowledgements. Jonas ?gren carried out the numerical calculations and gave some critical and constructive remarks on a draft version of
the paper. This support is cordially acknowledged. Also, the thorough work performed by one unknown reviewer is very much
appreciated. 相似文献
2.
3.
Truncated geoid and gravity inversion for one point-mass anomaly 总被引:1,自引:0,他引:1
The truncated geoid, defined by the truncated Stokes' integral transform, an integral convolution of gravity anomalies with
the Stokes' function on a spherical cap, is often used as a mathematical tool in geoid computations via Stokes' integral to
overcome computational difficulties, particularly the need to integrate over the entire boundary spheroid. The objective of
this paper is to demonstrate that the truncated geoid does, besides having mathematical applications, have physical interpretation,
and thus may be used in gravity inversion. A very simple model of one point-mass anomaly is chosen and a method for inverting
its synthetic gravity field with the use of the truncated geoid is presented. The method of inverting the synthetic field
generated by one point-mass anomaly has become fundamental for the authors' inversion studies for sets of point-mass anomalies,
which are published in a separate paper. More general applications are currently under investigation. Since an inversion technique
for physically meaningful mass distributions based on the truncated geoid has not yet been developed, this work is not related
to any of the existing gravity inversion techniques. The inversion for one point mass is based on the onset of the so-called
dimple event, which occurs in the sequence of surfaces (or profiles) of the first derivative of the truncated geoid with respect
to the truncation parameter (radius of the integration cap), its only free parameter. Computing the truncated geoid at various
values of the truncation parameter may be understood as spatial filtering of surface gravity data, a type of weighted spherical
windowing method. Studying the change of the truncated geoid represented by its first derivative may be understood as a data
enhancement method. The instant of the dimple onset is practically independent of the mass of the point anomaly and linearly
dependent on its depth.
Received: 26 September 1996 /Accepted: 28 September 1998 相似文献
4.
Y. M. Wang 《Journal of Geodesy》1990,64(3):231-246
The method of analytical downward continuation has been used for solving Molodensky’s problem. This method can also be used
to reduce the surface free air anomaly to the ellipsoid for the determination of the coefficients of the spherical harmonic
expansion of the geopotential. In the reduction of airborne or satellite gradiometry data, if the sea level is chosen as reference
surface, we will encounter the problem of the analytical downward continuation of the disturbing potential into the earth,
too. The goal of this paper is to find out the topographic effect of solving Stoke’sboundary value problem (determination
of the geoid) by using the method of analytical downward continuation.
It is shown that the disturbing potential obtained by using the analytical downward continuation is different from the true
disturbing potential on the sea level mostly by a −2πGρh 2/p. This correction is important and it is very easy to compute
and add to the final results. A terrain effect (effect of the topography from the Bouguer plate) is found to be much smaller
than the correction of the Bouguer plate and can be neglected in most cases.
It is also shown that the geoid determined by using the Helmert’s second condensation (including the indirect effect) and
using the analytical downward continuation procedure (including the topographic effect) are identical. They are different
procedures and may be used in different environments, e.g., the analytical downward continuation procedure is also more convenient
for processing the aerial gravity gradient data.
A numerical test was completed in a rough mountain area, 35°<ϕ<38°, 240°<λ<243°. A digital height model in 30″×30″ point value
was used. The test indicated that the terrain effect in the test area has theRMS value ±0.2−0.3 cm for geoid. The topographic effect on the deflections of the vertical is around1 arc second. 相似文献
5.
6.
Cheng Qiao Jiancheng Luo Yongwei Sheng Zhanfeng Shen Zhiwen Zhu Dongping Ming 《Journal of the Indian Society of Remote Sensing》2012,40(3):421-433
Water is one of the most common and important objects on the earth, and its extraction is of great significance to many related researches in remote sensing domain. However, water always appears diversely, which makes its extraction not so simple. Many former methods are developed to extract water, which mainly based on a single model and only use spectral information, but the results are not so satisfying. An adaptive extraction method based on normalized difference water index (NDWI) is proposed here to extract water completely and accurately from remote sensing image. This study first compute NDWI to enhance water??s spectral information, and then it is redefined so as to use the modified histogram auto-segmentation method to initially separate water from background; next, after segmentation, water pixels can be searched out and are taken as seed points to proceed region growing to get the local area of water; last, the edge of the local area is searched by a window template, and iterative classification within it is employed to precisely extract water??s precise partition. Experiments are carried out here on an ETM+ image of a paralic area to extract water. Through comparison with other commonly used methods, it shows that the performance of the proposed method is superior to the others. 相似文献
7.
J. Y. Chen 《Journal of Geodesy》1982,56(1):9-26
Summary The application of combined data (satellite and terrestrial data) to the practical computation of height anomalies or the
deflections of the vertical was originally suggested by (Molodensky et al. 1962). This idea usually leads to the modification
of Stokes' or Vening-Meinesz' functions in the integration procedure. In the recent decade there were various suggestions
in this regard especially for the computation of height anomalies. For example, a considerable mathematical insight into the
modification of Stokes' function and the truncation of its integral has been provided by (Meissl 1971, Houtze et al. 1979,
Rapp 1980, Jekeli 1980).
Five different methods for computing deflections of the vertical by modifying Vening-Meinesz' function are studied and compared
with each other. The corresponding formulae, the values of the coefficients in each method and the estimations of their corresponding
potential coefficient error and truncation error are given in this article.
This paper was written at the Institut f. Angewandte Geod?sie, Technische Universit?t Graz, Austria. 相似文献
8.
M. K. Paul 《Journal of Geodesy》1973,47(4):413-425
Neglecting distant zones in the computation of geoidal height using Stokes' formula gives rise to some truncation error. This
truncation error is expressible as a weighted summation of the zonal harmonic components of the gravity anomaly. Making use
of the well-known properties of Legendre polynomials, a compact method of computing these theoretical coefficients has been
developed in this paper. 相似文献
9.
《测量评论》2013,45(52):242-245
AbstractMy attention has recently been drawn to an article by G. H. Menzies (E.5.R., vi, 46, 474) on this subject. The present note is intended to point out improvements in the Anér method, which he favours, and to refute some of his criticisms of the Relaxation method. References are to page and table numbers in Menzies's paper. 相似文献
10.
Some modifications of Stokes' formula that account for truncation and potential coefficient errors 总被引:1,自引:1,他引:0
Stokes' formula from 1849 is still the basis for the gravimetric determination of the geoid. The modification of the formula,
originating with Molodensky, aims at reducing the truncation error outside a spherical cap of integration. This goal is still
prevalent among various modifications. In contrast to these approaches, some least-squares types of modification that aim
at reducing the truncation error, as well as the error stemming from the potential coefficients, are demonstrated. The least-squares
estimators are provided in the two cases that (1) Stokes' kernel is a priori modified (e.g. according to Molodensky's approach)
and (2) Stokes' kernel is optimally modified to minimize the global mean square error. Meissl-type modifications are also
studied. In addition, the use of a higher than second-degree reference field versus the original (Pizzetti-type) reference
field is discussed, and it is concluded that the former choice of reference field implies increased computer labour to achieve
the same result as with the original reference field.
Received: 14 December 1998 / Accepted: 4 October 1999 相似文献