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1.
M. K. Paul 《Journal of Geodesy》1978,52(3):177-190
Recurrence relations for the evaluation of the integrals of associated Legendre functions over an arbitrary interval within
(0°, 90°) have been derived which yield sufficiently accurate results throughout the entire range of their possible applications.
These recurrence relations have been used to compute integrals up to degree 100 and similar computations can be carried out
without any difficulty up to a degree as high as the memory in a computer permits. The computed values have been tested with
independent check formulae, also derived in this work; the corresponding relative errors never exceed 10−23 in magnitude.
Contribution from the Earth Physics Branch No. 719 相似文献
2.
R. H. Rapp 《Journal of Geodesy》1977,51(4):301-323
A set of 38406 1°×1° mean free air anomalies were used to derive a set of 1507 5° equal area anomalies that were supplemented
by 147 predicted anomalies to form a global coverage of 1654 anomalies. These anomalies were used to derive potential coefficients
to degree 52 using the summation formulae. In these computations, a smoothing operator was introduced and found to significantly
effect the results at higher degrees. In addition, the effects of the atmosphere, spherical approximation and terrain were
studied. It was found that the atmospheric effects and spherical approximation effects were about 0.3% of the actual coefficients.
The terrain correction effects amounted to 10 to 25% of the low degree coefficients depending on a specific terrain correction
model chosen; however, the correction terms found from the models did not yield solutions that agreed better with current
satellite derived potential coefficient determinations.
Anomalies were computed from the derived potential coefficients for comparison to the original anomalies. These comparisons
showed that the agreement between the two anomalies became significantly better as the degree of expansion increased to the
maximum considered. These comparisons shed some doubt on the rule of thumb that a block of size θ° can be represented by a spherical harmonic expansion to 180°/θ°. 相似文献
3.
Christopher Jekeli 《Journal of Geodesy》1980,54(2):137-147
Errors are considered in the outer zone contribution to oceanic undulation differences as obtained from a set of potential
coefficients complete to degree 180. It is assumed that the gravity data of the inner zone (a spherical cap), consisting of
either gravity anomalies or gravity disturbances, has negligible error. This implies that error estimates of the total undulation
difference are analyzed. If the potential coefficients are derived from a global field of 1°×1° mean anomalies accurate to
εΔg=10 mgal, then for a cap radius of 10°, the undulation difference error (for separations between 100 km and 2000 km) ranges
from 13 cm to 55 cm in the gravity anomaly case and from 6 cm to 36 cm in the gravity disturbance case. If εΔg is reduced to 1 mgal, these errors in both cases are less than 10 cm. In the absence of a spherical cap, both cases yield
identical error estimates: about 68 cm if εΔg=1 mgal (for most separations) and ranging from 93 cm to 160 cm if εΔg=10 mgal. Introducing a perfect 30-degree reference field, the latter errors are reduced to about 110 cm for most separations. 相似文献
4.
简述了原国家重力基本网(57网)的历史和技术要点,详细地分析了57网的各类误差和产生原因,提出了新、旧系统转换的方法。本文研究表明,57网基本点相对观测的实际精度约为±0.06mgal,仪器平均值误差为1.6×10~(-4)。基本点重力值含-13.56mgal的基准误差和1.9×10~(-4)的尺度差,并含有±0.1~0.2mgal的非线性系统误差。对基本点实行新、旧系统转换误差为±0.05mgal,基本不损失其观测精度。 相似文献
5.
R. H. Rapp 《Journal of Geodesy》1967,41(1):55-65
In support of requirements for the U.S. Air Force Cambridge Research Laboratories, gravity anomalies have been upward continued
to several elevations in different areas of the United States. One area was 340 to 400 N in latitude and 960 to 1030 W in longitude, generally called the Oklahoma area. The computations proceeded from 26, 032 point anomalies to the prediction
of mean anomalies in 14, 704, 2.5′×2.5′ blocks and 9,284, 5′×5′ blocks. These anomalies were upward continued along 28 profiles
at 5′ intervals for every 30′ in latitude and longitude. These anomalies at elevations were meaned in various patterns to
form mean 30′×30″, 10×10, 50×50 blocks. Comparisons were then made to the corresponding ground values. The results of these comparisons lead to practical
recommendations on the arrangement of flight profiles in airborne gravimetry. 相似文献
6.
G. Blaha 《Journal of Geodesy》1978,52(1):19-23
A simple formula is presented giving the value of γ−γ
r
to better than 0.001 mgal associated with an arbitrary reference ellipsoid, where γ is the normal gravity and γ
r
is its radial component. Further simplifications of this formula are possible, depending on the desired accuracy. Since in
the actual field g−gr equals γ−γ
r
to a good approximation, this formula makes it possible to work in terms of gr rather than in terms of the measured quantity g. Such a choice is attractive mainly because the spherical harmonic expansion
of gr is very simple. 相似文献
7.
AbstractFor the sake of the junior reader we may repeat an old and simple investigation. Let us suppose that the paper on which a map is printed undergoes a regular expansion p in one direction, say the X direction, and another regular expansion q in the Y direction, perpendicular to the former; it is required to know the effect of these expansions on the area of any parcel on the map. Note that, so far as the mathematics are affected, X and Y are not necessarily parallel to the margins of the sheet; we shall take them here as axes of any rectangular coordinate system. The symbols p and q are regarded as ratios, so that 100p and 100p represent the percentage expansions; if the paper contracts instead of expanding, no more is necessary than to change the sign. 相似文献
8.
D. Tsoulis 《Journal of Geodesy》2004,78(1-2):7-11
The spherical harmonic analysis of the 2×2 density and stratification information contained in the global crustal model CRUST 2.0 is presented from the viewpoint of gravity field recovery and interpretation. Using a standard Airy/Heiskanen (A/H) isostatic hypothesis and a radially distributed compensation mechanism, two models of topographic/isostatic (t/i) potential harmonic coefficients are obtained up to degree and order 90. The CRUST-derived coefficients are compared with the spectrum of uncompensated topography, with the EGM96 observed gravity field, and with the t/i spectrum based on an A/H hypothesis with a constant compensation depth of 30 km. The signal degree variances of both CRUST models decrease quite smoothly towards degree 90, while the seven-layer model approaches the EGM96 spectrum for degrees 80–90. The significant deviation of the CRUST spectra from the A/H combined spectrum may prove of relevance to local and regional applications investigating the validity of current isostatic hypotheses.Acknowledgments. Sincere thanks go to Nikolaos Pavlis and three unknown referees for their thoughtful comments. Figure 1 has been produced using the mapping package m_map by R. Pawlowicz, which is a MATLAB toolbox that can be freely downloaded from http://www2.ocgy.ubc.ca/~rich/map.html 相似文献
9.
Richard H. Rapp 《Journal of Geodesy》1975,49(1):57-63
For proper computation of the Stokes’ constants, or the evaluation of potential coefficients from terrestrial gravity data,
surface free-air anomalies should be corrected to sea level. Such a correction is composed of two parts; the first, the Molodensky
correction, G1, and a second, a term depending on the degree (n) and the expansion of (hΔg). This paper examines these terms numerically,
computing for 1654 5° equal area blocks values of G1 and the total correction based on spherical harmonic expansions to degree 20. The largest correction found was 0.37 mgals.
Corrections to potential coefficients caused by the anomaly correction were computed and compared to the original coefficients.
The ratio between the coefficient corrections and the full coefficients generally increased by degree having a maximum ratio
of 0.21 percent at degree 14 indicating that at the present time the corrections considered are negligible up to at least
degree 20. 相似文献
10.
Christopher Jekeli 《Journal of Geodesy》1983,57(1-4):10-28
The problem of the divergence of the geopotential spherical harmonic series at the earth's surface is investigated from a numerical, rather than a theoretical, approach. A representative model of the earth's potential is devised on the basis of a density layer, which, in the spherical approximation, generates a gravity field whose harmonic constituents decay according to an accepted degree variance model. This field, expanded to degree 300, and a topographic surface specified to a corresponding resolution of 67 km are used to compute the differences between truncated inner and outer series of the gravity and height anomalies at the surface of the earth model. Up to degree 300, these differences attain RMS values from 0.33 μgal to 86 μgal for the gravity anomaly and from 0.32 μm to 410 μm for the height anomaly, in areas ranging respectively from near the equator to the vicinity of the pole. In addition to these values, there is an expected truncation effect, caused by the neglect of higher degree components of the inner series, of about 30 mgal and 36 cm, respectively. The field is then subjected to a Gaussian filter which effectively cuts off information at degree 300 (at the 5% level). The RMS error to degree 300 is thereby reduced by factors of 10 to 20, with a concomitant reduction in the truncation effect to about 0.3 mgal and 0.7 cm. 相似文献
11.
G. Balmino J. Barriot R. Koop B. Middel N. C. Thong M. Vermeer 《Journal of Geodesy》1991,65(4):218-229
At different European institutes software has been developed for evaluation of the gravitational potential of the Earth using high degree spherical harmonic expansions. In this report the results of a comparison of a number of these software packages are presented. We compared the results for the second order derivatives (gravity gradients). It appeared that one of the most critical points in these computations is the definition of the coordinates, which should be as accurate as possible. Machine dependency and algorithm setup were of less importance, the former being only reflected in CPU timing results. 相似文献
12.
An operational algorithm for computation of terrain correction (or local gravity field modeling) based on application of closed-form solution of the Newton integral in terms of Cartesian coordinates in multi-cylindrical equal-area map projection of the reference ellipsoid is presented. Multi-cylindrical equal-area map projection of the reference ellipsoid has been derived and is described in detail for the first time. Ellipsoidal mass elements with various sizes on the surface of the reference ellipsoid are selected and the gravitational potential and vector of gravitational intensity (i.e. gravitational acceleration) of the mass elements are computed via numerical solution of the Newton integral in terms of geodetic coordinates {,,h}. Four base- edge points of the ellipsoidal mass elements are transformed into a multi-cylindrical equal-area map projection surface to build Cartesian mass elements by associating the height of the corresponding ellipsoidal mass elements to the transformed area elements. Using the closed-form solution of the Newton integral in terms of Cartesian coordinates, the gravitational potential and vector of gravitational intensity of the transformed Cartesian mass elements are computed and compared with those of the numerical solution of the Newton integral for the ellipsoidal mass elements in terms of geodetic coordinates. Numerical tests indicate that the difference between the two computations, i.e. numerical solution of the Newton integral for ellipsoidal mass elements in terms of geodetic coordinates and closed-form solution of the Newton integral in terms of Cartesian coordinates, in a multi-cylindrical equal-area map projection, is less than 1.6×10–8 m2/s2 for a mass element with a cross section area of 10×10 m and a height of 10,000 m. For a mass element with a cross section area of 1×1 km and a height of 10,000 m the difference is less than 1.5×10–4m2/s2. Since 1.5× 10–4 m2/s2 is equivalent to 1.5×10–5m in the vertical direction, it can be concluded that a method for terrain correction (or local gravity field modeling) based on closed-form solution of the Newton integral in terms of Cartesian coordinates of a multi-cylindrical equal-area map projection of the reference ellipsoid has been developed which has the accuracy of terrain correction (or local gravity field modeling) based on the Newton integral in terms of ellipsoidal coordinates.Acknowledgments. This research has been financially supported by the University of Tehran based on grant number 621/4/859. This support is gratefully acknowledged. The authors are also grateful for the comments and corrections made to the initial version of the paper by Dr. S. Petrovic from GFZ Potsdam and the other two anonymous reviewers. Their comments helped to improve the structure of the paper significantly. 相似文献
13.
Spherical harmonic series, commonly used to represent the Earth’s gravitational field, are now routinely expanded to ultra-high
degree (> 2,000), where the computations of the associated Legendre functions exhibit extremely large ranges (thousands of
orders) of magnitudes with varying latitude. We show that in the degree-and-order domain, (ℓ,m), of these functions (with full ortho-normalization), their rather stable oscillatory behavior is distinctly separated from
a region of very strong attenuation by a simple linear relationship: , where θ is the polar angle. Derivatives and integrals of associated Legendre functions have these same characteristics.
This leads to an operational approach to the computation of spherical harmonic series, including derivatives and integrals
of such series, that neglects the numerically insignificant functions on the basis of the above empirical relationship and
obviates any concern about their broad range of magnitudes in the recursion formulas that are used to compute them. Tests
with a simulated gravitational field show that the errors in so doing can be made less than the data noise at all latitudes
and up to expansion degree of at least 10,800. Neglecting numerically insignificant terms in the spherical harmonic series
also offers a computational savings of at least one third. 相似文献
14.
S. M. Nakiboglu 《Journal of Geodesy》1978,52(2):93-100
A variational principle for the Stokesian boundary value problem is derived using the Euler-Lagrange theory. The resulting
variational principle is then transformed into an equation determining the semi-major axis of the best fitting ellipsoid which
fulfills the conditionU
0
=W
0
.
The computations using three different geopotential models yields the semi-major axis of the earth ellipsoid asa=6378145.4 metres for the flatteningf=1/298.2564. The corresponding equatorial gravity and the geopotential number are computed as γa=978029.59 mgals andU
0=W
0=6.26367371 106 kgalmeters respectively. 相似文献
15.
本文联合T/P数据、T/P新轨道数据、ERS数据、GFO数据、GeosatGM数据和ERS-1/168数据,用测高卫星记录点的位置信息直接计算沿轨大地水准面的方向导数,结合测线轨迹方向的方位角在交叉点处推求垂线偏差,然后利用逆Vening-Meinesz公式计算了中国近海(0o~41oN,105o~132oN)2′×2′格网分辨率的海域重力异常模型。将其与CLS_SHOW99重力异常模型比较,统计结果表示与该模型差异的RMS为8.15mgal,在剔除差值大于20mgal的点(剔除3.3%)以后,RMS为4.72mgal;与某海区船测重力异常比较的RMS为8.91mgal。 相似文献
16.
A new gravity map, a new marine geoid around Japan and the detection of the Kuroshio current 总被引:3,自引:0,他引:3
About half a million marine gravity measurements over a 30∘×30∘ area centered on Japan have been processed and adjusted to produce a new free-air gravity map from a 5′×5′ grid. This map
seems to have a better resolution than those previously published as measured by its correlation with bathymetry. The grid
was used together with a high-degree and -order spherical harmonics geopotential model to compute a detailed geoid with two
methods: Stokes integral and collocation. Comparisons with other available geoidal surfaces derived either from gravity or
from satellite altimetry were made especially to test the ability of this new geoid at showing the sea surface topography
as mapped by the Topex/Poseidon satellite. Over 2 months (6 cycles) the dynamic topography at ascending passes in the region
(23∘47∘N and 123∘147∘E) was mapped to study the variability of the Kuroshio current.
Received: 15 July 1994 / Accepted: 17 February 1997 相似文献
17.
Richard H. Rapp 《Journal of Geodesy》1982,56(2):84-94
A spherical harmonic expansion of the earth's gravitational potential and equivalent rock topography to degree and order 180
is described. The potential implied by the topography considered as uncompensated and with isostatic compensation has been
computed. Good agreement with the observed potential field is found when the depth of compensation in the Airy theory is assumed
to be 50 km. At the higher degrees the correlation coefficient between the potential expansion and the equivalent rock topography
is about 0.5.
The Lachapelle equations for the topographic isostatic potential were tested using 1ox1o equivalent rock topography. The degree variances agree at the lower degrees but at degree 36 the Lachapelle results using
5o data underestimate the potential degree variances by about one-third. 相似文献
18.
A methodology for precise determination of the fundamental geodetic parameter w
0, the potential value of the Gauss–Listing geoid, as well as its time derivative 0, is presented. The method is based on: (1) ellipsoidal harmonic expansion of the external gravitational field of the Earth
to degree/order 360/360 (130 321 coefficients; http://www.uni-stuttgard.de/gi/research/ index.html projects) with respect
to the International Reference Ellipsoid WGD2000, at the GPS positioned stations; and (2) ellipsoidal free-air gravity reduction
of degree/order 360/360, based on orthometric heights of the GPS-positioned stations. The method has been numerically tested
for the data of three GPS campaigns of the Baltic Sea Level project (epochs 1990.8,1993.4 and 1997.4). New w
0 and 0 values (w
0=62 636 855.75 ± 0.21 m2/s2, 0=−0.0099±0.00079 m2/s2 per year, w
0/&γmacr;=6 379 781.502 m,0/&γmacr;=1.0 mm/year, and &γmacr;= −9.81802523 m2/s2) for the test region (Baltic Sea) were obtained. As by-products of the main study, the following were also determined: (1)
the high-resolution sea surface topography map for the Baltic Sea; (2) the most accurate regional geoid amongst four different
regional Gauss–Listing geoids currently proposed for the Baltic Sea; and (3) the difference between the national height datums
of countries around the Baltic Sea.
Received: 14 August 2000 / Accepted: 19 June 2001 相似文献
19.
Spherical harmonic expansions of the geopotential are frequently used for modelling the earth’s gravity field. Degree and
order of recently available models go up to 360, corresponding to a resolution of about50 km. Thus, the high degree potential coefficients can be verified nowadays even by locally distributed sets of terrestrial gravity
anomalies. These verifications are important when combining the short wavelength model impact, e.g. for regional geoid determinations
by means of collocation solutions. A method based on integral formulae is presented, enabling the improvement of geopotential
models with respect to non-global distributed gravity anomalies. To illustrate the foregoing, geoid computations are carried
out for the area of Iran, introducing theGPM2 geopotential model in combination with available regional gravity data. The accuracy of the geoid determination is estimated
from a comparison with Doppler and levelling data to ±1.4m. 相似文献
20.
Defining the distortion of a conformal map projection as the oscillation of the logarithm of its infinitesimal-scale σ, Chebyshev’s principle states that the best (minimum distortion) conformal map projection over a given region Ω of the ellipsoid is characterized by the property that σ is constant on the boundary of that region. Starting from a first map of Ω, we show how to compute the distortion δ0(Ω) of this Chebyshev’s projection. We prove that this minimum possible conformal mapping distortion associated with Ω coincides with the absolute value of the minimum of the solution of a Dirichlet boundary-value problem for an elliptic partial differential equation in divergence form and with homogeneous boundary condition. If the first map is conformal, the partial differential equation becomes a Poisson equation for the Laplace operator. As an example, we compute the minimum conformal distortion associated with peninsular Spain. Using longitude and isometric latitude as coordinates, we solve the corresponding boundary-value problem with the finite element method, obtaining δ0(Ω)=0.74869×10−3. We also quantify the distortions δl and δutm of the best conformal conic and UTM (zone 30) projections over peninsular Spain respectively. We get δl=2.30202×10−3 and δutm=3.33784×10−3. 相似文献