Ellipsoidal Neumann geodetic boundary-value problem based on surface gravity disturbances: case study of Iran

An ellipsoidal Neumann type geodetic boundary-value problem (GBVP) for the computation of disturbing potential on the surface of the Earth based on the surface gravity disturbance as the boundary data is formulated. The solution methodology of the GBVP can be algorithmically summarized as follows: (i) using global navigation satellite systems (GNSS) coordinates of the gravity stations, the surface gravity disturbances are generated as the boundary data. (ii) Applying the deflection correction to the gravity disturbances to arrive at the derivative of the surface disturbing potential along the ellipsoidal normal. (iii) Removing the low frequencies part of the gravity field using harmonic expansion to degree and order 110. (iv) Using the short wavelength part of the corrected gravity disturbances derived in the previous section as the boundary data within the constructed GBVP to derive the short wavelength disturbing potential over the Earth surface. (v) The computed shortwave length signals of disturbing potentials are converted to disturbing potential values by restoring the removed effects.     

地球重力场恢复中的位旋转效应

分析了地球自转引起的位旋转效应公式中采用近似速度的影响. 对一组GFZ的快速科学轨道、一组TUM的约化动力法轨道以及一组GFZ的事后科学轨道,计算了星历提供的速度与只有地球引力场对卫星产生作用时的卫星速度的差值,其中参考重力场模型分别采用EGM96、EIGEN2和EIGEN_CG01C. 通过比较得出：轨道数据与EIGEN2地球重力场模型的自恰性优于EGM96和EIGEN_CG01C地球重力场模型. 速度差各分量的变化具有很明显的周期性且与卫星轨道的运行周期相吻合. 当要求在卫星轨迹处获得1m2/s2精度的扰动位时,也即要求位旋转效应公式中卫星速度的近似精度小于2mm/s时,GFZ的快速科学轨道、TUM的约化动力法轨道只需要剔除那些速度精度不满足要求的卫星轨迹点;当要求在卫星轨迹处获得05m2/s2精度的扰动位时,应当重新估算上述轨道的速度信息,或采用精度更高的GFZ事后科学轨道.     

Ellipsoidal vertical deflections and ellipsoidal gravity disturbance: Case studies

First, we present three different definitions of the vertical which relate to (i) astronomical longitude and astronomical latitude as spherical coordinates in gravity space, (ii) Gauss surface normal coordinates (also called geodetic coordinates) of type ellipsoidal longitude and ellipsoidal latitude and (iii) Jacobi ellipsoidal coordinates of type spheroidal longitude and spheroidal latitude in geometry space. Up to terms of second order those vertical deflections agree to each other. Vertical deflections and gravity disturbances relate to a reference gravity potential. In order to refer the horizontal and vertical components of the disturbing gravity field to a reference gravity field, which is physically meaningful, we have chosen the Somigliana-Pizzetti gravity potential as well as its gradient. Second, we give a new closed-form representation of Somigliana-Pizzetti gravity, accurate to the sub Nano Gal level. Third, we represent the gravitational disturbing potential in terms of Jacobi ellipsoidal harmonics. As soon as we take reference to a normal potential of Somigliana-Pizzetti type, the ellipsoidal harmonics of degree/order (0,0), (1,0), (1, − 1), (1,1) and (2,0) are eliminated from the gravitational disturbing potential. Fourth, we compute in all detail the gradient of the gravitational disturbing potential, in particular in orthonormal ellipsoidal vector harmonics. Proper weighting functions for orthonormality on the International Reference Ellipsoid are constructed and tabulated. In this way, we finally arrive at an ellipsoidal harmonic representation of vertical deflections and gravity disturbances. Fifth, for an ellipsoidal harmonic Gravity Earth Model (SEGEN: http://www.uni-stuttgart.de/gi/research/paper/coefficients/coefficients.zip) up to degree/order 360/360 we compute the global maps of ellipsoidal vertical deflections and ellipsoidal gravity disturbances which transfer a great amount of geophysical information in a properly chosen equiareal ellipsoidal map projection.     

Spectral Combination of Spherical Gradiometric Boundary-Value Problems: A Theoretical Study

The Earth’s gravity potential can be determined from its second-order partial derivatives using the spherical gradiometric boundary-value problems which have three integral solutions. The problem of merging these solutions by spectral combination is the main subject of this paper. Integral estimators of biased- and unbiased-types are presented for recovering the disturbing gravity potential from gravity gradients. It is shown that only kernels of the biased-type integral estimators are suitable for simultaneous downward continuation and combination of gravity gradients. Numerical results show insignificant practical difference between the biased and unbiased estimators at sea level and the contribution of far-zone gravity gradients remains significant for integration. These contributions depend on the noise level of the gravity gradients at higher levels than sea. In the cases of combining the gravity gradients, contaminated with Gaussian noise, at sea and 250?km levels the errors of the estimated geoid heights are about 10 and 3 times smaller than those obtained by each integral.     

Integral inversion of SST data of type GRACE

Satellite missions CHAMP and GRACE dedicated to global mapping of the Earth’s gravity field yield accurate satellite-to-satellite tracking (SST) data used for recovery of global geopotential models usually in a form of a finite set of Stokes’s coefficients. The US-German Gravity Recovery And Climate Experiment (GRACE) yields SST data in both the high-low and low-low mode. Observed satellite positions and changes in the intersatellite range can be inverted through the Newtonian equation of motion into values of the unknown geopotential. The geopotential is usually approximated in observation equations by a truncated harmonic series with unknown coefficients. An alternative approach based on integral inversion of the SST data of type GRACE into discrete values of the geopotential at a geocentric sphere is discussed in this article. In this approach, observation equations have a form of Green’s surface integrals with scalar-valued integral kernels. Despite their higher complexity, the kernel functions exhibit features typical for other integral kernels used in geodesy for inversion of gravity field data. The two approaches are discussed and compared based on their relative advantages and intended applications. The combination of heterogeneous gravity data through integral equations is also outlined in the article. panovak@kma.zcu.cz     

From Satellite Gradiometry Data to Subcrustal Stress Due to Mantle Convection

Subcrustal stress induced by mantle convection can be determined by the Earth’s gravitational potential. In this study, the spherical harmonic expansion of the simplified Navier–Stokes equation is developed further so satellite gradiometry data (SGD) can be used to determine the subcrustal stress. To do so, we present two methods for producing the stress components or an equivalent function thereof, the so-called S function, from which the stress components can be computed numerically. First, some integral estimators are presented to integrate the SGD and deliver the stress components and/or the S function. Second, integral equations are constructed for inversion of the SGD to the aforementioned quantities. The kernel functions of the integrals of both approaches are plotted and interpreted. The behaviour of the integral kernels is dependent on the signal and noise spectra in the first approach whilst it does not depend on extra information in the second method. It is shown that recovering the stress from the vertical–vertical gradients, using the integral estimators presented, is suitable, but when using the integral equations the vertical–vertical gradients are recommended for recovering the S function and the vertical–horizontal gradients for the stress components. This study is theoretical and numerical results using synthetic or real data are not given.     

Use of Green’s function for determining the disturbing potential of an ellipsoidal Earth

A spherical approximation makes the basis for a majority of formulas in physical geodesy. However, the present-day accuracy in determining the disturbing potential requires an ellipsoidal approximation. The paper deals with constructing Green’s function for an ellipsoidal Earth by an ellipsoidal harmonic expansion and using it for determining the disturbing potential. From the result obtained the part that corresponds to the spherical approximation has been extracted. Green’s function is known to depend just on the geometry of the surface where boundary values are given. Thus, it can be calculated irrespective of the gravity data completeness. No changes of gravity data have an effect on Green’s function and they can be easily taken into account if the function has already been constructed. Such a method, therefore, can be useful in determining the disturbing potential of an ellipsoidal Earth.     

On inversion of the second- and third-order gravitational tensors by Stokes’ integral formula for a regional gravity recovery

A regional recovery of the Earth’s gravity field from satellite observables has become particularly important in various geoscience studies in order to better localize stochastic properties of observed data, while allowing the inversion of a large amount of data, collected with a high spatial resolution only over the area of interest. One way of doing this is to use observables, which have a more localized support. As acquired in recent studies related to a regional inversion of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) data, the satellite gravity-gradient observables have a more localized support than the gravity observations. Following this principle, we compare here the performance of the second- and third-order derivatives of the gravitational potential in context of a regional gravity modeling, namely estimating the gravity anomalies. A functional relation between these two types of observables and the gravity anomalies is formulated by means of the extended Stokes’ integral formula (or more explicitly its second- and third-order derivatives) while the inverse solution is carried out by applying a least-squares technique and the ill-posed inverse problem is stabilized by applying Tikhonov’s regularization. Our results reveal that the third-order radial derivatives of the gravitational potential are the most suitable among investigated input data types for a regional gravity recovery, because these observables preserve more information on a higher-frequency part of the gravitational spectrum compared to the vertical gravitational gradients. We also demonstrate that the higher-order horizontal derivatives of the gravitational potential do not necessary improve the results. We explain this by the fact that most of the gravity signal is comprised in its radial component, while the horizontal components are considerably less sensitive to spatial variations of the gravity field.     

On the discrete outer gravity field

Summary In the present paper the gravity field of the earth in the neighbourhood of the local disturbing masses is studied. The object of the method presented consists of the approximation of the disturbing potentialT _h , which fulfils Laplace's equation outside disturbing masses, on the earth's surface the fundamental boundary value condition of gravity and in infinity it is to be regular by the approximation of the disturbing potential (or by the discrete disturbing potential)T _h , which fulfils the respective finite difference approximation of Laplace's equation and the boundary value conditions in infinity and on the earth's surface. It is also shown that the approximation of the disturbing potentialT _h has the same properties as the disturbing potentialT. The method under consideration will be derived quite generally without any hypothesis about the distribution of the mass between the earth's surface and the geoid. It commences from the gravity data related to the earth's surface only-from the given geodetic measurements.     

Regional gravity field modeling based on rectangular harmonic analysis

Regional gravity field modeling with high-precision and high-resolution is one of the most important scientific objectives in geodesy,and can provide fundamental information for geophysics,geodynamics,seismology,and mineral exploration.Rectangular harmonic analysis(RHA)is proposed for regional gravity field modeling in this paper.By solving the Laplace’s equation of gravitational potential in local Cartesian coordinate system,the rectangular harmonic expansions of disturbing potential,gravity anomaly,gravity disturbance,geoid undulation and deflection of the vertical are derived,and so are the formula for signal degree variance and error degree variance of the rectangular harmonic coefficients(RHC).We also present the mathematical model and detailed algorithm for the solution of RHC using RHA from gravity observations.In order to reduce the edge effects caused by periodic continuation in RHA,we propose the strategy of extending the size of computation domain.The RHA-based modeling method is validated by conducting numerical experiments based on simulated ground and airborne gravity data that are generated from geopotential model EGM2008 and contaminated by Gauss white noise with standard deviation of 2 mGal.The accuracy of the 2.5′×2.5′geoid undulations computed from ground and airborne gravity data is 1 and 1.4cm,respectively.The standard error of the gravity disturbances that downward continued from the flight height of 4 km to the geoid is only 3.1 mGal.Numerical results confirm that RHA is able to provide a reliable and accurate regional gravity field model,which may be a new option for the representation of the fine structure of regional gravity field.     

The boundary integral calculations of the forward problem for D.C. sounding and MMR methods for a 3-D body near a vertical contact

Summary The paper presents comprehensive theory based on the boundary integral method for calculations of the electric potential, electric field and corresponding magnetic field due to a pair of D.C. source electrodes near a vertical resistivity contact in the halfspace, indlucing a 3-D disturbing body in the vicinity of the contact. Special attention is paid to the case when the disturbing body touches the vertical contact. Results of numerical calculations are presented in the form of sounding curves and a set of isoline graphs for potential, components of the electric and magnetic field (total and anomalous) on the surface of the Earth. It is shown that the presence of the disturbing body at the contact is most pronounced in the electrical characteristics. Anomalies in the magnetic field are small in comparison to the field due to the electric current in the electrode cable and primary currents flowing from the electrodes.     

General inverse of Stokes, Vening-Meinesz and Molodensky formulae

The undulation of the geoid, the gravity anomaly and the deflection of the vertical are the three basic observations describing the shape and the gravity field of the earth. The Stokes’ formula that computes the undulation of the geoid using the gravity anomaly on the geoid under spherical approximate conditions was first put forward by Stokes[1]. According to Stokes’ theory, The Vening-Meinesz formula that computes the meridian and the prime vertical components of the deflection of the ve…     

On correct application of one-step inversion of gravity data

One of the main problems on the numerical solution of integral equations is the resolution of input data. Among the integral equations used in geodesy we have the “onestep inversion” based on the first derivative of the Poisson integral, which transforms gravity values on the Earth’s surface to the gravity potential on the reference ellipsoid. In this study, it is shown that the required spatial resolution of the input gravity data on the Earth’s surface for correct one-step inversion depends on the height of the computational region, the fact that if overlooked can cause totally wrong results. Consequently the following two major questions are posed: (i) How could one know whether the spatial resolution of the input gravity data for correct one-step inversion is sufficient? (ii) What should be done if the spatial resolution is not sufficient? By studying the behaviour of the integral kernel, an algorithm is presented which enables an appropriate answer to the former question. In order to address the latter question, a method is proposed to modify the integral kernel which overcomes the adverse effect of insufficient spatial resolution of the input gravity data. Our answers, which possess the novelty of the study, are numerically verified by means of real and simulated gravity data. The numerical results approve the efficiency of the proposed method in solving the problem of insufficient spatial resolution of the input gravity data for correct one-step inversion.     

A geoid solution for airborne gravity data

Airborne gravity data is usually attached with satellite positioning of data points, which allow for the direct determination of the gravity disturbance at flight level. Assuming a suitable gridding of such data, Hotine’s modified integral formula can be combined with an Earth Gravity Model for the computation of the disturbing potential (T) at flight level. Based on T and the gravity disturbance data, we directly downward continue T to the geoid, and we present the final solution for the geoid height, including topographic corrections. It can be proved that the Taylor expansion of T converges if the flight level is at least twice the height of the topography, and the terrain potential will not contribute to the topographic correction. Hence, the simple topographic bias of the Bouguer shell yields the only topographic correction. Some numerical results demonstrate the technique used for downward continuation and topographic correction.     

数值模拟估算低低卫-卫跟踪观测技术反演地球重力场的空间分辨率

从两个方面模拟研究了低低卫-卫跟踪观测技术恢复地球重力场的空间分辨率. 利用重力位系数作为扰动量，积分30天的轨道，研究重力位系数变化引起低低卫-卫跟踪星间距离和速率变化，结果表明，对于地球重力场模型EGM96的前120阶，998%和97%的位系数扰动引起星间距离和速率变化的均方差大于1×１０-５m和１×１０-７m/s，并且星间距离观测值对地球重力场的反应更为敏感. 不考虑非保守力误差的影响，用随机误差为１×１０－５m和１×１０－６m/s的星间距离和速率变化作模拟观测量，恢复了78阶地球重力场位系数，结果表明，采用随机误差为１×１０－５m的星间距离恢复地球重力场的精度明显高于１×１０－６m/s的星间速率结果，但是如果考虑非保守力误差影响，则星间测速的优越性大大增强.     

Optimal Model for Geoid Determination from Airborne Gravity

Two different approaches for transformation of airborne gravity disturbances, derived from gravity observations at low-elevation flying platforms, into geoidal undulations are formulated, tested and discussed in this contribution. Their mathematical models are based on Green's integral equations. They are in these two approaches defined at two different levels and also applied in a mutually reversed order. While one of these approaches corresponds to the classical method commonly applied in processing of ground gravity data, the other approach represents a new method for processing of gravity data in geoid determination that is unique to airborne gravimetry. Although theoretically equivalent in the continuous sense, both approaches are tested numerically for possible numerical advantages, especially due to the inverse of discretized Fredholm's integral equation of the first kind applied on different data. High-frequency synthetic gravity data burdened by the 2-mGal random noise, that are expected from current airborne gravity systems, are used for numerical testing. The results show that both approaches can deliver for the given data a comparable cm-level accuracy of the geoidal undulations. The new approach has, however, significantly higher computational efficiency. It would be thus recommended for real life geoid computations. Additional errors related to regularization of gravity data and the geoid, and to accuracy of the reference field, that would further deteriorate the quality of estimated geoidal undulations, are not considered in this study.     

基于GRACE卫星重力数据确定地球重力场模型WHU-GM-05

基于卫星轨道运动的能量积分方程,可导出利用卫星跟踪卫星数据求解地球重力场的实用公式.本文在Jekeli给出的公式基础上导出了基于能量守恒方程利用两颗低-低卫星跟踪的扰动位差求解重力位系数的严密关系式.基于两颗GRACE卫星的观测数据,采用本文导出的严密能量积分方法求解得到120阶的GRACE地球重力场模型,命名为WHU-GM-05;将WHU-GM-05模型与国际上同类重力场模型EIGEN-GRACE系列和GGM02S分别在阶方差和大地水准面高等方面作了比较,并与美国和中国的部分地区GPS水准观测值进行了精度分析.结果表明基于本文推导的严密双星能量守恒方程得到的WHU-GM-05重力场模型精度与国际上同类重力场模型的精度相当.     

Global Crust-Mantle Density Contrast Estimated from EGM2008, DTM2008, CRUST2.0, and ICE-5G

We compute globally the consolidated crust-stripped gravity disturbances/anomalies. These refined gravity field quantities are obtained from the EGM2008 gravity data after applying the topographic and crust density contrasts stripping corrections computed using the global topography/bathymetry model DTM2006.0, the global continental ice-thickness data ICE-5G, and the global crustal model CRUST2.0. All crust components density contrasts are defined relative to the reference crustal density of 2,670 kg/m³. We demonstrate that the consolidated crust-stripped gravity data have the strongest correlation with the crustal thickness. Therefore, they are the most suitable gravity data type for the recovery of the Moho density interface by means of the gravimetric modelling or inversion. The consolidated crust-stripped gravity data and the CRUST2.0 crust-thickness data are used to estimate the global average value of the crust-mantle density contrast. This is done by minimising the correlation between these refined gravity and crust-thickness data by adding the crust-mantle density contrast to the original reference crustal density of 2,670?kg/m³. The estimated values of 485 kg/m³ (for the refined gravity disturbances) and 481?kg/m³ (for the refined gravity anomalies) very closely agree with the value of the crust-mantle density contrast of 480?kg/m³, which is adopted in the definition of the Preliminary Reference Earth Model (PREM). This agreement is more likely due to the fact that our results of the gravimetric forward modelling are significantly constrained by the CRUST2.0 model density structure and crust-thickness data derived purely based on methods of seismic refraction.     

Far-zone contributions to the gravity field quantities by means of Molodensky’s truncation coefficients

To reduce the numerical complexity of inverse solutions to large systems of discretised integral equations in gravimetric geoid/quasigeoid modelling, the surface domain of Green’s integrals is subdivided into the near-zone and far-zone integration sub-domains. The inversion is performed for the near zone using regional detailed gravity data. The farzone contributions to the gravity field quantities are estimated from an available global geopotential model using techniques for a spherical harmonic analysis of the gravity field. For computing the far-zone contributions by means of Green’s integrals, truncation coefficients are applied. Different forms of truncation coefficients have been derived depending on a type of integrals in solving various geodetic boundary-value problems. In this study, we utilise Molodensky’s truncation coefficients to Green’s integrals for computing the far-zone contributions to the disturbing potential, the gravity disturbance, and the gravity anomaly. We also demonstrate that Molodensky’s truncation coefficients can be uniformly applied to all types of Green’s integrals used in solving the boundaryvalue problems. The numerical example of the far-zone contributions to the gravity field quantities is given over the area of study which comprises the Canadian Rocky Mountains. The coefficients of a global geopotential model and a detailed digital terrain model are used as input data.     

Discussion of Mean Gravity Along the Plumbline

According to the definition of the orthometric height, the mean value of gravity along the plumbline between the Earth's surface and the geoid is defined in an integral sense. In Helmert's (1890) definition of the orthometric height, a linear change of the gravity with depth is assumed. The mean gravity is determined so that the observed gravity at the Earth's surface is reduced to the approximate mid-point of the plumbline using Poincaré-Prey's gravity gradient. Niethammer (1932) and later Mader (1954) took into account the mean value of the gravimetric terrain correction within the topography considering the constant topographical density distribution along the plumbline (for more details see Heiskanen and Moritz, 1967). Vaníek et al. (1995) included the effect of the lateral variation of the topographical density into the definition of Helmert's orthometric height. Recently, Hwang and Hsiao (2003) discussed the influence of the vertical gradient of disturbing gravity on the orthometric heights. In this paper, the mean integral value of gravity along the plumbline within the topography is defined so that the actual topographical density distribution and the change of the disturbing gravity with depth are taken into account. Based on the definition of the mean gravity, the relation between the orthometric and normal heights is discussed.