An existence theorem for absolute geodetic singularities

Summary. It is shown that disturbing masses can be found which can give locally rise in the external gravity field of a spherically symmetric earth to singularities both in geodetic and in gravity space coordinates; in this connection the concept of absolute geodetic singularities is introduced. It is also shown that the previous results hold independently of the mass of the disturbing body and for densities of the disturbing mass which can be lower than the mean Earth density. The connection between the geodetic singularity problem and the study and design of gravity gradient compensators for orbital experiments is put into light.     

Choice of norm for the density distribution of the Earth

Summary. The determination of the density distribution of the Earth from gravity data is called the inverse gravimetric problem. A unique solution to this problem may be obtained by introducing a priori data concerning the covariance of density anomalies. This is equivalent to requiring the density to fulfil a minimum norm condition. The generally used norm is the one equal to the integral of the square of the density distribution ( L² -norm), the use of which implies that blocks of constant density are uncorrelated. It is shown that for harmonic anomalous density distributions this leads to an external gravity field with a power spectrum (degree-variances) which tends too slowly to zero, i.e. implying gravity anomalies much less correlated than actually observed. It is proposed to use a stronger norm, equal to the integral of the square sum of the derivatives of the density distribution. As a consequence of this, base functions which are constant within blocks, are no longer a natural choice when solving the inverse gravimetric problem. Instead a block with a linearly varying density may be used. A formula for the potential of such a block is derived.     

差分GPS在乌拉特后旗1:25万区域重力调查中的应用

测地工作精度的高低直接影响着重力测量的精度,研究并采取行之有效的测地方法来配合重力测量工作是十分必要的。乌拉特后旗1:25万区域重力调查GPS测地工作采用1954年北京坐标系,重力测点平面位置、正常高采用差分GPS测量方法。结果表明,采用此方法既能满足技术要求,又达到了省时省力,且达到了重力测量的精度要求。     

The multibody space-time geodetic boundary value problem and the Honkasalo term

Summary. The geodetic boundary value problem is solved by taking temporal changes of geometry and gravity into account. It is aimed to get complete information about the mechanical response of the Earth to an external tidal force which is only imperfectly known. The information is used to solve the inverse Love-Shida problem, e.g. by computing the Love -Shida operators. The general solution (4.25), (4.26) is specialized for a perfectly known external gravity field and for a free fixed boundary of the Earth.     

Modelling of ground deformation and gravity fields using finite element method: an application to Etna volcano

Elastic finite element models are applied to investigate the effects of topography and medium heterogeneities on the surface deformation and the gravity field produced by volcanic pressure sources. Changes in the gravity field cannot be interpreted only in terms of gain of mass disregarding the ground deformation of the rocks surrounding the source. Contributions to gravity changes depend also on surface and subsurface mass redistribution driven by dilation of the volcanic source. Both ground deformation and gravity changes were firstly evaluated by solving a coupled axisymmetric problem to estimate the effects of topography and medium heterogeneities. Numerical results show significant discrepancies in the ground deformation and gravity field compared to those predicted by analytical solutions, which disregard topography, elastic heterogeneities and density subsurface structures. With this in mind, we reviewed the expected gravity changes accompanying the 1993–1997 inflation phase on Mt Etna by setting up a fully 3-D finite element model in which we used the real topography, to include the geometry, and seismic tomography, to infer the crustal heterogeneities. The inflation phase was clearly detected by different geodetic techniques (EDM, GPS, SAR and levelling data) that showed a uniform expansion of the overall volcano edifice. When the gravity data are integrated with ground deformation data and a coupled FEM modelling was solved, a mass intrusion could have occurred at depth to justify both ground deformation and gravity observations.     

3D Geological modelling and gravity inversion of a structurally complex carbonate area: application for karstified massif localization

This paper proposes a new methodology to improve the location of potential karstified areas by gravity inversion of a 3D geological model. A geological 3D model is built from surface observations, 2D seismic reflection profiles and well data. The reliability of this geological 3D model obtained from integration, interpretation and interpolation of such data is first tested against the structural consistency of the model. Its theoretical gravimetric response is compared to gravity field during the forward problem in order to evaluate the validity/robustness of the geological model. The coherency between the gravity field and the gravimetric response is tested. The litho‐inversion modelling quantifies the distribution of rock density in a probabilistic way, taking into account the geology and physical properties of rocks, while respecting the geological structures represented in the 3D model. The result of the inversion process provides a density distribution within carbonate formations that can be discussed in term of karstification distribution. Thus, lower densities correlate with areas that are strongly karstified. Conversely, higher than mean densities are found in carbonate formations mostly located under marly and impervious formations, preserving carbonate from karstification and paleokarstification.     

Internal motions and the figures of planets

Summary. The problem discussed in this paper is the effect of specified internal motions upon the external figure of a rotating planet.
An equation for the perturbation of a surface of constant density is derived. The properties of motions which can be maintained in a gravitational field are established.     

旅行分布的重力模式与交通模型的关系

从交通问题的对偶规划出发, 引入由Beckmann 和Wallance, Golob 和Beckman 等提出的不确定性效用方法, 建立交通问题和双约束重力模型的关系。证明当旅行者效用概率分布密度函数的标准差趋于零时, 双约束重力模型中的距离摩擦系数趋于正无穷, 由双约束重力模型确定的旅行分布使得总的交通成本达到最小。在这种关系中, 双约束重力模型中平衡因子的作用是市场调整旅行终点服务价格进而调整旅行者消费者剩余的结果。通过建立的分 析方法对北京、上海、广州、西安、武汉、成都和昆明等七个城市间航空交通的应用, 发现多数航线的模拟较好。结果表明各终点的差异性、消费者偏好的不同、交通工具的替代性和旅行目的之差异性等可以导致一些较大的误差, 这些差异性可以采用对起终点对的单位交通费用的调整来体现, 从而达到较好的模拟和分析效果。     

Three-dimensional gravity inversion for Moho depth at rifted continental margins incorporating a lithosphere thermal gravity anomaly correction

This paper describes a method for determining Moho depth, lithosphere thinning factor (γ= 1 − 1/β) and the location of the ocean–continent transition at rifted continental margins using 3-D gravity inversion which includes a correction for the large negative lithosphere thermal gravity anomaly within continental margin lithosphere. The lateral density changes caused by the elevated geotherm in thinned continental margin and adjacent ocean basin lithosphere produce a significant lithosphere thermal gravity anomaly which may be in excess of −100 mGal, and for which a correction must be made in order to determine Moho depth accurately from gravity inversion. We describe a method of iteratively calculating the lithosphere thermal gravity anomaly using a lithosphere thermal model to give the present-day temperature field from which we calculate the lithosphere thermal density and gravity anomalies. For continental margin lithosphere, the lithosphere thermal perturbation is calculated from the lithosphere thinning factor (γ= 1 − 1/β) obtained from crustal thinning determined by gravity inversion and breakup age for thermal re-equilibration time. For oceanic lithosphere, the lithosphere thermal model used to predict the lithosphere thermal gravity anomaly may be conditioned using ocean isochrons from plate reconstruction models to provide the age and location of oceanic lithosphere. A correction is made for crustal melt addition due to decompression melting during continental breakup and seafloor spreading. We investigate the sensitivity of the lithosphere thermal gravity anomaly and the predicted Moho depth from gravity inversion at continental rifted margins to the methods used to calculate and condition the lithosphere thermal model using both synthetic models and examples from the North Atlantic.     

Core–mantle boundary deformations and J2 variations resulting from the 2004 Sumatra earthquake

The deformation at the core–mantle boundary produced by the 2004 Sumatra earthquake is investigated by means of a semi-analytic theoretical model of global coseismic and postseismic deformation, predicting a millimetric coseismic perturbation over a large portion of the core–mantle boundary. Spectral features of such deformations are analysed and discussed. The time-dependent postseismic evolution of the elliptical part of the gravity field ( J ₂) is also computed for different asthenosphere viscosity models. Our results show that, for asthenospheric viscosities smaller than 10¹⁸ Pa s, the postseismic J ₂ variation in the next years is expected to leave a detectable signal in geodetic observations.     

Insights into the lithospheric structure and tectonic setting of the Barents Sea region from isostatic considerations

We study the tectonic setting and lithospheric structure of the greater Barents Sea region by investigating its isostatic state and its gravity field. 3-D forward density modelling utilizing available information from seismic data and boreholes shows an apparent shift between the level of observed and modelled gravity anomalies. This difference cannot be solely explained by changes in crustal density. Furthermore, isostatic calculations show that the present crustal thickness of 35–37 km in the Eastern Barents Sea is greater than required to isostatically balance the deep basins of the area (>19 km). To isostatically compensate the missing masses from the thick crust and deep basins and to adequately explain the gravity field, high-density material (3300–3350 kg m⁻³) in the lithospheric mantle below the Eastern Barents Sea is needed. The distribution of mantle densities shows a regional division between the Western and Eastern Barents and Kara Seas. In addition, a band of high-densities is observed in the lower crust along the transition zone from the Eastern to Western Barents Sea. The distribution of high-density material in the crust and mantle suggests a connection to the Neoproterozoic Timanide orogen and argues against the presence of a Caledonian suture in the Eastern Barents Sea. Furthermore, the results indicate that the basins of the Western Barents Sea are mainly affected by rifting, while the Eastern Barents Sea basins are located on a stable continental platform.     

Topological approach to remote sensing

Summary. The topological problem underlying remote sensing is analysed by determining the geometric singularities that an unknown surface or structure generically impresses on a sensing wavefield. It is shown that the analytical singularities observed in scattering amplitudes and echograms are produced by the topological singularities of the scattering system. Imposing the principle of structural stability on the inverse scattering problem, the singularities that generically occur in recorded signals, travel-time curves, surface contour maps and Fresnel-zone topographies can, together with the associated high-intensity diffraction patterns, be classified into a few universal standard forms described by catastrophe polynomials. As the source-receiver positions vary, the patterns change their morphologies in terms of specific bifurcation sets. By applying singularity and bifurcation theory to allow the effects of caustics (both in ray and wave theory) to be incorporated into three-dimensional techniques for reconstructing surfaces and subsurface structures from their echoes, the interpretation process is considerably simplified and permits an on-site 3D survey. Universal power laws for singularity-dominated echo amplitude variations with the source frequency are deduced. The shape of a scattering surface is reconstructed using the high-frequency regime alone. Discontinuities in the surface, edges and faults, are discussed in terms of constraint catastrophes and the patterns they produce in echograms are classified.     

Local and regional components of western Aegean deformation extracted from 100 years of geodetic displacement measurements

Our objectives are as follows. First, we wish to develop a methodology to recover the long-term component of deformation from any set of distributed, time-averaged geodetic strain measurements that were subject to seismic disturbance, given a catalogue of local seismicity that occurred during the measurement period. Second, using seismic and geodetic data sets that span approximately 100 years, we apply this technique in the western Aegean to assess the role of local seismicity in regional deformation. The methodology is developed using a model for crustal deformation constructed from a long-term, smooth regional strain field combined with instantaneous, local perturbations from upper-crustal earthquakes approximated by static elastic dislocations. By inverting geodetic displacements for the smooth field while simultaneously floating influential but uncertain earthquake source parameters, an estimate of the regional component of deformation that is approximately independent of the seismicity can be made. In the western Aegean we find that the horizontal component of regional deformation can be described with minor inaccuracy by a quadratic relative displacement field. The principal horizontal extensional axes calculated from the regionally smooth displacement field agree in orientation with the T-axes of earthquakes in the region. These observations indicate that the instantaneous elastic strain of the 10 km thick seismogenic layer is driven by a stress field that is smooth on the scale of the geodetic network as a whole, 200-300 km.     

High-resolution mean sea surface computed with altimeter data of Ers-1 (geodetic mission) and topex-poseidon

The remote-sensing satellite ERS-1, launched in 1991 to study the Earth's environment, was placed on a geodetic (168-day repeat) orbit between 1994 April and 1995 March to map, through altimetric measurements, the gravity field over the whole oceanic domain with a resolution of 8 km at the equator in both along-track and cross-track directions. We have analysed the precise altimeter data of the geodetic mission, and, by also using one year of Topex-Poseidon altimeter data, we have computed a global high-resolution mean sea surface. The various steps involved in pre-processing the ERS-1 data consisted of correcting the data for environmental factors, editing, and reducing, through crossover analyses, the radial orbit error, which directly affects sea-surface height measurements. For this purpose, we adjusted sinusoids at 1 and 2 cycle rev⁻¹ along the ERS-1 profiles in order to minimize crossover differences between ERS-1 and yearly averaged Topex-Poseidon profiles. In effect, the orbit of Topex-Poseidon is very accurately determined (within 2–3 cm for the radial component), so Topex-Poseidon altimeter profiles can serve as a reference to reduce the ERS-1 radial orbit error. The ERS-1 residual orbit error was further reduced through a second crossover analysis between all ascending and descending profiles of the geodetic mission. The along-track ERS-1 and Topex-Poseidon data were then interpolated over the whole oceanic domain on a regular grid of 1/16°× 1/16° size. The mapping of the gridded sea-surface heights reveals the very fine structure of the marine geoid, up until now unknown at a global scale. This new data set will be most useful for marine geophysical and tectonic investigations.     

ESTABLISHMENT OF SURVEYING SYSTEM OF THE GREAT WALL STATION

It is the main task of the antarctic surveying expedition to establish the surveying system. In three times of the antarctic surveying research expeditions, a complete and accurate surveying system has been established in the Chinese Great Wall Station area. The surveying system includes geodetic coordinate system, elevation system and gravity reference system. In this paper, the surveying methods and the mathematical models for establishing these systems are discussed, and the accuracy of results is analysed.     

Source ambiguity from an estimation of the scaling exponent of potential field power spectra

An analysis of the field scaling power spectrum yields useful information about the source distribution, but it is uncertain whether deterministic, random, fractal or mixed approaches have to be used for the interpretation. To this end, the scaling properties of potential field spectra are analysed for a number of different source models of geological interest. Besides the models of Naidu (purely random sources) and Spector and Grant (gross block statistical ensembles) we consider other types of density and magnetization distributions with spectral exponents in the fractal range, such as a single homogeneous body with a random white source distribution. Spectral slopes in the fractal range are obtained.
We also study the effects of important natural sources, such as salt domes and sedimentary basins, representing them with simple Gaussians or combinations of Gaussian signals. The same spectral slopes as for gravity signals generated by 3-D fractal source distributions are found for them. Hence the power law decay of the field is not a characteristic only of fractal source models.
If a 3-D fractal source distribution is assumed a priori , a way of verifying the goodness of the model is to examine the whitened field at source level. The probability that the whitened field derives from a random white population is estimated for synthetic and real anomalies by applying the usual statistical tests.     

Tidal gravity in Britain: tidal loading and the spatial distribution of the marine tide

Summary. Tidal gravity measurements have been made at six sites in Britain with two nulled LaCoste and Romberg Earth tide gravitymeters. The M ₂ observations from these and two further sites are compared with calculations of the tidal loading from the seas around the British Isles and the major oceans. Models of the M ₂ marine tides are convolved with Green's functions for appropriate radially stratified Earth models. The differences between the M ₂ observations and the theoretical calculations are less than 0.6 μ gals and it is shown that these differences contain further information concerning the errors in the marine tide models. The M ₂ marine tides on the north-west European continental shelf are reasonably well known and this allows a useful test of the feasibility of using tidal gravity measurements for the inverse ocean tide problem in areas where the ocean tides are less well known. The differential gravity loading signal between pairs of gravity stations is shown to be important for considerations of the uniqueness and accuracy of the inverse problem. M ₂ tidal gravity loading maps for the British Isles and Europe have been produced which are of use in making corrections to various geodetic measurements.     

Surface motion of a fluid planet induced by impacts

In order to approximate the free-surface motion of an Earth-sized planet subjected to a giant impact, we have described the excitation of body and surface waves in a spherical compressible fluid planet without gravity or intrinsic material attenuation for a buried explosion source. Using the mode summation method, we obtained an analytical solution for the surface motion of the fluid planet in terms of an infinite series involving the products of spherical Bessel functions and Legendre polynomials. We established a closed form expression for the mode summation excitation coefficient for a spherical buried explosion source, and then calculated the surface motion for different spherical explosion source radii a (for cases of   a / R = 0.001  to 0.035, R is the radius of the Earth) We also studied the effect of placing the explosion source at different radii r ₀ (for cases of   r ₀/ R = 0.90  to 0.96) from the centre of the planet. The amplitude of the quasi-surface waves depends substantially on a / R , and slightly on   r ₀/ R   . For example, in our base-line case,   a / R = 0.03, r ₀/ R = 0.96  , the free-surface velocity above the source is 0.26 c , whereas antipodal to the source, the peak free surface velocity is 0.19 c . Here c is the acoustic velocity of the fluid planet. These results can then be applied to studies of atmosphere erosion via blow-off caused by asteroid impacts.     

社区人居环境吸引力研究——以大连市为例

城市人居环境研究中一个重要的理论问题就是人居环境吸引力。本文提出人居环境吸引力和引力场的概念,构建了社区人居环境满意度指标体系,运用引力势能模型计算出184个社区的人居环境引力势能,借助MapInfo软件中的Vertical Mapper工具得到大连市人居环境引力势能等值线图和空间分布图,研究表明:(1)运用引力场理论对人居环境评价研究,是一种有效的评价方法;(2)大连市社区人居环境供求关系在空间上形成了"两心一带"分布格局;(3)大连市社区人居环境引力势能空间分异明显,大致呈东北西南向"带状"分布,在中心广场、西安路锦辉商场等值线由同心圆状向外逐渐递减;(4)社区人居环境供求关系与引力势能成正比,与社区与引力势能中心的距离呈反比,城市空间发展结构、海岸带是大连人居环境引力势能空间分布的驱动机制。     

Analysis of gravity fields in the northwestern Himalayas and Kohistan region using deep seismic sounding data

A Bouguer gravity anomaly map of the NW Himalayas and parts of the Kohistan/Hindukush region has been prepared using all available gravity data. Analysis of the gravity field has been carried out along a profile extending from Gujranwala (located near the edge of the Indian shield) to the Haramosh massif in a NNE–SSW direction. The gravity profile is located close to the DSS profile shot under the USSR–India scientific collaborative programme. Velocity information available along different parts of the profile has been used to infer values of crustal and upper mantle density.
The observed gravity field (Bouguer) has been interpreted in terms of Moho depth and density contrast between the crust and the mantle. The Moho depth is interpreted as increasing from nearly 35 km near the edge of the Indian shield to 75 km (below sea-level) underneath the Haramosh massif. A similar model is applicable to a profile passing to the west of Nanga Parbat massif, from Gujranwala to Ghizar, through the Kohistan region. However, along this profile high-density lower-crustal rocks appear to have been emplaced in the upper part along the main mantle thrust. The nature of isostatic compensation prevailing underneath the Himalayas has been discussed, as has the theory of lithospheric flexure proposed by Karner & Watts and Lyon-Caen & Molnar. It is felt that although these ideas explain the broad features of the Moho configuration as observed in the NW Himalayas, there are significant departures. The role of tectonic forces in shaping the Moho and causing changes in the density of the crust cannot be denied.