Geodetic singularities in the gravity field of a non-homogeneous planet

Summary. An inverse geodetic singularity problem is considered for a non-homogeneous spherical planet. The singularity condition is expressed in terms of the density distribution and of the geometrical parameters of the configuration; the condition for the density distribution is deduced which gives rise to singularities of parabolic type in the external gravity field of the planet. The structure of the gravity field in the neighbourhood of the singularities is investigated in detail together with the behaviour of the gradients of the disturbances in the geodetic coordinates.     

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。     

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.     

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.     

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.     

Tidal gravity and ocean tide loading in Europe

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The results are presented from tidal gravity measurements at five sites in Europe using LaCoste and Romberg ET gravimeters. Improvements that we have made to the accuracies of these gravimeters are discussed. It is shown that the 'standard' calibration of the International Center for Earth Tides, used for worldwide tidal gravity profiles, is 1.2 per cent too high. The M₂ and O₁ observations are compared with model calculations of the Earth's body tide and ocean tide loading and it is shown that there is a very significant improvement in the agreement between observations and models compared to that obtained with previous tidal gravity measurements. For O₁, where the ocean tide loading and attraction in central Europe is only 0.4 per cent of the body tide, our measurements verify that the Dehant-Wahr anelastic body tide model gravimetric factor is accurate to 0.2 per cent. It is also shown that the effects of lateral heterogeneities in Earth structure on tidal gravity are too small to explain the large anomalies in previously published tidal gravity amplitudes. The observations clearly show the importance of conserving tidal mass in the Schwiderski ocean tide model. For sites in central Europe, the M₂ and O₁ observations and the models are in agreement at the 0.1 μgal (10⁻⁹ m s⁻²) level and tidal corrections to this accuracy can now be made to absolute gravity measurements.     

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.     

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.     

Generation of Long Internal Gravity Waves in Waveguides by Rising Buoyant Air Masses and Other Sources

Summary The displacement fields generated in an internal gravity wave waveguide between plane rigid walls are compared for two types of source: an explosive point source and a rising buoyant sphere moving at constant speed. It is concluded that for large enough spheres and comparable energy expenditures, the buoyant sphere is a far more efficient source of long internal gravity waves. In particular it appears possible to conclude that, in the case of large events such as nuclear or volcanic explosions in the atmosphere, the rising heated air mass can generate long wavelength (Λ > 500 km) internal gravity waves at ionospheric heights.     

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.     

Numerical aspects of a spline-based multiresolution recovery of the harmonic mass density out of gravity functionals

We show the numerical applicability of a multiresolution method based on harmonic splines on the 3-D ball which allows the regularized recovery of the harmonic part of the Earth's mass density distribution out of different types of gravity data, for example, different radial derivatives of the potential, at various positions which need not be located on a common sphere. This approximated harmonic density can be combined with its orthogonal anharmonic complement, for example, determined out of the splitting function of free oscillations, to an approximation of the whole mass density function. The applicability of the presented tool is demonstrated by several test calculations based on simulated gravity values derived from EGM96. The method yields a multiresolution in the sense that the localization of the constructed spline basis functions can be increased which yields in combination with more data a higher resolution of the resulting spline. Moreover, we show that a locally improved data situation allows a highly resolved recovery in this particular area in combination with a coarse approximation elsewhere which is an essential advantage of this method, for example, compared to polynomial approximation.     

Gravity constraints on the dynamics of the crust–mantle system during Calabrian subduction

The thermomechanic evolution of the lithosphere–upper mantle system during Calabrian subduction is analysed using a 2-D finite element approach, in which the lithosphere is compositionally stratified into crust and mantle. Gravity and topography predictions are cross-checked with observed gravity and topography patterns of the Calabrian region. Modelling results indicate that the gravity pattern in the arc-trench region is shaped by the sinking of light material, belonging to both the overriding and subduction plates. The sinking of light crustal material, up to depths of the order of 100–150 km is the ultimate responsible for the peculiar gravity signature of subduction, characterized by a minimum of gravity anomaly located at the trench, bounded by two highs located on the overriding and subducting plates, with a variation in magnitude of the order of 200 mGal along a wavelength of 200 km, in agreement with the isostatically compensated component of gravity anomaly observed along a transect crossing the Calabrian Arc, from the Tyrrhenian to the Ionian Seas. The striking agreement between the geodetic retrieved profiles and the modelled ones in the trench region confirms the crucial role of compositional stratification of the lithosphere in the subduction process and the correctness of the kinematic hypotheses considered in our modelling, that the present-day configuration of crust–mantle system below the Calabrian arc results from trench's retreat at a rate of about 3 cm yr⁻¹, followed by gravitational sinking of the subducted slab in the last 5 Myr.     

Analysis of Difference Between Direct and Geodetic Mass Balance Measurements at South Cascade Glacier, Washington

Net mass balance has been measured since 1958 at South Cascade Glacier using the 'direct method,' e.g. area averages of snow gain and firn and ice loss at stakes. Analysis of cartographic vertical photography has allowed measurement of mass balance using the 'geodetic method' in 1970, 1975, 1977, 1979–80, and 1985–97. Water equivalent change as measured by these nearly independent methods should give similar results. During 1970–97, the direct method shows a cumulative balance of about −15 m, and the geodetic method shows a cumulative balance of about −22 m. The deviation between the two methods is fairly consistent, suggesting no gross errors in either, but rather a cumulative systematic error. It is suspected that the cumulative error is in the direct method because the geodetic method is based on a non-changing reference, the bedrock control, whereas the direct method is measured with reference to only the previous year's summer surface. Possible sources of mass loss that are missing from the direct method are basal melt, internal melt, and ablation on crevasse walls. Possible systematic measurement errors include under-estimation of the density of lost material, sinking stakes, or poorly represented areas.     

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.     

A thermal history model for the Earth with parameterized convection

Summary. A thermal history model for the Earth is described in which the energetically important effects of convection are parameterized through the Nusselt number. The validity of the resulting quasi-steady-state thermal model is shown to depend upon the separation of two time-scales—a dynamic time-scale associated with the overturn time for an assumed mantle-wide convective circulation, and a thermal time-scale associated with the cooling of the planet. Provided the initial thermal state of the Earth was 'hot', the assumption of a time-scale separation can be shown under certain conditions, to be valid throughout the Earth's history. In this connection, the temperature-dependent mantle rheology plays a key role in regulating the thermal history. It is shown that the present-day, gross thermal structure of the Earth can be understood within the context of a quasi-steady-state model which is driven mainly by primordial heat. The notion of whole-mantle convection is shown to be consistent with several additional observational constraints, including the observed mean lithospheric thickness and the mean plate velocities. We briefly consider the extension of the parameterized thermal model to Venus.     

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.     

南极拉斯曼丘陵地区大地测量控制网

南极拉斯曼丘陵地区大地测量控制网的建立与施测前后共历经４年方始完成，总控制面积约１７７．６平方公里。本文主要讨论了该网分区布设情况及总体平差的原理与方法，并对成果进行了统计分析。经过整体平差的结果可以满足１：１００００测图的需要。该网也是我国东南极中山站地区的基础大地控制网     

Spatial analysis of tilt and gravity observations of Earth tides in western Europe

Summary. Tidal gravity and tilt data are available with a great density in western Europe, so that local and regional variability of Earth tides are well documented. O1 and M2 are the most reliable waves and the relevant ocean loads are deduced from some cotidal charts and from Green's functions given by Farrell. The reliability of the charts is proved a posteriori by the convergence of O1 and M2 with respect to body tides for tilt and gravity. It is shown that a spatial trend of the gravimetric factor exists but cannot be analysed because of scatter in the data. For tilt tides the importance of the load due to the Bay of Biscay is emphasized; by considering tilt and strain powers, it is shown in a global approach that the noise is in favour of a local tilt-strain coupling, and it is proved that in each station a quantitative estimate of this phenomenon can be obtained by a statistical analysis derived from a theoretical scheme (King et al. ) in as much as the ocean load is known with sufficient accuracy; an appraisal of the diminishing tilt factor is then deduced simultaneously.     

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.     

Applicability of an ultra-long-range terrestrial laser scanner to monitor the mass balance of Muz Taw Glacier, Sawir Mountains, China

Glacier mass balance is a key component of glacier monitoring programs. Information on the mass balance of Sawir Mountains is poor due to a dearth of in-situ measurements. This paper introduces the applicability of an ultra-long-range terrestrial laser scanner(TLS) to monitor the mass balance of Muz Taw Glacier, Sawir Mountains, China. The Riegl VZ?-6000 TLS is exceptionally well-suited for measuring snowy and icy terrain. Here, we use TLS to create repeated high spatiotemporal resolution DEMs, focusing on the annual mass balance(June 2, 2015 to July 25, 2016). According to TLS-derived high spatial resolution point clouds, the front variation(glacier retreat) of Muz Taw Glacier was 9.3 m. The mean geodetic elevation change was 4.55 m at the ablation area. By comparing with glaciological measurements, the glaciological elevation change of individual stakes and the TLS-derived geodetic elevation change of corresponding points matched closely, and the calculated balance was-3.864±0.378 m w.e.. This data indicates that TLS provides accurate results and is therefore suitable to monitor mass balance evolution of Muz Taw Glacier.