Steric Sea-Level Change and its Impact on the Gravity Field caused by Global Climate Change

It is sometimes assumed that steric sea-level variations do not produce a gravity signal as no net mass change, thus no change of ocean bottom pressure is associated with it. Analyzing the output of two CO₂ emission scenarios over a period of 2000 years in terms of steric sea-level changes, we try to quantify the gravitational effect of steric sea-level variations. The first scenario, computed with version 2.6 of the Earth System Climate Model developed at the University of Victoria, Canada (UVic ESCM), is implemented with a linear CO₂ increase of 1% of the initial concentration of 365 ppm and shows a globally averaged steric effect of 5.2 m after 2000 years. In the second scenario, computed with UVic ESCM version 2.7, the CO₂ concentration increases quasi-exponentially to a level of 3011 ppm and is hold fixed afterwards. The corresponding globally averaged steric effect in the first 2000 years is 2.3 m. We show, due to the (vertical) redistribution of ocean water masses (expansion or contraction), the steric effect results also in a small change in the Earth’s gravity field compared to usually larger changes associated with net mass changes. Maximum effects for computation points located on the initial ocean surface can be found in scenario 1, with the effect on gravitational attraction and potential ranging from 0.0 to −0.7·10⁻⁵ m s⁻² and −3·10⁻³ to 6·10⁻³ m² s⁻², respectively. As expected, the effect is not zero but negligible for practical applications.     

Comparison of two approaches for considering laterally varying density in topographic effect on satellite gravity gradiometric data

The satellite gravity gradiometric data are influenced by laterally varying density in topographic masses, while in most of studies a constant density for the masses was considered. This assumption causes an error in estimating the topographic effect. This paper theoretically and numerically investigates the methods of Sjöberg as well as Novák and Grafarend to consider the laterally varying density for topographic masses in formulation of topographic potential in spherical harmonics.     

A method for calculating the plumb line declination on the basis of S-approximations

The high-precision air navigation tools developed in recent decades enable airborne gravity measurements that are suitable for solving many problems of geophysical exploration. In Russia, experiments along these lines were initiated in 1997–1998. Profile and areal surveys of a rather complex gravity field could be carried out with an accuracy of about 1 mGal. Knowledge of the plumb line declination (PLD) is needed in various areas of geophysics and geodesy. On the basis of airborne gravity observations, it is possible to use gravity data for calculating PLDs in both plain and mountainous regions. In the latter case, a correction for the effect of topographic masses is introduced into calculation formulas. With the use of the S-approximation method, based on the representation of harmonic functions as the sum of the potentials of simple and double layers on a certain carrier (in particular, on a horizontal plane), the gravity field was reconstructed from measurements at any point of space (at any altitude of measurement), in particular, on the surface of a geoid. A software package was specially developed for the PLD calculation by the Vening Meinesz formulas (the PLD calculation in a zeroth approximation) and methods of reconstructing anomalous fields on basis of S-approximations. Results of real-data calculations for one area of Russia are presented.     

The 3-D Truncation Filtering Methodology Defined for Planar and Spherical Models: Interpreting Gravity Data Generated by Point Masses

This paper focuses on one particular way of linear filtering the gravity data to facilitate gravity inversion or interpretation. With the use of integral transforms the gravity anomalies are transformed into new quantities that allow an easier interpretation with the help of pattern recognition. As the integral transforms are in fact filters, and as the regions of integration are caps with a variable radius, which can be systematically changed as a free parameter, we refer to such methodology as the truncation filtering. Such filters may be understood as weighted spherical windows moving over the surface, on which the gravity anomaly is defined, the kernel of the transform being the weight function. The objective of this paper is to define and deploy the truncation filtering for a planar model, i.e. for a homogenous horizontally infinite layer with embedded anomalous masses, and for a spherical model, i.e., for a homogenous massive sphere with embedded anomalous masses. Instead of the original gravity anomaly, the quantities resulting from the truncation filtering are interpreted/inverted. As we shall see, this approach has certain benefits. The fundamental concept of the truncation filtering methodology is demonstrated here in terms of the model consisting of one point mass anomaly.The relationship between the depth of the point mass and the instant of the onset of the dimple pattern observed in sequences produced by truncation filtering the synthetic gravity data generated by point masses is, for both the planar and spherical models, compiled by computer simulations, as well as derived analytically. It is shown, that the dimple pattern is a consequence of truncating the domain of the filter and is free of the choice of the kernel of the filter. It is shown, that in terms of the mean earth and depths of point masses no greater than some 100 km the spherical model may be replaced by a planar model from the perspective of the truncation filtering methodology. It is also shown, that from the viewpoint of the truncation filtering methodology the rigorous gravity anomaly may be approximated by the vertical component of the gravity disturbance. The relationship between the instant of the dimple onset and the depth of the point mass thus becomes linear and independent of the magnitude (mass) of the point mass.     

Interpretation of general geophysical patterns in Iran based on GRACE gradient component analysis

Only with satellites it is possible to cover the entire Earth densely with gravity field related measurements of uniform quality within a short period of time. However, due to the altitude of the satellite orbits, the signals of individual local masses are strongly damped. Based on the approach of Petrovskaya and Vershkov we determine the gravity gradient tensor directly from the spherical harmonic coefficients of the recent EIGEN-GL04C combined model of the GRACE satellite mission. Satellite gradiometry can be used as a complementary tool to gravity and geoid information in interpreting the general geophysical and geodynamical features of the Earth. Due to the high altitude of the satellite, the effects of the topography and the internal masses of the Earth are strongly damped. However, the gradiometer data, which are nothing else than the second order spatial derivatives of the gravity potential, efficiently counteract signal attenuation at the low and medium frequencies. In this article we review the procedure for estimating the gravity gradient components directly from spherical harmonics coefficients. Then we apply this method as a case study for the interpretation of possible geophysical or geodynamical patterns in Iran. We found strong correlations between the cross-components of the gravity gradient tensor and the components of the deflection of vertical, and we show that this result agrees with theory. Also, strong correlations of the gravity anomaly, geoid model and a digital elevation model were found with the diagonal elements of the gradient tensor.     

AUTOMATIC INTERPRETATION OF GRAVITY ANOMALIES ASSOCIATED WITH TWO-DIMENSIONAL MASS DISTRIBUTIONS*

Several new features are described which facilitate an automatic, more meaningful, and more accurate interpretation of gravity anomalies associated with approximately two-dimensional mass distributions. These include a provision for fixed points on the bounds of a distribution, outward dipping boundaries, calculation of the gravity effect at individual elevations of stations, smoothing of models and end-corrections for distributions of limited strike length. It is possible to obtain distributions about a median plane; these distributions are sensitive to shape and allow estimates of optimum depths and minimum density contrasts for the anomalous masses.     

The European Gravity Field and Steady-State Ocean Circulation Explorer Satellite Mission Its Impact on Geophysics

Current knowledge of the Earth's gravity field and its geoid, as derived from various observing techniques and sources, is incomplete. Within a reasonable time, substantial improvement will come by exploiting new approaches based on spaceborne gravity observation. Among these, the European Space Agency (ESA) Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite mission concept has been conceived and designed taking into account multi-disciplinary research objectives in solid Earth physics, oceanography and geodesy. Based on the unique capability of a gravity gradiometer combined with satellite-to-satellite high-low tracking techniques, an accurate and detailed global model of the Earth's gravity field and its corresponding geoid will be recovered. The importance of this is demonstrated by a series of realistic simulation experiments. In particular, the quantitative impact of the new and accurate gravity field and geoid is examined in studies of tectonic composition and motion, Glaciological Isostatic Adjustment, ocean mesoscale variability, water mass transport, and unification of height systems. Improved knowledge in each of these fields will also ensure the accumulation of new understanding of past and present sea-level changes.     

Gravity reduction with three-dimensional atmospheric pressure data for precise ground gravity measurements

The redistribution of air masses induces gravity variations (atmospheric pressure effect) up to about 20 μgal. These variations are disturbing signals in gravity records and they must be removed very carefully for detecting weak gravity signals. In the past, different methods have been developed for modelling of the atmospheric pressure effect. These methods use local or two-dimensional (2D) surface atmospheric pressure data and a standard height-dependent air density distribution. The atmospheric pressure effect is consisting of the elastic deformation and attraction term. The deformation term can be well modelled with 2D surface atmospheric pressure data, for instance with the Green's function method. For modelling of the attraction term, three-dimensional (3D) data are required. Results with 2D data are insufficient.From European Centre for Medium-Range Weather Forecasts (ECMWF) 3D atmospheric pressure data are now available. The ECMWF data used here are characterised by a spacing of Δ and Δλ = 0.5°, 60 pressure levels up to a height of 60 km and an interval of 6 h. These data are used for modelling of the atmospheric attraction term. Two attraction models have been developed based on the point mass attraction of air segments and the gravity potential of the air masses. The modelling shows a surface pressure-independent part of gravity variations induced by mass redistribution of the atmosphere in the order of some μgal. This part can only be determined by using 3D atmospheric pressure data. It has been calculated for the Vienna Superconducting Gravimeter site.From this follows that the gravity reduction can be improved by applying the 3D atmospheric attraction model for analysing long-periodic tidal waves including the polar tide. The same improvement is expected for reduction of long-term absolute gravity measurements or comparison of gravity measurements at different seasonal times. By using 3D atmospheric pressure data, the gravity correction can be improved up to some μgal.     

Comparison of Daily GRACE Gravity Field and Numerical Water Storage Models for De-aliasing of Satellite Gravimetry Observations

Reducing aliasing effects of insufficiently modelled high-frequent, non-tidal mass variations of the atmosphere, the oceans and the hydrosphere in gravity field models derived from the Gravity Recovery and Climate Experiment (GRACE) satellite mission is the topic of this study. The signal content of the daily GRACE gravity field model series (ITG-Kalman) is compared to high-frequency bottom pressure variability and terrestrially stored water variations obtained from recent numerical simulations from an ocean circulation model (OMCT) and two hydrological models (WaterGAP Global Hydrology Model, Land Surface Discharge Model). Our results show that daily estimates of ocean bottom pressure from the most recent OMCT simulations and the daily ITG-Kalman solutions are able to explain up to 40 % of extra-tropical sea-level variability in the Southern Ocean. In contrast to this, the daily ITG-Kalman series and simulated continental total water storage variability largely disagree at periods below 30 days. Therefore, as long as no adequate hydrological model will become available, the daily ITG-Kalman series can be regarded as a good initial proxy for high-frequency mass variations at a global scale. As a second result of this study, based on monthly solutions as well as daily observation residuals, it is shown that applying this GRACE-derived de-aliasing model supports the determination of the time-variable gravity field from GRACE data and the subsequent geophysical interpretation. This leads us to the recommendation that future satellite concepts for determining mass variations in the Earth system should be capable of observing higher frequeny signals with sufficient spatial resolution.     

Atmospheric effects in the derivation of geoid-generated gravity anomalies

Parameters of the gravity field harmonics outside the geoid are sought in solving the Stokes boundary-value problem while harmonics outside the Earth in solving the Molodensky boundary-value problem. The gravitational field generated by the atmosphere is subtracted from the Earth’s gravity field in solving either the Stokes or Molodensky problem. The computation of the atmospheric effect on the ground gravity anomaly is of a particular interest in this study. In this paper in particular the effect of atmospheric masses is discussed for the Stokes problem. In this case the effect comprises two components, specifically the direct and secondary indirect atmospheric effects. The numerical investigation is conducted at the territory of Canada. Numerical results reveal that the complete effect of atmosphere on the ground gravity anomaly varies between 1.75 and 1.81 mGal. The error propagation indicates that precise determination of the atmospheric effect on the gravity anomaly depends mainly on the accuracy of the atmospheric mass density distribution model used for the computation.     

Constraints on Glacial Isostatic Adjustment from GOCE and Sea Level Data

Current constraints on the glacial isostatic adjustment (GIA) process are mainly provided by relative sea-level data and GPS measurements. Due to a lack of resolving power in the shallow earth (down to about 200 km), these data sets only provide weak constraints on the shallow viscosity structure and the thickness of the lithosphere. Future high-resolution gravity data, as expected from ESA’s Gravity field and steady-state Ocean Circulation Explorer (GOCE) launched on March 17, 2009, are predicted to provide additional information on the shallow earth, more specifically the viscosity structure. Here we present an overview of recent developments in extracting information on rheology and stratification of the shallow earth from high-resolution quasi-steady gravity and geoid data to be obtained from GOCE.     

The effect of the magnetic field on the accuracy of measurements with Worden and Sharpe gravity meters

Summary The effect of an additional homogeneous magnetic field with an intensity of 0–4.5 Oe on the Worden quartz gravity meter No. 961 and on Sharpe quartz gravity meters Nos 173 and 174 was tested. Whereas no effect was observed with the Worden gravity meter, the magnetic field had a measurable effect on both the Sharpe gravity meters. The largest deviation of the reading beam is caused by the horizontal component of the magnetic field which acts in the plane of oscillation of the gravity-meter arm. The Sharpe gravity meter No. 173 is considerably sensitive; a field of 0.2 Oe intensity, corresponding to the magnitude of the horizontal component of the geomagnetic field in mid-latitudes, causes an error in the measurement of gravity of as much as 0.08 mGal. With a view to the different behaviours of the individual quartz gravity meters of the same type in a magnetic field, it should prove expedient to carry out check measurements with all gravity meters and, with regard to the sensitivity of the gravity meter to the magnetic field and the required accuracy of the gravity determination, take into account this perturbing factor in field measurements, as well as laboratory tests of gravity meters.     

Gravity changes accompanying volcanic activity at Pacaya volcano, Guatemala

Gravity changes of up to 1.2 ± 0.1 mgal (1 standard deviation) were measured at three points within 400 m of an active vent on Pacaya volcano, Guatemala during eleven days of January, 1975. For five continuous days gravity varied inversely with the average muzzle velocity of ejecta, the frequency of volcanic explosions, and the frequency of volcanic earthquakes. The gravity changes are most reasonably interpreted as the product of intravolcanic movements of magma with masses one to two orders of magnitude larger than any flow ever erupted from the volcano. However, elevation changes and/or combination of elevation and mass distribution changes could also have been an important factor in effecting the observed gravity variations. Because we lack elevation control on the gravity stations, we are unable to unequivocally conclude which factor or which combination of factors produced the gravity changes. The study indicates the possibility of gravity monitoring of hazardous volcanoes as a predictive tool, and as an added means for investigating the internal mechanism of volcanic eruptions.     

Atmospheric effects on satellite gravity gradiometry data

Atmospheric masses play an important role in precise downward continuation and validation of satellite gravity gradiometry data. In this paper we present two alternative ways to formulate the atmospheric potential. Two density models for the atmosphere are proposed and used to formulate the external and internal atmospheric potentials in spherical harmonics. Based on the derived harmonic coefficients, the direct atmospheric effects on the satellite gravity gradiometry data are investigated and presented in the orbital frame over Fennoscandia. The formulas of the indirect atmospheric effects on gravity anomaly and geoid (downward continued quantities) are also derived using the proposed density models. The numerical results show that the atmospheric effect can only be significant for precise validation or inversion of the GOCE gradiometric data at the mE level.     

The effect of DHM resolution in computing the topographic-isostatic harmonic coefficients within the window technique

The window technique was suggested earlier to get rid of the double consideration of the topographic-isostatic masses within the data window in the framework of the remove-restore technique. Within the course of the window technique, one needs to compute the harmonic coefficients of the topographic-isostatic masses for the data window. The paper studies the effect of using Digital Height Models (DHMs) with different resolutions of the computed harmonic coefficients of the topographic-isostatic masses for the data window. Two different test areas, one in Austria and one in Egypt, are considered in this investigation. A set of DHMs with different resolutions is available for both test areas. The harmonic coefficients of the topographic-isostatic masses for the data window are computed for both test areas using the available DHMs with different resolutions. A comparison among the potential degree variances of the different DHMs is carried out. The computation of the window topographic-isostatic gravity anomalies for both data sets is performed using the set of the available DHMs with different resolutions. The results show that using a DHM with the grid size of about 5 km for smooth topography and of about 3 km for rough topography gives practically the same topographic-isostatic gravity anomalies for the data window in a significantly much less CPU time compared to that of using the finest DHM.     

基于泊松小波径向基函数融合多源数据的局部重力场建模

融合多源数据的高精度、高分辨率的局部重力场建模是物理大地测量学的前沿和热点问题.本文研究了基于径向基函数融合多源数据的局部重力场建模方法,利用Monte-Carlo方差分量估计实现了不同类型的观测数据的合理定权,引入了最小标准差法确定基函数的适宜网络,分析了地形因素对于基函数网络确定及局部重力场建模精度的影响.以泊松小波基函数为构造基函数,结合残差地形模型,融合实测的陆地重力异常、船载重力异常及航空重力扰动数据构建了局部区域陆海统一的似大地水准面模型.研究结果表明:引入残差地形模型平滑了地形质量引入的高频扰动信号,简化了基函数的网络设计;并提高了重力似大地水准面的精度,平原地区其精度提高了4mm,地形起伏较大的山区其精度提高了约5cm.总体而言,基于"三步法"构建的局部重力似大地水准面在荷兰、比利时及德国相关区域,其精度分别达到1.12cm、2.80cm以及2.92cm.     

环域大地逆边值问题

从确定大地水准面实际出发,提出环域大地逆边值问题．文中首先建立环域大地道边值问题的数理模型．由于环域内边界待定,属自由边界,本质上是非线性问题．循传统给出环域逆边值问题的线性化形式．重点讨论并构造了线性化问题的解式,包括谱域内的解．     

Experiences with the use of mass-density maps in residual gravity forward modelling

In many modern local and regional gravity field modelling concepts, the short-wavelength gravitational signal modeled by the residual terrain modelling (RTM) technique is used to augment global geopotential models, or to smooth observed gravity prior to data gridding. In practice, the evaluation of RTM effects mostly relies on a constant density assumption, because of the difficulty and complexity of obtaining information on the actual distribution of density of topographic masses. Where the actual density of topographic masses deviates from the adopted value, errors are present in the RTM mass-model, and hence, in the forward-modelled residual gravity field. In this paper we attempt to overcome this problem by combining the RTM technique with a high-resolution mass-density model. We compute RTM gravity quantities over New Zealand, with different combinations of elevation models and mass-density assumptions using gravity and GPS/levelling measurements, precise terrain and bathymetry models, a high-resolution mass-density model and constant density assumptions as main input databases. Based on gravity observations and the RTM technique, optimum densities are detected for North Island of ~2500 kg m^?3, South Island of ~2600 kg m^?3, and the whole New Zealand of ~2590 kg m^?3. Comparison among the three sets of residual gravity disturbances computed from different mass-density assumptions show that, together with a global potential model, the high-resolution New Zealand density model explains ~89.5% of gravitational signals, a constant density assumption of 2670 kg m^?3 explains ~90.2%, while a regionally optimum mass-density explains ~90.3%. Detailed comparison shows that the New Zealand density model works best over areas with small residual heights. Over areas with larger residual heights, subsurface density variations appear to affect the residual gravity disturbance. This effect is found to reach about 30 mGal over Southern Alpine Fault. In order to improve the RTM modelling with mass-density maps, a higher-quality mass-density model that provides radially varying mass-density data would be desirable.