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.     

The gravity field and GGOS

The gravity field of the earth is a natural element of the Global Geodetic Observing System (GGOS). Gravity field quantities are like spatial geodetic observations of potential very high accuracy, with measurements, currently at part-per-billion (ppb) accuracy, but gravity field quantities are also unique as they can be globally represented by harmonic functions (long-wavelength geopotential model primarily from satellite gravity field missions), or based on point sampling (airborne and in situ absolute and superconducting gravimetry). From a GGOS global perspective, one of the main challenges is to ensure the consistency of the global and regional geopotential and geoid models, and the temporal changes of the gravity field at large spatial scales. The International Gravity Field Service, an umbrella “level-2” IAG service (incorporating the International Gravity Bureau, International Geoid Service, International Center for Earth Tides, International Center for Global Earth models, and other future new services for, e.g., digital terrain models), would be a natural key element contributing to GGOS. Major parts of the work of the services would, however, remain complementary to the GGOS contributions, which focus on the long-wavelength components of the geopotential and its temporal variations, the consistent procedures for regional data processing in a unified vertical datum and Terrestrial Reference Frame, and the ensuring validations of long-wavelength gravity field data products.     

地形/均衡重力场模型构制及其拟稳分析

重力空白区数据填补的一个主要方法是基于地壳均衡理论进行的,该方法亦用于EGM系列模型的构建中.本文研究了地形数据在构制地形/均衡重力场模型中的应用,分析了补偿深度对Airy位模型和面凝聚位模型的影响,给出二者的最佳补偿深度分别为50 km和40 km.以纯卫星重力模型为参考,后者在前120阶的精度要高于前者,但在121~250阶的精度较低,组合模型精度高于单一模型精度.对地形/均衡地球重力场模型进行了EGM2008拟稳分析,研究了不同分辨率基准的拟稳效果,分析表明：30'分辨率的拟稳基准所得拟稳模型对应的阶方差与参考阶方差曲线直到360阶都有较好的一致性,以EGM2008为基准,其相对累计大地水准面高误差在140阶时为6.83cm,相对累计重力异常误差在220阶时为1.10 mGal.     

A synthetic Earth gravity model based on a topographic-isostatic model

The Earth??s gravity field is related to the topographic potential in medium and higher degrees, which is isostatically compensated. Hence, the topographic-isostatic (TI) data are indispensable for extending an available Earth Gravitational Model (EGM) to higher degrees. Here we use TI harmonic coefficients to construct a Synthetic Earth Gravitational Model (SEGM) to extend the EGMs to higher degrees. To achieve a high-quality SEGM, a global geopotential model (EGM96) is used to describe the low degrees, whereas the medium and high degrees are obtained from the TI or topographic potential. This study differes from others in that it uses a new gravimetric-isostatic model for determining the TI potential. We test different alternatives based on TI or only topographic data to determine the SEGM. Although the topography is isostatically compensated only to about degree 40?C60, our study shows that using a compensation model improves the SEGM in comparison with using only topographic data for higher degree harmonics. This is because the TI data better adjust the applied Butterworth filter, which bridges the known EGM and the new high-degree potential field than the topographic data alone.     

A theoretical study on terrestrial gravimetric data refinement by Earth gravity models

The idea of this paper is to present estimators for combining terrestrial gravity data with Earth gravity models and produce a high‐quality source of the Earth's gravity field data through all wavelengths. To do so, integral and point‐wise estimators are mathematically developed, based on the spectral combination theory, in such a way that they combine terrestrial data with one and/or two Earth gravity models. The integral estimators are developed so that they become biased or unbiased to a priori information. For testing the quality of the estimators, their global mean square errors are generated using an Earth gravity model08 model and one of the recent products of the gravity field and steady‐state ocean circulation explorer mission. Numerical results show that the integral estimators have smaller global root mean square errors than the point‐wise ones but they are not efficient practically. The integral estimator of the biased type is the most suited due to its smallest global root mean square error comparing to the rest of the estimators. Due largely to the omission errors of Earth gravity models the point‐wise estimators are not sensitive to the Earth gravity model commission error; therefore, the use of high‐degree Earth gravity models is very influential for reduction of their root mean square errors. Also it is shown that the use of the ocean circulation explorer Earth gravity model does not significantly reduce the root mean square errors of the presented estimators in the presence of Earth gravity model08. All estimators are applied in the region of Fennoscandia and a cap size of 2° for numerical integration and a maximum degree of 2500 for generation of band‐limited kernels are found suitable for the integral estimators.     

Comparison of spectral and spatial methods for a Moho recovery from gravity and vertical gravity-gradient data

In global studies investigating the Earth’s lithospheric structure, the spectral expressions for the gravimetric forward and inverse modeling of the global gravitational and crustal structure models are preferably used, because of their numerical efficiency. In regional studies, the applied numerical schemes typically utilize the expressions in spatial form. Since the gravity-gradient observations have a more localized support than the gravity measurements, the gravity-gradient data (such as products from the Gravity field and steady-state Ocean Circulation Explorer - GOCE - gravity-gradiometry satellite mission) could preferably be used in regional studies, because of reducing significantly the spatial data-coverage required for a regional inversion or interpretation. In this study, we investigate this aspect in context of a regional Moho recovery. In particular, we compare the numerical performance of solving the Vening Meinesz-Moritz’s (VMM) inverse problem of isostasy in spectral and spatial domains from the gravity and (vertical) gravity-gradient data. We demonstrate that the VMM spectral solutions from the gravity and gravity-gradient data are (almost) the same, while the VMM spatial solutions differ from the corresponding spectral solutions, especially when using the gravity-gradient data. The validation of the VMM solutions, however, reveals that the VMM spatial solution from the gravity-gradient data has a slightly better agreement with seismic models. A more detailed numerical analysis shows that the VMM spatial solution formulated for the gravity gradient is very sensitive to horizontal spatial variations of the vertical gravity gradient, especially in vicinity of the computation point. Consequently, this solution provides better results in regions with a relatively well-known crustal structure, while suppressing errors caused by crustal model uncertainties from distant zones. Based on these findings we argue that the gravity-gradient data are more suitable than the gravity data for a regional Moho recovery.     

Structure and State of Stress of the Chilean Subduction Zone from Terrestrial and Satellite-Derived Gravity and Gravity Gradient Data

It is well known that the quality of gravity modelling of the Earth’s lithosphere is heavily dependent on the limited number of available terrestrial gravity data. More recently, however, interest has grown within the geoscientific community to utilise the homogeneously measured satellite gravity and gravity gradient data for lithospheric scale modelling. Here, we present an interdisciplinary approach to determine the state of stress and rate of deformation in the Central Andean subduction system. We employed gravity data from terrestrial, satellite-based and combined sources using multiple methods to constrain stress, strain and gravitational potential energy (GPE). Well-constrained 3D density models, which were partly optimised using the combined regional gravity model IMOSAGA01C (Hosse et al. in Surv Geophys, 2014, this issue), were used as bases for the computation of stress anomalies on the top of the subducting oceanic Nazca plate and GPE relative to the base of the lithosphere. The geometries and physical parameters of the 3D density models were used for the computation of stresses and uplift rates in the dynamic modelling. The stress distributions, as derived from the static and dynamic modelling, reveal distinct positive anomalies of up to 80 MPa along the coastal Jurassic batholith belt. The anomalies correlate well with major seismicity in the shallow parts of the subduction system. Moreover, the pattern of stress distributions in the Andean convergent zone varies both along the north–south and west–east directions, suggesting that the continental fore-arc is highly segmented. Estimates of GPE show that the high Central Andes might be in a state of horizontal deviatoric tension. Models of gravity gradients from the Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite mission were used to compute Bouguer-like gradient anomalies at 8 km above sea level. The analysis suggests that data from GOCE add significant value to the interpretation of lithospheric structures, given that the appropriate topographic correction is applied.     

Impact of GOCE Level 1b data reprocessing on GOCE-only and combined gravity field models

The reprocessing of Gravity field and steady-state Ocean Circulation Explorer (GOCE) Level 1b gradiometer and star tracker data applying upgraded processing methods leads to improved gravity gradient and attitude products. The impact of these enhanced products on GOCE-only and combined GOCE+GRACE (Gravity Recovery and Climate Experiment) gravity field models is analyzed in detail, based on a two-months data period of Nov. and Dec. 2009, and applying a rigorous gravity field solution of full normal equations. Gravity field models that are based only on GOCE gradiometer data benefit most, especially in the low to medium degree range of the harmonic spectrum, but also for specific groups of harmonic coefficients around order 16 and its integer multiples, related to the satellite’s revolution frequency. However, due to the fact that also (near-)sectorial coefficients are significantly improved up to high degrees (which is caused mainly by an enhanced second derivative in Y direction of the gravitational potential — V_YY), also combined gravity field models, including either GOCE orbit information or GRACE data, show improvements of more than 10% compared to the use of original gravity gradient data. Finally, the resulting gradiometry-only, GOCE-only and GOCE+GRACE global gravity field models have been externally validated by independent GPS/levelling observations in selected regions. In conclusion, it can be expected that several applications will benefit from the better quality of data and resulting GOCE and combined gravity field models.     

Grim4-C1, -C2p: combination solutions of the global earth gravity field

On the basis of the GRIM4-S1 satellite-only Earth gravity model, being accomplished in a common effort by DGFI and GRGS, a combination solution, called GRIM4-C1, has been derivcd using 1° × 1° mean gravity anomalies and 1° × 1° Seasat altimeter derived mean geoid undulations. In the meantime improvements could be achieved by incorporating more tracking data (GEOSAT, SPOT2-DORIS) into the solution, resulting in the two new parallel versions, the satellite-only gravity model GRIM4-S2 and the combined solution GRIM4-C2p (preliminary). All GRIM4 Earth gravity models cover the spectral gravitational constituents complete up to degree and order 50.In this report the emphasis is on the discussion of the combined gravity models: combination and estimation techniques, capabilities for application in precise satellite orbit computation and accuracies in long wavelength geoid representation. It is shown that with the new generation of global gravity models general purpose satellite-only models are no longer inferior to combination solutions if applied to satellite orbit restitution.     

Band-Limited Analysis of Current Satellite-to-Satellite Tracking, Gradiometry and Combined Earth Gravity Models

Several satellite-only gravity models based on the analysis of satellite-to-satellite tracking (SST) data have become available in the course of the last decade. The realization of the satellite missions CHAllenging Minisatellite Payload (CHAMP) and Gravity Recovery And Climate Experiment (GRACE) enabled the practical implementation of two modes of the SST principle, namely the high–low and the low–low SST. Though similar in their fundamental idea, which is the indirect observation of the gravity field based on the position of two satellites orbiting the Earth, the different architecture and geometrical layout of these techniques capture different fingerprints of the observed field. In the last few years, satellite-only gravity models based on the analysis of satellite gravity gradiometry (SGG) data became available and led to a new insight into the gravity field. The implementation of the SGG principle became possible after the launch of Gravity field and steady-state Ocean Circulation Explorer (GOCE), the first gravitational gradiometry mission. Based on the principle of differential accelerometry, GOCE provides the gravitational gradients which can be used in gravity field retrieval as primary observations of the field at satellite altitude. In the present study, we consider some of the current satellite-only and combined gravity models based on the analysis of CHAMP, GRACE, GOCE, gravimetry and altimetry data. In order to perform a thorough analysis of the models, we present an overview of tools for their quality assessment both in an absolute and relative sense in terms of computing spectral quantities, such as correlation or smoothing coefficients per degree and per order, attempting to demonstrate possible non-isotropic features in the models. Furthermore, typical geodetic measures in computing second-order derivatives, such as degree and order variances and difference variances, have been also evaluated for the same models, using the combined model EGM2008 as reference. Apart from these standard spectral assessment quantities, a systematic spatial representation of the second derivatives at satellite altitude has been performed. The combination of the two analysis steps (spectral and spatial) permits a first detailed assessment of the models, focusing especially on the identification of characteristic interpretable bandwidths.     

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.     

中国大陆精密重力潮汐改正模型

利用理论和实验重力固体潮模型,充分考虑全球海潮和中国近海潮汐的负荷效应,建立了中国大陆的精密重力潮汐改正模型.结果表明,采用不同的固体潮模型会对重力潮汐结果产生相对变化幅度小于0.06％的差异;在沿海地区海潮负荷的影响约为整个潮汐的4％,而中部地区约为1％,其中中国近海潮汐模型的影响约占整个海潮负荷的10%,内插或外推潮波的负荷约占海潮负荷的3％.通过比较实测的重力数据表明,本文给出的重力潮汐改正模型的精度远远优于0.5×10-8 m·s-2,说明了本文构建的模型的实用性,可为中国大陆高精度重力测量提供有效参考和精密的改正模型.     

Comparison of different gravity field implied density models of the topography

Density within the Earth crust varies between 1.0 and 3.0 g/cm³. The Bouguer gravity field measured in south Iran is analyzed using four different regional-residual separation techniques to obtain a residual map of the gravity field suitable for density modeling of topography. A density model of topography with radial and lateral distribution of density is required for an accurate determination of the geoid, e.g., in the Stokes-Helmert approach. The apparent density mapping technique is used to convert the four residual Bouguer anomaly fields into the corresponding four gravity im-plied subsurface density (GRADEN) models. Although all four density models showed good correlation with the geological density (GEODEN) model of the region, the GRADEN models obtained by high-pass filter-ing and GGM high-pass filtering show better numerical correlation with GEODEN model than the other models.     

One technique for refining the global Earth gravity models

The results of the theoretical and experimental research on the technique for refining the global Earth geopotential models such as EGM2008 in the continental regions are presented. The discussed technique is based on the high-resolution satellite data for the Earth’s surface topography which enables the allowance for the fine structure of the Earth’s gravitational field without the additional gravimetry data. The experimental studies are conducted by the example of the new GGMplus global gravity model of the Earth with a resolution about 0.5 km, which is obtained by expanding the EGM2008 model to degree 2190 with the corrections for the topograohy calculated from the SRTM data. The GGMplus and EGM2008 models are compared with the regional geoid models in 21 regions of North America, Australia, Africa, and Europe. The obtained estimates largely support the possibility of refining the global geopotential models such as EGM2008 by the procedure implemented in GGMplus, particularly in the regions with relatively high elevation difference.     

Density,stress, and gravity anomalies in the interiors of the earth and mars and the probable geodynamical implications: Comparative analysis

The formulas that allow, within the quadratic approximation, for the contribution of the anomalous masses, distributed along the height relative to the reference ellipsoid, in the Stokes parameters are derived. It is shown that the contribution of the quadratic terms is largest and commensurate, by the order of magnitude, with the linear contribution if the anomalous masses have a dipole distribution along the height. The quadratic contribution is particularly significant for Mars, where the span of relative variations in the surface topography is by an order of magnitude larger than in the Earth. The problem is solved and the method is developed for finding the depths of compensation for the topographical harmonics of different order and degree. The most probable levels of compensation for topographic irregularities are determined by the analysis of the distribution histograms for the depths of compensation. The maps of lateral distributions of the compensating masses at the selected levels are calculated. It is shown that the observed anomalous structures generate the anomalies in the internal gravity field, which may serve as a cause for the convective motion in the mantle and core of the planet. Besides, the probable nonisostatic vertical stresses in the crust and mantle of the Earth and Mars are calculated.     

Full gravity gradient tensors from vertical gravity by cosine transform

We present a method to calculate the full gravity gradient tensors from pre-existing vertical gravity data using the cosine transform technique and discuss the calculated tensor accuracy when the gravity anomalies are contaminated by noise. Gravity gradient tensors computation on 2D infinite horizontal cylinder and 3D ??Y?? type dyke models show that the results computed with the DCT technique are more accurate than the FFT technique regardless if the gravity anomalies are contaminated by noise or not. The DCT precision has increased 2 to 3 times from the standard deviation. In application, the gravity gradient tensors of the Hulin basin calculated by DCT and FFT show that the two results are consistent with each other. However, the DCT results are smoother than results computed with FFT. This shows that the proposed method is less affected by noise and can better reflect the fault distribution.     

Study of the anomalous gravity field in the arctic based on modern geopotential models

The results of modeling the anomalous gravity field of the Earth in the high-latitude regions, which are of interest in the context of Arctic exploration, are analyzed. The structure and accuracy characteristics of quasi-geoid heights (QGH) and gravity anomalies (GAs) in the Arctic are studied. The analysis addresses fifteen modern Russian and foreign spherical harmonic models of the gravitational potential of the Earth, including the EGM-2008 model, which provides high-order spherical harmonic representations, and the first models based on the data of satellite gradiometry.     

The Earth's gravity field

The Earth's gravity field can be determined from gravity measurements made on the surface of the Earth, and through the analysis of the motion of Earth satellites. Gravity data can be used to solve the boundary value problem of gravimetric geodesy in various ways, from the classical formulation using a geoid to the concept of a reference surface interior to the masses of the Earth to a statistical method. We now have gravity information for 1⁰ data blocks over 46% of the Earth's surface and more than several million point measurements available.Satellite observations such as range, range-rate, and optical data have been analyzed to determine potential coefficients used to describe the Earth's gravitational potential field. Coefficients, in a spherical harmonic expansion to degree 12, can be determined from satellite data alone, and to at least degree 20 when the satellite data is combined with surface gravity material. Recent solutions for potential coefficients agree well to degree 4, but with increasing disagreement at higher degrees.     

Layer-Based Modelling of the Earth’s Gravitational Potential up to 10-km Scale in Spherical Harmonics in Spherical and Ellipsoidal Approximation

Global forward modelling of the Earth’s gravitational potential, a classical problem in geophysics and geodesy, is relevant for a range of applications such as gravity interpretation, isostatic hypothesis testing or combined gravity field modelling with high and ultra-high resolution. This study presents spectral forward modelling with volumetric mass layers to degree 2190 for the first time based on two different levels of approximation. In spherical approximation, the mass layers are referred to a sphere, yielding the spherical topographic potential. In ellipsoidal approximation where an ellipsoid of revolution provides the reference, the ellipsoidal topographic potential (ETP) is obtained. For both types of approximation, we derive a mass layer concept and study it with layered data from the Earth2014 topography model at 5-arc-min resolution. We show that the layer concept can be applied with either actual layer density or density contrasts w.r.t. a reference density, without discernible differences in the computed gravity functionals. To avoid aliasing and truncation errors, we carefully account for increased sampling requirements due to the exponentiation of the boundary functions and consider all numerically relevant terms of the involved binominal series expansions. The main outcome of our work is a set of new spectral models of the Earth’s topographic potential relying on mass layer modelling in spherical and in ellipsoidal approximation. We compare both levels of approximations geometrically, spectrally and numerically and quantify the benefits over the frequently used rock-equivalent topography (RET) method. We show that by using the ETP it is possible to avoid any displacement of masses and quantify also the benefit of mapping-free modelling. The layer-based forward modelling is corroborated by GOCE satellite gradiometry, by in-situ gravity observations from recently released Antarctic gravity anomaly grids and degree correlations with spectral models of the Earth’s observed geopotential. As the main conclusion of this work, the mass layer approach allows more accurate modelling of the topographic potential because it avoids 10–20-mGal approximation errors associated with RET techniques. The spherical approximation is suited for a range of geophysical applications, while the ellipsoidal approximation is preferable for applications requiring high accuracy or high resolution.