Comparison of methods to model the gravitational gradients from topographic data bases

A number of methods have been developed over the last few decades to model the gravitational gradients using digital elevation data. All methods are based on second-order derivatives of the Newtonian mass integral for the gravitational potential. Foremost are algorithms that divide the topographic masses into prisms or more general polyhedra and sum the corresponding gradient contributions. Other methods are designed for computational speed and make use of the fast Fourier transform (FFT), require a regular rectangular grid of data, and yield gradients on the entire grid, but only at constant altitude. We add to these the ordinary numerical integration (in horizontal coordinates) of the gradient integrals. In total we compare two prism, two FFT and two ordinary numerical integration methods using 1" elevation data in two topographic regimes (rough and moderate terrain). Prism methods depend on the type of finite elements that are generated with the elevation data; in particular, alternative triangulations can yield significant differences in the gradients (up to tens of Eötvös). The FFT methods depend on a series development of the topographic heights, requiring terms up to 14th order in rough terrain; and, one popular method has significant bias errors (e.g. 13 Eötvös in the vertical–vertical gradient) embedded in its practical realization. The straightforward numerical integrations, whether on a rectangular or triangulated grid, yield sub-Eötvös differences in the gradients when compared to the other methods (except near the edges of the integration area) and they are as efficient computationally as the finite element methods.     

Centennial to millennial geomagnetic secular variation

A time-varying spherical harmonic model of the palaeomagnetic field for 0–7 ka is used to investigate large-scale global geomagnetic secular variation on centennial to millennial scales. We study dipole moment evolution over the past 7 kyr, and estimate its rate of change using the Gauss coefficients of degree 1 (dipole coefficients) from the CALS7K.2 field model and by two alternative methods that confirm the robustness of the predicted variations. All methods show substantial dipole moment variation on timescales ranging from centennial to millennial. The dipole moment from CALS7K.2 has the best resolution and is able to resolve the general decrease in dipole moment seen in historical observations since about 1830. The currently observed rate of dipole decay is underestimated by CALS7K.2, but is still not extraordinarily strong in comparison to the rates of change shown by the model over the whole 7 kyr interval. Truly continuous phases of dipole decrease or increase are decadal to centennial in length rather than longer-term features. The general large-scale secular variation shows substantial changes in power in higher spherical harmonic degrees on similar timescales to the dipole. Comparisons are made between statistical variations calculated directly from CALS7K.2 and longer-term palaeosecular variation models: CALS7K.2 has lower overall variance in the dipole and quadrupole terms, but exhibits an imbalance between dispersion in   g ¹₂  and   h ¹₂  , suggestive of long-term non-zonal structure in the secular variations.     

内蒙古中部重、磁场特征与地壳密度结构

中亚造山带中的古生代——中生代花岗岩普遍具有正εNd值，在世界上是十分独特的。美国西部加利福尼亚中生代-新生代花岗岩同样具有正εNd值，并且其地壳的速度分布特征与内蒙古中部十分相似。本文通过与美国加利福尼亚的地球物理研究成果进行对比，来研究内蒙古中部的地壳密度结构，特别是下地壳的组成，试图探讨产生正￡。值花岗岩的深部地质原因。研究结果表明，在大兴安岭-内蒙古造山带下地壳中可能存在与美国加里福尼亚类似的洋壳物质。此外，还对重、磁异常进行了处理，以确定蛇绿岩带的延展情况。由于在西拉木伦河附近存在切割至莫霍面的深断裂，结合地表出现的蛇绿岩带，故提出温都尔庙-西拉木伦河一线可能是内蒙古中部最重要的地质构造界线。     

韧性剪切变形对岩石地球化学行为的制约——以北阿尔金巴什考供韧性剪切带为例

韧性剪切变形对岩石地球化学行为的制约一直是地质学家们探讨的课题。本文以构成北阿尔金红柳沟——拉配泉俯冲碰撞杂岩带与北阿尔金地块边界的巴什考供斜向逆冲型韧性剪切带为例，通过对韧性剪切带内花岗岩变形前后不同变形强度构造岩的地球化学组成进行对比，确定等比线斜率，探讨韧性变形对岩石体积和成分变异的影响。计算结果表明，在糜棱岩化过程中，糜棱岩化花岗岩体积亏损21％，花岗质糜棱岩体积亏损31％。质量平衡计算结果和等比线图表明，韧；陛剪切作用导致SiO2，流失量最大，A12O3、K2O及Ba、Rb、Sr等都有不同程度的丢失，显示出较强的活动性，MnO、P2O5、Sc位于等比线上或附近，表现出相对的稳定性。岩石中活动组分的变异是流体渗滤作用引起的，不活动组分的变异是体．积亏损造成的。     

Outlier detection algorithms and their performance in GOCE gravity field processing

The satellite missions CHAMP, GRACE, and GOCE mark the beginning of a new era in gravity field determination and modeling. They provide unique models of the global stationary gravity field and its variation in time. Due to inevitable measurement errors, sophisticated pre-processing steps have to be applied before further use of the satellite measurements. In the framework of the GOCE mission, this includes outlier detection, absolute calibration and validation of the SGG (satellite gravity gradiometry) measurements, and removal of temporal effects. In general, outliers are defined as observations that appear to be inconsistent with the remainder of the data set. One goal is to evaluate the effect of additive, innovative and bulk outliers on the estimates of the spherical harmonic coefficients. It can be shown that even a small number of undetected outliers (<0.2 of all data points) can have an adverse effect on the coefficient estimates. Consequently, concepts for the identification and removal of outliers have to be developed. Novel outlier detection algorithms are derived and statistical methods are presented that may be used for this purpose. The methods aim at high outlier identification rates as well as small failure rates. A combined algorithm, based on wavelets and a statistical method, shows best performance with an identification rate of about 99%. To further reduce the influence of undetected outliers, an outlier detection algorithm is implemented inside the gravity field solver (the Quick-Look Gravity Field Analysis tool was used). This results in spherical harmonic coefficient estimates that are of similar quality to those obtained without outliers in the input data.     

A parametric study on the impact of satellite attitude errors on GOCE gravity field recovery

In the recent design of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite mission, the gravity gradients are defined in the gradiometer reference frame (GRF), which deviates from the actual flight direction (local orbit reference frame, LORF) by up to 3–4°. The main objective of this paper is to investigate the effect of uncertainties in the knowledge of the gradiometer orientation due to attitude reconstitution errors on the gravity field solution. In the framework of several numerical simulations, which are based on a realistic mission configuration, different scenarios are investigated, to provide the accuracy requirements of the orientation information. It turns out that orientation errors have to be seriously considered, because they may represent a significant error component of the gravity field solution. While in a realistic mission scenario (colored gradiometer noise) the gravity field solutions are quite insensitive to small orientation biases, random noise applied to the attitude information can have a considerable impact on the accuracy of the resolved gravity field models.     

GGM02 – An improved Earth gravity field model from GRACE

A new generation of Earth gravity field models called GGM02 are derived using approximately 14 months of data spanning from April 2002 to December 2003 from the Gravity Recovery And Climate Experiment (GRACE). Relative to the preceding generation, GGM01, there have been improvements to the data products, the gravity estimation methods and the background models. Based on the calibrated covariances, GGM02 (both the GRACE-only model GGM02S and the combination model GGM02C) represents an improvement greater than a factor of two over the previous GGM01 models. Error estimates indicate a cumulative error less than 1 cm geoid height to spherical harmonic degree 70, which can be said to have met the GRACE minimum mission goals. Electronic Supplementary Material Supplementary material is available in the online version of this article at