Vening Meinesz inverse isostatic problem with local and global Bouguer anomalies

Inverse problems in isostasy will consist in making the isostatic anomalies to be zero under a certain isostatic hypothesis. In the case of the Vening Meinesz isostatic hypothesis, the density contrast is constant, while the Moho depth (depth of the Mohorovičić discontinuity) is variable. Hence, the Vening Meinesz inverse isostatic problem aims to determine a suitable variable Moho depth for a prescribed constant density contrast. The main idea is easy but the theoretical analysis is somewhat difficult. Moreover, the practical determination of the variable Moho depths based on the Vening Meinesz inverse problem is a laborious and time-consuming task. The formulas used for computing the inverse Vening Meinesz Moho depths are derived. The computational tricks essentially needed for computing the inverse Vening Meinesz Moho depths from a set of local and global Bouguer anomalies are described. The Moho depths for a test area are computed based on the inverse Vening Meinesz isostatic problem. These Moho depths fit the Moho depths derived from seismic observations with a good accuracy, in which the parameters used for the fitting agree well with those determined geophysically. Received: 4 February 1999 / Accepted: 4 October 1999     

采用重力异常的导纳理论推估海底地形

系统探讨了采用重力异常的导纳理论推估海底地形的方法。在频率域内对海底地形和重力异常的相关性进行了分析,确定在20~300 km波段内两者相关系数最高,适合通过重力异常反演。以皇帝海山链为例,分别采用未补偿导纳模型和顾及挠曲补偿的导纳模型开展研究,其中涉及的地壳厚度和有效弹性厚度等参数通过均衡响应函数法得到。引入余弦低通滤波器,有效地解决了向下延拓过程中产生的高频震荡问题。算例结果表明,采用重力异常的导纳理论能够有效地反演海底地形,结果真实可靠,反演模型的相对精度在6%左右,与ETOPO1模型相当,且能够刻画出ETOPO1模型中未显现的地形细节。由于地形变化剧烈,反演模型在海山链沿线附近精度不甚理想。     

A new isostatic model of the lithosphere and gravity field

Based on the analysis of various factors controlling isostatic gravity anomalies and geoid undulations, it is concluded that it is essential to model the lithospheric density structure as accurately as possible. Otherwise, if computed in the classical way (i.e. based on the surface topography and the simple Airy compensation scheme), isostatic anomalies mostly reflect differences of the real lithosphere structure from the simplified compensation model, and not necessarily the deviations from isostatic equilibrium. Starting with global gravity, topography and crustal density models, isostatic gravity anomalies and geoid undulations have been determined. The initial crust and upper-mantle density structure has been corrected in a least squares adjustment using gravity. To model the long-wavelength (>2000 km) features in the gravity field, the isostatic condition (i.e. equal mass for all columns above the compensation level) is applied in the adjustment to uncover the signals from the deep-Earth interior, including dynamic deformations of the Earths surface. The isostatic gravity anomalies and geoid undulations, rather than the observed fields, then represent the signals from mantle convection and deep density inhomogeneities including remnants of subducted slabs. The long-wavelength non-isostatic (i.e. the dynamic) topography was estimated to range from –0.4 to 0.5 km. For shorter wavelengths (<2000 km), the isostatic condition is not applied in the adjustment in order to obtain the non-isostatic topography due to regional deviations from classical Airy isostasy. The maximum deviations from Airy isostasy (–1.5 to 1 km) occur at currently active plate boundaries. As another result, a new global model of the lithosphere density distribution is generated. The most pronounced negative density anomalies in the upper mantle are found near large plume provinces, such as Iceland and East Africa, and in the vicinity of the mid-ocean ridge axes. Positive density anomalies in the upper mantle under the continents are not correlated with the cold and thick lithosphere of cratons, indicating a compensation mechanism due to thermal and compositional density.     

重力均衡与地形波长——兼论均衡理论不适宜于珠峰重力值的推估

150多年来，重力均衡的理论已得到很大的发展，均衡异常与大地水准面差距在地球科学诸多学科中已得到了广泛的应用，各种均衡理论及其相应的重力异常在各种文献中已作了比较和评论；不同波长地形的重力效应，包括短波长的地形不能构成补偿也作了进一步研究。因此，在局部场中不宜用均衡补偿的方法作山区重力点值的推估，而曾经仅用地形（高程）的数据推估珠穆朗玛峰顶上的重力倒是适合的。     

Comparison of the deflection of the vertical components computed by astro-geodetic, gravimetric and topographic-isostatic techniques

Three different types of deflection of the vertical components wre computed at a number of selected astronomical stations in the south central United States. Astro-geodetic deflections were obtained from the differences between the astronomical and geodetic coordinates of each station. Gravimetric deflections were computed from isostatic anomalies. The isostatic isoanomaly charts used in this computation were constructed from data obtained by correcting point values taken from existing free-air isoanomaly charts for the effects of the Bouguer reduction and the topographic-isostatic reduction. It is felt that this procedure substantially reduces the labor involved in making isostatic reductions without sacrificing accuracy appreciably. Topographic-isostatic deflections were computed by Hayford's Method as modified by Orlin. The differences between the three types of deflection components are compared and conclusions made concerning their accuracy and utility. Aerounautical Chart and Information Center     

Gravity field approximation in the area of Greece with emphasis on local characteristics

The accuracy of the gravity field approximation depends on the amount of the available data and their distribution as well as on the variation of the gravity field. The variation of the gravity field in the Greek mainland, which is the test area in this study, is very high (the variance of point free air gravity anomalies is 3191.5mgal ²). Among well known reductions used to smooth the gravity field, the complete isostatic reduction causes the best possible smoothing, however remain strong local anomalies which disturb the homogeneity of the gravity field in this area. The prediction of free air gravity anomalies using least squares collocation and regional covariance function is obtained within a ±4 ... ±19mgal accuracy depending on the local peculiarities of the free air gravity field. By taking into account the topography and its isostatic compensation with the usual remove-restore technique, the accuracy of the prediction mentioned obove was increased by about a factor of 4 and the prediction results become quite insensitive to the covariance function used (local or regional). But when predicting geoidal heights, in spite of using the smoothed field, the prediction results remain still depend on the covariance function used in such a way that differences up to about 50cm/100km result between relative geoidal heights computed with regional or local covariance functions.     

Geoid determination using adapted reference field, seismic Moho depths and variable density contrast

 The traditional remove-restore technique for geoid computation suffers from two main drawbacks. The first is the assumption of an isostatic hypothesis to compute the compensation masses. The second is the double consideration of the effect of the topographic–isostatic masses within the data window through removing the reference field and the terrain reduction process. To overcome the first disadvantage, the seismic Moho depths, representing, more or less, the actual compensating masses, have been used with variable density anomalies computed by employing the topographic–isostatic mass balance principle. In order to avoid the double consideration of the effect of the topographic–isostatic masses within the data window, the effect of these masses for the used fixed data window, in terms of potential coefficients, has been subtracted from the reference field, yielding an adapted reference field. This adapted reference field has been used for the remove–restore technique. The necessary harmonic analysis of the topographic–isostatic potential using seismic Moho depths with variable density anomalies is given. A wide comparison among geoids computed by the adapted reference field with both the Airy–Heiskanen isostatic model and seismic Moho depths with variable density anomaly and a geoid computed by the traditional remove–restore technique is made. The results show that using seismic Moho depths with variable density anomaly along with the adapted reference field gives the best relative geoid accuracy compared to the GPS/levelling geoid. Received: 3 October 2001 / Accepted: 20 September 2002 Correspondence to: H.A. Abd-Elmotaal     

Spherical harmonic modelling to ultra-high degree of Bouguer and isostatic anomalies

The availability of high-resolution global digital elevation data sets has raised a growing interest in the feasibility of obtaining their spherical harmonic representation at matching resolution, and from there in the modelling of induced gravity perturbations. We have therefore estimated spherical Bouguer and Airy isostatic anomalies whose spherical harmonic models are derived from the Earth’s topography harmonic expansion. These spherical anomalies differ from the classical planar ones and may be used in the context of new applications. We succeeded in meeting a number of challenges to build spherical harmonic models with no theoretical limitation on the resolution. A specific algorithm was developed to enable the computation of associated Legendre functions to any degree and order. It was successfully tested up to degree 32,400. All analyses and syntheses were performed, in 64 bits arithmetic and with semi-empirical control of the significant terms to prevent from calculus underflows and overflows, according to IEEE limitations, also in preserving the speed of a specific regular grid processing scheme. Finally, the continuation from the reference ellipsoid’s surface to the Earth’s surface was performed by high-order Taylor expansion with all grids of required partial derivatives being computed in parallel. The main application was the production of a 1′ × 1′ equiangular global Bouguer anomaly grid which was computed by spherical harmonic analysis of the Earth’s topography–bathymetry ETOPO1 data set up to degree and order 10,800, taking into account the precise boundaries and densities of major lakes and inner seas, with their own altitude, polar caps with bedrock information, and land areas below sea level. The harmonic coefficients for each entity were derived by analyzing the corresponding ETOPO1 part, and free surface data when required, at one arc minute resolution. The following approximations were made: the land, ocean and ice cap gravity spherical harmonic coefficients were computed up to the third degree of the altitude, and the harmonics of the other, smaller parts up to the second degree. Their sum constitutes what we call ETOPG1, the Earth’s TOPography derived Gravity model at 1′ resolution (half-wavelength). The EGM2008 gravity field model and ETOPG1 were then used to rigorously compute 1′ × 1′ point values of surface gravity anomalies and disturbances, respectively, worldwide, at the real Earth’s surface, i.e. at the lower limit of the atmosphere. The disturbance grid is the most interesting product of this study and can be used in various contexts. The surface gravity anomaly grid is an accurate product associated with EGM2008 and ETOPO1, but its gravity information contents are those of EGM2008. Our method was validated by comparison with a direct numerical integration approach applied to a test area in Morocco–South of Spain (Kuhn, private communication 2011) and the agreement was satisfactory. Finally isostatic corrections according to the Airy model, but in spherical geometry, with harmonic coefficients derived from the sets of the ETOPO1 different parts, were computed with a uniform depth of compensation of 30?km. The new world Bouguer and isostatic gravity maps and grids here produced will be made available through the Commission for the Geological Map of the World. Since gravity values are those of the EGM2008 model, geophysical interpretation from these products should not be done for spatial scales below 5 arc minutes (half-wavelength).     

Short-wavelength Spectral Properties of the Gravity Field from a Range of Regional Data Sets

The GRACE (gravity recovery and climate experiment) and GOCE (gravity field and steady-state ocean circulation explorer) dedicated gravity satellite missions are expected to deliver the long-wavelength scales of the Earth’s gravity field with extreme precision. For many applications in Earth sciences, future research activities will have to focus on a similar precision on shorter scales not recovered by satellite missions. Here, we investigate the signal power of gravity anomalies at such short scales. We derive an average degree variance and power spectral density model for topography-reduced gravity anomalies (residual terrain model anomalies and de-trended refined Bouguer anomalies), which is valid for wavelengths between 0.7 and 100  km. The model is based on the analysis of gravity anomalies from 13 test regions in various geographical areas and geophysical settings, using various power spectrum computation approaches. The power of the derived average topography-reduced model is considerably lower than the Tscherning–Rapp free air anomaly model. The signal power of the individual test regions deviates from the obtained average model by less than a factor of 4 in terms of square-root power spectral amplitudes. Despite the topographic reduction, the highest signal power is found in mountainous areas and the lowest signal power in flat terrain. For the derived average power spectral model, a validation procedure is developed based on least-squares prediction tests. The validation shows that the model leads to a good prediction quality and realistic error measures. Therefore, for least-squares prediction, the model could replace the use of autocovariance functions derived from local or regional data.     

基于Airy-Hesikannen均衡模型的莫霍面反演

本文主要针对利用Airy -Hesikannen均衡假说构制不同地形区域包括平原、丘陵、和小、中、大、特大山区以及高原地区的重力异常时 ,其均衡模型参数 -地壳正常厚度在任何地区均不变而导致的均衡归算精度不高的缺陷 ,推导了基于Airy -Hesikannen均衡模型反演莫霍面的公式 ,并进行了相应的数值检验 ,根据此公式可统计出不同地形区域的均衡模型参数 ,以期提高均衡异常的计算精度     

月球Airy均衡状态与月壳厚度估计

月球水准面异常和表面地形变化是其内部密度不均匀和各个界面的起伏变化的体现,因此利用水准面和地形之比(geoid to topography ratio,GTR)可估计月球均衡和月壳厚度。本文基于月球重力场模型SGM100h和地形模型STM359_grid-02,经过去除表面玄武岩填充和深层异常质量影响,并结合理论Airy均衡模型中GTR与参考月壳厚度的关系,计算得到了新的月壳厚度模型。该模型的月壳平均厚度为36.9 km,背面比正面平均厚约13.5 km,Apollo12/14登陆点的月壳厚度分别是28.3 km和29.1 km。在各月海盆地存在着中央较薄、四周逐渐增厚的趋势。     

复杂背景下的仿蝇视觉遥感异常检测

本文针对遥感影像复杂背景下,背景地物光谱特征与目标光谱特征之间存在较强相关性的问题,提出一种基于仿蝇视觉的复杂背景下遥感异常检测算法。首先构建并行多孔径背景模型,实现对复杂背景特征的自适应描述;然后基于异常目标的光谱特征相对异常性,采用相对马氏距离区分异常区域、不确定区域与无目标区域,消除背景与目标光谱相关性对检测结果干扰的同时,弥补了传统假设检验无法区分无目标和不确定问题的不足;最后融合多个背景模型的检测结果,实现异常目标检测。仿真实验将围绕多种背景地物并存复杂区域的异常检测验证本文算法的有效性。     

Geoid determination with density hypotheses from isostatic models and geological information

 Geoid determination by Stokes's formula requires a complete knowledge of the topographical mass density distribution in order to perform gravity reductions to the geoid boundary. However, deeper masses are also of interest, in order to produce a smooth field of gravity anomalies which will improve results from interpolation procedures. Until now, in most cases a constant mass density has been considered, which is a very rough approximation of reality. The influence on the geoid height coming from different mass density hypotheses given by the isostatic models of Pratt/Hayford, Airy/Heiskanen and Vening Meinesz is studied. Apart from a constant mass density value, additional density information deduced from geological maps and thick sedimentary layers is considered. An overview of how mass density distributions act within Stokes's theory is given. The isostatic models are considered in spherical and planar approximation, as well as with constant and lateral variable mass density of the topographical and deeper masses. Numerical results in a test area in south-west Germany show that the differences in the geoid height due to different density hypotheses can reach a magnitude of more than 1 decimetre, which is not negligible in a precise geoid determination with centimetre accuracy. Received: 7 January 2002 / Accepted: 20 September 2002 M. Kuhn now at: Western Australian Centre for Geodesy, Curtin University of Technology, GPO Box U1987, Perth, WA 6845, Australia Acknowledgements. The author would gratefully thank Prof. Dr.-Ing. B. Heck, who was the supervisor of my PhD thesis, and the second examiner Prof. Dr.-Ing. K.H. Ilk, as well as all other colleagues for their support of this work. Particular thanks go to the Landesvermessungsamt Baden–Württemberg (Survey Department of Baden–Württemberg), Bureau Gravimetrique International (BGI, France) for providing the gravity data and the Geologisches Landesamt Baden–Württemberg (Geological Department of Baden–Württemberg) for providing data and maps of the sediment layers within the Rhine Valley. Grateful thanks goes to Prof. W.E. Featherstone and the reviewers Prof. S.D. Pagiatakis, Dr. U. Marti as well as an unknown reviewer for their helpful comments on this paper.     

Contribution to the investigation of structure and origin of the east African graben by gravimetry

For our investigation we have 235 measurements done in the east part of Democratic Republic of Congo by P. Herrinck during a magnetic survey including the graben region from the parallel joining Goma city and Mahagi city, the region between Albert and Aka lakes, and the route from Aba to Kinsagani. During the surveys the density of recording points has been selected according to the importance of anomalies. In this way, the offset was 1 km where the disturbance was high in Goma city and 20 km have been sufficient along the route from Aba to Kinsagani. For the topographic and isostatic reductions only one cartographic document has been chosen that was the international map of the scale 1/1 000 000 which presents a certain characteristic of homogeneity.     

Contribution to the Investigation of Structure and Origin of the East African Graben by Gravimetry

IntroductionGeophysics as its nameindicates has to do withthe physics of the earth and its surrounding at-mosphere . Gilbert’s discovery that the earth be-haves as a great and rather irregular magnet andNewton’s theory of gravitation may be said toconstitute the beginning of geophysics . Miningand the search of metals date from the earliestti mes , but the scientific record began with thepublicationin 1956 of the famous treatise De remetallica by Georgius Agricola , whichfor manyyears was …     

Improving Orthometric Heights Using Precise GPS Measurements

The vertical component obtained from the Global Positioning System (GPS) observations is from the ellipsoid (a mathematical surface), and therefore needs to be converted to the orthometric height, which is from the geoid (represented by the mean sea level). The common practice is to use existing bench marks (around the four corners of a project area and interpolate for the rest of the area), but in many areas bench marks may not be available, in which case an existing geoid undulation is used. Present available global geoid undulation values are not generally as detailed as needed, and in many areas they are not known better than ±1 to ±5 m, because of many limitations. This article explains the difficulties encountered in obtaining precise geoid undulation with some example computations, and proposes a technique of applying corrections to the best available global geoid undulations using detailed free-air gravity anomalies (within a 2° × 2° area) to get relative centimeter accuracy. Several test computations have been performed to decide the optimal block sizes and the effective spherical distances to compute the regional and the local effects of gravity anomalies on geoid undulations by using the Stokes integral. In one test computation a 2° × 2° area was subdivided into smaller surface elements. A difference of 37.34 ± 1.6 cm in geoid undulation was obtained over the same 2° × 2° area when 1° × 1° block sizes were replaced by a combination of 5' × 5' and 1' × 1' subdivision integration elements (block sizes).     

我国东部某地油气微渗漏的遥感研究

我国东部某地一些油气田区的遥感图像(本文为国土卫星彩红外片)上,常见有一些灰白色、浅褐色云雾状色调异常,经过地植物、土壤的地面光谱测试和土壤气体化探检查,发现色调；异常范围内、外植物和土壤光谱存在一定差异,异常内还发现有烃类气体异常,这种色调异常极有可能与油气微渗漏有关。对这种遥感图像上的色调异常进行的计算机增强处理研究表明,正规化与频谱变量、时间变量标准化交换组合处理效果较好,处理后的图像上色调异常显示十分明显。据永安地区国土卫星彩红外片正规化、标准化组合处理图像上显示的色调异常与已知油气田十分吻合,说明该处理方法是有效的。     

Effects of topographic–isostatic masses on gravitational functionals at the Earth’s surface and at airborne and satellite altitudes

Topographic–isostatic masses represent an important source of gravity field information, especially in the high-frequency band, even if the detailed mass-density distribution inside the topographic masses is unknown. If this information is used within a remove-restore procedure, then the instability problems in downward continuation of gravity observations from aircraft or satellite altitudes can be reduced. In this article, integral formulae are derived for determination of gravitational effects of topographic–isostatic masses on the first- and second-order derivatives of the gravitational potential for three topographic–isostatic models. The application of these formulas is useful for airborne gravimetry/gradiometry and satellite gravity gradiometry. The formulas are presented in spherical approximation by separating the 3D integration in an analytical integration in the radial direction and 2D integration over the mean sphere. Therefore, spherical volume elements can be considered as being approximated by mass-lines located at the centre of the discretization compartments (the mass of the tesseroid is condensed mathematically along its vertical axis). The errors of this approximation are investigated for the second-order derivatives of the topographic–isostatic gravitational potential in the vicinity of the Earth’s surface. The formulas are then applied to various scenarios of airborne gravimetry/gradiometry and satellite gradiometry. The components of the gravitational vector at aircraft altitudes of 4 and 10 km have been determined, as well as the gravitational tensor components at a satellite altitude of 250 km envisaged for the forthcoming GOCE (gravity field and steady-state ocean-circulation explorer) mission. The numerical computations are based on digital elevation models with a 5-arc-minute resolution for satellite gravity gradiometry and 1-arc-minute resolution for airborne gravity/gradiometry.     

Integrating age in the detection and mapping of incongruous patches in coffee (Coffea arabica) plantations using multi-temporal Landsat 8 NDVI anomalies

The development of cost-effective, reliable and easy to implement crop condition monitoring methods is urgently required for perennial tree crops such as coffee (Coffea arabica), as they are grown over large areas and represent long term and higher levels of investment. These monitoring methods are useful in identifying farm areas that experience poor crop growth, pest infestation, diseases outbreaks and/or to monitor response to management interventions. This study compares field level coffee mean NDVI and LSWI anomalies and age-adjusted coffee mean NDVI and LSWI anomalies in identifying and mapping incongruous patches across perennial coffee plantations. To achieve this objective, we first derived deviation of coffee pixels from the global coffee mean NDVI and LSWI values of nine sequential Landsat 8 OLI image scenes. We then evaluated the influence of coffee age class (young, mature and old) on Landsat-scale NDVI and LSWI values using a one-way ANOVA and since results showed significant differences, we adjusted NDVI and LSWI anomalies for age-class. We then used the cumulative inverse distribution function (α ≤ 0.05) to identify fields and within field areas with excessive deviation of NDVI and LSWI from the global and the age-expected mean for each of the Landsat 8 OLI scene dates spanning three seasons. Results from accuracy assessment indicated that it was possible to separate incongruous and healthy patches using these anomalies and that using NDVI performed better than using LSWI for both global and age-adjusted mean anomalies. Using the age-adjusted anomalies performed better in separating incongruous and healthy patches than using the global mean for both NDVI (Overall accuracy = 80.9% and 68.1% respectively) and for LSWI (Overall accuracy = 68.1% and 48.9% respectively). When applied to other Landsat 8 OLI scenes, the results showed that the proportions of coffee fields that were modelled incongruent decreased with time for the young age category and while it increased for the mature and old age classes with time. We concluded that the method could be useful for the identification of anomalous patches using Landsat scale time series data to monitor large coffee plantations and provide an indication of areas requiring particular field attention.     

Attraction of the compensating masses produced by using an arbitrary isostatic model

The representation of the topography is usually made by digital height models and digital density models. Therefore, one can produce the so-called digital Moho model (DMM) by applying a certain isostatic hypothesis. The attraction of such compensating masses is deducted. Also, some special cases are treated. The effect of neglecting the height of the computational point only on calculating the attraction of the compensating masses is studied. The results show that the height of the computational point must be taken into account on calculating the attraction of the compensating masses specially for high mountainous areas.