The R-approximation method and its application to the interpretation of detailed gravity and magnetic data

The application of the R-approximation method based on the use of the Radon integral transform within the framework of the method of linear integral representations to the interpretation of detailed gravity and magnetic data is considered in this paper. The results are presented for some research areas located in Russia and in the Crimea (Ukraine). The method determines the distribution of anomalous field elements, construction of the analytical continuation of fields, and their linear transformations with a sufficiently high degree of accuracy.     

Accurate and Efficient Regularized Inversion Approach for the Interpretation of Isolated Gravity Anomalies

A very fast and efficient approach for gravity data inversion based on the regularized conjugate gradient method has been developed. This approach simultaneously inverts for the depth (z), and the amplitude coefficient (A) of a buried anomalous body from the gravity data measured along a profile. The developed algorithm fits the observed data by a class of some geometrically simple anomalous bodies, including the semi-infinite vertical cylinder, infinitely long horizontal cylinder, and sphere models using the logarithms of the model parameters [log(z) and log(|A|)] rather than the parameters themselves in its iterative minimization scheme. The presented numerical experiments have shown that the original (non-logarithmed) minimization scheme, which uses the parameters themselves (z and |A|) instead of their logarithms, encountered a variety of convergence problems. The aforementioned transformation of the objective functional subjected to minimization into the space of logarithms of z and |A| overcomes these convergence problems. The reliability and the applicability of the developed algorithm have been demonstrated on several synthetic data sets with and without noise. It is then successfully and carefully applied to seven real data examples with bodies buried in different complex geologic settings and at various depths inside the earth. The method is shown to be highly applicable for mineral exploration, and for both shallow and deep earth imaging, and is of particular value in cases where the observed gravity data is due to an isolated body embedded in the subsurface.     

Focusing iterative migration of gravity gradiometry data acquired in the Nordkapp Basin,Barents Sea

Geological interpretation based on gravity gradiometry data constitutes a very challenging problem. Rigorous 3D inversion is the main technique used in quantitative interpretation of the gravity gradiometry data. An alternative approach to the quantitative analysis of the gravity gradiometry data is based on 3D smooth potential field migration. This rapid imaging approach, however, has the shortcomings of providing smooth images since it is based on direct integral transformation of the observed gravity tensor data. Another limitation of migration transformation is related to the fact that, in a general case, the gravity data generated by the migration image do not fit the observed data well. In this paper, we describe a new approach to rapid imaging that allows us to produce the density distribution which adequately describes the observed data and, at the same time, images the structures with anomalous densities having sharp boundaries. This approach is based on the basic theory of potential field migration with a focusing stabilizer in the framework of regularized scheme, which iteratively transfers the observed gravity tensor field into an image of the density distribution in the subsurface formations. The results of gravity migration can also be considered as an a priori model for conventional inversion subsequently. We demonstrate the practical application of migration imaging using both synthetic and real gravity gradiometry data sets acquired for the Nordkapp Basin in the Barents Sea.     

3D inversion of gravity data by separation of sources and the method of local corrections: Kolarovo gravity high case study

We present a novel methodology for 3D gravity/magnetic data inversion. It combines two algorithms for preliminary separation of sources and an original approach to 3D inverse problem solution. The first algorithm is designed to separate sources in depth and to remove the shallow ones. It is based on subsequent upward and downward data continuation. For separation in the lateral sense, we approximate the given observed data by the field of several 3D line segments. For potential field data inversion we apply a new method of local corrections. The method is efficient and does not require trial-and-error forward modeling. It allows retrieving unknown 3D geometry of anomalous objects in terms of restricted bodies of arbitrary shape and contact surfaces. For restricted objects, we apply new integral equations of gravity and magnetic inverse problems. All steps of our methodology are demonstrated on the Kolarovo gravity anomaly in the Danube Basin of Slovakia.     

Linear mechanism of generation and intencification of internal gravity waves in the ionosphere at their interaction with a nonuniform zonal wind: 1. Model of the medium and initial dynamic equations

The specific features of the generation and intensification of internal gravity wave structures in different atmospheric-ionospheric regions, caused by zonal local nonuniform winds (shear flows), are studied. The model of the medium has been explained and an initial closed system of equations has been obtained in order to study the linear and nonlinear dynamics of internal gravity waves (IGWs) when they interact with the geomagnetic field in a dissipative ionosphere (for the D, E, and F regions).     

东北地区重磁场与地壳结构特征

分析了东北地区的重、磁场特征,同时对研究区的布格重力异常和航磁异常据进行了小波分析计算.根据分析与计算可知,东北地区重力场以北东走向为主,表现出该地区重力场的主要趋势.根据磁场的分布特征,可将研究区分为六个区域:呼和浩特以北磁场相对平静区;赤峰一带正负磁场交互变化区;海拉尔以南磁场缓变区;齐齐哈尔—依春磁场剧烈变化区;长春—沈阳负磁场平缓区;牡丹江—丹东磁场区.利用重力资料,应用调和级数法对研究区的莫霍界面进行了反演计算,得到了该地区莫霍界面深度分布.根据磁力资料采用遗传算法反演计算了研究区居里界面的深度分布.同时对研究区的地壳结构特征进行了探讨.     

Response of the nighttime midlatitude ionosphere to the passage of an atmospheric gravity wave

Using the data of the ionospheric vertical sounding in Almaty, the response of various parameters of the nighttime F layer to the passage of an atmospheric gravity wave, generated during the large magnetic storm on July 24–25, 2004, is studied. The analysis of the phase relations between the variations in the electron density at the F layer maximum (NmF), the layer maximum height (hmF), and the layer half-thickness showed that they are determined by the slope of the wave phase front. It is shown that the half-thickness of the layer changes in anti-phase with the variations in NmF2. The known fact that the amplitudes of variations in the critical frequencies of the F ₂ layer are smaller than the amplitudes of electron density variations at fixed heights is explained.     

Solar-terrestrial storm of November 18–20, 2003. 1. Near-Earth disturbances in the solar wind

The structure, configuration, dynamics, and solar sources of the near-Earth MHD disturbance of the solar wind on November 20, 2003, is considered. The disturbances of October 24 and November 22 after flares from the same AR 10484 (10501) are compared. The velocity field in the leading part of the sporadic disturbance is for the first time studied in the coordinate system stationary relative to the bow shock. A possible scenario of the physical processes in the course of this solar-terrestrial storm is discussed in comparison with the previously developed scenario for the storm of July 15, 2000. It has been indicated that (1) the near-Earth disturbance was observed at the sector boundary (HCS) and in its vicinities and (2) the disturbance MHD structure included: the complicated bow shock, wide boundary layer with reconnecting fields at a transition from the shock to the magnetic cloud, magnetic cloud with a magnetic cavity including packed substance of an active filament, and return shock layer (supposedly). It has been found out that the shock front configuration and the velocity field are reproduced at an identical position of AR and HCS relative to the Earth on November 20 and 24. It has been indicated that the maximal magnetic induction in the cloud satisfied the condition B _m = (8πn ₁ m _p)^1/2(D ? NV₁), i.e., depended on the dynamic impact on the cloud during all three storms [Ivanov et al., 1974]. When the disturbance was related to solar sources, the attention has been paid to the parallelism of the axes of symmetry of the active filament, transient coronal hole, coronal mass ejection, zero line of the open coronal field (HCS), and the axis of the near-Earth magnetic cloud: the regularity previously established in the scenario of the storm of July 15, 2000 [Ivanov et al., 2005]. It has been indicated that the extremely large B _m value in the cloud of October 20 was caused by a strong suppression of the series of postflare shocks reflected from the heliospheric streamer.     

A nonhydrostatic unstructured-mesh soundproof model for simulation of internal gravity waves

A semi-implicit edge-based unstructured-mesh model is developed that integrates nonhydrostatic soundproof equations, inclusive of anelastic and pseudo-incompressible systems of partial differential equations. The model builds on nonoscillatory forward-in-time MPDATA approach using finite-volume discretization and unstructured meshes with arbitrarily shaped cells. Implicit treatment of gravity waves benefits both accuracy and stability of the model. The unstructured-mesh solutions are compared to equivalent structured-grid results for intricate, multiscale internal-wave phenomenon of a non-Boussinesq amplification and breaking of deep stratospheric gravity waves. The departures of the anelastic and pseudoincompressible results are quantified in reference to a recent asymptotic theory [Achatz et al. 2010, J. Fluid Mech., 663, 120–147)].     

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.     

On the discrete outer gravity field

Summary In the present paper the gravity field of the earth in the neighbourhood of the local disturbing masses is studied. The object of the method presented consists of the approximation of the disturbing potentialT _h , which fulfils Laplace's equation outside disturbing masses, on the earth's surface the fundamental boundary value condition of gravity and in infinity it is to be regular by the approximation of the disturbing potential (or by the discrete disturbing potential)T _h , which fulfils the respective finite difference approximation of Laplace's equation and the boundary value conditions in infinity and on the earth's surface. It is also shown that the approximation of the disturbing potentialT _h has the same properties as the disturbing potentialT. The method under consideration will be derived quite generally without any hypothesis about the distribution of the mass between the earth's surface and the geoid. It commences from the gravity data related to the earth's surface only-from the given geodetic measurements.     

Mean-field concept and direct numerical simulations of rotating magnetoconvection and the geodynamo

Mean-field theory describes magnetohydrodynamic processes leading to large-scale magnetic fields in various cosmic objects. In this study magnetoconvection and dynamo processes in a rotating spherical shell are considered. Mean fields are defined by azimuthal averaging. In the framework of mean-field theory, the coefficients which determine the traditional representation of the mean electromotive force, including derivatives of the mean magnetic field up to the first order, are crucial for analyzing and simulating dynamo action. Two methods are developed to extract mean-field coefficients from direct numerical simulations of the mentioned processes. While the first method does not use intrinsic approximations, the second one is based on the second-order correlation approximation. There is satisfying agreement of the results of both methods for sufficiently slow fluid motions. Both methods are applied to simulations of rotating magnetoconvection and a quasi-stationary geodynamo. The mean-field induction effects described by these coefficients, e.g., the α-effect, are highly anisotropic in both examples. An α²-mechanism is suggested along with a strong γ-effect operating outside the inner core tangent cylinder. The turbulent diffusivity exceeds the molecular one by at least one order of magnitude in the geodynamo example. With the aim to compare mean-field simulations with corresponding direct numerical simulations, a two-dimensional mean-field model involving all previously determined mean-field coefficients was constructed. Various tests with different sets of mean-field coefficients reveal their action and significance. In the magnetoconvection and geodynamo examples considered here, the match between direct numerical simulations and mean-field simulations is only satisfying if a large number of mean-field coefficients are involved. In the magnetoconvection example, the azimuthally averaged magnetic field resulting from the numerical simulation is in good agreement with its counterpart in the mean-field model. However, this match is not completely satisfactory in the geodynamo case anymore. Here the traditional representation of the mean electromotive force ignoring higher than first-order spatial derivatives of the mean magnetic field is no longer a good approximation.     

Approximation constructions in interpretation problems of gravity and magnetic surveys: A review

Methods of approximation occupy a significant place at the interpretation stage of gravity and magnetic surveys. Two main directions are the approximation of the initial field by an analytic function and the approximation or parametrization of a geological model.     

有限长圆柱体磁异常场全空间正演方法

在经典位场理论中,许多简单形体位场异常难以通过积分得到全空间的解析式.圆柱体是一类很重要的理论模型体,常用于模拟圆柱状地质体或非地质体(如管线),但目前还不能用解析公式正演有限长圆柱体在三维空间里的磁异常,而多是采用近似简化为有限长磁偶极子或线模型代替.对于有限长圆柱体,特别是半径相对于上顶埋深较大时,这种近似的误差不可忽略.本文利用共轭复数变量替换法,推导出有限长圆柱体在全空间的引力位一阶、二阶导数,利用Poisson关系得到磁异常正演公式,进而利用有限长圆柱体磁异常正演公式求解管状体的磁异常,得到不同磁化方向、不同大小的管线产生的磁场的特征,并将其推广到截面为椭圆的情况.最后通过模拟计算定量给出了将圆柱体近似为线模型的条件.     

Analysis of gravity source moment inversion. Part II: Usinga priori information about the source

The method of moment inversion, based on the approximation of the gravity anomaly by thetruncated series obtained from its multipole expansion, uses, implicitly,a priori information about the anomalous body. The series truncation imposes a regularizing condition on the equipotential surfaces (produced by the anomalous body), allowing the unique determination of some moments and linear combinations of moments that are the coefficients of the basis functions in the multipole expansion series. These moments define a class of equivalent distributions of mass. The equivalence criterion is based on the misfit between the observations and the field produced by the series truncated at a prefixed maximum order for the moments. The estimates of the moments of the equivalent distribution are shown to compose the stationary solution of a system of first-order linear differential equations for which uniqueness and asymptotic stability are guaranteed. Specifically for the series retaining moments up to second order, the implicita priori information introduced requires that the source have finite volume, be sufficiently distant from the measurement plane and that its spatial distribution of mass present three orthogonal planes of symmetry intersecting at the center of mass. Subject to these hypotheses, it is possible to estimate uniquely and simultaneously the total excess of mass, the position of the center of mass and the directions of the three principal axes of the anomalous body.     

Some space gravity formulas and the dimensions and the mass of the Earth

Summary Adopting thePizzetti-Somigliana method and using elliptic integrals we have obtained closed formulas for the space gravity field in which one of the equipotential surfaces is a triaxial ellipsoid. The same formulas are also obtained in first approximation of the equatorial flattening avoiding the use of the elliptic integrals. Using data from satellites and Earth gravity data the gravitational and geometric bulge of the Earth's equator are computed. On the basis of these results and on the basis of recent gravity data taken around the equator between the longitudes 50° to 100° E, 155° to 180° E, and 145° to 180° W, we question the advantage of using a triaxial gravity formula and a triaxial ellipsoid in geodesy. Closed formulas for the space field in which a biaxial ellipsoid is an equipotential surface are also derived in polar coordinates and its parameters are specialized to give the international gravity formula values on the international ellipsoid. The possibility to compute the Earth's dimensions from the present Earth gravity data is the discussed and the value ofMG=(3.98603×10²⁰ cm³ sec^–2) (M mass of the Earth,G gravitational constant) is computed. The agreement of this value with others computed from the mean distance Earth-Moon is discussed. The Legendre polinomials series expansion of the gravitational potential is also added. In this series the coefficients of the polinomials are closed formulas in terms of the flattening andMG.Publication Number 327, and Istituto di Geodesia e Geofisica of Università di Trieste.     

Derivation of the CHAMP-only global gravity field model TUG-CHAMP04 applying the energy integral approach

A global gravity field model TUG-CHAMP04, derived from CHAMP (CHAllenging Minisatellite Payload) satellite-to-satellite GPS tracking observations in the high-low mode (SST-hl) in combination with CHAMP accelerometry, is presented and described in detail in this paper. For this purpose the energy integral approach was applied to precise kinematic orbits and accelerometer data. The advantage of these kinds of orbits is that they are derived from purely geometrical information, hence no external gravity field information is used for the determination of the positions. The disadvantage of precise kinematic orbit information is, that no velocities are delivered and hence a procedure has to be elaborated to deduce the velocities from kinematic positions. This work is done in preparation for ESA’s GOCE (Gravity field and steady state Ocean Circulation Explorer) satellite mission (scheduled launch November 2006), aiming at a high precision and high-resolution gravity field model on a global scale. This paper concentrates on the CHAMP data processing, where, in contrast to the usual standard method (processing in the Earth fixed frame), an approach in the inertial frame is chosen. Focus is taken on the data preprocessing of both accelerometer and orbit data, emphasising on the correct treatment of data-gaps and outlier detection. Furthermore an arc-wise weighting strategy is introduced and the advantages/disadvantages of this approach are discussed. Finally, the TUG-CHAMP04 model, calculated from one year of CHAMP data is compared with the official CHAMP gravity field model EIGEN-3p and terrestrial data (GPS levelling data).     

Geophysical researches (gravity and magnetic) of the Eratosthenes Seamount in the eastern Mediterranean Sea

New free-air gravity and magnetic maps of the Eratosthenes Seamount and its vicinity were regenerated from potential field data. Stages of data processing are power spectrum, upward continuation, filtering on the free-air gravity anomaly data. RTP, pseudo-gravity transformation map, power spectrum, upward continuation, filtering, AS, and HGAS were applied on the magnetic data. A HGAS map shows the images and locations of the Eratosthenes magnetic body. Spectral analysis of the gravity and magnetic anomalies indicates that there is an elliptical elongated structure of the Eratosthenes Seamount in the width of approx. 86 km NW-SE orientation and in the length of 138 km NE-SW orientation, with a strike of N40°E and inclined to NW. It is considered that 22.49 ± 0.08 km obtained from power spectrum of the gravity data may be related to the crust thickness. Also, 15.67 ± 0.02 km obtained from power spectrum of the magnetic data is considered to be related to the magmatic basement of the Eratosthenes Seamount.     

Determination of the maximum-depth to potential field sources by a maximum structural index method

A simple and fast determination of the limiting depth to the sources may represent a significant help to the data interpretation. To this end we explore the possibility of determining those source parameters shared by all the classes of models fitting the data. One approach is to determine the maximum depth-to-source compatible with the measured data, by using for example the well-known Bott–Smith rules. These rules involve only the knowledge of the field and its horizontal gradient maxima, and are independent from the density contrast.Thanks to the direct relationship between structural index and depth to sources we work out a simple and fast strategy to obtain the maximum depth by using the semi-automated methods, such as Euler deconvolution or depth-from-extreme-points method (DEXP).The proposed method consists in estimating the maximum depth as the one obtained for the highest allowable value of the structural index (N_max). N_max may be easily determined, since it depends only on the dimensionality of the problem (2D/3D) and on the nature of the analyzed field (e.g., gravity field or magnetic field). We tested our approach on synthetic models against the results obtained by the classical Bott–Smith formulas and the results are in fact very similar, confirming the validity of this method. However, while Bott–Smith formulas are restricted to the gravity field only, our method is applicable also to the magnetic field and to any derivative of the gravity and magnetic field. Our method yields a useful criterion to assess the source model based on the (∂f/∂x)_max/f_max ratio.The usefulness of the method in real cases is demonstrated for a salt wall in the Mississippi basin, where the estimation of the maximum depth agrees with the seismic information.