Several noniterative, imaging methods for potential field data have been proposed that provide an estimate of the 3D magnetization/density distribution within the subsurface or that produce images of quantities related or proportional to such distributions. They have been derived in various ways, using generalized linear inversion, Wiener filtering, wavelet and depth from extreme points (DEXP) transformations, crosscorrelation, and migration. We demonstrated that the resulting images from each of these approaches are equivalent to an upward continuation of the data, weighted by a (possibly) depth-dependent function. Source distributions or related quantities imaged by all of these methods are smeared, diffuse versions of the true distributions; but owing to the stability of upward continuation, resolution may be substantially increased by coupling derivative and upward continuation operators. These imaging techniques appeared most effective in the case of isolated, compact, and depth-limited sources. Because all the approaches were noniterative, computationally fast, and in some cases, produced a fit to the data, they did provide a quick, but approximate picture of physical property distributions. We have found that inherent or explicit depth-weighting is necessary to image sources at their correct depths, and that the best scaling law or weighting function has to be physically based, for instance, using the theory of homogeneous fields. A major advantage of these techniques was their speed, efficiently providing a basis for further detailed, follow-up modelling.

The geological interpretation of gravity gradiometry data is a very challenging problem. While maps of different gravity gradients may be correlated with geological structures present, maps alone cannot quantify 3D density distributions related to geological structures. 3D inversion represents the only practical tool for the quantitative interpretation of gravity gradiometry data. However, 3D inversion is a complicated and time-consuming procedure that is very dependent on the a priori model and constraints used. To overcome these difficulties for the initial stages of an interpretation workflow, we introduce the concept of potential field migration, and demonstrate how it can be applied for rapid 3D imaging of entire gravity gradiometry surveys. This method is based on a direct integral transformation of the observed gravity gradients into a subsurface density distribution that can be used for interpretation or as an a priori model for subsequent 3D regularized inversion. For large-scale surveys, we show how migration runs on the order of minutes compared to hours for 3D regularized inversion. Moreover, the results obtained from potential field migration are comparable to those obtained from regularized inversion with smooth stabilizers. We present a case study for the 3D imaging of FALCON airborne gravity gradiometry data from Broken Hill, Australia. We observe good agreement between results obtained from potential field migration and those generated by 3D regularized inversion.

Three-dimensional potential field migration for rapid imaging of entire total-magnetic-intensity (TMI) surveys is introduced, and real time applications are discussed. Potential field migration is based on a direct integral transformation of the measured TMI data into a 3D susceptibility model, which could be directly used for interpretation or as an a priori model for subsequent regularized inversion. The advantage of migration is that it does not require any a priori information about the type of the sources present, nor does it rely on regularization as per inversion. Migration is very stable with respect to noise in measured data because the transform is reduced to the downward continuation of a function that is analytical everywhere in the subsurface. The 3D migration of TMI data acquired over the Reid-Mahaffy test site in Ontario, Canada is used as a test study. Our results are shown to be consistent with those results obtained from 3D regularized inversion as well as the known geology of the area. Interestingly, the migration of raw TMI data produces results very similar to the inversion of diurnally corrected and microleveled TMI data, suggesting that migration could be applied directly to real-time imaging during the acquisition.

We used a fast method to interpret self-potential data: the depth from extreme points (DEXP) method. This is an imaging method transforming self-potential data, or their derivatives, into a quantity proportional to the source distribution. It is based on upward continuing of the field to a number of altitudes and then multiplying the continued data with a scaling law of those altitudes. The scaling law is in the form of a power law of the altitudes, with an exponent equal to half of the structural index, a source parameter related to the type of source. The method is autoconsistent because the structural index is basically determined by analyzing the scaling function, which is defined as the derivative of the logarithm of the self-potential (or of its pth derivative) with respect to the logarithm of the altitudes. So, the DEXP method does not need a priori information on the self-potential sources and yields effective information about their depth and shape/typology. Important features of the DEXP method are its high-resolution power and stability, resulting from the combined effect of a stable operator (upward continuation) and a high-order differentiation operator. We tested how to estimate the depth to the source in two ways: (1) at the positions of the extreme points in the DEXP transformed map and (2) at the intersection of the lines of the absolute values of the potential or of its derivative (geometrical method). The method was demonstrated using synthetic data of isolated sources and using a multisource model. The method is particularly suited to handle noisy data, because it is stable even using high-order derivatives of the self-potential. We discussed some real data sets: Malachite Mine, Colorado (USA), the Sariyer area (Turkey), and the Bender area (India). The estimated depths and structural indices agree well with the known information.

In this paper we present a new method for source parameter estimation, based on the local wavenumber function. We make use of the stable properties of the Depth from EXtreme Points (DEXP) method, in which the depth to the source is determined at the extreme points of the field scaled with a power-law of the altitude. Thus the method results particularly suited to deal with local wavenumber of high-order, as it is able to overcome its known instability caused by the use of high-order derivatives. The DEXP transformation enjoys a relevant feature when applied to the local wavenumber function: the scaling-law is in fact independent of the structural index. So, differently from the DEXP transformation applied directly to potential fields, the Local Wavenumber DEXP transformation is fully automatic and may be implemented as a very fast imaging method, mapping every kind of source at the correct depth. Also the simultaneous presence of sources with different homogeneity degree can be easily and correctly treated. The method was applied to synthetic and real examples from Bulgaria and Italy and the results agree well with known information about the causative sources. (C) 2014 Elsevier B.V.

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 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 Bolt 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 (partial derivative f/partial derivative 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. (C) 2012 Elsevier B.V.]]>

We developed a new method for interpretation of airborne gravity gradiometry data, based on cokriging inversion. The cokriging method that we evaluated minimized the theoretical estimation error variance by using auto- and crosscorrelations of several variables. It does not require iterations and can easily include complex a priori information. Moreover, the smoothing effects in the inverted density structure model can be reduced to a certain extent due to the anisotropy constrain in the covariance model. We compared the recovered models obtained by inverting the different combinations of gravity-gradient components to understand how different component combinations contributed to the resolution of the recovered model. The results indicated that including multiple components for inversion increased the resolution of the recovered density model and improved the structure delineation. Moreover, in the case in which the parameters of the variogram model are not well chosen, cokriging with multicomponent combinations can still correctly recover the geometry of the targeted sources. The survey data of the Vinton dome were considered as a case study. The results of the inversion were in good agreement with the known formation in the region. This supports the validity of our method.

针对传统 DEXP 快速成像结果依赖于地质体构造指数的缺点，笔者提出了利用重力异常不同阶垂向导数比值 DEXP 的方法。该方法使得成像结果独立于地质体构造指数。地质体的深度估计完全自动化，仅需要寻找比值 DEXP 成像结果的极大值点位置。通过该方法估计出的深度结果，可以进行地质体构造指数的估计。通过模型试验，证明该方法能准确地估计出地质体的深度位置以及地质体的类型。将该方法应用到 Bell Geospace 公司在 Vinton Dome 测得的 Air- -FTG 全张量重力梯度数据中取得了很好的结果。

Full Tensor Gravity Gradiometry (FTG) data are routinely used in exploration programmes to evaluate and explore geological complexities hosting hydrocarbon and mineral resources. FTG data are typically used to map a host structure and locate target responses of interest using a myriad of imaging techniques. Identified anomalies of interest are then examined using 2D and 3D forward and inverse modelling methods for depth estimation. However, such methods tend to be time consuming and reliant on an independent constraint for clarification. This paper presents a semi-automatic method to interpret FTG data using an adaptive tilt angle approach. The present method uses only the three vertical tensor components of the FTG data (T-zx, T-zy and T-zz) with a scale value that is related to the nature of the source (point anomaly or linear anomaly). With this adaptation, it is possible to estimate the location and depth of simple buried gravity sources such as point masses, line masses and vertical and horizontal thin sheets, provided that these sources exist in isolation and that the FTG data have been sufficiently filtered to minimize the influence of noise. Computation times are fast, producing plausible results of single solution depth estimates that relate directly to anomalies. For thick sheets, the method can resolve the thickness of these layers assuming the depth to the top is known from drilling or other independent geophysical data. We demonstrate the practical utility of the method using examples of FTG data acquired over the Vinton Salt Dome, Louisiana, USA and basalt flows in the Faeroe-Shetland Basin, UK. A major benefit of the method is the ability to quickly construct depth maps. Such results are used to produce best estimate initial depth to source maps that can act as initial models for any detailed quantitative modelling exercises using 2D/3D forward/inverse modelling techniques.

We have presented a joint inversion of all gravity-gradient tensor components to estimate the shape of an isolated 3-D geological body located in subsurface. The method assumes the knowledge about the depth to the top and density contrast of the source. The geological body is approximated by an interpretation model formed by an ensemble of vertically juxtaposed 3-D right prisms, each one with known thickness and density contrast. All prisms forming the interpretation model have a polygonal horizontal cross-section that approximates a depth slice of the body. Each polygon defining a horizontal cross-section has the same fixed number of vertices, which are equally spaced from 0 degrees to 360 degrees and have their horizontal locations described in polar coordinates referred to an arbitrary origin inside the polygon. Although the number of vertices forming each polygon is known, the horizontal coordinates of these vertices are unknown. To retrieve a set of juxtaposed depth slices of the body, and consequently, its shape, our method estimates the radii of all vertices and the horizontal Cartesian coordinates of all arbitrary origins defining the geometry of all polygons describing the horizontal cross-sections of the prisms forming the interpretation model. To obtain a stable estimate that fits the observed data, we impose constraints on the shape of the estimated body. These constraints are imposed through the well-known zeroth- and first-order Tikhonov regularizations allowing, for example, the estimate of vertical or dipping bodies. If the data do not have enough in-depth resolution, the proposed inverse method can obtain a set of stable estimates fitting the observed data with different maximum depths. To analyse the data resolution and deal with this possible ambiguity, we plot the l(2)-norm of the residuals (s) against the estimated volume (v(p)) produced by a set of estimated sources having different maximum depths. If this s x v(p) curve (s as a function of v(p)) shows a well-defined minimum of s, the data have enough resolution to recover the shape of the body entirely. Conversely, if the observed data do not have enough resolution, some estimates with different maximum depths produce practically the same minimum value of s on the s x v(p) curve. In this case, the best estimate among a suite of estimates producing equally data fits is the one fitting the gravity-gradient data and producing the minima of both the source's bottom depth and volume. The histograms of the residuals can be used to quantify and remove systematic errors in the data. After removing these errors, we confirmed the ability of our method to recover the source geometry entirely (or its upper part only), if the data have sufficient (or insufficient) in-depth resolution. By inverting the gravity-gradient data from a survey over the Vinton salt dome (Louisiana, USA) with a density contrast of 0.55 g cm(-3), we estimated a massive cap rock whose maximum depth attains 460 +/- 10 m and its shallowest portion is elongated in the northeast-southwest direction.

全张量重力梯度测量技术日趋成熟，已成为地球物理勘探的重要方法之一。本文将重力异常三维反演技术应用于重力梯度张量数据反演，并针对重力梯度张量数据，构建了联合反演目标函数，同时引入投影梯度算法对反演求解过程施加约束。通过理论模型试验和实际资料的应用，表明了方法的可行性和应用前景。