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In potential-field inversion problems, it can be difficult to obtain reliable information about the source distribution with respect to depth. Moreover, spatial resolution of the solution decreases with depth, and in fact the more ill-posed the problem – and the more noisy the data – the less reliable the depth information. Based on early work in depth resolution, defined in terms of the singular value decomposition, we introduce a tool APPROXDRP which uses an approximation of the singular vectors obtained by the iterative Lanczos bidiagonalization algorithm, making it well suited for large-scale problems. This tool allows a computational/visual analysis of how much the depth resolution in a computational potential-field inversion problem can be obtained from the given data.We show that when used in combination with a plot of the approximate SVD quantities, APPROXDRP may successfully show the limitations of depth resolution resulting from noise in the data. This allows a reliable analysis of the retrievable depth information and effectively guides the user in choosing the optimal number of iterations, for a given problem.

To investigate the crustal magnetic structure, it is important to assess the susceptibility and remanence properties of rocks and ores. In this paper, we propose a method to extract the contributions of induced and remanent magnetization from modeling of magnetic anomalies. We first estimate the direction of the total magnetization vector by studying the reduced-to-pole anomaly and its correlation with different magnitude magnetic transforms. Then we invert the magnetic data to obtain the volumetric distribution of the magnetization intensity. As the third step, based on a priori information about the Koenigsberger ratio derived from petrophysical measurements, we extract the distributions in the source volume of the induced and remanent magnetization intensities, based on a generalized relationship involving the total and remanent magnetizations, and the true susceptibility. In this way, we are able to produce separate maps of the anomaly fields attributed to the physical magnetic source parameters: remanent and induced magnetization. After validating the method with synthetic data, we analyze the data relative to the Mesozoic and Cenozoic igneous rocks in Yeshan region, eastern China. The analysis of the separated magnetization components reveals that the intrusion of dioritic and basaltic rocks occurred at different geological periods, and the basaltic rocks were magnetized by a reversed geomagnetic field. The uncertainty analysis shows that a larger Koenigsberger ratio is beneficial to extract more reliable remanence and susceptibility information

We present a nonlinear method to invert potential fields data, based on inverting the scaling function (τ) of the potential fields, a quantity that is independent on the source property, that is the mass density in the gravity case or the magnetic susceptibility in the magnetic case. So, no a priori prescription of the density contrast is needed and the source model geometry is determined independently on it. We assume Talwani's formula and generalize the Multi-HOmogeneity Depth Estimation (MHODE) method to the case of the inhomogeneous field generated by a general 2D source. The scaling function is calculated at different altitudes along the lines defined by the extreme points of the potential fields and the inversion of the scaling function yields the coordinates of the vertices of a multiple source body. Once the geometry is estimated, the source density is automatically computed from a simple regression of the scaling function of the gravity data vs. that generated from the estimated source body with a unit-density. We solve the above nonlinear problem by the Very Fast Simulated Annealing algorithm. The best performance is obtained when some vertices are constrained by either reasonable bounds or exact knowledge. In the salt-dome case we assumed that the top of the body is known from seismic observations and we solved for the lateral and bottom parts of the body. We applied the technique on three synthetic cases of complex sources and on the gravity anomalies over the Mors salt-dome (Denmark) and the Godavari Basin (India). In all these cases the method performed very well in terms of both geometrical and source-property definition. This article is protected by copyright. All rights reserved

Summary Gravity data inversion is a fundamental tool for geological exploration. A large amount of algorithms have been developed in the past, with different approaches. The choice of the inversion algorithm depends, mainly, on the geological contest, the kind of solution desired, its resolution at the end of the process, the availability of a priori information and on how they can be included in the inversion algorithm. A priori information has, often, a key-role in the inversion process. However, in case of reconstruction of salt and subsalt structures, seismic information may be poor and unfeasible. So we need to constrain the model with other kind of information. In this work, we present a part of recent Eni R&D activity focused on gravity data inversion. The shown results are mainly based on the Data Space Inversion algorithm (Pilkington 2009), originally presented for the magnetic case, and here extended to the gravity problem. We also putted a strong effort on the computational side of the problem, taking advantage from our experience in HPC (High Performance Computing), in order to speed up the inversion process and so enable its use at industrial level. In this paper we present and discuss some results regarding the application of the methodology to the SEAM (SEG Advance Modeling, 2007) demonstrating that the algorithm allows a consistent depth and density model.

Many methods have been used to upward continue potential field data. Most techniques employ the Fast Fourier transform, which is an accurate, quick way to compute level-to-level upward continuation or spatially varying scale filters for level-to-draped surfaces. We here propose a new continuation approach based on the minimum-length solution of the inverse potential field problem, which we call Volume Continuation (VOCO). For real data the VOCO is obtained as the regularized solution to the Tikhonov problem. We tested our method on several synthetic examples involving all types of upward continuation and downward continuation (level-to-level, level-to-draped, draped-to-level, draped-to-draped). We also employed the technique to upward continue to a constant height (2500 m a.s.l.), the high-resolution draped aeromagnetic data of the Ischia Island in Southern Italy. We found that, on the average, they are consistent with the aeromagnetic regional data measured at the same altitude. The main feature of our method is that it does not only provide continued data over a specified surface, but it yields a volume of upward continuation. For example, the continued data refers to a volume and thus, any surface may be easily picked up within the volume to get upward continuation to different surfaces. This approach, based on inversion of the measured data, tends to be especially advantageous over the classical techniques when dealing with draped-to-level upward continuation. It is also useful to obtain a more stable downward continuation and to continue noisy data. The inversion procedure involved in the method implies moderate computational costs, which are well compensated by getting a 3D set of upward continued data to achieve high quality results.

In this paper we present an improvement to the local wavenumber method able to overcome its instability caused by the use of high-order derivatives. 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 altitude. As DEXP transformation is applied to the local wavenumber function, we show that the scaling law is independent on the structural index. This allows the technique to be fully automatic, so that it may be implemented as a very fast imaging method, mapping every source, of different homogeneity degree, at the correct depth.

In potential field inversion problems we usually solve underdetermined systems and realistic solutions may be obtained by introducing a depth-weighting function in the objective function. The choice of the exponent of such power-law is crucial. It was suggested to determine it from the field-decay due to a single source-block; alternatively it has been defined as the structural index of the investigated source distribution. In both cases, when k -order derivatives of the potential field are considered, the depth-weighting exponent has to be increased by k with respect that of the potential field itself, in order to obtain consistent source model distributions. We show instead that invariant and realistic source-distribution models are obtained using the same depth-weighting exponent for the magnetic field and for its k -order derivatives. A similar behavior also occurs in the gravity case. In practice we found that the depth weighting-exponent is invariant for a given source-model and equal to that of the corresponding magnetic field, in the magnetic case, and of the 1st derivative of the gravity field, in the gravity case. In the case of the regularized inverse problem, with depth-weighting and general constraints, the mathematical demonstration of such invariance is difficult, because of its non-linearity, and of its variable form, due to the different constraints used. However, tests performed on a variety of synthetic cases seem to confirm the invariance of the depth-weighting exponent. A final consideration regards the role of the regularization parameter; we show that the regularization can severely affect the depth to the source because the estimated depth tends to increase proportionally with the size of the regularization parameter. Hence, some care is needed in handling the combined effect of the regularization parameter and depth weighting.

We illustrate a new imaging method to estimate the depth to the sources of potential fields and the structural index. The method consists of applying the Depth from EXtreme Point (DEXP) transformation to the ratio (R) between two different-order partial derivatives of the field. While the scaling function of the potential field depends on the structural index, we show that the scaling function of R merely depends on the difference between the two used orders of differentiation. This allows three main features to be established for the DEXP transformation of R: (1) it is independent from the structural index; (2) the estimation of the source depths is fully automatic, simply consisting in the search of position of the extreme points of the DEXP image and (3) the structural index of each source is finally determined from the scaling function or the extreme points using the estimated depth. Besides the well-known characteristics of the DEXP transformation, such as high-resolution and stability, the DEXP transformation of R enjoys one more relevant feature: it can be applied to multisource cases, yielding simultaneously correct estimations of structural index and depth for each source in the same image. However, while the DEXP transformation is a linear transformation of the field, the DEXP transformation of R is non-linear, and a procedure is described to circumvent the non-linear effects. The method is tested with synthetic examples and the estimated source parameters show a good agreement with the true values. The method was applied also to real magnetic data from the Pima copper mine, Arizona, USA, Hamrawien area, Egypt and Cataldere, Bala district of Turkey. The results are consistent with the known information about the causative sources.

In potential field inversion problems we usually solve underdetermined systems and this leads to a very shallow solution, typically known as minimum length solution. This may be avoided introducing a depth weighting function in the objective function (Li and Oldenburg, 1996). In this paper we derive invariance rules for either the minimum norm minimization and for the regularized inversion with depth weighting and positivity constraints. For a given source class, corresponding to a specific structural index N, the invariance rule assures that the same solution is obtained inverting the magnetic (or gravity) field or any of its nth order vertical derivatives. Although we demonstrate mathematically this invariance rule for the minimum norm minimization only, it is shown to occur also for the regularized inversion with depth weighting and positivity constraints. In this case, a source-class invariant form of depth weighting is derived, referring to that of the magnetic field, in the magnetic case, and to the 1st derivative of the gravity field, in the gravity case. We also illustrate how the combined effect of regularization parameter and depth weighting influences the estimated source model depth in the regularized inversion with depth weighting and positivity constraints.

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.