Practical applications of uniqueness theorems in gravimetry: Part I—Constructing sound interpretation methods

Geophysics ◽  
2002 ◽  
Vol 67 (3) ◽  
pp. 788-794 ◽  
Author(s):  
João B. C. Silva ◽  
Walter E. Medeiros ◽  
Valéria C. F. Barbosa
Keyword(s):  
Inverse Problem ◽  
A Priori ◽  
Stable Solution ◽  
Gravity Inversion ◽  
Uniqueness Theorems ◽  
A Priori Information ◽  
Practical Applications ◽  
Inversion Methods ◽  
Geological Information ◽  
Priori Information

To obtain a unique and stable solution to the gravity inverse problem, a priori information reflecting geological attributes of the gravity source must be used. Mathematical conditions to obtain stable solutions are established in Tikhonov's regularization method, where the a priori information is introduced via a stabilizing functional, which may be suitably designed to incorporate some relevant geological information. However, there is no unifying approach establishing general uniqueness conditions for a gravity inverse problem. Rather, there are many theorems, usually establishing just abstract mathematical conditions and making it difficult to devise the type of geological information needed to guarantee a unique solution. In Part I of these companion papers, we show that translating the mathematical uniqueness conditions into geological constraints is an important step not only in establishing the type of geological setting where a particular method may be applied but also in designing new gravity inversion methods. As an example, we analyze three uniqueness theorems in gravimetry restricted to the class of homogeneous bodies with known density and show that the uniqueness conditions established by them are more probably met if the solution is constrained to be a compact body without curled protrusions at their borders. These conditions, together with stabilizing conditions (assuming a simple shape for the source), form a guideline to construct sound gravity inversion methods. A historical review of the gravity interpretation methods shows that several methods implicitly follow this guideline. In Part II we use synthetic examples to illustrate the theoretical results derived in Part I. We also illustrate that the presented guideline is not the only way to design sound inversion methods for the class of homogeneous bodies. We present an alternative approach which produces good results but whose design requires a good dose of the interpreter's art.

scholarly journals The Neuroelectromagnetic Inverse Problem and the Zero Dipole Localization Error

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Rolando Grave de Peralta ◽  
Olaf Hauk ◽  
Sara L. Gonzalez
Keyword(s):  
Inverse Problem ◽  
Unique Solution ◽  
A Priori ◽  
Inverse Solution ◽  
Localization Error ◽  
A Priori Information ◽  
Dipole Localization ◽  
Inverse Solutions ◽  
Priori Information ◽  
Additional Constraints

A tomography of neural sources could be constructed from EEG/MEG recordings once the neuroelectromagnetic inverse problem (NIP) is solved. Unfortunately the NIP lacks a unique solution and therefore additional constraints are needed to achieve uniqueness. Researchers are then confronted with the dilemma of choosing one solution on the basis of the advantages publicized by their authors. This study aims to help researchers to better guide their choices by clarifying what is hidden behind inverse solutions oversold by their apparently optimal properties to localize single sources. Here, we introduce an inverse solution (ANA) attaining perfect localization of single sources to illustrate how spurious sources emerge and destroy the reconstruction of simultaneously active sources. Although ANA is probably the simplest and robust alternative for data generated by a single dominant source plus noise, the main contribution of this manuscript is to show that zero localization error of single sources is a trivial and largely uninformative property unable to predict the performance of an inverse solution in presence of simultaneously active sources. We recommend as the most logical strategy for solving the NIP the incorporation of sound additional a priori information about neural generators that supplements the information contained in the data.

Generalized compact gravity inversion

Geophysics ◽  
1994 ◽  
Vol 59 (1) ◽  
pp. 57-68 ◽  
Author(s):  
Valeria Cristina F. Barbosa ◽  
João B. C. Silva
Keyword(s):  
Body Shape ◽  
Sedimentary Basin ◽  
A Priori ◽  
Gravity Inversion ◽  
A Priori Information ◽  
Magnetite Ore ◽  
Inversion Technique ◽  
True Value ◽  
Ore Body ◽  
Priori Information

Extending the compact gravity inversion technique by incorporating a priori information about the maximum compactness of the anomalous sources along several axes provides versatility. Thus, the method may also incorporate information about limits in the axes lengths or greater concentration of mass along one or more directions. The judicious combination of different constraints on the anomalous mass distribution allows the introduction of several kinds of a priori information about the (arbitrary) shape of the sources. This method is particularly applicable to constant, linear density sources such as mineralizations along faults and intruded sills, dikes, and laccoliths in a sedimentary basin. The correct source density must be known with a maximum uncertainty of 40 percent; otherwise, the inversion produces thicker bodies for densities smaller than the true value and vice‐versa. Because of the limitations of the inverse gravity problem, the proposed technique requires an empirical technique to analyze the sensitivity of solutions to uncertainties in the a priori information. The proposed technique is based on a finite number of acceptable solutions, presumably representative of the ambiguity region. By using standard statistical techniques, each parameter is assigned a coefficient measuring its uncertainty. The known hematite and magnetite ore body shape, in the vicinity of Iron Mountain, MO, was reproduced quite well using this inversion technique.

Three‐dimensional gravity inversion using simulated annealing: Constraints on the diapiric roots of allochthonous salt structures

Geophysics ◽  
2001 ◽  
Vol 66 (5) ◽  
pp. 1438-1449 ◽  
Author(s):  
Seiichi Nagihara ◽  
Stuart A. Hall
Keyword(s):  
Simulated Annealing ◽  
Gulf Of Mexico ◽  
Gravity Data ◽  
A Priori ◽  
Gravity Inversion ◽  
Inversion Method ◽  
A Priori Information ◽  
Final Density ◽  
Smoothing Effects ◽  
Priori Information

In the northern continental slope of the Gulf of Mexico, large oil and gas reservoirs are often found beneath sheetlike, allochthonous salt structures that are laterally extensive. Some of these salt structures retain their diapiric feeders or roots beneath them. These hidden roots are difficult to image seismically. In this study, we develop a method to locate and constrain the geometry of such roots through 3‐D inverse modeling of the gravity anomalies observed over the salt structures. This inversion method utilizes a priori information such as the upper surface topography of the salt, which can be delineated by a limited coverage of 2‐D seismic data; the sediment compaction curve in the region; and the continuity of the salt body. The inversion computation is based on the simulated annealing (SA) global optimization algorithm. The SA‐based gravity inversion has some advantages over the approach based on damped least‐squares inversion. It is computationally efficient, can solve underdetermined inverse problems, can more easily implement complex a priori information, and does not introduce smoothing effects in the final density structure model. We test this inversion method using synthetic gravity data for a type of salt geometry that is common among the allochthonous salt structures in the Gulf of Mexico and show that it is highly effective in constraining the diapiric root. We also show that carrying out multiple inversion runs helps reduce the uncertainty in the final density model.

scholarly journals Practical Tips for 3D Regional Gravity Inversion

Geosciences ◽  
2019 ◽  
Vol 9 (8) ◽  
pp. 351 ◽  
Author(s):  
Daniele Sampietro ◽  
Martina Capponi
Keyword(s):  
Spatial Resolution ◽  
Mass Density ◽  
A Priori ◽  
Regional Scale ◽  
Gravity Inversion ◽  
A Priori Information ◽  
Numerical Examples ◽  
Priori Information ◽  
Regional Gravity ◽  
The Ideal

To solve the inverse gravimetric problem, i.e., to estimate the mass density distribution that generates a certain gravitational field, at local or regional scale, several parameters have to be defined such as the dimension of the 3D region to be considered for the inversion, its spatial resolution, the size of its border, etc. Determining the ideal setting for these parameters is in general difficult: theoretical solutions are usually not possible, while empirical ones strongly depend on the specific target of the inversion and on the experience of the user performing the computation. The aim of the present work is to discuss empirical strategies to set these parameters in such a way to avoid distortions and errors within the inversion. In particular, the discussion is focused on the choice of the volume of the model to be inverted, the size of its boundary, its spatial resolution, and the spatial resolution of the a-priori information to be used within the data reduction. The magnitude of the possible effects due to a wrong choice of the above parameters is also discussed by means of numerical examples.

scholarly journals Linear inverse Gaussian theory and geostatistics

Geophysics ◽  
2006 ◽  
Vol 71 (6) ◽  
pp. R101-R111 ◽  
Author(s):  
Thomas Mejer Hansen ◽  
Andre G. Journel ◽  
Albert Tarantola ◽  
Klaus Mosegaard
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Covariance Function ◽  
A Priori ◽  
Model Space ◽  
Sequential Gaussian Simulation ◽  
A Priori Information ◽  
Sequential Approach ◽  
Linear Inverse Problem ◽  
Priori Information

Inverse problems in geophysics require the introduction of complex a priori information and are solved using computationally expensive Monte Carlo techniques (where large portions of the model space are explored). The geostatistical method allows for fast integration of complex a priori information in the form of covariance functions and training images. We combine geostatistical methods and inverse problem theory to generate realizations of the posterior probability density function of any Gaussian linear inverse problem, honoring a priori information in the form of a covariance function describing the spatial connectivity of the model space parameters. This is achieved using sequential Gaussian simulation, a well-known, noniterative geostatisticalmethod for generating samples of a Gaussian random field with a given covariance function. This work is a contribution to both linear inverse problem theory and geostatistics. Our main result is an efficient method to generate realizations, actual solutions rather than the conventional least-squares-based approach, to any Gaussian linear inverse problem using a noniterative method. The sequential approach to solving linear and weakly nonlinear problems is computationally efficient compared with traditional least-squares-based inversion. The sequential approach also allows one to solve the inverse problem in only a small part of the model space while conditioned to all available data. From a geostatistical point of view, the method can be used to condition realizations of Gaussian random fields to the possibly noisy linear average observations of the model space.

On an inverse spectral problem for one integro-differential operator of fractional order

2019 ◽  
Vol 27 (1) ◽  
pp. 17-23 ◽  
Author(s):  
Mikhail Ignatiev
Keyword(s):  
Inverse Problem ◽  
Differential Operator ◽  
Spectral Problem ◽  
Fractional Order ◽  
Inverse Spectral Problem ◽  
A Priori ◽  
A Priori Information ◽  
Inverse Spectral ◽  
Priori Information

Abstract An inverse spectral problem for some integro-differential operator of fractional order {\alpha\in(1,2)} is studied. We show that the specification of the spectrum together with a certain a priori information about the structure of the operator determines such operator uniquely. The proof is constructive and provides a procedure for solving the inverse problem.

“Unexpected” shear‐wave behavior

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1879-1883 ◽  
Author(s):  
Steve Horne ◽  
Colin MacBeth ◽  
Enru Liu
Keyword(s):  
Shear Wave ◽  
Symmetry Axis ◽  
A Priori ◽  
Transversely Isotropic ◽  
Seismic Profile ◽  
Test Facility ◽  
A Priori Information ◽  
Fracture Systems ◽  
Geological Information ◽  
Priori Information

In a previous study, we inverted shear‐wave birefringence observations from an azimuthal vertical seismic profile (VSP) experiment conducted at the Conoco Borehole Test Facility, Oklahoma (Horne and MacBeth, 1994; Horne, 1995). Our results indicate that the observations can be interpreted in terms of two distinctly different transversely isotropic (TI) models (Figure 1). The first model predicts the symmetry axis to be at N165°E and dipping 10° to the northwest. This orientation coincides with geological information relating to the fracture system that strikes between N50°E and N75°E (Queen and Rizer, 1990). Thus, this first model is consistent with a priori information, so that a possible source of the anisotropy can be identified. However, the second model derived from the inversion results suggests the symmetry axis to be at N200°E and dipping 30° to the southwest. If we interpret this result in terms of an equivalent medium resulting from aligned cracks or fractures, then this inferred crack‐fracture strike would lie in a direction conflicting with the a priori measurements. The bimodal nature of this solution can be readily understood if we examine the shear‐wave behavior for the different models shown in Figure 2. In this plot, the symmetry axis is chosen to be the [Formula: see text]. If we consider the near‐vertical propagation directions that are typically measured in VSP experiments, it can be seen that the qS1 polarizations lie either perpendicular (model 1) or parallel (model 2) to the symmetry axis. Since these polarizations are usually interpreted in terms of aligned crack‐fracture systems, the inferred strike would lie in the [Formula: see text] plane for model 1 and the [Formula: see text] plane for model 2. This interpretation is completely incorrect for model 2, since this inferred alignment is actually orthogonal to the alignment implied by the symmetry of the TI system. This situation represents a worrying aspect to the interpretation of shear‐wave surveys used to characterize crack‐fracture systems. The question that we address is whether anisotropic materials that possess properties similar to those of model 2 can be constructed from equivalent media resulting from cracks or fractures. We also consider other sources of anisotropy that may lead to this behavior.

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