3-D inversion of magnetic data

Geophysics ◽  
1996 ◽  
Vol 61 (2) ◽  
pp. 394-408 ◽  
Author(s):  
Yaoguo Li ◽  
Douglas W. Oldenburg
Keyword(s):  
Objective Function ◽  
A Priori ◽  
Magnetic Data ◽  
A Priori Information ◽  
Data Set ◽  
Earth Models ◽  
Priori Information ◽  
Model Region ◽  
D Model ◽  
Physical Realizability

We present a method for inverting surface magnetic data to recover 3-D susceptibility models. To allow the maximum flexibility for the model to represent geologically realistic structures, we discretize the 3-D model region into a set of rectangular cells, each having a constant susceptibility. The number of cells is generally far greater than the number of the data available, and thus we solve an underdetermined problem. Solutions are obtained by minimizing a global objective function composed of the model objective function and data misfit. The algorithm can incorporate a priori information into the model objective function by using one or more appropriate weighting functions. The model for inversion can be either susceptibility or its logarithm. If susceptibility is chosen, a positivity constraint is imposed to reduce the nonuniqueness and to maintain physical realizability. Our algorithm assumes that there is no remanent magnetization and that the magnetic data are produced by induced magnetization only. All minimizations are carried out with a subspace approach where only a small number of search vectors is used at each iteration. This obviates the need to solve a large system of equations directly, and hence earth models with many cells can be solved on a deskside workstation. The algorithm is tested on synthetic examples and on a field data set.

Estimation of density, magnetization, and depth to source: A nonlinear and noniterative 3-D potential‐field method

Geophysics ◽  
1997 ◽  
Vol 62 (3) ◽  
pp. 814-830 ◽  
Author(s):  
Maurizio Fedi
Keyword(s):  
A Priori ◽  
Real Data ◽  
Field Method ◽  
Magnetic Data ◽  
System Of Equations ◽  
Source Depth ◽  
A Priori Information ◽  
Magnetization Direction ◽  
Priori Information ◽  
Magnetic Case

The depth to the top, or bottom, and the density of a 3-D homogeneous source can be estimated from its gravity or magnetic anomalies by using a priori information on the maximum and minimum source depths. For the magnetic case, the magnetization direction is assumed to be constant and known. The source is assumed to be within a layer of known depth to the top h and thickness t. A depth model, satisfying both the data and the a priori information is found, together with its associated density/magnetization contrast. The methodology first derives, from the measured data, a set of apparent densities [Formula: see text] (or magnetizations), which do not depend on the layer parameters h and t, but only on source thickness. A nonlinear system of equations based on [Formula: see text], with source thicknesses as unknowns, is constructed. To simplify the solution, a more practical system of equations is formed. Each equation depends on only one value of thickness. Solving for the thicknesses, taking into account the above a priori information, the source depth to the top (or to the bottom) is determined uniquely. Finally, the depth solutions allow a unit‐density gravity model to be computed, which is compared to the observed gravity to determine the density contrast. A similar procedure can be used for magnetic data. Tests on synthetic anomalies and on real data demonstrate the good performance of this method.

Particle swarm optimization of 2D magnetotelluric data

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. E125-E141 ◽  
Author(s):  
Francesca Pace ◽  
Alessandro Santilano ◽  
Alberto Godio
Keyword(s):  
Inverse Problem ◽  
Particle Swarm Optimization ◽  
A Priori ◽  
Particle Swarm ◽  
Stability And Convergence ◽  
Real Field ◽  
A Priori Information ◽  
Data Set ◽  
Swarm Optimization ◽  
Priori Information

We implement the particle swarm optimization (PSO) algorithm for the two-dimensional (2D) magnetotelluric (MT) inverse problem. We first validate PSO on two synthetic models of different complexity and then apply it to an MT benchmark for real-field data, the COPROD2 data set (Canada). We pay particular attention to the selection of the PSO input parameters to properly address the complexity of the 2D MT inverse problem. We enhance the stability and convergence of the solution of the geophysical problem by applying the hierarchical PSO with time-varying acceleration coefficients (HPSO-TVAC). Moreover, we parallelize the code to reduce the computation time because PSO is a computationally demanding global search algorithm. The inverse problem was solved for the synthetic data both by giving a priori information at the beginning and by using a random initialization. The a priori information was given to a small number of particles as the initial position within the search space of solutions, so that the swarming behavior was only slightly influenced. We have demonstrated that there is no need for the a priori initialization to obtain robust 2D models because the results are largely comparable with the results from randomly initialized PSO. The optimization of the COPROD2 data set provides a resistivity model of the earth in line with results from previous interpretations. Our results suggest that the 2D MT inverse problem can be successfully addressed by means of computational swarm intelligence.

A COMBINATION SCHEME FOR FUZZY CLUSTERING

2002 ◽  
Vol 16 (07) ◽  
pp. 901-912 ◽  
Author(s):  
EVGENIA DIMITRIADOU ◽  
ANDREAS WEINGESSEL ◽  
KURT HORNIK
Keyword(s):  
A Priori ◽  
Fuzzy Cluster ◽  
A Priori Information ◽  
Cluster Validity ◽  
Data Set ◽  
Cluster Algorithms ◽  
Combination Scheme ◽  
Priori Information ◽  
Theoretical Considerations ◽  
Structurally Stable

In this paper we present a voting scheme for fuzzy cluster algorithms. This voting method allows us to combine several runs of cluster algorithms resulting in a common partition. This helps us to tackle the problem of choosing the appropriate clustering method for a data set where we have no a priori information about it. We mathematically derive the algorithm from theoretical considerations. Experiments show that the voting algorithm finds structurally stable results. Several cluster validity indexes show the improvement of the voting result in comparison to simple fuzzy voting.

scholarly journals A method to derive the Site Atmospheric State Best Estimate (SASBE) of ozone profiles from radiosonde and passive microwave data

2015 ◽  
Vol 8 (3) ◽  
pp. 3399-3422 ◽  
Author(s):  
E. Maillard Barras ◽  
A. Haefele ◽  
R. Stübi ◽  
D. Ruffieux
Keyword(s):  
Brightness Temperature ◽  
A Priori ◽  
A Priori Information ◽  
Data Set ◽  
Ozone Data ◽  
Best Estimate ◽  
Vertical Range ◽  
Ozone Profile ◽  
Atmospheric State ◽  
Priori Information

Abstract. We present a method to derive the site atmospheric state best estimate (SASBE) of the ozone profile combining brightness temperature spectra around the 142 GHz absorption line of ozone measured by the microwave radiometer SOMORA and ozone profiles measured by the radiosonde (RS). The SASBE ozone profile is obtained using the radiosonde ozone profile as a priori information in an optimal estimation retrieval of the SOMORA radiometer. The resulting ozone profile ranges from ground up to 65 km altitude and makes optimal use of the available information at each altitude. The high vertical resolution of the radiosonde profile can be conserved and the uncertainty of the SASBE is well defined at each altitude. A SASBE ozone profile dataset has been generated for Payerne, Switzerland, with a temporal resolution of 3 profiles a week for the time period ranging from 2011 to 2013. The relative difference of the SASBE ozone profiles to the AURA/MLS ozone profiles lies between −3 to 6% over the vertical range of 20–65 km. Above 20 km, the agreement between the SASBE and AURA/MLS ozone profiles is better than the agreement between the operational SOMORA ozone data set and AURA/MLS. Below 20 km the SASBE ozone data are identical to the radiosonde measurements. The same method has been applied to ECWMF-ERA interim ozone profiles and SOMORA data to generate a SASBE dataset with a time resolution of 4 profiles per day. These SASBE ozone profiles agree between −4 and +8% with AURA/MLS. The improved agreement of the SASBE datasets with AURA/MLS above 20 km demonstrates the benefit of better a priori information in the retrieval of ozone from brightness temperature data.

2-D Bayesian deconvolution

Geophysics ◽  
1991 ◽  
Vol 56 (12) ◽  
pp. 2008-2018 ◽  
Author(s):  
Marc Lavielle
Keyword(s):  
Objective Function ◽  
Seismic Data ◽  
Layer Structure ◽  
A Priori ◽  
A Priori Information ◽  
The Earth ◽  
Markov Random ◽  
Posteriori Distribution ◽  
A Posteriori Distribution ◽  
Priori Information

Inverse problems can be solved in different ways. One way is to define natural criteria of good recovery and build an objective function to be minimized. If, instead, we prefer a Bayesian approach, inversion can be formulated as an estimation problem where a priori information is introduced and the a posteriori distribution of the unobserved variables is maximized. When this distribution is a Gibbs distribution, these two methods are equivalent. Furthermore, global optimization of the objective function can be performed with a Monte Carlo technique, in spite of the presence of numerous local minima. Application to multitrace deconvolution is proposed. In traditional 1-D deconvolution, a set of uni‐dimensional processes models the seismic data, while a Markov random field is used for 2-D deconvolution. In fact, the introduction of a neighborhood system permits one to model the layer structure that exists in the earth and to obtain solutions that present lateral coherency. Moreover, optimization of an appropriated objective function by simulated annealing allows one to control the fit with the input data as well as the spatial distribution of the reflectors. Extension to 3-D deconvolution is straightforward.

Structurally tailored 3D anisotropic controlled-source electromagnetic resistivity inversion with cross-gradient criterion and simultaneous model calibration

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. E387-E402 ◽  
Author(s):  
Max A. Meju ◽  
Randall L. Mackie ◽  
Federico Miorelli ◽  
Ahmad Shahir Saleh ◽  
Roger V. Miller
Keyword(s):  
Field Data ◽  
A Priori ◽  
Calibration Data ◽  
A Priori Information ◽  
Data Set ◽  
Low Contrast ◽  
Controlled Source ◽  
Low Resistivity ◽  
Band Limited ◽  
Priori Information

Geologic interpretation of 3D anisotropic resistivity models from conventional marine controlled-source electromagnetic (CSEM) data inversion faces difficulties in low-resistivity contrast sediments and structurally complex environments that typify the new frontiers for hydrocarbon exploration. Currently, the typically reconstructed horizontal resistivity [Formula: see text] and vertical resistivity [Formula: see text] models often have conflicting depth structures that are difficult to explain in terms of subsurface geology, and the resulting resistivities may not be close to the true formation resistivities required for estimating reservoir parameters. We have investigated the concept that an objective geologically oriented or structurally tailored inversion can be achieved by requiring that the cross-product of the gradient of horizontal resistivity and the gradient of the vertical resistivity is equal to zero at significant geologic boundaries. We incorporate this boundary-shape criterion in our 3D inverse problem formulations, implemented within nonlinear model-space and conjugate-gradient contexts, for cases in which a priori calibration data from wells and/or seismically derived subsurface boundaries are available and for cases in which these are lacking. The resulting fit-for-purpose solutions serve to better analyze the peculiarity of a given data set. We applied these algorithms to synthetic and field CSEM data sets representing a fold-thrust environment with low-resistivity and low-contrast sediments. The resulting [Formula: see text] and [Formula: see text] models from cross-gradient joint inversion of synthetic data of appropriate frequency bandwidth without a priori information are structurally similar and consistent with the test models, whereas those from the inversions of band-limited field data are consistent with the available seismic and resistivity well-log data. This particular approach will thus be useful for lithologic correlation in frontier regions with limited a priori information using broadband CSEM data. For these band-limited field data, we found that the anisotropic bulk resistivities of the low-contrast sediments are better determined by incorporating a priori calibration data from triaxial resistivity logs and seismic horizons.

Reconstruction of borehole radar images by a modified f–k migration

2020 ◽  
Vol 221 (3) ◽  
pp. 1626-1634
Author(s):  
Jianjian Huo ◽  
Binzhong Zhou ◽  
Qing Zhao ◽  
Iain M Mason
Keyword(s):  
Final Section ◽  
Early Stage ◽  
A Priori ◽  
A Priori Information ◽  
Data Set ◽  
Radar Images ◽  
Imaging Tool ◽  
Bedding Planes ◽  
Borehole Radar ◽  
Priori Information

SUMMARY Borehole radar (BHR) is an effective imaging tool. It can be used to detect and map faults, fractures, folds, domes, partings and mine workings. Most BHRs have azimuthally omnidirectional radiation patterns. The echoes sensed by such BHRs may come from any direction. Considering a radar in a straight borehole that passes through a stack of flat reflection planes, V-shaped events or crosses appear on the time section. One of the arms of each cross is a real image while the other is an ambiguity of known origin. Directional ambiguities such as these obstruct efforts to interpret the data. In this paper, we address this difficulty by using a modified f–k migration algorithm to translate crosses into lines on the final section that are consistent with a priori information about for example bedding. Compared with conventional strategies, for example migration + f–k dip filter, this approach integrates the two separated processes into one and is straightforward, computationally effective and simple to implement. The method is demonstrated using a synthetic model and a real BHR field data set. It allows the interpreter to use a priori information about fault swarms or plausible bedding planes at an early stage. The reconstructed BHR image helps the search for geological anomalies such as fractures, partings, domes and rolls that could be a hazard for mining.

scholarly journals Interpretation of Magnetic Anomalies over 2D Fault and Sheet-Type Mineralized Structures Using Very Fast Simulated Annealing Global Optimization: An Understanding of Uncertainty and Geological Implications

Lithosphere ◽  
2021 ◽  
Vol 2021 (Special 6) ◽  
Author(s):  
Arkoprovo Biswas ◽  
Khushwant Rao
Keyword(s):  
A Priori ◽  
Theoretical Models ◽  
Magnetic Anomalies ◽  
Type Structure ◽  
Equivalent Model ◽  
Magnetic Data ◽  
Model Parameters ◽  
A Priori Information ◽  
Wide Range ◽  
Priori Information

Abstract Identification of intraterrane dislocation zones and associated mineralized bodies is of immense importance in exploration geophysics. Understanding such structures from geophysical anomalies is challenging and cumbersome. In the present study, we present a fast and competent algorithm for interpreting magnetic anomalies from such dislocation and mineralized zones. Such dislocation and mineralized zones are well explained from 2D fault and sheet-type structures. The different parameters from 2D fault and sheet-type structures such as the intensity of magnetization (k), depth to the top (z1), depth to the bottom (z2), origin location (x0), and dip angle (θ) of the fault and sheet from magnetic anomalies are interpreted. The interpretation suggests that there is uncertainty in defining the model parameters z1 and z2 for the 2D inclined fault; k, z1, and z2 for the 2D vertical fault and finite sheet-type structure; and k and z for the infinite sheet-type structure. Here, it shows a wide range of solutions depicting an equivalent model with smaller misfits. However, the final interpreted mean model is close to the actual model with the least uncertainty. Histograms and crossplots for 2D fault and sheet-type structures also reveal the same. The present algorithm is demonstrated with four theoretical models, including the effect of noises. Furthermore, the investigation of magnetic data was also applied from three field examples from intraterrane dislocation zones (Australia), deep-seated dislocation zones (India) as a 2D fault plane, and mineralized zones (Canada) as sheet-type structures. The final estimated model parameters are in good agreement with the earlier methods applied for these field examples with a priori information wherever available in the literature. However, the present method can simultaneously interpret all model parameters without a priori information.

scholarly journals Inversion of phase times for hypocenters and shallow crustal velocities, Mojave Desert, California

1979 ◽  
Vol 69 (2) ◽  
pp. 387-396 ◽  
Author(s):  
James A. Hileman
Keyword(s):  
Mojave Desert ◽  
Velocity Structure ◽  
A Priori ◽  
Relative Importance ◽  
Local Velocity ◽  
A Priori Information ◽  
Linear Least Squares ◽  
Data Set ◽  
Velocity Models ◽  
Priori Information

abstract Phase times for an ensemble of 20 aftershocks of the June 1, 1975, Galway Lake earthquake were inverted to determine hypocenters and local velocity structure simultaneously. A maximum likelihood formulation of linear, least-squares inversion was used so that the data were weighted according to their estimated variances. A trade-off parameter controlled the relative importance of the RMS error and the amount by which the model changed at each iteration. Individual aftershocks must be selected so that a wide variety of travel paths are represented. Initial trials disclosed that careful constraints are necessary for allowed changes in station delays. Fixed velocity boundaries, based on a priori information, were used for each of several starting models, and only layer velocities were allowed to vary. Each of the trials indicated that shallow crustal velocities in the vincinity of Galway Lake are somewhat lower than those of the usual velocity models. The velocities are not strongly constrained by this data set, but the results are consistent with a subsequent, detailed refraction survey by other workers. Mapping the travel-time surface in time-depth-distance space helps clarify the limitations inherent in a given problem.

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