3D magnetic data-space inversion with sparseness constraints

I have developed an inversion approach that determines a 3D susceptibility distribution that produces a given magnetic anomaly. The subsurface model consists of a 3D, equally spaced array of dipoles. The inversion incorporates a model norm that enforces sparseness and depth weighting of the solution. Sparseness is imposed by using the Cauchy norm on model parameters. The inverse problem is posed in the data space, leading to a linear system of equations with dimensions based on the number of data, [Formula: see text]. This contrasts with the standard least-squares solution, derived through operations within the [Formula: see text]-dimensional model space ([Formula: see text] being the number of model parameters). Hence, the data-space method combined with a conjugate gradient algorithm leads to computational efficiency by dealing with an [Formula: see text] system versus an [Formula: see text] one, where [Formula: see text]. Tests on synthetic data show that sparse inversion produces a much more focused solution compared with a standard model-space, least-squares inversion. The inversion of aeromagnetic data collected over a Precambrian Shield area again shows that including the sparseness constraint leads to a simpler and better resolved solution. The degree of improvement in model resolution for the sparse case is quantified using the resolution matrix.

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Least-squares reverse time migration via linearized waveform inversion using a Wasserstein metric

Geophysics ◽

10.1190/geo2018-0619.1 ◽

2019 ◽

Vol 84 (5) ◽

pp. S411-S423

Author(s):

Peng Yong ◽

Jianping Huang ◽

Zhenchun Li ◽

Wenyuan Liao ◽

Luping Qu

Keyword(s):

Least Squares ◽

Waveform Inversion ◽

Synthetic Data ◽

Gradient Algorithm ◽

Reverse Time ◽

Reverse Time Migration ◽

Wasserstein Metric ◽

Time Migration ◽

Imaging Results ◽

Primal Dual

Least-squares reverse time migration (LSRTM), an effective tool for imaging the structures of the earth from seismograms, can be characterized as a linearized waveform inversion problem. We have investigated the performance of three minimization functionals as the [Formula: see text] norm, the hybrid [Formula: see text] norm, and the Wasserstein metric ([Formula: see text] metric) for LSRTM. The [Formula: see text] metric used in this study is based on the dynamic formulation of transport problems, and a primal-dual hybrid gradient algorithm is introduced to efficiently compute the [Formula: see text] metric between two seismograms. One-dimensional signal analysis has demonstrated that the [Formula: see text] metric behaves like the [Formula: see text] norm for two amplitude-varied signals. Unlike the [Formula: see text] norm, the [Formula: see text] metric does not suffer from the differentiability issue for null residuals. Numerical examples of the application of three misfit functions to LSRTM on synthetic data have demonstrated that, compared to the [Formula: see text] norm, the hybrid [Formula: see text] norm and [Formula: see text] metric can accelerate LSRTM and are less sensitive to non-Gaussian noise. For the field data application, the [Formula: see text] metric produces the most reliable imaging results. The hybrid [Formula: see text] norm requires tedious trial-and-error tests for the judicious threshold parameter selection. Hence, the more automatic [Formula: see text] metric is recommended as a robust alternative to the customary [Formula: see text] norm for time-domain LSRTM.

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AN AUTOMATIC LEAST‐SQUARES MULTIMODEL METHOD FOR MAGNETIC INTERPRETATION

Geophysics ◽

10.1190/1.1440345 ◽

1973 ◽

Vol 38 (2) ◽

pp. 349-358 ◽

Cited By ~ 38

Author(s):

Peter H. McGrath ◽

Peter J. Hood

Keyword(s):

Least Squares ◽

Magnetic Anomalies ◽

Magnetic Data ◽

Model Parameters ◽

Hudson Bay ◽

Northern Ontario ◽

Magnetic Interpretation ◽

Hudson Bay Lowlands ◽

Sverdrup Basin ◽

Best Fit

The magnetic anomalies caused by such diverse model shapes as the finite strike length thick dike, the vertical prism, thes loping step, the parallelepiped body, etc., may be obtained through an appropriate numerical integration of the expression for the magnetic effect produced by a finite thin plate. Using models generated in this manner, an automatic computer method has been developed at the Geological Survey of Canada for the interpretation of magnetic data. Because the magnetic anomalies produced by the various model shapes are nonlinear in parameters of shape and position, it is necessary to use an iterative procedure to obtain the values for the various model parameters which yield a least‐squares best‐fit anomaly curve to a set of discrete observed data. The interpretation method described in this paper uses the Powell algorithm for this purpose. The procedure can sometimes be made more efficient using a Marquardt modification to the Powell algorithm. Examples of the use of the method are presented for an elongated anomaly in the Moose River basin of the Hudson Bay lowlands in northern Ontario, and for an areally large elliptical anomaly in the Sverdrup basin of the Canadian Arctic Islands.

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2.5‐D inversion of frequency‐domain electromagnetic data generated by a grounded‐wire source

Geophysics ◽

10.1190/1.1527076 ◽

2002 ◽

Vol 67 (6) ◽

pp. 1753-1768 ◽

Cited By ~ 37

Author(s):

Yuji Mitsuhata ◽

Toshihiro Uchida ◽

Hiroshi Amano

Keyword(s):

Frequency Domain ◽

Regularization Parameter ◽

Synthetic Data ◽

Magnetic Data ◽

Statistical Criterion ◽

Model Parameters ◽

Inversion Algorithm ◽

Gas Field ◽

Data Set ◽

Transient Electromagnetic

Interpretation of controlled‐source electromagnetic (CSEM) data is usually based on 1‐D inversions, whereas data of direct current (dc) resistivity and magnetotelluric (MT) measurements are commonly interpreted by 2‐D inversions. We have developed an algorithm to invert frequency‐Domain vertical magnetic data generated by a grounded‐wire source for a 2‐D model of the earth—a so‐called 2.5‐D inversion. To stabilize the inversion, we adopt a smoothness constraint for the model parameters and adjust the regularization parameter objectively using a statistical criterion. A test using synthetic data from a realistic model reveals the insufficiency of only one source to recover an acceptable result. In contrast, the joint use of data generated by a left‐side source and a right‐side source dramatically improves the inversion result. We applied our inversion algorithm to a field data set, which was transformed from long‐offset transient electromagnetic (LOTEM) data acquired in a Japanese oil and gas field. As demonstrated by the synthetic data set, the inversion of the joint data set automatically converged and provided a better resultant model than that of the data generated by each source. In addition, our 2.5‐D inversion accounted for the reversals in the LOTEM measurements, which is impossible using 1‐D inversions. The shallow parts (above about 1 km depth) of the final model obtained by our 2.5‐D inversion agree well with those of a 2‐D inversion of MT data.

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Tomographic filtering of mantle circulation models via the generalised inverse: A way to account for seismic data uncertainty

10.5194/egusphere-egu2020-9728 ◽

2020 ◽

Author(s):

Bernhard S.A. Schuberth ◽

Roman Freissler ◽

Christophe Zaroli ◽

Sophie Lambotte

Keyword(s):

Seismic Data ◽

Inverse Operator ◽

Synthetic Data ◽

Circulation Model ◽

Model Space ◽

Seismic Inversion ◽

Data Uncertainty ◽

Earth Model ◽

Model Parameters ◽

Noise Component

For a comprehensive link between seismic tomography and geodynamic models, uncertainties in the seismic model space play a non-negligible role. More specifically, knowledge of the tomographic uncertainties is important for obtaining meaningful estimates of the present-day thermodynamic state of Earth's mantle, which form the basis of retrodictions of past mantle evolution using the geodynamic adjoint method. A standard tool in tomographic-geodynamic model comparisons nowadays is tomographic filtering of mantle circulation models using the resolution operator R associated with the particular seismic inversion of interest. However, in this classical approach it is not possible to consider tomographic uncertainties and their impact on the geodynamic interpretation.&#160;Here, we present a new method for 'filtering' synthetic Earth models, which makes use of the generalised inverse operator G&#8224;, instead of using R. In our case, G&#8224; is taken from a recent global SOLA Backus&#8211;Gilbert S-wave tomography. In contrast to classical tomographic filtering, the 'imaged' model is constructed by computing the Generalised-Inverse Projection (GIP) of synthetic data calculated in an Earth model of choice. This way, it is possible to include the effects of noise in the seismic data and thus to analyse uncertainties in the resulting model parameters. In order to demonstrate the viability of the method, we compute a set of travel times in an existing mantle circulation model, add specific realisations of Gaussian, zero-mean seismic noise to the synthetic data and apply G&#8224;.&#160; &#160; Our results show that the resulting GIP model without noise is equivalent to the mean model of all GIP realisations from the suite of synthetic 'noisy' data and also closely resembles the model tomographically filtered using R. Most important, GIP models that include noise in the data show a significant variability of the shape and amplitude of seismic anomalies in the mantle. The significant differences between the various GIP realisations highlight the importance of interpreting and assessing tomographic images in a prudent and cautious manner. With the GIP approach, we can moreover investigate the effect of systematic errors in the data, which we demonstrate by adding an extra term to the noise component that aims at mimicking the effects of uncertain crustal corrections. In our presentation, we will finally discuss ways to construct the model covariance matrix based on the GIP approach and point out possible research directions on how to make use of this information in future geodynamic modelling efforts.

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Joint inversion of gravity and magnetic data for two-layer models

Geophysics ◽

10.1190/1.2194514 ◽

2006 ◽

Vol 71 (3) ◽

pp. L35-L42 ◽

Cited By ~ 23

Author(s):

Mark Pilkington

Keyword(s):

Least Squares ◽

Joint Inversion ◽

Physical Property ◽

Magnetic Data ◽

Model Parameters ◽

Constant Density ◽

Central Uplift ◽

Gravity And Magnetic ◽

Damped Least Squares ◽

Gravity And Magnetic Data

Gravity and magnetic data are inverted jointly in terms of a model consisting of an interface separating two layers having a constant density and magnetization contrast. A damped least-squares inversion is used to determine the topography of the interface. The inversion requires knowledge of the physical property contrasts across the interface and its average depth. Since the relationship between model parameters and data is weakly nonlinear, a constant damped least-squares inverse is used during the iterative solution search. The effect of this inverse is closely related to a downward continuation of the field to the average interface depth. The method is used to map the base of the Sept-Iles mafic intrusion, Quebec, Canada, and the shape of the central uplift at the Chicxulub impact crater, Yucatan, Mexico. At Sept-Iles, the intrusion reaches a thickness of [Formula: see text], coincident with the maximum gravity anomaly, south of the intrusion center. At Chicxulub, the top of the central uplift is modeled to be [Formula: see text] deep and has a single peak form.

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Minerals and ore deposits exploration using meta-heuristic based optimization on magnetic data

Contributions to Geophysics and Geodesy ◽

10.31577/congeo.2020.50.2.1 ◽

2020 ◽

Vol 50 (2) ◽

pp. 161-199

Author(s):

Mohamed GOBASHY ◽

Maha ABDELAZEEM ◽

Mohamed ABDRABOU

Keyword(s):

Mineral Exploration ◽

Random Noise ◽

Synthetic Data ◽

Ore Deposits ◽

Mine Planning ◽

Magnetic Data ◽

Real Field ◽

Model Parameters ◽

Tectonic Structures ◽

Whale Optimization

The difficulties in unravelling the tectonic structures, in some cases, prevent the understanding of the ore bodies' geometry, leading to mistakes in mineral exploration, mine planning, evaluation of ore deposits, and even mineral exploitation. For that reason, many geophysical techniques are introduced to reveal the type, dimension, and geometry of these structures. Among them, electric methods, self-potential, electromagnetic, magnetic and gravity methods. Global meta-heuristic technique using Whale Optimization Algorithm (WOA) has been utilized for assessing model parameters from magnetic anomalies due to a thin dike, a dipping dike, and a vertical fault like/shear zone geological structure. These structures are commonly associated with mineralization. This modern algorithm was firstly applied on a free-noise synthetic data and to a noisy data with three different levels of random noise to simulate natural and artificial anomaly disturbances. Good results obtained through the inversion of such synthetic examples prove the validity and applicability of our algorithm. Thereafter, the method is applied to real case studies taken from different ore mineralization resembling different geologic conditions. Data are taken from Canada, United States, Sweden, Peru, India, and Australia. The obtained results revealed good correlation with previous interpretations of these real field examples.

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Interferometric redatuming by sparse inversion

Geophysical Journal International ◽

10.1093/gji/ggs052 ◽

2012 ◽

Vol 192 (2) ◽

pp. 666-670 ◽

Cited By ~ 20

Author(s):

Joost van der Neut ◽

Felix J. Herrmann

Keyword(s):

Least Squares ◽

Image Space ◽

Seismic Sources ◽

Transform Domain ◽

Frequency Content ◽

Depth Level ◽

Data Space ◽

Sparse Inversion ◽

Least Squares Migration ◽

Incomplete Acquisition

Abstract Assuming that transmission responses are known between the surface and a particular depth level in the subsurface, seismic sources can be effectively mapped to this level by a process called interferometric redatuming. After redatuming, the obtained wavefields can be used for imaging below this particular depth level. Interferometric redatuming consists of two steps, namely (i) the decomposition of the observed wavefields into downgoing and upgoing constituents and (ii) a multidimensional deconvolution of the upgoing constituents with the downgoing constituents. While this method works in theory, sensitivity to noise and artefacts due to incomplete acquisition require a different formulation. In this letter, we demonstrate the benefits of formulating the two steps that undergird interferometric redatuming in terms of a transform-domain sparsity-promoting program. By exploiting compressibility of seismic wavefields in the curvelet domain, the method not only becomes robust with respect to noise but we are also able to remove certain artefacts while preserving the frequency content. Although we observe improvements when we promote sparsity in the redatumed data space, we expect better results when interferometric redatuming would be combined or integrated with least-squares migration with sparsity promotion in the image space.

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High-resolution wave-equation AVA imaging: Algorithm and tests with a data set from the Western Canadian Sedimentary Basin

Geophysics ◽

10.1190/1.2076748 ◽

2005 ◽

Vol 70 (5) ◽

pp. S91-S99 ◽

Cited By ~ 29

Author(s):

Juefu Wang ◽

Henning Kuehl ◽

Mauricio D. Sacchi

Keyword(s):

Inverse Problem ◽

Wave Equation ◽

Least Squares ◽

Oil And Gas ◽

Sedimentary Basin ◽

Model Space ◽

Sampled Data ◽

Inversion Algorithm ◽

Data Set ◽

Data Space

This paper presents a 3D least-squares wave-equation migration method that yields regularized common-image gathers (CIGs) for amplitude-versus-angle (AVA) analysis. In least-squares migration, we pose seismic imaging as a linear inverse problem; this provides at least two advantages. First, we are able to incorporate model-space weighting operators that improve the amplitude fidelity of CIGs. Second, the influence of improperly sampled data (footprint noise) can be diminished by incorporating data-space weighting operators. To investigate the viability of this class of methods for oil and gas exploration, we test the algorithm with a real-data example from the Western Canadian Sedimentary Basin. To make our problem computationally feasible, we utilize the 3D common-azimuth approximation in the migration algorithm. The inversion algorithm uses the method of conjugate gradients with the addition of a ray-parameter-dependent smoothing constraint that minimizes sampling and aperture artifacts. We show that more robust AVA attributes can be obtained by properly selecting the model and data-space regularization operators. The algorithm is implemented in conjunction with a preconditioning strategy to accelerate convergence. Posing the migration problem as an inverse problem leads to enhanced event continuity in CIGs and, hence, more reliable AVA estimates. The vertical resolution of the inverted image also improves as a consequence of increased coherence in CIGs and, in addition, by implicitly introducing migration deconvolution in the inversion.

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Multivariable teaching-learning-based optimization (MTLBO) algorithm for estimating the structural parameters of the buried mass by magnetic data

Geofizika ◽

10.15233/gfz.2020.37.6 ◽

2020 ◽

Vol 37 (2) ◽

pp. 213-235

Author(s):

Ata Eshaghzadeh ◽

Sanaz Seyedi Sahebari

Keyword(s):

Structural Parameters ◽

Synthetic Data ◽

Optimal Solution ◽

Horizontal Cylinder ◽

Magnetic Data ◽

Model Parameters ◽

Major Purpose ◽

Learner Model ◽

Teaching Learning Based Optimization ◽

Teaching Learning

This paper presents a nature-based algorithm, titled multivariable teaching-learning-based optimization (MTLBO) algorithm. MTLBO algorithm during an iterative process can estimates the best values of the buried structure (model) parameters in a multi-objective problem. The algorithm works in two computational phases: the teacher phase and the learner phase. The major purpose of the MTLBO algorithm is to modify the value of the learners and thus, improving the value of the model parameters which leads to the optimal solution. The variables of each learner (model) are the depth (z), amplitude coefficient (k), shape factor (q), angle of effective magnetization (θ) and axis location (x0) parameters. We employ MTLBO method for the magnetic anomalies caused by the buried structures with a simple geometric shape such as sphere and horizontal cylinder. The efficiency of the MTLBO is also studied by noise corruption synthetic data, as the acceptable results were obtained. We have applied the MTLBO for the interpretation of the four magnetic anomaly profiles from Iran, Brazil and India.

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An Augmented Iteratively Re-weighted and Refined Least Squares Method for lp Minimization Problem in Inverse Modeling of Potential Field Magnetic Data

Journal of Geological Research ◽

10.30564/jgr.v1i3.1316 ◽

2020 ◽

Vol 1 (3) ◽

Author(s):

Maysam Abedi

Keyword(s):

Magnetic Field ◽

Magnetic Susceptibility ◽

Least Squares ◽

Minimization Problem ◽

Potential Field ◽

Field Data ◽

Least Squares Method ◽

Porphyry Copper ◽

Magnetic Data ◽

Magnetic Field Data

The presented work examines application of an Augmented Iteratively Re-weighted and Refined Least Squares method (AIRRLS) to construct a 3D magnetic susceptibility property from potential field magnetic anomalies. This algorithm replaces an lp minimization problem by a sequence of weighted linear systems in which the retrieved magnetic susceptibility model is successively converged to an optimum solution, while the regularization parameter is the stopping iteration numbers. To avoid the natural tendency of causative magnetic sources to concentrate at shallow depth, a prior depth weighting function is incorporated in the original formulation of the objective function. The speed of lp minimization problem is increased by inserting a pre-conditioner conjugate gradient method (PCCG) to solve the central system of equation in cases of large scale magnetic field data. It is assumed that there is no remanent magnetization since this study focuses on inversion of a geological structure with low magnetic susceptibility property. The method is applied on a multi-source noise-corrupted synthetic magnetic field data to demonstrate its suitability for 3D inversion, and then is applied to a real data pertaining to a geologically plausible porphyry copper unit. The real case study located in Semnan province of Iran consists of an arc-shaped porphyry andesite covered by sedimentary units which may have potential of mineral occurrences, especially porphyry copper. It is demonstrated that such structure extends down at depth, and consequently exploratory drilling is highly recommended for acquiring more pieces of information about its potential for ore-bearing mineralization.

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