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Numerical Methods of Geophysical Fields Inversion
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Numerical Methods of Geophysical Fields Inversion

A special issue of Geosciences (ISSN 2076-3263). This special issue belongs to the section "Geophysics".

Deadline for manuscript submissions: closed (16 November 2018) | Viewed by 23723

Special Issue Editor

Prof. Dr. Vladimir A. Cheverda

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Guest Editor
Department of Computational Methods in Geophysics, Trofimuk Institute of Petroleum Geology and Geophysics of Siberian Branch of Russian Academy of Sciences, 3, prosp. Koptyuga, 630090 Novosibirsk, Russia
Interests: numerical linear algebra; mathematical modelling; finite difference simulation; optimization techniques; nonlinear least squares
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Special Issue Information

Dear Colleagues,

The overwhelming volume of modern knowledge regarding the Earth’s interior became available due to the results of geophysical observations, on or near the surface. Inversion of geophysical fields within the framework of the corresponding mathematical model provides the most complete knowledge about subsurface distributions of desired parameters. However, it is necessary to stress, that we could never describe a real geological medium using such an abstract object as a system of partial differential equations. Any mathematical model would leave unaccounted a series of processes, phenomena and relationships between parameters, no matter how complex it is. On the one hand, neglecting some of them can significantly distort the important physical properties of the studied fields, while the desire to take into account the widest possible their features leads to excessive complication of mathematical models and, as a result, to a sharp increase in the cost of data processing. Hence, the proper mathematical model is necessary in providing reliable results of geophysical inversion.

It is worth mentioning that the emergence and development of such a direction of modern mathematics as inverse and ill-posed problems originates in geophysics. As early as 1907, Gustav Herglotz published the paper Über das Benndorfsche Problem Fortpfianzungsgeschwindigkeit der Erdbebenstrahlen (Zeitschr. fiir Geophys. 1907, 8, 145- 147) devoted to the inverse kinematic problem for the radially-symmetric Earth. The method was developed further by Emil Wiechert in Bestimmung des Weges der Erdbebenwell~n im Erdinnern. l. Theoretisches, (Phys. Z. 1910, 11, 294-304) and forms the basis for the development of modern computer tomography.

The key position in modern theory and numerical methods of inverse and ill-posed problems takes nonlinear least squares (Levenberg K. 1944. A method for the solution of certain nonlinear problems in least squares Quart. Appl. Math., 2, 1944, 164 – 168; Guy Chavent. Nonlinear least squares for inverse problems, Springer, 2009) and various regularization techniques (Tikhonov A. Solution of incorrectly formulated problems and the regularization method. Soviet Mathematics. 1963, 4: 1035 - 1038). It is these two components form the basis of modern methods of geophysical fields inversion (Albert Tarantola: Inverse Problem Theory and methods for model parameter estimation), which is dedicated to this issue.

Prof. Vladimir Cheverda
Guest Editor

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Keywords

  • numerical linear algebra
  • mathematical modelling
  • finite difference simulation
  • optimization techniques
  • nonlinear least squares

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Published Papers (5 papers)

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Research

20 pages, 6862 KiB  
Article
Constrained Full Waveform Inversion for Borehole Multicomponent Seismic Data
by Marwan Charara and Christophe Barnes
Geosciences 2019, 9(1), 45; https://doi.org/10.3390/geosciences9010045 - 16 Jan 2019
Cited by 3 | Viewed by 3513
Abstract
Full-waveform inversion for borehole seismic data is an ill-posed problem and constraining the problem is crucial. Constraints can be imposed on the data and model space through covariance matrices. Usually, they are set to a diagonal matrix. For the data space, signal polarization [...] Read more.
Full-waveform inversion for borehole seismic data is an ill-posed problem and constraining the problem is crucial. Constraints can be imposed on the data and model space through covariance matrices. Usually, they are set to a diagonal matrix. For the data space, signal polarization information can be used to evaluate the data uncertainties. The inversion forces the synthetic data to fit the polarization of observed data. A synthetic inversion for a 2D-2C data estimating a 1D elastic model shows a clear improvement, especially at the level of the receivers. For the model space, horizontal and vertical spatial correlations using a Laplace distribution can be used to fill the model space covariance matrix. This approach reduces the degree of freedom of the inverse problem, which can be quantitatively evaluated. Strong horizontal spatial correlation distances favor a tabular geological model whenever it does not contradict the data. The relaxation of the spatial correlation distances from large to small during the iterative inversion process allows the recovery of geological objects of the same size, which regularizes the inverse problem. Synthetic constrained and unconstrained inversions for 2D-2C crosswell data show the clear improvement of the inversion results when constraints are used. Full article
(This article belongs to the Special Issue Numerical Methods of Geophysical Fields Inversion)
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16 pages, 3028 KiB  
Article
High-Resolution Seismic Data Deconvolution by A0 Algorithm
by Fedor Krasnov and Alexander Butorin
Geosciences 2018, 8(12), 497; https://doi.org/10.3390/geosciences8120497 - 18 Dec 2018
Cited by 1 | Viewed by 4286
Abstract
Sparse spikes deconvolution is one of the oldest inverse problems, which is a stylized version of recovery in seismic imaging. The goal of sparse spike deconvolution is to recover an approximation of a given noisy measurement [...] Read more.
Sparse spikes deconvolution is one of the oldest inverse problems, which is a stylized version of recovery in seismic imaging. The goal of sparse spike deconvolution is to recover an approximation of a given noisy measurement T = W ∗ r + W 0 . Since the convolution destroys many low and high frequencies, this requires some prior information to regularize the inverse problem. In this paper, the authors continue to study the problem of searching for positions and amplitudes of the reflection coefficients of the medium (SP&ARCM). In previous research, the authors proposed a practical algorithm for solving the inverse problem of obtaining geological information from the seismic trace, which was named A 0 . In the current paper, the authors improved the method of the A 0 algorithm and applied it to the real (non-synthetic) data. Firstly, the authors considered the matrix approach and Differential Evolution approach to the SP&ARCM problem and showed that their efficiency is limited in the case. Secondly, the authors showed that the course to improve the A 0 lays in the direction of optimization with sequential regularization. The authors presented calculations for the accuracy of the A 0 for that case and experimental results of the convergence. The authors also considered different initialization parameters of the optimization process from the point of the acceleration of the convergence. Finally, the authors carried out successful approbation of the algorithm A 0 on synthetic and real data. Further practical development of the algorithm A 0 will be aimed at increasing the robustness of its operation, as well as in application in more complex models of real seismic data. The practical value of the research is to increase the resolving power of the wave field by reducing the contribution of interference, which gives new information for seismic-geological modeling. Full article
(This article belongs to the Special Issue Numerical Methods of Geophysical Fields Inversion)
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19 pages, 5506 KiB  
Article
Modeling and Imaging of Multiscale Geological Media: Exploding Reflection Revisited
by Evgeny Landa, Galina Reshetova and Vladimir Tcheverda
Geosciences 2018, 8(12), 476; https://doi.org/10.3390/geosciences8120476 - 12 Dec 2018
Cited by 6 | Viewed by 3226
Abstract
Computation of Common Middle Point seismic sections and their subsequent time migration and diffraction imaging provides very important knowledge about the internal structure of 3D heterogeneous geological media and are key elements for successive geological interpretation. Full-scale numerical simulation, that computes all single [...] Read more.
Computation of Common Middle Point seismic sections and their subsequent time migration and diffraction imaging provides very important knowledge about the internal structure of 3D heterogeneous geological media and are key elements for successive geological interpretation. Full-scale numerical simulation, that computes all single shot seismograms, provides a full understanding of how the features of the image reflect the properties of the subsurface prototype. Unfortunately, this kind of simulations of 3D seismic surveys for realistic geological media needs huge computer resources, especially for simulation of seismic waves’ propagation through multiscale media like cavernous fractured reservoirs. Really, we need to combine smooth overburden with microstructure of reservoirs, which forces us to use locally refined grids. However, to resolve realistic statements with huge multi-shot/multi-offset acquisitions it is still not enough to provide reasonable needs of computing resources. Therefore, we propose to model 3D Common Middle Point seismic cubes directly, rather than shot-by-shot simulation with subsequent stacking. To do that we modify the well-known "exploding reflectors principle" for 3D heterogeneous multiscale media by use of the finite-difference technique on the base of grids locally refined in time and space. We develop scalable parallel software, which needs reasonable computational costs to simulate realistic models and acquisition. Numerical results for simulation of Common Middle Points sections and their time migration are presented and discussed. Full article
(This article belongs to the Special Issue Numerical Methods of Geophysical Fields Inversion)
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16 pages, 9719 KiB  
Article
Gravity Data Inversion with Method of Local Corrections for Finite Elements Models
by Petr S. Martyshko, Igor V. Ladovskii, Denis D. Byzov and Alexander G. Tsidaev
Geosciences 2018, 8(10), 373; https://doi.org/10.3390/geosciences8100373 - 10 Oct 2018
Cited by 17 | Viewed by 4194
Abstract
We present a new method for gravity data inversion for the linear problem (reconstruction of density distribution by given gravity field). This is an iteration algorithm based on the ideas of local minimization (also known as local corrections method). Unlike the gradient methods, [...] Read more.
We present a new method for gravity data inversion for the linear problem (reconstruction of density distribution by given gravity field). This is an iteration algorithm based on the ideas of local minimization (also known as local corrections method). Unlike the gradient methods, it does not require a nonlinear minimization, is easier to implement and has better stability. The algorithm is based on the finite element method. The finite element approach in our study means that the medium (part of a lithosphere) is represented as a set of equal rectangular prisms, each with constant density. We also suggest a time-efficient optimization, which speeds up the inversion process. This optimization is applied on the gravity field calculation stage, which is a part of every inversion iteration. Its idea is to replace multiple calculations of the gravity field for all finite elements in all observation points with a pre-calculated set of uniform fields for all distances between finite element and observation point, which is possible for the current data set. Method is demonstrated on synthetic data and real-world cases. The case study area is located on the Timan-Pechora plate. This region is one of the promising oil- and gas-producing areas in Russia. Note that in this case we create a 3D density model using joint interpretation of seismic and gravity data. Full article
(This article belongs to the Special Issue Numerical Methods of Geophysical Fields Inversion)
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15 pages, 7454 KiB  
Article
Moho Depth and Crustal Architecture Beneath the Levant Basin from Global Gravity Field Model
by Daniele Sampietro, Ahmed Hamdi Mansi and Martina Capponi
Geosciences 2018, 8(6), 200; https://doi.org/10.3390/geosciences8060200 - 2 Jun 2018
Cited by 17 | Viewed by 7361
Abstract
The study of the discontinuity between the Earth crust and upper mantle, the so-called Moho, and of the lithospheric architecture in general, has several important applications in exploration geophysics. For instance, it is used to facilitate the inversion of seismic-related data, in order [...] Read more.
The study of the discontinuity between the Earth crust and upper mantle, the so-called Moho, and of the lithospheric architecture in general, has several important applications in exploration geophysics. For instance, it is used to facilitate the inversion of seismic-related data, in order to obtain important information on the sedimentary layers or to study the Earth’s heat flux. In this paper, the Levant crustal structure is being investigated starting from the inversion of gravity disturbances coming from a global geopotential field model based on ESA GOCE satellite mission integrated with seismic derived information. In the considered area, which is of particular interest because of its richness from the resources point of view, the deep crustal structure is still a matter of study due to the presence of a thick sequence of sedimentary layers, deposited within geological eras by the Nile River. Within the current work, the shape of the Oceanic domain in correspondence to the Herodotus Basin and the Cyprus Arc has been clearly defined. Moreover the nature of the Levantine Basin and of the Eratosthenes crust has been investigated by a set of ad hoc tests, finding the presence of continental crust. Finally, the Moho depth and the crustal density distribution have been retrieved. Several localized anomalies, in the Cyprus area, have been identified and modelled too, thus confirming the presence of heavy material, with a thickness up to 10 km, in the sedimentary layer and shallower part of the crust. Full article
(This article belongs to the Special Issue Numerical Methods of Geophysical Fields Inversion)
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