DIRECT INVERSION OF TRANSMISSION SYNTHETIC SEISMOGRAMS

Geophysics ◽  
1978 ◽  
Vol 43 (5) ◽  
pp. 886-898 ◽  
Author(s):  
D. Loewenthal ◽  
P. R. Gutowski ◽  
S. Treitel
Keyword(s):  
Inverse Problem ◽  
Forward Problem ◽  
Direct Inversion ◽  
Homogeneous Half Space ◽  
Incident Plane ◽  
Free Case ◽  
Levinson Algorithm ◽  
Iterative Inversion ◽  
Inversion Techniques ◽  
The Given

Seismograms are routinely used to extract information about the internal structure of the subsurface. The basic philosophy is to idealize the earth with a simple model characterized by resolvable parameters. Further, we assume that the seismograms obey simple mathematical relations, such as the wave equation, and that these seismograms are uniquely determined by the given parameters. The computation of seismograms for a given model whose parameters are specified gives rise to the “forward problem”. The determination of the parameters of the model from the seismogram generally constitutes the “inverse problem”. While the forward problem is often relatively simple, the inverse problem tends to be much more involved. One of the few cases for which the inverse problem has a simple direct solution is given for a model consisting of a homogeneous one‐dimensional layered medium. The layers overlie a homogeneous half‐space, and are excited by a normally incident plane wave. Such direct solutions to the inverse problem are in contrast to the iterative inversion techniques of the Gilbert‐Backus type. We have chosen to treat the inverse problem for the seismogram escaping into the homogeneous substratum, i.e., the transmission seismogram. We do so because this time series is a simple autoregressive, or all‐pole process. The problem is studied both with and without white noise. The Wiener‐Levinson algorithm has been found to be well suited for the direct inversion of the noise‐free case, while the “maximum entropy” algorithm is generally more appropriate for noisy data. Numerical examples serve to clarify and illustrate these points.

To Solve the Load of Beam by Shearing Force Diagram and Bending Moment Diagram - Reverse Thinking in Strength of Materials

2014 ◽  
Vol 488-489 ◽  
pp. 546-549
Author(s):  
Heng Wen Zhang ◽  
Yue Shen ◽  
Sen Wei Zhang
Keyword(s):  
Inverse Problem ◽  
Bending Moment ◽  
Forward Problem ◽  
Load Case ◽  
Shearing Force ◽  
Strength Of Materials ◽  
Force Diagram ◽  
The Given

The paper has a discussion of forward problem and inverse problem for beams in strength of materials. Known load case of a beam can certainly determine its shearing force diagram and bending moment diagram, but conversely, there may be a variety of statically determinate or statically indeterminate constraint conditions. Furthermore, the solution from statically indeterminate constraint conditions doesnt agree with the given shearing force diagram and bending moment diagram in a general way.

Framework for Flaw Detection: Application to Dynamic Crack Detection in Flat Membranes

2008 ◽  
Author(s):  
Daniel Rabinovich ◽  
Dan Givoli ◽  
Shmuel Vigdergauz
Keyword(s):  
Inverse Problem ◽  
Crack Detection ◽  
Flexible Structures ◽  
Flaw Detection ◽  
Forward Problem ◽  
Dynamic Crack ◽  
Computational Framework ◽  
Time Harmonic ◽  
Present Context ◽  
Flat Membranes

A computational framework is developed for the detection of flaws in flexible structures. The framework is based on posing the detection problem as an inverse problem, which requires the solution of many forward problems. Each forward problem is associated with a known flaw; an appropriate cost functional evaluates the quality of each candidate flaw based on the solution of the corresponding forward problem. On the higher level, the inverse problem is solved by a global optimization algorithm. The performance of the computational framework is evaluated by considering the detectability of various types of flaws. In the present context detectability is defined by introducing a measure of the distance between the sought flaw and trial flaws in the space of the parameters characterizing the configuration of the flaw. The framework is applied to crack detection in flat membranes subjected to time-harmonic and transient excitations. The detectability of cracks is compared for these two cases.

scholarly journals Deblending by direct inversion

Geophysics ◽  
2012 ◽  
Vol 77 (3) ◽  
pp. A9-A12 ◽  
Author(s):  
Kees Wapenaar ◽  
Joost van der Neut ◽  
Jan Thorbecke
Keyword(s):  
Inverse Problem ◽  
Seismic Data ◽  
Iterative Procedure ◽  
Inversion Process ◽  
Direct Inversion ◽  
Source Data ◽  
Band Limited ◽  
Blended Data ◽  
Simultaneous Source ◽  
Additional Constraints

Deblending of simultaneous-source data is usually considered to be an underdetermined inverse problem, which can be solved by an iterative procedure, assuming additional constraints like sparsity and coherency. By exploiting the fact that seismic data are spatially band-limited, deblending of densely sampled sources can be carried out as a direct inversion process without imposing these constraints. We applied the method with numerically modeled data and it suppressed the crosstalk well, when the blended data consisted of responses to adjacent, densely sampled sources.

scholarly journals Adjoint Numerical Method for a Multiphysical Inverse Problem of Two-Phase Well Testing in Petroleum Reservoirs

2021 ◽  
Vol 2090 (1) ◽  
pp. 012139
Author(s):  
OA Shishkina ◽  
I M Indrupskiy
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Objective Function ◽  
Petroleum Reservoirs ◽  
Forward Problem ◽  
Adjoint Equations ◽  
Well Testing ◽  
Adjoint Methods ◽  
Two Phase ◽  
Gradient Calculation

Abstract Inverse problem solution is an integral part of data interpretation for well testing in petroleum reservoirs. In case of two-phase well tests with water injection, forward problem is based on the multiphase flow model in porous media and solved numerically. The inverse problem is based on a misfit or likelihood objective function. Adjoint methods have proved robust and efficient for gradient calculation of the objective function in this type of problems. However, if time-lapse electrical resistivity measurements during the well test are included in the objective function, both the forward and inverse problems become multiphysical, and straightforward application of the adjoint method is problematic. In this paper we present a novel adjoint algorithm for the inverse problems considered. It takes into account the structure of cross dependencies between flow and electrical equations and variables, as well as specifics of the equations (mixed parabolic-hyperbolic for flow and elliptic for electricity), numerical discretizations and grids, and measurements in the inverse problem. Decomposition is proposed for the adjoint problem which makes possible step-wise solution of the electric adjoint equations, like in the forward problem, after which a cross-term is computed and added to the right-hand side of the flow adjoint equations at this timestep. The overall procedure provides accurate gradient calculation for the multiphysical objective function while preserving robustness and efficiency of the adjoint methods. Example cases of the adjoint gradient calculation are presented and compared to straightforward difference-based gradient calculation in terms of accuracy and efficiency.

scholarly journals Approaches to improve the robustness of the Comparison Model Method for the inverse problem of groundwater hydrology

2021 ◽  
Author(s):  
Alessandro Comunian ◽  
Mauro Giudici
Keyword(s):  
Geometric Mean ◽  
Optimization Methods ◽  
Forward Problem ◽  
Data Sets ◽  
Direct Inversion ◽  
Model Method ◽  
Comparison Model ◽  
Multiple Data ◽  
Hydraulic Gradients ◽  
Multiple Data Sets

<p>Indirect inversion approaches are widely used in Geosciences, and in particular also for the identification of the hydraulic properties of aquifers. Nevertheless, their application requires a substantial number of model evaluation (forward problem) runs, a task that for complex problems can be computationally intensive. Reducing this computational burden is an active research topic, and many solutions, including the use of hybrid optimization methods, the use of physical proxies or again machine-learning tools <span>allow to avoid</span> considering the full physics of the problem when running a numerical implementation of the forward problem.</p><p>Direct inversion approaches represent computationally frugal alternatives to indirect approaches, because in general they require a smaller number of runs of the forward problem. The classical drawbacks of these methods can be alleviated by some implementation approaches and in particular by using multiple sets of data, when available.</p><p>This work is an effort to improve the robustness of the Comparison Model Method (CMM), a direct inversion approach aimed at the identification of the hydraulic transmissivity of a confined aquifer. The robustness of the CMM is here ameliorated by (i) improving the parameterization required to handle small hydraulic gradients; (ii) investigating the role of different criteria aimed at merging multiple data-sets corresponding to different flow conditions.</p><p>On a synthetic case study, it is demonstrated that correcting a small percentage of the small hydraulic gradients (about 10%) allows to obtain reliable results, and that a criteria based on the geometric mean is adequate to merge the results coming from multiple data-sets. In addition, the use of multiple-data sets allows to noticeably improve the robustness of the CMM when the input data are affected by noise.</p><p>All the tests are performed by using open source and widely <span>used</span> tools like the USGS Modflow6 and its Python interface flopy to foster the application of the <span>CMM. The scripts and corresponding package</span>, named <em>cmmpy</em>, is available on the Python Package Index (PyPI) and on bitbucket at the following address: https://bitbucket.org/alecomunian/cmmpy.</p>

ON THE ADIABATICITY OF ACOUSTIC PROPAGATION THROUGH NONGRADUAL OCEAN STRUCTURES

2001 ◽  
Vol 09 (02) ◽  
pp. 359-365 ◽  
Author(s):  
E. C. SHANG ◽  
Y. Y. WANG ◽  
T. F. GAO
Keyword(s):  
Inverse Problem ◽  
Shallow Water ◽  
Acoustic Field ◽  
Solitary Waves ◽  
Sound Propagation ◽  
Acoustic Propagation ◽  
Forward Problem ◽  
Internal Solitary Waves ◽  
New Criterion

To assess the adiabaticity of sound propagation in the ocean is very important for acoustic field calculating (forward problem) and tomographic retrieving(inverse problem). Most of the criterion in the literature is too restrictive, specially for the nongradual ocean structures. A new criterion of adiabaticity is suggested in this paper. It works for nongradual ocean structures such as front and internal solitary waves in shallow-water.

Learning solutions to the source inverse problem of wave equations using LS-SVM

2019 ◽  
Vol 27 (5) ◽  
pp. 657-669 ◽  
Author(s):  
Ziku Wu ◽  
Chang Ding ◽  
Guofeng Li ◽  
Xiaoming Han ◽  
Juan Li
Keyword(s):  
Inverse Problem ◽  
Approximate Solution ◽  
Support Vector Machines ◽  
Wave Equations ◽  
Source Term ◽  
Initial Conditions ◽  
Support Vector ◽  
Vector Machines ◽  
Robustness To Noise ◽  
The Given

Abstract A method based on least squares support vector machines (LS-SVM) is proposed to solve the source inverse problem of wave equations. Contrary to the most existing methods, the proposed method provides a closed form approximate solution which satisfies the boundary conditions and the initial conditions. The proposed method can recover the unknown source term with the given additional conditions. Furthermore, it has reasonable robustness to noise. Numerical results show the proposed method can be used to solve the source inverse problem of wave equations.

A Comparative Study of Direct and Iterative Inversion Approaches to Determine the Spatial Shear Modulus Distribution of Elastic Solids

2019 ◽  
Vol 11 (10) ◽  
pp. 1950097 ◽  
Author(s):  
Zhi Liu ◽  
Yanli Sun ◽  
Jianwei Deng ◽  
Dongmei Zhao ◽  
Yue Mei ◽  
...  
Keyword(s):  
Shear Modulus ◽  
Comparative Study ◽  
Inversion Method ◽  
Noise Levels ◽  
Elastic Solids ◽  
Direct Inversion ◽  
Advantages And Disadvantages ◽  
Iterative Inversion ◽  
Very High ◽  
Selection Of

This paper presents a comparative study of two typical inverse algorithms, i.e., direct and iterative inversion methods, to reconstruct the shear modulus distribution of linearly elastic solids. Both approaches are based on the finite element framework and compared utilizing both the simulated and experimental data. The reconstruction results demonstrate that both approaches are capable of identifying the nonhomogeneous shear modulus distribution of solids well. It can also be found that the direct inversion method is much faster than the iterative inversion method, whereas the iterative inversion method is capable of yielding better shear modulus ratio between the stiff inclusion and the soft background even with very high noise levels. Afterwards, a thorough comparison on the advantages and disadvantages of these two approaches has been performed. This comparative study provides useful information on the selection of the proper inverse scheme in estimating nonhomogeneous elastic property distribution of soft solids nondestructively.

scholarly journals On an Identification Problem on the Determination of the Parameters of the Dynamic System

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Dossan Baigereyev ◽  
Nevazi Ismailov ◽  
Yusif Gasimov ◽  
Atif Namazov
Keyword(s):  
Inverse Problem ◽  
Dynamic System ◽  
Calculation Method ◽  
Identification Problem ◽  
Quadratic Functional ◽  
Nonlinear Ordinary Differential Equations ◽  
Quasilinearization Method ◽  
The Right ◽  
The Given

An inverse problem is considered for the determination of the parameters, involved in the right-hand side of the system of nonlinear ordinary differential equations by given initial and final conditions. The solution of the problem is reduced to the minimization of the quadratic functional, which indeed is a deviation of the value of the solution from the given values at the end points. Using the quasilinearization method a calculation method is proposed to the solution of the considered problem. The application of this method is demonstrated on the example of the determination of the hydraulic resistance in the tubes.

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