scholarly journals Estimating a shear modulus of a transversely isotropic formation

Geophysics ◽  
1992 ◽  
Vol 57 (11) ◽  
pp. 1428-1434 ◽  
Author(s):  
K. J. Ellefsen ◽  
M. N. Toksöz ◽  
K. M. Tubman ◽  
C. H. Cheng
Keyword(s):  
Shear Modulus ◽  
Cost Function ◽  
Symmetry Axis ◽  
Synthetic Data ◽  
Clay Content ◽  
Transversely Isotropic ◽  
Optimization Methods ◽  
Current Estimate ◽  
Tube Wave ◽  
The Cost

We have developed a method that estimates a shear modulus [Formula: see text] of a transversely isotropic formation using the tube wave generated during acoustic logging. (The symmetry axis of the anisotropy is assumed to parallel the borehole.) The inversion, which is implemented in the frequency‐wavenumber domain, is based upon a cost function that has three terms: a measure of the misfit between the observed and predicted wavenumbers of the tube wave, a measure of the misfit between the current estimate for [Formula: see text] and the most‐likely value for [Formula: see text], and penalty functions that constrain the estimate to physically acceptable values. The largest contribution to the value of the cost function ordinarily comes from the first term, indicating that the estimate for [Formula: see text] depends mostly on the data. Because the cost function only has one minimum, it can be found using standard optimization methods. The minimum is well defined indicating that the estimate for [Formula: see text] is well resolved. Estimates for [Formula: see text] from synthetic data are almost always within 1 percent of their correct value. Estimates for [Formula: see text] from field data that were collected in a formation with a high clay content are typical of transversely isotropic rocks.

Weak elastic anisotropy and the tube wave

Geophysics ◽  
1993 ◽  
Vol 58 (8) ◽  
pp. 1091-1098 ◽  
Author(s):  
Andrew N. Norris ◽  
Bikash K. Sinha
Keyword(s):  
Shear Modulus ◽  
Elastic Moduli ◽  
Wave Speed ◽  
Elastic Anisotropy ◽  
Transversely Isotropic ◽  
Anisotropic Formation ◽  
Inversion Procedure ◽  
Anisotropy Parameters ◽  
Tube Wave ◽  
Simple Inversion

Tube‐wave speed in the presence of a weakly anisotropic formation can be expressed in terms of an effective shear modulus for an equivalent isotropic formation. When combined with expressions for the speeds of the SH‐ and quasi‐SV‐waves along the borehole axis, a simple inversion procedure can be obtained to determine three of the five elasticities of a transversely isotropic (TI) formation tilted at some known angle with respect to the borehole axis. Subsequently, a fourth combination of elastic moduli can be estimated from the expression for the qP‐wave speed along the borehole axis. The possibility of determining all five elasticities of a TI formation based on an assumed correlation between two anisotropy parameters is discussed.

scholarly journals The Application of an Evolutionary Algorithm to the Optimization of a Mesoscale Meteorological Model

2009 ◽  
Vol 48 (2) ◽  
pp. 317-329 ◽  
Author(s):  
Lance O’Steen ◽  
David Werth
Keyword(s):  
Cost Function ◽  
Evolutionary Algorithm ◽  
Synthetic Data ◽  
Atmospheric Model ◽  
Atmospheric Modeling ◽  
Model Parameters ◽  
Meteorological Model ◽  
Rms Error ◽  
Mesoscale Atmospheric Model ◽  
The Cost

Abstract It is shown that a simple evolutionary algorithm can optimize a set of mesoscale atmospheric model parameters with respect to agreement between the mesoscale simulation and a limited set of synthetic observations. This is illustrated using the Regional Atmospheric Modeling System (RAMS). A set of 23 RAMS parameters is optimized by minimizing a cost function based on the root-mean-square (rms) error between the RAMS simulation and synthetic data (observations derived from a separate RAMS simulation). It is found that the optimization can be done with relatively modest computer resources; therefore, operational implementation is possible. The overall number of simulations needed to obtain a specific reduction of the cost function is found to depend strongly on the procedure used to perturb the “child” parameters relative to their “parents” within the evolutionary algorithm. In addition, the choice of meteorological variables that are included in the rms error and their relative weighting are also found to be important factors in the optimization.

Building tilted transversely isotropic depth models using localized anisotropic tomography with well information

Geophysics ◽  
2010 ◽  
Vol 75 (4) ◽  
pp. D27-D36 ◽  
Author(s):  
Andrey Bakulin ◽  
Marta Woodward ◽  
Dave Nichols ◽  
Konstantin Osypov ◽  
Olga Zdraveva
Keyword(s):  
Seismic Data ◽  
Model Building ◽  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Synthetic Data ◽  
Rock Physics ◽  
Transversely Isotropic ◽  
Data Set ◽  
Anisotropic Models ◽  
Well Data

Tilted transverse isotropy (TTI) is increasingly recognized as a more geologically plausible description of anisotropy in sedimentary formations than vertical transverse isotropy (VTI). Although model-building approaches for VTI media are well understood, similar approaches for TTI media are in their infancy, even when the symmetry-axis direction is assumed known. We describe a tomographic approach that builds localized anisotropic models by jointly inverting surface-seismic and well data. We present a synthetic data example of anisotropic tomography applied to a layered TTI model with a symmetry-axis tilt of 45 degrees. We demonstrate three scenarios for constraining the solution. In the first scenario, velocity along the symmetry axis is known and tomography inverts for Thomsen’s [Formula: see text] and [Formula: see text] parame-ters. In the second scenario, tomography inverts for [Formula: see text], [Formula: see text], and velocity, using surface-seismic data and vertical check-shot traveltimes. In contrast to the VTI case, both these inversions are nonunique. To combat nonuniqueness, in the third scenario, we supplement check-shot and seismic data with the [Formula: see text] profile from an offset well. This allows recovery of the correct profiles for velocity along the symmetry axis and [Formula: see text]. We conclude that TTI is more ambiguous than VTI for model building. Additional well data or rock-physics assumptions may be required to constrain the tomography and arrive at geologically plausible TTI models. Furthermore, we demonstrate that VTI models with atypical Thomsen parameters can also fit the same joint seismic and check-shot data set. In this case, although imaging with VTI models can focus the TTI data and match vertical event depths, it leads to substantial lateral mispositioning of the reflections.

Acoustic anisotropic wavefields through perturbation theory

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. WC41-WC50 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Perturbation Theory ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Symmetry Axis ◽  
Finite Difference Methods ◽  
Transversely Isotropic ◽  
Direct Access ◽  
Acoustic Wave Equation ◽  
Anisotropy Parameters ◽  
The Cost

Solving the anisotropic acoustic wave equation numerically using finite-difference methods introduces many problems and media restriction requirements, and it rarely contributes to the ability to resolve the anisotropy parameters. Among these restrictions are the inability to handle media with [Formula: see text] and the presence of shear-wave artifacts in the solution. Both limitations do not exist in the solution of the elliptical anisotropic acoustic wave equation. Using perturbation theory in developing the solution of the anisotropic acoustic wave equation allows direct access to the desired limitation-free solutions, that is, solutions perturbed from the elliptical anisotropic background medium. It also provides a platform for parameter estimation because of the ability to isolate the wavefield dependency on the perturbed anisotropy parameters. As a result, I derive partial differential equations that relate changes in the wavefield to perturbations in the anisotropy parameters. The solutions of the perturbation equations represented the coefficients of a Taylor-series-type expansion of the wavefield as a function of the perturbed parameter, which is in this case [Formula: see text] or the tilt of the symmetry axis. The expansion with respect to the symmetry axis allows use of an acoustic transversely isotropic media with a vertical symmetry axis (VTI) kernel to estimate the background wavefield and the corresponding perturbation coefficients. The VTI extrapolation kernel is about one-fourth the cost of the transversely isotropic model with a tilt in the symmetry axis kernel. Thus, for a small symmetry axis tilt, the cost of migration using a first-order expansion can be reduced. The effectiveness of the approach was demonstrated on the Marmousi model.

OPTIMIZATION OF THE COST FUNCTION IN THE MONGE-KANTOROVICH PROBLEM (MKP) UNDER THE MONGE CONDITION

2014 ◽  
Vol 28 (05) ◽  
pp. 1451006
Author(s):  
LAURENT OUDRE
Keyword(s):  
Linear Programming ◽  
Cost Function ◽  
Large Distance ◽  
Synthetic Data ◽  
Real Data ◽  
Small Distance ◽  
Optimal Cost ◽  
The Cost ◽  
Additional Constraints ◽  
Kantorovich Problem

This paper presents a method for adapting the cost function in the Monge–Kantorovich Problem (MKP) to a classification task. More specifically, we introduce a criterion that allows to learn a cost function which tends to produce large distance values for elements belonging to different classes and small distance values for elements belonging to the same class. Under some additional constraints (one of them being the well-known Monge condition), we show that the optimization of this criterion writes as a linear programming problem. Experimental results on synthetic data show that the output optimal cost function provides good retrieval performances in the presence of two types of perturbations commonly found in histograms. When compared to a set of various commonly used cost functions, our optimal cost function performs as good as the best cost function of the set, which shows that it can adapt well to the task. Promising results are also obtained on real data for two-class image retrieval based on grayscale intensity histograms.

Comparison of GA and PSO in Boundary Element Inverse Analysis for Rebar Corrosion Detection

2013 ◽  
Vol 471 ◽  
pp. 319-323 ◽  
Author(s):  
Syarizal Fonna ◽  
M. Ridha ◽  
Syifaul Huzni ◽  
Ahmad K. Ariffin
Keyword(s):  
Cost Function ◽  
Boundary Element ◽  
Inverse Analysis ◽  
Optimization Methods ◽  
Concrete Surface ◽  
Global Optimum ◽  
Rebar Corrosion ◽  
Optimum Solution ◽  
The Cost ◽  
Global Optimum Solution

This paper presents the comparison of the two optimization methods, particle swarm optimization (PSO) and genetic algorithm (GA) in boundary element inverse analysis that applied to detect the corrosion location of rebar in the concrete. This comparison focuses at analyzing the performance of both methods in reaching the global optimum, considering that both heuristics are based on population search techniques. The model of 2-dimension rectangular reinforced concrete was used as a case example to compare both methods in boundary element inverse analysis. The boundary element inverse analysis was developed by combining Boundary Element Method (BEM) and PSO or GA. The inverse analysis is carried out by means of minimizing a cost function. The cost function is a residual between the calculated and measured potentials on the concrete surface. The calculated potentials are obtained by solving the Laplaces equation using BEM. The GA or PSO is used to minimize the cost function. Thus, the corrosion location of reinforcing steel in concrete can be detected. The numerical simulation results showed that one of PSO or GA can be used for the inverse analysis for detecting rebar corrosion by combining with BEM. However, it shows that PSO seem numerically superior compared to GA in term of consistency and accuracy in finding global optimum solution for such a problem.

scholarly journals A Size-Dependent Cost Function to Solve the Inverse Elasticity Problem

2019 ◽  
Vol 9 (9) ◽  
pp. 1799
Author(s):  
Xinbo Zhao ◽  
Yanli Sun ◽  
Yue Mei
Keyword(s):  
Theoretical Analysis ◽  
Shear Modulus ◽  
Cost Function ◽  
Absolute Error ◽  
Inverse Approach ◽  
Size Dependent ◽  
L2 Norm ◽  
Novel Method ◽  
The Cost ◽  
Shear Modulus Reconstruction

Characterizing nonhomogeneous elastic property distribution of solids is of great significance in various engineering fields. In this paper, we observe that the solution to the inverse problem utilizing the standard optimization-based inverse approach is sensitive to the sizes of inclusions. The standard optimization-based inverse approach minimizes a cost function, containing the absolute error between the measured and computed displacements in L2 norm. To address this issue, we propose a novel inverse scheme to characterize nonhomogeneous shear modulus distribution of solids. In this novel method, the cost function is modified, and is dependent on the size of the inclusions. A number of simulated experiments are performed, and demonstrate that the proposed approach is capable of improving the shear modulus contrast in inclusions and reducing the size sensitivity. Furthermore, a theoretical analysis is conducted to validate what we have observed in simulated experiments. This theoretical analysis reveals that what we have observed in the simulated experiments is not induced by the numerical issues Instead, the size sensitivity issue is induced by regularization. The findings of this work encourage us to propose new cost functions for the optimization-based inverse approach to improve the quality of the shear modulus reconstruction.

Customizing constraint incorporation in direct current resistivity inverse problems: A comparison among three global optimization methods

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. E409-E422 ◽  
Author(s):  
Francisco M. Barboza ◽  
Walter E. Medeiros ◽  
Jerbeson M. Santana
Keyword(s):  
Global Optimization ◽  
Inverse Problems ◽  
Cost Function ◽  
Direct Current ◽  
Optimization Methods ◽  
Dc Resistivity ◽  
Quantitative Interpretation ◽  
Computationally Efficient ◽  
Global Optimization Methods ◽  
The Cost

One-dimensional forward modeling in direct current (DC) resistivity is actually computationally inexpensive, allowing the use of global optimization methods (GOMs) to solve 1.5D inverse problems with flexibility in constraint incorporation. GOMs can support computational environments for quantitative interpretation in which the comparison of solutions incorporating different constraints is a way to infer characteristics of the actual subsurface resistivity distribution. To this end, the chosen GOM must be robust to changes in the cost function and also be computationally efficient. The performance of the classic versions of the simulated annealing (SA), genetic algorithm (GA), and particle swarm optimization (PSO) methods for solving the 1.5D DC resistivity inverse problem is here compared using synthetic and field data. The main results are as follows: (1) All methods reproduce synthetic models quite well, (2) PSO and GA are comparatively more robust to changes in the cost function than SA, (3) PSO first and GA second present the best computational performances, requiring less forwarding modeling than SA, and (4) GA gives higher performance than PSO and SA with respect to the final attained value of the cost function and its standard deviation. From our experience, to put them into effective operation, the methods can be classified from easy to difficult in the order PSO, GA, and SA as a consequence of robustness to changes in the cost function and of the underlying simplicity of the associated equations. To exemplify a quantitative interpretation using GOMs, we compare solutions with least-absolute and least-squares norms of the discrepancies derived from the lateral continuity constraints of the log-resistivity and layer depth as a manner of detecting faults. GOMs additionally provide the important benefit of furnishing not only the best solution but also a set of suboptimal quasisolutions from which uncertainty analyses can be performed.

scholarly journals Migration Using a Transversely Isotropic Medium with Symmetry Normal to the Reflector Dip

2011 ◽  
Vol 2011 ◽  
pp. 1-5 ◽  
Author(s):  
Tariq Alkhalifah ◽  
Paul Sava
Keyword(s):  
Model Building ◽  
Symmetry Axis ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Velocity Model ◽  
Downward Continuation ◽  
Velocity Model Building ◽  
Lateral Inhomogeneity ◽  
Velocity Information ◽  
Model Phase

A transversely isotropic (TI) model in which the tilt is constrained to be normal to the dip (DTI model) allows for simplifications in the imaging and velocity model building efforts as compared to a general TI (TTI) model. Although this model cannot be represented physically in all situations, for example, in the case of conflicting dips, it handles arbitrary reflector orientations under the assumption of symmetry axis normal to the dip. Using this assumption, we obtain efficient downward continuation algorithms compared to the general TTI ones, by utilizing the reflection features of such a model. Phase-shift migration can be easily extended to approximately handle lateral inhomogeneity using, for example, the split-step approach. This is possible because, unlike the general TTI case, the DTI model reduces to VTI for zero dip. These features enable a process in which we can extract velocity information by including tools that expose inaccuracies in the velocity model in the downward continuation process. We test this model on synthetic data corresponding to a general TTI medium and show its resilience.

Do dipole sonic logs measure group or phase velocity (revisited)?

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. C311-C322
Author(s):  
Stephen Horne ◽  
Richard T. Coates ◽  
Alexei Bolshakov
Keyword(s):  
Symmetry Axis ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Flexural Waves ◽  
S Wave ◽  
Slowness Surface ◽  
Phase Velocities ◽  
Group And Phase Velocities ◽  
Sonic Logs

We have revisited the debate about whether flexural waves from dipole sonic tools and standard processing algorithms measure group or phase velocities in anisotropic formations. We observe that much of the confusion arises from a failure to understand the different meanings of group and phase velocities. Using a transversely isotropic medium with a vertical axis of symmetry that exhibits a triplication in its S-wave group slowness surface, we generate synthetic flexural sonic waveforms corresponding to boreholes at angles of 0°–90° with respect to the anisotropy symmetry axis in 1° increments. We processed these synthetic data using standard time- and frequency-domain semblance methods. The results conclusively demonstrate that dipole sonic logs measure the group slowness for the group angle corresponding to the angle between the borehole and the anisotropic symmetry axis. In addition, data that we have evaluated suggest that current tool geometries and semblance processing may not always be sensitive enough to resolve all branches of the group slowness triplication surface.

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