Kirchhoff migration and velocity analysis for converted and nonconverted waves in anisotropic media

Geophysics ◽  
1993 ◽  
Vol 58 (2) ◽  
pp. 265-276 ◽  
Author(s):  
Arcangelo G. Sena ◽  
M. Nafi Toksöz
Keyword(s):  
Anisotropic Media ◽  
Velocity Model ◽  
Velocity Analysis ◽  
Converted Wave ◽  
Anisotropic Models ◽  
Kirchhoff Migration ◽  
Analysis Technique ◽  
Analysis Scheme ◽  
Isotropic Media ◽  
And Migration

We develop asymptotic expressions and outline a procedure to perform Kirchhoff migration in anisotropic media. This technique is based on a new Green’s tensor representation for azimuthally isotropic media obtained by using analytical forms for the ray amplitudes and traveltimes. Since in real applications the usage of general anisotropy in a migration scheme will be limited by the availability and reliability of the velocity model considered, we also develop a new anisotropic velocity analysis scheme to generate realistic anisotropic models for migration in azimuthally isotropic media for nonconverted and converted qP-qSV waves. This velocity analysis technique is based on nonhyperbolic traveltime‐offset formulas explicitly given in terms of the five elastic constants of azimuthal isotropy. The imaging technique is applied to nonconverted‐as well as converted‐wave surface seismic data. In both cases the method provides accurate images of the subsurface. Even with a weak to moderate percentage of anisotropy, we show that an isotropic migration algorithm cannot properly image the subsurface. This paper provides new nonconventional techniques for the velocity analysis and migration in anisotropic media and shows the feasibility of exploiting converted and nonconverted waves.

Normal moveout from dipping reflectors in anisotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 268-284 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Weak Anisotropy ◽  
Anisotropic Models ◽  
Transversely Isotropic Media ◽  
Wide Range ◽  
Isotropic Media ◽  
The Difference

Description of reflection moveout from dipping interfaces is important in developing seismic processing methods for anisotropic media, as well as in the inversion of reflection data. Here, I present a concise analytic expression for normal‐moveout (NMO) velocities valid for a wide range of homogeneous anisotropic models including transverse isotropy with a tilted in‐plane symmetry axis and symmetry planes in orthorhombic media. In transversely isotropic media, NMO velocity for quasi‐P‐waves may deviate substantially from the isotropic cosine‐of‐dip dependence used in conventional constant‐velocity dip‐moveout (DMO) algorithms. However, numerical studies of NMO velocities have revealed no apparent correlation between the conventional measures of anisotropy and errors in the cosine‐of‐dip DMO correction (“DMO errors”). The analytic treatment developed here shows that for transverse isotropy with a vertical symmetry axis, the magnitude of DMO errors is dependent primarily on the difference between Thomsen parameters ε and δ. For the most common case, ε − δ > 0, the cosine‐of‐dip–corrected moveout velocity remains significantly larger than the moveout velocity for a horizontal reflector. DMO errors at a dip of 45 degrees may exceed 20–25 percent, even for weak anisotropy. By comparing analytically derived NMO velocities with moveout velocities calculated on finite spreads, I analyze anisotropy‐induced deviations from hyperbolic moveout for dipping reflectors. For transversely isotropic media with a vertical velocity gradient and typical (positive) values of the difference ε − δ, inhomogeneity tends to reduce (sometimes significantly) the influence of anisotropy on the dip dependence of moveout velocity.

2‐D wavepath migration

Geophysics ◽  
2001 ◽  
Vol 66 (5) ◽  
pp. 1528-1537 ◽  
Author(s):  
H. Sun ◽  
G. T. Schuster
Keyword(s):  
Fresnel Zone ◽  
Complex Structure ◽  
Computation Time ◽  
Velocity Model ◽  
Velocity Analysis ◽  
Kirchhoff Migration ◽  
Time Sample ◽  
Grid Points ◽  
Computationally Intensive ◽  
Single Trace

Prestack Kirchhoff migration (KM) is computationally intensive for iterative velocity analysis. This is partly because each time sample in a trace must be smeared along a quasi‐ellipsoid in the model. As a less costly alternative, we use the stationary phase approximation to the KM integral so that the time sample is smeared along a small Fresnel zone portion of the quasi‐ellipsoid. This is equivalent to smearing the time samples in a trace over a 1.5‐D fat ray (i.e., wavepath), so we call this “wavepath migration” (WM). This compares to standard KM, which smears the energy in a trace along a 3‐D volume of quasi‐concentric ellipsoids. In principle, single trace migration with WM has a computational count of [Formula: see text] compared to KM, which has a computational count of [Formula: see text], where N is the number of grid points along one side of a cubic velocity model. Our results with poststack data show that WM produces an image that in some places contains fewer migration artifacts and is about as well resolved as the KM image. For a 2‐D poststack migration example, the computation time of WM is less than one‐third that of KM. Our results with prestack data show that WM images contain fewer migration artifacts and can define the complex structure more accurately. It is also shown that WM can be significantly faster than KM if a slant stack technique is used in the migration. The drawback with WM is that it is sometimes less robust than KM because of its sensitivity to errors in estimating the incidence angles of the reflections.

Explicit expressions for prestack map time migration in isotropic and VTI media and the applicability of map depth migration in heterogeneous anisotropic media

Geophysics ◽  
2006 ◽  
Vol 71 (1) ◽  
pp. S13-S28 ◽  
Author(s):  
Huub Douma ◽  
Maarten V. de Hoop
Keyword(s):  
Closed Form ◽  
Anisotropic Media ◽  
Velocity Model ◽  
Pure Mode ◽  
Anisotropic Parameter ◽  
Depth Migration ◽  
Time Migration ◽  
Isotropic Media ◽  
The Common ◽  
Vti Media

We present 3D prestack map time migration in closed form for qP-, qSV-, and mode-converted waves in homogeneous transversely isotropic media with a vertical symmetry axis (VTI). As far as prestack time demigration is concerned, we present closed-form expressions for mapping in homogeneous isotropic media, while for homogeneous VTI media we present a system of four nonlinear equations with four unknowns to solve numerically. The expressions for prestack map time migration in VTI homogeneous media are directly applicable to the problem of anisotropic parameter estimation (i.e., the anellipticity parameter η) in the context of time-migration velocity analysis. In addition, we present closed-form expressions for both prestack map time migration and demigration in the common-offset domain for pure-mode (P-P or S-S) waves in homogeneous isotropic media that use only the slope in the common-offset domain as opposed to slopes in both the common-shot and common-receiver (or equivalently the common-offset and common-midpoint) domains. All time-migration and demigration equations presented can be used in media with mild lateral and vertical velocity variations, provided the velocity is replaced with the local rms velocity. Finally, we discuss the condition for applicability of prestack map depth migration and demigration in heterogeneous anisotropic media that allows the formation of caustics and explain that this condition is satisfied if, given a velocity model and acquisition geometry, one can map-depth-migrate without ambiguity in either the migrated location or the migrated orientation of reflectors in the image.

scholarly journals EMT simulation and effect of TTI anisotropic media in EMT signal

2021 ◽  
Author(s):  
Qing-Yun Di ◽  
Olalekan Fayemi ◽  
Qi-Hui Zhen ◽  
Tian Fei
Keyword(s):  
Gaussian Quadrature ◽  
Anisotropic Media ◽  
Quadrature Rule ◽  
Transverse Magnetic ◽  
Measured Signal ◽  
Anisotropic Parameter ◽  
Transverse Magnetic Mode ◽  
Anisotropic Models ◽  
Isotropic Media ◽  
Order Of Magnitude

AbstractAn axisymmetric finite difference method is employed for the simulations of electromagnetic telemetry in the homogeneous and layered underground formation. In this method, we defined the anisotropy property using extensive 2D conductivity tensor and solved it in the transverse magnetic mode. Significant simplification arises in the decoupling of the anisotropic parameter. The developed method is cost-efficient, more straightforward in modeling anisotropic media, and easy to be implemented. In addition, we solved the integral operation in the estimation of measured surface voltage using Gaussian quadrature technique. We performed a series of numerical modeling of EM telemetry signals in both isotropic and anisotropic models. Experiment with 2D tilt transverse isotropic media characterized by the tilt axis and anisotropy parameters shows an increase in the EMT signal with an increase in the angle of tilt of the principal axis for a moderate coefficient of anisotropy. We show that the effect of the tilt of the subsurface medium can be observed with sufficient accuracy and that it is an order of magnitude of 5 over the tilt of 90 degrees. Lastly, consistent results with existing field data were obtained by employing the Gaussian quadrature rule for the computation of surface measured signal.

Toward a stable iterative migration velocity analysis scheme

Geophysics ◽  
2018 ◽  
Vol 83 (5) ◽  
pp. R475-R495
Author(s):  
Emmanuel Cocher ◽  
Hervé Chauris ◽  
René-Édouard Plessix
Keyword(s):  
Objective Function ◽  
Large Scale ◽  
Velocity Model ◽  
Migration Velocity ◽  
Velocity Analysis ◽  
Outer Loop ◽  
Image Domain ◽  
Reflection Data ◽  
Migration Velocity Analysis ◽  
Analysis Scheme

Migration velocity analysis is a family of methods aiming at automatically recovering large-scale trends of the velocity model from primary reflection data. We studied an image-domain version, in which the model is extended with the subsurface offset and we use the differential semblance optimization objective function. To incorporate first-order surface multiples in this method, the standard migration step is replaced by a least-squares iterative scheme aiming at determining an extended reflectivity model explaining primaries and multiples. Hence, this iterative migration velocity analysis strategy takes the form of a nested optimization problem, with gradient-based minimization techniques for the inner (migration part) and outer loops (macromodel estimation). The behavior of the outer loop gradient is unstable, depending on the number of iterations of the inner loop. This problem is addressed by slightly modifying the outer loop objective function: A “filter” operator attenuating unwanted energy in the extended reflectivity is applied before evaluating the focusing of reflectivity images. Simple synthetic numerical examples illustrate that this modification improves the stability of the gradient. In addition, a less expensive outer gradient computation is proposed, without harming the background velocity updates.

scholarly journals Dip‐moveout processing by Fourier transform in anisotropic media

Geophysics ◽  
1997 ◽  
Vol 62 (4) ◽  
pp. 1260-1269 ◽  
Author(s):  
John E. Anderson ◽  
Ilya Tsvankin
Keyword(s):  
Transverse Isotropy ◽  
Computing Time ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
P Wave ◽  
Surface Reflection ◽  
Anisotropic Models ◽  
Reflection Data ◽  
Isotropic Media ◽  
Vti Media

Conventional dip‐moveout (DMO) processing is designed for isotropic media and cannot handle angle‐dependent velocity. We show that Hale's isotropic DMO algorithm remains valid for elliptical anisotropy but may lead to serious errors for nonelliptical models, even if velocity anisotropy is moderate. Here, Hale's constant‐velocity DMO method is extended to anisotropic media. The DMO operator, to be applied to common‐offset data corrected for normal moveout (NMO), is based on the analytic expression for dip‐dependent NMO velocity given by Tsvankin. Since DMO correction in anisotropic media requires knowledge of the velocity field, it should be preceded by an inversion procedure designed to obtain the normal‐moveout velocity as a function of ray parameter. For transversely isotropic models with a vertical symmetry axis (VTI media), P‐wave NMO velocity depends on a single anisotropic coefficient (η) that can be determined from surface reflection data. Impulse responses and synthetic examples for typical VTI media demonstrate the accuracy and efficiency of this DMO technique. Once the inversion step has been completed, the NMO-DMO sequence does not take any more computing time than the genetic Hale method in isotropic media. Our DMO operator is not limited to vertical transverse isotropy as it can be applied in the same fashion in symmetry planes of more complicated anisotropic models such as orthorhombic.

Modeling and imaging with the scalar generalized‐screen algorithms in isotropic media

Geophysics ◽  
2001 ◽  
Vol 66 (5) ◽  
pp. 1551-1568 ◽  
Author(s):  
Jéro⁁me H. Le Rousseau ◽  
Maarten V. de Hoop
Keyword(s):  
Heterogeneous Media ◽  
Fourier Method ◽  
Numerical Algorithms ◽  
Velocity Model ◽  
The Family ◽  
Isotropic Media ◽  
And Migration ◽  
The One ◽  
Lateral Variations ◽  
Synthetic Models

The phase‐screen and the split‐step Fourier methods, which allow modeling and migration in laterally heterogeneous media, are generalized here so as to increase their accuracies for media with large and rapid lateral variations. The medium is defined in terms of a background medium and a perturbation. Such a contrast formulation induces a series expansion of the vertical slowness in which we recognize the first term as the split‐step Fourier approximation and the addition of higher‐order terms of the expansion increases the accuracy. Employing this expansion in the one‐way scalar propagator yields the scalar one‐way generalized‐screen propagator. We also introduce a generalized‐screen representation of the reflection operator. The interaction between the upgoing and downgoing fields is taken into account by a Bremmer series. These results are then cast into numerical algorithms. We analyze the accuracy of the generalized‐screen method in complex structures using synthetic models that exhibit significant multipathing: the IFP 2‐D Marmousi model and the SEG‐EAGE 3‐D salt model. As compared with the split‐step Fourier method, in the presence of lateral medium variations, the generalized‐screen methods exhibit an increased accuracy at wider angles of propagation and scattering. As a result, in the process of migration, we can choose a member of the family of our generalized‐screen algorithms in accordance with the complexity of the medium (velocity model).

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