Traveltime calculation and prestack depth migration in tilted transversely isotropic media

Geophysics ◽  
2004 ◽  
Vol 69 (1) ◽  
pp. 37-44 ◽  
Author(s):  
Dhananjay Kumar ◽  
Mrinal K. Sen ◽  
Robert J. Ferguson
Keyword(s):  
Direct Method ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
P Wave ◽  
Fourier Series Expansion ◽  
Prestack Depth Migration ◽  
Thrust Sheet ◽  
Depth Migration ◽  
Axis Of Symmetry ◽  
Tti Media

The principal objective of our work is to develop a technique for prestack depth migration in tilted transversely isotropic (TTI) media in which the axis of symmetry is not vertical and may be spatially varying. Such models are required to image seismic data in geologically complex regions such as the Canadian Foothills. We have developed a 2D Kirchhoff integral‐based migration algorithm in which the traveltime computation comprises the major task. Among the existing traveltime computation algorithms such as ray tracing with interpolation, ray bending, and eikonal solvers, a direct or a brute force approach of traveltime computation is generally highly robust. We have modified a direct method of first‐arrival P‐wave traveltime computation in 2D media that accounts for TTI. The algorithm requires that the group velocity be computed at each gridpoint, using either an analytic solution or by an approximate Fourier series expansion. The P‐wave traveltime contours computed for complex geologic models show the pronounced effects from TTI media. Our results, using laboratory P‐wave data collected over a physical model of an anisotropic thrust sheet, reveal that a 2D Kirchhoff migration based on our traveltime algorithm (TTI model) images the structure beneath the thrust sheet very well. The vertical transversely isotropic (assuming a vertical axis of symmetry) or isotropic imaging introduces false anticlinal structures. We compare our results with those obtained by a recursive‐extrapolation method and find that our approach images the underside of one of the thrust sheets better.

Anisotropic parsimonious prestack depth migration

Geophysics ◽  
2013 ◽  
Vol 78 (1) ◽  
pp. S25-S36 ◽  
Author(s):  
Ernesto V. Oropeza ◽  
George A. McMechan
Keyword(s):  
Model Building ◽  
Computing Time ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Velocity Model ◽  
P Wave ◽  
Prestack Depth Migration ◽  
Depth Migration ◽  
Tti Media ◽  
Map Migration

An efficient Kirchhoff-style prestack depth migration, called “parsimonious” migration, was developed a decade ago for isotropic 2D and 3D media by using measured slownesses to reduce the amount of ray tracing by orders of magnitude. It is conceptually similar to “map” migration, but its implementation has some differences. We have extended this approach to 2D tilted transversely isotropic (TTI) media and illustrated it with synthetic P-wave data. Although the framework of isotropic parsimonious may be retained, the extension to TTI media requires redevelopment of each of the numerical components, calculation of the phase and group velocity for TTI media, development of a new two-point anisotropic ray tracer, and substitution of an initial-angle isotropic shooting ray-trace algorithm for an anisotropic one. The model parameterization consists of Thomsen’s parameters ([Formula: see text], [Formula: see text], [Formula: see text]) and the tilt angle of the symmetry axis of the TI medium. The parsimonious anisotropic migration algorithm is successfully applied to synthetic data from a TTI version of the Marmousi2 model. The quality of the image improves by weighting the impulse response by the calculation of the anisotropic Fresnel radius. The accuracy and speed of this migration makes it useful for anisotropic velocity model building. The elapsed computing time for 101 shots for the Marmousi2 TTI model is 35 s per shot (each with 501 traces) in 32 Opteron cores.

Reverse time migration in tilted transversely isotropic (TTI) media

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. WCA179-WCA187 ◽  
Author(s):  
Robin P. Fletcher ◽  
Xiang Du ◽  
Paul J. Fowler
Keyword(s):  
Wave Equation ◽  
Numerical Solutions ◽  
Transversely Isotropic ◽  
Shear Velocity ◽  
Vertical Axis ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Axis Of Symmetry ◽  
Tti Media

Reverse time migration (RTM) exhibits great advantages over other imaging methods because it is based on computing numerical solutions to a two-way wave equation. It does not suffer from dip limitation like one-way downward continuation techniques do, thus enabling overturned reflections to be imaged. As well as correctly handling multipathing, RTM has the potential to image internal multiples when the boundaries responsible for generating the multiples are present in the model. In isotropic media, one can use a scalar acoustic wave equation for RTM of pressure data. In anisotropic media, P- and SV-waves are coupled together so, formally, elastic wave equations must be used for RTM. A new wave equation for P-waves is proposed in tilted transversely isotropic (TTI) media that can be solved as part of an acoustic anisotropic RTM algorithm, using standard explicit finite differencing. If the shear velocity along the axis of symmetry is set to zero, stable numerical solutions can be computed for media with a vertical axis of symmetry and [Formula: see text] not less than [Formula: see text]. In TTI media with rapid variations in the direction of the axis of symmetry, setting the shear velocity along the axis of symmetry to zero can cause numerical solutions to become unstable. A solution to this problem is proposed that involves using a small amount of nonzero shear velocity. The amount of shear velocity added is chosen to remove triplications from the SV wavefront and to minimize the anisotropic term of the SV reflection coefficient. We show modeling and high-quality RTM results in complex TTI media using this equation.

Reflection moveout approximations for P-waves in a moderately anisotropic homogeneous tilted transverse isotropy layer

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. C175-C185 ◽  
Author(s):  
Ivan Pšenčík ◽  
Véronique Farra
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Weak Anisotropy ◽  
P Waves ◽  
3D Space ◽  
Axis Of Symmetry ◽  
Relative Errors ◽  
Symmetry Tests ◽  
Ti Media

We have developed approximate nonhyperbolic P-wave moveout formulas applicable to weakly or moderately anisotropic media of arbitrary anisotropy symmetry and orientation. Instead of the commonly used Taylor expansion of the square of the reflection traveltime in terms of the square of the offset, we expand the square of the reflection traveltime in terms of weak-anisotropy (WA) parameters. No acoustic approximation is used. We specify the formulas designed for anisotropy of arbitrary symmetry for the transversely isotropic (TI) media with the axis of symmetry oriented arbitrarily in the 3D space. Resulting formulas depend on three P-wave WA parameters specifying the TI symmetry and two angles specifying the orientation of the axis of symmetry. Tests of the accuracy of the more accurate of the approximate formulas indicate that maximum relative errors do not exceed 0.3% or 2.5% for weak or moderate P-wave anisotropy, respectively.

scholarly journals An efficient wavefield inversion for transversely isotropic media with a vertical axis of symmetry

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. R195-R206 ◽  
Author(s):  
Chao Song ◽  
Tariq Alkhalifah
Keyword(s):  
High Resolution ◽  
Optimization Problem ◽  
Source Function ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Trade Off ◽  
Transversely Isotropic Media ◽  
Highly Nonlinear ◽  
Isotropic Media ◽  
Axis Of Symmetry

Conventional full-waveform inversion (FWI) aims at retrieving a high-resolution velocity model directly from the wavefields measured at the sensor locations resulting in a highly nonlinear optimization problem. Due to the high nonlinearity of FWI (manifested in one form in the cycle-skipping problem), it is easy to fall into local minima. Considering that the earth is truly anisotropic, a multiparameter inversion imposes additional challenges in exacerbating the null-space problem and the parameter trade-off issue. We have formulated an optimization problem to reconstruct the wavefield in an efficient matter with background models by using an enhanced source function (which includes secondary sources) in combination with fitting the data. In this two-term optimization problem to fit the wavefield to the data and to the background wave equation, the inversion for the wavefield is linear. Because we keep the modeling operator stationary within each frequency, we only need one matrix inversion per frequency. The inversion for the anisotropic parameters is handled in a separate optimization using the wavefield and the enhanced source function. Because the velocity is the dominant parameter controlling the wave propagation, it is updated first. Thus, this reduces undesired updates for anisotropic parameters due to the velocity update leakage. We find the effectiveness of this approach in reducing parameter trade-off with a distinct Gaussian anomaly model. We find that in using the parameterization [Formula: see text], and [Formula: see text] to describe the transversely isotropic media with a vertical axis of symmetry model in the inversion, we end up with high resolution and minimal trade-off compared to conventional parameterizations for the anisotropic Marmousi model. Application on 2D real data also indicates the validity of our method.

Common-Reflection-Point-Based Prestack Depth Migration for Imaging Lithosphere in Python: Application to the Dense Warramunga Array in Northern Australia

2020 ◽  
Vol 91 (5) ◽  
pp. 2890-2899 ◽  
Author(s):  
Weijia Sun ◽  
Brian L. N. Kennett
Keyword(s):  
Open Source ◽  
Receiver Function ◽  
Function Analysis ◽  
Vertical Component ◽  
P Wave ◽  
Northern Australia ◽  
Reflection Point ◽  
Prestack Depth Migration ◽  
Depth Migration ◽  
Conversion Point

Abstract We exploit estimates of P-wave reflectivity from autocorrelation of transmitted teleseismic P arrivals and their coda in a common reflection point (CRP) migration technique. The approach employs the same portion of the vertical-component seismogram, as in standard Ps receiver function analysis. This CRP prestack depth migration approach has the potential to image lithospheric structures on scales as fine as 4 km or less. The P-wave autocorrelation process and migration are implemented in open-source software—the autocorrelogram calculation (ACC) package, which builds on the widely used the seismological Obspy toolbox. The ACC package is written in the open-source and free Python programming language (3.0 or newer) and has been extensively tested in an Anaconda Python environment. The package is simple and friendly to use and runs on all major operating systems (e.g., Windows, macOS, and Linux). We utilize Python multiprocessing parallelism to speed up the ACC on a personal computer system, or servers, with multiple cores and threads. The application of the ACC package is illustrated with application to the closely spaced Warramunga array in northern Australia. The results show how fine-scale structures in the lithospheric can be effectively imaged at relatively high frequencies. The Moho ties well with conventional H−κ receiver analysis and deeper structure inferred from stacked autocorrelograms for continuous data. CRP prestack depth migration provides an important complement to common conversion point receiver function stacks, since it is less affected by surface multiples at lithospheric depths.

scholarly journals Frequency domain multiparameter acoustic inversion for transversely isotropic media with a vertical axis of symmetry

Geophysics ◽  
2019 ◽  
Vol 84 (1) ◽  
pp. C1-C14 ◽  
Author(s):  
Ramzi Djebbi ◽  
Tariq Alkhalifah
Keyword(s):  
Frequency Domain ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
The Other ◽  
Sensitivity Analyses ◽  
Data Set ◽  
Trade Off ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Axis Of Symmetry

Multiparameter full-waveform inversion for transversely isotropic media with a vertical axis of symmetry (VTI) suffers from the trade-off between the parameters. The trade-off results in the leakage of one parameter’s update into the other. It affects the accuracy and convergence of the inversion. The sensitivity analyses suggested a parameterization using the horizontal velocity [Formula: see text], Thomsen’s parameter [Formula: see text], and the anelliptic parameter [Formula: see text] to reduce the trade-off for surface recorded seismic data. We aim to invert for this parameterization using the scattering integral (SI) method. The available Born sensitivity kernels, within this approach, can be used to calculate additional inversion information. We mainly compute the diagonal of the approximate Hessian, used as a conjugate-gradient preconditioner, and the gradients’ step lengths. We consider modeling in the frequency domain. The large computational cost of the SI method can be avoided with direct Helmholtz equation solvers. We applied our method to the VTI Marmousi II model for various inversion strategies. We found that we can invert the [Formula: see text] accurately. For the [Formula: see text] parameter, only the short wavelengths are well-recovered. On the other hand, the [Formula: see text] parameter impact is weak on the inversion results and can be fixed. However, a good background [Formula: see text], with accurate long wavelengths, is needed to correctly invert for [Formula: see text]. Furthermore, we invert a real data set acquired by CGG from offshore Australia. We simultaneously invert all three parameters using our inversion approach. The velocity model is improved, and additional layers are recovered. We confirm the accuracy of the results by comparing them with well-log information, as well as looking at the data and angle gathers.

The effect of transverse isotropy on isotropic traveltime inversion of vertical seismic profile data—A modeling study

Geophysics ◽  
1990 ◽  
Vol 55 (9) ◽  
pp. 1235-1241 ◽  
Author(s):  
Jan Douma
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Vertical Axis ◽  
Seismic Profile ◽  
Isotropic Layer ◽  
Low Velocity ◽  
Traveltime Inversion ◽  
Axis Of Symmetry ◽  
Transversely Isotropic Layer

Traveltime inversion of multioffset VSP data can be used to determine the depths of the interfaces in layered media. Many inversion schemes, however, assume isotropy and consequently may introduce erroneous structures for anisotropic media. Synthetic traveltime data are computed for layered anisotropic media and inverted assuming isotropic layers. Only the interfaces between these layers are inverted. For a medium consisting of a horizontal isotropic low‐velocity layer on top of a transversely isotropic layer with a horizontal axis of symmetry (e.g., anisotropy due to aligned vertical cracks), 2-D isotropic inversion results in an anticline. For a given axis of symmetry the form of this anticline depends on the azimuth of the source‐borehole direction. The inversion result is a syncline (in 3-D a “bowl” structure), regardless of the azimuth of the source‐borehole direction for a vertical axis of symmetry of the transversely isotropic layer (e.g., anisotropy due to horizontal bedding).

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