Velocity-independent layer stripping of PP and PS reflection traveltimes

Geophysics ◽  
2006 ◽  
Vol 71 (4) ◽  
pp. U59-U65 ◽  
Author(s):  
Pawan Dewangan ◽  
Ilya Tsvankin
Keyword(s):  
Symmetry Plane ◽  
Transversely Isotropic ◽  
P Wave ◽  
Isotropic Layer ◽  
Processing Parameter ◽  
Amplitude Variation With Offset ◽  
Interval Velocity ◽  
Velocity Models ◽  
Potential Applications ◽  
Stable Computation

Building accurate interval velocity models is critically important for seismic imaging and AVO (amplitude variation with offset) analysis. Here, we adapt the [Formula: see text] method to develop an exact technique for constructing the interval traveltime-offset function in a target zone beneath a horizontally layered overburden. All layers in the model can be anisotropic, with an essential assumption that the overburden has a horizontal symmetry plane (i.e., up-down symmetry). Our layer-stripping algorithm is entirely data-driven and, in contrast to the generalized Dix equations, does not require knowledge of the velocity field anywhere in the medium. Important advantages of our approach compared to the Dix-style formalism also include the ability to handle mode-converted waves, long-offset data, and laterally heterogeneous target layers with multiple, curved reflectors. Numerical tests confirm the high accuracy of the algorithm in computing the interval traveltimes of both PP- and PS-waves in a dipping, transversely isotropic layer with a tilted symmetry axis (TTI medium) beneath an anisotropic overburden. In combination with the inversion techniques developed for homogeneous TTI models, the proposed layer stripping of PP and PS data can be used to estimate the interval parameters of TTI formations in such important exploration areas as the Canadian Foothills. Potential applications of this methodology also include the dip-moveout inversion for the P-wave time-processing parameter [Formula: see text] and stable computation of the interval long-spread (nonhyperbolic) moveout for purposes of anisotropic velocity analysis.

Estimation of interval anisotropy parameters using velocity-independent layer stripping

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. WB117-WB127 ◽  
Author(s):  
Xiaoxiang Wang ◽  
Ilya Tsvankin
Keyword(s):  
Symmetry Plane ◽  
Synthetic Data ◽  
Single Layer ◽  
Transversely Isotropic ◽  
P Wave ◽  
Correlated Noise ◽  
Fractured Reservoir ◽  
Processing Parameter ◽  
Reflection Data ◽  
Error Amplification

Moveout analysis of long-spread P-wave data is widely used to estimate the key time-processing parameter [Formula: see text] in layered transversely isotropic media with a vertical symmetry axis (VTI). Inversion for interval [Formula: see text] values, however, suffers from instability caused by the trade-off between the effective moveout parameters and by subsequent error amplification during Dix-type layer stripping. We propose an alternative approach to nonhyperbolic moveout inversion based on the velocity-independent layer-stripping (VILS) method of Dewangan and Tsvankin. Also, we develop the 3D version of VILS and apply it to interval parameter estimation in orthorhombic media using wide-azimuth, long-spread data. If the overburden is laterally homogeneous and has a horizontal symmetry plane, VILS produces the exact interval traveltime-offset function in the target layer without knowledgeof the velocity field. Hence, Dix-type differentiation of moveout parameters used in existing techniques is replaced by the much more stable layer stripping of reflection traveltimes. The interval traveltimes are then inverted for the moveout parameters using the single-layer nonhyperbolic moveout equation. The superior accuracy and stability of the algorithm are illustrated on ray-traced synthetic data for typical VTI and orthorhombic models. Even small correlated noise in reflection traveltimes causes substantial distortions in the interval [Formula: see text] values computed by conventional Dix-type differentiation. In contrast, the output of VILS is insensitive to mild correlated traveltime errors. The algorithm is also tested on wide-azimuth P-wave reflection data recorded above a fractured reservoir at Rulison field in Colorado. The interval moveout parameters estimated by VILS in the shale layer above the reservoir are more plausible and less influenced by noise than those obtained by the Dix-type method.

An anelliptic approximation for geometric spreading in transversely isotropic and orthorhombic media

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. C37-C47 ◽  
Author(s):  
Shibo Xu ◽  
Alexey Stovas ◽  
Yanadet Sripanich
Keyword(s):  
Symmetry Plane ◽  
Transversely Isotropic ◽  
Seismic Wave Propagation ◽  
P Wave ◽  
Effective Parameters ◽  
Amplitude Variation With Offset ◽  
Acoustic Anisotropy ◽  
Geometric Spreading ◽  
Parametric Expression ◽  
Data Processing Methods

The relative geometric spreading along the raypath contributes to the amplitude decay of the seismic wave propagation that needs to be considered for amplitude variation with offset or other seismic data processing methods that require the true amplitude processing. Expressing the P-wave geometric spreading factor in terms of the offset-traveltime-based parameters is a more practical and convenient way because these parameters can be estimated from the nonhyperbolic velocity analysis. We have developed an anelliptic approximation for the relative geometric spreading of P-wave in a homogeneous transversely isotropic medium with vertical symmetry axis (VTI) and an orthorhombic (ORT) medium under the acoustic anisotropy assumption. The coefficients in our approximation are only defined within the symmetry planes and computed from fitting with the exact parametric expression. For an ORT model, due to the symmetric behavior in different symmetry planes, the other coefficients in the approximation can be easily obtained by making corresponding changes in indices from the computed coefficients in one symmetry plane. From the numerical examples, we found that for a homogeneous VTI model, the anelliptic approximation is more accurate than the generalized nonhyperbolic moveout approximation form for larger offset. For a homogeneous ORT model, our anelliptic approximation is more accurate than its traveltime-based counterparts. Using the Dix-type equations for the effective parameters, our anelliptic form approximation is extended to a multilayered VTI and ORT models and has accurate results in both models.

scholarly journals Analysis of fluid substitution in a porous and fractured medium

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. WA157-WA166 ◽  
Author(s):  
Samik Sil ◽  
Mrinal K. Sen ◽  
Boris Gurevich
Keyword(s):  
Water Saturation ◽  
Transversely Isotropic ◽  
P Wave ◽  
Reflection Coefficients ◽  
Fluid Substitution ◽  
Fracture Density ◽  
Maximum Increase ◽  
Amplitude Variation With Offset ◽  
Fractured Medium ◽  
Medium Change

To improve quantitative interpretation of seismic data, we analyze the effect of fluid substitution in a porous and fractured medium on elastic properties and reflection coefficients. This analysis uses closed-form expressions suitable for fluid substitution in transversely isotropic media with a horizontal symmetry axis (HTI). For the HTI medium, the effect of changing porosity and water saturation on (1) P-wave moduli, (2) horizontal and vertical velocities, (3) anisotropic parameters, and (4) reflection coefficients are examined. The effects of fracture density on these four parameters are also studied. For the model used in this study, a 35% increase in porosity lowers the value of P-wave moduli by maximum of 45%. Consistent with the reduction in P-wave moduli, P-wave velocities also decrease by maximum of 17% with a similar increment in porosity. The reduction is always larger for the horizontal P-wave modulus than for the vertical one and is nearly independent of fracture density. The magnitude of the anisotropic parameters of the fractured medium also changes with increased porosity depending on the changes in the value of P-wave moduli. The reflection coefficients at an interface of the fractured medium with an isotropic medium change in accordance with the above observations and lead to an increase in anisotropic amplitude variation with offset (AVO) gradient with porosity. Additionally, we observe a maximum increase in P-wave modulus and velocity by 30% and 8%, respectively, with a 100% increase in water saturation. Water saturation also changes the anisotropic parameters and reflection coefficients. Increase in water saturation considerably increases the magnitude of the anisotropic AVO gradient irrespective of fracture density. From this study, we conclude that porosity and water saturation have a significant impact on the four studied parameters and the impacts are seismically detectable.

Seismic traveltime analysis for azimuthally anisotropic media: Theory and experiment

Geophysics ◽  
1997 ◽  
Vol 62 (5) ◽  
pp. 1570-1582 ◽  
Author(s):  
Colin M. Sayers ◽  
Daniel A. Ebrom
Keyword(s):  
Model Simulation ◽  
Symmetry Plane ◽  
Transversely Isotropic ◽  
Bedding Plane ◽  
Seismic Profile ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Principal Axes ◽  
Physical Model Simulation ◽  
Single Set

Natural fractures in reservoirs, and in the caprock overlying the reservoir, play an important role in determining fluid flow during production. The density and orientation of sets of fractures is therefore of great interest. Rocks possessing an anisotropic fabric and a preferred orientation of fractures display both polar and azimuthal anisotropy. Sedimentary rocks containing several sets of vertical fractures may be approximated as having monoclinic symmetry with symmetry plane parallel to the layers if, in the absence of fractures, the rock is transversely isotropic with symmetry axis perpendicular to the bedding plane. A nonhyperbolic traveltime equation, which can be used in the presence of azimuthally anisotropic layered media, can be obtained from an expansion of the inverse‐squared ray velocity in spherical harmonics. For a single set of aligned fractures, application of this equation to traveltime data acquired at a sufficient number of azimuths allows the strike of the fractures to be estimated. Analysis of the traveltimes measured in a physical model simulation of a reverse vertical seismic profile in an azimuthally anisotropic medium shows the medium to be orthorhombic with principal axes in agreement with those given by an independent shear‐wave experiment. In contrast to previous work, no knowledge of the orientation of the symmetry planes is required. The method is therefore applicable to P‐wave data collected at multiple azimuths using multiple offset vertical seismic profiling (VSP) techniques.

Analytic calculation of phase and group velocities of P-waves in orthorhombic media

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. C79-C97 ◽  
Author(s):  
Qi Hao ◽  
Alexey Stovas
Keyword(s):  
Transversely Isotropic ◽  
P Wave ◽  
P Waves ◽  
Type Approximation ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Potential Applications ◽  
Phase And Group Velocities ◽  
And Migration ◽  
Group Velocities

We have developed an approximate method to calculate the P-wave phase and group velocities for orthorhombic media. Two forms of analytic approximations for P-wave velocities in orthorhombic media were built by analogy with the five-parameter moveout approximation and the four-parameter velocity approximation for transversely isotropic media, respectively. They are called the generalized moveout approximation (GMA)-type approximation and the Fomel approximation, respectively. We have developed approximations for elastic and acoustic orthorhombic media. We have characterized the elastic orthorhombic media in Voigt notation, and we can describe the acoustic orthorhombic media by introducing the modified Alkhalifah’s notation. Our numerical evaluations indicate that the GMA-type and Fomel approximations are accurate for elastic and acoustic orthorhombic media with strong anisotropy, and the GMA-type approximation is comparable with the approximation recently proposed by Sripanich and Fomel. Potential applications of the proposed approximations include forward modeling and migration based on the dispersion relation and the forward traveltime calculation for seismic tomography.

Effects of transverse isotropy on P‐wave AVO for gas sands

Geophysics ◽  
1993 ◽  
Vol 58 (6) ◽  
pp. 883-888 ◽  
Author(s):  
Ki Young Kim ◽  
Keith H. Wrolstad ◽  
Fred Aminzadeh
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Velocity Anisotropy ◽  
Special Cases ◽  
Class 1 ◽  
Zero Offset ◽  
Ti Media

Velocity anisotropy should be taken into account when analyzing the amplitude variation with offset (AVO) response of gas sands encased in shales. The anisotropic effects on the AVO of gas sands in transversely isotropic (TI) media are reviewed. Reflection coefficients in TI media are computed using a planewave formula based on ray theory. We present results of modeling special cases of exploration interest having positive reflectivity, near‐zero reflectivity, and negative reflectivity. The AVO reflectivity in anisotropic media can be decomposed into two parts; one for isotropy and the other for anisotropy. Zero‐offset reflectivity and Poisson’s ratio contrast are the most significant parameters for the isotropic component while the δ difference (Δδ) between shale and gas sand is the most important factor for the anisotropic component. For typical values of Tl anisotropy in shale (positive δ and ε), both δ difference (Δδ) and ε difference (Δε) amplify AVO effects. For small angles of incidence, Δδ plays an important role in AVO while Δε dominates for large angles of incidence. For typical values of δ and ε, the effects of anisotropy in shale are: (1) a more rapid increase in AVO for Class 3 and Class 2 gas sands, (2) a more rapid decrease in AVO for Class 1 gas sands, and (3) a shift in the offset of polarity reversal for some Class 1 and Class 2 gas sands.

3-D description of normal moveout in anisotropic inhomogeneous media

Geophysics ◽  
1998 ◽  
Vol 63 (3) ◽  
pp. 1079-1092 ◽  
Author(s):  
Vladimir Grechka ◽  
Ilya Tsvankin
Keyword(s):  
Fractured Reservoirs ◽  
Symmetry Plane ◽  
Transversely Isotropic ◽  
Inhomogeneous Media ◽  
P Wave ◽  
Orthorhombic Symmetry ◽  
Azimuthal Variation ◽  
Anisotropic Models ◽  
Transversely Isotropic Media ◽  
Isotropic Media

We present a new equation for normal‐moveout (NMO) velocity that describes azimuthally dependent reflection traveltimes of pure modes from both horizontal and dipping reflectors in arbitrary anisotropic inhomogeneous media. With the exception of anomalous areas such as those where common‐midpoint (CMP) reflection time decreases with offset, the azimuthal variation of NMO velocity represents an ellipse in the horizontal plane, with the orientation of the axes determined by the properties of the medium and the direction of the reflector normal. In general, a minimum of three azimuthal measurements is necessary to reconstruct the best‐fit ellipse and obtain NMO velocity in all azimuthal directions. This result provides a simple way to correct for the azimuthal variation in stacking velocity often observed in 3-D surveys. Even more importantly, analytic expressions for the parameters of the NMO ellipse can be used in the inversion of moveout data for the anisotropic coefficients of the medium. For homogeneous transversely isotropic media with a vertical axis of symmetry (VTI media), our equation for azimuthally dependent NMO velocity from dipping reflectors becomes a relatively simple function of phase velocity and its derivatives. We show that the zero‐dip NMO velocity Vnmo(0) and the anisotropic coefficient η are sufficient to describe the P-wave NMO velocity for any orientation of the CMP line with respect to the dip plane of the reflector. Using our formalism, Vnmo(0) and η (the only parameters needed for time processing) can be found from the dip‐dependent NMO velocity at any azimuth or, alternatively, from the azimuthally dependent NMO for a single dipping reflector. We also apply this theory to more complicated azimuthally anisotropic models with the orthorhombic symmetry used to describe fractured reservoirs. For reflections from horizontal interfaces in orthorhombic media, the axes of the normal moveout ellipse are aligned with the vertical symmetry planes. Therefore, azimuthal P-wave moveout measurements can be inverted for the orientation of the symmetry planes (typically determined by the fracture direction) and the NMO velocities within them. If the vertical velocity is known, symmetry‐plane NMO velocities make it possible to estimate two anisotropic parameters equivalent to Thomsen’s coefficient δ for transversely isotropic media.

Common-reflection-point migration velocity analysis of 2D P-wave data from TTI media

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. C65-C79 ◽  
Author(s):  
Ernesto V. Oropeza ◽  
George A. McMechan
Keyword(s):  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Migration Velocity ◽  
P Wave ◽  
Velocity Analysis ◽  
Reflection Point ◽  
Axis Orientation ◽  
Velocity Models ◽  
Migration Velocity Analysis ◽  
Free Data

We have developed a common-reflection-point (CRP)-based kinematic migration velocity analysis for 2D P-wave reflection data to estimate the four transversely isotropic (TI) parameters [Formula: see text], [Formula: see text], and [Formula: see text], and the tilt angle [Formula: see text] of the symmetry axis in a TI medium. In each iteration, the tomographic parameter was updated alternately with prestack anisotropic ray-based migration. Iterations initially used layer stripping to reduce the number of degrees of freedom; after convergence was reached, a couple of more iterations over all parameters and all CRPs ensured global interlayer coupling and parameter interaction. The TI symmetry axis orientation was constrained to be locally perpendicular to the reflectors. The [Formula: see text] dominated the inversion, and so it was weighted less than [Formula: see text] and [Formula: see text] in the parameter updates. Estimates of [Formula: see text] and [Formula: see text] were influenced if the error in [Formula: see text] was [Formula: see text]; estimates of [Formula: see text] were also influenced if the error in [Formula: see text] was [Formula: see text]. Examples included data for a simple model with a homogeneous TI layer whose dips allowed recovery of all anisotropy parameters from noise-free data, and a more realistic model (the BP tilted transversely isotropic (TTI) model) for which only [Formula: see text], [Formula: see text], and [Formula: see text] were recoverable. The adequacy of the traveltimes predicted by the inverted anisotropic models was tested by comparing migrated images and common image gathers, with those produced using the known velocity models.

Generalized Dix equations for a layered transversely isotropic medium

Geophysics ◽  
2005 ◽  
Vol 70 (6) ◽  
pp. D77-D81 ◽  
Author(s):  
Bjørn Ursin ◽  
Alexey Stovas
Keyword(s):  
Initial Conditions ◽  
Transversely Isotropic ◽  
Normal Incidence ◽  
P Wave ◽  
Ocean Bottom ◽  
Sh Wave ◽  
S Wave ◽  
Interval Velocity ◽  
Wave Normal ◽  
Horizontal Layers

In standard Dix equations, normal-incidence traveltimes and rms or NMO velocities for PP reflections are used to estimate interval velocity and thickness for homogeneous isotropic horizontal layers. For a stack of homogeneous horizontal layers that are transversely isotropic with a vertical symmetry axis (VTI), the qPqP traveltimes (and the qPqSV traveltimes) depend on four parameters and the thickness of each layer. These can be estimated from the traveltimes of the qPqP and qPqSV reflected waves. The estimated qPqP traveltime parameters are P-wave normal-incidence traveltime, P-wave NMO velocity, and a heterogeneity factor entering in the shifted hyperbolic traveltime approximation or in a continued fraction traveltime approximation. The estimated qPqSV traveltime parameters are PS normal-incidence traveltime and PS NMO velocity combined with the qPqP traveltime parameters. These are used to compute S-wave normal-incidence traveltime and SV-wave NMO velocity. The estimated traveltime parameters are entered into explicit expressions for the four layer parameters and the thickness of each layer. These layer-recursive formulas, except for slightly different initial conditions, are also valid for ocean-bottom seismic data. For reflected SH-wave traveltimes, there are two elastic parameters and thicknesses per layer, and we can estimate three traveltime parameters. However, two of these traveltime parameters are functionally related, and it is impossible to estimate the three layer parameters from SH-wave traveltimes only.

Velocity analysis for tilted transversely isotropic media: A physical modeling example

Geophysics ◽  
2001 ◽  
Vol 66 (3) ◽  
pp. 904-910 ◽  
Author(s):  
Vladimir Grechka ◽  
Andres Pech ◽  
Ilya Tsvankin ◽  
Baoniu Han
Keyword(s):  
Physical Modeling ◽  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Velocity Analysis ◽  
Isotropic Layer ◽  
Anisotropic Parameter ◽  
Data Set ◽  
Symmetry Direction

Transverse isotropy with a tilted symmetry axis (TTI media) has been recognized as a common feature of shale formations in overthrust areas, such as the Canadian Foothills. Since TTI layers cause serious problems in conventional imaging, it is important to be able to reconstruct the velocity model suitable for anisotropic depth migration. Here, we discuss the results of anisotropic parameter estimation on a physical‐modeling data set. The model represents a simplified version of a typical overthrust section from the Alberta Foothills, with a horizontal reflector overlaid by a bending transversely isotropic layer. Assuming that the TTI layer is homogeneous and the symmetry axis stays perpendicular to its boundaries, we invert P-wave normal‐moveout (NMO) velocities and zero‐offset traveltimes for the symmetry‐direction velocity V0 and the anisotropic parameters ε and δ. The coefficient ε is obtained using the traveltimes of a wave that crosses a dipping TTI block and reflects from the bottom of the model. The inversion for ε is based on analytic expressions for NMO velocity in media with intermediate dipping interfaces. Our estimates of both anisotropic coefficients are close to their actual values. The errors in the inversion, which are associated primarily with the uncertainties in picking the NMO velocities and traveltimes, can be reduced by a straighforward modification of the acquisition geometry. It should be emphasized that the moveout inversion also gives an accurate estimate of the thickness of the TTI layer, thus reconstructing the correct depth scale of the section. Although the physical model used here was relatively simple, our results demonstrate the principal feasibility of anisotropic velocity analysis and imaging in overthrust areas. The main problems in anisotropic processing for TTI models are likely to be caused by the lateral variation of the velocity field and overall structural complexity.

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