scholarly journals Anisotropic P-wave travel-time tomography implementing Thomsen's weak approximation in TOMO3D

Solid Earth ◽  
2019 ◽  
Vol 10 (6) ◽  
pp. 1857-1876
Author(s):  
Adrià Meléndez ◽  
Clara Estela Jiménez ◽  
Valentí Sallarès ◽  
César R. Ranero
Keyword(s):  
Travel Time ◽  
Transverse Isotropy ◽  
P Wave ◽  
Weak Approximation ◽  
Weak Anisotropy ◽  
Simultaneous Inversion ◽  
Advantages And Disadvantages ◽  
Parameter Combination ◽  
Rough Approximation ◽  
Vertical Transverse Isotropy

Abstract. We present the implementation of Thomsen's weak anisotropy approximation for vertical transverse isotropy (VTI) media within TOMO3D, our code for 2-D and 3-D joint refraction and reflection travel-time tomographic inversion. In addition to the inversion of seismic P-wave velocity and reflector depth, the code can now retrieve models of Thomsen's parameters (δ and ε). Here, we test this new implementation following four different strategies on a canonical synthetic experiment in ideal conditions with the purpose of estimating the maximum capabilities and potential weak points of our modeling tool and strategies. First, we study the sensitivity of travel times to the presence of a 25 % anomaly in each of the parameters. Next, we invert for two combinations of parameters (v, δ, ε and v, δ, v⊥), following two inversion strategies, simultaneous and sequential, and compare the results to study their performance and discuss their advantages and disadvantages. Simultaneous inversion is the preferred strategy and the parameter combination (v, δ, ε) produces the best overall results. The only advantage of the parameter combination (v, δ, v⊥) is a better recovery of the magnitude of v. In each case, we derive the fourth parameter from the equation relating ε, v⊥ and v. Recovery of v, ε and v⊥ is satisfactory, whereas δ proves to be impossible to recover even in the most favorable scenario. However, this does not hinder the recovery of the other parameters, and we show that it is still possible to obtain a rough approximation of the δ distribution in the medium by sampling a reasonable range of homogeneous initial δ models and averaging the final δ models that are satisfactory in terms of data fit.

scholarly journals Anisotropic P-wave traveltime tomography implementing Thomsen's weak approximation in TOMO3D

2019 ◽  
Author(s):  
Adrià Meléndez ◽  
Clara Estela Jiménez ◽  
Valentí Sallarès ◽  
César R. Ranero
Keyword(s):  
P Wave ◽  
Weak Approximation ◽  
Weak Anisotropy ◽  
Traveltime Tomography ◽  
Simultaneous Inversion ◽  
Advantages And Disadvantages ◽  
Parameter Combination ◽  
P Wave Velocity ◽  
Rough Approximation ◽  
Vti Media

Abstract. We present the implementation of Thomsen's weak anisotropy approximation for VTI media within TOMO3D, our code for 2-D and 3-D joint refraction and reflection traveltime tomographic inversion. In addition to the inversion of seismic P-wave velocity and reflector depth, the code can now retrieve models of the Thomsen's parameters δ and ε. Here we test this new implementation following four different strategies on a canonical synthetic experiment. First, we study the sensitivity of traveltimes to the presence of a 25 % anomaly in each of the parameters. Next, we invert for two combinations of parameters, (v, δ, ε) and (v, δ, v⟂), following two inversion strategies, simultaneous and sequential, and compare the results to study their performances and discuss their advantages and disadvantages. Simultaneous inversion is the preferred strategy and the parameter combination (v, δ, ε) produces the best overall results. The only advantage of the parameter combination (v, δ, v⟂) is a better recovery of the magnitude of v. In each case we derive the fourth parameter from the equation relating ε, v⟂ and v. Recovery of v, ε and v⟂ is satisfactory whereas δ proves to be impossible to recover even in the most favorable scenario. However, this does not hinder the recovery of the other parameters, and we show that it is still possible to obtain a rough approximation of δ distribution in the medium by sampling a reasonable range of homogeneous initial δ models and averaging the final δ models that are satisfactory in terms of data fit.

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.

Analytic study of the geometrical spreading of P-waves in a layered transversely isotropic medium with a vertical symmetry axis

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1305-1315 ◽  
Author(s):  
Hongbo Zhou ◽  
George A. McMechan
Keyword(s):  
Isotropic Medium ◽  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Analytical Formula ◽  
Transversely Isotropic ◽  
P Wave ◽  
Geometrical Spreading ◽  
Transversely Isotropic Medium ◽  
Weak Anisotropy ◽  
Special Cases

An analytical formula for geometrical spreading is derived for a horizontally layered transversely isotropic medium with a vertical symmetry axis (VTI). With this expression, geometrical spreading can be determined using only the anisotropy parameters in the first layer, the traveltime derivatives, and the source‐receiver offset. Explicit, numerically feasible expressions for geometrical spreading are obtained for special cases of transverse isotropy (weak anisotropy and elliptic anisotropy). Geometrical spreading can be calculated for transversly isotropic (TI) media by using picked traveltimes of primary nonhyperbolic P-wave reflections without having to know the actual parameters in the deeper subsurface; no ray tracing is needed. Synthetic examples verify the algorithm and show that it is numerically feasible for calculation of geometrical spreading. For media with a few (4–5) layers, relative errors in the computed geometrical spreading remain less than 0.5% for offset/depth ratios less than 1.0. Errors that change with offset are attributed to inaccuracy in the expression used for nonhyberbolic moveout. Geometrical spreading is most sensitive to errors in NMO velocity, followed by errors in zero‐offset reflection time, followed by errors in anisotropy of the surface layer. New relations between group and phase velocities and between group and phase angles are shown in appendices.

scholarly journals Prior Analysis in Determination of TI Anisotropy Type through Well-Based Modelling: A Case Study on the Deep Water Environment

2021 ◽  
Vol 873 (1) ◽  
pp. 012102
Author(s):  
Madaniya Oktariena ◽  
Wahyu Triyoso ◽  
Fatkhan Fatkhan ◽  
Sigit Sukmono ◽  
Erlangga Septama ◽  
...  
Keyword(s):  
Wave Velocity ◽  
Deep Water ◽  
Initial Conditions ◽  
Transverse Isotropy ◽  
Water Environment ◽  
Horizontal Stress ◽  
P Wave ◽  
S Wave ◽  
Geomechanical Analysis ◽  
Vertical Transverse Isotropy

Abstract The existence of anisotropy phenomena in the subsurface will affect the image quality of seismic data. Hence a prior knowledge of the type of anisotropy is quite essential, especially when dealing with deep water targets. The preliminary result of the anisotropy of the well-based modelling in deep water exploration and development is discussed in this study. Anisotropy types are modelled for Vertical Transverse Isotropy (VTI) and Horizontal Transverse Isotropy (HTI) based on Thomsen Parameters of ε and γ. The parameters are obtained from DSI Logging paired with reference δ value for modelling. Three initial conditions are then analysed. The first assumption is isotropic, in which the P-Wave Velocity, S-Wave Velocity, and Density Log modelled at their in-situ condition. The second and third assumptions are anisotropy models that are VTI and HTI. In terms of HTI, the result shows that the model of CDP Gather in the offset domain has a weak distortion in Amplitude Variation with Azimuth (AVAz). However, another finding shows a relatively strong hockey effect in far offset, which indicates that the target level is a VTI dominated type. It is supported by the geomechanical analysis result in which vertical stress acts as the maximum principal axis while horizontal stress is close to isotropic one. To sum up, this prior anisotropy knowledge obtained based on this study could guide the efficiency guidance in exploring the deep water environment.

Nonhyperbolic reflection moveout for horizontal transverse isotropy

Geophysics ◽  
1998 ◽  
Vol 63 (5) ◽  
pp. 1738-1753 ◽  
Author(s):  
AbdulFattah Al‐Dajani ◽  
Ilya Tsvankin
Keyword(s):  
Transverse Isotropy ◽  
Single Layer ◽  
Azimuthal Velocity ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Analytic Description ◽  
Pure Mode ◽  
Quartic Term ◽  
Hti Media ◽  
Vertical Transverse Isotropy

The transversely isotropic model with a horizontal axis of symmetry (HTI) has been used extensively in studies of shear‐wave splitting to describe fractured formations with a single system of parallel vertical penny‐shaped cracks. Here, we present an analytic description of longspread reflection moveout in horizontally layered HTI media with arbitrary strength of anisotropy. The hyperbolic moveout equation parameterized by the exact normal‐moveout (NMO) velocity is sufficiently accurate for P-waves on conventional‐length spreads (close to the reflector depth), although the NMO velocity is not, in general, usable for converting time to depth. However, the influence of anisotropy leads to the deviation of the moveout curve from a hyperbola with increasing spread length, even in a single‐layer model. To account for nonhyperbolic moveout, we have derived an exact expression for the azimuthally dependent quartic term of the Taylor series traveltime expansion [t2(x2)] valid for any pure mode in an HTI layer. The quartic moveout coefficient and the NMO velocity are then substituted into the nonhyperbolic moveout equation of Tsvankin and Thomsen, originally designed for vertical transverse isotropy (VTI). Numerical examples for media with both moderate and uncommonly strong nonhyperbolic moveout show that this equation accurately describes azimuthally dependent P-wave reflection traveltimes in an HTI layer, even for spread lengths twice as large as the reflector depth. In multilayered HTI media, the NMO velocity and the quartic moveout coefficient reflect the influence of layering as well as azimuthal anisotropy. We show that the conventional Dix equation for NMO velocity remains entirely valid for any azimuth in HTI media if the group‐velocity vectors (rays) for data in a common‐midpoint (CMP) gather do not deviate from the vertical incidence plane. Although this condition is not exactly satisfied in the presence of azimuthal velocity variations, rms averaging of the interval NMO velocities represents a good approximation for models with moderate azimuthal anisotropy. Furthermore, the quartic moveout coefficient for multilayered HTI media can also be calculated with acceptable accuracy using the known averaging equations for vertical transverse isotropy. This allows us to extend the nonhyperbolic moveout equation to horizontally stratified media composed of any combination of isotropic, VTI, and HTI layers. In addition to providing analytic insight into the behavior of nonhyperbolic moveout, these results can be used in modeling and inversion of reflection traveltimes in azimuthally anisotropic media.

An effective anisotropy parameter in transversely isotropic media

Geophysics ◽  
1987 ◽  
Vol 52 (12) ◽  
pp. 1654-1664 ◽  
Author(s):  
N. C. Banik
Keyword(s):  
Transverse Isotropy ◽  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
P Wave ◽  
Weak Anisotropy ◽  
Reflection Seismic ◽  
The North ◽  
Reflection Amplitude ◽  
Transversely Isotropic Media ◽  
Isotropic Media

An interesting physical meaning is presented for the anisotropy parameter δ, previously introduced by Thomsen to describe weak anisotropy in transversely isotropic media. Roughly, δ is the difference between the P-wave and SV-wave anisotropies of the medium. The observed systematic depth errors in the North Sea are reexamined in view of the new interpretation of the moveout velocity through δ. The changes in δ at an interface adequately describe the effects of transverse isotropy on the P-wave reflection amplitude, The reflection coefficient expression is linearized in terms of changes in elastic parameters. The linearized expression clearly shows that it is the variation of δ at the interface that gives the anisotropic effects at small incidence angles. Thus, δ effectively describes both the moveout velocity and the reflection amplitude variation, two very important pieces of information in reflection seismic prospecting, in the presence of transverse isotropy.

scholarly journals Reducing Residual Moveout for Long Offset Data in VTI Media Using Padé Approximation

2019 ◽  
Vol 125 ◽  
pp. 15005
Author(s):  
Indah Nur Pratiwi ◽  
Mohammad Syamsu Rosid ◽  
Humbang Purba
Keyword(s):  
Travel Time ◽  
Transverse Isotropy ◽  
Padé Approximation ◽  
Three Dimensional ◽  
Synthetic Data ◽  
Single Layer ◽  
Pade Approximation ◽  
Depth Ratio ◽  
Long Offset ◽  
Vertical Transverse Isotropy

Modification of the hyperbolic travel time equation into non-hyperbolic travel time equation is important to increase the reduction residual moveout for long offset data. Some researchers have modified hyperbolic travel time equation into a non-hyperbolic travel time equation to obtain a more accurate value NMO velocity and parameter an-ellipticity or etha on the large offset to depth ratio (ODR) so that the residual moveout value is smaller mainly in large offset to depth ratio. The aims of research is to increase the reduction value of error residue at long offset data using Padé approximation then compare with several approximations. The method used in this study is to conduct forward modeling of the subsurface coating structure. The results of the three-dimensional analysis show that the Padé approximation has the best accuracy compared to the other travel time equations for ODR value up to 4 with an-ellipticity parameter is varying from 0 to 0.5. Testing of synthetic data for single layer on vertical transverse isotropy (VTI) medium obtained the maximum residual error value produced by the Padé approximation is 0.25% in ODR=4. Therefore, Padé approximation is better than other methods for reducing residual moveout.

P-wave reflection-moveout approximation for horizontally layered media of arbitrary moderate anisotropy

Geophysics ◽  
2020 ◽  
Vol 85 (2) ◽  
pp. C61-C70 ◽  
Author(s):  
Véronique Farra ◽  
Ivan Pšenčík
Keyword(s):  
Coordinate System ◽  
Transverse Isotropy ◽  
Local Coordinate System ◽  
Surface Profile ◽  
Taylor Series Expansion ◽  
Layered Media ◽  
P Wave ◽  
Orthorhombic Symmetry ◽  
Weak Anisotropy ◽  
Cartesian Coordinate

We present an approximate nonhyperbolic P-wave moveout formula applicable to horizontally layered media of moderate or weak anisotropy of arbitrary symmetry and orientation. Anisotropy symmetry and its orientation may differ from layer to layer. Instead of commonly used Taylor-series expansion of the square of the reflection traveltime in terms of the square of the offset, we use the weak-anisotropy approximation, in which the square of the reflection traveltime is expanded in terms of weak-anisotropy (WA) parameters. The resulting formula is simple, and it provides a transparent relation between the traveltimes and WA parameters. Along an arbitrarily chosen single surface profile, it depends, in each layer, on the thickness of the layer, on the reference P-wave velocity used for the construction of reference rays, and on three WA parameters specified in the Cartesian coordinate system related to the profile. In each layer, these three “profile” WA parameters depend on “local” WA parameters specifying anisotropy of a given layer in a local coordinate system and on directional cosines specifying the orientation of the local coordinate system with respect to the profile one. The number of local P-wave WA parameters may vary from three for transverse isotropy or six for orthorhombic symmetry to nine for triclinic symmetry. Our tests of the accuracy indicate that the maximum relative traveltime errors do not exceed 0.5% or 2.5% for weak or moderate P-wave anisotropy, respectively.

Fowler DMO and time migration for transversely isotropic media

Geophysics ◽  
1996 ◽  
Vol 61 (3) ◽  
pp. 835-845 ◽  
Author(s):  
John Anderson ◽  
Tariq Alkhalifah ◽  
Ilya Tsvankin
Keyword(s):  
Transverse Isotropy ◽  
Computing Time ◽  
Transversely Isotropic ◽  
P Wave ◽  
Effective Parameters ◽  
Weak Anisotropy ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Zero Offset

The main advantage of Fowler’s dip‐moveout (DMO) method is the ability to perform velocity analysis along with the DMO removal. This feature of Fowler DMO becomes even more attractive in anisotropic media, where imaging methods are hampered by the difficulty in reconstructing the velocity field from surface data. We have devised a Fowler‐type DMO algorithm for transversely isotropic media using the analytic expression for normal‐moveout velocity. The parameter‐estimation procedure is based on the results of Alkhalifah and Tsvankin showing that in transversely isotropic media with a vertical axis of symmetry (VTI) the P‐wave normal‐moveout (NMO) velocity as a function of ray parameter can be described fully by just two coefficients: the zero‐dip NMO velocity [Formula: see text] and the anisotropic parameter η (η reduces to the difference between Thomsen parameters ε and δ in the limit of weak anisotropy). In this extension of Fowler DMO, resampling in the frequency‐wavenumber domain makes it possible to obtain the values of [Formula: see text] and η by inspecting zero‐offset (stacked) panels for different pairs of the two parameters. Since most of the computing time is spent on generating constant‐velocity stacks, the added computational effort caused by the presence of anisotropy is relatively minor. Synthetic and field‐data examples demonstrate that the isotropic Fowler DMO technique fails to generate an accurate zero‐offset section and to obtain the zero‐dip NMO velocity for nonelliptical VTI models. In contrast, this anisotropic algorithm allows one to find the values of the parameters [Formula: see text] and η (sufficient to perform time migration as well) and to correct for the influence of transverse isotropy in the DMO processing. When combined with poststack F-K Stolt migration, this method represents a complete inversion‐processing sequence capable of recovering the effective parameters of transversely isotropic media and producing migrated images for the best‐fit homogeneous anisotropic model.

scholarly journals Teleseismic P-wave travel time and amplitude anomalies observed in Hokkaido region, Japan.

1990 ◽  
Vol 38 (2) ◽  
pp. 163-177 ◽  
Author(s):  
Ichiro Nakanishi ◽  
Yoshinobu Motoya
Keyword(s):  
Travel Time ◽  
P Wave ◽  
Wave Travel
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