Moveout Formula Research Articles

PS reflection moveout in a homogeneous anisotropic layer of arbitrary symmetry and tilt

Use of unconverted reflected PP waves leads to moveout formulae, which relate the moveout to only a limited number of parameters specifying anisotropy. Use of converted reflected PS waves extends the number of parameters to, generally, a complete set of twenty-one parameters. It leads, on the other hand, to more complicated formulae and phenomena unknown from studies of unconverted PP waves. In anisotropic media one has to deal with the existence of two S waves and their singularities, acoustic approximation cannot be used. We present and test approximate formulae for the reflection moveout of PS waves converted at a horizontal reflector underlying a homogeneous layer of arbitrary anisotropy symmetry and orientation. For their derivation, we use the combination of weak-anisotropy approximation and A-parameters specifying anisotropy. The complete set of twenty-one A-parameters represents an alternative to twenty-one independent elements of the stiffness tensor. In addition to the moveout formulae for separate PS1 and PS2 waves, we also present moveout formula for a P to common S wave whose traveltimes represent an average of traveltimes of PS1 and PS2 waves. Use of the P to common S wave reflection leads to several simplifications. Presented tests indicate that for most offsets and models with P-wave anisotropy up-to to 26% and S-wave anisotropy over 25%, the relative traveltime errors do not exceed 2.5%.

Estimation of parameters and thickness of a horizontal layer of arbitrary anisotropy from P-wave moveout

We present an inversion scheme to estimate anisotropy parameters and thickness of a homogeneous layer of arbitrary anisotropy from traveltimes of a reflected P wave. The scheme is based on the weak-anisotropy approximation of P-wave moveout formula. For the description of anisotropy, we use the so-called A-parameters, which represent an alternative to standard elastic moduli specifying anisotropic media, and allow specification of anisotropy of any symmetry, orientation and strength. The proposed scheme allows estimation of, at most, nine of fifteen A-parameters, which describe P-wave propagation in the weak-anisotropy approximation. Therefore, generally, the reconstruction of the P-wave phase-velocity surface from the estimated parameters is impossible. If we deal, however, with higher-symmetry anisotropy (transversely isotropic, orthorhombic) and if information about the orientation of its symmetry elements is available, reconstruction of the phase-velocity surface is possible. Synthetic tests indicate that the inversion scheme works well even in situations, in which the use of other commonly used moveout formulae leads to unsatisfactory results.

P-wave reflection-moveout approximation for horizontally layered media of tilted weak-to-moderate orthorhombic symmetry

An approximate nonhyperbolic P-wave moveout formula applicable to horizontally layered media of weak-to-moderate orthorhombic or higher-symmetry anisotropy of arbitrary orientation is presented and tested. Weak-anisotropy approximation is used for the derivation of the formula, in which the square of the reflection traveltime is expanded in terms of weak-anisotropy (WA) parameters specifying the anisotropy of the layers. Due to the use of the weak-anisotropy approximation, the proposed formula is applicable to any offset. Its accuracy decreases with increasing strength of anisotropy. The relation between traveltimes and WA parameters in the formula is transparent and relatively simple. For any offset along an arbitrarily chosen single surface profile, the formula depends in each layer on its thickness, on the auxiliary reference P-wave velocity, on, at most, six WA parameters describing P-wave orthorhombic or higher-symmetry anisotropy, and on three Euler angles specifying the orientation of symmetry elements in the layer. Performed tests indicate that the maximum relative traveltime error does not exceed 2.5% for P-wave anisotropy about 25%. Often, the errors are considerably lower.

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

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.

Reflection moveout approximation for a P-SV wave in a moderately anisotropic homogeneous vertical transverse isotropic layer

A description of the subsurface is incomplete without the use of S-waves. Use of converted waves is one way to involve S-waves. We have developed and tested an approximate formula for the reflection moveout of a wave converted at a horizontal reflector underlying a homogeneous transversely isotropic layer with the vertical axis of symmetry. For its derivation, we use the weak-anisotropy approximation; i.e., we expand the square of the reflection traveltime in terms of weak-anisotropy (WA) parameters. Traveltimes are calculated along reference rays of converted reflected waves in a reference isotropic medium. This requires the determination of the point of reflection (the conversion point) of the reference ray, at which the conversion occurs. This can be done either by a numerical solution of a quartic equation or by using a simple approximate solution. Presented tests indicate that the accuracy of the proposed moveout formula is comparable with the accuracy of formulas derived in a weak-anisotropy approximation for pure-mode reflected waves. Specifically, the tests indicate that the maximum relative traveltime errors are well below 1% for models with P- and SV-wave anisotropy of approximately 10% and less than 2% for models with P- and SV-wave anisotropy of 25% and 12%, respectively. For isotropic media, the use of the conversion point obtained by numerical solution of the quartic equation yields exact results. The approximate moveout formula is used for the derivation of approximate expressions for the two-way zero-offset traveltime, the normal moveout velocity and the quartic term of the Taylor series expansion of the squared traveltime.

Moveout approximation for a P-SV wave in a moderately anisotropic homogeneous DTI layer

We derive and test the moveout formula for converted P-SV (or SV-P) waves in a homogeneous, weakly or moderately anisotropic, transversely isotropic layer with axis of symmetry perpendicular to a dipping reflector underlying it. Instead of commonly used Taylor series expansion of the squared traveltime with respect to the offset, we use weak-anisotropy approximation of the exact traveltime formula. We replace the exact traveltime along the actual ray of a converted wave by its first-order weak-anisotropy approximation along a ray in a nearby reference isotropic medium. The traveltime formula is derived in the plane defined by the source-receiver line and the normal to the reflector. The source and the receiver may be situated arbitrarily in the model, the flat reflector may have an arbitrary 3D orientation. The accuracy of the proposed moveout formula is tested on models with varying strength of anisotropy for a varying dip of the reflector. In isotropic media, the formula yields highly accurate results with relative traveltime errors not exceeding 0.5%. In anisotropic media of P- and SV-wave anisotropy of about 25% and 12%, respectively, relative traveltime errors do not exceed 2%.

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

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.

Weak-anisotropy moveout approximations for P-waves in homogeneous TOR layers

We have developed approximate P-wave reflection moveout formulas for a homogeneous layer of orthorhombic symmetry overlying a horizontal or dipping reflector. The symmetry planes of an orthorhombic medium in the layer can be arbitrarily oriented. The formulas are based on the weak-anisotropy approximation. Thus, their accuracy depends not only on the strength of anisotropy, but also on the deviations of the phase and ray velocities. Nevertheless, the performed tests indicate that for P-wave anisotropy of approximately 25%, the maximum relative traveltime errors of the formulas do not exceed 3%. The formulas are expressed in terms of six P-wave weak-anisotropy parameters and three Euler angles specifying the orientation of the symmetry planes of the orthorhombic medium. In the case of a dipping reflector, the formulas also depend on the apparent dip angle of the reflector.

Weak-anisotropy moveout approximations for P-waves in homogeneous layers of monoclinic or higher anisotropy symmetries

We have used so-called weak anisotropy (WA) parameterization as an alternative to the parameterization of generally anisotropic media by a stiffness tensor. WA parameters consist of linear combinations of normalized stiffness-tensor elements controlling various seismic signatures; hence, they are theoretically extractable from seismic data. They are dimensionless and can be designed to have the same order of magnitude. WA parameters, similarly to Thomsen-type parameters, have a clear physical interpretation. They are, however, applicable to anisotropy of any symmetry, strength, and orientation. They are defined in coordinate systems independent of the symmetry elements of the studied media. Expressions using WA parameters naturally simplify as the anisotropy becomes weaker or as the anisotropy symmetry increases. We expect that, due to these useful properties, WA parameterization can potentially provide a framework for seismic data processing in generally anisotropic media. Using the WA parameterization, we have derived and tested approximate P-wave moveout formulas for a homogeneous layer of up-to-monoclinic symmetry, underlain by a horizontal reflector coinciding with a symmetry plane. The derived traveltime formulas represent an expansion of the traveltime with respect to (small) WA parameters. For the comparison with standard moveout formulas, we expressed ours in the common form of nonhyperbolic moveout, containing normal moveout velocity and a quartic coefficient as functions of the WA parameters. The accuracy of our formulas depends strongly on the deviation of ray- and phase-velocity directions (controlled by the deviation of the ray and phase velocities). The errors do not generally increase with increasing offset, nor do they increase with decreasing anisotropy symmetry. The accuracy of our formulas is comparable with, or better than, the accuracy of commonly used formulas.

Generalized moveout approximation for P–SV converted waves in vertically inhomogeneous transversely isotropic media with a vertical symmetry axis

ABSTRACTWe present an overall description of moveout formulas of P–SV converted waves in vertically inhomogeneous transversely isotropic media with a vertical symmetry axis by using the generalized moveout approximation. The term “generalized” means that this approximation can be reduced to some existing approximations by specific selections of parameters, which provides flexibility in application depending on objectives. The generalized moveout approximation is separately expressed in the phase and group domains. All five parameters of the group domain (or phase domain) generalized moveout approximation are determined using the zero offset (or horizontal slowness) ray and an additional nonzero offset (or horizontal slowness) ray. We discuss the selection of parameters to link the generalized moveout approximation to some existing approximations. The approximations presented are tested on homogeneous, factorized, and layered transversely isotropic models. The results illustrate that utilizing an additional reference ray significantly improves the accuracy of phase‐domain and group‐domain moveout approximations for a large range of horizontal slownesses and source–receiver offsets.

Anisotropic Radon Transform and Its Application to Demultiple

AbstractFor the seismic data acquired in the anisotropic medium and with large offset, although we use the high‐resolution hyperbolic Radon transform, the energy of the Radon domain is still not convergent. In order to overcome this problem, the influence of the anisotropic factor is considered in the integral path of Radon transform. By introducing the anisotropic anellipticity parameters η from non‐hyperbolic moveout formula, the travel‐time curve can be accurately described in the horizontal layer for large offset. We derive the integral equation of forward and inverse Radon transform with three parameters, which are the offset, the slowness and the anellipticity. For time‐varying of anisotropic Radon transform, the fast algorithm in the frequency domain is not applicable in the process of discretization. This paper adopt the optimal semblance weighted Gauss‐Seidel iterative algorithm which not only keeps the high resolution but also has the high calculation efficiency. In the examples of the anisotropic model and real marine seismic data with large offset, this method shows the effective and efficient application to demultiple.

Moveout approximations for P- and SV-waves in dip-constrained transversely isotropic media

We generalize the P- and SV-wave moveout formulas obtained for transversely isotropic media with vertical axes of symmetry (VTI), based on the weak-anisotropy approximation. We generalize them for 3D dip-constrained transversely isotropic (DTI) media. A DTI medium is a transversely isotropic medium whose axis of symmetry is perpendicular to a dipping reflector. The formulas are derived in the plane defined by the source-receiver line and the normal to the reflector. In this configuration, they can be easily obtained from the corresponding VTI formulas. It is only necessary to replace the expression for the normalized offset by the expression containing the apparent dip angle. The final results apply to general 3D situations, in which the plane reflector may have arbitrary orientation, and the source and the receiver may be situated arbitrarily in the DTI medium. The accuracy of the proposed formulas is tested on models with varying dip of the reflector, and for several orientations of the horizontal source-receiver line with respect to the dipping reflector.

Moveout approximations for P- and SV-waves in VTI media

We tested alternative expressions for the P- and SV-wave moveout formulas in VTI media based on the weak-anisotropy (WA) approximation. Our moveout formulas represent expansions with respect to small parameters, which are related to deviations of anisotropy from isotropy. First-order P-wave formulas depend on four parameters, two-way zero-offset traveltime [Formula: see text] related to the vertical velocity [Formula: see text], the depth [Formula: see text] of the single horizontal reflector, and two WA parameters [Formula: see text] and [Formula: see text]. The first-order SV-wave formulas depend on three parameters, again on [Formula: see text], now related to the SV-wave vertical velocity [Formula: see text], depth [Formula: see text], and the WA version of parameter [Formula: see text]. The second-order formulas are slightly more complicated. The P- and SV-wave formulas depend on an additional parameter [Formula: see text], the ratio of the SV- and P-wave vertical velocities. The SV-wave formula depends, in addition, on the WA parameter [Formula: see text]. Because the dependence of the moveout formulas on [Formula: see text] is very weak, [Formula: see text] can be specified as a typical SV- to P-wave velocity ratio, and the number of parameters necessary to specify the second-order formulas is four for both waves. The formulas are relatively simple, highly accurate around zero offset, and yield an exact long-offset asymptote. Their accuracy at intermediate offsets depends on deviations of ray- and phase-velocity directions. These formulas are also applicable in cases in which the reflected ray is situated in a plane of symmetry of an orthorhombic medium, whose other symmetry plane is horizontal. This also includes any HTI medium with an axis of symmetry in the plane containing the reflected ray.

A transversely isotropic medium with a tilted symmetry axis normal to the reflector

The computational tools for imaging in transversely isotropic media with tilted axes of symmetry (TTI) are complex and in most cases do not have an explicit closed-form representation. Developing such tools for a TTI medium with tilt constrained to be normal to the reflector dip (DTI) reduces their complexity and allows for closed-form representations. The homogeneous-case zero-offset migration in such a medium can be performed using an isotropic operator scaled by the velocity of the medium in the tilt direction. For the nonzero-offset case, the reflection angle is always equal to the incidence angle, and thus, the velocities for the source and receiver waves at the reflection point are equal and explicitly dependent on the reflection angle. This fact allows for the development of explicit representations for angle decomposition as well as moveout formulas for analysis of extended images obtained by wave-equation migration. Although setting the tilt normal to the reflector dip may not be valid everywhere (i.e., on salt flanks), it can be used in the process of velocity model building, in which such constrains are useful and typically are used.

Stacking velocities in the presence of overburden velocity anomalies

ABSTRACTLateral velocity changes (velocity anomalies) in the overburden may cause significant oscillations in normal moveout velocities. Explicit analytical moveout formulas are presented and provide a direct explanation of these lateral fluctuations and other phenomena for a subsurface with gentle deep structures and shallow overburden anomalies. The analytical conditions for this have been derived for a depth‐velocity model with gentle structures with dips not exceeding 12°. The influence of lateral interval velocity changes and curvilinear overburden velocity boundaries can be estimated and analysed using these formulas. An analytical approach to normal moveout velocity analysis in a laterally inhomogeneous medium provides an understanding of the connection between lateral interval velocity changes and normal moveout velocities. In the presence of uncorrected shallow velocity anomalies, the difference between root‐mean‐square and stacking velocity can be arbitrarily large to the extent of reversing the normal moveout function around normal incidence traveltimes. The main reason for anomalous stacking velocity behaviour is non‐linear lateral variations in the shallow overburden interval velocities or the velocity boundaries.A special technique has been developed to determine and remove shallow velocity anomaly effects. This technique includes automatic continuous velocity picking, an inversion method for the determination of shallow velocity anomalies, improving the depth‐velocity model by an optimization approach to traveltime inversion (layered reflection tomography) and shallow velocity anomaly replacement. Model and field data examples are used to illustrate this technique.

Anisotropic semblance analysis and NMO corrections for long offset data

Sedimentary rocks can be regarded as transversely isotropic (TI) media. One difficulty in seismic processing is how to flatten the events for long offset data. P-wave reflections from horizontal reflectors in transversely isotropic media have non-hyperbolic moveout. For a multi-layered model, the reflection moveout formula is usually expressed as a Taylor series with higher order terms ignored. Alkhalifah and Tsvankin (1995) developed a three term Taylor series formula to calculate reflection travel times from a horizontal reflector in TI media with vertical symmetry axis (VTI). Using this formula, NMO correction works well for short spread lengths, but not so well for long spreads. Zhang and Uren (2001a, b) developed an approximate explicit analytical P-wave ray velocity function for transversely isotropic (TI) media. From this ray velocity function, a reflection travel time formula from a horizontal reflector in TI media was derived. This formula can be used for anisotropic NMO correction. It works well for both large offsets and small offsets. In order to obtain the unknown parameters required for seismic processing, a 3D-semblance analysis technique has been developed. We tested this method with numerical data from TI models with a single horizontal reflector and from multi-horizontal reflector models. The method was also tested with an isotropic model with multi-horizontal reflectors. The results show that the events can be completely flattened even for very large offsets and that both multi-layered TI and isotropic models may appear to be anisotropic.

Anisotropic semblance analysis and NMO corrections for long-offset data

Sedimentary rocks can be regarded as transversely isotropic (TI) media. P-wave reflections from horizontal reflectors in transversely isotropic media have non-hyperbolic moveout. One difficulty in seismic processing is how to flatten such reflection events for long-offset data. For a multi-layered model, the reflection moveout formula is usually expressed as a Taylor series with higher order terms ignored. Alkhalifah and Tsvankin (1995) developed a three-term Taylor-series formula to calculate reflection travel times from a horizontal reflector in TI media with a vertical symmetry axis (VTI). Using this formula, NMO correction works well for short spread lengths, but not so well for long spreads.Zhang and Uren (2001a, b) developed an approximate explicit analytical P-wave ray velocity function for transversely isotropic (TI) media. From this ray velocity function, a reflection travel-time formula for a horizontal reflector in TI media was derived. This formula can be used for anisotropic NMO correction. It works well for both large offsets and small offsets. In order to obtain the unknown parameters required for seismic processing, a 3D-semblance analysis technique has been developed. We tested this method on numerical data from TI models with a single horizontal reflector and from multiple horizontal reflector models. The method was also tested on an isotropic model with multiple horizontal reflectors. The results show that the events can be completely flattened even for very large offsets, and that both multi-layered TI and isotropic models may appear to be anisotropic.

Traveltime formulas of near‐zero‐offset primary reflections for a curved 2D measurement surface

Analytic moveout formulas for primary near‐zero‐offset reflections in various types of gathers (e.g., common midpoint, common shot, zero offset) play a significant role in the seismic reflection method. They are required in stacking methods like the common midpoint (CMP) or the common‐reflection‐surface (CRS) stack. They also play a very important role in Dix‐type traveltime inversions and are of prime interest for seismic imaging. They are particularly attractive if they can be given a physical interpretation, involving for instance the wavefront curvatures of specific waves. The new formulas presented here have such a form. They give particular attention to the influence that a smooth curved measurement surface has on the computation of the traveltime and the moveout in various gathers as well as on the normal‐moveout (NMO) velocity in the CMP gather. This influence should be accounted for in the CMP or CRS stack as well as in the Dix‐type inversion. In the computation of interval velocities and the recovery of the depth of reflectors, the new NMO velocity formula is therefore more suited than the root‐mean‐square or NMO velocity for a planar measurement surface. It can be extended to a rugged free surface (mountainous terrain), but this extension requires a different derivation and different considerations. The influence of the surface curvature on the NMO velocity can be estimated with the new formula given here.

Simplified Nonhyperbolic Moveout Formula for a Transversely Isotropic Medium

Simplified Nonhyperbolic Moveout Formula for a Transversely Isotropic Medium

Multifocusing imaging over an irregular topography

If seismic data are acquired over an irregular topography, standard elevation statics methods may be inaccurate because the assumption of vertical raypaths will no longer be valid. An effective solution to the problem of irregular topography can be found through the use of the multifocusing method, in which large supergathers of seismic traces are stacked, each of which can span many common midpoint (CMP) gathers. This can be done by extending the multifocusing moveout formula to explicitly account for nonzero elevations of the source and receiver, as well as their horizontal coordinates. Implementation of this formula into the multifocusing algorithm is straightforward because estimating the necessary raypath information (i.e., emergence angles) is an integral part of the algorithm. The extended multifocusing moveout correction can be applied directly to the data acquired in areas of irregular topography without the need for prior elevation static corrections. Synthetic tests on such data show that the proposed technique results in a better alignment of reflection events.