3D description and inversion of reflection moveout of PS‐waves in anisotropic media

Common‐midpoint moveout of converted waves is generally asymmetric with respect to zero offset and cannot be described by the traveltime series t²(x²) conventionally used for pure modes. Here, we present concise parametric expressions for both common‐midpoint (CMP) and common‐conversion‐point (CCP) gathers of PS‐waves for arbitrary anisotropic, horizontally layered media above a plane dipping reflector. This analytic representation can be used to model 3D (multi‐azimuth) CMP gathers without time‐consuming two‐point ray tracing and to compute attributes of PS moveout such as the slope of the traveltime surface at zero offset and the coordinates of the moveout minimum. In addition to providing an efficient tool for forward modelling, our formalism helps to carry out joint inversion of P and PS data for transverse isotropy with a vertical symmetry axis (VTI media). If the medium above the reflector is laterally homogeneous, P‐wave reflection moveout cannot constrain the depth scale of the model needed for depth migration. Extending our previous results for a single VTI layer, we show that the interval vertical velocities of the P‐ and S‐waves (V_P0 and V_S0) and the Thomsen parameters ε and δ can be found from surface data alone by combining P‐wave moveout with the traveltimes of the converted PS(PSV)‐wave. If the data are acquired only on the dip line (i.e. in 2D), stable parameter estimation requires including the moveout of P‐ and PS‐waves from both a horizontal and a dipping interface. At the first stage of the velocity‐analysis procedure, we build an initial anisotropic model by applying a layer‐stripping algorithm to CMP moveout of P‐ and PS‐waves. To overcome the distorting influence of conversion‐point dispersal on CMP gathers, the interval VTI parameters are refined by collecting the PS data into CCP gathers and repeating the inversion. For 3D surveys with a sufficiently wide range of source–receiver azimuths, it is possible to estimate all four relevant parameters (V_P0, V_S0, ε and δ) using reflections from a single mildly dipping interface. In this case, the P‐wave NMO ellipse determined by 3D (azimuthal) velocity analysis is combined with azimuthally dependent traveltimes of the PS‐wave. On the whole, the joint inversion of P and PS data yields a VTI model suitable for depth migration of P‐waves, as well as processing (e.g. transformation to zero offset) of converted waves.     

P‐wave stacking‐velocity tomography for VTI media

A major complication caused by anisotropy in velocity analysis and imaging is the uncertainty in estimating the vertical velocity and depth scale of the model from surface data. For laterally homogeneous VTI (transversely isotropic with a vertical symmetry axis) media above the target reflector, P‐wave moveout has to be combined with other information (e.g. borehole data or converted waves) to build velocity models for depth imaging. The presence of lateral heterogeneity in the overburden creates the dependence of P‐wave reflection data on all three relevant parameters (the vertical velocity V_P0 and the Thomsen coefficients ε and δ) and, therefore, may help to determine the depth scale of the velocity field. Here, we propose a tomographic algorithm designed to invert NMO ellipses (obtained from azimuthally varying stacking velocities) and zero‐offset traveltimes of P‐waves for the parameters of homogeneous VTI layers separated by either plane dipping or curved interfaces. For plane non‐intersecting layer boundaries, the interval parameters cannot be recovered from P‐wave moveout in a unique way. Nonetheless, if the reflectors have sufficiently different azimuths, a priori knowledge of any single interval parameter makes it possible to reconstruct the whole model in depth. For example, the parameter estimation becomes unique if the subsurface layer is known to be isotropic. In the case of 2D inversion on the dip line of co‐orientated reflectors, it is necessary to specify one parameter (e.g. the vertical velocity) per layer. Despite the higher complexity of models with curved interfaces, the increased angle coverage of reflected rays helps to resolve the trade‐offs between the medium parameters. Singular value decomposition (SVD) shows that in the presence of sufficient interface curvature all parameters needed for anisotropic depth processing can be obtained solely from conventional‐spread P‐wave moveout. By performing tests on noise‐contaminated data we demonstrate that the tomographic inversion procedure reconstructs both the interfaces and the VTI parameters with high accuracy. Both SVD analysis and moveout inversion are implemented using an efficient modelling technique based on the theory of NMO‐velocity surfaces generalized for wave propagation through curved interfaces.     

The analysis and application of simplified two-parameter moveout equation for C-waves in VTI anisotropy media

Several parameters are needed to describe the converted-wave （C-wave） moveout in processing multi-component seismic data, because of asymmetric raypaths and anisotropy. As the number of parameters increases, the converted wave data processing and analysis becomes more complex. This paper develops a new moveout equation with two parameters for C-waves in vertical transverse isotropy （VTI） media. The two parameters are the C-wave stacking velocity （Vc2） and the squared velocity ratio （7v,i） between the horizontal P-wave velocity and C-wave stacking velocity. The new equation has fewer parameters, but retains the same applicability as previous ones. The applicability of the new equation and the accuracy of the parameter estimation are checked using model and real data. The form of the new equation is the same as that for layered isotropic media. The new equation can simplify the procedure for C-wave processing and parameter estimation in VTI media, and can be applied to real C-wave processing and interpretation. Accurate Vc2 and Yvti can be deduced from C-wave data alone using the double-scanning method, and the velocity ratio model is suitable for event matching between P- and C-wave data.     

Prestack migration velocity analysis based on simplified two-parameter moveout equation

Stacking velocity V _C2, vertical velocity ratio γ ₀, effective velocity ratio γ _eff, and anisotropic parameter χ _eff are correlated in the PS-converted-wave (PS-wave) anisotropic prestack Kirchhoff time migration (PKTM) velocity model and are thus difficult to independently determine. We extended the simplified two-parameter (stacking velocity V _C2 and anisotropic parameter k _eff) moveout equation from stacking velocity analysis to PKTM velocity model updating and formed a new four-parameter (stacking velocity V _C2, vertical velocity ratio γ ₀, effective velocity ratio γ _eff, and anisotropic parameter k _eff) PS-wave anisotropic PKTM velocity model updating and process flow based on the simplified two-parameter moveout equation. In the proposed method, first, the PS-wave two-parameter stacking velocity is analyzed to obtain the anisotropic PKTM initial velocity and anisotropic parameters; then, the velocity and anisotropic parameters are corrected by analyzing the residual moveout on common imaging point gathers after prestack time migration. The vertical velocity ratio γ ₀ of the prestack time migration velocity model is obtained with an appropriate method utilizing the P- and PS-wave stacked sections after level calibration. The initial effective velocity ratio γ _eff is calculated using the Thomsen (1999) equation in combination with the P-wave velocity analysis; ultimately, the final velocity model of the effective velocity ratio γ _eff is obtained by percentage scanning migration. This method simplifies the PS-wave parameter estimation in high-quality imaging, reduces the uncertainty of multiparameter estimations, and obtains good imaging results in practice.     

Converted-wave seismology in anisotropic media Revisited,Part II: Application to parameter estimation

In transversely isotropic media with a vertical symmetry axis (VTI), the converted-wave (C-wave) moveout over intermediate-to-far offsets is determined by four parameters. These are the C-wave stacking velocity V _C2, the vertical and effective velocity ratios γ ₀and γ _eff, and the anisotropic parameter X _eff. We refer to the four parameters as the C-wave stacking velocity model. The purpose of C-wave velocity analysis is to determine this stacking velocity model. The C-wave stacking velocity model V _C2, γ ₀, γ _geff, and X _eff can be determined from P- and C-wave reflection moveout data. However, error propagation is a severe problem in C-wave reflection-moveout inversion. The current short-spread stacking velocity as deduced from hyperbolic moveout does not provide sufficient accuracy to yield meaningful inverted values for the anisotropic parameters. The non-hyperbolic moveout over intermediate-offsets (x/z from 1.0 to 1.5) is no longer negligible and can be quantified using a background γ. Non-hyperbolic analysis with a γ correction over the intermediate offsets can yield V _C2 with errors less than 1% for noise free data. The procedure is very robust, allowing initial guesses of γ with up to 20% errors. It is also applicable for vertically inhomogeneous anisotropic media. This improved accuracy makes it possible to estimate anisotropic parameters using 4C seismic data. Two practical work flows are presented for this purpose: the double-scanning flow and the single-scanning flow. Applications to synthetic and real data show that the two flows yield results with similar accuracy but the single-scanning flow is more efficient than the double-scanning flow. This work is funded by the Edinburgh Anisotropy Project of the British Geological Survey. First Author Li Xiangyang, he is currently a professorial research seismologist (Grade 6) and technical director of the Edinburgh Anisotropy Project in the British Geological Survey. He also holds a honorary professorship multicomponent seismology at the School of Geosciences, University of Edinburgh. He received his BSc(1982) in Geophysics from Changchun Geological Institute, China, an MSc (1984) in applied geophysics from East China Petroleum Institute (now known as the China University of Petroleum), and a PhD (1992) in seismology from the University of Edinburgh. During 1984–1987, he worked as a lecturer with the East China Petroleum Institute. Since 1991, he has been employed by the British Geological Survey. His research interests include seismic anisotropy and multicomponent seismology.     

再论各向异性介质转换波地震学,第二部分:参数计算研究

在具有垂直对称轴横向各向同性介质中，利用四种参数来确定中间至远偏移距转换波（C-波）动校正。它们是C-波叠加速度Vc2，垂直速度比和有效速度比γ0和γeff以及各向异性参数χeff。我们将这四种参数作为C波叠加速度模型。C-波速度分析的目的就是确定这种叠加速度模型。C-波叠加速度模型Vc2，γ0，γeff，和χeff可以由P-波和C-波反射动校正资料获得。然而错误的传播是C-波反射动校正反演中的严重问题。当前短排列叠加速度由于是从双曲线动校正推算而得，因而其精度不足以为各向异性参数提供有意义的反演值。中间偏移非双曲线动校正不再被人们所勿略，而是可以用一个背景γ加以量化。非双曲线分析通过中间偏移距的γ校正量可以产生Vc2，若数据不含燥音，其误差小于1％。方法稳健，允许γ启始假定值的误差达20％。该方法也适用垂直非均匀各向异性介质。精度的提高使能够用4分量地震资料计算各向异性参数。为此提出了两种工作流程：双扫描和单扫描流程。理论数据和实际数据的应用表明这两种流程得出的结果其精度相似，但是单扫描流程比双扫描更有效。     

Velocity model updating in prestack Kirchhoff time migration for PS converted waves: Part I – Theory

A velocity model updating approach is developed based on moveout analysis of the diffraction curve of PS converted waves in prestack Kirchhoff time migration. The diffraction curve can be expressed as a product of two factors: one factor depending on the PS converted‐wave velocity only, and the other factor depending on all parameters. The velocity‐dependent factor represents the hyperbolic behaviour of the moveout and the other is a scale factor that represents the non‐hyperbolic behaviour of the moveout. This non‐hyperbolic behaviour of the moveout can be corrected in prestack Kirchhoff time migration to form an inverse normal‐moveout common‐image‐point gather in which only the hyperbolic moveout is retained. This hyperbolic moveout is the moveout that would be obtained in an isotropic equivalent medium. A hyperbolic velocity is then estimated from this gather by applying hyperbolic moveout analysis. Theoretical analysis shows that for any given initial velocity, the estimated hyperbolic velocity converges by an iterative procedure to the optimal velocity if the velocity ratio is optimal or to a value closer to the optimal velocity if the velocity ratio is not optimal. The velocity ratio (V_P/V_S) has little effect on the estimation of the velocity. Applying this technique to a synthetic seismic data set confirms the theoretical findings. This work provides a practical method to obtain the velocity model for prestack Kirchhoff time migration.     

Peculiarities of Rayleigh wave displacement in a transversely isotropic half space

We consider a transversely isotropic medium with vertical axis of symmetry (VTI). Rayleigh wave displacement components in a homogeneous VTI medium contain exp(±kr_jz), where z is the vertical coordinate, k is the wave number, and j?=?1, 2; r_j may be considered as depth-decay factor. In a VTI medium, r_j can be a real or imaginary as in an isotropic medium, or it can be a complex depending on the elastic parameters of the VTI medium; if complex, r₁ and r₂ are complex conjugates. In a homogeneous VTI half space, Rayleigh wave displacement is significantly modified with a phase shift when r_j changes from real to complex with variation of VTI parameters; at the transition, the displacement becomes zero when r₁?=?r₂. In a liquid layer over a VTI half space, Rayleigh waves are dispersive. Here, also Rayleigh wave displacement significantly modified with a phase shift when r_j changes from real to complex with a decrease of period. At very low period, phase velocity of Rayleigh waves becomes less than P-wave velocity in the liquid layer giving rise to Scholte waves (interface waves). The amplitudes of Scholte waves with a VTI half space are found to be significantly larger than those with an isotropic half space.     

Migration velocity analysis for tilted transversely isotropic media

Tilted transversely isotropic formations cause serious imaging distortions in active tectonic areas (e.g., fold‐and‐thrust belts) and in subsalt exploration. Here, we introduce a methodology for P‐wave prestack depth imaging in tilted transversely isotropic media that properly accounts for the tilt of the symmetry axis as well as for spatial velocity variations. For purposes of migration velocity analysis, the model is divided into blocks with constant values of the anisotropy parameters ε and δ and linearly varying symmetry‐direction velocity V_P0 controlled by the vertical (k_z) and lateral (k_x) gradients. Since determination of tilt from P‐wave data is generally unstable, the symmetry axis is kept orthogonal to the reflectors in all trial velocity models. It is also assumed that the velocity V_P0 is either known at the top of each block or remains continuous in the vertical direction. The velocity analysis algorithm estimates the velocity gradients k_z and k_x and the anisotropy parameters ε and δ in the layer‐stripping mode using a generalized version of the method introduced by Sarkar and Tsvankin for factorized transverse isotropy with a vertical symmetry axis. Synthetic tests for several models typical in exploration (a syncline, uptilted shale layers near a salt dome and a bending shale layer) confirm that if the symmetry‐axis direction is fixed and V_P0 is known, the parameters k_z, k_x, ε and δ can be resolved from reflection data. It should be emphasized that estimation of ε in tilted transversely isotropic media requires using nonhyperbolic moveout for long offsets reaching at least twice the reflector depth. We also demonstrate that application of processing algorithms designed for a vertical symmetry axis to data from tilted transversely isotropic media may lead to significant misfocusing of reflectors and errors in parameter estimation, even when the tilt is moderate (30°). The ability of our velocity analysis algorithm to separate the anisotropy parameters from the velocity gradients can be also used in lithology discrimination and geologic interpretation of seismic data in complex areas.     

三维倾斜界面PS转换波CMP道集时距及参数估计

在PS转换波资料处理过程中,往往需要联合P波资料提供相应的模型.在实际应用中存在P波和PS转换波层位对比困难.本文仅利用PS转换波数据,通过三维倾斜界面PS转换波CMP道集精确时距关系推导了近似时距解析表达式;分析了PS波的精确与近似时距关系随测线方位、界面倾角与倾向的变化规律及其拟合误差;并讨论了近似时距关系的三个时距参数随方位的变化特征;理论上给出描述时距的三维倾斜界面倾角、倾向、深度、纵波速度和横波速度这5个独立参数的估计方法,并通过理论模拟数据证明了该方法的可行性.     

Converted-wave imaging in anisotropic media: theory and case studies

Common‐conversion‐point binning associated with converted‐wave (C‐wave) processing complicates the task of parameter estimation, especially in anisotropic media. To overcome this problem, we derive new expressions for converted‐wave prestack time migration (PSTM) in anisotropic media and illustrate their applications using both 2D and 3D data examples. The converted‐wave kinematic response in inhomogeneous media with vertical transverse isotropy is separated into two parts: the response in horizontally layered vertical transverse isotrophy media and the response from a point‐scatterer. The former controls the stacking process and the latter controls the process of PSTM. The C‐wave traveltime in horizontally layered vertical transverse isotrophy media is determined by four parameters: the C‐wave stacking velocity V_C2, the vertical and effective velocity ratios γ₀ and γ_eff, and the C‐wave anisotropic parameter χ_eff. These four parameters are referred to as the C‐wave stacking velocity model. In contrast, the C‐wave diffraction time from a point‐scatterer is determined by five parameters: γ₀, V_P2, V_S2, η_eff and ζ_eff, where η_eff and ζ_eff are, respectively, the P‐ and S‐wave anisotropic parameters, and V_P2 and V_S2 are the corresponding stacking velocities. V_P2, V_S2, η_eff and ζ_eff are referred to as the C‐wave PSTM velocity model. There is a one‐to‐one analytical link between the stacking velocity model and the PSTM velocity model. There is also a simple analytical link between the C‐wave stacking velocities V_C2 and the migration velocity V_Cmig, which is in turn linked to V_P2 and V_S2. Based on the above, we have developed an interactive processing scheme to build the stacking and PSTM velocity models and to perform 2D and 3D C‐wave anisotropic PSTM. Real data applications show that the PSTM scheme substantially improves the quality of C‐wave imaging compared with the dip‐moveout scheme, and these improvements have been confirmed by drilling.     

Normal moveout velocity for pure‐mode and converted waves in layered orthorhombic medium

We study the azimuthally dependent hyperbolic moveout approximation for small angles (or offsets) for quasi‐compressional, quasi‐shear, and converted waves in one‐dimensional multi‐layer orthorhombic media. The vertical orthorhombic axis is the same for all layers, but the azimuthal orientation of the horizontal orthorhombic axes at each layer may be different. By starting with the known equation for normal moveout velocity with respect to the surface‐offset azimuth and applying our derived relationship between the surface‐offset azimuth and phase‐velocity azimuth, we obtain the normal moveout velocity versus the phase‐velocity azimuth. As the surface offset/azimuth moveout dependence is required for analysing azimuthally dependent moveout parameters directly from time‐domain rich azimuth gathers, our phase angle/azimuth formulas are required for analysing azimuthally dependent residual moveout along the migrated local‐angle‐domain common image gathers. The angle and azimuth parameters of the local‐angle‐domain gathers represent the opening angle between the incidence and reflection slowness vectors and the azimuth of the phase velocity ψ_phs at the image points in the specular direction. Our derivation of the effective velocity parameters for a multi‐layer structure is based on the fact that, for a one‐dimensional model assumption, the horizontal slowness and the azimuth of the phase velocity ψ_phs remain constant along the entire ray (wave) path. We introduce a special set of auxiliary parameters that allow us to establish equivalent effective model parameters in a simple summation manner. We then transform this set of parameters into three widely used effective parameters: fast and slow normal moveout velocities and azimuth of the slow one. For completeness, we show that these three effective normal moveout velocity parameters can be equivalently obtained in both surface‐offset azimuth and phase‐velocity azimuth domains.     

Non-hyperbolic moveout inversion of wide-azimuth P-wave data for orthorhombic media

The azimuthally varying non‐hyperbolic moveout of P‐waves in orthorhombic media can provide valuable information for characterization of fractured reservoirs and seismic processing. Here, we present a technique to invert long‐spread, wide‐azimuth P‐wave data for the orientation of the vertical symmetry planes and five key moveout parameters: the symmetry‐plane NMO velocities, V⁽¹⁾_nmo and V⁽²⁾_nmo , and the anellipticity parameters, η⁽¹⁾, η⁽²⁾ and η⁽³⁾ . The inversion algorithm is based on a coherence operator that computes the semblance for the full range of offsets and azimuths using a generalized version of the Alkhalifah–Tsvankin non‐hyperbolic moveout equation. The moveout equation provides a close approximation to the reflection traveltimes in layered anisotropic media with a uniform orientation of the vertical symmetry planes. Numerical tests on noise‐contaminated data for a single orthorhombic layer show that the best‐constrained parameters are the azimuth ? of one of the symmetry planes and the velocities V⁽¹⁾_nmo and V⁽²⁾_nmo , while the resolution in η⁽¹⁾ and η⁽²⁾ is somewhat compromised by the trade‐off between the quadratic and quartic moveout terms. The largest uncertainty is observed in the parameter η⁽³⁾ , which influences only long‐spread moveout in off‐symmetry directions. For stratified orthorhombic models with depth‐dependent symmetry‐plane azimuths, the moveout equation has to be modified by allowing the orientation of the effective NMO ellipse to differ from the principal azimuthal direction of the effective quartic moveout term. The algorithm was successfully tested on wide‐azimuth P‐wave reflections recorded at the Weyburn Field in Canada. Taking azimuthal anisotropy into account increased the semblance values for most long‐offset reflection events in the overburden, which indicates that fracturing is not limited to the reservoir level. The inverted symmetry‐plane directions are close to the azimuths of the off‐trend fracture sets determined from borehole data and shear‐wave splitting analysis. The effective moveout parameters estimated by our algorithm provide input for P‐wave time imaging and geometrical‐spreading correction in layered orthorhombic media.     

LIMITS OF STRUCTURAL INVERSION OF SEISMIC HORIZONS*

Time horizons can be depth-migrated when interval velocities are known; on the other hand, the velocity distribution can be found when traveltimes and NMO velocities at zero offset are known (wavefront curvatures; Shah 1973). Using these concepts, exact recursive inversion formulae for the calculation of interval velocities are given. The assumption of rectilinear raypath propagation within each layer is made; interval velocities and curvatures of the interfaces between layers can be found if traveltimes together with their gradients and curvatures and very precise V_NMO velocities at zero offset are known. However, the available stacking velocity is a numerical quantity which has no direct physical significance; its deviation from zero offset NMO velocity is examined in terms of horizon curvatures, cable length and lateral velocity inhomogeneities. A method has been derived to estimate the geological depth model by searching, iteratively, for the best solution that minimizes the difference between stacking velocities from the real data and from the structural model. Results show the limits and capabilities of the approach; perhaps, owing to the low resolution of conventional velocity analyses, a simplified version of the given formulae would be more robust.     

Surface-to-base amplification evaluated from KiK-net vertical array strong motion records

Site amplification defined as the peak value of spectrum ratio was investigated using surface and base accelerations recorded in a number of KiK-net down-hole arrays in Japan during three major earthquakes. An important task was to determine the spectral amplifications relative to outcropping motions with the aid of the down-hole array records. Based on soil data available for individual arrays, theoretical amplifications were calculated and adjusted to coincide with the peak amplifications of the array records. A good and unique correlation was found between the peak amplifications thus obtained and S-wave velocity ratios, defined by S-wave velocity in base layer divided by average S-wave velocity , for different sites and different earthquakes. The value of was evaluated from fundamental mode frequency and the thickness of an equivalent surface layer in which peak amplification is exerted. The conventional parameter V_s30; averaged shear wave velocity in the top 30 m used in current design codes, did not correlate well with the obtained amplifications. It is suggested that may be determined not only from V_s-logging data but also from microtremor measurements.     

Controls on sonic velocity in carbonates

Compressional and shear-wave velocities (V _p andV _s) of 210 minicores of carbonates from different areas and ages were measured under variable confining and pore-fluid pressures. The lithologies of the samples range from unconsolidated carbonate mud to completely lithified limestones. The velocity measurements enable us to relate velocity variations in carbonates to factors such as mineralogy, porosity, pore types and density and to quantify the velocity effects of compaction and other diagenetic alterations.Pure carbonate rocks show, unlike siliciclastic or shaly sediments, little direct correlation between acoustic properties (V _p andV _s) with age or burial depth of the sediments so that velocity inversions with increasing depth are common. Rather, sonic velocity in carbonates is controlled by the combined effect of depositional lithology and several post-depositional processes, such as cementation or dissolution, which results in fabrics specific to carbonates. These diagenetic fabrics can be directly correlated to the sonic velocity of the rocks.At 8 MPa effective pressureV _p ranges from 1700 to 6500 m/s, andV _s ranges from 800 to 3400 m/s. This range is mainly caused by variations in the amount and type of porosity and not by variations in mineralogy. In general, the measured velocities show a positive correlation with density and an inverse correlation with porosity, but departures from the general trends of correlation can be as high as 2500 m/s. These deviations can be explained by the occurrence of different pore types that form during specific diagenetic phases. Our data set further suggests that commonly used correlations like Gardner's Law (V _p-density) or the time-average-equation (V _p-porosity) should be significantly modified towards higher velocities before being applied to carbonates.The velocity measurements of unconsolidated carbonate mud at different stages of experimental compaction show that the velocity increase due to compaction is lower than the observed velocity increase at decreasing porosities in natural rocks. This discrepancy shows that diagenetic changes that accompany compaction influence velocity more than solely compaction at increasing overburden pressure.The susceptibility of carbonates to diagenetic changes, that occur far more quickly than compaction, causes a special velocity distribution in carbonates and complicates velocity estimations. By assigning characteristic velocity patterns to the observed diagenetic processes, we are able to link sonic velocity to the diagenetic stage of the rock.     

Reverse time migration of CMP-gathers an effective tool for the determination of interval velocities

One of the most important steps in the conventional processing of reflection seismic data is common midpoint (CMP) stacking. However, this step has considerable deficiencies. For instance the reflection or diffraction time curves used for normal moveout corrections must be hyperbolae. Furthermore, undesirable frequency changes by stretching are produced on account of the dependence of the normal moveout corrections on reflection times. Still other drawbacks of conventional CMP stacking could be listed.One possibility to avoid these disadvantages is to replace conventional CMP stacking by a process of migration to be discussed in this paper. For this purpose the Sherwood-Loewenthal model of the exploding reflector has to be extended to an exploding point model with symmetry to the lineP _EX M whereP _EX is the exploding point, alias common reflection point, andM the common midpoint of receiver and source pairs.Kirchhoff summation is that kind of migration which is practically identical with conventional CMP stacking with the exception that Kirchhoff summation provides more than one resulting trace.In this paper reverse time migration (RTM) was adopted as a tool to replace conventional CMP stacking. This method has the merit that it uses the full wave equation and that a direct depth migration is obtained, the velocityv can be any function of the local coordinatesx, y, z. Since the quality of the reverse time migration is highly dependent on the correct choice of interval velocities such interval velocities can be determined stepwise from layer to layer, and there is no need to compute interval velocities from normal moveout velocities by sophisticated mathematics or time consuming modelling. It will be shown that curve velocity interfaces do not impair the correct determination of interval velocities and that more precise velocity values are obtained by avoiding or restricting muting due to non-hyperbolic normal moveout curves.Finally it is discussed how in the case of complicated structures the reverse time migration of CMP gathers can be modified in such a manner that the combination of all reverse time migrated CMP gathers yields a correct depth migrated section. This presupposes, however, a preliminary data processing and interpretation.     

Converted-wave seismology in anisotropic media revisited,Part I: Basic theory

We have developed new basic theories for calculating the conversion point and the travel time of the P-SV converted wave (C-wave) in anisotropic, inhomogeneous media. This enables the use of conventional procedures such as semblance analysis, Dix-type model building and Kirchhoff summation, to implement anisotropic processing, and makes anisotropic processing affordable. Here we present these new developments in two parts: basic theory and application to velocity analysis and parameter estimation. This part deals with the basic theory, including both conversion-point calculation and moveout analysis. Existing equations for calculating the PS-wave (C-wave) conversion point in layered media with vertical transverse isotropy (VTI) are strictly limited to offsets about half the reflector depth (an offset-depth ratio, xlz, of 0.5), and those for calculating the C-wave traveltimes are limited to offsets equal to the reflector depth (x/z=l.0). In contrast, the new equations for calculating the conversion-point extend into offsets about three-times the reflector depth (x/z=3.0), those for calculating the C-wave traveltimes extend into offsets twice the reflector depth (x/z=2.0). With the improved accuracy, the equations can help in C-wave data processing and parameter estimation in anisotropic, inhomogeneous media. This work is funded by the Edinburgh Anisotropy Project (EAP) of the British Geological Survey. First author: Xiangyang Li, Mr. Li is currently a professorial research seismologist (Grade 6) and technical director of the Edinburgh Anisotropy Project in the British Geological Survey. He also holds a honorary professorship in multicomponent seismology at the School of Geosciences, University of Edinburgh. He received his BSc(1982) in Geophysics from Changchun Geological Institute, China, an MSc (1984) in applied geophysics from East China Petroleum Institute (now known as the China University of Petroleum), and a PhD (1992) in seismology from the University of Edinburgh. During 1984–1987, he worked as a lecturer with the East China Petroleum Institute. Since 1991, he has been employed by the British Geological Survey. His research interests include seismic anisotropy and multicomponent seismology.     

MAPPING NON-REFLECTING VELOCITY INTERFACES BY NORMAL MOVEOUT VELOCITIES OF UNDERLYING HORIZONS*

The normal moveout velocity of a reflecting bed is a function of the dips and curvatures of all overlying velocity interfaces. Now let the (N– 1)th velocity interface be a non- (or badly) reflecting bed, whereas the other interfaces, including the base of the Nth layer, reflect satisfactorily, and let the velocities U_{N– 1} and U_N of the (N– 1)th and Nth layer, respectively, be known. Then the normal moveout velocity for the base of the Nth layer, if known in one direction at a certain part of the surface of the earth, provides a second order differential equation in the horizontal coordinates x and y for the depth Z_{N – 1}(x, y) of the unknown interface. The mathematics becomes rather simple in the case of two-dimensional geological structures. For this case and N= 2 the differential equation mentioned can be solved by stepwise integration or by iteration. One of the many possible applications of the new concept is the determination of the structure of the base of an overthrusting sheet.