Practical VTI approximations: a systematic anatomy

Transverse isotropy (TI) with a vertical symmetry axis (VTI) often provides an appropriate earth model for prestack imaging of steep-dip reflection seismic data. Exact P-wave and SV-wave phase velocities in VTI media are described by complicated equations requiring four independent parameters. Estimating appropriate multiparameter earth models can be difficult and time-consuming, so it is often useful to replace the exact VTI equations with simpler approximations requiring fewer parameters. The accuracy limits of different previously published VTI approximations are not always clear, nor is it always obvious how these different approximations relate to each other. Here I present a systematic framework for deriving a variety of useful VTI approximations. I develop first a sequence of well-defined approximations to the exact P-wave and SV-wave phase velocities. In doing so, I show how the useful but physically questionable heuristic of setting shear velocities identically to zero can be replaced with a more precise and quantifiable approximation. The key here to deriving accurate approximations is to replace the stiffness a₁₃ with an appropriate factorization in terms of velocity parameters. Two different specific parameter choices lead to the P-wave approximations of Alkhalifah (Geophysics 63 (1998) 623) and Schoenberg and de Hoop (Geophysics 65 (2000) 919), but there are actually an infinite number of reasonable parametrizations possible. Further approximations then lead to a variety of other useful phase velocity expressions, including those of Thomsen (Geophysics 51 (1986) 1954), Dellinger et al. (Journal of Seismic Exploration 2 (1993) 23), Harlan (Stanford Exploration Project Report 89 (1995) 145), and Stopin (Stopin, A., 2001. Comparison of v(θ) equations in TI medium. 9th International Workshop on Seismic Anisotropy). Each P-wave phase velocity approximation derived this way can be paired naturally with a corresponding SV-wave approximation. Each P-wave or SV-wave phase velocity approximation can then be converted into an equivalent dispersion relation in terms of horizontal and vertical slownesses. A simple heuristic substitution also allows each phase velocity approximation to be converted into an explicit group velocity approximation. From these, in turn, travel time or moveout approximations can also be derived. The group velocity and travel time approximations derived this way include ones previously used by Byun et al. (Geophysics 54 (1989) 1564), Dellinger et al. (Journal of Seismic Exploration 2 (1993) 23), Tsvankin and Thomsen (Geophysics 59 (1994) 1290), Harlan (89 (1995) 145), and Zhang and Uren (Zhang, F. and Uren, N., 2001. Approximate explicit ray velocity functions and travel times for P-waves in TI media. 71st Annual International Meeting, Society of Exploration Geophysicists, Expanded Abstracts, 106–109).     

Dynamic model of mesoscale eddies. Eddy parameterization for coarse resolution ocean circulation models

In the framework of the eddy dynamic model developed in two previous papers (Dubovikov, M.S., Dynamical model of mesoscale eddies, Geophys. Astophys. Fluid Dyn., 2003, 97, 311–358; Canuto, V.M. and Dubovikov, M.S., Modeling mesoscale eddies, Ocean Modelling, 2004, 8, 1–30 referred as I–II), we compute the contribution of unresolved mesoscale eddies to the large-scale dynamic equations of the ocean. In isopycnal coordinates, in addition to the bolus velocity discussed in I–II, the mesoscale contribution to the large scale momentum equation is derived. Its form is quite different from the traditional down-gradient parameterization. The model solutions in isopycnal coordinates are transformed to level coordinates to parameterize the eddy contributions to the corresponding large scale density and momentum equations. In the former, the contributions due to the eddy induced velocity and to the residual density flux across mean isopycnals (so called Σ-term) are derived, both contributions being shown to be of the same order. As for the large scale momentum equation, as well as in isopycnal coordinates, the eddy contribution has a form which is quite different from the down-gradient expression.     

REFRACTION LE LONG DES BANCS MINCES RAPIDES ET EFFET D'ECRAN POUR LES MARQUEURS PROFONDS *

Refraction along thin high velocity layers and along basement is investigated in two cases. a) high velocity layer just on the basement. b) high velocity layer higher above. Period and attenuation of refracted waves are givers as a function of the layer thickness H. Refracted arrivals along thin high velocity layers are visible at significant distances if the layer thickness is not smaller than A/6, where A is the longitudinal wavelength in high velocity medium. The pseudoperiod is proportional to the layer thickness H. The attenuation at large distance follows an x^-ne^-k1x law, where n is close to I and k₁ is inversely proportional to H. Refracted arrivals along the basement are observable even in the case of thin high velocity layers situated in the overburden; their intensity is smaller and their pseudo-period larger than when no layer exists in the overburden. The intensity of the basement arrival decreases and the pseudoperiod increases with increasing layer thickness. The pseudoperiod and the attenuation of refracted arrivals along high velocity layers and along the basement are also highly dependent on acoustic contrasts. Both arrivals from a high velocity layer and from the basement can be recorded simultaneously, provided the frequency spectrum of the seismic chain is sufficiently broad. In all cases layer arrivals show a character very different from basement arrivals.     

The Effect of Velocity Inversions on H/V

We analyzed the phenomenology of microtremor H/V curves under inversions in the shear-wave velocity (V_s) profile in the subsoil. Under no V_s inversion the spectral signature of the H/V peaks is found to be ‘eye-shaped’ with the horizontal components higher than the vertical. Conversely, under negative velocity gradients, numerous of differences emerge. I) A H/V ratio below 1 is observed for a wide range of frequencies, due to the decrease of the horizontal components below the vertical one. II) In the presence of persistent H/V < 1, small bumps in the H/V ratio given by local minima in the vertical spectral component may represent the relics of the peaks indicating resonances and stratigraphic discontinuities. As a consequence, in the presence of velocity inversions the H/V > 2 SESAME (2004) criterion fails but a stratigraphic interpretation may still be possible. III) The H/V curves should always be interpreted together with the single component spectra. IV) Microtremor H/V measurements for stratigraphic/microzonation purposes on stiff artificial soils, (asphalt, concrete, cement, pavements) should always be avoided since the latter often produce velocity inversions. This may have consequences in the intermediate to high frequency domain ( > 1 Hz) also in the application of reference site methods, like H_site/H_bedrock, to microtremor. Theoretical modeling confirms these experimental findings.     

Influence of subsurface injection on time‐lapse seismic: laboratory studies at seismic and ultrasonic frequencies

Seismic monitoring of reservoir and overburden performance during subsurface CO₂ storage plays a key role in ensuring efficiency and safety. Proper interpretation of monitoring data requires knowledge about the rock physical phenomena occurring in the subsurface formations. This work focuses on rock stiffness and elastic velocity changes of a shale overburden formation caused by both reservoir inflation induced stress changes and leakage of CO₂ into the overburden. In laboratory experiments, Pierre shale I core plugs were loaded along the stress path representative for the in situ stress changes experienced by caprock during reservoir inflation. Tests were carried out in a triaxial compaction cell combining three measurement techniques and permitting for determination of (i) ultrasonic velocities, (ii) quasistatic rock deformations, and (iii) dynamic elastic stiffness at seismic frequencies within a single test, which allowed to quantify effects of seismic dispersion. In addition, fluid substitution effects connected with possible CO₂ leakage into the caprock formation were modelled by the modified anisotropic Gassmann model. Results of this work indicate that (i) stress sensitivity of Pierre shale I is frequency dependent; (ii) reservoir inflation leads to the increase of the overburden Young's modulus and Poisson's ratio; (iii) in situ stress changes mostly affect the P‐wave velocities; (iv) small leakage of the CO₂ into the overburden may lead to the velocity changes, which are comparable with one associated with geomechanical influence; (v) non‐elastic effects increase stress sensitivity of an acoustic waves; (iv) and both geomechanical and fluid substitution effects would create significant time shifts, which should be detectable by time‐lapse seismic.     

First-Order Perturbation Theory for Seismic Isochrons

A first-order perturbation theory for seismic isochrons is presented in a model independent form. Two ray concepts are fundamental in this theory, the isochron ray and the velocity ray, for which I obtain first-order approximations to position vectors and slowness vectors. Furthermore, isochron points are connected to a shot and receiver by conventional ray fields. Based on independent perturbation of the shot and receiver ray I obtain first-order approximations to velocity rays. The theory is applicable for 3D inhomogeneous anisotropic media, given that the shot and receiver rays, as well as their perturbations, can be generated with such model generality. The theory has applications in sensitivity analysis of prestack depth migration and in velocity model updating. Numerical examples of isochron and velocity rays are shown for a 2D homogeneous VTI model. The general impression is that the first-order approximation is, with some exceptions, sufficiently accurate for practical applications using an anisotropic velocity model.     

红河(元江)流域的典型鱼类游泳能力探究及在过鱼设施流速设计中的应用

为探讨红河（元江）流域的典型洄游鱼类游泳能力,给过鱼设施修建设计和鱼类行为学研究提供基础参数,本研究以该流域的野生暗色唇鲮（Semilabeo obscurus）、华南鲤（Cyprinus carpio rubrofuscus）和红魾（Bagarius rutilus）3种鱼类（体长范围0.05~0.45 m）为研究对象,在野外现场布置游泳能力测试水槽分别测试了感应流速、临界游泳速度和突进游泳速度.研究结果表明：（1）对比绝对游泳能力与体长相关性,3种鱼的相对游泳能力与体长的相关性更高,均呈显著负相关;（2）3种鱼之间的游泳能力存在差异,红魾的感应流速最大,为（0.10±0.01）m/s,华南鲤的临界游泳速度最大,为（0.66±0.01）m/s,华南鲤的突进游泳速度最大,为（1.48±0.18）m/s;（3）拟合趋流率与水流速度的相关曲线,所有测试鱼的趋流率与流速的函数关系为Pre=19.80V-1.13（R²=0.96,P<0.05）,得出95%目标鱼的趋流速度为0.11 m/s;利用生存分析方法,得出95%目标鱼累积疲劳的临界和突进游泳速度分别为0.58、1.01 m/s.（4）当以暗色唇鲮、华南鲤和红魾为主要过鱼对象时,建议过鱼设施的内部流速不应小于0.1 m/s,入口的设计流速范围为0.6~1.0 m/s,休息区域的流速范围为0.1~0.6 m/s,孔口或竖缝等高流速区的设计流速不宜超过0.8 m/s,若过鱼设施孔口边壁或底部加糙,可适当增至0.8~1.0 m/s.本研究中红河（元江）流域鱼类游泳能力等资料为过鱼设施修建及珍稀鱼类保护提供了数据支撑.     

On occurrence of point singularities in orthorhombic media

Degeneracies of the slowness surfaces of shear (and compressional) waves in low-symmetry anisotropic media (such as orthorhombic), known as point singularities, pose difficulties during modelling and inversion, but can be potentially used in the latter as model parameter constraints. I analyse the quantity and spatial arrangement of point singularities in orthorhombic media, as well as their relation to the overall strength of velocity anisotropy. A classification scheme based on the number and spatial distribution of singularity directions is proposed. In normal orthorhombic models (where the principal shear moduli are smaller than the principal compressional moduli), point singularities can only be arranged in three distinct patterns, and media with the theoretical minimum (0) and maximum (16) number of singularities are not possible. In orthorhombic models resulting from embedding vertical fractures in transversely isotropic background, only two singularity distributions are possible, in contrast to what was previously thought. Although the total number of singularities is independent of the overall anisotropy strength, for general (non-normal) orthorhombic models, different spatial distributions of singularities become more probable with increasing magnitude of anisotropy.     

Estimates of Horizontal Groundwater Flow Velocities in Boreholes

Currently, monitoring tools can be deployed in observation boreholes to better assess groundwater flow, flux of dissolved contaminants and their mass discharge in an aquifer. The relationship between horizontal water velocity in observation boreholes and Darcy fluxes in the surrounding aquifer has been studied for natural flow conditions (i.e., no pumping). Interpretation of measurements taken with dilution tests, the colloidal borescope, the Heat Pulse Flowmeter, and other techniques require the conversion of observed borehole velocity u to aquifer Darcy flux q_∞ . This conversion is typically done through a proportionality factor α = u/q_∞ . In experimental studies as well as in theoretical developments, reported values of α vary almost three orders of magnitude (from 0.5 to 10). This large variability in reported values of α could be explained by: (1) unclear distinction between Darcy flux and water seepage velocity, (2) unclear definition of water velocity in the borehole, (3) effects of well screen and the presence of the measurement device itself on the observable velocities, and (4) hydraulic conditions in the borehole annulus. We address (1), (2) from a conceptual/theoretical perspective, and (3) by means of numerical simulations. We show that issue (1) in low porosity aquifers can yield to order-of-magnitude discrepancies in estimates of q_∞ ; (2) may result in discrepancies of up to 50%, and (3) can cause differences up to 20% of water velocity in the borehole void space compared to the theoretical case of an open borehole.     

Selection of Momentum Variables for a Three-Dimensional Variational Analysis

Three choices of control variables for meteorological variational analysis (3DVAR or 4DVAR) are associated with horizontal wind: (1) streamfunction and velocity potential, (2) eastward and northward velocity, and (3) vorticity and divergence. This study shows theoretical and numerical differences of these variables in practical 3DVAR data assimilation through statistical analysis and numerical experiments. This paper demonstrates that (a) streamfunction and velocity potential could potentially introduce analysis errors; (b) A 3DVAR using velocity or vorticity and divergence provides a natural scale dependent influence radius in addition to the covariance; (c) for a regional analysis, streamfunction and velocity potential are retrieved from the background velocity field with Neumann boundary condition. Improper boundary conditions could result in further analysis errors; (d) a variational data assimilation or an inverse problem using derivatives as control variables yields smoother analyses, for example, a 3DVAR using vorticity and divergence as controls yields smoother wind analyses than those analyses obtained by a 3DVAR using either velocity or streamfunction/velocity potential as control variables; and (e) statistical errors of higher order derivatives of variables are more independent, e.g., the statistical correlation between U and V is smaller than the one between streamfunction and velocity potential, and thus the variables in higher derivatives are more appropriate for a variational system when a cross-correlation between variables is neglected for efficiency or other reasons. In summary, eastward and northward velocity, or vorticity and divergence are preferable control variables for variational systems and the former is more attractive because of its numerical efficiency. Numerical experiments are presented using analytic functions and real atmospheric observations.     

Palaeomagnetic timing of the rotation and translation of Cyprus

The geological evolution of the Mesozoic Troodos Ophiolite Complex in Cyprus, and the tectonic nature and timing of the palaeomagnetically indicated anticlockwise rotation of Cyprus of some 80° and ca. 15° northward translation, have been open for debate for some time. New palaeomagnetic data from 18 sites ( 180samples) in the post-ophiolite sediments, ranging in age from Upper Cretaceous to Upper Miocene, are presented. Most of the sites are of normal geomagnetic polarity, but indications of reversed polarity have been found in an older group of sediments (the Lefkara Formation of Upper Palaeocene age).Six sites from the older group of sediments (Upper Cretaceous to Eocene in age) give a site mean direction of the AF cleaned sediments of (D, I) = (323°, 29°) with α₉₅ = 18°, while 5 sites from a younger group of sediments (Oligocene to Miocene in age) give a cleaned site mean direction of (D, I) = (334°, 58°) with α₉₅ = 9°. These and published data suggest that an anticlockwise rotation of Cyprus of 60 ± 10° occurred early during the post-igneous evolution of the Cyprus oceanic crust between 90 and 50Ma, leaving only a minor anticlockwise rotation of 20 ± 10° to occur during the last 50 Ma. It is furthermore concluded that the northward translation of Cyprus of 15° mostly took place during the last 30Ma.It thus appears that a fairly rapid rotation of the Cyprus microplate first took place in the Late Cretaceous and Early Tertiary time with an average angular velocity of 1–2°/Ma, during which the northward translation was minor or negligible. In the latter half of the Tertiary, the sense of movement appears to have radically changed, the northward translation now being dominant with an average velocity of 5–6cm/yr. This temporal evolution is found to be in good agreement with the Mesozoic and Tertiary movements of the African lithospheric plate relative to Europe, as evidenced from the Atlantic sea-floor magnetic anomaly spreading history.     

Hydrological characteristics of vegetated river flows: a laboratory flume study

Abstract

Laboratory flume experiments were undertaken to measure the vertical profiles of mean flow velocity for three different flow discharges and four different stem densities of Hydrilla verticillata. The data were used to calculate three parameters, namely Manning's roughness coefficient, the Reynolds number and the Froude number. In addition, empirical equations were obtained for the vertical distribution of measured flow velocity within the transitional zone and above the plant canopy. The results show that: (a) the vertical distribution of measured flow velocity exhibits three zone profiles; (b) Manning's roughness coefficient decreases with increasing depth-averaged flow velocity; (c) the relationship between Manning's roughness coefficient and the depth-averaged flow velocity is within the smooth left inverse curve; (d) Manning's roughness coefficient significantly changes with increasing density of Hydrilla; (e) the Froude number is independent of the density of Hydrilla; and (f) both the Reynolds number and the Froude number increase with increasing depth-averaged flow velocity.

Citation Shi, J.Z., Li, Y.-H., Hughes, J.M.R., and Zhao, M., 2013. Hydrological characteristics of vegetated river flows: a laboratory flume study. Hydrological Sciences Journal, 58 (5), 1047–1058.

Editor Z.W. Kundzewicz     

CUBIC SPLINE APPROXIMATION OF INACCURATE RMS-VELOCITY DATA*

The reliable determination of the root-mean-square seismic velocity from multiple coverage seismic reflection data requires coherent horizontal reflections equally spaced in time. Observational error in the rms-velocity data produces significantly greater error in the instantaneous or interval velocity. A cubic spline approximation technique developed by Powell (1967) is designed to minimize the rapid velocity variation due to observational error and retain the significant variation in the rms-velocity data. The approximation algorithm requires the difference between the observed and approximated rms-velocity to be statistically random, and seeks to follow the “real” trends in the rms-velocity measurements and reject observational error. Provided the trends of the rms-velocity are reproduced, the instantaneous velocity derived from the spline approximation to rms-velocity data with significant scatter is more reliable than the interval velocity determined from the inaccurate rms-velocity values using the Dix (1955) approximation. This interval velocity is also a more reliable indicator of variations of lithology. The utility of the spline approximation of rms-velocity is illustrated for a special common depth point (CDP) expanding reflection spread test recorded in Hardeman County, Texas by the Consortium for Continental Reflection Profiling (COCORP). In this wave test reflections are observed from several CDPs which are closely spaced. It is desired to combine the inaccurate observations from a collection of nearby CDPs. The approximation algorithm for the rms-velocity determination is applied to both the rms-velocity data for the individual CDPs and the collection of rms-velocity data for eight nearby CDPs.     

Quantification and spatial distribution of pore-filling materials through constrained rock physics template and fluid response modelling in Paleogene clastic reservoir from Cauvery basin,India

Sands belonging to Kamalapuram Formation of Paleocene-Eocene age are deposited in Cauvery basin as incised valley fill during a regressive cycle. Here we attempt to quantify the influence of diagenesis on pore-filling materials using rock physics template constrained by geohistory modelling. Primarily, porosity–velocity and acoustic impedance – the ratio of P-wave and S-wave velocity (V_P/V_s) cross-plots are used as rock physics templates. Rock physics template has efficiently quantified pore-filling materials namely; contact cement and non-contact cement. The estimated contact cement and non-contact cement are correlated with conventional petrophysical logs within the selected depth interval. Further, this correlation is used to interpret the composition of pore-filling materials. Shallower depth intervals (I and II) exhibit moderate non-contact cement (4–5%) and insignificant contact cement (1–2% approx.) depositions. However, deeper interval (III) records a significant amount of pore-filling materials amounting average of 12% non-contact cement and 4% contact cement. Pore-filling materials demonstrate a positive correlation with the depth of burial. The fluid response is substantially affected by the degree of diagenesis, composition and spatial distribution of pore-filling materials. Shallower depth intervals (1770–1786 m and 1858–1878 m) are relatively more sensitive to fluid changes as it is affected by insignificant contact cement. The depth interval 1770–1786 m shows class II (oil) and class III (gas) amplitude variation with offset anomalies. The sand occurring in depth interval 1858–1878 m demonstrates class IIP (oil) and II (gas) anomaly. The deeper interval (2118–2170 m) is comparatively stiffer and demonstrates class I amplitude variation with offset (oil and gas sand) anomaly.     

Numerical analysis of one-dimensional nonlinear acoustic wave

Numerical investigations on one-dimensional nonlinear acoustic wave with third and fourth order nonlinearities are presented using high-order finite-difference (HFD) operators with a simple flux-limiter (SFL) algorithm. As shown by our numerical tests, the HFDSFL method is able to produce more stable, accurate and conservative solutions to the nonlinear acoustic waves than those computed by finite-difference combined with the flux-corrected-transport algorithm. Unlike the linear acoustic waves, the nonlinear acoustic waves have variable phase velocity and waveform both in time-space (t-x) domain and frequency-wavenumber (f-k) domain; of our special interest is the behaviour during the propagation of nonlinear acoustic waves: the waveforms are strongly linked to the type of medium nonlinearities, generation of harmonics, frequency and wavenumber peak shifts. In seismic sense, these characteristics of nonlinear wave will introduce new issues during such seismic processing as Normal Moveout and f-k filter. Moreover, as shown by our numerical experiment for a four-layer model, the nonlinearities of media will introduce extra velocity errors in seismic velocity inversion.     

Tunnel effect of fractal fault and transient S-wave velocity rupture (TSVR) of in-plane shear fault

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Effects of patterned Artemisia capillaris on overland flow velocity under simulated rainfall

Vegetation cover is an important factor for erosion control. Laboratory‐simulated rainfall experiments were conducted to quantify the effectiveness of patchy distributed Artemisia capillaris in retarding overland flow velocity. Simulated storms (60, 90, 120, and 150 mm h^?1) were applied on a bare plot (CK) and four different plant patterns, a banded pattern perpendicular to the slope direction (BP), a single long strip parallel to slope direction (LP), small patches distributed like a checkerboard (SP1), and small patches distributed like a letter “X” (SP2). All treatments had three replicates. Each plot underwent two sets of experiments, intact plant plots and root plots (the above‐ground parts were removed, only roots were reserved), respectively. Results showed that flow velocity increased with rainfall intensity, and the lower slope velocity (V_l) was higher than the upper slope velocity (V_u). The removal of grass shoots increased flow velocity. Compared with bare soil plot, intact plants reduced mean flow velocity by 14%–60%, whereas the reduction declined to <40% for the root plots. BP and both SP treatments performed more effectively than LP in retarding flow velocity, whereas no significant differences were identified between BP and SP. The contributions of A. capillaris shoots and roots to the reductions in flow velocity under different rainfall intensities were different. The shoots made greater contribution of 53%–97% at 60 and 90 mm h^–1, and the roots contributed more (51%–81%) at 120 and 150 mm h^–1. Runoff and sediment rate had significant (p < 0.05) linear correlations with mean flow velocity. Copyright © 2012 John Wiley & Sons, Ltd.     

Detection and reconstruction of large‐scale coherent flow structures in gravel‐bed rivers

Large‐scale flow structures (LSFS) in the streamwise direction are important features of gravel‐bed river flows, because they may contribute to sediment transport and gas exchange. In the present study, these structures are detected using Huang's empirical mode decomposition and reconstructed with phase‐averaging techniques based on a Hilbert transform of the velocity signal. The analysis is based on the fluctuating component of 15 quasi‐instantaneous velocity profiles measured with a three‐dimensional (3D) acoustic Doppler velocity profiler (ADVP) in an armoured gravel‐bed river with a low relative submergence of 2.9 (ratio between flow depth and bed grain diameter). LSFS were identified in most of the measured profiles and consistently showed similar features. We were able to characterize the geometry of these large‐scale coherent structures: the front has a vertical linear shift in the time domain and a vertical profile corresponding to a first quarter moon with the apex situated at z/h ≈ 0.4. In the vertical, the front scales with flow depth h, and in the streamwise direction, LSFS scale with three to seven times the mean flow depth. On the bed, the effect of LSFS is a periodic non‐linear variation of the friction velocity on average between 0.90 and 1.10 times the mean value. A model for the friction velocity cycle resulting from LSFS oscillation is presented. Copyright © 2014 John Wiley & Sons, Ltd.     

Nonparametric estimation of wave dispersion in high‐rise buildings by seismic interferometry

Interferometric identification and health monitoring of high‐rise buildings has been gaining increasing interest in recent years. The wave dispersion in the structure has been largely ignored in these efforts but needs to be considered to further develop these methods. In this paper, (i) the goodness of estimation of vertical wave velocity in buildings, as function of frequency, by two nonparametric interferometric techniques is examined, using realistic fixed‐base Timoshenko beam benchmark models. Such models are convenient because the variation of phase and group velocities with frequency can be derived theoretically. The models are those of the NS and EW responses of Millikan Library. One of the techniques, deconvolution interferometry, estimates the phase velocity on a frequency band from phase difference between motions at two locations in the structure, while the other one estimates it approximately at the resonant frequencies based on standing wave patterns. The paper also (ii) examines the modeling error in wave velocity profiles identified by fitting layered shear beam in broader band impulse response functions of buildings with significant bending flexibility. This error may affect inferences on the spatial distribution of damage from detected changes in such velocity profiles. Copyright © 2014 John Wiley & Sons, Ltd.     

Anisotropic migration velocity analysis: Application to a data set from West Africa

Although it is widely recognized that anisotropy can have a significant influence on the focusing and positioning of migrated reflection events, conventional depth imaging methods still operate with isotropic velocity fields. Here, we present an application of a 2D migration velocity analysis (MVA) algorithm, designed for factorized v(x, z) VTI (transversely isotropic with a vertical symmetry axis) media, to an offshore data set from West Africa. By approximating the subsurface with factorized VTI blocks, it is possible to decouple the spatial variations in the vertical velocity from the anisotropic parameters with minimal a priori information. Since our method accounts for lateral velocity variation, it produces more accurate estimates of the anisotropic parameters than those previously obtained with time‐domain techniques. The values of the anellipticity parameter η found for the massive shales exceed 0.2, which confirms that ignoring anisotropy in the study area can lead to substantial imaging distortions, such as mis‐stacking and mispositioning of dipping events. While some of these distortions can be removed by using anisotropic time processing, further marked improvement in image quality is achieved by prestack depth migration with the estimated factorized VTI model. In particular, many fault planes, including antithetic faults in the shallow part of the section, are better focused by the anisotropic depth‐migration algorithm and appear more continuous. Anisotropic depth migration facilitates structural interpretation by eliminating false dips at the bottom of the section and improving the images of a number of gently dipping features. One of the main difficulties in anisotropic MVA is the need to use a priori information for constraining the vertical velocity. In this case study, we successfully reconstructed the time–depth curve from reflection data by assuming that the vertical velocity is a continuous function of depth and estimating the vertical and lateral velocity gradients in each factorized block. If the subsurface contains strong boundaries with jumps in velocity, knowledge of the vertical velocity at a single point in a layer is sufficient for our algorithm to determine all relevant layer parameters.