AVO investigations of shallow marine sediments

Amplitude‐variation‐with‐offset (AVO) analysis is based on the Zoeppritz equations, which enable the computation of reflection and transmission coefficients as a function of offset or angle of incidence. High‐frequency (up to 700 Hz) AVO studies, presented here, have been used to determine the physical properties of sediments in a shallow marine environment (20 m water depth). The properties that can be constrained are P‐ and S‐wave velocities, bulk density and acoustic attenuation. The use of higher frequencies requires special analysis including careful geometry and source and receiver directivity corrections. In the past, marine sediments have been modelled as elastic materials. However, viscoelastic models which include absorption are more realistic. At angles of incidence greater than 40°, AVO functions derived from viscoelastic models differ from those with purely elastic properties in the absence of a critical angle of incidence. The influence of S‐wave velocity on the reflection coefficient is small (especially for low S‐wave velocities encountered at the sea‐floor). Thus, it is difficult to extract the S‐wave parameter from AVO trends. On the other hand, P‐wave velocity and density show a considerably stronger effect. Attenuation (described by the quality factor Q) influences the reflection coefficient but could not be determined uniquely from the AVO functions. In order to measure the reflection coefficient in a seismogram, the amplitudes of the direct wave and the sea‐floor reflection in a common‐midpoint (CMP) gather are determined and corrected for spherical divergence as well as source and streamer directivity. At CMP locations showing the different AVO characteristics of a mud and a boulder clay, the sediment physical properties are determined by using a sequential‐quadratic‐programming (SQP) inversion technique. The inverted sediment physical properties for the mud are: P‐wave velocity α=1450±25 m/s, S‐wave velocity β=90±35 m/s, density ρ=1220±45 kg/m³, quality factor for P‐wave Q_P=15±200, quality factor for S‐wave Q_S=10±30. The inverted sediment physical properties for the boulder clay are: α=1620±45 m/s,β=360±200 m/s,ρ=1380±85 kg/m³,Q_P=790±660,Q_S=25±10.     

Source/receiver array directivity effects on marine seismic attenuation measurements

Attenuation of seismic waves, quantified by the seismic quality factor Q, holds important information for seismic interpretation, due to its sensitivity to rock and fluid properties. A recently published study of Q, based on surface seismic reflection data, used a modified spectral ratio approach (QVO), but both source and receiver responses were treated as isotropic, based on simple raypath arguments. Here, this assumption has been tested by computing apparent attenuation generated by frequency-dependent directivity of typical marine source and receiver arrays and acquisition geometries. Synthetic wavelet spectra were computed for reflected rays, summed over the first Fresnel zone, from the base of a single interval, 50–3000 m thick and velocity 2000 m/s, overlying a 2200 m/s half-space, and for offsets of 71–2071 m. The source and receiver geometry were those of an actual survey. The modelled spectra are clearly affected by directivity, most strongly because of surface ghosts. In general, the strong high-frequency component, produced by the array design, leads to apparently negative attenuation in individual reflection events, though this is dependent on offset and target depth. For shallow targets (less than 400–500 ms two-way traveltime (TWT) depth), apparent Q-values as extreme as ?50 to ?100 were obtained. For deeper target depths, the directivity effect is far smaller. The implications of the model study were tested on real data. QVO was applied to 20 true-spectrum-processed CMPs, in a shallow (405–730 ms TWT) and a deeper (1000–1300 ms TWT) interval, firstly using a measured far-field source signature (effectively isotropic), and secondly using computed directivity effects instead. Mean interval Q^?1-values for the deeper interval, 0.029 ± 0.011 and 0.027 ± 0.018 for conventional and directional processing, respectively, suggested no directivity influence on attenuation estimation. For the shallow interval (despite poor spectral signal-to-noise ratios and hence scattered attenuation estimates), directional processing removed directivity-generated irregularities from the spectral ratios, resulting in an improvement from Q^?1_int = ?0.036 ± 0.130 to a realistic Q^?1_int = 0.012 ± 0.030: different at 94% confidence level. Equivalent Q-values are: for the deeper interval, 35 and 37 for conventional and directional processing, respectively, and ?28 and 86 for the shallow interval. These results support the conclusions of the model studies, i.e. that source/receiver directivity has a negligible effect except for shallow targets (e.g. TWT depth ≤ 500 ms) imaged with conventional acquisition geometry. In such cases directivity corrections to spectra are strongly recommended.     

A laboratory study of seismic velocity and attenuation anisotropy in near-surface sedimentary rocks

The laboratory ultrasonic pulse‐echo method was used to collect accurate P‐ and S‐wave velocity (±0.3%) and attenuation (±10%) data at differential pressures of 5–50 MPa on water‐saturated core samples of sandstone, limestone and siltstone that were cut parallel and perpendicular to the vertical borehole axis. The results, when expressed in terms of the P‐ and S‐wave velocity and attenuation anisotropy parameters for weakly transversely isotropic media (ɛ, γ, ɛ_Q, γ_Q) show complex variations with pressure and lithology. In general, attenuation anisotropy is stronger and more sensitive to pressure changes than velocity anisotropy, regardless of lithology. Anisotropy is greatest (over 20% for velocity, over 70% for attenuation) in rocks with visible clay/organic matter laminations in hand specimens. Pressure sensitivities are attributed to the opening of microcracks with decreasing pressure. Changes in magnitude of velocity and attenuation anisotropy with effective pressure show similar trends, although they can show different signs (positive or negative values of ɛ, ɛ_Q, γ, γ_Q). We conclude that attenuation anisotropy in particular could prove useful to seismic monitoring of reservoir pressure changes if frequency‐dependent effects can be quantified and modelled.     

Scattering and intrinsic attenuation in Cairo metropolitan area using genetic algorithm

A total number of 46 local earthquakes (2.0≤ML≤4.0) recorded in the period 2000–2011 by the Egyptian seismographic network (ENSN) were used to estimate the total (Q_t⁻¹), intrinsic (Q_i⁻¹) and scattering attenuation (Q_sc⁻¹) in Cairo metropolitan area, Egypt. The multiple lapse time window analysis (MLTWA) under the assumption of multiple isotropic scattering with uniform distribution of scatters was firstly applied to estimate the pair of L_e⁻¹, the extinction length inverse, and B₀, the seismic albedo, in the frequency range 3–24 Hz. To take into account the effect of a depth-dependent earth model, the obtained values of B₀ and L_e⁻¹ were corrected for an earth structure characterized by a transparent upper mantle and a heterogeneous crust. The estimated values of Q_t⁻¹, Q_sc⁻¹ and Q_i⁻¹ exhibited frequency dependences. The average frequency-dependent relationships of attenuation characteristics estimated for the region are found to be: Q_t⁻¹=(0.015±0.008)f ^{(−1.02±0.02)}, Q_sc⁻¹=(0.006±0.001)f ^{(−1.01±0.02)}, and Q_i⁻¹=(0.009±0.008)f ^{(−1.03±0.02)}; showing a predominance of intrinsic absorption over scattering attenuation. This finding implies that the pore-fluid contents may have great effect on the attenuation mechanism in the upper crust where the River Nile is passing through the study area. The obtained results are comparable with those obtained in other tectonic regions.     

Modeling of Wide-Angle Seismic Attributes Using the Gaussian Beam Method

In this study, observed seismic attributes from shot gather 11 of the SAREX experiment are used to derive a preliminary velocity and attenuation model for the northern end of the profile in southern Alberta. Shot gather 11 was selected because of its prominent Pn arrivals and good signal to noise ratio. The 2-D Gaussian beam method was used to perform the modeling of the seismic attributes including travel times, peak envelope amplitudes and pulse instantaneous frequencies for selected phases. The preliminary model was obtained from the seismic attributes from shot gather 11 starting from prior tomographic results. The amplitudes and instantaneous frequencies were used to constrain the velocity and attenuation structure, with the amplitudes being more sensitive to the velocity gradients and the instantaneous frequencies more sensitive to the attenuation structure. The resulting velocity model has a velocity discontinuity between the upper and lower crust, and lower velocity gradients in the upper and lower crust compared to earlier studies. The attenuation model has Q _p ^-1 values between 0.011 and 0.004 in the upper crust, 0.0019 in the lower crust and a laterally variable Q _p ^-1 in the upper mantle. The Q _p ^-1 values are similar to those found in Archean terranes from other studies. Although the results from a single gather are non-unique, the initial model derived here provides a self-consistent starting point for a more complete seismic attribute inversion for the velocity and attenuation structure.     

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.     

Investigation on regional attenuation of Vrancea (Romania) intermediate-depth earthquakes

This paper aims at investigating possible regional attenuation patterns in the case of Vrancea(Romania) intermediate-depth earthquakes.Almost 500 pairs of horizontal components recorded during 13 intermediate-depth Vrancea earthquakes are employed in order to evaluate the regional attenuation patterns.The recordings are grouped according to the azimuth with regard to the Vrancea seismic source and subsequently,Q models are computed for each azimuthal zone assuming similar geometrical spreading.Moreover,the local soil amplification which was disregarded in a previous analysis performed for Vrancea intermediate-depth earthquakes is now clearly evaluated.The results show minor differences between the four regions situated in front of the Carpathian Mountains and considerable differences in attenuation of seismic waves between the forearc and backarc regions(with regard to the Carpathian Mountains).Consequently,an average Q model of the type Q(f) = 115×f~(1.25) is obtained for the four forearc regions,while a separate Q model of the type Q(f) = 70×f~(0.90) is computed for the backarc region.These results highlight the need to evaluate the seismic hazard of Romania by using ground motion models which take into account the different attenuation between the forearc/backarc regions.     

Attenuation of High Frequency P and S Waves in the Gujarat Region,India

The local earthquake waveforms recorded on broadband seismograph network of Institute of Seismological Research in Gujarat, India have been analyzed to understand the attenuation of high frequency (2–25 Hz) P and S waves in the region. The frequency dependent relationships for quality factors for P (Q _P) and S (Q _S) waves have been obtained using the spectral ratio method for three regions namely, Kachchh, Saurashtra and Mainland Gujarat. The earthquakes recorded at nine stations of Kachchh, five stations of Saurashtra and one station in mainland Gujarat have been used for this analysis. The estimated relations for average Q _P and Q _S are: Q _P = (105 ± 2) f ^{0.82 ± 0.01}, Q _S = (74 ± 2) f ^{1.06 ± 0.01} for Kachchh region; Q _P = (148 ± 2) f ^{0.92 ± 0.01}, Q _S = (149 ± 14) f ^{1.43 ± 0.05} for Saurashtra region and Q _P = (163 ± 7) f ^{0.77 ± 0.03}, Q _S = (118 ± 34) f ^{0.65 ± 0.14} for mainland Gujarat region. The low Q (<200) and high exponent of f (>0.5) as obtained from present analysis indicate the predominant seismic activities in the region. The lowest Q values obtained for the Kachchh region implies that the area is relatively more attenuative and heterogeneous than other two regions. A comparison between Q _S estimated in this study and coda Q (Qc) previously reported by others for Kachchh region shows that Q _C > Q _S for the frequency range of interest showing the enrichment of coda waves and the importance of scattering attenuation to the attenuation of S waves in the Kachchh region infested with faults and fractures. The Q _S/Q _P ratio is found to be less than 1 for Kachchh and Mainland Gujarat regions and close to unity for Saurashtra region. This reflects the difference in the geological composition of rocks in the regions. The frequency dependent relations developed in this study could be used for the estimation of earthquake source parameters as well as for simulating the strong earthquake ground motions in the region.     

Observation of shear‐wave splitting in the multicomponent node data from Atlantis field,Gulf of Mexico

In 2005, a multicomponent ocean bottom node data set was collected by BP and BHP Billiton in the Atlantis field in the Gulf of Mexico. Our results are based on data from a few sparse nodes with millions of shots that were analysed as common receiver azimuthal gathers. A first‐order look at P‐wave arrivals on a common receiver gather at a constant offset reveals variation of P‐wave arrival time as a function of azimuth indicating the presence of azimuthal anisotropy at the top few layers. This prompted us to investigate shear arrivals on the horizontal component data. After preliminary processing, including a static correction, the data were optimally rotated to radial (R) and transverse (T) components. The R component shows azimuthal variation of traveltime indicating variation of velocity with azimuth; the corresponding T component shows azimuthal variation of amplitude and phase (polarity reversal). The observed shear‐wave (S‐wave) splitting, previously observed azimuthal P‐wave velocity variation and azimuthal P‐wave amplitude variation, all indicate the occurrence of anisotropy in the shallow (just below the seafloor) subsea sediment in the area. From the radial component azimuthal gather, we analysed the PP‐ and PS‐wave amplitude variation for the first few layers and determined corresponding anisotropy parameter and V_P/V_S values. Since fracture at this depth is not likely to occur, we attribute the observed azimuthal anisotropy to the presence of microcracks and grain boundary orientation due to stress. The evidence of anisotropy is ubiquitous in this data set and thus it argues strongly in favour of considering anisotropy in depth imaging for obtaining realistic subsurface images, at the least.     

Borehole study of compressional and shear attenuation of basalt flows penetrated by the Brugdan and William wells on the Faroes shelf

We investigated the seismic attenuation of compressional (P‐) and converted shear (S‐) waves through stacked basalt flows using short‐offset vertical seismic profile (VSP) recordings from the Brugdan (6104/21–1) and William (6005/13–1A) wells in the Faroe‐Shetland Trough. The seismic quality factors (Q) were evaluated with the classical spectral ratio method and a root‐mean‐square time‐domain amplitude technique. We found the latter method showed more robust results when analysing signals within the basalt sequence. For the Brugdan well we calculated effective Q estimates of 22–26 and 13–17 for P‐ and S‐waves, respectively, and 25–33 for P‐waves in the William well. An effective Q_S/Q_P ratio of 0.50–0.77 was found from a depth interval in the basalt flow sequence where we expect fully saturated rocks. P‐wave quality factor estimates are consistent with results from other VSP experiments in the North Atlantic Margin, while the S‐wave quality factor is one of the first estimates from a stacked basalt formation using VSP data. Synthetic modelling demonstrates that seismic attenuation for P‐ and S‐waves in the stacked basalt flow sequence is mainly caused by one‐dimensional scattering, while intrinsic absorption is small.     

VISCOELASTIC SEISMIC RESPONSES OF 2D RESERVOIR MODELS1

The use of relaxation mechanisms has recently made it possible to simulate viscoelastic (Q) effects accurately in time-domain numerical computations of seismic responses. As a result, seismograms may now be synthesized for models with arbitrary spatial variations in compressional- and shear-wave quality factors (Q₉, and Q_s, as well as in density (ρ) and compressional- and shear-wave velocities (V_p, and V_s). Reflections produced by Q contrasts alone may have amplitudes as large as those produced by velocity contrasts. Q effects, including their interaction with V_p, V_s and p, contribute significantly to the seismic response of reservoirs. For band-limited data at typical seismic frequencies, the effects of Q on reflectivity and attenuation are more visible than those on dispersion. Synthetic examples include practical applications to reservoir exploration, evaluation and monitoring. Q effects are clearly visible in both surface and offset vertical seismic profile data. Thus, AVO analyses that neglect Q may produce erroneous conclusions.     

Attenuation of P- and S-waves in the Chamoli Region, Himalaya, India

The attenuation properties of the crust in the Chamoli region of Himalaya have been examined by estimating the frequency-dependent relationships of quality factors for P waves (Q_α) and for S waves (Q_β) in the frequency range 1.5–24 Hz. The extended coda normalization method has been applied on the waveforms of 25 aftershocks of the 1999 Chamoli earthquake (M 6.4) recorded at five stations. The average value of Q_α is found to be varied from 68 at 1.5 Hz to 588 at 24 Hz while it varies from 126 at 1.5 Hz to 868 at 24 Hz for Q_β. The estimated frequency-dependent relations for quality factors are Q_α = (44 ± 1)f^(0.82±.04) and Q_β = (87 ± 3)f^(0.71±.03). The rate of increase of Q(f) for P and S waves in the Chamoli region is comparable with the other regions of the world. The ratio Q_β/Q_α is greater than one in the region which along with the frequency dependence of quality factors indicates that scattering is an important factor contributing to the attenuation of body waves in the region. A comparison of attenuation relation for S wave estimated here (Q_β = 87f^0.71) with that of coda waves (Q_c = 30f^1.21) obtained by Mandal et al. (2001) for the same region shows that Q_c > Q_β for higher frequencies (>8 Hz) in the region. This indicates a possible high frequency coda enrichment which suggests that the scattering attenuation significantly influences the attenuation of S waves at frequencies >8 Hz. This observation may be further investigated using multiple scattering models. The attenuation relations for quality factors obtained here may be used for the estimation of source parameters and near-source simulation of earthquake ground motion of the earthquakes, which in turn are required for the assessment of seismic hazard in the region.     

SEISMIC ATTENUATION IN THE VICINITY OF THE GEOTHERMAL ANOMALY AT URACH OBTAINED FROM NEAR-VERTICAL REFLECTION PROFILES1

The attenuation in the vicinity of the geothermal anomaly at Urach was determined by means of two near-vertical reflection profiles. The attenuation in the sediments and in the upper crust (3-4 km depth) was estimated by interpretation of the first (refracted) arrivals. For calculating the attenuation, the amplitude decay with respect to distance was used. Corrections for the spread factor, i.e. the geometric amplitude divergence was deduced from the traveltime curves. Below the anomaly, higher attenuation values (Q^?1～ 0.008) were observed compared with those in the undisturbed crust (Q^?1～ 0.002). This effect is probably due to the cracks and fissures in the upper part of the crystalline basement. The attenuation in the middle and lower crust was determined using near-vertical reflections from this depth interval. The use of the spectral ratio method leads to higher values of the effective attenuation Q^?1_eff below the heat flow anomaly compared to those of the‘ normal’crust. This zone of high Q^?1_eff coincides with the low velocity body below the heat flow anomaly. Both effects, the higher attenuation and the lower velocities, could be caused by high temperatures, cracks and fissures in the crust.     

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.     

Azimuthal variation in AVO response for fractured gas sands

Natural fractures in reservoirs play an important role in determining fluid flow during production, and hence the density and orientation of fractures is of great interest. In the presence of aligned vertical fractures, the reflection amplitude at finite offset varies with azimuth. The effect of natural fractures on the azimuthal AVO response from a gas-sandstone reservoir encased within shale is investigated. A simple expression for the difference in P-wave reflection coefficient from the top of the reservoir parallel and perpendicular to the strike of the fractures is obtained in terms of the normal and tangential compliances, Z_N and Z_T, of the fractures. This expression is valid for small anisotropy and material contrasts and is compared with the results of numerical modelling. For a given value of Z_T, the azimuthal variation in reflection coefficient at moderate offsets is found to increase with decreasing Z_N/Z_T. For gas-filled open fractures Z_N/Z_T ≈ 1, but a lower ratio of Z_N/Z_T may result from the presence of cement or clay within the fractures, or from the presence of a fluid with non-zero bulk modulus. For Z_N/Z_T = 1 and moderate offsets, the variation with offset of the reflection coefficient from the top of the fractured unit is dominated by the contrast in Poisson's ratio between the gas sand and the overlying shale, the effect of fractures only becoming noticeable as the critical angle for the unfractured sandstone is approached. However, for reflections from the base of the fractured unit, the variation in reflection amplitude with azimuth is much greater at conventional seismic offsets than for the reflection from the top. Azimuthal variations in the strength of the reflection from the top of the reservoir depend only on the variation in reflection coefficient, whereas the raypath is also a function of azimuth for reflections from the base of the fractured unit, leading to stronger, more visible, variations of AVO with azimuth. It follows that an azimuthal variation in AVO due to fractures in the overburden may be misinterpreted as due to the presence of aligned fractures in the reservoir.     

Seismic Wave Attenuation in the Greater Cairo Region, Egypt

In the present study, a digital waveform dataset of 216 local earthquakes recorded by the Egyptian National Seismic Network (ENSN) was used to estimate the attenuation of seismic wave energy in the greater Cairo region. The quality factor and the frequency dependence for Coda waves and S-waves were estimated and clarified. The Coda waves (Q _c) and S-waves (Q _d) quality factor were estimated by applying the single scattering model and Coda Normalization method, respectively, to bandpass-filtered seismograms of frequency bands centering at 1.5, 3, 6, 12, 18 and 24?Hz. Lapse time dependence was also studied for the area, with the Coda waves analyzed through four lapse time windows (10, 20, 30 and 40?s). The average quality factor as function of frequency is found to be Q _c?=?35?±?9f ^0.9±0.02 and Q _d?=?10?±?2f ^0.9±0.02 for Coda and S-waves, respectively. This behavior is usually correlated with the degree of tectonic complexity and the presence of heterogeneities at several scales. The variation of Q _c with frequency and lapse time shows that the lithosphere becomes more homogeneous with depth. In fact, by using the Coda Normalization method we obtained low Q _d values as expected for a heterogeneous and active zone. The intrinsic quality factor (Q _i ^?1 ) was separated from the scattering quality factor (Q _s ^?1 ) by applying the Multiple Lapse Time Domain Window Analysis (MLTWA) method under the assumption of multiple isotropic scattering with uniform distribution of scatters. The obtained results suggest that the contribution of the intrinsic attenuation (Q _i ^?1 ) prevails on the scattering attenuation (Q _s ^?1 ) at frequencies higher than 3?Hz.     

Comments on: “The partitioning of nickel between olivine and silicate melt” by S.R. Hart and K.E. Davis

We study a set of very high-quality records of first-order overtone Rayleigh waves from the deep-focus earthquake of September 29, 1973, in the Japan Sea. Standard surface wave techniques are used with these overtones, treated as individual seismic phases, to retrieve radiation pattern, Q, moment and phase velocity. A figure of M₀ = (6.7 ± 1.4) × 10²⁷dyn-cm is obtained, in total agreement with published values computed from either P waves, or fundamental Rayleigh waves. We also demonstrate the feasibility of using overtones as individual seismic phases in order to investigate their dispersion and attenuation properties.     

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.     

The estimation of spectra from time‐frequency transforms for use in attenuation studies

We introduce the signal dependent time–frequency distribution, which is a time–frequency distribution that allows the user to optimize the tradeoff between joint time–frequency resolution and suppression of transform artefacts. The signal‐dependent time–frequency distribution, as well as the short‐time Fourier transform, Stockwell transform, and the Fourier transform are analysed for their ability to estimate the spectrum of a known wavelet used in a tuning wedge model. Next, the signal‐dependent time–frequency distribution, and fixed‐ and variable‐window transforms are used to estimate spectra from a zero‐offset synthetic seismogram. Attenuation is estimated from the associated spectral ratio curves, and the accuracy of the results is compared. The synthetic consisted of six pairs of strong reflections, based on real well‐log data, with a modeled intrinsic attenuation value of 1000/Q = 20. The signal‐dependent time–frequency distribution was the only time–frequency transform found to produce spectra that estimated consistent attenuation values, with an average of 1000/Q = 26±2; results from the fixed‐ and variable‐window transforms were 24±17 and 39±10, respectively. Finally, all three time–frequency transforms were used in a pre‐stack attenuation estimation method (the pre‐stack Q inversion algorithm) applied to a gather from a North Sea seismic dataset, to estimate attenuation between nine different strong reflections. In this case, the signal‐dependent time‐frequency distribution produced spectra more consistent with the constant‐Q model of attenuation assumed in the pre‐stack attenuation estimation algorithm: the average L1 residuals of the spectral ratio surfaces from the theoretical constant‐Q expectation for the signal‐dependent time‐frequency distribution, short‐time Fourier transform, and Stockwell transform were 0.12, 0.21, and 0.33, respectively. Based on the results shown, the signal‐dependent time‐frequency distribution is a time–frequency distribution that can provide more accurate and precise estimations of the amplitude spectrum of a reflection, due to a higher attainable time–frequency resolution.