Heat transfer and thermal boundary layers in high amplitude annulus waves

Abstract

The heat transfer by a rotating, differentially-heated annulus of fluid is measured throughout the high amplitude wave regime. Only Δr_w T was varied (although v(T₁₅ ).K(T₁₅ ) varied by 46%), and it is found that Nu = C₁(λ)Ra^? away from the symmetry and low amplitude to wave transition curves and this is independent of ω. (λ is the wavelength.) On the wave side of these transition curves a region exists in which Nu (symmetry) λ Nu λ C₁(λ)Ra^?. The local heat transfer rate also varies strongly with wave phase.

Using a selection of measured internal thermal fields in the steady, high amplitude wave regime, the side-wall thermal boundary layer structure is examined. It is found that Nu, = C₂·Gr₂ ^A2; both C ₂ and A ₂ are independent of ω and λ to first order. For the time mean profiles, A ₂ ≈ 0.25; in the high heat transfer portion of the wave A ₂ < ¼ and in the low heat transfer portion of the wave A ₂ > ?. These relations hold over most of the vertical extent of the side walls. The deviations of the boundary layers from the above behavior which occur on the remainder of the walls is illustrated. The average thicknesses of the wall boundary layers ∞ Ra^?¼ except in that phase of the wave in which the wall to mid-gap temperature difference is the largest.     

Study of Basin-edge Effects on the Ground Motion Characteristics Using 2.5-D Modelling

This paper presents the role of basin-edge geometry in the generation of surface waves using 2.5-D modelling. The simulated responses of various basin-edge models revealed surface wave generation near the basin edge and their propagation normal to the edge. Seismic responses of basin-edge models using different fundamental frequency of soil along with spectral analysis of differential ground motion confirmed that surface waves start generating near the basin edge when body-wave frequency exceeds the fundamental frequency of soil. Spectral analysis of differential ground motion also confirmed the generation of high frequency surface wave. An increase of surface-wave amplitude with soil thickness was obtained. Large ground displacement observed near the basin edge may be due to the interference of surface/diffracted waves with the direct waves and their multiples. The effect of edge roughness on the surface-wave characteristics was found to be negligible as compared with the edge geometry. Simulated results revealed a decrease of surface-wave amplitude with edge slope, particularly in the case of surface waves caused by S waves. Surface wave generation near the basin edge was obtained for all four considered angles of incidence. At the same time, it was also inferred that the characteristics of these surface waves depend on the angle of incidence to some extent. The findings of this paper reveal that basin-edge effects deserve a particular attention for the purpose of earthquake-resistant design and seismic microzonation.Acknowledgement Financial assistance by the Department of Science and Technology (DST) New Delhi and the Indian National Science Academy (INSA), New Delhi is gratefully acknowledged.     

ACOUSTICAL IMAGING OF SOURCE RECEIVER COINCIDENT PROFILES*

The first part of this paper examines a special case of acoustical imaging in which the source and the receiver coincide. The benefits of weighting and muting are studied in detail by means of computer modeling. The test model consists of a single planar interface z=z₁, abruptly terminated at x= o. The amplitude and phase responses are computed in the plane z=z₀= o for two separations of neighboring stations, Δx=λ/10 and Δx=λ/2. Six different weighting factors are used in the test. However, in this source-receiver coincident case, three of the weighting factors produce identical responses, so that all six test factors may be represented by only four curves. It is found that when the spatial sampling at the aperture approaches the condition of critical sampling, i.e. Δx=λ/2, only the weighting factor which implicitly takes into account beam steering along the specular reflection path is acceptable. This factor alone keeps the amplitude and the phase curves undistorted until the difference 2 ·ΔR between two neighboring paths reaches approximately λ/2. If we set 2 ·ΔR=λ/2, we may construct a set of curves which we may call quite appropriately muting curves. These curves are physically interpretable only for station separation Δx > λ/4. The muting curves are symmetrical about the line x= 0 and their angular opening depends on spatial separation Δx, depth z, and wavelength λ (which may vary with depth). The second part of this paper suggests how the weighting factor with implicit beam steering can be applied to reconstruction of two and three-dimensional wavefields. Seismic migration of common depth point (CDP) stacked line data is also discussed. This is a hybrid case which presents certain theoretical difficulties. We shall also mention the velocity problem which is inherent to migration of CDP stacked data. The third and final part concerns implementation of the migration of CDP stacked data. When the spatial sampling is between λ/4 and λ/2, the migration process will benefit from beam steering and from muting. The benefits are more subtle when the separation of the traces is less than λ/4. However, in that case the cost of data collection is considerable and often prohibitive. In either case the migration of seismic data can be expedited by use of precalculated tables of migration velocities, ray path distances, and weights (including muting).     

A note on tsunami amplitudes above submarine slides and slumps

Tsunami generated by submarine slumps and slides are investigated in the near-field, using simple source models, which consider the effects of source finiteness and directivity. Five simple two-dimensional kinematic models of submarine slumps and slides are described mathematically as combinations of spreading constant or slopping uplift functions. Tsunami waveforms for these models are computed using linearized shallow water theory for constant water depth and transform method of solution (Laplace in time and Fourier in space). Results for tsunami waveforms and tsunami peak amplitudes are presented for selected model parameters, for a time window of the order of the source duration.The results show that, at the time when the source process is completed, for slides that spread rapidly (c_R/c_T≥20, where c_R is the velocity of predominant spreading), the displacement of the free water surface above the source resembles the displacement of the ocean floor. As the velocity of spreading approaches the long wavelength tsunami velocity the tsunami waveform has progressively larger amplitude, and higher frequency content, in the direction of slide spreading. These large amplitudes are caused by wave focusing. For velocities of spreading smaller than the tsunami long wavelength velocity, the tsunami amplitudes in the direction of source propagation become small, but the high frequency (short) waves continue to be present. The large amplification for c_R/c_T1 is a near-field phenomenon, and at distances greater than several times the source dimension, the large amplitude and short wavelength pulse becomes dispersed.A comparison of peak tsunami amplitudes for five models plotted versus L/h (where L is characteristic length of the slide and h is the water depth) shows that for similar slide dimensions the peak tsunami amplitude is essentially model independent.     

ON THE SIGNIFICANCE OF AMPLITUDE STUDIES IN SHALLOW REFRACTION SEISMICS*

An analysis of amplitudes of refraction records of some shallow refraction profiles shot primarily for detailing the near-surface structure in a granitic terrain has yielded information on refractor properties: reduced amplitudes are plotted on amplitude-distance graphs. The negative power n to which distance should be raised to represent (elastic) amplitude decay with respect to distance due to spreading of the critically refracted wave involved is examined. Computed values of this “spreading index”n are close to n = 2 as predicted by the theory. With this value of n, amplitude data are processed to determine residual attenuation attributable to elastic absorption in the bedrock. A graphical approach for this purpose from comparison of amplitude-distance graphs with the plots of amplitude decay due to spreading which is applicable to flat and horizontal refractor situations is suggested. Assuming residual attenuation to represent absorption in the granite bedrock, the computed coefficients of absorption, which vary from 0.5 to 3.90 km^?1 for a frequency of 50 Hz, are obtained. From amplitude graphs of reversed profiles it is shown that the amplitude differences plot bears a relation to lateral velocity changes in the refractor. From comparison of practical amplitude decay graphs with those computed for different subsurface models, it appears possible to detect fractured rock occurrences in the refractor.     

SEISMIC REFRACTION AND SCREENING BY THIN HIGH-VELOCITY LAYERS *

The screening effect of thin, relatively shallow high-velocity layers often presents considerable problems in seismic exploration. Such layers prevent the greater part of the seismic energy from travelling to greater depths and introduce additional refraction arrivals, confusing the seismogram still further. In order to investigate both the screening and refractive properties of high-velocity layers, scale-model experiments have been made over a wide range of layer-thickness/ wavelength ratios (0.05 < d/λ < 2) for suitably chosen material contrasts. The results may be summarised as follows. Refraction arrivals from thin layers in the field may be recognised by their relatively rapid amplitude decay. Furthermore, the “echeloning”-effect observed for refraction first arrivals may be due to the presence of a (thin) layered structure. Since the apparent refraction velocity varies with d/λ when d/λ < 1, differences between vertical well-log velocities and velocities observed along the surface may be expected, making time/depth conversion using surface velocity data inaccurate. Transmission of elastic energy may be expected, if anywhere, only near the shotpoint, at small geophone offset, and for relatively thin screens (d/λ < 0.1). The transmitted signal shape is then independent of the layer thickness. This transmitted energy may be registered either in a reflection set-up with geophones near the shotpoint, or in long-distance refraction work. Three possibilities are offered for overcoming the screening effect of thin high-velocity layers: Use longer-wavelength signals Apply short-spread reflection shooting Apply long-distance refraction shooting The experimental results obtained in scale-model arrangements of such set-ups confirm the potentialities of these methods.     

利用CAP方法和瑞利面波振幅谱联合反演宁强5.3级地震震源深度

2018年9月12日19时6分,陕西省汉中市宁强县发生5.3级地震,不同机构给出的震源深度结果相差较大。为进一步确定宁强5.3级地震的震源深度,基于区域速度模型,首先利用CAP方法反演得到该地震的震源机制解,然后采用瑞利面波振幅谱和CAP深度误差函数联合反演,进一步测定了此次地震的矩心深度。结果显示:CAP方法得到的陕西宁强5.3级地震矩心深度约为12km,瑞利面波振幅谱测定的矩心深度为13km,结合引入的误差函数联合反演,最终确定陕西宁强5.3级地震的矩心深度为13km左右,表明此次地震仍属于发生于上地壳的地震。     

The Selection of Field Acquisition Parameters for Dispersion Images from Multichannel Surface Wave Data

The accuracy and resolution of surface wave dispersion results depend on the parameters used for acquiring data in the field. The optimized field parameters for acquiring multichannel analysis of surface wave (MASW) dispersion images can be determined if preliminary information on the phase velocity range and interface depth is available. In a case study on a fill slope in Hong Kong, the optimal acquisition parameters were first determined from a preliminary seismic survey prior to a MASW survey. Field tests using different sets of receiver distances and array lengths showed that the most consistent and useful dispersion images were obtained from the optimal acquisition parameters predicted. The inverted S-wave velocities from the dispersion curve obtained at the optimal offset distance range also agreed with those obtained by using direct refraction survey.     

Revisiting the Basin-edge Effect at Kobe During the 1995 Hyogo-Ken Nanbu Earthquake

The basin edge effect, i.e., the interference of the direct S wave with the surface wave diffracted off the basin edge has been invoked by many authors to explain the damage distribution during the January 17, 1995 Hyogo-Ken Nanbu (Kobe) earthquake. Here we present the results of numerical experiments obtained with the spectral element method in 2-D geometry. Our results confirm that the amplification of horizontal motion close to the basin edge can be twice as large as the one measured in the center of the basin. This additional amplification is shown to depend strongly on the edge geometry and on frequency, due to physical dispersion of diffracted surface waves. In particular we obtain maximal amplification around 3 Hz, at frequencies critical for buildings.     

Tidal mixing in sill regions: influence of sill depth and aspect ratio

A non-hydrostatic model in cross-sectional form with an idealized sill is used to examine the influence of sill depth (h _s) and aspect ratio upon internal motion. The model is forced with a barotropic tide and internal waves and mixing occurs at the sill. Calculations using a wide sill and quantifying the response using power spectra show that for a given tidal forcing namely Froude number F _r as the sill depth (h _s) increases the lee wave response and vertical mixing decrease. This is because of a reduction in across sill velocity U _s due to increased depth. Calculations show that the sill Froude number F _s based on sill depth and across sill velocity is one parameter that controls the response at the sill. At low F _s (namely F _s ≪ 1) in the wide sill case, there is little lee wave production, and the response is in terms of internal tides. At high F _s, calculations with a narrow sill show that for a given F _s value, the lee wave response and internal mixing increase with increasing aspect ratio. Calculations using a narrow sill with constant U _s show that for small values of h _s, a near surface mixed layer can occur on the downstream side of the sill. For large values of h _s, a thick well-mixed bottom boundary layer occurs due to turbulence produced by the lee waves at the seabed. For intermediate values of h _s, “internal mixing” dominates the solution and controls across thermocline mixing.     

A simple two‐parameter model for scaling hillslope surface runoff

The objective of this research was to develop and parameterise a physically justified yet low‐parameter model to quantify observed changes in surface runoff ratios with hillslope length. The approach starts with the assumption that a unit of rainfall‐excess runoff generated at a point is a fraction β of precipitation P (m) which travels some variable distance down a slope before reinfiltrating, depending on the local rainfall, climate, soils, etc. If this random distance travelled Y is represented by a distribution, then a survival function will describe the probability of this unit of runoff travelling further than some distance x (m). The total amount of per unit width runoff Q (m²) flowing across the lower boundary of a slope of length λ (m) may be considered the sum of all the proportions of the units of rainfall excess runoff integrated from the lower boundary x = 0 to the upper boundary x = λ of the slope. Using these assumptions we derive a model Q(λ) = βPμλ/(μ + λ), (μ > 0, 0 ≤ β ≤ 1, λ ≥ 0) that describes the change in surface runoff with slope length, where μ (m) is the mean of the random variable Y. Dividing both sides of this equation by Pλ yields a simple two‐parameter equation for the dimensionless hillslope runoff ratio Q_h(λ) = βμ/(μ + λ). The model was parameterised with new rainfall and runoff data collected from three replicates of bounded 2 m wide plots of four different lengths (0.5, 1.0, 2.0 and 4.0 m) for 2 years from a forested SE Australian site, and with 32 slope length–runoff data sets from 12 other published studies undertaken between 1934 and 2010. Using the parameterised model resulted in a Nash and Sutcliffe statistic between observed and predicted runoff ratio (for all data sets combined) of 0.93, compared with –2.1 when the runoff ratio was fixed at the value measured from the shortest plot. Copyright © 2014 John Wiley & Sons, Ltd.     

Effect of the ponderomotive force caused by Alfvén waves on a background plasma in the dayside magnetosphere

The effect of the ponderomotive force on the background plasma modification near magnetic holes, which form at the dayside magnetospheric boundary under the action of the solar wind, has been studied. It was shown that this effect results in a substantial increase in a nonlinear plasma density disturbance. The dependence of the ponderomotive force on the magnetospheric parameters (the magnetic longitude, distance from the Earth’s surface, ratio of the wave frequency to the proton gyrofrequency, and ionospheric ion cyclotron wave amplitude) has been studied. Nonlinear plasma density disturbances will be maximal in the region of magnetic holes, which are located in the dayside magnetosphere at λ ~ 0°?30° geomagnetic longitudes (λ = 0° corresponds to noon), where the effect of the solar wind pressure is maximal. A similar effect is also observed in the dependence of a nonlinear plasma density disturbance on other magnetospheric parameters.     

THE VERTICAL MAGNETIC FIELD OF AN ALTERNATING CURRENT DIPOLE FOR HORIZONTALLY STRATIFIED MEDIA *

For the computation of the vertical component H_z of the magnetic field of a horizontal A.C. dipole lying on the earth's surface, a recurrence formula is presented for a horizontally stratified half space, to obtain the (n+ 1)-layer case from the w-layer case. By means of several computed diagrams for the two-layer case, H_z can be determined for different ratios of conductivity of the subsoil and that of the overburden. Thereby the distance from the dipole as well as the layer thickness h are expressed in terms of the wave length A of a plain wave in the overburden. Assuming a sufficiently large conductivity difference, the results show that evidence about the subsurface conditions can be obtained if the distance between the measuring coil and the dipole is of the order of A/3, and if the thickness h of the layer varies within the range A/100 < h < A/6. As an example for the 3-layer case, a nonconducting intermediate layer is assumed.     

sPL,一个近距离确定震源深度的震相

实际地震波形观测表明,对于大陆结构相对简单的地壳中的地震而言,有一震相出现在P 波和S波之间.一般在30~50 km附近发育得较好,其能量主要集中在径向分量,而垂向分量的振幅相对径向要小,切向分量上的振幅很弱,且波形以低频为主,通常没有P波尖锐.在利用FK方法计算合成地震图的基础上,发现该震相是由S波入射到自由地表形成水平传播的P波(文献称为surface P wave,自由地表P波)或者包括S波入射到地表后形成的多次P波或其散射震相.由于该震相是由S波和P波之间耦合而形成,本文将其定义为sPL(s coupled into P) 震相.理论波形研究表明,sPL相对直达P波的到时差对震中距离不敏感,而随着震源深度的增加几乎呈线性增加,因此可以很好的约束震源深度.本文以2005年江西九江地震为例,证实了sPL确定震源深度的可行性和可靠性.在观测到sPL震相的情况下,离震源50 km以内的一个三分量地震台站的波形就可以帮助获得可靠的震源深度,而不需要精确的震中距离.由于sPL震相出现距离较近,对于较小(三级以上)的地震也可以应用,因此在稀疏台网布局情形下sPL对于确定中小地震深度应该具有很好的应用意义.     

HEAD WAVES IN TWO-DIMENSIONAL SEISMIC MODELS*

Studies made on head waves in a large number of two-dimensional horizontal and inclined layer seismic models are described. The assumption that the amplitude of head wave decreases exponentially with distance is found to be valid in horizontal layers. For the correct estimation of the head wave decay coefficient, the decrease of amplitude due to the divergence of wave energy should always be considered. The variations of head wave amplitude at large distances from the source appear to follow the theory of Heelan (1953). It is, however, concluded that the theory can only be verified from the measurements in models which have large velocity contrast and thick layer at the top. A prominent interference phenomenon between the direct and head wave has been observed near the critical distance. This is more marked for up-dip profile in the case of inclined layer models. The rate of head wave amplitude decrease with distance beyond the zone of interference is found to depend on dip angle, direction of the dip, and velocity contrast.     

Response of vertical deformation on faults to remote strong earthquakes

Vertical coseismic deformation on non-causative fault caused by remote strong earthquakes(epicentral distance≥1500 km,MS≥7.0)are observed by fault-monitoring instruments of new type during recent two years.The monitor-ing result shows,delay time,maximum amplitude and duration of vertical deformation on the non-causative faulthave remarkable close relationship with earthquakes magnitude and epicentral distance.The delay time of verticalcoseismic deformation have positive linear relationship with epicentral distance.The velocity of coseismic defor-mation is 5.5 km/s,close to the velocity of surface wave in granite.The logarithms of maximum amplitude of co-seismic deformation and epicentral distance have remarkable linear relationship with magnitude.The greater themagnitude and the closer the epicentral distance are,the bigger the maximum amplitude of coseismic deformationon non-causative fault will be.Relative to the epicentral distance,the magnitude is the most important factor to theduration of coseismic vertical deformation on the non-causative fault.Stronger earthquake causes longer vibrationduration of coseismic deformation.The experiential equation of co-seismic deformation faults obtained by thiswork is significant on the coseismic deformation research.     

A review of numerical experiments on seismic wave scattering

This paper reviews applications of the finite-difference and finite-element methods to the study of seismic wave scattering in both simple and complex velocity models. These numerical simulations have improved our understanding of seismic scattering in portions of the earth where there is significant lateral heterogeneity, such as the crust. The methods propagate complete seismic wavefields through highly complex media and include multiply scattered waves and converted phases (e.g.,P toSV, SV toP, body wave to surface wave). The numerical methods have been especially useful in cases of moderate and strong scattering in complex media where multiple scattering becomes important. Progress has been made with numerical methods in understanding how near-surface, low-velocity basin structures scatter surface waves and vertically-incident body waves. The numerical methods have proven useful in evaluating scattering of surface waves and body waves from topography of both the free surface and interfaces buried at depth. Numerical studies have demonstrated the importance of conversions from body waves to surface waves (andvice versa) when lateral heterogeneities and topographic relief are present in the uppermost crust. Recently, several investigations have applied numerical methods to study seismic wave propagation in velocity models which vary randomly in space. This stochastic approach seeks to understand the effects of small-scale complexity in the earth which cannot be resolved deterministically. These experiments have quantified the relationships between the statistical properties of the random heterogeneity and the measurable properties of high-frequency (1 Hz) seismograms. These simulations have been applied to the study of many features observed in actual high-frequency seismic waves, including: the amplitude and time decay of seismic coda, the apparent attenuation from scattering, the dispersion of waveforms, and the travel time and waveform variations across arrays of receivers.     

Theoretical and Observed Depth Correction for M s

—?Modal summation technique is used to generate 5000, three-component theoretical seismograms of Love and Rayleigh waves, assuming modified PREM (PREM-C) and AK135F global earth models. The focal depth h and the geometrical fault parameters are randomly chosen so as to uniformly cover possible source mechanisms and obtain uniform distribution of log h in the interval 1?h?h?M _s of the form:¶ΔM _s (h)=0 forh< 20km, ΔM _s (h)=0.314log(h)-0.409 for 20≠h< 60km, ΔM _s (h)=1.351log(h)-2.253 for 60≠h< 100km, ΔM _s (h)=0.400log(h)-0.350 for 100≠h< 600km .¶After applying the above correction, the relationship between the surface wave magnitude and the scalar seismic moment for the observational data set significantly improves, and becomes independent of the source depth. In relation to CTBT, no depth correction is needed for M _S when the m _b ???M _S discriminant is computed, because the proposed correction is zero for earthquakes with foci above 20?km.     

SEISMIC WAVES IN ANELASTIC NON-LINEAR MEDIA—A THEORETICAL CONTRIBUTION *

Numerical solutions of the wave equation for a particular type of non-linear “constant Q” medium were carried out. These solutions were obtained after the transformation of the space derivatives in finite differences; power series of the time are used to express the solutions. The medium is characterized by a not single valued stress-strain relation; the stresses are greater when the absolute values of strain are increasing (passive work), and are less when they are decreasing (active work). A loss of energy follows which is constant for every cycle and independent of frequency. This model represents the simplest type of medium in agreement with the laboratory data on rock samples. For a similar medium the stress’values do not depend only on the instantaneous value of the strain, but also on the previous strain values, i.e. the history of the medium. All these characteristics are not compatible with linearity and require particular types of computing procedures similar to the one shown in this paper. The results of calculations here shown refer both to the propagation of an isolated wave and to the generation of a wave in a spherical hole by change of pressure. They refer particularly to the shape, the amplitude and the width of the isolated wave along the propagation path. The most important results for this type of medium are the following: a) for a plane single isolated wave, the displacement amplitude wave does not change along the propagated distance. The width increases linearly as function of the distance; b) the corresponding particle velocity decreases in amplitude; c) for single isolated spherical waves the displacement amplitude decreases with propagated distance only due to the geometric factor, i.e. inversely proportional to the propagated distance; its width increases in the same way as for plane waves. The comparison between these theoretical results with the field and seismological data show a sufficiently good agreement as far as the value interval of wave width and frequencies is concerned. Less satisfactory is the comparison regarding a linear dependence of the wave width on the distance. This fact happens probably because in the field often long trains of waves and not isolated waves occur. In effect, for trains of waves the behaviour is different than that of an isolated wave; particularly, for the former the frequency variations along the travelled path is less and the displacement variations greater. However, it seems likely that a further similar theoretical research for trains of waves propagating in this type of non-linear medium might be carried out to complete the present research.     

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.