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).     

Piecewise‐continuous velocity inversion

We suggest a new method to determine the piecewise‐continuous vertical distribution of instantaneous velocities within sediment layers, using different order time‐domain effective velocities on their top and bottom points. We demonstrate our method using a synthetic model that consists of different compacted sediment layers characterized by monotonously increasing velocity, combined with hard rock layers, such as salt or basalt, characterized by constant fast velocities, and low velocity layers, such as gas pockets. We first show that, by using only the root‐mean‐square velocities and the corresponding vertical travel times (computed from the original instantaneous velocity in depth) as input for a Dix‐type inversion, many different vertical distributions of the instantaneous velocities can be obtained (inverted). Some geological constraints, such as limiting the values of the inverted vertical velocity gradients, should be applied in order to obtain more geologically plausible velocity profiles. In order to limit the non‐uniqueness of the inverted velocities, additional information should be added. We have derived three different inversion solutions that yield the correct instantaneous velocity, avoiding any a priori geological constraints. The additional data at the interface points contain either the average velocities (or depths) or the fourth‐order average velocities, or both. Practically, average velocities can be obtained from nearby wells, whereas the fourth‐order average velocity can be estimated from the quartic moveout term during velocity analysis. Along with the three different types of input, we consider two types of vertical velocity models within each interval: distribution with a constant velocity gradient and an exponential asymptotically bounded velocity model, which is in particular important for modelling thick layers. It has been shown that, in the case of thin intervals, both models lead to similar results. The method allows us to establish the instantaneous velocities at the top and bottom interfaces, where the velocity profile inside the intervals is given by either the linear or the exponential asymptotically bounded velocity models. Since the velocity parameters of each interval are independently inverted, discontinuities of the instantaneous velocity at the interfaces occur naturally. The improved accuracy of the inverted instantaneous velocities is particularly important for accurate time‐to‐depth conversion.     

SEISMIC MIGRATION IN ELLIPTICALLY ANISOTROPIC MEDIA1

The study of wave propagation in media with elliptical velocity anisotropy shows that seismic energy is focused according to the horizontal component of the velocity field while the vertical component controls the time-to-depth relation. This implies that the vertical component cannot be determined from surface seismic velocity analysis but must be obtained using borehole or regional geological information. Both components of the velocity field are required to produce a correctly focused depth image. A paraxial wave equation is developed for elliptical anisotropic wave propagation which can be used for modelling or migration. This equation is then transformed by a change of variable to a second paraxial equation which only depends on one effective velocity field. A complete anisotropic depth migration using this transformed equation involves an imaging step followed by a depth stretching operation. This allows an approximate separation or splitting of the focusing and depth conversion steps of depth migration allowing a different velocity model to be used for each step. This split anisotropic depth migration produces a more accurate result than that obtained by a time migration using the horizontal velocity field followed by an image-ray depth conversion using the vertical velocity field. The results are also more accurate than isotropic depth migration and yield accurate imaging in depth as long as the lateral variations in the anisotropy are slow.     

基于成像射线的偏移剖面和速度时深转换评述

深度域地震波速度模型是地震勘探中最重要的参数之一,是获得精确深度域偏移剖面的基础.本文基于成像射线的概念,阐述了时间偏移域和深度域之间的关系,明确了时间偏移后的像点是通过成像射线与深度域的散射点联系起来的.在有横向变速的情况下,时间偏移剖面需要经过时深转换才能真实反映深度域层位,同样,从时间偏移剖面上提取的速度也需要经过时深转换才能得到深度域的速度.与前者不同的是,在速度的时深转换中,不仅速度的位置需要改变,且其数值也需要进行校正.该校正因子有几种等价的描述形式：度规量、速度扩散因子和几何扩散因子.     

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

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

THREE-TERM TAYLOR SERIES FOR t2 - x2-CURVES OF P- AND S-WAVES OVER LAYERED TRANSVERSELY ISOTROPIC GROUND*

The arrival-time curve of a reflection from a horizontal interface, beneath a homogeneous isotropic layer, is a hyperbola in the x - t-domain. If the subsurface is one-dimensionally inhomogeneous (horizontally layered), or if some or all of the layers are transversely isotropic with vertical axis of symmetry, the statement is no longer strictly true, though the arrival-time curves are still hyperbola-like. In the case of transverse isotropy, however, classical interpretation of these curves fails. Interval velocities calculated from t² - x²-curves do not always approximate vertical velocities and therefore cannot be used to calculate depths of reflectors. To study the relationship between velocities calculated from t² - x²-curves and the true velocities of a transversely isotropic layer, we approximate t² - x²-curves over a vertically inhomogeneous transversely isotropic medium by a three-term Taylor series and calculate expressions for these terms as a function of the elastic parameters. It is shown that both inhomogeneity and transverse isotropy affect slope and curvature of t² - x²-curves. For P-waves the effect of transverse isotropy is that the t² - x²-curves are convex upwards; for SV-waves the curves are convex downwards. For SH-waves transverse isotropy has no effect on curvature.     

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

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

3D KINEMATIC INVERSION FROM A SET OF LINE PROFILES1

In the case of 3D multilayered structures the 2D interval velocity analysis may be inaccurate. This fact is illustrated by synthetic examples. The method proposed solves the 3D inverse problem within the scope of the ray approach. The solution, i.e. the interval velocities and the reflection interface position, is obtained using data from conventional 2D line profiles arbitrarily located and from normal incidence time maps. Although the input information is essentially limited, the method presented reveals only minor biased velocity estimates. In order to implement the proposed 3D inversion method, we developed a processing procedure. The procedure performs the evaluation of reflection time and ray parameters along line profiles, 3D interval velocity estimation, and time-to-depth map migration. Tools to stabilize the 3D inversion are investigated. The application of the 3D inversion technique to synthetic and real data is compared with results of the 2D inversion.     

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.     

Estimation of Thomsen＇s anisotropic parameters from walkaway VSP and applications

Estimation of Thomsen＇s anisotropic parameters is very important for accuratetime-to-depth conversion and depth migration data processing. Compared with othermethods, it is much easier and more reliable to estimate anisotropic parameters that arerequired for surface seismic depth imaging from vertical seismic profile （VSP） data, becausethe first arrivals of VSP data can be picked with much higher accuracy. In this study, wedeveloped a method for estimating Thomsen＇s P-wave anisotropic parameters in VTImedia using the first arrivals from walkaway VSP data. Model first-arrival travel times arecalculated on the basis of the near-offset normal moveout correction velocity in VTI mediaand ray tracing using Thomsen＇s P-wave velocity approximation. Then, the anisotropicparameters 0 and e are determined by minimizing the difference between the calculatedand observed travel times for the near and far offsets. Numerical forward modeling, usingthe proposed method indicates that errors between the estimated and measured anisotropicparameters are small. Using field data from an eight-azimuth walkaway VSP in TarimBasin, we estimated the parameters 0 and e and built an anisotropic depth-velocity modelfor prestack depth migration processing of surface 3D seismic data. The results showimprovement in imaging the carbonate reservoirs and minimizing the depth errors of thegeological targets.     

Migration velocity analysis for tilted transversely isotropic media

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

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.     

RADON DEPTH MIGRATION1

A depth migration method is presented that uses Radon-transformed common-source seismograms as input. It is shown that the Radon depth migration method can be extended to spatially varying velocity depth models by using asymptotic ray theory (ART) to construct wavefield continuation operators. These operators downward continue an incident receiver-array plane wave and an assumed point-source wavefield into the subsurface. The migration velocity model is constrained to have longer characteristic wavelengths than the dominant source wavelength such that the ART approximations for the continuation operators are valid. This method is used successfully to migrate two synthetic data examples:

• 1 a point diffractor, and
• 2 a dipping layer and syncline interface model.

It is shown that the Radon migration method has a computational advantage over the standard Kirchhoff migration method in that fewer rays are computed in a main memory implementation.     

Inversion of normal moveout for monoclinic media1

Multiple vertical fracture sets, possibly combined with horizontal fine layering, produce an equivalent medium of monoclinic symmetry with a horizontal symmetry plane. Although monoclinic models may be rather common for fractured formations, they have hardly been used in seismic methods of fracture detection due to the large number of independent elements in the stiffness tensor. Here, we show that multicomponent wide-azimuth reflection data (combined with known vertical velocity or reflector depth) or multi-azimuth walkaway VSP surveys provide enough information to invert for all but one anisotropic parameters of monoclinic media. In order to facilitate the inversion procedure, we introduce a Thomsen-style parametrization for monoclinic media that includes the vertical velocities of the P-wave and one of the split S-waves and a set of dimensionless anisotropic coefficients. Our notation, defined for the coordinate frame associated with the polarization directions of the vertically propagating shear waves, captures the combinations of the stiffnesses responsible for the normal-moveout (NMO) ellipses of all three pure modes. The first group of the anisotropic parameters contains seven coefficients (ε^(1,2), δ^(1,2,3) and γ^(1,2)) analogous to those defined by Tsvankin for the higher-symmetry orthorhombic model. The parameters ε^(1,2), δ^(1,2) and γ^(1,2) are primarily responsible for the pure-mode NMO velocities along the coordinate axes x₁ and x₂ (i.e. in the shear-wave polarization directions). The remaining coefficient δ⁽³⁾ is not constrained by conventional-spread reflection traveltimes in a horizontal monoclinic layer. The second parameter group consists of the newly introduced coefficients ζ^(1,2,3) which control the rotation of the P-, S₁- and S₂-wave NMO ellipses with respect to the horizontal coordinate axes. Misalignment of the P-wave NMO ellipse and shear-wave polarization directions was recently observed on field data by Pérez et al. Our parameter-estimation algorithm, based on NMO equations valid for any strength of the anisotropy, is designed to obtain anisotropic parameters of monoclinic media by inverting the vertical velocities and NMO ellipses of the P-, S₁- and S₂-waves. A Dix-type representation of the NMO velocity of mode-converted waves makes it possible to replace the pure shear modes in reflection surveys with the PS₁- and PS₂-waves. Numerical tests show that our method yields stable estimates of all relevant parameters for both a single layer and a horizontally stratified monoclinic medium.     

COMMON OFFSET PLANE MIGRATION (COPMIG)*

The quality of results of migration before stack is sensitive to inaccuracies in the velocity field applied. This does not hold if only traces of similar sources-receiver distances (common offset traces) enter the migration process. In this case, velocity deviations generate minor shifts in travel times of migrated interfaces but no deterioration in quality. These time shifts are proportional to both the velocity error and the square of the source-receiver distance. The above observations suggest the following migration scheme: migrate separately the traces of the various common offset planes or groups of neighbouring common offset planes; for every common midpoint plane and as a function of travel-time perform a residual NMO search to find trajectories t) =t)_o+px)² of maximum coherency along which migrated events are aligned; correct for residual NMO and stack the migration results obtained in the various common offset planes to obtain the final migration result. This process not only takes care of inaccurate migration velocities but also corrects partly for effects of refraction. It is shown by means of an example that good migration results are generated even with a considerably deviating velocity field.     

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

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

COMPUTATION OF INTERVAL VELOCITIES FROM COMMON REFLECTION POINT MOVEOUT TIMES FOR n LAYERS WITH ARBITRARY DIPS AND CURVATURES IN THREE DIMENSIONS WHEN ASSUMING SMALL SHOT-GEOPHONE DISTANCES*

It is well known that interval velocities can be determined from common-reflection-point moveout times. However, the mathematics becomes complicated in the general case of n homogeneous layers with curved interfaces dipping in three dimensions. In this paper the problem is solved by mathematical induction using the second power terms only of the Taylor series which represents the moveout time as a function of the coordinate differences between shot and geophone points. Moreover, the zero-offset reflection times of the nth interface in a certain area surrounding the point of interest have to be known. The n—I upper interfaces and interval velocities are known too on account of the mathematical induction method applied. Thus, the zero-offset reflection raypath of the nth interface can be supposed to be known down to the intersection with the (n—1)th interface. The method applied consists mainly in transforming the second power terms of the moveout time from one interface to the next one. This is accomplished by matrix algebra. Some special cases are discussed as e.g. uniform strike and small curvatures.     

Seismic Wave Velocities in Deep Sediments in Poland: Borehole and Refraction Data Compilation

Sedimentary cover has significant influence on seismic wave travel times and knowing its structure is of great importance for studying deeper structures of the Earth. Seismic tomography is one of the methods that require good knowledge of seismic velocities in sediments and unfortunately by itself cannot provide detailed information about distribution of seismic velocities in sedimentary cover. This paper presents results of P-wave velocity analysis in the old Paleozoic sediments in area of Polish Lowland, Folded Area, and all sediments in complicated area of the Carpathian Mountains in Poland. Due to location on conjunction of three major tectonic units — the Precambrian East European Craton, the Paleozoic Platform of Central and Western Europe, and the Alpine orogen represented by the Carpathian Mountains the maximum depth of these sediments reaches up to 25 000 m in the Carpathian Mountains. Seismic velocities based on 492 deep boreholes with vertical seismic profiling and a total of 741 vertical seismic profiles taken from 29 seismic refraction profiles are analyzed separately for 14 geologically different units. For each unit, velocity versus depth relations are approximated by second or third order polynomials.