COMPUTATION OF THE NORMAL MOVEOUT VELOCITY IN 3D LATERALLY INHOMOGENEOUS MEDIA WITH CURVED INTERFACES*

For a 3D velocity model of curved first order interfaces and layer velocities which are arbitrary smooth functions of the space coordinates, the normal moveout (NMO)-velocity can be computed by numerically integrating a system of first order ordinary differential equations for a hypothetical wavefront that originates at the normal incidence point of the normal ray and moves up along the ray to the common mid-point of the common datum point (CDP) profile.     

PARAMETER OPTIMIZATION OF VELOCITY DEPTH FUNCTIONS OF GIVEN FORM BY USE OF ROOT-MEAN-SQUARE VELOCITIES*

From seismic surveys zero offset reflection times and root-mean-square velocities are obtained. By use of Dix-Krey's formula, the interval velocities can be calculated. If no well velocity survey exists, the interval velocities and T(o) times are the only available information. The suggested way to get a regionally valid velocity distribution is to select N“leading horizons”, where a major change in the velocity parameters occurs and to compute the parameters of the selected velocity depth function (in most cases linear increase with depth) by a special approximation for the interval between two adjacent “leading horizons”. Herewith all reflection horizons within the interval are taken into account.     

OBTAINING INTERVAL VELOCITIES FROM STACKING VELOCITIES WHEN DIPPING HORIZONS ARE INCLUDED *

Common-depth-point stacking velocities may differ from root-mean-square velocities because of large offset and because of dipping reflectors. This paper shows that the two effects may be treated separately, and proceeds to examine the effect of dip. If stacking velocities are assumed equal to rms velocities for the purpose of time to depth conversion, then errors are introduced comparable to the difference between migrated and unmigrated depths. Consequently, if the effect of dip on stacking velocity is ignored, there is no point in migrating the resulting depth data. For a multi-layered model having parallel dip, a formula is developed to compute interval velocities and depths from the stacking velocities, time picks, and time slope of the seismic section. It is shown that cross-dip need not be considered, if all the reflectors have the same dip azimuth. The problem becomes intractable if the dips are not parallel. But the inverse problem is soluble: to obtain, stacking velocities; time picks, and time slopes from a given depth and interval velocity model. Finally, the inverse solution is combined with an approximate forward solution. This provides an iterative method to obtain depths and interval velocities from stacking velocities, time picks and time slopes. It is assumed that the dip azimuth is the same for all reflectors, but not necessarily in the plane of the section, and that the curvature of the reflecting horizons is negligible. The effect of onset delay is examined. It is shown that onset corrections may be unnecessary when converting from time to depth.     

A FREQUENCY DOMAIN MAPPING APPROXIMATION OF MOVEOUT FILTERS WITH APPLICATIONS*

Wavenumber aliasing is the main limitation of conventional optimum least-squares linear moveout filters: it prevents adequate reject domain weighting for efficient coherent noise rejection. A general frequency domain multichannel filter design technique based on a one-to-one mapping method between two-dimensional (2D) space and one-dimensional (1D) space is presented. The 2D desired response is mapped to the 1D frequency axis after a suitable sorting of the coefficients. A min-max or Tchebycheff approximation to the desired response is obtained in the 1D frequency domain and mapped back to the 2D frequency domain. The algorithm is suitable for multiband 2D filter design. No aliasing damage is inherent in the linear moveout filters designed using this technique because the approximation is done in the frequency-wavenumber (f, k)-domain. Linear moveout filters designed by using the present coefficient mapping technique achieve better pass domain approximations than the corresponding conventional least-squares filters. Compatible reject domain approximations can be obtained from suitable mappings of the origin coefficient of the desired (f k)-response to the 1D frequency axis. The (fk)-responses of linear moveout filters designed by using the new technique show equi-ripple behavior. Synthetic and real data applications show that the present technique is superior to the optimum least-squares filters and straight stacking in recovering and enhancing the signal events with relatively high residual statics. Their outputs also show higher resolution than those of the optimum least-squares filters.     

MAPPING OF TECTONICS OF SEDIMENTS BY AIRBORNE MAGNETICS*

The considerable increase of sensitivity of new magnetometers allows the elaboration of magnetometric maps with contour intervals of fractions of one gamma. On these maps, small anomalies appear which had never been visible before. They are caused by susceptibility contrasts within the sediments. The areal distribution of these small undulations of the magnetic field depends on tectonics, therefore their analysis can give useful information about the tectonic pattern of the sediments. This paper discusses the automatic transformation of these small anomalies into tectonic units. The problem is solved by a method called Digital Template Analysis which has already yielded very useful results in gravity interpretation. The application of the method described is restricted to surveys executed by high sensitivity magnetometers. Therefore it is supposed that in most oil exploration problems these magnetometers will replace instruments with lower sensitivity. The improved magnetometer surveys in combination with the interpretation method described represent a most efficient, fast and unexpensive reconnaissance method.     

MIGRATION OF SEISMIC INTERFACES*

Most interpretation work is still based on stacked and not on migrated sections. In the case of heavy faulting and considerable velocity contrasts between formations, migration of interpreted interfaces poses a problem. In more detail, the problem may be specified as follows: — a given interpretation of a number of interfaces along with a given heterogeneous velocity field may not always have a plausible solution in the form of migrated interfaces in depth; — fault planes, salt boundaries, etc., are, in most cases, not directly interpretable in a section and are plotted by intuition using interface terminations as a guide; — the velocity field in fault zones is, in most cases, hard to determine. The interpreter may arrive at a plausible solution by repeating the migration process with various possible interpretations and various velocity assumptions. The subject of this paper is an algorithm based on ray-theory which allows one: — to handle faults and velocity variations at faults properly; — to perform migration in steps, working a particular geological unit at a time and proceeding to the next unit once the foregoing one has been properly migrated; — to display ray-paths, where necessary, for investigation of interface distortions, e.g., below fault areas. The algorithm is designed and implemented for application in an interactive environment. Inspection of intermediate and final results, investigation of interface distortions and modifications are performed on a graphics screen. Thus, various possible interpretations and velocity assumptions may be investigated within a short time. Interfaces interpreted on migrated sections may be over-migrated because of neglection of the influence of refraction in most section migration programs. This over-migration may also be corrected using the above algorithm in the “image ray” mode.     

LIMITS OF STRUCTURAL INVERSION OF SEISMIC HORIZONS*

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

EXPLORING DEEP INTERFACES BY SEISMIC WIDE ANGLE MEASUREMENTS*

Amplitudes of the vertical ground motion are larger in the critical angle region than in the near vertical region, especially amplitudes coming from deeper boundaries or interfaces with small velocity contrast. Four basic boundary models are used to derive specific travel time patterns for first order and higher order interfaces. The transition from reflected waves to diving waves in the supercritical angle region is shown for the different models. Examples from wide angle surveys in Bavaria and in the Buchara region in the USSR have many characteristic similarities. Travel time patterns and asymptotic velocities are almost identical. The most important boundaries of the continental crust - the Mohorov? I? discontinuity and the top of the crystalline basement - correspond to two boundary models mentioned before. Amplitude and frequency investigations have been used to derive items of the M-discontinuity. It is certainly a gradient zone and seems to have a lenticular structure with material of stepwise different velocities.     

ON TRACING SEISMIC RAYS WITH SPECIFIED END POINTS IN LAYERS OF CONSTANT VELOCITY AND PLANE INTERFACES*

A polygonal ray path connects the seismic source and detector positions when the intervening medium consists solely of constant velocity layers with plane interfaces which may have arbitrary orientation. The coordinates of the ray vertices satisfy a system of coupled equations resulting from the requirement that Fermat's principle be satisfied along the ray path. Solving the system of equations is equivalent to tracing the ray numerically. A notable feature of this approach is that a ray which is critically refracted over a segment of its path requires no special handling.     

INTERVAL VELOCITY ANALYSIS BY WAVE EXTRAPOLATION*

A method for interval velocity analysis is formulated on the basis of wavefield extrapolation, i.e., on the basis of wave-equation migration. When this scheme is applied to multioffset seismic sections or to an ensemble of CMP gathers, it allows for the proper treatment of dipping events. The underlying assumptions are that local velocities should be derived from data associated with events within the interval under consideration. To minimize the effect of the region above the layer of interest, the data are first extrapolated to the top of the analysis interval. Subsequent analysis of these data then pertains to the events within this interval. Velocity estimation consists of repeated wavefield extrapolations through the analysis interval using a set of trial velocities. The optimal velocity is chosen on the basis of coherency measures designed to express the collective phase agreement among a set of offset Fourier modes. The reliability of this approach to interval velocity estimation is demonstrated on synthetic multi-offset data.     

REFLECTION-WINDOW MAPPING OF SHALLOW BEDROCK*

A field study was undertaken to evaluate the effectiveness of the high frequency seismic reflection technique for mapping of shallow and irregular bedrock. Bedrock reflections were obtained using a hammer source with both in-line and common offset field layouts. The recording equipment included 12-channel enhancement seismographs, 28 Hz vertical geo-phones and a microcomputer. The latter increased the overall versatility of the seismic system. Field sites for this study are typical of the geological settings of the tin mining areas of Malaysia. The topographical ‘lows’ of the irregular bedrock control the localization of tin ore. The subsurface geology consists of a thin low velocity layer (± 300 m/s) overlying the compact overburden (± 1700 m/s) which in turn lies on bedrock. This paper discusses various criteria for designing an optimum window for obtaining usable reflections between the first arrival and the leading edge of the ground roll cone. Detailed mapping of the overburden and the bedrock interface by the reflection method can be useful in delineating areas for exploratory drilling and for optimum planning of mining operations.     

保角变换在区域面波群速度反演中的应用

通过保角变换，把一个球面局部区域扩展到球面上更大的球台域，再由球台域解析延拓到整个球面域.作为约束条件，变换过程中面波速度保持不变.在变换后的球面域上用球谐函数来拟合速度函数，达到降低球谐系数阶数的目的.面波群速度的反演变成了球谐系数的线性化反演，由球谐系数计算反演分辨核以及方差.该方法的优点是计算速度快，等值线光滑，构造界限清晰，该方法同样适用于各种局部球面区域的分析.     

SOIL CONSOLIDATION BEHAVIOR ASSESSED BY SEISMIC VELOCITY MEASUREMENTS*

Large gravity platforms are often used as alternatives to the more conventional pilesupported structures in hydrocarbon exploitation. A gravity structure, as opposed to the piled structure, is sitting on the sea floor by virtue of its weight and base width; as such it poses considerable problems for the site investigation engineer. One such problem is the calculation of the settlement of the structure and its time history; these depend upon the permeability and compressibility of the soil and its drainage conditions. The required data are usually obtained by sampling for subsequent laboratory testing. The collection of an undisturbed sample is beset by problems so that the consolidation behavior of the foundation material can only be inadequately assessed by laboratory testing. However, a series of laboratory consolidation experiments during which seismic velocities have been measured on the sample as consolidation proceeds shows that it is possible to reconstruct the stress-strain and time-dependent curves from the seismic data, once the initial void ratio and permeability of the soil are known. This leads the way to an in situ technique for predicting settlement using a combination of geophysical techniques (electrical resistivity and seismic velocities) to obtain the required engineering properties.     

PRACTICAL ASPECTS OF THE DETERMINATION OF 3-D STACKING VELOCITIES*

Proper stacking of three-dimensional seismic CDP-data generally requires the knowledge of normal moveout velocities in all source-receiver directions contributing to a CDP-gather. The azimuthal variation of the stacking velocities mainly depends on the dip of the seismic interfaces. For a single dipping plane a simple relation exists between the dip and the azimuthal variation of NMO-velocity. Varying strike and dip of subsequent reflectors, however, result in a complex dependency of the seismic parameters. Reliable information on the spatial distribution of the normal moveout (NMO)-velocity can be derived from a wavefront curvature estimation using a 3-D ray-tracing technique. These procedures require additional information, e.g. reflection time gradients or depth maps to show interval velocities between leading interfaces. Moreover, their application to an extended 3-D data volume is restricted by high costs. The need for a routine 3-D procedure resulted in a special data selection to create pseudo 2-D profiles and to apply existing velocity estimation routines to these profiles. At least three estimates in different directions are necessary to derive the full azimuthal velocity variation, characterized by the large and the small main axis and the orientation of the velocity ellipse. Errors are estimated by means of computer models. Stacking velocities obtained by mathematical routines (least-squares fit) and by seismic standard routines (NMO-correction and correlation) are compared. Finally, a general 3-D velocity procedure using cross-correlation of preliminarily NMO-corrected traces is proposed.     

VELOCITY DISPERSION OF SEISMIC WAVES*

In well velocity surveys made to calibrate Sonic (CV) Logs the calibration survey uses frequencies around 50 Hz whereas the Sonic Logging tool uses frequencies around 20 kHz. There thus exists the possibility of making a direct measure of velocity dispersion. In any one survey the disturbing factors, both instrumental and operational, will often mask any dispersive effect that might exist. Consequently this paper reports on a statistical analysis of the velocity differences resulting from calibration surveys and Sonic logs. Only Borehole Compensated Sonic Logs were used. Four areas were investigated: the North Sea, Abu Dhabi, Libya and Alaska. After rejecting logs and calibration records which were obviously in error there remained 424000 feet (about 130 km) of usable log distributed throughout 66 wells. The four areas were analysed separately and in no case was the estimated dispersion significantly different from zero. However, the mean values did correlate with lithology from (? 0.17 ± 0.18)% for the essentially carbonate section in Abu Dhabi to (+ 0.45 ± 0.25)% for the sand-shale section in Alaska, a positive sign meaning that the higher frequencies travelled faster. Except for Alaska the calibration surveys were made with a wall-clamp geophone, and for these areas amplitude measurements were made. After suitable corrections estimates of the absorption parameter Q were obtained. These varied from 20 to 200 with mean values of 63 for Libya, 70 for Abu Dhabi and 88 for the North Sea (excluding the Tertiary). If, as is usually assumed, the absorption mechanism is linear and is described by a Q which is independent of frequency, then these values would necessarily imply dispersion of several percent. As instanced above no such dispersion was observed. It is possible that the expected dispersion was compensated for by invasion of the mud filtrate into the borehole walls, but it is more likely that the absorption mechanism was substantially non-linear.     

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.     

保角变换在区域面波群速度反演中的应用

通过保角变换,把一个球面局部区域扩展到球面上更大的球台域,再由球台域解析延拓到整个球面域.作为约束条件,变换过程中面波速度保持不变.在变换后的球面域上用球谐函数来拟合速度函数,达到降低球谐系数阶数的目的.面波群速度的反演变成了球谐系数的线性化反演,由球谐系数计算反演分辨核以及方差.该方法的优点是计算速度快,等值线光滑,构造界限清晰,该方法同样适用于各种局部球面区域的分析.     

MIGRATION OF VSP DATA BY RAY EQUATION EXTRAPOLATION IN 2-D VARIABLE VELOCITY MEDIA*

The standard Kirchhoff algorithm can be generalized for migration of pre-stack finite-offset data from variable-velocity media. The concentric ellipses over which the data are spread in constant velocity media become significantly distorted (even multi-valued) in the variable velocity case. The specific shapes can be explicitly defined by kinematic extrapolation of the source and recorded wave fields with the ray equation. The use of Kirchhoff migration with a surface source and a subsurface recorder requires that two sets of Kirchhoff loci be superimposed. For each trace, the first set of loci is computed with the source and the actual recorder position as foci; the second set is computed with the source and the virtual recorder position as foci. This dual procedure explicitly incorporates the primary diffracted energy and the free-surface reflections, respectively. Implementation involves the construction of a virtual medium, lying above the free surface, with a velocity distribution that is the mirror image of the actual distribution below the free surface. Ray-equation extrapolation is performed through the real/virtual boundary. The resulting image is produced in a split form, with all the contributions of the primary reflected and diffracted energy lying in the lower ‘real’ half and all the contributions of the energy that was reflected at the free surface lying in the upper ‘virtual’ half. The final image is produced by folding the split image about the free surface and adding the two halves. A practical advantage is that the origin of various contributions (and artifacts) can be more readily identified (for interpretation or removal) in the split images. The ray-equation pre-stack migration algorithm is very general. It is applicable to all source-recorder geometries and variable velocity media and reduces exactly to the standard Kirchhoff algorithm when applied to zero or finite-offset surface survey data. The algorithm is illustrated by application to VSP data. For the VSP geometry, the algorithm does not require any specific trace spacing (in depth) and can be used for data from deviated as well as vertical holes.     

THE CORRELATION OF SEISMIC VELOCITIES WITH FORMATIONS IN THE SOUTH-WEST OF SCOTLAND *

In seismic studies being carried out to elucidate the structure at depth of the Midland Valley rift and the Tertiary igneous province in the West of Scotland, a lack of deep boreholes makes the collection of velocity data imperative for identification of seismic events and for determinations of depth. Three methods are used to investigate the correlation of seismic velocities and geological formations. The results show as strong a dependence of velocity on method of measurement as on lithology and the wide spread of values within a given formation makes the attribution of discrete velocity ranges to specific formations impossible. Changes of velocity are more significant than absolute values. The variations in both are discussed.     

INVERSION OF SEISMOGRAMS AND PSEUDO VELOCITY LOGS*

Pseudo velocity logs can be obtained by seismogram inversion, using true amplitude processing and detailed investigation of move-out velocities. The precision of the results depends on the quality of the seismic data and on the possibility of deconvolving without increasing the noise. An investigation is made of the deformation of pseudo logs due to seismic signal variations and to imperfections of deconvolution. Both marine and land examples are shown, in some cases with adjustment on well logs. When the dips are large, time sections must be migrated and pseudo velocity logs must be computed from migrated sections. Comparison of sonic logs with pseudo velocity logs obtained in the same area is usually good enough to obtain information on lithological parameter variations by adjustment of pseudo velocity logs on sonic logs. Even when no well is available, pseudo velocity logs can give some indications on the nature of sediments between seismic horizons.