Unstacked Data Research Articles

Estimation of kinematic wavefront characteristics and their use for multiple attenuation

Kinematic wavefront characteristics of primary reflected waves can be used to predict and attenuate surface-, as well as interbed multiples. In order to estimate kinematic wavefront characteristics directly from unstacked data the common-shot-point homeomorphic-imaging (CSP HI) method can be applied. This macro-model-independent method is based on a new local moveout correction that depends on two wavefront parameters: the emergence angle and the radius of wavefront curvature of a reflected primary wavefront. These parameters can be estimated by optimizing the semblance correlation measure, calculated in the common-shot-gather along the travel time curve defined by the new moveout correction. In the case of local maxima in the semblance functional, an automatic maximization procedure might lead to a wrong estimation of these parameters. In order to avoid such a situation we propose an interactive, horizon-based implementation of the CSP HI method, which allows manual picking of the optimal wavefront parameters along the seismic line. Afterwards, we use the estimated emergence angles for the prediction and attenuation of multiples, based on the simple but powerful idea that any multiple event can be represented as a combination of primaries. A synthetic example shows the viability of the parameter estimation, prediction and attenuation procedure.

Multiple prediction using the homeomorphic‐imaging technique

A new method for predicting different kinds of multiples and peg‐leg reflections in unstacked seismic data is discussed. The basis for this method is the fact that kinematic properties of multiples can be represented as a combination of kinematic properties of primary reflections. The prediction is made using a two‐step process. In the first step, the values for the angle of emergence and radius of curvature of the wavefront for primary reflections from ‘multiple‐generating’ interfaces are obtained. These parameters are estimated directly from unstacked data for every source point using the homeomorphic‐imaging technique. The second step consists of prediction of multiples from primary reflections that satisfy a so‐called ‘multiple condition’. This condition is the equality of the absolute values of the angles of emergence calculated from the first step. This method is effective even in complex media and information on the subsurface geology is not required. The parameters are estimated directly from the unstacked data and do not require any computational efforts such as in wavefield extrapolation of data.

Migration velocity analysis with the Kirchhoff integral

To obtain accurate migration velocities, we must estimate the velocity at migrated depth points. Wavefront focusing analysis with downward continuation yields the rms velocity at migrated depth points; however, the large amount of computation required for downward continuation limits use of this approach for routine processing. The purpose of this paper is to present an implementation of the Kirchhoff integral which makes the wavefront focusing analysis practical for time‐migration velocity analysis. Downward continuation focuses the wavefront to the zero offset at the depth controlled by the velocity used for the continuation. The migration velocity is then determined from the depth where the focused wavefront attains the maximum amplitude. The flexibility of the Kirchhoff integral allows us to compute only the zero‐offset trace at each depth point and lets us avoid most of the computation for the downward continuation of unstacked data. Furthermore, since the velocity is obtained from the location where the focused wavefront shows the maximum amplitude, prestack time migration with the velocity from this technique produces the maximum amplitude for the subsurface reflector.

Reference velocity model estimation from prestack waveforms: Coherency optimization by simulated annealing

Coherency inversion, which consists of maximizing a semblance function calculated from unstacked seismic waveforms, has the potential of estimating reliable velocity information without requiring traveltime picking on unstacked data. In this work, coherency inversion is based on the assumption that reflectors’ zero‐offset times are known and that the velocity in each layer may vary laterally. The method uses a type of Monte Carlo technique termed the generalized simulated annealing method for updating the velocity field. At each Monte Carlo step, time‐to‐depth conversion is performed. Although this procedure is slow at convergence to the global minimum, it is robust and does not depend on the initial model or topography of the objective function. Applications to both synthetic and field data demonstrate the efficiency of coherency inversion for estimating both lateral velocity variations and interface depth positions.

LITHOPROBE seismic reflection imaging of Rocky Mountain structures east of Canal Flats, British Columbia

LITHROPROBE seismic reflection data, coupled with information from industry seismic data and surface geology, image the thin-skinned structures of the western Rocky Mountains from the Main Ranges to the Rocky Mountain Trench near Canal Flats, British Columbia. Reprocessing of the LITHOPROBE seismic reflection line was conducted to improve resolution of upper-crustal features. Careful application of "conventional" processing techniques significantly improved the coherence of reflections from the first 6 s. A spatial semblance filter was applied to further enhance coherent signal, and residual-statics corrections were applied by cross correlation of unstacked data with semblance-filtered pilot traces.A near-basement reflection zone arising from Middle Cambrian strata is visible on an industry reflection profile at an approximate depth of 8 km beneath the Main Ranges. A similar reflection zone is imaged on the LITHOPROBE data at a depth of 11 km bsl but is interpreted as arising from Proterozoic strata. The autochthonous crystalline basement is interpreted as being below these layers and dipping about 2 °to the west. Geometric evidence is visible for several major thrust ramps involving the basal décollement and for an intermediate-level décollement that loses displacement into folds within the Porcupine Creek Anticlinorium. Reflections related to the Gypsum fault, the Redwall thrust, and the Lussier River normal fault are also imaged.

METHOD FOR DETERMINATION OF VELOCITY AND DEPTH FROM SEISMIC REFLECTION DATA1

ABSTRACTThe estimation of velocity and depth is an important stage in seismic data processing and interpretation. We present a method for velocity‐depth model estimation from unstacked data. This method is formulated as an iterative algorithm producing a model which maximizes some measure of coherency computed along traveltimes generated by tracing rays through the model. In the model the interfaces are represented as cubic splines and it is assumed that the velocity in each layer is constant. The inversion includes the determination of the velocities in all the layers and the location of the spline knots.The process input consists of unstacked seismic data and an initial velocity‐depth model. This model is often based on nearby well information and an interpretation of the stacked section.Inversion is performed iteratively layer after layer; during each iteration synthetic travel‐time curves are calculated for the interface under consideration. A functional characterizing the main correlation properties of the wavefield is then formed along the synthetic arrival times. It is assumed that the functional reaches a maximum value when the synthetic arrival time curves match the arrival times of the events on the field gathers. The maximum value of the functional is obtained by an effective algorithm of non‐linear programming.The present inversion algorithm has the advantages that event picking on the unstacked data is not required and is not based on curve fitting of hyperbolic approximations of the arrival times. The method has been successfully applied to both synthetic and field data.

Plane‐layer prestack inversion in the presence of surface reverberation

Experiments with synthetic data have indicated that generalized linear inversion may be used to estimate compressional velocities as a function of depth with high resolution directly from band‐limited, unstacked data. The ocean surface was not included in these experiments. In the presence of strong surface multiples, inversion is expected to take longer and be less accurate, because events from multiple surface reflections overlie primary events and normally have differing moveout. Existing velocity‐analysis techniques rely on the ability of an observer to make the difficult distinction between multiples and primaries. Equations are provided for adding the surface to the inversion procedure. This involves adding the surface effects to the Jacobian matrix as well as to the forward modeling procedure. To speed computation, the addition of the surface effects to the Jacobian matrix is delayed until after the matrix has been multiplied by a vector in the linear‐equation solution. Absorption is added to the inversion to represent the real world more closely and to improve computation speed by reducing sampling requirements. Realistic synthetic band‐limited data with high surface reverberation content were generated from a well‐log velocity profile. The inversion recovered the velocity profile to within 3 percent when a velocity increasing linearly with depth was used as a starting profile. The error in the model‐generated seismogram converges from 100 percent to within 2 percent of the reference data. The positions of interfaces are located more accurately at greater depths than at shallower depths because more sensors are observing deep strata than shallow strata. Convergence is to within 0.1 percent of that of the reference data at the maximum depth. The computation required 25 iterations and a total time of 66 hours on a DEC VAX 11/780. Reducing this time should be possible. In a preliminary study of the effects of noise, additive Gaussian noise was seen to limit the accuracy of the velocity estimate monotonically as the variance of the added noise was increased.

Three‐dimensional Born inversion with an arbitrary reference

Recent work of G. Beylkin helped set the stage for very general seismic inversions. We have combined these broad concepts for inversion with classical high‐frequency asymptotics and perturbation methods to bring them closer to practically implementable algorithms. Applications include inversion schemes for both stacked and unstacked seismic data. Basic assumptions are that the data have relative true amplitude, and that a reasonably accurate background velocity c(x, y, z) is available. The perturbation from this background is then sought. Since high‐frequency approximations are used throughout, the resulting algorithms essentially locate discontinuities in velocity. An expression for a full 3-D velocity inversion can be derived for a general data surface. In this degree of generality the formula does not represent a computationally feasible algorithm, primarily because a key Jacobian determinant is not expressed in practical terms. In several important cases, however, this shortcoming can be overcome and expressions can be obtained that lead to feasible computing schemes. Zero‐offsets, common‐sources, and common‐receivers are examples of such cases. Implementation of the final algorithms involves, first, processing the data by applying the FFT, making an amplitude adjustment and filtering, and applying an inverse FFT. Then, for each output point, a summation is performed over that portion of the processed data influencing the output point. This last summation involves an amplitude and traveltime along connecting rays. The resulting algorithms are computationally competitive with analogous migration schemes.

Linearized inversion of multioffset seismic reflection data in the ω-k domain

In the acoustic approximation, the Earth is described using only density and bulk modulus. Assuming smooth density variations, reflections can be described using a single function—the velocity of compressional waves. If a reference model which is close enough to the actual Earth is known, the problem of estimating the medium velocity from the observed data can be linearized. Using a least‐squares formulation and working in the ω-k domain, the linearized inverse problem for a homogeneous reference medium can be solved by a noniterative algorithm which is economically competitive with prestack migration. Numerical tests with synthetic and real data demonstrate the feasibility and the numerical stability of the method. The numerical results compare well with those obtained by migration of unstacked data, although superior results will only be obtained when the physics of the problem (including elastic versus acoustic effects, three‐dimensional propagation, and accurate source estimation) will realistically be taken into account.

Wave‐equation datuming before stack

This note describes the extension to unstacked seismic data of a computationally efficient form of the Kirchhoff integral published several years ago. In the previous paper (Berryhill, 1979), a wave‐equation procedure was developed to change the datum of a collection of zero‐offset seismic traces from one surface of arbitrary shape to another, even when the velocity for wave propagation is not constant. This procedure was designated “wave‐equation datuming,” and its applications to zero‐offset data were shown to include velocity‐replacement datum corrections and multilayer forward modeling. Extending this procedure to unstacked data requires no change in the mathematical algorithm. It is necessary only to recognize that operating on a common‐source group of seismic traces has the effect of extrapolating the receivers from one datum to another, and that, because of reciprocity, operating on a common‐receiver group changes the datum of the sources. Two passes through the data, common‐source computations, then common‐receiver computations, are required to change the datum of an entire seismic line before stack from one surface to another. Common‐source and common‐receiver trace groups must take the form of symmetric split spreads if both directions of dip are to be treated equally; reciprocity allows split spreads to be constructed artificially if the data were not actually recorded in the required form.

Migration velocity analysis by wave‐field extrapolation

Velocity information is essential to both common midpoint (CMP) stacking and migration. CMP stacking provides the basis for conventional velocity estimation techniques in that, for a number of trial velocities, the stack response of a CMP gather is computed and displayed in the form of a velocity table. An alternative approach to velocity estimation makes use of the basic ingredients of migration—downward extrapolation and imaging of seismic wave fields. The procedure involves migration of a CMP gather with a number of trial velocities and collection of the zero‐offset information, again in the form of a velocity table. Operating on a CMP gather, the migration‐based approach produces results similar to those of the conventional method. Analyses of synthetic CMP gathers using both methods show essentially equivalent treatments of seismic signal, and similar dependence of accuracy and resolving power on recording geometry. We have extended the migration‐based approach to include more than one CMP gather in each analysis. This extension allows proper treatment of dipping events and yields velocity information that is more appropriate for use in migration. By using the intermediate wave field at each step of downward extrapolation, we need only do a single constant‐velocity migration of the unstacked data followed by a simple mapping procedure in order to recover the velocity information.

Migration Velocity Analysis by Wave-Field Extrapolation

Velocity information is essential to both common midpoint (CMP) stacking and migration. CMP stacking provides the basis for conventional velocity estimation techniques in that, for a number of trial velocities, the stack response of a CMP gather is computed and displayed in the form of a velocity table. An alternative approach to velocity estimation makes use of the basic ingredients of migration–downward extrapolation and imaging of seismic wave fields. The procedure involves migration of a CMP gather with a number of trial velocities and collection of the zero-offset information, again in the form of a velocity table. Operating on a CMP gather, the migration-based approach produces results similar to those of the conventional method. Analyses of synthetic CMP gathers using both methods show essentially equivalent treatments of seismic signal, and similar dependence of accuracy and resolving power on recording geometry. We have extended the migration-based approach to include more than one CMP gather in each analysis. This extension allows proper treatment of dipping events and yields velocity information that is more appropriate for use in migration. By using the intermediate wave field at each step of downward extrapolation, we need only do a single constant-velocity migration of the unstacked data followed by a simple mapping procedure in order to recover the velocity information.

Inversion of seismic reflection data in the acoustic approximation

The nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation. The method is based on the generalized least‐squares criterion, and it can handle errors in the data set and a priori information on the model. Multiply reflected energy is naturally taken into account, as well as refracted energy or surface waves. The inverse problem can be solved using an iterative algorithm which gives, at each iteration, updated values of bulk modulus, density, and time source function. Each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation (backward in time) of the data residuals. The correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models. This shows, in particular, that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data, and commercially competitive with them.

Coherent noise in marine seismic data

Despite significant advances in marine streamer design, seismic data are often plagued by coherent noise having approximately linear moveout across stacked sections. With an understanding of the characteristics that distinguish such noise from signal, we can decide which noise‐suppression techniques to use and at what stages to apply them in acquisition and processing. Three general mechanisms that might produce such noise patterns on stacked sections are examined: direct and trapped waves that propagate outward from the seismic source, cable motion caused by the tugging action of the boat and tail buoy, and scattered energy from irregularities in the water bottom and sub‐bottom. Depending upon the mechanism, entirely different noise patterns can be observed on shot profiles and common‐midpoint (CMP) gathers; these patterns can be diagnostic of the dominant mechanism in a given set of data. Field data from Canada and Alaska suggest that the dominant noise is from waves scattered within the shallow sub‐buttom. This type of noise, while not obvious on the shot records, is actually enhanced by CMP stacking. Moreover, this noise is not confined to marine data; it can be as strong as surface wave noise on stacked land seismic data as well. Of the many processing tools available, moveout filtering is best for suppressing the noise while preserving signal. Since the scattered noise does not exhibit a linear moveout pattern on CMP‐sorted gathers, moveout filtering must be applied either to traces within shot records and common‐receiver gathers or to stacked traces. Our data example demonstrates that although it is more costly, moveout filtering of the unstacked data is particularly effective because it conditions the data for the critical data‐dependent processing steps of predictive deconvolution and velocity analysis.

Velocity Filtering of Seismic Reflection Data

The term 'velocity filtering' may be applied to any process which seeks to separate coherent energy incident upon a seismic array by using the apparent velocity of propagation across the array as a discriminant. The pie-slice process was carried out using a limited multitrace operator in the time-distance (t-x) domain. In order to economise on the convolutions involved, a typical operator consisted of 21 time-points in length, covered 12 traces and resulted in a rejection level about 20 dB down with a less than ideal response function. While the pie-slice or fan filter may be suitable for stacked profiles, we wish to propose the velocity filtering of unstacked data in the frequency-wavenumber (w-k) domain as a more suitable alternative. Providing certain precautions are taken in handling the dataset, several advantages of this procedure are obtained: -the number of output traces is equal to the number of input traces, -the ideal filter response is more closely achieved, -no filter coefficients need be computed beforehand, -implementation of the filter is by straightforward multiplication, -interactive examination of the data in the (w-k) domain permits greater flexibility in the design of the filter response, -since the eye follows group velocity in the (t-x) domain, the presentation in (w-k) space allows a better estimate to be made of phase veloeities and the degree of aliasing, -other wavefield operations may be performed on the data while in (w-k) space. Velocity filtering in the (w-k) domain is particularly important for the high-resolution, single geophone data recorded in the United Kingdom by the National Coal Board and examples of such data are presented in the paper. One of the objects of successful filter design is to achieve maximum separation between those signals which are to be retained and those which are to be suppressed. This may be realised if we can find a suitable representation or domain in which the data can be manipulated. The mute and low pass filter are simple examples of one-dimensional filters using time of arrival and frequency content respectively as discriminants. The two-dimensional velocity filter, as its name suggests, uses the apparent (horizontal) velocity of coherent energy across a seismic aperture to discriminate between wanted and unwanted signals. As such, the concept is not new and indeed the pie-slice or fan filter has been in use for many years now. However, until recently, the implementation of the velocity filter has been performed in the time-distance or (t-x) domain. The advent of truly fast, fast Fourier transtorms has enabled the advantages of working in the more natural (w- k) domain to be realised. The purposes of this paper are to discuss the implementation of the (w-k) algorithm and to point out its advantages with specific regard to the type of high-resolution data recorded by the National Coal Board in the United Kingdom.

Migration of seismic data from inhomogeneous media

Because conventional time‐migration algorithms are founded on the implicit assumption of locally lateral homogeneity, they leave events mispositioned when overburden velocity varies laterally. The ray‐theoretical depth migration procedure of Hubral often can provide adequate first‐order corrections for such position errors. Complex geologic structure, however, can so severely distort wavefronts that resulting time‐migrated sections may be barely interpretable and thus not readily correctable. A more accurate, wave‐theoretical approach to depth migration then becomes essential to image the subsurface properly. This approach, which transforms an unmigrated time section directly into migrated depth, more completely honors the wave equation for a medium in which variations in interval velocity and details of structural shape govern wave propagation. Where geologic structure is complicated, however, we usually lack an accurate velocity model. It is important, therefore, to understand the sensitivity of depth migration to velocity errors and, in particular, to assess whether it is justified to go to the added effort of doing depth migration. We show a synthetic data example in which the wave‐theoretical approach to depth migration properly images deep reflections that are poorly resolved and left distorted by either time migration or ray‐theoretical depth migration. These imaging results are, moreover, surprisingly insensitive to errors introduced into the velocity model. Application to one field data example demonstrates the superior treatment of amplitude and waveform by wave‐theoretical depth migration. In a second data example, deep reflections are so influenced by anomalous overburden structure that the only valid alternative to performing wave‐theoretical depth migration is simply to convert the unmigrated data to depth. When the overburden is laterally variable, conventional time migration of unstacked data can be as destructive to steeply dipping reflections as is CDP stacking prior to migration. A schematic example illustrates that when migration of unstacked data is judged necessary, it should normally be performed as a depth migration.

Depth migration before stack

When seismic data are migrated using operators derived from the scalar wave equation, an assumption is normally made that the seismic velocity in the propagating medium is locally laterally invariant. This simplifying assumption causes reflectors to be imaged incorrectly when lateral velocity gradients exist, irrespective of the degree of accuracy to which the subsurface velocity structure is known. A finite‐difference method has been implemented for migration of unstacked data in the presence of lateral velocity gradients, where the operation of wave field extrapolation is done in increments of depth rather than time. Performing this depth migration on unstacked data results in the imaging of reflectors on the zero‐offset trace, whereupon a zero‐offset section becomes a fully imaged‐in‐depth seismic section. Such a section, in addition to being a correctly migrated depth section, shows the same order of signal amplitude enhancement as in a normal stacking process.