Dip Moveout Research Articles

Imaging in 3-D DMO, Part I: Geometrical optics models

Systematic azimuthal variation in the input data for 3-D dip moveout (3DDMO) can compromise the illumination of the subsurface, produce imaging artifacts, and cause amplitude variation in the output 3DDMO traces. A simple graphical method allows easy visualization of the subsurface illumination at a dip moveout (DMO) output location for any set of input data. The same method permits selection or construction of sets of input data that illuminate all subsurface reflectors with no redundancy.

Imaging in 3-D DMO, Part II: Amplitude effects

The amplitude of the dip‐moveout (DMO) output for a planar reflector at a point P and half offset k is a function of the curvature of the midpoint curve of the DMO input traces, the dip magnitude, and the dip direction. No single weighting function will preserve the amplitude of dipping events. Seismic synthetics on a circle of midpoints minimal data set verify that this data set has 6 dB of amplitude variation with azimuth for a 15° dipping reflector and a 14 dB variation for a 45° reflector. The theory shows that the 3DDMO weight for an arbitrary minimal data set is equal to the impulse response weight multiplied by the density of the normal segments at the dip being imaged. Aliasing is also a function of the normal segment density at the dip being imaged. If the spatial sampling is fine enough to prevent aliasing, DMO can be performed on a dip by dip basis and, by correcting for the normal segment density, these amplitude variations may be corrected.

2-D continuation operators and their applications

Continuation to zero offset [better known as dip moveout (DMO)] is a standard tool for seismic data processing. In this paper, the concept of DMO is extended by introducing a set of operators: the continuation operators. These operators, which are implemented in integral form with a defined amplitude distribution, perform the mapping between common shot or common offset gathers for a given velocity model. The application of the shot continuation operator for dip‐independent velocity analysis allows a direct implementation in the acquisition domain by exploiting the comparison between real data and data continued in the shot domain. Shot and offset continuation allow the restoration of missing shot or missing offset by using a velocity model provided by common shot velocity analysis or another dip‐independent velocity analysis method.

An accurate formulation of log‐stretch dip moveout in the frequency‐wavenumber domain

Dip moveout (DMO) processing is a partial prestack migration procedure that has been widely used in seismic data processing. The DMO process has been described in Deregowski (1986), Hale (1991) and Liner (1990). Many different DMO algorithms have been developed over the past decade. These algorithms have been designed to improve either the accuracy or the computational speed of the DMO process. Hale (1984) developed a method for performing DMO via Fourier transforms that is accurate for all reflector dips (assuming constant velocity). Hale’s method is computationally expensive because his DMO operator is temporally nonstationary, but its accuracy and simplicity have made it an industry standard. It has become a benchmark by which results from other DMO algorithms are judged. Of all the methods used to make the frequency‐domain DMO computationally efficient, the technique of logarithmic time stretching, first suggested in Bolondi et al. (1982), is widely used. After logarithmic stretching of the time axis, the DMO operator becomes temporally stationary which enables replacement of the slow temporal Fourier integration with a fast Fourier transform combined with a simple phase shift. Bale and Jakubowicz (1987) presented a log‐stretch DMO operator (hereafter referred to as Bale’s DMO) in the frequency‐wavenumber (F-K) domain without approximations, while Notfors and Godfrey (1987) suggested an approximate version of log‐stretch DMO operator (hereafter referred to as Notfors’s DMO). Surprisingly, Bale’s full log‐stretch DMO operator produces a less satisfactory impulse response than Notfors’s approximate log‐stretch DMO scheme (see Liner, 1990). Liner (1990) attributed this characteristic to the fact that Bale’s DMO derivation implicitly assumes that the Fourier transform frequency in the log‐stretch domain is not time‐dependant. He presented an exact log‐stretch DMO operator (hereafter referred to as Liner’s DMO) which was derived by transforming the time log‐stretched Hale’s (t, x) DMO impulse response into the Fourier domain. Its derivation is relatively complicated, but Liner has shown that his DMO does generate good DMO impulse responses.

Fowler DMO and time migration for transversely isotropic media

The main advantage of Fowler’s dip‐moveout (DMO) method is the ability to perform velocity analysis along with the DMO removal. This feature of Fowler DMO becomes even more attractive in anisotropic media, where imaging methods are hampered by the difficulty in reconstructing the velocity field from surface data. We have devised a Fowler‐type DMO algorithm for transversely isotropic media using the analytic expression for normal‐moveout velocity. The parameter‐estimation procedure is based on the results of Alkhalifah and Tsvankin showing that in transversely isotropic media with a vertical axis of symmetry (VTI) the P‐wave normal‐moveout (NMO) velocity as a function of ray parameter can be described fully by just two coefficients: the zero‐dip NMO velocity [Formula: see text] and the anisotropic parameter η (η reduces to the difference between Thomsen parameters ε and δ in the limit of weak anisotropy). In this extension of Fowler DMO, resampling in the frequency‐wavenumber domain makes it possible to obtain the values of [Formula: see text] and η by inspecting zero‐offset (stacked) panels for different pairs of the two parameters. Since most of the computing time is spent on generating constant‐velocity stacks, the added computational effort caused by the presence of anisotropy is relatively minor. Synthetic and field‐data examples demonstrate that the isotropic Fowler DMO technique fails to generate an accurate zero‐offset section and to obtain the zero‐dip NMO velocity for nonelliptical VTI models. In contrast, this anisotropic algorithm allows one to find the values of the parameters [Formula: see text] and η (sufficient to perform time migration as well) and to correct for the influence of transverse isotropy in the DMO processing. When combined with poststack F-K Stolt migration, this method represents a complete inversion‐processing sequence capable of recovering the effective parameters of transversely isotropic media and producing migrated images for the best‐fit homogeneous anisotropic model.

Suppression of sea‐floor‐scattered energy using a dip‐moveout approach—Application to the mid‐ocean ridge environment

Multichannel seismic (MCS) images are often contaminated with in‐ and out‐of‐plane scattering from the sea floor. This problem is especially acute in the mid‐ocean ridge environment where sea‐floor roughness is pronounced. Energy shed from the unsedimented basaltic sea floor can obscure primary reflections such as Moho, and scattering off of elongated sea‐floor features like abyssal hills and fault scarps can produce linear events in the seismic data that could be misinterpreted as subsurface reflections. Moreover, stacking at normal subsurface velocities may enhance these water‐borne events, whose stacking velocity depends on azimuth and generally increases with time, making them indistinguishable from subsurface arrivals. To suppress scattered energy in deep water settings, we propose a processing scheme that invokes the application of dip moveout (DMO) to deliberately increase the differential moveout between sea‐floor‐scattered and subsurface events, thereby facilitating the removal of unwanted energy in the stacked section. After application of DMO, all sea‐floor scatterers stack at the water velocity, while subsurface reflections like Moho still stack at their original velocity. The application of DMO in this manner is contrary to the intended use that reduces the differential moveout between dipping events and allows a single stacking velocity to be used. Unlike previous approaches to suppress scattered energy, dip filtering is applied in the common‐midpoint (CMP) domain after DMO. Moveover, our DMO‐based approach suppresses out‐of‐plane scattering, and therefore is not limited to removal of in‐plane scattering as is the case with shot and receiver dip filtering techniques. The success of our DMO‐based suppression scheme is limited to deep water (a few kilometers of water depth for conventional offsets), where the traveltime moveout of energy scattered from the sea floor has a hyperbolic moveout with a stacking velocity that depends on the cosine of the scatterer steering angle in a manner analogous to how the moveout of a dipping reflector depends on the dip angle. The application of DMO‐based suppression to synthetics and MCS data collected along the southern East Pacific Rise demonstrates the effectiveness of our approach. Cleaner images of primary reflectors such as Moho are produced, even though present shot coverage along the East Pacific Rise is unduly sparse, resulting in a limited effective spatial bandwidth.

Accurate and efficient shot‐gather dip moveout processing in the log‐stretch domain1

AbstractAn accurate analytical expression for shot‐gather dip‐moveout (DMO) in the timespace log‐stretch domain has until now not been published. We present a simpler, alternative derivation of the exact DMO relationships of Black et al. which correctly take account of the repositioning of the midpoint. A new computationally efficient frequency‐wavenumber (F‐K) DMO operator for shot profiles is then derived, based on these DMO relationships in the time‐space log‐stretch domain. The newly derived DMO operator is, unlike most other log‐stretch DMO operators) accurate for the full range of reflector dips. Along with other schemes which are performed in the log‐stretch domain, it offers considerable time savings over conventional DMO processing. We have compared numerically the impulse response of the new operator with those of a number of other shot‐gather DMO operators, and found it to be superior and well match to the theoretical elliptical DMO response.

A partial DMO Operator for use with the Stacking Velocity Function

Dip moveout (DMO) processing is frequently used to enhance stacked seismic sections. All DMO methods strictly require that the input data have been normal-moveout (NMO) corrected using the true medium velocity rather than the stacking (or RMS) velocity function. Unfortunately this is rarely the case in that most data which are input to DMO have been NMO-corrected according to the stacking velocity, in order to maximise semblance on a CMP gather. Such data are incompatible with the basic DMO assumption. Under such circumstances only a partial DMO process, which produces a parabolic impulse response, should be used rather than a full DMO algorithm, such as that of Hale, which entails an elliptical impulse response. The partial DMO operator accounts for the inaccuracy of the velocity information used in the NMO correction, and leads to superior results.

A new approach to pre-stack seismic processing

SUMMARY Pre-stack processing of seismic reflection data is significantly simplified if the data organization is the same as that in which the data were acquired in the field; that is, in time slices through common-source gathers. For an impulsive source, the entire processing stream reduces to two elements: velocity analysis and reverse-time prc-stack migration. Many of the steps that are applied as independent operations in standard processing (including demultiplexing, sorting into common-midpoint gathers, elevation corrections, near-surface velocity static corrections, first break muting, ground roll removal, and both normal and dip move-out corrections) either are eliminated, or are applied implicitly during migration. This approach is ideally suited to parallel processing, and can be implemented in machines with small processor memories.

Normal moveout from dipping reflectors in anisotropic media

Description of reflection moveout from dipping interfaces is important in developing seismic processing methods for anisotropic media, as well as in the inversion of reflection data. Here, I present a concise analytic expression for normal‐moveout (NMO) velocities valid for a wide range of homogeneous anisotropic models including transverse isotropy with a tilted in‐plane symmetry axis and symmetry planes in orthorhombic media. In transversely isotropic media, NMO velocity for quasi‐P‐waves may deviate substantially from the isotropic cosine‐of‐dip dependence used in conventional constant‐velocity dip‐moveout (DMO) algorithms. However, numerical studies of NMO velocities have revealed no apparent correlation between the conventional measures of anisotropy and errors in the cosine‐of‐dip DMO correction (“DMO errors”). The analytic treatment developed here shows that for transverse isotropy with a vertical symmetry axis, the magnitude of DMO errors is dependent primarily on the difference between Thomsen parameters ε and δ. For the most common case, ε − δ > 0, the cosine‐of‐dip–corrected moveout velocity remains significantly larger than the moveout velocity for a horizontal reflector. DMO errors at a dip of 45 degrees may exceed 20–25 percent, even for weak anisotropy. By comparing analytically derived NMO velocities with moveout velocities calculated on finite spreads, I analyze anisotropy‐induced deviations from hyperbolic moveout for dipping reflectors. For transversely isotropic media with a vertical velocity gradient and typical (positive) values of the difference ε − δ, inhomogeneity tends to reduce (sometimes significantly) the influence of anisotropy on the dip dependence of moveout velocity.

Shot gather DMO in the double log domain

Dip moveout (DMO) correction is applied routinely now in seismic processing. Over the years, numerous DMO methods have been developed for the frequency‐wavenumber domain (Hale 1991).

Salt‐related structures in the Gulf of Mexico: A field guide for geophysicists

In finding economic quantities of oil and gas below a salt sheet, the Mahogany discovery well in Ship Shoal South Addition, offshore Louisiana, dramatically boosted exploration interest in the Gulf of Mexico’s subsalt play in shallow water. Interest was initially aroused by the earlier deep‐water discovery below a salt sheet forming part of “Mickey Mouse,” a cluster of pancake‐shaped salt sheets in Mississippi Canyon Block 211. A major technologic obstacle in this play has been the difficulty of imaging the base of salt sheets and the subsalt structures targeted for exploration. Some recent articles have showcased the spectacular fidelity of 3-D seismic data sets collected on closely spaced survey lines and processed by techniques such as 3-D prestack depth migration and 3-D dip moveout followed by 3-D turning‐ray Kirchhoff migration. However, at present the cost of this technology limits its use to specific, rigorously selected targets.

Comprehensive analysis of marine 3-D bin coverage

The increasing use of multisource, multistreamer, multiboat geometries, as well as stricter processing requirements, necessitates a review of current methods of marine 3-D bin coverage evaluation. In addition to traditional methods which often utilize only offset segmented fold maps for coverage evaluation, enhanced procedures have been developed to analyze multiple attributes that are pertinent to an assessment of 3-D marine bin coverage. These attributes include azimuths, midpoint scatter for various offsets within a bin, duplicate/missing offsets within a bin, uniformity of offset distribution, flex binning, dip moveout (DMO) effects, navigation accuracy, etc.

Dip‐moveout processing for depth‐variable velocity

In dip‐moveout (DMO) processing, the seismic velocity is often assumed to be constant. While several approximate techniques for handling vertical velocity variation have recently become available, here we propose a method for computing the kinematically exact DMO correction when velocity is an arbitrary function of the depth. While not known in advance, the raypath from source to receiver and the corresponding zero‐offset raypath satisfy several relationships, which are used to form a system of nonlinear equations. By simultaneously solving the equations via Newton‐Raphson iteration, we determine the mapping that transforms nonzero‐offset data to zero‐offset. Unlike previous schemes that approximately handle vertical velocity variation, this method makes no assumptions about the offset, dip, velocity function, or hyperbolic moveout. Tests using both synthetic and recorded seismic data demonstrate the effectiveness of this variablevelocity DMO. These tests show this method accurately handles vertical velocity variation, while use of constant‐velocity DMO can lead to significant errors. Comparing this technique to a formulation that approximately handles velocity variation, however, suggests that the improved accuracy of the exact technique may not be justified because of uncertainty in the velocity model and increased cost. While improved accuracy alone may not justify the use of this method in 2-D, its flexibility may in other cases. Changes could be made to handle 3-D DMO, DMO for mode‐converted waves, DMO in anisotropic media, or prestack divergence correction.

Offset dependent amplitudes: Fault curvature effects and dip move-out corrections

Equations have been presented previously which predict that reflector curvature can affect significantly seismic reflection amplitudes at both zero and nonzero source-receiver offsets. Here the fact that faults are generally concave-upward is used to examine the curvature effect for compaction-driven faults which the sediments have both exponential and logarithmic porosity decreases with increasing depth.

DMO velocity analysis with Jacubowicz’s dip‐decomposition method

Dip‐moveout (DMO) velocity analysis (VA) may be performed in several ways. Using the Fourier transform‐based DMO techniques (Hale 1984, Notfors and Godfrey 1987, Liner and Bleinstein 1988), VA is done iteratively where a sequence of VA, normal moveout (NMO), DMO, inverse NMO, and a second VA yields an estimate of the DMO velocities. Using an integral method for application of a DMO process, Forel and Gardner (1988) proposed a way for performing VA by transforming the data into the [Formula: see text] domain, where DMO velocities are obtained by any common VA technique. In another work, carried out by Chon and Gonzalez (1987) a velocity‐sensitivity analysis was added to a Kirchhoff integral DMO algorithm.

Dip‐moveout error in transversely isotropic media with linear velocity variation in depth

Levin modeled the moveout, within common‐midpoint (CMP) gathers, of reflections from plane‐dipping reflectors beneath homogeneous, transversely isotropic media. For some media, when the axis of symmetry for the anisotropy was vertical, he found departures in stacking velocity from predictions based upon the familiar cosine‐of‐dip correction for isotropic media. Here, I do similar tests, again with transversely isotropic models with vertical axis of symmetry, but now allowing the medium velocity to vary linearly with depth. Results for the same four anisotropic media studied by Levin show behavior of dip‐corrected stacking velocity with reflector dip that, for all velocity gradients considered, differs little from that for the counterpart homogeneous media. As with isotropic media, traveltimes in an inhomogeneous, transversely isotropic medium can be modeled adequately with a homogeneous model with vertical velocity equal to the vertical rms velocity of the inhomogeneous medium. In practice, dip‐moveout (DMO) is based on the assumption that either the medium is homogeneous or its velocity varies with depth, but in both cases isotropy is assumed. It turns out that for only one of the transversely isotropic media considered here—shale‐limestone—would v(z) DMO fail to give an adequate correction within CMP gathers. For the shale‐limestone, fortuitously the constant‐velocity DMO gives a better moveout correction than does the v(z) DMO.

3-D time‐variant dip moveout by the f-k method

The standard seismic imaging sequence consists of normal moveout (NMO), dip moveout (DMO), stack, and zero‐offset migration. Conventional NMO and DMO processes remove much of the effect of offset from prestack data, but the constant velocity assumption in most DMO algorithms can compromise the ultimate results. Time‐variant DMO avoids the constant velocity assumption to create better stacks, especially for steeply dipping events. Time‐variant DMO can be implemented as a 3-D, f-k domain process using the dip decomposition method. Prestack data are moved out with a set of NMO velocities corresponding to discrete values of in‐line and crossline dips. The dip‐dependent NMO velocity is computed to remove the trace offset and azimuth dependence of event times for an arbitrary velocity function of depth. After stacking the moved out CMP gathers, a three‐dimensional (3-D) dip filter is applied to select the particular in‐line and crossline dip. The final zero‐offset image is obtained by summing all the dip‐filtered sections. This process generates a saddle‐shaped 3-D impulse response for a constant velocity gradient. The impulse response is more complicated for a general depth‐variable velocity function, where the response exhibits secondary branches, or triplications, at steeper dips. These complicated impulse responses, including amplitude and phase effects, are implicitly produced by the f-k process. The dip‐decomposition method of 3-D time‐variant DMO is an efficient and accurate process to correct for the effect of offset in the presence of an arbitrary velocity variation with depth. The impulse response of this process implicitly contains complex features like a 3-D saddle shape, triplications, amplitude, and phase. Field data from the Gulf of Mexico shows significant improvement on a steep salt flank event.

MIGRATION TO ZERO OFFSET (DMO) FOR A CONSTANT VELOCITY GRADIENT: AN ANALYTICAL FORMULATION1

AbstractMigration to zero offset (MZO) is a prestack partial migration process that transforms finite‐offset seismic data into a close approximation to zero‐offset data, regardless of the reflector dips that are present in the data. MZO is an important step in the standard processing sequence of seismic data, but is usually restricted to constant velocity media. Thus, most MZO algorithms are unable to correct for the reflection point dispersal caused by ray bending in inhomogeneous media.We present an analytical formulation of the MZO operator for the simple possible variation of velocity within the earth, i.e. a constant gradient in the vertical direction. The derivation of the MZO operator is carried out in two steps. We first derive the equation of the constant traveltime surface for linear V(z) velocity functions and show that the isochron can be represented by a fourth‐degree polynomial in x, y and z. This surface reduces to the well‐known ellipsoid in the constant‐velocity case, and to the spherical wavefront obtained by Slotnick in the coincident source‐receiver case.We then derive the kinematic and dynamic zero‐offset corrections in parametric form by using the equation of the isochron. The weighting factors are obtained in the high‐frequency limit by means of a simple geometric spreading correction. Our analytical results show that the MZO operator is a multivalued, saddle‐shaped operator with marked dip moveout effects in the cross‐line direction. However, the amplitude analysis and the distribution of dips along the MZO impulse response show that the most important contributions of the MZO operator are concentrated in a narrow zone along the in‐line direction. In practice, MZO processing requires approximately the same trace spacing in the in‐line and cross‐line directions to avoid spatial aliasing effects.

Squeezing dip moveout for depth‐variable velocity

In dip‐moveout (DMO) processing, velocity variations with depth can be handled approximately by squeezing a constant‐velocity DMO operator to narrow its impulse response. This squeezed DMO approximation provides a computationally efficient and reasonably accurate method of DMO correction for depth‐variable velocity. DMO is squeezed by two modifications to constant‐velocity DMO. One modification is a squeezing function of time that depends only on simple time averages of velocity that are likely to be known before DMO is applied. This squeeze function ensures that squeezed DMO accurately handles moderately steep reflections and can be incorporated with simple time stretching before and after DMO, without any changes to existing constant‐velocity DMO methods. The second modification is a constant squeezing factor, which may be used to tune squeezed DMO to better handle steep reflections. This factor requires only trivial changes to constant‐velocity DMO methods. Tests with both synthetic and recorded seismic data suggest that squeezed DMO is an effective method for handling velocity variations with depth. These tests also show that the differences between constant‐velocity DMO and squeezed DMO can be significant.