Common Midpoint Gather Research Articles

Crooked‐line 2D seismic reflection imaging in crystalline terrains: Part 1, data processing

For cost and access reasons, most of the seismic reflection data collected in crystalline terrains have been acquired by 2D crooked‐line profiling. When the survey geometry is significantly irregular and the geologic structures have cross‐profile dip, several standard 2D imaging procedures severely underperform. As a result, reflection signal is poorly aligned across individual common midpoint (CMP) gathers, and much is lost during the CMP stack. To improve imaging, either the methods used to align signal before stack need to be modified or more tolerant methods of combining trace signals than the standard CMP stack need to be applied. Because a high‐fold 2D crooked‐line profile is really a 3D survey of a swath of terrain around the processing line, better signal alignment before CMP stacking may be achieved by revisiting the traveltime equation and including the cross‐dip terms into the moveout calculations. Therefore, in addition to the correction of NMO and in‐line dip moveout (DMO), we also locally compute and subsequently remove cross‐dip moveout (CDMO). This requires a procedure for estimating the amount of cross‐dip associated with each local reflection event. Stacking after the successful removal of the CDMO yields what we call an optimum cross‐dip stack—a seismic section that is significantly more complete and informative than the standard stack. Alternatively, amplitude stacking appears to be more robust to residual time anomalies. When little or no cross‐dip information can be extracted from the 2D crooked‐line data, we use it as a last resort to obtain a section that contains more structural information than the standard stack.

Estimation of quality factors from CMP records

Estimates of the quality, Q, factor are commonly obtained from vertical seismic data or stacked surface seismic data. This paper describes a method that allows Q‐factor to be estimated directly from common midpoint (CMP) gathers. Absorption of the wavefield is dependent on three parameters: frequency, traveltime in the medium, and medium Q‐factor. Assuming that the amplitude spectrum of the seismic source signature may be modeled by that of a Ricker wavelet, we derive an analytical relation between Q‐factor and seismic data peak frequency variation both along offset and vertical time direction. The Q‐factor is estimated from CMP gathers using a layer‐stripping approach.

Multifocusing imaging with controlled reflection‐point dispersal

Multifocusing imaging proposed by Gelchinsky (Gelchinsky et al., 1999a, b; Landa et al., 1999) belongs to a group of techniques that can be characterized as macro-model independent imaging methods. These methods, which also include the common-reflection surface method (Mann et al., 1999; Jager et al., 2001), and optical stack (de Bazelaire, 1988), are described in a collection of papers published as a special issue of the Journal of Applied Geophysics (Hubral, 1999). Macro-model independent imaging methods represent a new alternative to the classical processing sequence of NMO, dip moveout (DMO), and stacking. They are based upon a new transformation of 2-D multicoverage prestack data into a simulated zero-offset stack section. This transformation involves stacking large supergathers of seismic traces, each of which can span many common-midpoint (CMP) gathers. Stacking of large supergathers is made possible by the use of a generalized moveout correction. For a given source-receiver pair, this correction depends on three parameters: the wavefront curvatures of the normal wave, the normal incidence point wave (Hubral, 1983), and the emergence angle of the central ray. For each supergather and each zero-offset time, T , these parameters are obtained through a coherence analysis of the moveout corrected supergather.

Multifocusing imaging over an irregular topography

If seismic data are acquired over an irregular topography, standard elevation statics methods may be inaccurate because the assumption of vertical raypaths will no longer be valid. An effective solution to the problem of irregular topography can be found through the use of the multifocusing method, in which large supergathers of seismic traces are stacked, each of which can span many common midpoint (CMP) gathers. This can be done by extending the multifocusing moveout formula to explicitly account for nonzero elevations of the source and receiver, as well as their horizontal coordinates. Implementation of this formula into the multifocusing algorithm is straightforward because estimating the necessary raypath information (i.e., emergence angles) is an integral part of the algorithm. The extended multifocusing moveout correction can be applied directly to the data acquired in areas of irregular topography without the need for prior elevation static corrections. Synthetic tests on such data show that the proposed technique results in a better alignment of reflection events.

Comparison of seismic reflection and ground‐penetrating radar imaging at the controlled archaeological test site, Champaign, Illinois

AbstractShallow seismic reflection and ground‐penetrating radar images were collected at a replicated burial mound in the Controlled Archaeological Test Site (CATS) in Champaign, Illinois. The CATS mound contains a pig burial within a wood‐lined crypt at a depth of 1.6–2.4 m. Seismic reflection data were collected from two different energy sources: a small (0.5 kg) hammer for an impulsive source, and a vibrator for a frequency swept source. Seismic data were collected at densely spaced points (5 cm) along a line of 48 geophone receivers. These data were stacked in a common mid‐point gather, band‐pass filtered, and processed with frequency–wavenumber migration. The seismic image produced by the hammer source was dominated by bodywaves at 120 Hz, whereas the vibrator source image was dominated by surface waves at 70 Hz. Both seismic sources revealed clear reflections from the burial crypt, and placed the top of the crypt at the correct depth with a seismic velocity of 120 m s−1. The bottom of the crypt was poorly defined by the seismic data owing to multiple reflections within the crypt. The vibrator source also revealed a high‐frequency (360 Hz) reflector at 2.7 m depth within the mound, perhaps due to a resonant cavity within the pig's body. Single channel ground‐penetrating radar data were processed with the same approach, including band‐pass filtering and migration. The radar data reveal clear reflections from the burial crypt. Extremely fast radar velocities (260 mm ns−1) are required in the upper portion of the burial mound to place the top of the crypt at its correct depth. The bottom of the crypt was well defined by ground‐penetrating radar, and was located accurately with respect to the top of the crypt with a moderate radar velocity (170 mm ns−1). The application of both seismic reflection and ground‐penetrating radar to the same site may be beneficial for improved understanding of their abilities for shallow subsurface imaging. Copyright © 2002 John Wiley & Sons, Ltd.

Prestack Inverse-Ray Imaging of A 2D Homogeneous Layer: A Tutorial Study

Traveling directions of an inverse ray for the common shot, the common midpoint and the common offset gathers are explicitly determined from the geometry of reflected rays and an envelope of two reflected ellipses in a 2D homogeneous irregular layer. I find that a common shot gather and a common offset gather can be applied to image the structural velocity and interface. However, due to the symmetry of the travel-time hyperbola, a common midpoint gather is not suitable for structural imaging. Furthermore, from a common offset gather, the poststack inverse ray is proved as a special case of the prestack inverse ray. Error analysis of the prestack inverse ray indicates that the method of elliptic envelope provides more accurate imaging at far offsets than the method of ray geometry if the travel-time picks are limited along a reflected hyperbola. Alternatively, when the travel-time picks are sufficient, the method of ray geometry is superior to the method of elliptic envelope. The prestack inverse ray is also applied to image a sedimentary basin. The results suggest that the best way for applying the prestack inverse ray is to determine the layer velocity from ray geometry and to image the structural interfaces by considering both methods.

A Simplified form for Nonhyperbolic Equation in a TIM

According to Dix?s formula, for the isotropic and short spread case, the moveout curves of a common shotpoint (CSP) gather or a common midpoint (CMP) gather can be approximated by a hyperbola. As a matter of fact, the spread of the field survey may not always meet the requirement of Dix?s assumption and the real earth does not behave as a perfect isotropic medium. Releasing the limitation of spread length and considering the anisotropy the real earth, Byun et al (1989) derived the formula for nonhyperbolic analysis for horizontally layered, transversely isotropic, media. The formula is called skewed hyperbolic moveout formula and involves three measurement parameters: the average vertical velocity and horizontal and skew moveout velocities. Carry on Byun?s skewed hyperbolic moveout formula; Sena (1991) derived analytical expressions giving traveltime-offset curves for multilayered weakly azimuthally isotropic and anisotropic media on terms of the elastic properties of each layer. Inheriting to the idea of skewed hyperbola moveout that has been proposed, we assumed that the seismic ray follows an imaginary rectilinear path. Based on the fundamental concept of vector analysis, we decomposed the velocity of seismic wave into two components: vertical and horizontal components. Compare to Byun?s moveout formula, the velocity parameters had been reduced from three to two. We named the new formula as a simplified nonhyperbolic moveout formula. To ascertain our ideology, the new formula is tested by laboratory data. Laboratory walk away transmission experiments were done on both Plexiglas and phenolite that were homogenous and commonly adopted to stand for isotropic and anisotropic materials, respectively. A semblance analysis is adopted to process the observed data. Scanning the recorded gather using the new formula we proposed, laboratory works show the moveout velocity that was computed from reflection seismology could be decomposed into vertical and horizontal components with reasons. In the process of time-to-depth conversion, one can always estimate the depth or layer thickness by multiplying one-way traveltime to vertical velocity of the new moveout formula, and the horizontal velocity that can be used to diagnose the anisotropy of the explored area and to explore the lithology of the subsurface strata.

Recognition and reconstruction of coherent energy with application to deep seismic reflection data

Reflections in deep seismic reflection data tend to be visible on only a limited number of traces in a common midpoint gather. To prevent stack degeneration, any noncoherent reflection energy has to be removed. In this paper, a standard classification technique in remote sensing is presented to enhance data quality. It consists of a recognition technique to detect and extract coherent energy in both common shot gathers and final stacks. This technique uses the statistics of a picked seismic phase to obtain the likelihood distribution of its presence. Multiplication of this likelihood distribution with the original data results in a “cleaned up” section. Application of the technique to data from a deep seismic reflection experiment enhanced the visibility of all reflectors considerably. Because the recognition technique cannot produce an estimate of “missing” data, it is extended with a reconstruction method. Two methods are proposed: application of semblance weighted local slant stacks after recognition, and direct recognition in the linear τ-p domain. In both cases, the power of the stacking process to increase the signal‐to‐noise ratio is combined with the direct selection of only specific seismic phases. The joint application of recognition and reconstruction resulted in data images which showed reflectors more clearly than application of a single technique.

Seismics and optics: hyperbolae and curvatures

We rederive and generalize hyperbolic moveout formulae for the common-midpoint (CMP) gather and for the zero-offset (ZO) section that can be efficiently used for macro-model-independent reflection imaging in two-dimensional media. The hyperbolic moveout formulae for the common-midpoint gather are obtained from different Taylor series expansions of a particular parametric moveout surface defined in the multicoverage data space. Such a moveout surface involves three kinematic wave-field attributes of two hypothetical waves, which have to be determined by a coherency analysis. By using hyperbolic moveout curves in the CMP gather and in the ZO section one can determine these attributes in two steps. The relationships between the local shapes of the interfaces and the attributes of the hypothetical wave-fields attributes are considered by means of geometrical optics. The determination of these attributes allows to perform a macro-model-independent ZO simulation and a subsequent inversion.

Estimation ofQfrom surface seismic reflection data

Reliable estimates of the anelastic attenuation factor, Q, are desirable for improved resolution through inverse Q deconvolution and to facilitate amplitude analysis. Q is a useful petrophysical parameter itself, yet Q is rarely measured. Estimates must currently be made from borehole seismology. This paper presents a simple technique for determining Q from conventional surface seismic common midpoint (CMP) gathers. It is essentially the classic spectral ratio method applied on a trace‐by‐trace basis to a designatured and NMO stretch‐corrected CMP gather. The variation of apparent Q versus offset (QVO) is extrapolated to give a zero‐offset Q estimate. Studies on synthetics suggest that, for reasonable data quality (S/N ratios better than 3:1, shallow (<5°) dips, and stacking velocity accuracy <5%), source‐to‐reflector average Q is recoverable to within some 3% and Q for a specific interval (depending on its two‐way time thickness and depth) is recoverable to 15–20%. Three case studies are reported. First, Q versus offset and vertical seismic profiling (VSP) Q estimates for a southern North Sea line were in close agreement, validating the method. For Chalk, Mushelkalk‐Keuper, and Bunter‐Zechstein, Q was estimated as 130 ± 15, 47 ± 8, and 156 ± 18, respectively. Next, two alternative lithological interpretations of a structure seen in a frontier area were discriminated between when Q estimates of 680 to 820 were obtained (compared to some 130–170 in the overlying units), favoring a metamorphic/crystalline lithology rather than (prospective) sediments. This was later confirmed by drilling. Third, a profile of Q estimates along a 200-ms-thick interval, known to include a gas reservoir, showed a clear and systematic reduction in Q to a low of 50 ± 11, coincident with the maximum reservoir thickness, from some 90–105 outside the reservoir. Q for the reservoir interval itself was estimated at 17 ± 7.

The equivalent offset method of prestack time migration

A prestack time migration is presented that is simple, efficient, and provides detailed velocity information. It is based on Kirchhoff prestack time migration and can be applied to both 2-D and 3-D data. The method is divided into two steps: the first is a gathering process that forms common scatterpoint (CSP) gathers; the second is a focusing process that applies a simplified Kirchhoff migration on the CSP gathers, and consists of scaling, filtering, normal moveout (NMO) correction, and stacking. A key concept of the method is a reformulation of the double square‐root equation (of source‐scatterpoint‐receiver traveltimes) into a single square root. The single square root uses an equivalent offset that is the surface distance from the scatterpoint to a colocated source and receiver. Input samples are mapped into offset bins of a CSP gather, without time shifting, to an offset defined by the equivalent offset. The single square‐root reformulation gathers scattered energy to hyperbolic paths on the appropriate CSP gathers. A CSP gather is similar to a common midpoint (CMP) gather as both are focused by NMO and stacking. However, the CSP stack is a complete Kirchhoff prestack migrated section, whereas the CMP stack still requires poststack migration. In addition, the CSP gather has higher fold in the offset bins and a much larger offset range due to the gathering of all input traces within the migration aperture. The new method gains computational efficiency by delaying the Kirchhoff computations until after the CSP gather has been formed. The high fold and large offsets of the CSP gather enables precise focusing of the velocity semblance and accurate velocity analysis. Our algorithm is formulated in the space‐time domain, which enables prestack migration velocity analysis to be performed at selected locations and permits prestack migration of a 3-D volume into an arbitrarily located 2-D line.

A bootstrap procedure for high‐resolution velocity analysis

A method is given for further improving velocity estimates derived from high‐resolution velocity analysis. In conventional velocity analysis, a set of tentative velocities is used to apply a normal moveout (NMO) correction to a set of spatio‐temporal windows, the coherence measure is evaluated for each velocity and finally, the velocity estimate is retrieved from the peak of the coherence measure. Because analytical expressions for velocity uncertainties are difficult to derive, I propose an intensive statistical procedure, the bootstrap method, to assess the accuracy of the velocity estimate. In the bootstrap method, I create a data sample by randomly drawing seismic traces with replacement from a window of the common midpoint gather (CMP). Next, I calculate the velocity that maximizes the coherence measure for each bootstrap realization. The variation of this velocity provides a means to compute standard errors. I also use the bootstrap method to construct an average coherence measure and a kernel density estimator of the velocity that maximizes the coherence. The average coherence exhibits an important attenuation of spurious events while retaining enough resolution to model reflections properly with similar moveout curves. The latter is illustrated with synthetic and field data examples.

A residual NMO analysis method for multiple attenuation

The method of residual normal moveout (RNMO) analysis for multiples described in this paper is based on the NMO correction of common midpoint (CMP) gathers.

AVA analysis after velocity-independent DMO and imaging

A common midpoint (CMP) gather usually provides amplitude variation with offset (AVO) information by displaying the reflectivity as the peak amplitude of symmetrical deconvolved wavelets. This puts a reflection coefficient R at every offset h, giving a function R(h). But how do we link h with the angle of incidence, θ, to get the reflectivity function, R(θ)? This is necessary for amplitude versus angle-of-incidence (AVA) analysis. One purpose of this paper is to derive formulas for this linkage after velocity-independent dip-moveout (DMO), done by migrating radial sections, and prestack zero-offset migration. Related studies of amplitude-preserving DMO in the past have dealt with constant-offset DMO but have not given the connection between offset and angle of incidence after processing. The results in the present paper show that the same reflectivity function can be extracted from the imaged volume whether it is produced using radial-trace DMO plus zero-offset migration, constant-offset DMO plus zero-offset migration, or directly by prestack, common-offset migration. The data acquisition geometry for this study consists of parallel, regularly spaced, multifold lines, and the velocity of propagation is constant. Events in the data are caused by an arbitrarily oriented 3-D plane reflector with any reflectivity function. The DMO operation transforms each line of data (m, h, t), i.e., midpoint, half-offset, and time, into an (m1, k, t1) space by Stolt-migrating each radial-plane section of the data, 2h = Ut, with constant velocity U/2. Merging the (m1, k, t1) spaces for all the lines forms an (x, y, k, t1) space, where the first two coordinates are the midpoint location, the third is the new half-offset, and the fourth is the time. Normal moveout (NMO) plus 3-D zero-offset migration of the subspace (x, y, t1) for each k creates a true-amplitude imaged volume (X, Y, k, T). Each peak amplitude in the volume is a reflection coefficient linked to an angle of incidence.

Seeking Low Relief Structures: An Original Approach Applied in Western Oman

ABSTRACT A multiple elimination technique was tested in Elf’s Butabul Concession in Western Oman. The multiples mask the main exploration target, the Permian-Carboniferous Haushi reservoir. The technique is based on inversion in a curvature-time domain. The common mid-point gathers are transformed into this new domain and a mute is then applied to eliminate the multiples. The resulting function is then transformed back to the space-time (x-t) domain. The technique was tested on a newly-acquired, high-resolution seismic line. The survey is approximately 100 kilometers and traverses a low relief seismic prospect south of the Rabkha-1 well. Special high-resolution field acquisition parameters were used to maximize the effective separation of real events from multiples. To prevent aliasing the source and trace spacing were 12.5 and 25 meters, respectively. The Rabkha-1 well was used to select the best mute in the curvature-time domain. The mute required only minor adjustments over the line. The technique proved effective in eliminating the multiples and the resulting interpretation convinced the explorationists not to drill the prospect.

Bayesian inference, Gibbs' sampler and uncertainty estimation in geophysical inversion1

AbstractThe posterior probability density function (PPD), σ(m|dobs), of earth model m, where dobs are the measured data, describes the solution of a geophysical inverse problem, when a Bayesian inference model is used to describe the problem. In many applications, the PPD is neither analytically tractable nor easily approximated and simple analytic expressions for the mean and variance of the PPD are not available. Since the complete description of the PPD is impossible in the highly multi‐dimensional model space of many geophysical applications, several measures such as the highest posterior density regions, marginal PPD and several orders of moments are often used to describe the solutions. Calculation of such quantities requires evaluation of multidimensional integrals. A faster alternative to enumeration and blind Monte‐Carlo integration is importance sampling which may be useful in several applications. Thus how to draw samples of m from the PPD becomes an important aspect of geophysical inversion such that importance sampling can be used in the evaluation of these multi‐dimensional integrals. Importance sampling can be carried out most efficiently by a Gibbs' sampler (GS). We also introduce a method which we called parallel Gibbs' sampler (PGS) based on genetic algorithms (GA) and show numerically that the results from the two samplers are nearly identical.We first investigate the performance of enumeration and several sampling based techniques such as a GS, PGS and several multiple maximum a posteriori (MAP) algorithms for a simple geophysical problem of inversion of resistivity sounding data. Several non‐linear optimization methods based on simulated annealing (SA), GA and some of their variants can be devised which can be made to reach very close to the maximum of the PPD. Such MAP estimation algorithms also sample different points in the model space. By repeating these MAP inversions several times, it is possible to sample adequately the most significant portion(s) of the PPD and all these models can be used to construct the marginal PPD, mean) covariance, etc. We observe that the GS and PGS results are identical and indistinguishable from the enumeration scheme. Multiple MAP algorithms slightly underestimate the posterior variances although the correlation values obtained by all the methods agree very well. Multiple MAP estimation required 0.3% of the computational effort of enumeration and 40% of the effort of a GS or PGS for this problem. Next, we apply GS to the inversion of a marine seismic data set to quantify uncertainties in the derived model, given the prior distribution determined from several common midpoint gathers.

Combining reflection tomography with layer replacement for velocity analysis of near‐surface reefs

Reefs in the Browse Basin on the Australian North West Shelf typically have strong lateral velocity inhomogeneity on the reef flanks and produce an irregular sea‐floor topography. The combination of these two factors reduces the effectiveness of conventional velocity analysis as moveout on common midpoint (CMP) gathers is no longer hyperbolic. Consequently, the velocity structure on the reef flanks cannot be resolved properly. An accurate velocity structure is needed for these reefs if seismic data acquired in the vicinity are to be migrated correctly. Tomographic velocity analysis, using a simultaneous iterative reconstruction technique (SIRT), was found to improve the velocity analysis of the reef when compared to conventional analysis. However, the reef’s profile and rapid internal velocity gradients prevent accurate raypath determination. This technique can be further improved by preconditioning the data with wave‐equation datuming. The water is replaced with an average reef velocity by downward and upward continuation. This effectively removes most of the raypath distortions resulting from ray bending at the sea floor, thus reducing the nonlinearity of the forward modeling and improving the convergence of an iteratively linearized inversion. The velocities obtained from this method give an accurate picture of the reef’s internal velocity structure, as well as the lower portion of the seismic section below the reef.

Accuracy study of velocity estimation in 3-D layered models

Velocity estimation is examined in 3-D layered structures formed by plane and curved interfaces. The applied technique of coherency inversion tests the layer velocity through the repeating sequence of ray migration/coherency measurement. The reconstructed velocity‐depth model fits zero‐offset reflection times and maximizes semblance on input common midpoint (CMP) gathers. The correctness of layer velocity analysis disregarding the three‐dimensionality of the structures is under consideration. Using the 2-D coherency inversion technique, velocity is correctly determined in the upper layer of the examined structures. Two‐dimensional analysis in the deeper layer gives biased velocity estimates. The errors in the 2-D velocity estimates vary with the profile azimuth and appear in the form of the apparent velocity anisotropy. The inaccuracy of 2-D velocity estimation is analytically considered for the profile oriented along the refractor strike direction. The derived equation relates the velocity error to structure geometry and to the velocity contrast above and below the refractor. Three‐dimensional velocity analysis in the examined structures reveals that the layer velocity resolution is affected by the refractor shape. Below the convex refractor the velocity resolution deteriorates compared with that below the plane.

Recognizing surface scattering in ground‐penetrating radar data

Ground‐penetrating radar (GPR) data may show strong noise events as a result of scattering by surface objects on the ground or above the survey line. The relative strength of these events can be large in comparison to reflections from geologic features, because radar signals in the ground attenuate exponentially whereas signals that travel in the air attenuate geometrically. Migration of GPR field data from clastic and carbonate sequences in central Oklahoma distinguishes between scattered events and geologic events because the former are focused at the air‐wave velocity, while the latter are focused at the ground‐wave velocity. Forward modeling using locations of scatterers derived from migration confirms the presence of scattered events, and common midpoint (CMP) gathers are helpful in identifying surface scattering. Scattered events displayed at a horizontal/vertical scale of 1:1 are easily mistaken for subhorizontal, geologic reflections. Methods of recognizing scattered events and removing them, if possible, are therefore crucial to correct geological interpretation of GPR data.

Processing of pre-critical wide-angle seismic reflection data from the BABEL project

In addition to using the now widely recognised technique of simultaneously acquiring deep reflection profiles at sea and long-range refraction profiles using three-component land stations, the BABEL seismic experiment also recorded P- and S-wave energy from the large airgun array at four linear 38-component land spreads. The pioneering nature of this work led, in hindsight, to several deficiences in design, but also pointed the way towards substantial advances in analysis and interpretation of integrated seismic observations. The multifold nature of the recordings at offsets providing pre-critical reflections from Moho depths resulted in interleaved receiver gathers with trace spacings of 4–5 m, thus providing some of the most densely spaced deep-seismic wide-angle sections to date. With such a trace density, the signal-to-noise ratio was substantially increased by summation of adjacent traces, without degrading spatial resolution. Stacking the land array data using conventional common midpoint (CMP) gather techniques over source-receiver offsets of 7–78 km doubled the depth range constrained by numerically precise stacking velocity analysis. Offsets > 10 km provide normal moveouts > 1 s, although the 1.5 km of differential moveout available here limited velocity discrimination and required additional constraints to define zero-offset traveltimes. For one array, stacking velocity errors of ±200 m s−1 imposed a resulting error in zero-offset time of up to ±400 ms without use of any marine data. With receiver response functions matched, stacked sections using data from geophone arrays exhibit similar features to stacks made using hydrophone arrays at 0–3 km offsets in the same area, but also significant differences. The differences in reflection amplitudes cannot be simply attributed to theoretical amplitude-offset variations and probably represent variation in the reflectivity of the crustal rocks of the Baltic shield.