Fifty years of stacking

Acta Geophysica DOI: 10.2478/s11600-013-0191-4

________________________________________________ © 2014 Institute of Geophysics, Polish Academy of Sciences

Fifty Years of Stacking

Mohamed RASHED

Department of Geophysics, Faculty of Earth Sciences, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia

Geology Department, Faculty of Science, Suez Canal University, Ismailia, Egypt e-mail: [email protected]

A b s t r a c t

Common-Mid-Point (CMP) stacking is a major process to enhance signal-to-noise ratio in seismic data. Since its appearance fifty years ago, CMP stacking has gone through different phases of prosperity and negli-gence within the geophysical community. During those times, CMP stacking developed from a simple process of averaging into a sophisti-cated process that involves complicated mathematics and state-of-the-art computation. This article summarizes the basic principles, assumptions, and violations related to the CMP stacking technique, presents a histori-cal overview on the development stages of CMP stacking, and discusses its future potentiality.

Key words: CMP, stacking, seismic, processing.

1. INTRODUCTION In the field of geophysical exploration, the term stacking generally refers to averaging a number of geophysical measurements instead of taking one measurement in each field station in the hope of enhancing the coherent sig-nal and attenuating the random noise. In the world of seismic exploration, however, stacking is used extensively in different stages of data acquisition and processing. The simplest and earliest stacking in seismic exploration is vertical stacking or the use of multiple sources and/or receivers data acquisi-tion. In seismic data processing, the concept of stacking is used on several occasions. The summation of traces in velocity semblance, constant velocity

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scans, dynamic stack during velocity analysis, and the summation of diffrac-tion hyperbola in some sorts of migration are other forms of stacking (Mohanty et al. 2000, Wolf et al. 2004, Kostecki and Półchłopek 2006). This article, however, concentrates only on CMP stacking, which is the binding of Normal Move-Out (NMO) corrected traces across a CMP gather into a sin-gle trace whose signal-to-noise (S/N) ratio should be higher than that of in-dividual traces within the gather. In particular, the article’s focus is the way these traces and time samples are arranged, modified, and/or altered before averaging.

The CMP stacking technique was introduced by William Harry Mayne in 1962 and has since then been the basis for acquisition and processing of seismic reflection data. Mayne named his method common reflection point horizontal stacking. During the past 50 years a wide variety of names were used to describe that method, such as Common-Reflection-Point (CRP), Common-Bounce-Point (CBP), Common-Datum-Point (CDP), Common-Reference-Point (CRP), roll-along, and Common-Depth-Point (CDP). Re-cently, the geophysical community seems to settle down with the name Common-Mid-Point (CMP) stacking.

2. PRINCIPLES, ASSUMPTIONS, AND VIOLATIONS The concept of CMP stacking is quite simple. The distance between seismic source and receiver is varied systematically to survey the same subsurface point several times with different source-receiver offsets (Fig. 1a). The result is a CMP gather that has a number of traces imaging the same midpoint with different raypaths (Fig. 1b). The collected seismic traces are then corrected for normal move-out (Fig. 1c), and then summed (Fig. 1d) and normalized (Fig. 1e) to obtain the stacked trace. The stacking process is based solely on the assumption that signal is coherent while noise is often random and under proper conditions, stacking will remove several types of noise but it is most effective against random noise (Fig. 1). Mathematically, the signal-to-noise ratio is improved by a factor of N where N is the number of stacked traces or fold.

This traditional stacking technique assumes that all traces in a CMP gather have equal validity and that all signals are coherent while all noises are random. These assumptions, however, are valid only if earlier processes resulted in an ideal CMP gather in which all seismic events are perfectly aligned and their amplitudes are uniformly distributed while all noises are uncorrelated, stationary, normally distributed, and of close magnitudes. Clearly, this ideal case never exists in reality. On the contrary, imperfectly aligned reflections, coherent noise and noise bursts are both common and acceptable in all sorts of seismic data.

FIFTY YEARS OF STACKING

Fig. 1. Principles of CMP stacking.

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The assumption that all seismic reflections on the NMO-corrected CMP gather are impeccably aligned is not always true because of many theoretical and practical reasons. In applying the NMO correction, the travel time as a function of offset forms a series of plane horizontal iso-velocity layers that is approximated by a hyperbola. This approximation is better applied at small offsets than in large offsets (Yilmaz 2001). The departure of large-offset traces from hyperbolic equation is well known but it is usually ignored in seismic data processing. Moreover, estimating stacking velocities is based on picking a small number of velocities on some selected CMP gathers. Not on-ly the picking process has a good portion of human judgment and hence er-ror, but also it leaves huge temporal and spatial gaps in seismic data where velocities are estimated by simple interpolation between smoothed velocity curves. Despite of how sophisticated the processing software and how expert the data processing geophysicist, NMO correction never yield perfectly aligned seismic events along the entire seismic data volume.

The assumption of uniform distribution of amplitude, phase, and fre-quency of reflections along the NMO corrected CMP gather is also violated in many ways. One of the causes of the violation is the offset-related varia-tions in amplitude, waveform, reflection coefficient, and wave type conver-sion. Another cause of the violation is the frequency stretch associated with NMO correction. As a result of NMO correction, traces are stretched in a time-variant manner, causing their frequency content to shift to the low end of the spectrum. Frequency distortion increases at shallow depth and with large offsets (Yilmaz 2001). The problem of frequency stretch associated with NMO correction has long been known (Dunkin and Levin 1973, Rupert and Chun 1975) and has been discussed intensively in literature. There have been many attempts to attenuate the effect of NMO stretch (Miller 1992, Byun and Nelan 1997, Claerbout 1992, Shatilo and Aminzadeh 2000, Castoro et al. 2001, Trickett 2003). In today’s practice, however, the most common solution is stretch muting, where samples at the beginning of a trace that suffer severe NMO stretch are zeroed out (Miller 1992). Severe muting introduces several problems to seismic data such as decreasing fold, leaving gaps in 3-D volumes, reducing multiple suppression, and destroying information necessary for Amplitude Variation with Offset (AVO) analysis. Usually a compromise is made and some stretch is left which creates a stacked signal with lower frequency and hence low temporal resolution when conventional straight mean stacking is applied to the data.

Another problem that is related to offset variations is the gain-related complications. In the early stages of seismic data processing, and sometimes during data recording, a gain recovery function is applied to the data set in order to compensate for the amplitude loss caused by wave front divergence in late arrival times and large offsets. Since far offset traces are attenuated


more than near offset traces, they are altered by the gain correction more than small offset ones. When a gain recovery function is applied, just as geo-logically significant reflection signals are amplified, so too is noise (Reyn-olds 1997). Large-offset traces contain more gain-introduced noise than small-offset ones and they have bad influence on the final stack when they are included in the straight mean stacking process.

Another invalid assumption is that seismic events on NMO-corrected gathers have no residual static shifts. Static corrections are applied to land seismic data in order to compensate for the effect of variations in elevation, weathering thickness, weathering velocity, or reference to a datum. It is a common practice to correct for static variations using a combination of field refraction and residual static corrections. For these corrections to be per-formed in a satisfactory manner, we need an accurate normal move-out cor-rection and a relatively high-fold data with good S/N ratio (Pugin and Pullan 2000). Consequently, one should expect a considerable amount of static shifts remaining in seismic data. These remaining static shifts reduce the amplitude of the stacked signal and lower its frequency resulting in a lower resolution stack. The residual statics and their harmful effect on the stacked data have been under intensive studies since the early days of CMP stacking and until present (Hileman et al. 1968, Taner et al. 1974, Kirchheimer 1983, Stein et al. 2009, Gholami 2013).

In fact, the basic assumption that increasing source-receiver offset sys-tematically around a centered point on the surface would provide traces im-aging the corresponding point in depth is valid only if subsurface layers are perfectly horizontal and have no lateral velocity variation. That is why the concept of Dip-Move-Out (DMO) was intensively discussed in literatures and several strategies of DMO correction were proposed and tested (Levin 1971, Deregowski 1982, Jakubowicz 1990, Larner and Hale 1992, Liner 1999, Malcolm et al. 2005).

In conclusion, straight mean stacking or simple averaging of a group of seismic traces recorded at different offsets in a CMP gather to produce a sin-gle stacked trace is not accurate simply because it is based on a group of as-sumptions that are not perfectly valid when dealing with seismic data in reality. Moreover, straight mean stacking does not deal well with minor im-perfections in pre-stack processing steps that are common in seismic explo-ration, such as residual statics, imperfect NMO correction, and insufficient stretch muting. This inaccuracy may result in a final stack with lower ampli-tude, lower resolution, and hence less interpretability. These reasons were the driving force for the large number of alternative stacking techniques and algorithms proposed and applied in both the academia and the industry dur-ing the past 50 years.

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3. A CHRONICLE OF CMP STACKING The year 1962 may mark the official birth of CMP stacking as we know it today but the roots of the idea go back to 1938 when Cecil Green proposed and applied multiple reflection paths centered at a common depth point to eliminate the effect of dip on velocity determination in seismic profiles (Green 1938). Green’s work as well as others’ paved the road for William Mayne who applied for a US patent in 1950 proposing a short and vague idea of CMP stacking. His idea, patented in January 1956, explained that in-formation associated with a given reflection point but recorded using differ-ent shot and recording locations can be combined algebraically after applying appropriate time corrections and that the combining process will reduce noise (Mayne 1956). These pioneer preliminary ideas along with the introduction of magnetic tape recording instruments in seismic exploration (Loper and Pittman 1954) and digital recording and processing instruments in the early 1960’s have led to the outbreak of the CMP stacking technique (Mayne 1962).

It was not long before the CMP stacking technique became in the spot-light of the geophysical community and received different sorts of appraise and criticism. Many studies adopted and applied the CMP stacking technique with different degrees of success in enhancing S/N ratio (Musgrave 1962, Foster et al. 1964, Schneider et al. 1965, Zagst 1965, Meyerhoff 1966, Cour-tier and Mendenhall 1967, Marr and Zagst 1967). Other studies criticized the technique (Cressman 1968) or proposed alternative or modified techniques (Embree 1968, Robinson 1968). Actually, in 1967, Mayne himself wrote an article about the limitations of the CMP technique stating that its efficiency is conditioned by the appropriate alignment of primary reflections before stacking. He admitted that inaccurate move-out correction, residual statics, and waveform variations with offset could severely harm the output of the stacking process (Mayne 1967). Cressman (1968) shed some light on the ef-fect of velocity layering and dipping interfaces on the accuracy of normal move-out correction and hence on the output of the stacking process.

In 1966, Peter Embree applied for a patent for the diversity stack as an alternative stacking technique that was patented in August 1968. Diversity stack guarantees, through few computation steps, that the amplitudes exceed-ing certain threshold are excluded from the stacking process while ampli-tudes less than this threshold remain unchanged (Embree 1968). Diversity stack remains popular until today in land seismic data processing because of its effective discrimination against noise bursts, ground roll, and similar high-amplitude wave trains. Another reason for the attractiveness of the di-versity stack is its ability to preserve amplitude variations of the input rec-ords (Kirk 1981). In the same year, John Robinson proposed a modified


semi-deterministic approach to attenuate noise in CMP records. His idea was to come up with algebraic solutions for the frequency response of the prima-ry signal in the frequency domain and then perform statistical averaging of CMP trace pairs. Robinson stated that his method provides better results than the normal stacking process (Robinson 1968). Another related idea, patented to Norris Harris in 1968, is the common-offset gather stack. Harris claimed that his idea reduces random noise and helps identifying coherent noise so they can be filtered out (Harris 1968).

The early 1970’s continued witnessing loud criticism of the newly born CMP stacking technique as several alternative techniques had been proposed and tested, including the statistically optimal stack (Robinson 1970) and the Nth-root stack (Kanasewich et al. 1973). The optimum stack is based on ap-plying a weighting function to individual traces in a CMP gather to increase S/N ratio in the resultant stack. Robinson admitted that the output of the op-timum stack is not significantly better than that of the straight mean stack but it is worth further investigation (Robinson 1970). The Nth-root stacking gained its popularity because of its simplicity and its capability to discrimi-nate against noise spikes. In Nth-root stack, all the amplitudes within a time sample are N-rooted before stack and the stacked output is then raised to the power of N, where N equals 2 or any of its doubles, with the signs main-tained during the entire process. The idea was first introduced by Muirhead (1968) to facilitate avoiding false alarms in automatic detection of natural seismic events and was then adapted to CMP stacking by Kanasewich et al. (1973).

The concept of weighted stack that was introduced by Robinson in 1970 continues to be the core of many alternative stacking algorithms proposed till present. Many of these stacking techniques are based on trace or sample weighting using different criteria (White 1977, Brown et al. 1977, Rietsch 1980, Inguva and Schick 1981, Katz et al. 1985, Waltham and Boyce 1986, Anderson and McMechan 1990, Rashed et al. 2002, Rashed 2008, Liu et al. 2009, Sanchis and Hanssen 2011). Most of these techniques assign specific weights for traces or time samples based on certain criteria including offset, S/N ratio, variance, and correlation to a pilot trace.

The idea of median stack was first suggested by Claerbout and Muir (1973). Their experiments on several sorts of geophysical data showed that when data amplitudes are not uniform, the median could be a better and more robust representative than the mean (Claerbout and Muir 1973). In me-dian stack, for each time level, the median value is selected instead of the mean value. The median stack is still in use until today because its capability to exclude noise bursts and because it is little influenced by coherent noise occurring in the same time level of a primary signal on less than half of the input traces. The median stack’s main disadvantage is its dependence on a

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single value which compromises the discrimination against random noise provided by the averaging. Another disadvantage of the median stack is the abrupt change between amplitudes of neighboring samples both temporally and spatially, which may cause some frequency distortion and a band pass filter must be applied afterwards. The frequency distortion problem can also be overcome by the alpha-trimmed stack technique that will be discussed later (Watt and Bednar 1983). Modified median stack techniques such as the weighted mean stack and the iterative mean stack can also be applied. The weighted mean stack is the mean value of the output of a gapped median fil-ter applied to time samples whereas iterative weighted stack reduces the number of samples successively by repeated filtering (Naess and Bruland 1989). These modified median stacks are most useful in coherent noise sup-pression such as multiples. At that time, many studies were in favor of medi-an stack rather than mean stack (Stewart 1985, Woodward 1985).

The late 1970’s and early 1980’s witnessed the introduction of a large number of alternative stacking techniques that still form the cornerstone of all stacking algorithms proposed until present. In 1978 William Ruehle filed to patent a new idea to reduce the harmful effect of static shifts on the final stack and was granted a US patent in 1980 (Ruehle 1980). Ruehle’s idea was to form a pilot trace from the CMP gather and to time-shift traces within the gather by cross-correlation with the pilot trace to get maximum coherency of primary reflections. The idea is similar to the concept of the superstack pro-posed by Naess (1979). Superstack is based on separating positive and nega-tive amplitudes at each time level. The two sets are then treated iteratively to suppress abnormal amplitudes in each set. This treatment reduces the influ-ence of the total set of the least represented polarity on the final stack. The disadvantage of this treatment is the gradual distortion of the waveform that occurs with iteration and hence iteration should be kept minimal. Further studies, however, proved that the superstack can be effective in improving velocity analysis (Naess and Bruland 1981). Superstack is so efficient that it is still in use and subject to discussions until today despite of the large num-ber of stacking algorithms proposed afterwards (Cao et al. 2012, Rückemann 2012a, b).

Naess (1982) came up with a modified superstack technique to avoid the low lateral and vertical resolution caused by the averaging process. He named his technique single-trace iterative stack. In the single trace iterative stack, the same procedures of the superstack are applied to the NMO-corrected CMP gather but instead of averaging these traces at the end, an op-tional trace is selected to represent the gather. Since the S/N ratio enhance-ment made by the superstack occurs mainly before averaging, selecting a single trace instead of averaging should enhance S/N ratio without suffering distortion caused by the averaging process. However, practical application of


this technique showed that it is sensitive to data with excessive noise and to lack of lateral consistency across the CMP gather. Interestingly, in his arti-cle, Naess also attracted the attention to the fact that the so-called common midpoint is, in most cases, an area of a certain extension. This will set the basis of what we know today as Common-Reflection-Surface (CRS) stack.

In the annual meeting of the Society of Exploration Geophysicists held in Dallas, USA, in 1982 another two interesting stacking techniques were presented; these are the iterative weighted stack (Pruett 1982) and the ran-dom stack (Currie 1982). In the iterative weighted stack, a straight mean stack of the CMP gather is used as a pilot trace and time samples of traces within the gather are given normalized smoothed weights based on the vari-ance between their amplitudes and corresponding sample’s amplitude of the pilot trace. The weighted traces are stacked into a single trace that can be considered a final stack or a new pilot trace for a new iteration. The iterative weighted stack is described as a stable process where after several iterations; a little change is introduced to the stack. Another advantage of this stacking algorithm that it causes little signal distortion while effectively suppresses the effect of high amplitude noise. However, when extreme noise amplitudes are distributed on both sides of the mean value, the output of the procedure can be biased (Naess and Bruland 1985).

The idea of the random stack is quite interesting. Instead of averaging traces in an NMO-corrected CMP gather, an optional number of random samples is picked at each time level. These samples are used to construct a number of random sample traces, which are stacked in a special way. At each time level, if samples have the same sign, their average is assigned to the corresponding sample in the stacked trace; otherwise, the amplitude of the stacked trace is set to zero. Since primary reflections are in-phase while random and coherent noises are out-of-phase, this stacking procedure should keep the reflections intact while noise and multiples are zeroed out (Currie 1982). However, this procedure inherently results in a severe non-linear fil-tering of data. This non-linear filter is highly sensitive to noise in the data and can hardly be expected to give good results when applied to real data (Naess and Bruland 1985).

The following year, one of the simplest but most efficient stacking tech-niques called the alpha-trimmed stack was proposed by Watt and Bednar (1983). The technique is a modification of the median stack that overcomes the frequency distortion problem. In the alpha-trimmed stack, the average of a group of values situated around the middle of the sorted input values is taken instead of the single median value. The attractiveness of the alpha-trimmed stack is the flexibility, provided by the variable parameter alpha, to get a stack that has the desired portion of the properties of the mean stack or the median stack (Bednar and Watt 1984). The alpha-trimmed stack does not

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only discriminate against transient noise but can also attenuate interference noise from other crews in marine seismic data (Haldorsen and Farmer 1989). The main problem associated with the alpha-trimmed stack is the fold reduc-tion caused by the exclusion of a percent of extreme amplitudes. Another problem is the fact that the exclusion process is done blindly which may lead to excluding samples having positive contribution to the stack (Rashed 2008).

In the same year, David Buchanan and others described a special method of stacking seismic data collected to detect faults in seam layers. Their method was based on selecting traces having a common reflection segment of the reflector. A reflector is divided into a number of equal length seg-ments and each trace is assigned to the segment, which contains the point at which reflection would occur. Traces belonging to the same line segment are stacked into distance-distance space or distance-slowness space (Buchanan et al. 1983). Although primitive, this method might have set the base for a technique that is widely used today, which is Common-Reflection-Surface stack or CRS.

Some well-known stacking methods, used occasionally in the industry, appeared around that period such as minimum/maximum exclusion stack and sign-bit stack. In the minimum/maximum exclusion stack, the highest and lowest amplitudes at each time level are excluded from the stacking process in order to avoid stacking noise bursts. The sign bit stack adds +1 to the stacked sum if the absolute amplitude value at a sample is positive or zero and adds –1 if the amplitude value is negative. The actual sample values are ignored. This stacking technique is suitable where the main objective is to discriminate between positive and negative reflections in the final stack but it destroys the real amplitude and distorts the frequency content of seismic data. To my knowledge, these two methods have no documentation in litera-ture. Accordingly, I assume that these stacking methods were proposed and implemented within the industry without publication references.

Anderson and McMechan (1990) introduced a new weighted stack based on weighting traces using their amplitude decay rates and noise amplitudes. The amplitude-decay rates are measured relative to the median amplitude-decay rate as a function of midpoint and offset while the noise amplitudes are measured using the data before the first seismic arrivals or at late record times. This method worked fine with synthetic data but, compared to straight mean stacking method, it did not bring much enhancement when applied to real data (Anderson and McMechan 1990).

For some reason, the rest of 1990’s went by with neither major ad-vancements in the existing stacking techniques nor the introduction of new stacking algorithms. The only new stacking algorithm proposed was the multifocusing, which is a technique for move-out correction and stacking


that is based on the homeomorphic imaging (Berkovitch et al. 1994). In multifocusing, a zero-offset trace is constructed by stacking traces that do not necessarily belong to the same CMP gather. Instead, traces whose sources and receivers are within a specific super-base in the vicinity of a central point, are stacked together. The size of the super-base is determined by the size of the Fresnel zone. The move-out correction is performed based on the Fresnel zone and the curvature radii of two waveforms. Multifocusing results in a much higher resolution stacks due to the high fold and due to avoiding the ordinary NMO stretch effect (Berkovitch et al. 1998).

Figure 2 shows a seismic section stacked using the conventional CMP stack and the multifocusing stack. The comparison clearly shows the great enhancement multifocusing stack brings to the lateral and temporal resolu-tion of seismic data. Multifocusing is still an interesting topic of research un-til today (Berkovitch et al. 2008, Belfer et al. 2008, Landa et al. 2010).

A couple of years to the new millennium, Muller (1998) and Hubral et al. (1998) presented the revolutionary idea of the new Common-Reflection-Surface (CRS) stack in two separate talks presented in the EAGE conven-tion. The CRS stack can be looked at as just a spatial extension of the CMP stack. The CRS stack argues that since we all know and accept the fact that traces within a single CMP gather do not belong to the same reflection point but rather to an area, this area or aperture can be extended spatially to in-clude traces from the CMP neighborhood into the stacking process. The CRS stack assumes the subsurface to be set up by reflector segments with arbitrary location, orientation, and curvature and the CRS stacking operator approxi-

Fig. 2. Seismic section obtained by CMP stacking to the left and by multifocusing stacking to the right (after Belfer et al. 2008).

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mates the kinematic reflection response of such a reflector segment. Due to the larger number of parameters involved in the CRS stacking process, many publications investigating different parameters and experiments appeared since the introduction of CRS stack in 1998 and until today (e.g., Mann et al. 1999, Zhang et al. 2001, Gierse et al. 2003, Hertweck et al. 2005, 2007, Yoon et al. 2009, Baykulov et al. 2011, Santos et al. 2011, Fomel and Kazinnik 2013). The CRS stack is based on the same ideas and principles as the conventional CMP stack. The basic difference is that a CRS stack uses far more traces than those present in a CMP gather. For this reason, already the first CRS stack examples showed a significant increase in S/N ratio as compared to a conventional stack (Fig. 3). Therefore, the CRS stack tech-nique was regarded in the industry as a radically new concept of stacking da-ta beyond CMP.

The CRS brought much enhancement to the reflection continuity and resolution in final stacks but this issue is too large to be discussed in this ar-ticle.

Back to the CMP stacking, the new millennium triggered a new stacking algorithms rush that started with the beginning of the year 2000 and contin-ued until today. In 2000, Shatilo and Aminzadah patented a new idea of Constant Normal Move-out (CNMO) correction. The method calculates the NMO correction at the middle of the wavelet Constant Normal Move-Out (CNMO) correction, and then applies this CNMO correction to a time win-dow equals to the wavelet length at zero offset. Although Shatilo and Aminzadeh (2000) admitted that their method is imperfect, it definitely gave better results than the conventional NMO correction method and undoubted-ly paved the ground for other ideas such as the stretch-free stack proposed three years later by Trickett.

Fig. 3. Seismic section obtained by CMP stacking to the left and by CRS stacking to the right (after Hertweck et al. 2007).


In the same year, Sugimoto and others proposed a new method for stack-ing reflected scattered shear waves for ultra-shallow investigations. They considered the stacking using a virtual reflection point that is located either under the source or the receiver. They argued that their method produces good results for ultra-shallow shear wave imaging (Sugimoto et al. 2000).

Runnestrand et al. (2002) patented the idea of coherency stack, in which a pilot trace is formed by averaging traces within the CMP gather. At each time level, samples of the gather are correlated with the corresponding sam-ple of the pilot trace using an extended correlation algorithm, called RB cor-relation, which takes in consideration both the amplitude and the shape of waveforms. Based on this correlation, a specific RB factor is assigned to every sample in the CMP gather and samples are ranked or weighted accord-ingly. Samples that do not meet a user-defined threshold are muted and the rest of weighted samples are stacked. The coherency stack resulted in a great deal of enhancement when applied to real data.

Rashed (2003) combined the old idea of trace weighting with the rela-tively new idea of optimum window single trace data acquisition of Pullan and Hunter (1990) and came up with the optimum-offset weighted stack. In this stacking technique, an optimum trace in each CMP gather is selected based on S/N ratio. Traces are then weighted based on their offset with the optimum trace using the inverse distance weighting equation (Shepard 1968). The optimum-offset weighted stack was proved efficient in stacking shallow seismic data collected in urban areas where far-offset traces are af-fected by gain-introduced noise while near-offset traces are infested with ground-roll (Rashed 2003).

Trickett (2003) proposed a simple and robust idea, which he called stretch-free stacking, to avoid frequency stretch caused by NMO correction. The method replaces NMO correction and stacking by a single-step inver-sion to zero offset. Stretch-free stacking is just a combination of two ideas presented earlier, which are the block move sum NMO (Rupert and Chun 1975) and the inverse NMO stack (Claerbout 1992). Results of applying stretch-free stack to both synthetic and real data produced primary reflec-tions with significantly higher frequency than the conventional stack. One major drawback of this method is that NMO-corrected gathers are never cre-ated and hence some important processes, such as pre-stack residual static correction, are not possible. Trickett (2007) came up with another stacking algorithm called the Maximum-Likelihood-Estimation (MLE) stacking to treat seismic data with erratic noise. The algorithm estimates the probability distribution of noise as it varies with time and CMP, and then stacks traces using a maximum-likelihood estimator for that distribution. Application to real data showed that MLE stacking resulted in a cleaner stack that was easi-er to interpret than straight mean stacking. Another attractive subject of the

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MLE stacking is that it provides the estimated shape of the probability dis-tribution of the noise as it varies with time and CMP which might be useful in detecting the presence of multiples (Trickett 2007).

Zhang et al. (2004) introduced the High Order Correlative Weighted Stacking (HOCWS) technique that is based on applying weights that are cal-culated through high order correlative statistics in the wavelet domain. Ap-plication of HOCWS to synthetic data demonstrates its advantages in improving S/N ratio and suppressing correlative random noise. However, no tests were conducted on real data.

Tyapkin and Ursin (2005) proposed an optimum weighted stacking tech-nique that utilizes weighting traces before stacking based on signal ampli-tude and noise autocorrelation. The technique is based on the fact that signal maintains similar shape from one trace to another even if its amplitude var-ies. Accordingly, the signal shape is estimated using the principle of maxi-mum likelihood. The signal amplitude and the noise autocorrelation functions are then determined. Finally, each trace is weighted before stack-ing with a weighting factor that is proportional to the signal amplitude divid-ed by the noise variance. The method apparently provides better results than the straight mean stacking and can be used for both CMP and vertical stack-ing.

Neelamani et al. (2006) introduced a new method to stack seismic data called Stack-And-Denoise (SAD). SAD first estimates the S/N ratio for each trace using a new approach called Leave-Me-Out (LMO). In LMO, a pilot trace is constructed using all traces in a gather except one trace. The noise variance is then estimated for the left-out trace. After the noise variance is estimated for all traces in a CMP gather, these traces are weighted according to their S/N ratios and stacked. The residual noises in the weighted stack are then attenuated using denoising in the wavelet, Fourier, or curvelet transform domain. The data are then inversely transformed into a SAD stack. Results of application to both synthetic and real data show that SAD is effective in attenuating multiples as well as random noise.

In 2008 Rashed presented another stacking algorithm called the smart stack as an alternative stacking technique. Smart stack is based on automatic detection and rejection of harmful amplitudes and weighting up amplitudes representing the central part at each time level. In the smart stack, the alpha-trimmed average of each time level within the CMP gather is calculated and samples having different sign from this mean are excluded. The remaining samples are weighted according to their closeness to their alpha-trimmed mean. Application of smart stacking to different types of seismic data proved its capability to treat errors in the pre-stack processing such as residual stat-ics, insufficient stretch mute, and inaccuracies in NMO correction (Rashed 2008).


Liu et al. (2009) used local correlation to achieve a better stacking out-put. Liu’s idea was simple. Traces within the CMP gather are stacked using the conventional straight mean stacking to form a reference trace. Local cor-relation coefficients between the reference trace and traces in the CMP gath-er are computed. The smoothed local correlation curves are then used to weight traces before averaging. Comparing the results of the mean stack, smart stack, leave-me-out stack, and local correlation stack showed that stacking using local correlation yielded best results. However, the results of local correlation stacking can be unsatisfactory if the quality of the reference trace is poor (Liu et al. 2009).

Two years later, Sanchis and Hanssen (2011) proposed an enhanced lo-cal correlation stacking method to overcome the reference trace quality prob-lem. In their study, they used Kalman filter and S/N ratio-based stack to produce the reference trace and also proposed a new weight normalization scheme to be used with the local correlation weighting functions proposed by Liu et al. (2009). The results of applying their algorithm to both synthetic data and real subsalt marine data, show that both S/N estimation and Kalman reference stacking methods yielded better results than conventional stacking. Sanchis and Hanssen (2011) also found that the Kalman reference method produces the best overall seismic image contrast and reveals many more re-flected events, but at the expense of a higher noise level and a longer pro-cessing time. Thus, enhanced stacking using S/N estimation as reference method is a possible alternative that has the advantages of running faster, but also emphasizes some reflected events under the subsalt structure.

4. GLIMPSE INTO FUTURE For five decades, stacking has been holding its ground as a cornerstone in seismic data acquisition and processing and it seems that stacking will main-tain this ground for years to come. The main reason for the endurance and the development of stacking is due to the fact that it is the only process ca-pable of separating signal from noise even when they both have the same frequency. Another reason is that other major seismic data processing steps, such as velocity analysis and migration, some way or another involve the use of stacking concept. Analyzing future challenges and possibilities of seismic exploration, one can confidently state that some sort of stacking will remain the backbone of seismic exploration and will be in use for a while.

Future challenges in the field of seismic exploration will be related to the volume and the nature of seismic data. In the near future, we may be over-whelmed by huge amounts of unconventional types of seismic data that need to be recorded, processed, and displayed. Acquiring larger amounts of seis-mic data in the near future is not surprising because of the world increasing demand for resources and because of the continuous decrease in the cost of

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technology. These two factors combined will drive towards the collection of larger amounts of data from remote areas or over complex subsurface condi-tions in the hope of discovering new resources or developing the existing ones. In addition, wide-azimuth long-offset and fully sampled seismic wavefield data are expected to be collected more frequently.

The real challenge will be processing the unconventional types of seis-mic data that may be collected in the future. These data types may include data collected using ambient noise as a source, multi-component data, and extraterrestrial data that will be collected over other planets. These data would need special processing routines and surely unconventional stacking techniques. In addition, the near future may lead to the invention of new types of seismic sources, receivers, and field layouts. Ideas like fully wire-less field equipment is just around the corner, given today’s technological rate of advancing. A wireless seismic field survey will open a completely new world of possibilities once we get rid of the restrictions and limitations forced by the use of cables. It will not be surprising if future land seismic da-ta are conducted in a continuous mode, the same way marine seismic data and ground penetrating radar data are collected today. Actually, the land streamer is just a step towards this approach. These new seismic devices and files layouts would demand the application of unconventional processing routines.

The demand for subsurface images with better quality and higher resolu-tion is expected to continue. The need to illuminate deeper targets and to ex-tract as much information as possible will also be pushed to the limits. This information may include more details on the petrophysical properties of im-aged subsurface rocks, their fluid contents, fluid types, movement, and be-havior. They may also include stratigraphic and depositional information as well as a notion of the stress regime. This continuous demand would in-crease the need for alternative processing techniques including those related to stacking. Collecting multi-component seismic data is at its first steps to-day but it is expected to expand in the near future to meet these increasing demands. Multi-component seismic data certainly need specific processing and stacking algorithms to emphasis specific attributes. 4D or monitoring seismic data is another relatively new trend that is expected to continue for the next years. These monitoring studies are expected to lead to more capa-bility to extract more information about reservoir characterization and changes. This information is expected to be very sensitive and can be totally destroyed if biased or non-linear filtering algorithms are used. The need to develop processing and hence stacking algorithms that are efficient in en-hancing S/N ratio but unbiased and maintain data characteristics will be in its peak.


The most promising area for research in the near future is the develop-ment of CRS stacking. Until today, the CRS stacking is conducted by simple averaging because the process is complicated, time consuming, and compu-tationally heavy enough. With the giant steps currently taking place in the field of increasing computational power, and with the large number of litera-tures discussing different parameters of CRS, researchers will soon think of implementing unconventional stacking procedures to CRS to enhance its fi-nal output. Because of the large number of parameters involved in the CRS calculations, compared to a simple CMP, new algorithms that are more com-plicated may arise in the future. Weighting traces based on their location in the aperture, reflection angle, or any other criterion in CRS stacking is an is-sue that should be under investigations in the very near future.

Having a deep look at the present situation of stacking and at the future demands and challenges as well as the future possibilities and capabilities, it seems that stacking will continue to be a major part of seismic exploration and it will keep evolving and becoming more sophisticated to meet challeng-es of the future and to benefit from its potentials.

R e f e r e n c e s

Anderson, R.G., and G.A. McMechan (1990), Weighted stacking of seismic data us-ing amplitude-decay rates and noise amplitudes, Geophys. Prospect. 38, 4, 365-380, DOI: 10.1111/j.1365-2478.1990.tb01851.x.

Baykulov, M., S. Dümmong, and D. Gajewski (2011), From time to depth with CRS attributes, Geophysics 76, 4, S151-S155, DOI: 10.1190/1.3580607.

Bednar, J., and T. Watt (1984), Alpha-trimmed means and their relationship to me-dian filters, IEEE Trans. Acoust. Speech Signal Process. 32, 1, 145-153, DOI: 10.1109/TASSP.1984.1164279.

Belfer, I., A. Berkovitch, and K. Sydykov (2008), Multifocusing: A new method of multifold seismic data processing, CSEG Recorder April 2008, 30-32.

Berkovitch, A., B. Gelchinsky, and S. Keydar (1994), Basic formulae for multifo-cusing stack. In: Proc. 56th Meeting of European Association of Explora-tion Geophysics, Extended Abstracts.

Berkovitch, A., S. Keydar, E. Landa, and P. Trachtman (1998), Multifocusing in practice. In: Proc. 68th SEG Annual Conference and Exhibition, Expanded Abstracts, Society of Exploration Geophysicists.

Berkovitch, A., I. Belfer, and E. Landa (2008), Multifocusing as a method of im-proving subsurface imaging, The Leading Edge 27, 2, 250-256, DOI: 10.1190/1.2840374.

M. RASHED

Brown, R.J., G.H. Friesen, D.H. Hall, and O.G. Stephenson (1977), Weighted verti-cal stacking in crustal seismic reflection studies on the Canadian Shield, Geophys. Prospect. 25, 2, 251-268, DOI: 10.1111/j.1365-2478.1977. tb01166.x.

Buchanan, D.J., R. Davis, and P.J. Jackson (1983) Method of stacking seismic data, United States Patent, 4,393,484.

Byun, B.S., and E.S. Nelan (1997), Method and system for correcting seismic traces for normal move-out stretch effects, United States Patent, 5,684,754.

Cao, W., S.M. Hanafy, G.T. Schuster, G. Zhan, and C. Boonyasiriwat (2012), High-resolution and super stacking of time-reversal mirrors in locating seismic sources, Geophys. Prospect. 60, 1, 1-17, DOI: 10.1111/j.1365-2478.2011. 00957.x.

Castoro, A., R.E. White, and R.D. Thomas (2001), Thin-bed AVO: Compensating for the effects of NMO on reflectivity sequences, Geophysics 66, 6, 1714-1720, DOI: 10.1190/1.1487113.

Claerbout, J. (1992), Earth Soundings Analysis: Processing versus Inversion, Blackwell Scientific Publs., Cambridge, 314 pp.

Claerbout, J.F., and F. Muir (1973), Robust modeling with erratic data, Geophysics 38, 5, 826-844, DOI: 10.1190/1.1440378.

Courtier, W.H., and H.L. Mendenhall (1967), Experiences with multiple coverage seismic methods, Geophysics 32, 2, 230-258, DOI: 10.1190/1.1439864.

Cressman, K. (1968), How velocity layering and steep dip affect CDP, Geophysics 33, 3, 399-411, DOI: 10.1190/1.1439938.

Currie, W.S. (1982), Seismic velocity determination using random and nonlinear processes. In: Proc. SEG Annual Meeting, 17-21 October 1982, Dallas, USA, Conf. paper 1982-0029, Society of Exploration Geophysicists, Dallas, USA.

Deregowski, S.M. (1982), Dip-moveout and reflector dispersal, Geophys. Prospect. 30, 3, 318-322, DOI: 10.1111/j.1365-2478.1982.tb01308.x.

Dunkin, J.W., and F.K. Levin (1973), Effect of normal moveout on a seismic pulse, Geophysics 38, 4, 635-642, DOI: 10.1190/1.1440363.

Embree, P. (1968), Diversity seismic record stacking method and system, United States Patent Office, 3,339,396.

Fomel, S., and R. Kazinnik (2013), Non-hyperbolic common reflection surface, Geophys. Prospect. 61, 1, 21-27, DOI: 10.1111/j.1365-2478.2012.01055.x.

Foster, M.R., R.L. Sengbush, and R.J. Watson (1964), Design of sub-optimum filter systems for multi-trace seismic data processing, Geophys. Prospect. 12, 2, 173-191, DOI: 10.1111/j.1365-2478.1964.tb01896.x.

Gholami, A. (2013), Residual statics estimation by sparsity maximization, Geophys-ics 78, 1, V11-V19, DOI: 10.1190/geo2012-0035.1.


Gierse, G., J. Pruessmann, E. Laggiard, C. Boennemann, and H. Meyer (2003), Im-proved imaging of 3D marine seismic data from offshore Costa Rica with CRS processing, First Break 21, 12, 45-49.

Green, C.H. (1938), Velocity determinations by means of reflection profiles, Geo-physics 3, 4, 295-305, DOI: 10.1190/1.1439508.

Haldorsen, J.B.U., and P.A. Farmer (1989), Suppression of high-energy noise using alternative stacking procedure, Geophysics 54, 2, 181-190, DOI: 10.1190/ 1.1442642.

Harris, N.R. (1968), Stacking of seismic traces having common offset distances, United States Patent Office, 3,381,266.

Hertweck, T., J. Mann, and T. Kluver (2005), Event-consistent smoothing in the context of the CRS stack method, J. Seism. Explor. 14, 2-3, 197-215.

Hertweck, T., J. Schleicher, and J. Mann (2007), Data stacking beyond CMP, The Leading Edge 26, 7, 818-827, DOI: 10.1190/1.2756859.

Hileman, J.A., P. Embree, and J.C. Pflueger (1968), Automated static corrections, Geophys. Prospect. 16, 3, 326-358, DOI: 10.1111/j.1365-2478.1968. tb01980.x.

Hubral, P., G. Höcht, and R. Jäger (1998), An introduction to the common-reflection surface stack. In: Proc. 60th EAGE Conference and Exhibition, Extended Abstracts, 01-19, European Association of Geoscientists and Engineers.

Inguva, R., and L.H. Schick (1981), Information theoretic processing of seismic data, Geophys. Res. Lett. 8, 12, 1199-1202, DOI: 10.1029/GL008i012p 01199.

Jakubowicz, H. (1990), A simple efficient method of dip-moveout correction, Geo-phys. Prospect. 38, 3, 221-245, DOI: 10.1111/j.1365-2478.1990.tb01843.x.

Kanasewich, E.R., C.D. Hemmings, and I. Alpasian (1973), Nth-root stack nonlinear multichannel filter, Geophysics 38, 2, 327-338, DOI: 10.1190/1.1440343.

Katz, D., M. Landrum, and L. Schick (1985), Stacking of noisy seismic traces via maximum entropy with a correlation coefficient constraint, IEEE Trans. Acoust. Speech Signal Process. 33, 5, 1331-1333, DOI: 10.1109/TASSP. 1985.1164693.

Kirchheimer, F. (1983), Long-period static analysis by trigonometric approximation. In: Proc. 53rd SEG Annual International Meeting of, 11-15 September 1983, Las Vegas, USA, Conf. paper 1983-0317, Society of Exploration Geophysicists.

Kirk, P. (1981), Vibroseis processing. In: A.A. Fitch (ed.), Developments in Geo-physical Exploration Methods, Applied Science Publishers Ltd, London, 37-52, DOI: 10.1007/978-94-009-8105-8_2.

Kostecki, A., and A. Półchłopek (2006), Method for correction of the prestack mi-gration amplitude of converted waves, Acta Geophys. 54, 2, 113-125, DOI: 10.2478/s11600-006-0010-2.

M. RASHED

Landa, E., S. Keydar, and T.J. Moser (2010), Multifocusing revisited – inhomoge-neous media and curved interfaces, Geophys. Prospect. 58, 6, 925-938, DOI: 10.1111/j.1365-2478.2010.00865.x.

Larner, K., and D. Hale (1992), Dip-moveout error in transversely isotropic media with linear velocity. In: Proc. 62nd SEG Annual International Meeting, Expanded Abstracts, 979-983, Society of Exploration Geophysicists.

Levin, F.K. (1971), Apparent velocity from dipping interface reflections, Geophys-ics 36, 3, 510-516, DOI: 10.1190/1.1440188.

Liner, C.L. (1999), Concepts of normal and dip moveout, Geophysics 64, 5, 1637-1647, DOI: 10.1190/1.1444669.

Liu, G., S. Fomel, L. Jin, and X. Chen (2009), Stacking seismic data using local cor-relation, Geophysics 74, 3, V43-V48, DOI: 10.1190/1.3085643.

Loper, G.B., and R.R. Pittman (1954), Seismic recording on magnetic tape, Geo-physics 19, 1, 104-115, DOI: 10.1190/1.1437953.

Malcolm, A.E., M.V. de Hoop, and J.H. LeRousseau (2005), The applicability of dip moveout/azimuth moveout in the presence of caustics, Geophysics 70, 1, S1-S17, DOI: 10.1190/1.1852785.

Mann, J., R. Jäger, T. Müller, G. Höcht, and P. Hubral (1999), Common-reflection-surface stack – a real data example, J. Appl. Geophys. 42, 3-4, 301-318, DOI: 10.1016/S0926-9851(99)00042-7.

Marr, J., and E. Zagst (1967), Exploration horizons from new seismic concepts of CDP and digital processing, Geophysics 32, 2, 207-224, DOI: 10.1190/ 1.1439862.

Mayne, W.H. (1956), Seismic surveying, United States Patent Office, 2,732,906. Mayne, W.H. (1962), Common reflection point horizontal data stacking techniques,

Geophysics 27, 6, 927-938, DOI: 10.1190/1.1439118. Mayne, W.H. (1967), Practical considerations in the use of common reflection point

techniques, Geophysics 32, 2, 225-229, DOI: 10.1190/1.1439863. Meyerhoff, H.J. (1966), Horizontal stacking and multichannel filtering applied to

common depth point seismic data, Geophys. Prospect. 14, 4, 441-454, DOI: 10.1111/j.1365-2478.1966.tb02247.x.

Miller, R.D. (1992), Normal moveout stretch mute on shallow-reflection data, Geo-physics 57, 11, 1502-1507, DOI: 10.1190/1.1443217.

Mohanty, P.R., S. Phadke, and B.B. Bhattacharya (2000), Seismic resolution over coal seams: A numerical study, Acta Geophys. Pol. 48, 2, 215-222.

Muirhead, K.J. (1968), Eliminating false alarms when detecting seismic events automatically, Nature 217, 5128, 533-534, DOI: 10.1038/217533a0.

Müller, T. (1998), Common reflection surface stack versus NMO/stack and NMO/DMO stack. In: Proc. 60th EAGE Conference and Exhibition, Ex-tended Abstracts, 1-20, European Association of Geoscientists and Engi-neers.


Musgrave, A.W. (1962), Applications of the expanding reflection spread, Geophys-ics 27, 6, 981-993, DOI: 10.1190/1.1439126.

Naess, O.E. (1979), Superstack – an iterative stacking algorithm, Geophys. Prospect. 27, 1, 16-28, DOI: 10.1111/j.1365-2478.1979.tb00956.x.

Naess, O.E. (1982), Single-trace processing using iterative CDP-stacking, Geophys. Prospect. 30, 5, 641-652, DOI: 10.1111/j.1365-2478.1982.tb01331.x.

Naess, O.E., and L. Bruland (1981), Velocity analysis using iterative stacking, Geo-phys. Prospect. 29, 1, 1-20, DOI: 10.1111/j.1365-2478.1981.tb01007.x.

Naess, O.E., and L. Bruland (1985), Stacking methods other than simple summation, Dev. Geophys. Explor. Meth. 6, 189-223.

Naess, O., and L. Bruland (1989), Improvement of multichannel seismic data through application of the median concept, Geophys. Prospect. 37, 3, 225-241, DOI: 10.1111/j.1365-2478.1989.tb02204.x.

Neelamani, R., T.A. Dickens, and M. Deffenbaugh (2006), Stack-and-denoise: A new method to stack seismic datasets. In: Proc. 76th SEG Annual Interna-tional Meeting, Expanded Abstracts, 2827-2831, Society of Exploration Geophysicists.

Pruett, R. (1982), Long period multiple reflection suppression and enhanced velocity discrimination using a weighted stack, Technical Program Abstracts and Biographies, Society of Exploration Geophysicists, Dallas, USA.

Pugin, A., and S.E. Pullan (2000) First-arrival alignment static corrections applied to shallow seismic reflection data 1, J. Environ. Eng. Geoph. 5, 1, 7-15, DOI: 10.4133/JEEG5.1.7.

Pullan, S.E., and J.A. Hunter (1990), Delineation of buried bedrock valleys using the optimum offset shallow seismic reflection technique. In: S.H. Ward (ed.), Geotechnical and Environmental Geophysics. Vol. III: Geotechnical, Ch. 5, 75-88, Society of Exploration Geophysicists, DOI: 10.1190/ 1.9781560802785.3.ch5.

Rashed, M.A. (2003), Optimum-offset weighted stacking: a novel processing tech-nique to enhance signal-to-noise ratio in seismic data acquired in urban ar-eas and its application on Uemachi fault, Osaka, Japan, Ph.D. Thesis, Osaka City University, Osaka, Japan, 147 pp.

Rashed, M.A. (2008), Smart stacking: A new CMP stacking technique for seismic data, The Leading Edge 27, 4, 462-467, DOI: 10.1190/1.2907176.

Rashed, M., E. Yamamoto, M. Mitamura, S. Toda, T. Nishida, Y. Terada, H. Uda, H. Yokota, H. Nemoto, and K. Nakagawa (2002), Weighted stack of shal-low seismic reflection line acquired in downtown Osaka City, Japan, J. Appl. Geophys. 50, 3, 231-246, DOI: 10.1016/S0926-9851(02)00144-1.

Reynolds, J.M. (1997), An Introduction to Applied and Environmental Geophysics, John Wiley & Sons Ltd., London, 796 pp.

M. RASHED

Rietsch, E. (1980), Estimation of the signal-to-noise ratio of seismic data with an application to stacking, Geophys. Prospect. 28, 4, 531-550, DOI: 10.1111/ j.1365-2478.1980.tb01241.x.

Robinson, J.C. (1968), Noise attenuation on common-depth-point seismic records by a semideterministic approach, Geophysics 33, 5, 723-733, DOI: 10.1190/ 1.1439967.

Robinson, J.C. (1970), Statistically optimal stacking of seismic data, Geophysics 35, 3, 436-446, DOI: 10.1190/1.1440105.

Rückemann, C.-P. (2012a), Supercomputing resources empowering superstack with interactive and integrated systems, AIP Conf. Proc. 1479, 873-876, DOI: 10.1063/1.4756279.

Rückemann, C.-P. (2012b), Comparison of stacking methods regarding processing and computing of geoscientific depth data. In: Proc. Fourth International Conference on Advanced Geographic Information Systems, Applications, and Services (GEOProcessing 2012), 30 January – 4 February 2012, Va-lencia, Spain, 35-40.

Ruehle, W.H. (1980), Method of enhancing seismic reflection signals for nonsur-face-consistent static time shifts, United States Patent, 4,206,509.

Runnestrand, S., E.L. Butler, and D.B. Neff (2002), Coherency stack of seismic traces, United States Patent, 6,393,365 B1.

Rupert, G.B., and J.H. Chun (1975), The block move sum normal moveout correc-tion, Geophysics 40, 1, 17-24, DOI: 10.1190/1.1440511.

Sanchis, C., and A. Hanssen (2011), Enhanced local correlation stacking method, Geophysics 76, 3, V33-V45, DOI: 10.1190/1.3552687.

Santos, L., J. Schleicher, J.C. Costa, and A. Novais (2011), Fast estimation of com-mon-reflection-surface parameters using local slopes, Geophysics 76, 2, U23-U34, DOI: 10.1190/1.3553001.

Schneider, WA., E.R. Prince, and B.F. Giles (1965), A new data-processing tech-nique for multiple attenuation exploiting differential normal moveout, Geo-physics 30, 3, 348-362, DOI: 10.1190/1.1439582.

Shatilo, A., and F. Aminzadeh (2000), Constant normal-moveout (CNMO) correc-tion: A technique and test results, Geophys. Prospect. 48, 3, 473-488, DOI: 10.1046/j.1365-2478.2000.00190.x.

Shepard, D. (1968), A two-dimensional interpolation function for irregularly-spaced data. In: Proc. 23rd ACM National Conference, 517-524, Association for Computing Machinery, New York, DOI: 10.1145/ 800186.810616.

Stein, J.A., T. Langston, and S.E. Larson (2009), A successful statics methodology for land data, The Leading Edge 28, 2, 222-226, DOI: 10.1190/1.3086061.

Stewart, R.R. (1985), Median filtering: Review and a new F/K analogue design, J. Can. Soc. Explor. Geophys. 21, 1, 54-63.

Sugimoto, T., H. Saitou, and M. Okujima (2000), Study of underground imaging us-ing shear waves: the stacking method of the reflected scattered wave, Ar-


cheol. Prospect. 7, 4, 249-261, DOI: 10.1002/1099-0763(200012)7:4<249:: AID-ARP155>3.0.CO;2-#.

Taner, M.T., F. Koehler, and K.A. Alhilali (1974), Estimation and correction of near-surface time anomalies, Geophysics 39, 4, 441-463, DOI: 10.1190/ 1.1440441.

Trickett, S.R. (2003), Stretch-free stacking. In: Proc. 73rd SEG Annual Interna-tional Meeting, Expanded Abstracts, 2008-2011, Society of Exploration Geophysicists.

Trickett, S. (2007), Maximum-likelihood-estimation stacking. In: Proc. 77th SEG Annual International Meeting, Expanded Abstracts, 2640-2643, Society of Exploration Geophysicists.

Tyapkin, Y., and B. Ursin (2005), Optimum stacking of seismic records with irregu-lar noise, J. Geophys. Eng. 2, 3, 177-187, DOI: 10.1088/1742-2132/2/3/ 001.

Waltham, D.A., and J.F. Boyce (1986), Signal-to-noise ratio enhancement in seismic multifold data using Bayesian statistics, Geophys. Prospect. 34, 1, 56-72, DOI: 10.1111/j.1365-2478.1986.tb00452.x.

Watt, T., and J.B. Bednar (1983), Role of the alpha-trimmed mean in combining and analyzing seismic common-depth-point gathers, Technical Program Ab-stracts and Biographies, S3.5, Society of Exploration Geophysicists, Las Vegas, USA.

White, R.E. (1977), The performance of optimum stacking filters in suppressing un-correlated noise, Geophys. Prospect. 25, 1, 165-178, DOI: 10.1111/j.1365-2478.1977.tb01159.x.

Wolf, K., D. Rosales, A. Guitton, and J. Claerbout (2004), Robust moveout without velocity picking, Stanford Explor. Project 115, 273-282.

Woodward, M.J. (1985), Statistical averages for velocity analysis and stack: median vs. mean, Stanford Explor. Project 42, 97-111.

Yilmaz, Ö. (2001), Seismic Data Analysis: Processing, Inversion, and Interpretation of Seismic Data, 2nd ed., Society of Exploration Geophysicists Books, Tulsa, 2027 pp.

Yoon, M.-K., M. Baykulov, S. Dümmong, H.-J. Brink., and D. Gajewski (2009), Reprocessing of deep seismic reflection data from the North German Basin with the Common Reflection Surface stack, Tectonophysics 472, 1-4, 273-283, DOI: 10.1016/j.tecto.2008.05.010.

Zagst, E.F. (1965), Horizontal stacking improves seismic data, The Oil Gas J. 63, 33, 97-104.

Zhang, S.Z., Y.X. Xu, and H.H. Xia (2004), Correlative weighted stacking for seis-mic data in the wavelet domain. In: Proc. International Conference on En-vironmental and Engineering Geophysics (ICEEG), 6-9 June 2004, Wuhan, China, 161-165.

M. RASHED

Zhang, Y., S. Bergler, and P. Hubral (2001), Common-reflection-surface (CRS) stack for common offset, Geophys. Prospect. 49, 6, 709-718, DOI: 10.1046/ j.1365-2478.2001.00292.x.

Received 31 March 2013 Received in revised form 20 August 2013

Accepted 22 August 2013

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Fifty years of stacking