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Collaborating Authors
2008 SEG Annual Meeting
Summary We describe in this paper a new way of solving the twoway wave equation called the two-step Explicit Marching method. Compared to the conventional explicit finitedi fference algorithms, which can be second or fourth order but are subject to stability conditions and dispersion problems that limit the magnitude of the time steps used to propagate the wavefieds, the proposed method is based on a high order differential operator and allows arbitrary large time steps with guaranteed numerical stability and minimized dispersion. Synthetic and real data examples show that it allows the reverse time migration to be performed with the Nyquist time step, based on the maximum frequency of the input data, which is the maximum time step that can be used for proper imaging. Introduction Reverse-time migration has historically been more expensive than one-way wave equation migration because of large memory requirements and a larger number of computations. In the industry, the most popular way to implement reverse-time migration is explicit finite difference. Although easy to solve, it is only conditionally stable which imposes a limit on the marching time step size. On the other hand, all the finite-difference methods suffer from numerical dispersion problems. To overcome these problems, either high-order schemes are used and/or time step size is reduced. In either case there is an increase in the computational cost. This paper presents a new method to solve the twoway wave equation which we call the "Two-step Explicit Marching" algorithm. The new method is based on a high order polynomial expansion that allows Nyquist-based time steps to be used while ensuring any user-defined accuracy. Unlike the conventional explicit finite-difference schemes, it does not suffer from numerical stability or numerical dispersion problems. Therefore, it can be used to design a cost-effective and high quality reverse-time migration. Two-step explicit marching algorithm Zhang et al. (2007a) show how this equation can be transformed in a first-order in time equation involving the analytical signal of p, P, which can then be solved by integration. The interesting thing about equation (2) is that it is accurate for arbitrary time-steps. The drawbacks are that the pseudo-differential operator ? is diffcult to synthesize, especially when large velocity contrasts are present and introducing the complex wavefield P. In this paper, we design a solution to equation (3) based on an optimized expansion of the cosine operator, which allows a large extrapolation time step and does not suffer from numerical stability and dispersion problems as the convntional explicit finite-difference algorithms do. The problem of adding a source term to equation (3) is also considered. Numerical examples To show how the two-step explicit marching method works, we first apply to the 2004 BP 2D data set (Billette and Brandsberg-Dahl, 2005). This is a high quality dataset generated by finite-difference modeling with 1348 shot records and 15000m maximum offset. In the migration, we pre-filter the input data to 30hz and set the downward extrapolation time step to 0:015s. For such a data set, the new method handles complex velocity fairly well and gives good delineation of the salt boundaries (Figure 6) especially the steeply dipping salt anks and the overturned salt edges, which require high angle propagation or turning waves to image clearly.
Summary An inversion procedure is described wherein microseismic data recorded by a network of three component geophones are assumed to be represented as the sum of a compressional (P) and one or two shear (S) arrivals. The inversion operates in the frequency-space domain and includes a linear inversion for source waveforms and a nonlinear inversion for model properties or source locations. The linear inversion effectively reverses time using a ray trace Green function to recover the source-time functions. For the nonlinear inversion two waveform fitting functionals are constructed; one captures moveout and polarization information through a reconstructed data misfit, another captures information from arrival time differences through a spectral coherence functional. The two may be scaled and summed to form a joint X2 misfit which may be combined with soft prior information in a Bayesian posterior. This is then maximized using global search techniques. Model calibration is accomplished by inverting waveform data from known locations (e.g. perforation shots) for anisotropy and optionally for model smoothness and Q. Micro-earthquake event locations are determined by inverting waveform data given the calibrated model. Since the procedure involves fitting waveforms, time picking is not required. The beam-forming property of the receiver array and the complete polarization vector are used to enhance the signal to noise ratio of arrivals. The presence of a P arrival is not necessary to determine a location. The algorithm implementation uses layered VTI models, includes losses due to spreading, transmission and Q and handles an arbitrary distribution of receivers (e.g. from horizontal or multiple wells or surface locations). The inversion permits automated, objective data analysis with quantified uncertainties in estimated unknowns. Introduction Microseismic data analysis traditionally makes use of the difference between picked S and P arrival times to compute the distance and depth of the source; azimuthal polarizations are then used for direction. Inversions typically make use of a modified Geiger’s method, based on classical Levenberg-Marcquardt nonlinear least squares, to determine optimum locations with uncertainties (e.g. Lee and Stewart, 1981). Rapid grid search approaches using approximated travel times have also been proposed (Aldridge et. al. (2003)). Location methods that require manual event picking are subjective and time consuming and automated picking approaches, while able to handle large volumes of data, are often misled by noisy data. Most automatic picking algorithms also do not make use of the noise rejection potential of the receiver array. More recently waveform-based approaches have been presented. Kao and Shan (2004) showed results of a source scanning algorithm applied to earthquake data, Drew et. al. (2005) employed a characteristic function based on the product of P and S onset energy ratios to locate events, Druzhinin (2005) showed results of a 2D elastic migration approach with event locations inferred where P+S focusing occurred, Gajewski et. al. (2006) presented results of a similar acoustic technique using time reversal or diffraction stack focusing of recorded waveforms and Fuller et. al. (2007) also presented a diffraction stack approach to microearthquake event location.
SUMMARY Waveform inversion in the Laplace domain has the advantage of finding a smooth velocity structure including salt domes having a large velocity contrast to the background medium. In exploration, noises contaminate the data before and near the direct wave. The early time noises cause inaccuracy and instability of the Laplace transform of time signal, when we use the data for Laplace-domain waveform inversion. We developed an inversion algorithm using a method which removes the direct wave from common shot gather data. We performed velocity structure inversion and source estimation with a direct removed wavefield and we obtained the velocity model that was close to the true model. In this study, we developed an algorithm of waveform inversion in the Laplace domain which can be applied to a deep-sea survey and successfully applied to synthetic and field data. INTRODUCTION Waveform inversion in the Laplace domain has been studied recently (Shin and Cha, 768x). Using the inversion, they obtained high-reliability velocity model even if it contains salt domes. Waveform inversion in the Laplace domain is useful for obtaining the long-wavelength component of the velocity of the background medium. The resulting smooth velocity model is comparable to that achieved by refraction tomography inversion. Moreover, it is robust for selecting an initial velocity model or a space interval. However, waveform inversion in the Laplace domain with a conventional logarithmic norm had not succeeded in the field data that is acquired in the deep sea environment. In the case of the deep sea, the amplitude of a direct wave decays rapidly along the offset distance, until it vanishes in the noise. This decay is due to the Lloyd mirror effect on the surface of the water, which suppresses the amplitude of a direct wave. Even with near offset, the noises appear in the early time positioned between the direct wave and the first arrival reflection wave make the inversion difficult. Thus, a method to mute the direct wave is beneficial for inverting the velocity structure. In this study, we developed a waveform inversion in the Laplace domain using a direct wave removal method. The equation was built for source estimation without a direct wave. We applied the method to the synthetic model with a salt dome, and compared it with a waveform inversion using a conventional logarithmic norm. The method was tested with field data within a deep sea environment, with a result that corresponded to the migration result. INVERSION RESULTS Synthetic data Figure 1 shows a BP model (Billette and Brandsberg-Dhal, 2005). This model has a complex structure, in which the salt dome interpenetrates the background velocity structure. The salt dome and the subsalt structure were the targets of our test. For the BP model, the Laplace damping constant s = 1;2;3; ; ; ; 10 was used, and the grid size was 400 m for both the horizontal and vertical coordinates. The synthetic seismogram was generated in the Laplace domain and the data were used as the observed data for the inversion.
- Geophysics > Seismic Surveying > Seismic Processing (1.00)
- Geophysics > Seismic Surveying > Seismic Modeling > Velocity Modeling (1.00)
Introduction High-resolution seismic reflection has been successfully used to delineate preferential pathways within groundwater systems that represent corridors for contaminant movement in a wide variety of challenging hydrologic settings, many in settings where other methods and/or monitor wells have not been enlightening. For more than four decades, seismic reflection has found utility addressing near-surface groundwater problems where lateral variability of confining layers have inhibited even the most sophisticated flow models developed from monitor wells (Shepers, 1975). Optimizing acquisition and processing parameters while limiting interprettations to only what can be validated on shot records and consistent with wave propagation and reflection theory is essential for making meaningful and reliable contributions to a groundwater flow model and associated monitoring and remediation programs (Steeples and Miller, 1990). Years of application of the method in a wide range of settings and with a diverse set of imaging objectives has resulted in an excellent collection of case studies that provides guidance for current and future applications and developments as well as evaluations of method feasibility. The seismic reflection method has been used to establish lateral continuity in confining units with thick dry sandy overburdens as well as fine-grained unconsolidated and saturated near-surface settings. Mapping bedrock is an important application where percolation rates in the vadose zone are high and bedrock units are impermeable. Seismic reflection is a viable tool for studying sitewide variability in unconsolidated alluvial sediments ;where complex vertical migration paths can allow contaminants to easily move between local "confining" layers, leaving zones directly beneath a source contaminant free, while deeper layers, seemingly protected by several aquicludes, are rich in contaminant. The complexity of many depositional settings results in rapid vertical changes in material properties and therefore a need for high-resolution imaging that does not require a priori information or assumptions about the sequential nature of the vertical property changes. Even with seismic reflection''s many positive and sometimes amazing attributes and capabilities, it is imperative that an awareness of the method''s limitations and true potential in real-world settings be maintained. Near-surface seismic methods do not lend themselves to distinguishing different types of liquids within a groundwater system. For example, distinguishing DNAPLs or LNAPLs from within a saturated interval is beyond the resolution of the seismic tool in real-world settings. However, interrogation of the subsurface in search of lithologies or structures that might represent traps for contaminants has proven very effective. In fact, based on the properties of liquids in the subsurface, it is many times possible to infer traps and likely areas of high concentrations based on mapped reflector structures in conjunction with well control. Interruption in Confining Layers Monterey, California Clearing U.S. Military facilities for public use after base closure requires the surface and subsurface be issued a clean bill of health. At the former Fort Ord Army Post near Monterey, California, as much as 100 m of highly permeable dune sands overlay several aquicludes responsible for perching and confining the groundwater supply (both fresh and brackish) in the area.
- Government > Regional Government > North America Government > United States Government (1.00)
- Government > Military (1.00)
- Energy > Oil & Gas > Upstream (1.00)
Identification And Interpretation of Solution Mining Features In Bedded Salt Deposits On a Crosswell Reflection Profile
Turperning, R. (Michigan Technical University) | Morgan, T. (Z-Seis Reservoir Imaging) | Bryans, B. (Z-Seis Reservoir Imaging) | Tandon, K. (Z-Seis Reservoir Imaging) | Williams, M. Monier (Golder Associates) | Boone, S. (Golder Associates)
Summary Direct evidence of solution salt mining cavities and cavity collapse features have been detected on high resolution (order of 1-2 m) crosswell reflection seismic data in surveys for site evaluation for bridge construction. High frequency source crosswell surveys enable the detection of subtle features that aid in decision making, as in this case, for geotechnical engineering supporting bridge siting and construction. The inferred features on the reflection seismic profiles are compared with numerical mechanical displacement modeling and seismic wave propagation modeling results for hypothesis testing. Introduction Solution salt mining was conducted from the early 1920''s into the 1950''s via a series of wells in an area immediately east of the Detroit River in Windsor, Ontario. The area is near a proposed site for the Canadian end of the Detroit River International Crossing Bridge. Identification and mapping of subsidence and related activity down to the total depth of mining is a key part of the assessment of the site. Solution mining occurred in three major salt intervals separated by Dolomite and Shale layers and include within them shale stringers of varying thickness. Individual salt layers range from a few meters to more than ten meters in thickness. The creation of extensive solution cavities is well documented and a surface sinkhole occurred on the solution mining site in 1954 (Fernandez and Castro, 1996, Russell, 1983, Terzaghi, 1970). The cavities approximate inverted cones, but can also be cylindrical in shape. The major shale stringers control the vertical propagation of dissolution both upward and downward. Method Crosswell reflection seismic profiles were acquired among a number of wells surrounding the former solution mining site using Z-Seis Reservoir Imaging technology. Wells were logged with natural gamma ray and sonic tools prior to the reflection crosswell survey. Acquisition parameters were a 100 to 2000 Hz sweep with a 1500 ms record length and a 400 ms correlation record length. Data were sampled at .125 ms. 8 sweeps per 1.5 meter source interval were done. Receivers were also spaced 1.5 meters apart. A conventional crosswell reflection seismic data processing flow was followed (see Yu, 2002, Yu et. al., 2003, or Antonelli, et. al., 2004, for details). Coherent modes other than the upgoing P-wave reflections were removed from the field data using spatial filters. VSPCDP mapping was then performed followed by migration to collapse diffractions (Byun et. al., 2001, 2002). The migrated data were angle transformed and stacked over a reflection incidence angle range of 40 to 65 degrees to produce images suitable for interpretation. Interpretation of the reflection crosswell profiles was conducted to map the major geologic layers present at the site as well as other horizons and features noted on the seismic profiles. Raw field data and intermediate processing results were also referred to during the course of the seismic interpretation and evaluation of the geologic nature of noted features. Interpretation of the seismic data is supported by numerical displacement modeling, and acoustic wave propagation modeling. We used a 2D acoustic finite difference method for forward modeling of seismic wave propagation.
Summary We propose a new approximate partial differential equation for qP waves in transversely isotropic (TI) media. We analyse its relationship to other published "pseudoacoustic" TI equations. All pseudo-acoustic TI wave equations are coupled systems of second-order PDEs in time, derived from the same dispersion relation for qP waves by introducing different auxiliary functions. The new method combines efficient implementation and low artifacts. Modeling and reverse-time migration are shown to validate the wave equation. Introduction Seismic anisotropy is observed in many exploration areas (e.g., the North Sea, offshore West Africa, Canadian Foothills and the Gulf of Mexico). Conventional isotropic modeling and migration methods for seismic imaging are insufficient in these areas. Solving the Christoffel equations for homogeneous TI media gives three distinct wave modes: qP, qSV and qSH. The qSH mode decouples, leaving a fourth-order dispersion equation describing propagation of the coupled qP and qSV modes. Correctly, this dispersion relation describes wavespeeds for the distinguished vector polarizations corresponding to the qP or qSV waves. However, we can also treat it as defining the propagation of a scalar, "pseudo-acoustic" wavefield. Many researchers have implemented two-way wave-equation modeling and migration in anisotropic media with pseudo-acoustic approximations (Alkhalifah, 2000; Klie and Toro, 2001; Zhou et al., 2006a; Zhou et al., 2006b; Hestholm, 2007; Du et al., 2008). Alkhalifah (2000) introduced a pseudo-acoustic approximation for vertical transversely isotropic (VTI) media. Although this dispersion relation for a scalar wavefield has kinematics that are close to those of the qP arrivals in the real elastic vector wavefield, it allows spurious qSV-like events (Grechka et al., 2004). Based on Alkhalifah''s approximation, different space-time domain VTI wave equations have been proposed (Alkhalifah, 2000; Zhou et al., 2006a; Du et al., 2008). Tilting the symmetry axis relative to the coordinates does not add any new physics, just more algebraic complexity. The following description focuses on a new pseudoacoustic tilted transversely isotropic (TTI) wave equation also derived using Alkhalifah''s approximation. Method Direct solution of the corresponding fourth-order partial differential equation (PDE) in time, following immediately from equation 2, is cumbersome. Mixed spatial derivatives generally require more computation than derivatives in a single spatial variable because differencing operators become two or three dimensional convolutions rather than one dimensional. For this reason, we often seek to find equivalent coupled lower-order systems. The following three wave equations are all derived from equation (2). Each is a coupled system of second-order PDEs in time applicable when ? - ? is positive. This condition was shown by Grechka et al. (2004). The third wave equation is newly proposed here. Examples In a 2D modeling experiment, time snapshots of wave propagation in a homogenous TI medium ( Vpz= 3000 pz v m/s, ? = 0.24 , and ?= 0.1) are simulated using finite differencing. Figures 1-3 correspond to the same time snapshot from modeling with each wave equation using a vertical axis of symmetry (?=0°). The qP wave wavefront and a diamond-shaped spurious qSV wave wavefront can clearly be seen. Figure 1 displays the p wavefield of wave equation 1.
- North America > United States (0.36)
- North America > Mexico (0.24)
- Europe > United Kingdom > North Sea (0.24)
- (5 more...)
Summary The sensitivity of a controlled-source EM (CSEM) method to various heterogeneities at reservoir level has been investigated. The simulation approach employed here is based on the combined use of a 2.5-D EM finite-element modeling program and a rock-physics simulator. Firstly, lateral variations in lithological parameters like porosity and shaliness were considered for a 2-D reservoir structure. In both cases the simulations indicated that the sensitivity of possible lateral gradients was poor when compared with the homogeneous average-value case. Hence, reservoir heterogeneities caused by lateral variations in lithology are not picked up easily by the CSEM technique due to its low frequency limitation. However, in the second set of simulations the cause of the heterogeneities were due to variations in saturations in connection with a producing reservoir. Both the effect of secondary recovery representedby water flooding as well as EOR employing steam injection was simulated. The potential of the CSEM technique for such monitoring purposes was clearly demonstrated. Introduction The marine CSEM technique denoted Sea bed logging (SBL) (Eidesmo et al., 2002) has rapidly evolved as a complementary method to seismic when it comes to discrimination of pore fluid content. It is now applied routinely worldwide, but further knowledge of its limitations and areas of application is still in need. Forward modeling is an important tool both for survey planning and data interpretation as well as serving as an engine for inversion. However, standard modeling of SBL data utilize a simple and homogenous description of the potential hydrocarbon reservoir. This recognition has motivated the development of rock-physics based EM-modeling and a 1.5-D formulation has already been introduced and discussed by Wang et al. (2007a, 2007b). In this paper a more refined version of this modeling approach is introduced. A new rock-physics simulator based on a differential effective medium approach (Gelius and Wang, 2008) has been interfaced to a 2.5-D anisotropic finite element EM-code (Kong et al., 2008). This combined tool is tailored for investigating the effect of various reservoir heterogeneities. Such in homogeneities can be caused by distributions of temperature, pressure and saturation as wellas spatial variations in lithology (shaliness, grain size distribution and alignment a.s.o.). Example of use of this modeling tool is demonstrated employing a layered test model and a 2-D hydrocarbon target. By introducing reservoir heterogeneities caused both by lithology and saturations, the sensitivity of the SBL technique can be studied carefully. 2-D test model employed in the sensitivity analysis The test model is shown schematically in Fig.1, with the source being a horizontal electric dipole (HED) antenna with unit dipole moment and operating frequency of 1 Hz. It was placed 50m above the seafloor at the left boundary of the model. The water depth is 1000 m and its resistivity 0.3 Ohm-m. The model is gridded employing rectangular cells with dimension 100m x 50m. The hydrocarbon reservoir is 3000 m long and 150 m thick. Overlying the hydrocarbon layer, there is an anisotropic cap rock of shale characterized by a horizontal resistivity of 1 Ohm-m and a vertical resistivity of 2.5 Ohm-m (e.g. an anisotropy coefficient of 2.5).
- Geophysics > Borehole Geophysics (1.00)
- Geophysics > Electromagnetic Surveying > Electromagnetic Modeling (0.35)
Summary The acoustic wave equation has been widely used for the modeling and reverse-time migration of seismic data. The finite-difference method has long been the favored approach to solve this equation. To ensure quality results, accurate approximations are required for the spatial and time derivatives. This can be achieved numerically by using either very fine computation grids or very long finite-difference operators. Otherwise, the numerical error, called numerical dispersion, will be present in the data and contaminate the signals. However, either approach increases the computation cost dramatically. In this paper, we propose a new approach to address this problem by constructing a new wave equation, which we call the anti-dispersion wave equation. It involves introducing a dispersion attenuation term to the standard wave equation. When it is solved using finite difference, numerical dispersion in the original wave equation is attenuated significantly, leading to a much more accurate finite difference scheme with little additional computation cost. Introduction Pre-stack reverse-time migration includes forward extrapolation of the source wavefield and backward extrapolation of the receiver field in time (Whitmore, 1983). The finite difference method is the most popular scheme to extrapolate the two wave fields (Baysal, et.al. 1983), though a method, called pseudo-spectrum method, is also used, which combines the finite difference approximation to the time derivative with a Fourier method to compute the spatial derivatives (Loewenthal and Mufti, 1983). Traditionally, the image is constructed by cross correlating the extrapolated source and receiver wave-fields. It provides the correct kinematics, but often produces a large amount of noise contaminating the image. A more reliable imaging condition has been proposed by Liu et. al. (2007). In addition, the traditional imaging condition does not properly handle the amplitude, this problem has been better addressed by Zhang et.al. (2007). Even in an isotropic medium, the finite difference solution of the wave equation in two or three dimensions has numerical phase and group velocity that are different from those of the true medium. Different spatial frequency (wavenumber) components propagate at different velocities. This leads to increasing distortion of the waveform as propagation proceeds in time and distance. The error, called numerical dispersion, can be attenuated by either reducing the spatial sampling interval or using higher order difference schemes (Dablain, 1986). However, either approach will increase the computation cost dramatically, even though the coarser grid allowed by a longer difference operator generally brings a net computation saving compared to a lower order method, which requires a much finer grid. Another alternative is to optimize the coefficients in a lower order finite difference scheme to match the group velocity (Holberg, 1987) or the phase velocity (Etgen, 2007) of a higher order scheme. Fei and Larner (1995) applied the flux-corrected transport scheme proposed by Boris and Book (1973) to attenuate the dispersion in finite difference modeling and migration. As indicated in the paper, this method may result in resolution loss at locations where the correction is not necessary. In this paper, we pursue a new approach that cures the numerical dispersion by introducing a dispersion attenuator and constructing a modified wave equation, called the antidispersion wave equation.
Introduction Summary Almost all conventional pre-processing is conceived of with one-way wave propagation in-mind. If we take into account the existence of two-way wave propagation arrival events, then many of the underlying assumptions of moveout behavior implicit in some pre-processing techniques must be re-evaluated. Using 2D synthetic data, we demonstrate that the moveout behavior of double bounce arrivals (a class of two-way propagating events) can be compromised by pre-processing designed to remove events exhibiting "anomalous" moveout behavior . These observations are of interest to us, as we are now beginning to employ two-way migration schemes to image complex structures. However, if we continue to use conventional pre-processing techniques, we run the risk of removing the very events we are trying to image. The observations made on the basis of synthetic modeled data, are extended in this work to real data examples, all from the North Sea, where in the central graben, we commonly have steep piercement salt diapir structures, which are good candidates for producing useful double bounce arrivals, which can be imaged using RTM. The speed and cost effectiveness of contemporary computer systems now permits us to implement more general algorithmic solutions of the wave equation (Whitmore, 1983, Baysal et al, 1983, McMechan, 1984, Bednar et al 2003, Yoon et al 2003, Shan & Biondi 2004, Zhou et al, 2006, Zhang et al, 2006). The restriction to oneway propagation can be lifted, and data migrated so as to take advantage of more esoteric propagation paths, such as turned rays, double bounce arrivals, and potentially multiples (Mittet, 2006). However, in order to take advantage of these improved algorithms, we must ensure that the data input to migration have not been compromised in any way. Specifically, in this work we address the moveout behavior of double bounce events (Hawkins et al, 1995, Bernitsas et al, 1997, Cavalca & Lailly, 2005), and note how many conventional pre-processing algorithms can damage these arrivals, thus rendering some aspects of any subsequent high-end migration superfluous. We commence our analysis by reviewing the conclusions of some preliminary work studying synthetic data (Jones, 2008a), showing the moveout behavior of some simple double bounce events (also referred to as "prism waves" by some authors). For ease of demonstration, we firstly employ a ray-trace package, with which we can model individual selected arrivals, and later create more complex synthetic data using an elastic finite difference (FD) package. After investigating the moveout behavior of the simple models, we move-on to a model representing a complex North Sea salt dome structure (Davison, et al 2000, Thomson, 2004; Farmer, et al 2006). We show the effect of various conventional pre-processing steps on double bounce arrivals, and carry these analyses trough to migration with an 2D RTM algorithm capable of imaging the double bounce arrivals. We then extend this analysis and demonstration-ofprinciple from the 2D synthetic data to real data (Jones, 2008b), were we see similar classes of event, and the same degradation of double bounce arrivals shown in the synthetic trials.
- Europe > North Sea (0.58)
- Europe > Netherlands > North Sea (0.48)
- Europe > Denmark > North Sea (0.48)
- (2 more...)
SUMMARY The quality of seismic images obtained by reverse time migration strongly depend on the employed image condition. We propose a new imaging condition, which is motivated by stationary phase analysis of the classical cross-correlation imaging condition. Its implementation requires the Poynting vector of the source and receiver wavefields at the imaging point. An obliquity correction is added to compensate for the reflector dip effect on amplitudes of reverse time migration. Numerical experiments show that using an imaging condition with obliquity compensation improves reverse time migration by reducing backscattering artifacts and improving the illumination compensation. INTRODUCTION Pre-stack reverse time migration (RTM) is based on the time reversal property of the two-way wave equation and the crosscorrelation imaging condition proposed by Claerbout (1985). Several implementations of RTM using this imaging condition have been reported (McMechan, 1983; Kosloff and Baysal, 1983; Baysal et al., 1983). The computational demand for RTM is high compared to wave equation migration by downward extrapolation of the wavefield (Biondi, 2006). However, low cost parallel computing and more efficient storage hardware is making RTM feasible. The difficulties of seismic imaging below the salt column and in areas of high lateral velocity variation have drawn attention to RTM, which, at least theoretically, is able to meet those challenges. RTM has its limitations, though. Two major drawbacks are the artifacts produced by backscattering and the amplitudes of the migrated images, which are not proportional to the subsurface reflectivity (Biondi, 2006). To reduce the artifacts due to backscattering, several approaches have been recently proposed. Guitton et al. (2007) use a least-squares regularization; Fletcher et al. (2005) introduced a new forward waveequation to attenuate backscattering events and Yoon and Marfurt (2006) introduced the Poynting vector imaging condition. Several attempts to improve the amplitudes in RTM are based on illumination compensation with different kinds of regularization (Valenciano and Biondi, 2003; Kaelin and Guitton, 2006). Attempting to better understand the amplitudes in RTM, Haney et al. (2005) performed an asymptotic analysis of the crosscorrelation imaging condition. Their analysis assumes a single planar reflector in a 3D homogeneous medium, full coverage, and infinite aperture. They demonstrate that the amplitudes of RTM are affected by an obliquity factor that depends on the reflector dip. Based on this result, we propose an imaging condition which can asymptotically correct for this obliquity factor in RTM. We present numerical experiments that show the improvement of RTM images when the obliquity and illumination compensation are applied in the imaging condition. Numerical experiments demonstrate the improvement of the images when the obliquity factor and illumination compensation are included in the imaging condition for RTM. Here, we use a finite-difference implementation, but the correction can also be incorporated into Fourier-domain implementations like the one of Tessmer (2003). METHOD We start by revisiting the asymptotic analysis of the crosscorrelation imaging condition (Haney et al., 2005). Based on this result we propose an imaging condition for RTM which, asymptotically, improves the amplitude of RTM images. Asymptotic analysis of cross-correlation imaging condition The cross-correlation imaging condition for shot-profile migration (Claerbout, 1985)