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Henri Calandra

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Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2016 SEG International Exposition and Annual Meeting, October 16–21, 2016

Paper Number: SEG-2016-13961828

Abstract

ABSTRACT We consider the seismic inverse problem in the case of the time-harmonic elastic isotropic wave equation, in particular for the recovery of the Lamé parameters. We employ full waveform inversion (FWI) where the reconstruction is based upon iterative minimization techniques. We apply a multilevel scheme to stabilize our iterative reconstruction. We illustrate this idea using both Continuous Galerkin finite element method on unstructured tetrahedral meshes with surface and body waves and finite difference approximation on the regular meshes with body waves only. Presentation Date: Tuesday, October 18, 2016 Start Time: 10:45:00 AM Location: 146 Presentation Type: ORAL

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2015 SEG Annual Meeting, October 18–23, 2015

Paper Number: SEG-2015-5910360

Abstract

Summary In this paper, we propose stable low-order Absorbing Boundary Conditions (ABC) for elastic TTI modeling. Their derivation is justified in elliptic TTI media but it turns out that they are directly usable to non-elliptic TTI configurations. Numerical experiments are performed by using a new elastic tensor source formula which generates P-waves only in an elliptic TTI medium. Numerical results have been performed in 3D to illustrate the performance of the ABCs. Introduction Seismic Imaging is still progressing by taking advantage of advanced computational techniques constantly renewed. Nowadays simulations consider more realistic representations of the subsurface, typically moving from Acoustics to Elastodynamics and from Isotropy to Tilted Transverse Isotropy (TTI). Literature is rich in references about "pseudo-acoustic TTI" RTM. First attempts chose to simplify the elastic TTI approximation, initially depicted in Alkhalifah (1998), leading to several TTI formulations in e.g. Du et al. (2007); Fletcher et al. (2009); Zhang et al. (2011); Duveneck and Bakker (2011), while others investigated equation decoupling, see for instance Zhan et al. (2012). All these references target acoustic TTI RTM only except in Yan and Sava (2011) dealing with elastic TTI RTM. In any case, nothing is mentioned about boundary conditions which are supposed to be non-reflecting for keeping the numerical solution from pollution generated by the boundaries of the computational domain. We propose here to address this issue which is critical when considering elastic TTI modeling. Isotropic codes are usually based on Perfectly Matched Layers (PML) surrounding the domain of interest, see for instance Collino and Tsogka (2001). Unfortunately, it has been demonstrated in B´ecache et al. (2003) that PMLs are unstable in TTI media. Moreover, the numerical cost of the additional layer is prohibitive in 3D, especially in a RTM framework which is already computationally intensive. Besides, PML also impacts on parallel efficiency, since they requires heterogeneous computations on a large set of data. Hence, the design of stable Absorbing Boundary Condition (ABC) for elastic TTI is an effective alternative that should be considered for RTM. In a previous work, see Barucq et al. (2014), we have proposed a new elastic TTI ABC in 2D and we focus here on the extension to 3D.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2013 SEG Annual Meeting, September 22–27, 2013

Paper Number: SEG-2013-0607

Abstract

Summary We have developed an integrated method to obtain high-resolution subsurface elastic parameters using combined wave equation tomography (WET) and full waveform inversion (FWI). Both refraction and reflection data are used. During parameterization, long wavelength and short wavelength structures are separated and mapped into velocity and density to account for kinematics and dynamics, respectively. Full wavefield modeling is used to compute synthetic data that include all reflection and refraction arrivals. To better constrain the reflection amplitude, the near offset data are first inverted using FWI where all the model perturbations are mapped into density. The short wavelength density structure is then converted into vertical travel time domain where it is independent of long wavelength velocity model. As long wavelength structure (velocity) is updated, short wavelength structure is converted back into depth domain for wavefield computation. Finally FWI is applied all the data to retrieve short wavelength structures with resolution up to a quarter wavelength. The method is applied to two synthetic examples; our results shows that one can recover detailed velocity information starting from a model far from the true model.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2013 SEG Annual Meeting, September 22–27, 2013

Paper Number: SEG-2013-1203

Abstract

SUMMARY We analyze the correlation focusing objective functional introduced by van Leeuwen and Mulder to avoid the cycle-skipping problem in full waveform inversion. While some encouraging numerical experiments were reported in the transmission setting, we explain why the method cannot be expected to work for general reflection data. We characterize the form that the adjoint source needs to take for model velocity updates to generate a time delay or a time advance. We show that the adjoint source of correlation focusing takes this desired form in the case of a single primary reflection, but not otherwise. Ultimately, failure owes to the specific form of the normalization present in the correlation focusing objective.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2013 SEG Annual Meeting, September 22–27, 2013

Paper Number: SEG-2013-1089

Abstract

Summary The full waveform inversion (FWI) of land data are becoming increasingly necessary in hydrocarbon exploration. However, strong surface waves and the existence of complex topography make it difficult to recover the subsurface structure using refraction tomography. To obtain subsurface velocity models of complex topography, we propose the Laplace-Fourier domain FWI with a finite element method (FEM). The mesh was designed to avoid several problems that could affect the inverted results and to minimize the computational cost. Laplace-Fourier domain inversion can recover subsurface P- and S-wave velocities with starting simple initial velocities generated without any prior information.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2012 SEG Annual Meeting, November 4–9, 2012

Paper Number: SEG-2012-0472

Abstract

Summary We propose to use wave-equation tomography (WET) method to build long-wavelength velocity structure for full waveform inversion (FWI). In WET, full wavefield modeling is performed and cross-correlation time delay between the arrivals from synthetic and real waveforms is used as objective function. Adjoint method is used to calculate the gradient in each iteration efficiently. Since WET and FWI share similar inversion structure, we use a hybrid misfit function to combine the two methods as an integrated workflow that is able to estimate high-resolution structure from poor starting model. To stabilize WET and make it converge to global minimum, we precondition the time delay measures with maximum cross-correlation coefficients and perform adaptive scale smoothing to the gradients. By exploring the band-limited feature of seismic wavefield, WET can provide better resolution than ray-based travel time tomography, which is under high frequency approximation. To illustrate the advantage of wave-equation tomography, we show in a 2D synthetic test that WET provides subsurface information that is critical for successful FWI. We also test 2D Marmousi model and satisfactory inversion results are achieved without much manual manipulating.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2012 SEG Annual Meeting, November 4–9, 2012

Paper Number: SEG-2012-0502

Abstract

SUMMARY Although wave equation based migration techniques, such as Reverse Time Migration (RTM), have been popular for years, the acoustic approximation is still applied frequently. Even when considering anisotropic behavior, modifications to the acoustic wave equation are invoked to facilitate changes in wavespeed along different directions. In a classical marine survey, seismic waves are generated in the water layer and observations are recorded in the form of pressure fluctuations. In that case, using a purely acoustic wave equation is a reasonable assumption, since conversions between compressional and shear motions along the ocean bottom are weak. However, with the availability of Ocean Bottom Cables (OBCs) or in land surveys, shear waves do play an important role. On one hand, incorporating elastic information in imaging may enhance the coherence of arrivals and thus provide better images. On the other hand, recorded shear signals might contaminate the image if falsely interpreted as reflected compressional waves. Following these considerations, we carry out a 3D elastic experiment investigating proper imaging conditions for elastic migration.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2012 SEG Annual Meeting, November 4–9, 2012

Paper Number: SEG-2012-0138

Abstract

Summary When waveform inversion is performed in the Laplace-Fourier domain, wave propagation should be described through Laplace-Fourier domain modeling. However, because the modeling operator matrix organized by a complex-valued angular frequency is not satisfied with the positive definite, direct matrix solvers or iterative matrix solvers supporting nonsymmetrical linear systems should be used. In this study, 3D 2 nd -order time-8 th -order space-domain wave modeling with finite-difference stencils is employed in the waveform inversion instead of 3D complex-valued frequency-domain modeling, and the Graphic Processing Unit (GPU) architecture is used for higher speedup instead of the traditional CPU architecture. In this paper, Laplace-Fourier-domain waveform inversion with time-space wave modeling is called hybrid waveform inversion. To verify the feasibility of this technique, the waveform inversion is performed on the A1 line of the synthetic SEG/EAGE 3D salt model and 3D wide-azimuth real exploration data.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2012 SEG Annual Meeting, November 4–9, 2012

Paper Number: SEG-2012-1262

Abstract

SUMMARY We present a method for approximately inverting the Hessian of full waveform inversion as a dip-dependent and scaledependent amplitude correction. The terms in the expansion of this correction are determined by least-squares fitting from a handful of applications of the Hessian to random models - a procedure called matrix probing. We show numerical indications that randomness is important for generating a robust preconditioner, i.e., one that works regardless of the model to be corrected. To be successful, matrix probing requires an accurate determination of the nullspace of the Hessian, which we propose to implement as a local dip-dependent mask in curvelet space. Numerical experiments show that the novel preconditioner fits 70% of the inverse Hessian (in Frobenius norm) for the 1-parameter acoustic 2D Marmousi model.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2011 SEG Annual Meeting, September 18–23, 2011

Paper Number: SEG-2011-2471

Abstract

ABSTRACT This is the first study that applies a diffuse optical tomography point of view to the full waveform inversion. Focusing on the fact that the diffusion equation has a similar structure to the Laplace-domain wave equation, we relate the inverse problem in the Laplace-domain to the inverse scattering problem which is frequently used in studies of diffuse optical tomography. In this study, instead of the Born series, we suggest the Rytov series in order to deal with the logarithmic objective function of the Laplace-domain waveform inversion. It is also shown here that the inverse Rytov series is equivalent to the Newton-Kantorovich algorithm for the Laplace-domain inversion. Finally, we discuss about the convergence condition of the forward and inverse Rytov series, employing the contraction mapping concept.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2011 SEG Annual Meeting, September 18–23, 2011

Paper Number: SEG-2011-2549

Abstract

ABSTRACT The key for a high quality seismic depth imaging is the velocity model used in the seismic migration. Nonetheless constructing such models is still very challenging. Seismic velocity analysis tools have limitations when dealing with complex earth structures. Recent progress in high performance computing allows using Full Waveform Inversion (FWI) formulation to build velocity models. This technology is highly challenging due to the non-linearity and the non-uniqueness of the solution. It is based on the minimization of an objective function measuring the misfit between modeled and recorded data. There are many obstacles to apply waveform inversion to field seismic data. One of the most critical factors is the lack of low-frequency components in the recorded data needed for reconstructing long-wavelength structure. FWI in the Laplace domain is less sensitive to the lack of low frequencies in seismic data than FWI in time or Fourier domain. It is shown that starting from a very simple initial model it can recover the long wavelength of the model.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2011 SEG Annual Meeting, September 18–23, 2011

Paper Number: SEG-2011-2915

Abstract

ABSTRACT 3D Laplace-domain waveform inversion can recover a large velocity model for successive waveform inversion in the frequency domain. However, the grid interval in 3D Laplace-domain modeling and inversion cannot be sufficiently small because of the heavy computational cost. Therefore, we cannot assess whether or not the modeled wavefield is reliable if our model has an abruptly undulated sea bottom surface. The irregular finite element method can provide a solution; however, it increases the number of bands of the impedance matrix. Instead, we applied the Gaussian quadrature integration method in order to reflect two properties on one element at the irregular sea bottom. In order to verify this modeling algorithm, we compared our modeled wavefield with the analytic solutions for an unbounded homogeneous model, an unbounded two-layer model and an obliquely-inclined two-layer model. The results of the verification tests show that our modeling algorithm better describes a wavefield with an irregular sea bottom in the 3D Laplace domain than the conventional modeling algorithm.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2010 SEG Annual Meeting, October 17–22, 2010

Paper Number: SEG-2010-2820

Abstract

Summary Although the exploration of the foothills regions is of great importance for the oil industry, subsurface velocity cannot be estimated by refraction tomography due to the complex geological environment. To obtain subsurface velocity, we have proposed Laplace-domain waveform inversion with the finite element method, which uses both the refraction and reflection energy. Delaunay triangulation and an adaptive method for mesh generation are used to depict the complex topography and reduce the computational cost, respectively. By comparing the Laplace-domain analytic solution with the wavefield of numerical calculation at several damping constants, we confirmed that numerical solutions were in good agreement with analytical solutions in homogeneous unbounded media, which proved that Laplace-domain modeling using this mesh generation algorithm could be adapted to the seismic data. We demonstrated our algorithm for waveform inversion through a numerical example of the 2D synthetic Foothills model in the Laplace domain. Introduction Because a number of reservoirs have been found in folded and thrust belt areas, seismic images of these areas have been pursued as a way to find reservoirs among many oil companies. However, rugged surface topography can be a hindrance to establishing the accurate velocity model required for seismic imaging: (1) it is not easy to describe wave propagation in complex geography through numerical wave modeling, and (2) refraction tomography cannot give macro velocity information because the refraction energy is trapped by complex near-surface structures (Chaiwoot et al., 2009). To solve these problems, we exploit the Laplace-domain waveform inversion using finite elements generated by Delaunay triangulation and the adaptive method. Wave modeling on the basis of the finite element method can express the wave-propagation in complex geological structures by using Delaunay triangulation, and velocity information at the target depth can be extracted because Laplace-domain waveform inversion uses reflection and refraction energy both, which can penetrate the deeper part of the area. Because element size rarely affects numerical dispersion in the Laplace domain (Shin and Cha, 2008), adopting the adaptive method (as the depth of the model deepens from the surface, larger meshes are used) can reduce computational cost. This is especially true compared to using the frequency domain. Because acoustic waveform inversion in the Laplace domain is less sensitive to the effects of the Rayleigh and the converted waves, the acoustic inversion scheme can be applied to the land seismic data, which leads to uninvolved data processing in contrast to the elastic waveform inversion. We attempted to compare numerical Green’s functions on the homogeneous background medium with their analytic solutions in the synthetic Foothills model having with rugged surface topography, and we tested our waveform inversion algorithm on the synthetic model with complex geological structure. Laplace waveform inversion algorithm Shin and Cha (2008) proposed that the objective function using the logarithmic wavefields could be effective in Laplace-domain waveform inversion. Therefore, in this study, we used the logarithmic objective function. Then, the final descent direction can be established by summing the directions.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2010 SEG Annual Meeting, October 17–22, 2010

Paper Number: SEG-2010-0967

Abstract

Summary The waveform inversion in the time or frequency domain has a highly nonlinear characteristic resulting from many local minima. Furthermore, in the absence of low frequencies in real exploration data, it is difficult to restore long-wavelength structures using a time or frequency domain waveform inversion algorithm. Recently, an acoustic waveform inversion algorithm in the Laplace domain emerged to restore long-wavelength structures, playing a key role in imaging subsurface structures and velocity inversions, even if low frequencies are absent. This Laplace-domain inversion algorithm mitigates local minima problems in the inversion procedure. We extended the Laplace-domain waveform inversion to acoustic-elastic coupled media. We applied our algorithm to the synthetic example of a 2D elastic model containing a flat water layer, which is modified from the SEG/EAGE salt model. Introduction Recently, Shin and Cha (2008) proposed a 2D acoustic waveform inversion in the Laplace domain to recover a long-wavelength velocity model from seismic data. The Laplace-domain wavefield can be assumed to be the zerofrequency component of the damped wavefield. The residual wavefields in the Laplace domain show longwavelength features for all damping constants, unlike those in the frequency domain. Furthermore, the objective function shows few local minima compared with those in the frequency domain. Thus, the Laplace-domain waveform inversion could successfully generate long-wavelength (smooth) velocity models for field data whose lowfrequency components might be unreliable (Shin and Cha, 2008). We extended the Laplace-domain waveform inversion to acoustic-elastic coupled media. Though pressure data in the Laplace domain were used, our coupled Laplace-domain waveform inversion could generate long-wavelength models of both P- and S-wave velocity. The inverted velocity models in the Laplace domain could be used as good initial models for frequency-domain inversion because they might contain long-wavelength structural information of the true models. We demonstrated our algorithm by showing an inversion example for a 2D elastic model containing a flat water layer, which is modified from the SEG/EAGE salt model (Aminzadeh et al., 1997). Forward modeling in the Laplace domain for coupled media A marine seismic survey is usually performed for an acoustic-elastic coupled medium. The sea can be regarded as the acoustic medium and the bottom layer as the elastic medium, which is our subsurface target. The pressure by acoustic wave equation is measured in the acoustic medium and displacement by elastic wave equation is detected in the elastic medium. Since wave propagation in the coupled media is affected by both wave equations, the continuous condition of pressure and vertical stress at the interface between acoustic and elastic media should be considered. In this study, we assumed that each medium is heterogeneous and isotropic and the interface between fluid and solid medium is flat. To exploit the finite element method for modeling the coupled media, the 2D acoustic and elastic wave equations in the Laplace domain were first derived in matrix forms. Then, we combined the acoustic and elastic finite element formulas and coupled the pressure and displacement by the boundary condition at the interface condition (Zienkiewicz and Taylor, 2000).

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2010 SEG Annual Meeting, October 17–22, 2010

Paper Number: SEG-2010-0993

Abstract

Summary We suggest a method for selecting damping constants in the Laplace-domain waveform inversion by analyzing the relationship between the damping constant and the detectable depth of a high-velocity structure. The result indicates that the Laplace damping constant and the detectable depth of a high-velocity structure are almost in inverse proportion to each other, and the relationship between the two variables is dependent on the maximum offset and subsurface velocity. Given the maximum offset and approximate underground velocity, the maximum value of the Laplace damping constant can be determined. The minimum value can also be selected, given the maximum recording time of a field data set. Introduction For waveform inversion in the frequency domain, there have been many research papers, including Sirgue and Pratt (2004), that enable us to set the minimum value, maximum value and interval of frequency efficiently in the frequency domain inversion algorithm. However, for waveform inversion in the Laplace domain (Shin and Cha, 2008), the rule for determining the Laplace damping constants has not yet been established. The Laplace damping constant is very closely related to the penetration depth, which is one of the most important factors in inversion results. Therefore, it is necessary to analyze the relationship between the Laplace damping constant and the penetration depth. In this study, we introduce a method of choosing the Laplace damping constants in the Laplace-domain inversion based on the relationship between the Laplace damping constant and the detectable depth of a highvelocity structure. We first reveal the relationship given the maximum offset distance and subsurface velocity. Then, based on that relationship, we propose a method for choosing the damping constants in the Laplace-domain waveform inversion to find a high-velocity structure. Method of clarifying the relationship between the Laplace damping constant and the detectable depth of a high-velocity structure It is important to find high-velocity structures such as salt domes in exploration seismology because those structures can form traps that confine hydrocarbons. To locate a highvelocity structure, we have to use data that contains information about the structure. In other words, the wavefield we measure must be a function of the depth of the high-velocity structure. However, if the structure is located very deep, it cannot affect the wavefield and cannot be detected. In this context, we consider the detectable depth of a high-velocity structure as the maximum depth of the structure that influences the wavefield we measure near ground level, i.e., that can make the structure be detected. Of course, the penetration depth varies with the change of the shape of the high-velocity structure. But in this study, we set up an acoustic two-layer model and let the second layer represent the high-velocity structure to analyze for general cases. Also, the Laplace-domain 3D modeling algorithm for acoustic media was used. Figure 1 shows the relationship between the amplitude of the Laplace-domain wavefield and the depth of a highvelocity structure when the offset distance is 10 km and the damping constant is 5.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2010 SEG Annual Meeting, October 17–22, 2010

Paper Number: SEG-2010-1059

Abstract

Summary There are many obstacles to applying waveform inversion to seismic data. However, the most critical factor is the absence of the low-frequency components that are needed for constructing long-wavelength structure. This problem stems from the highly nonlinear property of waveform inversion, which causes the algorithm to be trapped in a local minimum. The waveform inversion in the Laplace domain, rather than the usual frequency domain, is capable of producing velocity models with long-wavelength information. A study on this method was recently published, which was limited to the problem of acoustic media. In this paper, we extend Laplace-domain waveform inversion to elastic media. Unlike acoustic inversion, elastic inversion requires sophisticated manipulation of the gradient direction. We suggest a method to modify pseudo-Hessian matrices by using a heuristic weighting function. We test our inversion algorithm on synthetic seismic data generated using the SEG/EAGE salt dome model and the CCSS model. Inversion results using these data sets also produce the long-wavelength velocity model and demonstrate that Laplace-domain waveform inversion is robust to the initial velocity model. Furthermore, we provide an example that shows that our inverted result is a suitable initial model for the frequency domain waveform inversion. Introduction Seismic waveform inversion methods (Lailly, 1983; Tarantola, 1984; Pratt et al, 1998; Shin et al. 2001; Operto et al., 2004) have the problem of local minima. To eliminate some of the obstacles making waveform inversion unsatisfactory for real data, Shin and Cha (2008) proposes 2-D acoustic waveform inversion in the Laplace domain. They demonstrate that this method recovers longwavelength information on the underlying velocity model, using both synthetic examples and field data. Furthermore, regardless of the damping constants chosen, the objective function in the Laplace domain has almost no local minima. While Laplace-domain waveform inversion has many advantages over frequency-domain inversion, the numerical examples so far provided have been limited to acoustic media. In the context of land seismic data, however, the fact that Rayleigh waves and converted waves do not exist in the acoustic approximation poses a serious problem. These waves could carry important information such as the S-wave velocity and the density of the medium (Mora, 1987). As an example, Shin and Min (2006) showed that there are significant differences between real seismic data and acoustically approximated synthetic data at large offsets. The main reason for the discrepancy is simply that the real marine environment includes a fluid-solid interface and solid earth below the sea floor. In this paper, we verify the accuracy of forward modeling in the Laplace domain using analytic solutions for an unbounded homogeneous model. As recent studies on the frequency-domain full waveform inversion adopt the back propagation algorithm, we exploit this algorithm to calculate the gradient direction without explicitly computing partial derivative wavefields (Shin and Min, 2006). Our inversion results demonstrate that our new algorithm can generate the long-wavelength velocity model even if a homogeneous initial model is used for inversion; furthermore it is suitable for generation of initial velocity model of subsequent frequency-domain full waveform inversion.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2010 SEG Annual Meeting, October 17–22, 2010

Paper Number: SEG-2010-4400

Abstract

SUMMARY Finding the optimal subsurface model, in the sense that the computed seismograms most closely resemble the observed ones, is a well-studied problem with few secrets. The main difficulty lies in finding the velocity model that best describes the observed data and particularly, recovering the long wavelength content of it. Nowadays, migration velocity analysis and tomography techniques have been used to obtain the most realistic velocity model for accurate 3D depth imaginig. In order to have both long and small wavelength information, we suggest using a misfit function optimizing simultaneously the amplitudes (the classical misfit function) and the traveltimes (the traveltime misfit function, Luo & Schuster, 1999; Tromp et al., 2005). Hence, the optimization problem will be divided into two optimization problems. The de-migration operator, which is the transpose of the migration operator, has the potential of quickly simulating seismograms to be compared to the observed ones. The originality in this work consists of the use of simple de-migration instead of the wave equation for waveform fitting, which will allow to fit not only the phases but also the amplitudes.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2008 SEG Annual Meeting, November 9–14, 2008

Paper Number: SEG-2008-2097

Abstract

Summary: In this article, we present some wave-field snapshots and synthetic seismograms by using a refined algorithm of the nearly analytic discrete (NAD) method that was proposed recently in BSSA, investigate the numerical dispersion of the refined algorithm through numerical simulations, and compare the wave-field results computed by the refined algorithm against those of the 8th-order finite-difference (FD) method. Numerical results show that the refined algorithm has no visible numerical dispersion for any space grid increments and can automatically suppress the numerical dispersion caused by discretizing the wave equation when too few samples per wavelength are used or when models have large velocity contrast. Introduction: Conventional explicit finite-difference and finite element methods for solving the acoustic- and elastic- wave equation suffer from numerical dispersion. The numerical dispersion can lower the resolution of synthetic seismograms. In order to eliminate the numerical dispersion, one way is to use sufficient grid points per upper half-power wavelength. For example, ten or more grid points per wavelength at the frequency of the upper half-power point should be adequate when the usual 2nd-order accuracy FD scheme is employed, while the 4th-order scheme seems to produce accurate results at five or six grid points per wavelength at the frequency of the upper half-power point (Alford et al., 1974). However, this way using more grid points per wavelength results in needing more computational costs and storages for computer code. Another way of attacking the numerical dispersion is to use high-order FD schemes (e.g., 8th-order FD method, Dablain, 1986) or staggered-grid FD methods (Virieux, 1986; Igel et al., 1995) or the pseudo-spectral method (PSM) (Kosloff and Baysal, 1982) to reduce the numerical dispersion. The higher-order FD or staggered-grid FD methods can further reduce the numerical dispersion, but they still suffer from the numerical dispersion when too few samples per wavelength are used (Sei and Symes, 1994). The PSM is attractive as the space operators are exact up to the Nyquist frequency. In other words, the PSM only requires 2 grid points per wavelength for eliminating the spatial numerical dispersion (Dablain, 1986). However, it also suffers from numerical dispersion in the time, and its numerical dispersion increases with increasing the time increment (Yang et al., 2006). The so-called "nearly analytic discrete method (NADM)" (Yang et al., 2003a) and it''s improved algorithm (Yang et al., 2007) for solving the acoustic and elastic equations is another kinds of effective methods for decreasing the numerical dispersion. These methods, based on the truncated Taylor expansion and the local interpolation compensation for the truncated Taylor series, use the wave displacement-, the velocity- and their gradient-fields to restructure the wave displacement fields. Hence it enables effectively to suppress the numerical dispersion. This paper is to present a refined algorithm of the NADM and investigate the efficient implementation of the refined NAD for these cases of heterogeneous media and very coarse space steps. Refined NAD Algorithm We first review and summarize the key ideas in it. The same notations as that in the original NADM (Yang et al., 2003a) in our present study,.

Proceedings Papers

Linbin Zhang, Paul Williamson, Elive Menyoli, Biaolong Hua, Eric Dussaud, Henri Calandra, Alexandre Khoury

Publisher: Society of Exploration Geophysicists

Paper presented at the 2007 SEG Annual Meeting, September 23–28, 2007

Paper Number: SEG-2007-2446

Abstract

ABSTRACT Conventional one-way wave equation used in seismic migration is not energy conserving. As a result, it can not provide correct amplitudes for the reflector. To correct image amplitudes the one-way energy-conserving wave equation is proposed to be applied in seismic migration. Since the new wave equation is similar to the conventional one-way wave equation, it is easy to implement without increasing too much computational time. In order to compensate for the acquisition aperture, the Hessian is applied to the final image. This method is tested with both synthetic and real dataset.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2007 SEG Annual Meeting, September 23–28, 2007

Paper Number: SEG-2007-2270

Abstract

Summary We present a new implementation of 3D Fourier finite-difference (FFD) depth migration method based on survey sinking (DSR) operator in source-geophone and midpoint-offset hybrid domain. The 3-D DSR downwardcontinuation operator is decomposed into two orthogonal downward-continuation operators: a 2-D inline prestack operator which is implemented in source-geophone coordinates and a 2-D crossline poststack operator which is implemented in midpoint-offset coordinates. This new implementation has much higher accuracy compared to the conventional FFD-based DSR implementation in midpoint-offset domain only. Numerical tests show that the new implementation is very efficient and provides much better migrated images than conventional implementation and the image quality from the new implementation is almost the same as that from shot profile migration. Introduction Two categories exist in the classification of one-way wave equation based depth migration: shot profile migration and survey sinking (DSR) migration (Claerbout, 1985) . In shot profile migration, source and receiver wave fields are extrapolated independently in depth and an imaging condition, such as the cross correlation of source/receiver wavefields, is applied to get the migrated image; In survey sinking migration, source and receiver wave fields are downward continued simultaneously and the zero-time and zero-offset imaging condition is applied to obtain the migrated image. Conventional survey sinking migration works in midpoint-offset domain to avoid sorting seismic data from common shot gathers to common receiver gathers. It has the advantage of unlimited apertures, contrast to limited apertures as in shot profile migration for the purpose of limiting computation time and memory requirements. In survey sinking migration, preprocessing of seismic data is required: The field recorded common shot gathers are first sorted into midpoint-offset domain (CMP bins) and then are regularized along offset. The input data for full DSR are 5-D, including arrival time; inline/crossline midpoints; and inline/crossline offsets. Taking advantages of the limited azimuthal range of conventional marine survey, a 5-D DSR extrapolation problem is reduced into a 4-D computation problem by making common azimuth assumption and applying stationary phase condition in cross line offset direction. This is the widely used common azimuth migration (CAM). (Biondi and Palacharia, 1996). In case of wide azimuth marine data, it is possible to group the wide azimuth data into different azimuth subsections and each subsection is then applied with CAM. Over years, different algorithms have been developed for DSR migration. Popovici (1996) developed a split-step DSR method and Biondi and Palacharia (1996) used phase shift plus interpolation (PSPI) approach. To achieve higher accuracy for strong lateral velocity variations, the generalized screen propagators (GSP) and Fourier finite difference (FFD) method (Zhang et al., 2005) have been proposed. All these methods work in midpoint-offset domain. However, detailed mathematical analysis shows that GSP/FFD implementation in midpoint-offset domain alone neglects the coupling term between midpoint wave number and offset wave number which has hugh compact on image quality when there are strong lateral velocity variations. In this paper, we present a new implementation of FFD based DSR depth migration in source-geophone and midpoint-offset hybrid domain.

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