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Keywords: temporal frequency

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Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2015-5917235

... Summary We present a survey sinking based framework for residual migration velocity analysis. By using a migration algorithm in the

**temporal****frequency**domain, our migration algorithm is efficient and allows us to pick velocity perturbations directly. Picking velocity perturbation directly...
Abstract

Summary We present a survey sinking based framework for residual migration velocity analysis. By using a migration algorithm in the temporal frequency domain, our migration algorithm is efficient and allows us to pick velocity perturbations directly. Picking velocity perturbation directly removes the issues associated with estimating velocity perturbations from indirect picking parameters, such as time lags. Our methodology is based on a wave-equation migration method that combines high accuracy with computational efficiency. By relying on survey sinking migration, we avoid the problem of estimating the source wavelet, which is a rarely mentioned problem associated with RTM and shot-gather migration. Our survey sinking framework allows us to robustly estimate the velocity sequentially in relatively thin layers. Introduction In order to produce a useful image of the subsurface, migration algorithms require an accurate estimate of the wave propagation velocity. Inaccurate velocity estimates typically result in a degraded image, and the effect of poor velocity estimates rapidly degrades the image at increasing depths. To address this, a wide variety of Migration Velocity Analysis (MVA) tools have been proposed. The strategy is to use some type of localized quality ("focusing") measure applied to the migrated image and, based on such meausure, estimate how the migration velocity should be modified to generate a more geophysically probable image. In MVA it is natural to use some type of extended imaging condition (see, e.g., Sava and Vlad, 2008, and Sava and Vasconcelos, 2010). A common way of extending the imaging condition is to study the migrated image for a range of time shift lags (temporal offsets), and pick the time lag that locally generates the best-focused migrated image. The use of time lags is popular as it typically can be applied efficiently without having to re-migrate the data. Migrating the data can be computationally expensive for accurate wave-equation based migration algorithms. For this reason using time lags is often the method of choice for MVA in connection with wave-equation based migration methods. The major disadvantage of using time lags (or spatial offsets) as parameter for MVA, is that the connection between the time lag and the velocity perturbation can be complicated to estimate in a complex medium (Yang and Sava, 2001). Even if a relation between time lag and velocity perturbation can be formally established, the resulting equations can be ill-conditioned, and lead to severe numerical problems. Furthermore, such solutions are typically only valid if the time lag (and the associated associated velocity perturbation) is small.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2011-3095

... the f-p domain. data corresponding seismic event synthetic data linear event curved event weighting function radon domain upstream oil & gas reference list

**temporal****frequency**fk spectrum seismic data interpolation spectrum domain spectra weighting function extrapolation spectra...
Abstract

ABSTRACT Seismic trace interpolation can be formulated as an underdetermined least squares inverse problem. In order to force interpolation to follow the directions of seismic events, the weighting function is designed to build the interpolated energy along the desired direction. Usually, such a weighting function is built in the Fourier domain. However, when Fourier spectra are aliased, especially in the case of up-sampling seismic traces (creating more output than input), the design of the weighting function faces problems. In this paper, we discuss these problems and a proposed solution using the f-p domain.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2009 SEG Annual Meeting, October 25–30, 2009

Paper Number: SEG-2009-2248

... sequential order of complex-valued frequencies in 2-D plane to perform the sequential inversion. An example of the sequential order is shown in Figure 2(b). Fourier

**temporal****frequency**( )f La pl ac e da m pi ng c on st an t ( ) Laplace-domain waveform inversion Laplace-Fourier-domain waveform inversion...
Abstract

Summary We propose a sequentially ordered single-frequency 2-D acoustic waveform inversion using a logarithmic objective function in the Laplace-Fourier domain. While previous Laplace- and Laplace-Fourier-domain waveform inversions were implemented as a simultaneous inversion of each frequency component, our algorithm sequentially inverts single-frequency data in the Laplace-Fourier domain, thus reducing computational resources. Most conventional waveform inversion methods need an initial velocity model close to true model. We overcome this problem by proposing a one-step waveform inversion method seeking to find a final velocity structure from simple initial velocity model such as a two-layer velocity model through hybrid combination of both Laplace domain inversion and Fourier domain inversion. The proposed algorithm is validated by the numerical experiments of both synthetic seismic data and field dataset. Introduction The non-linear waveform inversion seeks to derive subsurface physical properties such as velocity, density, Q values, and so on by minimizing iteratively the differences between the observed data and the synthetic data, usually in a least squares sense. Although waveform inversion can be implemented either in the time domain or in the frequency domain, frequency-domain waveform inversion might be more practical because the multiple shot acquisition can be simulated efficiently and the unknown source wavelet can be estimated simultaneously during waveform inversion (Pratt et al., 1998; Shin and Min, 2006). In frequency-domain waveform inversion, three inversion approaches have been tried: a multi-frequency simultaneous inversion, a sequential single-frequency inversion, and a combination of both approaches. The simultaneous inversion process requires large computation resources to handle the large number of shots or frequencies effectively. From the viewpoint of computation resources, a sequentially ordered single-frequency inversion might be most efficient. When we use a local descent approach and the initial model is far from the true model, the gradient will converge not to the global minimum but to the nearest local minimum. Especially when the low frequency information is missing in the seismic data, we cannot recover the low-wavenumber or long-wavelength of background velocity. Therefore, the previous waveform inversion researches have been carried out as a two-step process: (1) estimating the macro-model close to the true model, and (2) adding short-wavelength characteristics. Most commonly, a traveltime tomography result from the first process or a smoothed version of the true velocity model is used as a starting model for the waveform inversion as a second process (Sirgue and Pratt, 2004; Operto et al., 2006; Brenders and Pratt, 2007; Ben- Hadj-Ali et al., 2008; Bleibinhaus et al., 2008; Jaiswal et al., 2008). Recently, a novel waveform inversion technique, the waveform inversion in the Laplace and Laplace-Fourier domains, was proposed to recover a background velocity model even from the data suffering from lack of low-frequency information (Shin and Cha 2008, 2009). However, previous Laplace- and Laplace-Fourier-domain waveform inversions have been implemented as a simultaneous inversion for several Laplace damping constants and frequencies, thus requiring large computation resources. In this paper, we propose an efficient one-step waveform inversion algorithm by applying the sequential inversion to the Laplace- and Laplace-Fourier-domain waveform inversions and combining all Laplace-, Laplace-Fourier-, and conventional frequency-domain logarithmic waveform inversions.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2008-3280

... such as the amplitude, phase or the Hartley spectrum. Computations of the eigenvalue spectra of the Hessian matrix and the model resolution matrices reveal the benefits of simultaneously inverting several frequencies. We also show that (i)

**temporal****frequency**bandwidth is more important than dense spatial sampling (i.e...
Abstract

Summary Analyses of synthetic frequency-domain acoustic waveform data provide new insights into the design and imaging capability of cross-hole surveys. We show that the full complex Fourier spectral data offer significantly more information than other data representations such as the amplitude, phase or the Hartley spectrum. Computations of the eigenvalue spectra of the Hessian matrix and the model resolution matrices reveal the benefits of simultaneously inverting several frequencies. We also show that (i) temporal frequency bandwidth is more important than dense spatial sampling (i.e., many source and receiver positions), and (ii) trade-offs exist between the choice of temporal frequencies and spatial sampling strategies. Introduction Seismic tomography is a powerful and versatile tool for a wide range of imaging applications in the earth sciences. Crosshole techniques are of particular interest for shallow and intermediate target depths. The vast majority of applications reported in the literature employ ray-based methods, in which arrival times and possibly amplitudes are inverted for subsurface velocity and attenuation parameters. In the middle 1980''s full waveform inversion schemes that exploit the full information content offered by the seismic data were proposed (e.g. Tarantola, 1984; Mora, 1987). Unfortunately, the computing resources at that time did not allow these time-domain schemes to be applied to realistic problems. To ease the computational burden, a number of frequency-domain inversion schemes were proposed (e.g. Pratt, 1999; Zhou and Greenhalgh, 2003; Greenhalgh and Zhou, 2004). For such schemes only a few frequencies need to be considered during the inversions. This approach provided tomographic images that were comparable to their time-domain counterparts, but at substantially lower computer costs. The success of frequency-domain inversions is primarily based on the fact that most seismic data are band-limited, such that only a limited number of frequencies are required for characterizing the entire waveforms. There are several studies on how to set up optimized travel time tomography experiments (e.g. Curtis and Maurer, 2000), but the literature on quantitative assessments of waveform tomography experiments is sparse. This may be due to the fact that most waveform tomography algorithms employ backpropagation techniques (e.g. Mora, 1987; Pratt, 1999), which do not require the sensitivities in the Jacobian matrix to be computed explicitly. In this study, we investigate various aspects of acoustic waveform tomography. As a first, step the information content offered by different data representations is examined. Then, an eigenvalue analysis provides a means to asses the information content offered by different choices of temporal frequencies. Finally, the possible tradeoffs between spatial sampling and choice of temporal frequencies are analyzed. Results of these investigations are illustrated through the inversion of synthetic data. Acoustic waveform inversion algorithm, quality measures and experimental setup Our 2D acoustic frequency-domain waveform inversion code is based on a finite element forward solver (Zhou and Greenhalgh, 1998). For our synthetic experiments we considered a crosshole configuration involving two parallel boreholes of 30 m length and 20 m separation. A source and receiver spacing of 1 m was chosen for the initial experiments. The seven discrete frequencies of 100, 200, 500, 750, 1000, 1250 and 1500 Hz were considered.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2007-2762

... velocity model average velocity deviation phase factor geophysics parameter model horizontal wavenumber

**temporal****frequency**layer thickness Kinematically equivalent velocity distributions Alexey Stovas*, NTNU, Trondheim, Norway Summary Interpretation of the estimated traveltime parameters depends...
Abstract

ABSTRACT Interpretation of the estimated traveltime parameters depends on the chosen velocity model. The family of different velocity distributions results to the same traveltime parameters, while the freedom within the family depends on the number of traveltime parameters we estimate. To evaluate the criterion of this freedom we use the two-way propagator operator.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2006 SEG Annual Meeting, October 1–6, 2006

Paper Number: SEG-2006-2146

... numerical results in 1D are shown for several test cases. geophysics

**temporal****frequency**pratt upstream oil & gas inversion artificial intelligence reservoir characterization waveform inversion inverse propagation seismic waveform inversion equation wave velocity frequency domain...
Abstract

ABSTRACT Seismic waves are the most sensitive probe of the Earth's interior we have. With the dense data sets available in exploration, images of subsurface structures can be obtained through processes such as migration. Unfortunately, relating these surface recordings to actual Earth properties is non-trivial. Tomographic techniques use only a small amount of the information contained in the full seismogram and result in relatively low resolution images. Other methods use a larger amount of the seismogram but are based on either linearization of the problem, an expensive statistical search over a limited range of models, or both. We present the development of a new approach to full waveform inversion, i.e., inversion which uses the complete seismogram. This new method, which falls under the general category of inverse scattering, is based on a highly non-linear Fredholm integral equation relating the Earth structure to itself and to the recorded seismograms. An iterative solution to this equation is proposed. The resulting algorithm is numerically intensive but is deterministic, i.e., random searches of model space are not required and no misfit function is needed. Impressive numerical results in 1D are shown for several test cases.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2002 SEG Annual Meeting, October 6–11, 2002

Paper Number: SEG-2002-1034

..., illustrating these effects by modeling absorption for homogeneous and layered models. We find that when S and P-wave attenuation filters are compared in depth, they are exactly equal for the same Q value, in the homogeneous case. equation dispersion reservoir characterization

**temporal****frequency**...
Abstract

Summary An important practical question for multicomponent seismic surveys is how absorption impacts shear or converted wave resolution compared with that of P-waves. In this paper we undertake a comparative analysis of the expected effect of constant Q absorption on different modes, illustrating these effects by modeling absorption for homogeneous and layered models. We find that when S and P-wave attenuation filters are compared in depth, they are exactly equal for the same Q value, in the homogeneous case.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2001 SEG Annual Meeting, September 9–14, 2001

Paper Number: SEG-2001-0698

...An optimal choice of

**temporal****frequencies**for imaging: application to waveform inversion. L. Sirgue*, Ecole Normale Supérieure de Paris/ CGG R & D, R. G. Pratt, Queen s University. Summary We present a methodology defining an optimal**temporal****frequency**sequence in the reconstruction of velocity...
Abstract

ABSTRACT We present a methodology defining an optimal temporal frequency sequence in the reconstruction of velocity perturbations from a surface seismic survey. We show that, in the case of a 1-D perturbation of an otherwise homogeneous medium, this sequence depends only on the depth of the perturbation and on the offset range present in the acquisition. The main idea, derived from diffraction tomography, is that the larger the offset range, the fewer frequencies are needed. We use this approach to find a set of temporal frequencies that provides a continuous coverage of the wavenumber representation of the perturbation. Although this assumes a homogeneous background velocity field and a one-dimensional (1-D) velocity perturbation, we also show that this strategy can be effectively extended to 2-D complex structures with nonhomogeneous background media, using the Marmousi 2-D model to demonstrate this.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 1998 SEG Annual Meeting, September 13–18, 1998

Paper Number: SEG-1998-1515

... ABSTRACT No preview is available for this paper. correction

**temporal****frequency**attenuation upstream oil & gas free-surface multiple seismic data reservoir characterization geophysics signature inverse source signature source signature inverse multiple attenuation...
Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 1997 SEG Annual Meeting, November 2–7, 1997

Paper Number: SEG-1997-1100

... ABSTRACT No preview is available for this paper. reservoir characterization upstream oil & gas sinc interpolation fourier transform original trace amplitude spectrum frequency gulunay trace interpolation

**temporal****frequency**interpolator wavenumber interpolation factor...
Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 1996 SEG Annual Meeting, November 10–15, 1996

Paper Number: SEG-1996-1438

... ABSTRACT No preview is available for this paper.

**temporal****frequency**midpoint reservoir characterization transformation noise spatial domain irregularly forward model fourier transform transform domain parabolic radon transform sparse radon fourier domain noise event radon...
Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 1984 SEG Annual Meeting, December 2–6, 1984

Paper Number: SEG-1984-0812

... between the velocities determined and the actual sub- surface velocity field. The procedure is implemented in the three-dimensional frequency domain (

**temporal****frequency**, offset spatial frequency, common-midpoint spatial frequen- cy), and because not all frequency points contain seismic signal, only...