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Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2009-3386

... reconstruct signal

**singular****value****decomposition**coherence seismic wave field separation apparent velocity international exposition noise attenuation separation target signal reservoir characterization**singular****value**seismic data**svd**upstream oil & gas geophysics valid signal...
Abstract

Summary Singular value decomposition(SVD)Filtering is one seismic data processing technique using the lateral coherence difference of the seismic wave to achieve wavefield separation and eliminate noises. As the differences of propagation characteristics, apparent velocity and coherence in seismic signals, transformation of the valid signals may be transformed into those with better coherence by some mathematic transformations. In this paper, based on apparent velocities, we can align target signals into those of best horizontal coherence by linear transform methods (such as NMO, linear moveout correction). By means of the singular value decomposition(SVD), we may reconstruct signal with extracting the singular value of target signals. Then inverse linear transforming, we might accomplish seismic wave field separation as well as noise attenuation afterwards. Introduction Owing to the complicated civilization circumstances in data acquisition, irregular interference signals were brought into the field data. On the other hand, different data processing methods, such as deconvolution, migration, and so on, might also introduce random noise. The separation of wave field and noise attenuation are very important for data processing, since they are the precondition and guarantee to improve the SNR(Signal to Noise Ration) of the seismic data. As a result, we have to always struggle with the irregular signals during data processing. There exit a few methods for wave field separation and noise attenuation, one of which is based on the coherence of multi-channel seismic signals and the randomicity of the noise and so-called KL filtering firstly was put forward by Jones and Levy (1987). The KL filtering is to reconstruct signals according to the eigenvalues of valid signals in the multi-channel seismic records, and the method is effective in improving SNR (signal to noise ratio). Al-Yahya(1993) applied the KL filtering in reducing random noise . While SVD filtering is to extract coherent information from the seismic records for the separation of wave field and noise attenuation based on the coherence of the seismic waves. If big singular values are brought into data reconstruction, the weak signals and coherent noise will be eliminated. In order to reject specific event, we may remove the rather big singular value to reconstruct signals. Sergio (1988) firstly applied the method of SVD in the seismic processing. Jackson(1991) analysized the principle of the SVD filtering for seismic data processing in detail. Chen Z. D. (1996) improved the SVD filter successfully. Li W. J. (2004) applied the SVD filtering to attenuate linear direct wave and refracted wave effectively, and to highlighten the primary-reflection energy and avoid the loss of valid reflection information. Niu B. H. (1999) reduced the random noise by means of the multi-channel coherent analysis to seismic data with SVD.Shen H. Y.(2008) attenuated the random noise in frequency domain via SVD . The previous works above promoted the seismic wave field separation and noise attenuation obviously. In this paper, based on the differences of propagation characteristics, apparent velocity and coherence, the seismic wave field will be separated and denoised by linear transform methods with SVD.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2010-3717

... SUMMARY We present an adaptive

**singular****value****decomposition**(**SVD**) filtering method for enhancement of the spacial coherence of the reflections and for the attenuation of the uncorrelated noise. The**SVD**filtering is performed on a small number of traces and a small number of samples collected...
Abstract

SUMMARY We present an adaptive singular value decomposition (SVD) filtering method for enhancement of the spacial coherence of the reflections and for the attenuation of the uncorrelated noise. The SVD filtering is performed on a small number of traces and a small number of samples collected around each data component. The method uses the local slope of the reflections to re-sample the data set surrounding each data component and the SVD filtering is locally applied to compute the filtered data. The filtered data component is obtained by stacking the components of the first K eigenimages along the slope. The method is applied in two steps: (i) before the SVD computation, the normal move-out (NMO) correction is applied to the seismograms, with the purpose of flattening the reflections. We use the local slopes equal to 90_ to preserve the horizontal coherence of the primary reflections and (ii) for the second step the SVD filtering uses as input the filtered data of step-1 and the method is applied in the common-offset domain. Now the local slopes of the reflections are used in order to drive the SVD filtering. We illustrate the method using land seismic data of the Tacutu basin, located in the Northeast of Brazil. The results show that the proposed method is effective and is able to reveal reflections masked by the ground-roll. Introduction Singular value decomposition (SVD) is a coherency-based technique that provides both signal enhancement and noise suppression. It has been implemented in a variety of seismic applications. Freire and Ulrych (1988) apply the SVD filtering to the separation of upgoing and downgoing waves in vertical seismic profiling. Tyapkin et al. (2003) proposed to use the data alignment method of Liu (1999) to make the coherent noise horizontally aligned in one or more time sections of a common shot gather. The noise is represented by the first eigenimages and the remaining eigenimages represent the signal. Chiu and Howell (2008) proposed a method that uses SVD to compute eigenimages that represent coherent noise in a localized time-space windows. The data in the local windows is transformed into analytic signal and followed by a complex SVD to decompose the analytic signal into eigenimages that represent the coherent noise model. Melo et al. (2009) presented a filtering method for ground-roll attenuation that uses a 2-D time-derivative filter. Bekara and Baan (2007) proposed a local SVD approach to noise removal. In each data window the signal is horizontally aligned in time, and after SVD only the first eigenimage is retained. Then the procedure is repeated in the next data window using sliding windows with 50% overlap. Porsani et al. (2009) use SVD filtering to attenuate the ground roll. Before the SVD computation, normal move-out (NMO) correction is applied to the seismograms, with the purpose of flattening the reflections. SVD is performed on a small number of traces in a sliding window filtering approach. The output trace is the central trace of the first few eigenimages.

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-2016-13866459

... ABSTRACT Normally the

**singular****value****decomposition**(**SVD**) filtering is applied in the tx (time x distance) domain, exploring the spatial correlation between a set of seismic traces. In this paper we present a single trace**decomposition**approach based on the**SVD**method. This approach works on...
Abstract

ABSTRACT Normally the singular value decomposition (SVD) filtering is applied in the tx (time x distance) domain, exploring the spatial correlation between a set of seismic traces. In this paper we present a single trace decomposition approach based on the SVD method. This approach works on single traces decomposing each seismic trace based on the temporal correlation of the reflected events. We illustrate the method in the prediction and removal of the ground-roll. Presentation Date: Tuesday, October 18, 2016 Start Time: 8:50:00 AM Location: 140 Presentation Type: ORAL

Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2011-2694

..., we apply

**singular**-**value****decomposition**(**SVD**) to evaluate pseudo-inverse solution. Introducing**SVD**into MDD opens the way of interpreting the effect of the source-receiver configuration in the inversion procedure by linear mapping theory. We numerically simulate the wavefield with two-dimensional...
Abstract

ABSTRACT Multidimensional deconvolution (MDD) is an alternative method for seismic interferometry which retrieves new wavefield with desired source-receiver configuration from observed wavefield without source information. Because this method involves inverse problems to estimate new wavefield, we apply singular-value decomposition (SVD) to evaluate pseudo-inverse solution. Introducing SVD into MDD opens the way of interpreting the effect of the source-receiver configuration in the inversion procedure by linear mapping theory. We numerically simulate the wavefield with two-dimensional homogeneous model and investigate the rank of the data kernel of inverse problem for MDD. The sparse source distribution and the dense source distribution show almost same number of rank and also retrieve same wavefield when the spatial distribution is identical. Therefore analyzing the rank of the data kernel of inverse problem can be used for the determination of optimum source distribution. Furthermore, we show that the ambiguity of the wavefield which is inferred from the model resolution matrix constructed by the matrices from SVD showed the same trend with the discrepancy of the inverted wavefield from true wavefield. Therefore the evaluation of the reliability of the inverted wavefield could be possible by evaluating the model resolution matrix.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2009-1688

... the conventional crosscorrelation method (Wapenaar et al., 2008a; Wapenaar et al., 2008b). We apply MDD to crosswell geometry in order to retrieve crosswell impulse responses from surface sources using numerical modeling and field data. We adopted

**singular**-**value****decomposition**(**SVD**) for obtaining the...
Abstract

Introduction Summary Seismic interferometry is a process of generating new seismic data from existing wavefields. This enables us to expand the degree of freedom of source-receiver configuration. Seismic interferometry by multidimensional deconvolution (MDD) is proposed as an alternative to the conventional crosscorrelation method (Wapenaar et al., 2008a; Wapenaar et al., 2008b). We apply MDD to crosswell geometry in order to retrieve crosswell impulse responses from surface sources using numerical modeling and field data. We adopted singular-value decomposition (SVD) for obtaining the pseudoinverse solution to achieve MDD. Since the SVD pseudoinverse is highly dependent on the rank of the MDD matrix, Akaike''s information criterion (AIC) is adopted in order to determine the rank of the MDD matrix. We see that the MDD produces higher-resolution data compared with the conventional crosscorrelation method. Furthermore, amplitudes of downgoing reflection events are improved in MDD while downgoing reflection events are not recognizable in the crosscorrelation method. Seismic interferometry can be defined as the process of generating new seismic data from the crosscorrelation of existing wavefields from controlled sources around receivers of interest. The literature on interferometric techniques has grown spectacularly in recent years (e.g., Wapenaar et al., 2004; Wapenaar et al., 2006; Schuster et al., 2004; Snieder 2004; Bakulin and Calbert 2006). We have studied the application of seismic interferometry to crosswell geometry (Minato et al., 2007). In this application, the receiver arrays are placed in vertical boreholes and the controlled sources are placed along the surface. From this configuration, crosswell wavefields can be retrieved using seismic interferometry. Therefore, there is a possibility to perform crosswell seismic reflection without physical borehole sources. Furthermore, this method enables us to expand the investigation area using high-energy sources on the surface. Recently, seismic interferometry by multidimensional deconvolution (MDD) has been proposed as an alternative to the conventional crosscorrelation method (Wapenaar et al., 2008a; Wapenaar et al., 2008b). Advantages of seismic interferometry by MDD are that it compensates for the characteristics of the source wavelet, that may compensate for inhomogeneous source distribution, and that it is valid in dissipative media (the crosscorrelation method assumes lossless medium). In this study, we show that the crosswell seismic reflection method can be performed without borehole sources using seismic interferometry by MDD and that the amplitudes and resolution of imaged reflection boundaries are improved compared with results from the conventional crosscorrelation method. These applications were examined using numerical modeling and field data. Because our source-receiver configuration was an ill-posed problem for solving interferometry relation, we adopted an SVD pseudoinverse solution to achieve multidimensional deconvolution. Furthermore, we applied AIC (Akaike''s information criterion) in order to determine the available number of singular values. Numerical modeling results Data acquisition We applied seismic interferometry by MDD (equation 4) to numerical-modeling data acquired in two parallel boreholes from transient surface sources. The velocity model was inspired from logging P-wave velocity of our field data discussed in the following section. Two vertical boreholes are placed: well-1 (left well) and well-2 (right well) are represented as the receiver arrays in Figure 1 (triangles).

Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2013-1234

... events, with low frequency, and conserves the shape and frequency of sub-horizontal events. The

**Singular****Value****Decomposition**(**SVD**) can be applied as a horizontal coherence filter, usually being used after the NMO correction, in the XT domain. This paper discusses the joint of the radial transform with...
Abstract

SUMMARY The Radial Transform is a technique that maps the amplitudes of the seismogram from the usual distance-time domain (XT) to the angle-time, or velocity-time domain (RT). This rearrangement of the data turns linear events, as ground roll, direct waves and refracted waves into vertical events, with low frequency, and conserves the shape and frequency of sub-horizontal events. The Singular Value Decomposition (SVD) can be applied as a horizontal coherence filter, usually being used after the NMO correction, in the XT domain. This paper discusses the joint of the radial transform with the SVD power method for ground roll filtering.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2006-2584

...

**decomposition**upstream oil & gas**svd****value****decomposition**seismic migration fir digital filter mcclellan transformation implementation extrapolation reservoir characterization impulse response convolution fir digital filter parallel section**singular****value**digital filter transformation...
Abstract

ABSTRACT In this paper, we propose a new scheme for implementing predesigned two-dimensional (2-D) complex-valued explicit depth migration fi nite impulse response (FIR) digital filters, which are used for three dimensional (3-D) migration. The implementation is based on Singular Value Decomposition (SVD) of such quadrantal symmetrical 2-D FIR filters. In order to simplify the SVD computations for such impulse response structure, we apply a special matrix transformation on the migration FIR filters where we guarantee the retention of their wavenumber phase response. Unlike the existing explicit depth implementation methods, this implementation via SVD results in perfect circularly symmetrical responses. It also saves 53.3% of the total implementation cost per output sample when compared to direct implementation with symmetry via true 2-D convolution.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2003 SEG Annual Meeting, October 26–31, 2003

Paper Number: SEG-2003-1398

... Summary In this paper we present a new method for time-lapse signal enhancement using

**singular****value****decomposition**(**SVD**).**Singular****value****decomposition**is used to separate a 4D signal into its constituent parts: common geology, time-lapse response and noise. This signal enhancement technique...
Abstract

Summary In this paper we present a new method for time-lapse signal enhancement using singular value decomposition (SVD). Singular value decomposition is used to separate a 4D signal into its constituent parts: common geology, time-lapse response and noise. This signal enhancement technique is used to map out both the original and moved oil-water contact across the Nelson field. The SVD technique allows the oil-water contact (OWC) to be mapped across regions which would have been missed using traditional methods. The oil-water contact is observed to move upwards across the field with the largest movements being associated as anticipated with natural production. The results obtained are consistent with those predicted by the reservoir simulator model. Singular value decomposition is demonstrated to be a useful tool for enhancing the time-lapse signal and for gaining confidence in areas where traditional differencing fails.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2003 SEG Annual Meeting, October 26–31, 2003

Paper Number: SEG-2003-2044

.... Conventional processing methods, such as the high- pass, f – k and hyperbolic velocity filters, however, have certain disadvantages when handling actual seismic data. We present a new hybrid method combining the

**singular****value****decomposition**(**SVD**) with a special linear transformation of the common- shot...
Abstract

Summary Source-generated noise, such as air, refracted, guided waves, near-surface multiples, and ground roll, is one of the most challenging problems in the land seismic method. To suppress the noise, geophysicists have devised various techniques in both acquisition and processing stages. Conventional processing methods, such as the high- pass, f – k and hyperbolic velocity filters, however, have certain disadvantages when handling actual seismic data. We present a new hybrid method combining the singular value decomposition (SVD) with a special linear transformation of the common- shot gather. The method is aimed at effectively removing the noise while minimizing harm to the signal. As compared with other methods, the SVD- based one gives a denser approximation to source-generated noise before its subtraction from the seismic data, due to the use of more appropriate basis functions.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2012-0119

... sweep of the encoder. This harmonic distortion reduces the quality of raw vibroseis data. Here, we present a new approach to eliminate the integer and fractional harmonic inferences produced by the coupling between base plate and ground. The method is based on

**singular****value****decomposition**(**SVD**) in time...
Abstract

Summary Distortion exists in most vibrator operations on land because nonlinear effects cause the actual base-plate signal to differ from that of the pilot sweep of the encoder. This harmonic distortion reduces the quality of raw vibroseis data. Here, we present a new approach to eliminate the integer and fractional harmonic inferences produced by the coupling between base plate and ground. The method is based on singular value decomposition (SVD) in time frequency (TF) domain. The process is implemented in the base-plate signal and the uncorrelated data, and harmonic interference can be suppressed after correlation. Application of this method to actual data from Western China shows it to be capable of improving the quality of vibroseis data. In particular, this method helps increase the signal to noise ratio (SNR) and resolution of small-offset vibroseis data.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2012-0057

.... This paper analyzes the cause that leads to ill-conditioned matrix during data separation, introduces the basic principles of the

**singular****value****decomposition**(**SVD**) method and gives a real case of data separation using the method. Other methods such as Gauss-Jordan elimination method can also achieve...
Abstract

Summary Using different phase encoding schemes, a composite record comprising of different single-shot data can be acquired at different locations through simultaneous sweep with multiple vibrators. There are many ways to decompose this composite record into independent single-shot records. This paper analyzes the cause that leads to ill-conditioned matrix during data separation, introduces the basic principles of the singular value decomposition(SVD) method and gives a real case of data separation using the method. Other methods such as Gauss-Jordan elimination method can also achieve data separation; however, comparison of the data separation results between the singular value decomposition method and Gauss-Jordan elimination method in this paper shows that compared to the later, the S/N ratio of the single-shot record obtained with the former is much higher. A 90-degree phase shift of the original data is performed using Hilbert transform before data separation operation.

Journal Articles

Journal:
SPE Journal

Publisher: Society of Petroleum Engineers (SPE)

*SPE Journal*15 (02): 495–508.

Paper Number: SPE-118952-PA

Published: 17 December 2009

... of individual sensitivity coefficients. Explicit computation of the sensitivities is avoided by using a partial

**singular****value****decomposition**(**SVD**) based on a form of the Lanczos algorithm. At least for all synthetic problems that we have considered, the reliability, computational efficiency, and...
Abstract

Summary In gradient-based automatic history matching, calculation of the derivatives (sensitivities) of all production data with respect to gridblock rock properties and other model parameters is not feasible for large-scale problems. Thus, the Gauss-Newton (GN) method and Levenberg-Marquardt (LM) algorithm, which require calculation of all sensitivities to form the Hessian, are seldom viable. For such problems, the quasi-Newton and nonlinear conjugate gradient algorithms present reasonable alternatives because these two methods do not require explicit calculation of the complete sensitivity matrix or the Hessian. Another possibility, the one explored here, is to define a new parameterization to radically reduce the number of model parameters. We provide a theoretical argument that indicates that reparameterization based on the principal right singular vectors of the dimensionless sensitivity matrix provides an optimal basis for reparameterization of the vector of model parameters. We develop and illustrate algorithms based on this parameterization. Like limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS), these algorithms avoid explicit computation of individual sensitivity coefficients. Explicit computation of the sensitivities is avoided by using a partial singular value decomposition (SVD) based on a form of the Lanczos algorithm. At least for all synthetic problems that we have considered, the reliability, computational efficiency, and robustness of the methods presented here are as good as those obtained with quasi-Newton methods.

Proceedings Papers

Paper presented at the International Petroleum Technology Conference, December 7–9, 2009

Paper Number: IPTC-14042-MS

... a

**singular****value****decomposition**(**SVD**) of the representative azimuthal AVO responses for the target reservoir. By means of**singular****value****decomposition**it is possible to calculate seismic attributes that provide a direct mapping with fracture density. This mapping is then used in the inversion of...
Abstract

Abstract This work presents a pilot study of the application of a seismic inversion technique for fracture density and fracture orientation in an area where available well data revealed the presence of open fractures. The inversion method is based on a singular value decomposition of azimuthal AVO data. This decomposition allows us to calculate seismic attributes which are linked to the fracture density of the fracture network through anisotropic rock physics modeling. The outcome of this pilot study is promising since the obtained results are consistent with existing image log data; however, further testing and research on a larger area are needed to assess if the fracture characterization results are consistent with geological interpretations and other analysises of fracture networks in the reservoir. Introduction The characterization of fracture networks in naturally fractured reservoirs can potentially play an important role in the optimization and enhancement of hydrocarbon production. Accurate maps of fracture density can aid in identifying areas of high occurrence of fractures and sweet spots. However, the determination of fracture density is a difficult and challenging problem that often involves the integration of different types of data at different scales. At the log scale, image logs and dipole sonic logs are employed to characterize fractures locally, while at the seismic scale azimuthal AVO and shear wave splitting techniques are used in seismic fracture characterization studies. In this work we study the application of an inversion method to obtain the distribution of fracture density and fracture orientation in a pilot study covering a small area of a tight gas sand reservoir known to be fractured at least at one well location. The inversion method is based on a singular value decomposition (SVD) of the representative azimuthal AVO responses for the target reservoir. By means of singular value decomposition it is possible to calculate seismic attributes that provide a direct mapping with fracture density. This mapping is then used in the inversion of fracture density. The singular value decomposition method was originally implemented by Causse and coauthors for conventional AVO inversion (Causse et al., 2007). Varela and coauthors extended this methodology to invert for fracture density using azimuthal AVO and applied it both to synthetic data (Varela et al., 2007) and to laboratory experiments in a geological scale model (Varela et al., 2009). It is worthwhile to mention that one of the main advantages of this inversion method is that it is based on exact solutions of the AVOAz response calculated according to a full anisotropic modelling of the fractured reservoir and does not rely on the standard AVOAz approximations (Ruger et al., 1997). The seismic modelling in this work is based on the anisotropic rock physics model of Chapman (Chapman, 2003).

Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2007-0259

... the amplitude changes with azimuth in terms of principal components, which are calculated via

**singular****value****decomposition**(**SVD**). We present the method and illustrate its application to surface seismic with a synthetic example of multi-azimuth common-offset gathers with different fracture densities...
Abstract

ABSTRACT We develop a technique for inversion of fracture properties from azimuthal AVO. The method is based on the decomposition and reconstruction of the amplitude changes with azimuth in terms of principal components, which are calculated via singular value decomposition (SVD). We present the method and illustrate its application to surface seismic with a synthetic example of multi-azimuth common-offset gathers with different fracture densities.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2008-3244

... SUMMARY Tomographic velocity model building has become an industry standard for depth migration. Anisotropy of the Earth challenges tomography because the inverse problem becomes severely ill-posed.

**Singular****value****decomposition**(**SVD**) of tomographic operators or, similarly, eigendecomposition...
Abstract

SUMMARY Tomographic velocity model building has become an industry standard for depth migration. Anisotropy of the Earth challenges tomography because the inverse problem becomes severely ill-posed. Singular value decomposition (SVD) of tomographic operators or, similarly, eigendecomposition of the corresponding normal equations, are well known as a useful framework for analysis of the most significant dependencies between model and data. However, application of this approach in velocity model building has been limited, primarily because of the perception that it is computationally prohibitively expensive, especially for the anisotropic case. In this paper, we extend our prior work (Osypov et al., 2008) to VTI tomography, modify the process of regularization optimization, and propose an updated way for uncertainty and resolution quantification using the apparatus of eigendecomposition. We demonstrate the simultaneous tomographic estimation of VTI parameters on a real dataset. Our approach provides extra capabilities for regularization optimization and uncertainty analysis in anisotropic model parameter space which can be further translated into the structural uncertainty within the image. INTRODUCTION Velocity model building is one of the most challenging problems in modern depth imaging. 3D tomographic analysis has become the key technology to tackle this problem (Woodward et al., 1998). However, the tomographic inverse problem is ill-posed, which leads to big uncertainties and ambiguities in the reconstruction. The necessity to account for anisotropy in seismic velocities complicates this issue even further (Stunff et al., 1999; Grubb et al., 2001; Liu et al., 2004; Zhou et al., 2004). To circumvent this, various forms of regularization are used (e.g. (Tikhonov and Arsenin, 1977; Scales, 1987; Osypov and Scales, 1996; Yao et al., 1999)). However, regularizing tomography still remains a subjective virtue, if not black magic. An alternative way to solving the inverse problem by means of a Bayesian framework for model estimation requires knowledge of prior information and data uncertainty (Tarantola, 1987; Scales and Tenorio, 2001). For 1-D problems, Malinverno and Parker (2006) proposed using an empirical Bayes approach, when model priors and data uncertainties are considered as hyperparameters to be optimized. In general, for seismic 3D seismic tomography, prior information can come from other geophysical and borehole data, or more often as a geoscientist''s input (Bear et al., 2005). In the latter approach it becomes difficult to rigorously quantify this input in terms of probability functions. Singular value decomposition (SVD) provides an elegant framework for analysis of most significant dependencies between model and data. Truncated SVD can be used as a form of regularization (Jackson, 1972; Lines and Treitel, 1984; Scales et al., 1990) and resolution-covariance trade-off can be analyzed (Kalscheuer and Pedersen, 2007). For big problems when direct SVD solution is not computationally feasible, various iterative schemes for partial SVD have been proposed, first pioneered by Lanczos (1950) and Arnoldi (1951), and then developed further by Parlett (1980), Cullum and Willoughby (1986), Sorensen (1992), to name a few. Instead of SVD of the tomographic operator one can do eigendecomposition of the corresponding symmetric positive-definite matrix from normal equations.

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*Journal of Canadian Petroleum Technology*34 (05): 55–62.

Paper Number: PETSOC-95-05-07

Published: 01 May 1995

... reservoir boundary and the appropriate boundaryconditions are written (no flow or isopotential). This leads to M linear, equations In N unknowns, which can be conveniently solved, using the

**Singular****Value****Decomposition**(**SVD**) technique for the unknown rates-of the image wells. The iso-potentials, streamlines...
Abstract

A new method is proposed to obtain streamline distribution in a horizontalplane of a homogeneous reservoir. The method is a generalization of theclassical method of images used to obtain the potential and velocitydistribution in a symmetric pattern flood. However, it is extremely difficultto use the classical method of images, as shapes of reservoirs and locations ofwells are generally arbitrary in nature. Further, methods based on, finitedifference techniques are difficult to implement, because they requiresubstantial post-processing for front tracking and cannot handle singularitiesnear the well. In the proposed method, a large number of image wells (N) are arbitrarilylocated outside the reservoir boundary, which can be either sealing? type oruniform pressure (flow potential type). A large number of points M, (M>N)are selected along the reservoir boundary and the appropriate boundaryconditions are written (no flow or isopotential). This leads to M linear, equations In N unknowns, which can be conveniently solved, using the Singular Value Decomposition (SVD) technique for the unknown rates-of the image wells. The iso-potentials, streamlines and single-fluid fronts can then be obtainedusing, the principle of superposition. Introduction Knowledge of fluid front locations at various times plays a key role indesigning a suitable development plan for better exploitation of oilreservoirs. Depending on the nature of the drive mechanism, the problem may beone of suggesting a suitable pattern of wells for pressure maintenance in adepletion-drive pool or of suggesting locations of production wells in awater-drive reservoir to achieve a uniform movement of water-oil contact tomaximize areal sweep. The streamtube model was first introduced by Higgins and Leighton (1) . They described a computational technique to predict twophase flow through streamtubes generated from a potentiometric model for afive-spot pattern. The streamtubes (which lie between single-fluid streamlines)are divided into cells of equal volume, and average penneabilities andgeometrical shape factors determined for each cell are used to compute a Buckley-Leverett displacement. The overall performance is obtained by summingup the production from all streamtubes. Morel Seytoux 2 demonstratedthe use of conformal mapping techniques to obtain potential distribution insymmetry elements of staggered line drive and direct line drive patternsassuming unit-mobility ratio displacement. Abbaszadeh-Dehghani and Brigham (3) presented use of conformal mapping technique to obtainpotential and velocity distributions for symmetric pattern floods assuming unitmobility ratio displacements. In addition, they performed tracer dispersioncalculations in individual streamtubes as the injected tracer slug propagatesin the reservoir. Le Blanc and Caudle (4) suggested use of a methodof images technique to handle arbitrary shaped reservoirs, However. theirmethod involves a trial and error approach. Martin and Wegner (5) utilized a finite difference model instreamtube-isobar coordinate system with streamtube locations beingperiodically updated. Their paper presented justification to the assumptionmade by Higgins and Leighton, and Le Blanc and Caudle that waterfloodperformance can be calculated by holding streamlines constant as the floodfront progresses.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2010-3539

... method. Bear in mind that the acceleration strategies for computing the

**singular****value****decomposition**(**SVD**) that are proposed in this abstract are applicable to any rank reduc- tion filtering method. The method that we propose is based on the work of Rokhlin et al. (2009) and entails computing the**SVD**of...
Abstract

SUMMARY Multichannel random noise attenuation via rank reduction methods require the estimation of the singular values and vectors of large matrices. In general, this leads to algorithms that are computationally less attractive than classical methods for seismic noise attenuation based on f -x deconvolution. In order to make rank reduction methods more efficient, we investigate algorithms for fast estimation of the singular values and singular vectors. We study the problem of estimating the eigen-spectra of large matrices using randomized singular value decomposition. In particular, we apply a randomized singular value decomposition method to estimate rank reduced Hankel matrices that arise in multichannel singular spectrum analysis noise attenuation. INTRODUCTION The attenuation of random and coherent noise in seismic records is an important subject in seismic data processing. Classical methods for random noise attenuation exploit the predictability of the seismic signal in small spatio-temporal windows. Examples of the aforementioned concepts are f -x deconvolution (Canales, 1984) and t -x prediction error filtering methods (Abma and Claerbout, 1995). Another category of methods rely on rank reduction techniques to decompose a window of seismic data in coherent and incoherent components (Ulrych et al., 1999). Examples in this category abound in the geophysical literature. Freire and Ulrych (1988), for instance, proposed to carry out rank reduction of seismic images in the t - x domain via the so-called eigen-image decomposition. They also showed the connection of the eigen-image method to the Karhunen-Lo´eve transform (KL) and principal component analysis (PC) methods (Jones and Levy, 1987; Al-Yahya, 1991). The eigen-image approach works well for horizontal events. Chiu and Howell (2008) and Cary and Zhang (2009) extended this idea for the elimination of ground roll. For this purpose the offending event (ground roll) is flattened via a linear moveout correction. A rank reduction method that is independent of dip, and therefore, does not require flattening, has been proposed by Mari and Glangeaud (1990). This type of method operates in the f -x domain and requires the eigen-decomposition of the spectral matrix of the data. Connected to the latter are Cazdow filtering (Cadzow, 1988; Trickett, 2008) and multichannel singular spectrum analysis (MSSA) (Vautard et al., 1992; Sacchi, 2009; Oropeza and Sacchi, 2009). MSSA and Cazdow filtering are equivalent. They arise, however, from different signal analysis subfields. For instance, Cadzow method was proposed as a general framework for denosing images, whereas MSSA has been proposed to decompose time series arising in the study of dynamical systems (Broomhead and King, 1986). In this abstract we will consider filtering incoherent noise via the MSSA method. Bear in mind that the acceleration strategies for computing the singular value decomposition (SVD) that are proposed in this abstract are applicable to any rank reduction filtering method. The method that we propose is based on the work of Rokhlin et al. (2009) and entails computing the SVD of randomly compressed data matrices. In essence, we replace the SVD of large matrices by the SVD of two reduced matrices.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2011-3622

... Toeplitz matrix and rank re- duce this matrix via the Lanczos bidiagonalization algorithm, rather than using

**Singular****Value****Decomposition**(**SVD**). The computational cost of the Lanczos bidiagonalization is dom- inated by the cost of multiplying a block Toeplitz matrix by a vector. The latter can be...
Abstract

ABSTRACT Rank reduction strategies can be employed to attenuate noise and as a basic template for pre-stack regularization of seismic data. We propose to utilize the rank reduction method of Multichannel Singular Spectrum Analysis (MSSA) to implement a fast 5D seismic data reconstruction method by embedding 4D spatial data into a block Toeplitz matrix and rank reduce this matrix via the Lanczos bidiagonalization algorithm, rather than using Singular Value Decomposition (SVD). The computational cost of the Lanczos bidiagonalization is dominated by the cost of multiplying a block Toeplitz matrix by a vector. The latter can be efficiently implemented via multidimensional Fast Fourier Transforms. The proposed algorithm significantly decreases the computational cost of the rank-reduction stage needed for de-noising and reconstruction with respect to algorithms that utilize the Singular Value Decomposition (SVD). In essence, our algorithm exploits the structure of block Toeplitz matrices to accelerate the rank-reduction step of de-noising and reconstruction strategies used by Multichannel Singular Spectrum Analysis (MSSA) or Cadzow matrix completion methods. Synthetic data examples and a field data test were used to examine the proposed algorithm.

Journal Articles

Journal:
SPE Journal

Publisher: Society of Petroleum Engineers (SPE)

*SPE Journal*1 (01): 3–9.

Paper Number: SPE-26428-PA

Published: 01 March 1996

... the

**singular****value****decomposition**(**SVD**) methods. The minimization of the L1-norm included the least absolute**values**(LAV), the modified least absolute**values**(MLAV), the combination MLAV - LAV and the Nelder-Mead (polytope) method. Important characteristics of these were discussed, such as: constrained...
Abstract

This paper presents a comparison of several nonlinear parameter estimation methods for automated well test analysis. Methods based on either the L2-norm or the L1-norm were considered. The techniques based on the minimization of the L2-norm were the least squares (LS) and the singular value decomposition (SVD) methods. The minimization of the L1-norm included the least absolute values (LAV), the modified least absolute values (MLAV), the combination MLAV - LAV and the Nelder-Mead (polytope) method. Important characteristics of these were discussed, such as: constrained optimization, statistical analysis of the estimation results and error probability distributions. A new approach to constrain the optimization process was proposed and a modification in the original domains of the parameters to be estimated was also proposed in order to avoid divergence. An analysis of the use of pressure data, pressure derivative data and pressure + pressure derivative data as model functions was performed. Introduction Automated well test analysis is today an important and commonly used tool for formation evaluation. This technique uses numerical methods to solve the inverse problem of estimating reservoir parameters from the analysis of pressure and rate data. In the present work the performances of six basic methods are compared: least squares (LS), singular value decomposition (SVD), least absolute values (LAV), modified least absolute values (MLAV) the combination MLAV-LAV and the Nelder-Mead method. The first two methods (LS and SVD) are based on L2-norm and minimize the sum of the squared residuals. The last four methods either minimize the sum of the absolute values of the residuals or are based on the L1-norm. The performances of the various methods were compared using four examples of application: two sets of field data from a homogeneous single porosity reservoir, one synthetic data set and one field data set from a homogeneous double-porosity reservoir. P. 125^

Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2010-3986

... gathers by applying

**Singular****Value****Decomposition**(**SVD**) to the crosscorelograms before stacking. The**SVD**approach preserves energy that is stationary in the crosscorrelogram, thus enhancing energy from sources in stationary positions, which interfere constructively, and attenuating energy from non...
Abstract

SUMMARY Seismic interferometry is a technique used to estimate the Green’s function (GF) between two receiver locations, as if there were a source at one of the locations. By crosscorrelating the recorded seismic signals at the two locations we generate a crosscorrelogram. Stacking the crosscorrelogram over sources generates an estimate of the inter-receiver GF. However, in most applications, the requirements to recover the exact GF are not satisfied and stacking the crosscorrelograms yields a poor estimate of the GF. For these non-ideal cases, we enhance the real events in the virtual shot gathers by applying Singular Value Decomposition (SVD) to the crosscorelograms before stacking. The SVD approach preserves energy that is stationary in the crosscorrelogram, thus enhancing energy from sources in stationary positions, which interfere constructively, and attenuating energy from non-stationary sources that interfere distructively. We apply this method to virtual gathers containing the virtual refraction artifact and find that using SVD enhances physical arrivals. We also find that SVD is quite robust in recovering physical arrivals from noisy data when these arrivals are obscured by or even lost in the noise in the standard seismic interferometry technique.

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