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Keywords: pseudodifferential

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Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2017 SEG International Exposition and Annual Meeting, September 24–29, 2017

Paper Number: SEG-2017-17747338

... ABSTRACT We have developed a novel Q compensation approach for adjoint-based seismic imaging by

**pseudodifferential**scaling. The algorithm is stable, because it doesn’t involve amplitude amplification during wavefield extrapolations. We consider an image has correct amplitudes if, with the...
Abstract

ABSTRACT We have developed a novel Q compensation approach for adjoint-based seismic imaging by pseudodifferential scaling. The algorithm is stable, because it doesn’t involve amplitude amplification during wavefield extrapolations. We consider an image has correct amplitudes if, with the image as input, linearized Born modeling approximately produces the data. This can be achieved with the application of the inverse Hessian to the RTM image, to compensate propagation effects, including the Q effects. Pseudodifferential scaling (a dip and space dependent filter) is used to efficiently approximate the action of the inverse Hessian, and is applied to the viscoacoustic RTM image to compensate attenuation loss, and approximately recover the model perturbation. We evaluate the performance of the Q compensation using the Marmousi model. Numerical examples indicate that the adjoint RTM images with pseudodifferential scaling approximate the true model perturbation, and can be used as well-conditioned gradients for least-squares imaging. Presentation Date: Wednesday, September 27, 2017 Start Time: 2:15 PM Location: 371A Presentation Type: ORAL

Proceedings Papers

Publisher: Society of Exploration Geophysicists

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

Paper Number: SEG-2012-1262

... 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.

**pseudodifferential**hessian reservoir...
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 2009 SEG Annual Meeting, October 25–30, 2009

Paper Number: SEG-2009-2347

... amplitude algorithm reservoir characterization application upstream oil & gas syme operator multiple dip event artificial intelligence

**pseudodifferential**operator**pseudodifferential**input vector inversion dip-dependent scaling rami nammour international exposition normal...
Abstract

SUMMARY This abstract presents a computationally efficient method to approximate the inverse of the Hessian or normal operator arising in a linearized inverse problem for constant density acoustics model of reflection seismology. Solution of the linearized inverse problem problem involves construction of an image via prestack depth migration, then correction of the image amplitudes via application of the inverse of the normal operator. The normal operator acts by dip-dependent scaling of the amplitudes of its input vector. This property permits us to efficiently approximate the normal operator, and its inverse, from the result of its application to a single input vector, for example the image, and thereby approximately solve the linearized inverse scattering problem. We validate the method on a 2D section of the Marmousi model to correct the amplitudes of the migrated image. INTRODUCTION The linearized inverse scattering problem assumes a known background velocity field. The background velocity field controls the kinematics of the problem: it governs the relationship between the travel times of acoustic signals and the positions of reflectors. In the case of constant density acoustics the linearized inverse problem aims at calculating the velocity perturbation. The linearized scattering operator F approximately maps the true model m (velocity perturbation) to the measured data d (perturbation of the pressure field measured at the surface), Note that F depends on the background velocity field, but this dependence is suppressed for brevity. Interpreting (1) in a least squares sense yields the normal equations, Both linearized modeling and migration require the solution of a large scale PDE problems. The scale of these problems prohibits explicitly storing the normal operator as a matrix, and the use of direct matrix methods to invert it. Moreover, the expensive application of the normal operator limits the number of affordable iterations of iterative methods, as they require the application of the normal operator at each step. The main result of this paper is a numerically efficient approximate inversion of the normal operator, based on its most important theoretical property: the normal operator preserves the discontinuities (events) in the model m to which it is applied, provided that the background velocity is smooth and under some additional conditions (Beylkin, 1985; Rakesh, 1988; Stolk, 2000). The migrated image thus contains the same events as the model and serves as a first approximation to the real model. The disagreement between the model and migrated image is due to amplitude differences. We shall show how to correct these. Example Figure 1 the differences between the migrated image and the true model on the Marmousi benchmark model (Versteeg and Grau, 1991). The background velocity is obtained by smoothing the velocity field, and the model (velocity perturbation) is obtained as the difference between the velocity field and its smooth part (Figure 1(a)). The model is input to a finite difference time domain Born modeling code to obtain the data. Sources and receivers positions and the recording times were similar to those used in the original synthetic model (Versteeg and Grau, 1991).