Imaging by target-oriented wave-equation inversion

Abstract

A target-oriented strategy can be applied to estimate the wave-equation least-squares inverse image (m) by explicitly computing the Hessian (H). The least-squares inverse image is obtained as the solution, using a conjugate gradient algorithm, of a non-stationary leastsquares filtering problem Hm = m mig (where m mig is the migration image, and the rows of the Hessian are non-stationary filters). This approach allows us to perform the number of iterations necessary to achieve the convergence, by exploiting the sparsity and structure of the Hessian matrix. The results on a constant velocity model and a model with a velocity Gaussian anomaly show the validity of the method.

Imaging and velocity analysis by target-oriented wavefield inversion

“…B C D Reflector A Source an approach similar to one discussed by Valenciano (2008). The image-domain formulation allows a target-oriented application of the proposed inversion technique, enabling accurate reflectivity imaging at the reservoir level.…”

“…For a finite range of frequencies and limited acquisition geometry, however, the Hessian is no longer diagonal and not even diagonally dominant (Pratt et al, 1998;Chavent and Plessix, 1999;Plessix and Mulder, 2004;Valenciano et al, 2006). It has been shown by Albertin et al (2004) and Valenciano (2008) that, in areas of poor illumination, e.g., subsalt regions, the Hessian's main diagonal energy is smeared along its off-diagonals. Therefore, a diagonal matrix has limited effect in deblurring the migrated image, especially in poorly illuminated areas.…”

“…The distortion can be quantitatively measured by the resolution function (or point-spread function) of the imaging system (Lecomte and Gelius, 1998;Gelius et al, 2002;Valenciano, 2008), which is highly non-stationary, and which has a shape that depends on the subsurface illumination. The non-stationary property of the resolution function is illustrated in Figure 1.2.…”

“…Therefore, a diagonal matrix has limited effect in deblurring the migrated image, especially in poorly illuminated areas. That is why several authors, e.g., Albertin et al (2004) and Valenciano (2008), suggest computing a limited number of the Hessian off-diagonals to compensate for poor illumination and improve the inversion/deblurring result.…”

Migration-velocity analysis using image-space generalized wavefields

“…An eventual velocity update using this unbalanced amplitude gradient can originate a velocity model that violates the smoothness assumption implied by the Born approximation. Since these amplitude variations are not related to velocity inaccuracy, we should ideally attenuate them using some sort of illumination compensation scheme (Valenciano et al, 2009;Tang, 2009). Instead, to prevent these amplitude variations we apply a B-spline smoothing to the gradient, which consists of representing the gradient as B-spline basis functions, using the adjoint operator B , and transforming it back to the Cartesian space, using the forward operator B.…”