Curvelet-domain least-squares migration with sparseness constraints.

A non-linear edge-preserving solution to the least-squares migration problem with sparseness constraints is introduced. The applied formalism explores Curvelets as basis functions that, by virtue of their sparseness and locality, not only allow for a reduction of the dimensionality of the imaging problem but which also naturally lead to a non-linear solution with significantly improved signal-to-noise ratio. Additional conditions on the image are imposed by solving a constrained optimization problem on the estimated Curvelet coefficients initialized by thresholding. This optimization is designed to also restore the amplitudes by (approximately) inverting the normal operator, which is like-wise the (de)-migration operators, almost diagonalized by the Curvelet transform. Introduction Least-squares migration and migration deconvolution has been a topic that received a recent flare of interest [9] [10]. This interest is for a good reason because inverting for the normal operator (the demigration-migration operator) restores many of the amplitude artifacts related to acquisition and illumination imprints. However, the downside to this approach is that least-squares tends to smear the energy leading to a loss of resolution. By imposing certain sparseness constraints on the imaged reflectivity, progress has been made to boost the frequency content of the image [13] [10]. This paper comes up with an alternative formulation for the imaging problem designed to (i) deal with substantial amounts of noise; (ii) use the optimal (sparse \& local) representation properties of Curvelets and their almost diagonalization of the imaging operators; (iii) use non-linear thresholding techniques, supplemented by constrained optimization on the estimated coefficients, imposing additional sparseness on the model. This paper borrows from ideas by [4] and is an extension of earlier work by [7], in which Contourlets [5] were used to denoise and approximately least-square migrate without having access to demigration operators. The paper is organized as follows. First, we briefly discuss the imaging problem and motivate why Curvelets are the appropriate choice for seismic imaging and processing. We proceed by introducing the non-linear estimation procedure with thresholding in the image space. We show that Monte-Carlo sampling techniques can be used to compute a correction for the threshold that incorporates the coloring of the noise due to migration. We conclude by introducing a constrained optimization approach aimed to (approximately, via the diagonal/symbol of the normal operator in the Curvelet domain) invert for the normal operator while imposing sparseness. The optimization is designed to reduce possible imaging and estimation artifacts, such as side-band effects, erroneous thresholding and bad illumination. The method will be illustrated by a synthetic example using a Kirchoff migration operator. The seismic imaging problem In all generality, the seismic imaging problem can after linearization and high-frequency approximation be cast into the following form for the forward model describing our data: d = Km + n (1) where K is the (Kirchoff) (de)-migration/scattering operator, the model with the reflectivity and white Gaussian noise. The pertaining inverse problem has the following general form [see e.g. 11, 14] m n J(m) + | Km d | 2 1 min : m 2 μ (2) m where J(m) is an additional penalty function that contains prior information on the model, such as particular sparseness constraints. The control-parameter μ rules how much emphasis one would like to give to the prior information on the model. How can we find the appropriate domain to solve this inverse problem? By hitting the data with the migration operator (the adjoint of the scattering operator denoted by *), followed by sandwiching the normal operator between basis-function (de)-compositions, collapses the energy onto a limited number of coefficients. Question is how to recover these coefficients from the now colored noise n ~ + m~ A~ = u ~ or n BK + Bm KB BK = Bu * * * DO FIO 8 7 6 48 47 6 ψ