Peyman P. Moghaddam

Curvelet-based seismic data processing: A multiscale and nonlinear approach

Mitigating missing data, multiples, and erroneous migration amplitudes are key factors that determine image quality. Curvelets, little “plane waves,” complete with oscillations in one direction and smoothness in the other directions, sparsify a property we leverage explicitly with sparsity promotion. With this principle, we recover seismic data with high fidelity from a small subset (20%) of randomly selected traces. Similarly, sparsity leads to a natural decorrelation and hence to a robust curvelet-domain primary-multiple separation for North Sea data. Finally, sparsity helps to recover migration amplitudes from noisy data. With these examples, we show that exploiting the curvelets ability to sparsify wavefrontlike features is powerful, and our results are a clear indication of the broad applicability of this transform to exploration seismology.

Curvelet-based migration preconditioning and scaling

The extremely large size of typical seismic imaging problems has been a major stumbling block for iterative techniques to attain accurate migration amplitudes. These iterative methods are important because they complement theoretical approaches hampered by difficulties controlling problems such as finite-acquisition aperture, source-receiver frequency response, and directivity. To solve these problems, we apply preconditioning, which significantly improves convergence of least-squares migration. We discuss different levels of preconditioning: corrections for the order of the migration operator, corrections for spherical spreading, and position- and reflector-dip-dependent amplitude errors. Although the first two corrections correspond to simple scalings in the Fourier and physical domain, the third correction requires phase-space (space spanned by location and dip) scaling, which we carry out with curvelets. Our combined preconditioner significantly improves the convergence of least-squares wave-equation ...

Industrial-Scale Reverse Time Migration on GPU Hardware

We give an overview of an implementation of Reverse Time Migration (RTM) on heterogeneous multi-core hardware based on Graphics Processing Units (GPUs). We demonstrate the clear advantages of GPU-based hardware for RTM, not only in terms of performance for small to mid-scale problem sizes, but also for imaging 3D volumes on a scale appropriate to full Wide-Azimuth (WAZ) or Rich-Azimuth (RAZ) surveys in areas such as the Gulf of Mexico. We also point to advantages of GPU-based hardware in terms of energy efficiency and datacenter footprint.

Randomized full‐waveform inversion: a dimenstionality‐reduction approach

Full-waveform inversion relies on the collection of large multiexperiment data volumes in combination with a sophisticated back-end to create high-fidelity inversion results. While improvements in acquisition and inversion have been extremely successful, the current trend of incessantly pushing for higher quality models in increasingly complicated regions of the Earth reveals fundamental shortcomings in our ability to handle increasing problem sizes numerically. Two main culprits can be identified. First, there is the so-called “curse of dimensionality” exemplified by Nyquist’s sampling criterion, which puts disproportionate strain on current acquisition and processing systems as the size and desired resolution increases. Secondly, there is the recent “departure from Moore’s law” that forces us to develop algorithms that are amenable to parallelization. In this paper, we discuss different strategies that address these issues via randomized dimensionality reduction.

Curvelet-based Migration Preconditioning

The extreme large size of typical seismic imaging problems has been one of the major stumbling blocks for a successful application of iterative techniques from numerical linear algebra to attain accurate migration amplitudes. These iterative methods are important because they complement theoretically-driven approaches that are hampered by mundane differences to control problems such as finite-acquisition aperture, source-receiver frequency response, and directivity. To solve this problem, we apply the well-know technique of preconditioning that significantly increases the convergence of iterative solvers, making least-squares migration more tangible. First, we discuss different levels of preconditioning that range from corrections for the order of the migration operator to corrections for spherical spreading and position and reflector-dip dependent amplitude errors. While the first two corrections correspond to simple scalings in the Fourier and physical space, the third correction requires an intricate phase-space scaling, which we carry out with curvelets. Aside from providing the appropriate domain for the scaling, curvelets have the additional Seismic Laboratory for Imaging and Modeling, Department of Earth and Ocean Sciences, University of British Columbia, 6339 Stores Road, Vancouver, V6T 1Z4, BC, Canada.

Seismic imaging and processing with curvelets

B030 Seismic Imaging and Processing with Curvelets F.J. Herrmann* (University of British Columbia) G. Hennenfent (EOS-UBC) & P.P. Moghaddam (EOS-UBC) SUMMARY In this paper we present a nonlinear curvelet-based sparsity-promoting formulation for three problems in seismic processing and imaging namely seismic data regularization from data with large percentages of traces missing; seismic amplitude recovery for subsalt images obtained by reverse-time migration and primary-multiple separation given an inaccurate multiple prediction. We argue why these nonlinear formulations are beneficial. EAGE 69 th Conference & Exhibition — London UK 11 - 14 June 2007 In this paper we report recent developments on the

Seismic data processing with curvelets: a multiscale and nonlinear approach.

In this abstract, we present a nonlinear curvelet-based sparsitypromoting formulation of a seismic processing flow, consisting of the following steps: seismic data regularization and the restoration of migration amplitudes. We show that the curvelet’s wavefront detection capability and invariance under the migration-demigration operator lead to a formulation that is stable under noise and missing data.

Migration preconditioning with curvelets.

In this paper, the property of Curvelet transforms for preconditioning the migration and normal operators is investigated. These operators belong to the class of Fourier integral operators and pseudo-differential operators, respectively. The effect of this preconditioner is shown in term of improvement of sparsity, convergence rate, number of iteration for the Krylov-subspace solver and clustering of singular(eigen) values. The migration operator, which we employed in this work is the common-offset Kirchoff-Born migration.

Seismic Amplitude Recovery with Curvelets

A shock absorber includes a shock rod and piston which are disposed within a fluid chamber within a shock body. The piston separates the shock body fluid chamber into a compression fluid chamber and a rebound fluid chamber. A reservoir fluid chamber accommodates the entry of the shock rod into the fluid chamber as the shock absorber compresses under shock forces. The compression fluid chamber is in fluid communication with the reservoir fluid chamber through a third chamber or passage. A first valve may control passage of fluid from the compression fluid chamber into the passage. Second and third valves may be disposed within the passage in parallel with each other and in series with the first valve. The third valve may include an easily accessible knob disposed outside the shock body.

Optimization strategies for sparseness- and continuity- enhanced imaging : Theory

Two complementary solution strategies to the least-squares migration problem with sparseness& continuity constraints are proposed. The applied formalism explores the sparseness of curvelets on the reflectivity and their invariance under the demigrationmigration operator. Sparseness is enhanced by (approximately) minimizing a (weighted) `-norm on the curvelet coefficients. Continuity along imaged reflectors is brought out by minimizing the anisotropic diffusion or total variation norm which penalizes variations along and in between reflectors. A brief sketch of the theory is provided as well as a number of synthetic examples. Technical details on the implementation of the optimization strategies are deferred to an accompanying paper: implementation Introduction Least-squares migration and migration deconvolution have been topics that received a recent flare of interest [7, 8]. 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 unregularized least-squares tends to fit noise and smear the energy. In addition, artifacts may be created due to imperfections in the model and possible null space of the normal operator [11]. Regularization by minimizing an energy functional on the reflectivity can alleviate some of these problems, but may go at the expense of resolution. Non-linear functionals such as ` minimization partly deal with the resolution problem but ignore bandwidth-limitation and continuity along the reflectors [12]. Independent of above efforts, attempts have been made to enhance the continuity along imaged reflectors by applying anisotropic diffusion to the image [4]. The beauty of this approach is that it brings out the continuity along the reflectors. However, the way this method is applied now leaves room for improvement regarding (i) the loss of resolution; (ii) the non-integral and non-data constrained aspects, i.e. this method is not constrained by the data which may lead to unnatural results and ’overfiltering’. In this paper, we make a first attempt to bring these techniques together under the umbrella of optimization theory and modern image processing with basis functions such as curvelet frames [3, 2]. Our approach is designed to (i) deal with substantial amounts of noise (SNR ≤ 0); (ii) use the optimal (sparse & local) representation properties of curvelets for reflectivity; (iii) exploit the near diagonalization of the normal operator by curvelets [2]; and (iv) use non-linear estimation, norm minimization and optimization techniques to enhance the continuity along reflectors [5]. Optimization strategies for seismic imaging After linearization the forward model has the following form d = Km+ n, (1) where K is a demigration operator given by the adjoint of the migration operator; m the model wih the reflectivity and n white Gaussian noise with standard deviation σn (colored noise can be accounted for). Irrespective of the type of migration operator (our discussion is independent of the type of migration operator and we allow for post-stack Kirchoff as well as ’wave-equation’ operators), two complementary optimization strategies are being investigated in our group. These strategies are designed to exploit the sparseness & invariance properties of curvelet frames in conjunction with the enhancement of the overall continuity along reflectors. More specifically, the first method [6, 5] preconditions the migration operator, yielding a reformulation of the normal equations into a standard denoising problem Fd = ≈Id z}|{ FFx+ Fn (2) y = x+ (3) with ∗ the adjoint; F· = KC∗Γ−1· the curvelet-frame preconditioned migration operator with Γ· = diag(CK∗KC∗)· ≈ CK∗KC∗· by virtue Theorem 1.1 of [2], which states that Green’s functions are nearly diagonalized by curvelets; C, C∗ the curvelet transform and its transpose; x the preconditioned model related to the reflectivity according m = CΓ−1x and close to white Gaussian noise (by virtue of the preconditioning). Applying a soft thresholding to Eq. 2 with a threshold proportional to the standard deviation σn of the noise on the data, gives an estimate for the preconditioned model with some sparseness [see for details