Browsing by Author "Nammour, Rami"

Now showing 1 - 6 of 6
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    Approximate Constant Density Acoustic Inverse Scattering Using Dip-Dependent Scaling
    (2009-04) Nammour, Rami; Symes, William W.
    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.
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    Approximate Inverse Scattering Using Pseudodifferential Scaling
    (2009-04) Nammour, Rami
    This thesis proposes a computationally efficient method for approximating the inverse of the normal operator arising in the linearized inverse problem for reflection seismology. The inversion of the normal operator using direct matrix methods is computationally infeasible. Approximate inverses estimate the solution of the inverse problem or precondition iterative methods. Application of the normal operator requires an expensive solution of large scale PDE problems. However, the normal operator approximately commutes with pseudodifferential operators, hence shares their near diagonality in a frame of localized monochromatic pulses. Estimation of a diagonal representation in this frame encoded in the symbol of the normal operator: (1) follows from its application to a single input vector; (2) suffices to approximate its inverse. I use an efficient algorithm to apply pseudodifferential operators, given their symbol, to construct a rapidly converging optimization algorithm that estimates the symbol of an inverse for the normal operator, thereby approximately solving the inverse problem.
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    Approximate inverse scattering using pseudodifferential scaling
    (2009) Nammour, Rami; Symes, William W.
    This thesis proposes a computationally efficient method for approximating the inverse of the normal operator arising in the linearized inverse problem for reflection seismology. The inversion of the normal operator using direct matrix methods is computationally infeasible. Approximate inverses estimate the solution of the inverse problem or precondition iterative methods. Application of the normal operator requires an expensive solution of large scale PDE problems. However, the normal operator approximately commutes with pseudodifferential operators, hence shares their near diagonality in a frame of localized monochromatic pulses. Estimation of a diagonal representation in this frame encoded in the symbol of the normal operator: (1) follows from its application to a single input vector; (2) suffices to approximate its inverse. I use an efficient algorithm to apply pseudodifferential operators, given their symbol, to construct a rapidly converging optimization algorithm that estimates the symbol of an inverse for the normal operator, thereby approximately solving the inverse problem.
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    Approximate Multi-Parameter Inverse Scattering Using Pseudodifferential Scaling
    (2011) Nammour, Rami; Symes, William W.
    I propose a computationally efficient method to approximate the inverse of the normal operator arising in the multi-parameter linearized inverse problem for reflection seismology in two and three spatial dimensions. Solving the inverse problem using direct matrix methods like Gaussian elimination is computationally infeasible. In fact, the application of the normal operator requires solving large scale PDE problems. However, under certain conditions, the normal operator is a matrix of pseudodifferential operators. This manuscript shows how to generalize Cramer's rule for matrices to approximate the inverse of a matrix of pseudodifferential operators. Approximating the solution to the normal equations proceeds in two steps: (1) First, a series of applications of the normal operator to specific permutations of the right hand side. This step yields a phase-space scaling of the solution. Phase space scalings are scalings in both physical space and Fourier space. Second, a correction for the phase space scaling. This step requires applying the normal operator once more. The cost of approximating the inverse is a few applications of the normal operator (one for one parameter, two for two parameters, six for three parameters). The approximate inverse is an adequately accurate solution to the linearized inverse problem when it is capable of fitting the data to a prescribed precision. Otherwise, the approximate inverse of the normal operator might be used to precondition Krylov subspace methods in order to refine the data fit. I validate the method on a linearized version of the Marmousi model for constant density acoustics for the one-parameter problem. For the two parameter problem, the inversion of a variable density acoustics layered model corroborates the success of the proposed method. Furthermore, this example details the various steps of the method. I also apply the method to a 1D section of the Marmousi model to test the behavior of the method on complex two-parameter layered models.
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    Approximate Multi-Parameter Inverse Scattering Using Pseudodifferential Scaling
    (2011-04) Nammour, Rami
    I propose a computationally efficient method to approximate the inverse of the normal operator arising in the multi-parameter linearized inverse problem for reflection seismology in two and three spatial dimensions. Solving the inverse problem using direct matrix methods like Gaussian elimination is computationally infeasible. In fact, the application of the normal operator requires solving large scale PDE problems. However, under certain conditions, the normal operator is a matrix of pseudodifferential operators. This manuscript shows how to generalize Cramer's rule for matrices to approximate the inverse of a matrix of pseudodifferential operators. Approximating the solution to the normal equations proceeds in two steps: (1) First, a series of applications of the normal operator to specific permutations of the right hand side. This step yields a phase-space scaling of the solution. Phase space scalings are scalings in both physical space and Fourier space. Second, a correction for the phase space scaling. This step requires applying the normal operator once more. The cost of approximating the inverse is a few applications of the normal operator (one for one parameter, two for two parameters, six for three parameters). The approximate inverse is an adequately accurate solution to the linearized inverse problem when it is capable of fitting the data to a prescribed precision. Otherwise, the approximate inverse of the normal operator might be used to precondition Krylov subspace methods in order to refine the data fit. I validate the method on a linearized version of the Marmousi model for constant density acoustics for the one-parameter problem. For the two parameter problem, the inversion of a variable density acoustics layered model corroborates the success of the proposed method. Furthermore, this example details the various steps of the method. I also apply the method to a 1D section of the Marmousi model to test the behavior of the method on complex two-parameter layered models.
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    Full-waveform inversion via source-receiver extension
    (Society of Exploration Geophysicists, 2017) Huang, Guanghui; Nammour, Rami; Symes, William
    Full-waveform inversion produces highly resolved images of the subsurface and quantitative estimation of seismic wave velocity, provided that its initial model is kinematically accurate at the longest data wavelengths. If this initialization constraint is not satisfied, iterative model updating tends to stagnate at kinematically incorrect velocity models producing suboptimal images. The source-receiver extension overcomes this “cycle-skip” pathology by modeling each trace with its own proper source wavelet, permitting a good data fit throughout the inversion process. Because source wavelets should be constant (or vary systematically) across a shot gather, a measure of source trace dependence, for example, the mean square of the signature-deconvolved wavelet scaled by time lag, can be minimized to update the velocity model. For kinematically simple data, such measures of wavelet variance are mathematically equivalent to traveltime misfit. Thus, the model obtained by source-receiver extended inversion is close to that produced by traveltime tomography, even though the process uses no picked times. For more complex data, in which energy travels from source to receiver by multiple raypaths, Green’s function spectral notches may lead to slowly decaying trace-dependent wavelets with energy at time lags unrelated to traveltime error. Tikhonov regularization of the data-fitting problem suppresses these large-lag signals. Numerical examples suggest that this regularized formulation of source-receiver extended inversion is capable of recovering reasonably good velocity models from synthetic transmission and reflection data without stagnation at suboptimal models encountered by standard full-waveform inversion, but with essentially the same computational cost.