Let us denote by tobs the observed traveltime. By applying the expansion 2. This is called a least squares problem, which can be formally stated as finding the minimum of the following objective function:. Least-squares solutions very often do not provide good results and sometimes they do not even exist. In this work, we have considered linear tomography, where the geometry of the rays does not depend on the distribution of velocities 7.
In other words, the rays are straight, and there is no need to do further ray tracing, nor to update the vector of estimated slowness. In general, this approximation is valid only for small velocity or slowness contrasts.
Iterative methods for approximate solution of inverse problems - Ghent University Library
We consider the classical conjugate gradient method for normal equations to solve the least-squares linear system 2. Given a initial guess s0 and a maximum number of iterations kmax,. The matrix GTG is not explicitly computed, since it usually has a higher number of nonzero entries and a higher condition number than G Monte-Carlo GCV 15 :. Recalling the penalization of the first and second derivatives of the estimated model parameters proposed in 20 , we modify the minimal product criterion from 3 by introducing an additional term that penalizes large oscillations in the current iterate:.
We consider the model shown in Figure 1 left , which is described as a non symmetric anticlinal with tectonic origin.
Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems
Such situation has great relevance in oil exploration, since it has structural traps with folds that could accumulate hydrocarbons. In the following we evaluate the following relative mean square errors: the error of the perturbed data with respect to the observed data, the calculated traveltimes with respect to the perturbed data, and the estimated velocities and slownesses with respect to the true model. These errors are defined as follows:. In analogy with 15 , the error is normalized so that its scale is comparable with the scale of the stopping criterion functions.
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The iterative scheme 2. Table 1 shows the number of iterations and the mean square errors of each inversion. The estimated models are satisfactory, allowing the identification location and velocity of the different layers, including the target reservoir.
We considered the solution of an ill-conditioned least-squares problem arising from the discretization of a model for linear seismic tomography, which is a tool with wide application in reservoir geophysics.
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- Iterative Methods for Approximate Solution of Inverse Problems?
- Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems?
The discrete problem was presented as a system of linear equations, which we solved by the standard conjugate gradient method regularized by stopping in an early stage. In the numerical experiments with a synthetic geological model, we perturbed the observed data with different levels of noise. The estimated model parameters were all satisfactory, although the resolution in the reconstructed velocity tomograms were compromised when the noise level was higher.
Monte Carlo Methods and Applications
We expect that the effectiveness of the stopping criteria would be better in the absence of noise or for low noise if an appropriate preconditioning of the linear system is employed. We have also performed numerical experiments with conjugate gradient methods with Tikhonov regularization and noticed that, also in this case, a large number of iterations is needed for convergence.
Some of the preconditioners proposed in the literature that might be appropriate are based on the Fast Fourier Transform 4 and on the Algebraic Reconstruction Technique This motivates further testing and an error analysis in future works.
Oliveira and A. Digital ray tracing in two-dimensional refractive fields. Journal of the Acoustic Society of America, 72 5 , Fixed-point iterations in determining the Tikhonov regularization parameter. Inverse Problems, 24 3 , Numerical Linear Algebra with Applications, 21 3 , Chan, J. FFT-based preconditioners for Toeplitz-block least squares problems.
Equivalence of regularization and truncated iteration in the solution of ill-posed image reconstruction problems. Linear Algebra and its Appliations, , Rank-deficient and discrete ill-posed problems. SIAM, Philadelphia, Seismic borehole tomography - theory and computational methods. Proceedings of the IEEE, 74 2 , Lee, H. Huang, J. Dennis, P. An optimized parallel LSQR algorithm for seismic tomography.
Computers and Geosciences, 61 , Solving or resolving inadequate and noisy tomographic systems. Journal of Computational Physics, 61 3 , LSQR: An algorithm for sparse linear equations and sparse least squares. Computers and Geosciences, 33 5 , Santos, A. Journal of Seismic Exploration, 15 , Equivalence of regularization and truncated iteration for general ill-posed problems. Linear Algebra and its Applications, , Preconditioning conjugate gradient with symmetric algebraic reconstruction technique ART in computerized tomography.
As long as you attribute the data sets to the source, publish your adapted database with ODbL license, and keep the dataset open don't use technical measures such as DRM to restrict access to the database. The datasets are also available as weekly exports. NL EN. Ju Kokurin. More about Inverse problems Differential equations Iterative methods Mathematics.
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Numerical Methods for Solving Inverse Problems of Mathematical Physics
Thu 26 Sep Study Resto S5 : u. Fri 27 Sep Study Resto S5 : u. Sat 28 Sep closed Sun 29 Sep closed More opening hours. Bakushinsky, M. Kokurin, M. Publisher: Dordrecht : Springer,