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Excellent Approximation (excellent + approximation)
Selected AbstractsAdaptive subtraction of multiples using the L1 -normGEOPHYSICAL PROSPECTING, Issue 1 2004A. Guitton ABSTRACT A strategy for multiple removal consists of estimating a model of the multiples and then adaptively subtracting this model from the data by estimating shaping filters. A possible and efficient way of computing these filters is by minimizing the difference or misfit between the input data and the filtered multiples in a least-squares sense. Therefore, the signal is assumed to have minimum energy and to be orthogonal to the noise. Some problems arise when these conditions are not met. For instance, for strong primaries with weak multiples, we might fit the multiple model to the signal (primaries) and not to the noise (multiples). Consequently, when the signal does not exhibit minimum energy, we propose using the L1 -norm, as opposed to the L2 -norm, for the filter estimation step. This choice comes from the well-known fact that the L1 -norm is robust to ,large' amplitude differences when measuring data misfit. The L1 -norm is approximated by a hybrid L1/L2 -norm minimized with an iteratively reweighted least-squares (IRLS) method. The hybrid norm is obtained by applying a simple weight to the data residual. This technique is an excellent approximation to the L1 -norm. We illustrate our method with synthetic and field data where internal multiples are attenuated. We show that the L1 -norm leads to much improved attenuation of the multiples when the minimum energy assumption is violated. In particular, the multiple model is fitted to the multiples in the data only, while preserving the primaries. [source] General two-electron exponential type orbital integrals in polyatomics without orbital translations,INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 13 2009Philip E. Hoggan Abstract This article advocates the use of atomic orbitals which have direct physical interpretation, i.e., hydrogen-like orbitals. They are exponential type orbitals (ETOs). Convenient nodeless linear combinations are used, namely Slater type orbitals (STOs) (with a product of a single power of r and an exponential as radial factor). Until 2008, such orbital products on different atoms were difficult to manipulate for the evaluation of two-electron integrals. The difficulty was mostly due to cumbersome orbital translations involving slowly convergent infinite sums. These are completely eliminated using Coulomb resolutions. They provide an excellent approximation that reduces these integrals to a sum of one-electron overlap-like integral products that each involve orbitals on at most two centers. Such two-center integrals are separable in prolate spheroidal coordinates. They are thus readily evaluated. Only these integrals need to be re-evaluated to change basis functions. The above is still valid or three-center integrals. In four-center integrals, the resolutions require translating one potential term per product. This is outlined here and detailed elsewhere. Numerical results are reported for the H2 dimer and CH3F molecule. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009 [source] Electric field-derived point charges to mimic the electrostatics in molecular crystalsJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 10 2006Andrew E. Whitten Abstract Because of the way the electrostatic potential is defined in a crystal, it is not possible to determine potential-derived charges for atoms in a crystal. To overcome this limitation, we present a novel method for determining atomic charges for a molecule in a crystal based on a fit to the electric field at points on a surface around the molecule. Examples of fits to the electric field at points on a Hirshfeld surface, using crystal Hartree,Fock electron densities computed with a DZP basis set are presented for several organic molecular crystals. The field-derived charges for common functional groups are transferable, and reflect chemical functionality as well as the subtle effects of intermolecular interactions. The charges also yield an excellent approximation to the electric field surrounding a molecule in a crystal for use in cluster calculations on molecules in solids. © 2006 Wiley Periodicals, Inc. J Comput Chem 27: 1063,1070, 2006 [source] Testing for lack of dependence in the functional linear modelTHE CANADIAN JOURNAL OF STATISTICS, Issue 2 2008Piotr Kokoszka Abstract The authors consider the linear model Yn = ,Xn + ,n relating a functional response with explanatory variables. They propose a simple test of the nullity of , based on the principal component decomposition. The limiting distribution of their test statistic is chi-squared, but this distribution is also an excellent approximation in finite samples. The authors illustrate their method using data from terrestrial magnetic observatories. Un test d'absence de dépendance dans un modéle fonctionnel linéaire Les auteurs s'intéressent au modèle Yn = ,Xn + ,n linéaire , liant une variable réponse fonctionnelle à des variables explicatives. Ils proposent un test simple de nullité, de fondé sur la décomposition en composantes principales. La loi limite de leur statistique est une khi-deux, mais cette loi fournit aussi une excellente approximation à taille finie. Les auteurs illustrent leur méthode au moyen de données provenant d'observatoires du champ magnétique terrestre. [source] | ||||||||||