Inversion algorithm

Meyer M, Hermand JP, Asch M et al (2006) An analytic multiple frequency adjoint-based inversion algorithm for parabolic-type approximations in ocean acoustics. Inverse Probl Sci Eng 14 245-265... 

Figure 5. Normalized S mass distribution, China Lake, CA—average of 8 samples. Aerosol segregated by LPl and analyzed by FVFPD. The solid histogram is the mass distribution with respect to the 50% aerodynamic cutoff diameter the dashed histogram is the inverted distribution obtained from LPl calibration data and Twomey (1975) inversion algorithm (a) July2-4, 1978 ("M = 0.765 fig/m ) (b) October 20-22, 1978 (M = 1.702 /ig/m ) (c) September 5-7, 1979 (M =...

Fussen, D., A Critical Analysis of the Stratospheric Aerosol and Gas Experiment II Spectral Inversion Algorithm, J. Geophys. Res., 103, 8455-8464 (1998). 

Reinsel, G. C., W.-K. Tam, and L. H. Ying, "Comparison of Trend Analyses for Umkehr Data Using New and Previous Inversion Algorithms, Geophys. Res. Lett., 21, 1007-1010 (1994a). 

H. Rabitz The goals of Hamiltonian information extraction are clear, but prior means for this purpose are generally unsatisfactory in many respects. Various sources of data are available for exploitation. In some applications, reduced models will suffice, while for others, only high accuracy detailed potentials can meet the needs. What is necessary is a rigorous inversion algorithm that can incorporate appropriate physical constraints to reliably extract the Hamiltonian information. 

Bockhom et al. (2002) have introduced an approach, by which a calculated signal course is fitted to the whole experimental signal decay under the variation of different distribution parameters such as the width or the mean primary particle diameter. A major drawback of this method is the relative high computing effort for the evaluation which makes the application for online measurements questionable. Roth and Filippov (1996) have introduced an inversion algorithm that also suffers from a complicated and time-consuming calculation procedure and does not lead to unique solutions. 

Commercially available instruments usually employ both approaches. For highly skewed distributions or distributions having more than one mode, an inversion algorithm must be used [280] whereas for narrowly classified mono-modal distributions the cumulants approach is satisfactory. 

In indirect methods, the resonance parameters are determined from the energy dependence of the absorption spectrum. An important extra step — the non-linear fit of (t E) to a Lorentzian line shape — is required, in addition to the extensive dynamical calculations. The procedure is flawless for isolated resonances, especially if the harmonic inversion algorithms are employed, but the uncertainty of the fit grows as the resonances broaden, start to overlap and melt into the unresolved spectral background. The unimolecular dissociations of most molecules with a deep potential well feature overlapping resonances [133]. It is desirable, therefore, to have robust computational approaches which yield resonance parameters and wave functions without an intermediate fitting procedure, irrespective of whether the resonances are narrow or broad, overlapped or isolated. 

In the cited works [6,11-13], only the ability of several numerical inversion algorithms for recuperating w%V) was evaluated, and the MWD calculation was not considered. In the present article, the purely numerical Methods I-III are compared with Methods IV and V which require a linear molecular-weight calibration. For this reason, the calibration log M V) is included in Fig. lb. From that calibration and w%V), the true MWD w (log M) of Figs. Ic-lf was obtained. Note that the selection of a uniform and Gaussian broadening is not a limitation for an adequate evaluation of the (more general) Methods I-III. 

O. Dubovik, M.D. King (2000). A flexible inversion algorithm for retrieval of aerosol optical properties from sun and sky radiance measurements. J. Geophys. Res., 105, 20673-20696. 

Four necessary properties of the moment-inversion algorithm are the following. 

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