Statistical analysis of gas chromatographic peaks by the Gram-Charlier series of type A and the Edgeworth-Cramer series

S U M M A R YWe propose a new method for the quantitative resolution analysis in full seismic waveform inversion that overcomes the limitations of classical synthetic inversions while being computationally more efficient and applicable to any misfit measure. The method rests on (1) the local quadratic approximation of the misfit functional in the vicinity of an optimal earth model, (2) the parametrization of the Hessian in terms of a parent function and its successive derivatives and (3) the computation of the space-dependent parameters via Fourier transforms of the Hessian, calculated with the help of adjoint techniques. In the simplest case of a Gaussian approximation, we can infer rigorously defined 3-D distributions of direction-dependent resolution lengths and the image distortion introduced by the tomographic method. We illustrate these concepts with a realistic full waveform inversion for upper-mantle structure beneath Europe. As a corollary to the method for resolution analysis, we propose several improvements to full waveform inversion techniques. These include a pre-conditioner for optimization schemes of the conjugate-gradient type, a new family of Newton-like methods, an approach to adaptive parametrization independent from ray theory and a strategy for objective functional design that aims at maximizing resolution. The computational requirements of our approach are less than for a typical synthetic inversion, but yield a much more complete picture of resolution and trade-offs. While the examples presented in this paper are rather specific, the underlying idea is very general. It allows for problem-dependent variations of the theme and for adaptations to exploration scenarios and other wave-equation-based tomography techniques that employ, for instance, georadar or microwave data.

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“…Sargan 1975), analytical chemistry (e.g. Dondi et al 1981) and mineral physics (e.g. Pavese et al 1995).…”

Section: A P P E N D I X B : M U Lt I Va R I At E G R a M -C H A R L mentioning

confidence: 99%

Resolution analysis in full waveform inversion

Fichtner

Trampert

2011

Geophysical Journal International

204

123

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“…The maximum occurs at x CD around 0.6. To improve the precision in the value of the maximum, a Gram-Charlier peak function analysis [48] (using the Origin 8.0 software) was performed. This form was chosen for convenience as it is easy to use and test calculations indicated that it gives reliable results.…”

Section: Stoichiometry Of B-cyclodextrin:gemini Complexesmentioning

confidence: 99%

The effect of the head-group spacer length of 12-s-12 gemini surfactants in the host–guest association with β-cyclodextrin

Carvalho

Correia

Valente

et al. 2011

Journal of Colloid and Interface Science

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“…The basic principles followed were first to measure peak parameters and then extract from these the band extracolumn contribution on the basis of hypotheses regarding either the column or the band-broadening contribution. The numerical procedure employed here to determine the peak parameters-the Edgeworth-Cramér (EC) series fitting method-not only more powerful because it is able to give peak parameters with definite precision but, in addition, is also a self-validating method, able to give accurate peak parameter values under definite conditions (10)(11)(12)(13)(14). The purpose of this study is to 0003-2700/89/0361-1489S01.50/0 © 1989 American Chemical Society characterize the above mentioned injection systems in order to assess the most suitable experimental procedures to use in column efficiency measurements.…”

mentioning

confidence: 99%

Fluidic and syringe injection study by peak shape analysis

Remelli¹,

Blo²,

Dondi³

et al. 1989

Anal. Chem.

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Fluidic and syringe injections with different splitting conditions are compared by using the Edgeworth-Cramer peak shape analysis. Non retained component peaks on capillary and packed columns were considered. Skewness values as low as 0.02, with precision and repeatability of ±0.005, were obtained when using the capillary column with fluidic injection. Extracolumn contributions coming from the injection system were evaluated by using peak skewness under the hypothesis of exponential decay extracolumn factor. The corrected column efficiency values are in good agreement with the theoretically expected van Deemter equation data. A syringe injection system with high split rate can produce, for unretained samples, peak fidelity values greater than 0.8, which are comparable with those observed with the fluidic injection system. Applications in column performance studies are discussed

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Statistical analysis of gas chromatographic peaks by the Gram-Charlier series of type A and the Edgeworth-Cramer series

Cited by 52 publications

References 15 publications

Resolution analysis in full waveform inversion

Resolution analysis in full waveform inversion

The effect of the head-group spacer length of 12-s-12 gemini surfactants in the host–guest association with β-cyclodextrin

Fluidic and syringe injection study by peak shape analysis

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