Spectral modeling

Fast Fourier Transformation is widely used in many fields of science, among them chemoractrics. The Fast Fourier Transformation (FFT) algorithm transforms the data from the "wavelength" domain into the "frequency" domain. The method is almost compulsorily used in spectral analysis, e, g., when near-infrared spectroscopy data arc employed as independent variables. Next, the spectral model is built between the responses and the Fourier coefficients of the transformation, which substitute the original Y-matrix. 

Note that the Kolmogorov power spectrum is unphysical at low frequencies— the variance is infinite at k = 0. In fact the turbulence is only homogeneous within a finite range—the inertial subrange. The modified von Karman spectral model includes effects of finite inner and outer scales. 

Figure 1. Kolmogorov and modified von Karman spectral models. (Lo = 10 m and o 0.01 m for the von Karman plot.)...

FIGURE 58-2 Hydrogen ( H) spectrum from a normal human brain at 4 Tesla field strength. The spectrum is very complicated, comprised of many overlapping peaks which are difficult to resolve from each other. Spectral analysis routines make use of spectral models which are constructed from the individual spectra acquired from each biomolecule from in vitro solutions. This model is then fitted to the raw data and approximate concentrations for each biomolecule are extracted... 

This relation shows that for homogeneous turbulence, working in terms of the two-point spatial correlation function or in terms of the velocity spectrum tensor is entirely equivalent. In the turbulence literature, models formulated in terms of the velocity spectrum tensor are referred to as spectral models (for further details, see McComb (1990) or Lesieur (1997)). 

The spectral transport equation can also be used to generate a spectral model for the dissipation rate e. Multiplying (2.61) by 2vk2 yields the spectral transport equation for... 

A spectral model similar to (3.82) can be derived from (3.75) for the joint scalar dissipation rate eap defined by (3.139), p. 90. We will use these models in Section 3.4 to understand the importance of spectral transport in determining differential-diffusion effects. As we shall see in the next section, the spectral interpretation of scalar energy transport has important ramifications on the transport equations for one-point scalar statistics for inhomogeneous turbulent mixing. 

A key assumption in deriving the SR model (as well as earlier spectral models see Batchelor (1959), Saffman (1963), Kraichnan (1968), and Kraichnan (1974)) is that the transfer spectrum is a linear operator with respect to the scalar spectrum (e.g., a linear convection-diffusion model) which has a characteristic time constant that depends only on the velocity spectrum. The linearity assumption (which is consistent with the linear form of (A.l)) ensures not only that the scalar transfer spectra are conservative, but also that if Scap = Scr in (A.4), then Eap ic, t) = Eyy k, t) for all t when it is true for t = 0. In the SR model, the linearity assumption implies that the forward and backscatter rate constants (defined below) have the same form for both the variance and covariance spectra, and that for the covariance spectrum the rate constants depend on the molecular diffusivities only through Scap (i.e., not independently on Sc or Sep). 

Note that at spectral equilibrium the integral in (A.33) will be constant and proportional to ea (i.e., the scalar spectral energy transfer rate in the inertial-convective sub-range will be constant). The forward rate constants a j will thus depend on the chosen cut-off wavenumbers through their effect on (computationally efficient spectral model possible, the total number of wavenumber bands is minimized subject to the condition that... 

Clark, T. T. and C. Zemach (1995). A spectral model applied to homogeneous turbulence. 

J.M. Boone, T.R. FeweU and R.J. Jennings, Molybdenum, rhodium and tungsten anode spectral models using interpolated polynomials with application to mammography, Med. Phys. 24(12) (1997) 1863-1874. 

S. TriadaphUlon, E. Martin, G. Montagne, A. Norden, P. Jeffkins and S. Stimpson, Fermentation process tracking through enhanced spectral modeling, Biotechnol Eng., 97(3), 554—567 (2007). 

One way that probability p may be estimated is by statistical analysis of previous experimental data. Various spectral models could also be used to specify p. Both regular models such as that of Elsasser and random models... 

Model correlation functions. Certain model correlation functions have been found that model the intracollisional process fairly closely. These satisfy a number of physical and mathematical requirements and their Fourier transforms provide a simple analytical model of the spectral profile. The model functions depend on the choice of two or three parameters which may be related to the physics (i.e., the spectral moments) of the system. Sears [363, 362] expanded the classical correlation function as a series in powers of time squared, assuming an exponential overlap-induced dipole moment as in Eq. 4.1. The series was truncated at the second term and the parameters of the dipole model were related to the spectral moments [79]. The spectral model profile was obtained by Fourier transform. Levine and Birnbaum [232] developed a classical line shape, assuming straight trajectories and a Gaussian dipole function. The model was successful in reproducing measured He-Ar [232] and other [189, 245] spectra. Moreover, the quantum effect associated with the straight path approximation could also be estimated. We will be interested in such three-parameter model correlation functions below whose Fourier transforms fit measured spectra and the computed quantum profiles closely see Section 5.10. Intracollisional model correlation functions were discussed by Birnbaum et a/., (1982). 

Bird, R.E. (1984) A simple, solar spectral model for direct-normal and diffuse horizontal irradiance, Solar Energy 32, 461. 

Carvalho. F.R.S (1998). Radiative Transfer Simplified Spectral Model for Clear Sky conditions in Portugal, MSC Thesis dissertation. Faculty of Sciences - University of Lisbon. 

Gallagher, R. T., Wilson, I. D., and Hobby, K. (2008). New approach for identification of metabolites of a model drug Partial isotope-enrichment combined with novel mass spectral modeling software. In Proceedings of the 56th ASMS Conference on Mass Spectrometry and Allied Topics. ASMS, Denver, CO. 

Serra and Smith, 1989] Serra, X. and Smith, J. (1989). Spectral modeling synthesis A sound analysis/synthesis system based on a deterministic plus stochastic decomposition. In Proc. oflnt. Computer Music Conf., pages 281-284, San Francisco, CA. 

The spectral models may be regarded as a series of 20 probabilities of absorbance at each wavelength. Hence if the total absorbance over 20 wavelengths is summed to x, then the probability at each wavelength is simply the absorbance divided by x. Convert the three models into three probability vectors. 

These criteria resulted in the choice of a relatively simple, but accurate spectral model of Gueymard (SMARTS Simple Model for Atmospheric Transmission of Sunshine) to compute the reference standard terrestrial spectra.7... 

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