Fourier amplitude method

The direct method (DM) for solution of this set of equations was proposed by Atherton et al. [5], and in a somewhat a modified form by Dickinson and Gelinas [4] who solved r sets of equations each of size In consisting of Eq. (1) coupled with a particular j—value of Eq. (2). Shuler and coworkers [5] took an alternative approach in the Fourier Amplitude method in which a characteristic periodic variation is ascribed to each a, and the resulting solution of (1) is Fourier analyzed for the component frequencies. These authors estimate that 1.2r2 5 solutions of Eq. (1) together with the appropriate Fourier analyses are required for the complete determination of the problem. Since even a modest reaction mechanism (e.g. in atmospheric chemistry or hydrocarbon cracking or oxidation) may easily involve 100 reactions with several tens of species, it is seen that a formidable amount of computation can result. 

From the computational point of view the Fourier space approach requires less variables to minimize for, but the speed of calculations is significantly decreased by the evaluation of trigonometric function, which is computationally expensive. However, the minimization in the Fourier space does not lead to the structures shown in Figs. 10-12. They have been obtained only in the real-space minimization. Most probably the landscape of the local minima of F as a function of the Fourier amplitudes A,- is completely different from the landscape of F as a function of the field real space. In other words, the basin of attraction of the local minima representing surfaces of complex topology is much larger in the latter case. As far as the minima corresponding to the simple surfaces are concerned (P, D, G etc.), both methods lead to the same results [21-23,119]. 

Application to small amplitude methods the operational impedance the Fast Fourier Transform... 

Examples of mathematical methods include nominal range sensitivity analysis (Cullen Frey, 1999) and differential sensitivity analysis (Hwang et al., 1997 Isukapalli et al., 2000). Examples of statistical sensitivity analysis methods include sample (Pearson) and rank (Spearman) correlation analysis (Edwards, 1976), sample and rank regression analysis (Iman Conover, 1979), analysis of variance (Neter et al., 1996), classification and regression tree (Breiman et al., 1984), response surface method (Khuri Cornell, 1987), Fourier amplitude sensitivity test (FAST) (Saltelli et al., 2000), mutual information index (Jelinek, 1970) and Sobol s indices (Sobol, 1993). Examples of graphical sensitivity analysis methods include scatter plots (Kleijnen Helton, 1999) and conditional sensitivity analysis (Frey et al., 2003). Further discussion of these methods is provided in Frey Patil (2002) and Frey et al. (2003, 2004). 

The first widely used global method was the Fourier Amplitude Sensitivity Test (FAST) (for a review see [83]). In the FAST method, all rate parameters were simultaneously perturbed by sine functions with incommensurate frequencies. Fourier analysis of the solution of the model provided the variance crf(t) of concentration i, and also the variance o- (t) of c, arising from the uncertainty in the /th parameter. Their ratio... 

Many commodity prices exhibit cyclic behavior due to the investment cycle, so in some cases nonlinear models can be used, as in Figure 6.4c. Unfortunately, both the amplitude and the frequency of the price peaks usually vary somewhat erratically, making it difficult to fit the cyclic price behavior with simple wave models or even advanced Fourier transform methods. 

Fourier Transform Method. Another method of data reduction is to take a fast Fourier transform (FFT) of the wave (10). As indicated in Figure 7, the Fourier transform of a damped sine wave with a single frequency is a single maximum in the frequency domain at the frequency of the oscillation. The amplitude (H) of the transformed data as a function of angular frequency ([Pg.344]

Figure 7. Fourier transform method. The Fourier transform of an exponentially damped sine wave of period P and damping coefficient a is a single maximum at the oscillation frequency whose amplitude is inversely proportional to the damping...

In time domain measurements, the electrochemical system is subjected to a potential variation that is the resultant of many frequencies, like a pulse or white noise signal, and the time-dependent current from the cell is recorded. The stimulus and the response can be converted via Fourier transform methods to spectral representations of amplitude and phase angle frequency, from which the desired impedance can be computed as a function of frequency. 

Fourier-based techniques. For these methods, the input parameter realisations have to fulfil special frequency properties. Then a fi quency decomposition of the output maps different frequencies attributed to the input factors to different fractions of the variance of the output. Different frequency selection schemes have been developed, named Fourier Amplitude Sensitivity Test (FAST), Extended FAST (EFAST), and Random Balance Design (RBD). They can estimate first and/or total effects. 

Finally, we should note that each point in the amplitude record A(r) contains information about every point in the spectmm /(v). If A(t) can be recorded rapidly, as is the case in FT NMR experiments, the Fourier transform method could enable spectra to be recorded much more rapidly than by direct observation of absorption. 

One family of widely used synthetic ground motions are the source-based stochastic simulations based on the work of David Boore and a number of other researchers in the past several decades (e.g., Boore 2003 Beresnev and Atkinson 1998 Motazedian and Atkinson 2005 Boore 2009). This simulation method roots in the work of McGuire and Hanks (1980), which identifies the Fourier amplitude spectrum (FAS) of a ground motion considering the source, path, and... 

Figure 7.10 shows the resulting Fourier amplitude vs. time periods for the data with time periods up to 30 years. The figure shows a strong peak in the Fourier components at a time interval of around 11 years as expected. The identification of frequency components composing a signal is one of the most important applications of the Fourier series method. However in this example the identification of... 

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

The Fourier method is not a requirement, and direct sinusoidal fitting procedures are also used to fit the data from a set of images. A number of specialized procedures have been described over the years and it is worth noting that extracting the amplitude and phase may be done as a simple extension to conventional linear regression. 

The usual methods for characterization of anisotropic patterns involve the calculations of the scattering patterns and their further analysis [167,168] in the Fourier space. Thus, for an isotropic system, the scattering intensity, S(k) is symmetric, its maxima of the same amplitude are arranged at the circle k = k ... 

The electron crystallography method (21) has been used to characterize three-dimensional structures of siliceous mesoporous catalyst materials, and the three-dimensional structural solutions of MCM-48 (mentioned above) and of SBA-1, -6, and -16. The method gives a unique structural solution through the Fourier sum of the three-dimensional structure factors, both amplitude and phases, obtained from Fourier analysis of a set of HRTEM images. The topological nature of the siliceous walls that define the pore structure of MCM-48 is shown in Fig. 28. 

Even without having the stmcture factor phases, e.g. from electron microscopy images, it is possible to get some insight into the atomic architecture of a crystal. A simple but powerful method to get this information was introduced hy A.L. Patterson about 70 years ago. Following Patterson the Fourier synthesis is carried out using the squared stmcture factor amplitudes Fha which are equal to the measured intensities for the reflections with index hkl. Moreover, all phase values must be set to zero, which leads to the following (auto-correlation) function ... 

---