Phillips-Twomey method

Riebel and Ldffler [240] obtained an acoustic attenuation spectrum with one transducer pair to infer the particle size distribution. Solids concentrations and particle size distribution were obtained using both the Phillips-Twomey algorithm (PTA) and the relaxation method. The PTA gives a least squares solution by simple linear matrix operations to yield a numerical inversion from the attenuation spectra to the size distribution... 

As a whole, the procedure of measurement and mathematical evaluation is closely analogous to laser diffraction measurements in the Fraunhofer domain 5, but the problems arising from the numerical instability of the linear equation system are even more severe. Thus the Phillips-Twomey-Algorithm (PTA), successfully applied in laser diffraction spectrometry, will perform poorly in the presence of systematic errors, as they may arise from unknown particle shape or particle material. A nonlinear iterative procedure however, the relaxation method, has proven to yield excellent results even under difficult conditions. 

There are a few methods to solve Fredholm integral equations, which are not that prone to measurement faults [3, 12]. One of them is the inversion with linear smoothing according to Phillips-Twomey, which is used, for example, to analyze the measurement data of laser diffraction systems. 

The demand that the solution 6 be consistent with the data i results in the improved resolution that we expect from a deconvolution method. As we have explained, however, it also results in the amplification of high-frequency noise. The smoothing of this noise to some extent defeats the purpose of deconvolution. The tradeoff between smoothness and consistency is explicit in the formulation of a method first described by Phillips (1962) and further developed by Twomey (1965). In this method, we minimize the quantity... 

In general, problems having solutions that vary radically or discon-tinuously for small input changes are said to be ill-posed. Deconvolution is an example of such a problem. Tikhonov was one of the earliest workers to deal with ill-posed problems in a mathematically precise way. He developed the approach of regularization (Tikhonov, 1963 Tikhonov and Arsenin, 1977) that has been applied to deconvolution by a number of workers. See, for example, papers by Abbiss et al (1983), Chambless and Broadway (1981), Nashed (1981), and Bertero et al. (1978). Some of the methods that we have previously described fall within the context of regularization (e.g., the method of Phillips and Twomey, discussed in Section V of Chapter 3). Amplitude bounds, such as positivity, are frequently used as key elements of regularization methods. 

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