Reconstruction of acoustic source strength distributions and their interactions by inverse techniques

Yoon, Seongho (1998) Reconstruction of acoustic source strength distributions and their interactions by inverse techniques. University of Southampton, Doctoral Thesis.

Abstract

This thesis is devoted to the study of a discrete inverse problem in acoustics. In particular, this work is aimed at the reconstruction of the complex strengths of a number of acoustic sources when given the acoustic pressures measured at a number of points in the acoustic field. Since in the discrete inverse problem the condition number of the transfer function matrix to be inverted plays a crucial role in producing a satisfactory reconstruction, this is investigated in detail both analytically and empirically. In formulating our problem, we employ the least squares method with minimal norm. This approach reconstructs acoustic source strength distributions and their interactions with extremely good agreement with the true values in the ideal situation in which no measurement noise and no error involved in the transfer functions are assumed. In practice, however, their effect is unavoidable and in such cases the accuracy of the reconstruction is determined by the conditioning of the matrix to be inverted and the amount of measurement noise and error contained in the transfer functions.

Accordingly, regularisation methods such as Tikhonov regularisation, singular value discarding and singular value decomposition based regularisation are used to filter our the influence of the noise by improving the conditioning. In order to choose appropriately the regularisation parameter and the singular values to be discarded, we suggest two main methods. One method requires prior knowledge of either the acoustic source or the noise. The other method does not demand such prior knowledge and the techniques of ordinary cross-validation and generalised cross-validation are investigated. In our problem it is necessary to measure the acoustic pressure auto- and cross-spectra at a number of positions in the acoustic field. Direct measurement of these quantities, however, requires considerable experimental effort, especially when the number of field points to be measured is large. For this reason, a technique using a few reference microphones is proposed which enables the construction of the full cross-spectral matrix of acoustic pressures with the minimum number of measurements.

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