Spatial resolution limits for the reconstruction of acoustic source distribution by inverse techniques

Abstract

One technique for deducing the strength of an acoustic source distribution from measurements of the radiated field involves the inversion of the matrix of frequency response functions relating the field measurement points to the strengths of a number of point sources used to represent the source distribution.  In practice, the frequency response function matrix to be inverted may very often be ill-conditioned.  This ill-conditioning will also often result in an ill-posed problem and thus regularisation algorithms are used to produce reasonable solutions.  For this purpose, Tikhonov regularisation has been applied, and Generalised Cross Validation (GCV) and the L-curve method have been introduced as an effective method for determining of the proper amount of regularisation without prior knowledge of either the source distribution or the contaminating errors.  In this thesis, the emphasis is placed on the relationship between the spatial resolution of the reconstructed source distribution and the small singular values of the frequency response function matrix to be inverted.  However, the use of Tikhonov regularisation often suppresses the effect of small singular values and these are in turn often associated with high spatial frequencies of the source distribution.  Thus the process of regularisation produces a useful estimate of the acoustic source strength distribution but with a limited spatial resolution.  Furthermore, in the field of Fourier acoustics, the spatial resolution of the reconstructed source distribution is usually limited by the wavelength of the radiation.  This thesis expresses the relationship between estimation accuracy, spatial resolution, noise-level and source/sensor geometry, when a range of inverse sound radiation problems are regularised using Tikhonov regularisation.  The results presented form the basis of guidelines that enable the reconstruction of acoustic source strength with a resolution, under some circumstances, that is finer than the classical half-wavelength limit.

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