Optimal regularisation for acoustic source reconstruction by inverse methods

Abstract

An important inverse problem in the field of acoustics is that of reconstructing the strengths of a number of sources given a model of transmission paths from the sources to a number of sensors at which measurements are made. In dealing with this kind of the acoustical inverse problem, the strength of the discretized source distribution can be simply deduced from the measured pressure field and the inversion of corresponding matrix of frequency response functions. Hence, the accuracy of reconstruction of the source strength is crucially dependent on the conditioning of the matrix to be inverted. However, the problem of reconstructing acoustic source distributions from field measurement is very often ill-posed. In such cases, by using only the simple least-squares method, one cannot ensure a successful reconstruction of the acoustic source strength distribution. Therefore, Tikhonov regularisation is widely employed in order to produce reasonable solutions. However, determination of the amount of regularisation is not straightforward in practical applications without prior knowledge of either the strength of the acoustic sources or the contaminating measurement noise. Thus, two methods have been introduced, Generalised Cross Validation (GCV) and the L-curve method, which do not require prior information in order to determine the optimal regularisation parameter. In the present work, the abilities of the two methods are illustrated when these kinds of inverse sound radiation problems are dealt with using Tikhonov regularisation. Finally, through experimental demonstrations, some guidelines are proposed for determining the optimal degree of regularisation.