Time domain nearfield acoustical holography with three-dimensional linear deconvolution

In this paper, a formulation is proposed to improve the time domain reconstruction of non-stationary acoustic fields with nearfield acoustical holography. The formulation involves applying three-dimensional (3D) linear deconvolution using a Green's function sampled in the time and spatial domains. Because linear deconvolution does not assume periodic signals, it more appropriately describes the decaying behavior of sources that begin and end at null amplitude and that radiate over a finite amount of time. The proposed method outperforms standard circular convolution-based nearfield acoustical holography by up to a factor three in relative root-mean-square error, when compared using a transient baffled piston model, and its reconstructions remain accurate over large back-propagation distances. Furthermore, it is shown that truncation errors in linear deconvolution can be reduced by applying a 3D patch extrapolation algorithm; however, convergence depends on the choice of an adequate Tikhonov's regularization parameter. Three methods for predicting the optimal parameter are compared: the L-curve, the generalized cross-validation, and the empirical Bayesian method. It is shown that with the proposed formulation applied to reconstructing the field radiated by a transient baffled piston, the generalized cross-validation gives the overall best prediction for the noise levels and back-propagation distances studied.