An Analysis of Regularization Errors in Generalized Nearfield Acoustical Holography

Nearfield Acoustical Holography (NAH) has traditionally been utilized in the identification of noise sources on planar structures. The planar NAH was subsequently extended to handle noise source identification on separable geometry such as cylindrical and spherical surfaces using measurements taken on a conforming surface. Recent advances have replaced the mathematics of separable wave propagation with a Boundary Element Method (BEM) based numerical formulation, enabling NAH to reconstruct sources on arbitrarily complex geometry with arbitrarily shaped measurement surfaces. However, this generalized NAH leads to the solution of a discrete ill-posed problem that requires solution through singular value decomposition (SVD) or iterative strategies. Various regularization schemes have been proposed in the literature of inverse problems to be used in conjunction with SVD for robust inversion. Applications of these schemes to generalized NAH problems are beginning to appear in literature. We present a rigorous comparison of the regularization errors introduced by the Tikhonov regularization method used in conjunction with three techniques for the selection of the regularization parameter; Generalized Cross Validation (GCV), the L Curve criterion and the Morozov discrepancy principle. A model problem that encompasses all the complexities of a real world generalized NAH application is presented to aid the evaluation of these methods. The regularization errors introduced by each technique are contrasted for different levels of measurement noise in numerically synthesized holograms.