Regularization Techniques Applied to Nearfield Acoustical Holography

Nearfield Acoustical Holography (NAH) is an inverse problem in wave propagation which has found applications to both interior and exterior noise control problems. We can view the fundamental equation of NAH in the spatial frequency domain as a linear equation, p=Gv, where p is the measured pressure, v is the unknown normal velocity (usually on the surface of a vibrator), and G is the known transfer function. NAH inverts this equation solving for the velocity. However, this equation is ill-posed since small changes in p usually lead to large changes in v. Thus the need for regularization of the inversion. We will discuss regularization techniques applied to NAH, and will compare the errors associated with several regularization schemes; Tikhonov, conjugate gradient, Landweber iteration and a simple exponential filter approach (which appears to provide the best results). Furthermore, in an effort to illuminate the physical propagation mechanisms of NAH, we will discuss these approaches in the light of k-space.