Surface Reconstruction of a Three-Dimensional Ultrasonic Flaw

Abstract

In three-dimensional inverse scattering problems, the reconstruction of a solid scatterer is often difficult, if not impossible, and computationally expensive due to the dimensionality. To obtain only the geometrical information, a surface reconstruction algorithm is nattwally more, desirable since no additional knowledge can be gained ft'ore doing the solid reconstruction and the computation is reduced to two dimensions. With the application of the first Born approximation, this paper proposes a simple surface reconstruction technique for a three-dimensional target. In general, this method is ill-posed. However, the numerical instability part of the ill-posedness is removable when the surface has a twofold symmetry with respect to a plane. To demonstrate this approach, three analytical examples are shown. I_q.tr.__octi?n Reconstruction of the geomeu3, of an embedded flaw has been one of the major areas in nondestructive evaluation (NDE) research. Some of the algorithms published in the last decade were based on the elastic wave inverse (gust) Born approximation, which is essentially a low frequency approximation to an isolated weak scatterer in an otherwise isotropic and homogeneous medium whose material properties are similar to the host's [ 1,2]. A key factor in the development of these inverse algorithms is the spatial-frequency function "shape factor" that appears in the formulation of the inverse Bom approximation. This shape, factor embodies all information about the size, shape, and orientation of the scatterer and, therefore, one can develop inverse procedures to extract the geomeu3, in terms of a characteristic function (which has the value 1 inside the scatterer and 0 outside). The direct approach to recovering this characteristic function requires three-dimensional information in the spatial-frequency domain [3], which is computationally expensive. However, the basic equation of the general three-dimensional inverse Born approximation c,'m be.further cast to form a set of equations for the surface of a three-dimensional flaw. As a consequence, one can develop a "surface reconstruction" algorithm based upon these equations. Since the surface of a three-dimensional flaw is two-dimensional, the algorithm requires o_tly 'knowledge of a two-dimensional hyper-plane of the shape factor instead of a !.