Underwater Acoustic Imaging: Exact Geometric-Acoustics Treatment of the Image due to a Specular Reflector

Abstract

In underwater acoustic imaging, used to produce high-resolution images in turbid waters, a specular reflector in general produces a 'pseudoimage' of the receiving array, located on the reflecting surface; the pseudoimage is of considerable use since it reveals the shape of the surface. A system is considered in which a spherical transmitter together with a 2D receiving array give a 3D image in a single 'ping'. A treatment predicting the shape of the pseudoimage - in particular, its lateral extent - is given that is exact within geometrical acoustics. The surfaces to which the treatment is applied are the paraboloid (with two principal radii of curvature) - provided that the transmitter lies on the paraboloid's axis - the sphere, the cylinder and the plane. The treatment involves a ray-tracing algorithm based on the equation of the surface, and an algorithm to invert that procedure using the Levenberg-Marquardt method. Pseudoimages of lines in the array are graphed and discussed, along with, more interestingly, pseudoimages of squares. While the latter pseudoimages are parallelograms when the square is small, in general they are not parallelograms, since all four sides are curved. Further features found are that an 'object' in the array may produce multiple pseudoimages, no pseudoimage, 'local optima' and/or 'blockage points'. Such an exact determination of the resulting pseudoimage for selected surfaces gives useful insight into pseudoimages that occur in practice. Conditions of validity (arising because wave effects are neglected) are given. The report also contains a preliminary discussion of the extension that would be needed to include wave effects. In addition it is shown that, subject to more restrictive conditions, the results apply also to a general smooth surface. A 'paraxial' approximation (similar to the 'large-range approximation' of an earlier paper but somewhat more general) is described and found to be useful.