DETERMINATION OF TIME-OF-FLIGHT SURFACES USING THE METHOD OF MOMENTS

To produce an image from backscattered signals requires a knowledge of the time of flight from each source to each sensor in a transducer array. We reduced the determination of the rectangular coordinates of a new point on any time-of-flight surface, which requires finding the square root of the sum of squares, to the time of a single addition. To do this we represented the time-of-flight surface by a two-dimensional, positive-integer-degree polynomial and then implemented that polynomial in its forward-difference form. Two solutions for such a polynomial were found using the method of moments. A minimum-mean-square-error constraint yields polynomial coefficients from: 1) a numeric evaluation of the moments of arbitrary surfaces over rectangular regions; and 2) an analytical solution for moments of time-of-flight surfaces over sectorshaped regions. For a 6 x 6 cm region at a minimum range of 4 cm centered in a 74-degree sector, the maximum error for a second-degree polynomial was 0.30% of the average time of flight over the region for the numeric solution and 0.32% for the analytic solution, assuming a 1500 m/s background velocity. These time-of-flight approximations, when determined for a medium with a constant background velocity, convert cylindrical coordinates into rectangular coordinates.