Higher order corrections to an iterative approach for approximating bubble distributions from attenuation measurements

A formal theory exists for determining sound attenuation from a known distribution of bubble sizes in the ocean; however, an integral equation must be inverted if attenuation is given and the distribution of bubbles is not. An approximate distribution can be determined based on the resonant bubble approximation (RBA). An iterative approach, for which the RBA represents the zeroth iteration, was proposed and carried out to the first iteration in a previous paper . It was suggested that additional iterations would improve the bubble-distribution results. Here we formulate the procedure to carry the results to a higher order and demonstrate, based on a theoretical distribution of a multiple power law form, the improvements in successive approximations of the bubble distribution to the fourth iteration level. A recursion relation is developed that allows one to carry the iteration out to an arbitrary order. It is shown that regions of the distribution that change in the power-law exponent are places where the higher order corrections improve the results the most.