Well I'm amazed at all the work done already!  Sorry for the incomplete equation.  Note it is a direct copy from the Berktay paper mentioned, so it could be hunted down there.  The copy I have is Where Audible secondary pressure wave misc. physical parameters SPL of the ultrasonic carrier wave Envelope function (such as DSB-AM)This equation says that the audible demodulated ultrasonic pressure wave (output signal) is proportional to the twice differentiated, squared version of the envelope function (input signal).  Precompensation refers to the trick of anticipating these transforms and applying the inverse transforms on the input, hoping that the output is then closer to the untransformed input.