Modifying sound speed to reduce phase errors in parabolic equations by a Toda flow method

Abstract

It is well known that the standard parabolic wave equation, in the absence of range dependence, has the correct normal nodes (amplitudes) but incorrect eigenvalues (phase). This is regrettable because the desired information is generally more sensitive to phase errors than to amplitude errors. Several authors have attempted to increase the accuracy of the phases by modifying the sound-speed profile, even though this decreases the accuracy of the amplitudes. Attempts have been made to eliminate the phase errors (accepting the increase in amplitude errors) by a Toda flow method. From a given sound-speed profile, one can easily generate a matrix with the desired eigenvalues. The spectral mapping theorem shows that a certain square root of this matrix would have eigenvalues that are distorted in the opposite way from the parabolic wave equation. The modified sound-speed profile that corresponds to the last matrix is sought: A system of spectrum-preserving differential equations is solved giving the evolution of this matrix into one where the modified sound-speed profile can be recovered.