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\subsection{minimising the Jensen-Shannon distance between $P$ and $dP_+ + dP_-$ and $\rho d c  dU$ and $dP_+ - dP_-$} The separation of the measured $dP$ and $dU$ can be done for any arbitrary value of $c$ usng the water hammer equations and the assumption that the forward and backward waves are additive. Thus for any value of $c$ (assuming that $\rho$ is a constant) we obtain distributions of $dP_+$ and $dP_-$. For noiseless measurements and the 'true' value of $c$, the distribution of the measured $dP$ should equal the sum of the distributions of $dP_+$ and $dP_-$ and the distribution of $\rho c dU$ should equal the sum of the distributions of $dP_+$ and $-dP_-$. For noisy measurements these distributions will never be equal. We argue, however, that the difference between the differences in these distribution will be minimum when we use the true value of $c$.