deletions | additions       diff --git a/subsection_minimising_the_Jensen_Shannon__.tex b/subsection_minimising_the_Jensen_Shannon__.tex  index 783b2d7..8e21c10 100644  --- a/subsection_minimising_the_Jensen_Shannon__.tex  +++ b/subsection_minimising_the_Jensen_Shannon__.tex 
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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 $-dP_-$. That is, we minimise the sum of the two Jensen-Shannon distances  \[  \Delta = \Delta_{JS}(dP;\big(dP_+ \Delta_{JS}\big(dP;(dP_+  + dP_-)\big) + \Delta_{JS}(\rho cdU;\big(dP_+ \Delta_{JS}\big(\rho cdU;(dP_+  - dP_-)\big) \]  The Jensen-Shannon distance is zero only when the two distributions are identical and so we expect $\Delta > 0$ For noisy measurements. We argue, however, that the errors arising from the use of the incorrect wave speed $c$ will contribute to $\Delta$ so that it will be minimum when we use the true value of $c$.