deletions | additions       diff --git a/subsection_minimising_the_Jensen_Shannon__.tex b/subsection_minimising_the_Jensen_Shannon__.tex  index d6c8696..04e56d2 100644  --- a/subsection_minimising_the_Jensen_Shannon__.tex  +++ b/subsection_minimising_the_Jensen_Shannon__.tex 
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\subsection{minimising the Jensen-Shannon distance between $P$ and $dP_+ ($dP_+  + dP_-$ dP_-$)  and $\rho d dU$ and $dP_+ ($dP_+  - 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 sum of the distances between the these distributions will be minimum when we use the true value of $c$.  This argument follows from looking at what happens as our estimate of $c$ deviates from its 'true' value. For Consider first  $c$ smaller than the true value value. In this case,  the calculated values of $dP_\pm$ will be influenced more by $dP$ and less by $dU$. In the extreme $c = 0$ with $dP_+ = dP_- = \frac{1}{2}dP$ and the separated pressure changes depend only on $dP$. As the separated pressure changes become more like $dP$, the Jensen-Shannon distance between the sum of the separated pressures and $dP$ will decrease, being zero in the limit of $c=0$. The distance between the difference of the separated pressures and $\rho c dU$, however, will increase until it is equal to $\frac{1}{2}H(dU/\rho c)$ in the limit $c=0$.  For $c$ larger than the true value the opposite is true with $dP_\pm$ being more influenced by $dU$ and less by $dP$. In the extreme $c \rightarrow \infty$, $dP_+ = - dP_- = dU/\rho c$ and the separated pressure changes depend only on $dU$. -------- edited to here --------