Fitting a polynomial to a ridge in an array
Just a couple of thoughts:

That sounds like the classical application for a Kalman filter: you have one gaussian-overlaid signal that &quot;develops&quot; over the columns
If you have a signal model for your signal of interest, a parametric estimator might indeed be the best choice here.
You could write a likelihood function for each observation column $\mathbf c$, like $f_{\mu|\mathbf c}(\mu)=\ldots$ that describes how likely a given $\mu$ is given the observation in each column.
Then, you could state $\hat \mu = \arg\max\limits_\mu f_{\mu|\mathbf C}(\mu)$.
If you can assume your columns to be independent: $f_{\mu|\mathbf C}=\prod\limits_{\mathbf c \in \mathbf C}f_{\mu|\mathbf c}$, so that
\begin{align}\hat \mu &amp;= \arg\max\limits_\mu f_{\mu|\mathbf C}(\mu)\\
&amp;= \arg\max\limits_\mu\;\log\left(f_{\mu|\mathbf C}\right)\\
&amp;= \arg\max\limits_\mu\;\log \left(\prod\limits_{\mathbf c \in \mathbf C}f_{\mu|\mathbf c} \right)\\
&amp;= \arg\max\limits_\mu\;\sum\limits_{\mathbf c \in \mathbf C}\log\;f_{\mu|\mathbf c}\text,\end{align}(the logarithm might be very handy, considering the exponential nature your signal of interest seems to have). From there, it's an optimization problem.