the sum of these variables is also Gaussian, and the likelihood of observing a configuration of measured signals $\{M^i\}$ as ${\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace};{\ensuremath{\{M^{i}\}\xspace}}) = P({\ensuremath{\{M^{i}\}\xspace}}| {\ensuremath{{ {\mathbf g}}}\xspace})$, where $P({\ensuremath{\{M^{i}\}\xspace}}| {\ensuremath{{ {\mathbf g}}}\xspace})$ is the probability density function of observing the configuration [$\{M^{i}\}\xspace$]{}given parameters ${\ensuremath{{ {\mathbf g}}}\xspace}$. $${\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace};{\ensuremath{\{M^{i}\}\xspace}}) = \prod_{i} \frac{1}{\sqrt{2\pi h^i}} \exp\left[-\frac{(M^i - \mu^i)^2}{2 h^i} \right]$$ where $\mu^i = 1 + {\ensuremath{{ {\mathbf g}}}\xspace}\cdot{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}$ is the mean value of $M^i$ and $h^i = (\delta c^i / c_0)^2 + {\ensuremath{\sigma_{\Delta}}}^2$ its variance. We want to apply the method of maximum likelihood by finding the gradient parameters $\hat{{\ensuremath{{ {\mathbf g}}}\xspace}}$ that maximize this likelihood, i.e. $$\hat{{\ensuremath{{ {\mathbf g}}}\xspace}} = \textrm{arg} \, \max\limits_{{\ensuremath{{ {\mathbf g}}}\xspace}} {\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace};{\ensuremath{\{M^{i}\}\xspace}})$$ However, because of the complex dependence of $h^i$ on the gradient ${\ensuremath{{ {\mathbf g}}}\xspace}$, this is not possible analytically. We perform this optimization numerically using a Nelder-Mead method (Matlab’s fminsearch), with an initial guess set by the maximum for $n_r \to \infty$ (i.e. neglecting concentration sensing noise), which can be found exactly. For numerical convenience, we maximize the log-likelihood $\ln {\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace};{\ensuremath{\{M^{i}\}\xspace}})$, $\ln {\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace};{\ensuremath{\{M^{i}\}\xspace}}) = -\frac{1}{2}\sum_{i} \ln h^i - \sum_i \frac{(M^i - \mu^i)^2}{2 h^i}$ up to 