## 2 Methods
### Analog to Photon-Counting Signal Gluing
#### 2.3.2 Likelihood-Based Gluing
Defining \(m_{i}=N_{sh}\mathcal{R}_{i,\text{obs}}/f_{s}\) and using Poissonian statistics for the PC part and Gaussian statistics for the amplitudes, we define the following likelihood:  
\[\ln\mathcal{L}= \sum_{i=j_{\text{on}}}^{j_{\text{on}}+n_{\text{on}}}\Big{[}- \frac{\epsilon p_{i}}{1+\delta\epsilon p_{i}}+m_{i}\ln\!\left(\frac{\epsilon p _{i}}{1+\delta\epsilon p_{i}}\right)\] \[-\frac{\ln\!\left(N_{sh}\gamma^{2}+F^{2}g^{2}p_{i}\right)}{2}- \frac{(a_{i}-g(p_{i}-\beta r_{b})-a_{b})^{2}}{2\!\left(N_{sh}\gamma^{2}+F^{2} g^{2}p_{i}\right)}\Big{]}\] \[+\sum_{i=j_{\text{off}}}^{j_{\text{off}}+n_{\text{off}}}\Big{[}- \frac{\epsilon r_{b}}{1+\delta\epsilon p_{b}}+m_{i}\ln\!\left(\frac{\epsilon r _{b}}{1+\delta\epsilon p_{b}}\right)\Big{]}-\frac{(a_{b}-N_{sh}\langle A_{b} \rangle)^{2}}{2N_{sh}s_{A_{b}}^{2}} \tag{20}\]  
This likelihood has \(n_{\text{on}}\) independent parameters \(p_{i}\), apart from the outer parameters \(g,\delta,\epsilon,r_{b}\) and \(a_{b}\). Like in Veberic [49], the parameter \(\gamma\) can be previously determined from the residuals of a linear fit to \(m_{i}/N_{sh}\) as a function of \(A_{i,\text{obs}}\) over a range, where \(m_{i}/N_{sh}\) contains their lowest 2%. We checked that \(\gamma^{2}\) obtained this way is almost identical to \(s_{A_{b}}^{2}\). Then it was fixed for the next steps. To numerically maximize the likelihood, we follow the same approach as Veberic [49], by minimizing \(-2\ln\mathcal{L}\) with respect to the set of \(p_{i}\) (with the minimum located at \(\widehat{p}_{i}\)), for a given set of external parameters and minimize then the \"outer\" negative likelihood to find the solutions \((\widehat{g},\widehat{\delta},\widehat{\epsilon},\widehat{p},\widehat{a}_{b },\widehat{r}_{b}|\widehat{p}_{i})\), which provide a global maximum of Equation (20). Initialized with the prescriptions of Veberic [49], \(\epsilon_{\text{init}}=0.95\), \(\beta_{\text{init}}=1\), \(a_{\text{init}}=N_{sh}\langle A_{b}\rangle\), and \(r_{b\text{init}}=N_{sh}\langle\mathcal{R}_{b}\rangle\), the likelihood always converges in less than forty evaluations of the outer function. An issue with Equation (20) needs extra attention: By construction, the combination of parameters \(\epsilon,p_{i},g\) and \(\beta\) is degenerate or may lead to \(p_{i}\to\infty\), when simultaneously \(g\to 0\) and \(\epsilon\to 0\). This has to do with the fact that the PMT excess noise variance multiplies with \(g^{2}\), but only with \(p_{i}\), while all other ingredients of the likelihood, Equation (20), show a linear combination of \(\epsilon\) or \(g\) and \(p_{i}\). We suspect that this may be the reason why Veberic [49]did not treat PMT excess noise at all in their likelihood. To remedy this behavior, we maximize \(\ln\mathcal{L}\) twice: first, in a setup similar to Veberic [49]'s approach, in which the excess noise contribution \(F\) is set to zero. With the corresponding results, we pick \(\widehat{e}\) and fix it during a second maximization of \(\ln\mathcal{L}\), this time with \(F\) set to its correct value. The latter maximization does not diverge anymore to infinite values of \(p_{i}\). Finally, our data themselves have shown ringing of the PC baseline, after injecting signal pulses into the LOTR, which exceed the maximum amplitude of 500 mV, recommended by the provider. To highlight this behavior, Figure 11 shows residuals of \(a_{i}/N_{sh}\) and \(m_{i}/N_{sh}\) at the location of the likelihood maximum:  
\[r(a_{i}) =\frac{(a_{i}-\widehat{a}_{b})-\widehat{g}\Big{(}\widehat{p}_{i} -\widehat{\beta}\widehat{p}_{b}\Big{)}}{N_{sh}} \tag{21}\] \[r(m_{i}) =\frac{m_{i}-\widehat{e}\widehat{p}_{i}/\Big{(}1+\widehat{ \delta}\widehat{e}\widehat{p}_{i}\Big{)}}{N_{sh}} \tag{22}\]  
as a function of range \(R\).