## 2 Methods
### Analog to Photon-Counting Signal Gluing
#### 2.3.2 Likelihood-Based Gluing
The residuals have been fitted to a damped oscillation function  
\[r=A\cdot\exp\left(-\frac{2b}{c}\cdot R\right)\cdot\cos\left(\frac{4\pi f_{r}}{ c}\cdot R+\phi\right)\quad, \tag{23}\]  
where \(A\) the initial ringing amplitude, \(b\) the damping factor, \(c\) the speed of light, \(f_{r}\) the ringing frequency and \(\phi\) the phase. We tested a subtraction of either of the fitted residuals from the amplitude or photon-counting signal, but only the latter could effectively remove the oscillation from _both_ residuals (Figure 11, right). For this reason, we believe that the amplitude channel of the LOTR is still sufficiently linear, even in the case of strong over-driving of the LOTR, but the photon-counting baseline gets affected by ringing. Nonetheless, ringing affects the PC signal over a range in which the amplitude signal is normally used; hence, only the gluing procedure is slightly affected, but to a much smaller extent, the glued signal itself. \[\qquad\times PDF\big{(}r_{b}|\langle\mathcal{R}_{b}\rangle,s_{ \mathcal{R}_{b}}\big{)}\times PDF\big{(}a_{b}|\langle A_{b}\rangle,s_{A_{b}} \big{)} \tag{18}\]0 Visualization of the PC oscillations found during the likelihood-based gluing process on the 355 nm elastic channel with a strongly over-driven LOTR. Only the PC counting residuals on the lower left side had been corrected to obtain both figures on the right side. Whenever a fit (Equation (23)) to the photon-counting residuals after gluing (Equation (22)) shows an amplitude \(A\) larger than 0.2 photo-electrons per shot, the detected ringing is therefore subtracted from the photon-counting signal, and the gluing procedure is repeated. During the oscillation fits, the damping factor \(b\) is limited from below to \(b>0.01\)\(\mu\)s\({}^{-1}\), ensuring that the posterior corrections alter the signal at high ranges by always less than \(0.2\cdot\exp(-0.07\cdot R(\text{km}))\) photoelectrons per shot. However, in the cases studied with our data, the final amplitude and damping factors are orders of magnitude smaller (see Figure 11). The complete fitting procedure leads to single photoelectron gains compatible with those obtained from the \(\chi^{2}\)-based gluing procedure, but the dead times appear to be significantly larger (8-9 ns) than those provided by Licel (3-4 ns). The discrepancy is independent of the ringing correction and probably due to the limited bandwidth of a 10 m coaxial cable signal line connecting each PMT with its LOTR and the resulting increase in the pulse width of a single photoelectron. Figures 12 and 13 show scaled amplitudes \(a_{i}/N_{\text{slt}}\), dead-time-corrected photon counts \(m_{i}/N_{sh}\), and fitted photon count expectations \(\widehat{p}_{i}/N_{\text{sh}}\), before and after the ringing correction has been applied and together with the different fits (from initial guesses to the likelihood-based solutions before and after ringing corrections applied). One can observe that the solution before ringing correction is rather poor and does not describe well the data for amplitudes larger than \(\sim\)20 mV. After correction, the bias for large amplitudes has disappeared. On the right-hand side, the different signals are shown, which appear to match well, particularly after the ringing correction has been applied. A rather lengthy and CPU-intensive construction of a Hessian matrix \(H_{ij}=\partial\ln\mathcal{L}/\partial x_{i}\partial x_{j}\) from the likelihood Equation (20), with \(x_{i}\) and \(x_{j}\) being combinations of the parameters (\(g,\delta,\epsilon,\beta,a_{b}\) and \(r_{b}\)) around the likelihood maximum (\(\widehat{g},\widehat{\delta},\widehat{\epsilon},\widehat{\beta},\widehat{a}_ {b},\widehat{r}_{b}\)), profiled on all \(p_{i}\), allows to obtain an estimate of the covariance matrix \(H_{ij}^{-1}\) of the outer parameters. In elastic channels, we find uncertainties \(\sqrt{H_{ii}^{-1}}/\widehat{x}_{i}<0.2\%\) for all parameters, except \(\widehat{\beta}\), which exhibits an uncertainty of \(\approx\)0.5%.