## 2 Methods
### Analog to Photon-Counting Signal Gluing
#### 2.3.1 \(\chi^{2}\)-Based Gluing  
In this procedure, the photon-counting channel is first corrected for pulse pile-up [46; 50; 51], assuming a non-paralysable [52] system, following:  
\[\mathcal{R}_{i,\text{corr}}=\frac{\mathcal{R}_{i,\text{obs}}}{1-\tau\mathcal{ R}_{i,\text{obs}}}, \tag{12}\]  
where \(\mathcal{R}_{i,\text{obs}}\) is the recorded photon rate (normally expressed in units of MHz) in range bin \(i\), \(\mathcal{R}_{i,\text{corr}}\) is the pileup-corrected one, and \(\tau\) is the resolving time of the discriminator-counter combination (the \"dead time\") of the system. According to the recommendations of Liel [53], we used \(\tau=3.70\times 10^{-3}\) us for TR-20 LOTR modules operated at a sampling rate of \(f_{\text{s}}=20\) MSamples/s and \(\tau=3.06\times 10^{-3}\) us for the more recent TR-40 models, with \(f_{\text{s}}=40\) MSamples/s [16]. However, with the more elaborate likelihood-gluing method (Section 2.3.2), we found that the deadtime was rather of the order of \(\tau\approx 8\) ns, which we use in the following. The constant photon background \(\mathcal{R}_{b}\) is subtracted _after_ the pileup correction, and a general PC efficiency \(\epsilon\) can be introduced, leading to a background- and efficiency-corrected photon count rate and variance of the following:  
\[\mathcal{R}_{i,\text{chi2}} =\frac{1}{\epsilon}\cdot\left(\frac{\mathcal{R}_{i,\text{obs}}}{ 1-\tau\mathcal{R}_{i,\text{obs}}}-\frac{\mathcal{R}_{b}}{1-\tau\mathcal{R}_{ b}}\right) \tag{13}\] \[s_{U,L}^{2}(\mathcal{R}_{i,\text{chi2}}) =\frac{1}{\epsilon^{2}}\cdot\left(\frac{f_{s}^{2}}{N_{\text{ soft}}^{2}}\cdot\frac{|X_{i}-\mu_{UL}(X_{i})|^{2}}{\left(1-\tau\mathcal{R}_{i, \text{obs}}\right)^{4}}+\frac{s_{\mathcal{R}_{b}}^{2}}{\left(1-\tau\mathcal{R} _{b}\right)^{4}}\right)\quad, \tag{14}\]  
where Poissonian statistics have been assumed for the total accumulated number of photoelectrons \(X_{i}=N_{\text{sh}}\mathcal{R}_{i,\text{obs}}/f_{\text{s}}\) for the (asymmetric) variances \(s_{U,L}^{2}\). Some care must be taken when calculating the Poissonian variances of the measured photo-electron counts \(X_{i}\) in a given sampling bin \(i\). The 68.3% Confidence Interval (CI) does not correspond to \([X_{i}-\sqrt{X_{i}},X_{i}+\sqrt{X_{i}}]\), due to the asymmetry of the Poissonian probability mass function, particularly for low rates. We use the prescription of Garwood [54] (implemented as method exact-c in statsmodel's function confint_poisson) for an improved asymmetric CI \([\mu_{L},\mu_{U}]\), so that the Poissonian probability \(P_{L}=P(X\geq X_{i}|\mu=\mu_{L})=0.1587\) and \(P_{U}=P(X\leq X_{i}|\mu=\mu_{U})=0.1587\) and \(\mu_{L}\) and \(\mu_{U}\) are obtained from the \([0.1587,1-0.1587]\) quantiles of the gamma-function with \(X\) as shape parameter (although that method still suffers from slight over-coverage [55; 56]). The raw amplitudes \(A_{i,\text{obs}}\) have a contribution from the same photon background, scaled to AN voltages: \(g\,\mathcal{R}_{b}\), where \(g\) is the signal photo-electron gain, and the electronic baseline voltage \(B\). The sum of both has been measured as background in the AN channel as \(A_{b}=B+g\,\mathcal{R}_{b}\), hence: \(B=A_{b}-g\,\mathcal{R}_{b}\). For the amplitude part, we assume linear scaling between registered amplitudes \(A_{i,\text{obs}}\) above measured offset \(A_{b}\pm s_{A_{b}}\) and the background-corrected photon count rates, leading to the \"virtual amplitude count rates\":  
\[\mathcal{R}^{\prime}_{i,\text{chi2}} =f_{s}\cdot\frac{(A_{i,\text{obs}}-A_{b}-O)}{g} \tag{15}\] \[s^{2}(\mathcal{R}^{\prime}_{i,\text{chi2}}) =f_{s}^{2}\cdot\frac{(A_{i,\text{obs}}-A_{b}+g\,\mathcal{R}_{b}) }{g\,N_{sh}}\cdot ENF^{2}+f_{s}^{2}\cdot\frac{s_{A_{b}}^{2}}{g^{2}}\quad, \tag{16}\]  
with \(ENF\approx 1.08\) denoting the excess noise factor of the PMTs used [39; 40].