## 2 Methods
### Performance Simulation Using the Return Power Budget Method
#### 2.1.2 Signal to Noise Ratio  
We compute the SNR for one single laser shot for the case of a photon-counting LOTR adapted from Rocadenbosch et al. [38]:  
\[SNR_{1}(R)=\left(\mathcal{R}_{\text{sig}}(R)-\mathcal{R}_{\text{bg}}(R)\right) \cdot\frac{1}{ENF_{\text{PMT}}}\cdot\sqrt{\frac{2\Delta r}{c}\frac{1}{\left( \mathcal{R}_{\text{sig}}(R)+\mathcal{R}_{\text{bg}}(R)\right)}}\quad, \tag{5}\]  
where \(ENF_{PMT}\approx 1.08\) is the excess noise factor of our PMTs [39; 40], which accounts for the fluctuations in the signal produced by the PMT for each photoelectron, and \(\Delta r\) is the range resolution required (which might include rebinning of the data).  
In the far range, where the signal becomes background-dominated, the time required to reach a fixed minimum \(SNR_{\text{goal}}\) can be written as follows:  
\[T_{SNR_{\text{goal}}}(R)=\left(\frac{SNR_{\text{goal}}}{SNR_{1}(R)}\right)^{ 2}\frac{1}{PRF}\quad, \tag{6}\]  
where \(SNR\) is computed with Equation (5) and \(PRF\) is the laser pulse repetition frequency.  
In Figure 4, we report the \(T_{SNR_{\text{goal}}}(R)\) needed to reach \(SNR_{\text{goal}}\geq 10\) for the four wavelengths of interest and for two standard observation cases: that of a clear night and that for a night sky illuminated by a first or third quarter moon, a typical limit for IACT science data taking. One can see that in either case, the desired \(SNR_{\text{goal}}\) is reached within a minute up to a 25 km range. In both cases, a \(PRF\) of 10 Hz was assumed, according to the specifications of the pBRL laser [16].  
\begin{table}
\begin{tabular}{l c l} \hline
**Parameter** & **Value** & **Comments** \\ \hline
607 nm channel & & \\ \(E\) & 128 mJ & energy per pulse (at 532 nm) \\ \(N_{0}\) & 3.4\(\cdot\)10\({}^{17}\) & photons per pulse \\ \(l\) & 3.75 m & digitization length for 16-bit, 40 MS/s sampling rate \\ \(\rho\) & 0.97 & mirror reflectivity, after re-aluminization, otherwise \(<\)0.3 \\ \(\zeta\) & 0.14 \(\pm\) 0.02 & combined LLG and polychromator transmissions [16] \\ \(PDE\) & 0.04 \(\pm\) 0.01 & photon-detection efficiency [37] \\ \(\xi\) & 0.05 \(\pm\) 0.01 & combined channel efficiency \(\xi=\rho\cdot\xi\cdot PDE\) \\ \hline \end{tabular}
\end{table}
Table 1: _Cont._  
Figure 4: Required observation times to reach a \(SNR_{\text{goal}}\geq 10\) with a range resolution of 300 m for the pBRL with 10 Hz PRF under two example conditions: a clear night at the ORM (**Left**), and a moon-lit night with the maximum moon phase allowed by CTAO for science data taking (**Right**). The requirement for observation times less than one minute is marked as a horizontal dashed line.  
In Figure 5 we focus on \(T_{SNR_{\text{goal}}}(R)\) as a function of height for the weak 387 nm Raman line under different atmospheric conditions, laser power and pointing elevation: a vertically pointing RL during the clear night, the presence of _calima_, and moderate moon is shown, and one that points at the lowest required elevation of 25\({}^{\circ}\). We evaluated a case with a slow PRF = 10 Hz and a large laser pulse power of 70 mJ with that of a faster PRF = 200 Hz and a lower laser pulse power of 10 mJ. One can see again that within a minute of integration time, both solutions reached the required \(SNR_{\text{goal}}\geq 10\), with the latter proposal performing slightly better. For very low-elevation observations, this time interval allows for a precise determination of the return power only up to about (15-20) km.