## 2 Literature Review  
### Mathematical Models for Lidar Applications  
As described above, lidar systems emit laser beams with which they scan their surroundings. Here, the laser beam travels through the air, hits an object, is reflected back, passes through the lidar's optical detection system, and is registered by photo diodes. The detected laser power \(P_{\lambda}(r)\) depends on the sensor efficiency \(P_{\text{Sensor}}(r)\), the reflection characteristics of the object \(\beta(r)\), and the atmospheric losses \(P_{\text{loss}}(r)\), which depend on the distance \(r\) of the object to the lidar. Accordingly, the detected laser power \(P_{\lambda}(r)\) can be described in Equation (1):  
\[P_{\lambda}(r)=P_{\text{Sensor}}(r)\beta(r)P_{\text{loss}}(r). \tag{1}\]  
The influence of the three factors impacting the detected laser power \(P_{\lambda}(r)\) are discussed below. The sensor efficiency \(P_{\text{Sensor}}(r)\) equation is shown in Equation (2). \[P_{\text{Sensor}}(r)=P_{0}\frac{c\tau}{2}A\eta\frac{\xi(r)}{r^{2}} \tag{2}\]  
The first group of factors in Equation (2) denotes the intrinsic characteristics of the lidar system. Here, \(P_{0}\) is the initially emitted laser power, \(c\) is the speed of light, \(\tau\) stands for the spatial pulse width, the active sensor surface is described by \(A\), and the internal sensor losses are indicated by \(\eta\). \(\xi(r)\) stands for a sensor-specific function, which describes the overlap of two cones that originate from the emitter and detector in a bistatic beam setup (see Figure 2). These factors can be viewed as constant for this consideration and are not of deeper interest. The backscattering coefficient \(\beta(r)\) is an external factor, which is responsible for two things: firstly, if the laser beam hits an real target, some photons will be scattered back to the lidar, depending on the objects reflective characteristics; secondly, it will add noise to the point cloud by reflecting photons back to the lidar before the beam hits a real target. Small airborne particles cause scattering effects and scatter some photons back to the lidar sensor and will be detected by the detection system. The atmospheric loss factor, \(P_{\text{loss}}(r)\), shown in Equation (3), stands for the distance-related signal loss based on the extinction coefficient \(\alpha\). Photons will get absorbed or scattered by air molecules or particles. Here, the physical characteristics, namely the refractive index \(n\) and the particle diameter, have a major impact on the coefficient. The equation considers the photon beam's journey to the object and back to the sensor, hence the \"2\" factor in the exponent term. \[P_{\text{loss}}(r)=\exp\left(-2\int_{0}^{r}\alpha_{\text{ext},}(r^{\prime})\; dr^{\prime}\right) \tag{3}\]  
Plugging the Equations (2) and (3) in Equation (1) yields the laser equation as stated by [13]:  
\[P_{\lambda}(r)=P_{0}\frac{c\tau}{2}A\eta\frac{\xi(r)}{r^{2}}\beta(r)\exp\left( -2\int_{o}^{r}\alpha_{\text{ext},}(r^{\prime})\;dr^{\prime}\right), \tag{4}\]  
Although Equation (4) incorporates the effect of relevant environmental factors, the equation is not sufficient in a technical context, due to the fact that the laser diode array and the detector are in practice spatially separated as illustrated in Figure 2. This configuration is known as a bistatic beam setup. Depending on the distance \(r\) to the lidar, a potential target is only partially illuminated (\(R_{1}\leq r\leq R_{2}\)) or even not at all (\(r\leq R_{1}\)). Additionally, lidar uses pulsed lasers to scan the surroundings, which is also not considered in Equation (4). To address this issue, [9] have proposed lidar Equation (5), which considers the constraints of a physical device.