**4. Applications**

Models provide a tool to predict light behavior in different situations. There exist two basic analytical models that correspond to two particular situations in which the RTE has an analytical solution. The solutions can be used to compare the results obtained with stochastic models, such as MC, and verify that MC can be also applied to solve RTT without solving the RTE.

On the one hand, it is well known that when absorption prevails over scattering (*μ<sup>a</sup>* ≫ *μs*), Beer-Lambert law can be applied [22]. On the other hand, under certain conditions of the diffusion regime (10*μ<sup>a</sup>* ≈*μs*), in source-free and homogeneous media conditions, there is an analytical solution based on the theory of photon density wave in steady conditions [38]. **Figure 4** shows, in both cases, results obtained from MC simulations (red dots) and the analytical solution (blue line). They are superimposed showing the validity of our implementation of the MC approach.

As discussed, if one wants to study pulsed light propagation of a LiDAR system, time-resolved simulations have to be used. The MC code needs to be modified by adding a record of the time history of each photon to have the time-tracking of light propagation. By using this history, it is possible to generate the temporal profile of

#### **Figure 4.**

*Left: Beer-Lambert Law. Propagation of light through a medium with <sup>μ</sup><sup>a</sup>* <sup>¼</sup> <sup>10</sup>*cm*�1,*μ<sup>s</sup>* <sup>¼</sup> <sup>0</sup>*:*05*cm*�1,*<sup>g</sup>* <sup>¼</sup> <sup>0</sup>*:*9*. Right: Diffusion regime. Propagation of light through a medium with <sup>μ</sup><sup>a</sup>* <sup>¼</sup> <sup>0</sup>*:*05*cm*�1,*μ<sup>s</sup>* <sup>¼</sup> <sup>20</sup> *cm*�1,*<sup>g</sup>* <sup>¼</sup> <sup>0</sup>*:*2*.*

optical power. For this case, there also exist situations in which there is an analytical solution to the RTE [39]. For example, in the case of an infinite diffusing medium with a point source, as shown in **Figure 5**, the setting between the analytical and the simulated solution is almost perfect.

Finally, **Figure 6** shows how models based on MC methods are able to predict pulsed light interactions, which is, in fact, the topic of interest in this chapter. It shows the temporal profile of a pulse of light sent of Gaussian shape and 1 mJ of energy (in orange) and light received back to the lighting (in blue). The medium is characterized by the following properties: *<sup>μ</sup><sup>a</sup>* <sup>¼</sup> <sup>0</sup> *cm*�1,*μ<sup>s</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>01</sup> *cm*�1,*<sup>g</sup>* <sup>¼</sup> <sup>0</sup>*:*9, which would correspond to an almost clear media, and a 100% reflective plane is placed at a distance of

**Figure 5.** *Time-resolved propagation of light through a medium with <sup>μ</sup><sup>a</sup>* <sup>¼</sup> <sup>0</sup>*cm*�1,*μ<sup>s</sup>* <sup>¼</sup> <sup>2</sup>*cm*�1,*<sup>g</sup>* <sup>¼</sup> <sup>0</sup>*:*9*:*

**Figure 6.** *Light signal sent and received through a medium with <sup>μ</sup><sup>s</sup>* <sup>¼</sup> <sup>0</sup>*:*01*cm*�1,*<sup>g</sup>* <sup>¼</sup> <sup>0</sup>*:*9*.*

*Modeling the Use of LiDAR through Adverse Weather DOI: http://dx.doi.org/10.5772/intechopen.109079*

0.5 m. It can be observed from this simple figure the working principle of LiDAR. The position of the reflected light peak, with respect to the initial pulse, allows us to obtain information on the position of the object. The shape of the pulse is maintained and the integral of both pulse profiles meets the expected energy of the propagation calculated from the range equation.

One of the problems shown by LiDAR technology when working through adverse weather conditions is the saturation of the sensor. This saturation is the result of the backscattering that light undergoes when just entering the media, and it may saturate

**Figure 7.**

*a) Detected backscattering in a medium with g* ¼ 0*:*9 *and different μs, for a light pulse with an initial energy of 1 mJ. b) Experimental results of detected backscattering in a foggy medium for different visibilities.*

the sensor. Thus, to start with, it is interesting to analyze simple backscattering interactions in different media in order to have an idea of how the first light that would arrive at our detection in foggy conditions would influence our sensor. As the final objective is to use these MC methods to improve a long-range active sensing technique that interacts with a target, we also focused on studying the effect of reflected light that reaches back the plane of illumination, i.e., the light that comes back from an object hidden behind the turbid environment. Therefore, we want to know how the pulse may be attenuated and what the range of the system will be.

**Figure 7a** shows different profiles of backscattered light simulated using an MC code. The greater the scattering coefficient, the greater the amount of backscattered energy and the more extended in time. It can be also observed that the backscattering signal does not have the same shape as the initial pulse, which is defined as a perfect Gaussian-shaped pulse [40]. **Figure 7b** shows experimental data of a pulsed LiDAR working under different artificial fog densities, simulating different meteorological visibilities. One can see that stochastic models (**Figure 7a**) reproduce the expected behavior of light (**Figure 7b**).

Using the simulations and knowing the specifications of our sensor, one can adjust electronic gain and optical power to avoid saturation of it, or even come up with solutions related to the optical design of the system.

Next, in **Figure 8** we present the temporal profiles under the same conditions as in **Figure 7a**, but with the presence of an object placed at 0.5 m with a reflectivity of 100%. If one supposes that the sensor is not saturated under any of the presented conditions, it is observed that around the expected position of the object appears the peak that corresponds to its presence along with the backscattering contribution. Using the simulations, it is possible to evaluate what is the level of scattering at which the target is no longer detected (it is not distinguishable from the backscattering signal), so it would not be possible to obtain its image or compute its distance.

The same type of study can be performed to establish the range limit of the system taken into account between visibility and surface properties of the object. Data from an experimental test are shown in **Figure 9**, and it can be seen how both parameters may influence the results.

**Figure 8.**

*Detected backscattering in a medium with g* ¼ 0*:*9 *and different μs, for a light pulse with an initial energy of 1 mJ, with the presence of a perfect reflecting target.*

*Modeling the Use of LiDAR through Adverse Weather DOI: http://dx.doi.org/10.5772/intechopen.109079*

#### **Figure 9.**

*Signal detected from a returning pulse of light pointing toward (LEFT) a metallic object at 12 m of distance and (RIGHT) a diffusive object at 14 m of distance, for two different visibilities (20 m and 75 m).*
