# Monte Carlo Simulations in Aviation Contrail Study: A Review
## 4 Statistical Approaches and Monte Carlo Simulations for Global Evaluation of Contrail Effect  
At the end of the last century, the Intergovernmental Panel for Climate Change (IPCC) promoted an assessment of aviation's contribution to global warming [94], from which a series of uncertainties in the quantitative evaluation arise. Apart from the microphysical characteristics of cirrus that directly come from air transport, the second source of uncertainties can be found in the dynamics of the aircraft-generated cloudiness. An interesting statistical approach to derive the actual contrail lifetimes, based on geostationary satellite observations, was presented in [95] and is briefly reviewed here. The method in [95] exploits the availability of an automatic contrail tracking algorithm, capable of extracting a trail's evolution information from remote-sensing images. Two examples of such algorithms are available in [96; 97]. Contrail tracking is limited by satellite spatial and wave-length resolutions, which affect the initial detection of a trail and its disappearance, respectively. In fact, both determine a time bias between (i) a contrail's first observation and its actual genesis (pre-detection time) and (ii) contrail undetectability in images and its radiative forcing contribution becoming negligible. Analyzing the ACTA (automatic contrail tracking algorithm) database [98], the authors of [95] found that the contrails' survival function,  
\[S(t)=\frac{n(t)}{n(0)}, \tag{7}\]  
with \(n(t)\) representing the number of contrails still observable in satellite images at time \(t\), is well fitted by a generalized ([PERSON] [99]) exponential law  
\[S(t)=\exp\Bigl{[}(-\lambda\ t)^{k}\Bigr{]}. \tag{8}\]  
Apart from reproducing the decreasing number of observed contrails, Equation (8) can be used for estimating their unobserved persistency times. Consider that the percentage of contrails observed at time \(T=\tau\) survives for a further time lapse \(\delta\) given by  
\[\frac{n(\tau+\delta)}{n(\tau)}=\frac{S(\tau+\delta)}{S(\tau)}=P\ (T=\tau+\delta\mid T=\tau), \tag{9}\]  
from which the conditional probability density function  
\[f(\delta|\tau)=-\frac{d}{d\delta}\biggl{(}\frac{S(\tau+\delta)}{dS(\tau)} \biggr{)}, \tag{10}\]  
can be derived. Via exploitation of a regression to the database with Equation (8), and via a rule-of-thumb that suggests a slow (\(\sim 5\) km/h), constant contrail spreading speed--from which a Gaussian-like distribution of their width before detection can be postulated--it has been possible to build up a Monte Carlo calculation for extracting the actual life-time of observed contrails. In fact, three random, uniformly distributed numbers are generated and used to sample the three available distributions, that is, the time in which a contrail is revealed by satellite, the pre-detection time, and the time a contrail survives after its disappearance from satellite. The example discussed above shows once again the preeminent role of statistics and a Monte Carlo simulation in the study of contrails in aviation, even on a different length scale. Apart from microphysical characteristics and the actual lifetime of aircraft-induced cloudiness, another source of the uncertainty discussed in the IPCC report lies in the contrails' contribution to climate. As a matter of fact, assessing mitigation options for contrail-induced climate effects, in terms of the adoption of both technology and local diversion strategies despite large uncertainties regarding atmospheric processes, is currently a theme of scientific debate [100; 101; 102], involving different future air-traffic scenarios. A detailed, fully-coupled calculation evaluating the effect of aviation-induced cloudiness on a global scale presents the same complexity of the quantitative estimation of the whole radiative forcing due to natural stratiform clouds.