## 3 Methodology for Evaluating the Effectiveness of Rock Fall Protections
### Re-Evaluation of Hazard at the Study Site
#### 3.3.2 Protection Measures in the Propagation Zone  
As far as rock fall protections in the propagation zone are concerned, starting from Equation (4), it can be written:  
\[\begin{cases}\text{E}<\text{E}_{\text{opt}}\\ \text{d}<\text{D}_{\text{opt}}\end{cases}\rightarrow\text{H}=\text{H}(\text{E }_{\text{m}},\text{T}_{\text{m}})<\text{H}(\text{E},\text{T}) \tag{15}\]  
However, it must be pointed out that, despite location, energy absorption capacity and height being selected in an optimal way, it can still occur in some instances that rock fall modelling tools (such as trajectory simulation software) might prove that the measures would intercept the vast majority, but not all of the possible paths followed by the blocks. On the other hand, since the optimal design of the measure is expected to reduce this phenomenon considerably, even in those cases in which a certain percentage \(\text{P}_{\text{r}}{}^{\star}\) of trajectories could develop above the measure (d \(>\text{D}_{\text{opt}}\rightarrow\text{P}_{\text{r}}{}^{\star}>0\)), this percentage would be most likely very small and, thus, related to a very high value of return period, according to Equation (2):  
\[\text{T}=1/(\text{P}_{\text{f}}\cdot\text{P}_{\text{r}})\rightarrow\text{T}^{ \star}=1/(\text{P}_{\text{f}}\cdot\text{P}_{\text{r}}{}^{\star})>\text{T}=1/( \text{P}_{\text{f}}\cdot\text{P}_{\text{r}})\qquad\text{with }\text{P}_{\text{r}}{}^{\star}<\text{P}_{\text{r}} \tag{16}\]  
In this relation, T* is the return period associated to the frequency of reach \(\text{P}_{\text{r}}{}^{\star}\), computed after the effects of the protection are considered and accounting for a few potential trajectories travelling beyond the measures; \(\text{P}_{\text{r}}\) and T are the frequency of reach and return period without the effects of the protection measures.  
A very high value of return period could imply such a rare occurrence that the resulting hazard level could still be compatible with the safety criteria established for the exposed areas, i.e., T* \(\geq\text{T}_{\text{m}}\) (even regardless of the energy along the trajectories), which means it can be written T* = Topt.  
If the condition \(d<D_{opt}\) cannot strictly be satisfied, therefore, a control to verify whether \(T^{*}\geq T_{m}\) must still be performed, to check whether the design of the protection can still be considered as optimal, despite not all trajectories being intercepted by the protection.  
Based on these considerations, the condition for hazard mitigation expressed by Equation (4) can be restructured as:  
\[\begin{cases}E<E_{opt}\\ d<D_{opt}\end{cases}\ \rightarrow\ \begin{cases}E<E_{opt}\\ T=T_{opt}\end{cases}\text{with }T_{opt}\ \geq\ T_{m} \tag{17}\]  
where \(T_{opt}=1/(P_{f}\cdot P_{r,opt})\) and \(P_{r,opt}\) is the frequency of reach at the location of the protection in perfect working order.  
In line with the structure of Equation (17), for hazard mitigation in the presence of existing protection measures, it must result:  
\[\begin{cases}E<E_{red}\\ d<D_{red}\end{cases}\ \rightarrow\ \begin{cases}E<E_{red}\\ T=T_{red}\end{cases}\text{with }T_{red}\ \geq\ T_{m} \tag{18}\]  
where \(T_{red}=\)1/(\(P_{f}\cdot P_{r,aug}\)) and \(P_{r,aug}\) is the frequency of reach at the location of the protection, augmented by the reduction of the effective height of the measure.  
The criteria for establishing the effects of existing protections in the propagation zone on hazard assessment and zoning are, therefore, for barriers, embankments and walls:  
\[H=\left\{\begin{array}{ll}H(E_{m},T_{m})&\text{if }\ \ E<E_{red}\text{ and }(d<D_{red}\text{ or }d\ |\ T_{red}\geq T_{m})\\ H(E,T)&\text{if }\ \ E\geq E_{red}\text{ or }(d\geq D_{red}\text{ and }d\ |\ T_{red}<T_{m})\end{array}\right. \tag{19}\]  
whereas for ditches and berms:  
\[\left\{\begin{array}{ll}H(E_{m},T_{m})&\text{if }\ \ d<D_{red}\text{ or }d\ |\ T_{red}\geq T_{m}\\ H(E,T)&\text{if }\ \ d\geq D_{red}\text{ and }d\ |\ T_{red}<T_{m}\end{array}\right. \tag{20}\]