## 5 The Methodology  
The first step of the proposed methodology consists of defining a Unified Climatic Parameter (UCP) for a specific building material, in this case SP calcarenite. This UCP is a linear combination of the climatic parameters (cp\({}_{i}\)) affecting the deterioration process of the building material, calculated with Equation (2), where a\({}_{i}\) is the weighting coefficients for the studied material and for all the deterioration parameters (cp\({}_{i}\)) considered. In the case of SP calcarenite in the province of Granada, two climatic parameters are considered: the number of freezing days per year (n\({}_{\text{fdy}}\)) and the mean relative humidity (RH) (see Figure 5) [23]. To obtain a simple relation for the evolution of the recession (R, in mm) that is easy to compute, and considering that Tan[x] is nearly linear for values of x in the interval [\(-0.5\), \(0.5\)], the expression proposed in [23] is adopted to calculate the recession value:  
\[\text{R}=\text{t-}\xi\text{Tan}[\text{UCP}]=\text{t-}\xi\text{Tan}[\sum_{i=1}^ {n}a_{i}\text{cp}_{i}] \tag{3}\]  
Here, t represents the number of exposed years, and \(\xi\) is the parameter used to adjust the UCP to the photogrammetric value observed on site. Equation (3) applies to a single climate condition. If climatic parameters change over time, Equation (4) should be used:  
\[\text{R}=\sum_{j=1}^{N}\Delta_{j}\cdot\xi\text{Tan}[\sum_{i=1}^{n}a_{ij}\text {cp}_{i}] \tag{4}\]  
The recession obtained with photogrammetry is shown in the seventh column of Table 2 (column 6 divided by column 5). For the case studied, Equation (3) is expanded to a non-linear system of 3 equations with three unknowns (\(\xi\), a\({}_{1}\), and a\({}_{2}\)) obtained from \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**HISTORICAL** & **Year of** & **Masl** & **Location** & **Affected** & **Average** & **R** & **Standard** \\
**BUILDING** & **Construction** & **(m)** & & **Height (m)** & **Area (m2)** & **(mm)** & **Deviation** \\ \hline Del Carmen & 1589 & 895 & Main facade (NW) & 1.70 & 0.0323 & 19.0 & 0.012 \\ \cline{3-7} Church & & & Lateral facade (NE) & 1.00 & 0.0238 & 23.8 & 0.006 \\ \hline Nusetra Señora & 1778/1573 & 173 & Main facade (SE) & 2.75 & 0.0003 & 0.11 & 0.002 \\ \cline{3-7} del Rosario & & & Corner (S) & 2.50 & 0.0013 & 0.52 & 0.002 \\ \hline San Jerónimo & 1519 & 689 & Main facade (NW) & 2.00 & 0.0084 & 4.2 & 0.0001 \\ \cline{3-7} Monastery & & & Buttress (SE) & 2.50 & 0.0500 & 20.0 & 0.0079 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average area affected by deterioration. Figure 5: Flow chart of the methodology proposed for SP calcarenite in Granada. observation. Each of the equations corresponds to one of the buildings studied, as specified in the previous section:  
\[ \begin{array}{l}21.4=434.\xi\mathrm{Tan}[49.77\mathrm{a}_{1}+67.6\mathrm{a}_{2 }]\\ 0.315=347.5\cdot\xi\mathrm{Tan}[0.36\mathrm{a}_{1}+71.74\mathrm{a}_{2}]\\ 12.1=502\cdot\xi\mathrm{Tan}[49.13\mathrm{a}_{1}+63.95\mathrm{a}_{2}]\end{array}  \tag{5}\]  
The system of equations in Equation (5) is solved by using weighted least squares, with the condition that the UCP is \(\pi/2\) for the worst-case scenario of the climatic parameters (i.e., 365 days per year with temperatures below zero and 100% relative humidity). Using this approach, the values obtained for the three parameters are as follows: \(\xi\) = 0.1, \(\mathrm{a}_{1}\) = 4.27 \(\times\) 10\({}^{-3}\), and \(\mathrm{a}_{2}\) = 1.02 \(\times\) 10\({}^{-4}\). To validate the method, the data obtained from the laboratory are incorporated into the set of non-linear equations (Equation (5)). The number of freeze-thaw cycles is considered as the number of freezing days. Experimental data corresponding to 50 freeze-thaw cycles in Figure 2 are used in the weighted least squares through Equation (1). In this case, the values of the unknowns in Equation (3) change slightly: \(\xi\) = 0.109, \(\mathrm{a}_{1}\) = 4.28 \(\times\) 10\({}^{-3}\), and \(\mathrm{a}_{2}\) = 0.95 \(\times\) 10\({}^{-4}\).