## 2 Materials and Methods
### Potential Ecosystem Services Favorability Map  
The provision of potential ecosystem services is a synthetic decision combining different factors whose relative importance is prioritized according to a value judgment involving the evaluators. From a geospatial perspective, one of the approaches to perform this synthesis is to use a multi-criteria analysis method incorporated in a GIS forming a system called Spatial Multicriteria Analysis (SMCA) or Spatial Multi-Criteria Decision Support System (MC-SDSS) [75]. It was therefore decided to adopt such a system to map the potential ES of the Ouled Hanneche forest by combining the three quantitative indicators described above. Specifically, these three factors are the average NDVI between 8 May 2017 and 25 June 2017, the multidimensional Rao entropy between these same dates, and the Shannon entropy of the geomorphon classes. All three are masked to keep only the pixels located on the land cover classes corresponding to the GV-GV-GV, GV-NPV-NPV, SNOW-GV-GV, and NPV-GV-GV classes. The MC-SDSS procedure implies that the indicators are standardized to be established in a common evaluation scale [75]. A linear normalization according to the rank method (min-max) was applied to the three rasters using the spatialEco package [76]. This transformation is applied to data without negative values. The formula is the following:  
\[v(a_{ik})=\left((a_{ik}-min_{i}\{a_{ik}\})/(max_{i}\{a_{ik}\}-min_{i}\{a_{ik}\})\right)\]  
where \(v(a_{ik})\) is the value for pixel \(i\) of the \(k\)th criterion, \(a\), normalized to a score from 0 to 1. \(min_{i}\{a_{ik}\}\) and \(max_{i}\{a_{ik}\}\) are the minimum and maximum values of the \(k\)th indicator, respectively. The decision rule combining the normalized raster criteria is then performed according to the linear weighted combination (WLC), assigning a relative importance weight according to each indicator [75; 77]. The evaluation of the importance of the criteria weights is a primordial step for which we used the analytic hierarchy process (AHP) of [PERSON] (1980, 2004) [78; 79]. It should be noted at this point that we decided to apply a single weight per criterion, which corresponds to the global approach of weighting, implying a certain spatial homogeneity of the criterion. Otherwise, a local weighting adjusted spatially within the criterion can be applied. [PERSON]'s pairwise comparison uses a scale of 1 to 9 to assess the preferences of one indicator over another per pair of criteria. The relative importance of each factor in the assessment of ES is thus transcribed into a numerical value scale and organized in matrix form (Table 2). From this matrix, a vector containing the criteria weights is obtained by calculating its largest eigenvalue. The weights are then normalized so their sum is unitary (Table 3). The coherence of the judgement deciding the attribution of the weights is estimated thanks to a ratio whose value must not be higher than 0.1 (Table 4). These steps were performed using the ahpsurvey package of R [80]. Finally, the WLC decision rule is translated as an algebraic combination of maps by the relation \(V(A_{i})=\sum_{k}^{1}w_{k}v(a_{ik})\). \(V(A_{i})\) denotes the overall evaluated value associated with each pixel i; \(w_{k}\) is the weight relative to the \(k\)th attribute; and \(v(a_{ik})\) is the value at the location of the same \(k\)th criterion. Our decision synthesis classifies each indicator according to the degree of importance in the global estimation of the ES supply. This relative degree of importance, summarized in Table 2, is justified as follows. We consider that the average NDVI over the dates in May and June is the most important factor in characterizing the integrity of the forest ecosystem, which gives it an essential role in the delivery of provisioning, regulating, and supporting services.