# Seismic Assessment of Historical Masonry Buildings at Different Scale Levels: A Review
## 2 Large-Scale Approaches
### Empirical Methods
#### 2.1.1 Damage Probability Matrix Methods (DPM)  
The DPM methods express the conditional probability of reaching a certain damage level D during an earthquake of intensity : P[D = j i], deriving from observing post-event damages in a specific site after a seismic event of a given intensity. The DPM methods gather the damage grade distribution for each building vulnerability class in the analysed area according to different macro-intensities of the seismic event. Thus, the same probability of reaching a specific damage state for a determined seismic intensity is associated with each element of the vulnerability class. Figure 1: Categories of large-scale approaches (the figure is based on [10]). DPM methods--first introduced by [PERSON] and [PERSON] [45]--allow a rapid and relatively cheap evaluation of the seismic risk of a large number of buildings. They show some limits related to the need for a huge amount of data concerning all the macroseismic intensities and building typologies of a specific area: this means, practically, that they are suitable for regions with extensive historical seismicity records and the availability of a huge data report [46]. To be sufficiently reliable, DPM methods should therefore be used in the same region where they were originally calibrated, since they are influenced by structural and architectural features [47]. In the Italian national scenario, the devastating Friuli (1976) and Irpinia (1980) seismic events promoted the first systematic application of large-scale vulnerability assessment methods [12; 13; 48; 49]. In Braga, [PERSON] [12], the statistical analysis of the damages caused by the Irpinia seismic event led to an elaborate DPM-based procedure defining vulnerability classes accounting for vertical and horizontal structural typologies. Each class had the same conditional probability of reaching a given damage grade--expressed through a binomial distribution ranging between 0 and 1--for a determined seismic intensity defined in terms of the Medvedev-Sponheuer-Karnik scale (MSK scale) [50]. Similar applications, just to cite a few, can be found in [PERSON], [PERSON] [13], [PERSON], [PERSON] [14] and [PERSON] and [PERSON] [15]. The definition of a new European Macroseismic Scale, the EMS98 [51], allowed the first relevant evolution of the method proposed by [PERSON], [PERSON] [12]. Depending on materials and the code design level, the EMS grouped buildings into six classes (from A to F) based on the expected behaviour during a similar seismic event. Each class damage attitude was depicted through the same fragility curve, and five discrete damage-related levels were proposed, differentiated as a function of the seriousness and extent of the damage that occurred within the structural elements [21; 52]. The EMS98 was used to integrate the DPM method, such as in [PERSON], [PERSON] [16], introducing an additional vulnerability class to account for buildings realised since 1980 and showing a lower vulnerability since retrofitted or designed in agreement with more recent seismic codes. [PERSON] and [PERSON] [17]--later refined in [PERSON], [PERSON] [18]--derived a complete DPM from the EMS definition according to six vulnerability classes, six damage grades and the macroseismic intensity from V to XII. Fuzzy sets were used to associate numerical ranges to the damage frequencies expressed through linguistic expression in the EMS98, and the achieved results were used, for instance, in [PERSON], [PERSON] [19]. A similar attempt was made by [PERSON], [PERSON] [53], replacing the original MSK with the MCS ([PERSON], 1930)--the Mercalli-Cancani-Sieberg scale--widely used in the Italian building catalogues. Further, the building's number was replaced by the dwelling's number to fit with the 1991 Italian National Statistical Office (ISTAT) data. In Bernardini