## 2 Intensity Attenuation: State of the Art and Definitions  
Several IPEs are available in the literature, whose field of applicability depends on the dataset on which they are built. IPEs provide an estimation of expected intensity as a function of distance; available formulations differ in terms of the distance metric (e.g., distance from epicenter, hypocenter, rupture plane), adopted functional form, and methodological choices (e.g., input data, data censoring, statistical estimator). [PERSON] and [PERSON] (1997) [27] developed an IPE from a training set of 22 California earthquakes; their equation is computed as a function of distance from the macrosiseismic epicenter (MM scale) and adopted a linear functional form. [PERSON] et al. (1999; 2010) [28, 29] estimated the axial orientation, location of the seismogenic source, and earthquake size from macroseismic data in Italy. The epicenter location was calculated using the concept of macroseismic intensity attenuation as a function of distance. [PERSON] and [PERSON] (2006) [30] computed regional intensity attenuation models for 33 earthquakes in France, providing region-specific coefficients. [PERSON] et al. (2020) [PERSON] et al. (2020) calculated an IPE by analyzing 14 normal faulting earthquakes that occurred in Central and Southern Apennines (Italy) between 1688 and 2016; their equation adopts the epicentral distance and log-linear functional form. The methodological aspects related to the derivation of IPEs are sketched in Figure 1. The map of the rupture area (Figure 1a) shows the epicenter (red star) and intensity data points (IDPs), which are all the points where EEEs have been observed and an ESI-07 value has been assigned. In this example, the earthquake generated a surface rupture along the principal and distributed faults (red segments) and other effects in the nearby region. Two ESI-07 hypothetical isoseismals, corresponding to intensities X (dotted blue line) and IX (dotted purple line), have been represented. The dotted black line is the trace of the main fault, intended as the rupture linework of the primary fault or tectonic/seismogenic feature responsible for the earthquake [PERSON] et al. (2020). The different ways to compute the source-to-site distance for a normal fault are represented in Figure 1b: \(\mathrm{D_{Epi}}\) represents the distance from the epicenter (adopted, for instance, by [PERSON] and [PERSON], 1997; [PERSON] et al., 2020, 2020); \(\mathrm{D_{Hypo}}\) is the distance from the hypocenter (i.e., [PERSON], 2001; [PERSON] and [PERSON], 2004 [31; 32]); \(\mathrm{D_{Rup}}\) is the distance from the fault rupture ([33; 34]); and finally, \(\mathrm{D_{RupX}}\) is the closest horizontal distance between each IDP and the main fault rupture. IPEs can be derived using point-based (Figure 1c; e.g., [PERSON] and [PERSON], 1997, [PERSON] et al., 2020 [10; 27]) and grid-based approaches (Figure 1d; e.g., [PERSON] et al., 2018 [35]). The former requires the availability of point data, while the latter requires the availability of isoseismals. In the IDP-based analysis, each point (A and B in the example)  
Figure 1: Sketch of the input parameters for IPE calculations: (**a**) map showing the surface faulting (red lines), epicenter (star), intensity data points (IDPs, black points), and isoseismal lines; (**b**) definition of horizontal distance from the rupture (\(\mathrm{D_{RupX}}\)); (**c**) illustration of the epicentral distance (\(\mathrm{D_{Epi}}\)) and distance from surface faulting (\(\mathrm{D_{RupX}}\)), calculated with the IDP method; and (**d**) grid method (the values shown in the figures are only an example and do not refer to any specific case). is labeled with the intensity and distance values; then, data are analyzed statistically. In the grid-based analysis, firstly, a grid of cells of a given dimension (1 km\({}^{2}\) in the example) is created and the cell center is extracted. The corresponding ESI-07 value is assigned to each cell according to the isoseismal value; distance values are assigned too.