## 4 Source Modeling  
The scarcity of records belonging to the 1654 event is reasonably due to the ca. 350-year age of the earthquake itself, more than due to the real absence of effects on the landscape and villages. However, we are still able to use the collected records to infer the 1654 source parameters. Indeed, an analysis of the geographic distribution and the type of earthquake effect (i.e., building damage, ground failures, and hydrological change) is a way to provide constraints on both the fault location and the deformation style. In particular, the coseismic hydrological changes (increase or decrease in the discharge of springs and streamflows, the water level in wells, turbid flow from springs and rivers, and liquefaction) can be a valid alternative method to provide further constraints to estimate (or to confirm related hypotheses) the faulting style of major historical earthquakes of which the seismogenic source is unknown or in dispute, as is the case of the 1654 earthquake. The basic rationale is that such hydrological variations are explained by the coseismic static strain and pore pressure changes predicted by the poroelastic theory, as first proposed by [PERSON] [24]. Following this theory, an earthquake imposes a coseismic strain field that causes rocks to dilate or contract; the opening or closing of saturated cracks in rocks result in decreases or increases in the ground water discharge from springs and streams. The amplitude of the hydrological changes is proportional to the volumetric strain field, so that the groundwater discharge increases in areas that contract and decreases in areas that extend. Following this rationale, in recent decades, the character of the coseismic hydrological changes has often been found to be related to the style of faulting ([PERSON] [25] and references therein). The most important caveat regarding the use of hydrological changes in this kind of study is that local precipitation can influence the effect that is observed and it must be carefully investigated. In the case of the 1654 earthquake, the available reports confirm the absence of rainfall in the days preceding the event, which is reasonable as the earthquake occurred at the end of July--the driest period in peninsular Italy. For a complete review of the application and of the limits of this theory in seismogenic studies, see [PERSON] [25]. It is possible now to perform the calculations of the coseismic strain for the 1654 earthquake produced by the potential sources listed in Table 1 to verify the best fault solution fitting with the observed hydrological and geological effects. The static strain change induced by an earthquake can be calculated using a fault dislocation model. The calculations of the strain were made in an elastic half-space with uniform isotropic elastic properties following Okada [26], and using Coulomb 3.4 [27; 28]. In particular, we investigated the deformation imposed by the BF, SFa, and PF [12] and by the Sora fault as proposed by [PERSON] et al. [13], referred to as SFb. The fifth modeled source (macroseismic fault, referred as MF) is derived from Boxer 4.0 [7], a code that computes the quantitative parameters of earthquakes from the inversion of macroseismic intensity data, which is routinely used for the parametrization of the historical events of the Catalogue. The considered faults are generally reported by the authors with dip-slip to normal-oblique kinematics; this is also confirmed by the focal mechanisms displayed in Figure 4. Thus, we first performed our modeling on pure normal sources (rake \(-90^{\circ}\), Figure 6a-e); then, a second round of calculation was carried out considering left-lateral oblique slip (rake \(-50^{\circ}\), Figure 6f-i,I). Finally, we tested the strain calculations on a set of normal sources with oblique right-lateral slip component (rake \(-130^{\circ}\), Figure 6m-q).