## 2 Improved KF Version  
This work is a new step toward the development of the quantitative treatment of the so-called macroseismic intensity in order to obtain geophysical information on the source of mostly pre-instrumental earthquakes on Italian territory [17; 18]. To retrieve this information, we use the method of inversion of kinematic function, KF. This method of inversion has already been used in the Italian territory. The first inversion was carried out for the Noto earthquake M\({}_{\rm wp}\)7.4 of 11 January 1693, which struck southeastern Sicily [17]. Despite many qualitative works localizing the source offshore on the Ibleo Maltese structure, the source of January 11 and its ninth foreshock is inland and linked to the Scichi Monte Lauro structure, which was recently confirmed by the work in [19]. In recent years we have performed the inversion of the earthquake of 7 September 1920 [20] and the earthquake of 14 April 1895 at M\({}_{\rm wp}\)6.1 in Ljubljana, which struck part of northeastern Italy. The former confirms the hypothesis of a normal fault linked with the Garfagnana graben, the latter, despite the complexity of the macroseismic dataset, leads to the source being associated with the Dinaric style of the area. The aim of this article is to retrieve some information on the source of the earthquake M\({}_{\rm wp}\)6 of 27 March 1928 in the Friuli Venezia Giulia (FVG) region in Northeast Italy. In the same region, the KF method was used for the M\({}_{\rm wp}\)6.2 earthquake of 18 October 1936 [21] (see in Section 2) and the previously cited M\({}_{\rm w}\)6.45 of 6 May 1976 [10]. Information on the sources of historical earthquakes is important for hazard assessment and the development of damage scenarios for engineering planning [22]. The KF inversion method is based on a kinematic function (KF) [23], developed in [24]. The kinematic function, KF, has already been presented in other publications, and the details can be found in Appendix A. The first version of the KF has already been validated using the regional intensity patterns of Californian earthquakes, which have been provided with instrumental solutions. Following these validations, we applied the method in other regions of the world [25]. Our hypothesis of the M8.2 Bihar-Nepal 1934 earthquake is confirmed in the recent study of [26] and the epicenter of M8.5 Shillong 1895 is confirmed in [27]. The algorithm is also useful for earthquakes that occurred offshore [28], overcoming the concept of other macroseismic methods where a surface projection corresponds to that of the area of maximum intensities. Indeed, the KF inversion method is based on the radiation pattern of S-waves, as in the example of [PERSON] et al. (2009) [22]. Regarding the KF formula (see Appendix A, Formula (A1)), we only want to emphasize here that it agrees with the asymptotic approximation [29; 30; 31]. This means that the KF uses as a propagation medium an elastic half-space in the distance range of about 5 to 100 km from the source and wavelengths shorter than the shortest distance between the observer and source. If a site is within 5 km from the epicenter, the method assigns the maximum intensity value of the computed dataset. We also recall that in our convention, the fault plane is dipping to the right of the positive direction of the strike; the rake angle is seen on the fault plane from the hanging wall of the fault and measured anticlockwise from 0\({}^{\circ}\) to 360\({}^{\circ}\) between the positive direction of the strike and the direction of the slip vector. It should also be kept in mind that since the intensity has no polarity, our KF method is unable to distinguish between the results of mechanisms that differ in the rake angle by 180\({}^{\circ}\), since it produces the same radiation but with reversed polarities in both cases. This ambiguity can only be resolved by instrumental records or additional tectonic/geodynamic information. Secondly, our KF inversion problem is close to bimodality in the case of almost pure dip-slip mechanisms [21].