## 2 Improved KF Version
In this paper, we propose a simplified version of the standard method that allows us to systematically explore the whole parameter space. This need arises from a simple fact, that intensity data are discrete and can assume only 11 possible values; this strongly limits the amount of possible free parameters. It is true that intensity spatial distribution plays a role as well, as we will see. This means that there are many possible equivalent solutions unless we somehow limit the free parameters making use of as much a priori information as possible. This is even more dramatic if we consider that often, only a subset of intensities is available, that is, there are only seven or eight intensity values out of the possible eleven. With 12 parameters, even in principle, this would lead to an underdetermined problem. Most of the Italian events, however, are small enough (M \(<\) 6.5) such that a uniform rupture velocity is sufficient. Moreover, excellent S-wave velocity models are now available. This means that we can use one of these [32] instead of inverting for the S-wave velocity. In this case, we choose a simplified approach, in line with the general simplicity of the original method, that is, to select the average velocity in the area comprised between the fault and the intensity point. Finally, a magnitude estimate usually exists a priori. The \"boxer\" method in fact has a good feature that we will make use of, that is, the magnitude estimation. It is in fact the main tool to compute macroseismic magnitudes in the CPTI15 catalog that are scaled to correspond to M\({}_{\rm w}\) proxies (M\({}_{\rm wp}\)). It is also possible to constrain the fault to have a fixed length that depends on the assumed magnitude, e.g., in [33]. In our case, however, we do not straightforwardly use the RLD, the subsurface rupture length (Formula (A3) of Appendix A) general relation, but a function of it, namely 80% of its value. We tentatively selected this value on the base of admittedly very few available observations. In Italy, in fact, only four events occurred after 1976 that (1) yielded M\({}_{\rm w}\)\(>\) 6, (2) occurred on land, (3) were not intermixed with other events that occurred nearby, and 4) were investigated in terms of slip distribution. We show these in Table 1. We roughly estimated the along strike length of the principal asperities using the literature data, namely [15] for the 6 May 1976 Friuli event; [34] for the 6 April 2009 L'Aquila event; [35] for the 24 August 2016 Accumoli event; and [16] for the 20 May 2012 Mirandola event. The above reasoning reduces the number of free parameters to eight and consequently helps stabilize the solution. We also rewrote the inversion program to optimize it as much as possible. This program is now fully parallel with respect to the epicentral location, that is, the computation is performed in parallel with respect to the grid of possible epicentral locations. Its main limit, however, is not related to the computation workload but to the speed of memory access in the CPU cache. The final improvement is in the way intensity data are treated. As is known, real valued intensities do not exist; whenever we obtain an uncertainty between two intensity values or, even worse, a non-numeric value such as a generic damage assessment, we need to evaluate the probability that this value assumes a given intensity. Since we use the current version of the [1,2] historical catalog, we pre-computed the probability of uncertain intensities on the assumption that they are distributed in such a way that the relative proportion between the intensity values does not change. In the case of non-numeric values, we normalized the values proposed in [2]. This means that for a calculated intensity \(I_{c}=M\) and an observed intensity \(I_{o}=N\div N+1\), the residual value will be:  
\[R(M)^{2}=\left[P_{N}(M-N)^{2}\right]+\left[P_{N+1}(M-N-1)^{2}\right] \tag{1}\]  
where \(P_{N}\)_P()_ and \(P_{N+1}\) are the normalized partition coefficients. The partition coefficients are reported in Appendix B.