## 3 Validation of the Improved KF Version  
To give a sense of what is the resolution of our approach, we tested it on the macroseismic field of the 20 May 2012 event. The macroseismic field of this event is extremely poor (53 intensity points), since only localities within the immediate surrounding area are reported, even though it was felt over a wide area. This earthquake source is, however, extremely well investigated and reasonably simple, so that we can be sure that an along strike representation of the main asperity will be satisfactory. Following [16], we selected a source with a length of 11 km, unilateral rupture, a 12 km depth, 114\({}^{\circ}\) strike, 30\({}^{\circ}\) dip, 90\({}^{\circ}\) rake, and an M\({}_{\rm w}\) = 6.09, according to [1,2]. So, we basically left the program as free to choose the epicenter and the Mach value (rupture propagation velocity with respect to S-wave velocity). This is because the intensity (MCS) points only give three possible values (I\({}_{\rm MCS}\) = 7, 6, 5) and eight points out of fifty-three are uncertain, while another 12 are expressed as non-numerical values (I\({}_{\rm MCS}\) = D), an overall situation that is even worse than most historical events available in [1,2]. In Figure 2, we show the two best solutions, SA and SB, of two different families. As the results show quite evidently, it is SB, the second best, which is most similar to the real one. This means that there is a distinct probability that in many cases, we will need to choose one family over another not only on the basis of the absolute best residual but also on the basis of tectonic considerations. As far as \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Date** & **M\({}_{\rm w}\)** & **80\% RLD** & **Observed** \\ \hline
6 May 1976 & 6.45 & 18.6 & 18 \\
6 April 2009 & 6.29 & 14.9 & 15 \\
24 August 2016 & 6.18 & 12.9 & 13 \\
20 May 2012 & 6.09 & 11.4 & 11 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Four modern Italian events where total length of asperities is known compared to 0.8*RLD as obtained in [35]. They are organized in descending order of magnitude. (Magnitude from CPTII5. )the choice between SA and SB is concerned, in case we did not know the source's true geometry and location, we could only prefer SB on the basis that it is nearer to the area of maximum damage. However, we note that the epicentral error is still about 6 km, and we should never forget that due to variations in vulnerability, local effects, and improper and simplified modeling, we cannot hope to obtain a better accuracy. There is another crucial point, which is that it is necessary to fix a priori the correct fault length and magnitude, otherwise weird solutions start to appear, particularly in an area of about 20 km to the southwest of the true epicenter. These latter solutions allow for much better matching of the macroseismic data obviously, with completely wrong fault lengths. We also need to understand what the meaning of the macroseismic epicenter is. In general, according to the Boxer method [6; 7], this is in fact conceptually very similar to a centroid in a moment expansion representation. Conversely, we consider it as the surface projection of the point where the rupture of the main asperities starts. Another issue we need to address is how to perform inversion. Since our data are quantized, there is no derivative continuity. This means that none of the classic derivative-based inversion methods can be applied. Even random approaches such as the Monte Carlo method will fail, since the quantified data will induce strong nonlinearity effects, as small variations in a parameter can induce large variations in the residual. The only way we found is to resolve the issue by brute force, that is, to explore the whole space parameter and find possible epicentral areas and a family of solutions, progressively refining them. This approach needs to be necessarily coarse at the beginning, since an initial in-depth exploration would require exploring a number of solutions that could easily be in excess of 1015.