To compose our learning set (Table ) we searched the whole Italian territory, selecting all instrumentally well-documented earthquakes, i.e., events whose location uncertainties are small and that also feature good-quality macroseismic information. For each event the best available source was chosen by expert evaluation based on all available literature sources; Table  reports the exact bibliographic source of its depth and magnitude. Whenever a specific study about a given earthquake exists, we used the relocated depth (if available). We built a learning set comprising macroseismic data either obtained from a direct survey or collected through the web so as to gather information to be used as a sort of “Rosetta Stone” for obtaining the parameters of historical earthquakes. For pre-2008 events we used the data stored in the DBMI15 v2.0 catalogue Italian Macroseismic Database;, a compilation of macroseismic intensities for Italian earthquakes that occurred in the time window 1000–2017 CE, whereas for more recent events of Mw≤5.9 we used either intensity data from the web-based HSIT catalogue or the MDPs collected by targeted post-earthquake surveys conducted by experienced INGV QUEST personnel (Istituto Nazionale di Geofisica e Vulcanologia QUick Earthquake Survey Team; https://quest.ingv.it/index.php/rilievi-macrosismici; last access: 21 February 2023). Only for the 18 January 2017 Mw 5.5 event we had to make an exception to this rule, due to the incompleteness of HSIT data caused by the evacuations following the Mw 6.0 mainshock of 24 August 2016.

The use of web-based data was fundamental to accomplishing our goals because these data were almost always the only observations available, especially for deeper earthquakes (>30 km). Furthermore, the use of macroseismic data obtained from direct surveys of earthquake damage was fundamental for the correct analysis of the attenuation curves, especially in the epicentral area. The combination of web-based HSIT data and dedicated traditional studies does not affect the results of the learning set because the earthquakes that we considered are all relatively recent and their macroseismic field was estimated through a direct field survey. In general, intensity maps drawn for historical earthquakes show more scattered patterns of damage than those revealed by spatially rich, web-based intensity data for similar-sized, recent events . This problem particularly affects earthquakes whose effects are estimated through written sources. The same is true if only written sources (e.g., newspapers) are used to estimate the intensities of recent earthquakes; they will inevitably end up being overestimated .

The events comprising the learning set were further selected based on the following criteria: 1.

Pre-2007 earthquakes must have Mw≥5.0, but events of Mw≥4.5 are also accepted if they are backed by a targeted study.

All 42 earthquakes listed in Table  fulfill these rather strict criteria with the only exception being no. 6 and no. 17 (MDP <60), two deeper events that were already included in the learning set of as they are crucial for characterizing lower-crustal and subcrustal seismicity.

Notice that selection criteria 1–4 had already been adopted by . Additional criteria 5 through 9 were added in consideration that the present work deals with the entire Italian territory and hence with a much larger diversity of the potentially concerned earthquakes, more specifically the following:

Criterion no. 5 was added due to the recurring lack of data in the epicentral areas of the main aftershocks of the 2016–2017 central Italy sequence, due to the widespread evacuations following the Mw 6.0 mainshock of 24 August 2016 and to the superposition of the effects of subsequent shocks.

We analyzed separately two data subsets, respectively, comprising only earthquakes located in northern Italy and earthquakes located in the rest of the Italian Peninsula. We made this choice because the dataset used in included only earthquakes from a region whose lithospheric structure and wave propagation properties are homogeneous . Conversely, in this work we wanted to evaluate the possible influence of variable attenuation properties resulting from the full range of tectonic and geodynamic diversity characterizing the Italian Peninsula.

Due to the intervening minor updates in our methodology – and specifically in the calculation of the starting point of our moving window, which implies a slightly different steepness for the first 50 km of the attenuation curve (mean: -0.00015; max: 0.007) – we first recalculated the attenuation steepness for all the 20 earthquakes comprising the learning set used by (Table , IDs 1 to 20; see Figs.  and ). We added to this dataset the 1996 Emilia earthquake (ID 23), originally rejected because its depth from the ISIDe catalogue was fixed at 10 km; for this event we now use the depth evaluated by . We then analyzed the earthquakes we selected for the rest of Italy and calculated their intensity attenuation steepness (Figs. , ; Table , from no. 21 to no. 42, except for no. 23).

As discussed earlier, in both datasets, which together form our new learning set, we observed a distinct break in steepness at an epicentral distance of about 50 km (see Figs. , ). In describing this feature of the Italian attenuation curves, contended that within a ∼50 km epicentral distance the ground shaking is dominated by direct seismic phases, whose propagation is highly sensitive to earthquake depth, whereas Moho-reflected phases dominate at larger distances. According to this hypothesis, the exact distance of the transition would be controlled by the average Moho depth along the source–receiver path.

For all the earthquakes in the learning set we then plotted the steepness (S; intensity per kilometer), i.e., the absolute value of the attenuation slope, versus focal depth (D; km) and found two separate but very similar best-fitting logarithmic functions (Fig. ). For northern Italy we found 1S=(-0.020±0.006)ln⁡D+0.093±0.018, whereas for central and southern Italy we found 2S=(-0.016±0.007)ln⁡D+0.079±0.019.

The coefficients of each function fall within the 95 % confidence interval of the other function, suggesting that our method does not detect any statistically significant change in the attenuation of macroseismic intensity between the two domains, at least over the first 50 km of epicentral distance. This finding also suggests that an approach based on averaging the intensity values distributed over circular search areas has the ability to smooth out most of the inevitable azimuthal differences in crustal propagation properties.

We decided to calculate a new logarithmic function using all 42 earthquakes in the learning set so as to obtain a law that may be used over the whole Italian region (green line in Fig. ): 3S=(-0.018±0.004)ln⁡D+0.087±0.013.

The regression F test of the three regressions is acceptable at a significant probability level of p<0.0001. As we expected, Eq. () is similar to the previous two equations and exhibits narrower 95 % confidence bands resulting from the larger number of available data points. Equation () can be applied for a steepness interval of 0.058≤S≤0.010, which corresponds to a depth interval of 5≤D≤73 km. Notice that the function is not constrained beyond these limits and hence should not be used for shallower or deeper events. For depths greater than 35 km and steepness less than 0.02, the uncertainty is larger. Consequently, the confidence bands of Eq. () in Fig.  exhibit wider limits, yet they still provide valuable information on depth estimation, albeit within a wider error range. Notice also that for epicentral distances >50 km the curves shown in Figs.  and exhibit slightly different steepness between the northern and the central and southern Italian datasets (respectively, 0.01 and 0.02), in agreement with the observations made by . Conversely, as mentioned above, in the first 50 km of the attenuation curve there does not seem to be any influence by crustal attenuation properties; hence in this case a trade-off exists only between magnitude and depth. In contrast, the intensity attenuation for epicentral distances >50 km for earthquakes occurring in northern Italy, where the crust is generally thicker than in the rest of the country, frequently shows a characteristic, very gently sloping plateau that has been interpreted as due to Moho-reflected seismic waves by . A further element to be taken into account is the difference in seismic-wave propagation between the Tyrrhenian and Adriatic sides of Italy, most likely resulting from a rather different efficiency of the seismic energy propagation of the crust–upper-mantle system . But again, no differences are appreciated in our analysis for earthquakes located on the Tyrrhenian or Adriatic sides with epicentral distances up to 50 km (see Figs. , , and ).

The steepness of the first 50 km of the attenuation curves calculated for the earthquakes of our learning set (Figs. , ) is independent from magnitude, as already empirically observed by for a smaller sample of events. To prove this statement, we correlated the steepness with Mw in addition to the natural logarithm of depth (see Eq. ) and found that its coefficients are not significant (95 % confidence interval includes the null value). As an example of the independence of the steepness from magnitude we plotted in Fig. a the attenuation curves for four earthquakes falling in the rather wide Mw range 4.8–6.5, with a similar instrumental depth, in the range 7.3–8.7 km (no. 4, no. 16, no. 24, no. 39; see Table ). Figure a shows that all the calculated steepness values fall in a rather narrow range (0.045–0.051), regardless of magnitude. Figure b shows that the same behavior is observed also for four deeper earthquakes (no. 8, no. 10, no. 11, no. 26; see Table ), which share a similar instrumental depth (24.5–29.2 km) but exhibit a different Mw value (4.0–5.6).

The invariance of the attenuation steepness with magnitude for events in the learning set is a key point as it makes our approach suitable for analyzing historical earthquakes even if their size is not well constrained. Instead, other methodologies are subject to a trade-off between depth and magnitude, as both parameters are treated as unknown. Our approach is similar to that of previous investigators , who based their analyses on manually drawn isoseismals, but directly uses the fit of the attenuation curve computed on averages of the original MDP falling inside moving circular windows.

We analyzed the possibility of reproducing the empirical trend of Fig.  through predictive models, expressing the macroseismic intensity (intensity prediction equations, IPEs) or the peak ground acceleration (PGA) as a function of the earthquake magnitude and distance. It is worth noting that many of the IPEs and GMPEs (ground motion prediction equations) proposed in the literature (e.g., ) assume a predetermined depth for all earthquakes considered. The difficulty in considering this parameter is due to the uncertainty associated with the depth itself, not only for historical earthquakes but also for recent events located in areas that are geologically complex or not monitored by a dense seismic network. To explore the variation in the attenuation steepness with depth, we therefore used three of the models that explicitly include this parameter: the IPE by , the IPE by , and the GMPE by . We chose these functions because they feature a simple functional form which determines a magnitude-independent attenuation steepness, as suggested by real earthquake data. Conversely, a functional form containing a term combining magnitude and distance would lead to a change in the shape of the attenuation curve with distance and to a variation in the steepness for a variable magnitude.

We then used two different conversion equations to convert the PGA values obtained with the adopted GMPEs into MCS (Mercalli–Cancani–Sieberg intensity scale) macroseismic intensities so as to test also the influence of the conversion process. We used these equations to compute the macroseismic intensities caused at several epicentral distances by a hypothetical Mw 5.0 earthquake located at variable depth. We then applied the same 10 km moving-window average method used for the analyzed earthquakes and calculated the regression line within a distance of 50 km.

Figure a shows some of the average-intensity values obtained using the IPE proposed by for a magnitude of Mw 5.0, along with their regression lines. We remark that, although the macroseismic intensity is proportional to the logarithm of the hypocentral distance, the linear regression of intensity versus epicentral distance gives statistically significant results in the adopted distance range. In addition we show that, even using an IPE, 50 km is a reasonable limit for a linear regression that approximates the first part of the attenuation curve well. Figure b shows the steepness of the regression lines, thus calculated as a function of the earthquake focal depth, and the values, calculated with the same method, derived from PGA using the GMPE proposed by , converted into macroseismic intensity. It is worth noting that differences caused by the use of two different conversion equations are greater than the differences caused by the use of the IPE in place of the GMPE. At any rate, in all three cases the trend of the values as a function of the depth is similar to the trend we found empirically. The greater difference is observed for depth larger than 35 km, probably because the empirical regression is less constrained for such deep events. This is reflected in the wider confidence bands of Eq. () (see Fig. ), due to fewer earthquakes in the learning set at those depths. Having been obtained with a completely different kind of data, this result suggests that the approach followed for deriving Eq. () is adequate and reliable.

The reliability of the steepness of the first 50 km of the attenuation curve depends on the quality and spatial distribution of the available MDPs and on the accuracy of the epicentral locations. Italian macroseismic data are systematically stored in the DBMI v4.0 database ; as a rule of thumb, the older the earthquake is, the less complete and reliable the historical sources from which macroseismic intensities were derived are (e.g., ).

To test our procedure we investigated the minimum number of MDPs of the macroseismic field that are needed to obtain an estimate of the attenuation steepness. To this end we intentionally and randomly depleted the macroseismic field of the 20 May 2012 Mw 5.8 Emilia earthquake (no. 13), a well-recorded event for which over 200 spatially well-distributed MDPs are available, using data from the HSIT database (; see Fig. a). For each of the 10 ring-shaped search areas (see Fig. b, c) we performed a gradual reduction in the number of MDPs (from 1 % to 99 %) and did the same for each step and for all areas (see Fig. ). For example, let us consider three adjacent ring-shaped areas: the first with 32 MDPs, the second with 18 MDPs, and the third with 9 MDPs. The depletion procedure would lead (among others) to the following steps: a 35 % depletion will leave 21 MDPs in the first area, 12 MDPs in the second, and 6 MDPs in the third; a 68 % depletion will leave 10, 6, and 3, respectively; and a 97 % of depletion will leave 1, 1, and 0 MDPs.

The linear fit of the attenuation trend was calculated 1000 times for each depletion step so as to evaluate the steepness variability through its standard deviation.

Figure  shows the number of MDPs versus the standard deviation of the steepness, which is equal to 0.01 when only 14 % of the total data are left, corresponding to 30 MDPs; this implies that the most likely steepness values (68 %) fall within a standard deviation equal to ±0.01.

Our depletion test shows that we may obtain an acceptable attenuation steepness even for historical earthquakes for which at least 30 MDPs are available, provided that they are homogeneously distributed for each distance window. Moreover we calculated the depth reliability by estimating the depths corresponding to the confidence bands of Eq. () for each calculated steepness of the earthquakes in the analyzed set (see Fig. ); these values are shown in Table S1 in the Supplement. We also calculated the depth of the 42 events in our learning set using the proposed methodology (Table S2, Fig. S1 in the Supplement).

While the inferred hypocentral depth is independent of magnitude and can be obtained simply based on the steepness of the line that best fits the first 50 km of the attenuation curve, the estimation of the magnitude itself affects the y intercept (the expected intensity at the epicenter, IE) of the linear fit (Fig. ): for a constant depth the y intercept increases for an increasing magnitude (Fig. ) and decreases if depth increases for a constant magnitude. Therefore, a reliable magnitude determination based on macroseismic data must necessarily take into account earthquake depth.

We devised a two-step procedure where depth is estimated first (step 1) and then Mw is empirically estimated using our data from the learning set to derive a standard least-squares regression equation among D (depth), IE, and Mw (step 2): 4Mw=(0.18±0.19)ln⁡D+(0.56±0.11)IE+(1.44±1.06).

Equation () can be applied for a depth interval of 5≤D≤73 km and 3.5≤IE≤8.1. Figure  shows the data of the learning set used in the regression, together with the magnitude isolines of Eq. (). This relationship shows the extent of geometric attenuation of intensity due to the propagation of seismic waves from the hypocenter to the epicenter.

For ln⁡D the 95 % confidence interval of the coefficient includes the null value; however, the coefficient becomes significant at a slightly smaller confidence level (93 %). Magnitudes obtained through this procedure are referred to as y-intercept Mw. In this perspective the attenuation curve becomes a sort of “earthquake identity card”, as it contains all the elements needed to retrieve magnitude and depth from the observed intensities, provided that a reliable calibration scheme is available. Such calibration can be regarded as an application to seismology of the principle of actualism popularized by British naturalists in the late 18th century: “observing modern earthquakes to understand those of the past”.

Deriving magnitude using only well-studied earthquakes with their expected epicentral intensities provides a better estimate of Mw because it is based on larger intensities than that obtained by inverting an IPE .

The method summarized by Eqs. () and () is simple and intuitive, and it may allow for a geological verification of the depth before estimating the magnitude. Our step 2 uses a method similar to that proposed by , who applied it only to carefully selected datasets so as to minimize the bias caused by a poorly constrained depth or by an incomplete macroseismic field.

In conclusion, starting from our empirical observations of the independence of the attenuation steepness from magnitude, we were able to mitigate the trade-off between magnitude and depth when estimating both these parameters from macroseismic data.

It is hard to estimate macroseismic intensities for individual earthquakes occurring close in time and space (multiple events, strong aftershocks, etc.; e.g., ), particularly in the case of historical events. Macroseismic data may then reflect accumulated effects, a circumstance that would ultimately affect the attenuation steepness and hence contaminate the inferred earthquake depth and magnitude. This is a recurring problem in historical earthquake catalogues, a condition that is hard to overcome even for modern earthquakes and even if a very rapid damage survey is carried out because the first large shock inevitably causes an increase in the vulnerability of the building stock whose effects on later shocks are virtually impossible to identify.

To quantify the effect of multiple events on the determination of earthquake depth, we analyzed the 29 May 2012 Mw 5.7 Emilia earthquake, one of the events in the learning set, using the MDPs from DBMI15 instead of those supplied by HSIT (ID 14 in Table 1). For this event the DBMI macroseismic field (, https://emidius.mi.ingv.it/ASMI/event/20120529_0700_000, last access: 22 December 2022) includes the effects of the 20 May event (Mw 5.9; https://emidius.mi.ingv.it/ASMI/event/20120520_0203_000, last access: 22 December 2022), which occurred nearby. As expected, these circumstances misled our method, causing a drastic overestimation of the earthquake depth (36.8 km). Conversely, thanks to the rapidity in the response given by citizens and to the ensuing lack of contamination, the HSIT dataset method returned a depth ≤5 km, much closer to instrumental estimate (8.1 km).

A similar case of contamination could be that of two earthquakes that occurred 7 months apart in two distinct but relatively close areas of the northern Apennines: the 10 November 1918 Mw 6.0 Appennino forlivese and the 29 June 1919 Mw 6.4 Mugello shocks (IDs 101 and 102 in Table S1, respectively). Our work estimates a depth range of 19–27 km for the 1919 earthquake (see Table S1). This result is incompatible with the estimated depth of the DISS seismogenic source that is deemed responsible for the 1919 event (fault ID ITIS086; depth range of 1–7 km based on seismotectonic evidence; ). We suspect that this anomalous depth estimate (19–27 km) could be explained by a sort of overlap between the two macroseismic fields because a portion of the intensity pattern of the 1919 earthquake overlaps the region struck by the 1918 shock . As a result, the (apparent) intensity pattern of the 1919 earthquake is likely contaminated by the 1918 event. Both in the case of the 2012 sequence and in the case of the 1918–1919 earthquakes, the inferred depth of the second mainshock is deeper than that shown by instrumental data or suggested by geological observations, due to the overlap between the two macroseismic fields. The intensity fields for 29 May 2012 and 29 June 1919 appear more spread out than they should, due to the contamination from the previous earthquakes; this entails a lower steepness of the attenuation curve – corresponding to an apparently lower attenuation of macroseismic intensity – which ultimately translates into a deeper depth.

Our approach works well if the size of the seismic source is negligible relative to the epicentral distances, but it may not be immediately applicable to estimate the attenuation of the macroseismic intensity for a high-magnitude earthquake . To test the validity and possible limitations of this assumption, we evaluated the maximum magnitude for which the use of a point-source approximation is granted, using both our learning set and our analyzed set.

We used the empirical relationships proposed by to calculate the rupture area and the expected length of the seismogenic source based on the y-intercept Mw (Eq. ). Assuming a dip angle of 45∘ for every fault, irrespective of its kinematics and tectonic setting, we calculated the surface projection area of each rectangular source and the radius Re of the equivalent circle (i.e., a circle with the same projected area as the fault; thus Re is a function of Mw). We found Re >10 km only for earthquakes of Mw≥6.75 (10 km is our standard radius of the moving circular search areas), but not having any such earthquakes in the learning set (Table ), we used a geometric correction only to infer the depth of 21 earthquakes in the analyzed set (see discussion at the end of this section and in Table S1).

Then we applied to this group of higher-magnitude earthquakes a procedure that we call “variable moving windows”. More specifically, we used as the first search area a circular window of radius Re, inside which we averaged the MDP intensities, while for the subsequent windows – each one shifted by 5 km, as usual – we adopted the standard 10 km radius increase. For the 13 January 1915 Mw 7.1 Marsica (central Apennines) earthquake, one of the largest in Italian history and for which there exists a very reliable macroseismic dataset, we made a test using the RJB distance following the method implemented by and calculating the average of the MDPs inside a rectangular rather than a ring-shaped area. We singled out this earthquake because it was a single mainshock event and its macroseismic field should not have been contaminated by the effects of previous or later significant events (see Sect. ).

The comparison of the attenuation steepness calculated using the RJB distance or using the moving-window or variable-moving-window methodology proposed here shows only modest fluctuations. For the 1915 earthquake we found a steepness of 0.044, 0.044, and 0.047, respectively, using the RJB approach, the moving-window approach, and the variable-moving-window approach. These steepness fluctuations imply a difference of about 1.5 km in the expected depth. Given the uncertainty about our knowledge of the source geometry for historical earthquakes and the limited impact of using the RJB distance in the window approach, we decided to recalculate all distances for earthquakes with Mw≥6.75 using the variable-moving-window method only, which analyses the MDPs over circular search areas.

In conclusion, using an extended source approach for the largest earthquakes has a minimal influence on the steepness. Conversely, the effect on the y intercept (IE) is not negligible. The correct way of calculating their magnitude would be using RJB distances, but due to the lack of information on source geometry we use the variable-moving-window method and apply a geometric correction to the intercept value. As a result, for 21 earthquakes with Mw≥6.75 (see Table S1) we assumed an extended source with a radius of Re. Consequently, the distances of the relevant MDPs were systematically reduced by Re, leading to a geometric correction of the regression line and of its intercept: IE=IE-S×Re, where S is the steepness.

Finally, we recalculated the magnitude of these 21 earthquakes using Eq. ().

Since ours is a two-step method and magnitude is calculated after estimating depth, we provided the estimate of the error associated with the magnitude of the earthquakes in the analyzed set, based on the confidence limits of depth, by applying Eq. () for the lower and higher depth limit based on the 95 % confidence band (Table S1).

As a countercheck we used our method to calculate the depth first (Eq. ) and then the magnitude (Eq. ) of the 42 events of our learning set (Fig. , Table S2; see also magnitude residuals in Fig. S2) so as to analyze the difference from the instrumental magnitudes listed in Table . We obtained differences in the range 0.68 to -0.41 magnitude units (Fig. S2), respectively, with an average of -0.03 and a standard deviation of 0.28.

We then compared the macroseismic magnitudes calculated through our method with those calculated through the Boxer method , using the very same intensity dataset from DBMI15. Notice, however, that the parameters of the earthquakes comprising our learning set were computed using also data from other sources, such as HSIT and CFTI5Med (Table ). Table  lists the magnitude of all earthquakes in the learning set for which the comparison was possible. Notwithstanding the significant differences between the two methods, the root mean squared error between instrumental magnitudes and those estimated with Boxer and our method turned out to be comparable at 0.38 and 0.35, respectively.

Finally, the reliability of the y-intercept Mw is a function primarily of the accuracy of the macroseismic field from which IE is derived but also of the estimated depth. For instance, we examined the 13 January 1909 northern Italy earthquake (https://emidius.mi.ingv.it/ASMI/event/19090113_0045_000, last access: 22 December 2022), whose macroseismic field is suggestive of a rather deep source. We obtained a depth of 44 km, yielding Mw 5.5; were this earthquake to be much shallower (e.g., 5 km), Eq. () would return a substantially lower Mw value (5.1).