Investigating Spectral Estimates of Stress Drop for Small to Moderate Earthquakes With Heterogeneous Slip Distribution: Examples From the 2016–2017 Amatrice Earthquake Sequence
Abstract
Estimates of spectral stress drop are fundamental to understanding the factors controlling earthquake rupture and high frequency ground motion, but are known to include large, poorlyconstrained uncertainties. We use earthquakes from the 2016–2017 sequence in the Italian Appenines (largest event at Norcia, M_{w} 6.3) to investigate these uncertainties and their causes. The similarlysized events near Amatrice (M_{w} 6.0) and Visso (M_{w} 5.9) enable better constrained relative analysis. We calculate S wave source spectra, corner frequencies, and spectral stress drop for 30 of the larger events. We compare both empirical and modeling approaches to isolate the source spectra and calculate source parameters; we also compare our results with those from published studies. Both random and systematic interstudy variations are larger than the standard errors reported by any individual study. The reported magnitude dependence of stress drop varies between studies, being largest for generalized inversions and smallest for more individual event based approaches. The relative spectral estimates of interevent stress drop are more consistent; all approaches estimated higher stress drop in the Amatrice earthquake than the similarsized Visso earthquake. In contrast, finite fault inversions of these two earthquakes found that the Visso earthquake had the larger region of concentrated, higher slip, whereas the Amatrice earthquake had multiple, lower slip, subevents. The Amatrice spectra contain more high frequency energy than those of the Visso earthquake. This comparison suggests that consistent measurement of a higher spectral stress drop indicates greater highfrequency ground motion but may correspond to greater rupture complexity rather than higher stress drop.
Key Points

Spectral stress drop estimates from different methods do not agree, largely due to ambiguities in separating source and path

Relative stress drop estimates within each study are more reliable and reflect the high frequency radiation and ground motions

Finite fault inversions show that corner frequency and stress drop can reflect source complexity rather than actual stress release
Plain Language Summary
The stress release (or stress drop) during an earthquake is an important element of seismic hazard forecasting; high stress drop earthquakes radiate more high frequency energy, causing stronger ground shaking. The stress drop also provides information about the energy budget, and the size of fault ruptured, and consequently, earthquake triggering and rupture dynamics. Reliable estimates of stress release are difficult to make, largely because of the ambiguity in removing the distorting propagation effects experienced by waves traveling from earthquake to seismometer from recorded seismograms. Most measurements are made using frequency amplitude spectra. We use two methods to estimate earthquake stress drop for 30 of the larger earthquakes in central Italy (2016–2017) and compare them with the results of previous studies. We find that the variation between absolute values estimated in different studies is much larger than the reported formal inversion errors. The relative values are more reliable, with different studies consistently finding a particular earthquake has relatively high or low stress drop. Direct comparison of the similarsized, damaging Amatrice and Visso earthquakes reveals that the relative spectral stress drop estimates reflect the relative strength of highfrequency ground motion, but may indicate more complex rupture rather than higher average stress release.
1 Introduction
The stress drop in an earthquake is a fundamental factor controlling the rupture dynamics, highfrequency radiation and ground motions (e.g., Kanamori & Brodsky, 2004). Unfortunately, it has become increasingly clear that widelyused spectral stress drop estimates involve uncertainties (random and systematic) that are both large and poorly understood (e.g., Abercrombie, 2021; Baltay et al., 2022). The differences between studies, which depend on the method used and many assumed parameters and data selections, are often much larger than the formal model and inversion errors (e.g., Pennington et al., 2021; Shearer et al., 2019). The main sources of uncertainties are thought to be the separation of source, path and site effects using frequencylimited seismic data, and the assumptions of simplistic source and attenuation models (e.g., Abercrombie, 2021; Ide & Beroza, 2001; Pennington et al., 2021; Shearer et al., 2019). Consistent relative variations in spectral stress drop estimates within an earthquake population from independent studies are more robust than the results of any individual study (e.g., Pennington et al., 2021).
Spectral source models all assume a simple, typically circular, rupture, that does not represent the heterogeneous ruptures of real earthquakes, as noted by Madariaga (1979). The slip heterogeneity, also referred to as source complexity, of large earthquakes is well known from finite fault inversions (e.g., Cocco et al., 2016; Mai & Beroza, 2000; Mai & Thingbaijam, 2014; Ye et al., 2016) but a simpler source model was generally thought to be more appropriate for smaller earthquakes. Improved recording is revealing that similar complexity is common in smaller earthquakes (e.g., Abercrombie et al., 2020; Pennington, Chang, et al., 2022; Pennington, Uchide, & Chen, 2022; Uchide & Imanishi, 2016; Yamada et al., 2005), as would be expected from selfsimilarity. The spectrum of a complex earthquake source will also exhibit a more complex shape than that of a simple circular rupture, with peaks and troughs at frequencies corresponding to the duration and timing of the slip heterogeneity (e.g., Gallovič & Valentová, 2020; Ye et al., 2016). This prompts the question of what simple spectral stress drop estimates are really measuring if an earthquake involves slip heterogeneity. How should these estimates be used to probe earthquake source physics or predict future ground motions? Yoshimitsu et al. (2019) simply excluded any earthquakes with complex spectra from their source parameter analysis which resulted in highly consistent, robust measurements, but the results from such a highly selected subset cannot be considered representative of the earthquake population. Ruhl et al. (2017) and Abercrombie (2014) also tried to separate simple and complex earthquakes in their analysis.
To compare spectral estimates with finite fault models, requires estimating an average stress drop from the heterogeneous slip distributions. Noda et al. (2013) compared various approaches to doing this for a single distribution. Brown et al. (2015) discussed the difficulties of assigning a single “average” stress drop, with uncertainties, for a large, complex earthquake even using a single approach when multiple finite fault inversions are available. How these measurements compare to the results of simple spectral fitting is not well known. Cocco et al. (2016) and others combined results from spectral modeling and finite fault inversion but did not compare the two for individual events; Ye et al. (2016) compared the spectra of their source time functions to standard source spectra, finding significant deviation for more heterogeneous slip distributions, but did not model the spectra. Lin and Lapusta (2018) and Gallovič and Valentová (2020) showed that seismic spectral estimates of stress drop can be poor indications of stress release in an event with a highly complex slip distribution. Abercrombie (2021) demonstrated that the combination of a limited frequency bandwidth and a complex source spectrum can lead to large, systematic variation in stress drop estimates and an artificial dependence of spectral stress drop on seismic moment, if a simple source model is used. This follows from the fact that in spectral source studies covering earthquakes over a range of magnitudes, the signal frequency range above the corner frequency for the large earthquakes is much larger than that below the corner frequency, but the opposite is true for the smaller earthquakes potentially leading to systematic differences in the modeling.
The 2016–2017 damaging central Italy seismic sequence represents an excellent opportunity for investigating earthquake source properties given the high number of earthquakes, the large number of seismic stations, the extended range of magnitude and the extensive area affected by the sequence. The sequence included three mainshocks (Amatrice: M6.0, 24 August 2016; Visso: M5.9, 16 August 2016 and Norcia: M6.3, 30 October 2016) and six other earthquakes with magnitude M_{w} > 5 (Figure 1). Each mainshock ruptured different portions of the NWSE trending, SW dipping, normalfault segments that run parallel to the axis of the Apennines Mountain belt (Chiaraluce et al., 2017; Michele et al., 2020). The sequence started with the M_{w} 6.0 earthquake on 24 August 2016 at 01:36 (UTC) close to the town of Amatrice; the earthquake caused intense ground shaking, the deaths of 299 people and the evacuation of 31,764 more, and also the total destruction of Amatrice's old town. Following two months of continuous aftershock activity, on the 26 October 2016 at 19:18 (UTC) the M_{w} 5.9 Visso earthquake occurred approximately 30 km NNW of the M_{w} 6.0 Amatrice earthquake. Four days later, on the 30 October 2016 at 06:40 (UTC), the strongest event of the sequence with M_{w} 6.3 occurred close to Norcia, roughly halfway between the towns of Amatrice and Visso.
On 18 January 2017, the seismic activity migrated at the south of Amatrice causing further significant damage. In this area eight earthquakes with M_{w} ≥ 4.0, four with M_{w} ≥ 5.0, occurred within a few hours. The seismicity pattern of the 2016–2017 Amatrice seismic sequence showed the strong interaction between the inherited compressional thrusts and the younger and active normal faults (Barchi et al., 2021).
The importance of understanding these earthquakes, and mitigating future hazards led to studies of the stress drop of earthquakes in the 2016–2017 sequence, including those of Morasca et al. (2019), Wang et al. (2019), Bindi, Spallarossa, et al. (2020), and Kemna et al. (2021). There is considerable variability between the results of these previous studies. Morasca et al. (2019), Wang et al. (2019), and Bindi, Spallarossa, et al. (2020) focused on the magnitude scaling dependence, all finding that the larger magnitude earthquakes tended to have higher stress drops, to varying degrees. The exact magnitude range over which this dependence is resolved depends on the limitations of the frequency range of the available data. Previous work has shown that a limited frequency band can lead to scaling artifacts (e.g., Abercrombie, 2021; Chen & Abercrombie, 2020; Ruhl et al., 2017). Kemna et al. (2021) used a different approach and found no magnitude dependence, partly due to different methods, and partly due to different interpretations of the reliable frequency and magnitude ranges. None of these individual analyses investigated specifically why their results differed or agreed with the others.
Kemna et al. (2021) also looked at the spatial and temporal variation of their stress drop estimates; they found them to be related to the segmentation of the fault system with the earthquakes to the south of the OlevanoAntrodocoSibillini (OAS) thrust front having relatively low stress drops. The other studies did not investigate either spatial or temporal variability.
Calderoni et al. (2017), Wang et al. (2019), and Colavitti et al. (2022) all investigated the azimuthal directivity of the larger earthquakes in the 2016–2017 Italy sequence and found them to be predominantly unilateral ruptures, distinctly different from the assumed bilateral circular models of the stress drop studies. Calderoni et al. (2013) and Calderoni et al. (2015) found significant alongstrike directivity for earthquakes in the region as small as M3.3, indicating that neglecting directivity in empirical Green's function (EGF) events could also affect results of studies using them to isolate source radiation for larger earthquakes. The kinematic finitefault inversions of the largest three earthquakes (Chiarluce et al., 2017; Tinti et al., 2016) also revealed significant slip heterogeneity that is likely to affect spectral stress drop estimates obtained assuming simple models. The existence of these models, and also dynamic inversions of the two largest earthquakes (Gallovič et al., 2019; Scognamiglio et al., 2018; Tinti et al., 2021) provide the opportunity to investigate what information absolute and relative spectral stress drops are able to provide about complex ruptures.
Here we use two distinct approaches to measure spectral stress drop from recorded S waves and perform a series of tests to investigate the size and relative importance of the uncertainties resulting from the various assumptions made. We then compare our results to those from the previous published studies to quantify variability and look for consistencies. Finally, we compare the spectral measurements of stress drop to the finitefault models of Tinti et al. (2016) and Chiaraluce et al. (2017) to improve our understanding of what the spectral corner frequency and “stress drop” parameters actually represent when an earthquake is not a simple circular rupture.
2 Spectral Source Parameters
2.1 Theoretical Background
Estimating earthquake source parameters requires isolating the source term from the other factors. We do this using two different, existing, approaches so we can compare the results; see Abercrombie (2021) for a discussion of some of the advantages and disadvantages of the different approaches.
After estimating κ and H(f) we can fit the source spectral shape with Equation 2 to obtain the source parameters.
The constants in Equations 5 and 6 depend on the assumed source model and rupture velocity (e.g., Kaneko & Shearer, 2014; Madariaga, 1976) and change the absolute values of stress drop (e.g., Abercrombie & Rice, 2005). Using the same values to convert M_{0} and f_{0} into stress drop allows comparison of relative values between studies, even if the absolute values remain unknown.
2.2 Data
We select for analysis 30 of the largest (M_{w} > 3.5) earthquakes in the 2016–2017 sequence, that are well recorded at multiple stations (Figure 1). This selection includes the 16 M_{w} ≥ 4.4 earthquakes previously analyzed for directivity by Calderoni et al. (2017), and also included in most previous spectral stress drop studies. We search for closely located, wellrecorded candidate EGF events (following Abercrombie, 2014, 2015) to use in the SR analysis for these target earthquakes, under the additional constraint that for both target and EGF pairs of events the published moment tensors (when calculated) are similar. There are two moment tensor catalogs available for the 2016–2017 sequence, one following Herrmann et al. (2011) and the other following Scognamiglio et al. (2010). We identify 42 potential EGF events (2.2 ≤ M_{L} ≤ 4.05, between 1 and 5 EGFs for each individual target event) with a magnitude difference range for the pairs between 0.79 and 3.09. For four EGFtarget event pairs the source separation is between 7 and 10 km, and for all the others (92%) it is ≤5 km. Table S1 in Supporting Information S2 includes the hypocentral information about the starting data base.
We use the seismograms recorded by the mostly 24bit broadband 40s Nanometrics Trillium seismometers of the Italian Seismic Network run by the Istituto Nazionale di Geofisica e Vulcanologia (INGV). We correct all the seismograms using the factory supplied seismometer transfer functions.
From each instrumentcorrected velocity seismogram we select a time window for the spectral analysis. For the spectral fitting the time window is calculated to include 95% of the signal (cumulative integral of squared velocity of the horizontal components) duration above noise, after the S wave arrival; the time windows range from 10 to 30 s, increasing with event magnitude, and 10% of the window (1–3 s) is before the S arrival. For the SR analysis, we select a window of 10 s, starting 1 s before the S wave arrivals for all events to facilitate calculation of ratios between earthquakes of different magnitude.
A noise segment having the same duration as the corresponding signal time window is extracted before the Pwave arrival. We taper both the noise and signal windows with a 10% cosine filter to calculate the fast Fourier transform. We calculate the geometrical mean of both the noise and signal spectra on the two horizontal components, and then the signaltonoise ratio (SNR) of the target events and EGFs at each frequency sample. We compute the SNR at all the available stations for each of the selected events, and we only use frequencies at which SNR >3 (Figure S1 in Supporting Information S1).
2.3 Seismic Moment
Seismic moment can be estimated during the spectral fitting (e.g., Bindi, Spallarossa, et al., 2020; Bindi, Zaccarelli, & Kotha, 2020) or independent estimates can be used. Using independent measurements of moment removes tradeoff between corner frequency and moment in both spectral fitting and EGF analyses (e.g., Abercrombie, 2013; Moyer et al., 2018; Walter et al., 2017). As the 30 earthquakes that we analyze are relatively large, we use independently estimated values from regional moment tensor modeling; these methods use the longest period seismic data available and are thought to be most robust. Figure 2 shows that the two available moment tensor catalogs for the sequence have similar dependence on M_{L}, and no systematic bias. We primarily use the solutions following Herrmann et al. (2011), denoted M_{wMH}, in our analysis. We repeat some of our subsequent analysis for earthquakes also included in the Time Domain Moment Tensor catalog (Scognamiglio et al., 2010), which we denote as M_{wTDMT}, using these solutions to investigate the effects of moment uncertainty. If an EGF earthquake is not included in a particular catalog, we calculate its moment from that of the corresponding target event selected on the basis of the greatest number of stations that recorded the event, and the long period SR, as described below. Figure 2 also compares the moment estimates by Bindi, Spallarossa, et al. (2020); they are systematically higher than the moment tensor values, and show a different dependence on M_{L}; M_{w} − M_{L} decreases with increasing M_{L.}
2.4 Spectral Fitting
This approach follows that of Calderoni et al. (2019) and involves first calculating the attenuation correction, and then modeling the resulting source spectra.
2.4.1 Calculate Attenuation Correction, κ
To calculate κ for each station we use recordings of the 9 largest earthquakes (M_{w} > 5) in the 2016–2017 sequence, and the strongest earthquake from the 2009 L’ Aquila seismic sequence that occurred in the same source region (Figure 1), as they have the best signal to noise over the widest frequency range. Following Calderoni et al. (2019) we model the loglinear spectra of the S wave windows on each horizontal component for each event using Equation 2 (Figure 3).

FR_{0}—the widest frequency range: between the low frequency plateau of the displacement spectrum and the limit of SNR (F_{min}; F_{max}) Hz

FR_{1}—the higher frequency range: (F_{min} + 1/3 ΔF; F_{max}) Hz with ΔF = (F_{max} − F_{min})

FR_{2}—the lower frequency range: (F_{min}; F_{max} − 1/3 ΔF) Hz
The κ estimates in the three range of frequencies (Figures 3a, 3b and 4a) are similar, although FR_{2} yields systematically slightly higher values. We prefer the FR_{0} results as they benefit from the stability of the largest frequency range. This approach to estimating κ assumes that the earthquake source acceleration spectrum is constant in the measurement range. The frequency range bands FR_{0} and FR_{2} may include sufficiently low frequencies to be affected by the finite duration of the sources, but as our preferred results are within 1 standard deviation of those from the higher frequency range (FR_{1}), we consider this negligible.
2.4.2 Calculate Corner Frequency and Stress Drop
We correct each recorded velocity spectrum using the κ (attenuation parameter) values and the site effect calculated at each station, following the approach developed by Calderoni et al. (2010) and Rovelli and Calderoni (2014). We assume the seismic moments following (Herrmann et al., 2011) and fit the attenuationcorrected spectra with Equation 2, assuming F_{S} = 2, an average R_{θ,ф} of 0.55 (Boore & Boatwright, 1984), ρ = 2800 kg/m^{3} and β = 3,100 m/s from Herrmann et al. (2011). We calculate stress drop using Equations 5 and 6.
Figures 4b and 4c compares the stress drops obtained using the estimates of κ from the three different frequency ranges; Δσ_{FR0} (using our preferred κ) and Δσ_{FR1} are the same within a factor of 1.2, but Δσ_{FR2} is systematically higher, by a factor of 1.8 with respect Δσ_{FR0}. Using the higher frequency κ (FR_{1}) results in a smaller dependence of stress drop (Δσ_{FR2}) with moment, demonstrating how tradeoffs between attenuation and source affect inferences of source scaling. This systematic variation reflects the influence of the radiation and attenuation of high frequency waves on the stress drop estimates; Figure 5 shows the model fits to the spectra using FR_{0}, for the Amatrice and Visso earthquakes, and a smaller event as examples. Using a different moment estimate for these calculations would not affect the shape of the amplitude spectra, or the relative stress drop estimates, but simply cause a linear change in absolute stress drop estimated, similar to assuming an alternative source model (e.g., Abercrombie & Rice, 2005) and constants in Equations 5 and 6.
2.5 Spectral Ratios
To investigate any dependence of calculated stress drop on analysis method, we recalculate the stress drops for the same 30 earthquakes, using the same raw S wave spectra, using an EGF approach. Before computing the SR, we interpolate the numerator and denominator spectra using logarithmic sampling to decrease the weighting toward the higherfrequency part of the spectra (e.g., Abercrombie, 2021; Ide et al., 2003) and smoothed with a 0.2 Hz wide triangular operator. Calculating the SR between target and EGF events eliminates the propagation and site terms (Equations 1 and 4) and provides the ratio of the source terms.
Rather than inverting the spectral ratios for the moment ratio, we use independent measurements of seismic moment (made at longer periods for the larger event) to decrease the number of unknowns, and improve the stability of the results (e.g., Walter et al., 2017). The ratio of the moment values affects the corner frequency measurements and so we repeat the SR fitting using both moment tensor catalogs (Figure 2) to investigate the uncertainties.
Some EGF events are too small to be included in either one or both moment tensor catalogs. We could attempt to fit the three unknowns (M_{02}, f_{01}, f_{02}) together in these instances but we prefer to first estimate the moment ratio, using different frequency ranges, and making various specified assumptions about the corner frequencies, to understand better the tradeoffs between the parameters.
If the EGF event is not included in the momenttensor catalog, we estimate M_{02} from the ratio of the lowfrequency displacement plateau to the known M_{01} of the target event. We use the frequency range F_{min,EGF} to F_{cTargHP}, where F_{min,EGF} is the minimum frequency with SNR >3, and F_{cTargHP} is an estimate, by visual inspection, of where the amplitude of the SR starts to decrease. We estimate the best fit values of M_{02} through a grid search procedure (Equation 6) where M_{01} is known and f_{01} and f_{02} are written in terms of seismic moment and stress drop (Equations 4 and 5). As a starting approximation we estimate Δσ_{SR,2} and Δσ_{SR,1} using the scaling law deduced by Calderoni et al. (2013). To eliminate the problem due to the bandwidth limitations, we choose a seismic event located within approximately one source dimension of the EGF with magnitude in the range from 4 to 5, except for the M_{02} of the Visso earthquake EGFs.
We then use these moment values to estimate Δσ_{SR,1} in the frequency range in which both spectra are above the signal to noise threshold, F_{minNEGF} to F_{maxNEGF}, by performing a double loop inversion (Equation 4). In the first step, the best fit values of Δσ_{SR,1} and Δσ_{SR,2} are estimated through a grid search procedure where M_{01} and M_{02} are fixed; in the second step, the best fit value of Δσ_{SR,1} is estimated through a grid search procedure where M_{01}, M_{02} and Δσ_{SR,2} are fixed. Δσ_{SR,1} is computed as the weighted average over targetEGF pairs. This approach is similar to applying a constraint to f_{02} (e.g., Pennington et al., 2021) to avoid the tradeoffs observed by Shearer et al. (2019).

We repeat the analysis using both moment tensor catalogs to investigate how the choice of the moment magnitude (M_{wTDMT,1} M_{wMH,1}) affects the Δσ_{SR,1}; examples of the results for specific events are shown in Figure S2 in Supporting Information S1. Figure 2 compares the different moment estimates, and Figures 6a and 6b shows their impacts on stress drop estimates.

We find that varying the choice of frequency range used to calculate the moment of the EGF event corresponds to changes in M_{w2} within 0.2 units (see examples in Figure S2 in Supporting Information S1). We investigate the effects of such variation in the moment ratio on the resulting stress drop by adding and subtracting 0.2 M units to M_{w2} and repeating the analysis (Figures 6c and 6d). Increasing M_{w2} by 0.2 units can decrease the stress drop of the target stress drop by up to a factor of 2 (Figure 6b).

The choice of the EGF used in the SR also affects the resulting Δσ_{SR,1} estimates (e.g., Abercrombie, 2015). We consider the potential EGFs and select the best based on the frequency range with good SNR, and how well the spectral ratios and displacement spectra fit the circular source model. Examples of the effects of different EGF are shown in Figures S3 and S4 in Supporting Information S1.

EGF directivity effect (Figures S4 and S5 in Supporting Information S1). When the difference between the Δσ_{SR,1} and Δσ_{SP,1} is greater than a factor of 2.5 for an individual EGF it is likely that the EGF itself involves significant directivity, and so these EGF events are excluded (e.g., see Figure S4 in Supporting Information S1).

Choice of frequency range SNR ≥ 0.2 Hz and SNR ≥ 0.05 Hz on the Δσ_{SR,1} and Δσ_{SP,1} (Figures S6 and S7 in Supporting Information S1). The frequency range has a greater influence on Δσ_{SR,1} estimates especially for the larger magnitude events (M > 4) which need lower frequency signal to resolve the corner frequency.
These tests show that the uncertainties resulting from the input assumptions, and choices of data and analysis are larger than the formal model errors; the error bars from the different tests do not overlap in Figures 4, 1, and 7. From our starting database of 30 large events and 49 EGFs (all targetEGF pairs in Table S1 in Supporting Information S2), we obtain results for 29 target events and 41 EGFs using the SR approach (targetEGF pairs in bold text in Table S1 in Supporting Information S2).
We also check the influence of the focal mechanisms of the targetEGF pairs on the stress drop estimates (Table S2 in Supporting Information S2); all the earthquakes involve predominantly normal faulting. As shown in the Table S2 in Supporting Information S2 the stress drop estimates are quite similar even when there is some difference between focal mechanisms of the targetEGF pairs. Differing focal mechanisms will have most effect for stations close to nodal planes and so we suspect the large number of stations and good azimuth coverage (Figure 1) reduces the effect of the focal mechanism of EGFs on stress drop estimates in this study.
3 Results of Spectral Source Analysis, and Comparison With Previous Studies
We compare the stress drop measurements we make using the spectral fitting and SR approaches in Figure 7. There is a lot of variability and, as shown in the preceding sections, data selection choices and assumptions used in the different methods have a significant effect both on absolute values, and the relative values and magnitude dependence. There is a spread of approximately a factor of 4 in stress drop for each event that does not include any dependence on source model constants, or rupture velocity which are assumed the same throughout. All the approaches reveal some increase in spectral stress drop with moment; the SR method results in a lower scaling dependence than the spectral fitting, although using only the higher frequency estimate of attenuation would decrease the scaling dependence from the latter (Section 2.4).
An artificial apparent increase in estimates of spectral stress drop with moment is known to result from imperfect correction for attenuation (e.g., Ide et al., 2003) and also from the use of the limited frequency of the available data (e.g., Ruhl et al., 2017). We indicate the stress drops corresponding to the minimum and maximum frequencies used in the SR fitting (solid gray lines), and also use dashed gray lines to indicate the likely resolvable limits for corner frequency based on the work of Ruhl et al. (2017) and Chen and Abercrombie (2020). Our spectral measurements are wellresolved by these recommended criteria, but the possibility of systematic bias in a data set, remains, as a direct consequence of the modeling of the larger earthquakes being dominated by frequencies above the corner frequency and that of the smaller events being dominated by frequencies below the corner frequency (Abercrombie, 2021).
3.1 Comparison With Previous Studies: Stress Drop Scaling With Moment
In Figure 8 we compare our preferred results using each of the two methods (Table S3 in Supporting Information S2), to those of the previous published studies; Figure 5 and Figures S8 and S9 in Supporting Information S1 show examples of how the different models match the observed data. To ensure meaningful comparisons in Figure 8 we recompute the spectral stress drops from the published moment and corner frequency values, using the same constants in Equations 5 and 6 as we use in our own analysis (Table S4 in Supporting Information S2). Figure S10 in Supporting Information S1 includes the effects of using different moment estimates. The systematic and random uncertainties in both moment and corner frequency are far larger than the formal reported uncertainties (which do not overlap for individual events between studies) from the individual inversion methods. The different approaches result in different average stress drop values, varying over a factor of 10, and also different dependence on magnitude.
Morasca et al. (2019) analyzed 83 earthquakes using the coda wave ratio approach of Mayeda et al. (2007). They fixed the moments to those of Herrmann et al. (2011) as we do but obtained higher corner frequencies and stress drops (Figure 8 and Figure S10 in Supporting Information S1). Bindi, Spallarossa, et al. (2020) performed a largescale generalized inversion of over 4,111 earthquakes (M 1.5–6.5) in which they inverted for attenuation together with moment and corner frequency. This approach yields the strongest increase in stress drop with magnitude of those shown in Figure 8. The seismic moments estimated by Bindi, Spallarossa, et al. (2020) are systematically larger than those from the moment tensor catalogs (Figure 2) and also exhibit a different relationship to the INGV catalog magnitude. The generalized inversions tend to be relatively stable as they include large numbers of earthquakes but are subject to tradeoffs between moment and corner frequency, and also between spatial variation in attenuation and source effects within the inversion volume (e.g., Abercrombie et al., 2021; Bindi et al., 2021). In a followup study, Morasca et al. (2022) combined the generalized inversion technique with the coda wave approach of Mayeda et al. (2003) to investigate the tradeoff between moment and cornerfrequency. They obtained moment estimates comparable with those of Herrmann et al. (2011), and stress drops more comparable to the ones determined here for the largest events.
Kemna et al. (2021) used three increasingly wellconstrained approaches to obtain independent estimates of stress drop for earthquakes from the sequence: individually fitting the spectra for source and attenuation (16,000 earthquakes M_{L} 0.6–6.5), then a clustering approach with spatial constraints on attenuation, and a SR method for smaller subsets of events. In contrast to the other studies, they obtained stress drops that are independent of the seismic moment, above M_{L}3, and interpret the apparent decrease in stress drop at smaller magnitudes as a consequence of measurement bias using bandlimited data.
In addition to comparing the estimated stress drop values directly, we also investigate how well the different source models fit the recorded data. We calculate the model spectra using the parameters calculated in the different studies and compare them to the attenuationcorrected spectra and the spectral ratios; Figure 5, Figure S8, and S9 in Supporting Information S1 show some examples. The variability of the observed spectra and ratios is large compared to the difference between the different model predictions suggesting the real uncertainties are much larger than predicted. All the SR models (assuming simple circular rupture) with nonconstant stress drop fall within 1 standard deviation of the interstation variation for these earthquakes.
The same moments for the target and EGF earthquakes are used for both the SR (Δσ_{SR,1}) and spectral fitting (Δσ_{SP,1}) methods applied in this study. The seismic moment estimates from Bindi, Spallarossa, et al. (2020) are systematically different (Figure 2 and Figure S11 in Supporting Information S1) leading to systematically lower amplitudes for the predicted lowfrequency spectral ratios (see the blue curves at low frequency in Figure 5b). This could result from tradeoffs between the attenuation correction, and the seismic moment and corner frequency, and hence resolution of scaling.
In common with most previous studies of this earthquake sequence, we find that the earthquakes M > 5 tend to have higher spectral stress drops than the smaller events. In Figure 8, most studies, including ours and that of Bindi, Spallarossa, et al. (2020) also exhibit an increase in spectral stress drop with moment in the M3–5 range. In their larger data set, however, Bindi, Spallarossa, et al. (2020) observed little dependence of spectral stress drop on magnitude in the range M_{w}3–5, suggesting that the increase seen in Figure 8 in this magnitude range may be an artifact of the selection of the relatively small number of events we study here.
To investigate resolution of stress drop scale dependence, we also fit the spectral ratios assuming identical stress drop for target and EGF earthquakes (Figure 5, Figures S8 and S9 in Supporting Information S1); these models have consistently poorer fits (and also one less degree of freedom), with the model high frequency level systematically toward the lower range of the data, but they are within the range of the data for many event pairs. The results from both methods used here show an increase in spectral stress drop with moment that varies with analysis choices; the relationship is not obviously an artifact of observational limitations, but the tradeoffs are large and assumptions significant (e.g., Abercrombie, 2021) so we interpret the trends in Figure 8 and Figure S10 in Supporting Information S1 with caution.
Increase in spectral stress drop with moment has been reported within specific magnitude ranges, in individual studies, but trends like the ones observed here cannot extend over a large magnitude range without producing values for smaller and larger earthquakes far outside the observations (e.g., Abercrombie, 1995; Ide & Beroza, 2001; Salvadurai, 2019). The relation between the spectral measurements and the actual stress release during an earthquake is also not straightforward if the source deviates from a simple, circular rupture, for example, a different aspect ratio producing multiple corner frequencies (Ji & Archuleta, 2020, 2022), or other slip heterogeneity (Abercrombie, 2021). The availability of finitefault models for the largest earthquakes in our analysis enables us to investigate this further in the Section 4.
3.2 Spatial Variation in Stress Drop
Despite all this variability, there are also some strong consistencies observable in Figure 8 with individual events having relatively high (or low) stress drop estimates in all studies. For example, in Figure 8c all studies show event 13 to have a higher stress drop than event 15. Following Pennington et al. (2021) we consider relative values that are consistent using multiple methods and studies to be the most reliable parameters.
Kemna et al. (2021) reported distinct spatial variation in stress drop during the 2016–2017 sequence, but Bindi, Spallarossa, et al. (2020) did not investigate the spatial variation of stress drop and the other studies analyzed relatively few events. We plot the spatial variation of our calculated spectral stress drop values in Figure 9 and observe that they are consistent with the pattern reported by Kemna et al. (2021). Earthquakes to the north of Visso, and also just to the south of the OAS thrust front, have lower stress drops. The seismicity pattern of the 2016–2017 Amatrice seismic sequence also shows strong interaction between the inherited compressional thrusts and the younger and active normal faults. To investigate whether the fault geometry and segmentation have direct effects on the earthquake sources would require more detailed analysis of a larger number of earthquakes, with careful consideration of the tradeoff between depthdependent attenuation and source (Abercrombie et al., 2021).
4 Comparison With Finite Fault Inversions
The three largest earthquakes have all been modeled extensively using finite fault analyses, providing the opportunity to investigate how spectral stress drop estimates compare to independent estimates of rupture area, slip heterogeneity and stress release. We can compare the spectral stress drop estimates directly with the average stress drop in the dynamic rupture models of the two largest earthquakes. Gallovič et al. (2019) obtained an average stress drop of about 4 MPa for the Amatrice earthquake, and Tinti et al. (2021) determined an average stress drop of 2–4 MPa for the Norcia earthquake. These values are comparable to the spectral stress drops obtained here, and by Kemna et al. (2021) and Morasca et al. (2022), but significantly smaller than those of Bindi, Spallarossa, et al. (2020) and Morasca et al. (2019). As we assume the Brune (1970, 1971) source model, which has the highest value for k in Equation 5 of standard spectral models (e.g., Kaneko & Shearer, 2015; Madariaga, 1976) simply using a different spectral source model would increase all the spectral stress drop values.
The Amatrice and Visso earthquakes are very similar in magnitude, and so present a good opportunity to compare relative spectral estimates of source dimension and stress drop, with existing finite fault models without needing to consider magnitude dependence or different levels of resolution. All the spectral studies report a lower stress drop, by a factor of between 1.5 and 2, for the Visso mainshock than the Amatrice mainshock (events 28 and 29 respectively in Figure 8c). The difference is small compared to the known uncertainties, but it is consistent across multiple methods and studies.
Tinti et al. (2016) and Chiaraluce et al. (2017) inverted strong motion seismograms recorded at distances less than 45 km for kinematic rupture models of the Amatrice and Visso earthquakes, respectively. They followed the approach of Hartzell and Heaton (1983), Dreger and Kaverina (2000), and Dreger et al. (2005), and bandpass filtered the data between 0.02 and 0.5 Hz. The relatively low maximum frequency was selected to reduce the possible contributions of local site effects (Bindi et al., 2011) to the source inversion modeling. They interpolated their resulting slip distributions to a 0.5 × 0.5 km subfault grid. The resulting moment rate functions (calculated by summing the slip over the fault in each time step), their displacement spectra, and the slip distributions are plotted in Figures 10 and 11. Both of these very similar magnitudes, M ∼ 6, earthquakes exhibit some source complexity. These two earthquakes therefore provide an excellent opportunity to investigate what a spectral stress drop estimate represents when an earthquake is not a simple circular source. The Amatrice earthquake consists of two distinct, subevents and the total rupture area is larger than that of the Visso earthquake. The slip in the Visso earthquake was more focused into one region, and the maximum slip slightly higher than that in the Amatrice earthquake. These observations are the opposite to those predicted by the spectral stress drop modeling; the Amatrice earthquake has a higher spectral stress drop than the Visso earthquake which would imply a smaller rupture area and a higher average slip because of their similar moment.
Estimating the area, mean slip and stress drop of a heterogeneous slip distribution is not simple as it depends on the area selected and the weighting used (e.g., Noda et al., 2013; Ye et al., 2016). Here we follow the approach of Brown et al. (2015) to calculate the average values of slip within a range of different slip contours to ensure we have comparable reliable relative estimates. The mean slip for the areas defined by all the selected slip contours is higher for the Visso earthquake than the Amatrice, see Table S5 in Supporting Information S2, and the corresponding areas are larger for the Amatrice earthquake.
The moment rate function of the Amatrice earthquake has a longer duration (confirmed by the dynamic modeling, Gallovič et al., 2019), consistent with the larger rupture area of slip in the finite fault inversion, compared to the Visso earthquake. The effective source dimensions calculated following Mai and Beroza (2000) and Thingbaijam and Mai (2016) are also larger for the Amatrice earthquake than the Visso. Again, these observations are the opposite of the relative values predicted from the spectral stress drop modeling. However, the spectrum of the model of the Amatrice earthquake contains more highfrequency energy than the model of the Visso earthquake, which is consistent with the higher corner frequency calculated for the Amatrice earthquake in almost all spectral studies (Figure 8) and indicates how source complexity can affect spectral modeling. The circular rupture areas predicted by the different spectral models are also shown, with that predicted for the Visso earthquake typically larger than for Amatrice in contrast to the relative size of the finite fault rupture areas.
To investigate further the relationship between the different models, we use the method of Ripperger and Mai (2004) to calculate the stress release on the fault plane in the two earthquakes from the finite fault models. As expected, the Visso earthquake has a larger, more concentrated area of high stress drop (Figure 11), and a higher slipweighted mean stress drop (following Noda et al., 2013) of 4.8 MPa compared to 2.9 Mpa for the Amatrice earthquake.
This comparison is consistent with the work of Lin and Lapusta (2018), Gallovič and Valentová (2020) and Liu et al. (2023), implying that source complexity can significantly distort spectral measurements of absolute, and relative stress drop that depend on the assumption of simple source models. In this case of two similarsized earthquakes, the spectral source estimates find the Amatrice to have a smaller rupture area, and higher average slip, whereas the kinematic finite fault inversions reveal the opposite. The consistency of the measurements is more reliable than the absolute numbers. This implies that a higher corner frequency and higher spectral stress drop may indicate slip heterogeneity and other source complexity rather than high localized stress release, In such cases, a higher corner frequency and higher spectral stress drop cannot be simply interpreted in terms of source physics. Finite fault analysis of exceptionally wellrecorded smaller earthquakes (e.g., Pennington, Chang, et al., 2022; Pennington, Uchide, & Chen, 2022) also reveal similar slip indicating that even our smaller magnitude measurements may not be well approximated by the circular or point source approximation. The complex spectral shapes we observe in the spectral ratios, and the evidence for azimuthal directivity in the smaller, and even the EGF earthquakes (Figures S4 and S5 in Supporting Information S1) also support the inference that source complexity cannot be ignored for any of the earthquakes considered here. Abercrombie (2021) also showed how a complex source recorded in a limited frequency bandwidth, typical of many spectral studies, can produce an artificial increase in stress drop with seismic moment. This supports the need to keep an open mind concerning the resolution of stress drop scaling in the 2016–2017 Italy earthquake sequence. However, since it is the high frequency radiation which causes the peak ground accelerations, a high Brunestyle stress drop from spectral modeling may be a more reliable indicator of strong ground motions, whether they be caused by a concentrated area of high slip, or a more distributed complex rupture.
5 Conclusions
We analyze 30 of the larger earthquakes in the central Italy sequence of 2016–2017, to estimate spectral stress drop using two different approaches, and varying the various data selections and assumptions involved. We then compare and interpret our results in comparison with the previously published results, allowing us to separate more reliably real source variability from the large uncertainties.
We find significant uncertainties in stress drop estimates are the product of many different causes, including choice of the magnitude catalog, frequency and width, ambiguities in separating source and path, tradeoffs between source, path and site effects, and assumption of a simplified source model. For these reasons, we find wide variation in absolute stress drop estimates, even when assuming the same source model constants.
Relative parameter observations that are consistent across multiple studies are the most reliable. It is encouraging that some events are found to have higher or lower spectral stress drop estimates across multiple studies implying that they represent real variability. Our results support the previously reported spatial variation (Kemna et al., 2021), with decreased stress drop to the south of the OAS thrust front.
Varying levels of correlation between stress drop and magnitude are observed using different methods, magnitude ranges and selections of earthquakes. All results presented here show some increase in stress drop with magnitude; the SR method results in a lower scaling dependence than the spectral fitting. Whether this dependence is real or a consequence of unmodelled source complexity combined with limited frequency range is still unclear.
Through the comparison between the two earthquakes with similar magnitude, Amatrice and Visso seismic events, we deduce that source complexity can significantly distort spectral measurements of absolute, and relative stress drop that assume simple source models. Higher spectral stress drop estimates may represent better strong ground movements and be less directly related to source physics.
There is clearly still much work to do to understand what a spectral stress drop estimate really means, and what the real uncertainties are. Using multiple approaches, and only interpreting consistent results (e.g., Pennington et al., 2021) remains the most reliable way forward at present. The ongoing community stress drop validation study should also provide clarification of these important questions (e.g., Baltay et al., 2022).
Acknowledgments
This study was supported by the “Realtime Earthquake Risk Reduction for a Resilient Europe” (RISE) project, funded by the European Union's Horizon 2020 research and innovation program (Grant Agreement No. 821115). This material is based upon work supported by the National Science Foundation under Grant EAR 2043281 to Abercrombie.
Open Research
Data Availability Statement
The seismic data used here are recorded by the INGV and are accessible from the European Integrated Data Archive (EIDA), INGV Seismological Data Centre (2006, https://doi.org/10.13127/SD/X0FXNH7QFY). The local magnitudes are from the Italian Seismological Instrumental and Parametric DataBase (http://iside.rm.ingv.it/) through the event selection via interactive map for region shown in Figure 1. The Moment tensors from Herrmann and Malagnini are available from https://www.eas.slu.edu/eqc/eqc_mt/MECH.IT/#DETAILS, and the moment tensors from Scognamiglio are available at http://terremoti.ingv.it/tdmt. The finite fault models from Chiaraluce et al. (2017), were provided by E. Tinti, and we gratefully acknowledge her assistance in analyzing and interpreting these models. The SAC software (Goldstein et al., 2003) was used for routine seismogram processing.