Here I compare estimates of tectonic strain rates from dense Global Positioning System measurements with the seismicity released in the last ~500 years in the Apennines (Italy). The rates of seismic moment accumulation from geodesy and of historical seismic release by earthquakes agree within the uncertainties, ruling out significant aseismic deformation. Within the considered 400 km long section of the Apennines, this balance yields an average recurrence interval of 30–75 years for MW≥6.5 events without requiring a future earthquake larger than those observed historically (MW~7). A minimum estimate of unreleased strain allows MW≥6.5 and MW≥6.9 events to be released in ~35% and ~10% of the central-southern Apennines, respectively. The definition of the seismic potential for smaller events is more uncertain, and their occurrence remains a significant threat throughout the Apennines.
Assessment of seismic hazard is, in most places, mainly based on past, historical seismicity. Short-term fluctuations are an inevitable consequence of the episodic earthquake release. It is thus of primary importance to evaluate whether the sampled seismicity is representative of the effective rate of tectonic deformation or, in other words, stationary relative to the steady state tectonic loading [Main, 1996]. The theory of the elastic rebound [Reid, 1910] offered a conceptual framework to link the rate of tectonic loading and the intermittent earthquake release, but the successful application of quasi-periodic, deterministic models of recurrence has been challenging [Scholz, 2002]. The constrain that, given enough time, the rate of strain released by earthquakes should mirror the rate of tectonic deformation provides, however, a powerful tool for the assessment of seismic hazard. Measurements of crustal deformation can, under simple assumptions, be translated in estimates of the average frequency and magnitude of the largest events [Molnar, 1979; Kagan, 2002]. Lack of or inaccurate geodetic data and short or incomplete seismic catalogues have limited the accuracy of this approach, especially in slowly deforming continental regions. Any comparison between geodetic measurements of tectonic deformation and seismicity should take special care to avoid discrepancies that may originate from short seismic catalogues and small sampled areas [Ward, 1998; Pancha et al., 2006]. Further advances in this field are thus likely to come from regions, such as the Apennines, combining a long historical record of earthquakes [Rovida et al., 2011] and accurately measurable geodetic strain rates.
The Apennines are an actively extending mountain belt running for ~700 km in a NW-SE direction along the Italian peninsula. Active deformation is well documented by the instrumental and historical seismic catalogues [Selvaggi, 1998; Rovida et al., 2011; Chiarabba et al., 2005] and by geological and geodetic data [Hunstad et al., 2003; D'Agostino et al., 2011]. NE-SW extension overprints previous contractional structures, now active in the Northern Adriatic Sea and in the Po Plain. Active extension now appears to be driven by a combination of relative NE-SW motion between the Adriatic and Tyrrhenian domains and by gradients of gravitational potential energy localizing strain along the crest of the Apennines [D'Agostino et al., 2011]. Here I compare estimates of tectonic strain from dense Global Positioning System (GPS) measurements with the seismicity released in the last ~500 years in the Apennines.
2 GPS Velocity and Strain Rate Field
In this work I use an original GPS velocity field composed of velocity vectors and associated uncertainties from 201 continuous stations, of which 75 belong to the Rete Integrata Nazionale GPS network [Avallone et al., 2010], and 40 survey-type repeatedly measured benchmarks (see supporting information). To emphasize the extensional deformation across the Apennines, the velocity field (Figure 1a) is shown in a reference frame defined by stations on the Tyrrhenian coastal region (D'Agostino et al.  and supporting information). Extension across the Apennines is clearly defined by the differential ~3 mm/yr motion between the Tyrrhenian and Adriatic coasts. The principal axes and second invariant value of the strain rate tensor (Figures 1b and 1c; supporting information) show that the Apennines are characterized by a ~50 km wide continuous belt deforming at 40–60 nstrain/yr that remarkably follows the regional highest elevations, and the drainage divides between the Adriatic and Tyrrhenian Sea.
I estimate the rate of seismic moment accumulation using the strain rate field in a 400 × 130 km2 area parallel to the Apennines (Figure 1c). The relationship between the strain rate tensor and scalar moment rate is not unique [Savage and Simpson, 1997]. Various methods, listed in Pancha et al. , have been proposed. In case of a dominant uniaxial strain, as in the Apennines, these methods provide very similar results, and I use the scalar version of Kostrov's formula [Kostrov, 1974] proposed by Ward 
where Aiand are the area and the largest absolute eigenvalue of the strain rate tensor in each of the ith equidimensional 30 × 30 km cell, respectively. Hs is the seismogenic thickness and μ is the rigidity modulus (3.0 × 1010N/m2). Equation (1) furnishes a estimate of 76.9±15.6×1016 Nm/yr. I use a homogeneous value of Hs=10±2.5 km which include the ipocentrals depth of instrumentally recorded M>5 earthquakes [Chiarabba et al., 2005].
3 Historical Seismic Moment Release
Seismic events with MW≥5.0 in the time interval 1550–2010 (Table S4 in the supporting information) have been extracted from the CPTI11 catalogue of Italian earthquakes [Rovida et al., 2011]. Each event in the CPTI11 catalogue has an estimate of equivalent moment magnitude obtained from all available intensities and equated to MWusing regression coefficients calibrated with intensity data available for instrumentally recorded earthquakes [Rovida et al., 2011]. Assessments of catalogue completeness [Stucchi et al., 2011] suggest that in the axial belt of the Apennines, the catalogue is complete for MW≥6 events starting approximately from the beginning of sixteenth century. Macroseismic estimates of MW and epicentral locations have been used for consistency also for those events having instrumentally determined values. The moment magnitude estimates are then converted in seismic moment using the formula [Hanks and Kanamori, 1979]. The largest selected event (1857 Val D'Agri MW 7.0) contributes for 14% to the total seismic moment, whereas the contribution of the first three biggest events (1857, 1915, and 1688) amounts to 39%. My estimate of seismic moment release is thus unlikely to be critically affected by few, large events in the catalogue. Since the instrumentally recorded earthquakes and the pattern of geodetic deformation consistently indicate uniaxial extension in a NE-SW direction, I assume that all seismic events have released NE-SW extension by slip on 45°-dipping NW-SE trending normal faults. Using this approximation, the eigenvectors of the seismic moment tensors are horizontal or vertical and the rate of seismic moment release can be simply obtained by summing the scalar moments of each individual event. There are two sources of errors which affect estimates of long-term rates of seismic moment release. The first is the error of scalar moment of individual earthquakes, whereas the second is the spatiotemporal randomness of earthquake occurrence. Considering earthquake occurrence as a random, Poisson-like process, I assign to the jth event an uncertainty equal to its seismic moment . The rate of seismic moment released in the time interval T and its variance are then evaluated with the following equations:
Assuming stationary seismicity, independent events and absence of aseismic deformation the moment rate can be viewed as a random variable which, for a sufficiently large time interval, tends asymptotically to the value of the long-term tectonic loading moment rate. The cumulative seismic moment since 1550 is shown in Figure 2a and agrees within the uncertainties with the seismic moment predicted by the geodetic accumulation rate assuming a stable behavior in the last centuries. This is most clearly seen in Figure 2b where the rate of seismic moment release (equation (2)), calculated in progressively longer intervals, has large initial fluctuations but converges asymptotically toward a stable value in agreement with the rate of tectonic loading . The historical seismic moment rate depends on the assumed dip of the faults as 1/sin[2×dip]. Assuming a 60°-dipping normal faults thus reduce the estimated rate by 14%, but the rate of seismic moment release and accumulation still agree within the uncertainties.
4 Earthquake Recurrence
Empirical observations show that small earthquakes occur more frequently than large ones following a Gutenberg-Richter (G-R) distribution [Gutenberg and Richter, 1944] that relates the cumulative number of earthquakes N above a given magnitude M, by log(N)=a−bM where a is a constant and the b value is generally approximately 1. I verify the hypothesis that the total seismic moment distributes into earthquake sizes following a G-R distribution that conserves [Kagan, 2002] the rate of tectonic loading without requiring events larger than those observed historically (. By truncating the cumulative G-R distribution above magnitudes corresponding to moment (the seismic moment of the event), the frequency distribution [Molnar, 1979] of earthquakes of moment ≥M0 is
Figure 3 shows the frequency distribution from (4) using the geodetic moment rate (76.9±15.6×1016 Nm/yr), from the largest event in the catalogue, and a b value of 1, as suggested by recent analyses of the instrumental catalogue [Gasperini et al., 2013]. The G-R distribution so obtained (Figure 3) agrees within uncertainties with the observed seismicity for MW>6. At lower magnitudes, observed frequencies are systematically lower than the G-R model consistently with the completeness's analysis of the CPTI11 catalogue [Stucchi et al., 2011]. The G-R model predicts that (1) a MW≥6.5 event must occur, on average, every 30–75 years (95% confidence range) to seismically balance the rate of tectonic deformation and (2) that the fraction of seismic moment released by MW≤6 events is ~30% of the total. A significant part of this amount can be released by MW≤5 events (not extracted from the CPTI11 catalogue) and by missing events in the magnitude range 5≤MW≤6. By comparing observed and calculated frequencies (Figure S1, in the supporting information), I estimate the moment rate contribution from MW≤6 “missing” events as 16.8×1016 Nm/yr (see supporting information). This correction term is added to the historical seismic release and reinforces the agreement between seismic and geodetic moment rates (Figure 2).
5 Spatial Distribution of Strain Release
The occurrence of an earthquake locally depends on the absolute level of strain, but a complete knowledge of past earthquakes history for a sufficiently long time interval is, in most cases, unaccessible. A minimum estimate of the local accumulated strain can, however, be obtained by the pointwise integration of seismic moment accumulation and release, for the time interval covered by sufficiently complete information. Figures 4a and 4b show the spatiotemporal distribution of the earthquakes that occurred since 1550 along the Apennines. The macroseismic epicenter of each event is centered on a horizontal bar whose length is calculated from an empirical magnitude-fault length scaling relationship for normal faults [Wells and Coppersmith, 1994]. To obtain the spatial distribution of seismic moment release the moment of each event is smoothed using a Gaussian function, centered on the macroseismically determined epicenter, whose 4σ width equals the estimated fault length. Unless propagation effects dominate the radiation field, the macroseismic epicenter is the best indicator of the barycenter of the seismic moment release [Bakun et al., 2011]. The contributions of all earthquakes are then integrated numerically to obtain a one-dimensional estimate of seismic moment release at various time frames and compared with the value of the 1550–2010 cumulated tectonic strain in Figure 4c. It appears that some regions released the equivalent cumulated strain at a relatively early stage and remained relatively silent afterward (i.e., the epicentral area of the two 1703 events). Other regions have been affected by multiple MW≥6.5 events that completely depleted the 1550–2010 strain budget (epicentral area of the 1980 event). In other cases, a seismic event released the accumulated strain in the form of a " gap-filling" event late in the considered period (1857 and 1915 events). I observe an area of reduced seismic release, supposedly associated with a high level of accumulated deformation, comprised between the epicentral areas of the 1915 and 1805 events. Moving spatial windows of 25 and 50 km are used to assess the distribution of accumulated moment and the earthquake potential of MW~6.5 and ~6.9 events, respectively. In ~35% and ~10% of the along-strike length, there is enough accumulated strain to release a MW~6.5 and ~6.9 event, respectively. As expected, the potential of a MW~6.9 event is concentrated in the ~50 km wide region between the 1805 and 1915 events, where previous large events date back to 1349 and 1456 [Rovida et al., 2011]. A significant potential of MW~6.5 also extend in the northern part where enough strain appears to have accumulated following the two 1703 events. The estimate of earthquake potential of MW≤6.5 events is subject to considerably higher uncertainty. It is likely that many short (~10–20 km) fault segments, loaded by both regional strain rate and stress transfer by nearby events, have stored enough deformation to fail in MW~6 events.
6 Discussion and Conclusions
It is of interest to characterize the average recurrence for the smallest possible dimension, namely, for an area large enough to contain the fault length associated to the event (i.e., ~50 km for a MW~7 event). Since the rate of extension is relatively constant along strike, I simply estimate the recurrence of MW≥6.5 events by scaling by a factor 8 (400/50) the recurrence in the total 400 km long section. Balancing the tectonic loading by earthquakes thus requires a MW≥6.5 event to locally strike, on average, every 240–600 years. Elapsed times in regions of unreleased strain (Figure 4d) have a comparable duration, possibly pointing to heightened seismic hazard in these regions. Two major sources of uncertainty affect, however, estimates of seismic hazard based on seismic moment deficit. The first arises from the poorly known seismic history and uncertain level of strain prior to 1550. The second source comes from the intrinsic uncertainty of earthquake recurrence. In particular, it is not clear if the distribution of recurrence intervals is best described by a renewal model with a strong central tendency [Matthews et al., 2002], by the view that repeated large earthquakes can happen in rapid succession without requiring time for stress regeneration [Jackson and Kagan, 2006] or that strain is released by clustered events in a time span considerably shorter than their mean recurrence interval [Wallace, 1987]. A recent 36Cl exposure dating study of the fault scarps of the Fucino fault system [Benedetti et al., 2013] suggests that over the last 12,000 kyr, >30 large earthquakes broke the faults in synchrony in 3–6 kyr spaced clusters of large events releasing most of the strain in <1–2 kyr long episodes. If these results are generalized, there emerges a general scenario where migrating pulses of activity are temporarily focused for a timescale of few thousand years, similarly to what is proposed for the Basin and Range province [Wallace, 1987]. It thus appears that the reliability of the estimates of moment deficit in Figure 4d to be indicative of high seismic hazard depends on the extent to which accumulated strain is locally reset by individual large events [Weldon et al., 2004]. If the strain is accumulated over timescales several times the average recurrence interval and released by temporal grouping of large events, classical schemes of earthquake predictability employed in the assessment of seismic hazard [Shimazaki and Nakata, 1980] would be of limited use.
In summary, the convincing evidence of complete seismic release of tectonic deformation in the Apennines argues for a stationary response of seismicity to tectonic loading at the timescale of ~500 years. Previously reported discrepancies [Hunstad et al., 2003] are thus likely to originate from the limited temporal (126 years) and spatial dimensions used for comparison of seismic and geodetic strain rates. Although starting from a similar cumulative value of historical seismic moment release, Selvaggi  obtained an average extension rate (~1.6 mm/yr) across the central-southern Apennines a factor of 2 smaller than the one measured geodetically (~3 mm/yr). The adoption of a longer time interval (700 years), likely to be significantly incomplete before the sixteenth century, is probably the reason for the observed discrepancy. These comparisons emphasize the need for an accurate selection of appropriate spatial and temporal dimensions when comparing geodetic and seismic deformation data [Ward, 1998; Pancha et al., 2006].
The results of this work provide the basis (1) to evaluate the completeness of the seismic catalogue, (2) to guide the spatial design of the seismogenetic zoning in probabilistic seismic hazard assessments, (3) to improve the definition of frequency and magnitude of the largest events, and (4) to motivate a wider integration of geodetic data in seismic hazard practices. A minimum estimate of unreleased strain since 1550 shows that the potential of MW~6.9 events concentrates in a single region, whereas the potential of smaller events is more widely distributed. The rapidly deforming (>25 nstrain/yr) part of the Apennines hosts ~1.6 million people, and several urban communities with >10,000 inhabitants lie potentially close to active faults. Taking into account the uncertainties in (1) the strain level prior to 1550, (2) our limited knowledge of mechanisms governing earthquake recurrence, and (3) the significant potential impact of a MW~6.0 event in the Apennines, a conservative approach is to consider the rapidly deforming area as equally exposed to destructive seismic events.
I am grateful to Giulio Selvaggi for several helpful discussions. I thank Tim Wright and Sigurjón Jónsson for having stimulated the writing of this paper and Steven Ward for sharing insights on geodetic and seismic deformation. Comments by the reviewers Corné Kreemer and Philippe Vernant were appreciated. Figures were produced with the GMT software. Data supporting Figure 1 is available as in Table S1 in the supporting information.
The Editor thanks Corne' Kreemer and an anonymous reviewer for their assistance in evaluating this paper.