Source parameters of the 1908 Messina Straits, Italy, earthquake from geodetic and seismic data

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. B4, 2080, /2001JB000434, 2002 Source parameters of the 1908 Messina Straits, Italy, earthquake from geodetic and seismic data Antonella Amoruso Dipartimento di Fisica, Università dell Aquila, L Aquila, Italy Luca Crescentini Dipartimento di Scienze della Terra, Università di Camerino, Camerino, Italy Roberto Scarpa Dipartimento di Fisica, Università dell Aquila, L Aquila, Italy Received 12 May 2000; revised 14 October 2001; accepted 19 October 2001; published 27 April [1] We carry out a nonlinear joint inversion of P wave first-motion polarities and coseismic surface displacement data of the 1908 Messina earthquake. We model the earthquake using a single planar fault: Slip is at first assumed to be uniform across the whole fault, then independent in a small set of coplanar subfaults, and finally smoothly variable across the fault. The first two steps are accomplished using a global minimization technique. The main features of the retrieved model are very robust and independent of the seismic velocity profile, of the presence of questionable bench marks, and of the assumed residual distribution. The along-strike component of slip is about half the along-dip component, in agreement with the direction of the extensional stress axis retrieved from geological observations. Surface vertical displacement is consistent with tsunami data and with the morphology of the Messina Straits. INDEX TERMS: 7215 Seismology: Earthquake parameters; 1242 Geodesy and Gravity: Seismic deformations (7205); 7260 Seismology: Theory and modeling; KEYWORDS: Seismic deformations, seismicity and seismotectonics, earthquake parameters 1. Introduction [2] The Calabrian arc evidences a very intense historical seismic activity and fast movements closely related to the opening of the southern Tyrrhenian Sea. It is marked by both crustal events and intermediate and deep focus earthquakes located along the inner side of the arc [McKenzie, 1972]. The southernmost segment of the arc is characterized by the highest seismicity; earthquakes having magnitudes over 7 occurred at least in 1905 and 1908 and during a complex sequence of five large earthquakes from February to March 1783 [e.g., Tortorici et al., 1995]. [3] According to several authors [e.g., Cucci et al., 1996], Calabria and its surroundings are uplifting at a rate of 1.0 ± 0.1 mm yr 1. Regional uplift probably began shortly before 0.7 Ma, and its rate could be explained as the terminal velocity of isostatic viscous relaxation following removal of loading due to detachment of the Tyrrhenial Benioff zone from beneath Calabria [Westaway, 1993]. The fault segments in the Southern Apennines seismogenic belt, which extends almost continuously along the inner side of the arc (Figure 1), often accommodate NE-SW to E-W extension perpendicular to the axis of the chain by almost pure dip-slip large movements, while the transverse zones which usually mark the boundaries of the main faults generate mainly strike-slip intermediate magnitude earthquakes. Probably most of the almost pure normal faults are blind [e.g., Cucci et al., 1996]. The southern segment of the Calabrian arc is cut by the Messina Straits (Figure 1), which is formed by a graben with N-S oriented axis. The Messina Straits is bounded by two systems of antithetic NNE-SSW trending normal faults [Ghisetti, 1992; Tortorici et al., 1995] that indicate an extensional stress axis oriented about N115 E. Precision levelings [Mulargia et al., 1984] and trilateration measurements [Caputo et Copyright 2002 by the American Geophysical Union /02/2001JB000434$09.00 al., 1981] suggest that the straits is subsiding at a rate of 1 mm yr 1 and that Sicily is moving northward with respect to Calabria. [4] The 1908 Messina earthquake is one of the strongest historical seismic events that ever occurred in Italy, with more than 60,000 casualties and extensive damage. It was felt by people in a radius of 300 km, with maximum damage (XII degree Mercalli intensity scale) occurring in the cities of Messina and Reggio Calabria. This earthquake occurred 3 years after another large event in the area, located at sea, and was followed by a tsunami with sea waves as high as 12 m, entering locally up to 200 m inland [Platania, 1909]. Smaller earthquakes were felt in the Messina region from 1 month before the main event to 10 years after. The 1908 earthquake was accompanied by visible ground cracks [Baratta, 1910]: the coastline road close to Messina showed a graben-type collapse, with fractures up to 100 m long and slip around 0.6 m. It is not clear if these fractures are related to the seismogenic fault or are landslides, and there is no clear evidence of further surface breaking. [5] Information about land movements at Messina, associated to some extent with the 1908 earthquake, comes from data recorded by a tide gauge operating in the Messina harbor since They suggest uplift of the coast at a rate of 16 mm yr 1 in the period before 1900, subsidence of 3 mmyr 1 in the period , and subsidence of more than 2 cm yr 1 in the periods and [Mulargia and Boschi, 1983; Baldi et al., 1983; Bottari et al., 1992]. No data exist for the period because of the damage due to the main shock. However, the data from the Palermo harbor for the same period do not show this trend, suggesting that the subsidence of Messina was a local event. This preseismic subsidence may indicate that aseismic slip on the main fault had been occurring since 1906, whereas the subsidence for the period could be associated with fault-slip episodes as well as with viscoelastic effects following the main shock. ESE 4-1

2 ESE 4-2 AMORUSO ET AL.: THE 1908 MESSINA STRAITS EARTHQUAKE [9] Data from leveling surveys have been first employed by Mulargia and Boschi [1983], who suggested a mechanism involving two normal faults, a northernmost one with a low dip angle and striking NNE-SSW and a southernmost one, antithetic to the former and with a high dip angle, causing a relative subsidence of the Calabrian block. [10] Capuano et al. [1988] interpreted vertical displacement data using an inverse method with an initial model and a priori information, which in principle strongly influence the final solution. At first, they modeled the earthquake with a single fault, initially located where proposed by Schick [1977] (Figure 2), and whose initial values of strike, dip, and rake are from Martini and Scarpa [1983]. Displacements of the bench marks located south of Reggio Calabria were not well modeled and the total square residual was 2400 cm 2. Capuano et al. introduced a higher slip area close to Reggio Calabria; the total square residual lowered to 1840 cm 2, but according to them the small improvement suggests the lack of any statistical basis for a variable dislocation model. They also considered two faults, starting from the model by Mulargia and Boschi [1983]. The improvement in the fit (total square residual was 1940 cm 2 ) was considered statistically not significant, and they concluded that the seismic source was a single normal fault (strike 4, dip39, rake 133 ) located beneath the Messina Straits (Figure 2). The surface projection of its upper side crosses the city of Messina. Figure 1. Schematic map of the Messina Straits. The axis origin is located at N, E. Solid circles, bench mark sites; crosses, numbered bench marks; solid diamond, area of maximum energy release according to Omori [1909]; solid diamond and square, areas of maximum energy release according to Baratta [1910]; triangle, epicenter of the 1975 earthquake; inverted triangle, area of the 1985 swarm. Locations of major fault systems are from Capuano et al. [1988]. [6] Seismograms, geodetic measurements, tsunami data, macroseismic observations, and geological surveys have led to a plethora of published models, including nearly any source mechanism and location. From 23 to 30 seismic records have been used by Schick [1977], Martini and Scarpa [1983], and Gasparini et al. [1982, 1985] to retrieve focal mechanisms by P wave polarities. Schick [1977] suggested a pure normal faulting process on a fault striking 195 and dipping 70. Martini and Scarpa [1983] and Gasparini et al. [1985] proposed a solution with strike 11, dip42, and rake 121. [7] Capuano et al. [1988] pointed out that uncertainties in the depth location and in the chosen P wave velocity model might account for the different strikes and dips of the fault plane determined by different authors. Inversion of P wave polarities alone consequently does not allow retrieval of details of the focal mechanism, and it is obviously unable to locate the seismogenic fault. They used surface wave magnitudes computed from seismograms recorded at 16 observatories distributed worldwide to estimate M s = 7.1 ± 0.2 and M 0 = (5.6 ± 2.8) Nm. [8] Several regional seismograms recorded in central Europe have been recently digitized and analyzed [Pino et al., 2000]. Pino et al. confirmed the seismic moment estimate by Capuano et al. [1988], giving M 0 = (5.4 ± 2.2) N m. They also assessed the unilateral northward propagation of the rupture along an 43-kmlong fault and the existence of a maximum slip of 4 m in the central part of the slipped area (assuming a fault width of 20 km). Figure 2. Surface projection of the seismogenic fault and focal mechanism (lower hemisphere Schmidt projection; stations as used in this work) using solutions for fault A from Schick [1977], fault B from Bottari et al. [1986], fault C from Capuano et al. [1988], fault D from Boschi et al. [1989], and fault E from De Natale and Pingue [1991] (effectively slipped area according to De Natale and Pingue). Thick sides are the upper ones.

3 AMORUSO ET AL.: THE 1908 MESSINA STRAITS EARTHQUAKE ESE 4-3 [11] Boschi et al. [1989] questioned the previous solutions and used geological observations and macroseismic and seismometric data to constrain the main features of the geodetic model. They used a variable slip technique [Ward and Valensise, 1989] to refine the uniform-slip model retrieved by Valensise [1988] using a priori hypothesis and concluded that the earthquake occurred along a 11 -striking, 30 -dipping normal fault (Figure 2). Slip is concentrated into two regions: The former is near Villa San Giovanni at a depth of 5 km and could be an effect of the bench mark locations, and the latter is near Reggio Calabria at a depth of 7 km and is considered genuine. Boschi et al. excluded the four Messina bench marks from the analysis, used the remaining Sicilian ones only as indicative of local tilt, and at first also excluded 16 Calabrian bench marks later included to refine the model. The total square residual of Calabrian bench marks is 884 cm 2. From the analysis of the sensitivity kernels they concluded that 30% of the fault, i.e., the southernmost shallower part of it, is not resolved by the leveling lines and that in the other parts the actual slip could differ up to 70% from the computed one. De Natale and Pingue [1991] used a similar technique to refine the Capuano et al. [1988] uniform-slip model. The total square residual is 800 cm 2. The slip distribution is characterized by a higher slip area that is located close to the city of Reggio Calabria at a depth of 8 km. Another higher slip area is located in the part of the fault crossing the city of Messina, but according to De Natale and Pingue, its reality cannot be assessed unequivocally, and the analysis by Pino et al. [2000] did not confirm its existence. [12] Valensise and Pantosti [1992] pointed out that the model proposed by Capuano et al. [1988] and De Natale and Pingue [1991] agrees with the observations everywhere except north of Messina, where repetition of a 1908-type earthquake would rapidly erase the Ganzirri peninsula (see Figure 1) and straighten the straits opening toward the Tyrrhenian Sea. The repetition of events like that of Boschi et al. [1989] fits nearly all the young geological features of the straits, e.g., the morphology of the Messina gravels on the Calabrian side of the straits [see Valensise and Pantosti, 1992, Figure 4]. However, as recognized by Valensise and Pantosti, the seaward tilt produced by the faulting process would tend to align the Messina Straits coastline with the fault trend. The observation that both the Tyrrhenian and the modern shorelines strike close to N-S and not N11 E like the fault in their model should give a possible objection to the effectiveness of the superposition of regional uplift and repeated 1908-type events, as modeled by Boschi et al. [1989], in building up the present Messina Straits setting. According to Valensise and Pantosti [1992], there is no evidence of the graben structure envisioned, e.g., by Ghisetti [1984], and its action would produce tilting of the terraces and of young deposits in a direction opposite to what is seen in the field. [13] Piatanesi et al. [1999], Tinti et al. [1999], and Tinti and Armigliato [1999] considered the tsunami produced by the earthquake. They took into account the polarity of the first impacting wave and the run-up heights in some coastal places, and they considered as tsunamigenic sources the faults proposed by Capuano et al. [1988] and Boschi et al. [1989]. Piatanesi et al. [1999] show that the source of Boschi et al. [1989] gives a better, even if poor, fit with the data of maximum water elevation. Tinti et al. [1999] extended the previous work looking for the effects of bathymetry. They divided the fault of Capuano et al. [1988] (see Figure 2, fault C) into three subfaults, each 18.9 km in length, and considered two possible faulting mechanisms: a pure normal dislocation and a dislocation with relevant strike-slip component, as proposed by Capuano et al. [1988]. They found that bathymetry plays a minor role and that the pure normal mechanism gives better results for the Calabrian coasts while giving worse results for the Sicilian ones. Tinti et al. suggested that the major part of the seismic moment (73%) should have been released in the southern part of the fault, 20% in the central part, and 7% in the northern part. Tinti and Armigliato [1999] also considered the fault proposed by Boschi et al. [1989] and suggested a seismogenic fault similar to that of Capuano et al. [1988] but displaced southward; moment release mainly occurred in the central and southernmost parts. They also remarked that this feature is not consistent with geodetic data. [14] Completely different models have been published on the basis of macroseismic and geological observations. Bottari et al. [1986] considered shape and position of the isoseismals of higher grade. They stated that the focal mechanism should be a dip-slip one with relative downthrust movement of the NW block (the Sicilian one) along a plane striking 222 and dipping 59 (Figure 2). [15] Tortorici et al. [1995] pointed out that the Messina Straits is characterized by NE-SW trending normal faults which dip at high angles and that no low-angle normal fault has been reported in geological observations. They concluded that the 1908 Messina earthquake may be related to the offshore segment of the west dipping NNE-SSW trending Reggio Calabria fault [e.g., Ghisetti, 1992; Westaway, 1992]. [16] We use a nonlinear approach for deriving the faulting mechanism by inverting simultaneously both P wave first arrivals and vertical displacements to overcome the limitations of the different data sets. This joint nonlinear method has been recently applied to the 1915 Fucino (Italy) earthquake [Amoruso et al., 1998]. 2. Data [17] Precision double-run levelings were carried out a few years before the 1908 earthquake, and some lines were resurveyed just after the seismic event. Measurements were performed in in Sicily, in in Calabria, and in March 1909 both in Sicily and in Calabria. p The uncertainties (in cm) of the two older surveys are ffiffiffi p ffiffiffi d p and d, respectively, and that for the 1909 survey is ffiffiffi d, where d is the along-route distance (in km). Observed circuit misclosure for the whole 1908 Calabrian leveling (367 km in length) was 4.17 cm, lower than the field tolerance limits given by the International Geodetic Commission (5.7 cm) [Loperfido, 1909]. [18] Resurveyed leveling lines are located along the Calabrian coast (82 bench marks) and in Sicily (32 bench marks) (Figure 1 and Table 1). Comparison between bench marks as described by Loperfido [1909] and maps allowed Capuano et al. [1988] to estimate their locations. Uncertainties are mostly of the order of tens of meters, up to hundreds of meters in few cases. [19] A major question concerns the large displacements suffered by the four bench marks located close to the harbor of Messina, due to the possible effects of tsunami and soil liquefaction (see photographs of Baratta [1910]). As previously mentioned, Boschi et al. [1989] asserted that these bench marks certainly suffered slumping and excluded them from the data set. According to De Natale and Pingue [1991], there is no clear justification for eliminating these data, even if the possibility that they can be misinterpreted cannot be rejected. After performing several tests to investigate the effects of excluding Messina bench marks, they concluded that the only noteworthy variation in the retrieved solution is the disappearance of the shallow slip patch around Messina but also that their exclusion makes fault location and strike direction more uncertain. [20] In the case of leveling surveys, random errors associated with vertical heights are cumulative along the route, representing a Gaussian random walk. Height differences between adjacent bench marks are treated as uncorrelated data [e.g., Árnadóttir et al., 1992; Murray et al., 1996]. However, residuals of the best fitting model are due to random leveling errors as well as to basically uncorrelated local response effects (soil compaction, landslides, topographic focusing, etc.), small features (from a geometrical point

4 ESE 4-4 AMORUSO ET AL.: THE 1908 MESSINA STRAITS EARTHQUAKE Table 1. Bench Mark Location, Observed Uplifts, and Along- Root Distance a Bench Mark x, km y, km Uplift, cm Distance, km Table 1. (continued) Bench Mark x, km y, km Uplift, cm Distance, km a Locations from Figure 1 of Capuano et al. [1988]; axis origin is at N, E; x axis is from west to east, y axis is from south to north. Uplifts and along-route distances are after Loperfido [1909]. Bench marks 1 82 are in Calabria, and bench marks are in Sicily. Distances of bench marks 84, 98, and 104 are from bench mark 83, which belongs to all three Sicilian leveling lines. of view) of the actual slip distribution, and bench mark instability. Uncorrelated errors due to interseismic bench mark instability have been taken into account by Pollitz et al. [1998] by adding a known additional error term to the diagonal of the covariance matrix. In our case, large fluctuations of the measured changes of the height differences between adjacent bench marks are evident in the central part of the Calabrian leveling line (Figure 3, top). Several tests show that such changes disconcert the inversion procedure if height differences between adjacent bench marks are used, while elevation change models are able to fit the main features of the deformation pattern (Figure 3, middle). In what follows, we assume (less rigorously) that observed uplifts are uncorrelated. [21] The 1908 Messina earthquake was recorded by 110 seismographic stations around the world [Rizzo, 1910]. Since it occurred before a global homogeneous network was organized and installed, seismological data are far from being precise, and

5 AMORUSO ET AL.: THE 1908 MESSINA STRAITS EARTHQUAKE ESE 4-5 Figure 3. (top) Measured uplift differences between adjacent bench marks (circles) and predicted values (dotted line) for the best fitting uniform-slip model if height differences are assumed to be uncorrelated. Sections and are out of the frame. (middle) Observed uplifts along the leveling lines (circle); predicted uplifts from model A, uniform slip (solid line) and distributed slip (dashed line), and for the best fitting uniform-slip model if height differences are assumed uncorrelated (dotted line). (bottom) Residuals from model A, uniform slip (squares) and distributed slip (triangles). many original records are no longer available. As a consequence, seismological analyses have to rely mostly upon polarity determinations dating back to that period, and the focal mechanism is not well constrained because of the poor coverage of the focal sphere by the stations for which data are available. Here we use 20 P wave polarities, and unlike other studies [e.g., Martini and Scarpa, 1983], we do not consider the three seismic stations that are closest to the earthquake epicenter (Messina, Catania, and Mileto) since their position on the focal sphere is strongly dependent on the fault location and on where the rupture started. 3. Inversion Technique [22] Slip distribution is usually estimated by nonnegative least squares algorithms or gradient techniques, using a given fixed fault geometry. In some cases, fault geometry is obtained from geological studies or from aftershock locations; in other cases, the best fitting uniform-slip model is used. However, the best fitting variable slipping fault could differ from the best fitting uniform slipping fault. [23] We assume that the fault is a rectangular dislocation embedded in a homogeneous, elastic, isotropic half-space; the top edge of the fault is parallel to the surface. We consider a small number of coplanar subfaults (n along strike and m along dip), which slip independently at the same rake angle. Assuming a pure double-couple mechanism, the complete formulation of faulting involves a parameter set a which consists of the strike and dip angles of the fault, the rake angle of the slip vector, two geometrical fault dimensions, location of the center point of the intersection of the fault plane with the surface (x 0, y 0 ), depth of the upper side of the fault (z 0 ), and magnitude of the slip for each subfault. Increasing the number of subfaults evidences the main features of the slip pattern and possible changes in fault geometry while relaxing the assumption of uniform slip. Location and geometry of the fault and slip distribution are retrieved by a global optimization technique. Finally, the best fitting fault is divided into 450 (30 along strike and 15 along dip) subfaults, and the model is refined to approximate a continuous slip distribution with a fixed rake angle. [24] The misfit function is obtained from maximum likelihood arguments (for details, see Amoruso et al. [1998]). Readings of first-motion polarity take on discrete values h i (0.5 for compression and 0.5 for dilatation, i =1,..., N p ). They contribute to the misfit function through a weighted sum of absolute deviations between observed and predicted values. Levelings take on continuous values v ji ( j = 1 for the Calabrian bench marks and j =2forthe Sicilian ones, i =1,..., N j ); errors s ji can sometimes be considered normally distributed, but sometimes the presence of outliers suggests the use of an error distribution with larger tails than the Gaussian one, like the two-sided exponential distribution. This leads to minimizing two different misfit functions. The reliability of the best fitting model is tested by considering the sensitivity of the model parameters to the error distribution and possibly by checking the actual distribution of the residuals against the assumptions. [25] The misfit function for normally distributed uplift errors is M 2 ¼ X2 j¼1 X Nj i¼1 v ji þ u j0 v ji ðaþ 2þ b 2 XNp s ji i¼1 jh i h i ðaþjp i ; ð1þ and the misfit function for two-sided-exponential distributed uplift errors is M 1 ¼ X2 j¼1 X Nj i¼1 v ji þ u j0 v ji ðaþ s ji þ b2 XNp i¼1 jh i h i ðaþjp i ; ð2þ where v ji (a) and h i (a) represent the model predictions for n ji and h i, respectively; u j0 is the static baseline correction for baseline j, necessary because displacement measurements are not absolute and we cannot assume any bench mark unaffected by the earthquake: u j0 behaves like a further free parameter of the model for each leveling line. Uncertainties s ji can be different from one another. We assign the same uncertainty s to all uplifts, but sensitivity of the results on assigning a 10 times larger uncertainty to bench marks (central part of the Calabrian survey) and (Messina harbor) and a 100 times larger uncertainty to bench marks is tested. [26] Since it has not been possible to deeply study and to compare the few seismograms still available, in this work the weighting factor p i is merely proportional to the expected amplitude of the first P pulse from the radiation pattern [Aki and Richards, 1980]: p i ¼ jsin 2q i cos f i j; ð3þ where q and f are the polar and azimuthal angles of the departing seismic ray in polar coordinates centered on the source. This choice of p i gives small relevance to near-nodal stations, thus minimizing the effects of lateral refraction and uncertainties in the velocity profile.

6 ESE 4-6 AMORUSO ET AL.: THE 1908 MESSINA STRAITS EARTHQUAKE [27] The factor b 2 controls the relative importance of the weighted polarity deviation (WPD) versus the weighted squared residual (WSR, equation (1)) or the weighted absolute residual (WAR, equation (2)) of uplifts. The comparison of these components versus b 2 and the sensitivity of the best fitting model allow the comparison of startup properties of the seismic source to average properties of the faulting process. [28] Theoretical vertical displacements have been computed using relationships given by Okada [1985] for finite rectangular sources in a half-space, while first-motion polarities have been computed as by Aki and Richards [1980]. We use the velocity model of Herrin [1968] (hereinafter referred to as VMA) and the IASPEI91 velocity model [Kennett and Engdahl, 1991] (hereinafter referred to as VMB). [29] Several optimization techniques have been developed to find the global minimum in the presence of multiple local minima. We use a variant (ASA, adaptive simulating annealing) of true simulated annealing (SA) [e.g., Ingber, 1993; Ingber and Rosen, 1992], a method able to offer at least a statistical proof of global optimization within finite times [Ingber, 1996]. SA is inspired by the physical annealing of solids to a state of minimum energy [Metropolis et al., 1953; Kirkpatrick et al., 1983] and processes cost functions possessing arbitrary degrees of nonlinearities, discontinuities, stochasticity, and arbitrary boundary conditions and constraints, after choosing a cooling schedule appropriate for each problem. It is a very powerful tool, but it is often considered to be too computationally intensive to be applied in strict adherence to sufficiency conditions to statistically guarantee finding an optimal solution. ASA relies on randomly importance sampling the parameter space and considers that in a D-dimensional parameter space, different parameters have different finite ranges and different annealing-time-dependent sensitivities, measured by the curvature of the cost function at local minima, which might require different annealing schedules. The exponential annealing schedules permit resources to be spent adaptively on reannealing and on pacing the convergence in all dimensions, ensuring ample global searching in the first phases of search and ample quick convergence in the final phases. We also use the ability of ASA to self-optimize its own cooling parameters recursively. [30] In the case of n m independently slipping subfaults the misfit function is minimized in a (8 + n m +2)D space. When the misfit function M 2 is used, we take advantage of the linear relationship between surface displacements and fault slips, and we compute the n m slips and the two static baseline corrections by the algorithm for the least squares problem with linear inequality constraints of Lawson and Hanson [1995] for each point in the 8D space sampled by ASA. This makes minimization faster and more reliable. [31] When we estimate the slip distribution, we use both the geometry and mechanism of the fault plane of the best fitting uniform-slip model and of the best fitting n m model, but we increase the along-strike length and the downdip width of the fault. The fault is divided into an even grid of N s =30 15 subfaults, with variable slip magnitude and constant rake. From Harris and Segall [1987] we add the term g 2 A XNs i¼1 r 2 2 s ð4þ i to the misfit function M 2 (equation (1)), and we remove the P wave polarity term, which is independent of the slip magnitude s; r 2 s is a finite difference approximation to the Laplacian of s and A is the area of each subfault. It would be interesting to also use the misfit function M 1 (equation (2)), but it leads to the prohibitive problem of nonlinear minimization in a huge space. [32] The adimensional smoothing parameter g 2 controls the importance of minimizing the roughness of the slip. We choose Figure 4. Trade-off curves between the fit to the displacement data (WSR) and to the P wave polarity data (WPD) if misfit M 2 is minimized. All points but those labeled with crosses have been obtained using VMA. Crosses, VMB, uniform slip; circles, uniform slip; triangles, 2 1 subfaults; inverted triangles, 2 2 subfaults; squares, 3 2 subfaults; and diamonds, 4 2 subfaults. Values of b 2, WSR, and WPD for VMA, uniform slip and 4 2 subfaults are given in Table 2; in the other cases, values of b 2 vary in similar ranges. Arrows indicate points obtained using b 2 = 850. its value from the trade-off curve between WSR and the roughness of the slip distribution. Cross validation is a more rigorous criterion [e.g., Matthews and Segall, 1993]. Tests performed by Árnadóttir and Segall [1994] show that cross validation gives an optimal estimate of g 2, while the trade-off curve gives a slightly smoother solution. We use the trade-off estimation since it is somewhat more conservative in smoothness and computionally less intensive than the cross-validation technique. The slip distribution is retrieved by the algorithm for the least squares problem with linear inequality constraints of Lawson and Hanson [1995]. 4. Results 4.1. Uniform Slipping Fault [33] The presence of the weighting factor b 2 in equations (1) and (2) makes the value of the bench mark-independent uncertainty s unimportant. In what follows, we arbitrarily assign to s the value s ¼ 1cm; so that WSR in M 2 equals the total square residual of uplifts in cm 2.Ifb 2 = 0, then M 2 reduces to standard least squares. The number of degrees of freedom g, left after fitting the 114 bench mark uplifts to the 11 parameters, i.e., nine fault parameters and two baseline corrections, is 103. We compare M 2 to g to get information about the a priori scaling of the uncertainties. Seismic stations are placed on the focal sphere for a hypocenter at 10 km depth. To avoid useless searches in regions which are very distant from the Messina Straits, the center point of the surface projection of the fault upper side is allowed to vary inside a 2500 km 2 square centered 5 km SW of Reggio Calabria. [34] At first we minimize M. WSR and WPD are plotted in Figure 4 for different values of b 2 and for both velocity models. If b 2 = 0, then M 2 = 1576 and M 2 s 2 s 2 ¼ g when s 3.9 cm.

7 AMORUSO ET AL.: THE 1908 MESSINA STRAITS EARTHQUAKE ESE 4-7 Table 2. Best Fitting Model Parameters (VMA) at Different Values of b 2, After Minimizing M 2 b 2 x 0,km y 0,km z 0, km Length, km Width, km Strike, deg Dip, deg Rake, deg Moment, N m WSR WPD Uniform Slipping Faults a , , , Subfaults b , , , a Model A is labeled as b 2 = 850. b Model F is labeled as b 2 = 850. pffiffiffiffiffi Elevation random error for bench mark i, route j, iss ji = a j d ji where d ji is theq along-route ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidistance (in km). Combined survey precision a = a 2 pre þ a2 post depends on the precisions of preearthquake and postearthquake levelings; thus a 1 = cm km 1/2 (Calabrian leveling) and a 2 = cm km 1/2 (Sicilian levelings). Expected s from random leveling errors is vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 P u j¼1 a 2 Nj t j i¼1 d ji s ¼ ¼ 1:2cm: N 1 þ N 2 [35] The geometry of the best fitting fault is quite similar for VMA and VMB. The parameters of the best fitting models slightly change with b 2 (see Table 2 for the VMA case). Both trade-off curves suggest choosing b 2 = 850, where they evidence an abrupt change in slope. WPD is lower in the case of VMA, suggesting that the local velocity profile is closer to VMA than to VMB. Unless otherwise specified, hereinafter we always use VMA; predicted uplifts and residuals are shown in Figure 3. Table 3 gives parameters of the best fitting models using s = s and b 2 = 850 (VMA, model A, and VMB, model B), as well as assigning a 10 times larger uncertainty to bench marks and (model C) and a 100 times larger uncertainty to bench marks (model D). The most noteworthy differences of models C and D with respect to model A are larger uncertainties in strike, dip, and W-E location of the fault. Residuals are not normally distributed; ð5þ however, estimates based on minimization of M 1 are not significantly different from model A (see Table 3, model E, and Figures 5a and 5b). Confidence intervals are determined using the bootstrap percentile method [Árnadóttir and Segall, 1994], which does not make assumptions about the underlying statistics of the errors. It applies the best fitting technique to a large number of synthetic data sets, generated from the actual data set using random resampling with replacement. Because of replacement, typically 1/e of the data points are replaced by duplicated original points. It is computationally intensive, and we repeat the resampling and estimation procedures 500 times. [36] Following Tortorici et al. [1995], we have checked the plausibility of a west dipping NNE-SSW trending fault, allowing strike to vary only from 90 to 270. As a result, this constraint is inconsistent with coseismic vertical displacement data and with P wave polarities. WSR of the best fitting uniform-slip model is larger than 5300, and WPD is larger than The n m Subfaults [37] We slowly relax the uniform-slip assumption, increasing the number of subfaults, in order 2 1, 2 2, 3 2, and 4 2, in an attempt to fit small-scale features of the uplift pattern. WSR and WPD in M 2 for different values of b 2 are plotted in Figure 4. Trade-off curves are smoother and smoother while increasing the number of subfaults, and this makes the choice of b 2 more subjective. Table 2 lists fault parameters in the 4 2 case. They strongly depend on b 2, suggesting that relaxing the uniform-slip Table 3. Best Fitting Model Parameters in the Case of Uniform Slipping Faults, Using b 2 = 850 a Model x 0,km y 0,km z 0, km Length, km Width, km Strike, deg Dip, deg Rake, deg A B C D E Moment, Nm a Models A to D have been obtained after minimizing M 2. Model A, VMA; model B, VMB; model C, VMA, 10 times larger uncertainty to bench marks and 94 97; model D, VMA, 100 times larger uncertainty to bench marks 94 97; model E, VMA, after minimizing M 1. Uncertainties are at the 95% confidence level.

8 ESE 4-8 AMORUSO ET AL.: THE 1908 MESSINA STRAITS EARTHQUAKE Figure 5. Surface projection of the seismogenic fault and focal mechanism (lower hemisphere Schmidt projection) for the best fitting models using b 2 = 850. All plots but Figure 5b are for model obtained using VMA. (a) Uniform slip; (b) uniform slip, VMB; (c) 2 1 subfaults; (d) 22 subfaults; (e) 3 2 subfaults; (f ) 4 2 subfaults; (g) 5 2 subfaults; and (h) 6 2 subfaults. See also Table 3, model B, and Table 4. For the question mark on the southernmost shallower subfault in Figure 5h, see text. Arrows indicate surface projection of the slip vectors (the scale is that the longest arrow in Figure 5d is 4 m). assumption makes the choice of b 2 more and more critical. However, the value b 2 = 850, chosen from the trade-off curve in the case of uniform slipping, still seems reasonable, and we also use it for 5 2 and 6 2 subfaults to test the stability of the best fitting parameters while further increasing the number of subfaults. Best fitting models using b 2 = 850 are shown in Figure 5 and in Table 4. No uncertainty is given: Bootstrapping is computationally too intensive in the n m case, since auto-optimization of cooling parameters is needed to assure reliable inversion. Several tests suggest a larger uncertainty in strike and rake with respect to the uniform-slip model. Fault location and dip angle are approximately constant, but strike, rake, and seismic moment depend on the number of subfaults. Strike is 8 in the case of 2 1 subfaults and rotates westward while increasing the number of subfaults. Strike and rake seem correlated, but forcing strike to be negative increases misfit. Strike becomes approximately S-N in the case of 4 2 subfaults (model F) and negative when the number of subfaults is further increased, approaching the best fitting uniformslip model. Features of the slip pattern are more and more resolved (Figure 5). If the number of subfaults is large, spurious slip arise in subfaults not well resolved by leveling lines. The problem is already evident in the case of 6 2 subfaults: a slip as large as 17 m is predicted in the southernmost shallower subfault. We use models A and F as reference models for further analysis since fault parameters in the case of 5 2 subfaults are almost identical to model A and in the case of 6 2 subfaults are in the ranges delimited by models A and F. WSR is smaller and smaller while increasing the number of subfaults. It is difficult to assess how significant the WSR improvements are. Owing to linear inequality constraints, the actual number of degrees of freedom g is lower than expected (10 + n m). We use g =10+n m and the F test to give a conservative estimate of the significance. The F test indicates that WSR improvements are statistically significant at more than 95% confidence level in all but one case (92% from the best fitting 2 2 model to the best fitting 3 2 model). WPD is usually larger than in model A. No noteworthy change in the best fitting faults occurs when we use the IASPEI91 P wave velocity profile, or we weight Messina bench marks one hundredth the other ones. In this case, for 4 2 subfaults and using b 2 = 850, best fitting parameters are practically the same as in model C apart from the seismic moment ( Nm). [38] When we minimize misfit M 1, the dimension of the parameter space where the nonlinear search is performed is much higher and minimization is much slower, making best fitting somewhat less reliable in reasonable computational time. Several tests suggest that best fitting models are similar to those obtained by minimizing M Slip Distribution [39] The behavior of fault parameters while increasing the number of subfaults and the large value of WPD for positive values of strike suggests that the fault responsible for the 1908 Messina earthquake is somewhat between model A and model F. We estimate the slip distribution using both models as references, but we increase the fault length along strike to 100 km in order to let slipping occur north of the Ganzirri peninsula (taking into account the model proposed by Capuano et al. [1988]) and south of Calabria (taking into account suggestions from tsunami data [Tinti and Armigliato, 1999]), and we increase the downdip width to 30 km, starting from the surface. The fault is divided into subfaults. Since location of the center point of the fault upper side does not affect WPD, we use a simple gridding technique to Table 4. Best Fitting Model Parameters in the Case of n m Subfaults (VMA) Using b 2 = 850 and Minimizing M 2 a n m x 0,km y 0,km z 0, km Length, km Width, km Strike, deg Dip, deg Rake, deg Moment, Nm WSR WPD M a Models A and F from Table 2 are labeled as 1 1 and 4 2, respectively.

9 AMORUSO ET AL.: THE 1908 MESSINA STRAITS EARTHQUAKE ESE 4-9 Figure 6. (left) Trade-off curves between the fit to the displacement data (WSR) and the roughness of the model (see text for details) at three different values of the seismic moment. Values of the smoothing parameter g 2 are indicated in the plot. (right) Fit to the displacement data (WSR) versus the seismic moment if g 2 = for models A and F. retrieve the best value of x 0 for each reference model, obtaining again the values of models A and F. The extended faults are still referred to as models A and F. The fault is very long, and the spatial smoothing parameter g is unable to limit slip in subfaults not resolved by leveling lines. We generate trade-off curves between the fit to the data (WSR) and the roughness of the best fitting slip distribution for eight maximum allowed values of the seismic moment in the range N m. All curves suggest g 2 = Three trade-off curves for model A are shown in Figure 6 (left); curves obtained for model F are analogous and are not shown for the sake of clarity. When M 0 = N m, WSR is 592 for model A and 608 for model F. Seismic moment is chosen from the trade-off curve with WSR, using g 2 = (see Figure 6, right). Larger values of the seismic moment give models that do not fit the data significantly better. Models A and F give comparable WSRs, but model A could be preferred on the basis of WPDs. The slip distributions for the two models are in Figure 7. In both cases the majority of the seismic moment is released in a region, 600 km 2 in area, extending south of the Calabrian coastline (the highest slip is 6 m at a depth of 6 km); a smaller slip patch is 25 km NE of the Ganzirri pensinsula, at a depth of 17 km. In the case of model A the main slipping region exhibits a smoother southern tail, and a minor shallow slip patch is located north of the Ganzirri peninsula. Predicted bench mark uplifts for the two models are practically indistinguishable; those for model A are shown in Figure 3. Predicted surface uplifts for model A are shown in Figure 8. If we remove the seven bench marks whose residuals are larger than twice the standard deviation of the distribution, WSR decreases to 251 for model A and to 248 for model F, but the slip distribution does not change substantially. Removing Messina bench marks from the analysis gives no significant difference in the retrieved slip distributions, and WSR is 571 for model A and 583 for model F. 5. Discussion [40] Even if our models are not optimal with respect to geodetic data alone (we use trade-off curves with seismic data), WSR is always lower than previously published values obtained for models with comparable degrees of freedom (see section 1). On the basis of the results described above, we suggest that the fault responsible for the 1908 Messina earthquake is well represented by model A, possibly with a slight eastward rotation of strike toward model F. Dip angle (40 ) is smaller than usual in normal faults. Boschi et al. [1989] already suggested that the fault responsible for the 1908 Messina earthquake was a low-angle (29 ) fault, and Valensise and Pantosti [1992] stressed the importance of this feature in constructing the Messina Straits by repeated occurrence of the 1908 Figure 7. (top) Slip distribution for model A. (bottom) Slip distribution for model F. The seismic moment is Nm;g 2 = Gray scale is not linear in slip.

10 ESE 4-10 AMORUSO ET AL.: THE 1908 MESSINA STRAITS EARTHQUAKE close to the second center of the 1908 earthquake according to Baratta [1910] (the first Baratta center is consistent with the Omori epicenter). The high-slip region extends southwest of the southernmost Calabrian coastline, in agreement with suggestions from the analysis of tsunami data [Piatanesi et al., 1999; Tinti et al., 1999; Tinti and Armigliato, 1999]. We have tested the robustness of the southern extension of slip and the agreement with the threesubfault model by Tinti et al. [1999] using the bootstrapping technique and 1000 random data sets. Seismic moment released south of the dashed line in Figure 8 is in the range 25 35% of the total released moment at the 50% confidence level and in the range 13 44% at the 95% confidence level for model A (14 25% and 5 33% for model F). The two dotted lines in Figure 8 delimit the central subfault of Tinti et al. [1999] and divide the Messina Straits area into three regions. From south to north the percentage of the seismic moment released in the three regions is in the range 45 54%, 25 29%, and 20 27%, respectively, at the 50% confidence level and is in the range 35 61%, 22 35%, and 13 35% at the 95% confidence level for model A (40 49%, 38 44%, 11 18%, and 30 57%, 33 51%, and 7 27% for model F). [42] The main geomorphological features of the Messina Straits are extensively discussed by Valensise and Pantosti [1992]. Here we note that both their model and our models are consistent with the shape of the lowest terrace and with the deformation of the Messina gravels on the Calabrian side of the straits. In our models, surface uplift seems more consistent with the orientation of the modern shorelines and with the deformation of the Messina gravels on the Sicilian side of the straits, particularly in the northern area [see Valensise and Pantosti, 1992, Figure 4]. A realistic amount of subsidence is predicted along the axis of the straits. Figure 8. Contour map of surface vertical displacements (in cm) for the best fitting variable slip solution from model A. Contour step is 20 cm. Thick arrow indicates surface projection of the slip vector (not in scale); its direction is in agreement with the extensional stress axis from geological observations (double-ended arrow) [Tortorici et al., 1995]. event. Normal faults dipping at 35 have been recently recognized as responsible for the 1997 Umbria-Marche earthquakes; these faults might have reactivated thrust planes of the Pliocene compressional tectonics [Amato et al., 1998]. A similar process could have occurred in the case of the 1908 Messina earthquake. The existence of a right-lateral component is a robust feature of our model and is consistent with trilateration measurements by Caputo et al. [1981] and with the occurrence of the 1975 M L = 4.7 rightlateral earthquake [Bottari and Lo Giudice, 1975]. The direction of the surface projection of the slip vector is in agreement with the direction of the extensional stress axis retrieved from geological observations [Tortorici et al., 1995] (see Figure 8). [41] Slip distribution and predicted surface uplift for models A and F vary in minor detail. In the case of model A the Ganzirri peninsula separates the northernmost (minor) patch from the main one and, consequently, would not be erased by repetitions of type events (see section 1). The high dislocation patch beneath Messina (3 m) suggested by De Natale and Pingue [1991] is never present in our inversions, in agreement with the results by Pino et al. [2000]. The main patch is in the Messina Straits area, where Omori [1909] and Schick [1977] located the epicenter looking at macroseismic data. The deepest part of the main highslip region is in the area where the most energetic event which occurred after the 1908 earthquake, the 1975 M L = 4.7 right-lateral earthquake, took place (Figures 1 and 7, and Bottari and Lo Giudice [1975]). The northern part of the same region is in the source zone of the 1985 M L < 4 seismic events [Moia, 1987] and is 6. Conclusions [43] P wave first-motion polarities and coseismic surface uplifts of the 1908 Messina earthquake have been jointly analyzed using a nonlinear inversion procedure. We started from a uniform-slip model, but we have relaxed the uniform-slip assumption by increasing the number of subfaults up to 6 2 subfaults. The behavior of optimal fault parameters while increasing the number of subfaults suggests that the fault responsible for the earthquake is well approximated by the uniform-slip model, possibly with a slight eastward rotation of the strike. [44] Slip distribution exhibits a high-slip region extending southwest of the southernmost Calabrian coastline (in agreement with suggestions from the analysis of tsunami data) and a smaller deeper high-slip region located NE of the Ganzirri peninsula. [45] Our results are consistent with the main geomorphological features of the Messina Straits, i.e., the shape of the lowest terrace, the deformation of the Messina gravels, and the orientation of modern shorelines. A realistic amount of subsidence is predicted along the axis of the straits. [46] Acknowledgments. We thank C. Cimoroni for help in developing numerical codes, E. Turco for useful discussions about the geological setting of the Messina Straits, and A. Tarantola for useful discussions about global minimization techniques. We are very grateful to Roland Burgmann, Mark Murray, and Susan Schwartz for suggestions improved the manuscript. References Aki, K., and P. G. Richards, Quantitative Seismology: Theory and Methods, vol. I, 557 pp., W. H. Freeman, New York, Amato, A., et al., The 1997 Umbria-Marche, Italy, earthquake sequence: A first look at the main shocks and aftershocks, Geophys. Res. Lett., 25, , Amoruso, A., L. Crescentini, and R. Scarpa, Inversion of source parameters from near- and far-field observations: An application to the 1915 Fucino earthquake, Central Apennines, Italy, J. Geophys. Res., 103, 29,989 29,999, 1998.