A probabilistic method for the estimation of earthquake. source parameters from spectral inversion : application to the

Transcription

1 A probabilisti method for the estimation of earthquake soure parameters from spetral inversion : appliation to the Central Italy seismi sequene Authors : Mariano Supino (1), Gaetano Festa (1) and Aldo Zollo (1) (1) Department of Physis Ettore Panini, Università degli Studi di Napoli Federio II, Italy Corresponding author : Mariano Supino; mariano.supino@unina.it; Tel. : ; Fax :

2 Summary We develop a probabilisti framework based on the onjuntion of states of information between data and model, to jointly retrieve earthquake soure parameters and anelasti attenuation fator from inversion of displaement amplitude spetra. The evaluation of the joint probability density funtions (PDFs) enables us to take into aount between-parameter orrelations in the final estimates of the parameters and related unertainties. Following this approah, we first searh for the maximum of the a-posteriori PDF through the basin hopping tehnique that ouples a global exploration built on a Markov hain with a loal deterministi maximization. Then we ompute statistial indiators (mean, variane and orrelation oeffiients) on soure parameters and anelasti attenuation through integration of the PDF in the viinity of the maximum likelihood solution. Definition of quality riteria based on the signal to noise ratio and the similarity of the marginal PDFs with a Gaussian funtion enables us to the define the frequeny domain for the inversion and to get rid of unonstrained solutions. We perform syntheti tests to assess theoretial orrelations as a funtion of the signal to noise ratio and to define the minimum bandwidth around the orner frequeny for onsistent parameter resolution. As an appliation, we finally estimate the soure parameters for the Central Italy seismi sequene. We found that the lassial saling between the seismi moment and the orner frequeny holds, with an average stress drop of MPa. However, the main events in the sequene exhibit a stress drop larger than the average value. Finally, the small seismi effiieny indiates a stress overshoot, possibly due to dynami effets or large fritional effiieny. Short title : Probabilisti soure parameters estimation 2

3 Keywords : Earthquake soure observations, Probability distributions, Joint inversion, Body waves, Wave propagation, Computational seismology. 3

4 1. Introdution The haraterization of the soure parameters of small to moderate earthquakes is an important step in understanding the general mehanisms of earthquake nuleation and propagation, sine it shines a light on the physial proesses involving faults over different spae and time sales. The major issue in understanding the physis of suh ruptures is the orret haraterization of the energy budget assoiated with the different mehanisms whih take plae during the earthquake nuleation, unstable propagation, short wavelength radiation and arrest. Although the seismi rupture non-linearly ombines several spae and time sales, a few marosopi parameters an provide insights on its evolution, suh as the earthquake size and the stress drop released during a seismi event. However, their estimation is very unertain (Cotton et al. 2013), owing to unertainties in data and models and to the oupling between soure and wave propagation up to the observation sites. Indeed, atual estimates of stress drop and apparent stress do not allow to distinguish if these parameters are independent of or they sale with the earthquake size (e.g. Sholz 1994; Ide & Beroza 2001; Shaw 2009; Coo et al. 2016). Different kinemati and dynami soure models, built on irular raks, have been proposed to infer soure parameters of small to moderate earthquakes from observations. They are derived from different initial hypotheses, suh as an instantaneously stress pulse applied on the fault (Brune 1970), a rupture evolving at onstant veloity (Sato & Hirasawa 1973) or a rupture spontaneously propagating under pre-defined stress onditions (Madariaga 1976), but they all provide a desription of the far-field displaement spetrum oherent with observations. In these models, the displaement amplitude spetrum is almost onstant at low-frequenies and proportional to the seismi moment, and deays as 2 f at large frequenies. The ut-off frequeny of this spetrum is related to the soure radius and an be onneted to the average stress drop using the stati solution of Eshelby (1957) and Keilis-Borok (1959). 4

5 Several methods for the haraterization of earthquake soures using a spetral analysis have been proposed in literature. Parametri approahes provide a representation of the soure and propagation through a limited set of parameters. Single station solutions make use of theoretial Green s funtions with a frequeny-independent attenuation fator (Boatwright et al. 1991; Aberrombie 1995). To redue the epistemi unertainty related to the wave propagation, theoretial Green s funtion an be replaed by reords of small earthquakes, playing the role of empirial Green s funtions (EGFs); in this ase the spetral ratio between ouples of stations is modelled (Prieto et al. 2004; Aberrombie & Rie 2005). However, EGFs are required to have more than one point of magnitude smaller than the main event to haraterize, to resolve soure parameters (Aberrombie & Rie 2005). On the ontrary, non-parametri tehniques do not assume the funtional shape of the Green s funtion, but they remove it from the spetra using a data-driven approah (Castro et al. 1990; Oth et al. 2007). Sine the stress drop is onneted to the ube of the orner frequeny, haraterization of earthquake saling laws requires a orret assessment of the unertainties and orrelations among parameters. Confidene intervals for soure parameters estimated from single station spetra are usually derived from the fitting proedure, using, for instane, a jakknife analysis (Prieto et al. 2007) or by a linearization of the problem (Zollo et al. 2014). When using an EGF-based approah, unertainties also rely on the quality and number of EGFs, and on their loation and size with respet to the main event. Unertainties an be larger than 50% for single EGFs loated outside the soure radius of the main event (Kane et al. 2011; Aberrombie 2015). Large unertainties in earthquake soure parameters and Q arise from orrelations (Aberrombie 1995), and solutions have been proposed with the goal of reduing them, suh as a multi-step, iterative approah (Zollo et al. 2014). However, a systemati omparison between different methodologies highlighted the dependene of the results on the fitting model (Oye et al. 2005; Sonley & Aberrombie 2006). A probabilisti approah to this inverse problem an allow to investigate suh orrelations, defining a probability density funtion (PDF) in the parameter spae and providing a onsistent estimate of 5

6 the unertainties. This approah is beoming more and more widespread in seismologial appliations, e.g., for earthquake loation (Lomax et al. 2000), kinemati soure haraterization (Song & Somerville 2010) and kinemati soure inversion (Piatanesi et al. 2007; Minson et al. 2013). A Bayesian approah has also been proposed for the estimation of earthquake soure parameters (Garia-Aristizabal et al. 2016). However, this approah is foused on prior states based on expert opinions of seismologists that require visual inspetion of displaement spetra and thus it is not suitable for massive analysis of large databases. Moreover, the tehnique is built fixing the quality fator and then diretly mixing the estimations from single spetra in a single PDF. On the one hand it neglets existing orrelations among soure parameters and quality fator, on the other hand final estimates are poorly onstrained and show a large variability, with retrieved moment magnitude for the analyzed earthquake ranging between 5 and 9. Here we develop a probabilisti approah using the framework defined by Tarantola (2005), based on the onjuntion of states of information between data and model, to jointly retrieve soure parameters and Q with unertainties and to analyze the between-parameter orrelations. Following this approah, we first searh for the maximum of the a-posteriori PDF, then we ompute statistial parameters through integration of the PDF. Several exploration tehniques have been proposed to seek for the maximum (or minimum) of an objetive, non-linear funtion. With the development of omputational resoures, more often global exploration methods are applied to inverse problems in geophysis, to limit the possibility to fall in loal maxima or minima, suh as the simulated annealing (Kirkpatrik et al. 1983; Mosegaard & Vestergaard 1991), the geneti algorithm (Goldberg & Holland 1988; Sen & Stoffa 1992; Festa & Zollo 2006), the neighborhood algorithm (Sambridge 1999; Marson-Pidgeon et al. 2000; Lua et al. 2012) and the Montearlo exploration (Sambridge & Mosegaard 2002). Sometimes they are oupled with loal solvers to speed up the onvergene, suh as the Simplex (Zollo et al. 2000). Here we adopt the basin hopping tehnique (Wales & Doye 1997; Wales 2003), whih ouples a global exploration built on a Markov hain with a loal deterministi maximization/minimization, based on a quasi-newton algorithm. The tehnique improves the 6

7 onvergene toward the maximum or the minimum of the seleted objetive/ost funtion while maintaining a global view on the funtion; in addition, it is based on sole three parameters, allowing to shape the tehnique to the speifi problem with fine tuning of the parameters. In this work we present a probabilisti approah for the haraterization of the earthquake soure parameters. In the first setion we setup the physial quantities to be estimated and define the forward operator, built on the generalized Brune s model to be used for the inverse problem. In the next setion, we summarize the main ingredients of the probabilisti framework and we adapt it to the speifi problem of retrieving the soure parameters. Here, we detail the strategy for the exploration and sampling of the a-posteriori PDF. In a first step we searh for the maximum of the PDF and then we analyze its shape in a domain entered around it. After alulation of the marginal PDF for single parameters and for ouples of parameters, we define unertainties and orrelation oeffiients, and we establish a quality riterion allowing to automatially rejet unonstrained solutions. Then we perform syntheti tests to assess theoretial orrelations and define the minimum bandwidth for parameter resolution. As an appliation, we finally estimate the soure parameters for the Central Italy seismi sequene. 2. Generalized Brune s model In linear elastiity, the displaement produed by a point soure and reorded at a given reeiver is the onvolution of the soure time funtion by the Green s propagator. Thus, the displaement spetral amplitude in the frequeny domain u( f ) an be fatorized as u( f ) S( f ) G( f ), where f is the frequeny, S( f ) is the modulus of the Fourier transform of the soure time funtion and G( f ) the modulus of the Green s propagator. We separately model the far field P- and S-waves. For the soure ontribution, we onsider the generalized Brune s model (Brune 1970) : 7

8 M 0 S( M 0, f, ; f ) f 1 f (1) with S depending on three parameters: the seismi moment M 0, the orner frequeny f and the high frequeny spetral deay fator. Under the hypothesis of an attenuation quality fator independent of the frequeny, the Green s funtion in the spetral domain an be simplified to (Aki & Rihards 1980) Q G( Q' ; f ) K A (, ) e ft ( rr, 0 ) Q' rr 0 (2) In the above formula, K is a onstant, depending on the soure-reeiver geometrial onfiguration and the elasti properties of the medium rossed by the waves, A ( rr, ) 0 and T ( rr, ) 0 are the geometrial spreading and the travel-time related to the seleted wave from the soure at r0 to the 1 reeiver at r, respetively, and Q ' is the reiproal of the quality fator. For a 1D layered model Q the analytial representation of the onstant K is (Aki & Rihards 1980) 1/2 1/2 5/2 1/2 0 0 R F K S 4 ( r ) ( r) ( r ) ( r) (3) Here R is the radiation pattern ontribution, depending on the phase (P or S wave), F S is the freesurfae orretion oeffiient, is the density and the wave veloity. The geometrial spreading desribes how the amplitude deays as a funtion of the distane from the soure; both the geometrial spreading and the travel time an be omputed using the ray theory. They simplify to A ( rr, ) 0 1 r r 0 and T (, ) r r 0 rr 0 for a homogeneous medium, and the geometrial spreading is independent of the phase. In our modelling, we assume A ( rr, ) and T ( rr, 0) known; they are omputed either in a K, 0 homogeneous or in a 1D horizontally layered medium. Unertainties in these terms ontribute to the inrease of the epistemi unertainties on the soure parameters and 8 Q ' estimations. Both the

9 geometrial spreading and the onstant K are sale fators for the seismi moment. They depend on the relative loation of the soure and the reeiver and on the veloity struture rossed by the waves. However, beause of the logarithmi sale of the seismi moment, their unertainties poorly affet the estimation of the event magnitude. The travel-time appears in formula (2) through the produt Q' T ( rr, ), where the 0 Q ' fator is retrieved from the inversion of the displaement spetra. Unertainty on T ( rr, 0) only affets the final estimate of inversion of the quality fator. Q ' and an be ompletely absorbed in the Beause of the exponential nature of the seismi moment, we define the forward operator as the logarithm of the displaement spetral amplitude : f log u log M0 log 1 log ft ( rr, 0) Q' log e (4) f where K A ( rr, 0) ; log u(log M 0, f,, Q ' ) depends on four unknown parameters that will be inverted analyzing the spetra obtained from seismi reords. For sake of simpliity, we summarize the set of parameters to be estimated through the vetor m (log M 0, f,, Q ' ) and we indiate with M the model spae, the subdomain of model parameters. h R, h 4, whih individuates the range of variability of the 3. Soure parameters inversion 3.1. Probabilisti framework for the inverse problem Although the equation (4) provides a ontinuous mapping between the parameter spae and the theoretial amplitude spetrum, the displaement spetra obtained from observations are sampled at a disrete, finite set of points. Let us indiate with log u ( f ), k 0,1,.., n d the logarithm of obs obs k 9

10 the disrete Fourier amplitude spetrum omputed from the observed displaement; the vetor dobs belongs to the data spae D. In the above relationship, f k k kfmin, where T is the window length T of the seleted signal in time, f min 1 T the minimum frequeny in the spetrum, n is half of the number of samples in the time domain, and fn 1 nfmin the Nyquist frequeny, where t is the 2 t time step of the reorded signal. It is worth to note that when using the FFT for spetral omputation, the signal is padded to zero to satisfy the ondition that n is a power of two. To ompare theoretial and observed spetra, we then ompute the theoretial predition at the same disrete set of frequenies. Let us indiate with ( ) log u( ; f ), k 0,.., n g m m the disrete forward operator. The k solution of the inverse problem is indeed the set of parameters at best. * m gm, suh as * approahes d obs We introdue a probabilisti framework for the resolution of the inverse problem (Tarantola 2005). Let us indiate the a-priori probability density funtion over the model spae M, M ( m ), representing the information available for the model parameters independently of the observations. Analogously, the a-priori probability density over the data spae D is indiated as ( dd, ), representing the results of the measurement operation. Finally, we define the onditional probability density of obtaining a data vetor d for the given set of parameters m, ( dm ), that represents the information about the model predition and its unertainties and is thus onneted to the physial theory we are using to model the observations. Combining the two states of information ( m) ( d, d ) and ( dm ) through the onjuntion operation leads to the a-posteriori PDF M D obs over the parameter spae, whih is the solution of the inverse problem for the speifi observed data d obs : ( m) K ( m) L( m, d ) (5) M M obs D obs 10

11 where L( m, d ) ( d, d ) ( d m) dd is the likelihood funtion and obs D obs D K is a normalization onstant. Within this approah the best model * m is the value that maximizes the PDF (5); however, we an also integrate ( m ) to infer statistial indiators, suh as the mean value of the distribution, M the standard deviation and the orrelation oeffiients. We assume that the a priori PDF over model spae is uniform; moreover, we assume that both modelization and data unertainties are normally distributed : n 1/2 1 T 1 m m ( d m) 2 det C exp ( g( m) d) C ( g( m) d ) 2 and n 1/2 1 T 1 D ( d, dobs ) 2 det Cd exp ( d dobs ) Cd ( d d obs) 2, with C m the ovariane matrix related to modelization unertainties and C d the ovariane matrix related to measurement unertainties. Under this hypothesis the likelihood writes (Tarantola 2005): 1 where L( m, d ) exp S( m, d ) (6) obs 1 T S( m, dobs ) g( m) dobs CD ( g( m) d obs ) is the ost funtion, and CD Cd C m is the 2 total ovariane matrix. It is worth to note that this relation is general, regardless the linearity of the gm ( ) operator. We assume that the ovariane matrix is diagonal and it has the following form CD 1 MSE, where MSE n i1 ( dobs ) i gi n h * m 2 obs. Under this assumption maximization of the likelihood funtion also orresponds to the minimization of the L 2 distane between data and preditions, T S '( m, d ) g( m) d ( g( m) d ), whih does not depend on the MSE and an be omputed obs obs obs independently of the knowledge of the solution m *. In the following appliation on real data we found a-posteriori that the MSE is in the range The data unertainties are assoiated to the S/N ratio. When seleting data with a S/N ratio > 10 on average in the seleted frequeny range, the 11

12 ontribution of the noise to the MSE is at least one order of magnitude smaller. Thus, the MSE is dominated by the modelization unertainties within this range of S/N values and it is retrieved to be independent of the speifi S/N value. The solution of the inverse problem is omputed in two steps : we first ompute the minimum of the ost funtion S '( md, obs ), using the basin hopping tehnique, as desribed in the next setion, then we evaluate the MSE, whih is used for the estimation of the a-posteriori PDF estimation of the unertainties requires the integration of ( ) M m. ( ) M m. Finally, the If the forward operator gm ( ) is linear, S( md, obs ) is quadrati and ( ) M m is normal; the more nonlinear gm ( ), the farther ( ) M m from a Gaussian PDF. However, though strongly non-linear, the forward operator gm ( ) an be linearized in the viinity of the best model, in a subdomain * M entered around * m. If the value of ( ) M m is enough small outside * M, to not signifiantly ontribute to the marginal PDFs related to the single parameters, we an extrat the mean and the variane for eah parameter, and the orrelation oeffiients for all ouples of parameters, limiting the exploration to the domain * M. We define * M as the hyperube M * I.. I m1 m h, where I mi is a 1D interval ontaining the value * m i. Let us define the marginal PDF for the parameter m i as ( m) ( m) dm, and the marginal PDF for the ouple ( mi, m j) as M i M * Mi ( m, m ) ( m) dm, where M i j M * Mij M I.. I I.. I is the hyperube built * i m m 1 m 1 m 1 i i h aounting for all the parameters exept and m i and m j. Mean value, variane and orrelation are finally omputed as: M the hyperube built exluding the parameters m * ij i m ( m ) dm i i M i i * * M \ Mi ( m ) ( m ) dm 2 2 i i i M i i * * M \ Mi ov ( m )( m ) ( m, m ) dm dm i, j i i j j M i j i j * * M \ Mij (7) 12

13 3.2. The basin hopping algorithm for the searh of the global minimum The searh for the minimum of the ost funtion S '( md, obs ) is performed through the global optimization tehnique of the basin hopping. It uses a random sampling of the model spae, based on a Markov hain with a transition probability given by the Metropolis riterion. Here we shortly summarize the searhing strategy of the tehnique. If after j iterations the exploration has reahed the point m j, at the (j+1)-th iteration a random perturbation of the oordinates is performed, moving the model in the point (0) m j1 ; this latter is onsidered as the starting point for a loal minimization, whih brings the exploration in the point m j1.the minimization is performed using the Broyden Flether Goldfarb Shanno algorithm (Flether 1987). The point m j1 is then ompared with m j. If the ost funtion at the end of the (j+1)-th iteration is smaller than the ost funtion at the end of the j-th iteration, i.e. S'( m 1, d ) S'( m, d ), the transition from j obs j obs m j to m j1 is aepted, else it is aepted with a probability Ptrans ( m j, m j1) given by the Metropolis riterion: P trans S'( m j1, dobs ) S'( m j, dobs ) exp, where the temperature T of the Metropolis T sheme is fixed all along the exploration. It is worth to note that the solution at the end of eah iteration m j omes from a loal minimization proess, speeding up the searh for the final solution, while maintaining a onstant temperature failitates the hopping out of ost funtion basins whih ontain loal minima. The method is based on sole three parameters, allowing for simple tuning. The first one is related to the initial modulus of the random perturbation. The perturbation is assumed to be the same fration of the range of variability for all the parameters. If the exploration of the parameter m i is onstrained in the domain ( mi,min, m i,max ), the size of the initial perturbation is thus 13

14 i,0 i,max i,min m m m. It is worth to note that the magnitude of the perturbation dynamially hanges during the exploration. It is based on the fration of the transitions a r from the point m j to m j1, evaluated every 50 iterations; if ar 0.5, is inreased dividing the previous value by 0.9; if ar 0.5, is dereased multiplying the previous value by 0.9. This ondition allows to explore farther and farther regions when the solution does not move from the same loation in the model spae for many iterations. The seond parameter is the temperature T of the Metropolis riterion, whih is hosen by balaning the ability to onverge toward the final solution and the possibility to esape from a loal minimum. Its magnitude order should be omparable with the average differene between the loal minima, and thus it requires preliminary investigation. Finally, the exploration stops when the maximum number of iterations n iter is reahed. The tuning of the parameters is problem dependent. From syntheti tests on theoretial spetra, we obtained onvergene to the global minimum with 0.1, n iter and T of the order of the unity Definition of M* for unertainty omputation The use of the joint PDF allows not only to seek for the best solution, but also to ompute the unertainties related to the best model, via integration of ( m ). We annot use the parameter spae exploration from the basin hopping tehnique, beause it does not rely on a Monte Carlo sampling and thus, onvergene of integrals is not guaranteed when inreasing the iteration number. On the other hand, a omplete desription of ( m ) in the whole parameter spae is omputationally M expensive and, in many ases, unneessary, sine this funtion very often rapidly dereases to zero when moving away from the maximum. For this reason, we limit the omputation of the joint PDF M 14

15 in the hyperube * M entered on the best fit model m *. For the definition of the hyperube we explore the 1D onditional distributions ( m ) ( m,.., m,.., m ) i 1,..., h (8) * * Mi i M 1 i h where all parameters are fixed to the value that they have in the global maximum of the PDF while the parameter m i an vary. We then define the interval I [ m, m ], ontaining the value * i idown iup * m i, suh that. In the ase in whih the onditional PDF an be * M ( mi down ) M ( mi up) 0.05 M ( mi ) i i i desribed by a Gaussian funtion, the interval * I i is symmetri around * m i and its length is four times the standard deviation of the Gaussian funtion. Sine the marginal PDF has usually a larger standard deviation than the onditional PDF beause of the orrelations among parameters, we onsider the enlarged interval Ii [ midown, miup ]. m max m,(1 ) m m * idown imin i idown being 2.5 the saling fator between the marginal and the onditional PDF standard deviations; analogously * mi up min mi max,(1 ) mi m iup. Finally, the domain * M is obtained by tensorization : * M I1.. Ih The quality of the solution We an finally hek a-posteriori the assumption of Gaussian unertainties, evaluating the quality of the retrieved marginal PDFs ( m ) in terms of similarity with a normal distribution. As similarity riterion, we adopt the normalized ross-orrelation funtion: M i ( ) ( m ) ( m ) dm * Mi M i exp i i 2 where ( ) ( ;, ) exp mi N mi i i is the expeted, normal distribution having median i and variane (9) 2 i. We seleted a quality threshold ; if the zero-lag orrelation (0) In our analysis we hose 0.95., the solution is aepted. 15

16 3.5. Dataset features and final results expression In the previous setions, we desribed how to retrieve soure parameters from the inversion of a single spetrum. However, an earthquake is reorded at several stations, usually on the three omponents of a seismi instrument. For a single station, we invert for one spetrum for the P wave, obtained from the vertial omponent, and one spetrum for the S wave, obtained as the geometrial mean of the two spetra, omputed on the horizontal omponents (Flether et al. 1984). When ombining information from diverse stations we should be aware of that eah station provides a different image of the earthquake soure, depending on the diretivity, the radiation pattern and propagation effets. For that reason, we annot onsider eah spetrum as a repeated measure of the same soure parameters. After obtaining an estimation of the soure parameter i k i k with unertainty from the inversion of a single spetrum at the k-th station, the final estimation of this parameter is given by the weighted mean i K w i k 1 K k 1 k w i k i k (10) and the weighted unertainty is given by i K k1 1 w i k (11) 1 where ( wi) k and K is the total number of stations ontributing to the mean. 2 ( ) i k 4. Syntheti tests 4.1. Large signal to noise ratio 16

17 As a first example we want to show the reliability of the method on a signal poorly affeted by the noise. This example will also enable us to understand how the unertainties are omputed and what are the intrinsi orrelations between ouples of parameters. For the aim, we assume an exat knowledge of geometrial spreading, elasti properties and travel time of the wave (we assume and T ( rr, ) 0 known in formula (4)). We onsider a theoretial Brune s spetrum with the following parameters log M 0 10, f 10 Hz, 2 and Q 100. We then pollute the signal with noise, having a signal to noise ratio RS/N = 100. First, RS/N is defined in the time domain. Here the displaement u is the sum of the signal st () and the noise nt () : u( t) s( t) n( t). The displaement n amplitude spetrum an be written as (see Appendix A) log( u) log( s) os( S N), where s s and n are the amplitude spetra and S and N the phase spetra of the signal and the noise, respetively; they are all funtions of the frequeny. The ratio n s sales as 1 R S / N and at low frequenies, for a flat noise spetrum the ratio n s 1 R S/ N. However, the noise spetrum is usually not flat in the displaement, but it dereases as a funtion of the frequeny as well as the soure spetrum does, eventually with different slopes. In addition, the noise spetrum is not unoherent, but it presents peaks and holes related to ambient and site effets. In these tests we impose the following perturbation on the Brune s spetrum u Brune to aount for the noise effet in the displaement: 1 2 f log u log ubrune sin 1 RS / N fn (12) The noise has a oherent ontribution having a sinusoidal shape with amplitude equal to the reiproal of the signal to noise ratio, modulated by a random ontribution. Here we have f 1 Hz and a random variable over the interval [ 0.5,0.5]. The spetrum has been inverted in the frequeny band [0.1 Hz -100 Hz]. 17 N

18 Within this high value of the signal to noise ratio, the solution almost perfetly fits the initial spetrum (Figure 1). The 1-D marginal PDFs are Gaussian-like distributions, entered on the values imposed to generate the syntheti spetrum (Figure 2). We have the following estimates for the soure parameters : log M , f Hz and , and the final estimate for the quality fator is Q , whose unertainty has been obtained by propagating the error on Q '. We have very small unertainties on the seismi moment and the quality fator (< 0.1%), while the unertainty is about at 1% on both f and. In Figure 3 we represent the 2-D marginal PDFs as heatmaps. Sine the data unertainty is negligible in this ase, the maps represent the intrinsi orrelation among the parameters. This orrelation is due to the modelization unertainty, and annot be redued. The absolute value of all the orrelation oeffiients is above 0.6; larger (anti-)orrelations an be found between f and log M 0, and Q ', with values lose to -1. These large orrelations justify the use of the fator, to define the exploration interval for the omputation of the marginal PDFs from the onditional PDFs (Setion 3.3) Signal to noise ratio RS/N = 5 We perform a test similar to the previous example, with the syntheti spetrum to be retrieved having the same parameters as above and polluted with a perturbation having the same funtional shape. In this ase we use a smaller signal to noise ratio (RS/N = 5), this value being representative of the average RS/N value in the analyzed dataset. The spetrum is inverted in the frequeny band [0.1 Hz -100 Hz]. In Figure 4 we represent the omparison between the retrieved solution (red urve) and the syntheti spetrum (blue urve). We still have a good estimate of the parameters, but unertainties are larger. The final estimates are log M , f Hz, and Q

19 Although the smallest unertainties are still retrieved for the seismi moment and the quality fator, now they have inreased to 0.8% and 1.1% respetively. The perentage error is inreased to 18% on f and to 16% on. In Figure 4 we ompare the marginal distribution for this ase with the ase of RS/N = 100 for the orner frequeny, on the same sale. The distribution still maintains its Gaussian shape, but its width is signifiantly inreased. This also ours for the other three marginal distributions related to log M 0, and Q '. It is worth to note that, on the ontrary, orrelation oeffiients do not signifiantly hange (Figure 5): thus, orrelation among parameters is mostly due to the modelization unertainty, and the data effet on it is negligible Frequeny bandwidth for the inversion The resolution of the soure parameters and of the quality fator depends on the frequeny bandwidth available for the inversion. In the previous examples the bandwidth for the inversion is enough large, with two deades before the orner frequeny and one deade after it, to allow the resolution of the parameters. We want to investigate how the resolution degrades when we shrink the bandwidth around the orner frequeny. We use the same parametrization of the syntheti spetrum as in the previous examples and we use a value of RS/N = 5. We analyze the redution of the bandwidth with three different approahes: we redue the band at frequenies larger than f (ase 1), smaller than f (ase 2) and symmetrially around f (ase 3). The results are shown in Figure 6, panels A, B and C respetively. In Figure 6 we represent the relative differene between the expeted and the retrieved values, for all the parameters. In all ases, when we redue the band, the unertainty inreases and eventually a bias in the estimation of two or more parameters an emerge beause of the unresolved orrelations. In Figure 6A (ase 1) we an observe that the parameters are well resolved down to a frequeny band as large as 0.4 deade. The seismi moment and the quality fator are almost unsensitive to the redution of the high-frequeny band, while the unertainty in both f and signifiantly inreases as the bandwidth dereases. The orrelations start to be poorly resolved at a 19

20 width of 0.4 deade beyond the orner frequeny. The mean value of all the parameters is less sensitive to the redution of the bandwidth at the left of the orner frequeny (ase 2, Figure 6B). Still larger errors arise from the orrelation between f and but with mean values well onstrained down to a 0.1 deade. When a symmetri restrition is performed the quality of the solution is ontrolled by the high-frequeny region and again we need a bandwidth of 0.4 deade to have proper resolution on the parameters (ase 3, Figure 4.6C). Finally, in Figure 7 we show the heatmaps of the orrelation between Q and f for a symmetri bandwidth size of 0.3 deade and 0.4 deade around the orner frequeny. When reduing the bandwidth, we see a migration of the maximum of the marginal PDF toward the upper limit of the explored frequeny band, with the smoother deay of the spetrum around f being instead explained through a slightly lower Q value. It is worth to note that the auray in the quality fator estimate depends on the value of Q itself: the smaller the Q the more relevant its effet on the spetrum. The unertainty on the Q estimate inreases as Q inreases. If we onsider the solutions for Q = 100 and Q = 800 obtained from a theoretial spetrum inverted in the same frequeny band, symmetri around f with 0.4 deade available on the two sides of the orner frequeny, the perentage error moves from 5 % to 37 %, with the estimates for the quality fator of Q 98 5 and Q , respetively. However, the hange in the Q value does not affet signifiantly the auray and the quality of the solutions for the other parameters. 5. Appliation to the Central Italy sequene 5.1. Data A major earthquake sequene interested the Central Italy region from August, 2016 to January, The first event of the sequene, the M w 6.0 Amatrie earthquake ourred on August 24, 2016 with epienter in the village of Aumoli and aused 298 asualties and more than displaed 20

21 persons. The largest earthquake of the sequene the M w 6.5, Noria earthquake - ourred on Otober 30, 2016 with epienter in the village of Noria. It generated large slip at the surfae, with a maximum observed disloation of about 2 m. We omputed the soure parameters for the major events of this sequene. The dataset onsists of aelerometri reords for all the events of the sequene with M L 4.0 (56 events), reorded by stations within 100 km from the epienter. In Figure 8 we represent the loation of the events and the stations : we have at maximum 62 stations per event and the hypoentral distane ranges between 9 km and 100 km. The total number of reords is The sampling frequeny is 100 Hz, 125 Hz and 200 Hz, depending on the station. The waveforms were downloaded from ESM (Engineering Strong-Motion database) (Luzi et al. 2016); the metadata were aquired from INGV bulletin (ISIDe working group 2016) Proessing For eah reord, the definition of the S-wave (signal) duration motion duration (Trifuna & Brady 1975): t was based on the expeted ground 0.02exp(0.74 M L ) 0.3 t H (13) a where H is the hypoentral distane and a is a fator introdued to resale the ground motion duration to the S-wave duration; we hose a 4 for the largest events in the dataset ( M 5.9) and a 2 for the remaining events. The S-wave time-window TS was defined using the theoretial S- wave arrival time T S obtained from the 1-D veloity model of Chiarabba (2009); the time window starts before S A noise time-window T suh that the seleted window is T T 0.1 t, T 0.9 t S S S TN of the same duration t was seleted before the event origin time T 0 as T T t, T. With this hoie, we do not inlude the P-wave within the noise window. Sine N L

22 the diret P wave does not usually perturb the S waveform, but eventually only the P oda may affet it, using the P wave as noise to be ompared to the S signal artifiially amplifies the noise ontribution. On both signal and noise waveforms we removed onstant and linear trends and applied a Hannfuntion tapering on the first and last 5 % of the data. FFT was omputed for pre-proessed signal and noise; multipliation by f 2 yields displaement amplitude spetra. Eah spetrum has been finally smoothed with a 5-points moving average filter. When analyzing a single spetrum, we omputed the signal-to-noise ratio for eah point in the frequeny domain; starting from the enter of the domain, we extrated the first points at the left and the right of f with S/N = These points beome the bounds of the frequeny domain used for the fit. As a result, this allows us to rule out the regions of the frequeny domain where the noise is omparable to the signal, and to rejet reords having no information about the earthquake. An example of spetrum passing this first hekpoint is shown in Figure 9, where the region in the frequeny domain seleted for the fit is evidened. In the same Figure, we also represent the noise spetra with a gray line. The S/N value is larger than 10 in a large portion of the domain seleted for the fit. For these data we ompute the a-posteriori PDF, by first evaluating the maximum of the likelihood funtion and then the marginal a-posteriori PDF on the single parameters. In Figure 9 we superimpose to the observed data the theoretial spetrum omputed using the solutions obtained from the PDFs (red urve). When analyzing the marginal a-posteriori PDF for the orner frequeny, we see that it has a long tail, up to the maximum frequeny of the data, indiating that this parameter is unonstrained. Thus, the solution for the soure parameters from this spetrum annot be onsidered. When omparing the shape of this PDF with a Gaussian funtion, we found the degree of similarity of 0.93, aording to the riterion defined in Setion 3.4. With a quality threshold of 0.95 this solution is automatially disarded. Applying the S/N and the similarity riteria we ruled out 701 solutions (37 % of the available reords). The final estimates are based on 1208 solutions, and we have from 6 to 37 solutions per event in the dataset. 22

23 5.3. Results We investigated the earthquake soure properties for the main events of the Central Italy sequene. In Figure 10 we show an example of S waveform in time domain and the related displaement amplitude spetrum reorded at the station FIAM for the M L 4.0 event ourred the day at 23:22:05 (UTC). Superimposed to the observed spetrum (blue urve), we also plot the noise spetrum (gray urve) and the theoretial spetrum (red urve), obtained using the mean estimate of the parameters. It is worth to note that the final solution well desribes the observed spetrum. In Figure 11, we show the 1-D marginal PDFs for all the parameters whih exhibit a peaked, Gaussian-like behavior. For those funtions, the similarity ranges between and 1. These solutions are thus aepted aording to the pre-defined threshold. We plot 2-D marginal PDFs as heatmaps in Figure 12. We learly reognize the orrelations among the parameters and they have almost the same shape as the heatmaps omputed for the syntheti waveforms with large S/N ratio (Figure 3). Thus, the orrelation is mainly governed by the modelization unertainties in this ase. This shape an be observed for most of the data providing soure parameters for this ase study. We finally represent in Figure 13 the solutions for all the events, where we plot the orner frequeny 1 as a funtion of the seismi moment. We observe on average that the standard saling M 0 3 f holds (Aki 1967) with an average stati stress drop MPa, although we have a large variability in the orner frequeny estimates for events with similar seismi moment. Speifially, for events with moment magnitude M W between 4 and 5 we report a variability in the orner frequeny of a fator 5, with the stress drop jumping from few hundreds kpa to 10 MPa. However, the majority of the events has a stress drop lose to the average values within unertainties. As the magnitude inreases the stress drop on average inreases. For the Noria event ( MW ), we have a orner frequeny f Hz and a high-frequeny deay slope The 23

24 estimated soure radius is r km and the stati stress drop is 4 3MPa. For the Amatrie event, we have a moment magnitude of MW , a orner frequeny f Hz and a high-frequeny deay slope The estimated soure radius is r km, the stati stress drop is 9 4MPa. Finally, for the Visso earthquake, we have a moment magnitude of MW , a orner frequeny f Hz and a highfrequeny deay slope The estimated soure radius is r km, the stati stress drop is MPa. Results for and Q are shown in Figure 14. The distribution has a median value equal to 2.1, with the 60 % of events exhibiting a value between 1.9. and 2.3 as expeted from the standard Brune s model (Brune 1970). The Q distribution has a mean value of 230, and a standard deviation of 50; this an be interpreted as a mean value for the anelasti attenuation fator of the whole explored region. It is onsistent, at a onfidene level of two standard deviations, with the estimate of Bindi et al. (2004). They found an average S wave anelasti attenuation fator of 318 for the Central Italy region; it is worth to note that they used a different Green s funtion with a frequeny-dependent Q fator and a onstant Q value was found only for frequenies above 8 Hz Disussion The average stress drop obtained in this study is similar to what retrieved for the two major seismi sequenes, that have interested the Central Italy region (Umbria-Marhe , L Aquila 2009) in the last two deades. Bindi et al. (2004) found an average stress drop of 2 1MPa for the Umbria- Marhe sequene; for L Aquila sequene Paor et al. (2015) showed a stress drop variability that spans two orders of magnitude, ranging in the interval (0.1 25) MPa, with an average value of 2.6 MPa, and a higher value - 10 MPa - for the largest event ( MW 6.3). They also observed a 24

25 stress drop inrease from 1 to 10 MPa with the moment magnitude ranging from 3 to 5.8. Del Gaudio et al. (2015) reanalyzed the soure parameters for some events of the L Aquila sequene, to extrat appropriate empirial Green s funtions for numerial simulations. They also found a selfsimilarity in the seleted dataset, inluding events with magnitude ranging between 3.5 and 6.3, with an average stress drop of 3 MPa. We report a larger stress drop for the largest magnitude events in the sequene, as generally observed for events in Central Italy (Bindi et al. 2017; Bindi et al. 2018), although the retrieved values are slightly smaller as ompared to the values obtained in literature (Bindi et al. 2018). Kinemati inversions of the major events in this sequene revealed that most of the slip has been produed in asperities whose size is signifiantly smaller than the fault size (Tinti et al. 2016; Liu et al. 2017; Chiaralue et al. 2017). Speifially, in the ase of the Amatrie earthquake, Chiaralue et al. (2017) retrieved a small size slip path ( 33 x Km 2 ) with a maximum slip larger than 1 m, just updip of the hypoenter and a seondary slip path of about x Km, with an average slip of about 50 2 m. On the rest of the fault ( 20x15 Km ) the slip level is smaller than 30 m. Also, for the M 6.5 Noria and the M 5.9 Visso earthquakes the slip is onentrated in few pathes whose size is smaller than the rupture area expeted for events of suh a magnitude (Wells & Coppersmith 1994). Hene, larger magnitude events have been mainly generated by asperities where the stress is onentrated and thus exhibiting a large yield strength, within a fault system with an average lower level of fritional strength. We finally omputed the Savage-Wood seismi effiieny SW (Savage & Wood 1971) as the ratio between the apparent stress a and the stress drop. The apparent stress is omputed from the radiated energy E R as (e.g. Wyss 1979) : E R a (14) M 0 25

26 The radiated energy is the sum of P and S wave ontributions : E E E. The S-wave radiated ( P) ( S) R R R energy an be obtained for eah station-event ouple from the theoretial soure spetrum S( f ), built from the retrieved soure parameters aording to the formula (Boatwright & Flether 1984) : Finally, we use the relationship E ( S ) R 5 S 0 E f S ( f ) df (15) 3 (Boatwright & Flether 1984), where is the ratio 5 ( P) S ( S) R E 5 3 R 2P S P between the P and the S orner frequenies. We fixed suh that the ratio E / E 13.7 (Boatwright & Flether 1984), and we found that the seismi effiieny only depends on. In our study, the high-frequeny saling follows on average a f -2 deay, as proposed in the lassial models of irular ruptures (Madariaga 1976), leading to an almost onstant seismi effiieny with seismi moment (Figure 15), whih ranges between Nm and Nm. Suh low values of seismi effiieny are usually retrieved for earthquakes (Beeler et al. 2003; Zollo et al. 2014) and indiate an overshoot in the stati final stress and ompared to the dynami fritional level. This an be related to a dynami overshoot, indiating a fritional weakening of the interfae, while the rupture is propagating on the fault, or large energy required to reate new rupture surfae (high frature effiieny). Higher values of SW are haraterized by large errors. This behavior is expeted beause R R a large seismi effiieny is retrieved for values lose to 1.5. Sine we have onvergene in the estimation of the radiated energy (15) for 1.5, nearby the onvergene limit of the variability in E R is large, thus generating a large unertainty in the SW estimation. 6. Conlusions We developed a probabilisti framework based on the onjuntion of states of information between data and model, to jointly retrieve earthquake soure parameters and the anelasti attenuation fator. 26

27 We modeled the observed far field displaement spetrum assuming a irular rupture model (Brune 1970) for the soure and a Green s funtion haraterized by a frequeny-independent quality fator for the propagation. The forward operator is therefore defined on a set of 4 parameters: three parameters for the soure the seismi moment M 0, the orner frequeny f and the high-frequeny deay exponent and one parameter the Q-fator for the propagation. These parameters are strongly orrelated among eah other. We estimated the joint probability density funtion (PDF) over the 4-D model spae to extrat the orrelation matrix of the parameters; this allows to obtain estimates and unertainties from the PDF, taking into aount the orrelations. Sine we modeled the observations with a non-linear operator, a global exploration of the model spae is required in order to find the best solution to desribe the data. We used a global optimization tehnique that relies on the definition of a Markov hain in the model spae and on the ombination of a deterministi minimization with a random exploration of the spae (Wales & Doye 1997; Wales 2003). In order to validate the developed methodology, we performed syntheti tests on spetra with different signal to noise ratios, defined on different frequeny domains. The method proved its effiay with all the syntheti ases analyzed here. The resolution of the estimates depends both on the S/N and the frequeny bandwidth available for the inversion; at least 0.1 deades on the left and 0.4 deades on the right of the f are required in the frequeny domain to obtain reliable estimations for the inversion. Moreover, we showed that the unertainty on the quality fator estimate depends on the value of Q itself. The method has been applied to the main events of the Central Italy sequene ( M L 4.0 1, 56 events). We found that the standard saling M 0 holds with an average stati stress drop 3 f MPa. For the main event ( M ), we estimated a orner frequeny W 27

28 f Hz and a high-frequeny deay slope ; the soure radius is r km and the stati stress drop is 4 3MPa. The average stress drop retrieved in this study is onsistent with the values inferred from the two major seismi sequenes of the Central Italy region in the last two deades. The Savage-Wood seismi effiieny ranges between 0.04 and 0.25, and is almost onstant with the explored seismi moment ( MPa ). Suh low values may indiate an overshoot in the stati final stress when ompared to the dynami fritional level. 28

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37 Figures Figure 1 : Syntheti spetrum (blue urve) and solution retrieved from the inverse problem (red urve) for the ase of RS/N =

38 Figure 2 : 1-D marginal PDFs omputed for a RS/N = 100. In the four panels the PDF are represented for log(m0), f, γ and Q. The blak vertial line is the true value of the parameter. All the distributions show a Gaussian-like behavior. 38

39 Figure 3 : 2-D marginal PDFs (heatmaps) omputed for a RS/N = 100. Correlation oeffiients are at the top of eah heatmap. We see that large antiorrelations our for the ouples log(m0)-f, and γ- Q, with orrelation oeffiients lose to

40 Figure 4 : Left panel : Syntheti spetrum generated with a RS/N = 5 (blue urve) and the solution of the inverse problem (red urve). Right panel : omparison between two marginal PDFs as a funtion of f, for RS/N = 100 (green urve) and RS/N = 5 (red urve). 40

41 Figure 5 : 2-D marginal PDFs (heatmaps) omputed for a RS/N = 5. Correlation oeffiients are at the top of eah heatmap; both the shape and orrelation oeffiients do not signifiantly hange as ompared to the ase of a RS/N = 100 (Figure 3). 41

42 Figure 6 : Differene between the expeted and the retrieved values for the parameters log(m0), f, γ and Q as a funtion of the frequeny bandwidth used for the inversion. Panels A, B and C orresponds to a hange in the bandwidth from 0.4 to 1 deade to the right of f, to the left of f and symmetrially around f, respetively. 42

43 Figure 7 : 2-D marginal PDFs (heatmaps) for the parameters f and Q. In the left panel we represent the solution obtained using a bandwidth of 0.4 deade on the two sides of the orner frequeny; in the right panel we show the solution with a bandwidth of 0.3 deade. We see that the maximum of the PDF migrates in the right panel to the upper limit of the explored f range, far from the true value Q = 0.1, f = 10 Hz. 43

44 Figure 8 : Map representing the loation of the events (red dots) and stations (yellow triangles). 44

45 Figure 9 : Here we show an example of spetrum (Event ID-INGV: , Station: SSFR) that not provided a solution in terms of soure parameters. In the left panel, we represent the spetrum of the signal (blue urve), the noise spetrum (gray urve) and the best-fit solution (red urve). In the right panel, we plot the 1-D marginal PDF for the parameter f. The solution is rejeted beause this marginal PDF does not overome the threshold riterion for Gaussian similarity. 45

46 Figure 10 : An example of aepted solution. In the left panel we plot the 2 horizontal omponents of the signal; in the right panel we represent the signal spetrum (blue urve), the noise spetrum (gray urve) and the best-fit solution (red urve). The traes are shown for the event ID-INGV and the station FIAM. 46

47 Figure 11 : 1-D marginal PDFs for the parameters log(m0), f, γ and Q. All the urves show a Gaussian-like behavior for the same event-station as shown in Figure

48 Figure 12 : 2-D marginal PDFs for the parameters log(m0), f, γ and Q, for the same event-station as shown in Figure 10. These heatmaps look similar to the theoretial ones of Figure 3, indiating that orrelations are mainly driven by model unertainties. 48