GNGTS 2021 - Atti del 39° Convegno Nazionale

21 GNGTS 2021 S essione 1.1 We analyze the azimuthal dependence of the residual, corrected for repeatable source, site and path contributions. This term is  W 0 that represents the residual variability, not captured by the other corrective terms. Some events show that the ground motion increases at certain source-to-site azimuths and decreases for stations located in the opposite direction, suggesting the presence of directivity effects. Therefore, in order to detect the directivity effects, for each event we decompose the aleatory residual  W 0 term in: [4] We analyze the azimuthal dependence of the residual, corrected for repeatable source, site and path contributions. This term is δW 0 that represents the residual variability, not captured by the other corrective terms. Some events show that the ground motion increases at certain source-to-site azimuths and decreases for stations located in he opposi e direction, suggesting the presence of directivity effects. Therefore, in order to detect the directivity effects, for each event we decompose the aleatory residual δW 0 term in: � = 2 ( ) + ����� [4] where 2 ( ) is the fitting function of the aleatory residuals that depends on the source to-site azimuth , and ����� is the remaining residual. We use two functions for fitting the D2D term: i. A simple fitting model based on a cosinusoidal function (Sommerville, 1997) ii. A fitting model based on the directivity coefficient � which grounds on a simple theoretical rupture propagation model. According to Boatwright (2007), the general expression of � for a bilateral rupture is the following: � = � � � [��� � � � ���� (��� � ) ] � + (���) � [��� � � � ���� (��� � ) ] � [5] where  D 2 D θ is the fitting function of the aleatory residuals that depends on the source to-site azimuth θ , and  W nodir is the remaining residual. We use two functions for fitting the  D 2 D term:  i. A simple fitting model based on a cosinusoidal function (Sommerville, 1997) ii. A fitting model based on the directivity coefficient C d which grounds on a simple theoretical rupture propagation model. According to Boatwright (2007), the general expression of C d for a bilateral rupture is the following: [5] directivity effects. Therefore, in order to detect the directivity effects, for each event we decompose the aleatory residual δW 0 term in: � = 2 ( ) + ����� [4] where 2 ( ) is the fitting function of the aleatory residuals that depends on the source to-site azimuth , and ����� is the remaining residual. We use two functions for fitting the D2D term: i. A simple fitting model based on a cosinusoidal function (Sommerville, 1997) ii. A fitting model based on the directivity coefficient � which grounds on a simple theoretical rupture propagation model. According to Boatwright (2007), the general expression of � for a bilateral rupture is the following: � = � � � [��� � � � ���� (��� � ) ] � + (���) � [��� � � � ���� (��� � ) ] � [5] Where Where � � � is the Mach number (ratio between rupture and shear-wave velocities), while and � are the azimuths of the station and of the rupture direction, respectively. Parameter ∈< 0; 1 > and represents the relative portion of rupture length in the direction � . So, for a simple homogeneous source model that ruptures in a bilateral way, the directivity amplification of the spectral ordinate should be proportional to �� . In the case of a unilateral rupture propagating along the nearest fault at a constant velocity according to the Haskell model (1964), the full expression of � , obtained for = 1, is: � = � ��( � � � )��� (��� � ) [6] For frequency larger than the corner frequency of the event, the spectral ordinates scale proportionally to �� , where the exponent depends on the source model considered. Omega- squared kinematic models with single corner frequency ( -squared model by Herrero and Bernard, 1994) suggest =1, while models with two corner frequencies (Haskell model with constant slip and rise time) suggest =1. Here, we calibrate the C d model using the Levenberg-Marquardt optimization algorithm (Marquardt, 1963), which is commonly used to solve non-linear least squares problems. The explanatory parameters can vary within a fixed range of values: n in the interval between 0 and 2, between 0.6 and 1, from 0.5 to 1 and 0 in the range 0-360°. In Fig.2 we show the trend of of M4.0, 30 October 2016 at 11:58:17 which has a strong is the Mach number (ratio between rupture and shear-wave velocities), while θ and θ 0 are the azimuths of the station and of the rupture direction, respectively. Parameter k ε <0;1> and represents the relative portion of rupture length in the direction θ 0 . So, for a simple homogeneous source model that rupt res in a bilateral way, the directivity amplification of the spectral ordinate should be proportional to Where � � � is the Mach number (ratio between rupture and shear-wave velocities), while the azim ths of the station and of the rupture direction, resp ctively. Par meter ∈< represents the relative portion of rupture length in the direction � . So, for a simple ho source model that ruptures in a bilateral way, the directivity amplification of the spectr should be proportional to �� . In the case of a unilateral rupture propagating along the nearest fault at a consta according to the Haskell model (1964), the full expression of � , obtained for = 1, is: � = � ��( � � � )��� (��� � ) For frequency larger than the corner frequency of the event, the spectral ordin proportionally to �� , where the exponent depends on the source model considere squared kinematic models with single corner frequency ( -squared model by Herrero an 1994) suggest =1, while models with two corner frequencies (Haskell model with co and rise time) suggest =1. Here, we calibrate the C d odel using the Levenberg- optimization algorithm (Marquardt, 1963), which is commonly used to solve non-l squares problems. The explanatory parameters can vary within a fixed range of value interval between 0 and 2, between 0.6 and 1, from 0.5 to 1 and 0 in the range 0-360°. In Fig.2 we show the trend of � of M4.0, 30 October 2016 at 11:58:17 which ha directional effect within 4 frequency bands, which are 1, 5, 10 and 20 Hz. After severa the purposes of this work, we decide to use a functional with =0.85 and =0.5, and inv In the case of a unilateral rupture propagating along the nearest fault at a constant velocity according to the Haskell mod l (1964), the full expression of C d , obtained for k = 1, is: Where � � � is the Mach number (ratio between rupture and shear-wave velocities), while and � are the azimuths of the station and of the rupture direction, respectively. Parameter ∈< 0; 1 > and represents the relative portion of rupture length in the direction � . So, for a simple homogeneous source model that ruptures in a bilateral way, the directivity amplification of the spectral ordinate should be proportional to �� . In the case of a unilateral rupture ropagating along the nearest fault at a constant velocity according to the Haskell model (1964), the full expression of � , obtained for = 1, is: � = � ��( � � � )��� (��� � ) [6] For frequency larger than the corner frequency of the event, the spectral ordinates scale proportionally to �� , where the exponent depends on the source model considered. Omega- squared kinematic models with single corner frequency ( -squared model by Herrero and Bernard, 1994) suggest =1, while models with two corner frequencies (Haskell model with constant slip and rise time) suggest =1. Here, we calibrate the C d model using the Levenberg-Marquardt optimization algorithm (Marquardt, 1963), which is commonly used to solve non-linear least squares problems. The explanatory parameters can vary within a fixed range of values: n in the interval between 0 and 2, between 0.6 and 1, from 0.5 to 1 and 0 in the range 0-360°. In Fig.2 we show the tre d of � of M4.0, 30 Octob r 2016 at 11:58:17 which h s a strong directional effect within 4 frequency bands, which are 1, 5, 10 and 20 Hz. After several trials, for the purposes of this work, we decide to use a functional with =0.85 and =0.5, and invert only for and in the regression (blue curve). [6] For frequency larger than the corner frequency of the event, the spectral ordinates scale proportionally to Where � � � is the M ch number (ratio between rupture and shear-wave velocities), while and � are the azimuths of the station and of the rupture direction, respectively. Parameter ∈< 0; 1 > and represents the relative portion of rupture length in the direction � . So, for a simple homogeneous source model that ruptures in a bilateral way, the directivity amplification of the spectral ordinate should be r portional to �� . In the case of a unilateral rupture propagating along the nearest fault at a constant velocity according to the Haskell model (1964), the full expression of � , obtained for = 1, is: � = � ��( � � � )��� (��� � ) [6] For frequency larger than the corner frequency of the event, the spectral ordinates scale proportionally to �� , where the exponent depends on the source model considered. Omega- squared kinematic models with single corner frequency ( -squared model by Herrero and Bernard, 1994) suggest =1, while models with two corner frequencies (Haskell model with constant slip and rise time) suggest =1. Here, we calibrate the C d model using the Levenberg-Marquardt optimization algorithm (Marquardt, 1963), which is commonly used to solve non-linear least , where the exponent depends on the source model considered. Omega- squared kinematicmodels with single corner freque cy ( k -sq ar dmodel by Herrero and Bernard, 1994) suggest η =1, while models with wo corner fr qu ncies (Haskell model with constant slip and rise time) suggest η =1. Here, we calibrate th C d model using the Levenberg-Marquardt optimization algorith (Marquardt, 1963), which is commonly used to solve n n-linear least squares problems. The explanatory parameters can vary within a fixed range of values: n in the interval b tween 0 and 2, k bet een 0.6 and 1, a om 0.5 to 1 and θ 0 in the range 0-360°. In Fig.2 we show the trend of  W 0 of M4.0, 30 October 2016 at 11:58:17 which has a strong directional effect within 4 frequency bands, which are 1, 5, 10 and 20 Hz. After several trials, for the purpos s of this work, we decide to use a functional with k =0.85 and a =0.5, and i vert only f r η and θ 0 in the C d regression (blue curve). By doing so, we assume a unilateral rupture for all events, unless we then verify (through the stati tic l coefficient of det rminati n, R ) that is satisfying. As already parti lly expected and observed (Chen et al ., 2014; Pacor et al ., 2016), the directivity is dependent on the frequency band investigated.