5:15 PM - 6:30 PM
[SSS11-P04] Formulas for using simple array to determine Love-wave characteristics based on rotational component of microtremors
Keywords:microtremor, Love wave , array, rotational component, spatial differentiation, phase velocity
Yoshida & Uebayashi (2018, 2020) demonstrated the effectiveness of using a rotational component (i.e., using spatial derivatives) in determining phase velocities of Love waves. Their method is based on a double triangular microtremor array. In this study, we present formulas to determine Love-wave characteristics using a rotational component based on a three-point array.
Cho et al. (2006) developed a general theory to analyze circular-array data of three-component microtremors. The titled formulation begins with equations (16) and (17) of their paper. These are equations representing the radial- and tangential-component waveforms recorded with a three-component seismometer located at (r, θ), where r is the distance from the origin and θ is the azimuthal angle from the x-axis to the y-axis. Taking the divergence and rotation of these waveforms, we can obtain
D(t, r, θ) = -∫∫∫ ik exp[-iωt-irkcos(φ-θ)] ζR(dω,dk,dφ), (divergent component) (1)
C(t, r, θ) = -∫∫∫ ik exp[-iωt-irkcos(φ-θ)] ζL(dω,dk,dφ), (rotational component) (2)
where t is time, i is the imaginary unit, k is the wavenumber, ω is the angular frequency, and φ is a wave arriving direction. We set r and θ to 0 in the above divergent- and rotational-component waveforms (D and C, respectively). Taking autocorrelation functions and then taking Fourier transforms yields
PSDR(ω) = [cR(ω)]2PSDD(ω)/ω2, (3), PSDL(ω) = [cL(ω)]2PSDC(ω)/ω2, (4)
where PSDR(ω) and PSDL(ω) is the power spectral densities (PSDs) for integrated spectraζR(dω,dk,dφ) andζL(dω,dk,dφ), respectively. cR(ω) and cL(ω) are the phase velocities of Rayleigh and Love waves, respectively.Taking the sum of equations (3) and (4), we obtain
PSDR(ω)+PSDL(ω) = [cR(ω)]2PSDD(ω)/ω2 + [cL(ω)]2PSDC(ω)/ω2. (5)
We define the total powers of the horizontal components, PSDH(ω), as the sum of the PSDs for Rayleigh and Love waves. i.e.,
PSDH(ω) = PSDR(ω)+PSDL(ω). (6)
By the use of equations (5) and (6), we have
cL(ω)=[(PSDH(ω)ω2 - PSDD(ω)[cR(ω)]2)/PSDC(ω)]1/2. (7)
The use of equation (7) assumes that the observation of three-component microtremors was conducted using a triangular (three-point) array. On the right-hand side of equation (7), the phase velocity of Rayleigh waves, cR(ω), is obtained by applying the SPAC method to the vertical-component data from a three-point array, and PSDC(ω) and PSDD(ω) are obtained by processing two horizontal components of the three-point array (YU2020). PSDH(ω) can be obtained from the two horizontal components at a single point (i.e., the sum of the PSDs for either the radial and tangential components or the east-west and north-south components). Therefore, the Love-wave phase velocity is obtained from a three-component three-point array. Once cL(ω) is obtained, R/L (Amplitude ratio of Rayleigh to Love waves) of Arai and Tokimatsu (2004) can be obtained using equations (3) and (4). Alternatively, since cR(ω) is already obtained, we can obtain the H/V ratio of Rayleigh waves via equation (3). It may be more convenient to use this quantity to model a velocity structure than to use R/L, as the calculation of theoretical values for a fitting becomes simple.
This research was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers 19H02287.
Arai & Tokimatsu, 2004. Bull. Seismol. Soc. Am., 94, 53–63.
Cho, Tada, & Shinozaki, 2006. Geophys. J. Int., 165, 236–258.
Yoshida & Uebayashi, 2020. Bull. Seismol. Soc. Am., doi://10.1785/0120200139.
Yoshida & Uebayashi, 2018. BUTSURI-TANSA, 71, 15-23. (in Japanese)
Cho et al. (2006) developed a general theory to analyze circular-array data of three-component microtremors. The titled formulation begins with equations (16) and (17) of their paper. These are equations representing the radial- and tangential-component waveforms recorded with a three-component seismometer located at (r, θ), where r is the distance from the origin and θ is the azimuthal angle from the x-axis to the y-axis. Taking the divergence and rotation of these waveforms, we can obtain
D(t, r, θ) = -∫∫∫ ik exp[-iωt-irkcos(φ-θ)] ζR(dω,dk,dφ), (divergent component) (1)
C(t, r, θ) = -∫∫∫ ik exp[-iωt-irkcos(φ-θ)] ζL(dω,dk,dφ), (rotational component) (2)
where t is time, i is the imaginary unit, k is the wavenumber, ω is the angular frequency, and φ is a wave arriving direction. We set r and θ to 0 in the above divergent- and rotational-component waveforms (D and C, respectively). Taking autocorrelation functions and then taking Fourier transforms yields
PSDR(ω) = [cR(ω)]2PSDD(ω)/ω2, (3), PSDL(ω) = [cL(ω)]2PSDC(ω)/ω2, (4)
where PSDR(ω) and PSDL(ω) is the power spectral densities (PSDs) for integrated spectraζR(dω,dk,dφ) andζL(dω,dk,dφ), respectively. cR(ω) and cL(ω) are the phase velocities of Rayleigh and Love waves, respectively.Taking the sum of equations (3) and (4), we obtain
PSDR(ω)+PSDL(ω) = [cR(ω)]2PSDD(ω)/ω2 + [cL(ω)]2PSDC(ω)/ω2. (5)
We define the total powers of the horizontal components, PSDH(ω), as the sum of the PSDs for Rayleigh and Love waves. i.e.,
PSDH(ω) = PSDR(ω)+PSDL(ω). (6)
By the use of equations (5) and (6), we have
cL(ω)=[(PSDH(ω)ω2 - PSDD(ω)[cR(ω)]2)/PSDC(ω)]1/2. (7)
The use of equation (7) assumes that the observation of three-component microtremors was conducted using a triangular (three-point) array. On the right-hand side of equation (7), the phase velocity of Rayleigh waves, cR(ω), is obtained by applying the SPAC method to the vertical-component data from a three-point array, and PSDC(ω) and PSDD(ω) are obtained by processing two horizontal components of the three-point array (YU2020). PSDH(ω) can be obtained from the two horizontal components at a single point (i.e., the sum of the PSDs for either the radial and tangential components or the east-west and north-south components). Therefore, the Love-wave phase velocity is obtained from a three-component three-point array. Once cL(ω) is obtained, R/L (Amplitude ratio of Rayleigh to Love waves) of Arai and Tokimatsu (2004) can be obtained using equations (3) and (4). Alternatively, since cR(ω) is already obtained, we can obtain the H/V ratio of Rayleigh waves via equation (3). It may be more convenient to use this quantity to model a velocity structure than to use R/L, as the calculation of theoretical values for a fitting becomes simple.
This research was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers 19H02287.
Arai & Tokimatsu, 2004. Bull. Seismol. Soc. Am., 94, 53–63.
Cho, Tada, & Shinozaki, 2006. Geophys. J. Int., 165, 236–258.
Yoshida & Uebayashi, 2020. Bull. Seismol. Soc. Am., doi://10.1785/0120200139.
Yoshida & Uebayashi, 2018. BUTSURI-TANSA, 71, 15-23. (in Japanese)