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[SSS10-08] Use of the spatial autocorrelation method in microtremor survey: the robust use, the analyzable wavelength range
Keywords:microtremor, spatial autocorrelation , array
This paper discusses the titled subject based on a recent study by the author on the performance of the spatial autocorrelation (SPAC) method (Cho et al., 2021). It would call for reconsideration of existing guidelines for the use of the SPAC method. This abstract is an excerpted-and-rearranged version of the discussion in Cho (2022ab). For the details, readers are referred to this and the above papers.
(Standard SPAC method)
In this study, we define the standard SPAC method as follows.
i) A triangular array (Fig. 1a or b) is used.
ii) The following basic equation is used:
ρ(f)=J0(rk(f)),
where ρ is the SPAC coefficient, f is the frequency, J0 is the zeroth-order Bessel function of the first kind, r is the array radius, and k is the wavenumber.
iii) After obtaining ρ from an array observation, at each frequency, the value of rk is inverted using the above equation in the rk range less than 3, which is converted to the phase velocity using the relation c=2πrf/rk.
According to existing guidelines, the upper limit of the analyzable wavelength range is about 10r (Sato and Okada, 2016) or 4r-6r (Foti et al., 2017).
(Robust SPAC method)
In the robust SPAC method as defined in this paper, we calculate the phase velocity c1 by using equation c1=2πrf1/2.40, where f1 is the frequency at which a SPAC coefficient curve crosses zero for the first time (at the lowest frequency/ at the longest wavelength of 2.6r). This method is equivalent to using only the first zero-crossing point in the method of Aki (1957) to determine phase velocities. Therefore, this method is not novel but has been rarely noticed in the microtremor array survey in usual earthquake engineering. It should be emphasized that this approach is useful as being very robust against incoherent noise. This is because the frequency corresponding to the zero-crossing point is unaffected even when the signal-to-noise ratio (SNR) is low enough to generate devastating biases in the estimates of the SPAC coefficients. This method is also very robust against the biases due to the microtremor wavefield because the frequency at the first zero-crossing point is not affected by the wavefield as long as the triangular array is used. An illustrative example of applying the zero-crossing method is shown in Fig. 2.
(Upper limit of the analyzable wavelength range)
A reference phase-velocity dispersion curve (RPVDC) is defined as that obtained by deploying microtremor arrays of various sizes, applying the "robust SPAC method" to each array, and then connecting phase velocities thus obtained. The upper limit of the analyzable wavelength range (upper limit wavelength) is defined as the wavelength at which the dispersion curve obtained by the standard SPAC method deviates from the RPVDC. Based on a large number of array data sets, we take the statistics of the upper limit wavelength normalized by the array radius (normalized upper limit wavelength, NULW) for each group categorized by array sizes.
The analysis results show that, for very small arrays with a radius of about 1 m, the NULW reaches few-to-several tens, but rapidly decreases to about 3 to 5 as the array radius increases to reach several tens of meters (Fig. 3). The NULW stays around 3 to 5 for larger arrays. In other words, the NULW strongly depends on the array radius when the array size is small. The guidelines of Sato and Okada (2016) and Foti et al. (2017) can be understood as corresponding to arrays of radius around 10 m and tens of meters or more, respectively.
Acknowledgments: This work was supported by JSPS Grants-in-Aid for Scientific Research JP19H02287 and 20K04118.
Aki, K. (1957): BERI, Univ. Tokyo, 35, 415–457.
Cho, I. (2022a): Array-size dependency of the upper limit wavelength normalized by array radius for the standard spatial autocorrelation method, submitted to Earth, Planets and Space.
Cho, I. (2022b): The earth monthly, in press (in Japanese).
Cho et al. (2021): GJI, 226, 1676–1694.
Foti et al. (2017): Bull. Earthq. Engineer., 16, 2367-2420.
Sato, H., Okada, H. (2016): Geophysical exploration handbook (revised edition), SEGJ, Chap. 4, 229-248 (in Japanese).
(Standard SPAC method)
In this study, we define the standard SPAC method as follows.
i) A triangular array (Fig. 1a or b) is used.
ii) The following basic equation is used:
ρ(f)=J0(rk(f)),
where ρ is the SPAC coefficient, f is the frequency, J0 is the zeroth-order Bessel function of the first kind, r is the array radius, and k is the wavenumber.
iii) After obtaining ρ from an array observation, at each frequency, the value of rk is inverted using the above equation in the rk range less than 3, which is converted to the phase velocity using the relation c=2πrf/rk.
According to existing guidelines, the upper limit of the analyzable wavelength range is about 10r (Sato and Okada, 2016) or 4r-6r (Foti et al., 2017).
(Robust SPAC method)
In the robust SPAC method as defined in this paper, we calculate the phase velocity c1 by using equation c1=2πrf1/2.40, where f1 is the frequency at which a SPAC coefficient curve crosses zero for the first time (at the lowest frequency/ at the longest wavelength of 2.6r). This method is equivalent to using only the first zero-crossing point in the method of Aki (1957) to determine phase velocities. Therefore, this method is not novel but has been rarely noticed in the microtremor array survey in usual earthquake engineering. It should be emphasized that this approach is useful as being very robust against incoherent noise. This is because the frequency corresponding to the zero-crossing point is unaffected even when the signal-to-noise ratio (SNR) is low enough to generate devastating biases in the estimates of the SPAC coefficients. This method is also very robust against the biases due to the microtremor wavefield because the frequency at the first zero-crossing point is not affected by the wavefield as long as the triangular array is used. An illustrative example of applying the zero-crossing method is shown in Fig. 2.
(Upper limit of the analyzable wavelength range)
A reference phase-velocity dispersion curve (RPVDC) is defined as that obtained by deploying microtremor arrays of various sizes, applying the "robust SPAC method" to each array, and then connecting phase velocities thus obtained. The upper limit of the analyzable wavelength range (upper limit wavelength) is defined as the wavelength at which the dispersion curve obtained by the standard SPAC method deviates from the RPVDC. Based on a large number of array data sets, we take the statistics of the upper limit wavelength normalized by the array radius (normalized upper limit wavelength, NULW) for each group categorized by array sizes.
The analysis results show that, for very small arrays with a radius of about 1 m, the NULW reaches few-to-several tens, but rapidly decreases to about 3 to 5 as the array radius increases to reach several tens of meters (Fig. 3). The NULW stays around 3 to 5 for larger arrays. In other words, the NULW strongly depends on the array radius when the array size is small. The guidelines of Sato and Okada (2016) and Foti et al. (2017) can be understood as corresponding to arrays of radius around 10 m and tens of meters or more, respectively.
Acknowledgments: This work was supported by JSPS Grants-in-Aid for Scientific Research JP19H02287 and 20K04118.
Aki, K. (1957): BERI, Univ. Tokyo, 35, 415–457.
Cho, I. (2022a): Array-size dependency of the upper limit wavelength normalized by array radius for the standard spatial autocorrelation method, submitted to Earth, Planets and Space.
Cho, I. (2022b): The earth monthly, in press (in Japanese).
Cho et al. (2021): GJI, 226, 1676–1694.
Foti et al. (2017): Bull. Earthq. Engineer., 16, 2367-2420.
Sato, H., Okada, H. (2016): Geophysical exploration handbook (revised edition), SEGJ, Chap. 4, 229-248 (in Japanese).