Application of scattering theory to P-wave amplitude fluctuations in the crust
- Kazuo Yoshimoto^{1}Email author,
- Shunsuke Takemura^{2} and
- Manabu Kobayashi^{1}
Received: 17 October 2015
Accepted: 28 November 2015
Published: 10 December 2015
Abstract
The amplitudes of high-frequency seismic waves generated by local and/or regional earthquakes vary from site to site, even at similar hypocentral distances. It had been suggested that, in addition to local site effects (e.g., variable attenuation and amplification in surficial layers), complex wave propagation in inhomogeneous crustal media is responsible for this observation. To quantitatively investigate this effect, we performed observational, theoretical, and numerical studies on the characteristics of seismic amplitude fluctuations in inhomogeneous crust. Our observations of P-wave amplitude for small to moderately sized crustal earthquakes revealed that fluctuations in P-wave amplitude increase with increasing frequency and hypocentral distance, with large fluctuations showing up to ten-times difference between the largest and the smallest P-wave amplitudes. Based on our theoretical investigation, we developed an equation to evaluate the amplitude fluctuations of time-harmonic waves that radiated isotropically from a point source and propagated spherically in acoustic von Kármán-type random media. Our equation predicted relationships between amplitude fluctuations and observational parameters (e.g., wave frequency and hypocentral distance). Our numerical investigation, which was based on the finite difference method, enabled us to investigate the characteristics of wave propagation in both acoustic and elastic random inhomogeneous media using a variety of source time functions. The numerical simulations indicate that amplitude fluctuation characteristics differ a little between medium types (i.e., acoustic or elastic) or source time function durations. These results confirm the applicability of our analytical equation to practical seismic data analysis.
Keywords
Introduction
To predict the strong ground motion of future destructive earthquakes, empirical ground motion prediction equations have been developed and applied to many regions of the world (e.g., Si and Midorikawa 1999; Douglas 2003). These empirical equations are useful for estimating the average characteristics of seismic amplitude variation with hypocentral distance; however, the observed ground motions (e.g., peak ground velocity) show large scatter around the predictions (e.g., Strasser et al. 2009). It has been suggested that path effects (e.g., Atkinson 2006; Anderson and Uchiyama 2011) are one of the key factors causing this data scatter; source effects (e.g., Brillinger and Preisler 1984; Ripperger et al. 2008) and site effects (e.g., Chen and Tsai 2002; Lin et al. 2011; Yoshimoto and Takemura 2014) also play a part. Since path effects are governed by lithospheric inhomogeneities, they can be quantitatively estimated using appropriate physical models of wave propagation in inhomogeneous media. However, few studies have tackled this issue (e.g., Nikolaev 1975), mainly because of the difficulties in acquiring high-quality observational data.
In order to investigate the stochastic properties of the Earth’s internal inhomogeneities, short-wavelength random inhomogeneities are commonly characterized using stochastic medium models introducing characteristic correlation distances and mean square fractional fluctuations (e.g., Sato et al. 2012). At frequencies below ~1 Hz, spatio-temporal variations in teleseismic P-waves, which are considered as plane incident waves into seismic observation arrays, have been analyzed to investigate short-wavelength random inhomogeneities beneath seismic stations. Flatté and Wu (1988) investigated short-wavelength inhomogeneities in the lithosphere and asthenosphere by analyzing fluctuations in the travel time and amplitudes of teleseismic P-waves observed by the Norwegian Seismic Array (NORSAR). Ritter et al. (1998) estimated lithospheric inhomogeneities in the Massif Central region of France, by analyzing frequency-dependent intensities of the mean wavefield and the fluctuation wavefield of teleseismic P-waves.
A number of theoretical and numerical studies have focused on the amplitudes and phase fluctuations of transmitted waves in random inhomogeneous media (e.g., Rytov et al. 1989; Ishimaru 1997). In these studies, the Rytov approximation has commonly been used to estimate amplitude variance and the phase fluctuations of transmitted waves. Shapiro and Kneib (1993) estimated scattering attenuation in acoustic inhomogeneous media resulting from amplitude fluctuations using the Rytov approximation for the wavefield in a weak fluctuation regime. Hoshiba (2000) demonstrated that the amplitude fluctuations predicted by the Rytov approximation successfully explain the numerical results obtained for scalar plane waves propagating in three-dimensional (3D) Gaussian-type random media. Müller and Shapiro (2003) proposed a formulation to calculate the log-amplitude variance of plane-transmitted waves in anisotropic Gaussian-type random media. However, despite these studies, analyzing the seismograms of local and/or regional earthquakes requires the development of a more realistic physical model that can adequately model spherical wave propagation in realistic random inhomogeneous structures (e.g., von Kármán-type random media).
In this study, we first analyzed the seismograms of local earthquakes observed by a dense seismic network in order to reveal the characteristics of P-wave amplitude fluctuations in the crust. We then derived an analytical equation for calculating the amplitude fluctuation of spherical waves radiating from a point source in acoustic von Kármán-type random media. The results obtained by this equation were checked using numerical results from finite-difference method (FDM) simulations of seismic wave propagation. Finally, we considered the extent to which observed P-waves amplitude fluctuations in the crust were explained by the predictions of our analytical equation. The results confirmed the applicability of our analytical equation to practical seismic data analysis.
Seismic observations
Data
Earthquake catalog
No. | Date/time (JST) | Lat. (°N) | Log. (°E) | Depth (km) | Strike (°) | Dip (°) | Rake (°) | M_{W} |
---|---|---|---|---|---|---|---|---|
1 | 2005/03/19, 15:03:46.92 | 132.80 | 35.01 | 5 | 149 | 86 | −20 | 3.6 |
2 | 2005/07/28, 09:11:11.30 | 132.72 | 34.91 | 8 | 138 | 88 | 3 | 3.5 |
3 | 2006/11/26, 12:58:35.01 | 132.88 | 35.15 | 11 | 277 | 89 | 165 | 3.5 |
4 | 2007/05/13, 08:13:54.79 | 132.79 | 35.01 | 8 | 56 | 86 | 152 | 4.3 |
5 | 2007/10/14, 23:38:50.08 | 133.20 | 35.43 | 5 | 157 | 85 | −35 | 3.5 |
6 | 2010/01/21, 08:50:36.79 | 132.81 | 34.99 | 5 | 338 | 83 | 36 | 3.4 |
7 | 2010/06/07, 07:43:01.09 | 132.74 | 35.12 | 8 | 349 | 87 | −14 | 3.4 |
8 | 2011/02/07, 00:36:15.77 | 133.03 | 34.89 | 5 | 211 | 81 | 149 | 3.6 |
9 | 2011/06/04, 01:57:31.01 | 132.67 | 35.10 | 11 | 336 | 87 | 22 | 4.9 |
10 | 2011/06/16, 23:32:21.15 | 132.67 | 35.10 | 8 | 161 | 89 | −12 | 3.8 |
11 | 2011/11/21, 19:16:29.59 | 132.89 | 34.87 | 11 | 241 | 89 | 170 | 5.2 |
12 | 2011/11/25, 04:52:24.33 | 132.90 | 34.87 | 5 | 64 | 76 | −150 | 4.0 |
13 | 2013/01/08, 20:19:32.76 | 132.42 | 34.46 | 20 | 216 | 84 | 172 | 3.5 |
P-wave amplitude fluctuations
We measured maximum P-wave amplitudes from the three-component vector amplitude of the filtered velocity seismograms. Following the method of Kobayashi et al. (2015), in order to eliminate the effects of different earthquake source sizes and site amplifications, the observed maximum P-wave amplitudes were normalized by the averaged S-wave coda amplitude at lapse times of 60–70 s, which were based on the coda normalization method (e.g., Yoshimoto et al. 1993). Hereafter, we refer to the coda-normalized maximum P-wave amplitude as “P-wave amplitude”.
By selecting only data with similar and large P-wave radiation pattern coefficients (|F _{ P }| ≥ 0.7), the increase in data scatter with increasing hypocentral distance, along with the frequency changes, became clearer (Fig. 2b). Using an observational property of the small data scatter at short hypocentral distances (except for one outlier at ~15 km in the 1–2-Hz band), which is likely to indicate well-calibrated source magnitudes, we were able to investigate the propagation dependence of P-wave amplitude fluctuations in the inhomogeneous crust.
Analytical methods
Analytical equation for amplitude fluctuations
where c _{0} is the background velocity and ξ(r) is the fractional fluctuation of velocity. The fractional fluctuation should be small or |ξ(r)| ≪ 1 and 〈ξ(r)〉 = 0, where the symbol 〈〉 denotes the statistical ensemble average.
The 3D integral of Eq. (4) becomes non-zero only for the volume element d V′, for which a fractional fluctuation of velocity exists. Equation (4) is identical to equation (17–23) of Ishimaru (1997), except for the negative sign arising from the different definitions of the fractional fluctuation (Eq. (1)) and the fluctuation of the refractive index of Ishimaru (1997).
This relationship is useful for estimating \( {\sigma}_{\chi}^2 \) when we do not have rigorous information on u _{0}(r).
From the same geometric consideration of wave propagation as in Ishimaru (1997), Eqs. (12) and (13) can be used for a plane wave case when γ = 1.
Evaluation of amplitude fluctuations
FDM simulations
Types of inhomogeneous media
No. | Medium-type | Randomness | Background parameters | Variable parameters | |||
---|---|---|---|---|---|---|---|
a(km) | ε | P-wave velocity (km/s) | S-wave velocity (km/s) | Density (g/cm^{3}) | |||
A | Acoustic | 1.0 | 0.03 | 6.0 | 0 | 2.6 | P-wave velocity |
B | Elastic | 1.0 | 0.03 | 6.0 | 3.5 | 2.6 | P- and S-wave velocity |
Density | |||||||
C | Elastic | 1.0 | 0.03 | 6.0 | 3.5 | 2.6 | P- and S-wave velocity |
Source time functions
No. | Type | Frequency (Hz) | Duration (s) |
---|---|---|---|
1 | Sin | 1.5 | 9 |
2 | Sin | 3.0 | 1 |
3 | Sin | 3.0 | 2 |
4 | Sin | 3.0 | 3 |
5 | Sin | 3.0 | 6 |
6 | Sin | 3.0 | 9 |
7 | Ricker | 1.5 | ~0.67 |
8 | Ricker | 3.0 | ~0.33 |
To construct realistic inhomogeneous crustal models, we employed stochastic random elastic fluctuations characterized by an exponential-type power spectral density function in the wavenumber domain (β = 2 of the von Kármán-type power spectral density function). We assumed a = 1 km and ε = 0.03, which were estimated for the crust in the Chugoku region by Kobayashi et al. (2015). Constructed stochastic random elastic fluctuations were embedded over mean background elastic parameters. We assumed a linear relationship of fluctuations among the P-wave velocity, S-wave velocity, and the density from Birch’s law (Birch 1961) for medium model B, whereas density fluctuations was not allowed in medium model C. For the construction of the acoustic random inhomogeneous medium, which we used in the numerical verification of the analytical equation, we set the S-wave velocity of medium model C to 0 km/s.
Results and discussion
Analytical predictions and FDM simulation results
The hypocentral distance variations of the maximum amplitudes in the random inhomogeneous acoustic media (model A) are shown in Fig. 6b. To collect sufficient data, we conducted four FDM simulations using four different realizations of random media. The apparent attenuation of maximum amplitudes was due to geometrical spreading and scattering effects. An increase in the variation of amplitude was observed with increasing hypocentral distance; however, the variation in amplitude was barely detectable at small hypocentral distances. As a result, we observed up to ten-times difference between the largest and the smallest P-wave amplitudes. The increasing variation in amplitude with increasing hypocentral distance appeared to saturate at a certain distance (~30 km) in the 3-Hz simulation.
Amplitude fluctuations: source wavelet dependence
Earthquake sources radiate non-time-harmonic waves, especially at low frequencies below the corner frequency. Thus, for the application of Eq. (13) to substantial earthquake data, it was necessary to check the extent to which the time-harmonic wave predictions explained the amplitude level variance for non-time-harmonic waves.
Our findings showed that Eq. (13) with an adequate saturation limit may be practically effective for estimating the amplitude level variance of not only time-harmonic waves, but also non-time-harmonic waves, despite the slight under estimation in the strong wavefield fluctuation regime for non-time-harmonic waves.
Amplitude fluctuations: medium-type dependence
In addition to numerical analyses using random acoustic media, we conducted additional FDM numerical simulations using elastic random media in order to investigate the dependence of elastic parameter fluctuations on the amplitude level variance.
Predictions for spherical waves and plane waves
Analytical predictions and observational results
Despite the small data volume, the log-amplitude distribution of P-waves observed at hypocentral distance of 70 km appeared to show a normal distribution rather than a uniform distribution (Fig. 14b). This characteristic was consistent with the results obtained by the FDM simulations (Figs. 7 and 10b); furthermore, the wider distribution for high-frequency waves was also consistent with the FDM simulation results.
Focus of future analyses
The FDM simulations in this study demonstrated that variations of wave amplitude in random inhomogeneous media depend on not only the inhomogeneity of the medium but also on the duration of the source wavelet. Thus, to use analytical Eq. (13) for the analysis of P-waves from local and/or regional earthquakes, it is important to consider the source duration of a target earthquake, which should change by earthquake size or magnitude. In this study, in order to minimize this effect, we selected similar size earthquakes for analysis. We anticipate that predictions by Eq. (13), which deals with quasi-time-harmonic waves, will be more suitable for data from large earthquakes with long source time durations. However, a detailed quantitative analysis of earthquake-size dependence on P-wave amplitude variations will be the focus of future work.
In this study, we also restricted our focus to P-wave analysis. Thus, applying analytical estimations using Eq. (13) to strong ground motion prediction is not straightforward, in particular because peak ground motions (e.g., peak ground velocity) are mainly excited by S-waves (e.g., Atkinson and Mereu 1992). However, we believe that estimations from Eq. (13) are practically useful for the evaluation of S-wave amplitude variations in inhomogeneous crust because the Rytov approximation has been successfully applied to the analysis of vector waves (e.g., electromagnetic wave propagation; Ishimaru 1997). From the analysis of strong motion data, Midorikawa and Ohtake (2003) reported that the variance of peak horizontal accelerations and velocities increased with increasing hypocentral distance over a short range (<50 km), but decreased with increasing earthquake magnitude. These characteristics are consistent with those found for P-waves in this analysis, suggesting the applicability of our methodology to strong ground motion prediction in earthquake engineering.
Conclusions
To better understand the amplitude fluctuations of high-frequency seismic waves, we carried out observational, theoretical, and numerical analyses of waves propagating in random inhomogeneous media.
Our seismic observations of the P-wave amplitudes of small to moderately sized crustal earthquakes revealed that the fluctuation of P-wave amplitude increased with increasing hypocentral distance, with up to ten-times difference between the largest and the smallest P-wave amplitudes. The saturation of this phenomenon occurred at certain hypocentral distances. Furthermore, the observed amplitude fluctuations and their rate of increase with the hypocentral distance increased with increasing frequency.
Our theoretical studies presented an analytical equation for evaluating the amplitude fluctuation of time-harmonic waves isotropically radiated by a point source and spherically propagating in acoustic von Kármán-type random media. The equation approximately explained the characteristics of P-wave amplitude fluctuations observed from local crustal earthquake as a function of frequency and hypocentral distance. Using this equation, we confirmed the differences in amplitude fluctuation between spherical and plane waves. It was verified that the amplitude fluctuations of plane waves are greater than those of spherical waves when compared at the same travel distance from source.
Our numerical analyses based on FDM simulations enabled us to investigate the characteristics of wave propagation in both acoustic and elastic random media using a variety of source time functions. The results of the simulation showed that the hypocentral distance variations of the amplitude fluctuations obtained for the elastic random media were similar to those obtained for the acoustic random media, except for some differences in absolute values (i.e., the amplitude fluctuation was slightly smaller for the elastic random medium than for the acoustic random medium). We found that the values of the variance of the amplitude level in the weak wavefield fluctuation regime were very similar among the different source time function durations; however, the values in the strong wavefield fluctuation regime increased with decreasing duration. These results indicate that our analytical equation for amplitude fluctuations may be useful in the analysis of substantial seismic data.
Declarations
Acknowledgements
We thank two anonymous reviewers, and the editor, Takuto Maeda, for constructive comments that improved an earlier draft of this manuscript. We acknowledge the National Research Institute for Earth Science and Disaster Prevention, Japan for providing the Hi-net/F-net waveform data and the CMT solutions from the F-net. We also used the unified hypocentral catalog provided by the Japan Meteorological Agency. The FDM simulations of seismic wave propagation were conducted on the computer system of the Earthquake and Volcano Information Center at the Earthquake Research Institute, The University of Tokyo. The authors are particularly indebted to Haruo Sato and Hisashi Nakahara for valuable comments and discussions concerning wave propagation in inhomogeneous media. All figures were drawn using the Generic Mapping Tools software package developed by Wessel and Smith (1998).
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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