Retrieval of long-wave tsunami Green’s function from the cross-correlation of continuous ocean waves excited by far-field random noise sources on the basis of a first-order Born approximation
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2012
Received: 1 April 2011
Accepted: 31 August 2011
Published: 2 March 2012
We investigate the theoretical background for the retrieval of the tsunami Green’s function from the cross-correlation of continuous ocean waves. Considering that a tsunami is a long-wavelength ocean wave described by 2-D linear long-wave equations, and that the sea-bottom topography acts as a set of point-like scatterers, we use a first-order Born approximation in deriving the tsunami Green’s function having coda waves. The scattering pattern is non-isotropic and symmetrical with respect to the forward and backward directions. We indicate a retrieval process which shows that the derivative of the cross-correlation function of wavefields at two receivers with respect to the lag time gives the tsunami Green’s function when point noise sources generating continuous ocean waves are distributed far from, and surrounding, the two receivers. Note that this relation between the cross-correlation and the Green’s function is different from the case in which uncorrelated plane-wave incidence from all directions is assumed to be continuous ocean waves. The Green’s function retrieved from continuous ocean waves will be used as a reference to examine the validity of the Green’s function obtained by numerical simulations.
The technique of seismic interferometry, whereby the seismic Green’s function between two points is extracted from the cross-correlation of ambient seismic noise, has received a great deal of attention from seismologists (e.g., Campillo and Paul, 2003). Using this technique, seismologists can calculate the Green’s function, or estimate subsurface structures, by analyzing ambient noise, without the need for natural or artificial earthquakes (e.g., Shapiro et al., 2005; Nishida et al., 2009). For tsunami researchers, it is necessary to use the correct tsunami Green’s functions for tsunami source inversion analysis (e.g., Satake, 1989; Saito et al., 2010) or for simulating disastrous tsunamis generated by past, and anticipated future, huge earthquakes (e.g., Furumura et al., 2011). The correct tsunami Green’s function should be obtainable by numerical simulation using accurate and high-resolution bathymetry data. However, we cannot always obtain the correct Green’s function through numerical simulations. For example, in the 2010 Maule, Chile, earthquake tsunami, there was a significant discrepancy (∼30 min) between the observed, and simulated, arrival times around Japan (Satake et al., 2010). Therefore, it would be very useful if we are able to synthesize the tsunami Green’s function from observed “ambient” ocean waves; in other words, if we can retrieve the tsunami Green’s function from the cross-correlation of continuous ocean waves observed by, for example, ocean-bottom pressure gauges. Therefore, we investigate the theoretical background for the retrieval of the tsunami Green’s function from the cross-correlation of continuous ocean waves. This is the first attempt to demonstrate a retrieval process of the tsunami Green’s function which, in particular, includes coda waves. Considering that the tsunami has a long wavelength and the sea-bottom topography acts as a set of pointlike scatterers, we use a first-order Born approximation. The framework of the present study follows Sato (2009), who dealt with the case of 3-D scalar waves with isotropic scattering. For the application to a tsunami, the present study extends the approach of Sato to the case of 2-D waves with special non-isotropic scattering.
2. Tsunami Green’s Function in a Scattering Medium: Single Scattering
3. Cross-correlation Function of Waves in a Scattering Medium Illuminated by Surrounding Noise Sources
Equation (16) for the 2-D tsunami is consistent with the counterpart for the 3-D scalar wave (e.g. Roux et al., 2005; Sato, 2009). Note, however, that this form is different from that derived by Nakahara (2006) for 2-D scalar waves, in which the Hilbert transform of the cross-correlation function gives the Green’s function. The difference comes from the assumption of the noise sources. The present study assumes point noise sources, whereas Nakahara (2006) assumes uncorrelated plane-wave incidence as a noise wave-field. In 2-D space, a cylindrical wave impulsively radiated from a point source exhibits waveform broadening with a lapse time even in homogeneous media. By contrast, a plane wave maintains its initial shape during the propagation. This causes the difference between the results obtained in the present study and those of Nakahara (2006), in which the broadening nature of the Green’s function is recovered by Hilbert-transforming the cross-correlation function.
4. Numerical Examples
In order to validate the theoretical results of the present study, we performed numerical simulations based on the boundary integral method of Yomogida and Benites (1995), which was originally developed for simulating SH wave scattering in 2-D elastic media with cavities. We herein apply this method to tsunami scattering. This is justified by the fact that the 2-D wave equation of a tsunami (Eq. (1)), and that of SH waves in an elastic media of constant density are mathematically equivalent. In this relationship between the tsunami and the SH waves, a cavity in the elastic media corresponds to an island (a small area with zero sea depth).
As a more realistic example, we placed land (half space with zero sea depth) beside the receivers (Fig. 5(c)). The distance between the land s coast and the nearest receiver is 150 km. Although the number of noise sources is unchanged, there are no noise sources in the land. The tsunami reflected from the coast is recognized at ±4, 000 s in Fig. 5(d). Compared with the case of Fig. 5(b), the waveform retrieved from the cross-correlation function partly disagrees with the synthesized Green’s function. In particular, the amplitude with respect to the time axis is significantly asymmetric in the Green’s function retrieved from the cross-correlation function, because of the uneven distribution of the noise sources in space. Nevertheless, the arrival times of the maximum-amplitude wave, the scattered wave, and the wave reflected from the land in the Green’s function retrieved from the cross-correlation function explain well these arrival times for the synthesized Green function. This indicates that the retrieved Green’s function can be used as a reference in examining the validity of the travel time as calculated by numerical tsunami simulations.
5. Concluding Remarks
The present paper describes a retrieval process for the 2-D tsunami Green’s function from the cross-correlation of continuous ocean waves based on a first-order Born approximation. Equation (16) indicates that the derivative of the cross-correlation function for two receivers with respect to lag time gives the tsunami Green’s function including coda waves. Furthermore, we have presented numerical examples of the Green’s function retrieved from the cross-correlation function of the continuous waves to confirm the validity of the obtained results.
We herein assumed single scattering for the coda excitation. On the other hand, Margerin and Sato (2011) demonstrated the effects of a multiple scattering process in the retrieval of the 3-D scalar-wave Green’s function using Feynman diagrams. Some studies have also taken the higher-order perturbation of waves into consideration for the retrieval process (Sanchez-Sesma et al., 2006; Snieder et al., 2008; Wapenaar et al., 2010). A general theory of Green’s function retrieval for linear partial differential equations was recently presented by Fleury et al. (2010). Such general theories, which are not (or, at least, not strongly) restricted by the approximation conditions, are obviously important. On the other hand, the simplicity of analyzing specific situations, such as tsunami propagation with single scattering, as considered in the present study, may also be useful. For example, such an analysis would be helpful in interpreting the numerical simulation results, or practical analyses, of observed tsunami records. For the case of tsunami propagation, since accurate high-resolution bathymetric data is available, it would be interesting to simulate the retrieval of the tsunami Green’s function using actual bathymetry data. The results reported herein will be helpful in interpreting the features of actual records and the retrieved Green’s functions.
The authors would like to thank K. Yomogida for allowing the use of his computer code for the boundary integral simulations. We would also like to thank H. Sato and H. Nakahara for their useful discussions and advice. The manuscript was greatly improved by the careful reading and comments provided by T. Hara and the two anonymous reviewers.
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