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# [HDS10-24] Reciprocity and representation theorems for linear long and dispersive waves and its application to the tsunami amplification problem nearby the coastline

When tsunami waves attack the coastline, its amplitude is amplified significantly at around the shallow coastline, affected by its shallower bathymetry. Because including a short wavelength tsunami at shallow bathymetry into numerical simulation directly is not economical, theoretical/empirical amplification factors are frequently used for estimating tsunami amplitude nearby the coast from the simulation result at offshore. One of the most widely used relations is the Green's law. This law is based on the energy flux conservation, and it predicts changes in amplitude by changes in bathymetry. However, it implicitly assumes that the bathymetry depths do not change in directions perpendicular to the tsunami ray direction. Another drawback is that it does not tell anything about arrival times. Recently, Tsushima (2016, 2017) proposed to use frequency-dependent empirical amplification factors formed as a recursive time-domain digital filter. In this study, I propose an alternative way of evaluating tsunami amplification factor, based on Betti's theorem.

Betti's theorem describes a relationship between elastic motions due to two sets of forces in elastodynamics. The reciprocity relation, which allows exchanging the source and receiver position in Green's function, can be derived from that under the homogeneous boundary condition. In tsunami studies, Korolev (2011, 2012) showed numerically that the linear long waves satisfy approximately, and Hossen et al. (2015) utilized this relation to time-reverse imaging for the inverse problem of the tsunami source model.

I first confirmed the linear long wave tsunami satisfies a relation corresponding to the Betti's theorem in elastodynamics. I also confirmed its extension to the linear dispersive model under the Boussinesq approximation (e.g., Saito et al., 2004) satisfies the similar relationship. In this formula, tsunami wave height at a point can be expressed as the source radiation term and the contribution from the closed boundary surrounding the point, which corresponds to the representation theorem in elastodynamics. For the tsunami amplification problem, I only use the latter term by choosing the closed boundary without source inside. In that case, tsunami wave at the point is expressed as a convolution between tsunami waves observed at boundaries and Green's function of tsunami radiated from the point. The Green's function consists of tsunami height and tsunami flow velocity terms radiated at a point as a delta function of space and time. Notice that this formulation requires tsunami flow velocity terms in addition to tsunami height at the boundary.

In this formulation, tsunami waveform nearby the coast can be reconstructed from tsunami waves at outside. I tested the proposed method via numerical simulation with a synthetic bay-like bathymetry model. The target station is located an inner portion of the bay, while the boundary is set at the entrance of the bay. Plane-wave tsunami incidence to the bay is assumed, and synthetic tsunami waveforms are calculated at the point. Also, the Green's function at the boundary points from the target station is pre-calculated. The integration along the boundary is approximated as a sum of waveforms at the boundary points. The synthesized tsunami inside the bay agrees quite well with the assumed tsunami waves.

The proposed method has an advantage that tsunami waves as a time series can be forecasted. On the other hand, the formulation suggests flow velocity terms are mandatory for forecasting tsunami amplitude, which makes it difficult to utilize this formula directly to the observed tsunami heights. Recent tsunami data assimilation (Maeda et al., 2015) enables us to forecast tsunami height and flow velocities based on the assimilation of dense offshore data. Hybridizing data assimilation with the proposed method would contribute to more quick and accurate tsunami forecasting.

Betti's theorem describes a relationship between elastic motions due to two sets of forces in elastodynamics. The reciprocity relation, which allows exchanging the source and receiver position in Green's function, can be derived from that under the homogeneous boundary condition. In tsunami studies, Korolev (2011, 2012) showed numerically that the linear long waves satisfy approximately, and Hossen et al. (2015) utilized this relation to time-reverse imaging for the inverse problem of the tsunami source model.

I first confirmed the linear long wave tsunami satisfies a relation corresponding to the Betti's theorem in elastodynamics. I also confirmed its extension to the linear dispersive model under the Boussinesq approximation (e.g., Saito et al., 2004) satisfies the similar relationship. In this formula, tsunami wave height at a point can be expressed as the source radiation term and the contribution from the closed boundary surrounding the point, which corresponds to the representation theorem in elastodynamics. For the tsunami amplification problem, I only use the latter term by choosing the closed boundary without source inside. In that case, tsunami wave at the point is expressed as a convolution between tsunami waves observed at boundaries and Green's function of tsunami radiated from the point. The Green's function consists of tsunami height and tsunami flow velocity terms radiated at a point as a delta function of space and time. Notice that this formulation requires tsunami flow velocity terms in addition to tsunami height at the boundary.

In this formulation, tsunami waveform nearby the coast can be reconstructed from tsunami waves at outside. I tested the proposed method via numerical simulation with a synthetic bay-like bathymetry model. The target station is located an inner portion of the bay, while the boundary is set at the entrance of the bay. Plane-wave tsunami incidence to the bay is assumed, and synthetic tsunami waveforms are calculated at the point. Also, the Green's function at the boundary points from the target station is pre-calculated. The integration along the boundary is approximated as a sum of waveforms at the boundary points. The synthesized tsunami inside the bay agrees quite well with the assumed tsunami waves.

The proposed method has an advantage that tsunami waves as a time series can be forecasted. On the other hand, the formulation suggests flow velocity terms are mandatory for forecasting tsunami amplitude, which makes it difficult to utilize this formula directly to the observed tsunami heights. Recent tsunami data assimilation (Maeda et al., 2015) enables us to forecast tsunami height and flow velocities based on the assimilation of dense offshore data. Hybridizing data assimilation with the proposed method would contribute to more quick and accurate tsunami forecasting.