Scattering of linear long-wave tsunamis due to randomly fluctuating sea-bottom topography: coda excitation and scattering attenuation

The scattering coefficient is one of the most fundamental parameters by which to quantify the scattering intensity for waves as a function of scattering angle and wave frequency. This study presents a derivation of the scattering coefficient for linear long-wave tsunami equations in randomly fluctuating sea-bottom topography using the first-order Born approximation. The scattering coefficient is directly related to the power spectrum density function of fluctuations in the bottom topography and shows a strong tsunami-wavelength dependence. The scattering regime is determined by the normalized wavenumber ak, where k is the tsunami wavenumber and a is the correlation distance of the fluctuating sea-bottom topography. The scattering pattern for small wavenumbers, ak << 1, is symmetric in the forward and backward directions, whereas the pattern for large wavenumbers, ak >> 1, shows small-angle scattering around the forward direction. Based on the theoretically derived scattering coefficient, we evaluate the excitation of tsunami coda and leading-wave attenuation as a function of the normalized wavenumber. The coda energy for small wavenumber ak << 1 is proportional to k(3), whereas the energy for large wavenumber ak >> 1 is proportional to k(-p+3) when the power spectral density function of the sea-bottom fluctuation is characterized by a power law with the exponent of -p in large wavenumber. The scattering attenuation represented by the inverse of the quality factor (Sc)Q(-1) is proportional to k(2) for small wavenumber ak << 1, whereas (Sc)Q(-1) is proportional to k(-p+2) for large wavenumber ak >> 1.