*Earth Planets Space,* Vol. 64 (No. 1), pp. 43-48, 2012

doi:10.5047/eps.2011.08.020

*LETTER*
## 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

Tatsuhiko Saito^{1} and Jun Kawahara^{2}

^{1}National Research Institute for Earth Science and Disaster Prevention, 3-1 Tennodai, Tsukuba, Ibaraki 305-0006, Japan

^{2}Faculty of Science, Ibaraki University, 2-1-1 Bunkyo, Mito, Ibaraki 310-8512, Japan

(Received April 1, 2011; Revised August 29, 2011; Accepted August 31, 2011; Online published March 2, 2012)

**Abstract: **
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.

**Key words: **
Tsunami, theory, interferometry.

Corresponding author E-mail: saito-ta@bosai.go.jp

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