From earthquake size to far-field tsunami amplitude: development of a simple formula and application to DART buoy data

We derive a simple formula relating tsunami amplitude in the far field to seismic moment, distance and azimuth from propagating rupture. Our formula is obtained from a comparison of a set of 4650 Pacific-wide simulations, computed for a series of sources spread over 10 subduction zones and four order of magnitudes in seismic moments. Our simulations are run both for a real grid reproducing the true bathymetry of the Pacific Basin and for an idealized one featuring a constant depth of 4000 m and no shorelines. This enables us to study and model separately the influence on the final amplitude of a tsunami wave of effects such as directivity and irregular bathymetry. The contribution of source size directivity and propagation over the sphere are studied using the constant-depth simulations. The influence of distance does not require any dispersive term and is properly modelled by geometrical spreading on the sphere. The directivity term, described classically in the frequency domain by Ben-Menahem & Rosenman can be approximated in the time domain by a moment-dependent linear regression as a function of azimuth. Finally, and after an allowance is made for the effect of receiver bathymetry using Green's law, the effect of irregular bathymetry is found to be generally defocusing, and can be modelled as a linear regression with distance. Once an estimate of the seismic moment of the parent earthquake is known, and under the assumption of a subduction mechanism along a fault of known azimuth, the resulting formula allows to forecast far-field tsunami amplitudes on the high seas. We use a data set of 116 tsunami amplitudes recorded at 51 past and present DART buoys following 21 tsunamigenic events to compare the estimates predicted by our algorithm to the amplitudes actually recorded. The average values of the residuals are 0.00 +/- 0.25 logarithmic units, and 0.02 +/- 0.20 at distances greater than 20 degrees. An important aspect of our algorithm is that it correctly predicts the DART amplitudes for the 2011 Tohoku tsunami (0.10 +/- 0.15 logarithmic units), even though its region was not included when building the algorithm.