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Abstract

Tsunami generation by offshore earthquakes is a problem of scientific interest and practical
relevance, and one that requires numerical modelling for data interpretation and hazard
assessment. Most numerical models utilize two-step methods with one-way coupling between
separate earthquake and tsunami models, based on approximations that might limit
the applicability and accuracy of the resulting solution. In particular, standard methods focus
exclusively on tsunami wave modelling, neglecting larger amplitude ocean acoustic and
seismic waves that are superimposed on tsunami waves in the source region. In this study, we
compare four earthquake-tsunami modelling methods. We identify dimensionless parameters
to quantitatively approximate dominant wave modes in the earthquake-tsunami source region,
highlighting how the method assumptions affect the results and discuss which methods are
appropriate for various applications such as interpretation of data from offshore instruments in
the source region. Most methods couple a 3-D solid earth model, which provides the seismic
wavefield or at least the static elastic displacements, with a 2-D depth-averaged shallow water
tsunami model. Assuming the ocean is incompressible and tsunami propagation is negligible
over the earthquake duration leads to the instantaneous source method, which equates the
static earthquake seafloor uplift with the initial tsunami sea surface height. For longer duration
earthquakes, it is appropriate to follow the time-dependent source method, which uses
time-dependent earthquake seafloor velocity as a forcing term in the tsunami mass balance.
Neither method captures ocean acoustic or seismic waves, motivating more advanced methods
that capture the full wavefield. The superposition method of Saito et al. solves the 3-D
elastic and acoustic equations to model the seismic wavefield and response of a compressible
ocean without gravity. Then, changes in sea surface height from the zero-gravity solution are
used as a forcing term in a separate tsunami simulation, typically run with a shallow water
solver. A superposition of the earthquake and tsunami solutions provides an approximation
to the complete wavefield. This method is algorithmically a two-step method. The complete
wavefield is captured in the fully coupled method, which utilizes a coupled solid Earth and
compressible ocean model with gravity. The fully coupled method, recently incorporated into
the 3-D open-source code SeisSol, simultaneously solves earthquake rupture, seismic waves
and ocean response (including gravity). We show that the superposition method emerges as
an approximation to the fully coupled method subject to often well-justified assumptions.
Furthermore, using the fully coupled method, we examine how the source spectrum and ocean
depth influence the expression of oceanic Rayleigh waves. Understanding the range of validity
of each method, as well as its computational expense, facilitates the selection of modelling
methods for the accurate assessment of earthquake and tsunami hazards and the interpretation
of data from offshore instruments.