Elucidating the Mechanics of Tsunami Generating Earthquake Rupture with Long Period Seismology

Current techniques used to determine whether slip reached the seafloor in submarine earthquakes are expensive and cannot be applied to most earthquakes of interest. This lack of observations has hindered the understanding of the physical conditions controlling whether an earthquake will displace the seafloor and increase the likelihood of generating a significant tsunami. We develop here a new a theory which overcomes the limitations of current seismological frameworks and fully considers the effect of fault slip that reaches the surface.

The upper 5-10~km of the megathrust fault plane is thought to be frictionally weak and not capable of storing elastic strain energy, and thus does not typically host slip during earthquake rupture. However, the 2011 Tohoku-oki Mw 9.1 earthquake exhibited an unprecedented ~50m of peak fault slip which extended to the trench. The Japanese trench is one of the best instrumented subduction zones in the world, and seafloor geodesy, repeat pass multibeam bathymetric imaging, ocean-bottom pressure measurements, and seismic reflection studies all showed clear evidence that there was significant slip that extended to the trench/seafloor. The elastic strain energy stored and released near the Earth's surface (seafloor; Figure 1 region A) substantially alters calculations of the peak ground motions and the occurrence of highly hazardous tsunamis which are generated by underwater landslides and the movement of the seafloor. The devastating tsunamis resulting from the 2004 Sumatra and 2011 Tohoku-oki earthquakes, are both painful reminders of the serious hazard posed by these events.

Figure 1: Schematic cross-sectional view of the megathrust fault zone within a generalized subduction zone modified from Lay et al., (2012). The label A shows the near‐trench domain, B shows the central megathrust domain, C shows the downdip domain, and D shows the transitional domain.

Specifically we are interested in answering the question ``Under what physical conditions will rapid slip during an earthquake displace the seafloor and increase the likelihood of generating a significant tsunami?'' To do this, we will distinguish events for which slip reached and did not reach the surface. This requires developing a theory which involves the careful application of Gauss' Theorem in the presence of a free-surface (seafloor) and an intersecting fault slip discontinuity. The effect of the free-surface is not included in conventional seismic source theory and has only been considered for point-sources in the high-frequency limit that asymptotically approach the free-surface.

Cartoon Schematic of volumes, interfaces, and source region involved in derivation of surface rupture term in excitation coefficients.
⊕ denotes the entire integration volume and is composed of the union of the the solid (⊕_{S}) and fluid (⊕_{F}) regions.

Consider a source volume S^{t} (Figure 2) which represents any indigenous source at time t (a source which occurs within or upon the surface of the Earth that does not involve any forcing from external sources).
The source region (S^{t}) represents a localized, transient breakdown of the linearized elastic constitutive relation (i.e., frictional failure on a fault interface).
Since the equivalent force densities from an indigenous source do not exert a net force or torque upon the Earth, it cannot alter its linear or angular momentum, and the solution s(**x**,t), the displacement at location **x** at time t, to this inhomogeneous excitation problem can be expressed as a sum of normal modes **s**_{k}

where the subscript k denotes the k-th normal mode, ω_{k} is the characteristic frequency of the k-th mode, and C_{k} is the normal-mode excitation coefficient.
The normal-mode excitation coefficients can be expressed in terms of the stress glut (**S**) or equivalent force densitities which capture the effects of the earthquake source.
By applying a more general form of Gauss' Theorem than is ussually encountered, which is valid for integration of a piecewise continuously differentiable tensor field over a composite volume, we can derive an expression for the normal mode excitation coefficients C_{k}.
In Figure 3 we show the geometry and excitation coefficents for a moment-tensor density source (**m**) of an idealized fault which does and does not intersect the free surface (note that ε_{k} is the strain associated with the eigenfunction **s**_{k}).

Cartoon schematic of buried and surface-rupture faults. The buried fault surface shown in grey represent the source elements which are used in typical finite fault inversion. The outcropping fault elements shown in red require the additional (Note that the domains of integration have been changed from the general volume S_{t} to be over the idealized 2D fault surface (Σ_{t}).

In order to illustrate the importance of this effect, we approximate our finite fault to be a point source.
This calculation will allow us to estimate a lower bound for the size of the effect we are proposing to calculate.
To compute seismograms, the eigenfunctions (s_{k}) were calculated for the Preliminary Reference Earth Model (PREM) using MINEOS.
Modes that have periods longer than 200~seconds were used to generate synthetic seismograms.
We initially consider the effect of a 45°-dip thrust fault located at 10~km depth with and without surface rupture recorded by a station that is located at an epicentral distance of 90° with the azimuth of 60° (Figure 4a).
The synthetic seismogram that includes the surface-rupture term shows larger absolute amplitude than for a buried source, indicating that the term that is conventionally ignored contributes significantly to the amplitude of long-period arrivals.
Furthermore, if we examine the fundamental Rayleigh waves, the magnitude of their excitation decreases with decreasing period, demonstrating that the surface-rupture preferentially affect longer-period waves.
This increase in the long-period energy may help explain the observations of tsunami generating earthquakes which have been inferred to have long source durations or slow rupture velocities.
To investigate the dependence of the surface-rupture term on the focal mechanism, we recompute synthetic seismograms using vertical strike slip, and vertical dip slip mechanisms (Figure 4b and 4c).
The results clearly show that the amplitude difference between events with and without surface rupture becomes smaller with mechanisms with less vertical motion, and in the case of the vertical strike-slip mechanism, the two synthetic seismograms are identical.

Preliminary results for a variety of source mechanisms for a source-station pair with an epicentral distance of 90 degrees and azimuth of 60 degrees. Normalized vertical displacement seismograms for a 45°-dip normal fault (a), a vertical strike slip (b), and vertical dip-slip (c). Traditional sources which do include surface rupture are shown in cyan, and sources which do include the free-surface term are shown in black. The difference between these two curves is shown in red.

Note that the results shown here are generated with a number of assumptions and represent the minimum effect that will be exhibited by including the surface interaction term, and that a full theory still needs to be developed to understand this effect completely. In addition to changing the absolute amplitude of the effect, the inclusion of finite fault effects via the full theory will result in a more complex response as a function of frequency. For example, by including specific length scales related to the fault geometry, different modes will be excited more strongly than others as opposed to those calculated using the point source approximation.