Tsunami Propagation Uncertainties:

The uncertainties in the tusnami propagation algorithms is to some extent due to the efficiency and structure of simulation algorithms. Thus, as is customary in fluid mechanics, these algorithms are routinely benchmarcked via laboratory and field observations. However, tsunami simulations are often far from ideal, that is from exactly what happened in the real world. This is mostly due uncertainties in both the tsunami source and propagation medium, i.e., bathymetry.

Our knowledge of tsunami sources in any form (earthquake, landslide, atmospheric pressure gradient, etc), is never complete. For instance, in the case of earthquakes the entire extent of underwater faulting is never fully resolved. More importantly, in the case of megathrust events the constraints on earthquake size are sketchy, particularly in the few hours or even days after the earthquakes.

Bathymetry model (which strongly affect tsunami propagation) are estimates of the real world. This depends on the source scale as well as the target coastal resolution, but it is important to obtain the most accurate models for the best simulations.

However, two legitimate questions are:

1. How can we quantify the source/bathymetry resolution effect?
2. How much source/bathymetry resolution do we really need to reasonably predict the main propagation features?

An effective way to answer these questions is decomposing source and bathymetry complexities into simpler components and then comparing tsunami simulations (using a highly benchmarked algorithm) from the simplified versions to the real deal.

A. Bathymetry Resolution

Since bathymetry (especially on the global scale) is distributed on a sphere (i.e., Earth), it is appropriate to use spherical harmonics to mathematically expand bathymetry. The individual terms of expansion, or Y_l^ms represent eigenfunctions of Laplace's equation (l and m are called degree and order). In this formalism, -l ≤ m ≤ l and therefore, l is the dominant factor in determining resolution. In this fashion, l=0 corresponds to a flat ocean, and l=∞ represents the original field.

By expanding the bathymetry in the harmonics space and isolating various numbers, we can obtain smoothed bathymetry fields various levels of physical resolution.





By simulating tsunamis at each resolution step, we gain insight into how physical resolution of bathymetry affects tsunami propagation. The figure below shows maximum tsunami wave heights calculated for each level of bathymetry resolution (see the figure above) for a M₀=10²⁹ dyn-cm, pure-thrust earthquake (φ=355, δ=15, λ=90) in Chile.




We can easily see that by adding to the maximum l used in bathymetry expansion, we get closer and closer to the real scenario. Using the MT parameter (Salaree, 2019) as the misfit between the smoothed simulation and the actual case, and more statistical analyses (such as the F-test), we can see that the l=40 (~1000 km) denotes the bathymetry resolution necessary to reproduce the main features necessary for a M₀=10³⁰ dyn-cm source.