Abstract: Tsunami inundation is rare on most coastlines, but large events can have devasting consequences for life and infrastructure. There is demand for inundation hazard maps to guide risk-management actions, such as the design of tsunami evacuation zones, tsunami-resilient infrastructure, and insurance. But the frequency of tsunami-generating processes (e.g., large earthquakes, landslides, and volcanic collapses) is usually very uncertain. This reflects limitations in scientific knowledge, and the short duration of historical records compared to the long inter-event times of dangerous tsunamis. Consequently, tsunami hazards are subject to large uncertainties which should be clearly communicated to inform risk-management decisions.

Probabilistic Tsunami Hazard Assessment (PTHA) offers a structured approach to quantifying tsunami hazards and the associated uncertainties, while integrating data, models, and expert opinion. For earthquake-generated tsunamis, several national and global-scale PTHAs provide databases of hypothetical scenarios, scenario occurrence-rates and their uncertainties. Because these “offshore PTHAs” represent the coast at coarse spatial resolutions (~ 1-2 km) they are not directly suitable for onshore risk management and can only simulate tsunami waveforms accurately in deep-water, far from the coast. Yet because offshore PTHAs can use earthquake and tsunami data at global scales, they offer relatively well tested representations of earthquake-tsunami sources, occurrence-rates, and uncertainties. Furthermore, by combining an offshore PTHA with a high-resolution coastal inundation model, the resulting onshore tsunami hazard can in-principle be derived at spatial resolutions appropriate for risk management (~ 10 m) for any site of interest.

This study considers the computational problem of rigorously transforming offshore PTHAs into site-specific onshore PTHAs. In theory this can be done by using a high-resolution hydrodynamic model to simulate inundation for every scenario in the offshore PTHA. In practice this is computationally prohibitive, because modern offshore PTHAs contain too many scenarios (on the order of 1 million) and inundation models are computationally demanding. Monte-Carlo sampling offers a rigorous alternative that requires less computation, because inundation simulations are only required for a random subset of scenarios. It is also known to converge to the correct solution as the number of scenarios is increased.

This study develops several approaches to reduce Monte-Carlo errors at the onshore site of interest, for a given computational cost. As compared to existing Monte-Carlo approaches for offshore-to-onshore PTHA, the key novel idea is to use deep-water tsunami wave heights (modelled by the offshore PTHA) to estimate the relative “importance” of each scenario near the onshore site of interest, prior to inundation simulation. Scenarios are randomly sampled from the offshore PTHA in a way that over-represents the “important” scenarios, and the theory of importance sampling enables weighting these scenarios so as to correct for the sampling bias. This can greatly reduce Monte-Carlo errors for a given sampling effort. In addition, because importance-sampling is analytically tractable, the variance of the Monte-Carlo errors can be estimated at offshore sites prior to sampling. This helps modellers to estimate the adequacy of a proposed Monte-Carlo sampling scheme prior to expensive inundation computation. The analytical variance result also enables the theory of optimal-sampling to be applied in a way that to reduces the Monte-Carlo variance, by non-uniformly sampling from earthquakes of different magnitudes.

The new techniques are applied to an onshore earthquake-tsunami PTHA in Tongatapu, the main island of Tonga. In combination the new techniques lead to efficiency improvements equivalent to simulating 4-18 times more scenarios, as compared with commonly used Monte-Carlo methods for onshore PTHA. They also enable the hazard uncertainties in the offshore PTHA to be translated onshore, where they are of most significance to risk management decision-making. The greatest accuracy improvements occur for large tsunamis, and for computations that represent uncertainties in the hazard.