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The uncertainty quantification of tsunami hazard assessments due to uncertain earthquake characteristics is not straightforward and faces two important challenges. First, it needs the generation of earthquake samples with consistent probability laws, based on past event observations. Second, It needs the propagation of uncertainty from the earthquake to the tsunami response with a feasible computational cost. In this study we first propose a methodology which generates consistent earthquake samples with respect to the probability properties of the slip distribution and location. We use a Karhunen-Loeve (K-L) expansion and a translation process. Second, we propose the application of an uncertainty propagation method, known as Stochastic Reduced Order Model (SROM), which is more accurate than Monte Carlo simulations.
Earthquake sample generation
The earthquakes are characterized with random distance to the trench and slip distribution. While the distance is simply modeled as a random value, the slip distribution is modeled as a random field. The generation of samples of this latter is performed by a K-L expansion. Since the marginal probability distribution of slip has been observed to be non-Gaussian, we combine the K-L expansion with a translation process, making the methodology flexible enough to consider any marginal distribution. The methodology is proved to be more consistent than former methodologies to generate earthquake samples.
The upper panels of Figure 1 show three samples of Mw 9.0 earthquakes in the north end of Manila trench. The slip distribution is modeled with a Log-normal marginal distribution and a Von Karman covariance function. Scaling relations are used to define shape parameters of these probability properties. The distance to the trench, on the other hand, is modeled with a uniform distribution, which is bounded by the allowable earthquake depths. The lower panels in Fig. 1 show the vertical seafloor displacements caused by the earthquakes, which are adopted as initial conditions for tsunami simulations.
Figure 1: Upper panels: Horizontal projection of the slip distribution of three earthquake samples in meters. Lower panels: Seafloor vertical displacement used as initial condition in tsunami models in meters.
Uncertainty propagation to tsunami response
The SROM is implemented for the uncertainty propagation. This model is similar to the classic Monte Carlo simulations in the sense that it uses a set of samples and a deterministic tsunami model to estimate statistics of the tsunami uncertainties. The relevant characteristic of SROM is that selects the samples and their probabilities by means of an optimization problem. The optimization problems minimizes the discrepancies between the target probability properties and the statistics of the set of samples.
Figure 2: Finest computational grids of 40 meters in Hong Kong (left) and Kao Hsiung (right) with depths in meters. Numbers indicate assessed locations.
We illustrate the method by analyzing the uncertainties of tsunami assessments in Kao Hsiung, Taiwan, and Hong Kong, China. The topo-bathymetry in these regions and the assessed locations are shown in Fig.2. We use a set of 200 earthquake samples obtained with the optimization of SROM. Once the initial conditions (equal to the vertical deformation) are determined, the tsunami model COMCOT is utilized to obtain the statistics of uncertainties of tsunami height in the assessed locations.
Fig.3 shows the probability of exceedance of the maximum tsunami height minus the coseismic displacement for the four assessed locations. This metric measures the effective tsunami height relative to the shoreline elevation before the earthquake. Above each graph we indicate the coefficient of variation (CoV, standard deviation divided by the mean) which is a measure of the uncertainty. As we observe, the tsunami heights closer to the earthquake are more uncertain than those in the far field.
Figure 3: Exceedance curve of the maximum tsunami height for the four locations depicted in Figure 2.
Relevant contributions of this study
Our methodology has two relevant contributions:
- The proposed sample generation method computes random samples in such a way that they are consistent with the target probability properties, defined for the slip distribution and location. We have observed that former methodologies apply inconsistent procedures which modify probability properties. Another important feature of the methodology is that admits non-rectangular rupture areas, which are more appropriate for large earthquake events.
- We demonstrated that uncertainty estimations of SROM are more accurate than those of the classic Monte Carlo method with the same number of simulations. This aspect is valuable, since tsunami simulation models are computationally very demanding.
By the end of this research, we expect to adopt this methodology in worst case scenario and probabilistic tsunami hazard (PTHA) assessments.
This research is supported by Fulbright Commission and Becas-Chile scholarships (Ignacio Sepulveda) and research grants from NSF to Cornell University. The topographic and bathymetric data used in this study was kindly shared by the Civil Engineering and Development Department of Hong Kong, the Marine Department of Hong Kong and the National Central University of Taiwan.
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