Implementation/optimization of moving least squares response surfaces for approximation of hurricane/storm surge and wave responses

Abstract

One of the important recent advances in the field of hurricane/storm modelling has been the development of high-fidelity numerical simulation models for reliable and accurate prediction of wave and surge responses. The computational cost associated with these models has simultaneously created an incentive for researchers to investigate surrogate modelling (i.e. metamodeling) and interpolation/regression methodologies to efficiently approximate hurricane/storm responses exploiting existing databases of high-fidelity simulations. Moving least squares (MLS) response surfaces were recently proposed as such an approximation methodology, providing the ability to efficiently describe different responses of interest (such as surge and wave heights) in a large coastal region that may involve thousands of points for which the hurricane impact needs to be estimated. This paper discusses further implementation details and focuses on optimization characteristics of this surrogate modelling approach. The approximation of different response characteristics is considered, and special attention is given to predicting the storm surge for inland locations, for which the possibility of the location remaining dry needs to be additionally addressed. The optimal selection of the basis functions for the response surface and of the parameters of the MLS character of the approximation is discussed in detail, and the impact of the number of high-fidelity simulations informing the surrogate model is also investigated. Different normalizations of the response as well as choices for the objective function for the optimization problem are considered, and their impact on the accuracy of the resultant (under these choices) surrogate model is examined. Details for implementation of the methodology for efficient coastal risk assessment are reviewed, and the influence in the analysis of the model prediction error introduced through the surrogate modelling is discussed. A case study is provided, utilizing a recently developed database of high-fidelity simulations for the Hawaiian Islands.

Appendices

Appendix 1: Implementation in coastal risk assessment

Within the JPM setting, hurricane/storm coastal risk may be quantified in terms of the response ${\hat{\mathbf{z}}}({\mathbf{x}})$ provided by the surrogate model and a probability density function p(x) describing the uncertainty in the input hurricane parameters x. For real-time risk evaluation, that is, when predicting the risk due to an approaching hurricane, p(x) may be constructed through the estimates provided by the National Hurricane Center (http://www.nhc.noaa.gov); each component of x can be selected to follow an independent Gaussian distribution with mean equal to the forecast quantities and standard deviation equal to the associated statistical estimation error (Taflanidis et al. 2012). For long-term hurricane risk evaluation for a region, p(x) is selected based on statistical data, and it further incorporates information on occurrence rates for hurricanes, not just on relative plausibility of the model parameters (Resio et al. 2007).

Risk is ultimately expressed as some desired statistic of the response z, for example the probability that the wave height will exceed some specific threshold or the median surge. The exact selection used for these statistics leads to definition of the risk consequence measure h _i [.]. Ultimately, for any component z _i of the response vector, the risk, denoted H _i , is provided by the multi-dimensional probabilistic integral (Taflanidis et al. 2012)

$$ H_{i} = \int\limits_{X} {h_{i} \left[ {\hat{z}_{i} ({\mathbf{x}})} \right]\,p({\mathbf{x}}){\text{d}}{\mathbf{x}}} $$

(25)

where X corresponds to the region of possible values for x. The risk consequence measure depends on the definition for H _i , and it additionally addresses the prediction error ε _i . Through its appropriate selection different potential hurricane risk quantifications can be addressed. For example, if H _i corresponds to the expected (mean) value for some z _i , ${H}_{i} = {E}[z_{i} ]$ (where E[] denotes expectation), then (Taflanidis et al. 2012)

$$ h_{i} \left[ {\hat{z}_{i} ({\mathbf{x}})} \right] = \hat{z}_{i} ({\mathbf{x}}) $$

(26)

and the model prediction error has no impact on the risk consequence measure. If alternatively, H _i corresponds to the probability that some z _i will exceed some threshold β _i , ${H}_{i} = P[z_{i} > \beta_{i} ]$ (where P[] denotes probability), then (Taflanidis et al. 2012)

$$ h_{i} [\hat{z}_{i} ({\mathbf{x}})] = F_{{_{i} }} (\hat{z}_{i} - \beta_{i} ) $$

(27)

where F _i corresponds to the cumulative distribution function for the model prediction error for z _i . In this case, the statistics of the prediction error do have an impact on the risk quantification. This simplifies to

$$ h_i\left[ {\hat{z}_{i} ({\mathbf{x}})} \right] = {{\Upphi}}\left[ {\frac{{\hat{z}_{i} ({\mathbf{x}}) - \beta_{i}}}{{\sigma_{{\varepsilon_{i} }} }}} \right] $$

(28)

for the proposed case of Gaussian distribution for the model prediction error, where Φ[.] denotes the standard Gaussian cumulative distribution function.

Once risk for z _i has been quantified by the proper selection of the consequence measure (dependent only on $\hat{z}_{i}$), the probabilistic integral in Eq. (25) can be estimated by stochastic simulation (Robert and Casella 2004). For the simplest approach (direct Monte Carlo), and using N samples of x randomly selected from p(x), the estimate for H _i is given by

$$ \tilde{H}_{i} = \frac{1}{N}\sum\limits_{m = 1}^{N} {h_i\left[{\hat{z}_{i} ({\mathbf{x}}^{m} )} \right]} \, $$

(29)

where vector x ^m denotes the sample of the uncertain parameters used in the mth simulation. The quality of this estimate is assessed through its coefficient of variation, δ obtained by

$$ \delta \approx \frac{1}{\sqrt N }\sqrt{\frac{{\frac{1}{N}\sum\nolimits_{m = 1}^{N} {\left( {h_i\left[{\hat{z}_{i} ({\mathbf{x}}^{m} )} \right]} \right)^{ 2} }}}{{\tilde{H}_{i}^{ 2} }} - 1} \, $$

(30)

which decreases (i.e. estimation improves) proportionally to$\sqrt N$. Thus, estimation of risk may be efficiently and accurately performed using the established surrogate model, as evaluation of $\hat{z}_{i} ({\mathbf{x}})$ requires minimal computational effort [thus a large number of samples N can be used for (29)].

Appendix 2: Summary of cases considered for response surfaces and corresponding optimal parameters

The following two tables present a summary of all cases considered (Table 6) in the case study along with the optimal parameters, that is, type of basis functions, values for c and k and for weights r _k for them (Table 7). These correspond to all unique cases that were examined in Sects. 5.4, 5.5 and 5.6, by considering the full- and sub-optimization problems for the different outputs and the different normalization of response and objective function selections (more information on the selection is provided in Sect. 5.3). Each case is referenced here by its identification index, ID. Note that for parameters in Table 7 for which their basis function are not explicitly referenced as linear, the chosen polynomial degree is quadratic.

Table 6 Summary of cases considered for response surfaces

Table 7 Optimal values for model parameters of response surfaces