Dynamic parameter sensitivity in numerical modelling of cyclone-induced waves: a multi-look approach using advanced meta-modelling techniques

Abstract

The knowledge and prediction of cyclones as well as wave models experienced significant improvements in this last decade, opening the perspective of a better understanding of the wave sensitivity to the cyclone characteristics (e.g. track angle of approach θ, forward speed V _f, radius of maximum wind R _m, landfall position x _o, etc.). Physically, waves are strongly linked to the time-varying evolution of the relative cyclone position. Thus, even assuming the main cyclone characteristics to be stationary, exploring the role played by each of them should necessarily be conducted in a dynamic manner. This problem is investigated using the advanced statistical tools of variance-based global sensitivity analysis (VBSA) in different ways to provide an overall view of wave height sensitivity to cyclone characteristics: (1) step-by-step: by computing the time series of sensitivity measures; (2) aggregated: by summarising the time-varying information into a single sensitivity indicator; (3). mode-based: by studying the sensitivity with respect to the occurrence of specific temporal patterns (e.g. up-down translation of the overall series). Yet, applying this multi-look dynamic sensitivity analysis faces two major difficulties: (1) VBSA requires a large number of simulations (typically > 10,000), which appears to be incompatible with the large computation time cost of numerical codes (>several hours for a single run); (2) integrating the time dimension imposes to process a large amount of information via vectors of large size (e.g. series of significant wave height H _S discretised over several hundreds of time steps). In this study, we propose a joint procedure combining kriging meta-modelling (to overcome the 1st issue) and principal component analysis techniques (to overcome the 2nd issue by summarising the time information into a limited number of components). The applicability of this strategy is tested and demonstrated on a real case (Sainte-Suzanne city, located at Reunion Island) using a set of 100 cyclone-induced H _S series, each of them being computed for different scenarios of cyclone characteristics, i.e. using only 100 long-running simulations. The key role of R _m over the whole evolution of H _S is shown by means of the aggregated option, with a more specific influence in the vicinity of Sainte-Suzanne (when the cyclone eye is located less than 200 km away from the site) as highlighted by the step-by-step option. The step-by-step option also highlights the influence of the landfall position on the H _S peak reached in strong interaction with θ and R _m. Finally, the role of V _f in the occurrence of a turning point marking a shift near landfall between regimes of low-to-high H _S values is also identified. The above results provide guidelines for future research efforts on cyclone characteristics prediction.

Appendices

Appendix 1: generalisation of Sobol’ indices for multivariate (functional) outputs

Consider the functional output H _S(s) discretised on a regular grid of track positions s viewed as vectors of large but finite dimension, and X the vector of input parameters following Gamboa et al. (2014), the generalised Sobol’ index S _agg,i for the ith parameter holds as:

$$S_{agg,i} = \frac{{Tr({\mathbf{C}}_{i} )}}{{Tr({\mathbf{C}})}}$$

(6)

where Tr is the trace; ${\mathbf{C}} = \text{cov} \left( {{\mathbf{H}}_{S} } \right)$ and ${\mathbf{C}}_{i} = \text{cov} \left( {E\left( {\left. {{\mathbf{H}}_{S} } \right|X_{i} } \right)} \right)$ are covariance matrices.

Equation 6 is the index of first order, which measures the relative contribution of the ith input parameter. Higher order indices as well as total effect can also be defined in the same way than the Sobol’ indices for scalar output. These sensitivity measures can be estimated using a pick-and-freeze as proposed by Gamboa et al. (2014).

Appendix 2: correlation function adapted to categorical inputs

This appendix is mainly based on Storlie et al. (2013): Sect. 2.1. A variety of correlation (and covariance) functions have been proposed in the literature (see e.g. Stein 1999). The commonly used model is the exponential correlation function defined as follows:

$$R({\text{u}};{\text{v}}) = \exp \left( - \sum\limits_{i = 1}^{d} {\frac{{\left| {u_{i} - v_{i} } \right|^{{\rho_{i} }} }}{{\beta_{i} }}}\right)$$

(7)

where ρ is the vector of power parameters (typically between 0 and 2) controlling the shape of the correlation function, and the vector β determines the rate at which the correlation decreases as one moves in the ith direction (with i from 1 to d). Intuitively, if u = v then the correlation is 1, whereas if the distance between both vectors tends to the infinity, then the correlation tends to 0. In this article, the difficulty is to handle continuous and categorical input variables (in our case, these correspond to the limited number of scenarios of cyclone tracks θ): to do so, we chose to a covariance function, which is adapted to this case, as described by Storlie et al. (2013). Consider x ₁,…, x _q the continuous input parameters, x _q+1,…, x _d the unordered categorical ones. Consider first the case of one categorical variable x _j, a possible correlation function is:

$$R({\text{x}}_{j} ;x_{j}^{*} ) = \exp \left( { - \frac{{Ind\left( {x_{j} \ne x_{j}^{*} } \right)}}{\beta }} \right)$$

(8)

where Ind is the indicator function so that Ind = 1 if ${\text{x}}_{j} \neq x_{j}^{*}$ and 0 otherwise; β is the corresponding length-scale parameter.

By using Eq. 7 for continuous variables, a separable correlation function (i.e. product of one-dimensional correlation) can be defined:

$$R({\text{x}}_{j} ;x_{j}^{*} ) = \exp \left( - \sum\limits_{i = 1}^{q} {\frac{{\left| {x_{i} - x_{i}^{*} } \right|^{{\rho_{i} }} }}{{\beta_{i} }} - \sum\limits_{i = q + 1}^{d} {\frac{{Ind\left( {x_{j} \ne x_{j}^{*} } \right)}}{{\beta_{i} }}} }\right)$$

(9)

As underlined by Storlie et al. (2013), the correlation as aforedescribed is isotropic, which is a reasonable assumption in many cases. More sophisticated approaches may rely on Qian et al. (2008).

Appendix 3: Principal Component Analysis

The PCA decomposition is based on the empirical eigenfunctions and vectors of the variance-covariance matrix ${\varvec{\Sigma}} = {}^{t}{\mathbf{H}}_{S}^{C} \cdot {\mathbf{H}}_{S}^{C}$ with ${\mathbf{H}}_{S}^{C}$ the matrix of n ₀ N-dimensional H _S series centred around the mean function ${\bar{\text{H}}}_{S}^{{}}$. Let us define the eigenvalues $\lambda_{1} > \lambda_{2} > \cdots > \lambda_{N}$ of ${\varvec{\Sigma}}$ ordered in increasing order and V a N×N matrix of normalised eigenvectors v of ${\varvec{\Sigma}}$, each column being associated with a given eigenvalue λ. The n ₀×N matrix ${\mathbf{H}}^{PC}$ of principal components PCs holds as follows:

$${\mathbf{H}}^{PC} = {\mathbf{H}}_{S}^{C} \cdot{\mathbf{V}}$$

(10)

The column h of ${\mathbf{H}}^{PC}$ are mutually orthogonal linear combination of the columns of ${\mathbf{H}}_{S}^{C}$ so that ${\mathbf{h}}_{k} = {\mathbf{H}}_{S}^{C} \cdot{\mathbf{v}}_{k}$ and $\left\| {{\mathbf{h}}_{k} } \right\|^{2} = \lambda_{k}$. By construction the trace of ${\mathbf{H}}^{PC}$ is the same as Σ so that the d first PCs concentrate a given level of explained variance, aka inertia, i.e. of a given amount of information.