Quantification of predictive uncertainty with a metamodel: toward more efficient hydrologic simulations

Abstract

Hydrologic flood prediction has been a quite complex and difficult task because of various sources of inherent uncertainty. Accurately quantifying these uncertainties plays a significant role in providing flood warnings and mitigating risk, but it is time-consuming. To offset the cost of quantifying the uncertainty, we adopted a highly efficient metamodel based on polynomial chaos expansion (PCE) theory and applied it to a lumped, deterministic rainfall–runoff model (Nedbør–Afstrømnings model, NAM) combined with generalized likelihood uncertainty estimation (GLUE). The central conclusions are: (1) the subjective aspects of GLUE (e.g., the cutoff threshold values of likelihood function) are investigated for 8 flood events that occurred in the Thu bon river watershed in Vietnam, resulting that the values of 0.82 for Nash–Sutcliffe efficiency, 4.05% for peak error, and 4.35% for volume error are determined as the acceptance thresholds. Moreover, the number of ensemble behavioral sets required to maintain the sufficient range of uncertainty but to avoid any unnecessary computation was set to 500. (2) The number of experiment designs (N) and degree of polynomial (p) are key factors in estimating PCE coefficients, and values of N = 50 and p = 4 are preferred. (3) The results computed using a PCE model consisting of polynomial bases are as good as those given by the NAM, while the total times required for making an ensemble in the PCE model are approximately seventeen times faster. (4) Two parameters (“CQOF” and “CK12”) turned out to be most dominant based on a visual inspection of the posterior distribution and the mathematical computations of the Sobol’ and Morris sensitivity analysis. Identification of the posterior parameter distributions from the calibration process helps to find the behavioral sets even faster. The unified framework that presents the most efficient ways of predicting flow regime and quantifying the uncertainty without deteriorating accuracy will ultimately be helpful for providing warnings and mitigating flood risk in a timely manner.

Appendix: PCE definition and PCE coefficients determination

The following is a brief summary of PCE theory (Blatman and Sudret 2010; Sudret 2008). Consider a deterministic rainfall–runoff model denoted by ${\boldsymbol{\mathcal{M}}}$. The input space of the model is represented by random vectors of input parameters $\varvec{X}$, and the model response, $\varvec{Y}$ is:

$$ \varvec{Y} = {\boldsymbol{\mathcal{M}}}\left( \varvec{X} \right) $$

(4)

The goal of this theory is to approximate the computational model, ${\boldsymbol{\mathcal{M}}}$ with the PCE model, ${\boldsymbol{\mathcal{M}}}^{PCE} \left( \varvec{X} \right)$. The latter is computed with a finite sum of orthonormal polynomials for the input parameters.

$$ \varvec{Y} \approx {\boldsymbol{\mathcal{M}}}^{PCE} \left( \varvec{X} \right) = \mathop \sum \limits_{ \in A} y\;\Psi \;\left( \varvec{X} \right) $$

(5)

where $y_{{}}$$\left\{ {\varvec{\alpha}\in A} \right\}$ are PCE model coefficients to be determined for all multi-indices, $\varvec{\alpha}= \left\{ {\alpha_{1} , \ldots ,\alpha_{M} } \right\}$ belonging to a set of candidate polynomials $A$, $M$ is the number of deterministic model parameters $\varvec{X} = \left\{ {X_{i} , i = 1, \ldots ,M} \right\}$. $\Psi \;\left( \varvec{X} \right)$ are the corresponding multivariate orthonormal polynomials given as the input parameters. The multi-dimensional polynomials are constructed as the product of univariate orthonormal polynomials:

$$ \Psi \;\left( \varvec{X} \right) = \mathop \prod \limits_{i = 1}^{M}\Psi _{{_{i} }}^{\left( i \right)} \left( {X_{i} } \right) $$

(6)

Here, $\Psi _{{_{i} }}^{\left( i \right)}$ is the univariate orthonormal polynomials of the i-th parameter of degree $\alpha_{i}$. The set $A$ can be determined by the number of $M$ input parameters and the degree p of PCE model as:

$$ A = A^{M,p} = \left\{ {\varvec{\alpha}\in {\mathbb{N}}^{M} : \left|\varvec{\alpha}\right| \le p} \right\},\quad card A^{M,p} = \left( {\begin{array}{*{20}c} {M + p} \\ p \\ \end{array} } \right) $$

(7)

Depending on the probabilistic characteristics of $\varvec{ X}$, different polynomial bases can be used for $\Psi \left( \varvec{X} \right)$. Polynomial basis functions based on the Weiner-Askey scheme (Xiu and Karniadakis 2002) are illustrated in Table 12 for the commonly-used distributions of random variables.

Table 12 Polynomial basis functions for probability distributions of uncertain parameters

In this study, the regression methods is employed to establish PCE coefficients (Berveiller et al. 2006; Sudret 2008):

$$ y = {\text{argmin}}_{{y \in {\mathbb{R}}^{\left| A \right|} }} {\mathbb{E}}\left[ {\left( {\varvec{Y} - \mathop \sum \limits_{{\varvec{\alpha}\in A}} y_{\varvec{\alpha}}\Psi _{\varvec{\alpha}} \left( \varvec{X} \right)} \right)^{2} } \right] $$

(8)

Given a collection $\varvec{x} = \left\{ {x^{\left( 1 \right)} , \ldots ,x^{\left( N \right)} } \right\}$ consisting of the number of N sets of the parameters $\varvec{X}$ (the set $\varvec{x}$ is called the experimental design), ${\mathbf{\mathcal{Y}}} = \left\{ {{\boldsymbol{\mathcal{M}}}\left( {x^{\left( 1 \right)} } \right), \ldots ,{\boldsymbol{\mathcal{M}}}\left( {x^{\left( N \right)} } \right)} \right\}$ is the corresponding model evaluation $\left\{ {{\mathcal{Y}}^{\left( j \right)} = {\boldsymbol{\mathcal{M}}}\left( {x^{\left( j \right)} } \right), j = 1, \ldots ,N} \right\}$. The estimates of the PCE coefficients are thus given by:

$$ \hat{y} = {\text{argmin}}_{{y \in {\mathbb{R}}^{\left| A \right|} }} \frac{1}{N}\mathop \sum \limits_{j = 1}^{N} \left( {{\mathcal{Y}}^{\left( j \right)} - \mathop \sum \limits_{{\varvec{\alpha}\in A}} y_{\varvec{\alpha}}\Psi _{\varvec{\alpha}} \left( {x^{\left( j \right)} } \right)} \right)^{2} $$

(9)

which is equivalent to:

$$ \hat{y} = \left( {{\mathbf{F}}^{{\mathbf{T}}} {\mathbf{F}}} \right)^{ - 1} {\mathbf{F}}^{\text{T}} {\mathcal{Y}} $$

(10)

where ${\mathbf{F}}$ is so-called the information matrix of size $N \times \left| A \right|$ whose generic term reads:

$$ {\mathbf{F}}_{jl} =\Psi _{l} \left( {x^{\left( j \right)} } \right)\quad j = 1, \ldots ,N;l = 0, \ldots , card \,A - 1 $$

(11)

Once a PCE model is derived, the prediction using the model is extremely simple and straightforward. We input the values of parameters $\varvec{X}$ to Eq. (5) and then obtain the values of model response $\varvec{Y}$.