Hydrological signatures

Several signatures describing and quantifying properties of discharge time series are introduced in view to analyze and calibrate hydrological models [Westerberg and McMillan, 2015]. These signatures permit to describe various aspects of the rainfall-runoff behavior such as: flow distribution (based for instance on flow percentiles), flow dynamics (based for instance on base-flow separation [Lyne and Hollick, 1979, Nathan and McMahon, 1990]), flow timing, etc.. A so-called continuous signature is a signature that can be computed on the whole study period. Flood event signatures on the other hand focus on the behavior of the high flows that are observed in the flood events. These flood event signatures are calculated via a proposed segmentation algorithm as depicted in Hydrograph segmentation.

Denote \(P(t)\) and \(Q(t)\) are the rainfall and runoff at time \(t\in\mathbf{U}\), where \(\mathbf{U}\) is the study period. Then \(Qb(t)\) and \(Qq(t)\) are the baseflow and quickflow computed using a classical technique for streamflow separation (please refer to [Lyne and Hollick, 1979] and [Nathan and McMahon, 1990] for more details). Considering a flood event in a period denoted \(\mathbf{E} \subset \mathbf{U}\), so all studied continuous signatures (denoted by the letter C) and flood event signatures (denoted by the letter E) are given in the table below.

List of all studied signatures

Notation	Signature	Description	Formula	Unit
Crc	Continuous runoff coefficients	Coefficient relating the amount of runoff to the amount of precipitation received	\(\frac{\int^{t\in\mathbf{U}} Q(t)dt}{\int^{t\in\mathbf{U}} P(t)dt}\)	–
Crchf		Coefficient relating the amount of high-flow to the amount of precipitation received	\(\frac{\int^{t\in\mathbf{U}} Qq(t)dt}{\int^{t\in\mathbf{U}} P(t)dt}\)	–
Crclf		Coefficient relating the amount of low-flow to the amount of precipitation received	\(\frac{\int^{t\in\mathbf{U}} Qb(t)dt}{\int^{t\in\mathbf{U}} P(t)dt}\)	–
Crch2r		Coefficient relating the amount of high-flow to the amount of runoff	\(\frac{\int^{t\in\mathbf{U}} Qq(t)dt}{\int^{t\in\mathbf{U}} Q(t)dt}\)	–
Cfp2	Flow percentiles	0.02-quantile from flow duration curve	\(\text{quant}(Q(t), 0.02)\)	mm
Cfp10		0.1-quantile from flow duration curve	\(\text{quant}(Q(t), 0.1)\)	mm
Cfp50		0.5-quantile from flow duration curve	\(\text{quant}(Q(t), 0.5)\)	mm
Cfp90		0.9-quantile from flow duration curve	\(\text{quant}(Q(t), 0.9)\)	mm
Eff	Flood flow	Amount of quickflow in flood event	\(\int^{t\in\mathbf{E}} Qq(t)dt\)	mm
Ebf	Base flow	Amount of baseflow in flood event	\(\int^{t\in\mathbf{E}} Qb(t)dt\)	mm
Erc	Flood event runoff coefficients	Coefficient relating the amount of runoff to the amount of precipitation received	\(\frac{\int^{t\in\mathbf{E}} Q(t)dt}{\int^{t\in\mathbf{E}} P(t)dt}\)	–
Erchf		Coefficient relating the amount of high-flow to the amount of precipitation received	\(\frac{\int^{t\in\mathbf{E}} Qq(t)dt}{\int^{t\in\mathbf{E}} P(t)dt}\)	–
Erclf		Coefficient relating the amount of low-flow to the amount of precipitation received	\(\frac{\int^{t\in\mathbf{E}} Qb(t)dt}{\int^{t\in\mathbf{E}} P(t)dt}\)	–
Erch2r		Coefficient relating the amount of high-flow to the amount of runoff	\(\frac{\int^{t\in\mathbf{E}} Qq(t)dt}{\int^{t\in\mathbf{E}} Q(t)dt}\)	–
Elt	Lag time	Difference time between the peak runoff and the peak rainfall	\(\arg\max_{t\in\mathbf{E}} Q(t)\) \(-\arg\max_{t\in\mathbf{E}} P(t)\)	dt
Epf	Peak flow	Peak runoff in flood event	\(\max_{t\in\mathbf{E}} Q(t)\)	mm

where \(dt\) is the timestep.

Next we are interested in investigating the simulation uncertainty, in term of signatures, depending on the input parameters of the model. Let us consider the \(m\)-parameters set of the model \(\theta=(x_{1},...,x_{m})\). Then a signature type \(i\) is represented as \(S_{i}=f_i(\theta)\). We are interested at several variance-based sensitivity indices (Sobol indices), called first-order and total-order indices. The first- (depending on \(x_{j}\)), and the total-order (depending on \(x_{\sim j}\), i.e. all parameters except \(x_{j}\)) Sobol indices of the simulated signature \(S_{i}\) are respectively defined as follows:

\[s_{i}^{1j}=\frac{\mathbb{\mathbb{V}}[\mathbb{E}[S_{i}|x_{j}]]}{\mathbb{\mathbb{V}}[S_{i}]}\]

and:

\[s_{i}^{1 \sim j}=\frac{\mathbb{\mathbb{E}}[\mathbb{V}[S_{i}|x_{\sim j}]]}{\mathbb{\mathbb{V}}[S_{i}]}=1-\frac{\mathbb{\mathbb{V}}[\mathbb{E}[S_{i}|x_{\sim j}]]}{\mathbb{\mathbb{V}}[S_{i}]}.\]

In such a way, [Azzini et al., 2021] proposed a method to estimate these indices on parameter sets of Monte-Carlo simulations via Saltelli generator [Saltelli, 2002], which is implemented in the SALib Python library [Herman and Usher, 2017, Iwanaga et al., 2022].