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Hydrological and Climate Data
Discharge time series were extracted from the observation dataset introduced by Bourgin et al. (2010a, b). This data set is composed of 4496 watersheds, their main river daily time series and their hydrologic descriptions. This data set was initially subset to low anthropogenic influenced and low groundwater support watershed, comprising 662 stations. We further reduced the data to 152 stream gauges, by keeping only continuous time series from January 1968 to December 2008 (Figure 1.1).
Continuous wavelet transforms
Scale-Time patterns have first been extracted for each watershed, and each variable, using continuous wavelet transform (cf. Figure 1.2, (a)). For any finite energy signal 𝑥, it is possible to obtain a scale-time representation by mapping it to a series of subspaces spawned by a generating function, the mother wavelet, and its scaled versions (Torrence and Compo 1998; Grinsted et al. 2004). The time series is then represented in terms of a given scale and time location. The first subspace is generated by a mother wavelet at scale 1 and its time translations. Then, other subspaces are generated by scaling the mother wavelet up, referring to daughter wavelets, and time translating it. For each scale, one subspace is constructed. Daughter wavelets are usually calculated as: 𝜓𝑎,𝑏=1√𝑎𝜓(𝑡−𝑏𝑎).
The left hand side (LHS) term is the daughter wavelet of scale 𝑎 and time translation 𝑏 at time 𝑡. The first right hand side (RHS) term is the scaling of the mother wavelet 𝜓 and the last one is the time translation.
Image Euclidean Distance Clustering
As shown in Figure 1.2(b), the similarities between wavelet spectra of each watershed, and, separately, on each variable, have been estimated. Distances between 2-dimensional data, such as maps or wavelet spectra, are commonly estimated using Euclidean distance between pairwise points (pED; i.e computing 𝑓2(𝑥1,𝑦1)−𝑓1(𝑥1,𝑦1)). However, such a procedure has no neighborhood notion, making it impossible to account for globally similar shapes (cf. definition of global and local similarity in Wang et al. 2005).
To avoid this issue, Wang et al. (2005) developed the Image Euclidean distance calculation method (hereinafter IEDC). The IEDC method modifies the pED equation in two ways (Wang et al. 2005): i) the distance between pixels values is computed not only pairwise, but for all indices; ii) a Gaussian filter, function of the spatial distance between pixels, is applied. The Gaussian filter then applies less weight to the computed distance between very close and far apart pixels, while emphasizing on medium spaced ones (Wang et al. 2005).
Fuzzy clustering
Fuzzy clustering has then been used to cluster the different watershed based on their similarities (Figure 1.2, (b)). Fuzzy clustering is a soft clustering method (Dunn 1973). While soft clustering spreads membership over all clusters but with varying probability, hard clustering attributes each station one and only one cluster membership. Soft clustering is therefore better-suited when the spatial variability, originating from different stations’ characteristics, is smooth, such as in hydroclimatic data.
For instance, precipitation and temperature patterns are unlikely to change suddenly from one station to a neighboring one, and in turn, markedly different from the next neighbor (Moron et al., 2007; Lloyd-Hughes et al., 2009; Rahiz & New, 2012). As such, several stations tend to show transitional or hybrid patterns, and can potentially be member of different clusters, limiting the robustness of hard clustering procedure (Liu and Graham 2018). In this study, we used the FANNY algorithm (Kaufman & Rousseeuw, 1990). Fuzzy clustering performance is determined by the ability of the algorithms to recognize hybrid stations (i.e. stations incorporating multiple features from different patterns observed in other coherent regions), while allowing for a clear determination of the membership of stations with unique features (Kaufman & Rousseeuw, 1990). Fanny clustering has been shown to be flexible with the modification of its exponent 𝑟, offering the possibility to adapt the clustering to the data, and to enhance performance (Liu and Graham 2018). A critical part is however the selection of the optimal number of clusters. Rather than setting the number arbitrarily, we use an estimation of the optimum number of clusters by first computing a hard clustering method: the consensus clustering (Monti et al. 2003). Thus, the number of clusters providing the best stability (i.e. the minimal changes of membership when new individuals are added) is considered optimal as recommended in Şenbabaoǧlu et al. (2014). The different clusters’ memberships are then mapped to discuss the spatial coherency of each hydroclimate variable (cf. Figure 1.2).
Cross scales interactions
For each variable and each cluster, cross-scale interactions have also been explored (Figure 1.2, (c)). Cross-scale interactions refer to phase-phase and phase-amplitude couplings between time scales of a given time series (Paluš, 2014; hereinafter PS14). Here, coupling means that the state (either phase or amplitude) of a signal 𝑦 is dependent on the state of a signal 𝑥. Such couplings are non-linear, and, therefore, if the amplitude of 𝑦 varies with the phase of 𝑥, the reverse is not necessarily true. Thus, in the classical setting, for any directional coupling (e.g. 𝑥→𝑦), there must be a lag in the relationship between 𝑥 and 𝑦 . When both oscillations are synchronized, changes in 𝑥 affect 𝑦 in the same way changes in 𝑦 affect 𝑥, and a symmetry takes place. In that case, the 𝑥→𝑦 relationship may either have a lag shorter than the time series time step or be part of a larger system (Pikovsky et al., 2001). The different couplings describe causality relationship, such as described in (Granger 1969), referring to information transfer from one part of the signal to another. Granger causality is the potential for improving the prediction of the future of 𝑦 knowing information about the past of 𝑥. Following PS14 and Jajcay et al. (2018), who compares the most used methods when studying causality, we choose the conditional mutual information (CMI) surrogates method, combined with wavelet transforms.
First, using a Morlet mother wavelet, the instantaneous phase and amplitude at time 𝑡 and scale 𝑠 of the signal are obtained. Next, the conditional mutual information, 𝐼(𝜙𝑥(𝑡);𝜙𝑦(𝑡+𝜏)−𝜙𝑦(𝑡)|𝜙𝑦(𝑡)) for the phase and 𝐼(𝜙𝑥(𝑡);𝐴𝑦(𝑡+𝜏)|𝐴𝑦(𝑡)),𝐴𝑦(𝑡−𝜂)),𝐴𝑦(𝑡−2𝜂)) for amplitude is computed. In the case of the phase-phase, the CMI measures how much the present phase of 𝑥 contains information about the future phase of 𝑦 knowing the present value of 𝑦. For the amplitude, CMI measures how much the present phase of 𝑥 contains information of the future amplitude of 𝑦 knowing the present and past values of 𝑦. The statistical significance of the CMI measure is assessed using 5000 phase-randomized surrogates, having the same Fourier spectrum, mean and standard deviation as the original time series, as in Ebisuzaki (1997).
Spatiotemporal clustering of hydrological variability
The wavelet transforms of each station have been computed and checked for similarities using IEDC fuzzy clustering to identify homogeneous regions and characterize critical scales of hydroclimate variability over France. Cross scale couplings were then investigated for each homogeneous region.
Scale-time patterns
Six regions with homogeneous scale-time patterns are identified (Figure 1.9a): North-western (#CL1-Q, black), North-eastern (#CL2-Q, blue), North-centre (#CL3-Q, red), Centre-western (#CL4-Q, green), South-eastern (#CL5-Q, yellow) and South-western (#CL6-Q, pink). Using monthly data, discharge is mainly fluctuating at annual time scales, as determined through the global wavelet spectra (Figure 1.9b). One cluster, i.e. the South-eastern regions, however, shows significant intra-seasonal variability (Figure 1.9b).
Continuous wavelet spectra show that both annual and intra-seasonal variability can be non-stationary, with temporal changes in terms of amplitude discriminating the regions (Figure 1.9c). For instance, while other watersheds show almost continuous significant annual variability, annual variability is only significant for specific periods in South-eastern watersheds (Figure 1.9c). Similarly, South-eastern region, intra-seasonal variability sparsely appears significant from the 1980’s (Figure 1.9b).
Cross-scale interactions
As shown in Figure 1.11a, North-western watersheds are characterized by a quasi-self-interaction between 6.5-7yr and 6yr time scales (Figure 1.11a). North-eastern watersheds show phase-phase causality of the 4-7yr on the 2-4yr time scales, and of the 3-5yr on 7-8yr time scales (Figure 1.11a), similarly than in precipitation (Figure 1.5a). Bi-direction interaction also occurs between 6-7.5yr and 8yr time scale in the North-eastern regions (Figure 1.11a). The specificity of the north-eastern watersheds’ wavelet spectra (cf. Figure 1.10a-b) probably result from the complex phase interactions between the different time scales. Centre-western watersheds displays phase-phase causality of the 4-5yr scale on the 2yr, 3.5yr and 5yr time scales (Figure 1.11a). As in precipitation (Figure 1.5a), south-eastern regions show phase-phase causality of the 2, 4 and 5-8yr scales on the 4-5yr time scale (Figure 1.11a). South-western watersheds show phase-phase causality of the 4yr on the 7-8yr time scales. Meanwhile, north-centre watersheds do not show any phase-phase causality (Figure 1.11a).
Hydrological Data
Discharge time series come from the observation dataset introduced by Bourgin et al. (2010a; 2010b). This data set is composed of 4496 watersheds and their main river daily time series and is accompanied with hydrologic descriptions of each watershed. This data set was subset to low anthropogenic influenced and low groundwater support watershed which accounts for 662 stations. We further reduce the data by keeping only continuous time series from January 1968 to December 2008. This results in 152 watershed stations. Figure 2.1a shows the locations for the 152 stations with their respective river.
Six regions of homogeneous scale-time variability were identified (Figure 2.1b; cf. Part I, Figure 1.9). We took the average time series of each region as the base data set for this study. The six regions are identified as follows: North-western (#CL1-Q, black), North-eastern (#CL2-Q, blue), North-centre (#CL3-Q, red), Centre-western (#CL4-Q, green), South-eastern (#CL5-Q, yellow) and South-western (#CL6-Q, pink).
Precipitation-Discharge spectral similarity with z500
In this section, we aim at identifying the regions in which atmospheric circulation show similar spectral characteristics (i.e. variance, timing at different time scales) to precipitation and discharge variability over the different coherent regions (cf. Figure 2.1b). Similar spectral characteristics in z500 could then be interpreted as potential source of precipitation or discharge variability at all time scales, with quasi-linear relationship. Meanwhile dissimilar spectral characteristics in z500 would relate to non-linear interactions: i) relationship restricted to a single time scale; or ii) non-linear interactions across time scale.
Figure 2.9 shows z500 spectral similarity maps associated with precipitation variability. Precipitation variability in the North-western, North-eastern, North-centre and Centre-western shows strong spectral similarity in z500 over the North Sea, off North America east coast and the great Lakes region (Figure 2.8). Spectral similarity between precipitation and z500 are however more pronounced over the North Sea for the North-centre and -eastern clusters (Figure 2.9). This is consistent with the low pressure centred on the North Sea, favouring wet conditions over those regions (Figures 2.5-8). Precipitation-z500 spectral similarities tend to span over a larger area, including the North Atlantic Ocean, for North-western regions (Figure 2.9). In this region, precipitation-z500 spectral similarities are consistent with the low pressure areas identified over the North Seas and off North America east coast, but not over the Great Lakes, the Atlantic Ocean, the Mediterranean Sea (Figures 2.5-9). Spectral similarity patterns are different in the southern Clusters (South-western and –eastern watersheds), which are particularly pronounced between the sub-tropical North Atlantic and Western Europe regions (Figure 2.9). South-western watersheds also show strong spectral similarities with z500 South of Greenland, while strong spectral similarities are found over Greenland for South-eastern watersheds (Figure 2.9). In South-western regions, precipitation-z500 spectral similarities are consistent with the low pressure located over the sub-tropical North Atlantic and Western Europe, as well as with the high pressure area identified South of Greenland (Figures 2.5-9). There are however no clear consistency between the z500-precipitation spectral similarity and the associated composite circulation pattern for South-eastern patterns (Figures 2.5-9), suggesting more complex relationships over this region.
Table of contents :
Avant-propos
Table of contents
Introduction
Part I – Spatiotemporal Scales of Hydroclimate Variability in France
Part II – Spatiotemporal Scales of the Large-scale Hydroclimate Variability
Part III – Dynamics of the North Atlantic atmospheric circulation
Part IV – General Discussions & Conclusion
References
Supplementary material
List of Figures
