The Odon catchment area extends over 217 km². The river travels 49 km before ending in the Orne in Caen (Figure 1). The Odon flows on the Armorican massif for the most part and the Parisian massif close to the confluence with the Orne. It drains successive hills mainly composed of schist and sandstone. Three temperature sensors are installed on this river.

The Odon starts in a Variscan syncline where bar sandstones form high points in the Normandy region. The river then flows along a narrow valley in a SE/NW orientation. In Od1, the corridor is around 40 m with a main channel around 1.5 m and quite shallow. The rivers flows between sills and pools in a sinuous way. Riverbanks are densely vegetated. The watertable is relatively low, varying from a few centimeters to a few decimeters at the end of winter. The riverbed is mainly composed of rough sediments. After the Variscan syncline, the Odon follows a meander from West to East then continues along a N/S orientation in Brioverian lands (hills and open valleys made of shale and sandstone).

In Od2, the corridor is 100 m wide and the main channel is an artificial 2.5 m wide section bordered with high trees. The river flows between pools and high waters with a mix of mud and stone riverbed. The watertable varies quickly up to several decimetres. Finally, near the city of Caen, the Odon drops in a chalk plateau at the west of the Parisian basin. The river takes a SW/NE orientation until its confluence with the Orne. Slopes are gentle except at the bottom of the valley where they can reach 30% (Supplementary Figure S1).

In Od3, the site is shaded from the direct influence of the Sun (especially in winter). The south side is densely forested with conifers which limits the sunlight all year long. However, the main river channel reaches 5 m wide so the middle of the river is exposed to direct sunlight in summer.

The Sélune is one of the three coastal rivers ending in the Mont-St-Michel bay in the North-West of France (Figure 2). The catchment area is close to 1100  km². It drains several hills on the schist-sandstone and granite Armorican massif. The Sélune travels around 76 km in total along its longitudinal profile. In the first half of the XX ^st century, two hydroelectric dams were built at a few kilometers from the mouth of the river: the Vezins dam and la Roche-Qui-Boit dam, respectively 36 and 16 m high. To reestablish a natural flow regime and provide ecological benefits, the French government decided to remove some facilities which did not allow certain fish species, such as European eels, to cross them. The Vezins dam was removed in 2019 and the destruction of la Roche-Qui-Boit dam is scheduled for 2022.

The river water temperature monitoring was performed using three temperature sensors installed along the river and respectively named: Se1, Se2 and Se3. Se1 is located upriver the Vezins dam and its lake (Figure 2). At this location, the river is an open site (Supplementary Figure S2), about 20 m wide and strongly influenced by atmospheric factors. Se2 is located about a hundred meters downriver the dam. At this place, the vegetation is denser and the riverbed is 19 m wide (before the dam’s removal). Se3 is located a few kilometers below la Roche-Qui-Boit dam. At this location, the river is 22 m wide and densely vegetated on the banks.

The Sélune and the Odon were equipped by the DREAL of Normandy with temperature sensors HOBO water Temp Pro v2 (U22-001) with an accuracy of ±0.2°C. The sensors were fixed on riparian trees’ roots, mostly in shaded areas and at depths which ensure a continuous immersion (except for Se2). Temperature is measured every 2 h, interpolated to a hourly basis in order to get a homogeneous time frame with other measurements (T _a , streamflow etc). In this study, for each river, a dataset of 10 years was analysed. A common time series was chosen between the 27 May 2011 and the 7 June 2021 (Table 1). Each dataset is composed of 89,490 time steps.

The temperature on both rivers evolves according to a seasonal cycle (Figure 3). Colder in winter (November to February) and warmer in summer (June to September). Minimal temperatures get close to 0°C for the Odon and the Sélune upriver and around 3°C for the Sélune downriver. The mean temperature of both rivers tends to increase downriver. This rise reaches 1°C in the Odon and 0.7°C in the Sélune. Measured maxima are higher for the Sélune which is a wider river and less shaded than the Odon. Stations on the Sélune are further from the source than stations on the Odon. The distance has an impact on the sensibility of the thermal regime to climatic factors (Le Lay et al., 2019).

Concerning Se2, as the temperature sensor was located at the foot of the Vezins dam, it was regularly emerged during reservoir fillings to produce electricity. Therefore, the temperature at this station could reach thermal minima way below 0°C when it was emerged because then, it measured the air temperature. After the destruction of the dam in 2019, Se2 was covered by a thick sediment layer and its thermal amplitude was reduced (Figure 3).

Due to occasional sensor failures and/or maintenance operations, river temperature time series are not perfectly continuous. In total, less than 0.2% of the time steps was filtered as outliers or filled with the methods described below.

Missing data: it can affect a very specific period (a few hours only) due to maintenance operation or longer periods (several days, weeks or months) due to sensor failure. To address this defect, several strategies were considered. If the number of missing points remained low enough (less than 20 h), linear interpolation was used to fill the gap without compromising the physical evolution of the series. If more points were missing, three options were chosen depending on the needs: 1) averaging over upriver and downriver stations (if possible and if coherent); 2) simply removing this part of data from all series to perform ICA or 3) replace with the mean value of the series (in order to avoid missed raw data presented as NaN; acronym for Not a Number).

In the following sections, all data presented were pre-processed with the methods mentioned above for the period 2011–2021.

ICA can identify hidden factors which influence the observed data without knowing mixing mechanisms. It can decompose one or several random non Gaussian signals X ∈ R ^m in a linear combination of independent signals such as: X = A ⋅ S (1) with X = [ x 1 , x 2 , … , x m ] T gathering all the signals, S an independent factor matrix S = [ s 1 , s 2 , … , s m ] T gathering all the sources with a standard distribution and A, a coefficient matrix or mixing matrix.

Decomposing observations in several independent sources can estimate A and S with two main hypothesis. On the one hand, the sources are statistically independent. On the other hand, not more than one independent component can have a Gaussian distribution.

Each independent component S is represented by a graph and a weight matrix A. The graph shows the variation of S with time whereas the matrix shows the variation of S with the stations.

ICA analysis was performed using the Fast-ICA package from the R software (Hyvärinen and Oja, 2000; Marchini et al., 2021). This algorithm is suited to estimate the mixing matrix A and the source matrix S. Data are centered so the initial observations X can be obtained by adding A ⋅ S to the mean of the X series. X i = ∑ A i j ⋅ S j + x ̄ i (2) with x ̄ the mean value of the initial X _i th serie. The main limit of this method is the number of sources than can be extracted. In our case, three temperature sensors limit the number of sources to three for each river.

In order to overcome the limit of the available datasets, we applied a successive ICA method to extract more information. This method uses the following steps.

1) From initial temperature datasets, run an ICA.

This successive ICA method allows to extract further information from the first ICA components. By removing an obvious contribution after the first ICA, one can get to more complex phenomena. Choosing which ICA component to remove relies on several criteria.

• When a physical signal is clearly identified (for example a seasonal variation correlated with the air temperature).

In all cases, the new dataset’s distribution should be non-Gaussian. Ultimately, the decomposition process ends when the remaining ICA components are reduced to white noise or when there is only one exploitable signal left. Therefore, after each ICA, Box-Pierce tests were performed to remove useless white noise signals in the process. In this study, signals were considered exploitable when their amplitude was larger than the sensor’s accuracy (±0.2°C). Signals with smaller amplitudes were either considered as residual signals or identified as white noise with Box-Pierce tests.

The example below shows how after extracting ICA components, a non-Gaussian component can be refined by subtracting noise signals. The main calculation steps are described below.

1) Run a first ICA with the measured time series. Identify a component to work on (here the seasonal component).

The Box-Pierce test indicates if a signal is characteristic of a white noise with a p-value > 0.05 (diamond boxes in Figure 4). This p-value measures the probability of auto-correlation in the signal. The higher the p-value, the higher the probability for the signal to be a white noise.

The diagram below (Figure 4) shows the process of the successive ICA method applied to a measured time series. All ICA components in this figure were multiplied by the mean of their matrix coefficients on the three stations. From the first ICA, only the component X1 is kept (black frame). The second time series (black frame middle left) are composed from X1 only. The second ICA (red and blue frames middle) displays a seasonal component (X1’) and two other signals with very low amplitudes ( < 0.04°C). In the 1) red-framed path, only the seasonal component is kept and the third datasets 1) are formed (red bottom middle). The third ICA (red frame bottom right) shows a seasonal component and two residual signals X2a and X3a with amplitudes at the limit of the sensor’s accuracy. In the 2) blue-framed path, the seasonal component is removed and the third time series 2) are composed with the residual ICA components X2’ and X3’. The third ICA (blue frame upper right) displays three components among which X2b and X3b are tested as white noises (p-values respectively 0.08 and 0.274). X1b is negative to the Box-Pierce test but its amplitude remains much lower than the sensor’s accuracy.

In the end, the last seasonal component X1a can be considered “cleaner” as its predecessors X1 and X1’. In this case, the residual signals show very low amplitudes so the differences between the different stages of the seasonal components remains subtle.

This study takes advantage of the successive ICA method to generate a hypothetical temperature time series on the Sélune. This new signal is a proposition of what the temperature time series in Se2 would have been if the Vezins dam had not been constructed. On a long time scale, the dam operation can be considered as temporary. For example, when confronting the river thermal regime with climate change, it could be useful to remove the effect of the dam. To do this, the following steps were applied.

1) Run a first ICA with the original temperature time series. Identify the dam component.

The chart below (Figure 5) shows in 1) the new time series on the three stations (solid black) compared with the original time series. We see that for stations Se1 and Se3, there is almost no difference as they are little affected by the dam component. The new signal is shifted by −2 °C in average. However, in Se2, we can see that the signal is more regular in the new time series. Irregular peaks and variations originating from the dam have disappeared. After 2019 and the dam’s removal, the two signals are closer and there is less difference between them.

In (b), the chart shows the ICA components obtained with the new temperature datasets (black curves in Figure 5). We can recognise the seasonal component (XS1 in black) as well as the daily component (XS2 in orange). The third one (XS3 in green) is negative to the Box-Pierce noise test but shows an amplitude under the accuracy of the measuring sensor. Therefore, it is likely coming from the sensor itself and considered as a residual signal.

The interest of the successive method was tested in water temperature prediction using a regression linear model. In this test, two runs were made. First, the model was applied on original T _W time series with a separation of 70–30% between the train and test data. Then, the model was applied on corrected data for the train part, which means without the dam component. The test data was kept identical to the first test. Among the three stations, the difference between the two runs was most visible on station Se2 where the dam component is the most important. Removing the dam component from the time series improved the fit between predicted and measured test data by 34.4% and contributed to reduce the RMSE by 8.9% (Table 2).

Although this test used a simple regression model, it shows the interest of removing a particular signal which affects a temporary period. In this case, the dam itself is not supposed to affect the river thermal regime several years after its removal. By using the successive ICA method, it is easier to choose the influence of environmental factors for future thermal regimes.

From the temperature time series, at least three components were extracted for each river to take advantage of the number of stations in the range 2011–2021. Corresponding ICA components will herein be referred with their code names in the tables below (Table 3). For each ICA component, the mixing coefficients make the link with each station. Signs indicate the relative relationships between the components and the station. For example, XOdA shows the same kind of relationship for all stations as all the coefficients have the same sign. Then, the absolute value of the coefficients shows the impact of each component on the different stations. A high absolute value indicates a strong influence of this component on the thermal regime of the station. On the contrary, a value close to 0 means that the component has little effect on the thermal regime at this station. For example, XOdC has a strong influence in Od2 whereas it has a negligible effect in Od3.

From the matrix coefficients, we can see that the first components (XOdA for the Odon and XSeA for the Sélune) clearly have a dominant effect on the thermal regimes and affect all stations quite equally. On the contrary, looking at the coefficients values, XOdC affects mainly Od2 and XOdB seems correlated to the upriver stations Od1 and Od2. On the Sélune, XSeB affects the upriver station Se1 more and XSeC is clearly correlated to Se2.

At this stage, we can separate the first components which are regular and affect all stations with the same magnitude (including an increasing effect downriver) on the one hand, and other components which affect some stations in particular on the other hand. Complementary to the mixing coefficients, graphic representations of the ICA components (Figure 6) show their impacts on the stations. The graphs represent the time variation of each ICA component for each station weighted with the corresponding matrix coefficient.5b

On the Odon, the maximum amplitude is obtained with the first component XOdA (±10 °C). XOdB and XOdC represent secondary components of the signal with smaller amplitudes than XOdA (±2°C). Finally, Od3 displays a behaviour quite different from Od1 and Od2 based on the three ICA components. In Od3, XOdA’s matrix coefficient is the strongest of all stations whereas XOdC is almost nil and XOdB takes an opposite sign compared to Od1 and Od2.

On the Sélune however, the three components take high amplitudes depending on the stations. The maximum amplitude is observed for XSeA for all stations. Then, XSeB shows a signal half the amplitude of XSeA for station Se1 and XSeC shows an amplitude almost as high as XSeA in the period 2011–2013 for station Se3.

In the following subsections, physical factors are attributed to the different components. As the number of ICA components is limited to 3, all components can contain several factors’ signatures. Therefore, the physical factor attributed to the ICA component will concern the most visible ones. Physical factors can be separated between climatic factors at the catchment or regional scale and environmental factor at the measuring site scale.

In addition to climatic daily variations, ICA can also highlight a more general influence of environmental factors. This section focuses on the temperature ICA components containing environmental factors’ signatures such as the X _B and X _C components for each river. The most visible factors are attributed to ICA components. Complementary analysis with PCA will refine the determination of the factors.

When analysing time series, most commonly methods such as Fourier or Morlet wavelets are frequency dependent. The initial signal is separated into several contributions based on the frequency of the phenomenon. This first approach can separate different physical phenomena which operate on very different time scale (seasonal, daily etc…). ICA, however, does not discriminate frequency scaled phenomena. Therefore, it can separate signals with similar frequency signatures. Therefore, these methods are complementary: frequency based methods are sensitive to the sampling frequency whereas ICA is sensitive to the homogeneity of the time series. The main limit of ICA lies in the separation of signals. On the two rivers, ICA could extract only three mathematically independent components on the one hand and residual signals and white noise on the other hand. Physical factors influencing the temperature are never completely independent and therefore, ICA alone can not fully separate all the contributions, unless the number of sensors is important.

However, as a first analysis step, ICA provides useful information and draws a first study frame for the physical phenomenon to look into. Also, it can be used in a backwards methodology where a factor is known to have an impact on the thermal regime but the way it impacts remains to be determined. ICA can highlight a specific signal which can be identified as the specific factor. In this study, the Vezins dam was known to influence the thermal regime of the Sélune and ICA could extract a specific signal to remove its temporary effect.