Lake Geneva (locally known as Le Léman) is a perialpine lake located between Switzerland and France (46.458∘ N, 6.528∘ E) at an altitude of 372 m (Fig. 1). It is the largest freshwater lake in western Europe (surface area and volume of 580 and 89 km3, respectively), with a retention time of 11.4 years. Due to relatively mild winter temperatures and its large depth of 309 m, complete deep convective mixing occurs only every 5 to 10 winters (Schwefel et al., 2016). The lake is composed of two parts: the large eastern basin (Grand Lac), with a maximum depth of 309 m, mean depth of 160 m, and mean width of 10 km in which gyres are frequently observed, and the Petit Lac, the narrow and shallow western basin (maximum depth of 70 m, mean width of 4.5 km). The centers of the two basins are some 30 km apart, which defines the cutoff distance of the EnKF (more details in Sect. 3). The surrounding topography is mountainous, mainly in the southeast, hence affecting the wind circulation above the basin. Lake Geneva is mesotrophic, with strong variation in turbidity and light penetration depth over the year (ranging from 3.6 to 14 m).

The primary purpose of a 3D hydrodynamic model is to solve the time-dependent, nonlinear differential equations of the hydrostatic free-surface flows in a computational grid. Various modeling suites have been developed to solve those equations accounting for momentum (Reynolds-averaged Navier–Stokes – RANS) and fluid mass (continuity), as well as heat and mass transfer. The open-source Delft3D-FLOW software is used in this study.

OpenDA is an open interface standard. It provides access to a set of open-source tools, allowing for the integration of arbitrary numerical models and observations through calibration and data assimilation algorithms. Its goal is to minimize algorithmic development by promoting the exchange of software solutions among researchers and users (Deltares, 2019; http://www.openda.org, last access: 9 March 2020).

An OpenDA interface has recently been developed for the z-layer Delft3D-FLOW using the black-box wrapper (file-based) approach (Baracchini et al., 2019a). This interface has been further expanded for DA in this study. Additions include extended modifications of the Delft3D-FLOW model definition file, model forcing files (on an equidistant grid only) for OpenDA's noise models, and support for localization, which allows users to limit the area of influence of an observation. The entire source code is available on GitHub (https://github.com/OpenDA-Association/OpenDA, last access: 9 March 2020).

The EnKF is an attractive alternative for nonlinear dynamics and systems with high dimensionality. Reichle et al. (2002a) found that the EnKF is more robust than the EKF while being more flexible to obtain system covariances, a core element of the DA problem (Bertino et al., 2007). Indeed, whereas the careful estimation of covariances often required a lot of effort (De Lannoy et al., 2007b), in the EnKF they are derived dynamically from a small ensemble of model trajectories (and therefore take into account the physics of the model), which grasps the essential parts of the error structure (Reichle et al., 2002b). The EnKF only considers a sample of the state variable to represent the processes modeled. The covariance matrix becomes a sampled covariance matrix, and predictive probability density functions of the state vectors are approximated by Monte Carlo simulations (Nakamura et al., 2006). It nonlinearly propagates a finite ensemble of model trajectories instead of using a linearized equation for the error covariance, so no computation of derivatives is required. The EnKF still considers a linear correction procedure and assumes Gaussian distributions of the random variables. When this is not the case, the filter still produces a variance-minimizing solution, though it is not the optimal estimate (Bertino et al., 2007).

We develop below the fundamentals behind the algorithm. We first define the true model state (corresponding to the actual physical state of the lake) vector x of the system at time t as xt (in our case temperature for the entire 3D model grid), M the nonlinear lake system operator, η the process noise, and u the forcing vector (here meteorological forcing) for a time t. The state propagation equation reads 1xt=Mtxt-1,ut-1+ηt-1. In this study, the noise is added in the forcing term u and subsequently dropped in the notation of Eq. (2). The state space vector, noted x^, is an approximation (done by the hydrodynamic model Delft3D-FLOW) of the true state x. The forecast state (the input information for DA at time t) is defined by x^f and the analysis state obtained after DA as x^a. The model propagation equation now reads 2x^tf=Mtx^t-1a,ut-1. As we do not measure the true state of the system (x), the observation (y) equation is defined by the following, with H an operator relating the system state to the observation and ε is some measurement noise: 3yt=Htxt+εt, with the observation prediction given by 4y^t=Ht(x^tf). Note that in our case, we directly observe what we compute (i.e., surface temperature and profiles at computed grid points), and therefore in this study H is an identity matrix. The resulting data assimilation estimate of the state vector (x^a), which will be used in the next cycle as restart condition, is given by 5x^ta=x^tf+Kt(yt-y^t). That last equation (Eq. 5) is a central concept of DA; it introduces the weighting factor K, also referred to as Kalman gain. The Kalman gain can be viewed as a balance of the model and observation uncertainties, together with the error correlation of all the elements of the state vector. It aims to minimize the error covariance of the state estimate during the analysis time Eq. (5). It is defined as 6Kt=PtfHtTHtPtfHtT+Rt-1, with Rt the measurement error covariance matrix (in this study we assume no cross-correlation between observation errors, and hence Rt is diagonal and determined from the uncertainty of the measurements; Sect. 2.4) and Pf the a priori state error covariance matrix. Error covariance is a key component of DA. The EnKF is able to compute a time-varying covariance error based on the dynamics of the system. This is a critical property when considering variables with short decorrelation spatiotemporal scales (Kuragano and Kamachi, 2000). In addition to the probability density function of the state (when in the presence of process noise), covariance estimation is achieved considering ensemble members. For an ensemble of forecasts (j=1,…,N), each subject to a disturbance (e.g., in model processes, forcing, or initial conditions), P is obtained from 7Ptf=1N-1∑j=1N(xt,jf-x¯tf)(xt,jf-x¯tf)T. From Eq. (7) we can conclude that the error-spreading pattern across the domain is indeed derived from the ensemble members in a systematic way. This is not the case for some variational methods such as 3D-VAR, whereby the statistics are considered isotropic with little variation over time. In the EnKF each ensemble member is then updated individually (based on Eq. 5). The state average over the ensemble provides the a posteriori state estimate. Additionally, in contrast to the extended (or traditional) Kalman filter, there is no need to propagate the state covariance nor to estimate the initial state covariance and model error covariance matrices. The EnKF only uses the first and second moments to construct the probability density functions; it cannot ensure higher-order statistics by opposition to the particle filter (Nakamura et al., 2006).

The EnKF is widely used for large systems with uncertain initial states, and variants are still being developed to leverage its limitations (Hoel et al., 2016). Several authors (Bertino et al., 2007; Evensen, 1994; Verlaan and Heemink, 2001) found better performance for highly nonlinear systems in comparison to the EKF. This approach can accommodate large datasets or missing observations, and it can incorporate correlated nonlinear and error measurement models. Moreover, the ensembles are easy to implement in parallel fashion. Models with high dimensionality are well suited for this type of assimilation, which requires a relatively low number of ensemble members to produce stable and accurate results (detailed in the Results section and Discussion section). We used this algorithm for the results presented in this study.

The aim of this section is to detail the various properties of the EnKF and DA setup, which is specific to this study.

The benefit of DA is shown with four examples in Fig. 3, comparing LSWT from AVHRR measurements with LSWT from the control run model and DA experiment. We first highlight (top panels) the fact that DA assimilation can perform correctly even in the case of missing observations over the lake surface. The state covariance matrix could update the model in areas where no data were available. Model accuracy is thereby improved at the basin scale rather than at observation locations. This is particularly relevant as large lakes are often partly cloudy. The second example demonstrates the potential of DA to correct the state variable – a cold bias in the present case – while maintaining the coherent structure of the complex spatial thermal gradient (Fig. 3, second row). The third example shows gyre-like flow structures. Such rotating structures are difficult to observe from AVHRR LSWT data (third row of Fig. 3), partly due to their limited spatial resolution and weak signature at the surface. However, a gyre created by a NNE wind on 12 August is better visible in the model results (clockwise in the western part of the main basin and counterclockwise in the center). In that case, the DA updated the LSWT while keeping the physical structure and flow spatial coherence of the control run. Finally, the lowest panels in Fig. 3 show how DA improves observations and the future quantification of transient upwelling. While the upwelling in the Petit Lac was partially already caught by the control run, the DA allowed for a much better adjustment of its intensity and extent. Another similar case is presented in Appendix B.

The benefit from DA is also evident when looking at the temporal evolution of LSWT (Figs. 4 and 5). In Figs. 4 and 5, the AVHRR LSWT is again compared at two locations with the simulations with and without DA. The observed strong summer temporal variability with biweekly temperature variations exceeding 5 ∘C is not well resolved in the control run (Fig. 4); however, it is much better reproduced by applying DA. The warming phase also benefits significantly from the assimilation. The control run surface temperature started to increase in the second part of March, while the warming occurred in early March in the observations and DA. Both models are in good agreement during the cooling phase after August. While few observations were available during this late period, not much improvement is obtained for the baseline, which was already accurate. Overall, every point of the DA run is close to or at least within the ∼1 ∘C uncertainty of the AVHRR observations (see Sect. 2.4 for more information on uncertainty).

Similar conclusions arise from Fig. 5, which provides a close-up of time series of the summer period in the western basin (Petit Lac). Ensemble spread is smaller during the period of strongest stratification from late July to late August. Overall, the model uncertainty arising from perturbed wind fields reaches 2 ∘C in the summer, when it is the highest and 1 ∘C on average. Major upwellings in June and July are caught by both model runs, although the intensity is too weak in the control run. Again, data–model discrepancies and temperature variability are largest from late May to early August. Starting in August, the models with and without DA exhibit similar dynamics, both close to observations.

We further compared the upwelling of 15 September with river temperature data with a model surface grid point located 3 km away (Fig. 6). The upwelling has indeed been observed in the lake outflow, dropping from 21 to 12 ∘C in 6 d. The figure shows that the control run underestimated the upwelling by 5 ∘C, while the DA run underestimated it by ∼2.5 ∘C. The AVHRR observation was 1.5 ∘C warmer than the river temperature. Figure 6 also confirms that the model does not suffer from spurious behavior after an assimilation. Model shocks are not observed and numerical equilibrium is reached in a sub-daily period.

We investigated how the vertical structure and subsurface dynamics are affected by the DA. Figure 7 provides a comparison of the DA performance over depth with in situ data instead of AVHRR measurements. Overall, for both stations significant improvements are obtained over the entire water column and throughout the year. Major improvements are observed at the thermocline depth, correctly represented in the DA experiment. Its strong vertical gradient significantly benefited from the assimilation of temperature profiles. The warm bias between 5 and 25 m of depth, resulting in an overestimation of the mixed layer depth in the control run, is effectively eliminated.

Finally, we evaluated the EnKF ensemble size needed by a convergence analysis. A period of 1.5 months, from June to mid-July with a spin-up time of 2 weeks (without DA), is selected for assessment. This period of weak spring thermal stratification has been selected, as it is the time of the year with the most complex and broadest range of dynamics (Fig. 4).

The results indicate that for an increasing number of ensembles (N) the analysis error decreases. Figure 8 provides RMSEs and MAEs for different ensemble sizes, also differentiating between assimilated data sources. We conclude that major gains are achieved with 10 ensemble members. For in situ data only, 20 ensembles seem to be the sweet spot. Due to the much larger number of AVHRR observations (i.e., one image provides thousands of observations since it covers the entire spatial extent of the computational grid), the red (AVHRR) and black (all measurements) lines are confounded. Finally, assessment of the ensemble spread showed that few gains in second-order moments were found with larger ensemble sizes. Indeed, in the scope of this study, the additional benefits for N>20 are limited. At this stage, the 1 ∘C uncertainty of the AVHRR LSWT data might become a limiting factor hindering further improvements.

With a vision towards DA for operational lake forecasting systems and the computational constraints associated with real-time hydrodynamics, we conclude that 20 members provide a satisfactory compromise for the system considered in this study.

Figure 3 shows that various physical processes, such as upwellings and gyres, are better resolved with the use of EnKF. Upwellings typically occur more prominently at the beginning or end of the season, when stratification is weaker. The better identification of such processes is of prime importance for various water quality aspects (e.g., heat extraction, wastewater discharge, water intakes; Gaudard et al., 2019). Yet the magnitude of such events has rarely been quantified due to difficulties with their large-scale identification. Through the combination of remote sensing observations and 3D hydrodynamic modeling, we open new possibilities for monitoring and predicting such phenomena. In this study we found that upwellings are better reproduced in both intensity and spatial extent. Comparing temperature measurements with a surface model grid point 3 km away from the outflow showed good agreements after DA. An underestimation of the upwelling of 2.5 ∘C after DA is observed (compared to 5 ∘C with the control run). Most of this remaining difference can be attributed to the satellite underestimating the event as well (by 1.5 ∘C, with an uncertainty of 1 ∘C) and the remoteness and depth (surface) of the pixel compared. For Lake Geneva, this is of particular interest when an upwelling occurs in the western basin (Petit Lac), dropping the outflow temperature for millions of downstream residents. In terms of gyres, those structures are repeatedly observed in Lake Geneva (Bouffard et al., 2018; Kiefer et al., 2015). Because of the Coriolis force, subsequent strong uplifts to down-lifts of the thermocline occur, which structure the lateral dispersion of primary productivity (Soomets et al., 2019).

Among the various ensemble sizes assessed for this study (Fig. 8), we found that relatively small ensemble sizes (∼20) are enough to derive suitable time-varying covariances and error-spreading patterns. This is particularly important in the presence of variables with short decorrelation time and spatial scales. Studies indicated that relatively small ensembles fail to accurately estimate the small correlation patterns of remote observations (Houtekamer and Mitchell, 2001). The localization scheme implemented (Sect. 3.1), defining a cutoff radius around each observation, allows users to circumvent this limitation. Houtekamer and Mitchell (2001) found that for an increasing ensemble size, the optimal cutoff value increases as well. Larger ensemble sizes not only restrain the underestimation of ensemble spread and accuracy, but also allow for the use of more remote observations. For DA experiments with limited data, larger ensemble sizes may be a requirement to maximize observational coverage.

A main limitation of the EnKF is the Gaussian assumption, which in the case of large data–model mismatches could have led to artifacts and unrealistic a posteriori state values. This has not been observed in this analysis with the provided noise definition and observational stochastic setup. Furthermore, while we did not systematically study the physics after each analysis step, we think the method can still be used for the study of physical processes, provided the user assesses the intensity of those physical discontinuities. Out of the 152 assimilations, only 8 created some numerical instabilities in the model, though they were small enough to prevent solver failure. The existence of an upper limit to the amount of information assimilated was not investigated here, as the aim of this work is to provide an operational system with data assimilation in lakes.

Other difficulties arise in the presence of bias, whereby Kalman filtering performs suboptimal corrections (Dee and Da Silva, 1998), as observations and the model are assumed unbiased. Solutions for dealing with biases in EnKF may become necessary (De Lannoy et al., 2007a). In the present approach, however, occasionally occurring model biases have been effectively handled by the update. The DA model did not drift back to its biased or control run state. We believe that this is a result of the adequate initial parameterization of the model (Baracchini et al., 2019a). This further highlights the crucial importance of accurate model calibration and formulation before applying DA experiments. It is worth noting that the EnKF is able to also provide updates to parameters and forcing conditions, which in some cases may provide more persistent improvements (for example, when time-varying parameters are needed).

This DA experiment is time-consuming from a computational aspect. For example, it took nearly 1 month to compute the present setup on a dual Intel Xeon E5-2697v4 processor with 256 GB of memory, generating close to a terabyte of data. While the analysis time for in situ data has been reasonable (∼1 h), the immense number of observations generated by an AVHRR image (entire coverage of the surface layer of the computational grid) brought the analysis time up to 3 h for a single image. This is largely due to the current lack of multi-core support for the analysis step. A multi-core local analysis has been implemented in the scope of this study for multi-variable (e.g., temperature with water levels and/or flow velocities) assimilations, but gains can further be achieved from a local analysis based on the observation localization scheme.