The study site, Lake Erken, is a mesotrophic lake located in east-central Sweden that has a surface area of 24 km2, a maximum depth of 21 m, and an average retention time of 7 years. The lake is dimictic, with seasonal stratification commonly beginning in May–June and ending in August–September. The onset of ice cover usually begins in December–February, and the loss of ice occurs in March–April (Persson and Jones, 2008). Located near the Baltic coast, Lake Erken is wind-exposed and susceptible to periodic wind-induced turbulent mixing.

Changes in algal Chl in Lake Erken have a typical seasonal pattern, with spring and summer peaks in concentration (Pettersson et al., 2003). Spring blooms are dominated by dinoflagellates and diatoms (Pettersson, 1985) and initiated by overwinter species from the last autumn (Yang et al., 2016). Cyanobacteria dominate summer peaks in Chl, given that they can optimize their vertical position with regard to nutrients and light (Paerl, 1988; Pierson et al., 1992).

Lake Erken has a long-running automated monitoring programme that provides hourly meteorological data, water temperature profiles between 0.5 and 15 m at 0.5 m intervals, and the flow from the inflow and outflow (Fig. 1). A manual sampling programme collects samples during ice-free time at 5–7 d intervals for all major nutrient concentrations (e.g. NOx, NH4, PO4, total P, and Si, etc.), dissolved oxygen (O2), and Chl concentration. The timing of the onset and loss of ice cover are also monitored yearly by the lab. More detailed information on the sampling programme is in the Supplement (see Sect. S1) and Moras et al. (2019).

In this study, we tested three workflows using a dataset split for training (years 2004–2016) and testing (years 2017–2020). In all three workflows, a 5-fold cross-validation using the training dataset was used to optimize the hyperparameters in the ML models. Workflow 1 directly predicts Chl concentration based on available environmental observations (Table 1). The training and testing datasets were limited by the frequency of lake nutrient observations, which resulted in 5–7 d gaps between data points. The time step of LSTM was set to 1; that is, the environmental factors on the target date and previous observation date, which may be 5–7 d ago, were used to train the model and make predictions.

In workflow 2 and 3, a two-step approach was applied (Table 1). Daily measurements of physical factors were used to pre-generate daily variations in lake nutrients via separate ML models, and the ML models were trained at a daily time step using the measured environmental factors and pre-generated nutrient concentrations. The time step of LSTM was then set to 7 d.

In workflow 3, three hydrodynamic features, i.e. mixing layer depth (ze), Wedderburn number (Wn), and the seasonal thermocline depth (thermD), derived from the GOTM model were regarded as daily training features in the two-step ML approach. The definitions and calculations of these features are explained in the next section, Sect. 2.5, “Feature selection and processing for ML models”, and the Supplement (Sect. S3)

Following the two-step approach and using workflow 3, we set up two tests. (1) To assess the uncertainty induced by variations in the data used to train the ML models, we shuffled the training years, randomly taking 13 years out of the 2004–2018 dataset 30 times, and tested the model predictions of Chl during 2019–2020. And, (2) to test if the workflow could be used for other water systems which may have less frequent lake nutrient monitoring data, we conducted a data sparsity test that evaluated the sensitivity of models to the lake nutrient and Chl sampling interval. For this test the lake nutrient and Chl concentration observations in the training dataset were downsampled to a 7, 14, 21, 28, and 35 d sampling interval. Then for each sampling interval using the 2004–2020 dataset, Chl was predicted for different consecutive 4-year periods when the ML models were trained by the remaining 13 years of data. Data shuffling was conducted 13 times so that every 4-year period in our dataset was tested.

The feature selection process is based on some a priori knowledge of the underlying phenomena related to algal blooms. All workflows made use of the daily automated monitoring data. In addition, the temperature difference (ΔT) between surface water (averaged over the upper 3 m) and bottom water (15 m) was also used to represent the thermal structure of the lake, and the duration of ice cover in the previous winter and the number of days from ice-off date were used.

In workflow 2 and 3 nutrients are predicted sequentially, with each pre-generated nutrient prediction included in the training data of the next nutrient prediction (Table 1). Workflow 3 added ze, computed using the GOTM-simulated vertical eddy diffusivity (Kz) profiles; thermD, estimated using Lake Analyzer (Read et al., 2011) based on GOTM-simulated temperature profiles; and Wn, a dimensionless parameter measuring the balance between wind stress and the pressure gradient resulting from the slope of the interface (see Sect. S3), as additional daily training features.

Model performance was evaluated by comparing the simulated and measured Chl concentrations and by calculating the mean absolute error (MAE), root mean square error (RMSE), and correlation coefficient (R2). To evaluate the accuracy of the model in detecting the onset of an algal bloom, we calculated a confusion matrix in workflows 2 and 3, where the observations were linearly interpolated to daily values, and predicted daily Chl concentration were smoothed with a 7 d rolling mean. Using these data, the onset of a bloom was categorized as occurring when the daily change of Chl (ΔChl) exceeded a threshold, 0.35 mg m-3 d-1. This works well in Lake Erken where Chl concentrations are frequently monitored (near weekly), and the linear interpolation can be expected to be reasonably representative of the Chl concentrations between measured samples. Considering the randomization in the ML models, we also add a 3 d window on the bloom onset prediction; that is, we considered the prediction of a bloom valid if the measured data suggested a bloom the day before or after the simulated onset. We used the true positive rate (TPR), false positive rate (FPR), and modified accuracy (kappa), which considers the possibility of the agreement occurring by chance (McHugh, 2012), to identify the potential of ML models to correctly capture the algal bloom onset (see Table S1). A model with 100 % TPR, 0 % FPR, and 100 % kappa would constitute a perfect fit.

In workflow 1, both the GBR and LSTM clearly reproduced spring and summer blooms (Fig. 2a) but underestimated the intensity of blooms (Fig. 2a, b). Neither ML model captured the extraordinarily high Chl (∼ 15–30 mg m-3) in the summer of 2019. Although the abnormal summer bloom in 2019 could contribute to the higher RMSE and MAE in the testing dataset than the mean values in the training dataset, the cross-validation on the training dataset (see Table S2) shows what appears possibly to be an overfitting issue in both models. The achieved accuracy of models is attributed to the daily availability of physical inputs and the fact that in Lake Erken water samples are collected frequently at 5–7 d intervals. Workflow 1 may be most valuable in reconstructing previous variations in algal Chl, filling the gaps between measured Chl observations and feature importance ranking (see Fig. S4). But when using this workflow, future forecasts will be limited by the absence of future nutrient data.

As in workflow 1, both ML models in workflow 2 had poor fit in the summer of 2019 and suffered from overfitting leading to higher MAE and RMSE and lower R2 in testing datasets than training datasets (see Table S2).

Overall, both the GBR and LSTM showed slightly higher MAE (4.22 mg m-3 vs. 3.87 mg m-3) and RMSE (6.27 mg m-3 vs. 6.00 mg m-3) when compared to workflow 1 (Table 2). But they also showed improved performance in terms of capturing the peak values of Chl during spring blooms (Figs. 2, S5). Both workflows outperformed the SELMAPROTBAS PB model in simulating concentrations of lake nutrients (see Fig. S6). The ML models were more accurate in predicting the low values of NOx and peak values of PO4 and total P. However, both ML models and the PB model failed in predicting the extremely high values of measured lake nutrients, such as the autumn peak of NH4 in 2017 (Fig. S6e) and the spring peak of O2 in 2018 (Fig. S6c), Thus, higher workflow 2 MAE and RMSE (Table 2) are presumably due to the inaccuracies in the pre-generated nutrient training data, but the improved daily predictions that better capture the bloom events overshadow these flaws.

Including hydrodynamic training information in workflow 3 did not significantly improve lake nutrient predictions compared to workflow 2 (see Fig. S6), and when using workflow 3 both ML models showed comparable performance in Chl predictions compared to workflow 1. However, the predictions of the spring bloom in all years improved compared to workflows 1 and 2, in terms of the magnitude and timing of the spring bloom (Fig. 2e). This was the case in 2019–2020 (Fig. 2a), which was an abnormally warm winter with only 5 d ice cover and had an unusually early spring algal bloom. Both the GBR and LSTM in workflows 2 and 3 did not capture the extremely intensive bloom (with peak values close to 30 mg m-3) in summer of 2019, and neither did the PB model.

Furthermore, adding hydrodynamic features derived from the PB model improved predictions of the onset of algal blooms (Figs. 2e and 4), with the overall TPR increasing by 15 % and 5 % and FPR increasing around 5 % and 3 % in the GBR and LSTM models, respectively. Compared with the PB model, which showed lower TPR (15 %) and FPR (6 %), ML models are more likely to predict algal bloom at the correct time. The optimal TPR was from LSTM in workflow 3, which could detect the onset of algal blooms with TPR closed to 50 %. However, the concomitant higher FPRs indicating an incorrect warning of algal bloom is also more likely to occur in the ML models, since the PB model is more like to miss the bloom entirely. The kappa values of both ML models and the PB model are close to 80 %, showing that all models simulated the entire period (blooms and the periods between blooms) to a moderate–strong level (McHugh, 2012).

The results presented so far are based on a typical strategy of training ML models for a historical period, in this case 2004–2016, and then accessing model performance in a second period between 2017–2020. The accuracies of the model predictions were to some extent related to the range and variability in the training data. To evaluate the importance of this, we randomly removed 2 years from a 2004–2018 training dataset and made 30 different predictions of Chl during 2019–2020 when the models had difficulties predicting spring and summer blooms (Fig. 5). When trained with the various shuffled combinations, both ML models were capable of reproducing the seasonal variations in algal Chl with a 4.5 % and 5.8 % coefficient of variation (CV) in MAE and a 24.0 % and 16.4 % CV in TPR of GBR and LSTM, respectively (see Table S3 in the Supplement). This provides an indication of the uncertainty that may arise as a consequence of differences in the training datasets used for in our workflows. And, it also shows that even a relatively long training period of 13 years can not totally capture the system behaviour in such a way as to lead to nearly similar bloom predictions.

Although none of the model runs captured the intensive summer bloom in 2019, the spring bloom in both years was well represented, especially by LSTM, in terms of timing and magnitude.

Despite comparable RMSE and MAE in LSTM and the GBR (Fig. 4c), both higher TPRs (with median of 60 %) and FPRs (with median of 18 %) in LSTM indicate that the LSTM model was more aggressive in making algal bloom predictions. The GBR model's apparent advantage in FPRs (with median 10 %) is largely the result of it making a lower number of bloom predictions since the low concentrations between spring and summer blooms in 2020 were not well represented (Fig. 4b).

To examine the possible use of workflow 3 when data are less frequently available, lake nutrient and Chl data were downsampled so that the effects of sampling frequency on model predictions could be evaluated. Each downsampled dataset was also rearranged into 13 different 13-year training periods and 4-year testing periods. The variability in predictions provided a measure of model performance and uncertainty. Figure 5 shows the uncertainty in model predictions as a consequence of the chosen sampling intervals.

The MAEs and RMSEs of both GBR and LSTM models tended to increase with the longer sample intervals. The median MAE was always slightly higher for the LSTM model, except when trained with the original dataset (Fig. 5a). While our initial evaluation of TPR using 2017–2020 as the testing period and 2004–2016 as the training period suggested the LSTM model was more accurate in turns of detection of algal bloom onsets (Fig. 3), Fig. 5c showed the median TPR of GBR model calculated by the shuffling year test was over 50 %, higher than that found when using the original testing and training periods. This can be explained by the fact that the 2017–2020 testing period as in Fig. 3 and shown as large points in Fig. 5 was unusually difficult for the GBR to simulate. Consequently, even though the GBR model usually performs better in the shuffled data test in Fig. 5, Fig. 3, which show the results of the 2017–2020 testing period, presented the opposite result. This illustrates the importance of the sequence of training and testing years for evaluating model performance.

For the first three sampling intervals the GBR model clearly had better TPR values than the LSTM model. The median TPRs of GBR model started to drop below 30 % once the sample interval reached 21 d. For LSTM, medium TPRs remained lower than 30 %, for all sampling intervals, but also showed a much wider range of variability (Table S4) dependent on the training and tested datasets used. In general, both models preformed best at the original and 7 d sampling interval but then showed slightly worse performance that was consistent up to a sample interval of 21 d. In terms of the errors evaluated over the entire 4-year testing period (Fig. 5a, b) the GBR model had lower errors and, therefore, better predicted the seasonal variations of Chl concentration. The time series comparison of observed and predicted Chl from this shuffling year data sparsity test can be found in the Supplement (Figs. S7–S9).

In three workflows, the ML models successfully reproduced the Chl seasonal patterns, capturing the spring and summer bloom events, with lower averaged RMSEs and MAEs than a PB model simulation that was previously calibrated for Lake Erken. And in all three workflows, LSTM model always showed slightly lower RMSE and MAE and higher R2 in predicting Chl concentrations than the GBR model and higher TPR in detecting the onset of algal bloom events. Workflow 1, which predicted Chl based on all available environmental factors including lake nutrient observations, showed that both ML models can reproduce the seasonal dynamics of algal Chl with promising accuracy (MAE = 3.55 and 3.58 mg m-3, RMSE = 5.77 and 5.64 mg m-3, and R2 = 0.13 and 0.20, for GBR and LSTM, respectively) via the direct input of available environmental observations. These ML models can be applied to reconstruct past patterns of algal Chl, fill the gaps between measured Chl observations, and interpret the mechanisms that drive phytoplankton dynamics. Workflows 2 and 3 adopted a two-step approach, first using separate ML models to estimating daily changes in lake nutrient concentration and in Workflow 3 also including PB model derived physical factors as training features of the algal ML model. These two workflows allowed for daily predictions of changes in algal Chl concentration using both observations and pre-generated lake nutrient concentrations at a consistent daily time step, and at only a minor decrease in performance compared to workflow 1, workflow 2 and 3 demonstrated a wider potential range of applications (e.g. interpolation, reconstruct historical data, and algal bloom forecast) via making daily forecasts with less-than-daily measured nutrient observations.

The one clear failure of both the ML- and PB-based model predictions was that during July–August 2019, Chl concentrations in integrated samples collected between the surface and 6–12 m exceeded 20 mg m-3 over a 5-week period. Neither the PB model nor ML models captured this unusually persistent bloom (Figs. 2, S3). At this time the phytoplankton were dominated by the Cyanobacteria Gloeotrichia and Anabaena, that form a resting akinete life stage at the end of their yearly bloom, which can initiate the following year's bloom as they are transformed to vegetative cells that migrate from the sediment to the upper water column. We hypothesize that the large summer bloom in 2019 was the result of unusually large recruitment of akinetes in this year (Karlsson-Elfgren et al., 2005, 2004). The life cycle of Cyanobacteria is not a process included in the PB model (but see Hense and Beckmann, 2006, and Jöhnk et al., 2011), so increased recruitment of akinetes could explain the underestimation of the 2019 summer bloom. Even the LSTM algorithms could not account for previous conditions so far back in time as to affect the formation and deposition of Cyanobacteria akinetes (this may require the memory of the last ice-free season). The consequent poor fit of summer bloom in 2019 partially led to the higher MAE and RMSE in the testing dataset compared to the training dataset in all three workflows, in both GBR and LSTM models.

Warm winters can initiate a chain of events, i.e. shortening the ice cover duration, extending spring circulation, affecting nutrient availability, and causing an earlier spring bloom (Adrian et al., 2006; Yang et al., 2016). According to the ice record in Lake Erken (see Fig. S1), in 2020, the lake was covered by very thin ice for only 5 d, which is the shortest duration since observations were first recorded in 1954. The spring bloom in 2020 did occur earlier than other years (see Fig. S3), and both ML models which considered the timing of lake ice show fairly good performance in predicting the timing and magnitude of this abnormally early spring bloom (Figs. 2, 5)

To reach the goal of incorporating ML models into operational forecasts either for short-term management support or longer-term evaluation and planning, two steps must occur. First the ML model must be developed, trained, and evaluated on the water body of interest due to the unique physical characteristics and water quality dynamics in different systems. Secondly, future forcing data for the model must be obtained and integrated into a workflow that makes the future predications. In regards to the second point, a lack of frequent water monitoring (Stanley et al., 2019) is a major deterrence to applying ML models to many lakes. The data sparsity test (Fig. 5) showed that, at least for Lake Erken, the ML models can still detect the seasonal algal dynamics even for sample intervals approaching 1 month (Figs. S7–S9). If this result holds for other lakes, the use of the two-step ML workflow could offer a method of forecasting seasonal variations in algal Chl, even in lakes with relatively infrequent nutrient monitoring but higher frequency meteorological and hydrological data.

The hybrid PB/ML models have the potential to provide reasonably accurate and timely short-term algal bloom forecasts, working as part of an early-warning system for the water resource management (Baracchini et al., 2020), and clearly have the ability to predict border seasonal variations in algal Chl concentration. However, since a large number of water temperature and water quality samples are required for ML training, and since our results only apply to one well-studied lake, obtaining more datasets to test and evaluate the workflows developed here is necessary. Monitoring networks (e.g. Global Lake Ecological Observatory Network, GLEON; https://gleon.org/, last access: 19 September 2022), could provide the data to allow more extensive testing and application of hybrid PB/ML models, and we are presently working in the GLEON network to test the methods developed in this paper on many other lakes.