Before you can fill in the missing values, you must first identify their position in the processed series. That can be achieved using the NumPy library in Python [57]. Then, a moving window-based function is used to fill in these indices one by one. Starting with the index of the first missing value and continuing to the end, this loop function takes data windows of a specific size called Look-Back (LB), before each index as input for the model to predict the next missing values. The predicted value is added to the original data and can be used in future data entry windows, especially when there are many consecutive missing data points. Then the window advances one step so that it can be used as a basis for predicting the next missing value clue, if any.

One of the difficulties lies in the choice of the window size (LB) which will allow all the complexity of the series to be captured. This size depends not only on the phenomenon studied, but also on the size and distribution of the missing values in the series. The choice of a small LB increases the calculation time and decreases the precision of the imputed values as the size of the missing consecutive data increases, at the same time before choosing a large LB it is necessary to be reassured that it exists enough samples of this size without missing values to sufficiently train the model [20]. The search for the solution to the problem of precision even on data missing for several months, as well as optimization of the calculation time led Imad and [20] three scenarios for structuring model training data:

1)Scenario 1:which is described as basic or traditional. With this approach, hourly data with a Look-Back defined based on the size of the missing consecutive data in each station is sent into the model. We vary LB is varied to optimize the precision. The Look-Back can vary from one hour to several months or even years. The result and the consequences of this strategy are discussed in Section 3. But one can already say that this strategy poses problems of error accumulation with consecutive missing data of long duration and is also computationally intensive;

2)Scenario 2: This approach attempts to solve the problems of error accumulation and calculation time by introducing the concept of mini-Look-Back (mLB) in addition to the LB. Here, each series of size LB is restructured into sub-series (Si) of size mLB. The S_i are seen as new variables. Figure S1 in Appendix shows the division of the series following this approach. This strategy optimizes scenario 1 but introduces large errors after a period of sudden measurement variation;

3)Scenario 3: This methodology is based on daily data, i.e., a new series made up of values measured at the same time of day over the entire period of the series. For a year we will therefore have, 365 values which will be used to train the model and the trained model will be used to fill in the missing values in all hours of the day of the entire series (see Figure 3).

It should be noted that all of these approaches can be used with good precision depending on the size of gaps in the time series and this is what is will done in this study. Depending on the distribution of gaps, one will combine several approaches on the same station.

The data used in this work come from various sources and at different acquisition steps (see Table 1). Prior to model development, all tide gauge records were harmonized to ensure temporal and vertical consistency across datasets. The original measurements were acquired from different instruments and periods, with sampling intervals ranging from 10 minutes to 1 hour depending on the station and acquisition system. All-time series were resampled to a uniform hourly resolution using linear interpolation, which provides a balance between temporal resolution and data completeness.

To ensure vertical consistency, all records were referenced to a common tidal datum. When necessary, datum adjustments were applied based on available metadata and benchmark information. Particular attention was given to the Wouri estuary dataset (where the original data was sampled at 10-minute intervals), which consists of measurements from multiple instruments deployed over different periods. For this station, consistency checks were performed to identify potential discontinuities between successive records.

Offset detection was conducted by analyzing overlapping periods, when available, or by comparing statistical properties (mean sea level and variance) before and after instrument changes. Detected offsets were corrected by applying constant adjustments to align the time series segments. Visual inspection and statistical validation were performed to confirm the continuity and physical consistency of the corrected records. A first treatment consisted of standardizing the sampling intervals at one hour for all stations.

It has been noted above that there are many models that can be used to reconstruct missing data. The choice of the appropriate model, as well as its effectiveness, depends on the nature, availability and complexity of the data, as well as the size of the missing data. In the present case, the gaps range from one hour to several months in addition to being randomly distributed in the series (see Figure 3). It therefore appears that traditional/classical models such as interpolation cannot provide reliable results, especially when the missing period is long. However, they can be used to fill in missing data over short periods of a few hours to prepare enough chronicles to perform LSTM network training in a second step to fill in missing data over longer time periods which are more difficult. In order to select the appropriate method to fill in missing data points while retaining data trends and shape, we set a threshold for the size of missing data to choose between interpolation methods and LSTM. This threshold is set at six hours, based on the natural periodic components of tidal signals, which include both long and short periods [20]. When the duration of missing data is less than half of the tidal period, the trends and shape of the data do not show a significant change. It was therefore possible to use linear interpolation to fill them. However, for longer durations, exceeding six hours, the LSTM model is necessary to capture complex temporal dependencies and fill gaps accurately

After interpolation, the next step is to normalize our data between −1 and 1 since the convergence function of the DNN (Deep Neural Network) model uses the hyperbolic tangent function, which makes it easier to train the model [58]. This normalization is done according to the equation:

where max ( y ) and min ( y ) are respectively the maximum and minimum values of the time series.

Figure 3. State of the time series of the tide gauge stations of Pointe-Noire, Wouri Estuary and Dakar (from top to bottom) that need to be reconstructed.

The implementation and characteristics of the data filling model developed in this work is described in the following paragraphs. The using programming environment is Python version 3.7.9 with the following packages: TensorFlow version 2.4.1, Kéras version 2.4.0, Scikit-learn| version 0.23.2.

3.3.1. Implementation

The model of a neural network carried out in this work was established with several LSTM units in which many parameters must be specified and optimized before the training phase, such as the number of layers L, the number of neurons N (=LSTM units), batch size, dropout fraction and learning rate.

3.3.2. Preparing Model Training Data

To be able to feed the LSTM model, the time series must be restructured in the form of supervised learning. Thus, a function following the concept of a moving window was implemented with a loop allowing the data to be restructured in the form of two matrices ( X , Y ) = ( { x 1 , ⋅ ⋅ ⋅ , x i , ⋅ ⋅ ⋅ , x n } , { y 1 , ⋅ ⋅ ⋅ , y i , ⋅ ⋅ ⋅ , y n } ) . Only pairs ( X , Y ) not containing missing values are kept. Hence the importance of interpolation to fill gaps less than 6 hours in order to maximize the number of valid pairs ( X , Y ) for training and testing the model.

All preprocessing steps were designed to prevent data leakage. In particular, normalization was performed using parameters (mean and standard deviation) computed exclusively from the training dataset. These parameters were then applied unchanged to the validation and test sets. Once the series is transformed into supervised learning, it is divided it into three groups where:

1) 60% of the data is used for training the model; i.e., to optimize the parameters of the model;

2) 20% for validation in order to evaluate the generalization capacity of the model to data that it does not know et also diagnose overfitting problems;

3) The remaining 20% is used as test data to evaluate the relevance of the trained model.

The loss function L used in the model is the mean square error (MSE) with a regularization term L2:

where n is the number of data, y ^ the predicted value, y the measured value and λ the adjustable regularization parameter for the 2-norm ‖ W ‖ 2 of the weight matrices W. L2 regularization was used to reduce overfitting in the models [59][60].

The quality of the model is measured during the training phase by calculating the root mean square error (RMSE) and the mean absolute percentage error (MAPE), which are common statistical methods, used to compare the deviation of the value actual versus predicted value in order to evaluate the predicted performance of DNNs. They are described by the following equations:

where r i is the actual observed data, p i the predicted data and N the number of samples.

To compare the performance of the model in different stations, the relative RMSE metric is used defined as follows:

where max value and min value are respectively the maximum and minimum of the values of the time series and the RMSE is the error presented in equation (4). Relative error is used because Ω can vary from station to station. An RMSE value of 0.5 m for example, in a time series which extends from 1 to 10 m does not have the same interpretation as in another station where the range is 4 to 9 m. So, using a relative error in each time series, such as a percentage measurement, provides a better understanding of model performance.

The coefficient of multiple determination (R²) whose equation is presented below is also used to compare the degree of variance of the predicted values. It can take negative values and its maximum value (corresponding to perfect predictions) is 1.

To evaluate the robustness of the proposed model, synthetic gaps were introduced into the observed time series. A total of 30 - 100 gap injections were performed for each gap duration scenario. The gap lengths were selected to represent short-, medium-, and long-duration missing data conditions.

The temporal locations of the synthetic gaps were randomly selected within the evaluation period, following a uniform distribution, while ensuring that injected gaps did not overlap and remained within the bounds of the available data. This approach allows for testing the model under diverse conditions representative of real-world data loss.

To ensure statistical reliability, the experiment was repeated over 5 - 10 independent trials with different random gap positions. The reported performance metrics (RMSE, MAPE, and R²) correspond to the average values across all trials. In addition, the standard deviation was computed to assess the variability of the model performance.

3.3.3. Tuning Model Hyperparameters

In order to find the best hyperparameters of the proposed model, a Bayesian hyperparameter optimization was performed on the training dataset [61][62]. The number of LSTM layers, the number of LSTM nodes in each layer, the activation function, the mini-batch size b, the learning rate α, the learning rate decay α, the dropout rate between LSTM layers and the regularization parameter λ were adjusted. Bayesian optimization manages the LSTM model as a random black box function on the hyperparameters, which is expensive to evaluate. Therefore, the algorithm estimates the best hyperparameters with minimal target function evaluations [63]. Beforehand, a Gaussian process is built on the target functions and the best hyperparameters are selected according to their likelihood. Table 2 presents the result of the Bayesian optimization for the present model.

3.4.1. Choice of Scenarios

As explained in section 2.3, the scenarios rely on those of Janbian [20] for this reconstruction. Although the advantages and disadvantages of each scenario are well described in Janbian [20] this has the advantage of assessing for each approach the continuous size of maximum gaps that could be filled with the metrics that go with it. For this, a reference station with more than one continuous year without missing data was chosen. The training of the model is done without this part of data which will serve as a test. Then continuous and increasingly larger gaps were created and each time after imputation of data by the model, the RMSE, the MAPE and the R² were computed to assess the maximum size of continuous missing data can be filled by each strategy and with what precision.

3.4.2. Evaluation of the Quality of the Reconstructed Series

Here, it is evaluated whether the reconstructed series still remains sufficiently reliable or even better for studies of characterization of the tide of the station or for the evaluation of sea level for example. To do this, the script T_Tide developed by [64] is used to extract the harmonic components of the original series and the reconstructed one. The principle of the analysis is that given a place and a time t, the height of the tide can be expressed by the following formula:

where z 0 represents the average level, A i the amplitude of wave i, w : pulsation of wave i, ∅ i : Phase of the wave i [65].

This part is ended by calculating the complex errors between the 10 most significant constituents of the reconstructed tide and the unreconstructed one. Knowing that these account for more than 80% of tidal energy. This error is obtained by the formula

δ Z is the complex difference defined in [66] by the equation:

Once the input data is ready, the hyperparameters of the model are defined so that it properly adjusts the weights and biases of the LSTM cells, an iteration (epoch) of 500 is defined with a patience of 10. Table 2 and Table 3 below respectively presents the parameters and hyperparameters of the LSTM model. It should be noted that with the exception of the input data, the same configurations will be used for all stations.

Table 2. layers and training parameters of the LSTM model.

Table 3. Result of hyperparameters from model adjustment.

Once the model is implemented and parameterized, its training and validation are initiated to first evaluate its ability to predict water levels. The Pointe-Noire station (Congo) only contains short-term gaps (less than a month) distributed throughout the series with a missing data rate of 12%. It has served as the reference for the application of scenario 1. The measurements of the SM2 station (Wouri Cameroon) are the most affected with nearly 43% of missing data of varying duration. The measurements from this station can be divided into two parts with a gap of almost a year in the middle (between 2020 and 2022). Scenario 1 will also be applied to fill short-term gaps.

Figure 5. Visual division of the data set between training data (60%) in blue, validation data (20%) in green and test data (20%) in yellow.

The Pointe-Noire and Wouri series (resampled into hourly data) are interpolated and prepared as described in Section 2.6.2 into training, validation and test data (see Figure 5), are sent to the model with a one-week LB. Figure 6 allows you to visualize the evolution curve of the loss function. One can see in Figure 7, the good correlation that there is between the measurements and the predictions of the training and test water levels for the Pointe-Noire station.

Figure 6. evolution curves of the loss function during training (Blue) and validation (red) with the best reached after 120 iterations.

The best model resulting from training is then tested. Figure 7 shows the result of the tests with the performance metrics (Summary in Table 4) which makes it possible to confirm through prediction the adequacy of the model to the data.

Table 4. The recap of performance of training and testing of forecasting by the LSTM model applied to short-term missing data size series (One month Maximum) from scenario 1.

Figure 7. Results of the adequacy of the prediction with the measured data with a high level of precision achieved and a low MSE for training and testing.

Model training was also done with the scenario 2 strategy on data from the Dakar station, with the aim of filling gaps of more than a month. Scenario 3 was applied to the Wouri and Dakar stations for gaps of more than six months. This strategy consists first of decomposing the series into daily hourly data, see Figure 8.

Figure 8. Breakdown into daily hourly data over the series (SM2 at the top) and over one year (the one below).

Figure 9. From top to bottom, the first image represents the evolution curve of the MSE during the training and validation phase of the Wouri station; and the one below the model test result after training the Dakar station for scenario 3 applies at 11 p.m.

Figure 9 illustrates the evolution of the mean squared error (MSE) during the training and validation phases of the LSTM model, as well as an example of the prediction results obtained for the Dakar station using Scenario 3 at 11:00 PM.

After this decomposition, the authors choose one hour per day for training the model. The objective is to reduce the accumulation of errors in scenarios 1 and 2 on long-term gaps and also reduce calculation times for training and imputation. one can see in Figure 10 that a good correlation is obtained.

It is shown that the model well reproduces data with few uncertainties at the extrema.

It is noted that given the disparity of missing data in the series, one had to combine approaches to fill the gaps while keeping the trends and seasonality of the series. Table 5 presents the summary of the result’s strategy 3 used by station Dakar and Estuary Woury.

Table 5. Results of performance of training and testing of forecasting by the LSTM model applied to short-term missing data size series (One month Maximum) from scenario 3.

To be able to study this behavior, one year of continuous data without gaps was removed from the initial measurements and was not integrated into the training and testing of the model. To better understand the comparison of results between the approaches we chose a station to serve as a reference for this section. Our choice fell on the Pointe-Noire station for the quality of this chronological series. Indeed, in addition to having the year 2013 complete towards the end of the series (see Figure 10), its gaps range from one hour to less than a year unlike the other two stations (Wouri Estuary and Dakar) which practically have the series divided into several parts with continuous gaps over several years.

The methodology consists of evaluating the performance metrics, namely RMSE, R², and MAPE, over increasingly large continuous gaps in order to determine the conditions under which each reconstruction approach should be used, as shown in Table 6. It appears that:

Scenario 1 is suitable for short-term gaps with a multiple determination coefficient which remains significant up to gaps of 84 days and an RMSE of 0.15 m.Scenario 2 and 3 with our conditions can be used for gaps of up to one year with an RMSE of 0.14 and 0.15 m respectively. But the values of the coefficient of multiple determination rather suggest using scenario 2 for gaps of longer duration.

It should be noted that the result of this investigation depends on the quality and quantity of the model training data as well as the torque values (LB, mLB). Globally, this approach has the advantage, depending on the quality of each series, to let know which strategy is best to fill in its missing data. Depending on the desired precision, which in general will depend on what the reconstructed series will be used for, this approach is preferred to another or even make combinations.

Figure 10. Time series for the year 2013 Pointe-Noire used to evaluate the behavior of the model.

Table 6. Summary of metrics for evaluating the behavior of the model in relation to the size of the gaps at the Pointe-Noire station.

The LSTM model validated in the previous step for the reconstruction of tidal time series can now be used to fill real gaps. This phase start by reconstructing gaps of different sizes, in order to analyze the behavior of the model. The model predicts one water level at a time. The imputation is done by a loop which allows the prediction to be repeated n times (n being the number of missing values in the series). Figures 11-13 show some results by scenarios 1,3 respectively.

Figure 11. Focus on the first 2 months with gaps filled by the strategy one of the Pointe-Noire station (Congo).

Figure 12. Result of the imputation of the 23 PM after daily breakdown decomposition of the Wouri station.

Scenario 1 was also used at the Wouri station to fill small gaps and then scenario 3 was used for longer gaps at the station. that was also applied to Dakar. Figure 12and Figure 13 below show the result of a daily breakdown.

LSTM neural networks are quite successful in reconstructing water level variations, when the right approach to structuring the input data is found. Figure 14 shows the three stations after imputation of missing data. Table 7 also sums up the strategies used for each station.

Table 7. Summary of the different scenarios used for the imputation of each station.

Figure 13. Another example of scenario 3 applies to the Dakar station with at the top the 7AM series of the entire station and at the bottom the same series with filled gaps.

Figure 14. Final results of the tide gauge series reconstructed from the high to the low respectively present, Dakar (Senegal), Wouri (Cameron) and Pointe-Noire (Congo).

The training of the model is more or less long depending on the choice of the size of LB and mLB.

This section aims to make sure that the reconstructed tide gauge series still has enough information for the study of the tidal phenomenon. The analysis of Pointe-Noire reference station is carried out. This series, which had nearly 12% missing data, was reconstructed based on scenario 1. Figure 15 shows the result of harmonic decomposition of all series after reconstruction. To do this, harmonic decomposition was carried out as presented in section 2.6.2 using T_Tide which succeeded in extracting 71 harmonics components with a confidence interval of 95%. Figure 15 shows the components of different classes. It is found that the majority of the components which characterize the tide in the Gulf of Guinea with a dominance of the semi-diurnal waves M2 and the long-term components (Sa, SSa, MF, etc.) and even the M10 wave signifying the proximity of the tide gauge station with the coast (shallow environment) is one of the significant components.

Finally, Figure 16, presents the comparison of the amplitudes of 10 main harmonic components of the reconstructed and non-reconstructed series with the complex error associated with each of the waves. The maximum error is 5cm for the M2 wave at the Pointe-Noire Station.

Figure 15. Harmonic constituents of the reconstructed Pointe-Noire(a), Estuary Wouri(b) and Dakar(c) series grouped by class with the most significant in blue and the least significant in red.

Figure 16. From top to bottom, (a) is the amplitude of the unfilled harmonic constituents, and (b) is the amplitude of the reconstructed series, and (c) is the complex error.