Enhancing Tidal Wave Predictions for the Estuary of the Nakdong River Using a Fixed-Lag Smoother

Abstract

The prediction of tidal waves is essential for improving not only our understanding of the hydrological cycle at the boundary between the land and ocean but also energy production in coastal areas. As tidal waves are affected by various factors, such as astronomical, meteorological, and hydrological effects, the prediction of tidal waves in estuaries remains uncertain. In this study, we present a novel method that can be used to improve short-term tidal wave prediction using a fixed-lag smoother based on sequential data assimilation (DA). The proposed method was implemented for tidal wave predictions of the estuary of the Nakdong River. As a result, the prediction accuracy was improved by 63.9% through DA and calibration using regression. Although the accuracy of the DA diminished with the increasing forecast lead times, the 1 h lead forecast based on DA still showed a 44.4% improvement compared to the open loop without DA. Moreover, the optimal conditions for the fixed-lag smoother were analyzed in terms of the order of the smoothing function and the length of the assimilation window and forecast leads time. It was suggested that the optimal DA configuration could be obtained with the 8th-order polynomial as the smoothing function using past and future DA assimilation windows assimilated 6 h or longer.

1. Introduction

The accurate prediction of tidal waves is important not only for improving our understanding of the hydrological cycle at the boundary between the land and ocean but also for improving energy production in coastal areas. One of the conventional techniques for tidal level prediction is the harmonic analysis method based on least squares estimation. Harmonic analysis based on least square regression uses the superposition of sinusoidal functions describing the amplitude, phase, and frequency of the oscillatory components, such as tides and sound. After Darwin’s pioneering work [1], dating back to the late 19th century, harmonic analysis was further improved by several researchers [2,3]. In addition, various approaches, including artificial neural networks, backpropagation neural networks, wavelet transformation, inaction methods, and hybrid models, have been proposed to improve tidal wave prediction [4,5,6,7,8,9,10,11,12,13]. However, one of the limitations of conventional tide level prediction is that it does not consider hydrological and meteorological factors. In particular, in the case of an estuary, the river discharge and storm surge can significantly affect the water level, which may exacerbate the accuracy of the tidal wave prediction. One of the techniques used to fill the gap in model prediction is data assimilation (DA), which updates the model states or parameters using newly obtained observations [14]. There have been also advancing in tidal wave prediction using DA approaches, such as Kalman filtering [15,16,17,18,19]. Nevertheless, it is expensive to implement DA methods for specific models and applications since the modeling procedure needs to be reorganized or reformulated to adjust the model states or parameters when new observations become available [20]. Post-processing is another option that can be employed to reduce model biases [21]. While DA dynamically changes the model states or parameters, post-processing adjusts the simulation results using an additional statistical treatment with no changes in the prediction model. However, post-processing has rarely been implemented for the purpose of tidal wave prediction.

In this study, we present a novel method that can improve short-term tidal wave prediction using a smoother fixed-lag. While this method originates from DA, we used the proposed smoother as a post-processor to produce the optimal solution within the assimilation window, with no changes in the model states. Before applying the smoother, the tidal wave simulation was adjusted using a regression equation to consider the impacts of the river discharge and atmospheric pressure. The proposed method was implemented for tidal wave predictions of the estuary of the Nakdong River. Additionally, the optimal conditions for the fixed-lag smoother were analyzed in terms of the order of function and the length of the assimilation window and forecast lead time.

2. Study Area and Data

The Nakdong River (basin area: 23,860${\mathrm{km}}^2$, length: 525.15$\mathrm{km}$) is one of the longest rivers on the Korean Peninsula, flowing through the entire Yeongnam region and meeting the South Sea at the estuary [22]. The estuary of the Nakdong River is characterized by a complex branch channel network, where the movement of water is mainly driven by tidal water and freshwater discharge. In 1987, the estuary gate was constructed with the aim of reducing the saltwater damage to the irrigation area in the Gimhae Plain and supplying water to the industrial areas [23]. However, as the ecological diversity and water quality have decreased since the construction of the estuary bank, in recent years, a partial re-opening of the Nakdong Estuary Bank gate was planned and tested [24]. The movement of fresh and saltwater is affected by both the river discharge and tidal sea level in the estuary. Therefore, accurate tide prediction is an important factor for the control of the concentration and location of salt water during the operation of the estuary gate.

As shown in Figure 1, the tidal observation station is located outside the Nakdong River estuary gate, measuring the water level at frequency intervals of 10 min. Figure 2 illustrates the time series of the water level inside the gate, referring to the river (internal water level), and outside the gate, referring to the sea (external water level), from 1 June 2021 to 30 June 2021, including all the observations (1 June 2021 to 24 August 2021) to enhance the visual interpretation of the figure. As observed inside and outside the river gates, the internal water level is affected by the inflow and outflow of the river, while the external water level mainly changes due to the tide level. Excepting the complete opening of the estuary bank floodgate due to flooding, the river water level varies between EL.0.51 m and EL.1.01 m, with an average of EL.0.76 m.

3. Methodology

In this section, we describe the tidal wave simulation method, an adjustment technique using regression, the DA-based smoothing method, and the configuration of the DA experiments.

3.1. Tidal Wave Simulation Using TASK2K and Regression without DA

The tidal analysis software kit 2000 (TASK2K), based on harmonic decomposition, was applied for the operational prediction of the tidal level of the estuary of the Nakdong River. However, it was found that the difference between the observed and predicted water surface elevations became significantly larger, reaching up to approximately 13 cm when the Nakdong River estuary gate was opened during the pilot operation in 2021. The large simulation error was considered to be affected by temporal changes in discharge during the opening of the gate. Therefore, additional adjustment was implemented, considering the effects of the discharge and atmospheric pressure via a regression relationship curve (Figure 3).

Figure 3 shows the relationship curve between the water surface elevation and the inland discharge from the Nakdong River. The black dotted lines represent the relationship and the time series of the observed water level (Tidal_Obs). The red lines represent those of TASK2K (Tidal_Cal_Orig), while the blue lines refer to those based on the additional calibration using regression (Tidal_Cal_Re). The regression equation is shown below. The simulation results obtained via each method are analyzed in Section 4.

3.2. Fixed-Lag Smoother

The water level of the estuary is tightly controlled by the local physiography, in addition to the tidal effect. Even after the adjustment using the regression model, the tidal prediction may not fully capture the effects of the hydro-meteorological conditions. In this work, we investigate the way in which assimilating the water level observations of the estuary could further improve the tidal wave predictions. For the DA technique, we use the fixed-lag smoother method to combine the base simulations based on TASK2K with the observations undertaken within an assimilation window. The fixed-lag smoother minimizes the objective function J as follows:

$$\mathrm{J}=\frac{1}{n}{{\displaystyle \sum }}_i^n{\left(O_i-P_i\right)}^2+\frac{1}{m}{{\displaystyle \sum }}_{j=n+1}^{n+m}{\left(S_j-P_j\right)}^2$$

(1)

where $O_i$ denotes the observed water level at timestep $i$, $P_i$(or $P_j$) denotes the predicted water level at timestep $i$ (or $j$) using the fixed-lag smoother, $S_j$ denotes the original simulation by TASK2K at timestep $j$, and $n$ and $m$ denote the size of the assimilation window in the past and future periods of time, respectively. Although any mathematical formulation can be used to produce the optimal prediction $P$ within the assimilation window, a $k$-th polynomial function f(t), as shown below, is selected, considering the patterns of the tidal wave and the numerical feasibility:

$$f\left(t\right)={\mathrm{a}}_0+{\mathrm{a}}_1{t}^1+{\mathrm{a}}_2{t}^2+\cdots +{\mathrm{a}}_{k-1}{t}^{k-1}+{\mathrm{a}}_k{t}^{\mathrm{k}}$$

(2)

where ${\mathrm{a}}_0$, ${\mathrm{a}}_1$, …, and ${\mathrm{a}}_k$ denote the coefficients of a polynomial function f (t), and $t$ denotes the timestep within the assimilation window ranging from 1 to $m+n$.

Figure 4 illustrates the way in which the fixed-lag smoother works to enable improved tidal wave prediction. The assimilation window consists of two windows: the past and the future. For part of the past timesteps in the assimilation window, the observations are used as information to construct the smoother function. On the other hand, for part of the future timesteps, in which no observations are yet available, the simulation results provide information to the smoother. The blue line, represented by a polynomial function, is the optimal solution between the observations and simulations within an assimilation window. The least squares regression is utilized to obtain the optimal polynomial function. Since the smoother is optimal only within the given window, the coefficients of the polynomial function are recursively updated at every assimilation step. Unlike the conventional DA methods, which usually require the updating of the model states and parameters, the fixed-lag smoothing technique of tidal wave prediction can provide improved predictions via the optimal polynomial function with no changes in the simulation models. The simplified implementation procedure of the smoother is partly the result of the fact that the errors in tidal wave prediction are mainly associated with the magnitude, and the timing errors are relatively small in number. While benefiting from information based on observations, the polynomial function asymptotically approaches the base simulation, whereby the frequency of the tidal wave is mainly determined by another force, such as the gravity of the moon.

3.3. Configuration of the DA Experiments

The fixed-lag smoother, used for tidal wave prediction, provides an optimal solution within an assimilation window for a smoothing function and a prediction lead time. Since the tidal wave prediction is highly nonlinear, the configuration of the DA may affect the prediction performance. In this study, the effects of the DA conditions were analyzed through multiple experiments, considering the polynomial order, DA window period, and lead time. In this sub-section, the configuration of the DA experiments is presented.

The DA simulation experiments were implemented using 10-min-interval observation data collected from 1 June 2021 to 24 August 2021. In the experiments, 1 step equated to 10 min. For example, a 24-step-ahead lead time equated to a 4 h ahead lead time.

Table 1. Summarized configuration of the DA experiments.

Experiments	Polynomial Order	Size of DA Window for Past Timesteps	Size of DA Window for Future Timesteps	Prediction Lead Time
DA experiment 1		Ⅹ	Ⅹ	Ⅹ
DA experiment 2-1	Ⅹ		Ⅹ	Ⅹ
DA experiment 2-2	Ⅹ	Ⅹ		Ⅹ
DA experiment 3	Ⅹ	Ⅹ	Ⅹ

shows the simulated conditions of each DA experiment. The fixed condition was expressed as Ⅹ, and the variable condition was expressed as. The polynomial order is a non-linear function that integrates observational and predictive data, which can affect the accuracy of the tide prediction depending on the shape and order of the function. In DA experiment 1, the effects of various polynomial orders were assessed, while the other factors, such as sizes of the DA window for the past and future timesteps, were fixed. Additionally, the accuracy of DA may vary depending on the sizes of the past and future assimilation windows, which would be assessed in DA experiment 2. The effects of the assimilation window were assessed through two sub-experiments, separating the past observation interval and the future prediction interval. In DA experiment 2-1, the future prediction interval was fixed, and the effects of varying the size of the past observation interval were assessed. In contrast, in DA experiment 2-2, the past observation interval was fixed, while the future prediction interval was changed. For DA experiment 2, the polynomial order and lead time were fixed under the same conditions. In DA experiment 3, we assessed how long the effects of DA lasted for varying prediction lead times while the other conditions were fixed.

4. Results

In this section, the results of the tidal wave predictions and DA experiments are analyzed.

4.1. Comparison of Tidal Wave Predictions

Figure 5 illustrates the time series of the observed and predicted surface water elevations. While Figure 5a shows the time series of the full simulation period, Figure 5b–d presents enlarged views of the different time slots. We found that the original prediction (Tidal_Cal_Orig) was improved by the calibration using regression (Tidal_Cal_Re) and further improved by the fixed-lag smoother (DA). The 1 h lead forecast by DA (DA_1 h) fell between two simulations: the regression (Tidal_Cal_Re) and the fixed-lag smoother (DA). As shown in Figure 5c,d, the improvement provided by DA became more apparent when the discrepancy between the observed and predicted water levels was large, usually at the crests and troughs of the tidal wave. In Figure 5, the results of the fixed-lag smoother (DA) and the 1 h lead forecast by DA (DA_1 h) is shown, which were obtained using the following configuration: 36 steps (6 h) for the past and future DA windows with the smoothing function of an 8th polynomial line. The sensitivity of the DA configuration, analyzed through multiple DA experiments, is described in Section 4.2.

Table 2. Comparison of the performance of the water surface elevation predictions.

Evaluation Index	Evaluation Range	RMSE of Predicted Water Surface Elevation (m)
		Open Loop (Tidal_Cal_Orig)	Calibrated (Tidal_Cal_Re)	DA Analysis (DA)	1 h Lead Forecast by DA (DA_1 h)
RMSE (m) (percentile improvement compared to open loop)	All data	0.108 (-)	0.071 (34.3% improved)	0.039 (63.9% improved)	0.060 (44.4% improved)
	Data above 90% and below 10% quantiles	0.129 (-)	0.077 (40.3% improved)	0.041 (68.2% improved)	0.065 (49.6% improved)

compares the performance of the predicted water surface elevation shown in Figure 5, assessed by the root mean square error (RMSE). The values of the RMSE are as follows: 0.108 m, 0.071 m, 0.039 m, and 0.060 m for Tidal_Cal_Orig, Tidal_Cal_Re, DA, and DA_1 h, respectively. Compared to the original open-loop prediction (Tidal_Cal_Orig), as a baseline, the results of the regression (Tidal_Cal_Re) and the fixed-lag smoother (DA) correspond to an improvement of 34.3% and 63.9%. The 1 h lead forecast by DA (DA_1 h) shows 44.4% improvement compared to the open-loop prediction, which means that there are benefits to be gained from DA, even in terms of the forecast, while the accuracy of DA diminishes as the forecast lead times increase. In order to assess the performance under extreme conditions, the RMSE values are estimated for the observation ranges above 90% and below 10% quantiles. In the case of the open loop (Tidal_Cal_Orig), the error for the extreme ranges is 0.129, which is 19.4% larger than the error for the ranges collectively. However, the predictions are successfully enhanced by calibration using regression and DA. The percentile improvement compared to the open loop is 40.3%, 68.2%, and 49.6% for the calibrated prediction (Tidal_Cal_Re), DA analysis (DA), and 1 h lead DA forecast (DA_1 h), respectively. It is worth noting that an additional improvement amounting to approximately 10% is achieved in the case of the 1 h lead forecast when DA is applied to the calibration result based on regression.

Figure 6 compares the observed and simulated surface water elevation measurements. While the open loop by TASK2K (Tidal_Cal_Orig) is the most dispersed, deviating from a 1:1 line (Figure 6a, the fixed-lag smoother (DA) shows the most significantly improved result, demonstrating that the use of the fixed-lag smoother can further improve the prediction based on regression.

4.2. DA Experiments for Improved Tidal Wave Predictions

Since a tidal wave generally has a time series pattern, it is important to select the time-lag value appropriately in order to identify the data portion that is closely correlated with the observations used by the model [25]. In this section, we investigate the prediction accuracy based on the selection of the appropriate polynomial order, lead time length, and time-lag values using the fixed-lag smoother techniques.

4.2.1. DA experiment 1: Impact of the Order of the Polynomial

The effect of the polynomial order of the fixed-lag smoother function on the tidal wave forecast accuracy was analyzed. Since there are numerous combinations of DA configurations, some representative cases were examined to evaluate the performance of the DA with different polynomial orders of the smoother function over varying forecast lead times. In Figure 7, the size of the DA window for the “future” timesteps is fixed at 60 steps (10 h), while the size of the DA window for the past timesteps varies from 12 to 84 steps (2 h, 6 h, 10 h, and 14 h). On the contrary, in Figure 8, the size of the DA window for the “past” timesteps is fixed, while the size of the DA window for the future timesteps varies.

When the size of the DA window for the past timesteps is relatively small (e.g., Cases 1 and 2), the highest predictive performance is obtained with the 6th polynomial order. In Case 1, at the forecast lead time of six steps (1 h), the values of the RMSE are 0.062 m, 0.064 m, and 0.069 m for the 6th, 8th, and 10th polynomial orders, while in Case 2, the values are 0.056 m, 0.058 m, and 0.06 m for the 6th, 8th, and 10th polynomial orders, respectively. However, the performance increases as the size of the past DA window becomes larger and when the order of the polynomial is higher than 6th (e.g., Cases 3 and 4). It was also found that, with the 6th polynomial order, the accuracy deteriorates quickly when the size of the past DA window is greater than 10 h. In Case 3, at the forecast lead time of 6 steps (1 h), the values of the RMSE are 0.063 m, 0.056 m, and 0.057 m for the 6th, 8th, and 10th polynomial orders, while in Case 4, they are 0.1 m, 0.059 m, and 0.057 m for the 6th, 8th, and 10th polynomial orders, respectively. Among the varying past DA windows, the smoothing function with the 8th order shows stable and improved results.

In Figure 8, the predictive performance of each polynomial order is depicted according to the future DA windows. Similar to the sensitivity of the “past” DA window (Figure 7), as the size of the “future” DA window increases, the performance of the polynomial with the 6th order deteriorates more quickly. Although the performances of the 8th- and 10th-order polynomials are similar, the 8th-order polynomial is slightly better in terms of performance when the forecast lead time is longer than 1 h.

In Case 5, at the forecast lead time of six steps (1 h), the values of the RMSE are 0.064 m, 0.067 m, and 0.073 m for the 6th, 8th, and 10th polynomial orders, while in Case 6, they are 0.058 m, 0.058 m, and 0.06 m for the same polynomial orders, respectively. In Cases 7 and 8, the performance was the best using the 8th polynomial at the lead time of 6 steps (1 h). In Case 7, at the forecast lead time of six steps (1 h), the values of the RMSE are 0.063 m, 0.056 m, and 0.057 m for the 6th, 8th, and 10th polynomial orders, while in Case 8, they are 0.075 m, 0.057 m, and 0.057 m for the same polynomial orders, respectively.

4.2.2. DA Experiment 2: Impact of the DA Windows

The prediction accuracy of the DA depends on the window size and future prediction size of the past observations used. Therefore, to analyze the effect of the DA window size on the short-term tidal wave prediction accuracy, we performed two simulations separating the past observation interval and the future prediction interval. In the first case, the size of the past DA window changed, while the future DA window was fixed (

Table 3. The conditions for DA experiment 2-1: varying the past DA window.

Case	DA Window Past Step	DA Window Future Step	Lead Time Step	Polynomial Order
DA_Past_2 h	12	72	12	8
DA_Past_6 h	36	72	12	8
DA_Past_10 h	60	72	12	8
DA_Past_14 h	84	72	12	8

and Figure 9). In the second case, the size of the future DA window changed, while the past DA window was fixed (

Table 4. The conditions for DA experiment 2-1: varying the future DA window.

Case	DA Window Past Step	DA Window Future Step	Lead Time Step	Polynomial Order
DA_future_2 h	60	12	12	8
DA_future_6 h	60	36	12	8
DA_future_10 h	60	60	12	8
DA_future_14 h	60	84	12	8

and Figure 10). The same values for the polynomial order and forecast lead time conditions were applied in DA experiment 2.

Not surprisingly, it was found that, for the most part, the prediction accuracy decreased as the lead time increased. In both simulations, the prediction accuracy, according to the window size, decreased with the increasing lead time, and the RMSE (m) value asymptotically converged on the base simulation, Tidal_Cal_Re (0.071 m). When taken together, the accuracy of short-term tidal wave predictions using a fixed-lag smoother needs to be carefully selected, as it shows a sensitive response to the lead time length.

Figure 9 shows the performance of the tidal wave prediction with varying past DA window sizes. DA_past_2 h to DA_past_14 h indicate that the length of the past observation period is set to 24-step intervals, ranging from 12 steps to 84 steps (2 h, 6 h, 10 h, and 14 h). The RMSE values of Tidal_Cal_Orig and Tidal_Cal_Re show prediction errors of 0.108 m and 0.071 m. The best performance at the 6-step lead (1 h) forecast is obtained using DA_past_6 h (0.056 m), followed by DA_past_10 h (0.057 m), DA_past_14 h (0.063 m), and DA_past_2 h (0.063 m). As a result of the simulation of the control of the past DA window size at the 8th polynomial, it was found that the optimal past DA window size for a 1 h lead time was 6 h.

Figure 10 shows the results of the simulation of the future prediction period and evaluation using the RMSE evaluation index for each lead timestep. The past observation period was fixed at 60 steps, the 8th polynomial, and 12 steps (2 h) for the lead time. In Figure 10, DA_future_2 h to DA_future_14 h indicate that the length of the future prediction interval is set to 24-step intervals, ranging from 12 steps to 84 steps (2 h, 6 h, 10 h, and 14 h). The ranking for the 1 h lead time according to the RMSE (m) values in the graph shows the order of 1. DA_future_10 h (0.056 m), 2. DA_future_14 h (0.057 m), 3. DA_future_6 h (0.058 m), 4. DA_future_2 h (0.067 m). As a result of the simulation of the control of the past DA window past size at the 8th polynomial, it was found that the optimal past DA window size for the 1 h lead time was 10 h.

4.2.3. DA Experiment 3: Impact of the Lead Time Length

In order to analyze the effect of the lead time length on the fixed-lag smoother accuracy, the lead time length was adjusted and simulated. Figure 11 shows the results of the RMSE evaluation of the prediction performance, corresponding to the length of the lead time interval. As shown in (a), (b), and (c) in Figure 11, The lead time stages were divided into 3 h, 6 h, and 12 h. In this simulation, the length of the past observation period was adjusted to 2 h, 6 h, 10 h, and 14 h. The length of the future prediction interval was fixed at 12 h, and the 8th polynomial was applied.

The results of the estimation accuracy evaluation according to the lead time length are as follows. As shown by the previous results, the longer the lead time is, in general, the more significantly reduced the prediction accuracy will be. Furthermore, the increase in the lead time was approximated to the RMSE (0.071 m) of Tidal_Cal_Re. Interestingly, when the lead time was 12 h, with the DA window exceeding 14 h, the value of the RMSE evaluation index showed variations in the manner of a waveform. Overall, these results suggest that setting the appropriate lead time and DA window length for the data has a significant impact on prediction accuracy. As a result of the three experimental simulations according to the lead time length, it seems appropriate to set the lead time length to be within 2 h in terms of the prediction accuracy. Furthermore, it is proposed that the length of the past DA window is reduced as the prediction period becomes longer and the degree of polynomial increases.

5. Conclusions

Predicting tidal waves is essential in order to not only better understand the hydrological cycle at the boundary between the land and ocean but also to improve energy production in coastal areas. In this study, the fixed-lag smoother method was proposed, implemented, and evaluated to improve short-term tidal wave prediction. The proposed fixed-lag smoother, based on sequential DA, can be used as a post-processor with no changes in the model structure. This study analyzed the optimal conditions for DA in terms of the polynomial order, DA window length, and lead time length. As a result, the prediction accuracy was improved by 63.9% through DA and the calibration using regression. Although the accuracy of DA diminished with increasing forecast lead times, the 1 h lead forecast by DA (DA_1 h) still showed a 44.4% improvement compared to the open-loop method without data assimilation. For the extreme ranges of the tidal wave, the benefits gained from data assimilation were more apparent, as indicated the fact that 40.3%, 68.2%, and 49.6% relative improvements were obtained by the regression (Tidal_Cal_Re), DA analysis (DA), and 1 h lead DA forecast (DA_1 h). An additional improvement of approximately 10% can be expected in the case of the 1 h lead forecast in addition to the calibration using regression.

In addition, the optimal conditions for the fixed-lag smoother were analyzed in terms of the order of the smoothing function and the length of the assimilation window and forecast lead time. It was suggested that the optimal DA configuration could be obtained with the 8th-order polynomial. Additionally, it was appropriate to set the DA window length to 6 h or more. Attention should be paid to decisions regarding the appropriate lead time when aiming to improve short-term wave prediction accuracy.

The empirical results of this study provide a new understanding of polynomial order determination and optimal lead time proposal, as well as the determination of the optimal observation interval length for past and future DA windows. In this study, the fixed-lag smoother was applied to a three-month observation data period collected from June to August 2021. However, in future study, it is necessary to verify the findings based on additional tidal observation data. In addition, since DA techniques can be linked to data-based models as well as physical models, the application of such links with various models that can optimally improve the accuracy of tidal predictions should be considered. This would be advantageous for determining the optimal DA window length by applying the optimal parameter extraction technique and will be analyzed in further studies.