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Neural network for the prediction and supplement of tidal record in Taichung Harbor, Taiwan
Advances in Engineering Software 33 (2002) 329–338 www.elsevier.com/locate/advengsoft
Neural network for the prediction and supplement of tidal record in Taichung Harbor, Taiwan T.L. Leea,*, C.P. Tsaib, D.S. Jengc, R.J. Shiehb a Department of Construction and Planning, Leader University, Tainan 709, Taiwan, ROC Department of Civil Engineering, National Chung-Hsing University, Taichung 402, Taiwan, ROC c School of Engineering, Griffith University, Gold Coast Campus, Qld 9726, Australia
b
Accepted 1 August 2002
Abstract Accurate tidal prediction and supplement is an important task in determining constructions and human activities in coastal and oceanic areas. The harmonic tidal level is conventionally used to predict tide levels. However, determination of the tidal components using the spectral analysis requires a long-term tidal level record (more than one year [Handbook of coastal and ocean engineering 1 (1990) 534]). In addition, calculating the coefficients abbreviated of tide component using the least-squares method also requires a large database of tide measurements. This paper presents an application of the artificial neural network for predicting and supplementing the long-term tidal-level using the short term observed data. On site, tidal-level data at Taichung Harbor in Taiwan will be used to test the performance of the artificial neural network model. The results show that the tidal levels over a long duration can be efficiently predicted or supplemented using only a short-term tidal record. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Tidal prediction and supplement; Spectral analysis; Harmonic analysis; Artificial neural network
1. Introduction Tidal level record is an important factor in determining constructions or activities in maritime areas. To describe the property of the tidal-level variations for an open sea, Darwin [2] proposed the equilibrium tidal theory, but it did not accurately estimate the tidal level for the complex bottom topography in the near-shore area. Later, Doodson [5,6] employed the least-squares method to determine harmonic constants. Since then, the least-squares analysis in determining harmonic parameters has been widely used to predict the tidal level. However, the shortcoming of this method is that the parameters of the tidal constituents are determined by using a long-term tidal record in site. Kalman [10] proposed the Kalman filtering method to calculate the harmonic parameters instead of the leastsquares method. In this model, a large tidal data was not required. Gelb [8] and Mizumura [15] also proved that the harmonic parameters using the Kalman filtering method [10] could be easily determined from only a small amount of * Corresponding author. Tel.: þ 886-6-255-2689; fax: þ886-6-255-2669. E-mail address: [email protected] (T.L. Lee).
historical tidal records. Yen et al. [20] utilized the Kalman filtering method in determination of parameters in the harmonic tide-level model as well. The estimation of harmonic parameters could predict accurately the tidal level using the Kalman filtering method, which is solved by the covariance matrix. However, it is necessary to determine the available parameters of the local tide before predicating the tidal level. Tsai and Lee [19] applied the backpropagation neural network (BPN) to forecast the tidal level using the historical observations of water levels without determining the harmonic parameters. However, their model is used only for the instant forecasting of tidal levels, not a long-term prediction. Besides the prediction of tidal level, supplement of tidal record is also important for a complete observation of tide database. The discontinuous observations may come from the damage of recording facilities, natural disasters or inappropriate operation and so on. The discontinuous record could either be short-term (few hours) or long-term (few months even up to one year). Thus, establishing a simple and executable supplementary model for tidal record is desired. Recently, the artificial neural network (ANN) has been
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widely applied to various areas to overcome the problem of exclusive and the non-linear relationships. For example, in water resources, French et al. [7] used the ANN to predict the rainfall intensity. Their results indicate that ANN is capable of learning the complicated relationship describing the space – time evolution of rainfall. Campolo et al. [1] applied ANN for river flood forecasting. Zhang and Stanley [21] forecasted raw-water quality parameter for the North Sakatchewan River by neural network modeling. Other successful examples of the application of neural network in water resources and hydraulics have been reported in Refs. [4,12]. In coastal engineering, Mase and Kianto [13,14] applied the ANN algorithm to assess the stability of the armor unit and the rubble-mound breakwater and estimate the wave forces acting on the structures. Later, ANN models have been further applied to calculate the tide and wave heights [3,18]. Recently, Tsai and Lee [19] used neural network for tide forecasting by using the field data of diurnal and semidiurnal tide. However, their model is only applicable for instant prediction, not long-term predication. The BPN developed by Rumelhart et al. [17] is the most representative learning model for the ANN. The procedure of the BPN repeatedly adjusts the weights of the connections in the network so as to minimize the measure of the difference between the actual output vector of the net and the desired output vector. The BPN is widely applied in a variety of scientific areas—especially in applications involving diagnosis and forecasting. The aim of this paper is to establish an ANN model for the long-term prediction and supplement of tidal data. The database of Taichung Harbor in Taiwan is used as an example to demonstrate the compatibility of the proposed model.
been well documented in the literature, only a brief is given in this section. A typical three-layered network with an input layer (I ), a hidden layer (H ) and an output layer (O ) (Fig. 1) is adopted in this study. Each layer consists of several neurons and the layers are interconnected by sets of correlation weights. The neurons receive inputs from the initial inputs or the interconnections and produce outputs by transformation using an adequate non-linear transfer function. A common transfer function is the sigmoid function expressed by f ðxÞ ¼ ð1 þ e2x Þ21 ; it has a characteristics of df =dx ¼ f ðxÞ ½1 2 f ðxÞ: The training process of neural network is essentially executed through a series of patterns. In the learning process, the interconnection weights are adjusted within input and output value. BPN is the most representative learning model for the ANN. The procedure of the BPN is the error, as the output layer propagates backward to the input layer through the hidden layer in the network to obtain the final desired outputs. The gradient descent method is utilized to calculate the weight of the network and adjust the weight of interconnections to minimize the output error. The error function at the output neuron is defined as E¼
1X ðT 2 Ok Þ2 2 k k
ð1Þ
where Tk and Ok are separately the value of target and output. Further details of the BPN algorithm can be found in Ref. [17].
3. Case study The hourly tidal data collected at Taichung Harbor, Taiwan during 1996– 1999 were used to test the accuracy of
2. Neural networks ANN is an information-processing system mimicking the biological neural network of the brain by interconnecting many artificial neurons. Since the principle of ANN has
Fig. 1. Structure of an ANN.
Fig. 2. Locations of Taichung Harbor, Taiwan (Location: (1208310 3100 E, 248170 2200 N).
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Fig. 3. Structure of the major tidal components for ANN without the hidden layer.
the proposed ANN’s model. The location of the Taichung Harbor station (1208310 3100 E, 248170 2200 N) is indicated in Fig. 2. The main component of tides in the Taichung Harbor is M2. According to the past records, its highest water level is 5.86 m, the lowest water level is 0.55 m and the average tidal range is 3.54 m. The relative root mean squared error (RMS) and correlation coefficient (CC) were used for the agreement index to present the accuracy of the present model. They are defined by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX u n u ðYðtÞ 2 Yt Þ2 u u k¼1 ð2Þ RMS ¼ u n u X 2 t ðYt Þ k¼1 n X
ðYðtÞ 2 YðtÞÞðYt 2 Yt Þ
k¼1 CC ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n X X ðYðtÞ 2 YðtÞÞ2 ðYt 2 Yt Þ2 k¼1
ð3Þ
k¼1
in which YðtÞ is the value of prediction; Yt denotes the value of observation and n is the total number of hourly tide levels. YðtÞ is the mean of prediction and Yt is the mean value of observation. It is noted that RMS is a nondimensional parameter.
However, the inclusion of additional unnecessary constituents does not significantly improve the accuracy of prediction. Therefore, appropriate tidal components must be determined at the beginning stage. As reported by Reid [16], determination of the major tidal component using the spectral methods required at least a year tidal record for shallow water with significant meteorological noise. In this study, we propose to use the corresponding weighting relations in ANN to determine the tidal components. For a tidal component with heavy weight, its effect is more important for procedure of prediction than others. To represent the relations of input layer and weighting function, the back-propagating neural network without hidden layer is used (Fig. 3). In the network structure, 69 tidal components and their corresponding cosðvi tÞ and sinðvi tÞ are used, while the tidal levels YðtÞ are used in the output layer (Table 1). Based on one-month (January 1998) and two-month (January and February 1998) tidal records, the relations between weighting function and tidal components can be determined. For example, the learning outcomes of one month and two months tidal level are presented in Figs. 4 and 5, respectively.
3.1. Determination of the tidal components In general, the prediction of tidal level can satisfy the practical requirement, if the numbers of tidal components are sufficient. Using more tidal components, the accuracy of the prediction of tidal level will certainly be enhanced.
Fig. 4. The results of the main components of tide using one month of tidal data (January 1998) for the ANN model.
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Table 1 The 69 tidal components [11] No.
Tidal component
Hz (1/hr)
Angular velocity (deg/hr)
No.
Tidal component
Hz (1/hr)
Angular velocity (deg/hr)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Z0 SA SSA MSM MM MSF MF ALP1 2Q1 SIG1 Q1 RHO1 O1 TAU1 BET1 NO1 CHI1 PI1 P1 S1 K1 PSI1 PHI1 THE1 J1 SO1 OO1 UPS1 OQ2 EPS2 2N2 MU2 N2 NU2 H1
0.00000000 0.00011407 0.00022816 0.00130978 0.00151215 0.00282193 0.00305009 0.03439657 0.03570635 0.03590872 0.03721850 0.03742087 0.03873065 0.03895881 0.04004043 0.04026859 0.04047097 0.04143851 0.04155259 0.04166667 0.04178075 0.04189482 0.04200891 0.04309053 0.04329290 0.04460268 0.04483084 0.04634299 0.07597494 0.07617731 0.07748710 0.07768947 0.07899925 0.07920162 0.08039733
0.0000000 0.0410652 0.0821376 0.4715208 0.5443740 1.0158948 1.0980324 12.3827652 12.8542860 12.9271392 13.3986600 13.4715132 13.9430340 14.0251716 14.4145548 14.4966924 14.5695492 14.9178636 14.9589324 15.0000012 15.0410700 15.0821352 15.1232076 15.5125908 15.5854440 16.0569648 16.1391024 16.6834764 27.3509784 27.4238316 27.8953560 27.9682092 28.4397300 28.5125832 28.9430388
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69
M2 H2 MKS2 LDA2 L2 T2 S2 R2 K2 MSN2 ETA2 MO3 M3 SO3 MK3 SK3 MN4 M4 SN4 MS4 MK4 S4 SK4 2MK5 2SK5 2MN6 M6 2MS6 2MK6 2SM6 MSK6 3MK7 M8 M10
0.08051140 0.08062547 0.08073957 0.08182118 0.08202355 0.08321926 0.08333334 0.08344740 0.08356149 0.08484548 0.08507364 0.11924210 0.12076710 0.12206400 0.12229210 0.12511410 0.15951060 0.16102280 0.16233260 0.16384470 0.16407290 0.16666670 0.16689480 0.20280360 0.20844740 0.24002200 0.24153420 0.24435610 0.24458430 0.24717810 0.24740620 0.28331490 0.32204560 0.40255700
28.9841040 29.0251692 29.0662452 29.4556248 29.5284780 29.9589336 30.0000024 30.0410640 30.0821364 30.5443728 30.6265104 42.9271560 43.4761560 43.9430400 44.0251560 45.0410760 57.4238160 57.9682080 58.4397360 58.9840920 59.0662440 60.0000120 60.0821280 73.0092960 75.0410640 86.4079200 86.9523120 87.9681960 88.0503480 88.9841160 89.0662320 101.9933640 115.9364160 144.9205200
As seen in Fig. 4, the one-month tidal level record can only roughly indicate if the tidal level is dominated by diurnal or semi-diurnal tides, but it cannot indicate the major tidal components. However, the learning outcome of two-month data (Fig. 5) indicates 5 –7 significant tidal components. They are: M2 (Luni-solar sei-diurnal), K1 (Luni-solar diurnal), O1 (Principal lunar diurnal), P1 (Principal solar diurnal), S2 (Principal solar), N2 (Larger lunar elliptic) and NO1 (Compound tides). These are similar
to the results from spectral analysis with two years tidal record (1995,1996), as shown in Fig. 6 [11]. In general, the number of main tidal components will directly affect the accuracy of tidal forecasting. Thus, the influence of the number of tidal components on the accuracy is examined through a parametric study here. Using the data in Taichung Harbor as an example, the RMS values for various numbers of tidal components are tabulated in Table 2. As seen in the table, the RMS is 0.0884 with five
Fig. 5. The results of the main components of tide using two months of tidal data (January and February 1998) for the ANN model.
Fig. 6. The results of the main components of tide using the spectral method based on two year data (1995 and 1996).
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Table 2 Test of various tidal components using the ANN Harbor
The name of input tidal components
Number of tidal components
RMS
Taichung Harbor
M2, S2, N2, K1 M2, S2, N2, K1, O1 M2, S2, N2, K1, O1, P1 M2, S2, N2, K1, O1, P1, K2
4 5 6 7
0.0895 0.0844 0.0860 0.1175
Table 3 The effect of the number of hidden layers Harbor
Hidden layers
RMS
Taichung Harbor
0 1 2
0.0856 0.0844 0.0859
tidal components (M2, K1, O1, S2 and N2). However, the errors increase to 0.1175, if we include another two tidal components, P1 and K2. This implies that P1 and K2 cannot improve the accuracy of the tidal forecasting. Thus, M2, K1, O1, S2 and N2 are considered as the five main tidal components for Taichung Harbor. It is noted that the determination of the above five tidal components is based on linear ANN model. Here, the linear ANN model is used as the first approximation. If a nonlinear ANN model is used to identify the tidal component, the results may be different. 3.2. The ANN’s tide model Doodson [5,6] proposed a harmonic analysis for tide forecasting. In his model, the least-squares method was used to determine the harmonic constants. These constants are further substituted into harmonic equation to determine the
tidal level. This model has been widely used because of its simplicity. Based on the harmonic theory, the vertical tidal level YðtÞ at time t at any place is expressed as following YðtÞ ¼ A0 þ
N X
ðAi cos vi t þ Bi sin vi tÞ
ð4Þ
i¼1
where A0 is the mean water level, Ai and Bi are coefficients of tide components, vi is the angular frequency of the tidal components and N is the total number of component tides. Fig. 7 illustrates the basic structure of the ANN tidal forecasting model with one hidden layer. Each tidal component corresponds to cosðvi tÞ and sinðvi tÞ: There is only one variable for output layer, i.e. the tidal level YðtÞ: 3.3. Effects of neural network structure In general, the factors which directly affect the ANN model include the number of hidden layers, learning factor (h ), momentum factor (a ), the number of training iterations (Epochs) and the number of neurons in each layer. Since the neural network is a non-linear procedure and the network parameters will affect each other, the adjustment of each parameter to optimize the whole network is not an easy task. This section discusses how the neural network structures affect the performance of the forecasting model.
Fig. 7. Structure of the tidal forecasting for an ANN.
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Table 4 The effect of the number of the hidden neurons
Table 5 The recommended structure of the ANN
Harbor
Neurons of hidden layer
RMS
Taichung Harbor
1 3 5 7 10
0.0866 0.0851 0.0847 0.0844 0.0846
The performance of the neural network structures with no hidden layers, one hidden layer, and two hidden layers with the same training parameters h ¼ 0.01 and a ¼ 0.8 and 1000 training iterations is tabulated in Table 3. The results clearly indicate that one hidden layer has the better performance. However, the influence of the number of hidden layer is insignificant. For example, the difference of RMS between without hidden layer and with one hidden layer is only 0.0012. However, the input neurons without hidden layer are not mapped with non-linear transformation, thus resulting in a slightly larger error compared to the single hidden layer. In contrast, more hidden layers lead to more complex operations for neural networks. Thus, one hidden layer is used in this study. The number of neurons in the hidden layer also affects the performance of the ANN. Table 4 shows the RMS error for various neurons structures. It shows that the forecasting performance of the ANN is improved when the number of
Harbor
Number of hidden layer
Number of neurons in hidden layer
h
a
Epoch
Taichung Harbor
1
7
0.01
0.8
1000
neurons increases. However, a large number of neurons in the hidden layer decreases the accuracy owing to overlearning. Thus, the number of neurons in the hidden layer is recommended to be seven because of the satisfactory prediction performance. The value of the learning rate (h ) will significantly affect the convergence of neural network learning algorithm and the momentum factor (a ) is used to avoid stopping the learning process at a local minimum instead of global minimum [9]. To have a better performance, either low h or high a is expected to accelerate the convergence of the training process. After some preliminary tests, we use a learning rate of 0.01 and a momentum factor of 0.8 in all training cases. The number of training iterations of 1000 is selected in all cases and the network parameters is also listed in Table 5. 3.4. Long term tide forecasting In this study, based on the aforementioned optimal neural
Fig. 8. Comparison of observed tide levels with those predicted over one year for Taichung Harbor (4/1996, 10/1996, 2/1997).
T.L. Lee et al. / Advances in Engineering Software 33 (2002) 329–338 Table 6 The performance of the one year using the different day’s measurements Harbor
Training sets
RMS
CC
Taichung Harbor
1 day (12/4/1996) 7 days (12–8/4/1996) 15 days (12–26/4/1996) 30 days (12/4–11/5/1996)
0.3815 0.2133 0.0844 0.1835
0.5136 0.8775 0.9822 0.9309
Table 7 The survey of the supplement for Taichung Harbor Case
Date of supplement
The number of days
1 2 3 4 5
1998/03/31–1998/04/01 1998/08/14–1998/08/21 1998/11/04–1998/11/11 1999/06/08–1999/06/09 1999/10/19–1999/12/31
2 8 8 2 63
network, we use different data base in the training procedure to predict the one-year tidal level in Taichung Harbor (Fig. 8). Table 6 shows a year of hourly tidal predictions at the Taichung Harbor with different training periods. The results indicate that the tidal ANN forecasting model is able to predict one-year tidal level with 15-day hourly tidal observations. Based on the 15-day collected data (12–26 April 1996), the one-year prediction of tidal level (April 1996–March 1997) against the observation is illustrated in Fig. 9. In the figure, solid lines denote the observation data, and dashed lines are the predicted values. The prediction of the present model overall agree with the observation. The correlation coefficient over one year is 0.9182, which is reasonable good. 3.5. Supplement of tidal data Besides the long-term prediction of tidal level, the
335
Table 8 The performance of the supplement using the different day’s measurements, Case 2 Harbor
Training sets
RMS
CC
Taichung Harbor
1 day (1998/8/13) 7 days (1998/8/7–1998/8/13) 15 days (1998/7/30–1998/8/13) 30 days (1998/7/13–1998/8/13)
0.1809 0.0669 0.0734 0.1052
0.5396 0.9878 0.9653 0.9371
supplement of tidal data is also important for engineering practice. In this section, we will demonstrate the application of the proposed ANN model in the supplement of tidal data. Table 7 summaries the missing data of Taichung Harbor during 1998 and 1999. As seen in the table, five sets of data are incomplete (Case 1 – Case 5 in Table 8). Using Case 2 as an example (Table 8), the verification of the proposed model is discussed in detail here. In the case, 8 days hourly tidal level data (14/08/1998 – 21/08/1998) is missing. To supply the missing data, we choose one-day, seven-day, 15-day and 30-day data for the training procedure. The results of tidal supplement are given in Table 8. This table indicates that a longer training procedure may not provide a better estimation. For example, seven-day training data provide a best estimation for Case 2. Similarly, the learning data and verification of other cases are listed in Table 9, and the results of learning procedure and prediction are tabulated in Table 10. Based on the numerical results, it is found that seven-day learning data is sufficient to provide excellent supplementary tidal data for short-term missing data (less than eight days). However, to reach reasonable accuracy (e.g. CC . 0.9, and RMS , 0.1) of supplementary data for longer period (e.g. 63 days in Case 5), a 15-day training data is required. As shown in Table 10, the correlation coefficient of Case 2 for training procedure is 0.9980, and that for the
Fig. 9. Comparison of observed tide levels with those predicted in the learning process for Taichung Harbor, Case 2 (RMS ¼ 0.0311).
Fig. 10. Comparison of observed tide levels with those supplemented in the supplementing process for Taichung Harbor, Case 2 (RMS ¼ 0.0669).
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Fig. 11. Supplement of tidal levels for Taichung Harbor, Case 1–Case 5.
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Fig. 11 (continued )
verification can reach 0.9878. The time series of learning and estimation of tidal level (including the observation data and prediction) are presented in Figs. 9 and 10. The figures clearly indicate that both learning (Fig. 9, RMS ¼ 0.031) and verification (Fig. 10, RMS ¼ 0.0669) can reach a high accuracy. This demonstrates the compatibility of the proposed ANN mode, in the learning and supplement of tidal data. Based on the aforementioned procedure, we further predict and back-estimate the missing data in Taichung
Harbor during 1998 and 1999 (Fig. 11). In these figures, the solid lines represent the supplementary data from the proposed model, and symbols are the observation data. Fig. 11 demonstrates the capacity of the proposed ANN model in supplement of tidal data for Case 1– Case 5.
4. Conclusions The conventional method of the harmonic analysis
Table 9 The performance of the supplement and test for Taichung Harbor Case
Supplement dates
Days
Training dates
Days
Test dates
Days
1 2 3 4 5
1998/03/31–1998/04/01 1998/08/14–1998/08/21 1998/11/04–1998/11/11 1999/06/08–1999/06/09 1999/10/19–1999/12/31
2 8 8 2 63
1998/03/24–1998/03/30 1998/08/07–1998/08/13 1998/10/28–1998/11/03 1999/06/01–1999/06/07 1999/10/04–1999/10/18
7 7 7 7 15
1998/04/02–1998/04/16 1998/08/22–1998/9/20 1998/11/12–1998/12/11 1999/06/10–1999/06/24 1999/07/21–1999/10/03
15 37 37 15 75
Table 10 The performance of the learning and test for Taichung Harbor Case
1 2 3 4 5
Learning
Test
RMS
CC
RMS
CC
0.0354 0.0311 0.0305 0.0374 0.0533
0.9979 0.9980 0.9984 0.9960 0.9933
0.0932 0.0669 0.0945 0.0724 0.0785
0.9778 0.9878 0.9799 0.9904 0.9934
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requires a large amount of observed tidal data in determining the total number of component tides and the harmonic parameters. In this paper, we proposed an alternative method (ANN) for forecasting and supplementing the tidal level. The case study shows that the major constituents can be obtained by using a two-month measured data. The learning rate of 0.01 and a momentum factor of 0.8 are used in all training cases. In the proposed ANN model, one hidden layer is used. The results also demonstrate that the prediction of one-year tidal level forecasting can be satisfactorily achieved with a 15-day observed data. As for the supplement of tidal data, a seven-day training data is sufficient for short-period missing data (less than eight days), while a 15-day training data is required for the supplement of a long-period missing data. Based on the examples presented, the proposed ANN for predicting and supplementing the tidal-level can be further applied to other locations in Taiwan or in the world.
Acknowledgements The authors are grateful to the Prediction Center of Central Weather Bureau, Taiwan, for supporting valuable field data.
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Neural network for the prediction and supplement of tidal record in Taichung Harbor, Taiwan T.L. Leea,*, C.P. Tsaib, D.S. Jengc, R.J. Shiehb a Department of Construction and Planning, Leader University, Tainan 709, Taiwan, ROC Department of Civil Engineering, National Chung-Hsing University, Taichung 402, Taiwan, ROC c School of Engineering, Griffith University, Gold Coast Campus, Qld 9726, Australia
b
Accepted 1 August 2002
Abstract Accurate tidal prediction and supplement is an important task in determining constructions and human activities in coastal and oceanic areas. The harmonic tidal level is conventionally used to predict tide levels. However, determination of the tidal components using the spectral analysis requires a long-term tidal level record (more than one year [Handbook of coastal and ocean engineering 1 (1990) 534]). In addition, calculating the coefficients abbreviated of tide component using the least-squares method also requires a large database of tide measurements. This paper presents an application of the artificial neural network for predicting and supplementing the long-term tidal-level using the short term observed data. On site, tidal-level data at Taichung Harbor in Taiwan will be used to test the performance of the artificial neural network model. The results show that the tidal levels over a long duration can be efficiently predicted or supplemented using only a short-term tidal record. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Tidal prediction and supplement; Spectral analysis; Harmonic analysis; Artificial neural network
1. Introduction Tidal level record is an important factor in determining constructions or activities in maritime areas. To describe the property of the tidal-level variations for an open sea, Darwin [2] proposed the equilibrium tidal theory, but it did not accurately estimate the tidal level for the complex bottom topography in the near-shore area. Later, Doodson [5,6] employed the least-squares method to determine harmonic constants. Since then, the least-squares analysis in determining harmonic parameters has been widely used to predict the tidal level. However, the shortcoming of this method is that the parameters of the tidal constituents are determined by using a long-term tidal record in site. Kalman [10] proposed the Kalman filtering method to calculate the harmonic parameters instead of the leastsquares method. In this model, a large tidal data was not required. Gelb [8] and Mizumura [15] also proved that the harmonic parameters using the Kalman filtering method [10] could be easily determined from only a small amount of * Corresponding author. Tel.: þ 886-6-255-2689; fax: þ886-6-255-2669. E-mail address: [email protected] (T.L. Lee).
historical tidal records. Yen et al. [20] utilized the Kalman filtering method in determination of parameters in the harmonic tide-level model as well. The estimation of harmonic parameters could predict accurately the tidal level using the Kalman filtering method, which is solved by the covariance matrix. However, it is necessary to determine the available parameters of the local tide before predicating the tidal level. Tsai and Lee [19] applied the backpropagation neural network (BPN) to forecast the tidal level using the historical observations of water levels without determining the harmonic parameters. However, their model is used only for the instant forecasting of tidal levels, not a long-term prediction. Besides the prediction of tidal level, supplement of tidal record is also important for a complete observation of tide database. The discontinuous observations may come from the damage of recording facilities, natural disasters or inappropriate operation and so on. The discontinuous record could either be short-term (few hours) or long-term (few months even up to one year). Thus, establishing a simple and executable supplementary model for tidal record is desired. Recently, the artificial neural network (ANN) has been
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widely applied to various areas to overcome the problem of exclusive and the non-linear relationships. For example, in water resources, French et al. [7] used the ANN to predict the rainfall intensity. Their results indicate that ANN is capable of learning the complicated relationship describing the space – time evolution of rainfall. Campolo et al. [1] applied ANN for river flood forecasting. Zhang and Stanley [21] forecasted raw-water quality parameter for the North Sakatchewan River by neural network modeling. Other successful examples of the application of neural network in water resources and hydraulics have been reported in Refs. [4,12]. In coastal engineering, Mase and Kianto [13,14] applied the ANN algorithm to assess the stability of the armor unit and the rubble-mound breakwater and estimate the wave forces acting on the structures. Later, ANN models have been further applied to calculate the tide and wave heights [3,18]. Recently, Tsai and Lee [19] used neural network for tide forecasting by using the field data of diurnal and semidiurnal tide. However, their model is only applicable for instant prediction, not long-term predication. The BPN developed by Rumelhart et al. [17] is the most representative learning model for the ANN. The procedure of the BPN repeatedly adjusts the weights of the connections in the network so as to minimize the measure of the difference between the actual output vector of the net and the desired output vector. The BPN is widely applied in a variety of scientific areas—especially in applications involving diagnosis and forecasting. The aim of this paper is to establish an ANN model for the long-term prediction and supplement of tidal data. The database of Taichung Harbor in Taiwan is used as an example to demonstrate the compatibility of the proposed model.
been well documented in the literature, only a brief is given in this section. A typical three-layered network with an input layer (I ), a hidden layer (H ) and an output layer (O ) (Fig. 1) is adopted in this study. Each layer consists of several neurons and the layers are interconnected by sets of correlation weights. The neurons receive inputs from the initial inputs or the interconnections and produce outputs by transformation using an adequate non-linear transfer function. A common transfer function is the sigmoid function expressed by f ðxÞ ¼ ð1 þ e2x Þ21 ; it has a characteristics of df =dx ¼ f ðxÞ ½1 2 f ðxÞ: The training process of neural network is essentially executed through a series of patterns. In the learning process, the interconnection weights are adjusted within input and output value. BPN is the most representative learning model for the ANN. The procedure of the BPN is the error, as the output layer propagates backward to the input layer through the hidden layer in the network to obtain the final desired outputs. The gradient descent method is utilized to calculate the weight of the network and adjust the weight of interconnections to minimize the output error. The error function at the output neuron is defined as E¼
1X ðT 2 Ok Þ2 2 k k
ð1Þ
where Tk and Ok are separately the value of target and output. Further details of the BPN algorithm can be found in Ref. [17].
3. Case study The hourly tidal data collected at Taichung Harbor, Taiwan during 1996– 1999 were used to test the accuracy of
2. Neural networks ANN is an information-processing system mimicking the biological neural network of the brain by interconnecting many artificial neurons. Since the principle of ANN has
Fig. 1. Structure of an ANN.
Fig. 2. Locations of Taichung Harbor, Taiwan (Location: (1208310 3100 E, 248170 2200 N).
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Fig. 3. Structure of the major tidal components for ANN without the hidden layer.
the proposed ANN’s model. The location of the Taichung Harbor station (1208310 3100 E, 248170 2200 N) is indicated in Fig. 2. The main component of tides in the Taichung Harbor is M2. According to the past records, its highest water level is 5.86 m, the lowest water level is 0.55 m and the average tidal range is 3.54 m. The relative root mean squared error (RMS) and correlation coefficient (CC) were used for the agreement index to present the accuracy of the present model. They are defined by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX u n u ðYðtÞ 2 Yt Þ2 u u k¼1 ð2Þ RMS ¼ u n u X 2 t ðYt Þ k¼1 n X
ðYðtÞ 2 YðtÞÞðYt 2 Yt Þ
k¼1 CC ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n X X ðYðtÞ 2 YðtÞÞ2 ðYt 2 Yt Þ2 k¼1
ð3Þ
k¼1
in which YðtÞ is the value of prediction; Yt denotes the value of observation and n is the total number of hourly tide levels. YðtÞ is the mean of prediction and Yt is the mean value of observation. It is noted that RMS is a nondimensional parameter.
However, the inclusion of additional unnecessary constituents does not significantly improve the accuracy of prediction. Therefore, appropriate tidal components must be determined at the beginning stage. As reported by Reid [16], determination of the major tidal component using the spectral methods required at least a year tidal record for shallow water with significant meteorological noise. In this study, we propose to use the corresponding weighting relations in ANN to determine the tidal components. For a tidal component with heavy weight, its effect is more important for procedure of prediction than others. To represent the relations of input layer and weighting function, the back-propagating neural network without hidden layer is used (Fig. 3). In the network structure, 69 tidal components and their corresponding cosðvi tÞ and sinðvi tÞ are used, while the tidal levels YðtÞ are used in the output layer (Table 1). Based on one-month (January 1998) and two-month (January and February 1998) tidal records, the relations between weighting function and tidal components can be determined. For example, the learning outcomes of one month and two months tidal level are presented in Figs. 4 and 5, respectively.
3.1. Determination of the tidal components In general, the prediction of tidal level can satisfy the practical requirement, if the numbers of tidal components are sufficient. Using more tidal components, the accuracy of the prediction of tidal level will certainly be enhanced.
Fig. 4. The results of the main components of tide using one month of tidal data (January 1998) for the ANN model.
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Table 1 The 69 tidal components [11] No.
Tidal component
Hz (1/hr)
Angular velocity (deg/hr)
No.
Tidal component
Hz (1/hr)
Angular velocity (deg/hr)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Z0 SA SSA MSM MM MSF MF ALP1 2Q1 SIG1 Q1 RHO1 O1 TAU1 BET1 NO1 CHI1 PI1 P1 S1 K1 PSI1 PHI1 THE1 J1 SO1 OO1 UPS1 OQ2 EPS2 2N2 MU2 N2 NU2 H1
0.00000000 0.00011407 0.00022816 0.00130978 0.00151215 0.00282193 0.00305009 0.03439657 0.03570635 0.03590872 0.03721850 0.03742087 0.03873065 0.03895881 0.04004043 0.04026859 0.04047097 0.04143851 0.04155259 0.04166667 0.04178075 0.04189482 0.04200891 0.04309053 0.04329290 0.04460268 0.04483084 0.04634299 0.07597494 0.07617731 0.07748710 0.07768947 0.07899925 0.07920162 0.08039733
0.0000000 0.0410652 0.0821376 0.4715208 0.5443740 1.0158948 1.0980324 12.3827652 12.8542860 12.9271392 13.3986600 13.4715132 13.9430340 14.0251716 14.4145548 14.4966924 14.5695492 14.9178636 14.9589324 15.0000012 15.0410700 15.0821352 15.1232076 15.5125908 15.5854440 16.0569648 16.1391024 16.6834764 27.3509784 27.4238316 27.8953560 27.9682092 28.4397300 28.5125832 28.9430388
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69
M2 H2 MKS2 LDA2 L2 T2 S2 R2 K2 MSN2 ETA2 MO3 M3 SO3 MK3 SK3 MN4 M4 SN4 MS4 MK4 S4 SK4 2MK5 2SK5 2MN6 M6 2MS6 2MK6 2SM6 MSK6 3MK7 M8 M10
0.08051140 0.08062547 0.08073957 0.08182118 0.08202355 0.08321926 0.08333334 0.08344740 0.08356149 0.08484548 0.08507364 0.11924210 0.12076710 0.12206400 0.12229210 0.12511410 0.15951060 0.16102280 0.16233260 0.16384470 0.16407290 0.16666670 0.16689480 0.20280360 0.20844740 0.24002200 0.24153420 0.24435610 0.24458430 0.24717810 0.24740620 0.28331490 0.32204560 0.40255700
28.9841040 29.0251692 29.0662452 29.4556248 29.5284780 29.9589336 30.0000024 30.0410640 30.0821364 30.5443728 30.6265104 42.9271560 43.4761560 43.9430400 44.0251560 45.0410760 57.4238160 57.9682080 58.4397360 58.9840920 59.0662440 60.0000120 60.0821280 73.0092960 75.0410640 86.4079200 86.9523120 87.9681960 88.0503480 88.9841160 89.0662320 101.9933640 115.9364160 144.9205200
As seen in Fig. 4, the one-month tidal level record can only roughly indicate if the tidal level is dominated by diurnal or semi-diurnal tides, but it cannot indicate the major tidal components. However, the learning outcome of two-month data (Fig. 5) indicates 5 –7 significant tidal components. They are: M2 (Luni-solar sei-diurnal), K1 (Luni-solar diurnal), O1 (Principal lunar diurnal), P1 (Principal solar diurnal), S2 (Principal solar), N2 (Larger lunar elliptic) and NO1 (Compound tides). These are similar
to the results from spectral analysis with two years tidal record (1995,1996), as shown in Fig. 6 [11]. In general, the number of main tidal components will directly affect the accuracy of tidal forecasting. Thus, the influence of the number of tidal components on the accuracy is examined through a parametric study here. Using the data in Taichung Harbor as an example, the RMS values for various numbers of tidal components are tabulated in Table 2. As seen in the table, the RMS is 0.0884 with five
Fig. 5. The results of the main components of tide using two months of tidal data (January and February 1998) for the ANN model.
Fig. 6. The results of the main components of tide using the spectral method based on two year data (1995 and 1996).
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Table 2 Test of various tidal components using the ANN Harbor
The name of input tidal components
Number of tidal components
RMS
Taichung Harbor
M2, S2, N2, K1 M2, S2, N2, K1, O1 M2, S2, N2, K1, O1, P1 M2, S2, N2, K1, O1, P1, K2
4 5 6 7
0.0895 0.0844 0.0860 0.1175
Table 3 The effect of the number of hidden layers Harbor
Hidden layers
RMS
Taichung Harbor
0 1 2
0.0856 0.0844 0.0859
tidal components (M2, K1, O1, S2 and N2). However, the errors increase to 0.1175, if we include another two tidal components, P1 and K2. This implies that P1 and K2 cannot improve the accuracy of the tidal forecasting. Thus, M2, K1, O1, S2 and N2 are considered as the five main tidal components for Taichung Harbor. It is noted that the determination of the above five tidal components is based on linear ANN model. Here, the linear ANN model is used as the first approximation. If a nonlinear ANN model is used to identify the tidal component, the results may be different. 3.2. The ANN’s tide model Doodson [5,6] proposed a harmonic analysis for tide forecasting. In his model, the least-squares method was used to determine the harmonic constants. These constants are further substituted into harmonic equation to determine the
tidal level. This model has been widely used because of its simplicity. Based on the harmonic theory, the vertical tidal level YðtÞ at time t at any place is expressed as following YðtÞ ¼ A0 þ
N X
ðAi cos vi t þ Bi sin vi tÞ
ð4Þ
i¼1
where A0 is the mean water level, Ai and Bi are coefficients of tide components, vi is the angular frequency of the tidal components and N is the total number of component tides. Fig. 7 illustrates the basic structure of the ANN tidal forecasting model with one hidden layer. Each tidal component corresponds to cosðvi tÞ and sinðvi tÞ: There is only one variable for output layer, i.e. the tidal level YðtÞ: 3.3. Effects of neural network structure In general, the factors which directly affect the ANN model include the number of hidden layers, learning factor (h ), momentum factor (a ), the number of training iterations (Epochs) and the number of neurons in each layer. Since the neural network is a non-linear procedure and the network parameters will affect each other, the adjustment of each parameter to optimize the whole network is not an easy task. This section discusses how the neural network structures affect the performance of the forecasting model.
Fig. 7. Structure of the tidal forecasting for an ANN.
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Table 4 The effect of the number of the hidden neurons
Table 5 The recommended structure of the ANN
Harbor
Neurons of hidden layer
RMS
Taichung Harbor
1 3 5 7 10
0.0866 0.0851 0.0847 0.0844 0.0846
The performance of the neural network structures with no hidden layers, one hidden layer, and two hidden layers with the same training parameters h ¼ 0.01 and a ¼ 0.8 and 1000 training iterations is tabulated in Table 3. The results clearly indicate that one hidden layer has the better performance. However, the influence of the number of hidden layer is insignificant. For example, the difference of RMS between without hidden layer and with one hidden layer is only 0.0012. However, the input neurons without hidden layer are not mapped with non-linear transformation, thus resulting in a slightly larger error compared to the single hidden layer. In contrast, more hidden layers lead to more complex operations for neural networks. Thus, one hidden layer is used in this study. The number of neurons in the hidden layer also affects the performance of the ANN. Table 4 shows the RMS error for various neurons structures. It shows that the forecasting performance of the ANN is improved when the number of
Harbor
Number of hidden layer
Number of neurons in hidden layer
h
a
Epoch
Taichung Harbor
1
7
0.01
0.8
1000
neurons increases. However, a large number of neurons in the hidden layer decreases the accuracy owing to overlearning. Thus, the number of neurons in the hidden layer is recommended to be seven because of the satisfactory prediction performance. The value of the learning rate (h ) will significantly affect the convergence of neural network learning algorithm and the momentum factor (a ) is used to avoid stopping the learning process at a local minimum instead of global minimum [9]. To have a better performance, either low h or high a is expected to accelerate the convergence of the training process. After some preliminary tests, we use a learning rate of 0.01 and a momentum factor of 0.8 in all training cases. The number of training iterations of 1000 is selected in all cases and the network parameters is also listed in Table 5. 3.4. Long term tide forecasting In this study, based on the aforementioned optimal neural
Fig. 8. Comparison of observed tide levels with those predicted over one year for Taichung Harbor (4/1996, 10/1996, 2/1997).
T.L. Lee et al. / Advances in Engineering Software 33 (2002) 329–338 Table 6 The performance of the one year using the different day’s measurements Harbor
Training sets
RMS
CC
Taichung Harbor
1 day (12/4/1996) 7 days (12–8/4/1996) 15 days (12–26/4/1996) 30 days (12/4–11/5/1996)
0.3815 0.2133 0.0844 0.1835
0.5136 0.8775 0.9822 0.9309
Table 7 The survey of the supplement for Taichung Harbor Case
Date of supplement
The number of days
1 2 3 4 5
1998/03/31–1998/04/01 1998/08/14–1998/08/21 1998/11/04–1998/11/11 1999/06/08–1999/06/09 1999/10/19–1999/12/31
2 8 8 2 63
network, we use different data base in the training procedure to predict the one-year tidal level in Taichung Harbor (Fig. 8). Table 6 shows a year of hourly tidal predictions at the Taichung Harbor with different training periods. The results indicate that the tidal ANN forecasting model is able to predict one-year tidal level with 15-day hourly tidal observations. Based on the 15-day collected data (12–26 April 1996), the one-year prediction of tidal level (April 1996–March 1997) against the observation is illustrated in Fig. 9. In the figure, solid lines denote the observation data, and dashed lines are the predicted values. The prediction of the present model overall agree with the observation. The correlation coefficient over one year is 0.9182, which is reasonable good. 3.5. Supplement of tidal data Besides the long-term prediction of tidal level, the
335
Table 8 The performance of the supplement using the different day’s measurements, Case 2 Harbor
Training sets
RMS
CC
Taichung Harbor
1 day (1998/8/13) 7 days (1998/8/7–1998/8/13) 15 days (1998/7/30–1998/8/13) 30 days (1998/7/13–1998/8/13)
0.1809 0.0669 0.0734 0.1052
0.5396 0.9878 0.9653 0.9371
supplement of tidal data is also important for engineering practice. In this section, we will demonstrate the application of the proposed ANN model in the supplement of tidal data. Table 7 summaries the missing data of Taichung Harbor during 1998 and 1999. As seen in the table, five sets of data are incomplete (Case 1 – Case 5 in Table 8). Using Case 2 as an example (Table 8), the verification of the proposed model is discussed in detail here. In the case, 8 days hourly tidal level data (14/08/1998 – 21/08/1998) is missing. To supply the missing data, we choose one-day, seven-day, 15-day and 30-day data for the training procedure. The results of tidal supplement are given in Table 8. This table indicates that a longer training procedure may not provide a better estimation. For example, seven-day training data provide a best estimation for Case 2. Similarly, the learning data and verification of other cases are listed in Table 9, and the results of learning procedure and prediction are tabulated in Table 10. Based on the numerical results, it is found that seven-day learning data is sufficient to provide excellent supplementary tidal data for short-term missing data (less than eight days). However, to reach reasonable accuracy (e.g. CC . 0.9, and RMS , 0.1) of supplementary data for longer period (e.g. 63 days in Case 5), a 15-day training data is required. As shown in Table 10, the correlation coefficient of Case 2 for training procedure is 0.9980, and that for the
Fig. 9. Comparison of observed tide levels with those predicted in the learning process for Taichung Harbor, Case 2 (RMS ¼ 0.0311).
Fig. 10. Comparison of observed tide levels with those supplemented in the supplementing process for Taichung Harbor, Case 2 (RMS ¼ 0.0669).
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Fig. 11. Supplement of tidal levels for Taichung Harbor, Case 1–Case 5.
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Fig. 11 (continued )
verification can reach 0.9878. The time series of learning and estimation of tidal level (including the observation data and prediction) are presented in Figs. 9 and 10. The figures clearly indicate that both learning (Fig. 9, RMS ¼ 0.031) and verification (Fig. 10, RMS ¼ 0.0669) can reach a high accuracy. This demonstrates the compatibility of the proposed ANN mode, in the learning and supplement of tidal data. Based on the aforementioned procedure, we further predict and back-estimate the missing data in Taichung
Harbor during 1998 and 1999 (Fig. 11). In these figures, the solid lines represent the supplementary data from the proposed model, and symbols are the observation data. Fig. 11 demonstrates the capacity of the proposed ANN model in supplement of tidal data for Case 1– Case 5.
4. Conclusions The conventional method of the harmonic analysis
Table 9 The performance of the supplement and test for Taichung Harbor Case
Supplement dates
Days
Training dates
Days
Test dates
Days
1 2 3 4 5
1998/03/31–1998/04/01 1998/08/14–1998/08/21 1998/11/04–1998/11/11 1999/06/08–1999/06/09 1999/10/19–1999/12/31
2 8 8 2 63
1998/03/24–1998/03/30 1998/08/07–1998/08/13 1998/10/28–1998/11/03 1999/06/01–1999/06/07 1999/10/04–1999/10/18
7 7 7 7 15
1998/04/02–1998/04/16 1998/08/22–1998/9/20 1998/11/12–1998/12/11 1999/06/10–1999/06/24 1999/07/21–1999/10/03
15 37 37 15 75
Table 10 The performance of the learning and test for Taichung Harbor Case
1 2 3 4 5
Learning
Test
RMS
CC
RMS
CC
0.0354 0.0311 0.0305 0.0374 0.0533
0.9979 0.9980 0.9984 0.9960 0.9933
0.0932 0.0669 0.0945 0.0724 0.0785
0.9778 0.9878 0.9799 0.9904 0.9934
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requires a large amount of observed tidal data in determining the total number of component tides and the harmonic parameters. In this paper, we proposed an alternative method (ANN) for forecasting and supplementing the tidal level. The case study shows that the major constituents can be obtained by using a two-month measured data. The learning rate of 0.01 and a momentum factor of 0.8 are used in all training cases. In the proposed ANN model, one hidden layer is used. The results also demonstrate that the prediction of one-year tidal level forecasting can be satisfactorily achieved with a 15-day observed data. As for the supplement of tidal data, a seven-day training data is sufficient for short-period missing data (less than eight days), while a 15-day training data is required for the supplement of a long-period missing data. Based on the examples presented, the proposed ANN for predicting and supplementing the tidal-level can be further applied to other locations in Taiwan or in the world.
Acknowledgements The authors are grateful to the Prediction Center of Central Weather Bureau, Taiwan, for supporting valuable field data.
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