Artificial neural networks in merging wind wave forecasts with field observations
O. Makarynskyy*
Western Australian Centre for Geodesy, Curtin University of Technology, GPO Box U1987, Perth 6845, Australia
*[E-mail: O.Makarynskyy@curtin.edu.au ]
Received 9 January 2006, revised 2 August 2006
An attempt to improve wind wave short-term forecasts based on artificial neural networks is reported. The novelty of the study consists in the use of relatively short time series of wave observations collected over 2 consecutive months to accomplish the tasks of wave predictions and data assimilation. Separate neural networks were developed to predict five wind wave parameters, namely, the significant wave height, zero-up-crossing wave period, peak wave period, mean direction at the peak period and directional spreading over intervals of 3, 6, 12 and 24 hours, and to correct these predictions. Data from a directional buoy were used to train and validate the networks. The results of the simulations carried out without and with the proposed methodology were favourably compared to time series of wave parameters estimated in the field. Moreover, time series plots and scatterplots of the wave characteristics as well as statistics show an improvement of the results achieved due to data merging.
[Key words: Neural methodology, wind wave parameters, directional buoy measurements, numerical wave model, wave predictions, data merging, time series ]
[IPC Code: Int.CI8. (2006) G06F 7/20; G06Q 99/00]
Introduction
The knowledge of wind wave characteristics is of great importance in ocean and coastal environmental studies as well as in industrial activities. Generally, the sea state is simulated using global and local numerical models, or with some time domain methods. Global spectral numerical models predict waves on the basis of wind forecasts using the wave action balance equation. Hence, low quality of the wind fields can lead to incorrect wave forecasts.
Moreover, a global system, which usually provides boundary conditions to a local wave model, might introduce interpolation and coarse grid errors1.
The methods of time series extension, which also find a wide use in the wave forecasting, simulate new members of time series considering solely the history of the wave parameters behavior. Despite the fact that the use of these methods is limited by presence of correlation patterns, they allow avoiding the errors typical to the numerical procedures of wind modeling and wave interpolation, and the uncertainties of wind- wave relations. To produce wave predictions stochastic models, employing the Auto Regressive Moving Average (ARMA) or the Auto Regressive
Integrated Moving Average (ARIMA)2, or involving the technique of Artificial Neural Networks (ANNs)3-5 could be used.
The ARMA and ARIMA models were evolved from the time series analysis technique. These models transform the signal into a weighted sum of previous observations and of the Gaussian white noise.
Consequently, predictions can be expressed in terms of model coefficients and residuals.
ANNs are able to solve different tasks. In particular, they allow approximation of the nonlinear mathematical dependencies between the elements of a system. After appropriate training over a number of input-output patterns belonging to this system, feeding a network with an independent input produces an appropriate output.
Comparing the predictions of significant wave heights obtained with the stochastic and neural network techniques, Deo & Sridhar Naidu6 and Agrawal & Deo2 noticed a better performance of the neural methodology, especially when the short prediction intervals of 3 and 6 hours were considered.
In order to improve the quality of the wave numerical forecasts, several data assimilation techniques were developed, like the optimum interpolation7 or the Kalman filtering8. Following the
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idea that a feed-forward artificial neural net with an arbitrary number of processing units in a single hidden layer is a universal function approximator9, Babovic10 presented a hybrid forecast model that produced accurate predictions of sea currents merging observations of the currents, sea levels and wind with neural networks.
Fig. 1⎯Location of the Sines buoy station (star) at the Portuguese coast
A methodology of wave predictions’ improvement based on artificial neural networks is presented and discussed in this paper. Fundamentals of artificial neural networks are available in Bishop11, and Haykin12 while some particular applications of the neural technique to the wave simulation problems are reported by Abrahart et al.13, or Makarynskyy et al.14-16.
The proposed methodology allows merging observations of the significant wave height Hs, the zero-up-crossing wave period T02, the peak wave period Tp, the mean direction at the peak period Dp, and of the directional spreading with short-term
“initial” or “basic” forecasts of these parameters.
These preliminary simulations are called so to discriminate them from the modified “final”
predictions produced as a result of this merging. In this particular study, the basic forecasts were also obtained using artificial neural networks. However, this implies that forecasts coming from different sources can be used and corrected.
Data used
Time series of the significant wave height, zero-up- crossing wave period, peak wave period, mean direction at the peak period, and of the directional spreading for the time interval 4 December, 1999, 0000 UTC – 31 January, 2000, 2100 UTC were employed in this study17. The observations were obtained from a “WAVEC” buoy, which is moored in 97 m depth offshore of Sines Harbor (37o 55’ 16” N, 8o 55’ 44” W) (Fig. 1) and is maintained by the Hydrographic Institute of the Portuguese Navy. The values of the wave characteristics were estimated18 every three hours starting from 0000 UTC (Fig. 2).
During the time under consideration, the Hs changed from 0.5m to 4.5m; the T02 and Tp oscillated from 3 to 12s and 4 to 18s, respectively; the Dp was between 210o and 330o; and the directional spreading stayed within the limits 15-50o.
In order to train two different sorts of neural networks and to test the modified forecasts, each time series was divided into three non-overlapping data sets containing from 150 to 160 data points. Data set 1
was used to train the neural networks producing the basic forecasts. Data set 2 served for training the nets, which were assimilating a current observation of wave parameter to correct the initial prediction. Data set 3 provided an independent validation of both the basic and the final forecasts.
Forecasting of wave parameters
Preliminary studies4,16 have already been performed using the above mentioned field measurements. These studies have shown that the commonly used three- layer feed-forward back-propagation networks with a non-linear differentiable log-sigmoid transfer function in the hidden layer and linear transfer function in the output layer are useful at simulating the wave parameters off of the Portuguese west coast.
Networks with the same structure were applied in this study.
Abrahart et al.13 proposed to use saliency analysis for determining the optimal architecture of a net developed to solve a particular task. Makarynskyy et al.15-16,19 demonstrated that networks of relatively simple architecture (for instance, with 4 input nodes : 17 hidden nodes : 4 output nodes used in the simulations of Hs) produce results of the same or even higher accuracy (in terms of the root mean square error RMSE, correlation coefficient R and scatter index SI) than more sophisticated nets (with 16 input neurons : 33 hidden neurons : 4 output neurons).
A separate network was developed to simulate each wave parameter with each lead-time (twenty artificial neural nets in total). Following the results obtained by Makarynskyy et al.16, the artificial neural networks designed to provide the basic forecast of the significant wave height and zero-up-crossing wave period over short periods (3 and 6 hours) had two
Fig. 2⎯Time series plots of Hs, T02, Tp, Dp and the directional spreading estimated from buoy observations for the period December 4, 1999-January 31, 2000
units in the input layer, incorporating the current and the previous observations. The networks devised to predict these integral wave parameters over longer intervals (12 and 24 hours) had four input nodes spanning 12 hours history, as well as all the nets simulating the peak wave periods, mean directions and directional spreading.
Twenty additional neural networks were developed to merge wave parameters estimated from field measurements with the results of predictions for four lead times. Each network assimilating (merging) the data had two nodes in the input layer employing the current observation of wave parameter and the basic forecast.
The number of nodes in the hidden layer h of all these networks was obtained using an empirical expression h = 2z + 114, where z is the number of input nodes. All these neural networks had one output node, producing only one value in both the basic and the modified predictions. Each network was trained in 500 epochs with the resilient backpropagation algorithm. After this number of epochs the gradient of the performance function (averaged squared error between the network outputs and the target outputs) was close to zero.
Figure 3 displays time series plots of the significant wave heights obtained from observations and predicted without and with data merging as well as scatter diagrams of the observed versus simulated values. The graphical representation of these results demonstrates that the assimilation of the current value improves the forecasts over all time periods considered. The verification statistics of the basic and
the final forecasts allow assessment of this correction quantitatively (Table 1). Values of the RMSE and the SI were halved with a 3-hour lead-time compared to the basic forecast. The differences diminish with larger prediction intervals. The increase in the correlation coefficient is 0.02-0.04 for warning times of 3-12 hours, reaching 0.07 for predictions over 24 hours in advance. The significant wave heights simulated in both the basic and in the final forecasts with leading times 12 and 24 hours are generally overestimated.
When the data merging procedure employed to modify the basic forecasts of the zero-up-crossing wave period (Fig. 4, Table 2), similar considerations apply. The values of RMSE and SI computed for the final forecast over 3 hours are three times lower than those obtained from the basic simulations. With an increase in the prediction interval, the differences in these statistics disappear, although the statistics themselves remain fairly good. The correlation coefficient from the assimilating data simulations with 3–12 hours lead time are larger by 0.08–0.09 than those coming from the initial predictions. This does not apply to the forecast with a 24-hour interval, where the simulated values hardly follow the observed peaks, and the improvement is limited to 0.03.
As noticed by Makarynskyy et al.16 and Makarynskyy & Makarynska5, the peak wave period is the most difficult variable to predict. The merging procedure application partially fixes the problem (Fig. 5), only when a 3-hour lead-time is concerned.
The root mean square errorand the scatter index of
Table 1⎯Verification statistics of Hs simulations with different lead times. Here RMSE is the root mean square error (metres, m), R is the correlation coefficient, SI is the scatter index. Subscripts “b”
and “t” stand for basic and target forecast, respectively
RMSEb RMSEt Rb Rt SIb SIt
+3 0.16 0.08 0.97 0.99 0.13 0.06
+6 0.28 0.16 0.93 0.97 0.23 0.13
+12 0.48 0.39 0.85 0.87 0.39 0.32
+24 0.77 0.77 0.67 0.74 0.62 0.62
Table 2⎯Verification statistics of T02 simulations with different lead times. Here RMSE is the root mean square error (seconds, s), R is the correlation coefficient, SI is the scatter index. Subscripts “b”
and “t” stand for basic and target forecast, respectively
RMSEb RMSEt Rb Rt SIb SIt
+3 0.87 0.29 0.90 0.99 0.13 0.04
+6 1.09 0.80 0.84 0.92 0.17 0.12
+12 1.42 1.24 0.72 0.80 0.22 0.19
+24 1.69 1.72 0.59 0.62 0.26 0.26
Fig. 3⎯Significant wave heights and its scatterplots obtained from Sines buoy and simulated by neural nets for different lead times for the period January 12-31, 2000
Fig. 4⎯Zero-up-crossing wave periods and its scatterplots obtained from Sines buoy and simulated by neural nets for different lead times for the period January 12-31, 2000
Fig. 5⎯Peak wave periods and its scatterplots obtained from Sines buoy and simulated by neural nets for different lead times for the period January 12-31, 2000
the final forecast are four times smaller compared to the basic prediction, while the correlation coefficient is 0.33 higher (Table 3).
Scatter diagrams of the observed versus simulated mean directions of the peak period and values of directional spreading are presented in Figs 6 and 7.
The statistics of the mean direction of the peak period forecasts (Table 4) show a significant improvement (from 0.12 to 0.51) in the correlation coefficient when employing the data assimilation procedure. That is not true for the root mean square error and scatter index.
Analysis of the statistics calculated from simulations of the directional spreading (Table 5) reveals considerable improvements for the predictions one step ahead only.
The relatively low accuracy of the predictions over larger lead times can probably be a result of abrupt changes in the observed values of the wave parameters occurring in the vicinity of the 350th data point (Fig. 2). In turn, these changes could be induced by a change of the direction of wave approach from northwest to southwest and west reflected in the mean direction of the peak period. Due to these variations, the third data set contains the lowest minima of all wave parameters as well as the lowest mean values of Hs and Dp, and the highest maxima of T02 and directional spreading (Table 6). The maxima of Tp
and Dp are on the upper observed limit. These peculiarities contribute to the largest differences between the maxima and minima of the zero-up-
Table 3⎯Verification statistics of Tp simulations with different lead times
RMSEb(s) RMSEt(s) Rb Rt SIb SIt
+3 1.57 0.42 0.65 0.98 0.12 0.03
+6 1.77 1.58 0.52 0.65 0.14 0.12
+12 1.94 1.96 0.33 0.38 0.15 0.15
+24 2.12 2.03 0.06 0.13 0.16 0.15
Fig. 6⎯Scatter plots of the mean direction of the peak period obtained from Sines buoy and simulated by neural nets for different lead times for the period January 12-31, 2000
Fig. 7⎯Scatter plots of the directional spreading obtained from Sines buoy and simulated by neural nets for different lead times for the period January 12-31, 2000
Table 4⎯Verification statistics of Dp simulations with different lead times. Here RMSE is the root mean square error (degree, o)
RMSEb RMSEt Rb Rt SIb SIt
+3 14.53 13.64 0.69 0.81 0.05 0.05 +6 19.96 17.65 0.58 0.71 0.07 0.06 +12 22.11 19.18 0.14 0.65 0.08 0.07 +24 17.67 19.55 0.38 0.51 0.06 0.07 Table 5⎯Verification statistics of directional spreading simulations with different lead-times
RMSEb(o) RMSEt(o) Rb Rt SIb SIt
+3 5.30 0.77 0.28 0.99 0.17 0.03
+6 4.88 5.11 0.38 0.39 0.16 0.16
+12 5.30 5.07 0.33 0.42 0.17 0.16
+24 5.70 6.50 0.15 0.17 0.18 0.21
Table 6⎯Values of the minimum (Min), mean and maximum (Max) calculated for three data sets, which were used for training and validation of neural networks
Data set 1 Data set 2 Data set 3
Min Mean Max Min Mean Max Min Mean Max
crossing wave period, peak wave period, mean wave direction at the peak period, and of the directional spreading.
Hence, the forecasting neural networks perform successfully with patterns not involved in the learning sessions. Taking into account the fact that feeding a network with a larger number of training patterns usually contributes to more accurate simulations, better results are expected when longer learning time series are engaged.
The described methodology could involve basic forecasts coming from artificial neural networks or from other source, such as a global wave model. The final forecasts of Hs, Tp, Dp and the directional spreading produced by merging real-time wave observations can serve as the open boundary wave conditions to form a parametric spectral input to, for instance, the SWAN model20. This third generation wave model is suitable for describing the propagation of wind-generated waves from the offshore to the surf-zone. Thus, following the steps of basic and final forecasts and, further, applying a wave model, improved predictions of the nearshore wave conditions can be obtained. Quality of such nearshore forecasts can be assessed by comparing the model output with observations.
Conclusion
The accuracy of the wave parameter predictions over shorter time periods of three and six hours was considerably improved by using the data merging procedure. However, compared to the basic simulations, the final forecasts over larger time scales of 12 and 24 hours generally do not show any superiority. This can be a consequence of an abrupt change in the values of wave parameters in the data sets used for validation purposes. Feeding the networks with a larger number of training patterns should elevate the quality of the simulations.
The methodology provides improved open boundary conditions for a wave model operating nearshore. The results of numerical wave model experiments are to be compared with field measurements.
Acknowledgement
The author is grateful to the Instituto Hidrografico of the Portuguese Navy for making available the buoy data. The author would like to thank Dr. Serge Sutulo and Prof. Paul Work for their helpful suggestions.
Some part of this work was completed during author’s Post-Doctoral Fellowship from the “Fundação para a Ciência e Tecnologia” of the Ministry of Science and High Education of Portugal held at the Instituto Superior Tecnico, Lisbon, Portugal.
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