A Tide Level Time Series Prediction Method Based on Deep Learning

Abstract

The present invention belongs to the field of time series prediction and relates to a tidal level time series prediction method based on deep learning. The present invention adopts the HA-DeepAR model. First, harmonic analysis is performed on the preprocessed tidal level time series to calculate tidal harmonic constants, and the astronomical tide information in the tidal level is calculated using the tidal harmonic constants. For the residual water level information, the DeepAR model is used for prediction. Finally, the predicted astronomical tide level and the residual water level are superimposed to obtain the total tidal level prediction value, and the confidence interval of the tidal level is given to facilitate risk assessment.

Description

Technical Field

[0001] The present invention belongs to the field of time series prediction, and relates to a tidal level time series prediction method based on deep learning. Background Art

[0002] Tides are a special type of seawater fluctuation, mainly affected by the sun and the moon, and have a certain periodicity. The tidal values at high water levels are usually up to several meters, posing a certain threat to the navigation of cargo ships and vessels. Tides not only affect the activities of people in coastal areas, but also affect the habitat environment of local organisms. Accurately estimating local tides can help people schedule port cargo handling, operate offshore platforms, monitor the marine environment and resource development at appropriate times. Therefore, accurate tide estimation is very important.

[0003] The theory of tidal equilibrium was proposed by Newton in his book "Principia" published in 1687. The proposal of this theory is based on the law of universal gravitation. Darwin used his theory to estimate the tides of a simple seabed environment in coastal areas through harmonic analysis based on tidal forces. Doodson further used a large amount of tidal observation data and adopted the method of least squares of the mean square to estimate the complex tides in shallow water. Kalman made a more effective tidal level estimation through the Kalman filtering method, in which only a small amount of historical tidal level data was used. Yen et al. tried to use the Kalman filter to estimate the actual sea conditions.

[0004] In the 1980s, with the development of machine learning, neural networks were adopted by many researchers as a new learning method for tide prediction. Vaziri used an artificial neural network (ANN) and an autoregressive integrated moving average (ARIMA) model to estimate the surface water level of the Caspian Sea. Deo and Chaudhari further found that the tidal values estimated at subordinate stations using an ANN model were more accurate than traditional linear regression methods. Tsai and Lee began using the BP algorithm for tide estimation, setting the input of the neural network to the observed tidal values and corresponding residual values for the previous two hours, and the output to the estimated tidal value for the next hour. However, this ANN method only guaranteed accuracy for short-term tide estimation. Lee therefore proposed a long-term tide estimation method that combined the BP neural network with traditional harmonic analysis, using the main harmonic components as the neural network input. Rajasekaran et al. introduced the FN (functional network) and SLNN (sequential learning neural network) models for long-term tide estimation based on the tidal input of the previous two time steps. Meena and Agrawal combined a feedforward-backward propagation (FFBP) network model with the modified Newton method (LM) to estimate tide values at distant tide stations based on data from local tide stations. El Diasty proposed a model combining harmonic analysis (HA) and wavelet networks (WN) for accurately estimating sea level, concluding that the HA and WN models were superior to either HA or WN. Riazi developed a well-adapted deep learning method that accurately predicts tide levels using the Earth's position, Earth's rotation, the Moon's position, and other factors as input.

[0005] While numerous studies have applied neural networks to tide prediction, achieving good accuracy, most of these efforts rely on direct tide prediction. Actual tide levels not only experience periodic fluctuations due to celestial motion but are also influenced by short-term meteorological factors. Meteorological factors are a primary component of tide levels, while meteorological factors manifest as disturbances to theoretical tide levels. For practical maritime activities, the primary risk comes from tidal surges caused by other nonlinear factors, so this aspect warrants separate discussion. Furthermore, most current tide predictions are point-based, lacking real-time accuracy, making it difficult to assess risk and provide more effective information for navigation and maritime operations. Summary of the Invention

[0006] To address these issues, this paper proposes a tide prediction method based on a harmonic analysis-deep autoregressive network (HA-DeepAR). Tide data consists of four components: mean sea level, astronomical tide level, residual water level, and observation error. Without considering observation error, mean sea level is the average of historical tide levels, so predicting tide level essentially involves predicting both the astronomical tide level and the residual water level.

[0007] In the present invention, first, the tidal observation data of the tide gauge station are processed to fill in the missing values, and then the astronomical tide level in the tide level is calculated by using the harmonic analysis method. The difference between the astronomical tide level and the actual tide level is called the residual water level. The residual water level, also known as the additional or abnormal water level, refers to the sea surface disturbance caused by random factors such as meteorology, and is mainly composed of two parts: the short-term water level anomaly caused by short-period meteorological factors such as wind, air pressure, and precipitation, and the seasonal anomaly of the sea surface caused by climate factors. The residual water level part is predicted by using the DeepAR network. Finally, the astronomical tide level and the residual water level are superimposed to obtain the final tide level prediction result.

[0008] The technical solution of the present invention is as follows:

[0009] A method for predicting tidal time series based on deep learning, the specific steps are as follows:

[0010] (1) Data preprocessing. The mean imputation method is used to fill in the possible missing values in the original tidal historical data by using the average value of the tide level.

[0011] (2) Tidal harmonic analysis. The purpose is to calculate the harmonic constants of each astronomical tidal component according to the tidal observation data; the expression of any astronomical tidal component is:

[0012] ξ = fH cos(σt + V + u)

[0013] where, H is the tidal component amplitude, σ is the tidal component frequency, V is the astronomical initial phase angle, and f and u respectively represent the correction values for the tidal component amplitude H and the astronomical initial phase angle V caused by the 18.61-year change of the lunar orbit. According to the equilibrium tide theory, at the upper culmination, that is, when the phase angle σt + V + u = 0, high tide should occur. However, due to reasons such as seabed friction and seawater inertia, the high tide of the actual tide near the shore will occur slightly later, that is, there is a high tide interval. Therefore, the retardation angle K needs to be additionally introduced into the phase angle.

[0014] Finally, the actual astronomical tidal component expression is obtained as:

[0015] ξ = fH cos(σt + V + u - K)

[0016] where, the tidal component amplitude H and the retardation angle K are called the harmonic constants of the astronomical tidal component and need to be solved according to the tide level data, and other parameters can be directly deduced from astronomical elements.

[0017] The actual tide level ξ(t) is composed of the mean sea level, multiple astronomical tidal components, and the residual water level superimposed together:

[0018] <00,00087>

[0019] where: a0 is the mean sea level during the observation period, R , , , ,

[0019] , , ,

[0016] , i ,

[0015] , ,

[0014] ,

[0018] ,

[0013] ,

[0017] is the astronomical tidal component amplitude, σi is the angular velocity of the astronomical partial tide, θ i is the initial phase angle of the astronomical partial tide, m is the number of astronomical partial tides, and γ(t) is the residual water level.

[0020] Substituting the astronomical partial tide expression into the actual tide level expression gives:

[0021]

[0022] Comparing the two actual tide level expressions gives:

[0023]

[0024] where f i , V i and u i can all be directly calculated from astronomical elements. Therefore, calculating the harmonic constants H i and K i actually means calculating R i and θ i . For convenient calculation and solution, using trigonometric identities, each astronomical partial tide is expressed as a sine function and a cosine function with the same frequency:

[0025]

[0026] where

[0027]

[0028] Finally, the astronomical tide level is decomposed into 2m sine functions and cosine functions with fixed frequencies, different amplitudes but initial phases of 0.

[0029] Finally, the least squares method is used to fit the above formula to the actual tide level data, and then a i and b i are solved. Finally, the tidal harmonic constants H i and K i of each partial tide are obtained, and the astronomical tide level can be calculated. By subtracting the astronomical tide level from the actual tide level, the residual water level information is obtained.

[0030] (3) Establish a deep autoregressive model.

[0031] z i,t represents the i-th time series at time t, that is, the residual water level time series, and x i,t represents the covariate of the i-th time series at time t, that is, other ocean meteorological information related to the tide level. The goal of the DeepAR model is to establish each residual water level time series at t0 and later on the premise of the historical residual water level data Conditional probability distribution:

[0032]

[0033] Since DeepAR adopts an autoregressive network architecture, it is assumed that the above distribution can be written in the following likelihood form:

[0034]

[0035] where h i,t = h(h i,t-1 , z i,t-1 , x i,t ) is the output of the autoregressive network, h(·) represents the function inside the autoregressive network. The autoregressive network structure of the present invention adopts a long short-term memory network LSTM (Long Short-Term Memory), and θ(·) represents the mapping of the output to the parameters of the given distribution, such as the mapping of the output to the mean μ and standard deviation σ of the Gaussian distribution. During training, the solution of the residual water level distribution function is achieved by maximizing the above likelihood function, that is, the loss function is:

[0036]

[0037] The difference in the prediction process compared with training is that z i,t-1 is not the actual data, but the sampling mean of the previous prediction distribution.

[0038] (4) Add the astronomical tide level obtained in step (2) to the residual water level information predicted at the same moment in step (3) to obtain complete tide level prediction information.

[0039] The beneficial effects of the present invention are as follows: (1) By combining the HA algorithm with the DeepAR model, the tide level information is split, and the astronomical tide and the residual water level are predicted separately. Compared with directly applying an artificial neural network to predict the entire tide level, the present invention can effectively perform feature decomposition and extraction to improve the prediction accuracy. (2) Applying the DeepAR model to predict the tide level can effectively handle the seasonal fluctuations of tides, and since the output of DeepAR is a probability distribution, it can well predict the fluctuations of the tide level, conduct a risk assessment on the accuracy of the tide level, and provide more valuable information for offshore operations and marine activities. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 is a flowchart of a method for predicting tide level time series based on deep learning according to the present invention.

[0041] Figure 2 is a composition diagram of the tide level.

[0042] Figure 3It is the structural diagram of the DeepAR model.

[0043] Figure 4(a) shows the predicted tide level results of the Pohnpei tide station with a prediction step of 12 hours.

[0044] Figure 4(b) shows the predicted tide level results of the Pohnpei tide station with a prediction step of 24 hours. Specific implementation manners

[0045] The specific implementation manners of the present invention will be further described below in combination with the accompanying drawings and technical solutions.

[0046] The present invention proposes a method for predicting tide level time series based on deep learning. The overall method flow chart is as Figure 1 shown, Figure 2 which is the tide level composition diagram. The specific steps are as follows:

[0047] (1) Data preprocessing. The historical tide level data can be obtained from the NOAA Data Center of the United States National Oceanic and Atmospheric Administration. The observation interval of the tidal data is 1 hour, and the total observation duration of the tidal data is 3 years. There may be missing values in the original data, and the missing values need to be filled. The present invention adopts the mean imputation method and uses the average value of the tide level for filling.

[0048] (2) Tidal harmonic analysis. The purpose is to calculate the harmonic constants of each astronomical partial tide according to the tidal observation data; the expression of any astronomical partial tide is:

[0049] ξ = fH cos(σt + V + u)

[0050] where, H is the partial tide amplitude, σ is the partial tide frequency, V is the astronomical initial phase angle, and f and u respectively represent the correction values for the partial tide amplitude H and the astronomical initial phase angle V caused by the 18.61-year change of the lunar orbit. According to the equilibrium tide theory, at the upper culmination, that is, when the phase angle σt + V + u = 0, a high tide should occur. However, due to reasons such as seabed friction and seawater inertia, the high tide of the actual tide near the shore will occur slightly later, that is, there is a high tide interval. Therefore, a retardation angle K needs to be additionally introduced into the phase angle.

[0051] Finally, the actual astronomical partial tide expression obtained is:

[0052] ξ = fH cos(σt + V + u - K)

[0053] where the partial tide amplitude H and the retardation angle K are called the harmonic constants of the tidal astronomical partial tide and need to be solved according to the tide level data, and other parameters can be directly deduced according to astronomical elements.

[0054] The actual tide level ξ(t) is composed of the mean sea level, multiple astronomical partial tides, and the residual water level superimposed:

[0055]

[0056] where: a0 is the average sea level during the observation period, R i is the amplitude of the astronomical partial tide, σ i is the angular velocity of the astronomical partial tide, θ i is the initial phase angle of the astronomical partial tide, m is the number of astronomical partial tides, and γ(t) is the residual water level, generally referring to the water level change caused by the change of the water level meteorological condition.

[0057] Substitute the previously obtained astronomical partial tide expression into the actual tide level expression to get:

[0058]

[0059] By comparing the above two equations, we can get:

[0060]

[0061] The f i , Vi i and u i in the above system of equations can all be directly calculated from astronomical elements. Therefore, calculating the harmonic constants H i and K i is actually to calculate R i and θ i . For the convenience of calculation and solution, using trigonometric identities, each astronomical partial tide is expressed by sine and cosine functions with the same frequency:

[0062]

[0063] where

[0064]

[0065] Finally, the astronomical tide level is decomposed into 2m sine and cosine functions with fixed frequencies, different amplitudes but initial phases of 0.

[0066] Finally, use the least squares method to make the above formula fit the actual tide level data, and then solve for the above a i and b i , and finally obtain the tidal harmonic constants H i and K i of each partial tide, and then the astronomical tide level can be calculated. Subtract the astronomical tide level from the actual tide level to obtain the residual water level information.

[0067] (3) Establish a deep autoregressive model. The prediction target of classical deep learning methods is usually the value of the sequence at each time step. In contrast, the prediction target of the deep autoregressive model (DeepAR) based on neural networks is the probability distribution of the value of the sequence at each time step. DeepAR is an autoregressive RNN time series model, with a recurrent neural network model containing hidden states inside. It can effectively learn the global model from relevant time series and can learn complex patterns, such as the seasonality of data and the growth of uncertainty over time. Tides are generated by the gravitational force of celestial bodies on seawater. As the seasons change, meteorological factors will also change accordingly, which in turn affects the residual water level. Therefore, the tide level change has strong seasonality, and DeepAR can well perceive and learn this feature.

[0068] The structure of the DeepAR model is as Figure 3 shown. By giving the observed values of the sequence, the future observed values are predicted. The left side is the training process, and the right side is the prediction process. In the figure, z i,t represents the i-th time series at time t, that is, the residual water level time series, and x i,t represents the covariates of the i-th time series at time t, that is, other marine meteorological information related to the tide level. The goal of the DeepAR model is to establish the conditional probability distribution of each time series after t0 on the premise of giving the historical residual water level data before t0:

[0069]

[0070] Because DeepAR adopts an autoregressive network architecture, it is assumed that the above distribution can be written in the following likelihood form:

[0071]

[0072] where h i,t = h(h i,t-1 , z i,t-1 , x i,t ) is the output of the autoregressive network. h(·) represents the function inside the autoregressive network. In this invention, the autoregressive network structure adopts the long short-term memory network LSTM (Long Short-Term Memory). θ(·) represents the mapping of the output to the parameters of the given distribution, such as the mapping of the output to the mean μ and standard deviation σ of the Gaussian distribution. During training, the solution of the residual water level distribution function is achieved by maximizing the above likelihood function, that is, the loss function is:

[0073]

[0074] The difference between the prediction process and training is that zi,t-1 Not the actual data, but the sampling mean of the previous predicted distribution.

[0075] (4) Add the astronomical tide levels obtained in step (2) to the residual water level information predicted at the same moment in step (3) to obtain the complete tide level prediction information.

[0076] The specific implementation steps of this embodiment are as follows:

[0077] 1. Experimental equipment and environment configuration

[0078] Software system: Linux Ubuntu 16.04 LTS Server system

[0079] Programming language: Python 3.8

[0080] Deep learning framework: GluonTS 0.8.1

[0081] 2. Experimental method

[0082] First, obtain the historical observation data of the tide gauge station from the NOAA Data Center of the United States. The tidal observation interval is 1 hour, the unit of tide height is millimeter, and the overall time range of the dataset is 3 years. First, perform statistics and collation on the original data, fill in the missing tidal observation data by the method of mean, and divide the dataset.

[0083] After data preprocessing, it is necessary to solve the astronomical tide information of the tide gauge station. ttide is a classic tidal analysis toolkit. It can decompose the tide level time series into cosine functions of different frequencies, that is, different astronomical partial tides, and solve the harmonic constants of the partial tides. Using the solved harmonic constants, by the harmonic analysis method, the partial tides are superimposed to obtain the historical astronomical tide level, and the difference is taken with the actual tide level observation data to obtain the historical residual water level. Taking the Pohnpei tide gauge station as an example, the tide level sequence from 2016 to 2017 is decomposed, and a total of 68 tidal harmonic constants of different partial tides are obtained. Then, using the harmonic analysis method, the residual water level information of the training set and the test set is calculated. The test set is used to train the DeepAR model, and the test set is used to evaluate the model effect.

[0084] Input the residual water level information into the DeepAR model for model training. DeepAR is a time series prediction method based on deep learning proposed by Amazon in 2017 and has been integrated into Amazon SageMaker and GluonTS. The former is the machine learning cloud platform of AWS, and the latter is the open-source time series prediction tool library of Amazon. In this invention, the GluonTS tool library is used to call the DeepAR model, and the input step is the tidal residual water level data of the previous 24 hours. Since the tidal rise cycle is generally 24 hours, the prediction steps are 12 hours and 24 hours respectively. Assume that the residual water level follows a normal distribution, that is, θ(h i,t ,Θ) = N(μ,σ), so the DeepAR likelihood function l(z i,t |θ(h i,t ,Θ)) is:

[0085]

[0086] where z is the residual water level, μ is the mean of the normal distribution, and σ is the standard deviation of the normal distribution.

[0087] Since the predicted value is a probability distribution, the predicted tide level value at the prediction point is represented by sampling and averaging. The number of sampling specimens is 100. Finally, the root mean square error RMSE is used as the model evaluation index:

[0088]

[0089] where Z true represents the actual tide level observation data, and Z prediction represents the model predicted value. The root mean square error measures the deviation between the predicted value and the true value. Compared with the absolute error, it is more sensitive to outliers and can better reflect the robustness of the model.

[0090] The experimental results are shown in Figure 4(a) and Figure 4(b). The tide level data is the tide level observation data of the Pohnpei tide station on December 31, 2018. The prediction durations of the two figures are 12 hours and 24 hours respectively, and the root mean square errors of the predictions are 0.09 and 0.14 respectively. It can be seen from the figure that the predicted tide level fits well with the actual tide level, and all actual tide level data are within the 90% confidence interval of the predicted distribution, indicating that the method of this invention can better predict the tide level and provide the tide level prediction interval, providing a basis for risk assessment of marine activities and offshore operations.

Claims

1. A tidal time series prediction method based on deep learning, characterized in that: The specific steps are as follows: (1) Data preprocessing: The mean interpolation method is used to fill in the missing values in the original tide level historical data using the average value of the tide level; (2) Tidal harmonic analysis: The purpose is to calculate the harmonic constant of each astronomical tidal component based on tidal observation data. The expression for any astronomical tidal component is: ξ=fH cos(σt+V+u) Where H is the tidal amplitude, σ is the tidal frequency, V is the astronomical initial phase angle, and f and u represent the corrections to the tidal amplitude H and astronomical initial phase angle V, respectively, caused by the 18.61-year variation in the lunar orbit. According to equilibrium tide theory, high tide should occur at the upper transit, when the phase angle σt+V+u=0. However, due to seabed friction and seawater inertia, the actual high tide near the shore occurs slightly later, i.e., there is a high tide gap. Therefore, an additional lag angle K is introduced into the phase angle. Finally, the actual astronomical tidal expression is obtained as follows: ξ=fH cos(σt+V+uK) Among them, the tidal amplitude H and the retardation angle K are called the harmonic constants of the astronomical tidal components, which need to be solved based on the tide level data. Other parameters can be directly calculated based on astronomical elements. The actual tide level ξ(t) is the superposition of mean sea level, multiple astronomical tides and residual water level: Where: a0 is the average sea level during the observation period, R i is the astronomical tidal amplitude, σ i is the astronomical tidal angular velocity, θ i is the initial phase angle of the astronomical tide, m is the number of astronomical tides, and γ(t) is the residual water level; Substituting the astronomical tidal expression into the actual tidal expression yields: Comparing the two actual tide level expressions, we get: Among them, f i , V i and u i All of these can be directly calculated based on astronomical elements, so the calculation of the harmonic constant H i and K i In fact, we need to calculate R i and θ i To facilitate calculation and solution, trigonometric identities are used to represent each astronomical tide using sine and cosine functions of the same frequency: in Finally, the astronomical tide level was decomposed into 2m sine and cosine functions with fixed frequencies, different amplitudes but initial phases of 0; Finally, the least squares method is used to fit the above formula to the actual tide data, and then the solution is obtained. i and b i Finally, the tidal harmonic constant H of each tide is obtained i and K i , the astronomical tide level can be calculated, and the residual water level information can be obtained by subtracting the actual tide level from the astronomical tide level; (3) Establish a deep autoregressive model: z i,t represents the i-th time series at time t, i.e. the residual water level time series, x i,t Represents the covariate of the i-th time series at time t, that is, other ocean meteorological information related to the tide level; the goal of the DeepAR model is to obtain the historical residual water level data before the given time t0. Under the premise of establishing each residual water level time series from t0 onwards The conditional probability distribution of : Because DeepAR uses an autoregressive network architecture, it is assumed that the above distribution is written in the following likelihood form: where h i,t =h(h i,t-1 ,z i,t-1 ,x i,t ) is the output of the autoregressive network, h(·) represents the function inside the autoregressive network, and θ(·) represents the mapping of the output to the parameters of the given distribution. During training, the residual water level distribution function is solved by maximizing the likelihood function, that is, the loss function is: The difference between the prediction process and the training process is that z i,t-1 Not the actual data, but the sampling mean of the last predicted distribution; (4) The astronomical tide level obtained in step (2) is accumulated with the residual water level information predicted at the same time in step (3) to obtain complete tide level prediction information.

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