Application of a deep learning technique to the problem of oil spreading in the Gulf of Thailand

Abstract

One of the important mechanisms of the oil weathering processes (OWP) is spreading of oil spills. This mechanism is the horizontal expansion of the oil slick with inertia-gravity, gravity-viscosity, and viscous-surface tension. In the prediction of spreading, the surface of the slick can be considered as an ellipse where the major axis is in the direction of the wind. Ocean wave models, which account for the interaction between wind and waves, can be used to predict the state of the sea including wind direction in two dimensions where the wave spectrum is allowed to evolve freely with no constraints on the spectral shape. However, the wave model simulation for long duration is time-consuming. In this study, the technique of deep learning, a part of the machine learning method, is implemented to obtain a model used to get quick prediction of the wind direction. The technique uses outputs from an ocean wave model and applies the multivariate time series to obtain a linear relationship among multiple time series of wind prediction from the wave model. The wind forecast is taken as inputs to the deep learning model. Some of these inputs that are significant are selected by using the sigmoid function which is an activation function. The minimum error of prediction from the deep learning model is obtained by the gradient descent method. The numerical results of the prediction spreading of oil spill in the Gulf of Thailand based on the wind prediction by the deep learning technique are presented.

1 Introduction

After crude oil leaks on the sea surface due to the crude oil exploration, crude oil transportation or accidents from the explosion of the oil rig, it can affect and harm the environment for a long period of time. When crude oil spills into the sea, it can cause the oil weathering processes (OWP) [1] which produce natural, physical, and chemical changes. They are scattered in a thin layer called oil slicks.

A review of oil spill modeling, conducted by the American Society of Civil Engineers Task Committee on Modeling of Oil Spills (ASCE) [2], focuses on the oil spill processes for the real-time models, emergency planning, and risk assessment. When crude oil is spilled, protective measures need to be taken to reduce the impact. To take these measures, we must be able to predict the short-term and long-term behavior of the oil spill using basic analytical techniques.

The input data required by oil spill models are wave height, wave direction, wind speed, and wind direction. These data can be obtained from ocean wave models, for example, the wave modeling (WAM) model. The WAM model was first developed by a group of researchers [3] in Norway in 1988. In this study, a technique of deep learning is implemented by using output data from the ocean wave model. This would improve accuracy and reduce time consumption in wave simulation based on the ocean wave model. In [4], Kanbua and Chuai-Aree studied the sea wave generated by tropical cyclones in the Gulf of Thailand. Their work was carried out by using the cycle 4 version of the WAM model. The model domain covered latitudes 5N–15N and longitudes 95E-105E, and the model spatial resolution reached 0.25 degree. The model can somewhat reproduce the observed characteristics of the waves. One of the widely used methods for simulating ocean waves is making use of wind-wave spectrums. The ocean waves produced in this way can reflect the statistical characteristics of the real ocean well. The waves just look like superposition of significant wave heights.

Many research works in deep learning have been carried out recently. Hossain et al. [5] compared the accuracy of historical pressure, humidity, and temperature data gathered from meteorological sensors in Northwestern Nevada and deep learning network with Stacked Denoising Auto-Encoders (SDAE). In 2016, Gupta et al. [6] compared the performance of two optimization techniques in the linear regression model for an accurate weather prediction.

2 Deep learning method

The artificial neurons which imitate the human brain neurons are the basic building blocks of neural networks. As shown in Fig. 1, the processes in a neural network are described as follows. When input signals are weighted, they use an activation function to produce an output signal. These neurons are spread across several layers, including input layers, hidden layers, and output layers, on the neural network. A connection between two neurons refers to the strength or amplitude, and it is called neuron weights. Usually, the weights are initialized by small random numbers ranging from 0 to 1. The neuron weights get updated when the network is trained to be more predictive. To control the inception of a neuron, an activation function is used to map summed weighted input to the output of the neuron.

Figure 1

Neural networks scheme. A basic unit of neural networks

For a pre-activation function, we use a multiple linear regression technique [7] as follows:

$$\begin{aligned} g(\mathbf{x}) = \mathbf{w^{T} x} + b, \end{aligned}$$

(1)

where g represents a pre-activation function, x and b are input vector and bias values, and w is the vector of the connection weights which represent the strengths between the connections of neurons.

Next, we use the output from pre-activation function to compute activation function by

$$\begin{aligned} \hat{y}(\mathbf{x}) = a\bigl(g(\mathbf{x})\bigr) = a\bigl(\mathbf{w^{T} x} + b\bigr), \end{aligned}$$

(2)

where $\hat{y}(\mathbf{x})$ is the predicted output from neurons and $a(\cdot )$ is the activation function.

We use sigmoid activation function. This function gives a value between 0 and 1. The value is always positive, bounded, and strictly increasing. When the pre-activation function gives the higher value, there is more chance the neuron is activated. The function is given by

$$\begin{aligned} a(\cdot ) = \operatorname{sigm}(\cdot ) = \frac{1}{1+e^{-\cdot }}. \end{aligned}$$

(3)

We can rewrite Equation (2) and Equation (1) as

$$\begin{aligned} \hat{y}^{(i)} = a\bigl(g^{(i)}\bigr) = \frac{1}{1+e^{-g^{(i)}}} \end{aligned}$$

(4)

and

$$\begin{aligned} g_{j}^{i} = \sum_{k} \bigl(w_{jk}^{i} \cdot x_{k}^{i-1} \bigr) + b_{j}^{i}, \end{aligned}$$

(5)

where $w_{jk}^{i}$ is the weight from the kth neuron in the $(i-1)$th layer to the jth neuron in the ith layer, $b_{j}^{i}$ and $g_{j}^{i}$ are the bias and the pre-activation value of the jth neuron in the ith layer, respectively.

The cost function, which is derived from a loss function, is used to compute the error between the prediction and the actual data. Given $(x^{(1)},y^{(1)}),\ldots, (x^{(m)},y^{(m)})$, where $x^{(i)}$ is the actual input and $y^{(i)}$ is the actual output, we seek $\hat{y}^{(i)} \approx y^{(i)}$. For this purpose, the loss function is utilized to compute the error for a single training example by measuring the discrepancy between the prediction $\hat{y}^{(i)}$ and the desired output $y^{(i)}$. The cost function can be obtained using the loss function $L(\hat{y}^{(i)},y^{(i)})$ of the entire training set which is in the form of sum of squared errors (SSE):

$$\begin{aligned} L\bigl(\hat{y}^{(i)},y^{(i)}\bigr) = \frac{1}{2} \bigl(\hat{y}^{(i)} - y^{(i)}\bigr)^{2}. \end{aligned}$$

(6)

The values of the connection weight w and the bias value b can be found so that the overall cost function $J(\mathbf{w},b)$ is minimized, where

$$\begin{aligned} J(\mathbf{w},b) = \frac{1}{m}\sum_{i=1}^{m} \bigl[L\bigl(\hat{y}^{(i)},y^{(i)}\bigr)\bigr]. \end{aligned}$$

(7)

We can reduce the value of the cost function by adjusting the weight values between the neurons. The weight values may be changed until the cost function is minimized. We use the technique called the gradient descent method to find the minimum value of the cost function.

Let θ denote the parameters including the connection weights w and the bias values b. In the gradient descent method, the parameters are updated by

$$\begin{aligned} \theta _{j} = \theta _{j} - \alpha \frac{\partial }{\partial \theta _{j}} J(\mathbf{w},b). \end{aligned}$$

(8)

Using calculus, we obtain

$$\begin{aligned} \theta _{j} = \theta _{j} - \alpha \frac{1}{m} \sum_{i=1}^{m} \bigl[\bigl(h_{ \theta }\bigl(x^{(i)}\bigr) - y^{(i)} \bigr) x_{j}^{(i)}\bigr], \end{aligned}$$

(9)

where α is the learning rate, and $h_{\theta }(x)$ is $\sum_{j=0}^{n}[\theta _{j} x_{j}]$.

In this study, the input data include significant wave height, wind speed, and wind direction. The data are divided into train data sets and test data sets using the ratio 2 to 1.

3 Models

3.1 Ocean wave model

The WAM model [8] is a third-generation wave model which is based on the wave transport equation. The model does not need a requirement on the shape of the wave spectrum. Therefore, the governing equation is the wave energy equation which is given by, at any specific location on the sea surface,

$$\begin{aligned} \frac{\partial E}{\partial t}+\frac{\partial }{\partial \phi }( \dot{\phi }E)+\frac{\partial }{\partial \lambda }( \dot{\lambda }E) +\frac{ \partial }{\partial \theta }(\dot{\theta }E)=S_{\mathrm{in}}+S_{ \mathrm{ds}}+S_{\mathrm{nl}}, \end{aligned}$$

(10)

where E represents the spectral density such that ϕ, λ, θ, and t are latitudes, longitudes, directions, and times, respectively.

The terms ϕ̇, λ̇, and θ̇ are the rates of change of the position and propagation direction of a wave packet traveling along a great circle path. $S_{\mathrm{in}}$ is the wave energy influx from winds, $S_{\mathrm{ds}}$ denotes the dissipation of wave energy, and $S_{\mathrm{nl}}$ is the nonlinear effects caused by wind-wave interaction.

The model can be implemented for any given regional or global grid with a topographic data set. The wave propagation can be done on a latitudinal-longitudinal grid or on a Cartesian grid. The model yields various outputs including significant wave height, mean wave direction and frequency, swell wave height and mean direction, and wind direction.

3.2 Oil-spill model

After oil is spilled on the sea surface, oil slicks can be affected by the oil weathering processes (OWP). In [9], Fay described oil spreading, one process of OWP, as a horizontal expansion of the oil slick with inertia-gravity, gravity-viscous, and viscous-surface tension [2, 10]. Lehr [11] considered that the oil spill can be spread in an elliptical shape whose major axis is in the direction of the wind [12, 13]. The spreading equations are given by

$$\begin{aligned}& L_{\mathrm{min}} = 53.76 \biggl[\frac{\Delta \rho }{\rho _{0}} \biggr] ^{1/3} V_{\mathrm{oil}}^{1/3} t^{1/4}, \end{aligned}$$

(11)

$$\begin{aligned}& L_{\mathrm{max}} = L_{\mathrm{min}} + 0.95 U_{w}^{4/3} t^{3/4}, \end{aligned}$$

(12)

$$\begin{aligned}& A = 2270 \biggl[\frac{\Delta \rho }{\rho _{0}} \biggr]^{2/3} V_{ \mathrm{oil}}^{2/3} t^{1/2} + 40 \biggl[ \frac{\Delta \rho }{\rho _{0}} \biggr] ^{1/3} V_{\mathrm{oil}}^{1/3} U_{w}^{4/3} t, \end{aligned}$$

(13)

where $L_{\mathrm{min}}$ and $L_{\mathrm{max}}$ are the lengths (m) of the minor and major axes of an ellipse, respectively, and A is the area of oil slick (m²). The terms $\rho _{w}$, $\rho _{0}$ are the densities of water and oil, respectively, such that $\Delta \rho = \rho _{w} - \rho _{0}$. $V_{\mathrm{oil}}$ denotes the total volume of an oil spill in barrels, $U_{w}$ is the wind speed (knots) at 10 m over sea surface, and t is time (min.).

4 Numerical results

In this section, we present the simulation results of prediction of wind speed and direction in the movement of oil spill. The wind field forecast including the wind speed and the wind direction is based on the deep learning technique. The wind prediction is then used for studying the spreading of oil spill.

In this study, the outputs of wind data from WAM during 5 June to 31 October 2018 are used as a data set in the deep learning technique. The outputs are available every three hours each day, starting at 7.00 am (local time). There are totally 1,192 data points. Two thirds of the data set is taken as the training set, whereas one third of the data set is used as the test set as shown in Fig. 2.

Figure 2

Wind speed data sets. Time series of wind speed prediction from WAM during 5 June to 31 October 2018

For the wind speed prediction, Figs. 3(a) and 3(b) show the plots of mean square errors (MSE) of wind speed versus the number of iterations of training where we investigate the suitable values of the learning rate α and the number of hidden layers, respectively. The results show that MSE is lower when $\alpha = 1.2$ and 3.0. Therefore, $\alpha = 1.2$ is selected for our deep learning technique since there is no significant difference resulting from $\alpha = 3.0$ as shown in Fig. 3(a). Moreover, MSE is lowest when the number of hidden layers is 9 as shown in Fig. 3(b). As a result, the learning rate of $\alpha = 1.2$ and the number of hidden layers of 9 are used in the prediction model obtained from the deep learning technique.

Figure 3

MSE of wind speed (learning rates and hidden layers). The plot of MSE of wind speed versus the number of iterations (a) with various learning rates: $\alpha = 0.5$ (solid line), $\alpha = 1.2$ (dashed line), and $\alpha = 3.0$ (dotted line) and (b) with various numbers of hidden layers of 4 (solid line), 6 (dashed line), and 9 (dotted line)

Figure 4 shows the plot of MSE of wind speed versus the number of iterations. It can be seen that MSE decreases when the model is run through more iterations. Figures 5(a) and 5(b) present the plots of wind speed versus time for the training set and the test set, respectively, using the predictive model. It can be seen that the model gives good prediction.

Figure 4

MSE of wind speed. The plot of MSE of wind speed versus the number of iterations with the learning rates $\alpha = 1.2$ and the number of hidden layers = 9

Figure 5

Deep learning prediction of wind speed. The plot of wind speed versus time (a) using the training set (blue solid line) and the deep learning technique (red dashed line) and (b) using the test set (blue solid line) and the deep learning technique (red dashed line)

For the wind direction prediction, the results follow in the same manner as those of the wind speed prediction, as shown in Figs. 6–9.

Figure 6

Wind direction data sets. Time series of wind direction prediction from WAM during 5 June to 31 October 2018

Figure 7

MSE of wind direction (learning rates and hidden layers). The plot of MSE of wind direction versus the number of iterations (a) with various learning rates: $\alpha = 0.5$ (solid line), $\alpha = 1.2$ (dashed line), and $\alpha = 3.0$ (dotted line) and (b) with various numbers of hidden layers of 4 (solid line), 6 (dashed line), and 9 (dotted line)

Figure 8

MSE of wind direction. The plot of MSE of wind direction versus the number of iterations with the learning rates $\alpha = 1.2$ and the number of hidden layers = 9

Figure 9

Deep learning prediction of wind direction. The plot of wind direction versus time (a) using the training set (blue solid line) and the deep learning technique (red dashed line) and (b) using the test set (blue solid line) and the deep learning technique (red dashed line)

To investigate the computing time used in the deep learning technique against the number of iterations, we divide the data set into groups. The results show that when 50 or 100 iterations are applied, the computing time decreases very fast when we divide the data set into more groups, as shown in Tables 1–2.

Table 1 Comparison of computing time in wind speed prediction using various numbers of iterations and numbers of data groups

Table 2 Comparison of computing time in wind direction prediction using various numbers of iterations and numbers of data groups

When the wind forecast is applied in the spreading of oil spill, Fig. 10 depicts the spreading of oil spill using the wind prediction from the ocean wave model and from the deep learning technique. We observe that the two methods give good agreement on the spread of oil spills.

Figure 10

Oil spill spreading. The prediction of oil spill spreading at time t and $t+1$ on 6 December 2018 using the wind prediction (a) from the ocean wave model and (b) from the deep learning technique

5 Conclusions

In this paper, we apply a deep learning method to obtain a model that can be used to get quick prediction of wind speed and direction. The outputs from an ocean wave model, based on the wave transport equation, are used as the training data set to get the prediction model. The results show that the model can give good prediction on wind speed and direction with high accuracy when the learning rate is 1.2 and the number of hidden layers is 9. Furthermore, the prediction on the spreading of oil spill using the wind prediction from the deep learning technique agrees with that from the WAM model, while the deep learning technique takes less computing time on wind prediction.