Prediction of sea ice evolution in Liaodong Bay based on a back-propagation neural network model

Cold Regions Science and Technology 145 (2018) 65–75

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Cold Regions Science and Technology journal homepage: www.elsevier.com/locate/coldregions

Prediction of sea ice evolution in Liaodong Bay based on a back-propagation neural network model

MARK

Na Zhanga,b, Yuteng Maa, Qinghe Zhangb,⁎ a b

Tianjin Key Laboratory of Civil Structure Protection and Reinforcing, Tianjin Chengjian University, Tianjin 300384, China State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Sea ice Spatial evolution BP model Wind direction Wind duration

In the present study, a back-propagation neural network model (BP model) was developed with the aim of predicting the sea ice spatial evolution in Liaodong Bay. In addition to air temperature and wind speed, two new variables wind direction and wind duration were used to train the BP model. Validation of the BP model with measurements showed that the BP model can effectively predict the spatial evolution of sea ice. The sensitivity studies indicated that wind direction and wind duration can obviously improve the prediction accuracy of sea ice edge in heavy ice years. Moreover, the BP model was easy to set up as it only used four yearlong periods, 2003–2004, 2005–2006, 2006–2007 and 2009–2010, and the results were not very sensitive to the training dates over the four years. The BP model results were not very sensitive to the training algorithms as well. By comparison with a least-square-based method (LSM), the BP model clearly outperformed the LSM during the period of ice melt with nonlinear characteristics caused by the frequent appearance of cold waves. Furthermore, the BP model had a higher accuracy in estimating the spatial evolution of sea ice compared with a logit model, especially for the ice edge, which is more easily affected by the complex ocean environment.

1. Introduction The Bohai Sea is a seasonal sea ice area with different degrees of freezing every winter (Bai et al., 2011). There are three bays in the Bohai Sea, namely Liaodong Bay, Bohai Bay and Laizhou Bay. Liaodong Bay has the highest latitude and it is also the most serious ice area among the three bays, as can be seen from Fig. 1 (Ning et al., 2009; Shi and Wang, 2012a). Sea ice first appears in the northern top of Liaodong Bay at the beginning of December, and lasts until the end of mid-March next year. (Zhang et al., 2015). Heavy ice may lead to serious problems for offshore operations, ship navigation, port transportation and marine fisheries (Shi and Wang, 2012b; Su et al., 2013). In addition, sea ice is also hoped to be developed and used as a source of freshwater to alleviate the problem of freshwater shortages in coastal areas of northern China (Williams et al., 2013; Gu et al., 2013). Therefore, this study is important for reducing disasters due to sea ice and estimating the sea ice resources in Liaodong Bay. Many studies have been carried out to perform sea ice estimations. They have included numerical models and empirical models (Gao et al., 2011; Zhang et al., 2016). Numerical simulation is a commonly used method for predicting sea ice parameters, such as coupled sea ice-ocean models based on thermodynamics and the dynamics of sea ice (Hibler,

⁎

1979; Rae et al., 2014). However, the fact that there are so many input variables in the ice-ocean models leads to high uncertainties in simulations; therefore, it is not surprising that the results of numerical model may not be in good agreement with the measurements. An empirical model is another way to predict sea ice parameters based on the primary factors that influence sea ice cycles as derived from statistical methods. Previous studies about sea ice estimation have concentrated on sea ice thickness and sea ice area. For example, Dong and Liu (1989) indicated that sea ice thickness was mainly controlled by cumulative melting degree-days (CMDD) and cumulative freezing degree-days (CFDD). Zeng et al. (2016) estimated the ice thickness of Bohai Sea in 2010 based on the ice surface temperature from MODIS and meteorological data from ECMWF. Su et al. (2012) noted that the sea ice area was associated with CFDD highly based on statistical analysis in Liaodong Bay. In addition to CFDD and CMDD, Zhang et al. (2016) found that the wind was also an important factor influencing the prediction of sea ice, and they developed an empirical model to estimate the extent of sea ice using a non-linear least square method (LSM) and logit model based on measured wind speeds and air temperature. Compared to the data from MODIS, the predicted area and spatial evolution of sea ice area were able to account for 87% of the variability (Zhang et al., 2016). However, some limitations exist in the current

Corresponding author. E-mail address: [email protected] (Q. Zhang).

http://dx.doi.org/10.1016/j.coldregions.2017.10.002 Received 9 May 2017; Received in revised form 30 September 2017; Accepted 4 October 2017 Available online 05 October 2017 0165-232X/ © 2017 Published by Elsevier B.V.

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List of symbols DE

CFDD

cumulative freezing degree days, CFDD = ∑ Tair

Tair ≤ −5°C , [°C]

Ds DE

CMDD

cumulative melting degree days, CMDD = ∑ |Tair| Tair ≥ −2.5°C , [°C]

Tair Ta Tw N

daily average air temperature in Yingkou and Jinzhou, [°C] Tair with the effect of time lag. [°C] temperature integration over time with wind speeds higher than force 5, Tw = Ta × N. [°C] cumulative times of winds with speeds higher than force 5 that are accumulated over a winter.

Ds

The organization of the paper is as follows. Section 2 introduces the geographical location of the study area and the input data needed for model development. Section 3 describes the BP method. Section 4 discusses the sensitivity studies of the model input parameters. Model results are provided in Section 5. A comparison to other methods is discussed in Section 6. Finally, Section 7 presents the conclusion.

empirical model. For example, the nonlinear correlation between sea ice cover and meteorological data is not separable; therefore, it is hard to develop a simple empirical model to completely describe the complex nonlinear relationships. Further research is needed to improve the predictive precision of sea ice. In this paper, a method known as a back-propagation neural network (BP) is applied to predict the spatial evolution of sea ice in Liaodong Bay. More meteorological elements such as wind direction and wind duration will be used for model development and prediction, in addition to the meteorological variables mentioned in Zhang et al. (2016). BP is able to find solutions under complex conditions and produce ideal results in solving non-linear issues (Hsieh, 2009). Because of these advantages, BP has been widely used in various research fields to improve the nonlinear relationships, for instance, predicting optics communication (Li and Zhao, 2017), tide levels (Lee, 2008; Salim et al., 2015), sand ripple geometry under waves (Yan et al., 2008), air pollutants concentrations (Bai et al., 2016), carbon dioxide emissions (Sun and Xu, 2016) and water temperature (Liu et al., 2016).

41

2. Data The training data needed to develop the BP model are sea ice cover and meteorological data. The spatial cover of sea ice was extracted from MODIS images in Liaodong Bay using the Classification and Regression Tree (CART) method to reduce the error of sea ice due to suspended sediment (Zhang et al., 2015). We used 9 years' of data, from 2003 to 2012, to investigate the seasonal variation in sea ice in Liaodong Bay. For example, time-varying curves of the sea ice evolution in winter of 2003–2004 and 2009–2010 are plotted in Fig. 2(a) and Fig. 2(b), and the sea ice in 2010 was very serious with the maximum sea ice area of

Jinzhou

N

Fig. 1. Geographical location and sea ice distribution in the Bohai Sea.

20

10

o

Qing river estuary

Dalian

20

Tianjin

39

30

Li ao do ng

Qinhuangdao

40

Bohai Bay

30 Bohai Sea

Laotieshan

Yellow Sea

20

38 Laizhou Bay Shandong

37 117

118

119

120

121

122

123

Longitude (o)

Sea ice area (×103 km2)

30 25

30

a

Sea ice area

Sea ice area (×103 km2)

Latitude ( )

Ba y

Yingkou

20 15 10 5 0 2003/12/01

2004/01/01

2004/02/01

2004/03/01

25

b

20 15 10 5 0 2009/12/01

2004/04/01

Date

Sea ice area

2010/01/01

2010/02/01

Date

Fig. 2. Time-varying curves of the sea ice area from MODIS in 2003–2004 (a) and 2009–2010 (b).

66

2010/03/01

2010/04/01

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2.7 × 104 km2, which was much larger than the area of 1.38 × 104 km2 in 2004. Moreover, the duration of this sea ice in the winter of 2009–2010 was about three and half month, which was much longer than the winter of 2003–2004. The measured meteorological data was available from the official website of Chinese meteorological service (http://cdc.cma.gov.cn). In the paper, in addition to the daily air temperatures and maximum wind speed at two stations of Yingkou and Jinzhou mentioned in Zhang et al. (2016), two new variables, namely wind direction and wind duration with speed higher than force 5 (8 m/s) were used. For instance, two time-varying curves of the daily maximum wind speeds and daily average air temperature at Yingkou station (122°10′ E and 40°39′ N in Fig. 1) are plotted in Fig. 3(a–d) and Fig. 4(a–d). As seen from the graph, the average daily air temperature in the winter of 2010 was lower than in the winter of 2004. Moreover, the frequency of wind speed greater than 8 m/s in the winter of 2010 was much higher than in winter 2004.

data. For a more detailed description of x and y, please refer to Zhang et al. (2016). Taking the MODIS image from January 22nd, 2004, as an example, the red, green and blue circles represent the positions of sampling on the north, west and east coast, respectively (Fig. 6). The six meteorological data were CFDD, CMDD, Ta, Wspeed, Wdir and Wtime, respectively. The output node produces the output of spatial evolution of sea ice in Liaodong Bay. BP algorithm contains two learning processes: forward propagation of signal and back propagation of error. The former involves the input samples being transferred from the input layer to the output layer and processed by a hidden layer. If the actual output vector does not conform to the desired output vector, the back propagation of error is transferred by adjusting the weights and biases to minimize the error between the target value and the output value. The process of determining weights and biases is training. The goal of the training is to reduce the error, which is defined as follows (Cobaner et al., 2010):

3. Methods

E = (1/P)

P

∑ Ep (1)

p=1

The BP model was developed for forecasting the sea ice evolution in Liaodong Bay. BP algorithm is derived from the perspective of the correctness, which makes the learning algorithm have a theoretical basis (Rumelhart et al., 1986). Therefore, it has been widely used in the field of scientific research.

where E is the global error, P is the total training times and Ep is the pth training error, as shown below: N

Ep = (1/2)

∑ (Ok − Dk )2

(2)

k=1

3.1. Theory of the BP model

where N is the total output node number, Ok and Dk are the output and target output values corresponding to the kth output node, respectively (Cobaner et al., 2010). In the BP networks, the number of nodes in the hidden layers will directly affect the learning ability of the network. However, the relative error of the network output does not decrease with the increase of hidden nodes (Yan et al., 2014). When the hidden node increases to a certain number, the output error appears to oscillate, that is, the network output precision is no longer improved. Thus, it is necessary to

The BP model used for sea ice forecasting includes three layers of input, hidden and output, as shown in Fig. 5. The input layer has eight nodes (x1, x2, ⋯, x8) and the output layer has one node (O1). Eight variables were used for BP model training of the spatial evolution of sea ice, including two geographical elements, x and y (where x denotes the distance from the shore and y is the distance from the origin along the midline of sample points, as shown in Fig. 6), and six meteorological 10

20

a

Air temperature

b

Wind speed

5

Wind speed(m/s)

Air temperature(°C)

15 0

-5

-10

10

5 -15

-20 2003/12/01

2004/01/01

2004/02/01

0 2003/12/01

2004/03/01

2004/01/01

Date

420

2004/03/01

24

c

d

Wind direction

360

Wind duration

18

300

Wind duration(h)

Wind direction( o)

2004/02/01

Date

240 180 120

12

6

60 0 2003/12/01

2004/01/01

2004/02/01

0 2003/12/01

2004/03/01

Date

2004/01/01

2004/02/01

2004/03/01

Date

Fig. 3. Time-varying curves of the daily air temperature (a), maximum wind speed (b) and direction (c), and wind duration with speed higher than force 5 (d) at Yingkou station in 2003–2004.

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20

a

Air temperature

b

Wind speed

5

Wind speed(m/s)

Air temperature(°C)

15 0

-5

-10

10

5 -15

-20 2009/12/01

2010/01/01

2010/02/01

2010/03/01

0 2009/12/01

2010/04/01

2010/01/01

Date

420 360

2010/02/01

2010/03/01

2010/04/01

Date

30

c

Wind direction

d

Wind duration

300

Wind duration(h)

Wind direction( o)

24

240 180 120

18

12

6 60 0 2009/12/01

2010/01/01

2010/02/01

2010/03/01

0 2009/12/01

2010/04/01

Date

2010/01/01

2010/02/01

2010/03/01

2010/04/01

Date

Fig. 4. Time-varying curves of the daily air temperature (a), maximum wind speed (b) and direction (c), and wind duration with speed higher than force 5 (d) at Yingkou station in 2009–2010.

find the optimum number of hidden nodes to achieve a good balance between the output precision and operation cost. In this paper, the hidden nodes are determined by empirical formula from Xu et al. (2006):

of m and n are 8 and 1 respectively. According to Eq. (3), the optimum numbers of hidden nodes l are 5–6 for the model setup of spatial evolution of sea ice. 3.2. Performance testing of sea ice area

l=

0.43mn + 0.12n2 + 2.54m + 0.77n + 0.35 + 0.51

(3) The coefficient of determination, denoted as RCor2, was used to test the training and forecasting accuracy of the sea ice area. RCor2 is an indication of agreement between the measured data and models results (Cameron and Trivedi, 2005; Kurz-Kim and Loretan, 2014) with the

where l is the number of nodes in the hidden layers, m is the number of nodes in the input layers, n is the number of nodes in the output layers, and the decimal portion of the results needs to be rounded. The values

Fig. 5. Structure of a BP network.

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no rth

we st

N o Or rth igin co as t

X_

es tc oa st

X_

st

Ea

st co as t

ea

Th e

m id lin e

of Li ao do ng

Ba y

W

X_

and area of sea ice. According to Test 1, the overall accuracy of spatial distribution of sea ice on January 22nd, 2004 and February 12th, 2010 were 79.11% and 69.02%, respectively. This shows that the ideal results cannot be obtained if only the cumulative effect of air temperature is considered. The overall accuracy of the Test 2 has improved compared to Test 1 when Ta is included, the total accuracy of January 22nd, 2004 has been as high as 89.07%, while in February 12th, 2010 was only 78.69%. Based on the measured meteorological data, it is found that the frequency of strong winds above force 5 is very low in the ice age of 2004; therefore, the air temperature was the dominant factor in the evolution of sea ice in 2004. However, there was a high frequency of winds above force 5 in the ice age of 2010, which made the role of wind cannot be ignored. For this reason, wind speeds were added to Test 3, and the increase of the overall accuracy was not more than 0.5% in January 22nd, 2004 and in February 12th, 2010. Overall accuracy slightly increased no more than 1.03% with the wind direction was added in Test 4. When the duration of the strong wind above force 5 was added on the basis of the Test 4, the total accuracy of Test 5 was improved greatly, which was about 91.37% in January 22nd, 2004 and 93.11% in February 12th, 2010, respectively. In order to prove that this ascension is due to the effect of wind duration or both of wind duration and direction, we carried out test 6, which removed the wind direction on the basis of test 5. The results showed that the total accuracy of January 22nd, 2004 and February 12th, 2010 was reduced to almost the same as that of test 4. It can be concluded that adding both of the wind duration and the wind direction can effectively predict the spatial evolution of sea ice in Liaodong Bay. In order to make it clear that the influence of two factors of wind duration and wind direction on the spatial distribution of sea ice, the results of all factors (Test 5) are compared with those without considering the wind duration and wind direction (Test 3), as shown in Fig. 7. The total accuracy of Test 3 and Test 5 was only a difference of 2%, which was not significant due to the small wind speed in January 22nd, 2004. However, it was important to note that the ice edge of the west coast was influenced more due to the effect of NW wind, compared to the east side. (Fig. 7a). Fig. 7(b) shows a different result due to the role of strong wind. The total accuracy of Test 5 is 14% higher than that of Test 3, and the error is mainly located on the north and east edge of sea ice due to the strong wind from NE. It shows that the wind direction and the duration of the strong wind above force 5 can effectively improve the prediction accuracy of the sea ice edge, especially for the prediction of heavy ice years; therefore, they cannot be neglected in the prediction of spatial evolution of sea ice in Liaodong Bay.

Fig. 6. The location of the training sample.

following expression: 2

N

⎛ ∑ (y − y )(u ̂ − u ) ⎟⎞ i i ⎝i = 1 ⎠

⎜

2 RCor =

N

N

∑ (yi − y )2⋅ ∑ (uî − u )2 i=1

i=1

N

N

i=1

i=1

(4)

where y = N−1 ∑ yi , u = N−1 ∑ uî , yi is the measured data and uî is the model results.

3.3. Performance testing of spatial evolution of sea ice In this paper, the overall accuracy, denoted as p0, was used to test the training and forecasting accuracy of the spatial evolution of sea ice. p0 is the relative observed agreement among raters (Galton, 1892) and is expressed as follows:

p0 =

s n

(5)

where s is the number of pixels corresponding to the same value between the true color image of MODIS and an estimated remote sensing image of sea ice spatial evolution. n is the total number of pixels.

4.2. Training date

4. Sensitivity analysis

To determine the preferred date for building the BP model, the sea ice areas estimated from the BP model established by training variables

Sensitivity analysis is carried out in this paper to determine the effects of the parameters on the model results. The factors used in the sensitivity experiments include the training variables, training date and training algorithm.

Table 1 Sensitivity analysis of spatial evolution of sea ice based on the different training variables. Test number

4.1. Training variables A total of 8 variables mentioned above, including 2 geographic variables and 6 meteorological variables were used to train the BP model. In order to understand the role of the meteorological variables and their combinations in the spatial evolution of sea ice, 6 groups of tests were carried out. They are: Test 1 (CFDD, CMDD), Test 2 (CFDD, CMDD, Ta), Test 3 (CFDD, CMDD, Ta, Wspeed), Test 4 (CFDD, CMDD, Ta, Wspeed, Wdir), Test 5 (CFDD, CMDD, Ta, Wspeed, Wdir, Wduration), and Test 6 (CFDD, CMDD, Ta, Wspeed, Wduration). Taking two typical years as an example, the sensitivity analysis of spatial evolution of sea ice was shown in Table 1. Overall accuracy and error were used to measure the training accuracy of spatial distribution

1

2

3

4

5

6

2004-1-24 2010-2-12

79.11 69.02

89.07 78.69

89.22 79.07

89.57 80.10

91.37 93.11

89.42 81.35

Test number

Sea ice area (× 103 km2)

2004-1-24 error 2010-2-12 error

69

Overall accuracy of spatial evolution (%)

1

2

3

4

5

6

8.46 −0.28 19.5 −0.29

11.38 − 0.03 22.4 − 0.18

11.57 − 0.02 22.5 − 0.18

11.39 − 0.03 22.7 − 0.17

12.34 0.05 27.1 −0.01

11.36 − 0.03 23.4 − 0.15

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a

Fig. 7. The spatial evolution of sea ice estimated by Test 3 (blue line) and Test 5 (red line) in January 22, 2004 (a) and February 12, 2010 (b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

b

from different dates were tested. Following Zhang et al. (2016), RCor2 was used to verify the performance by contrasting the sea ice area from the MODIS and BP model. Ten tests with different dates were used to create the BP model. At the same time, the BP model can be used for other dates to estimate the sea ice area, a process that we call cross validation. The selected dates for cross validation are listed in Table 2. According to Table 2, using the BP model with a model creation date from a light ice year and heavy ice year in Test 1, it is not satisfactory for predicting the sea ice area in normal ice years, such as the years of 2003–2004, 2005–2006 and 2010–2011. Test 2 involves a normal ice year, 2005–2006, on the basis of Test 1. The prediction accuracy for the sea ice area in Test 2 is improved compared with Test 1, except for the years 2003–2004, which had a no wind event with a seaward wind speed higher than 9 m/s in 2003–2004. Test 3 adds the year 2003–2004 on the basis of Test 2. The average RCor2 increased from 0.85 to 0.91, and the forecasting accuracy significantly improved. To test the effect of heavy ice years, Test 4 only considers 4 years, including a light ice year and normal ice years. The RCor2 of the heavy ice year in 2009–2010 decreased from 0.94 in Test 3 to 0.72. However, the forecast for heavy ice years is very important for the prevention and control of sea ice disasters. Therefore, the data of heavy ice years are absolutely necessary in the process of constructing the BP model. Test 5 only considers 4 years, including normal ice years and a heavy ice year. The predicted RCor2 of the light ice years of 2004–2005, 2006–2007 and 2008–2009 were 0.66, 0.71 and 0.67, respectively. Test 6 adds a normal ice year, 2010–2011, on the basis of Test 3, and Test 7–Test 9 sequentially add one year of data to the previous test. The RCor2 of the sea ice area

between MODIS and that from the BP model ranged from 0.89 to 0.90. The comparison results indicate that a high accuracy can be achieved with the four years in test 3 (2003–2004, 2005–2006, 2006–2007 and 2009–2010), which account for 91% of the variability in the sea ice area and a 0.05 Standard Deviation. In other words, the BP model results are not very sensitive to the training dates over the four years. 4.3. Training algorithm The selection of a training algorithm was important for establishing the BP model and thus affected the forecasting precision of the sea ice area. To determine the influence of different training algorithms on the prediction results of the BP model, 8 types of training algorithms were selected as the BP neural network training function. They include the Bayesian regularization algorithm (1), adaptive gradient descent algorithm (2), LevenburgMarquardt algorithm (3), adaptive learning rate moment gradient decreased algorithm (4), BP elastic algorithm (5), Ploak-Ribiere conjugate gradient algorithm (6), Fletcher-Reeves conjugate gradient algorithm (7) and Scaled Conjugate Gradient algorithm (8). As shown in Table 3, the results based on various training algorithms of the BP model are similar; moreover, values of 0.92 to 0.93 for RCor2 indicate an excellent model fit. The predicted RCor2 were between 0.88 and 0.91, which shows that the forecasting data obtained by various training algorithms agree well with the measured data, especially for the Fletcher-Reeves conjugate gradient algorithm. Overall, the model results are not very sensitive to the training algorithms and the Fletcher-Reeves conjugate gradient algorithm was more suitable for sea ice area forecasting in Liaodong Bay.

Table 2 RCor2 for cross validation of the BP model based on the different training dates in Liaodong Bay. Estimation date

2003–2004 2004–2005 2005–2006 2006–2007 2007–2008 2008–2009 2009–2010 2010–2011 2011–2012 Average Standard Deviation

Model creation date Test 1

Test 2

Test 3

Test 4

Test 5

Test 6

Test 7

Test 8

Test 9

Test 10

0.65 0.84 0.67 0.92 0.89 0.77 0.97 0.73 0.84 0.81 0.11

0.67 0.84 0.90 0.80 0.91 0.83 0.94 0.87 0.86 0.85 0.08

0.94 0.90 0.91 0.80 0.96 0.92 0.94 0.91 0.91 0.91 0.05

0.93 0.83 0.93 0.80 0.83 0.83 0.72 0.95 0.90 0.86 0.08

0.79 0.66 0.91 0.71 0.89 0.67 0.95 0.93 0.92 0.83 0.12

0.94 0.90 0.91 0.78 0.93 0.92 0.93 0.95 0.86 0.90 0.05

0.94 0.87 0.91 0.73 0.92 0.89 0.93 0.95 0.87 0.89 0.07

0.91 0.85 0.92 0.77 0.98 0.92 0.95 0.91 0.90 0.90 0.06

0.92 0.87 0.91 0.76 0.97 0.92 0.94 0.93 0.88 0.90 0.06

0.92 0.80 0.91 0.77 0.95 0.92 0.93 0.91 0.92 0.89 0.06

Test 1 represents a model creation date of 2006–2007 and 2008–2009, representing light and heavy ice events, respectively. Test 2 represents the model creation dates of 2006–2007, 2005–2006 and 2009–2010, representing light, normal and heavy ice events, respectively. Test 3 adds the year 2003–2004 on the basis of Test 2. Test 4 represents the model creation dates of light and normal ice years in 2003–2004, 2006–2007, 2005–2006 and 2010–2011. Test 5 represents the model creation dates of normal and heavy ice years in 2005–2006, 2010–2011, 2011–2012 and 2009–2010. Test 6 adds a normal ice year of 2010–2011 on the basis of Test 3. Test 7 adds a light ice year of 2004–2005 on the basis of Test 6. Test 8 represents the model creation dates for 7 years from 2003 to 2010. Test 9 represents the model creation dates for 8 years from 2003 to 2011. Test 10 represents the model creation dates for all 9 years from 2003 to 2012.

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Table 3 RCor2 of the sea ice area for MODIS and the BP neural network training functions. Tests 1–8 represent the Bayesian regularization algorithm (1), adaptive gradient descent algorithm (2), Levenburg-Marquardt algorithm (3), adaptive learning rate moment gradient decreased algorithm (4), BP elastic algorithm (5), Ploak-Ribiere conjugate gradient algorithm (6), FletcherReeves conjugate gradient algorithm (7) and Scaled Conjugate Gradient algorithm (8), respectively. Test number

Ice training 1

2

3

4

5

6

7

8

RCor2

0.93

0.92

0.92

0.92

0.93

0.92

0.92

0.92

Test number

Ice forecasting 1

2

3

4

5

6

7

8

0.89

0.89

0.88

0.89

0.88

0.89

0.91

0.88

RCor2

5. Results

91% for the not training dates. It was worth noting that errors may occur at low latitudes in Liaodong Bay when the ice was heavy, such as in Fig. 8(d). However, this error will be reduced when the meteorological data obtains higher spatial and temporal resolution. Comparison between sea ice area estimated by BP model and MODIS is shown in Fig. 9, and the RCor2 is 0.92 for the training dates and 0.91 for the not training dates. The errors between annual maximum sea ice area by BP model and MODIS are not more than 10%.

In this section, we present the spatial evolution of sea ice that was determined by method introduced in Section 3. After a network is trained, the remaining five years will be used to validate the performance of the model. The Fletcher-Reeves conjugate gradient is selected as the training algorithm of the BP model. The partial results of spatial evolution of sea ice by BP model are shown in Fig. 8. The overall accuracy is 92% for the training dates and

a A

b

c

d

e

f

g

h

i

Fig. 8. Comparison of the marginal ice line from BP model (red line), and MODIS images on (a) January 22nd, 2004; (b) February 8th, 2006; (c) January 30th, 2007; (d) February 12th, 2010; (e) February 9th, 2005; (f) January 12th, 2008; (g) January 16th, 2009; (h) January 20th, 2011; and (i) February 15th, 2012. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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15 10 5

0 2003/12/01

2004/01/01

2004/02/01 2004/03/01

2004/04/01

20

25 T rained by BP model MODIS

15 10 5

0 2005/12/01

2006/01/01

Date

30

d

35 T rained by BP model MODIS

25 20 15 10 5

0 2009/12/01

2010/01/01

2010/02/01 2010/03/01

2010/04/01

30

e

20

10 5

0 2006/12/01

35

20 15 10 5

25 Forcasted by BP model MODIS

10 5

2005/01/01

MODIS

15

2007/01/01

2011/01/01

2011/02/01 2011/03/01

2011/04/01

30

f

2005/02/01 2005/03/01

2005/04/01

h

20

Forcasted by BP model

20 15 10 5

0 2011/12/01

25 Forcasted by BP model MODIS

10 5

2008/01/01

Date

2008/02/01 2008/03/01

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15 10 5

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Fig. 9. Time series of sea ice area estimated by BP model and MODIS from 2003 to 2012.

1.3 °C to − 6.05 °C. A similar situation occurred in the melting period of 2009 (Fig. 10d), which shows that the BP model has more advantages than LSM in dealing with non-linear problems.

6. Comparisons to other methods Two statistical methods, LSM and logit model, were applied to compare the performance of the BP model in predicting the area and spatial evolution of sea ice, respectively.

6.2. Comparisons to the logit model for predicting the spatial evolution of sea ice

6.1. Comparisons to LSM for predicting the sea ice area In this section, the BP model is further compared with a statistical model, logit, in which the spatial evolution of sea ice is estimated by Zhang et al., 2016. The logic model is one of the discrete choice models which belong to the category of multivariate analysis. The coefficients of the logit model are estimated by maximum likelihood (Hosmer and Lemeshow, 2000). The logit model setup by Zhang et al. (2016) is as follow:

LSM gives the best estimation by minimizing the square sum of the error between the calculated data and measured data (Legendre, 1805; Ghasemi et al., 2014). The method has been used to forecast the sea ice area of Liaodong Bay by Zhang et al. (2016). According to Fig. 10a and Fig. 10b, both the BP model and LSM coincide with the area from MODIS well for each year from 2003 to 2012. The RCor2 between the area from MODIS and that fitted from the BP model and LSM were 0.92 and 0.87, respectively. The RCor2 of the sea ice area between the area from MODIS and that forecasted from the BP model and LSM were 0.91 and 0.86, respectively. The results indicate that both the BP and LSM methods are effective in estimating the sea ice area and that the BP model could be used to provide a higher accuracy compared with the LSM method. According to Table 4, the RCor2 between the area from MODIS and that from the BP model and LSM are both 0.92 during the freezing period, which indicates strong agreement between the methods. However, the RCor2 between the area from MODIS and that from the BP model and LSM are 0.90 and 0.81 during the melting period, respectively. The results indicate that the area calculated by BP is better than that obtained by LSM during the ice melting stage, as shown in Fig. 10(c) and Fig. 10(d). The difference between the two graphs is caused by the frequent cold waves which has influenced the melting process with the characteristics of complexity, uncertainty and nonlinearity. For instance, the ice area increased from 13,281 km2 on March 1st to 15,204 km2 on March 9th, 2010 according to MODIS images (Fig. 10c). Northerly wind lasted four days beginning on March 5th at a speed of 8.9–10.3 m/s, and the air temperature decreases from

Logit(P (ice )) = f (x , y, CFDD 2, CMDD 2, CFDD, CMDD, Ta, Tw ) = − 0.0566865x − 0.0484839y − 0.0000291CFDD 2 − 0.0022995CMDD2 − 0.0265566CFDD + 0.0485007CMDD − 0.1394712Ta − 0.0496213Tw − 1.173737

(6)

where P(ice) is the probability of the sea ice, of which 0 indicates no ice, on the contrary there is ice. x, y, CFDD, CMDD, Ta, Tw are the explanatory variables, as previously mentioned. Based on the training data from 2003 to 2004, 2005–2006, 2006–2007 and 2009–2010, the sea ice covers from the BP and logit models on nine dates were obtained. Table 5 shows that the overall accuracy in the BP model is consistently higher than that of the logit model. The average overall accuracy of the fitted model of sea ice coverage is 93.4% for the BP model and 90.0% for the logit model, and the average overall accuracy of the forecasted sea ice cover is 93.2% for the BP model and 87.8% for the logit model. Moreover, the standard deviations of the fitted and forecasted sea ice coverages for the BP model are 1.2% and 1.4%, respectively, which are lower than those of the logit model, 2.0% and 1.5%. Overall, the spatial evolution of sea ice estimated from the logit 72

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Fig. 10. Comparison of the fitted (a) and forecasted sea ice area (b) from 2003 to 2012, fitted sea ice area in 2009–2010 (c) and forecasted sea ice area in 2008–2009 (d) from the BP and LSM methods in Liaodong Bay.

model was similar to the BP model (Fig. 11). However, in the ice marginal zone, the spatial evolution of sea ice estimated from the BP model was much closer to the remote-sensing image than that from the logit model, especially for severe sea ice conditions. Compared with the image from MODIS, sea ice from the logit model was obviously overestimated on February 12th, 2010, and January 20th, 2011 (Fig. 11a and Fig. 11b). By contrast, sea ice from the logit model was obviously underestimated on February 9th, 2005, and February 15th, 2012 (Fig. 11c and Fig. 11d). The results indicate that both the BP and logit models are effective for estimating the spatial evolution of sea ice and that the BP model has a higher forecasting accuracy compared with the logit model, especially for the ice edge, which indicates significant nonlinearities affected by the complex ocean environment.

Table 4 Comparison of the RCor2 of the sea ice area estimated by the BP and LSM methods in Liaodong Bay. Test name

RCor2

Ice freezing

Ice melting

BP model

LSM

BP model

LSM

0.92

0.92

0.90

0.81

Table 5 Total accuracy of the spatial evolution of sea ice according to the BP and logit models in Liaodong Bay. Date

2004-01-22 2006-02-08 2007-01-30 2010-02-12 – Average Standard Deviation

Overall accuracy for fitting (%) BP

logit

91.8 94.3 93.1 94.3 – 93.4 1.2

90.0 89.8 87.7 92.5 – 90.0 2.0

Date

2005–02-09 2008–01-12 2009–01-16 2011–01-29 2012–02-15

Overall accuracy for forecasting (%) BP

logit

92.0 93.8 93.2 95.2 91.7 93.2 1.4

85.7 88.1 87.8 90.0 87.4 87.8 1.5

7. Conclusions In this paper, a BP model for predicting the spatial evolution of sea ice was developed using measured meteorological data and sea ice extent extracted from MODIS for 9 years from 2003 to 2012, among which four yearlong datasets of 2003–2004, 2005–2006, 2006–2007 and 2009–2010 were used for model training and the remaining were used for model forecasting. The results of sensitivity analysis show that the air temperature is the main influencing factor for predicting the sea ice in light ice years, while the wind speed, wind direction and wind duration with speed higher than force 5 play important roles in the prediction of sea ice in heavy ice year. Furthermore, the results of 73

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a

Fig. 11. Comparison of the predicted ice edge from BP model (the red line) and the logit model (the yellow line) with MODIS remote sensing images on (a) February 12th, 2010; (b) January 20th, 2011; (c) February 9th, 2005; and (d) February 15th, 2012. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

b

f

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d

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Acknowledgments

sensitivity analysis are also presented and a high accuracy can be obtained by the BP model developed with only four years, 2003–2004, 2005–2006, 2006–2007 and 2009–2010, which account for 91% of the variability of the evolution of sea ice area and a 0.05 Standard Deviation. The BP model is not very sensitive to the training dates over the four years. It is also not very sensitive to the training algorithms, and the Fletcher-Reeves conjugate gradient algorithm is more suitable to forecast the evolution of sea ice extent. According to the model results, the overall accuracy of the spatial evolution trained by BP model is 92%, and the relative error of the maximum area and spatial distribution of sea ice forecasted by the BP model does not exceed 10%. Overall, these results suggest that the trained and forecasted area and spatial evolution of sea ice agree well with those of MODIS. Two statistical methods, LSM and a logit model, were used to compare the performance of the BP model in predicting the area and spatial evolution of sea ice, respectively. The results achieved indicate that both LSM and the BP model achieve similar results for sea ice model training and forecasting for the period of ice freezing. However, the BP model clearly outperforms LSM in the period of ice melting, with nonlinear characteristics caused by the frequent appearance of cold waves. Furthermore, compared with the logit model, the BP model has a higher accuracy in estimating the spatial evolution of sea ice, especially for the ice edge, which is more easily affected by the complex ocean environment. This indicates that the BP model is more advantageous than LSM and the logit model in dealing with non-linear problems. The BP model developed in this paper can be applied to forecast the evolution of sea ice extent according to the forecasted meteorological data, which can be obtained from numerical forecast products with higher resolution in time and space (Kioutsioukis et al., 2016).

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