Development of GMDH-Based Storm Surge Forecast Models for Sakaiminato, Tottori, Japan

Journal of Marine Science and Engineering

Article Development of GMDH-Based Storm Surge Forecast Models for Sakaiminato, Tottori, Japan

Sooyoul Kim 1 , Hajime Mase 2,3,4,5, Nguyen Ba Thuy 6,* , Masahide Takeda 7, Cao Truong Tran 8 and Vu Hai Dang 9 1 Center for Water Cycle Marine Environment and Disaster Management, Kumamoto University, 2-39-1, Kurokami, Chuo-ku, Kumamoto 860-8555, Japan; [email protected] 2 Professor Emeritus, Kyoto University, Kyoto 611-0011, Japan; [email protected] 3 Toa Corporation, Tokyo 163-1031, Japan 4 Hydro Technology Inst., Co., Ltd., Osaka 530-6126, Japan 5 Nikken Kogaku Co., Ltd., Tokyo 160-0023, Japan 6 Vietnam National Hydrometerological Forecasting Center Hanoi, No 8, Phao Dai Lang, Dong Da, Hanoi, Vietnam 7 Research and Development Center, Toa Corporation, Kanagawa 230-0035, Japan; [email protected] 8 Le Quy Don Technical University, 236 Hoang Quoc Viet St, Hanoi, Vietnam; [email protected] 9 Institute of Marine Geology and Geophysics, VAST, 18 Hoang Quoc Viet St, Hanoi, Vietnam; [email protected] * Correspondence: [email protected]

Received: 31 July 2020; Accepted: 4 October 2020; Published: 14 October 2020

Abstract: The current study developed storm surge hindcast/forecast models with lead times of 5, 12, and 24 h at the Sakaiminato port, Tottori, Japan, using the group method of data handling (GMDH) algorithm. For training, local meteorological and hydrodynamic data observed in Sakaiminato during Typhoons Maemi (2003), Songda (2004), and Megi (2004) were collected at six stations. In the forecast experiments, the two typhoons, Maemi and Megi, as well as the typhoon Songda, were used for training and testing, respectively. It was found that the essential input parameters varied with the lead time of the forecasts, and many types of input parameters relevant to training were necessary for near–far forecasting time-series of storm surge levels. In addition, it was seen that the inclusion of the storm surge level at the input layer was critical to the accuracy of the forecast model.

Keywords: storm surge; group method of data handling (GMDH); 24-h lead time forecast

1. Introduction Understanding flood mechanisms related to tropical cyclones (TCs) is vital for mitigating flood risks in low-lying coastal areas, planning early warning systems, and supporting decision-making with respect to evacuations, which is closely related to casualties and economic damage. In Japan, coastal floods generally occur due to either sole surges/waves or combined surges and waves during TC events, superimposed on tides and lower frequency components (e.g., Mori et al. [1]). On the coast facing the Pacific Ocean, the maximum surge level occurs when TCs makes landfall. Meanwhile, the maximum surge appears 15–18 h later at Sakaiminato, as well as on the Tottori coasts facing the Sea of Japan/East Sea, following the TCs passage (a so-called after-runner storm surge, according to Kim et al. [2]). In other words, when TCs move along a track similar to those of Typhoons Maemi (2003), Songda (2004), and Megi (2004), the maximum surge level is generated at Sakaiminato when the TC moves around the island of Hokkaido (see Figure1b) at a distance of approximately 1000 km from Sakaiminato. The maximum surge levels due to Typhoon Maemi and Megi were approximately 0.6 m, which isidentical to asurge level with a 100-year return period at Sakaiminato

J. Mar. Sci. Eng. 2020, 8, 797; doi:10.3390/jmse8100797 www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 3 of 26 the surge level is included in the training set. However, our question is that if the surge level is excluded in the training set, how accurate can storm surge forecasts be? Thus, surge level may be replaced with sea surface level at Sakaiminato where the annual averaged-tidal range is about 0.3 m. We expect that if the surge level is excluded from the input data set, we can skip the procedure for extracting it from the sea surface level. In addition, we are encouraged to look for an easier operating model in practice in comparison with the ANN-based model that is run on computers supported by storm surge experts and computer technicians. The group method of data handling (GMDH) is part of the neural network family, and it is similar to GP in terms of self-constructing features that build elements in hidden layers and derive second-order polynomials. The GMDH algorithm also represents a white-box forecasting because GMDHJ. Mar. Sci. derives Eng. 2020 formulas, 8, 797 to select for appropriate components among training data. However, storm2 of 27 surge forecasting that deals with GMDH is very low, even though several studies applied GMDH to the prediction of the wave, medical image recognition, and container transportation throughput (Kim (Hiyajo et al. [3]). Additionally, Sakaiminato can be characterized by its tidal range, the annual average et al. [10], Kondo [36], Mo et al. [37]). The benefit of the GMDH is that its operation is comprehensible of which is approximately 0.3 m along the Tottori coast. The linear superimposition of the surge in comparison to the ANN. For example, the surge level can be predicted by solving a series of one and tide can be easily carried out, because their interaction is weak, meaning that the sea surface to three closed-form equations using a calculator or spreadsheet if the GMDH-based model is level with the 100-year return period can be estimated to be approximately 0.9 m. This is relatively successively trained. On the other hand, the ANN-based model’s cost is relatively higher, which is lower than the peak sea surface level along the Paciﬁc Ocean coast; for example, Typhoon Jebi (2018) operated on a computer machine. generated a peak surge level of 2.78 m, resulting in a sea surface level of 3.29 m (Mori et al. [1]). As such, Therefore, the present study investigates the applicability of the GMDH algorithm to the storm people in Sakaiminato lack awareness of storm surge risks. Therefore, the surge information should be surge hindcasting and forecasting with the lead times of 5, 12, and 24 h. In addition, the effect of the provided early upon the TCs approach, as well as 15–18 h later, following the TCs passage, so that local inclusion of the surge level at the input layer on the accuracy of the surge forecast is examined. Section communities can make quick decisions for risk management in Sakaiminato. To provide the surge 2 describes GMDH data and methodology. In Section 3, results and discussion are provided. Section information, an early warning system is necessary. 4 concludes the findings of the study.

Ama Saigo Sakaiminato Matsue Yonago

Hamada

(a) Meteorological stations (b) Hydrodynamic stations

FigureFigure 1. 1.TyphoonTyphoon tracks tracks and and stations stations for for meteorological meteorological (Hamada, (Hamada, Yonago, Yonago, Matsue, Matsue, Ama, Ama, and and Saigo) Saigo) andand hydrodynamic hydrodynamic (Sakaiminato) (Sakaiminato) stations stations around around Sakaiminato, Sakaiminato, Tottori, Tottori, Japan. Japan.

2. DescriptionEarly warning of GMDH, systems Study for Area, storm Data, surge and forecasting Methodology consist of either process-based numerical prediction models or machine learning-based models. Numerical prediction models are based Herein, we describe the application of GMDH to storm surge forecasting and then explain data onshallow water equations. In early warning systems, a two-dimensional model consisting of taken for training and testing. After Ivankhnenko [38] initially introduced GMDH, many applications shallow water equations is generally preferable due toits light computational load (e.g., SLOSH; have used it in a variety of fields. For example, Kondo [36] employed the GMDH to recognize medical Jelesnianski et al. [4], ADCIRC; Luettich and Westerink [5], FVCOM; Chen et al. [6], SELFE; Zhang and images. Mo et al. [37] developed a GMDH-based hybrid model to forecast a container amount for Baptista [7], Delft3D; DELTARES [8], and SuWAT; Kim et al. [9]). Such modelsare forced by wind and portmanagement. Lee and Suh [16] derived wave overtopping formulas using GDMH. Kim et al. [10] pressure ﬁelds that are estimated/forecasted by either wind and pressure parameter formulae or a used the GMDH algorithm to derive a second-order polynomial for wave prediction. The detailed general circulation model. information on GMDH can be found in the study conducted by Onwubolu [39]. On the other hand, machine learning techniques that use artiﬁcial neural networks (ANNs)

or deep learning methods can also be used to develop machine learning-based forecast models. Such machine learning-based forecast models have to be supervised by training data. For instance, sets that consist ofwind speeds, wind directions, sea-level pressures, and sea surface levels observed near target sites can be used to supervise the model. Typhoon characteristics such as typhoon positions, sizes, forwarding speeds, and central pressures can also be used to supervise it. In recent years, machine learning-based models have been widely applied in many fields of Ocean and Coastal Engineering, such as forpredicting waves (Kimet al. [10], Deo and Sridhar [11], Deo et al. [12], Londhe and Panchang [13], Makarynskyy et al. [14], Oh and Suh [15], James et al. [16], Tom et al. [17], Tsai et al. [18], Tsuda et al. [19], Lee and Suh [20]), wave forces (Mase and Kitano [21], Mase et al. [22], Van Gent et al. [23]), storm surges(Kim et al. [24,25], Hien et al. [26], Bishnupriya and Bhaskaran [27], Oliveira et al. [28], Lee [29]), tides (Deo and Chaudhar [30]), sediment suspensions (Yoon et al. [31]), and debris flows (Chang and Chao [32]). In addition, conventional regression methods formed J. Mar. Sci. Eng. 2020, 8, 797 3 of 27 by linear regression have been employed to forecast storm surges, derived on the basis of wind speed, wind direction, sea level pressure, and typhoon approach direction (e.g., Mori et al. [33], Roberts et al. [34]). The linear regression method derives its results based on the relationship between storm surge and other factors. On the other hand, in contrast to the conventional regression method, the group method of data handling (GMDH) derivesa nonlinear second-order polynomial based on complicated storm surge phenomena. Among machine learning techniques, ANN shave been widely employed in developing storm surge forecast models for application in Sakaiminato (e.g., Kim et al. [24,25]). Kim et al. [24] investigated the best training sets for the storm surge forecast with lead times of up to 24 h in Sakaiminato, Japan. Kim et al. [25] proposed an experiential selection procedure for determining the best training set for storm surge forecasting by establishing two parameters of the unit number in the hidden layer and number of hidden layers in an ANN. Hien et al. [3] applied genetic programming (GP), which is an evolutionary procedure to derive solutions in the form of computer programs (Koza [35]), to forecast the storm surge at Sakaiminato. GP automatically determines relevant components among many different types of input parameters to evolve surge prediction formulas. Based on the GP-derived formulas, storm surge forecasting is able to be more interpretable. The studies mentioned above employed the locally observed meteorological and hydrodynamic components of the wind, sea-level pressure, sea surface level, and surge level, as well as the typhoon feature. In their studies, the surge level is included in the training set. However, our question is that if the surge level is excluded in the training set, how accurate can storm surge forecasts be? Thus, surge level may be replaced with sea surface level at Sakaiminato where the annual averaged-tidal range is about 0.3 m. We expect that if the surge level is excluded from the input data set, we can skip the procedure for extracting it from the sea surface level. In addition, we are encouraged to look for an easier operating model in practice in comparison with the ANN-based model that is run on computers supported by storm surge experts and computer technicians. The group method of data handling (GMDH) is part of the neural network family, and it is similar to GP in terms of self-constructing features that build elements in hidden layers and derive second-order polynomials. The GMDH algorithm also represents a white-box forecasting because GMDH derives formulas to select for appropriate components among training data. However, storm surge forecasting that deals with GMDH is very low, even though several studies applied GMDH to the prediction of the wave, medical image recognition, and container transportation throughput (Kim et al. [10], Kondo [36], Mo et al. [37]). The benefit of the GMDH is that its operation is comprehensible in comparison to the ANN. For example, the surge level can be predicted by solving a series of one to three closed-form equations using a calculator or spreadsheet if the GMDH-based model is successively trained. On the other hand, the ANN-based model’s cost is relatively higher, which is operated on a computer machine. Therefore, the present study investigates the applicability of the GMDH algorithm to the storm surge hindcasting and forecasting with the lead times of 5, 12, and 24 h. In addition, the effect of the inclusion of the surge level at the input layer on the accuracy of the surge forecast is examined. Section2 describes GMDH data and methodology. In Section3, results and discussion are provided. Section4 concludes the findings of the study.

2. Description of GMDH, Study Area, Data, and Methodology Herein, we describe the application of GMDH to storm surge forecasting and then explain data taken for training and testing. After Ivankhnenko [38] initially introduced GMDH, many applications have used it in a variety of ﬁelds. For example, Kondo [36] employed the GMDH to recognize medical images. Mo et al. [37] developed a GMDH-based hybrid model to forecast a container amount for portmanagement. Lee and Suh [16] derived wave overtopping formulas using GDMH. Kim et al. [10] used the GMDH algorithm to derive a second-order polynomial for wave prediction. The detailed information on GMDH can be found in the study conducted by Onwubolu [39]. J. Mar. Sci. Eng. 2020, 8, 797 4 of 27

2.1. GMDH-Based Surge Forecast Model The present study calculates true valuesusing a second-order polynomial, Equation (1), which was initially introduced by Ivankhnenko [38]:

y = a + a x + a x + a x2 + a x2 + a x x (1) n k0 k1 in k2 jn k3 in k4 jn k5 in jn where xi. and xj. are two inputs; yn. is an output; i, j, and k are the index of the input component; and n is the layer number. To determine the coeﬃcient, ak , in Equation (1), the partial derivatives of Equation (2) are taken in terms of each coeﬃcient, ak, and set equal to zero, as in Equation (3). The expansion of Equation (3) obtains the following system of the Gauss normal Equations of (4) to (9), which are solved using the training data set:

N X 2 g = (yn yon) (2) − n=1

∂g = 0 (3) ∂ak XN XN n o y = a + a x + a x + a x2 + a x2 + a x x (4) kn k0 k1 in k2 jn k3 in k4 jn k5 in jn n=1 n=1 XN XN y x = a x + a x2 + a x x + a x2 x + a x3 + a x x2 (5) kn in k0 in k1 in k2 jn in k3 in jn k4 jn k5 in jn n=1 n=1 XN XN y x = a x + a x x + a x2 + a x2 x + a x2 x + a x3 (6) kn jn k0 jn k1 jn in k2 jn k3 jn in k4 in jn k5 jn n=1 n=1

XN XN n o y x x = a x x + a x2 x + a x x2 + a x2 x2 + a x3 x + a x x3 (7) kn in jn k0 in jn k1 in jn k2 in jn k3 in jn k4 in jn k5 in in n=1 n=1 XN XN y x2 = a x2 + a x3 + a x2 x + a x3 x + a x4 + a x2 x3 (8) kn in k0 in k1 in k2 in jn k3 in jn k4 in k5 in jn n=1 n=1 XN XN y x2 = a x2 + a x x2 + a x3 + a x x3 + a x2 x2 + a x4 (9) kn jn k0 jn k1 in jn k2 in k3 in jn k4 in in k5 jn n=1 n=1 where yon is the observation. When applying the GMDH, yn is a storm surge level where xi and xj are either two input parameters or subsets of each element at the hidden layers through the iteration generated by Equation (1). Figure2 shows the training procedureto develop a GMDH-based storm surge forecast model. J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 5 of 26

• Step 8: Repeat Steps 1–7 using the subset data generated by Equation (1) for each element in Step 5. • Step 9: Compare the smallest value, 𝑔, in Step 8 to that in Step 6. • Step 10: Repeat Steps 8 and 9 until the present layer’s best accuracy is subservient to that of the previous layer. If the best performing element is determined from Step 6, its partial expression becomes a GMDH-based storm surge forecast model. Through the iteration, if the best performing element is

judgedat the first layer, two input components are assigned in Equation (1) as 𝑥 and 𝑥. Then, one equation consists of the GMDH-based storm surge model. On the other hand, if it is judged to be more than the 2nd layer, asubset generated at the hidden layer is allocated in Equation (2), resulting in several equations corresponding to the number of hidden layers where the procedure stopped. In the present study, all partial expressions are obtained in the more than just the 2nd layer, so that two input parameters in Equation (1) are generated by the subset at the hidden layer. As described above, the techniques of ANN, GP, multi-layer perceptron, and GMDH are in the form of a stochastic algorithm (Hien et al. [26]), while other data-driven models of the K-nearest neighbors, decision tree, J.and Mar. support Sci. Eng. vector2020, 8, 797regression are deterministic. Thus, the GMDH requires several iterations to derive5 of 27 the second-order polynomial.

Inputs Layer 1 Layer 2 Layer 3 Layer 4 Output

#1 #1 & #2 (#1L2) #1L2 & #2L2 (#1L3) #1L3 & #2L3 Input component #1 #2 #2 & #1 (#2L2) #1L2 & #3L2 (#2L3) #2L3 & #3L3 Storm surge Input level component #2 #1 #1 & #1 (#3L2) #1L2 & #3L2 (#3L3) #3L3 & #1L3 Input component #3 ......

#n-2&#n #n-2&#n-2 & #n-2&#n #n-2 & #n #n & #n-1 Input component #n Select the best Select E2 and Select E2 and Select E2 and performing element proceed to proceed to proceed to with its minimum the next layer the next layer the next layer least square error, if E2 < E1. if E2 < E3, if E4 > E3. E1 or terminate the iternation if E2 > E3. Figure 2. A procedure that develops the group method of data handling (GMDH)-based storm surge forecastforecast modelmodel (circle(circle indicatesindicates thethe element).element).

StepTable 1: Generate 1. Experiments pairs of for the training elements. and testing For a pair,the GMDH-based one element storm at the surge input forecast layer consistsmodel. of two • input parameters (see Table1) and another element at the output layer compounds the observation. Input Parameters Step 2: Transfer the pairSea-Level to layer 1. Maximum • Sea Sea- Output Pressure Wind Wind Typhoon Central Wind Step 3:Surface Determine Level the values of the coefficients, ak, through the Gauss normal EquationsParameter (4)–(9). • Model Drop Speed Direction Position Pressure Speed Near Step 4:Level Derive Pressure Equation (1) with the coefficient, a . • No. Rate k the Center Step 5: Predict the output, yn, using Equation (1) for each element. • TP Storm Step 6:SSL Calculate SLP the value,DSLPg, which WS is the diffWDerence between(=LOT + the predictionCP MWS and observationSurge for • each element. LAT) Level 1 O O O Step 7: Determine the element of the smallest value, g, of Equation (2) and rearrange the elements • 2 O O O O 3 in decreasingO order. O O O O 4 Step 8: RepeatO Steps O 1–7 using O the subset O data generated O by Equation (1) for each element in OStep 5. • 5 O O O O Step 9: Compare the smallest value, g, in Step 8 to that in Step 6. • Step 10: Repeat Steps 8 and 9 until the present layer’s best accuracy is subservient to that of the • previous layer.

If the best performing element is determined from Step 6, its partial expression becomes a GMDH-based storm surge forecast model. Through the iteration, if the best performing element is judgedat the ﬁrst layer, two input components are assigned in Equation (1) as xi and xj. Then, one equation consists of the GMDH-based storm surge model. On the other hand, if it is judged to be more than the 2nd layer, asubset generated at the hidden layer is allocated in Equation (2), resulting in several equations corresponding to the number of hidden layers where the procedure stopped. In the present study, all partial expressions are obtained in the more than just the 2nd layer, so that two input parameters in Equation (1) are generated by the subset at the hidden layer. As described above, the techniques of ANN, GP, multi-layer perceptron, and GMDH are in the form of a stochastic algorithm (Hien et al. [26]), while other data-driven models of the K-nearest neighbors, decision tree, and support vector regression are deterministic. Thus, the GMDH requires several iterations to derive the second-order polynomial. J. Mar. Sci. Eng. 2020, 8, 797 6 of 27

Table 1. Experiments for training and testing the GMDH-based storm surge forecast model.

Input Parameters Maximum Sea Sea-Level Sea-Level Wind Wind Typhoon Central Wind Speed Output Surface Pressure Pressure Speed Direction Position Pressure Near the Parameter Level Drop Rate Model Center No. TP Storm SSL SLP DSLP WS WD (=LOT CP MWS Surge + LAT) Level 1 O O O 2 O O O O 3 O O O O O 4 O O O O O O 5 O O O O 6 O O O O O 7 O O O O O O 8 O O O O O O O 9 O O O O O O 10 O O O O O O 11 O O O O O 12 O O O O O O 13 O O O O 14 O O O O O O O 15 O O O O O O O O 16 O O O O O O 17 O O O O O O O 18 O O O O O O O O 19 O O O O O O O O 20 O O O O O O O O 21 O O O O O O O O O 22 O O O 23 O O O O 24 O O O O O

2.2. Description of Study Area and Data

2.2.1. Study Area The Sakaiminato port in Tottori faces two continental shelf seas in the Sea of Japan/East Sea (SJES) and, from June to November, is at risk of storm surges due to typhoons. The 2.6 typhoons annually have made apass or landfall on the Tottori coast since 1975 (Hiyajo et al. [3]). Hiyajo et al. [3] estimated 0.63 m for the 100-year return period of the surge level in Sakaiminato from the available historical sea surface level measured by the Japan Meteorological Agency from 1924 to 2008. We selected three typhoons of Maemi 2003, Songda 2004, and Megi 2004, whose surge levels almost corresponded to the surge level of a 100-year return period. In particular, the Maemi and Megi surge levels were close to the 100-year return period level. These typhoons are treated as the representative and most intensive events on the Tottori coast, as they cause high surge levels in Sakaiminato (Kim et al. [2]). Hiyajo et al.[3] also showed that these typhoons pass along the track, as shown in Figure1b. Three tracks are in the range of the historical typhoon and inducestorm surge in Sakaiminato. As shown in Kim et al. [2], the peak surge level in Sakaiminato port appears 15~18 h later after the typhoon undergoes a change in shape and intensity as extratropical cyclones near Hokkaido. The annual tidal range is 0.3 m, resulting in it being ignorable in storm surge simulations due to a weak interaction betweensurge and tide. However, if the interaction of surge and tide is signiﬁcant, it should be considered in forecasting systems (e.g., Flowerdew et al. [40])

2.2.2. Input and Output Parameters The study aimed to forecast the storm surge level at Sakaiminato port, Tottoriand hence employedlocal meteorological and hydrodynamic data measured hourly around Sakaiminato. J. Mar. Sci. Eng. 2020, 8, 797 7 of 27

The present study used identical data sets to Kim et al. ([24,25]) and Hien et al. [26] to develop a GMDH-based storm surge forecast model for Sakaiminato, Tottori, Japan. For the meteorological data, the wind speed, wind direction, sea-level pressure, and its drop rate from background pressures (1013 hPa) were collected at ﬁve meteorological stations of Hamada, Matsue, Yonago, Ama, and Saigo (Figure1). For the hydrodynamic data, the sea surface level and surge level were gathered at one hydrodynamic station at Sakaiminato (see Figure1). We selected typhoon information data of the position, central atmospheric pressure, and maximum wind speed near the typhoon center that represented the typhoon characteristics. Those data described above weremeasured during three typhoons of Maemi 2003, Songda 2004, and Megi 2004. The typhoons generated the storm surge level at Sakaiminatoidentical to the surge level with a 100-yearreturn period (Hiyajo et al. [3]). According to Hiyajo [3], the three typhoons’ tracksfollowed a track with a high risk of generating higher surge levels at Sakaiminato (see Figure1a). Before training, input and output parameters were normalized as follows:

the surge level : ηi = eηi (10)

the sea surface level (= atmospheric tide + storm surge, SSL) : vSSL = evSSL / 2m (11)

the sea level pressure (SLP) : vSLP = evSLP / 1013 hPa (12) − the sea-level pressure dropfrom background sea-level pressure (= 1013 hPa, DSLP):

vDSLP = evDSLP / 100 hPa (13)

the wind speed (WS) : vWS = evWS / 100 m/s (14)

the wind direction (WD) : vWD = evWD / 360 degree (15)

the longitude of typhoon (LOT) : vLG = evLG / 150◦E (16)

the latitude of typhoon (LAT) : vLT = evLT / 50◦N (17)

the central atmospheric pressure of typhoon (CP) : vCAP = evCAP / 1013 hPa (18)

the highest wind speed near a typhoon center (MWS) : vHWS = evHWS / 100 m/s (19) where the tilde (~) on the right-hand side of Equations (10)–(19) indicate the raw parameters of normalization in the range of 1 to 1. For the normalization of the tide, the observed sea surface level − was divided by 2 m because the tidal range at Sakaiminato was, at maximum, 2 m. The data of wind, pressure, and water level observed at the station were collected from Japan’s Meteorological Agency. The typhoon data were collected from Digital Typhoon.

2.2.3. Methods for Training and Testing The GMDH used in this study was trained with the input and output parameters collected during two typhoons (Typhoons Maemi 2003 and Songda 2004), while it was tested with the input ones gathered during Typhoon Songda 2004. In detail, the parameters taken in the experiments are listed in Table1. One hindcast and three forecast models with 5-, 12-, and 24-h lead times were developed using the following equations, respectively: = st=1,...,5 st=1,...,5 st=1,...,5 st=1,...,5 p=1,2 ηt=i f SSLt=i, SLPt=i , DSLPt=i , WSt=i , WDt=i , TPt=i , CPt=i, MWSt=i (20) = st=1,...,5 st=1,...,5 st=1,...,5 st=1,...,5 st=1,2 ηt=i+5 f SSLt=i, SLPt=i , DSLPt=i , WSt=i , WDt=i , TPt=i , CPt=i, MWSt=i (21) = st=1,...,5 st=1,...,5 st=1,...,5 st=1,...,5 st=1,2 ηt=i+12 f SSLt=i, SLPt=i , DSLPt=i , WSt=i , WDt=i , TPt=i , CPt=i, MWSt=i (22) = st=1,...,5 st=1,...,5 st=1,...,5 st=1,...,5 st=1,2 ηt=i+24 f SSLt=i, SLPt=i , DSLPt=i , WSt=i , WDt=i , TPt=i , CPt=i, MWSt=i (23) J. Mar. Sci. Eng. 2020, 8, 797 8 of 27 where t is the time, st is the number of stations, p is the longitude and latitude, and η is the storm surge predicted by the GMDH-based model. For each model, 24 individual forecast models were developed with different sets consisting of eight parameters, with one at the input layer and the other at the output layer (Table1). The di fference between the present study and Kim et al. ([24,25]) is the exclusion of the storm surge level at the input layer. In addition, we conducted an additional experiment that added a surge level to each combination. Then, the best performing model for each lead time was determined based on the performance indices of the correlation coefficient (CC), normalized root mean square error (NRMSE), and standard deviation (STD). q 1 Pi=n 2 n i=1 ηobs,i η f ore,i NRMSE = − (24) η η obs,max − obs,min Pi=n i=1 ηobs,i ηobs η f ore,i η f ore CC = − − (25) q r P = 2 P = 2 i n η η i n η η i=1 obs,i − obs i=1 f ore,i − f ore uv t i=n 1 X 2 STD = η η (26) n 1 obs,i − f ore − i=1 where ηobs is the observed surge level, η f ore is the forecasted surge level, ηobs is the averaged observation, η f ore is the average forecast, ηobs, max is the observed highest surge level, and ηobs,min is the observed lowest surge level.

3. Results The GMDH-based storm surge hindcast/forecast model was developed by training and testing with the data set. In the training phase, the GMDH-based models were trained through either the ﬁrst layer or second layers that generated subsets and derived second-order polynomials of Equation (1) with the obtained coeﬃcients. If the second-order polynomial was determined, the surge level in the testing phase was assessed by calculating the performance indices of CC, NRMSE, and STD in comparison with the observed Typhoon Song dasurge level. Then, we could make a decision regarding which model was the best among the 24 models. Table2 shows computational times for training and testing the GMDH-based surge forecast model. The average run time was approximately 5.8 s and 6.8 s for hindcasts, and 5 h, 12 h, and 24 h for forecasts, respectively.

3.1. Performances of GMDH-Based Storm Surge Hindcast Models

3.1.1. Exclusion of Surge Levels The GMDH-based storm surge hindcast model corresponded to the zero-hour lead time and was trained with the data set of two typhoons and then tested with the set of Typhoon Songda. In the experiment, the surge level is not considered. Finally, it was assessed by using the two performance indices of CC and NRMSE, as shown in Figure3. J. Mar. Sci. Eng. 2020, 8, 797 9 of 27

Table 2. Computational elapsed times in second for training and testing the GMDH-sed surge model.

Model No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Hindcast 5.44 5.30 6.12 5.90 5.67 5.71 5.69 5.99 6.20 5.33 5.77 5.69 5.85 5.92 5.88 5.66 5.82 5.81 5.83 5.87 5.93 5.66 5.71 5.77 5 h forecast 5.85 5.69 5.72 5.76 5.75 5.62 5.65 5.74 5.88 5.73 5.73 5.73 5.64 5.98 5.71 5.76 5.74 6.02 5.72 5.86 5.66 5.62 5.78 5.63 12 h forecast 5.70 5.61 5.71 5.79 5.73 5.68 5.78 5.84 5.95 5.79 5.69 5.89 5.79 5.84 5.81 5.65 5.75 5.97 5.81 5.96 5.96 5.63 5.64 5.72 24 h forecast 5.73 6.50 5.76 5.96 5.82 5.84 27.44 7.23 5.95 6.09 5.59 5.60 6.01 5.68 5.89 5.59 5.83 5.78 5.59 5.65 5.85 5.53 5.62 5.63 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 10 of 26 J. Mar. Sci. Eng. 2020, 8, 797 10 of 27 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 10 of 26 1

0.81

0.80.6 CC 0.60.4

CC 0.40.2

0.20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 model no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

(a) Correlationmodel coefficient no. (CC).

(a) Correlation coefficient (CC).

0.2

0.150.2

0.150.1

0.050.1 NRMSE (%) NRMSE

0.050 NRMSE (%) NRMSE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 model no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

(b) Normalized root meanmodel square no. error (NRMSE).

FigureFigure 3. Performance 3. Performance indices indices(b) Normalized of of CC CC and and root NRMSENRMSE mean square were calculated calculatederror (NRMSE). between between Typhoon Typhoon Songda Songdasurgesurge levelslevels observed observed at Sakaiminato at Sakaiminato and and hindcasted hindcasted byby the GMDH GMDH-based-based storm storm surge surge hindcast hindcast models. models. FigureRed indicates 3. Performance the highest indices CC (or of the CC lowest and NRMSE NRMSE), were while calculated purple shows between the Typhoonworst. Songdasurge Red indicateslevels observed the highest at Sakaiminato CC (or the and lowest hindcasted NRMSE), by the while GMDH purple-based shows storm thesurge worst. hindcast models. Red indicates the highest CC (or the lowest NRMSE), while purple shows the worst. Before discussing the best model based on the performance index, it should be noted that, for each model, we derived models in the form of Equation (1). For example, model no. 4 was the best model for hindcasting and was obtained from the 3rd layer with the minimum lest residual of 0.014 (1.4%) in

)

the trainingm procedure. Its 2nd-order polynomial derived was given in Equation (27):

(

n )

t 2 2

(

= + + + y3c 0.00776 0.362533x1 0.598088x17 39.8235x 40.04x 79.9204x1 x17 (27) i 3 3 1 17 3 3

n 3 3

d − −

i where x1 andP x17 are the variables at the 3rd hidden layer. Model no. 4 was trained with the training 3 d 3

e data set presentedr in Equation (28):

P = st=1,...,5 st=1,...,5 st=1,...,5 st=1,...,5 ηt=i f SSLj=1,t=i, SLPj=2,6,t=i, DSLPj=7,11,t=i, WSj=12,16,t=i, WDj=17,21t=i (28) Observation (m) where j(ais) Comparisons the index of of thepredictions input components.and observations. At the 1st(b) T layer,ime-series a pair of predictions oSSL (x )and and observations. WD (x ) had the Observation (m) 11 171 lowest least residual (LR) of 0.028 (2.8%) next to a pair of SSL (x1 ) and WD (x21 ) with LR of 0.033 (a) ComparisonsFigure 4. Comparisons of predictions of observations and observations. and predictions (b hindcasted) Time-series by of m1 predictionsodel no. 4, which and observations. 1was the best (3.3%) amongperformance the 210 model pairs. at Sakaiminato. The two pairs derived Equations (29) and (30) as: Figure 4. Comparisons of observations and predictions hindcasted by model no. 4, which was the best performance model at Sakaiminato. 2 uOn1 = the2.15668 other hand,3.05022 the worst+ 0.0258085 performancex7 + 1.08922models 9x in terms0.00651275 of CC and0.00797044 NRMSE werex1 x 7model (29) − 1 11 − − 1 1 nos. 2 and 9, which are highlighted in purple in Figure 3. When comparing model no. 2′s CC with On the other hand, the worst performance models2 9 in terms of CC2 and NRMSE were model vmodel1 = 2.21427 nos. 1 and3.10306 3, we can0.0428193 see that ignoringx21 + 1.10669 SSL orx adding+ 0.0171726 WS to mxodel n0.00503917o. 2 was notx 1enoughx21 to (30) nos. 2 and 9, which are highlighted in purple1 in Figure11 3. When comparing211 model no. 2′s 1CC 1with predict the Typhoon− Songda− surge. In addition, removing SSL in the data set− of model no. 8 was not model nos. 1 and 3, we can see that ignoring SSL or adding WS to model no. 2 was not enough to Twoacceptable, outputs because of Equations the accuracy (29) of andmodel (30) no. would9 became be worse the subsetin comparison of x12 andwith modelx22 at no. the 8. 2nd layer, for instance.predict the After Typhoon rearranging Songdasurge. the pairs In addition, in a decreasing removing order SSL in estimated the data set by of models model derivedno. 8 was at not the 1st acceptable, because the accuracy of model no. 9 became worse in comparison with model no. 8. layer, two pairs of x12 and x22 (the lowest LR of 0.020 (2.0%)), and x32 and x102 (the second-lowest LR of

J. Mar. Sci. Eng. 2020, 8, 797 11 of 27

0.022 (2.2%)) were determined among the 210 pairs regenerated at the 2nd layer. The polynomials of Equations (31) and (32) are derived with the two pairs listed above.

2 2 u2 = 0.0127957 + 0.200435x1 + 0.999941x2 18.41x 20.2414x + 38.2968x1 x2 (31) − 2 2 − 12 − 22 2 2 2 2 v2 = 0.0126375 + 5.12335x3 3.90761x10 349.574x 335.071x + 684.252x3 x10 (32) − 2 − 2 − 32 − 102 2 2

Two outputs of u2 and v2 shown by Equations (31) and (32) are the subset of x13 and x23 at the 3rd layer, as seen above. Because the lowest LR of 2.0% at the 2nd layer was still smaller than that of 2.8% at the 1st layer, the development of the GMDH-based model proceeded. At the 3rd layer, two pairs of x13 and x173 (the lowest LR of 0.014 (1.4%)), and x13 and x143 (the second-lowest LR of 0.023 (2.3%)) were obtained and compounded by Equation (27). At the 4th layer, a pair of x14 and x24 had the lowest LR of 0.018 (1.8%). Since the minimum LR of the 4th was larger than that of the 3rd layer. As a result, the best model could be determined by Equation (27) derived at the 3rd layer, where u2 and v2 are represented by x13 and x173 in Equation (27). In the experiment, the CC and NRMSE indices coherently indicated the best model; therefore, Figure3 shows only two plots (STD is omitted). As a result, the averaged CC and NRMSE values of the 24 models were 0.64 and 10%, respectively. Among the CC values, the highest CCs (0.92) were obtained from model nos. 4 and 15, which were trained with the SLP, DSLP, SSL, WS, and WD data sets, as well as all types of input parameters except MWS. The smallest NRMSEs were also calculated using model J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 10 of 26 nos. 4 and 15 results, which were 0.05% in both models. The result indicated that the best model may be model no. 4, which used sea-level(a) pressure,Correlation dropcoefficient rate, (CC). tide, wind speed, and wind direction for training. It was0.2 more economically efficient in terms of data collection compared to the training set of model no. 15, which uses all input parameters except maximum wind speed around the typhoon 0.15 center. Model no. 40s training set revealed that the after-runner storm surge at Sakaiminato had a peak surge level0.1 of 15~18 h after the typhoon passage (Kim et al., [2]), which could be hindcasted by sea-level pressure, drop rate, tide, wind speed, and wind direction. Moreover, it should be noted that 0.05 the GMDH(%) NRMSE algorithm is self-constructed, i.e., it takes useful elements, while abandons ineffectual ones through the iteration0 in hidden layers. From the results, model nos. 4 and 15 had identical accuracy. Meanwhile, model1 no. 215 3 was 4 5 trained 6 7 with 8 9 more 10 11 parameters 12 13 14 15 compared 16 17 18 to 19 the 20 training 21 22 23 data 24 of model no. 4. This meant that the parameters of model no.model 15 no. were filtered through the training and some parameters were dropped from the useful parameter set. Thus, the two models’ surge levels were (b) Normalized root mean square error (NRMSE). almost homogeneous. Comparisons of the observed and predicted surge levels are given in Figure4, showingFigure that the 3. Performance overall GMDH-based indices of CC stormand NRMSE surge were hindcast calculated model between well Typhoon predicted Songdasurge the Songdasurge levels observed at Sakaiminato and hindcasted by the GMDH-based storm surge hindcast models. level, while the peak was slightly overestimated in comparison to the observed one. Red indicates the highest CC (or the lowest NRMSE), while purple shows the worst.

)

(

Observation (m)

(a) Comparisons of predictions and observations. (b) Time-series of predictions and observations.

FigureFigure 4. Comparisons 4. Comparisons of of observations observations andand predictions hi hindcastedndcasted by by model model no. no.4, which 4, which was the was best the best performanceperformance model model at Sakaiminato.at Sakaiminato.

On the other hand, the worst performance models 9 in terms of CC and NRMSE were model nos. 2 and 9, which are highlighted in purple in Figure 3. When comparing model no. 2′s CC with model nos. 1 and 3, we can see that ignoring SSL or adding WS to model no. 2 was not enough to predict the Typhoon Songdasurge. In addition, removing SSL in the data set of model no. 8 was not acceptable, because the accuracy of model no. 9 became worse in comparison with model no. 8.

3.1.2. Inclusion of Surge Levels The additional experiment was carried out with the surge level in each combination. In this experiment, the GMDH-based storm surge models perfectly hindcasted the Songda surge level, except for model nos. 4, 8, 9, 10, and 21, which were not converged because the input data set formed singular matrices. One of the prediction results obtained from model no. 1 is given in Figure 5. As a result, the surge level highly impacted the prediction accuracy of the GMDH-based surge hindcast model.

J. Mar. Sci. Eng. 2020, 8, 797 12 of 27

On the other hand, the worst performance models 9 in terms of CC and NRMSE were model nos. 2 and 9, which are highlighted in purple in Figure3. When comparing model no. 2 0s CC with model nos. 1 and 3, we can see that ignoring SSL or adding WS to model no. 2 was not enough to predict the Typhoon Songdasurge. In addition, removing SSL in the data set of model no. 8 was not acceptable, because the accuracy of model no. 9 became worse in comparison with model no. 8.

3.1.2. Inclusion of Surge Levels The additional experiment was carried out with the surge level in each combination. In this experiment, the GMDH-based storm surge models perfectly hindcasted the Songda surge level, except for model nos. 4, 8, 9, 10, and 21, which were not converged because the input data set formed singular matrices. One of the prediction results obtained from model no. 1 is given in Figure5. As a result, the surge level highly impacted the prediction accuracy of the GMDH-based surge hindcast model. J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 11 of 26

(a) Comparisons of predictions and (b) Time-series of predictions and observations observations

FigureFigure 5. Comparisons 5. Comparisons of of observations observations and predictions predictions hindcasted hindcasted by bymodel model no.1 no.1at Sakaiminato. at Sakaiminato.

3.2. Performances3.2. Performances of GMDH-Basedof GMDH-Based 5-h 5-h LeadLead Time Storm Storm Surge Surge Forecast Forecast Models Models

3.2.1.3.2.1. Exclusion Exclusion of of Surge Surge Levels Levels in in the the TrainingTraining Data Set Set TheThe GMDH-based GMDH-based 5-h 5-h lead lead time time stormstorm surge forecast forecast models models were were trained trained with with the data the sets data of sets of TableTable1. It 1. ignored It ignored the the surge surge level level and and was was assessedassessed with with Typhoon Typhoon Songda’s Songda’s data data in the in testing the testing phase. phase. The averagedThe averaged CC andCC and NRMSE NRMSE values values were were 0.78 0.78 and and 10%, 10%, respectively. respectively. Among Among the the models, models, the the highest highest CC (0.881) was obtained from three models (see Figure 6a): model no. 4 was trained with SLP, CC (0.881) was obtained from three models (see Figure6a): model no. 4 was trained with SLP, DSLP, DSLP, SSL, WS, and WD; model no. 12 was trained with SLP, DSLP, SSL, TP, and CP; model no. 17 SSL,was WS, trained and WD; with modelSLP, DSLP no., SSL, 12 wasTP, CP, trained and MWS, with which SLP, DSLP,were id SSL,entical TP, to the and training CP; model set of model no. 17 was trainedno. 12, with except SLP, for DSLP, MWS. SSL, When TP, comparing CP, and MWS, the CC which values were between identical model to nos. the 12 training and 17, setthe ofparameter model no. 12, exceptof MWS for MWS. had no When significant comparing impact on the the CC prediction values between of the Songda model surge nos. level. 12 andWhen 17, comparing the parameter the of MWSCC had value no of significant model no. 4 impact with model on the nos. prediction 1, 2, and 3, ofwe the found Songda that the surge inclusion level. of When sea surface comparing level, the CC valuewind speed, of model and no. wind 4 with direction model benefits nos. 1, for 2, andpredicting 3, we surge found level that and the inclusionimproving of the sea CC surface value level, windvaried speed, slightly. and wind Furthermore, direction model benefits no. for2 trained predicting with SLP, surge DSLP, level and and SSL improving nearly had the a CCsimilar value CC varied slightly.value Furthermore, of 0.87. In other model words no., the 2 parameters trained with of sea-level SLP, DSLP, pressu andre, SSLdrop nearly rate, and had sea a similarsurface level CC value of 0.87.had Ina major other effect words, on surge the parameters level prediction. of sea-level In addition, pressure, the parameters drop rate, of the and typhoon sea surface position level and had a majorits ecentralffect on pressure surge levelcould prediction.have a minor In impact, addition, thusthe slightly parameters improving of theprediction typhoon ability. position On the and its centralother pressure hand, the could worst have CC value a minor was impact, calculated thus from slightly model improvingno. 9 trained prediction with SLP, DSLP, ability. WS, On WD, the other and TP. In comparison, with the CC of model no. 8 trained with the same data set to model no. 9 hand, the worst CC value was calculated from model no. 9 trained with SLP, DSLP, WS, WD, and TP. except for SSL, model no. 9′s CC became substantially worse. This was found when comparing other In comparison, with the CC of model no. 8 trained with the same data set to model no. 9 except for models with the CC of model no. 10, which was trained with the data set including SSL and excluding SSL,WS model in the no. training 90s CC set becameof model substantiallyno. 9. The exclusion worse. of WS This had was no foundsignificant when impact comparing on the prediction. other models Thus, for the 5-h lead time forecast, it can be said that the major parameters were sea-level pressure, drop rate, and sea surface level for training. The best model no. 12 was derived with the minimum least residual of 8.6%at the 4th layer through the development procedure described below: 𝑦 = 0.00533 + 2.34286𝑥 − 1.42545𝑥 + 106.419𝑥 + 99.754𝑥 − 206.024𝑥𝑥 (33)

where 𝑥 and 𝑥 are the variables at the 4th hidden layer generated by a second-order polynomial derived at the 3rd hidden layer. As shown Figure 6b, a trend of NRMSE was coherently similar to the trend of CC, wherein it is clear that the best CC and NRMSE are obtained from model nos. 4, 12, and 17, while the worst CC and NRMSE are acquired from model no. 9. On average, the value of NRMSE is approximately 10%. Model nos. 12 and 17 revealed 7.6% of NRMSE, which was the best. On the other hand, model no. 9 returned22.9%, which was the worst and remarkable in comparison with other models’ accuracies.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 12 of 26

Figure 7 shows comparisons between the observations and predictions of model no. 12. Surge levels were slightly overestimated from the beginning of the time-series. The GMDH-based model wellJ. Mar. predicted Sci. Eng. 2020 the, 8 ,first 797 and second peaks. However, the model predicted the third peak next to13 ofthe 27 second one. withKim the CCet al. of [24] model reported no. 10, that which the training was trained set th withat compounded the data set the including surge level, SSL sea-level and excluding pressure, WS dropin the of training sea-level set pressure, of model longitude no. 9. The and exclusion latitude ofof WStyphoon’s had no location, signiﬁcant sea impact surface on level, the prediction.and wind speedThus, forwas the the 5-h best lead (see time model forecast, no. 25 it canin Figure be said 5). that On the the major other parameters hand, the current were sea-level study’s pressure,best set consisteddrop rate, of and sea-level sea surface pressure, level drop for training. rate, sea The surface best level, model typhoon no. 12 was position, derived and with central the minimumpressure. Theleast parameters residual of 8.6%atof sea-level the 4th pressure, layer through drop rate, the development sea surface level, procedure and describedtyphoon position below: were in common, while the inclusion of the surge level and wind speed was incoherent. In other words, the 2 2 cause of predictiony4 = 0.00533 accuracy+ 2.34286 mightx 4have1.42545 come xfrom7 + the106.419 discrepancyx + 99.754 in thex training206.024 set.x4 xIn7 particular,(33) 4 − 4 44 74 − 4 4 the surge level had a critical impact on the forecast accuracy. Both CC and NRMSE values obtained fromwhere modelx44 and no.x 2574 arewere the better variables than atthose the 4thfrom hidden model layer no. 12. generated In addition, by a another second-order possible polynomial reason is thatderived that atthe the two 3rd studies hidden applied layer. different neural network algorithms.

1.0 0.8 0.6

CC 0.4 0.2 0.0 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425 model no.

(a) Correlation coefficient. 60 50 40 30 20 NRMSE (%) 10 0 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425 model no.

(b) Normalized root mean square error.

FigureFigure 6. 6. PerformancePerformance indices indices of of CC CC and and NRMSE NRMSE calculat calculateded between between Typhoon Typhoon Songda Songda surge surge levels levels observedobserved at SakaiminatoSakaiminato and and predicted predicted by by the the GMDH-based GMDH-based 5-h lead5-h timelead stormtime storm surge forecastsurge forecast models. models.Model no.Model 25 is no. the 25 best is 05-hthe best lead 05-h time lead ANN-based time ANN-based model trained model by trained the input by datathe input set consisting data set consistingof the surge of level,the surge sea-level level, pressure,sea-level droppressure, of sea-level drop of pressure,sea-level pressure, longitude longitude and latitude and oflatitude typhoon, of typhoon,sea surface sea level, surface and level, wind and speed wind (Kim speed et al. (Kim [24]). et al. [24]).

As shown Figure6b, a trend of NRMSE was coherently similar to the trend of CC, wherein it is clear that the best CC and NRMSE are obtained from model nos. 4, 12, and 17, while the worst CC and NRMSE are acquired from model no. 9. On average, the value of NRMSE is approximately 10%. Model nos. 12 and 17 revealed 7.6% of NRMSE, which was the best. On the other hand, model no. 9 returned 22.9%, which was the worst and remarkable in comparison with other models’ accuracies. Figure7 shows comparisons between the observations and predictions of model no. 12. Surge levels were slightly overestimated from the beginning of the time-series. The GMDH-based model well J. Mar. Sci. Eng. 2020, 8, 797 14 of 27 predicted the ﬁrst and second peaks. However, the model predicted the third peak next to the secondJ. Mar. one. Sci. Eng. 2020, 8, x FOR PEER REVIEW 13 of 26

(a) Comparisons of predictions and (b) Time-series of predictions and observations. observations.

FigureFigure 7. Comparisons 7. Comparisons of of observations observations and and predictions predictions forecastedforecasted by by model model no.12, no.12, which which is isthe the best best for the 5for h-lead the 5 timeh-lead at time Sakaiminato. at Sakaiminato.

3.2.2.Kim Inclusion et al. [24 ]of reported Surge Levels that in the the training Training set Data that Set compounded the surge level, sea-level pressure, drop of sea-levelIn the additional pressure, training, longitude the surge and level latitude was ofinclud typhoon’sed in each location, data set, sea as surfacelisted in level,Table 1. and As wind speedshown was in the Figure best 8, (see model model nos. no.7, 8, 9, 25 13, in 15, Figure 16, 215,). and On 23 the were other not converged hand, the with current the data study’s sets that best set consistedincluded of sea-levelthe surge level pressure, because drop a matrix rate, of sea a data surface set was level, singular, typhoon resulting position, in an unsolvable and central inverse pressure. Thematrix. parameters The GMDH-based of sea-level 5-h pressure, lead time dropstorm rate,surge seaforecast surface models level, predicted and typhoon the Songda position surge with were in common,averaged while CC theand inclusionNRMSE to ofbe0.83 the surgeand 18.03%, level andrespectively. wind speed Compared was incoherent.to the values of In 0.78 other and words, the cause24.20% of from prediction the model accuracy trained might without have the comesurge leve froml, the the involvement discrepancy of in the the surge training level set.in the In input particular, data set deduced prediction accuracy improvement. On average, the improvement rates were0.02 the surge level had a critical impact on the forecast accuracy. Both CC and NRMSE values obtained and 6% for CC and NRMSE, respectively. The improvement rates of CC (=(Surge − fromNonsurge model no.)/Nonsurge 25 were, where better Surge than and those Nonsurge from modelis the predicted no. 12. Insurge addition, level by another the model possible trained reason by is thatthe that data the twoset studieswith and applied without diﬀ erentthe surg neurale level, network respectively) algorithms. and NRMSE (=(Nonsurge − Surge)/Nonsurge) are shown in Figure 9. The positive value indicates an improved surge level. The 3.2.2.CC Inclusion and NRMSE of Surge values Levels were in improved the Training by 0.13 Data (model Set no. 5) and 34% (model no. 6). In this experiment,In the additional model no. training, 12 was selected the surge as the level best was model included because init had each the data smallest set, NRMSE as listed and in the Table 1. As shownsecond in highest Figure CC.8, model In comparison, nos. 7, 8, 9,model 13, 15, no. 16, 12 21, trained and 23without were notthe convergedsurge level withand improved the data setsits that NRMSE by 15%, although it incurred a slightly worse CC value of-0.04. As a result, the surge level included the surge level because a matrix of a data set was singular, resulting in an unsolvable inverse was proven to be one of the key components for improving prediction accuracy. matrix. The GMDH-based 5-h lead time storm surge forecast models predicted the Songda surge with Compared with the accuracy of the best ANN-based 5-h lead time storm surge forecast model averaged(Kim et CC al. and[24]), NRMSE the GMDH-based to be0.83 model’s and 18.03%, accuracy respectively. was lower (see Compared model no. to 25 the in Figure values 8). of This 0.78 and 24.20%value from came the from model the trainedcharacteristic without differences the surge of the level, two the algorithms. involvement For the of thepractical surge application level in the of input datathe set GMDH deduced approach, prediction the accuracy of improvement. the GMDH-based On model average, should the be improvement improved by rates increasing were the 0.02 and 6% fornumber CC and of the NRMSE, typhoon. respectively. The improvement rates of CC (=(Surge Nonsurge)/Nonsurge, − where Surge and Nonsurge is the predicted surge level by the model trained by the data set with and without the surge level, respectively) and NRMSE (=(Nonsurge Surge)/Nonsurge) are shown in − Figure9. The positive value indicates an improved surge level. The CC and NRMSE values were improved by 0.13 (model no. 5) and 34% (model no. 6). In this experiment, model no. 12 was selected as the best model because it had the smallest NRMSE and the second highest CC. In comparison, model no. 12 trained without the surge level and improved its NRMSE by 15%, although it incurred a slightly worse CC value of-0.04. As a result, the surge level was proven to be one of the key components for improving prediction accuracy. Compared with the accuracy of the best ANN-based 5-h lead time storm surge forecast model (Kim et al. [24]), the GMDH-based model’s accuracy was lower (see model no. 25 in Figure8). This value came from the characteristic diﬀerences of the two algorithms. For the practical application

J. Mar. Sci. Eng. 2020, 8, 797 15 of 27 of the GMDH approach, the accuracy of the GMDH-based model should be improved by increasing the number of the typhoon. J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 14 of 26

1.0 0.8 0.6

CC 0.4 0.2 0.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Model No.

(a) Correlation coefficient. 60 50 40 30 20 NRMSE (%) 10 0 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425 Model No.

(b) Normalized root mean square error.

FigureFigure 8. 8.Performance Performance indicesindices ofof CCCC andand NRMSENRMSE calculatedcalculated betweenbetween TyphoonTyphoon SongdaSongda surgesurge levelslevels observedobserved at at Sakaiminato Sakaiminato and and predicted predicted by theby GMDH-basedthe GMDH-based 5-h lead 5-h time lead storm time surgestorm forecast surge modelsforecast trainedmodels withtrained the with data the set data that set included that included the surge the su level.rge level. Model Model no. no. 25 25 is theis the best best 05-h 05-h lead lead time time ANN-basedANN-based modelmodel trainedtrained byby thethe inputinput datadata setset consistingconsistingof ofsurge surgelevel, level,sea-level sea-levelpressure, pressure,drop dropof of sea-levelsea-level pressure, pressure, longitude longitude and and latitude latitude of typhoon,of typhoon, sea surfacesea surface level, level, and windand wind speed speed (Kim et(Kim al. [ 24et ]).al. [24]). 3.3. Performances of GMDH-Based 12-h Lead Time Storm Surge Forecast Models

3.3.1. Exclusion0.15 of Surge Levels in the Training Data Set

The testing0.10 results of the GMDH-based model are given in Figure 10. On average, the values of CC, NRMSE, and STD were 0.82, 12.2%, and 0.09 m, respectively. In this experiment, because CC and NRMSE0.05 were incoherent, STD was taken as an additional indicator. In CC, model nos. 14 and 20 were the highest. At the same time, NRMSE and STD indicated that the two models were the CC worst. When0.00 looking at the time-series of Songda’s surge, the models highly overestimated the surge levels around the1 peak 2 (not 3 4 shown 5 6 here). 7 8 This 9 101112131415161718192021222324 caused higher CCs and larger errors. On the other hand, model-0.05 no. 15 had one of the smallest errors of NRMSE. However, a time-series in model no. 15 showed large time lags between observations and predictions (not shown here). Interestingly, all models except-0.10 the model nos. 14, 20, 18, 16, 11, and 5 had a similar NRMSE order. Thus, we put our focus on STD, releasing the smallest one from modelModel nos. No. 22, 23, and 24. These were trained with WS, WD, TP, and CP. Furthermore, three models predicted almost identical surge levels, as shown in (a) Correlation coefficient.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 14 of 26

1.0 0.8 0.6

CC 0.4 0.2 0.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Model No.

(a) Correlation coefficient. 60 50 40 30 20 NRMSE (%) 10 J. Mar. Sci. Eng. 2020, 8, 797 16 of 27 0 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425 Figure 11, implying that the training set of WS andModel WD wasNo. enough to train the GMDH-based storm surge forecast model with a 12-h lead time. However, the surge level predicted by the best models was highly ampliﬁed around the peak. As a result, the best model no. 22 was acquired with the lowest (b) Normalized root mean square error. least residual of 0.046 (4.6%) after the 2nd iteration through the development procedure described as follows:Figure 8. Performance indices of CC and NRMSE calculated between Typhoon Songda surge levels 2 2 observed aty 2Sakaiminato= 0.0153805 and+ 0.153777predictedx 2by+ the0.599465 GMDH-basedx3 + 3.76317 5-h leadx time+ 1.04064 stormx surge forecast 2 2 22 32 (34) models trained with the data set that included the surge level. Model no. 25 is the best 05-h lead time 4.40019x22 x32 ANN-based model trained by the input data− set consisting of surge level, sea-level pressure, drop of where x and x are the variables at the 2nd hidden layer generated by a second-order polynomial sea-level22 pressure,32 longitude and latitude of typhoon, sea surface level, and wind speed (Kim et al. derived[24]). at the 1st hidden layer.

0.15

0.10

0.05 CC 0.00 1 2 3 4 5 6 7 8 9 101112131415161718192021222324 -0.05

-0.10 Model No.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 15 of 26 (a) Correlation coefficient.

0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324 NRMSE (%) -20

-40 Model No.

(b) Normalized root mean square error.

Figure 9. Improvement rates of the inclusion of surge levellevel in the input data set to thethe exclusionexclusion for CC and NRMSE, respectively. For CC and NRMSE, positivepositive means that the inclusion of the surge level improved.improved.

3.3. Performances of GMDH-Based 12-h Lead Time Storm Surge Forecast Models

3.3.1. Exclusion of Surge Levels in the Training Data Set The testing results of the GMDH-based model are given in Figure 10. On average, the values of CC, NRMSE, and STD were0.82, 12.2%, and 0.09 m, respectively. In this experiment, because CC and NRMSE were incoherent, STD was taken as an additional indicator. In CC, model nos. 14 and 20 were the highest. At the same time, NRMSE and STD indicated that the two models were the worst. When looking at the time-series of Songda’s surge, the models highly overestimated the surge levels around the peak (not shown here). This caused higher CCs and larger errors. On the other hand, model no. 15 had one of the smallest errors of NRMSE. However, a time-series in model no. 15 showed large time lags between observations and predictions (not shown here). Interestingly, all models except the model nos. 14, 20, 18, 16, 11, and 5 had a similar NRMSE order. Thus, we put our focus on STD, releasing the smallest one from model nos. 22, 23, and 24. These were trained with WS, WD, TP, and CP. Furthermore, three models predicted almost identical surge levels, as shown in Figure 11, implying that the training set of WS and WD was enough to train the GMDH-based storm surge forecast model with a 12-h lead time. However, the surge level predicted by the best models was highly amplified around the peak. As a result, the best model no. 22 was acquired with the lowest least residual of 0.046 (4.6%) after the 2nd iteration through the development procedure described as follows: 𝑦 = 0.0153805 + 0.153777𝑥 + 0.599465𝑥 + 3.76317𝑥 + 1.04064𝑥 (34) −4.40019𝑥𝑥 where 𝑥 and 𝑥 are the variables at the 2nd hidden layer generated by a second-order polynomial derived at the 1st hidden layer. In Kim et al. [24], the best training set was formed for the surge level, sea-level pressure, drop rate, and typhoon position (see model no. 25 in Figure 10). By including the surge level in the training set, the values of CC and NRMSE were improved, but STD was larger in comparison to the present result, which excluded it. However, the present result was not in line with Kim et al. [19]. It was believed that the reason that the current study did not include the surge level in the input training set was because the motivation of the study challenged the use of raw data observed at the station. Thus, the surge level should be calculated by extracting atmospheric tidal levels from the sea surface level. Further, because Kim et al. [19] did not test the best training set, which consisted of wind speed and wind direction, we were unable to directly compare the two results. One of the differences between the results of Kim et al. [24] and Hien et al. [26] was the implication of the different

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 16 of 26 algorithms. We found that the storm surge time-series forecasted by model no. 22 was similar to those predicted by the support vector regression model presented by Hien et al. [26]. However, Figure 11 shows that the validation result of surge prediction was similar between the ANN-based model (Kim et al. [24]) and GMDH-based model in the present study. Therefore, further studies should be conductedJ. Mar. Sci. Eng. to2020 assess, 8, 797 potential data sets that include surge levels, which improve forecast accuracy17 of 27 with a12-h lead time.

1.0 0.8 0.6 CC 0.4 0.2 0.0 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425 Model No.

(a) Correlation coefficient.

NRMSE (%) 20

0 12345678910111213141516171819202122232425 Model No.

(b) Normalized root mean square error.

0.30 0.25 0.20 0.15 STD 0.10 0.05 0.00 12345678910111213141516171819202122232425 Model No.

Figure 10. Performance indices of CC and NRMSE calculated between Typhoon Songda surge levels Figure 10. Performance indices of CC and NRMSE calculated between Typhoon Songda surge levels observed at Sakaiminato and predicted by the GMDH-based 12-h lead time storm surge forecast models. observed at Sakaiminato and predicted by the GMDH-based 12-h lead time storm surge forecast Model no. 25 is the best ANN-based 12-h lead time model trained by the input data set that consisted of models. Model no. 25 is the best ANN-based 12-h lead time model trained by the input data set that surge level, sea-level pressure, drop of sea-level pressure, longitude and latitude of typhoon, sea surface level, and wind speed (Kim et al. [24]). J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 17 of 26

J. Mar.consisted Sci. Eng. 2020of surge, 8, 797 level, sea-level pressure, drop of sea-level pressure, longitude and latitude of18 of 27 typhoon, sea surface level, and wind speed (Kim et al. [24]).

(a) Comparisons of predictions and (b) Time-series of predictions and observations. observations.

FigureFigure 11. 11. ComparisonsComparisons of of observations observations and and predictions predictions forecasted forecasted by by model no.22 no. 22 that that is is the the best best performanceperformance model model for for the the 12-h 12-h lead lead time time at at Sakaiminato. Sakaiminato.

3.3.2. InInclusion Kim et al.of [Surge24], the Levels best trainingin the Training set was formedData Set for the surge level, sea-level pressure, drop rate, and typhoon position (see model no. 25 in Figure 10). By including the surge level in the training set, In the previous experiment, the surge level was included in each data set. Some GMDH-derived the values of CC and NRMSE were improved, but STD was larger in comparison to the present result, equations were also not acquired because of the singular matrix (see Figure 12), i.e., model nos. 3, 4, which excluded it. However, the present result was not in line with Kim et al. [19]. It was believed that 7, 8, 15, 19, and 21. Among the GMDH-based models, model nos. 14 and 20 were the best models the reason that the current study did not include the surge level in the input training set was because because it had the highest CC (Figure 12a), while model no. 13 was the best because it had the lowest the motivation of the study challenged the use of raw data observed at the station. Thus, the surge level NRMSE (Figure 12b). In addition, when the surge level was included, model nos. 2 and 6were not should be calculated by extracting atmospheric tidal levels from the sea surface level. Further, because improved for CC (Figure 13a) and model nos. 1, 2, and 6 were not improved for NRMSE (Figure 13b). Kim et al. [19] did not test the best training set, which consisted of wind speed and wind direction, The rest of the models, however, were either improved or remained exactly the same in comparison we were unable to directly compare the two results. One of the diﬀerences between the results of to the models trained with the data set, including surge level. The averaged CC and NRMSE were0.83 Kim et al. [24] and Hien et al. [26] was the implication of the diﬀerent algorithms. We found that the and 26.3%, whereas for the surge level they were 0.83 and 29.05%. In other words, the surge level in storm surge time-series forecasted by model no. 22 was similar to those predicted by the support vector the input data set and GMDH-based model’s NRMSE was improved. When excluding the surge level, regression model presented by Hien et al. [26]. However, Figure 11 shows that the validation result of the best model was model no. 22, which was trained with the WS, WD, TP, and CP data sets. surge prediction was similar between the ANN-based model (Kim et al. [24]) and GMDH-based model However, when the surge level was included in the input layer, model nos. 14 and 20 demonstrated in the present study. Therefore, further studies should be conducted to assess potential data sets that improved accuracy. Further, CC values became slightly better, ranging from 0.9 to 0.91, while include surge levels, which improve forecast accuracy with a12-h lead time. NRMSE changed from 65% to 26%. As mentioned above, the best models were model nos. 14 and 20, which3.3.2. Inclusionhad the largest of Surge CC Levels and smallest in the Training NRMSE. Data ThisSet was determined by considering the surge level in the SLP, DSLP, WS, WD, TP, CP, and MWS data sets. In the previous experiment, the surge level was included in each data set. Some GMDH-derived Although the GMDH-based model was trained with a data set that included the surge level, the equations were also not acquired because of the singular matrix (see Figure 12), i.e., model nos. 3, 4, 7, GMDH-based 12-h lead time surge forecast model was unable to be more accurate than the ANN- 8, 15, 19, and 21. Among the GMDH-based models, model nos. 14 and 20 were the best models because based 12-h lead time surge forecast model (see model no. 25 in Figure 12). it had the highest CC (Figure 12a), while model no. 13 was the best because it had the lowest NRMSE (Figure 12b). In addition, when the surge level was included, model nos. 2 and 6were not improved for CC (Figure 13a) and model nos. 1, 2, and 6 were not improved for NRMSE (Figure 13b). The rest of the models, however, were either improved or remained exactly the same in comparison to the models trained with the data set, including surge level. The averaged CC and NRMSE were 0.83 and 26.3%, whereas for the surge level they were 0.83 and 29.05%. In other words, the surge level in the input data set and GMDH-based model’s NRMSE was improved. When excluding the surge level, the best model was model no. 22, which was trained with the WS, WD, TP, and CP data sets. However, when the surge level was included in the input layer, model nos. 14 and 20 demonstrated improved accuracy. Further, CC values became slightly better, ranging from 0.9 to 0.91, while NRMSE changed from 65% to 26%. As mentioned above, the best models were model nos. 14 and 20, which had the largest CC

J. Mar. Sci. Eng. 2020, 8, 797 19 of 27 and smallest NRMSE. This was determined by considering the surge level in the SLP, DSLP, WS, WD,

TP,J. Mar. CP, Sci. and Eng. MWS 2020, 8 data, x FOR sets. PEER REVIEW 18 of 26

1.0 0.8 0.6 CC 0.4 0.2 0.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Model No.

(a) Correlation coefficient. 80

NRMSE (%) 20

0 12345678910111213141516171819202122232425 Model No.

(b) Normalized root mean square error.

FigureFigure 12.12. PerformancePerformance indicesindices ofof CCCC andand NRMSENRMSE calculatedcalculated betweenbetween TyphoonTyphoon SongdaSongda surgesurge levelslevels observedobserved at at Sakaiminato Sakaiminato and and predicted predicted by theby GMDH-basedthe GMDH-based 12-h lead12-h time lead storm time surgestorm forecast surge models.forecast Modelmodels. no. Model 25 is theno. best 25 is ANN-based the best ANN-based 12-h lead time 12-h model lead time trained model bythe trained input by data the set input that data consisted set that of surgeconsisted level, of sea-level surge level, pressure, sea-level drop ofpressure, sea-level drop pressure, of sea-level longitude pressure, and latitude longitude of typhoon, and sealatitude surface of level,typhoon, and sea wind surface speed level, (Kim and et al. wi [24nd]). speed (Kim et al. [24]).

Although the GMDH-based model was trained with a data set that included the surge level, 0.05 the GMDH-based 12-h lead time surge forecast model was unable to be more accurate than the ANN-based 12-h lead time surge forecast model (see model no. 25 in Figure 12).

3.4. Performances0.00 of GMDH-Based 24-h Lead Time Storm Surge Forecast Models 123456789101112131415161718192021222324

3.4.1. ExclusionCC of Surge Levels in the Training Data Set Figure-0.05 14 depicts the performance indices of CC, NRMSE, and STD. The averaged indices were 0.79, 15.1%, and 0.08 m for CC, NRMSE, and STD, respectively. Among the 24 models, the highest CC values of 0.87 were calculated from model nos. 12 and 21, while the lowest NRMSE of 8% came from model no.-0.10 21. Model nos. 1 and 2 presented the worst CC (0.64) and NRMSE (27.5%), respectively. The STD value also indicated that model no.21 hadModel the smallest No. error of 0.04 m. As a result, the best performance for model no. 21 could be obtained, when training with the SLP, DSLP, SSL, WS, WD, TP, CP, and MWS data sets. Kim et al. [24] showed(a) Correlation that the coefficient. best training set compounds of the surge level

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 18 of 26

1.0 0.8 0.6 CC 0.4 0.2 0.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Model No.

(a) Correlation coefficient. J. Mar. Sci. Eng. 2020, 8, 797 20 of 27 80 were SLP,60 DSLP, WS, WD, and TP (see model no. 25 in Figure 14). Although the surge level was not taken as an input parameter in this study, model no. 8 was trained with the same parameter set as in Kim et al.40 [24]’s best set, except for the surge level, which revealed large errors in comparison to model no. 21. This could be a result of the surge level being the input layer and a vital factor for predicting surge levelNRMSE (%) 20 with the lead time. When comparing model nos. 12 and 21, the wind speed, wind direction, and maximum wind speed were thought to improve model prediction error. Of the worse performance models, model0 nos. 1 and 2 were trained with seal-level pressure, drop rate, and sea surface level, each of which12345678910111213141516171819202122232425 signiﬁcantly revealed more substantial CC, NRMSE, and STD errors. In other words, the too-small number of the input parameter caused an inaccurate prediction. Figure9 show son the right side along the x-axis that the errors of NRMSEModel and STDNo. became small. As a result, model no. 21 was secured with the lowest least residual of 0.032 (3.2%) in the 2nd iteration: (b) Normalized root mean square error. 2 2 y2 = 0.029 + 0.173109x3 0.186717x25 0.978694x + 0.596642x Figure 12. Performance indices of CC and2 − NRMSE calculated2 − between32 Typhoon Songda252 surge levels (35) observed at Sakaiminato and predicted by+5.26692 the GMxDH-based32 x252 12-h lead time storm surge forecast models. Model no. 25 is the best ANN-based 12-h lead time model trained by the input data set that where x and x are the variables at the 2nd hidden layer generated by a second-order polynomial consisted32 of25 surge2 level, sea-level pressure, drop of sea-level pressure, longitude and latitude of derivedtyphoon, at the sea 1st surface hidden level, layer. and wind speed (Kim et al. [24]).

0.05

0.00 123456789101112131415161718192021222324 CC -0.05

-0.10 Model No.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW (a) Correlation coefficient. 19 of 26

20 NRMSE (%) 0 123456789101112131415161718192021222324 -20 Model No.

(b) Normalized root mean square error.

Figure 13. Improvement rates of the inclusion of surge leve levell in the input data set to the exclusion for CC and NRMSE, respectively. For CC and NRMSE, po positivesitive means that the inclusion of the surge level improved.

3.4. Performances of GMDH-Based 24-h Lead Time Storm Surge Forecast Models

3.4.1. Exclusion of Surge Levels in the Training Data Set Figure 14 depicts the performance indices of CC, NRMSE, and STD. The averaged indices were0.79, 15.1%, and 0.08 m for CC, NRMSE, and STD, respectively. Among the 24 models, the highest CC values of 0.87 were calculated from model nos. 12 and 21, while the lowest NRMSE of 8% came from model no. 21. Model nos. 1 and 2 presented the worst CC (0.64) and NRMSE (27.5%), respectively. The STD value also indicated that model no.21 had the smallest error of 0.04 m. As a result, the best performance for model no. 21 could be obtained, when training with the SLP, DSLP, SSL, WS, WD, TP, CP, and MWS data sets. Kim et al. [24] showed that the best training set compounds of the surge level were SLP, DSLP, WS, WD, and TP (see model no. 25 in Figure 14). Although the surge level was not taken as an input parameter in this study, model no. 8 was trained with the same parameter set as in Kim et al. [24]’s best set, except for the surge level, which revealed large errors in comparison to model no. 21. This could be a result of the surge level being the input layer and a vital factor for predicting surge level with the lead time. When comparing model nos. 12 and 21, the wind speed, wind direction, and maximum wind speed were thought to improve model prediction error. Of the worse performance models, model nos. 1 and 2 were trained with seal-level pressure, drop rate, and sea surface level, each of which significantly revealed more substantial CC, NRMSE, and STD errors. In other words, the too-small number of the input parameter caused an inaccurate prediction. Figure 9 show son the right side along the x-axis that the errors of NRMSE and STD became small. As a result, model no. 21 was secured with the lowest least residual of 0.032 (3.2%) in the 2nd iteration: 𝑦 = 0.029 + 0.173109𝑥 − 0.186717𝑥 − 0.978694𝑥 + 0.596642𝑥 (35) +5.26692𝑥𝑥 where 𝑥 and 𝑥 are the variables at the 2nd hidden layer generated by a second-order polynomial derived at the 1st hidden layer. Figure 15 depicts comparisons of observations and predictions. Overall, the best 24-h lead time forecast model overestimated the Songdasurge level. While the peak level was well predicted, the surge levels around the first peak and after the second peak were overestimated. Compared with the ANN results in Kim et al. ([24,25]) and GP results in Hien et al. [26], we found that the accuracy of the best GMDH-based model was closer to the results obtained from the deterministic data-driven models, i.e., the multi-layer perceptron, k-nearest neighbors, decision tree, and support vector

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 20 of 26 regression, as shown in Hien et al. [26]. As mentioned above, the surge level was omitted in the presentJ. Mar. Sci. study, Eng. 2020 which, 8, 797 may have had an impact on surge prediction. 21 of 27

1.0 0.8 0.6

CC 0.4 0.2 0.0 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425 Model No.

(a) Correlation coefficient. 80 60 40

NRMSE (%) 20 0 12345678910111213141516171819202122232425 Model No.

(b) Normalized root mean square error. 0.25 0.20 0.15

STD 0.10 0.05 0.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Model No.

FigureFigure 14. 14. PerformancePerformance indices indices of of CC CC and and NRMSE NRMSE calculated calculated between between Typhoon Typhoon Songda Songda surge surge levels levels observedobserved at SakaiminatoSakaiminato andand predicted predicted by by the the GMDH-based GMDH-based 24-h 24-h lead lead time stormtime storm surge forecastsurge forecast models. models.Model no.Model 25 is no. the 25 best is ANN-basedthe best ANN-based 24-h lead 24-h time lead model time trained model by trained the input by data the input set that data consisted set that of consistedsurge level, of sea-levelsurge level, pressure, sea-level drop pressure, of sea-level drop pressure, of sea-level longitude pressure, and latitude longitude of typhoon, and latitude sea surface of typhoon,level, and sea wind surface speed level, (Kim and et al.wind [24 ]).speed (Kim et al. [24]).

Figure 15 depicts comparisons of observations and predictions. Overall, the best 24-h lead time forecast model overestimated the Songdasurge level. While the peak level was well predicted, the surge levels around the ﬁrst peak and after the second peak were overestimated. Compared with the ANN results in Kim et al. ([24,25]) and GP results in Hien et al. [26], we found that the accuracy of the best GMDH-based model was closer to the results obtained from the deterministic data-driven models, i.e., the multi-layer perceptron, k-nearest neighbors, decision tree, and support vector regression, J. Mar. Sci. Eng. 2020, 8, 797 22 of 27

as shown in Hien et al. [26]. As mentioned above, the surge level was omitted in the present study,

J. Mar.which Sci. mayEng. 2020 have, 8, hadx FOR an PEER impact REVIEW onsurge prediction. 21 of 26

(a) Comparisons of predictions and observations. (b) Time-series of predictions and observations.

FigureFigure 15. 15. ComparisonsComparisons of ofobservations observations and and predictions predictions forecasted forecasted by by model model no. no. 21, 21, which which is isthe the best best forfor the the 24 24 h-lead h-lead time time at atSakaiminato. Sakaiminato.

3.4.2.3.4.2. Inclusion Inclusion of ofSurge Surge Levels Levels in in the the Training Training Data Data Set Set TheThe additional additional experiment experiment investigated investigated the theeffect eﬀ ectof the of thesurge surge level level on prediction on prediction accuracy accuracy of the of 24-hthe lead 24-h time lead by time including by including the surge the level surge in level the data in the set. data As set.found As from found previous from previous experiments, experiments, some GMDH-basedsome GMDH-based models modelsdid not converge did not converge (i.e., model (i.e., nos. model 9, 10, nos. 15, 22, 9, 10, 23, 15, and 22, 24) 23, (Figure and 24) 16). (Figure The CC 16 ). andThe NRMSE CC and values NRMSE obtained values obtainedfrom models from trained models by trained the data by the set data without set without the surge the level surge were level 0.79 were and0.79 36% and on 36% average. on average. On the Onother the hand, other those hand, from those the from models the modelstrained trainedby the data by the setdata with set the with surge the levelsurge were0.73 level were and 0.7336.64%on and 36.64% average. on Overall, average. the Overall, CC of the the models CC ofthe was models similar was or worse similar when or worse the surgewhen level the surgewas involved level was (see involved Figure (see 17a). Figure Furthermore, 17a). Furthermore, when the when input the data input was data small, was small,the improvementthe improvement of the of prediction the prediction accuracy accuracy (NRMSE) (NRMSE) was wasapparent apparent (see (see model model nos. nos. 1, 2, 1, 3, 2, 3,and and 8 8in in FigureFigure 17b). 17b). When When comparing comparing the the improvement improvement rate ratess of of the the 5 5and and 12-h 12-h lead lead times, times, the the improvement improvement raterate apparently apparently declined declined during during the the 24-h 24-h lead lead time. time. In other In other words, words, the effect the e ﬀofect the of surge the surge level levelon the on 24-hthe lead 24-h leadtime timeforecast forecast accuracy accuracy was wasnot notcritical. critical. Among Among the theGMDH-based GMDH-based models, models, model model no. no. 5 5 showedshowed the the best best performance performance based based on on CC CC and and NRMSE NRMSE values. values. However, However, model model no. no. 21 21 trained trained withoutwithout the the surge surge level level was was more more accurate accurate than than model model no. no. 5 5trained trained with with the the surge surge level. level. InIn addition, addition, the the accuracy accuracy of of the the GMDH-based GMDH-based 24-h 24-h lead lead time time storm storm surge surge forecast forecast model model trained trained withwith the the data data set set including including the the surge surge level level did did no nott reach reach the the ANN-based ANN-based 24-h 24-h lead lead time time storm storm surge surge forecastforecast model model (see (see Figure Figure 16). 16 ).

1.0 0.8 0.6

CC 0.4 0.2 0.0 12345678910111213141516171819202122232425 Model No.

(a) Correlation coefficient.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 21 of 26

(a) Comparisons of predictions and observations. (b) Time-series of predictions and observations.

Figure 15. Comparisons of observations and predictions forecasted by model no. 21, which is the best for the 24 h-lead time at Sakaiminato.

3.4.2. Inclusion of Surge Levels in the Training Data Set The additional experiment investigated the effect of the surge level on prediction accuracy of the 24-h lead time by including the surge level in the data set. As found from previous experiments, some GMDH-based models did not converge (i.e., model nos. 9, 10, 15, 22, 23, and 24) (Figure 16). The CC and NRMSE values obtained from models trained by the data set without the surge level were 0.79 and 36% on average. On the other hand, those from the models trained by the data set with the surge level were0.73 and 36.64%on average. Overall, the CC of the models was similar or worse when the surge level was involved (see Figure 17a). Furthermore, when the input data was small, the improvement of the prediction accuracy (NRMSE) was apparent (see model nos. 1, 2, 3, and 8 in Figure 17b). When comparing the improvement rates of the 5 and 12-h lead times, the improvement rate apparently declined during the 24-h lead time. In other words, the effect of the surge level on the 24-h lead time forecast accuracy was not critical. Among the GMDH-based models, model no. 5 showed the best performance based on CC and NRMSE values. However, model no. 21 trained without the surge level was more accurate than model no. 5 trained with the surge level. In addition, the accuracy of the GMDH-based 24-h lead time storm surge forecast model trained J.with Mar. Sci.the Eng.data2020 set, 8 including, 797 the surge level did not reach the ANN-based 24-h lead time storm23 surge of 27 forecast model (see Figure 16).

1.0 0.8 0.6

CC 0.4 0.2 0.0 12345678910111213141516171819202122232425 Model No.

J.J. Mar. Mar. Sci. Sci. Eng. Eng. 2020 2020, ,8 8, ,x x FOR FOR PEER PEER REVIEW REVIEW 22 22 of of 26 26 (a) Correlation coefficient.

100100 8080

6060 4040 NRMSE (%) NRMSE (%) 2020 00 1234567891011121314151617181920212223242512345678910111213141516171819202122232425 ModelModel No. No.

((bb)) Normalized Normalized root root mean mean square square error. error.

FigureFigure 16. 16. Performance Performance indices indices of of CC CC and and NRMSE NRMSE calculated calculated between between Typhoon Typhoon SongdaSongda surgesurge levelslevels observedobserved at atat Sakaiminato SakaiminatoSakaiminato and andand predicted predictedpredicted by thebyby GMDH-basedthethe GMGMDH-basedDH-based 24-h lead24-h24-h time leadlead storm timetime surgestormstorm forecast surgesurge forecast models.forecast Modelmodels.models. no. Model Model 25 is theno. no. best25 25 is is ANN-based the the best best ANN-based ANN-based 24-h lead time24-h 24-h modellead lead time time trained model model by thetrained trained input by by data the theset input input that data consisteddata set set that that of surgeconsistedconsisted level, ofof sea-level surgesurge level,level, pressure, sea-levelsea-level drop of pressure,pressure, sea-level drop pressure,drop ofof sea-levelsea-level longitude pressure,pressure, and latitude longitudelongitude of typhoon, andand sealatitudelatitude surface ofof level,typhoon,typhoon, wind sea sea direction, surface surface level, andlevel, wind wind wind speed direction, direction, (Kim and and et al. wind wind [24]). speed speed (Kim (Kim et et al. al. [24]). [24]).

0.150.15 0.100.10 0.050.05

CC 0.00

CC 0.00 123456789101112131415161718192021222324 -0.05-0.05 123456789101112131415161718192021222324 -0.10-0.10 -0.15-0.15 ModelModel No. No.

((aa)) Correlation Correlation coefficient. coefficient. 100100 Figure 17. Cont. 5050 00 123456789101112131415161718192021222324 -50-50 123456789101112131415161718192021222324 -100-100 NRMSE (%) NRMSE (%) -150-150 -200-200 ModelModel No. No.

((bb)) Normalized Normalized root root mean mean square square error. error.

FigureFigure 17. 17. Improvement Improvement rates rates of of the the inclusion inclusion of of surge surge leve level lin in the the input input data data set set to to the the exclusion exclusion for for CCCC andand NRMSE,NRMSE, respectively.respectively. ForFor CCCC andand NRMSE,NRMSE, popositivesitive meansmeans thatthat thethe inclusioninclusion ofof thethe surgesurge levellevel improved. improved.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 22 of 26

100 80 60 40 NRMSE (%) 20 0 12345678910111213141516171819202122232425 Model No.

(b) Normalized root mean square error.

Figure 16. Performance indices of CC and NRMSE calculated between Typhoon Songda surge levels observed at Sakaiminato and predicted by the GMDH-based 24-h lead time storm surge forecast models. Model no. 25 is the best ANN-based 24-h lead time model trained by the input data set that consisted of surge level, sea-level pressure, drop of sea-level pressure, longitude and latitude of typhoon, sea surface level, wind direction, and wind speed (Kim et al. [24]).

0.15 0.10 0.05

CC 0.00 -0.05 123456789101112131415161718192021222324 -0.10 -0.15 Model No. J. Mar. Sci. Eng. 2020, 8, 797 24 of 27

(a) Correlation coefficient. 100 50 0 -50 123456789101112131415161718192021222324 -100 NRMSE (%) -150 -200 Model No.

(b) Normalized root mean square error.

Figure 17. Improvement rates of the inclusion of surge leve levell in the input data set to the exclusion for CC and NRMSE, respectively. For CC and NRMSE, po positivesitive means that the inclusion of the surge level improved.

4. Discussion

Sakaiminato’s storm surge has peak level characteristics that appear approximately 15 h after a typhoon’s passage. In addition, typhoon’s move offshore without landfall. The annual tidal range is approximately 0.3 m on average. We found that the best training data set compounds the site-specific typhoon-related component, depending on the lead time through the experiment. In the experiment that excluded surge level for the zero-hour lead time, the surge was generated via sea-level pressure, drop rate, wind speed, and wind direction after the typhoon passed offshore. When the surge was forecasted with the 5-h lead time, the indicators of the typhoon position and central pressure were necessary. The 12 h almost corresponded to the time of the peak surge level occurrence and was a time when the typhoon was located around Hokkaido Prefecture, which is more than 800 km away from Sakaiminato. Therefore, wind information should be added as an indicator. The Ekman setup led by Ekman transport was a component of wind-driven ocean currents that generatedanafter-runner storm surge at Sakaiminato in the presence of the Coriolis force over the Tottori coast (Kim et al. [2]). The 24-h lead time affected non-stormy weather and the ocean, resulting in arequirement of all the typhoon-related components. In the additional experiment where the data set included the surge level, we found that the surge level improved the near–far forecast’s prediction accuracy. However, we still need to further investigate the feature selection because we missed several combinations, i.e., eight components of the 28 data sets. Therefore, a feature selection that automatically decides critical components without making potential combinations should be proposed. Through the methodology presented in this study, it was clear that the GMDH enabled application to other sites. Thus, site-specific typhoon-related characteristics were assessed by the feature selection and the storm surge forecast model was developed by training with fully large data that could be produced via numerical simulations using long-term ensemble projections for historical and future climates (e.g., Mori et al. [33]) and stochastic tropical cyclone models (e.g., Nakajo et al. [41]). Even though the number of typhoons considered in the current study was small, they were still sufficient indemonstrating GMD Heffectiveness. The three typhoons generated sufficiently high surge levels that corresponded to a 100-year return period at Sakaiminato. Those typhoons can be treated ashistorically representative because they generate after-runner storm surges and record winds and pressures at local meteorological stations. However, the present study did not consider potential typhoons that generated either high or lowsurge levels, at least compared with the three typhoon-generated surge levels thatresulted in potential forecast errors. Therefore, the typhoon number should increase for practical forecasting. In addition to the typhoon number, Mori et al. [33] J. Mar. Sci. Eng. 2020, 8, 797 25 of 27 reported that the future 100-year return values of storm surge would increase by 20% on the East Asian coast due to climate change. From this perspective, we expect that the storm surge at Sakaiminato will increase by approximately 0.6 m in the future. Therefore, climate change’s impact should be considered when training the GMDH-based surge forecast model with experimental climate change data. Once the GMDH-based storm surge forecast model is successively trained, the storm surge forecast model with an ensemble prediction system (EPS) can estimate the uncertainty in a storm surge forecast. Therefore, the accuracy of the GMDH-based forecast model with EPS should be assessed in practice. Suppose its accuracy is satisfied with uncertainty. In that case, the GMDH-based forecast model could provide arrange of surge levels using typhoon information forecasted with JMA and observations at designated time intervals. Thus, the forecasted surge levels can serve as a pre-indicator for evacuation. The performance of the GMDH-based surge forecast model is highly relevant to the reliability of the present method. Hien et al. [25] compared machine learning methods that resulted in better GP and ANN stochastic methods performances, especially in comparison with multi-layer perceptron, k-nearest neighbors, decision tree, and support vector regression, which are ordinarily called data-driven models, as they can capture the mapping between input and output variables (forecast problems) without directly studying the mechanism of storm surges. The GMDH is one of these stochastic methods and thus has better reliability than the data-driven method. However, the present method used GMDH results with lower accuracy, as compared to the ANN and GP methods (Kim et al. [24], Hien et al. [26]). Moreover, the present method either included or excluded the surge level in data set results with higher accuracies for the 5- and 12-h lead times, and the same accuracy for the 24-h lead time. In other words, the accuracy of the present GMDH method can be improved with an appropriate training set and data set, as described above.

5. Conclusions The present study developed a storm surge forecast model for Sakaiminato, Tottori, Japan using the GMDH algorithm. The study challenged forecast storm surge levels with lead times of 5, 12, and 24 h, and then hind casted them. To train the GMDH-based model, local meteorological and hydrodynamic data was observed and collected at stations around Sakaiminato during Typhoons Maemi (2003), Songda (2004), and Megi (2004). In the forecast experiments, we used two typhoons—Maemi and Megi—and one typhoon (Songda) for training and testing, respectively. The present study’s remarkable conclusions are presented below:

For training the GMDH-based hind cast model, the best training set can be made by adding the • surge level to other components. The best set consisted of the surge level, sea-level pressure, drop rate, sea surface level, typhoon • position, and central pressure, all of which resulted in the most accurate GMDH-based 5-h lead time forecast model. The surge level, wind speed, and wind direction are required for the best set of GMDH-based • 12-h lead time forecast model. To forecast the surge level with a 24-h lead time, many types of input parameters are preferable, • and all input parameters were shown in this study. The inclusion of the storm surge level at the input layer is critical to the accuracy of the near–far • forecast model for the 5- and 12-h lead times, particularly.

Further studies should be done with large data sets to consider climate change and to improve the accuracy of the GMDH-based storm surge forecast model. Moreover, the uncertainty of the GMDH-based storm surge forecast model with an ensemble prediction system should be veriﬁed for the practical forecast. Furthermore, a selection procedure to automatically decide components should be investigated against other man-made data sets. J. Mar. Sci. Eng. 2020, 8, 797 26 of 27

Author Contributions: Conceptualization, S.K., H.M. and M.T.; methodology, S.K. and H.M.; software, S.K., N.B.T. and C.T.T.; validation, S.K., N.B.T., H.M., M.T. and C.T.T.; formal analysis, S.K. and N.B.T.; data curation, S.K.; writing—original draft preparation, S.K.; writing—review and editing, N.B.T., H.M., M.T. and C.T.T.; visualization, S.K.; Funding acquisition, S.K. and N.B.T. and V.H.D. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by JSPS KAKENHI Grant Number JP20H02260 and JP20K05046, Specialists Center of Port and Airport Engineering (SCOPE) in Japan, and Project code TNMT. 2018.05.28 of Vietnam Ministry of Natural Resources and Environment (MONRE), and partially supported by the VAST with the grant project code VAST06.05/20–21. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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