Storm surge model based on variational data assimilation method

Shi-li HUANG* , Jian XU, De-guan WANG, Dong-yan LU

1. College of Water Conservancy and Hydropower Engineering, Hohai University,Nanjing 210098, P. R. China

2. Shanghai Water Authority, Shanghai 200232, P. R. China

3. Shanghai Water Planning and Design Research Institute, Shanghai 200232, P. R. China

4. College of Environmental Science and Engineering, Hohai University, Nanjing 210098, P. R. China

1 Introduction

A storm surge, one of the most serious hazards in coastal areas, is an abnormal rising of the sea level caused by atmospheric disturbances like strong wind and sudden changes in atmospheric pressure (Feng 1982). Storm surges threaten human safety and social stability in China, especially in the coastal areas with high population density and more developed economies. Numerical forecasting is a useful method for the study of storm surge hazards.Significant progress has been made in the numerical forecasting of storm surges (Jain et al.2007; Huang et al. 2008; Dube et al. 2009; Lee 2006). Numerical forecasting accuracy is affected by many factors (Kong et al. 2008; Kim and Yamashita 2004), such as the uncertainty of model parameters, errors from the idealized control equations, and numerical discretization.On the other hand, many in situ stations supply large observed data sets. They are also important data sources for the forecasting of storm surges. A combination of observed data sets and a numerical model may lead to a better forecast. To do this, we used the variational data assimilation method to combine the observed data sets and a storm surge model based on unstructured grids.

2 Basic principles of variational data assimilation method

Variational data assimilation is a method for reducing the differences between simulation results and observation results by controlling the relevant model parameters. The purpose is to carry out practical and accurate analysis and forecasting.

For a given discrete model M and a model state variable x:

where xiis the state variable at the ith position, and p represents the control parameters of the discrete model as well as the control variables of the variational model. The cost function J of the differences between the observation and calculation results can be described by Eq. (2):

where t is time; T is the simulation time period, which is called as the assimilation time window; H is the observation operator; and yobsrepresents the observation variables.

According to the theory of functional variation, when the gradient satisfies the Euler-Lagrange optimal conditions, that is, when the gradient of J to p is zero, J is the minimum and the control variable p has the optimal value; this system has the optimal solution(Le Dimet and Talagrand 1986). The key to solving the problem is the computation of the gradient. Some algorithms are available to compute the gradient of the cost function (Cacuci 2003). In the forecasting of storm surges, the direct method of computing the gradient is not feasible, but the adjoint method can be used (Lai et al. 2008; Peng and Xie 2006; Zhang et al.2003; Griffith and Nichols 2000).

3 Numerical model of storm surge

3.1 Basic equations

The complex flow influenced by the upstream inflow, the tide, and the typhoon can be described by two-dimensional hydrodynamic equations. The control equations can be written in a conservative form as follows:

Given initial conditions and boundary conditions, one can compute the water level changes caused by combined effects of storm surges and astronomical tides. The typhoon wind field is calculated by the parameterized model. The central pressure is calculated by the Fujita formula (Fujita 1952), which is widely used, and the wind field is synthesized with the gradient and the transitional terms (Sha et al. 2004).

3.2 Wind stress drag coefficient calculation

The wind stress drag coefficient determines the momentum transfer rate between the air and water surface (Zhou et al. 2009). Whether the calculation of storm surges is reasonable or not depends on the accuracy of the wind drag coefficient. Its value has mostly remained constant in previous numerical simulations, meaning that the surface roughness does not change in the storm surge. With continuing study of wind stress and momentum transfer between air and water, it has been found that the coefficient relates to the water surface roughness height, which in turn relates to the wind speed.

The surface wind stress used in this study is calculated with a formula considering the influence of the tidal level in storm surge simulation (Kong et al. 2008):

where Cdis the wind drag coefficient, ζ is the height above the mean sea level, andis the conventional value of the wind drag coefficient, which is 0.002 6.

3.3 Numerical methods

In order to efficiently quantify the dynamic change of the water level in small-scale rivers and large-scale offshore waters during the typhoon, a storm surge model was established using the finite volume method on an unstructured grid (Lai et al. 2008). The grid combines with triangular cells and quadrilateral cells. This model can simulate the dynamic change of the floodplain, including its submergence and emergence.

4 Analysis and prediction tests of storm surge in Huangpu River and coastal areas of Shanghai

4.1 Study areas

The computation domains of the numerical prediction model of storm surges in the Huangpu River and the coastal areas of Shanghai are shown in Fig. 1. The upper boundaries of the Yangtze River, Huangpu River, and Hangzhou Bay are, respectively, Xuliujing, Mishidu,and Zhapu. The open sea extends to Xiangshan in the south, 40 km away from Lüsi in the north and 200 km away from Wusongkou in the east. The span of the study, which is from the Yangtze River to the Huangpu River, is rather large. The design of the study areas was reasonable and economical, both in economizing the computer memory (especially when using the data assimilation method)and enhancing convenience of practical prediction.

Fig. 1 Study area and unstructured computational grid

4.2 Synthetic data test

In order to eliminate errors in observation data and overcome the difficulties in verification caused by the uncertainty of parameters, numerical tests of data assimilation using synthetic data were performed first. The artificially synthetic observation data were presented with the forward model with certain model parameters. Based on the synthetic data, one can efficiently verify the reliability of the storm surge variational data assimilation model by eliminating the effects of observation and the numerical errors arising in practical applications.

Data from a storm surge event caused by a typhoon were used for data assimilation tests and validation of the variational data assimilation method. Given boundary conditions and initial conditions, the storm surge process caused by the typhoon was simulated directly, and the tidal level data from the observation stations (Shidongkou, Changxing, Hengsha, Zhongjun,Nancaodong, Dajishan, Majishan, Lühua, Jigujiao, Sheshan, and Wujing, shown in Fig. 1)were exported for the subsequent variational data assimilation tests. The data from Huangpu Park, Wusongkou, Beicaozhong, and Lühua were selected for model validation.

The data assimilation tests on the synthetic tidal level data from the observation stations were performed using the wind drag coefficient Cdas the control variable. Assuming a certain value of Cd(0 in this test), the on-the-hour tidal level data for the 12-hour storm surge period from the 11 stations were assimilated, while other parameters, such as roughness,initial conditions, and boundary conditions, were known. After six outside loop iterations, the normal gradient fell low enough (10-8in this experiment), and the optimizer decided that convergence has been obtained. The iteration-convergence process of the regularized cost function is shown in Fig. 2. The identified wind drag coefficient is= 0.002 6, which is the same as the true value of Cd.

Fig. 2 Iteration-convergence process of regularized cost function

Fig. 3 compares the tidal level processes from the four main stations. Tidal level processes at Lühua, one of the assimilation stations, indicate the principle of variation model,that is, the smaller the cost function J is, the greater the agreement between the assimilation results and actual results. Tidal level processes at the other three stations (Huangpu Park,Wusongkou, and Beicaozhong), which were not involved in the data assimilation, were also calculated correctly, and the calculated tidal process coincided with the actual process. The results show that the process of calculation of storm surges can be improved by identifying the wind drag coefficient in the variational data assimilation model.

Fig. 3 Water level comparisons based on synthetic data

4.3 Case study

The variational data assimilation model was used for the verification of tidal level prediction for Typhoon 0515 (Khanun), and prediction results using the forward simulation model were unsatisfactory.

In the tidal level process of Typhoon 0515, the high tidal level occurred at 16:00 on September 11, 2005 (86th hour in Fig. 4). For the sake of consistency with the actual situation,the data from 16:00 September 10 to 12:00 September 11 at Mishidu, Xuliujing, and Zhapu were used as a measured boundary, and the data from 13:00 to 19:00 September 11 were used as a forecasting boundary. Data from four stations were used for water level verification. The validation results are shown in Fig. 4. The mean square errors of the forward model forecasting results from Beicaozhong, Zhongjun, Wusongkou, and Huangpu Park are,respectively, 22.4 cm, 18.3 cm, 24.0 cm, and 23.6 cm, while the mean square errors of the variational data assimilation model are, respectively, 12.4 cm, 13.2 cm, 11.1 cm, and 9.0 cm.Compared with the results of the forward model, the forecast accuracy of the variational data assimilation model is further improved.

Fig. 4 Comparisons of water level of storm surge induced by Typhoon 0515

5 Conclusions

A storm surge forecasting model based on a high-resolution unstructured grid was established. Artificially synthesized data tests were carried out to verify the theoretical accuracy of the variational assimilation techniques. The variational assimilation numerical forecast of Typhoon 0515 showed that the mean square errors of the water level in Beicaozhong, Zhongjun, Wusongkou, and Huangpu Park improved by 44.6%, 27.9%, 53.8%,and 61.9%, respectively, through use of the developed variational storm surge model. The variational data assimilation method can significantly improve the accuracy of storm surge forecasting and provide bases for disaster prevention and mitigation.

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