## 1. Introduction

The upper-ocean circulation and sea surface waves are mainly driven by sea surface winds. Particularly, the strong winds of tropical cyclones (TC) inevitably induce severe storm surge. Through the wind forcing (wind stress) on the sea surface due to the friction, the momentum is transferred from the atmosphere to the ocean. Therefore, a successful storm surge modeling greatly depends on an accurate estimate of the wind stress (Doyle 2002; Moon 2005; Xie et al. 2008).

*a*and

*b*are empirical parameters. Since Sheppard (1958) first proposed the linear formula for

*C*in the middle of the last century, many scientists have made large efforts in estimating the values of parameters

_{d}*a*and

*b*and have come to significantly different results (Deacon and Webb 1962; Miller 1964; Zubkovskii and Kravchenko 1967; Brocks and Krugermeyer 1970; Sheppard et al. 1972; Wieringa 1974; Kondo 1975; Smith and Banke 1975; Smith 1980; Wu 1980; Large and Pond 1981; Donelan 1982; Geernaert et al. 1987; Yelland and Taylor 1996). Among all the formulas, the most widely used nowadays are those of Wu (1980), Smith (1980), and Large and Pond (1981). After extensive research, Wu (1980) provided the values for

*a*and

*b*under the condition of wind speed less than 15 m s

^{−1}: (

*a, b*) = (0.8, 0.065), and later extended it to the situation of strong wind, such as in a hurricane (Wu 1982). Smith (1980) analyzed wind speed, temperature, and wave height observed at a fixed platform, and obtained the values of

*a*and

*b*at the wind speed range of 6–22 m s

^{−1}: (

*a*,

*b*) = (0.61, 0.063). Based on the data over the deep-ocean area and the assumption of unlimited wind fetch, Large and Pond (1981) found (

*a, b*) = (0.49, 0.065) in the wind speed range of 10–26 m s

^{−1}and

^{−1}.

Although some recent studies have shown the nonlinear dependence of ^{−1} (Emanuel 2003; Powell et al. 2003; Donelan et al. 2004), the linear parameterization [Eq. (2)] is still widely used in storm surge forecasting. Therefore, we still employ the formula of ^{−1}). We will demonstrate that even with this simple linear parameterization, obtaining an optimal value for

One of the efficient ways to estimate the wind stress drag coefficient is through fitting the model output to the observations using adjoint technique (Derber 1987; Le Dimet and Talagrand 1986). Yu and O’Brien (1991) estimated the wind stress drag coefficient and the vertical profile of eddy viscosity coefficient by assimilating the observational data into a simple 1D Ekman ocean model using adjoint technique. Zhang et al. (2002, 2003) used a 2D Princeton Ocean Model (POM) and its adjoint model to estimate the wind stress drag coefficient by assimilating pseudo-observations of subtidal water level and coastal tidal elevation. Chen et al. (2008) performed an idealized numerical experiment on an estuary to derive the wind stress drag coefficient using a 2D adjoint model based on an unstructured grid with finite volume. The results of all these studies indicate that the adjoint data assimilation is an effective approach for estimating the wind stress drag coefficient and other parameters. However, all these studies used 1D or 2D models with simplified physics. Some previous studies have shown that 3D storm surge models can improve the storm surge forecasting considerably, compared to 1D or 2D models (Xie et al. 2004; Peng et al. 2006), since the 3D models can take into account the nonlinear processes such as the bottom friction and tide–current–wave interaction (although they have relatively smaller impacts on the storm surge compared to the wind forcing). Although 3D models require more computational resources, it will not be a big problem with the rapid development of the computer technology. Therefore, it is worth to explore the effects of optimizing the wind stress drag coefficient in storm surge simulation in the framework of 3D ocean models. Although a nongradient simple method may also work for optimizing only a few parameters, the adjoint approach is more effective and necessary when considering the spatial variation of *C _{d}*) we try to optimize is not only case dependent and temporally/spatially various but also model dependent. In the framework of 2D models, although the 2D adjoint model of a 2D forward model could be sufficient for obtaining an “optimal”

*C*for the 2D model, such an optimal

_{d}*C*may be not suitable or optimal to a 3D forward model, and vice versa. Actually, as a subsequent study of our previous study, which demonstrates the feasibility of optimizing the initial conditions in improving storm surge forecasts using the 3D adjoint model of Princeton Ocean Model developed by Peng and Xie (2006) and Peng et al. (2007), this study aims to demonstrate the feasibility of adjusting

_{d}Besides the uncertainties in the wind stress drag coefficient, there is still a number of uncertainties that may cause considerable systematic bias in storm surge forecasts, such as the wind speed forecasting, the resolution, and the simplification or parameterization of the physical/dynamical processes in a storm surge model, etc. In state of the art, these uncertainties are inevitable or cannot be removed completely because of imperfect forecasting of the storm track or its intensity, limited computer resource, and poor understanding of physical/dynamical processes of the ocean, etc. Ensemble forecasting is a practical way to offset the systematic bias in storm surge forecasts caused by these uncertainties (Stamey et al. 2007; Flowerdew et al. 2007; Kazuo et al. 2010). In addition to the ensemble forecasting, adjusting an appropriate parameter in the storm surge model could be another practical solution to the problem (Peng et al. 2007). In the study of Peng et al. (2007), the maximum wind radius in the calculation of wind stress based on the empirical Holland model was chosen as the parameter to be adjusted. However, when wind fields are obtained from a numerical weather model output rather than using an empirical formula that is the function of the maximum wind radius, it is invalid to adjust the maximum wind radius. As mentioned above,

In section 2, the Princeton Ocean Model and its adjoint model are briefly introduced. Section 3 describes the experimental setup design, including the selection of the storm surge case and model domain. The results are presented in section 4. Conclusions and discussion are given in the last section. The verification of the POM-4DVAR system using

## 2. The Princeton Ocean Model and its adjoint model

The models used in this study are the POM, 2002 version (Blumberg and Mellor 1987; Mellor 2004 and its adjoint model (Peng and Xie 2006; Peng et al. 2007). POM is a three-dimensional ocean model with primitive equations, embedded in a second-moment turbulence closure model (the level 2.5 Mellor–Yamada scheme; Mellor and Yamada 1982). The main features of POM include vertically terrain-following sigma coordinates, horizontally curvilinear orthogonal coordinates with Arakawa C-grid differencing scheme, a free surface, and time-splitting step for external (fast) and internal (slow) modes. The external mode is two dimensional and uses a short time step based on the Courant–Friedrichs–Lewy (CFL) condition and the external wave speed, whereas the internal mode is three dimensional and uses a long time step based on the CFL condition and the internal wave speed. In addition, while the horizontal time differencing is explicit, the vertical differencing is implicit, which eliminates time constraints for the vertical coordinate and allows for the use of fine vertical resolution in the surface and bottom boundary layers. The model state variables include 2D current velocity *UA* and *VA*, 3D current velocity *U* and *V*, temperature *T*, salinity *S*, surface elevation *η*, and turbulent kinetic energy, doubled *q*, and turbulent length scale *l*. Readers are referred to Blumberg and Mellor (1987) and Mellor (2004) for a full description of POM.

*A*and

*B*are the scaling parameters;

*r*is the distance from the storm center; and

*B*lies between 1 and 2.5. From (3) and (4), we can see that the accuracy of wind speed calculated by the Holland model is greatly dependent on the accuracy of predicted storm track and intensity.

The tangent linear and adjoint models of the 3D POM with turbulence closure scheme were developed by Peng and Xie (2006). Because of the high nonlinearity and discontinuity of vertical turbulence, a smoothing approximation was made in the linearization of the Mellor–Yamada turbulence scheme. As addressed in the first section, the purpose of this study is to reduce the bias in storm surge forecasts caused by various uncertainties through adjusting *a* and *b* of *a* and *b*, minor modification in the adjoint codes and corresponding correctness check are needed. The correctness test of the POM-4DVAR coding with parameters *a* and *b* being the control variables is given in the appendix. It shows that, in the idealized circumstances, the “true” values of the parameters *a* and *b* can be recovered by data assimilation based on the modified POM-4DVAR system.

*a*and

*b*being the control variables is defined as a misfit between the model and the observations, that is,where

*a*and

*b*, the minimization of cost function is performed. It is achieved by obtaining its gradient with respect to the control variables

*a*and

*b*by integrating the adjoint model of POM backward in time. The limited memory Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi-Newton minimization algorithm (Liu and Nocedal 1989) is employed to obtain the optimal control variables.

## 3. Experiment setup

The model domain is set to cover an area of 0°~30°N, 99°~125°E (Fig. 1) with a horizontal resolution of ⅕° × ⅕° (

### a. Identical twin experiments

To address whether the model errors from surface wind stress and other sources except ^{−1}. After crossing the Philippine Islands and entering the South China Sea, Ketsana (2009) swept all the way westward until landing at the east coast of Vietnam at 0700 UTC 29 September 2009 (Fig. 1). It caused at least 272 deaths and large damage in the Philippines, Vietnam, and China. The tidal station located at Yongxin Island (indicated in Fig. 1) reported a maximum water level increment of 2 m.

For ITEs, we assume that the surface wind speed is much larger than the ocean surface current speed, that is,

The design of ITEs. Here

For the nature run of each set, the ocean surface wind speed

### b. Real case experiment

To test whether the proposed method is efficient in practice, we apply this method to the surge forecasting for the storm of Hagupit (2008) by using real data. Typhoon Hagupit (2008) formed on 14 September 2008 in the western Pacific and moved westward toward the Philippines (Fig. 4). It entered the South China Sea on 21 September 2008 and intensified when moving northwest to China, and made landfall near Maoming in Guangdong Province of China at 0645 local time (LT) 24 September. It destroyed 14 333 houses and cost $824 million (U.S. dollars) in damages.

The experiment setup is given in Table 2. The observations of water level were collected from four stations along the coastline in Guangdong Province of China (Fig. 4). The surface winds are from the WRF model output. The model settings are the same as those for the second set of ITEs described in section 3a. Similar to the notation of ITEs, CTRL_RL1/CTRL_RL2 and DA_RL1/DA_RL2 denote the control run and data assimilation experiments,respectively, where “1” and “2” represent different a starting time of the model as shown in Table 2. For DA_RL1/DA_RL2, the observed water-level data of two stations (S1 and S3) are assimilated to adjust *C _{d}*, while those of other two stations are used for verification. A 24-h spinup is made before the control run or data assimilation experiments. A 3-h cycle within which the water-level data are assimilated every one hour is employed to get the optimal

*C*in DA_RL1/DA_RL2. A 24-h forecast starting at 0000 UTC 23 September (for DA_RL1) or 0600 UTC 23 September (for DA_RL2) with the corresponding optimal

_{d}*C*is made to compare with the control run without adjusting

_{d}*C*.

_{d}The design of real case experiments.

## 4. Results

### a. Results from the first set of ITEs

*a*,

*b*) of

*i*th error of the modeled variable

*n*is the sample volume of variable

*X*. It is obvious that the forecasts of both water level and surface currents are improved by the data assimilation. The improvements of the storm surge forecasts can be attributed to the correction of wind stress bias caused by the erroneous wind speed (CTRL_V− or CTRL_V+) through the optimal

### b. Results from the second set of ITEs

The optimal parameters (*a*, *b*) of

### c. Results from the third set of ITEs

As presented in Table 3, the optimal parameters (*a*, *b*) of the wind drag coefficient after 4DVAR are (0.7101, 0.1038). Figure 14 shows the 24-h forecasted water-level fields for NAT_2 (the “true”), CTRL_RS, and DA_RS. The errors of the water level caused by lower resolution are corrected effectively by adjusting

The optimal values of *a* and *b* obtained from different 4DVAR experiments.

### d. Results from the real case experiments

As presented in Table 2, the optimal parameters (*a*, *b*) of the wind stress drag coefficients for DA_RL1 (starting at 0000 UTC 23 September) and DA_RL2 (starting at 0600 UTC 23 September) are (0.5935, 0.081 55) and (0.516, 0.0771), respectively. Figures 18 and 19 show the time evolution of 24-h water level at sites 1–4 for CTRL_RL1/CTRL_RL2 and DA_RL1/DA_RL2. It is found that the simulated water-level values from DA_RL1/DA_RL2 are closer to the observations than those from CTRL_RL1/CTRL_RL2 most of the time during the 24-h period. The RMSE and the maximum values of the water level for each experiment are presented in Table 4. The errors of the 24-h forecasted water level are reduced effectively in DA_RL1/DA_RL2. The maximum values of the water level from DA_RL1/DA_RL2 are closer to the observed ones than those from CTRL_RL1/CTRL_RL2, too. These results indicate that adjusting *C _{d}* through 4DVAR is a feasible and practical solution to improve the storm surge forecasts in a real situation.

The RMSE and maximum values of the 24-h forecasted water level at stations S1–S4 compared to the observations (units: m).

It should be noted, however, that in a real situation, the improvements of storm surge forecasts by adjusting C_{d} thought 4DVAR are much less than those in the twin experiments. The primary reasons could be 1) the real observations are too sparse; 2) the real observations may contain large errors; and 3) the minimization process does not converge because of a number of factors, such as model instability, too many local minimum points, and nonpositive definite features of the background error covariance.

## 5. Conclusions and discussion

In this study, we employ the adjoint technique to adjust the parameters of wind stress drag coefficient

The errors of storm surge forecasts may come from several sources: 1) the true value of

One should be aware that the “optimal” value of

Finally, it should be pointed out that the adjustment of

This work was jointly supported by the Innovation Key Program of the Chinese Academy of Sciences (Grant KZCX2-EW-208), the National High Technology Research and Development Program of China (863 Program) (Grant 2010AA012304), the National Natural Science Foundation of China (Grant 41076009), the Ministry of Science and Technology of the People’s Republic of China (MOST) (Grant 2011CB403504), and the Hundred Talents Program of the Chinese Academy of Sciences.

# APPENDIX

## Correctness Test of the POM-4DVAR for Adjusting *Cd*

*C*is defined aswhere

*M*is the nonlinear model,

*M*is the tangent linear model,

_{T}*x*is the state variable,

*dx*is the perturbation of

*x*, and

Table A1 shows the result of the TLM check based on Eq. (A2). When using a 64-bit compiler, six-digit precision (in this case, six digits of “9” or “0”) is considered enough to confirm the correctness of the TLM coding. From Table A1, we can conclude that the TLM coding is correct.

The results of the TLM check.

*M*is the adjoint model and other variables are the same as in Eq. (A2). In our testing case, the value of left-hand side of Eq. (A3) is 42 045.333 953 763 2, while the right-hand side is 42 045.333 953 762 8. In a 64-bit compiler, we can conclude that the adjoint model coding is correct.

_{A}To verify whether the true wind drag coefficient can be retrieved in an idealized situation using the adjoint data assimilation approach, ITEs are designed, as shown in Table A2. The ocean surface wind speed *a*, *b*) to a half or a twice the “true” (

The design of numerical experiments. The asterisk means (*a*_{0}, *b*_{0}) = (0.49, 0.065).

Fig. A1 displays the variation of the cost function and the gradient values with the iteration numbers of the minimization procedure for experiments DA_Cd− and DA_Cd+. It is found that for both experiments, both the cost function and its gradient decrease dramatically by more than three orders within the first seven iterations, indicating that the minimization procedure converges very fast. Fig. A2 shows the optimal values of parameters *a* and *b* at each iteration for experiments DA_Cd− and DA_Cd+. It is found that both *a* and *b* reach their true values (i.e., *a* = 0.49 and *b* = 0.065) after the fourth iteration. Therefore, the adjoint model of the POM with parameters of the wind drag coefficient being control variables was correctly coded, and the POM-4DVAR approach is able to obtain the true values of the wind stress drag coefficient at the idealized situation (i.e., everything is perfect except the erroneous wind stress drag coefficient).

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