Chinese Journal of Oceanology and Limnology Vol. 30 No. 5, P. 868-875, 2012 http://dx.doi.org/10.1007/s00343-012-1185-9
Inversion of two-dimensional tidal open boundary conditions of M2 constituent in the Bohai and Yellow Seas* CAO Anzhou (曹安州)1, GUO Zheng (郭筝)1, 2, LÜ Xianqing (吕咸青)1, ** 1
Laboratory of Physical Oceanography, Ocean University of China, Qingdao 266003, China
2
National Marine Data and Information Service, Tianjin 300171, China
Received Sep. 13, 2011; accepted in principle Dec. 22, 2011; accepted for publication Apr. 25, 2012 © Chinese Society for Oceanology and Limnology, Science Press, and Springer-Verlag Berlin Heidelberg 2012
Abstract Two-dimensional tidal open boundary conditions of the M2 constituent in the Bohai and Yellow Seas (BYS) have been estimated by assimilating T/P altimeter data. During inversion, independent point (IP) strategy was used, in which several IPs on the open boundary is assumed, values at these IPs can be optimized with an adjoint method, and those at other grid points are determined by linearly interpolating the values at IPs. The reasonability and feasibility of the model are tested by ideal twin experiments. In the practical experiment (PE) after assimilation, the cost function may reach 1% or less of its initial value. Mean absolute errors in amplitude and phase can be less than 5 cm and 5°, respectively, and the obtained co-chart can show the character of the M2 constituent in the BYS. The results of the PE indicate that using only two IPs on the open boundary can yield better simulated results. Keyword: two-dimensional; adjoint method; T/P altimeter; open boundary conditions; independent points
1 INTRODUCTION Tides and tidal currents are basic motions of ocean water and are important in research on storm surges, circulation, and water mass, among others (Munk, 1997). The study on tides is much more important in coastal regions and marginal seas. As typical coastal and marginal seas, the Bohai, Yellow and East China Seas have been studied by many researchers. Fang (1986), Kang et al. (1998), Lefevre et al. (2000) and many others have done related work, addressing the characteristics of the main constituents. A major difficulty in tidal simulation is the determination of open boundary conditions (OBCs). In previous studies, OBCs were obtained by interpolating observations near the open boundary, or from numerical model results of a larger region. However, OBCs obtained by these two methods require adjustment to fit the model. Based on the theory of inverse problems, the adjoint method is a powerful tool for parameter estimation, and has the advantage of assimilating various observations distributed in time and space. Through assimilating satellite altimetry and tidal gauge data, OBCs of a tidal model can be optimized automatically using the
adjoint method. Lardner (1993, 1995) estimated OBCs in a 2-D tidal model, indicating that OBCs may be obtained by assimilating observations in this depth-averaged model. Seiler (1993) did a related study with a quasi-geostrophic, open-ocean model. He et al. (2004) investigated shallow water tidal constituents in the BYS by assimilating T/P altimeter data with the adjoint method. Han et al. (2006) used a 2-dimensional nonlinear numerical Princeton Ocean Model (POM) to describe tides in the China Seas, using data from tidal gauges and T/P altimeter satellite. Myers and Baptista (2001) developed a regional tide model for the eastern North Pacific Ocean, through inversion with two-dimensional finite element codes. They obtained OBCs of eight main constituents by assimilating tidal gauge data; parameter sensitivity was also examined. Zhang and Lu (2010) simulated three-dimensional tidal currents
* Supported by the State Ministry of Science and Technology of China (Nos. 2007AA09Z118, 2008AA09A402), the National Natural Science Foundation of China (No. 41076006), and the Ministry of Education’s 111 Project (No. B07036) ** Corresponding author:
[email protected]
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in the marginal seas by assimilating satellite altimetry data. They designed twin experiments (TE) to demonstrate model reasonability and feasibility, and OBCs were estimated in practical experiments (PE). In this paper, OBCs of the M2 constituent in the Bohai and Yellow Seas (BYS) are estimated by assimilating T/P altimeter data. Independent point (IP) strategy is used for the inversion. We assume several IPs on the open boundary. Values at the IPs can be optimized with the adjoint method, and those at other grid points on the open boundary are determined by linearly interpolating the values at IPs. Compared with the aforementioned studies, the aim here is to investigate the influence of different IP strategies on simulated results, and to increase simulation realism using the adjoint method in a 2-D tidal model. This paper is organized as follows. The tidal model is introduced in Section 2. In Section 3, TEs are carried out to examine model reasonability and feasibility. In Section 4, the M2 constituent in the BYS is simulated by optimizing OBCs. Finally, conclusions are given in Section 5.
2 TIDAL MODEL 2.1 2-D tidal model Assuming pressure is hydrostatic and density is constant, the 2-D, depth-averaged tidal model used is as follows:
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Initial conditions are such that the initial sea surface elevation, x- and y- velocities are zero. The closed boundary conditions are such that the normal velocity is zero, and OBCs are sea surface elevations on the open boundary, as follows: , (2) where a and b are the Fourier coefficients on the open boundary, and ω is the angular frequency of the M2 constituent. The cost function is constructed as ,
(3)
where Kis a constant (here, K=1), D is the set of the observation locations, is the simulated result, is the observation and ite is the number of time steps of the forward model. Our finite difference schemes are similar to Lu and Zhang (2006), except that we used the rectangular coordinate instead of their spherical coordinate. 2.2 Correction of Fourier coefficients at IPs We assume that {al,bl:l=1,…,L} are the Fourier coefficients of the M2 constituent at the IPs on the open boundary. and are linearly interpolated values of these Fourier coefficients at other points. Then, we obtain the following:
∑
,
(4)
∑
,
(5)
where ϕs,l is the weight of linear interpolation, and Ls is a set of IPs whose distances from the s-th grid point on the open boundary are less than a radius R. The magnitude of R is given beforehand.
∑
,
(6)
where Ws,l is the weight coefficient in the Cressman form: ,
(1)
where t is time, x and y are Cartesian coordinates (positive eastward and northward, respectively), h(x, y) is undisturbed water depth at location (x, y), (x, y) is sea surface elevation above the undisturbed sea level, u(x, y, t) and v(x, y, t) are velocity components in the horizontal x- and y-directions, f is the Coriolis parameter, k is the bottom friction coefficient, and A is the coefficient of horizontal eddy viscosity.
,
(7)
where rs,l is the distance from the s-th grid point to the l-th IP on the open boundary, and R represents the influence radius mentioned above. Then, we can adjust the Fourier coefficients at IPs:
′
∑
∑
′
∑
∑
,
,
(8)
(9)
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where Sl is a set of grid points on the open boundary, whose distances from the l-th IP are less than R. al′, bl′ and al, bl are prior and estimated values of the Fourier coefficients at IPs, respectively, and T j is defined as follows. , if (ml, nl) is on the left open boundary; , if (ml, nl) is on the right open boundary; , if (ml, nl) is on the lower open boundary;
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T/P altimeter data, tidal gauge data, bottom friction coefficient and horizontal eddy viscosity coefficient are the same as in Lu and Zhang (2006). A bathymetry map of the BYS and observation sites are shown in Fig.1.
3 TWIN EXPERIMENT (TE) Here, the aim of TE is to examine model reasonability and feasibility. The design and results of these experiments are shown below. 3.1 Calculation process of TE
, if (ml, nl) is on the upper open boundary, where g is the acceleration from gravity, μ and v are adjoint variables similar to those defined in Lu and Zhang (2006). Kb is calculated in the following: First, we define X = ( a1, a2,...,aL, b1, b2,...,bL)T, X ′ = ( a1′, a2′,...,aL′, b1′, b2′,...,bL′)T, and
The TE calculation process is designed as follows. (1) Run the forward tidal model with prescribed OBCs. The simulated results at grid points on T/P satellite tracks are taken as “observations”. Here, only the locations of T/P altimetry are used and these “observations” are not real, but the simulated results of the forward tidal model based on the prescribed OBCs. (2) Initial values of Fourier coefficients at IPs (taken as zero here), and those at other grid points obtained by interpolating the values at IPs, are assigned to the OBCs for running the forward tidal model. Values of state variables are obtained. (3) The difference of water elevation between simulated results and “observations” serves as the external force of the adjoint model. Values of adjoint state variables are obtained through backward integration of the adjoint equations. (4) After obtaining the values of state variables and adjoint variables, the Fourier coefficients at IPs can be optimized by Eqs.8 and 9. Repeat steps (2), (3) and (4) and, with the Fourier coefficients at IPs optimized, the difference of water
, Then, Eqs.8 and 9 can be rewritten as
′
,
(10)
where is the L2 norm of Y at the first iteration step, ckb is a constant (here, ckb=1), and NIP is the number of IPs. Thus, Kb can be obtained by ,
(11)
2.3 Model setting The region of interest is the BYS and the open boundary is at 34°N. Horizontal resolution of the model is 1010 and the time step is 372.618 seconds, which is 1/120 the period of the M2 constituent. The
Fig.1 Bathymetry map of BYS T/P altimeter tracks: +; tidal gauge positions: •
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CAO et al.: Inversion of open boundary conditions in the BYS
elevation between simulated results and “observations” decreases. Once a convergence criterion is met, the process of inversion terminates. The optimization algorithm used is the typical steepest descent method, which is the same as that used by Zhang and Lu (2010). 3.2 Settings of TE Seven TEs were done, corresponding to different OBCs (Fig.2). In TE1, the Fourier coefficients at two endpoints are prescribed, and values at the other grid points on the open boundary are acquired by linearly interpolating prescribed values at endpoints. For experiments TE2 to TE6, respectively 3, 5, 9, 13, 19 grid points are evenly distributed, at which Fourier coefficients are prescribed. In TE7, values at all 36 grid points are prescribed. During the inversion process, we used seven IP strategies (Fig.3), as follows. Strategy (A): Only the two endpoints are taken as IPs, and the radius of influence is the length of 35 grids. Strategy (B): Three evenly distributed grid points are taken as IPs, and the radius is the length of 18 grids. Strategy (C) to Strategy (F): The numbers of IPs
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are 5, 9, 13, 19, respectively, which are distributed evenly. Radiuses are the lengths of 9, 5, 3, and 2 grids, respectively. Strategy (G): All 36 grid points are set as IPs. In TE1, Strategies (A)−(G) are used to invert OBCs and simulate tides. In TE2, Strategies (B)−(G) are used; in TE3, Strategies (C)−(G); in TE4, Strategies (D)−(G); in TE5, Strategies (E)−(G); in TE6, Strategies (F)−(G); in TE7, only Strategy (G). 3.3 Result and discussion of TE Iteration is terminated once a convergence criterion is met. The criterion could be that the last two values of the cost function are sufficiently close, magnitude of the gradient is sufficiently small, discrepancy between updated and old parameters is satisfactorily small, or a combination of these. In our work, the criterion is that the number of iteration steps is exactly 100. The ratio of the cost function and its initial value is shown in Fig.4 as a function of iteration step. After 100 iteration steps, the cost function and gradient may reach 1% (or even 0.1%) and 10-4−10-8of their initial values, respectively. The value of the cost function after 30 steps is almost equal to its final value, as is that of the gradient. Fig.2 gives partial inversion
Fig.2 Prescribed OBCs and partially inverted OBCs of TE1-TE7 [TE1-(A), TE2-(B), TE3-(C), TE4-(D), TE5-(E), TE6-(F), and TE7-(G)]
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Fig.3 Seven IP strategies Larger points represent IPs, and smaller ones represent other grid points on the open boundary.
results of TEs. We conclude that by use of IP strategy, the prescribed OBCs can be inverted successfully by assimilating the “observations”. Mean absolute errors (MAEs) and correlation coefficients between the prescribed OBCs and inverted ones are shown in Fig.5. Obviously, when the number of IPs equals that of grid points for which Fourier coefficients are prescribed, the MAE and the correlation coefficient reach minimum and maximum in every TE, respectively. On one hand, this indicates that OBCs can be inverted successfully using IP strategies. On the other hand, it signifies that in a real situation where the appropriate number of IPs to use is unknown, we should try several IP strategies to attain the most accurate OBCs. Furthermore, from Fig.5 we see that as complexity of prescribed OBCs increases from TE1 to TE7, MAE increases and the correlation coefficient decreases, even with use of the most appropriate IP strategy. Independent “observations” were used to test the model. The central grid was chosen from every 66 grid as the position of independent observations. These observations were not used for assimilation, but for testing simulated results. Table 1 shows differences between simulated results and the independent “observations.” ΔH and Δg represent MAE in amplitude and phase, respectively. Another parameter Hs evaluates the simulated results, and is defined as follows:
Fig.4 Cost function and gradient of TEs
Table 1 Differences between simulated independent “observations”
results
TE-strategy
ΔH (cm)
Δg (°)
Hs (cm)
TE1-(A)
0.7
0.6
1.1
TE2-(B)
1.7
0.6
1.9
TE3-(C)
1.1
2.5
2.0
TE4-(D)
1.3
2.5
2.5
TE5-(E)
1.1
1.9
2.1
TE6-(F)
1.1
2.0
2.3
TE7-(G)
1.1
1.8
2.3
∑
,
and
(12)
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Fig.5 MAEs and correlation coefficients of given OBCs and inverted ones
where HC = A0 cos(g0) − Ac cos(gc) and HS = A0 sin(g0) − Ac sin(gc). A0 and g0 are observed tidal amplitude and phase, respectively, whereas Ac and gc are modeled ones. The differences in Table 1 indicate that simulated results match observations well, which demonstrates model reasonability and feasibility.
4 PRACTICAL EXPERIMENT In this section, OBCs are optimized by assimilating the T/P data to simulate the M2 constituent. As mentioned above, we do not know how many IPs are appropriate for use, so we used all seven IP strategies in Section 3. Figure 6 shows the cost function of the PE in log form. After 100 steps of iteration, the cost function may reach 1% of its initial value or less, and the gradient 10-410-8 of its initial value. Optimized OBCs are shown in Fig.7, from which we see that OBCs inverted with the seven IP strategies are very different. However, considering the magnitude of differences (Table 2), we regard all these inverted OBCs as acceptable. As shown in Table 2, MAEs in amplitude and phase between simulation and T/P data are less than 5 cm and 5°, respectively, and those between simulation and tidal gauge data are about 5 cm and 5°. This indicates that simulated results are consistent with observations, which further demonstrates model reasonability and feasibility. Differences between inverted OBCs and those of
Fig.6 Cost function and gradient of PE Table 2 Differences between simulated results and observations (T/P data and tidal gauge data) T/P data IP strategy ΔH (cm) Δg (°) H (cm) s
Tidal gauge data ΔH (cm)
Δg (°)
Hs (cm)
(A)
3.5
4.4
6.1
4.0
5.2
10.2
(B)
5.0
4.8
7.5
3.8
4.7
9.5
(C)
4.5
4.8
7.1
3.9
4.9
9.2
(D)
4.1
4.1
6.0
3.8
5.0
9.3
(E)
3.8
4.0
5.8
4.1
5.2
9.7
(F)
3.5
3.9
5.5
4.0
5.1
9.7
(G)
3.6
3.7
5.5
4.1
5.7
10.6
Lu and Zhang (2006) are shown in Table 3. Results using Strategy (A) are most consistent with that work, whereas ones obtained with Strategy (G) are least
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Table 3 Differences between inverted OBCs and those of Lü and Zhang (2006) IP strategy
ΔH (cm)
Δg (°)
Hs (cm)
(A)
3.8
4.8
7.4
(B)
10.9
3.8
12.9
(C)
11.1
4.5
13.0
(D)
12.1
2.5
12.9
(E)
12.3
4.7
14.7
(F)
14.9
6.2
18.8
(G)
18.9
7.6
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consistent. Co-charts obtained using Strategy (A) and (G) are shown in Fig.8. From this, it is evident that both co-tidal charts can describe the character of the M2 constituent in the BYS. There are two amphidromic points in the Bohai Sea, one of which is near Qinhuangdao and the other near the Yellow River delta. There are also two amphidromic points in the Yellow Sea, one of which is north of Chengshantou and the other is southeast of Qingdao. When excessive IPs are used (such as with Strategy (G) in PE), coamplitude and co-phase lines near the open boundary take on a disorganized pattern, which is the reason for the big difference with Strategy (G) in Table 3. Based
Fig.7 Optimized OBCs of PE, using seven IP strategies
Fig.8 Co-charts obtained using Strategy (A) and (G) Dashed line denotes co-amplitude line (m), and solid line denotes the co-phase line (°)
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on these results, we conclude that for inversion of the M2 constituent in the BYS, results from Strategy (A) are optimum.
5 SUMMARY AND CONCLUSION OBCs of the M2 constituent in the BYS were estimated by assimilating T/P altimeter data. IP strategies were used for the inversion. Through TEs, model reasonability and feasibility were tested. In the PE after assimilation, the cost function may reach 1% of its initial value or less. MAEs in amplitude and phase were less than 5 cm and 5°, respectively, and the obtained co-chart can describe the character of the M2 constituent in the BYS. Finally, considering the inversion results, we conclude that for inversion of the M2 constituent in the BYS, results obtained with Strategy (A) are superior. References Fang G H. 1986. Tide and tidal current charts for the marginal seas adjacent to China. Chin. J. Oceanol. Limn., 4(1): 1-16. Han G J, Li W, He Z J, L K, Ma J R. 2006. Assimilated tidal results of tidal gauge and TOPEX/POSEIDON data over the China Seas using a variational adjoint approach with a nonlinear numerical model. Adv. Atmos. Sci., 23: 449-460. He Y J, Lü X Q, Qiu Z F, Zhao J P. 2004. Shallow water tidal
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constituents in the Bohai and the Yellow Sea from a numerical adjoint model with TOPEX/POSEIDON altimeter data. Cont. Shelf Res., 24: 1 521-1 529. Kang S K, Lee S R, Lie H J. 1998. Fine grid tidal modeling of the Yellow and East China Seas. Cont. Shelf Res., 18: 739772. Lardner R W. 1993. Optimal control of an open boundary condition for a numerical tidal model. Comput. Method. Appl. M., 102: 367-387. Lardner R W. 1995. Optimal assimilation of current and surface elevation data in a two-dimensional numerical tidal model. Appl. Math. Model., 19: 30-38. Lefevre F, Le Provost C, Lyard F H. 2000. How can we improve a global ocean tide model at a redional scale? A test on the Yellow Sea and the East China Sea. J. Geophys. Res., 105: 8 707-8 725. Lü X Q, Zhang J C. 2006. Numerical study on spatially varying bottom friction coefficient of a 2D tidal model with adjoint method. Cont. Shelf Res., 26: 1 905-1 923. Munk W H. 1997. Once again: once again-tidal friction. Prog. Oceanogr., 40: 7-35. Myers P E, Baptista M A. 2001. Inversion for tides in the Eastern North Pacific Ocean. Adv. Water Resour., 24: 505-519. Seiler U. 1993. Estimation of open boundary conditions with the adjoint method. J. Geophys. Res., 98: 22 855-22 870. Zhang J C, Lü X Q. 2010. Inversion of three-dimensional tidal currents in marginal seas by assimilating satellite altimetry. Comput. Method. Appl. M., 199: 3 125-3 136.
