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 武汉大学学报·信息科学版  2017, Vol. 42 Issue (12): 1811-1817

#### 文章信息

ZHANG Hongmei, HUANG Jiayong, ZHAO Jianhu, CHEN Zhigao, ZHU Shifang

An Improved Tidal Current Separation Method of Radial Basis Function Using Gradient Training

Geomatics and Information Science of Wuhan University, 2017, 42(12): 1811-1817
http://dx.doi.org/10.13203/j.whugis20150309

### 文章历史

1. 武汉大学动力与机械学院, 湖北 武汉, 430072;
2. 武汉大学测绘学院, 湖北 武汉, 430079

An Improved Tidal Current Separation Method of Radial Basis Function Using Gradient Training
ZHANG Hongmei1, HUANG Jiayong2, ZHAO Jianhu2, CHEN Zhigao2, ZHU Shifang2
1. School of Power and Mechanical Engineering, Wuhan University, Wuhan 430072, China;
2. School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China
First author: ZHANG Hongmei, PhD, professor, specializes in the marine survey. E-mail: hmzhang@whu.edu.cn
Foundation support: The National Natural Science Foundation of China, Nos. 41376109, 41576107, 41176068; Sea and Island Use Dynamic Monitoring Research Program of Liaoning Province, No.LNZC20170900260
Abstract: Ship-mounted acoustic Doppler current profiler (ADCP) measurements have been used to obtain detailed observations of the spatial patterns of flows. To separate tidal and subtidal currents from the ship-mounted ADCP data at tidal reach, traditional harmonic analysis method require redundant measurements, and complex tidal current separation operations. Developments in radial basis function (RBF) interpolation theory are demonstrated to significantly improve the quality of the tidal velocity field extracted from the measurements. Tidal current separation method of RBF using greedy fit couldn't optimal centers for RBF, and the over-fitting of RBF would lead to the instability of separation model. To overcome these deficiencies, this paper proposes an improved tidal current separation method of radial basis function using gradient training. The tidal current separation results in Xuliujing section verified the feasibility of the tidal current separation method of gradient training RBF. The inner precision of reconstructed flow field based on the tidal current separation method using gradient training is better than 0.21 m/s, and the prediction accuracy is better than 0.25 m/s.

1 传统潮流分离方法 1.1 潮流调和分析

 $\left\{ \begin{array}{l} U\left( t \right) = {U_0} + \sum\limits_{i = 1}^m {{U_i}\cos \left( {{\omega _i}t - {\theta _i}} \right)} \\ V\left( t \right) = {V_0} + \sum\limits_{i = 1}^m {{V_i}\cos \left( {{\omega _i}t - {\xi _i}} \right)} \end{array} \right.$ (1)

 $U\left( t \right) = {U_0} + \sum\limits_{i = 1}^m {\left[ {{a_i}\cos \left( {{\omega _i}t} \right) + {b_i}\sin \left( {{\omega _i}t} \right)} \right]}$ (2)

1.2 径向基函数法

 $S\left( X \right) = P\left( X \right) + \sum\limits_{j = 1}^h {{\lambda _j}\mathit{\Phi }\left( {{r_j}} \right)}$ (3)

Vennell等研究中P(X)采用二阶多项式，基函数采用高斯函数：

 $\begin{array}{*{20}{c}} {P\left( X \right) = f\left( {x,y} \right) = }\\ {\left[ {\begin{array}{*{20}{c}} 1&x&y&{{x^2}}&{xy}&{{y^2}} \end{array}} \right]{{\left[ {\begin{array}{*{20}{c}} {{\beta _1}}& \cdots &{{\beta _6}} \end{array}} \right]}^{\rm{T}}}} \end{array}$ (4)
 $\mathit{\Phi }\left( {{r_j}} \right) = \exp \left( { - \frac{{r_j^2}}{{2{\delta ^2}}}} \right)$ (5)

 $\mathit{\boldsymbol{S}}\left( \mathit{\boldsymbol{X}} \right) = \left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{P}}&\mathit{\boldsymbol{A}} \end{array}} \right]\mathit{\boldsymbol{W}} = \mathit{\boldsymbol{DW}}$ (6)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{U}} = \left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{D}}&{\mathit{\boldsymbol{D}}\cos \left( {{\omega _1}t} \right)}&{\mathit{\boldsymbol{D}}\sin \left( {{\omega _1}t} \right)}& \cdots \end{array}} \right] \times }\\ {{{\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{W}}_0}}&{{\mathit{\boldsymbol{W}}_{C1}}}&{{\mathit{\boldsymbol{W}}_{S1}}}& \cdots \end{array}} \right]}^{\rm{T}}}} \end{array}$ (7)

 $\sum\limits_{j = 1}^h {{\lambda _j}p\left( {{c_j}} \right) = 0}$ (8)

2 基于梯度训练法的径向基函数潮流分离算法

 $U\left( {{X_i}} \right) = {F_0} + \sum\limits_{n = 1}^N {\left( {{F_n}\cos \left( {{\omega _n}{t_i}} \right) + {G_n}\sin \left( {{\omega _n}{t_i}} \right)} \right)}$ (9)

 $F\left( {{X_i}} \right) = \sum\limits_{j = 1}^h {{w_i}\mathit{\Phi }\left( {{X_i} - {c_j}} \right)} ,\mathit{\Phi }\left( {{r_j}} \right) = \exp \left( { - \frac{{r_j^2}}{{2{\delta ^2}}}} \right)$ (10)

 $U\left( {{X_i}} \right) = \sum\limits_{j = 1}^h {\mathit{\Phi }\left( {{X_i} - {c_j}} \right)\mathit{\Gamma }}$ (11)

 $E = \frac{1}{2}\sum\limits_{i = 1}^p {e_i^2} ,{e_i} = {U_i} - U\left( {{X_i}} \right)$ (12)

 $\begin{array}{l} \Delta {c_{j,x}} = - \eta \frac{{\partial E}}{{\partial {c_{j,x}}}} = \frac{\eta }{{\delta _j^2}}\sum\limits_{i = 1}^P {{e_i}\mathit{\Phi }\left( {{X_i} - {c_j}} \right)\left( {{X_{i,x}} - {c_{j,x}}} \right)} \left[ {{w_j} + \sum\limits_{n = 1}^N {\left( {{w_{\left[ {\left( {2n - 1} \right)h + j} \right]}}\cos \left( {{\omega _n}{t_i}} \right) + } \right.} } \right.\\ \;\;\;\;\;\;\;\;\;\;\left. {\left. {{w_{\left( {2nh + j} \right)}}\sin \left( {{\omega _n}{t_i}} \right)} \right)} \right] \end{array}$ (13)
 $\begin{array}{l} \Delta {c_{j,y}} = - \eta \frac{{\partial E}}{{\partial {c_{j,y}}}} = \frac{\eta }{{\delta _j^2}}\sum\limits_{i = 1}^P {{e_i}\mathit{\Phi }\left( {{X_i} - {c_j}} \right)\left( {{X_{i,y}} - {c_{j,y}}} \right)} \left[ {{w_j} + \sum\limits_{n = 1}^N {\left( {{w_{\left[ {\left( {2n - 1} \right)h + j} \right]}}\cos \left( {{\omega _n}{t_i}} \right) + } \right.} } \right.\\ \;\;\;\;\;\;\;\;\;\;\left. {\left. {{w_{\left( {2nh + j} \right)}}\sin \left( {{\omega _n}{t_i}} \right)} \right)} \right] \end{array}$ (14)
 $\begin{array}{l} \Delta {\delta _j} = - \eta \frac{{\partial E}}{{\partial {\delta _j}}} = \frac{\eta }{{\delta _j^3}}\sum\limits_{i = 1}^P {{e_i}\mathit{\Phi }\left( {{X_i} - {c_j}} \right){{\left( {{X_i} - {c_j}} \right)}^2}} \left[ {{w_j} + \sum\limits_{n = 1}^N {\left( {{w_{\left[ {\left( {2n - 1} \right)h + j} \right]}}\cos \left( {{\omega _n}{t_i}} \right) + } \right.} } \right.\\ \;\;\;\;\;\;\;\;\;\;\left. {\left. {{w_{\left( {2nh + j} \right)}}\sin \left( {{\omega _n}{t_i}} \right)} \right)} \right] \end{array}$ (15)
 $\Delta {w_j} = - \eta \frac{{\partial E}}{{\partial {w_j}}} =\\ \left\{ \begin{array}{l} \eta \sum\limits_{i = 1}^P {{e_i}\mathit{\Phi }\left( {{X_i} - {c_j}} \right)} ,j = 1,2, \cdots ,h\\ \eta \sum\limits_{i = 1}^P {{e_i}\mathit{\Phi }\left( {{X_i} - {c_j}} \right)\cos \left( {{\omega _n}{t_i}} \right)} ,j = \left( {2n + 1} \right)h + 1,\left( {2n - 1} \right)h + 2, \cdots ,\left( {2n - 1} \right)h + h\\ \eta \sum\limits_{i = 1}^P {{e_i}\mathit{\Phi }\left( {{X_i} - {c_j}} \right)\sin \left( {{\omega _n}{t_i}} \right)} ,j = 2nh + 1,2nh + 2, \cdots ,2nh + h \end{array} \right.$ (16)

 $\left\{ \begin{array}{l} {c_{j,x}}\left( {k + 1} \right) = {c_{j,x}}\left( k \right) + \Delta {c_{j,x}}\\ {c_{j,y}}\left( {k + 1} \right) = {c_{j,y}}\left( k \right) + \Delta {c_{j,y}}\\ {\delta _j}\left( {k + 1} \right) = {\delta _j}\left( k \right) + \Delta {\delta _j}\\ {w_j}\left( {k + 1} \right) = {w_j}\left( k \right) + \Delta {w_j} \end{array} \right.$ (17)

3 实验结果及分析 3.1 实验数据

 图 1 测量断面位置及流速频谱分析图 Figure 1 Location of Section and Frequency-Spectrum
3.2 潮流分离

 实验 拟合方法 内符合精度(±) 外推精度(±) 1 传统调和分析方法径向基函数法(贪婪拟合法)径向基函数法(梯度训练法) 0.1910.1450.202 /// 2 径向基函数法(贪婪拟合法)径向基函数法(梯度训练法) 0.1130.150 0.3080.215 3 径向基函数法(贪婪拟合法)径向基函数法(梯度训练法) 0.1080.150 0.3450.246

3.3 分离结果分析

(1) 余流分析

 图 2 传统方法与本文方法所得余流和M2分潮潮流椭圆长半轴比较 Figure 2 Comparison of Residual and Ellipse Semimajor of M2 with Traditional Harmonic Analysis Method and Gradient Training Method

(2) 潮流分析

4 结语