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Volume 5 Issue 2

Apr.  1988

Article Contents

THE LOCAL SPLINE VERTICAL INTERPOLATION METHOD OF TEMPERATURE AND GEOPOTENTIAL HEIGHT FIELDS AND THE TIME-DEPENDENT DIFFERENCE FORM OF THE HYDROSTATIC EQUATION


doi: 10.1007/BF02656778

  • In numerical weather prediction (NWP), the accuracy of vertical interpolation of the initial data is a problem which is greatly concerned by people. In this paper, we specify vertical distributions of the temperature and the geopotential height fields and examine three interpolation methods, i.e. the Lagrangian polynomial inter-polation method (hereafter abbreviated to LP method), the linear interpolation method (LN method) and the local spline interpolation method (LS method) proposed by the author. The examination shows that when the vertical resolution of the initial data is high enough, for example, the number of the given data levels N is 10 or more, all the three methods get good accuracy of interpolation, especially, the LP and the LS methods have very little errors almost tending to zero, while the LN method has a little larger errors than the two formers and the errors at various levels have the same sign. When N is reduced to 5, the LP and the LS methods still have quite good accuracy and similar error distributions, while the LN method has less accuracy. If the geopo-tential height field needs to be adjusted in order to satisfy the hydrostatic equilibrium with the temperature field which is assumed fixed, then the LS method has minimum errors. The examination also indicates that the vertical resolution with at least 5 levels of initial data can keep the interpolation accuracy. Otherwise the accuracy will not be guaranteed no matter which method is used.It is also pointed out in this paper that the temperature and the geopotential height fields can be given inde-pendently in numerical prediction models in order to keep higher interpolation accuracy. However, the hydro-static equation should be finite differenced in other way which is somewhat different from the conventional one. In other words, the time dependent difference form of the equation should be used, so that the initial interpola-tion accuracy could have influence on the time integration.
  • [1] Pengfei WANG, 2017: A High-Order Spatiotemporal Precision-Matching Taylor-Li Scheme for Time-Dependent Problems, ADVANCES IN ATMOSPHERIC SCIENCES, 34, 1461-1471.  doi: 10.1007/s00376-017-7018-1
    [2] Ji Zhengang, Chao Jiping, 1987: TELECONNECTIONS OF THE SEA SURFACE TEMPERATURE IN THE INDIAN OCEAN WTTH SEA SURFACE TEMPERATURE IN THE EASTERN EQUATORIAL PACIFIC, AND WITH THE 500 hPa GEOPOTENTIAL HEIGHT FIELD IN THE NORTHERN HEMISPHERE, ADVANCES IN ATMOSPHERIC SCIENCES, 4, 343-348.  doi: 10.1007/BF02663604
    [3] S.K. Sinha, S. Rajamani, 1995: Multivariate Objective Analysis of Wind and Height Fields in the Tropics, ADVANCES IN ATMOSPHERIC SCIENCES, 12, 233-244.  doi: 10.1007/BF02656836
    [4] Ji Zhongzhen, Wang Bin, 1997: Multispectrum Method and the Computation of Vapor Equation, ADVANCES IN ATMOSPHERIC SCIENCES, 14, 563-568.  doi: 10.1007/s00376-997-0074-1
    [5] Baofeng JIAO, Lingkun RAN, Na LI, Ren CAI, Tao QU, Yushu ZHOU, 2022: Comparative Analysis of the Generalized Omega Equation and Generalized Vertical Motion Equation, ADVANCES IN ATMOSPHERIC SCIENCES.  doi: 10.1007/s00376-022-1435-5
    [6] Shenming FU, Jie CAO, Xingwen JIANG, Jianhua SUN, 2017: On the Variation of Divergent Flow: An Eddy-flux Form Equation Based on the Quasi-geostrophic Balance and Its Application, ADVANCES IN ATMOSPHERIC SCIENCES, 34, 599-612.  doi: 10.1007/s00376-016-6212-x
    [7] Chen Wanlong, Chu Pao-Shin, 1990: On the Couplings between Chebyshev Coefficients as Derived from the Monthly Mean Geopotential Fields at 500 hPa over East Asia and the Southern Oscillation, ADVANCES IN ATMOSPHERIC SCIENCES, 7, 347-353.  doi: 10.1007/BF03179766
    [8] ZHU Congwen, Chung-Kyu PARK, Woo-Sung LEE, Won-Tae YUN, 2008: Statistical Downscaling for Multi-Model Ensemble Prediction of Summer Monsoon Rainfall in the Asia-Pacific Region Using Geopotential Height Field, ADVANCES IN ATMOSPHERIC SCIENCES, 25, 867-884.  doi: 10.1007/s00376-008-0867-x
    [9] Peng Yongqing, Yan Shaojin, Wang Tongmei, 1995: A Nonlinear Time-lag Differential Equation Model for Predicting Monthly Precipitation, ADVANCES IN ATMOSPHERIC SCIENCES, 12, 319-324.  doi: 10.1007/BF02656980
    [10] Zhou Jiabin, 1985: A NEW TYPE OF TIME-SERIES-FORECASTING METHOD, ADVANCES IN ATMOSPHERIC SCIENCES, 2, 385-401.  doi: 10.1007/BF02677255
    [11] Xiaojuan SUN, Siyan LI, Julian X. L WANG, Panxing WANG, Dong GUO, 2022: A New Method of Significance Testing for Correlation-Coefficient Fields and Its Application, ADVANCES IN ATMOSPHERIC SCIENCES, 39, 529-535.  doi: 10.1007/s00376-021-1196-6
    [12] YUAN Zhuojian, JIAN Maoqiu, 2003: A Linear Diagnostic Equation for the Nonhydrostatic Vertical Motion W in Severe Storms, ADVANCES IN ATMOSPHERIC SCIENCES, 20, 875-881.  doi: 10.1007/BF02915511
    [13] S. K. Sinha, D. R. Talwalkar, S. Rajamani, 1987: ON SOME ASPECTS OF OBJECTIVE ANALYSIS OF HUMI-DITY OVER INDIAN REGION BY THE OPTIMUM INTERPOLATION METHOD, ADVANCES IN ATMOSPHERIC SCIENCES, 4, 332-342.  doi: 10.1007/BF02663603
    [14] WANG Hesong, JIA Gensuo, 2013: Regional Estimates of Evapotranspiration over Northern China Using a Remote-sensing-based Triangle Interpolation Method, ADVANCES IN ATMOSPHERIC SCIENCES, 30, 1479-1490.  doi: 10.1007/s00376-013-2294-x
    [15] S.K. Sinha, D.R. Talwalkar, S.G. Narkhedkar, S. Rajamani, 1989: A Scheme for Objective Analysis of Wind Field Incorporating Multi-Weighting Functions in the Optimum Interpolation Method, ADVANCES IN ATMOSPHERIC SCIENCES, 6, 435-446.  doi: 10.1007/BF03342547
    [16] Da-Lin ZHANG, Xiaoxue WANG, 2003: Dependence of Hurricane Intensity and Structures on Vertical Resolution and Time-Step Size, ADVANCES IN ATMOSPHERIC SCIENCES, 20, 711-725.  doi: 10.1007/BF02915397
    [17] GUO Yanjun, DING Yihui, 2011: Impacts of Reference Time Series on the Homogenization of Radiosonde Temperature, ADVANCES IN ATMOSPHERIC SCIENCES, 28, 1011-1022.  doi: 10.1007/s00376-010-9211-3
    [18] Yi Zengxin, T. Warn, 1987: A NUMERICAL METHOD FOR SOLVING THE EVOLUTION EQUATION OF SOLITARY ROSSBY WAVES ON A WEAK SHEAR, ADVANCES IN ATMOSPHERIC SCIENCES, 4, 43-54.  doi: 10.1007/BF02656660
    [19] Zhang Banglin, Liu Jie, Sun Zhaobo, 1993: A New Multidimensional Time Series Forecasting Method Based on the EOF Iteration Scheme, ADVANCES IN ATMOSPHERIC SCIENCES, 10, 243-247.  doi: 10.1007/BF02919147
    [20] Daosheng XU, Jeremy Cheuk-Hin LEUNG, Banglin ZHANG, 2023: A Time Neighborhood Method for the Verification of Landfalling Typhoon Track Forecast, ADVANCES IN ATMOSPHERIC SCIENCES, 40, 273-284.  doi: 10.1007/s00376-022-1398-6

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Manuscript History

Manuscript received: 10 April 1988
Manuscript revised: 10 April 1988
通讯作者: 陈斌, bchen63@163.com
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    沈阳化工大学材料科学与工程学院 沈阳 110142

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THE LOCAL SPLINE VERTICAL INTERPOLATION METHOD OF TEMPERATURE AND GEOPOTENTIAL HEIGHT FIELDS AND THE TIME-DEPENDENT DIFFERENCE FORM OF THE HYDROSTATIC EQUATION

  • 1. Department of Atmospheric Sciences Nanjing University, Nanjing

Abstract: In numerical weather prediction (NWP), the accuracy of vertical interpolation of the initial data is a problem which is greatly concerned by people. In this paper, we specify vertical distributions of the temperature and the geopotential height fields and examine three interpolation methods, i.e. the Lagrangian polynomial inter-polation method (hereafter abbreviated to LP method), the linear interpolation method (LN method) and the local spline interpolation method (LS method) proposed by the author. The examination shows that when the vertical resolution of the initial data is high enough, for example, the number of the given data levels N is 10 or more, all the three methods get good accuracy of interpolation, especially, the LP and the LS methods have very little errors almost tending to zero, while the LN method has a little larger errors than the two formers and the errors at various levels have the same sign. When N is reduced to 5, the LP and the LS methods still have quite good accuracy and similar error distributions, while the LN method has less accuracy. If the geopo-tential height field needs to be adjusted in order to satisfy the hydrostatic equilibrium with the temperature field which is assumed fixed, then the LS method has minimum errors. The examination also indicates that the vertical resolution with at least 5 levels of initial data can keep the interpolation accuracy. Otherwise the accuracy will not be guaranteed no matter which method is used.It is also pointed out in this paper that the temperature and the geopotential height fields can be given inde-pendently in numerical prediction models in order to keep higher interpolation accuracy. However, the hydro-static equation should be finite differenced in other way which is somewhat different from the conventional one. In other words, the time dependent difference form of the equation should be used, so that the initial interpola-tion accuracy could have influence on the time integration.

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