低阶修正的Hotine截断核函数的频谱分析与应用
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摘要
传统截断核函数存在谱泄露问题,且实测数据在移去恢复频段的利用率低。本文以Hotine核函数为例引入了一种高低阶均修正的截断核函数,在其基础上进一步提出了仅低阶修正的截断核函数,具体包含余弦修正和线型修正两种类型。修正核函数能够有效地控制截断核函数存在的谱泄露问题,并且增大了实测数据在修正频段对高程异常的贡献率。试验结果表明,当低阶修正带宽一定时,低阶修正核函数计算的似大地水准面精度优于传统截断核函数计算的似大地水准面精度,并且与高低阶均修正的核函数的解算精度相当。但在计算效率上,低阶修正核函数明显优于高低阶均修正的核函数。本文的试验证实了在基于Helmert第二压缩法的边值问题(Stokes-Helmert或Hotine-Helmert边值问题)中低阶修正核函数是一种比较有效的核函数。
The traditional spheroidal kernel results in the spectrum leakage, and the utilization rate of the removed degrees of the measured data is low. Hence, a kind of spheroidal kernel whose high and low degrees are both modified is introduced in this research, which is exampled by the Hotine kernel. In addition, the low-degree modified spheroidal kernel is proposed. Either cosine or linear modification factors can be utilized. The modified kernel functions can effectively control the spectrum leakage compared with the traditional spheroidal kernel. Furthermore, the modified kernel augments the contribution rate of the measured data to the height anomaly in the modified frequency domain. The experimental results show that the accuracies of the quasi-geoids using the cosine and linear low-degree modified kernels are higher than the traditional spheroidal kernel, and differ little from the accuracies of the quasi-geoids using the kernel whose high and low degrees are both modified when the low-degree modification widths of these two kinds of kernels are the same. Since the computational efficiency of the low-degree modified kernel is improved obviously, the low-degree modified kernel behaves better in constructing the(quasi-) geoid based on Stokes-Helmert or Hotine-Helmert boundary value theory.
The traditional spheroidal kernel results in the spectrum leakage, and the utilization rate of the removed degrees of the measured data is low. Hence, a kind of spheroidal kernel whose high and low degrees are both modified is introduced in this research, which is exampled by the Hotine kernel. In addition, the low-degree modified spheroidal kernel is proposed. Either cosine or linear modification factors can be utilized. The modified kernel functions can effectively control the spectrum leakage compared with the traditional spheroidal kernel. Furthermore, the modified kernel augments the contribution rate of the measured data to the height anomaly in the modified frequency domain. The experimental results show that the accuracies of the quasi-geoids using the cosine and linear low-degree modified kernels are higher than the traditional spheroidal kernel, and differ little from the accuracies of the quasi-geoids using the kernel whose high and low degrees are both modified when the low-degree modification widths of these two kinds of kernels are the same. Since the computational efficiency of the low-degree modified kernel is improved obviously, the low-degree modified kernel behaves better in constructing the(quasi-) geoid based on Stokes-Helmert or Hotine-Helmert boundary value theory.
引文
[1]管泽霖,管铮,黄谟涛,等.局部重力场逼近理论和方法[M].北京:测绘出版社,1997.GUAN Zelin,GUAN Zheng,HUANG Motao,et al.Theory and method of regional gravity field approximation[M].Beijing:Surveying and Mapping Press,1997.
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[3]李建成.最新中国陆地数字高程基准模型:重力似大地水准面CNGG2011[J].测绘学报,2012,41(5):651-660,669.LI Jiancheng.The recent Chinese terrestrial digital height datum model:gravimetric quasi-geoid CNGG2011[J].Acta Geodaetica et Cartographica Sinica,2012,41(5):651-660,669.
[4]荣敏.Stokes-Helmert方法确定大地水准面的理论与实践[D].郑州:信息工程大学,2015.RONG Min.Stokes-Helmert method for geoid determination[D].Zhengzhou:Information Engineering University,2015.
[5]马健.Hotine-Helmert边值问题确定似大地水准面的理论与方法[D].郑州:信息工程大学,2018.MA Jian.Theory and methods of the Hotine-Helmert boundary value problem for the determination of the quasi-geoid[D].Zhengzhou:Information Engineering University,2018.
[6]邓标,洪绍明,宋雷.区域厘米级似大地水准面的精化[J].大地测量与地球动力学,2009,29(3):125-127.DENG Biao,HONG Shaoming,SONG Lei.Refining of centimeter-precise local quasi-geoid[J].Journal of Geodesy and Geodynamics,2009,29(3):125-127.
[7]荣敏,周巍,任红飞.Meissel-Stokes核函数应用于区域大地水准面分析[J].测绘工程,2015,24(9):5-10.RONG Min,ZHOU Wei,REN Hongfei.Meissel-Stokes function used in the regional geoid determination[J].Engineering of Surveying and Mapping,2015,24(9):5-10.
[8]SJBERG L E,HUNEGNAW A.Some modifications of Stokes’formula that account for truncation and potential coefficient errors[J].Journal of Geodesy,2000,74(2):232-238.
[9]WONG L,GORE R.Accuracy of geoid heights from modified Stokes kernels[J].Geophysical Journal International,1969,18(1):81-91.
[10]FEATHERSTONE W E,EVANS J D,OLLIVER J G.AMeissl-modified Vanícˇek and Kleusberg kernel to reduce the truncation error in gravimetric geoid computations[J].Journal of Geodesy,1998,72(3):154-160.
[11]FEATHERSTONE W E.Band-limited kernel modifications for regional geoid determination based on dedicated satellite gravity field missions[M]∥TZIAVOS I N.Gravity and Geoid.Thessaloniki:[s.n.],2003:341-346.
[12]王伟,李姗姗,马彪,等.基于均衡理论构建区域平均空间重力异常方法研究[J].大地测量与地球动力学,2013,33(4):146-150.WANG Wei,LI Shanshan,MA Biao,et al.A method for establishing mean free-air gravity anomaly based on isostatic theory[J].Journal of Geodesy and Geodynamics,2013,33(4):146-150.
[13]马健,魏子卿,武丽丽,等.有限范围的重力层间改正算法[J].测绘学报,2017,46(1):26-33.DOI:10.11947/j.AGCS.2017.20160173.MA Jian,WEI Ziqing,WU Lili,et al.The Bouguer correction algorithm for gravity with limited range[J].Acta Geodaetica et Cartographica Sinica,2017,46(1):26-33.DOI:10.11947/j.AGCS.2017.20160173.
[14]孙文,吴晓平,王庆宾,等.高精度重力数据格网化方法比较[J].大地测量与地球动力学,2015,35(2):342-345.SUN Wen,WU Xiaoping,WANG Qingbin,et al.Comparison and analysis of high-precision gravity data gridding methods[J].Journal of Geodesy and Geodynamics,2015,35(2):342-345.
[15]李姗姗,曲政豪.重力数据误差对大地水准面模型建立的影响[J].大地测量与地球动力学,2016,36(10):847-850,869.LI Shanshan,QU Zhenghao.Effect of error of gravity data on geoid determination[J].Journal of Geodesy and Geodynamics,2016,36(10):847-850,869.
[16]VANCˇEK P,KLEUSBERG A.The Canadian geoid-Stokesian approach[J].Manuscripta Geodaetica,1987,12(2):86-98.
[17]FEATHERSTONE W E,KIRBY J F,KEARSLEY A H W,et al.The AUSGeoid98geoid model of Australia:data treatment,computations and comparisons with GPS-levelling data[J].Journal of Geodesy,2001,75(5-6):313-330.
[18]VANCˇEK P,FEATHERSTONE W E.Performance of three types of Stokes’s kernel in the combined solution for the geoid[J].Journal of Geodesy,1998,72(12):684-697.
[19]李建成,晁定波.利用Poisson积分推导Hotine函数及Hotine公式应用问题[J].武汉大学学报(信息科学版),2003,28(S1):55-57.LI Jiancheng,CHAO Dingbo.Derivation of Hotine function using Poisson integral and application of Hotine formula[J].Geomatics and Information Science of Wuhan University,2003,28(S1):55-57.
[20]FEATHERSTONE W E.Deterministic,Stochastic,hybrid and band-limited modifications of Hotine’s integral[J].Journal of Geodesy,2013,87(5):487-500.
[21]程芦颖.不同扰动位泛函间的积分变换广义核函数[J].测绘学报,2013,42(2):203-210.CHENG Luying.General kernel functions based on integral transformation among different disturbing potential elements[J].Acta Geodaetica et Cartographica Sinica,2013,42(2):203-210.
[22]HUANG J L,VRONNEAU M.Canadian gravimetric geoid model 2010[J].Journal of Geodesy,2013,87(8):771-790.
[23]刘敏,黄谟涛,欧阳永忠,等.顾及地形效应的重力向下延拓模型分析与检验[J].测绘学报,2016,45(5):521-530.DOI:10.11947/j.AGCS.2016.20150453.LIU Min,HUANG Motao,OUYANG Yongzhong,et al.Test and analysis of downward continuation models for airborne gravity data with regard to the effect of topographic height[J].Acta Geodaetica et Cartographica Sinica,2016,45(5):521-530.DOI:10.11947/j.AGCS.2016.20150453.
[24]翟振和,王兴涛,李迎春.解析延拓高阶解的推导方法与比较分析[J].武汉大学学报(信息科学版),2015,40(1):134-138.ZHAI Zhenhe,WANG Xingtao,LI Yingchun.Solution and comparison of high order term of analytical continuation[J].Geomatics and Information Science of Wuhan University,2015,40(1):134-138.
[25]WEI Ziqing.High-order radial derivatives of harmonic function and gravity anomaly[J].Journal of Physical Science and Application,2014,4(7):454-467.
[26]马健,魏子卿,任红飞,等.顾及远区影响的向下延拓实用算法[J].地球物理学进展,2018,33(2):498-502.MA Jian,WEI Ziqing,REN Hongfei,et al.Practical algorithm of the downward continuation considering the far-zone effect[J].Progress in Geophysics,2018,33(2):498-502.
[27]马健,魏子卿,任红飞.确定似大地水准面的HotineHelmert边值解算模型[J].测绘学报,2019,48(2):153-160.DOI:10.11947/j.AGCS.2019.20170594.MA Jian,WEI Ziqing,REN Hongfei.Hotine-Helmert boundary-value calculation model for quasi-geoid determination[J].Acta Geodaetica et Cartographica Sinica,2019,48(2):153-160.DOI:10.11947/j.AGCS.2019.20170594.
[28]马健,魏子卿.近区地形直接与间接影响的棱柱模型算法[J].测绘学报,2018,47(11):1429-1436.DOI:10.11947/j.AGCS.2018.20170369.MA Jian,WEI Ziqing.Prism algorithms for the near-zone direct and indirect topographic effects[J].Acta Geodaetica et Cartographica Sinica,2018,47(11):1429-1436.DOI:10.11947/j.AGCS.2018.20170369.
[29]李建成.我国现代高程测定关键技术若干问题的研究及进展[J].武汉大学学报(信息科学版),2007,32(11):980-987.LI Jiancheng.Study and progress in theories and crucial techniques of modern height measurement in China[J].Geomatics and Information Science of Wuhan University,2007,32(11):980-987.
[2]MARTINEC Z.Boundary-value problems for gravimetric determination of a precise geoid[M].Berlin:Springer,1998.
[3]李建成.最新中国陆地数字高程基准模型:重力似大地水准面CNGG2011[J].测绘学报,2012,41(5):651-660,669.LI Jiancheng.The recent Chinese terrestrial digital height datum model:gravimetric quasi-geoid CNGG2011[J].Acta Geodaetica et Cartographica Sinica,2012,41(5):651-660,669.
[4]荣敏.Stokes-Helmert方法确定大地水准面的理论与实践[D].郑州:信息工程大学,2015.RONG Min.Stokes-Helmert method for geoid determination[D].Zhengzhou:Information Engineering University,2015.
[5]马健.Hotine-Helmert边值问题确定似大地水准面的理论与方法[D].郑州:信息工程大学,2018.MA Jian.Theory and methods of the Hotine-Helmert boundary value problem for the determination of the quasi-geoid[D].Zhengzhou:Information Engineering University,2018.
[6]邓标,洪绍明,宋雷.区域厘米级似大地水准面的精化[J].大地测量与地球动力学,2009,29(3):125-127.DENG Biao,HONG Shaoming,SONG Lei.Refining of centimeter-precise local quasi-geoid[J].Journal of Geodesy and Geodynamics,2009,29(3):125-127.
[7]荣敏,周巍,任红飞.Meissel-Stokes核函数应用于区域大地水准面分析[J].测绘工程,2015,24(9):5-10.RONG Min,ZHOU Wei,REN Hongfei.Meissel-Stokes function used in the regional geoid determination[J].Engineering of Surveying and Mapping,2015,24(9):5-10.
[8]SJBERG L E,HUNEGNAW A.Some modifications of Stokes’formula that account for truncation and potential coefficient errors[J].Journal of Geodesy,2000,74(2):232-238.
[9]WONG L,GORE R.Accuracy of geoid heights from modified Stokes kernels[J].Geophysical Journal International,1969,18(1):81-91.
[10]FEATHERSTONE W E,EVANS J D,OLLIVER J G.AMeissl-modified Vanícˇek and Kleusberg kernel to reduce the truncation error in gravimetric geoid computations[J].Journal of Geodesy,1998,72(3):154-160.
[11]FEATHERSTONE W E.Band-limited kernel modifications for regional geoid determination based on dedicated satellite gravity field missions[M]∥TZIAVOS I N.Gravity and Geoid.Thessaloniki:[s.n.],2003:341-346.
[12]王伟,李姗姗,马彪,等.基于均衡理论构建区域平均空间重力异常方法研究[J].大地测量与地球动力学,2013,33(4):146-150.WANG Wei,LI Shanshan,MA Biao,et al.A method for establishing mean free-air gravity anomaly based on isostatic theory[J].Journal of Geodesy and Geodynamics,2013,33(4):146-150.
[13]马健,魏子卿,武丽丽,等.有限范围的重力层间改正算法[J].测绘学报,2017,46(1):26-33.DOI:10.11947/j.AGCS.2017.20160173.MA Jian,WEI Ziqing,WU Lili,et al.The Bouguer correction algorithm for gravity with limited range[J].Acta Geodaetica et Cartographica Sinica,2017,46(1):26-33.DOI:10.11947/j.AGCS.2017.20160173.
[14]孙文,吴晓平,王庆宾,等.高精度重力数据格网化方法比较[J].大地测量与地球动力学,2015,35(2):342-345.SUN Wen,WU Xiaoping,WANG Qingbin,et al.Comparison and analysis of high-precision gravity data gridding methods[J].Journal of Geodesy and Geodynamics,2015,35(2):342-345.
[15]李姗姗,曲政豪.重力数据误差对大地水准面模型建立的影响[J].大地测量与地球动力学,2016,36(10):847-850,869.LI Shanshan,QU Zhenghao.Effect of error of gravity data on geoid determination[J].Journal of Geodesy and Geodynamics,2016,36(10):847-850,869.
[16]VANCˇEK P,KLEUSBERG A.The Canadian geoid-Stokesian approach[J].Manuscripta Geodaetica,1987,12(2):86-98.
[17]FEATHERSTONE W E,KIRBY J F,KEARSLEY A H W,et al.The AUSGeoid98geoid model of Australia:data treatment,computations and comparisons with GPS-levelling data[J].Journal of Geodesy,2001,75(5-6):313-330.
[18]VANCˇEK P,FEATHERSTONE W E.Performance of three types of Stokes’s kernel in the combined solution for the geoid[J].Journal of Geodesy,1998,72(12):684-697.
[19]李建成,晁定波.利用Poisson积分推导Hotine函数及Hotine公式应用问题[J].武汉大学学报(信息科学版),2003,28(S1):55-57.LI Jiancheng,CHAO Dingbo.Derivation of Hotine function using Poisson integral and application of Hotine formula[J].Geomatics and Information Science of Wuhan University,2003,28(S1):55-57.
[20]FEATHERSTONE W E.Deterministic,Stochastic,hybrid and band-limited modifications of Hotine’s integral[J].Journal of Geodesy,2013,87(5):487-500.
[21]程芦颖.不同扰动位泛函间的积分变换广义核函数[J].测绘学报,2013,42(2):203-210.CHENG Luying.General kernel functions based on integral transformation among different disturbing potential elements[J].Acta Geodaetica et Cartographica Sinica,2013,42(2):203-210.
[22]HUANG J L,VRONNEAU M.Canadian gravimetric geoid model 2010[J].Journal of Geodesy,2013,87(8):771-790.
[23]刘敏,黄谟涛,欧阳永忠,等.顾及地形效应的重力向下延拓模型分析与检验[J].测绘学报,2016,45(5):521-530.DOI:10.11947/j.AGCS.2016.20150453.LIU Min,HUANG Motao,OUYANG Yongzhong,et al.Test and analysis of downward continuation models for airborne gravity data with regard to the effect of topographic height[J].Acta Geodaetica et Cartographica Sinica,2016,45(5):521-530.DOI:10.11947/j.AGCS.2016.20150453.
[24]翟振和,王兴涛,李迎春.解析延拓高阶解的推导方法与比较分析[J].武汉大学学报(信息科学版),2015,40(1):134-138.ZHAI Zhenhe,WANG Xingtao,LI Yingchun.Solution and comparison of high order term of analytical continuation[J].Geomatics and Information Science of Wuhan University,2015,40(1):134-138.
[25]WEI Ziqing.High-order radial derivatives of harmonic function and gravity anomaly[J].Journal of Physical Science and Application,2014,4(7):454-467.
[26]马健,魏子卿,任红飞,等.顾及远区影响的向下延拓实用算法[J].地球物理学进展,2018,33(2):498-502.MA Jian,WEI Ziqing,REN Hongfei,et al.Practical algorithm of the downward continuation considering the far-zone effect[J].Progress in Geophysics,2018,33(2):498-502.
[27]马健,魏子卿,任红飞.确定似大地水准面的HotineHelmert边值解算模型[J].测绘学报,2019,48(2):153-160.DOI:10.11947/j.AGCS.2019.20170594.MA Jian,WEI Ziqing,REN Hongfei.Hotine-Helmert boundary-value calculation model for quasi-geoid determination[J].Acta Geodaetica et Cartographica Sinica,2019,48(2):153-160.DOI:10.11947/j.AGCS.2019.20170594.
[28]马健,魏子卿.近区地形直接与间接影响的棱柱模型算法[J].测绘学报,2018,47(11):1429-1436.DOI:10.11947/j.AGCS.2018.20170369.MA Jian,WEI Ziqing.Prism algorithms for the near-zone direct and indirect topographic effects[J].Acta Geodaetica et Cartographica Sinica,2018,47(11):1429-1436.DOI:10.11947/j.AGCS.2018.20170369.
[29]李建成.我国现代高程测定关键技术若干问题的研究及进展[J].武汉大学学报(信息科学版),2007,32(11):980-987.LI Jiancheng.Study and progress in theories and crucial techniques of modern height measurement in China[J].Geomatics and Information Science of Wuhan University,2007,32(11):980-987.
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