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In this paper, we propose a new method of numerical differentiation to determine the height of the top layer of the atmospheric boundary layer. In this method, a regularization technique is used to convert the problem of calculating the differential of the curve of the corners into the problem of finding the minimum value of the objective function. The two-parameter model function method is used to select the regularization parameters. Finally, the maximum gradient method is used to determine the top height of the boundary layer. Firstly, the effectiveness of the new method is validated through two numerical experiments. The experimental results show that as the noise of the occultation data increases, the fluctuation of the height of the boundary layer top obtained by the difference method and the numerical differentiation method combined with the L curve scheme increases. And the height obtained by the two-parameter model function method is very stable, which shows that the new method can filter the noise well, thereby retaining the main information about the profile. Then, based on the COSMIC angle data in January, April, July and October 2007－2011, the new method is used to analyze the seasonal characteristics of the height of the global oceanic and atmospheric boundary layer, compared with the seasonal distribution obtained by “zbalmax” with the occultation data. The results show that the seasonal distribution characteristics of the two data are very consistent: the height of the boundary layer is higher in the area where the sea surface temperature is higher than that in the surrounding sea area; on the contrary, the height of the boundary layer top is lower. In the sea area where the warm current passes, the height of the boundary layer is higher; in the sea area where the cold current passes, the height of the boundary layer is lower.
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Keywords:
- boundary layer height /
- regularization method /
- two-parameter model function method /
- bend angle gradient
[1] Stull R B 1999 An Introduction to Boundary Layer Meteorology (Vol. 13) (Dordrecht: Kluwer Academic Publishers) pp3, 4
[2] Seibert P, Beyrich F, Gryning S E, Joffre S, Rasmussen A, Tercier P 2000 Atmos. Environ. 34 1001
Google Scholar
[3] Zeng X, Brunke M A, Zhou M, Fairall C, Bond N A, Lenschow D H 2004 J. Clim. 17 4159
Google Scholar
[4] Chan K M, Wood R 2013 J. Geophys. Res. Atmos. 118 12
Google Scholar
[5] Ho S P, Peng L, Anthes R A, Kuo Y H, Lin H C 2015 J. Clim. 28 2856
Google Scholar
[6] Holzworth G C 1964 Mon. Weather Rev. 92 235
Google Scholar
[7] Coulter R L 1979 J. Appl. Meteorol. Climatol. 18 1495
Google Scholar
[8] Van Pul W A J, Holtslag A A M, Swart D P J 1994 Bound.-Layer Meteorol. 68 173
Google Scholar
[9] Bianco L, Wilczak J M 2002 J. Atmos. Oceanic Technol. 19 1745
Google Scholar
[10] Lokoshchenko M A 2002 J. Atmos. Oceanic Technol. 19 1151
Google Scholar
[11] Balsley B B, Frehlich R G, Jensen M L, Meillier Y 2006 J. Atmos. Sci. 63 1291
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[12] Sokolovskiy S, Kuo Y H, Rocken C, Schreiner W S, Hunt D, Anthes R A 2006 Geophys. Res. Lett. 33 12
Google Scholar
[13] Sokolovskiy S V, Rocken C, Lenschow D H, Kuo Y H, Anthes R A, Schreiner W S, Hunt D C 2007 Geophys. Res. Lett. 34 18
Google Scholar
[14] Baars H, Ansmann A, Engelmann R, Althausen D 2008 Atmos. Chem. Phys. 8 7281
Google Scholar
[15] Seidel D J, Ao C O, Li K 2010 J. Geophys. Res. Atmos. 115 16
Google Scholar
[16] Dai C, Wang Q, Kalogiros J A, Lenschow D H, Gao Z, Zhou M 2014 Bound.-Layer Meteorol. 152 277
Google Scholar
[17] Yan S, Xiang J, Du H 2019 Adv. Atmos. Sci. 36 303
Google Scholar
[18] Tikhonov A N 1963 Dokl. Akad. Nauk SSSR 151
[19] Willoughby R A 1979 SIAM Rev. Soc. Ind. Appl. Math. 21 266
Google Scholar
[20] Tautenhahn U 2002 Inverse Prob. 18 191
Google Scholar
[21] 张路寅 2011 硕士学位论文 (济南: 山东大学)
Zhang L Y 2011 M.S. Thesis (Jinan: Shandong University) (in Chinese)
[22] Hanke M, Neubauer A, Scherzer O 1995 Numer. Math. 72 21
Google Scholar
[23] Shirangi M G, Emerick A A 2016 J. Pet. Sci. Eng. 143 258
Google Scholar
[24] 姜祝辉, 黄思训, 何然, 周晨腾 2011 物理学报 60 068401
Google Scholar
Jiang Z H, Huang S X, He R, Zhou C T 2011 Acta Phys. Sin. 60 068401
Google Scholar
[25] 赵延来, 黄思训, 杜华栋, 仲跻芹 2011 物理学报 60 079202
Google Scholar
Zhao Y L, Huang S X, Du H D, Zhong Q Q 2011 Acta Phys. Sin. 60 079202
Google Scholar
[26] Zhang L, Huang S X, Shen C, Shi W L 2011 Chin. Phys. B 20 514
Google Scholar
[27] Zhong J, Huang S X, Fei J F, Du H D, Zhang L 2011 Chin. Phys. B 20 064301
Google Scholar
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Google Scholar
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Google Scholar
[29] Hansen P C 1992 SIAM Rev. Soc. Ind. Appl. Math. 34 561
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Google Scholar
[31] 黄威, 刘磊, 高太长, 李书磊, 胡帅 2016 光谱学与光谱分析 36 3620
Google Scholar
Huang w, Liu L, Gao T C, Li S L, Hu S 2016 Spectrosc. Spect. Anal. 36 3620
Google Scholar
[32] Golub G H, Heath M, Wahba G 1979 Technometrics 21 215
Google Scholar
[33] Wen Y-W, Chan R H 2018 Inverse Prob. Imag. 12 1103
Google Scholar
[34] Scherzer O 1993 Computing 51 45
Google Scholar
[35] Morozov V A 1984 Methods for Solving Incorrectly Posed Problems (New York: Springer-Verlag) pp65−70
[36] Kunisch K, Zou J 1998 Inverse Prob. 14 1247
Google Scholar
[37] Lu S, Pereverzev S V 2011 Numer. Math. 118 1
Google Scholar
[38] 王泽文 2010 博士学位论文 (南京: 东南大学)
Wang Z W 2010 Ph. D. Dissertation (Nanjing: Southeast University) (in Chinese)
[39] 刘继军 2005 不适定问题的正则化方法及应用 (北京: 科学出版社) 第83−89页
Liu J J 2005 Regularization Method and Application of Ill-posed Problems (Beijing: Science Press) pp83−89 (in Chinese)
[40] 汪代维, 杨修群 2002 气象学报 02 129
Google Scholar
Wang D W, Yang X Q 2002 Acta Meteorol. Sin. 02 129
Google Scholar
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图 1 2007−2011年1, 4, 7, 10月份掩星廓线数量的时间分布
Figure 1. Temporal distribution of the number of occultation profiles in January, April, July and October, 2007−2011.
图 2 双参数模型函数法流程图
Figure 2. Flow chart of two-parameter model function method.
图 3 由差分法和模型函数法得到的弯角梯度廓线(BA表示弯角) (a)弯角廓线添加随机误差
$\delta = 0.0025$ , 边界层顶高度${H_{\rm{M}}} = 1.2\;{\rm{km}}, {H_{\rm{d}}} = 1.4\;{\rm{km}}$ ; (b)弯角廓线添加随机误差$\delta = 0.005$ , 边界层顶高度${H_{\rm{M}}} = 1.1\;{\rm{km}}, {H_{\rm{d}}} = 1.4\;{\rm{km}}$ ; (c)弯角廓线添加随机误差$\delta = 0.0075$ , 边界层顶高度${H_{\rm{M}}} = 1.2\;{\rm{km}}, {H_{\rm{d}}} = 1.8\;{\rm{km}}$ ; (d)弯角廓线添加随机误差$\delta = 0.01$ , 边界层顶高度HM = 1.2 km, Hd = 3.7 kmFigure 3. Angle gradient profile obtained by the difference method and the model function method using the bending angle gradient profile (BA represents the bending angle): (a) Bending angle profile with uniform random error
$\delta = 0.0025$ , boundary layer top height${H_{\rm{M}}} = 1.2\;{\rm{km}}, {H_{\rm{d}}} = 1.4\;{\rm{km}}$ ; (b) bending angle profile with uniform random error$\delta = 0.005$ , boundary layer top height${H_{\rm{M}}} = 1.1\;{\rm{km}}, {H_{\rm{d}}} = 1.4\;{\rm{km}}$ ; (c) bending angle profile with uniform random error$\delta = 0.0075$ , boundary layer top height${H_{\rm{M}}} = 1.2\;{\rm{km}}, {H_{\rm{d}}} = 1.8\;{\rm{km}}$ ; (d) bending angle profile with uniform random error$\delta = 0.01$ , boundary layer top height${H_{\rm{M}}} = 1.2\;{\rm{km}}, {H_{\rm{d}}} = 3.7\;{\rm{km}}$ .
图 4 由差分法、L曲线法和模型函数法得到的边界层顶高度 (a) 三种方法基于廓线1得到的边界层顶高度, Htrue = 3.15 km, std(HM) = 0.013, std(HL) = 0.44, std(HM) = 0.61; (b) 三种方法基于廓线2得到的边界层顶高度, Htrue = 4.55 km, std(HM) = 0.020, std(HL) = 0.89, std(HM) = 1.19
Figure 4. Height of the boundary layer obtained by the three methods: (a) Three methods to get the height of the boundary layer top based on profile 1, Htrue = 3.15 km, std(HM) = 0.013, std(HL) = 0.44, std(HM) = 0.61; (b) three methods to get the height of the boundary layer top based on profile 2, Htrue = 4.55 km, std(HM) = 0.020, std(HL) = 0.89, std(HM) = 1.19
图 5 用模型函数法得到的海洋5年平均的边界层顶高度(所用资料为2007—2011年1, 4, 7, 10四个月份的掩星弯角的资料) (a) 1月份平均边界层顶高度; (b) 4月份平均边界层顶高度; (c) 7月份平均边界层顶高度; (d) 10月份平均边界层顶高度
Figure 5. The 5-year average boundary layer height of the ocean obtained by the model function method, the data used is the bending angle profile of the four months of 2007−2011 in January, April, July, and October: (a) The average height of the boundary layer in January; (b) the average height of the boundary layer in April; (c) the average height of the boundary layer in July; (d) the average height of the boundary layer in October.
图 6 海洋5年平均的边界层顶高度(所用资料为2007—2011年1, 4, 7, 10四个月份的掩星自带zbalmax的资料) (a) 1月份平均边界层顶高度; (b) 4月份平均边界层顶高度; (c) 7月份平均边界层顶高度; (d) 10月份平均边界层顶高度
Figure 6. The 5-year average boundary layer height of the ocean obtained by the model function method, the data used is the zbalmax provided by CDAAC of the four months of 2007−2011 in January, April, July, and October: (a) The average height of the boundary layer in January; (b) the average height of the boundary layer in April; (c) the average height of the boundary layer in July; (d) the average height of the boundary layer in October.
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[1] Stull R B 1999 An Introduction to Boundary Layer Meteorology (Vol. 13) (Dordrecht: Kluwer Academic Publishers) pp3, 4
[2] Seibert P, Beyrich F, Gryning S E, Joffre S, Rasmussen A, Tercier P 2000 Atmos. Environ. 34 1001
Google Scholar
[3] Zeng X, Brunke M A, Zhou M, Fairall C, Bond N A, Lenschow D H 2004 J. Clim. 17 4159
Google Scholar
[4] Chan K M, Wood R 2013 J. Geophys. Res. Atmos. 118 12
Google Scholar
[5] Ho S P, Peng L, Anthes R A, Kuo Y H, Lin H C 2015 J. Clim. 28 2856
Google Scholar
[6] Holzworth G C 1964 Mon. Weather Rev. 92 235
Google Scholar
[7] Coulter R L 1979 J. Appl. Meteorol. Climatol. 18 1495
Google Scholar
[8] Van Pul W A J, Holtslag A A M, Swart D P J 1994 Bound.-Layer Meteorol. 68 173
Google Scholar
[9] Bianco L, Wilczak J M 2002 J. Atmos. Oceanic Technol. 19 1745
Google Scholar
[10] Lokoshchenko M A 2002 J. Atmos. Oceanic Technol. 19 1151
Google Scholar
[11] Balsley B B, Frehlich R G, Jensen M L, Meillier Y 2006 J. Atmos. Sci. 63 1291
Google Scholar
[12] Sokolovskiy S, Kuo Y H, Rocken C, Schreiner W S, Hunt D, Anthes R A 2006 Geophys. Res. Lett. 33 12
Google Scholar
[13] Sokolovskiy S V, Rocken C, Lenschow D H, Kuo Y H, Anthes R A, Schreiner W S, Hunt D C 2007 Geophys. Res. Lett. 34 18
Google Scholar
[14] Baars H, Ansmann A, Engelmann R, Althausen D 2008 Atmos. Chem. Phys. 8 7281
Google Scholar
[15] Seidel D J, Ao C O, Li K 2010 J. Geophys. Res. Atmos. 115 16
Google Scholar
[16] Dai C, Wang Q, Kalogiros J A, Lenschow D H, Gao Z, Zhou M 2014 Bound.-Layer Meteorol. 152 277
Google Scholar
[17] Yan S, Xiang J, Du H 2019 Adv. Atmos. Sci. 36 303
Google Scholar
[18] Tikhonov A N 1963 Dokl. Akad. Nauk SSSR 151
[19] Willoughby R A 1979 SIAM Rev. Soc. Ind. Appl. Math. 21 266
Google Scholar
[20] Tautenhahn U 2002 Inverse Prob. 18 191
Google Scholar
[21] 张路寅 2011 硕士学位论文 (济南: 山东大学)
Zhang L Y 2011 M.S. Thesis (Jinan: Shandong University) (in Chinese)
[22] Hanke M, Neubauer A, Scherzer O 1995 Numer. Math. 72 21
Google Scholar
[23] Shirangi M G, Emerick A A 2016 J. Pet. Sci. Eng. 143 258
Google Scholar
[24] 姜祝辉, 黄思训, 何然, 周晨腾 2011 物理学报 60 068401
Google Scholar
Jiang Z H, Huang S X, He R, Zhou C T 2011 Acta Phys. Sin. 60 068401
Google Scholar
[25] 赵延来, 黄思训, 杜华栋, 仲跻芹 2011 物理学报 60 079202
Google Scholar
Zhao Y L, Huang S X, Du H D, Zhong Q Q 2011 Acta Phys. Sin. 60 079202
Google Scholar
[26] Zhang L, Huang S X, Shen C, Shi W L 2011 Chin. Phys. B 20 514
Google Scholar
[27] Zhong J, Huang S X, Fei J F, Du H D, Zhang L 2011 Chin. Phys. B 20 064301
Google Scholar
[28] 谢正超, 王飞, 严建华, 岑可法 2015 物理学报 64 240201
Google Scholar
Xie Z C, Wang F, Yan J H, Ling K F 2015 Acta Phys. Sin. 64 240201
Google Scholar
[29] Hansen P C 1992 SIAM Rev. Soc. Ind. Appl. Math. 34 561
Google Scholar
[30] Hansen P C 1994 Numer. Algorithms 6 1
Google Scholar
[31] 黄威, 刘磊, 高太长, 李书磊, 胡帅 2016 光谱学与光谱分析 36 3620
Google Scholar
Huang w, Liu L, Gao T C, Li S L, Hu S 2016 Spectrosc. Spect. Anal. 36 3620
Google Scholar
[32] Golub G H, Heath M, Wahba G 1979 Technometrics 21 215
Google Scholar
[33] Wen Y-W, Chan R H 2018 Inverse Prob. Imag. 12 1103
Google Scholar
[34] Scherzer O 1993 Computing 51 45
Google Scholar
[35] Morozov V A 1984 Methods for Solving Incorrectly Posed Problems (New York: Springer-Verlag) pp65−70
[36] Kunisch K, Zou J 1998 Inverse Prob. 14 1247
Google Scholar
[37] Lu S, Pereverzev S V 2011 Numer. Math. 118 1
Google Scholar
[38] 王泽文 2010 博士学位论文 (南京: 东南大学)
Wang Z W 2010 Ph. D. Dissertation (Nanjing: Southeast University) (in Chinese)
[39] 刘继军 2005 不适定问题的正则化方法及应用 (北京: 科学出版社) 第83−89页
Liu J J 2005 Regularization Method and Application of Ill-posed Problems (Beijing: Science Press) pp83−89 (in Chinese)
[40] 汪代维, 杨修群 2002 气象学报 02 129
Google Scholar
Wang D W, Yang X Q 2002 Acta Meteorol. Sin. 02 129
Google Scholar
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