Optimal Spectral Decomposition (OSD) for Ocean Data Analysis

Transcription

1 Optimal Spectral Decomposition (OSD) for Ocean Data Analysis Peter C Chu (1) and Charles Sun (2) (1) Naval Postgraduate School, Monterey, CA pcchu@nps.edu, (2) NOAA/NODC, Silver Spring, MD 291 Charles.Sun@noaa.gov GTSPP Meeting, Honolulu, Hawaii, 27 October 28

2 How can we effectively use observational ocean data to represent and to predict the ocean state?

3 Collaborators Leonid M. Ivanov (California State Univ) Chenwu Fan (NPS) Tateana Margolina (NPS) Oleg Melnichenko (Univ of Hawaii)

4 References Chu, P.C., L.M. Ivanov, T.P. Korzhova, T.M. Margolina, and O.M. Melnichenko, 23a: Analysis of sparse and noisy ocean current data using flow decomposition. Part 1: Theory. Journal of Atmospheric and Oceanic Technology, 2 (4), Chu, P.C., L.M. Ivanov, T.P. Korzhova, T.M. Margolina, and O.M. Melnichenko, 23b: Analysis of sparse and noisy ocean current data using flow decomposition. Part 2: Application to Eulerian and Lagrangian data. Journal of Atmospheric and Oceanic Technology, 2 (4), Chu, P.C., L.M. Ivanov, and T.M. Margolina, 24: Rotation method for reconstructing process and field from imperfect data. International Journal of Bifurcation and Chaos, 14 (4),

5 References Chu, P.C., L.M. Ivanov, and T.M. Margolina, 25: Seasonal variability of the Black Sea Chlorophyll-a concentration. Journal of Marine Systems, 56, Chu, P.C., L.M. Ivanov, and O.M. Melnichenko, 25: Fall-winter current reversals on the Texas-Lousiana continental shelf. Journal of Physical Oceanography, 35, Chu, P.C., L.M. Ivanov, O.V. Melnichenko, and N.C. Wells, 27: On long baroclinic Rossby waves in the tropical North Atlantic observed from profiling floats. Journal of Geophysical Research, 112, C532, doi:1.129/26jc3698. Chu, P. C., L. M. Ivanov, O. V. Melnichenko, and R.-F. Li, 28: Argo floats revealing bimodality of large-scale mid-depth circulation in the North Atlantic. Acta Oceanologica Sinica, 27 (2), 1-1. Chu, P.C., C. Sun, and C. Fan, 29: Variability in meridional overturning circulation and thermohaline structure detected from GTSPP/Argo/MOODS/OSCAR Data. Proceedings on 21th Symposium on Climate Variability, American Meteorological Society, Phoenix, January 11-15, 29.

6 Observational Data

7 A Popular Method for Ocean Data Analysis: Optimum Interpolation (OI)

8 OI Equation Grid point k, Observational Point j f Q k o Q j f Q j First guess field (gridded) Observation First guess interpolated on the observational point N a f o f k = k + αkj j j j= 1 Q Q ( Q Q ) a Q k Analyzed field at the grid point

9 OI Weight Coefficients α kj N j= 1 o ( η + δλ) α = η ij ij i kj kj η ij η kj Autocorrelation functions λ o i Signal-to-noise ratio

10 Three Requirements for the OI Method (1) First guess field (2) Autocorrelation functions (3) High signal-to-noise ratio

11 What happens if the three conditions are not satisfied?

12 Spectral Representation - a Possible Alternative Method Ψm Basis functions c any ocean variable

13 Flow Decomposition

14 Basis Functions (Closed Basin)

15 Basis Functions (Open Boundaries)

16 Boundary Conditions

17 Spectral Decomposition M = + m Φm m= 1 T( x, t) T ( x) c ( t) ( x, t) M = + m Φm m= 1 S( x, t) S ( x) d ( t) ( x, t)

18 Benefits of Using OSD (1) Don t need first guess field (2) Don t need autocorrelation functions (3) Don t require high signal-to-noise ratio (4) Basis functions are pre-determined before the data analysis.

19 Optimal Mode Truncation

20 Vapnik (1983) Cost Function

21 Optimal Truncation Gulf of Mexico, Monterey Bay, Louisiana- Texas Shelf, North Atlantic Kopt = 4, Mopt = 3

22 Determination of Spectral Coefficients (Ill- Posed Algebraic Equation) This is caused by the features of the matrix A.

23 Rotation Method (Chu et al., 24)

24 Example-1 Temporal and spatial variability of Pacific Ocean

25 T (1 m)

26 T (1 m)

27 T (5 m)

28 Seasonal Anomaly versus WOA 94 (1 m) Monthly mean ( ) minus WOA 94 Monthly Mean

29 Seasonal Anomaly versus WOA 94 (1 m) Monthly mean ( ) minus WOA 94 Monthly Mean

30 Seasonal Anomaly versus WOA 94 (25 m) Monthly mean ( ) minus WOA 94 Monthly Mean

31 Seasonal Anomaly versus WOA 94 (5 m) Monthly mean ( ) minus WOA 94 Monthly Mean

32 T: NINO-3 (5 o S-5 o N, 15 o W-9 o W)

33 Example-2 OSD for Analyzing ARGO Data Baroclinic Rossby Waves in the tropical North Atlantic

34 Tropical North Atlantic (4 o -24 o N) Important Transition Zone Meridional Overturning Circulation (MOC) (Rahmstorf 26)

35 MOC Variation Heat Transport Variation Climate Change

36 Are mid-depth (~1 m) ocean circulations steady? If not, what mechanisms cause the change? (Rossby wave propagation)

37

38 ARGO Observations (Oct-Nov 24) (a) Subsurface tracks (b) Float positions where (T,S) were measured 6 N 6 N 5 N 5 N 4 N 4 N 3 N 3 N 2 N 2 N 1 N 1 N N 1 S (a) 7 W 5 W 3 W 1 W N 1 S (b) 7 W 5 W 3 W 1 W

39 6 N Circulations at 1 m estimated from the original ARGO float tracks (bin method) April 24 April N 5 N N 4 N 32 4 N 28 3 N 24 3 N 2 2 N 16 2 N 12 1 N N 7 W 6 W 5 W 4 W 3 W 2 W 1 W N N 5 cm/s 7 W 5 W 3 W 1 W It is difficult to get physical insights and to use such noisy data into ocean numerical models.

40 Boundary Configuration Basis Functions for OSD 6 N 5 N 4 N 3 N Γ 2 Azores Islands 2 N Γ 1 1 N N 1 S Γ 3 Γ 3 / 7 W 5 W 3 W 1 W

41 Basis Functions for Streamfunction Mode-1 and Mode-2 6 N 6 N 5 N 5 N 4 N 3 N N 3 N N N 1 N 1 N N 1 2 N S 7 W 5 W 3 W 1 W 1 S 7 W 5 W 3 W 1 W

42 Circulations at 1 m (March 4 to May 5) Bin Method OSD 6 N 6 N 5 N 5 N 4 N 4 N 3 N 3 N 2 N 2 N 1 N 5 cm/s 1 N 5 cm/s N 7 W 5 W 3 W 1 W N 7 W 5 W 3 W 1 W

43 Mid-Depth Circulations (1 m) Mar-May 4 May Jul 4 Jul-Sep 4 Sep Nov 4 6 N 6 N 6 N 6 N 5 N 5 N 5 N 5 N 4 N 4 N 4 N 4 N 3 N 3 N 3 N 3 N 2 N 2 N 2 N 2 N 1 N 1 N 1 N 5 cm/s N 5 W 3 W 1 W 7 W Nov 4 Jan 5 5 cm/s N N 7 W 1 N 5 cm/s 5 cm/s 5 W 3 W N 7 W 1 W 5 W 3 W 1 W Mar May 5 Jan-Mar 5 6 N 6 N 6 N 5 N 5 N 5 N 4 N 4 N 4 N 3 N 3 N 3 N 2 N 2 N 2 N 1 N 1 N 1 N 5 cm/s 5 cm/s 5 cm/s N N 7 W 7 W 5 W 3 W 1 W 5 W 3 W 1 W N 7 W 5 W 3 W 1 W 7 W 5 W 3 W 1 W

44 Temperature at 95 m (March 4 to May 5) NOAA/WOA OSD 6 N 6 N 5 N N N 3 N N 3 N N N N 5 1 N 5 N 7 W 6 W 5 W 4 W 3 W 2 W 1 W N 7 W 6 W 5 W 4 W 3 W 2 W 1 W

45 8 4 Mid-Depth Temperature (95 m) May 4 Jul 4 Sep 4 Nov 4 6 N 6 N 6 N 6 N 5 N 4 N 3 N 2 N 1 N N W 6 W 5 W 4 W 3 W 2 W 1 W N 4 N 3 N 2 N 1 N N W 6 W 5 W 4 W 3 W 2 W 1 W N 4 N 3 N 2 N 1 N N W 6 W 5 W 4 W 3 W 2 W 1 W N 4 N 3 N 2 N 1 N N W 6 W 5 W 4 W 3 W 2 W 1 W Jan 5 Mar 5 May 5 6 N 5 N 5 6 N 5 N N 5 N N 3 N N 3 N N 3 N N N N N 5 1 N 5 1 N 5 N 7 W 6 W 5 W 4 W 3 W 2 W 1 W N 7 W 6 W 5 W 4 W 3 W 2 W 1 W N 7 W 6 W 5 W 4 W 3 W 2 W 1 W

46 Baroclinic Rossby Waves in Tropical North Atlantic

47 Fourier Expansion Temporal Annual and Semi-anuual

48 Fourier Expansion Temporal Annual and Semi-anuual

49 Optimization

50 Annual Component

51 Semi-annual Component

52 Time Longitude Diagrams of Meridional Velocity Along 11 o N Apr. Feb. Dec. Oct. Aug June 1 (a) (b) 5 W 4 W 3 W 2 W 5 W 4 W 3 W 2 W Annual Semi-Annual

53 Time Longitude Diagrams of temperature Along 11 o N Apr. Feb. Dec. Oct. Aug. June Apr. Feb. Annual Semi-Annual (c) (d) Dec. Oct. Aug. June (a) (b) 5 W 4 W 3 W 2 W 5 W 4 W 3 W 2 W 5 W 4 W 3 W 2 W 5 W 4 W 3 W 2 W Annual Semi-Annual 55 m 95 m

54 Annual Currents (1 m) May-Jun 24 Jul-Aug 24 3 N 5 cm/s 3 N 5 cm/s 2 N 2 N λ 1 λ 2 B 1 // A 1 1 N B 1 A 1 1 N B 1 / N 7 W 6 W 5 W 4 W 3 W 2 W 1 W (a) N 7 W 6 W 5 W 4 W 3 W 2 W 1 W (b) 3 N Sep-Oct 24 Nov-Dec 24 3 N 5 cm/s 5 cm/s 2 N B 1 // 2 N A 1 1 N 1 N B 2 A 2 A 2 N 7 W 6 W 5 W 4 W 3 W 2 W 1 W (c) N 7 W 6 W 5 W 4 W 3 W 2 W 1 W (d)

55 Characteristics of Annual Rossby Waves Western Basin Eastern Basin Western Basin Eastern Basin

56 (b) (a) 7 W 6 W 5 W 4 W 3 W 2 W 1 W N 7 W 6 W 5 W 4 W 3 W 2 W 1 W N 2 N 1 N N 3 N 2 N 1 N N 2 N 1 N N Annual Monthly Temperature Anomaly ( o C) at 95 m Depth Annual Rossby Waves (7-1 cm/s) Jun 4 Aug 4 Oct 4 Dec 4 3 N N 2 1 N (c) N 7 W 6 W 5 W 4 W 3 W 2 W 1 W 7 W 6 W 5 W 4 W 3 W 2 W 1 W (d) Temperature anomaly ( C)

57 Annual Monthly Temperature Anomaly ( o C) at 25 m Depth Equatorially Forced Coastal Kelvin waves (27-3 cm/s) Jun 4 Aug 4 3 N 3 N 2 N N N N 3 N (b) N (a) 7 W 6 W 5 W 4 W 3 W 2 W 1 W 7 W 6 W 5 W 4 W 3 W 2 W 1 W Oct 4 Dec 4 1 N 3 N N N N N N 7 W 6 W 5 W 4 W 3 W 2 W 1 W (c) N 7 W 6 W 5 W 4 W 3 W 2 W 1 W (d) Temperature anomaly ( C)

58 Zonal cross-sections of the annual component of the temperature anomaly ( o C) Depth (m) o N in Jun (a) 45 W 4 W 35 W 3 W 25 W 2 W 15 W Depth (m) o N in Oct, W A 1 A 1 (b) 45 W 4 W 35 W 3 W 25 W 2 W Depth (m).4 16 o N in Oct 4 7 A (c) 5 W 45 W 4 W 35 W 3 W 25 W 2 W

59 Baroclinic Modes Depth (m) Depth (m) (a) Φ l Φ l combination (b)

60 Annual Component in the Western Sub-Basin 3 Mean wind KE <E i > (m 2 /s 2 ) 2 1 (a) IV VI VIII X XII II IV Zonal: circle Mean KE for mid-depth currents <e i > (cm 2 /s 2 ) 1.5 (b) Meridional : Correlation between Winds and currents Correlation between wind Stress curl and streamfunction (solid: no-lag, dashed: 3 mon lag R ii R IV VI VIII X XII II IV (c) IV VI VIII X XII II IV (d) IV VI VIII X XII II IV Month square R (δt) 11

61 Annual Component in the Eastern Mean wind KE Mean KE for mid-depth currents Correlation between Winds and currents Correlation between wind Stress curl and streamfunction (solid: no-lag, dashed: 3 mon lag Sub-Basin <E i > (m 2 /s 2 ) <e i > (cm 2 /s 2 ) R ii R (a) IV VI VIII X XII II IV (b) IV VI VIII X XII II IV (c) IV VI VIII X XII II IV (d) IV VI VIII X XII II IV Month.5.5 R 11 (δt) Zonal: circle Meridional : square

62 Semi-annual currents at 1 m depth (24) (a)5/15 (b)5/3 (c)6/14 (d)6/29 (e) 7/13 3 N 2 N 1 N N 3 N 2 N C 2 5 cm/s (a) 7 W 6 W 5 W 4 W 3 W 2 W 1 W C 2 C 1 D 2 D 1 5 cm/s D 3 D 4 3 N 2 N 1 N N 3 N 2 N C 2 C 1 D2 5 cm/s D 3 D 4 7 W 6 W 5 W 4 W 3 W 2 W 1 W D 1 5 cm/s C 2 D 4 C 1 (d) 1 N C 1 1 N D 2 D 1 N D 2 D 1 7 W 6 W 5 W 4 W 3 W 2 W 1 W (b) N 7 W 6 W 5 W 4 W 3 W 2 W 1 W (e) 3 N 5 cm/s C 2 D 3 D 4 2 N 1 N C 1 D 2 D 1 N 7 W 6 W 5 W 4 W 3 W 2 W 1 W (c)

63 Semi-annual monthly temperature anomaly at 95m depth 3 N 3 N (a)jun 4 (b) Aug 4 (c) Oct 4 (d) Dec 4. 2 N 1 N N 3 N (a) 7 W 6 W 5 W 4 W 3 W 2 W 1 W 2 N 1 N N 3 N (b) 7 W 6 W 5 W 4 W 3 W 2 W 1 W 2 N 2 N 1 N 1 N N 7 W 6 W 5 W 4 W 3 W 2 W 1 W (c) N 7 W 6 W 5 W 4 W 3 W 2 W 1 W (d) Temperature anomaly ( C)

64 Semi-annual component of monthly temperature anomaly along 11 o N Depth (m) (a) 55 W 5 W 45 W 4 W 35 W 3 W 25 W 2 W.25 (a) 6/ Depth (m) (b) 7/ (b) 55 W 5 W 45 W 4 W 35 W 3 W 25 W 2 W.5 (c) 8/4 (24) (d) 9/ Depth (m) (c) 55 W 5 W 45 W 4 W 35 W 3 W 25 W 2 W Depth (m) (d) 55 W 5 W 45 W 4 W 35 W 3 W 25 W 2 W

65 Semi-annual temperature anomaly at 55m depth (24) 3 N (a) 5/15 2 N 1 N (b) 6/29 N 3 N 7 W 6 W 5 W 4 W 3 W 2 W 1 W (a) 2 N 1 N N 7 W 6 W 5 W 4 W 3 W 2 W 1 W (b) Temperature anomaly ( C)

66 Semiannual Component in the Western Sub-Basin (a).6.4 (d) (a) wind KE (b) current KE (c) corr wind stress and currents (d) corr between semi-annual currents and mean wind (e) corr between semiannual currents and annual wind stress. <E > (m 2 /s 2 ) i <e > (cm 2 /s 2 ) i IV VI VIII X XII II IV (b) IV VI VIII X XII II IV (c) r ii r ii IV VI VIII X XII II IV (e) IV VI VIII X XII II IV Month R ii.2.4 IV VI VIII X XII II IV Month

67 Semiannual Component in the Eastern Sub-Basin.8 (a).8 (d) <E i > (m 2 /s 2 ) (a) wind KE (b) current KE (c) corr wind stress and currents (d) corr between semi-annual currents and mean wind (e) corr between semiannual currents and annual wind stress. <e > (cm 2 /s 2 ) i IV VI VIII X XII II IV (b) IV VI VIII X XII II IV (c) r ii r ii IV VI VIII X XII II IV (e) IV VI VIII X XII II IV Month.2 R ii IV VI VIII X XII II IV Month

68 Results The annual and semi-annual unstable standing Rossby waves are detected in both the western and eastern sub-basins. The wind-driven Ekman pumping seems to be responsible for the standing wave generation in both the sub-basins.

69 Example-3 OSD for Analyzing Combined Current Meter and Surface Drifting Buoy Data

70 Ocean Velocity Observation 31 near-surface (1-14 m) current meter moorings during LATEX from April 1992 to November 1994 Drifting buoys deployed at the first segment of the Surface Current and Lagrangian-drift Program (SCULP-I) from October 1993 to July 1994.

71 Moorings and Buoys

72 LTCS current reversal detected from SCULP-I drift trajectories.

73

74 Reconstructed and observed circulations at Station-24.

75 Probability of TLCS Current Reversal for Given Period (T) n ~-current reversal n 1 ~ 1-current reversal n 2 ~ 2-current reversals m ~ all realizations

76 Fitting the Poison Distribution μ is the mean number of reversal for a single time interval μ ~.8

77 Dependence of P, P 1, P 2 on T For observational periods larger than 2 days, the probability for no current reversal is less than.2. For 15 day observational period, the probability for 1-reversal reaches.5 Data Solid Curve Poison Distribution Fitting Dashed Curve

78 Time Interval between Successive Current Reversals (not a Rare Event)

79 LTCS current reversal detected from the reconstructed velocity data December 3, 1993 January 3, 1994 January 6, 1994

80 EOF Analysis of the Reconstructed Velocity Filed Variance (%) EOF 1/21/93-5/21/93 12/19/93-4/17/94 1/5/94-11/29/

81 Mean and First EOF Mode

82 Mean Circulation 1. First Period (1/21-5/21/93) 2. Second Period 12/19/93-4/17/94) 3. Third Period (1/5-11/29/94)

83 EOF1 1. First Period (1/21-5/21/93) 2. Second Period 12/19/93-4/17/94) 3. Third Period (1/5-11/29/94)

84 Calculated A1(t) Using Current Meter Mooring (solid) and SCULP-1 Drifters (dashed)

85 8 total reversals observed Uals ~ alongshore wind

86 Morlet Wavelet A 1 (t) Uals

87 Surface Wind Data 7 buoys of the National Data Buoy Center (NDBC) and industry (C-MAN) around LATEX area

88 Regression between A1(t) and Surface Winds Solid Curve (reconstructed) Dashed Curve (predicted using winds)

89 Results Alongshore wind forcing is the major factor causing the synoptic current reversal. Other factors, such as the Mississippi- Atchafalaya River discharge and offshore eddies of Loop Current origin, may affect the reversal threshold, but can not cause the synoptic current reversal.

90 Part-4 OSD for Analyzing CODAR Data

91 CODAR

92 Monterey Bay

93 3 cm/s 37. o N 3 cm/s 36.9 o N 36.8 o N 36.7 o N 36.6 o N o W o W 122. o W o W o W o W o W 122. o W o W o W Place for comments: left - radar derived currents for 17: UT December 1, 1999 right reconstructed velocity field.

94

95 Conclusions OSD is a useful tool for processing realtime velocity data with short duration and limited-area sampling especially the ARGO data. OSD has wide application in ocean data assimilation.