Postdoctoral Final Report

37
Figure 1 D=26.7 km D=18.9 km D= 30 km

Transcript of Postdoctoral Final Report

Figure 1

D=26.7

km

D=18.9 km

D= 30 km

Puerto Rico

SW coast and shelf

Location of the shelf and offshelf ADCP.

Figure 1b

Offshelf

shelfbreakshelf

IMF sifting of u (tk , z) z=30 m, k = 1,2,3…K ; CE(100,5)Figure 2

270 275 280 285 290 295 300 305 310 315-10

0

10

F1

270 275 280 285 290 295 300 305 310 315-10

0

10

F2

270 275 280 285 290 295 300 305 310 315-10

0

10

F3

270 275 280 285 290 295 300 305 310 315-10

0

10

F4

270 275 280 285 290 295 300 305 310 315-10

0

10

F5

270 275 280 285 290 295 300 305 310 315-40

-20

0

20

Year day (2000)

cm/s

Original signal: u ( t , z ), Z = 30 m

Figure 3

270 275 280 285 290 295 300 305 310 315-15

-10

-5

0

5

10

15

Year day (2000)

Spe

ed (

cm/s

)

First IMF of u(t), F1, and its envelope, a(t); Z=30 m

Generate matrix F from original dataFigure 4

=

),(),(),(),(

),(),(),(),(

),(),(),(),(

),(),(),(),(

321

3332313

2322212

1312111

MKKKK

M

M

M

ztXztXztXztX

ztXztXztXztX

ztXztXztXztX

ztXztXztXztX

LMMMM

L

L

L

XOriginal Data

Time Series

L Intrinsic Mode Functions (IMF)

F1(tk,z1)F2(tk,z1)F3(tk,z1)...FL(tk,z1)

EMD

Matrix F containsL IMFs for each depth.

),(),(),(),(

),(),(),(),(),(),(),(),(

),(),(),(),(

321

3332313

2322212

1312111

MKLMLMLML

MKMMM

MKMMM

MKMMM

ztFztFztFztF

ztFztFztFztFztFztFztFztF

ztFztFztFztF

LMMMM

L

L

L

),(),(),(),(

),(),(),(),(

),(),(),(),(

),(),(),(),(

3333231

33333323313

32332322312

31331321311

ztFztFztFztF

ztFztFztFztF

ztFztFztFztF

ztFztFztFztF

KLLLL

K

K

K

LMMMM

L

L

L

),(),(),(),(

),(),(),(),(

),(),(),(),(

),(),(),(),(

2232221

23233223213

22232222212

21231221211

ztFztFztFztF

ztFztFztFztF

ztFztFztFztF

ztFztFztFztF

KLLLL

K

K

K

LMMMM

L

L

L

Repeat the analysison each time series.

zm

),(),(),(),(

),(),(),(),(

),(),(),(),(

),(),(),(),(

1131211

13133123113

12132122112

11131121111

ztFztFztFztF

ztFztFztFztF

ztFztFztFztF

ztFztFztFztF

KLLLL

K

K

K

LMMMM

L

L

L

l

tk

Decomposition of the 3D matrix F into a series of 2D matrices Fl slicing it at a particular component l where l=1, 2, 3…L

=

),(),(),(),(

),(),(),(),(

),(),(),(),(),(),(),(),(

321

3332313

2322212

1312111

MKLKLKLKL

MLLLL

MLLLL

MLLLL

ztFztFztFztF

ztFztFztFztF

ztFztFztFztFztFztFztFztF

LMMMM

L

L

L

LF

=

),(),(),(),(

),(),(),(),(

),(),(),(),(

),(),(),(),(

3332313

33333233133

23323223123

13313213113

MKKKK

M

M

M

ztFztFztFztF

ztFztFztFztF

ztFztFztFztF

ztFztFztFztF

LMMMM

L

L

L

3F

=

),(),(),(),(

),(),(),(),(

),(),(),(),(

),(),(),(),(

2322212

32332232132

22322222122

12312212112

MKKKK

M

M

M

ztFztFztFztF

ztFztFztFztF

ztFztFztFztF

ztFztFztFztF

LMMMM

L

L

L

2F

=

),(),(),(),(

),(),(),(),(

),(),(),(),(),(),(),(),(

1312111

31331231131

21321221121

11311211111

MKKKK

M

M

M

ztFztFztFztF

ztFztFztFztF

ztFztFztFztFztFztFztFztF

LMMMM

L

L

L

1F

IMF C

ompo

nent

l

Depth zm

Tim

e t k

Amplitude of IMF component L for time tk (k=1) and depth level zm (m=1)

Dimension of F= K x M x LDimension of Fl= K x M

Figure 5

EOF decomposition of the 2D matrices F1, F2, F3… FL and their rearrangement to generate the 3D matrix Fn (n=1,2,3…M).

Figure 6

2D F1

F2

F3

FL

EOF R2(zi,zj)

EOF R1(zi,zj)

EOF R3(zi,zj)

EOF RL(zi,zj)

F31 F3

2 F33 F3

M

F11 F1

2 F13 F1

M

F21 F2

2 F23 F2

M

FL1 FL

2 FL3 FL

M

Reconstruct F1 along each EOF mode

F1 F2 F3 FM3D

88.36%

2.43%

4.28%

7.45%

11.72%

13.34%

49.14%

Explained Variance

2nd ApproachAl

n

93.52%

2.34%

2.63%

6.48%

10.29%

15.87%

55.91%

Explained Variance

1st ApproachFl

n

First IMFl=1

6

Subtotal

5

4

3

2

1

Mode numbern

Table 1

n

n

λλ

Σ n

n

λλ

Σ

Year Day (2000)

Freq

uenc

y (C

PD

)

Hilbert Spectrum

275 280 285 290 295 300 305 310

0.5

1

1.5

2

2.5

3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Year Day (2000)

Freq

uenc

y (C

PD

)

Hilbert Spectrum

275 280 285 290 295 300 305 310

0.5

1

1.5

2

2.5

3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Year Day (2000)

Freq

uenc

y (C

PD

)

Hilbert Spectrum

275 280 285 290 295 300 305 310

0.5

1

1.5

2

2.5

3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Year Day (2000)

Freq

uenc

y (C

PD

)

Hilbert Spectrum

275 280 285 290 295 300 305 310

0.5

1

1.5

2

2.5

3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A

B

C

D

),),(( 3011 kk ttH lω

),),(( 3021 kk ttH lω

),),(( 3031 kk ttH lω

),),(( 301 kk ttH lω( ) 2

1

CPDcm/s

Figure 7

270 275 280 285 290 295 300 305 31010

12

14

16

18

20

22

24

26

28

30

32Barotropic currents range and lunar phases

Year day (2000)

Ran

ge (

cm/s

)

1.6 2.4 0.0 1.74 1.22 -0.81- 4 , 6 - 4- new moon

4 first Qt.

, full moon

6 last Qt.

Figure 8

Table 2

A11

2.74

3.83

2.25

Time lag between barotropic neap

tides and baroclinic

spring tides (days)

3.05

2.43

0.23

Time lagbetween

barotropicand baroclinic

spring tides (days)

FQ

LQ

FQ

Moon PhaseFQ=first quarter

LQ=last quarter

NM

FM

NM

Moon PhaseNM=new moon

FM=full moon

2.29311.6-0.45308.86309.31

4.37298.70.54294.87294.33

2.64282.10.39279.85279.46

Total time lagbetween phase andbaroclinic spring

tides (days)

BaroclinicTides

(spring)

Age ofBarotropicneap tides

BarotropicTides

(neap)

PhaseYD

2000

2.24303.57-0.81300.52301.33

3.65291.021.22288.59287.37

1.97273.81.74273.57271.83

Total time lagbetween phase andbaroclinic spring

tides (days)

BaroclinicTides

(spring)

Age ofBarotropic

spring tides

BarotropicTides

(spring)

PhaseYD

2000

Year Day (2000)

Dep

th (

m)

EOF 3

275 280 285 290 295 300 305 310

-60

-50

-40

-30

-20

-10

-4

-3

-2

-1

0

1

2

3

4

Year Day (2000)

Dep

th (

m)

EOF 2

275 280 285 290 295 300 305 310

-60

-50

-40

-30

-20

-10

-10

-8

-6

-4

-2

0

2

4

6

8

10

Year Day (2000)

Dep

th (

m)

F11

275 280 285 290 295 300 305 310

-60

-50

-40

-30

-20

-10

-10

-5

0

5

10

Year Day (2000)

Dep

th (

m)

275 280 285 290 295 300 305 310

-60

-50

-40

-30

-20

-10

-10

-5

0

5

10

F12

F13

F1: u

A

B

C

D

Figure 9

F11

cm/s

cm/s

cm/s

cm/s

Figure 10

cm/s

dept

h (m

)

F11

275 280 285 290 295 300 305 310

-60

-50

-40

-30

-20

-10

275 280 285 290 295 300 305 3100

5

10

15

20

25

30

year day (2000)

rang

e (c

m/s

)

Range of barotropic U at ADCP3

-10

-5

0

5

10

cm/s

cm/s

cm/s

cm/s

Year Day (2002)

Dep

th (

m)

EOF 3

275 280 285 290 295 300 305 310

-60

-50

-40

-30

-20

-10

-6

-4

-2

0

2

4

6

Year Day (2002)

Dep

th (

m)

EOF 2

275 280 285 290 295 300 305 310

-60

-50

-40

-30

-20

-10

-4

-3

-2

-1

0

1

2

3

4

Year Day (2002)

Dep

th (

m)

EOF 1

275 280 285 290 295 300 305 310

-60

-50

-40

-30

-20

-10

-10

-8

-6

-4

-2

0

2

4

6

8

10

Year Day (2000)

Dep

th (

m)

275 280 285 290 295 300 305 310

-60

-50

-40

-30

-20

-10

0

2

4

6

8

10

12

A11

A12

A13

A1: u

A

B

C

D

Figure 11

Figure 12

dept

h (m

)

A11

275 280 285 290 295 300 305 310

-60

-50

-40

-30

-20

-10

275 280 285 290 295 300 305 3100

5

10

15

20

25

30

year day (2000)

rang

e (c

m/s

)

Range of barotropic U at ADCP3

-10

-5

0

5

10cm/s

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

5

6PDF estimate of envelope frequency

frequency (CPD)

0.135 CPD

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

5

6PDF estimate of envelope frequency

frequency (CPD)

0.0706 CPD

Year Day (2000)

Freq

uenc

y (C

PD

)

Hilbert Spectrum: First four IMF's of the envelope A1(tk, 30 m)

275 280 285 290 295 300 305 3100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Year Day (2000)

Freq

uenc

y (C

PD

)

Hilbert Spectrum: all IMF's of the envelope A1(tk, 30 m)

275 280 285 290 295 300 305 3100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0

0.5

1

1.5

A

B

C

D

Figure 13

Year Day (2000)

Dep

th (

m)

Kinetic Energy of the internal tide, F1 ADCP2

280 285 290 295 300 305

-110

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

2

4

6

8

10

12

J/m3

Year Day (2000)

Dep

th (

m)

Kinetic Energy of the internal tide, F1 ADCP1

280 285 290 295 300 305

-110

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

2

4

6

8

10

12

14

16

18

20

22

J/m3

Year Day (2000)

Dep

th (

m)

Kinetic Energy of the internal tide, F1 ADCP3

280 285 290 295 300 305

-110

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

1

2

3

4

5

6

7

8

9

J/m3

Figure 14

2.3 d

2.4 d

1.45 d1.4 d

A

B

C

275 280 285 290 295 300 305 310-40

-20

0

20

40original signal for depth level 30.5m

spee

d cm

/s

275 280 285 290 295 300 305 310-10

0

10

SD bandpass filtered signal

275 280 285 290 295 300 305 310-10

0

10

F1 signal from EMD analysis

275 280 285 290 295 300 305 310-10

0

10

D3 signal from Wavelet analysis

YD (2000)

Figure 15

KE of the baroclinic tide on YD 279-284 2000FFT vs. EMD

Year Day (2000)

Dep

th (

m)

Kinetic Energy of the internal tide (SD bandpass filter)

279 279.5 280 280.5 281 281.5 282 282.5 283 283.5 284-30

-28

-26

-24

-22

-20

-18

-16

-14

-12

-10

0.5

1

1.5

2

2.5

3

3.5

J/m3

Year Day (2000)

Dep

th (

m)

Kinetic Energy of the internal tide, F1 ADCP3

279 279.5 280 280.5 281 281.5 282 282.5 283 283.5 284-30

-28

-26

-24

-22

-20

-18

-16

-14

-12

-10

1

2

3

4

5

6

7

8

9

J/m3

A B

Figure 16

KE of the internal tide on YD 279-284 2000Wavelet vs. EMD

Year Day (2000)

Dep

th (

m)

Kinetic Energy of the internal tide, F1 ADCP3

279 279.5 280 280.5 281 281.5 282 282.5 283 283.5 284-30

-28

-26

-24

-22

-20

-18

-16

-14

-12

-10

1

2

3

4

5

6

7

8

9

J/m3

A B

Year Day (2000)

Dep

th (

m)

Kinetic Energy of the internal tide (Wavelets: db2; detail 3)

279 279.5 280 280.5 281 281.5 282 282.5 283 283.5 284-30

-28

-26

-24

-22

-20

-18

-16

-14

-12

-10

2

4

6

8

10

12

14

16

18

J/m3

Figure 17

Results Parguera

ADCP offshelf

Figure 18

80 85 90 95 100 105 110 115 120 1251.8

2

2.2

2.4

2.6

2.8

year day (2002)

rang

e(cm

/s)

Range of barotropic U at the offshelf ADCP

Figure 19

A B

Figure 20

Year Day (2002)

Dep

th (

m)

F13 u

80 85 90 95 100 105 110 115 120 125-450

-400

-350

-300

-250

-200

-150

-100

-50

-5

-4

-3

-2

-1

0

1

2

3

4

5

Year Day (2002)

Dep

th (

m)

F12 u

80 85 90 95 100 105 110 115 120 125-450

-400

-350

-300

-250

-200

-150

-100

-50

-4

-3

-2

-1

0

1

2

3

4cm/s cm/s

cm/s cm/s

A

B

C

D

Figure 21

6.39 %4.56 %6

66.022 %5.66 %

6.47 %

9.17 %

9.79 %

11.94 %

16.60 %

Explained Variance

2nd ApproachAl

n

78.47 %3.82 %

5.99 %

9.08 %

11.51 %

16.25 %

27.26 %

Explained Variance

1st ApproachFl

n

First IMFl=1

7

Subtotal

5

4

3

2

1

Mode numbern

Table 3

n

n

λλ

Σ n

n

λλ

Σ

Year day 2002

cm/s cm/s

cm/s cm/s

A

B

C

D

Figure 22

A

B

C

D

Figure 23

Year Day (2000)

Dep

th (

m)

Kinetic Energy of the internal tide, F1

80 85 90 95 100 105 110 115 120 125-450

-400

-350

-300

-250

-200

-150

-100

-50

0

2

4

6

8

10

12J/m3

Year Day (2000)D

epth

(m

)80 85 90 95 100 105 110 115 120 125

-450

-400

-350

-300

-250

-200

-150

-100

-50

0

1

2

3

4

5

6

7

8

9Potential Energy of the internal tide, F1

J/m3

A B

Figure 24

Year day 2002 Year day 2002

Year Day (2000)

Dep

th (

m)

Kinetic Energy of the diurnal internal tide

80 85 90 95 100 105 110 115 120 125-450

-400

-350

-300

-250

-200

-150

-100

-50

1

2

3

4

5

6

7

8

9

10

J/m3

Year Day (2000)D

epth

(m

)

Potencial Energy of the diurnal internal tide

80 85 90 95 100 105 110 115 120 125-450

-400

-350

-300

-250

-200

-150

-100

-50

1

2

3

4

5

6

7

8

A B

Figure 25

Year day 2002 Year day 2002

Year Day (2000)D

epth

(m

)

Kinetic Energy of the diurnal internal tide,F2, offshelf

80 85 90 95 100 105 110 115 120 125-75

-70

-65

-60

-55

-50

-45

-40

-35

-30

-25

0

1

2

3

4

5

6

7

8

9

10

11J/m3

Year Day (2000)

Dep

th (

m)

Kinetic Energy of the internal tide, F1, offshelf

80 85 90 95 100 105 110 115 120 125-75

-70

-65

-60

-55

-50

-45

-40

-35

-30

-25

0

1

2

3

4

5

6

7

8

9

10

11J/m3

A B

Figure 26

Year day 2002Year day 2002

A B

Figure 27

A B

Figure 28

79.5417 81.6267 83.7117 85.7967 87.8817 89.9667 92.0517 94.1367

F1, u, shelf

5

6

7

8

9

1011

12

1314

Year day 2002

79.5417 81.6267 83.7117 85.7967 87.8817 89.9667 92.0517 94.1367

F1, v, shelf

Year day 2002

A B

Figure 29de

pth

(m)

80 82 84 86 88 90 92 94

-20

-15

-10

-5

-10

-8

-6

-4

-2

0

2

4

6

8

10F1: u shelf cm/sF1: u, shelf

Year day 2002de

pth

(m)

80 82 84 86 88 90 92 94

-20

-15

-10

-5

-6

-4

-2

0

2

4

6F1 shelf

cm/sF1: v, shelf

Year day 2002

A B

Figure 30

Year Day (2000)

Dep

th (

m)

Kinetic Energy of the internal tide, F1 shelf

80 82 84 86 88 90 92 94

-22

-20

-18

-16

-14

-12

-10

-8

-6

0.5

1

1.5

2

2.5

3

3.5

J/m3

Year day 2002Year Day (2000)

Dep

th (

m)

80 82 84 86 88 90 92 94

-22

-20

-18

-16

-14

-12

-10

-8

-6

0.5

1

1.5

2

2.5

3Potencial Energy of the internal tide, F1 shelf

J/m3

Year day 2002

80 85 90 95 100 105 110 115 120 125

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Tidal Height at Magueyes Island, La Parguera, Puerto Rico

Year day 2002

Wat

er L

evel

hei

ght (

m)

Figure 31

-0.020

0.02C

1

EMD sifting of Parguera Tides (Extrema,113,5, intermittency: 2,6,-1,20,40,63,-1,-1,-1,-1,-1)

-0.020

0.02

C2

-0.020

0.02

C3

-0.020

0.02

C4

-0.020

0.02

C5

-0.10

0.1

C6

-0.10

0.1

C7

-0.10

0.1

C8

-0.10

0.1

C9

-0.10

0.1

C10

85 90 95 100 105 110-0.1

00.1

C11

Year Day (2002)

Figure 32