Cross-spectral Analysis

Investigates the dependence or relationship between two time series.

yyxxN

LC Lk

LN

kkxy

11

1

Begin by reviewing cross-correlation function – how similar two time series x , y are

L = lagN = # of elements in time series

Covariance or cross-covariance

N

kk

N

kk

Lk

LN

kk

xy

yyN

xxN

yyxxN

L

1

2

1

2

1

11

11

11

yx

xyC

Cross-correlation

-1 < ψxy < 1

Ensenada de la Paz

Year Day

Current velocity in Ensenada de La Paz

Current velocity and water level (red) in Ensenada de La Paz

Year Day

Current velocity and water level (red) in Ensenada de La Paz

Year Day

N

kk

N

kk

Lk

LN

kk

xy

yyN

xxN

yyxxN

L

1

2

1

2

1

11

11

11

Cross-correlation between current velocity and water level in Ensenada de La Paz

N

kk

N

kk

Lk

LN

kk

xy

yyN

xxN

yyxxN

L

1

2

1

2

1

11

11

11

Cross-correlation between current velocity and water level in Ensenada de La Paz

N

kk

N

kk

Lk

LN

kk

xy

yyN

xxN

yyxxN

L

1

2

1

2

1

11

11

11

A generalization of:

ivuw

N

kkk

N

kkk

Lk

LN

kk

ww

w*wN

w*wN

w*wN

L

122

111

21

1

11

11

11

21

* is the complex conjugate

Complex Cross-correlation

N

kkk

N

kkk

Lk

LN

kk

ww

w*wN

w*wN

w*wN

L

122

111

21

1

11

11

11

21

L

w1w

2

Complex Cross-correlation between vectors at 2 depths in Ensenada de La Paz

N

kkk

N

kkk

Lk

LN

kk

ww

w*wN

w*wN

w*wN

L

122

111

21

1

11

11

11

21

Complex Cross-correlation between vectors at 2 depths in Ensenada de La Paz

N

kkk

N

kkk

Lk

LN

kk

ww

w*wN

w*wN

w*wN

L

122

111

21

1

11

11

11

21

Complex Cross-correlation between vectors at 2 depths in Ensenada de La Paz

Coherence spectrum (coherency)

kyykxx

kxykxy

fGfG

fGf

2

2

kykxkxy fXf*XtN

fG2

kxkxkxx fXf*XtN

fG2

kykykyy fXf*XtN

fG2

221

2 1 = 0.1, 0.05, 0.01

X is the Fourier Transform

< > = running mean

Current velocity and water level (red) in Ensenada de La Paz

Year Day

cpd

221

2 1

2

kyykxx

kxykxy

fGfG

fGf

2

2

= 0.1, 0.05, 0.01

Coh

eren

ce s

quar

ed

window size= 2

221

2 1

kyykxx

kxykxy

fGfG

fGf

2

2

= 0.1, 0.05, 0.01

xy

xyk GRe

GImtanaf

JAMES RIVER DATA

kyykxx

kxykxy

fGfG

fGf

2

2

xy

xyk GRe

GImtanaf

Coh

eren

ce 2

Pha

se (

deg)

95%

99%

Window size= 10

kyykxx

kxykxy

fGfG

fGf

2

2 C

oher

ence

2P

hase

(de

g)

95%99%

Window size= 24

xy

xyk GRe

GImtanaf

kyykxx

kxykxy

fGfG

fGf

2

2

kykxkxy fXf*XtN

fG2

kxkxkxx fXf*XtN

fG2

kykykyy fXf*XtN

fG2

221

2 1

kyykxx

kxykxy

fGfG

fGf

2

2

kykx fXfX

kyykxx

kxykxy

fGfG

fGf

2

2 C

oher

ence

2P

hase

(de

g)

95%

99%

Window size= 10

xy

xyk GRe

GImtanaf

kyykxx

kxykxy

fGfG

fGf

2

2 C

oher

ence

2P

hase

(de

g)

95%

99%

Window size= 10

xy

xyk GRe

GImtanaf

kyykxx

kxykxy

fGfG

fGf

2

2 C

oher

ence

2P

hase

(de

g)

95%99%

Window size= 24

xy

xyk GRe

GImtanaf

kyykxx

kxykxy

fGfG

fGf

2

2 C

oher

ence

2P

hase

(de

g)

95%99%

Window size= 24

xy

xyk GRe

GImtanaf