tUtU km

M

i

M

jjkimjikm fftatatUtU

1 1

M

iikimikm fftUtU

1

imi

M

kikkm fftUtU

1

0 C

Forming the co-variance matrix of the data

taftUM

iiimm

1

jiiji tata

Multiplying both sides times fik, summing over all k and using the orthogonality condition:

Canonical form of eigenvalue problemeigenvectors 2taii eigenvalues

0 C

tUtUtUtUtUtU

tUtUtUtUtUtUtUtUtUtUtUtU

MMMM

M

M

21

22212

12111

Mf

ff

2

1

00

0000

Mf

ff

2

1

0

00

221

2222112

1221111

MMMMMM

mM

mM

ftUtUftUtUftUtU

ftUtUftUtUftUtUftUtUftUtUftUtU

tUtUC kmmk

I is the unit matrix and are the EOFs

Eigenvalue problem corresponding to a linear system:

Matrix = [6637,18]

rows > columns

tUtUtUtUtUtU

tUtUtUtUtUtUtUtUtUtUtUtU

MMMM

M

M

21

22212

12111

Mf

ff

2

1

00

0000

Mf

ff

2

1

Matrix ul = [6637,18]

>> uc=cov(ul);>> u1=ul(:,1);>> sum((u1-mean(u1)).^2)/(length(u1)-1)

ans =

9.6143>> u2=ul(:,2);>> sum((u1-mean(u1)).*(u2-mean(u2)))/(length(u1)-1)

ans =

10.1154

N

iii uu

N 1

2

11

Covariance Matrix

Maximum covariance at surface

>> uc=cov(ul);>> [v,d]=eig(uc);

eigenvalues (or lambda)

>> lambda=diag(d)/sum(diag(d));

>> uc=cov(ul);>> [v,d]=eig(uc);

>> uc=cov(ul);>> [v,d]=eig(uc);>> v=fliplr(v); %flips matrix left to right

Mode 185.3%

Mode 213.2%

taftUM

iiimm

1

Mode 185.3%

Mode 213.2%

>> ts=ul*v;

taftUM

iiimm

1

ts=[6637,18]

Mode 185.3%

Mode 213.2%

>> for k=1:nzvt(k,:,:)=ts(:,k)*v(:,k)';end

vt=[18, 6637,18]

mode #evolution in time

time series #

>> v1=squeeze(vt(1,:,:))’;>> v2=squeeze(vt(2,:,:))’;

Dep

th (m

)

Dep

th (m

)

Dep

th (m

)

Complex Empirical Orthogonal Functions – James River Data

Linear combination of spatial predictors or modes that are normal or orthogonal to each other

taftUM

iiimm

1

u

v

taftUM

iiimm

1

Streamwise

Cross-stream

Rotated 49 degrees

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iiimmmm

1

0 C

>> ul=complex(u,v);>> uc=cov(ul);>> [v,d]=eig(uc);

Mode 196.5%

Mode 1

>> lambda=diag(d)/sum(diag(d));>> v=fliplr(v);

Mode 22.5%

Mode 2

Mode 196.5%

Streamwise

cross-stream

>> ts=ul*v;

Mode 196.5%

Principal-axis

cross-axis

Mod

e sc

alin

g

>> ts=ul*v;

Mode 22.5%

Streamwise

Cross-stream

Mod

e sc

alin

g

>> ts=ul*v;

Mod

e sc

alin

g

Mode 22.5%

Streamwise

cross-stream

>> ts=ul*v;

Low-pass filtered data in James River

m/s

m/s

Streamwise

cross-stream

Mode 175%

m/s

m/s

m/s

Streamwise

cross-stream

Mode 222%

m/s

m/s

m/s

Streamwise

cross-stream

Dep

th (m

)

radians

Phase of EOFS

Mode 1

Mode 2

Mode 1 Mode 2

streamwise

cross-stream

>> ts=ul*v;

streamwise

cross-stream

>> ts=ul*v;

Modes 1 + 2 (75% + 22%)

Original

>> for k=1:nzvt(k,:,:)=ts(:,k)*v(:,k)';end

>> v1=squeeze(vt(1,:,:))’;>> v2=squeeze(vt(2,:,:))’;

Modes 1 + 2 + 3 + 4 (77%+22% +2% + 0.6%)

Original