I is the unit matrix and are the EOFs

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:

taftUM

iiimm

1

taftUM

iiimm

1

Sometimes use another complex form of EOFS. Apply the Hilbert Transform of U to understand a bit more about the phase of propagation of the phenomenon

In essence, the Hilbert transform of a signal (time series) produces a complex variable with real part identical to the time series and the imaginary part is shifted 90º from the original (sometimes called the analytical signal):

>> t=[1:1:1000];>> u1=sin(2*pi*t/200);>> plot(t,u1,’LineWidth’,3)

u1

t

>> t=[1:1:1000];>> u1=sin(2*pi*t/200);>> plot(t,u1,’LineWidth’,3)>>>> u2=hilbert(u1); >> hold on;>> plot(real(u2),'r--','LineWidth',3)

u

t

>> t=[1:1:1000];>> u1=sin(2*pi*t/200);>> plot(t,u1,’LineWidth’,3)>>>> u2=hilbert(u1); >> hold on>> plot(real(u2),'r--','LineWidth',3)>> plot(imag(u2),'g','LineWidth',3)

u

t

Hilbert Transform of U can be regarded as the convolution of U with h(t) = 1/(t)

dthUPUH

d

tUP1P is the Cauchy principal value

(assigns values to improper integrals)

hilbert uses a four-step algorithm:

1. It calculates the FFT of the input sequence, storing the result in a vector x. 2. It creates a vector h whose elements h(i) have the values:

1 for i = 1, (n/2)+12 for i = 2, 3, ... , (n/2)0 for i = (n/2)+2, ... , n3. It calculates the element-wise product of x and h.4. It calculates the inverse FFT of the sequence obtained in step 3 and returns the first n elements of the result.

Echo Intensity in a Fjord

Mean Profile10 log (echo)

Anomaly from Mean

60%18%

9%

Hilbert EOF

64%

18%

6%