IX. EMPIRICAL ORTHOGONAL FUNCTIONS · 24 February 2009 Chapter IX. Empirical Orthogonal Functions 3...

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24 February 2009 Chapter IX EOFs Notes 8Wendell S. Brown 1 IX. EMPIRICAL ORTHOGONAL FUNCTIONS A. Empirical Orthogonal Functions (EOF): Time Domain Consider a spatial array of discrete time series observations (for example eastward current) at M locations - u jt , where j indicates station number (j = 1, 2 ... M) and t indicates the time step [t = 1, 2 ...N; where (N-1) Δt = T; length of series]. The zero-lag cross-covariance for the j th and k th elements of the array is given by ) u - u )( u - u ( N 1 = R k kt j jt N 1 = t jk , (IX.1) where R is a square, symmetric, real M x M matrix and overbar indicates a designated series mean value. The diagonal elements of R jk in (IX.1) (or where j = k) are the station record variances according to ) u - u ( N 1 = R 2 j jt N 1 = t jj (IX.2) The total variance of the “system” (or array) time series (also called trace of R or Tr R) is R = R Tr jj M 1 = j . (IX.3) A diagonalization of the M x M matrix R (more detailed explanation below) yields a set of M empirical orthogonal functions (EOF) or modes; in which the j th mode consists of a Real EIGENVECTOR e mj , with components m =1,2,….M; and for which the j th and k th modes are orthogonal to each other in the sense that = e e jk mk mj M 1 = m , (IX.4a) and a

Transcript of IX. EMPIRICAL ORTHOGONAL FUNCTIONS · 24 February 2009 Chapter IX. Empirical Orthogonal Functions 3...

Page 1: IX. EMPIRICAL ORTHOGONAL FUNCTIONS · 24 February 2009 Chapter IX. Empirical Orthogonal Functions 3 The variance of amt is the variance of the mth mode. The mth mode eigenvalue m

24 February 2009 Chapter IX EOFs Notes 8Wendell S. Brown

1

IX. EMPIRICAL ORTHOGONAL FUNCTIONS

A. Empirical Orthogonal Functions (EOF): Time Domain

Consider a spatial array of discrete time series observations (for example eastward current) at

M locations - u jt , where j indicates station number (j = 1, 2 ... M) and t indicates the time

step [t = 1, 2 ...N; where (N-1) Δt = T; length of series].

The zero-lag cross-covariance for the jth and kth elements of the array is given by

)u - u)(u - u( N

1 = R kktjjt

N

1=tjk , (IX.1)

where R is a square, symmetric, real M x M matrix and overbar indicates a designated

series mean value.

The diagonal elements of R jk in (IX.1) (or where j = k) are the station record variances

according to

)u - u( N

1 = R

2jjt

N

1=tjj (IX.2)

The total variance of the “system” (or array) time series (also called trace of R or Tr R) is

R = RTr jj

M

1=j . (IX.3)

A diagonalization of the M x M matrix R (more detailed explanation below) yields a set of

M empirical orthogonal functions (EOF) or modes; in which the jth mode consists of a

� Real EIGENVECTOR emj ,

with components m =1,2,….M; and for which the jth and kth modes are orthogonal

to each other in the sense that

= e e jkmkmj

M

1=m , (IX.4a)

and a

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� Positive EIGENVALUEm .

such that

e = e R mkmmjjk

M

1=j . (IX.4b)

(Note: The number of M-component eigenvectors is equal to the number of stations).

Under these circumstances, ujt can be expanded in terms of these eigenvectors emj according

to

e a = u mjmt

M

1=mjt (IX.5a)

where the amplitude time series of the mth eigenvector emjcan be expressed as

u e = a jtmj

M

1=jmt . (IX.5b)

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The variance of amt is the variance of the mth mode.

The mth mode eigenvalue m is that mode’s variance (i.e., energy). The percentage of the

total "system" (or array) variance, which from (IX.3) can be written

= RTr m

M

1=j , (IX.6a)

that is “explained” by that mode is given by the ratio ofm /Tr R.

The dimensional “first” EOF (or mode-1 EOF), as defined by the product of the normalized

eigenvector elj and amplitude )( 2

1

l or

)(e 2

1

llj (IX.6b)

has the largest variance or eigenvalue l .

This set of eigenvectors/eigenvalues – the solution - is determined by a least square fit

between the solution and the cross-covariance matrix R. The best fit to the cross-covariance

matrix is defined by

. minimum = )ee - R( 2lkljljk

kj (IX.7)

Mathematically, the “fitting-process” partitions the total “system” (or array) variability

variance into a set of M orthogonal modes; emj /m ; each “explaining” different patterns of

correlated information in the array, under the rather artificial constraint that the eigenvectors

be orthogonal (i.e., statistically independent from each other).

***************************

AN EXAMPLE OF THE APPLICATION OF THE T-EOF TECHNOLOGY

***************************

Consider the following example of an array of inverted echo sounder (IES) measurements

made across the continental slope into the deep ocean seaward of the Amazon River outflow

(Figure. IX.1a). The IES is an acoustic projector whose travel time π from the bottom to the

ocean to the surface (see Figure. IX.1b) and back is related to ocean properties – primarily

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the depth of the main thermocline - according to dz, t)C(Z, /C)(2 = ho where C is sound

speed and h is local water depth. Hydrographic measurements at the IES mooring site are

used to compute the dynamic heights; which are then correlated with the corresponding IES

travel times. The variability of the IES time-series of equivalent dynamic height for each of

the moored IES deployments is presented in (Figure. IX.2).

Figure. IX.1. (a-left) The location of four bottom-mounted inverted echo sounders (IES) and three current meter moorings that were deployed in the tropical Atlantic from 1989-1992. (b-right) The depths of the different instruments are illustrated in the bathymetric transect.

Figure. IX.2 Dynamic height time series derived from IES travel times measured by instruments located in Figure IX.1. The EOF decomposition of the cross-covariance matrix of these four (scalar) IES dynamic

height time series (i.e., equ IX.1) yields four time-domain EOFs (TD-EOF). The normalized

eigenvector structures of the two most energetic IES TD-EOFs are depicted in Figure IX.3.

Note that the time variability of either of these TD-EOFs can be derived by computing the

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product of the modal time series (Figure IX. 4) and the normalized eigenvector station

amplitude structure. The geostrophic transport inferred from the eigenmode IES differences

are indicated to the right.

Figure. IX.3 To the left are the normalized structures of the 2 most energetic IES eigenmodes (MODE-1

above and MODE-2 below). Also indicated to the right are the geostrophic transport distributions that are qualitatively consistent the IES dynamic height variability eigenmode structure.

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Figure. IX.4 The amplitude time series (in dyn-m) of the eigenmodes presented in Figure IX.3.

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B. Empirical Orthogonal Functions: Frequency Domain

Consider a 2-dimensional vector time series (t)V

, with components v}{u, , that have been

discretely sampled t N, to form

t)(nV = Vn

(IX.8)

where n = 1,2,...N.

The complex Fourier coefficients are

tN

mn2i-exp Vt

tN

2 = V n

n

^

m

(IX.9a)

where the m refers to the frequencies tN

m2 = f = f m

at which these are computed. and

where the vector details of V^

m

can be expressed a number of different ways ;

eu

eu

u

u

v

u V

2

1

i2

i1

2

1

m

m^

m

(IX.9b)

Complex Fourier coefficients for the components can be expressed in different ways as

discussed in the following.

***********************************

Define the general Cartesian cross-spectral energy density matrix (SDM) for an arbitrary

number of components of

>uu< S = S j*iij , (IX.10)

where the m index, referring to frequency harmonics, has been dropped.

The complex representation of cross-spectral energy density (SDM) is

iQ - C = S ijijij (IX.11)

The normalized SDM or spectrum coherence

)SS(S

2/1

jjii

ijij (IX.12)

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The 2-component Sij is a 2 x 2 Hermitian matrix, where Hermitian implies that γuv = γuv* , is

S

S

vvuv

uvuu

(IX.13)

Consider the following two simple kinds of motion in the Cartesian representation in which

there has been a two-fold decomposition; in (1) frequency and (2) along mutually-orthogonal

axes (cartesian e.g.).

1. Oscillatory Rectilinear Motion

- along an arbitrary line described by

j + i = r , (IX.14)

where α and β are direction cosines.

Figure. IX.4 Hodograph of rectilinear motion

Fourier components of the velocity vector [at some frequency fm] are

)e(a = v )e(a = u io

io

(IX.15)

where the phase of both components θ1 = θ2 = θ.

If θ = 0, then

a = v a = u oo (IX.16a)

or

a

a =v

o

o^

(IX.16b)

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and the SDM, which is real for this case because there is no phase lag between components, is

2*

2

2o >a< = S (IX.17)

Note: If a measured SDM is real, then r

may be deduced. For example, if α=1 and β=0, then

the motion will be only along the x-axis and onlyS uu 11remains.

1. Oscillatory Elliptical Motion

The Fourier components

, expa = v expa = u 2)/ i(o

io

(IX.18)

where the indicates the phase for rotation in either direction;

+ = ACW and - = CW

or V

=

ea

ea

)2

i(o

io

(IX.19)

Figure IX.5 Hodograph for elliptical motion.

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Then the SDM is purely imaginary because of 2

phase lag

2*

2

2o

)i(

i a >a< = S (IX.20)

Note that α = β => circular motion

Important: In general, since all elements of the cartesian SDM are non-zero the values α

and β depend on the cartesian axis orientation; which is a disadvantage. Thus we seek a

more general approach.

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General Fourier Component Representation

e a + e a = V 2211

^

, (IX.21)

where ai (i = 1,2) are complex amplitudes (as before), but now ei

are basis vectors

representing different types of motion (not orthogonal directions.

For example, rectilinear motion along 2 mutually-orthogonal axes are described in terms of

Cartesian system as follows:

1

0 = e ;

0

1 = e 21

(IX.22)

where

(1) the ei

represent motion in the two different directions;

(2) the ai now tell us the amplitude and phase of the type of motion described by unit

vectors.

The ai are found by finding the projection of V

on the respective ei

according to

V e = a

V e = a^

*22

^*11

(IX.23)

In this case it is trivial because there is no phase (i.e. imaginary part) so that

)a (= u = a o11 (IX.24a)

)a (= u = a o22 (IX.24b)

In this case, the generalized Fourier coefficient ai and the ui are the same.

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Figure . IX.6 Cartesian components of a typical three-dimensional vector time series measurement

(Calman, 1978).

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Figure. IX.7 The Cartesian spectrum density matrix for the data shown in Figure. IX.7 (Calman, 1978).

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Figure. IX.8 Rotary representation of the data shown in Figure. IX.7 (Calman, 1978).

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Rotary Representation of V

m

The decomposition in terms of Cartesian components can be rewritten as

expu

expu C +

expu

expu C =

eu

eu =

u

u = V

2)/-i(2

i1

22)/+i(

2

i1

1i

2

i1

2

1^

1

1

1

1

2

1

m (IX.25)

where the basis (or unit) vectors now are

; i-

1 2 = e

i

1 2 = e 2/1-

22/1-

1

(IX.26)

+ -

Hodograph

Figure IX.9 Hodograph for circular motion

Counter-clockwise and clockwise rotating circular rotating unit vector motions and are

orthonormal i.e. the scalar products are

. 0 = e e

1 = e e

2*1

1*1

(IX.27)

We can find the generalized Fourier coefficients ai by projecting the unit vectors onto the

original Cartesian Fourier component vector, i.e.

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} { i)- (1, 2 , )ui + u( 2 = V e = a = a

)u ,u( i-

1 2 .e.i )ui - u( 2 = V e = a = a

2/121

2/1-^

*2_2

212/1

212/1-

^*1+1

m

m

(IX.28a)

where

. eu = u

eu = u2

1

i22

i11

(IX.28b)

Computing the diagonal elements of the generalized spectral energy density matrix

S

S Q2 + >uu< + >uu<

2

1 = >aa< = S = S

- +

between

coherence

Q2 - >uu< + >uu< 2

1 = >aa< = S = S

_

+

122*21

*12

*2_22

122*21

*11

*1+11

(IX.29)

Note the relationship between the generalized and cartesian spectral densities.

A Generalized Spectral Density Matrix S

Then S = Sij = <ai*aj> in which the generalized coherences are:

. |>a*a><a*a|<

>a*a< = 2/1

jjii

jiij (IX.29)

It can be shown that a generalized SDM can be derived from the Cartesian SDM = Sc

according to

e S e = >a*a< S = S jc*

ijiij (IX.30)

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EMPIRICAL EIGENMODE SPECTRA

Let us now generalize the previous approach still further by determining an orthonormal set

of eigenvectors gi

of the Cartesian cross-spectral energy density matrix (SDM) such that

g = g S iiic

, (IX.31)

where the frequency-dependent gi

can be thought of as the kinematical normal modes for

the time series measurements.In the general case, the eigenvectors gi

be found by solving

the eigenvalue problem for each frequency.

Expanding a 2-D Fourier component vector ^

V

in the eigenvectors of the particular cross-

spectral density matrix for a specific frequency:

gg 2211

^

a + a = V

(IX.32)

! !

where λ11/2 λ2

1/2

are real eigenvalues, since the SDM is Hermitian.

H

ere spectrum amplitudes are the original elements and are Sii = λi. We have already shown

that for the generalized representation of the Fourier coefficient vector

ea = ... ea + ea = V iii

2211

^ , (IX.33)

the coherence between modes is computed in terms of the generalized cross-spectral density

matrix (or Cartesian SDM) according to

. e S e >a*a< S jc*

ijiij (IX.34)

By letting g = e ii and recalling the original eigenvalue problem above:

. g g = g S g = S jj*ij

c*iij

(IX.35)

Then from the orthonormal property of gi

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ijjij = S (IX.36)

Because

jifor 0

j=ifor 1 = ij , (IX.37)

Sii = λi (IX.38)

and

0 Sij (IX.39)

The latter showing that the different modes gi

are independent (i.e., all coherences are zero).

But this simplicity comes at the expense of a more complicated hodograph for each

eigenvector.

Empirical Orthogonal Functions for 2-D Linear Motion

;a

a = V SDMCartesian

o

o^

2

2

2 =

o

c aS

Solving for the eigenvalues of the above problem leads to

) + (>a< = 222o1

0 = 2

which shows that only one mode of motion present at this frequency.

The corresponding unit magnitude eigenvector is

; N = g 11

where . )) + (( N2/-122

1

Here g1

represents linear motion along r

- the only coherent motion. The second

eigenvalue is degenerate so that g2

is indeterminant; 0 = g2

.

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Empirical Orthogonal Functions for 2-D Elliptical Motion

ea

ea = V

2)/i(o

io^

Cartesian SDM

2*

2

2o

c

)i(

i >a< = S

Eigenvalues: λ1 = ao2 (α2 + β2) ;

λ2 = 0

Again only one mode of motion

Unit Magnitude Eigenvector:

i N = g 11

where ])+[( = N2/-1222/-1

11

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Figure. IX.10 Empirical mode spectra for the data shown in Figure. IX.7.

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Figure. IX.11 The three-dimensional Fourier vector holograph. The parameters which specify the shape of the hodograph are indicated.

THREE DIMENSIONAL VECTOR

tifme V Re = (t)V

^

m

1-2N/

1=m

where

w(t)

v(t)

u(t)

= (t)V

Complex Fourier Coefficients

T)mn/N2iexp(- (t)V t)]/(N[2 = V nn

^

at tN

m2 = f = f m

with

eu

eu

eu

u

u

u

w

v

u

V

3

2

1

i3

i2

i1

3

2

1

m

m

m^

m

Cartesian spectral energy density matrix (SDM) (subscript m has been dropped)

>uu< S j*iij

where < > � ensemble average.

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Off-diagonal elements of the SDM - complex representation:

Sij = Cij - iQij

co- quad- spectra

Complex Coherence

]SS[

S 2/1

jjii

ijij

Sij is a 3x3 Hermitian (3 real and 3 complex independent elements) matrix here

ww

vvuu

uwuvuu

S

S

S

(Hermitian � lower left off diagonal elements are complex

conjugates of upper right off diagonal elements.)

For Fourier component vectors from two measurements

u

u

u

= u

(k)3

(k)2

(k)1

(k)

with k = 1, 2, 3, ….

The cross-spectral density matrix (CSDM) is

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>u u< = S (1)j

(k)i

1)(kij

This is a 6x6 Hermitian matrix

vectornd2

>uu<

>ww<

>wv><vv<

>wu><vu><uu><wu><vu><uu<

2*2

1*1

1*11

*1

2*12

*12

*11

*11

*11

*1

ELLIPTIC MOTION - in Cartesian representation for one 2-D vector

where, if α = β, then the motion is circular.

In general, let the Fourier component vector, with complex ai, be represented

ea + ea + ea = V 332211

^ m

where ei

are unit vectors - each representing a particular type of motion.

For example, Cartesian

1

0

0

= e

0

1

0

= e

0

0

1

= e 321

In general,

V e = a^

*ii

lorthonorma = ee ijj*i

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where in this case, u = a ii

; which implies rectilinear motion.

Generalized Spectral Density Matrix

>aa< S j*iij

Generalized Coherence

]>aa><aa[<

>aa< 2/1

j*ji

*i

j*i

ij

It can be shown that any generalized SDM can be found from the Cartesian SDM, S(c), by

using the appropriate (orthonormal) unit vectors to transform the spectrum according to

. e S e >aa< S j(c)*

ij*iij

For a multiple-vector time series, the cross-spectrum density matrix generalizes in the same

way in this case, although e and e (2)(1) may be different

e a = U and e a = V(2)i

(2)i

^(1)i

(1)i

3

1=i

^ m

Generalized CSDM >a 2< = S (2)j

(1)*i

(1)(2)ij

where usually e = e (2)i

(1)i .

� Rotational Invariants

Trace KE x 2 = S TrS ii

3

KE x 2 = S STr ii

3

M

)(ui 2

1 = KE and |u| = >u u< = S

22ii

*iii

Determinant of SDM

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)Re2 + || = || - || - (1SSS = |S| 231

223

212332211

where Γ = γ12γ23γ31 and γij are complex coherences.

For 2-D, the Determinant is ))( - (1S S = |S| 2122211H and “a measure

of the incoherent noise”

Degree of Polarization |P|

)STr 2/(1

|S| - 1 P 2

H

M2

is the fraction of non-random, non-isotropic energy

|P| is real and varies between 0 and 1.

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Anisotropy Ratio |A|

P-1

P =A

is the ratio of non-random, non-isotropic energy to the random, isotropic energy (i.e.,

a “signal to noise “ ratio).

Sum of Principal Minors

The 2-D determinant in one of the coordinate planes

|| - 1 SS 2

1 = M M 2

ijjjiiji

ii

3

Mean degree of polarization

)(TrS

M - 1 = P 2

2

Total Squared Quadrature Spectrum

Q 2

1 )S - S( 2/1 Q 2

ijj=i

2jiij

ji

2

,

which is the net amount of rotating energy in a plane.

Rotary Coefficient

STr 2/1

Q C

H

HrH ,

which is the fraction of the net rotating

Mean Rotary Coefficient

TrS 2/1

Q Cr

Multiple Coherence

minor principal rd3 MS

|S| - 1

3333312

ROTARY REPRESENTATION

Fourier vector is decomposed into two counter-rotating circular motions in a plane (usually

the horizontal plane). In 3-D, a linear oscillation that is normal to the plane of rotary vectors

is added

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eu

0

0

C +

0

eu

eu

C +

0

eu

eu

C =

eu

eu

eu

= V3

11

1

3

2

1

i3

3)

2-i(

1

i1

22)/+i(

1

i1

1

i3

i2

i1

^

This can be written in terms of the following unit vectors

0

i

1

2 = e 2/1-1

Counter-Clockwise

0

i-

1

2 = e 2/1-2 Clockwise

1

0

0

e 3 Vertical

Generalized Fourier Coefficients

counterclockwise (+) - )u i-u(2 = V e - a = a 212/1-

^*1+1

clockwise (-) - )u i+u( 2 = V e = a = a 212/1-

^*2-2

vertical - , u = V e = a 3

^*33

where eu = u 1i11

etc.

ELEMENTS OF THE GENERALIZED SDM

can be found using Cartesian Spectral Density Matrix

>aa< S j*1ij

or

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24 February 2009 Chapter IX. Empirical Orthogonal Functions

28

matrixdensity spectralCartesian e S e j(c)*

i

Q2 - >uu< + >uu< 2/1 = >aa< = S = S 122*21

*11

*1+11

STr

Q =Cr Cr) - S(1Tr 2/1

H

uvH

computed from the

Cartesian SDM

Cr) + S(1Tr 2/1 = >aa< = S = S H2*2-22

S = >uu< = >aa< = S w3*33

*333

S+; S- are the rotary spectra

1973 Mooers, of

a"autospectrinner "

Page 29: IX. EMPIRICAL ORTHOGONAL FUNCTIONS · 24 February 2009 Chapter IX. Empirical Orthogonal Functions 3 The variance of amt is the variance of the mth mode. The mth mode eigenvalue m

24 February 2009 Chapter IX. Empirical Orthogonal Functions

29

Rotary coefficient

S + S

S - S = C-+

-+r

i.e. fraction of net rotating energy

Normalized Cross Spectrum

-+2/12211

1212 =

)SS(S =

-+e|| )Cr - S)(1Tr2(/1

C i - )S - S2(/1 = i

2/12H

uvvvuu

Stability )Cr - /(1)Cr - P( = || 2222-+

),S - S/(C 2 = tan uuvvuv-+

where (Mooers)) umautospectr(outer >aa< S S -*+-+12

Remaining Coherence Element for 3-D vectors

)S (S

)S i S( 2

1 =

>aa><aa<

>aa< = 2/1

w

vwuw

3*3

* 2/13

*+

w-+

Rotary Spectrum Representation for a 3-D Vector

S

S

S

w

w--

w+-++