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Transcript of IX. EMPIRICAL ORTHOGONAL FUNCTIONS · 24 February 2009 Chapter IX. Empirical Orthogonal Functions 3...
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
24 February 2009 Chapter IX EOFs Notes 8Wendell S. Brown
2
� 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)
24 February 2009 Chapter IX. Empirical Orthogonal Functions
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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
24 February 2009 Chapter IX. Empirical Orthogonal Functions
4
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
24 February 2009 Chapter IX. Empirical Orthogonal Functions
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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.
24 February 2009 Chapter IX. Empirical Orthogonal Functions
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Figure. IX.4 The amplitude time series (in dyn-m) of the eigenmodes presented in Figure IX.3.
24 February 2009 Chapter IX. Empirical Orthogonal Functions
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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)
24 February 2009 Chapter IX. Empirical Orthogonal Functions
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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)
24 February 2009 Chapter IX. Empirical Orthogonal Functions
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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.
24 February 2009 Chapter IX. Empirical Orthogonal Functions
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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.
24 February 2009 Chapter IX. Empirical Orthogonal Functions
11
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.
24 February 2009 Chapter IX. Empirical Orthogonal Functions
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Figure . IX.6 Cartesian components of a typical three-dimensional vector time series measurement
(Calman, 1978).
24 February 2009 Chapter IX. Empirical Orthogonal Functions
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Figure. IX.7 The Cartesian spectrum density matrix for the data shown in Figure. IX.7 (Calman, 1978).
24 February 2009 Chapter IX. Empirical Orthogonal Functions
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Figure. IX.8 Rotary representation of the data shown in Figure. IX.7 (Calman, 1978).
24 February 2009 Chapter IX. Empirical Orthogonal Functions
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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.
24 February 2009 Chapter IX. Empirical Orthogonal Functions
16
} { 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)
24 February 2009 Chapter IX. Empirical Orthogonal Functions
17
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
24 February 2009 Chapter IX. Empirical Orthogonal Functions
18
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
.
24 February 2009 Chapter IX. Empirical Orthogonal Functions
19
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
24 February 2009 Chapter IX. Empirical Orthogonal Functions
20
Figure. IX.10 Empirical mode spectra for the data shown in Figure. IX.7.
24 February 2009 Chapter IX. Empirical Orthogonal Functions
21
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.
24 February 2009 Chapter IX. Empirical Orthogonal Functions
22
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
24 February 2009 Chapter IX. Empirical Orthogonal Functions
23
>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
24 February 2009 Chapter IX. Empirical Orthogonal Functions
24
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
24 February 2009 Chapter IX. Empirical Orthogonal Functions
25
)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.
24 February 2009 Chapter IX. Empirical Orthogonal Functions
26
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
24 February 2009 Chapter IX. Empirical Orthogonal Functions
27
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
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 "
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+-++