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Modern Seismology – Data processing and inversion1
Fourier Theory
Fourier Series and Transforms
• Orthogonal functions• Fourier Series• Discrete Fourier Series• Fourier Transform• Chebyshev polynomials
Scope: we are trying to approximate an arbitrary function and obtain basis functions with appropriate coefficients.
Modern Seismology – Data processing and inversion2
Fourier Theory
Fourier Series
The Problem
we are trying to approximate a function f(x) by another function gn(x) which consists of a sum over N orthogonal functions (x) weighted by some coefficients an.
)()()(
0
xaxgxfN
iiiN
Modern Seismology – Data processing and inversion3
Fourier Theory
... and we are looking for optimal functions in a least squares (l2) sense ...
... a good choice for the basis functions (x) are orthogonal functions. What are orthogonal functions? Two functions f and g are said to be
orthogonal in the interval [a,b] if
b
a
dxxgxf 0)()(
How is this related to the more conceivable concept of orthogonal vectors? Let us look at the original definition of integrals:
The Problem
!Min)()()()(
2/12
2
b
a
NN dxxgxfxgxf
Modern Seismology – Data processing and inversion4
Fourier Theory
Orthogonal Functions
... where x0=a and xN=b, and xi-xi-1=x ...If we interpret f(xi) and g(xi) as the ith components of an N component
vector, then this sum corresponds directly to a scalar product of vectors. The vanishing of the scalar product is the condition for orthogonality of
vectors (or functions).
N
iii
b
aN
xxgxfdxxgxf1
)()(lim)()(
figi
0 ii
iii gfgf
Modern Seismology – Data processing and inversion5
Fourier Theory
Periodic functions
-15 -10 -5 0 5 10 15 200
10
20
30
40
Let us assume we have a piecewise continuous function of the form
)()2( xfxf
2)()2( xxfxf
... we want to approximate this function with a linear combination of 2 periodic functions:
)sin(),cos(),...,2sin(),2cos(),sin(),cos(,1 nxnxxxxx
N
kkkN kxbkxaaxgxf
10 )sin()cos(
2
1)()(
Modern Seismology – Data processing and inversion6
Fourier Theory
Orthogonality... are these functions orthogonal ?
0,00)sin()cos(
0
0,,0
)sin()sin(
0
02
0
)cos()cos(
kjdxkxjx
kj
kjkj
dxkxjx
kj
kj
kj
dxkxjx
... YES, and these relations are valid for any interval of length 2.Now we know that this is an orthogonal basis, but how can we obtain the
coefficients for the basis functions?
from minimising f(x)-g(x)
Modern Seismology – Data processing and inversion7
Fourier Theory
Fourier coefficientsoptimal functions g(x) are given if
0)()(!Min)()(22
xfxgorxfxg nan k
leading to
... with the definition of g(x) we get ...
dxxfkxbkxaaa
xfxga
N
kkk
kn
k
2
10
2)()sin()cos(
2
1)()(
2
Nkdxkxxfb
Nkdxkxxfa
kxbkxaaxg
k
k
N
kkkN
,...,2,1,)sin()(1
,...,1,0,)cos()(1
with)sin()cos(2
1)(
10
Modern Seismology – Data processing and inversion8
Fourier Theory
Fourier approximation of |x|... Example ...
.. and for n<4 g(x) looks like
leads to the Fourier Serie
...
5
)5cos(
3
)3cos(
1
)cos(4
2
1)(
222
xxxxg
xxxf ,)(
-20 -15 -10 -5 0 5 10 15 200
1
2
3
4
Modern Seismology – Data processing and inversion9
Fourier Theory
Fourier approximation of x2
... another Example ...
20,)( 2 xxxf
.. and for N<11, g(x) looks like
leads to the Fourier Serie
N
kN kx
kkx
kxg
12
2
)sin(4
)cos(4
3
4)(
-10 -5 0 5 10 15-10
0
10
20
30
40
Modern Seismology – Data processing and inversion10
Fourier Theory
Fourier - discrete functions
iN
xi2
.. the so-defined Fourier polynomial is the unique interpolating function to the function f(xj) with N=2m
it turns out that in this particular case the coefficients are given by
,...3,2,1,)sin()(2
,...2,1,0,)cos()(2
1
*
1
*
kkxxfN
b
kkxxfN
a
N
jjj
N
jjj
k
k
)cos(2
1)sin()cos(
2
1)( *
1
1
****0 kxakxbkxaaxg m
m
km kk
... what happens if we know our function f(x) only at the points
Modern Seismology – Data processing and inversion11
Fourier Theory
)()(*iim xfxg
Fourier - collocation points... with the important property that ...
... in our previous examples ...
-10 -5 0 5 100
0.5
1
1.5
2
2.5
3
3.5
f(x)=|x| => f(x) - blue ; g(x) - red; xi - ‘+’
Modern Seismology – Data processing and inversion12
Fourier Theory
Fourier series - convergencef(x)=x2 => f(x) - blue ; g(x) - red; xi - ‘+’
-10 -5 0 5 10 15-10
0
10
20
30
40
50 N = 4
-10 -5 0 5 10 15-10
0
10
20
30
40
50 N = 8
Modern Seismology – Data processing and inversion13
Fourier Theory
Fourier series - convergencef(x)=x2 => f(x) - blue ; g(x) - red; xi - ‘+’
-10 -5 0 5 10 15-10
0
10
20
30
40
50 N = 16
-10 -5 0 5 10 15-10
0
10
20
30
40
50 N = 32
Modern Seismology – Data processing and inversion14
Fourier Theory
Gibb’s phenomenonf(x)=x2 => f(x) - blue ; g(x) - red; xi - ‘+’
0 0.5 1 1.5-6
-4
-2
0
2
4
6
N = 32
0 0.5 1 1.5-6
-4
-2
0
2
4
6
N = 16
0 0.5 1 1.5-6
-4
-2
0
2
4
6
N = 64
0 0.5 1 1.5-6
-4
-2
0
2
4
6
N = 128
0 0.5 1 1.5-6
-4
-2
0
2
4
6
N = 256
The overshoot for equi-spaced Fourier
interpolations is 14% of the step height.
Modern Seismology – Data processing and inversion15
Fourier Theory
Chebyshev polynomialsWe have seen that Fourier series are excellent for interpolating (and differentiating) periodic functions defined on a regularly spaced grid. In many circumstances physical phenomena which are not periodic (in space) and occur in a limited area. This quest leads to the use of Chebyshev polynomials.
We depart by observing that cos(n) can be expressed by a polynomial in cos():
1cos8cos8)4cos(
cos3cos4)3cos(
1cos2)2cos(
24
3
2
... which leads us to the definition:
Modern Seismology – Data processing and inversion16
Fourier Theory
Chebyshev polynomials - definition
NnxxxTTn nn ],1,1[),cos(),())(cos()cos(
... for the Chebyshev polynomials Tn(x). Note that because of x=cos() they are defined in the interval [-1,1] (which - however - can be extended to ). The first polynomials are
0
244
33
22
1
0
and]1,1[for1)(
where188)(
34)(
12)(
)(
1)(
NnxxT
xxxT
xxxT
xxT
xxT
xT
n
Modern Seismology – Data processing and inversion17
Fourier Theory
Chebyshev polynomials - GraphicalThe first ten polynomials look like [0, -1]
The n-th polynomial has extrema with values 1 or -1 at
0 0.2 0.4 0.6 0.8 1-1
-0.5
0
0.5
1
x
T_
n(x)
nkn
kx extk ,...,3,2,1,0),cos()(
Modern Seismology – Data processing and inversion18
Fourier Theory
Chebyshev collocation pointsThese extrema are not equidistant (like the Fourier extrema)
nkn
kx extk ,...,3,2,1,0),cos()(
k
x(k)
Modern Seismology – Data processing and inversion19
Fourier Theory
Chebyshev polynomials - orthogonality... are the Chebyshev polynomials orthogonal?
02
1
1
,,
0
02/
0
1)()( Njk
jkfor
jkfor
jkfor
x
dxxTxT jk
Chebyshev polynomials are an orthogonal set of functions in the interval [-1,1] with respect to the weight functionsuch that
21/1 x
... this can be easily verified noting that
)cos()(),cos()(
sin,cos
jxTkxT
ddxx
jk
Modern Seismology – Data processing and inversion20
Fourier Theory
Chebyshev polynomials - interpolation... we are now faced with the same problem as with the Fourier series. We want to approximate a function f(x), this time not a
periodical function but a function which is defined between [-1,1]. We are looking for gn(x)
)()(2
1)()(
100 xTcxTcxgxf
n
kkkn
... and we are faced with the problem, how we can determine the coefficients ck. Again we obtain this by finding the extremum
(minimum)
01
)()(2
1
1
2
x
dxxfxg
c nk
Modern Seismology – Data processing and inversion21
Fourier Theory
Chebyshev polynomials - interpolation... to obtain ...
nkx
dxxTxfc kk ,...,2,1,0,
1)()(
2 1
12
... surprisingly these coefficients can be calculated with FFT techniques, noting that
nkdkfck ,...,2,1,0,cos)(cos2
0
... and the fact that f(cos) is a 2-periodic function ...
nkdkfck ,...,2,1,0,cos)(cos1
... which means that the coefficients ck are the Fourier coefficients ak of the periodic function F()=f(cos )!
Modern Seismology – Data processing and inversion22
Fourier Theory
Chebyshev - discrete functions
iN
xi
cos
... leading to the polynomial ...
in this particular case the coefficients are given by
2/,...2,1,0,)cos()(cos2
1
* NkkfN
cN
jjjk
m
kkkm xTcTcxg
1
**0
* )(2
1)( 0
... what happens if we know our function f(x) only at the points
... with the property
N0,1,2,...,jj/N)cos(xat)()( j* xfxgm
Modern Seismology – Data processing and inversion23
Fourier Theory
Chebyshev - collocation points - |x|
f(x)=|x| => f(x) - blue ; gn(x) - red; xi - ‘+’
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1 N = 8
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1 N = 16
8 points
16 points
Modern Seismology – Data processing and inversion24
Fourier Theory
Chebyshev - collocation points - |x|f(x)=|x| => f(x) - blue ; gn(x) - red; xi - ‘+’
32 points
128 points
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1 N = 32
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1 N = 128
Modern Seismology – Data processing and inversion25
Fourier Theory
Chebyshev - collocation points - x2
f(x)=x2 => f(x) - blue ; gn(x) - red; xi - ‘+’
8 points
64 points
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2 N = 8
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2 N = 64
The interpolating function gn(x) was shifted by a small amount to be visible at all!
Modern Seismology – Data processing and inversion26
Fourier Theory
Chebyshev vs. Fourier - numerical
f(x)=x2 => f(x) - blue ; gN(x) - red; xi - ‘+’
This graph speaks for itself ! Gibb’s phenomenon with Chebyshev?
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1 N = 16
0 2 4 6-5
0
5
10
15
20
25
30
35
N = 16
Chebyshev Fourier
Modern Seismology – Data processing and inversion27
Fourier Theory
Chebyshev vs. Fourier - Gibb’s
f(x)=sign(x-) => f(x) - blue ; gN(x) - red; xi - ‘+’
Gibb’s phenomenon with Chebyshev? YES!
Chebyshev Fourier
-1 -0.5 0 0.5 1-1.5
-1
-0.5
0
0.5
1
1.5 N = 16
0 2 4 6-1.5
-1
-0.5
0
0.5
1
1.5 N = 16
Modern Seismology – Data processing and inversion28
Fourier Theory
Chebyshev vs. Fourier - Gibb’s
f(x)=sign(x-) => f(x) - blue ; gN(x) - red; xi - ‘+’
Chebyshev Fourier
-1 -0.5 0 0.5 1-1.5
-1
-0.5
0
0.5
1
1.5 N = 64
0 2 4 6 8-1.5
-1
-0.5
0
0.5
1
1.5 N = 64
Modern Seismology – Data processing and inversion29
Fourier Theory
Fourier vs. Chebyshev Fourier Chebyshev
iN
xi2
iN
xi
cos
periodic functions limited area [-1,1]
)sin(),cos( nxnx
cos
),cos()(
x
nxTn
)cos(2
1
)sin()cos(
2
1)(
*
1
1
**
**0
kxa
kxbkxa
axg
m
m
k
m
kk
m
kkkm xTcTcxg
1
**0
* )(2
1)( 0
collocation points
domain
basis functions
interpolating function
Modern Seismology – Data processing and inversion30
Fourier Theory
Fourier vs. Chebyshev (cont’d)Fourier Chebyshev
coefficients
some properties
N
jjj
N
jjj
kxxfN
b
kxxfN
a
k
k
1
*
1
*
)sin()(2
)cos()(2
N
jjj kf
Nck
1
* )cos()(cos2
• Gibb’s phenomenon for discontinuous functions
• Efficient calculation via FFT
• infinite domain through periodicity
• limited area calculations
• grid densification at boundaries
• coefficients via FFT
• excellent convergence at boundaries
• Gibb’s phenomenon