fourier theorymodern seismology – data processing and inversion 1 fourier series and transforms...

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.


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


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


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


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)()(


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)


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


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


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


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


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 - ‘+’


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


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


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.


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:


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


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()(


Fourier Theory

Chebyshev collocation pointsThese extrema are not equidistant (like the Fourier extrema)

nkn

kx extk ,...,3,2,1,0),cos()(

k

x(k)


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


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


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 )!


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


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


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


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!


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


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


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


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


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

fourier theorymodern seismology – data processing and inversion 1 fourier series and transforms...

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