Seismic Data ProcessingLecture 3

Approximation SeriesPrepared by

Dr. Amin E. Khalil

Today's Agenda

• Complex Numbers

• Vectors• Linear vector spaces• Linear systems

• Matrices• Determinants• Eigenvalue problems• Singular values• Matrix inversion

• Series • Taylor• Fourier

• Delta Function

• Fourier integrals

How to determine Eigenvectors?

Matrix aplications

• Stress and strain tensors• Calculating interpolation or differential operators for finite-

difference methods• Eigenvectors and eigenvalues for deformation and stress problems

(e.g. boreholes)• Norm: how to compare data with theory• Matrix inversion: solving for tomographic images• Measuring strain and rotations

Taylor Series

Many (mildly or wildly nonlinear) physical systems are transformed to linear systems by using Taylor series

1

)(

32

!

)(

...'''6

1''

2

1')()(

i

ii

dxi

xf

dxfdxfdxfxfdxxf

provided that all derivatives of f(x) are continuous and exist in the interval [x,x+h]

Examples of Taylor series

!6!4!2

1)cos(642 xxx

x

!7!5!3

)sin(753 xxx

xx

!3!2

132 xx

xe x

Reference: http://numericalmethods.eng.usf.edu

What does this mean in plain English?

As Archimedes would have said, “Give me the value of the function at a single point, and the value of all (first, second, and so on) its derivatives at that single point, and I can give you the value of the function at any other point”

Example

Find the value of 6f given that ,1254 f ,744 f

,304 f 64 f and all other higher order derivativesof xf at 4x are zero.

Solution: !3!2

32 hxf

hxfhxfxfhxf

4x 246 h

!3

24

!2

2424424

32

fffff

!3

26

!2

2302741256

32

f

860148125

341

Fourier SeriesFourier series assume a periodic function …. (here:

symmetric, zero at both ends)

,1

22sin)( 0 n

L

nxaaxf n

n

L

n

L

dxL

xnxf

La

dxxfL

a

0

0

0

sin)(2

)(1

What is periodic function?

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function which is not periodic is called aperiodic.

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 F(x) weighted by some coefficients an.

)()()(0

xaxgxfN

iiiN

Strategy

... and we are looking for optimal functions in a least squares (l2) sense ...

... a good choice for the basis functions F(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:

!Min)()()()(

2/12

2

b

a

NN dxxgxfxgxf

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

Periodic function example

-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 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

... 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

... 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

Importance of Fourier Series

• Any filtering … low-, high-, bandpass• Generation of random media• Data analysis for periodic contributions • Tidal forcing• Earth’s rotation• Electromagnetic noise• Day-night variations

• Pseudospectral methods for function approximation and derivatives

Delta Function

… so weird but so useful …

00)(,1)(

)0()()(

tfürttdt

fdttft

det

ta

at

afattf

ti

2

1)(

)(1

)(

)()()(

for

Delta Function

As input to any system (the Earth, a seismometers …)

As description for seismic source signals in time and space, e.g., with Mij the source moment tensor

As input to any linear system -> response Function, Green’s function

)()(),( 00 xxMx ttts

Fourier Integrals

The basis for the spectral analysis (described in the continuous world) is the transform pair:

dtetfF

deFtf

ti

ti

)()(

)(2

1)(

Fourier Integral (transform)

• Any filtering … low-, high-, bandpass• Generation of random media• Data analysis for periodic contributions • Tidal forcing• Earth’s rotation• Electromagnetic noise• Day-night variations

• Pseudospectral methods for function approximation and derivatives

Thank you