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ِapproximation series
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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
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