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Modern Seismology – Data processing and inversion1

Fourier: Applications

Fourier Transform: Applications

• Seismograms• Eigenmodes of the Earth• Time derivatives of

seismograms• The pseudo-spectral

method for acoustic wave propagation



Fourier: Space and TimeSpace

x space

variableL spatial

wavelength

k=2π/λ

spatial

wavenumberF(k) wavenumber

spectrum

Spacex space

variable

L spatial

wavelengthk=2π/λ

spatial

wavenumber

F(k) wavenumber

spectrum

Timet Time variableT periodf frequencyω=2πf

angular

frequency

Timet Time variableT periodf frequencyω=2πf

angular

frequency

Fourier

integralsFourier

integrals

With

the

complex

representation

of sinusoidal

functions

eikx

(or (eiwt) the

Fourier

transformation

can

be

written

as:

With

the

complex

representation

of sinusoidal

functions

eikx

(or (eiwt) the

Fourier

transformation

can

be

written

as:

∫

∫∞

∞−

∞

∞−

−

=

=

dxexfkF

dxekFxf

ikx

ikx

)(21)(

)(21)(

π

π



The Fourier Transform discrete vs. continuous

∫

∫∞

∞−

∞

∞−

−

=

=

dxexfkF

dxekFxf

ikx

ikx

)(21)(

)(21)(

π

π

1,...,1,0,

1,...,1,0,1

/21

0

/21

0

−==

−==

∑

∑−

=

−−

=

NkeFf

NkefN

F

NikjN

jjk

NikjN

jjk

π

π

discrete

continuousWhatever

we

do on the

computer

with

data

will be

based

on the

discrete

Fourier

transform

Whatever

we

do on the computer

with

data

will

be

based

on the

discrete Fourier

transform



The Fast Fourier Transform

... the latter approach became interesting with the introduction of theFast Fourier Transform (FFT). What’s so fast about it ?

The FFT originates from a paper by Cooley and Tukey (1965, Math. Comp. vol 19 297-301) which revolutionised all fields where Fourier transforms where essential to progress.

The discrete Fourier Transform can be written as

1,...,1,0,ˆ

1,...,1,0,1ˆ

/21

0

/21

0

−==

−==

∑

∑−

=

−−

=

Nkeuu

NkeuN

u

NikjN

jjk

NikjN

jjk

π

π




... this can be written as matrix-vector products ...for example the inverse transform yields ...

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−

−

−

1

2

1

0

1

2

1

0

)1(1

22642

132

ˆ

ˆˆˆ

1

11

11111

2

NNNN

N

N

u

uuu

u

uuu

M

M

M

M

LLL

MMM

MMM

K

K

K

ωω

ωωωωωωωω

.. where ...

Nie /2πω =




... the FAST bit is recognising that the full matrix - vector multiplicationcan be written as a few sparse matrix - vector multiplications

(for details see for example Bracewell, the Fourier Transform and its applications, MacGraw-Hill) with the effect that:

Number of multiplicationsNumber of multiplications

full matrix FFT

N2 2Nlog2 N

this has enormous implications for large scale problems.Note: the factorisation becomes particularly simple and effective

when N is a highly composite number (power of 2).




.. the right column can be regarded as the speedup of an algorithm when the FFT is used instead of the full system.

Number of multiplicationsNumber of multiplications

Problem full matrix FFT Ratio full/FFT

1D (nx=512) 2.6x105 9.2x103 28.41D (nx=2096) 94.981D (nx=8384) 312.6



Spectral synthesis

The

red

trace

is

the

sum

of all blue

traces!The

red

trace

is

the

sum

of all blue

traces!



Phase and amplitude spectrum

)()()( ωωω Φ= ieFF

The spectrum consists of two real-valued functions of angular frequency, the amplitude spectrum mod (F(ω)) and the phase spectrum φ(ω)

In many cases the amplitude spectrum is the most important part to be considered. However there are cases where the phase spectrum plays an important role (-> resonance, seismometer)



… remember …

22 ))((*

)sin(cos)sin(cos*

ribaibazzz

rririribaz

i

=−+==

=−−−=

−=−=− φφφ

φφ



The spectrumAmplitude spectrumAmplitude spectrum Phase spectrumPhase spectrum

Four

ier

spac

eFo

urie

rsp

ace

Phys

ical

spac

ePh

ysic

alsp

ace



The Fast Fourier Transform (FFT)

Most processing

tools (e.g. octave, Matlab,

Mathematica, Fortran, etc) have

intrinsic

functions for

FFTs

Most processing

tools (e.g. octave, Matlab,

Mathematica, Fortran, etc) have

intrinsic

functions for

FFTs

>> help fft

FFT Discrete Fourier transform.FFT(X) is the discrete Fourier transform (DFT) of vector X. Formatrices, the FFT operation is applied to each column. For N-Darrays, the FFT operation operates on the first non-singletondimension.

FFT(X,N) is the N-point FFT, padded with zeros if X has lessthan N points and truncated if it has more.

FFT(X,[],DIM) or FFT(X,N,DIM) applies the FFT operation across thedimension DIM.

For length N input vector x, the DFT is a length N vector X,with elements

NX(k) = sum x(n)*exp(-j*2*pi*(k-1)*(n-1)/N), 1 <= k <= N.

n=1The inverse DFT (computed by IFFT) is given by

Nx(n) = (1/N) sum X(k)*exp( j*2*pi*(k-1)*(n-1)/N), 1 <= n <= N.

k=1

See also IFFT, FFT2, IFFT2, FFTSHIFT.

>> help fft

FFT Discrete Fourier transform.FFT(X) is the discrete Fourier transform (DFT) of vector X. Formatrices, the FFT operation is applied to each column. For N-Darrays, the FFT operation operates on the first non-singletondimension.

FFT(X,N) is the N-point FFT, padded with zeros if X has lessthan N points and truncated if it has more.

FFT(X,[],DIM) or FFT(X,N,DIM) applies the FFT operation across thedimension DIM.

For length N input vector x, the DFT is a length N vector X,with elements

NX(k) = sum x(n)*exp(-j*2*pi*(k-1)*(n-1)/N), 1 <= k <= N.

n=1The inverse DFT (computed by IFFT) is given by

Nx(n) = (1/N) sum X(k)*exp( j*2*pi*(k-1)*(n-1)/N), 1 <= n <= N.

k=1

See also IFFT, FFT2, IFFT2, FFTSHIFT.

Matlab

FFT



Frequencies in seismograms



Amplitude spectrum Eigenfrequencies



Sound of an instrument

a‘ - 440Hz



Instrument Earth

26.-29.12.2004 (FFB )

0 S2 – Earth‘s gravest tone T=3233.5s =53.9min

Theoretical eigenfrequencies



Fourier Spectra: Main Cases random signals

Random

signals

may

contain

all frequencies. A spectrum

with constant

contribution

of all frequencies

is

called

a white

spectrum

Random

signals

may

contain

all frequencies. A spectrum

with constant

contribution

of all frequencies

is

called

a white

spectrum



Fourier Spectra: Main Cases Gaussian signals

The

spectrum

of a Gaussian

function

will itself

be

a Gaussian function. How

does

the

spectrum

change, if

I make

the

Gaussian

narrower

and narrower?

The

spectrum

of a Gaussian

function

will itself

be

a Gaussian function. How

does

the

spectrum

change, if

I make

the

Gaussian

narrower

and narrower?



Fourier Spectra: Main Cases Transient waveform

A transient

wave

form is

a wave

form limited

in time (or

space) in comparison

with

a harmonic

wave

form that

is

infinite

A transient

wave

form is

a wave

form limited

in time (or

space) in comparison

with

a harmonic

wave

form that

is

infinite



Puls-width and Frequency Bandwidthtime (space) spectrum

Nar

rowi

ngph

ysic

alsi

gnal

Wid

enin

gfr

eque

ncy

band



Spectral analysis: an Example

24 hour

ground

motion, do you

see

any

signal?



Seismo-Weather

Running spectrum

of the

same

data



Some properties of FT• FT is linear

signals can be treated as the sum of several signals, the transform will be the sum of their transforms

• FT of a real signalshas symmetry properties

the negative frequencies can be obtained from symmetry properties

• Shifting corresponds to changing the phase (shift theorem)

• Derivative)(*)( ωω FF =−

)()()()(

tfeaFFeatfai

ai

ω

ω

ω

ω−

−

→−

→−

)()()( ωω Fitfdtd n

n

→



Fourier Derivatives

∫

∫∞

∞−

−

∞

∞−

−

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂=∂

dkekikF

dkekFxf

ikx

ikxxx

)(

)()(

.. let us recall the definition of the derivative using Fourier integrals ...

... we could either ...

1) perform this calculation in the space domain by convolution

2) actually transform the function f(x) in the k-domain and back



Acoustic Wave Equation - Fourier Method

let us take the acoustic wave equation with variable density

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂=∂ pp

c xxt ρρ11 2

2

the left hand side will be expressed with our standard centered finite-difference approach

[ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂=−+−+ pdttptpdttp

dtc xx ρρ1)()(2)(1

22

... leading to the extrapolation scheme ...



Acoustic Wave Equation - Fourier Method

where the space derivatives will be calculated using the Fourier Method. The highlighted term will be calculated as follows:

)()(21)( 22 dttptppdtcdttp xx −−+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂=+

ρρ

njx

nnnj PPikPP ∂→→→→→ − 1FFTˆˆFFT υυυ

multiply by 1/ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂→→⎟⎟

⎠

⎞⎜⎜⎝

⎛∂→⎟⎟

⎠

⎞⎜⎜⎝

⎛∂→→∂ − n

jxx

n

x

n

xnjx PPikPP

ρρρρ υυ

υ

1FFTˆ1ˆ1FFT1 1

... then extrapolate ...



... and the first derivative using FFTs ...

function df=sder1d(f,dx)% SDER1D(f,dx) spectral derivative of vectornx=max(size(f));

% initialize kkmax=pi/dx;dk=kmax/(nx/2);for i=1:nx/2, k(i)=(i)*dk; k(nx/2+i)=-kmax+(i)*dk; endk=sqrt(-1)*k;

% FFT and IFFTff=fft(f); ff=k.*ff; df=real(ifft(ff));

.. simple and elegant ...



Fourier Method - Comparison with FD - Table

0

20

40

60

80

100

120

140

160

5 Hz 10 Hz 20 Hz

3 point5 pointFourier

Difference (%) between numerical and analytical solution as a function of propagating frequency

Simulation time5.4s7.8s

33.0s



500 1000 15000

1

2

3

4

5

6

7

8

9

10

500 1000

1

2

3

4

5

6

7

8

9

10

500 1000 15000

1

2

3

4

5

6

7

8

9

10

Numerical solutions and Green’s Functions

3 point operator 5 point operator Fourier MethodFr

eque

ncy

incr

ease

s

Impu

lse

resp

onse

(ana

lytic

al) c

onco

lved

with

sou

rce

Impu

lse

resp

onse

(num

eric

al c

onvo

lved

with

sou

rce

fourier transform: applications - uni-muenchen.deigel/... · >> help fft fft discrete fourier...

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