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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
Modern Seismology – Data processing and inversion2
Fourier: Applications
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
Modern Seismology – Data processing and inversion3
Fourier: Applications
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
Modern Seismology – Data processing and inversion4
Fourier: Applications
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
Modern Seismology – Data processing and inversion5
Fourier: Applications
The Fast Fourier Transform
... 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 =
Modern Seismology – Data processing and inversion6
Fourier: Applications
The Fast Fourier Transform
... 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).
Modern Seismology – Data processing and inversion7
Fourier: Applications
The Fast Fourier Transform
.. 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
Modern Seismology – Data processing and inversion8
Fourier: Applications
Spectral synthesis
The
red
trace
is
the
sum
of all blue
traces!The
red
trace
is
the
sum
of all blue
traces!
Modern Seismology – Data processing and inversion9
Fourier: Applications
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)
Modern Seismology – Data processing and inversion10
Fourier: Applications
… remember …
22 ))((*
)sin(cos)sin(cos*
ribaibazzz
rririribaz
i
=−+==
=−−−=
−=−=− φφφ
φφ
Modern Seismology – Data processing and inversion11
Fourier: Applications
The spectrumAmplitude spectrumAmplitude spectrum Phase spectrumPhase spectrum
Four
ier
spac
eFo
urie
rsp
ace
Phys
ical
spac
ePh
ysic
alsp
ace
Modern Seismology – Data processing and inversion12
Fourier: Applications
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
Modern Seismology – Data processing and inversion13
Fourier: Applications
Frequencies in seismograms
Modern Seismology – Data processing and inversion14
Fourier: Applications
Amplitude spectrum Eigenfrequencies
Modern Seismology – Data processing and inversion15
Fourier: Applications
Sound of an instrument
a‘ - 440Hz
Modern Seismology – Data processing and inversion16
Fourier: Applications
Instrument Earth
26.-29.12.2004 (FFB )
0 S2 – Earth‘s gravest tone T=3233.5s =53.9min
Theoretical eigenfrequencies
Modern Seismology – Data processing and inversion17
Fourier: Applications
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
Modern Seismology – Data processing and inversion18
Fourier: Applications
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?
Modern Seismology – Data processing and inversion19
Fourier: Applications
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
Modern Seismology – Data processing and inversion20
Fourier: Applications
Puls-width and Frequency Bandwidthtime (space) spectrum
Nar
rowi
ngph
ysic
alsi
gnal
Wid
enin
gfr
eque
ncy
band
Modern Seismology – Data processing and inversion21
Fourier: Applications
Spectral analysis: an Example
24 hour
ground
motion, do you
see
any
signal?
Modern Seismology – Data processing and inversion22
Fourier: Applications
Seismo-Weather
Running spectrum
of the
same
data
Modern Seismology – Data processing and inversion23
Fourier: Applications
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
→
Modern Seismology – Data processing and inversion24
Fourier: Applications
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
Modern Seismology – Data processing and inversion25
Fourier: Applications
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 ...
Modern Seismology – Data processing and inversion26
Fourier: Applications
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 ...
Modern Seismology – Data processing and inversion27
Fourier: Applications
... 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 ...
Modern Seismology – Data processing and inversion28
Fourier: Applications
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
Modern Seismology – Data processing and inversion29
Fourier: Applications
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