choosing parameters for frequency domain forward modelling we will be running frequency domain...
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![Page 1: Choosing parameters for frequency domain forward modelling We will be running frequency domain software – part of the waveform tomography package At this](https://reader036.vdocuments.mx/reader036/viewer/2022083008/56649e9d5503460f94b9d6da/html5/thumbnails/1.jpg)
Choosing parameters for frequency domain forward modelling We will be running frequency domain software –
part of the waveform tomography package At this stage we only want to demonstrate
forward modelling Before we use the software, we will review the
basic design steps for frequency domain finite differences
Manual pages contain more detailed information
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Time domain wraparound (aliasing) in the time domain, if we undersample (use too large a
Δt) then high frequencies “wrap” around the frequency
axis and alias as low frequencies
in the frequency domain, if we don’t sample adequately
(i.e., use too large a Δf), then “time wraparound”, or “time
aliasing” occurs
we choose Δf=1/Tmax – if Tmax is large (due to high order
multiples, etc), then unless you use very small Δf you will
always be undersampling
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Time domain wraparound (aliasing)f(t)
DFT-1{ }F( )ω
T =1/Δmax f
F( )ω
Δf
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Time domain wraparound (aliasing) due to periodicity in any discrete Fourier series
if f(t) is non-zero for time greater than Tmax, the late time samples will alias at early time
prevent this by using a complex-valued frequency, i.e., we compute
differencing operators must use complex frequencies
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Time domain wraparound (aliasing) F(ω') is just the Fourier transform of f(t)e-t/τ
thus
the time function has effectively been multiplied by a decaying exponential to recover the desired function, we multiply by et/τ :
the unaliased components (n=0) are unaffected, the aliased components are suppressed
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Time domain wraparound (aliasing)f(t)
F( )ω
Δf
f(t)
e-t/τ
f(t)e-t/τ
DFT-1{ }F( )
e
ω'
+t/τ
T =1/Δmax f
DFT-1{ }F( )
e
ω'
+t/τ
T =1/Δmax f
DFT-1{ }
*
F( )
e
ω'
+t/τ
T =1/Δmax f