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

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

Time domain wraparound (aliasing)f(t)

DFT-1{ }F( )ω

T =1/Δmax f

F( )ω

Δf

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

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

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