1. 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

5. 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/T max – if T max is large (due to high order multiples, etc), then unless you use very small Δf you will always be undersampling

6. Time domain wraparound (aliasing)

7. 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

8. 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 e t/τ : the unaliased components (n=0) are unaffected, the aliased components are suppressed

9. Time domain wraparound (aliasing)