(COMPUTER) Recall equation (7.17) for the vibroseis sweep function:
(a) Solve for f0 and b in the case of a 20-s long sweep between 1 and 4 Hz. Hint:
b = 3/20 is incorrect! Think about how rapidly the phase changes with time.
(b) Compute and plot v(t) for this sweep function. Assume that A(t) =
sin2(πt/20) (this is termed a Hanning taper; note that it goes smoothly to
zero at t = 0 and t = 20 s). Check your results and make sure that you have
the right period at each end of the sweep.
(c) Compute and plot the autocorrelation of v(t) between −2 and 2 s.
(d) Repeat (b) and (c), but this time assume that A(t) is only a short 2-s long taper
at each end of the sweep, that is, A(t) = sin2(πt/4) for 0 <>
for 2 ≤ t ≤ 18, and A(t) = sin2[π(20 − t)/4] for 18 <>
milder taper leads to more extended sidelobes in the autocorrelation function.
(e) What happens to the pulse if autocorrelation is applied a second time to
the autocorrelation of v(t)? To answer this, compute and plot [v(t) v(t)]
[v(t) v(t)] using v(t) from part (b). Is this a way to produce a more impulsive
wavelet?