(COMPUTER) A common pulse shape used in reflection seismic modeling is the
Ricker wavelet, defined as
where fp is the peak frequency in the spectrum of the wavelet.
(a) Using a digitization rate of 200 samples/s and assuming fp = 4 Hz, make a
plot of sR(t) and its time derivative between −0.5 and 0.5 s.
(b) Following Huygens’principle, model the plane reflector shown in Figure 7.23
as a large number of point sources. Use a velocity of 4 km/s, and a coincident source and receiver located 2 km from the center of a 10 km by 10 km
plane, with secondary Huygens sources spaced every 0.1 km on the plane
(i.e., 101 points in each direction, for a total of 10 201 sources). Construct
and plot a synthetic seismogram representing the receiver response to the
Ricker wavelet from part (a) by summing the contribution from each secondary source. At each point, compute the two-way travel time from/to the
source/receiver and the geometric spreading factor 1/r2. Add the Ricker
wavelet, centered on the two-way time and scaled by the geometric spreading factor, to your synthetic time series for each of the 10 201 points. Note
that the resulting waveform for the reflected pulse is not the same shape as
the Ricker wavelet.
(c) Repeat part (b), but this time use the Kirchhoff result of (7.68) and Section
7.7.2 that the secondary sources are given by the derivative of the Ricker
wavelet. Assume that R(θ) = 1 for this example. Show that the synthetic
reflected pulse has the correct shape.
(d) Verify that the reflected pulse from part (c) has an amplitude of 0.25, the same
as the predicted amplitude of a pulse 4 km away from a point source that has
unit amplitude at r = 1.
(e) Note: Because convolution is a linear operation, parts (b) and (c) can be
performed more efficiently by summing over the 10 201 points assuming a
simple spike source (s(t) = 1/r2 at the t = 0 point only) and then convolving
the resulting time series with the Ricker wavelet or its time derivative to obtain
the final synthetic seismogram. The intermediate time series will contain
considerable high frequency noise but this is removed by the convolution.
(f) The main reflected pulse should arrive at t = 1 s. What is the origin of the
small pulse at about 2.7 s?