1. EXERCISE 3:
Convolution and deconvolution in seismic signal processing

2. Convolution
Convolution (Kanasewich 1981) is a mathematical operation defining the change of shape of a waveform resulting from its passage through a filter. Thus, for example, a seismic pulse generated by an explosion is altered in shape by filtering effects, both in the ground and in the recording system, so that the seismogram (the filtered output) differs significantly from the initial seismic pulse (the input).
As a simple example of filtering, consider a weight suspended from the end of a vertical spring. If the top of the spring is perturbed by a sharp up-and-down movement (the input), the motion of the weight (the filtered output) is a series of damped oscillations out of phase with the initial perturbation.

3. The effect of a filter is described mathematically by a convolution operation such that, if the input signal g(t) to the filter is convolved with the impulse response f(t) of the filter, known as the convolution operator, the filtered output y(t) is obtained (asterisk denotes a convolution operator):

4. The mathematical implementation of convolution involves time inversion (or folding) of one of the functions and its progressive sliding past the other function, the individual terms in the convolved output being de- rived by summation of the cross-multiplication products over the overlapping parts of the two functions.
In general, if gi (i = 1, 2, , m) is an input function and fj ( j = 1, 2, , n) is a convolution operator, then the convolu tion output function yk is given by

5. In Fig the individual steps in the convolution process are shown for two digital functions, a double spike function given by gi = g1,g2,g3 = 2,0,1 and an impulse response function given by f i = f1, f2, f3, f4 = 4, 3, 2, 1, where the numbers refer to discrete amplitude values at the sampling points of the two functions. From Fig it can be seen that the convolved output yi = y1, y2, y3, y4, y5, y6 = 8, 6, 8, 5, 2, 1. Note that the convolved output is longer than the input waveforms; if the func- tions to be convolved have lengths of m and n, the convolved output has a length of (m + n - 1).

6. Deconvolution or inverse filtering (Kanasewich 1981) is a process that counteracts a previous convolution (or filtering) action. Consider the convolution operation given in equation
y(t) is the filtered output derived by passing the input waveform g(t) through a filter of impulse response f(t).
Knowing y(t) and f(t), the recovery of g(t) represents a de- convolution operation.
In the seismic case, y(t) is the seismic record resulting from the passage of a seismic wave g(t) through a portion of the Earth, which acts as a filter with an impulse response f(t).

8. Reflectivity series describes the geological medium with reflectivity contrasts
This is the seismic wave propagating through the medium
TASK: Calculate and plot the recorded seismic waveform using convolution operator

9. TASK: recover the reflectivity series using spiking deconvolution operator

10. EXERCISE 4: Snell’s reflection and refraction law
Seismic source
Consider a P wave that leaves the source along the ray path as shown in the diagram and hits the boundary between the upper layer and the second layer with an angle of incidence of 30°. Given the transmitting velocities for a P wave in all the subsequent layers, sketch the path of the ray until it hits the bottom, and find all the angles of incidence and refraction along the way.