Space-domain convolution and vibroseis acquisition

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Problem

The techniques and concepts of convolution, aliasing, $ z $-transforms, and so on can be applied to other domains than the time-domain. Express the source and group patterns of Figure 9.2a as functions of $ x $ (horizontal coordinate) and convolve the two to verify the effective pattern shown.

Background

The convolution of two functions $ g(t) $ and $ h(t) $, written $ g(t)*h(t) $, is defined as


$ {\begin{aligned}g(t)*h(t)&=\int \limits _{-\infty }^{+\infty }g(\tau )h(t-\tau )\,\mathrm {d} {\tau }\\&=\int \limits _{-\infty }^{+\infty }h(\tau )g(t-\tau )\,\mathrm {d} {\tau }=h(t)*g(t).\end{aligned}} $ (9.2a)


For digital functions (see problem 9.4), the integrals become sums:


$ {\begin{aligned}g_{t}*h_{t}=\sum \limits _{k}^{}g_{k}\ h_{t-k}=\sum \limits _{k}^{}h_{k}\ g_{t-k}=g_{t}*g_{t}.\end{aligned}} $ (9.2b)


These equations state that convolution is equivalent to superposition in which each element of one function is replaced by the other function multiplied (weighted) by the element being replaced. The sum of all values at a given time is the value of $ g(t)*h(t) $ at that instant.

Figure 9.2a.  Vibroseis acquisition. (i) Field locations for six sweeps (shown displaced vertically); (ii), array pattern for four vibroseis trucks spaced 33 m apart moving forward 16.5 m for each sweep. Each receiver array consists of geophones spaced uniformly over 100 m. For the next location, an additional group of phones is added on one side and dropped on the other. The numbers in (ii) give the multiplicity.
Figure 9.2a.  Boxcar and its transform.
Figure 9.2b.  Two boxcars.

Another explanation of convolution is the following: We note that $ g(-t) $ is $ g(t) $ reflected in the $ t $-axis; in other words, the curve of $ g(-t) $ is the same as that of $ g(t) $ except that it is reversed in direction [keeping $ g(0) $ fixed]. The function $ g(t-\tau ) $ is $ g(t) $ displaced $ \tau $ units to the right. Thus, in equation (9.2a) the value of $ g(t)*h(t) $ at the time $ t=\tau $ is obtained by moving $ h(t) $ to the right $ \tau $ units, reflecting it in the $ t $-axis, then summing (integrating) the products of corresponding abscissas. The result is the same whichever function is displaced and reflected.

Arrays are discussed in problem 8.6, aliasing in problem 9.4, and $ z $-transforms in Sheriff and Geldart, 1995, section 15.5.3.

Solution

Taking the spatial sampling interval as 16.5 m, the source pattern consists of [1,1,2,2,3,3,3, 3,2,2,1,1] for a total array length of 200 m. The geophone group is an array 100 m long; to convolve the source and receiver arrays, they should have the same spatial intervals, so we take six receivers spaced 16.5 m apart. To convolve the two arrays, we replace each element of the receiver array with the source array (taking the weights as unity). The result is that shown at the bottom of Figure 9.2a.

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