Spatial sampling restrictions

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Problem 12.1a

Show that the maximum spatial sampling can be written


(12.1a)


where is the maximum frequency of interest, and is the maximum angle of approach.

Background

A wave is a function of time and space, e.g., (see problem 2.5); therefore it can be sampled in time at a fixed location (problem 9.4) or in space at a fixed time (see Sheriff and Geldart, 1995, section 8.3.10). In both cases the sampling theorem (see problem 9.4) states that the wave can be sampled at fixed intervals and be recovered exactly from the sampled data provided all frequencies are less than the Nyquist frequency , that is, less than half the sampling frequency:


(12.1b)


For spatial sampling, gives the number of waves per unit length and, hence, corresponds to frequency in the time domain. Therefore, for spatial sampling at intervals , the equivalent of equation (12.1b) is


(12.1c)

so


where , .


Solution

The maximum sampling interval is associated with the minimum apparent wavelength. From equation (12.1c) we have


(12.1d)

Problem 12.1b

Show that the maximum group spacing is


(12.1e)


where the dip moveout is in milliseconds/unit distance.


Solution

Assuming , we replace in equation (12.1d) with [see equation (4.2b)]:



where time difference in milliseconds between two geophones separated by a distance .

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