FIELD ARRAYS

In geophysical data recording there is usually a local array whose elements are added locally before a single channel is recorded. For example, the SEP student group once laid out more than 4056 geophones which they grouped and added into 169 separate recorders. We may think of the local superposition as an integration over a small interval of space to create a sampled space function from a continuous one. With vibrator sources, it is also customary to vibrate on various nearby source locations and sum them into a single signal. Figure 1 is a caricature of what happens. On the left a data field appears to be a continuous function of space (it is actually 500 spatial locations) with various impulsive signals at different times and distances.

 

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Figure 1 Quasicontinuous field (left) added in groups (right).

For simplicity, all signals have unit amplitude. The 500 signals are segregated into 10 groups of 50 and each group of 50 is summed into a single channel. The various signals sum to functions that could be called ``slump shouldered rectangles.'' If both x and t-meshes were refined further, the ``slump shoulders'' on the rectangles would diminish in importance and we would notice that the rectangles were still imperfect. This is because the rectangle approximation arises from the approximation that the hyperbola is a straight line over the recording array. In reality, there is curvature and the effect of curvature is strongest near the apex, so the rectangle approximation is poor at the apex.

Some of the rectangles are longer than others. The narrow ones are tall and the wide ones are short because the area of each rectangle must be 50 (being the sum of 50 channels each holding a 1). Since the rectangles all have the same area, were we to lowpass filter the recorded data we would recover the original characteristic that all these signals have the same amplitude.

If the 500 channels had been individually recorded we could convolve them along the horizontal axis with a 50 point long rectangle function. The output could be sampled once every 50 points to give the coarse-mesh data in Figure 1. Convolving with a rectangle function of x is dual to multiplying by a sinc function $\sin(k_x \Delta x) /(k_x \Delta x)$in the Fourier domain. The sinc function has its first zero where one wavelength of a spatial sinusoid precisely fits inside the rectangle function. When 49 out of 50 channels are abandoned we have spatial aliasing (spectral folding) of higher wavelengths into lower ones. Mathematically, we would prefer to have a rectangle in the k_x-domain which would erase the energy that would otherwise be aliased to lower frequencies and would preserve the lower frequencies without scaling them. Thus Fourier transform theory implies we should use the (unruly) sinc function as a weighting function for the field operations. As you might imagine, the subject of antenna design is the subject of many patents that address the fundamental issues in various ways.