8/11/2019 Re Sampling

1/3

Resampling

The proper digitization of a waveform requires two samples per period of the highest frequency present in thewaveform. If we use less than two samples, spurious (aliased) frequencies are introduced.

In discussing array dimensions, we think in terms of the highest frequency present in the waveform. In thiscontext, it is the highest reflection frequency that we may realistically hope to maintain through the field workand the processing. That this frequency defines the permissi le length of an array, the permissi le elevationchange across an array, and the effect of an array on the first reak.

The same frequency defines our maximum feathering angle, and it defines a myriad other varia les in processing.!onsequently, this frequency is the most asic of all the varia les to e decided, oth in the field and inprocessing.

"e know that this highest reflection frequency is defined y the emitted spectrum of the source, y the ghostreflection(s), and y the earth filter. #ecause of the progressive high$cut action of the earth filter, we know thatthe highest frequency present in the reflection waveform is likely to e associated with the shallow reflections. "ethen have to make a %udgement as to what the highest &significant& frequency is, or is likely to e, on theshallowest reflection of exploration interest. This %udgement is crucial to the entire seismic technique.

's if that were not difficult enough, the higher frequencies attenuated in the earth can e partially recovered (toan extent that depends on the signal$to$noise ratio) y the process of deconvolution. ur %udgement thereforehas to take into account a later improvement of unknown degree.

ot surprisingly, we seek to postpone this %udgement as late as possi le. In particular, we do not want to make itin the field (except that, in *i roseis, we are forced to do so). Therefore the standard practice is to assume afrequency a little higher than we think it could possi ly e, and to ase the recording sampling interval on that. If we set this frequency too low, we lose useful data+ if we set it too high, all it costs us is tape. o the decision isfairly easy, in the field.

Typically, we take the highest reflection frequency to e a out - / 0z, in the field. If this were the highestfrequency present, we could o tain two samples per period y sampling at /1 0z 2 every 3 ms. #ut there may

e noise present at higher frequencies, and possi ly also at higher amplitudes. Therefore we dare not digitize at 3ms.

The solution is to apply an antialias filter. In the field, we usually set the cutoff frequency of this filter to coincidewith the highest expected reflection frequency (in our example, - / 0z).

The slope of the analog filter used in the field is unlikely to e much less than 41 d#5octave+ therefore we canexpect the noise to e 41 d# down at /1 0z. In practice, such noise is negligi le. o /1 0z, or somethingapproaching it, is definitely the highest frequency present in the waveform. Therefore we must digitize at twosamples per period of this frequency+ our sampling frequency is /11 0z, and the sampling interval ms.

The field assumption that the highest reflection frequency is - / 0z therefore leads us to digitize at mssampling interval in the field. If it were /1 0z we would digitize at - ms+ if 67 0z, at 3 ms.

'fter the data are safely recorded, however, we can determine the highest reflection frequency present. 8urther,digital filters applied in processing can have slopes steeper than those of the analog filters in the field+ this meansthat we can reduce the noise to negligi le proportions at a frequency closer to the highest reflection frequency

( 8igure - ).

http://figurewin%28%27../asp/graphic.asp?code=4423&order=0%27,%270%27)http://figurewin%28%27../asp/graphic.asp?code=4423&order=0%27,%270%27)

8/11/2019 Re Sampling

2/3

Figure 1

o if we find, for example, that the highest reflection frequency is 91 0z, and if a digital antialias filter allows usto suppress the noise adequately y a very steep slope etween 91 0z and - / 0z, then our waveforms would ecompletely and safely specified y sampling at 3 ms. The volume of data would e halved, and our processingcosts would e much reduced.

The process of resampling thus involves the application of a digital antialias filter, followed y the dropping of

alternate sample values.

The antialias filter must e specified as to its cutoff frequency, its slope, and its phase response. The cutofffrequency is determined y our estimate (or measurement) of the highest significant reflection frequency. Theslope is that necessary to yield negligi le noise output at the new sampling frequency+ ideally, we should explorethis y tests, ut in practice we usually seek the safety of a steep slope. The phase response should normally ethat of minimum phase.

The economies o taina le y resampling are greater than the proportional reduction in data volume. This isecause several of the later processes involve convolutions and correlations, which take computer time in

proportion to the square of the num er of samples. :esampling, therefore, is a much$used process+ in the field, itavoids the need to make restrictive decisions a out frequency content, and in the processing it allows us tominimize the cost after the frequency content is known.

In 8igure , we see the com ined effect of data reduction y array formation and resampling.

http://figurewin%28%27../asp/graphic.asp?code=4423&order=1%27,%271%27)http://figurewin%28%27../asp/graphic.asp?code=4423&order=1%27,%271%27)

8/11/2019 Re Sampling

3/3

Figure 2

8igure illustrates a test segment of a section processed to preserve /$m arrays, /$m group intervals, 71$foldstack, and $ms sampling. 8igure illustrates the effect of forming -; ;- arrays at /1$m group intervals (thusreducing to -/$fold stack), and resampling to 3 ms. 8or the purposes of the survey, the economies of 8igure were considered %ustified.

http://figurewin%28%27../asp/graphic.asp?code=4423&order=1%27,%271%27)http://figurewin%28%27../asp/graphic.asp?code=4423&order=1%27,%271%27)http://figurewin%28%27../asp/graphic.asp?code=4423&order=1%27,%271%27)http://figurewin%28%27../asp/graphic.asp?code=4423&order=1%27,%271%27)http://figurewin%28%27../asp/graphic.asp?code=4423&order=1%27,%271%27)http://figurewin%28%27../asp/graphic.asp?code=4423&order=1%27,%271%27)