Figure 5.2: Temporal resolution. Left: Resolution increases with the maximum frequency. The number of octaves is 2,3 and 4 for the black, blue and red spectra, respectively. Right: Resolution is relatively insensitive to the minimum frequency. The number of octaves are 2,3 and 4, but the resolution is the same. A key learning is that side lobe reduction is obtained by adding low frequencies.
seismic? Te frequency f of an organ pipe is f = v/ λ, where v is the speed of sound in air (340 m/s) and λ is the wavelength. Let L be the length of the pipe. Te longest possible wavelength equals 2L and 4L for open and closed pipes respectively. Te maximum wavelength thus is λ = 4L, and the corresponding minimum frequency equals f = v/4L. One of the biggest organs in the world is the Boardwalk Hall
Auditorium organ in Atlantic City. It is equipped with 33,112 pipes, and the biggest pipe has a length of 64 ft. Tis is an open pipe so the corresponding lowest frequency is around 8 Hz. A closed pipe of the same length would give a lower frequency of 4 Hz. But the story does not stop at 4 Hz. Te lowest produced
note is obtained by combining a stopped 64 ft and stopped 422⁄3 ft pipe to produce a resultant 256 ft pipe which gives 2 Hz! Tis is far below the threshold of the human ear, which is approximately 16 Hz. So what is the point in this focus on low frequencies for organ pipes? Can we feel the low frequencies directly on our body, or is it a combination of hearing and body feeling? Anyhow, there is a strong similarity between the design of big
organ pipes and today’s developments in broadband seismic. As geophysicists, we would be thrilled if our marine seismic system produced frequencies truly from 2 Hz and upwards. We would want to activate all the pipes of the organ in Atlantic City, and especially the big pipes! Te low frequencies are of particular interest for deep imaging, inversion and high-end interpretation.
5.1.2 Temporal Resolution
Improving bandwidth and resolution has been a priority since the early days of the seismic method – to see thinner beds, to image smaller faults, and to detect lateral changes in lithology. Although sometimes used synonymously, the terms bandwidth and resolution actually represent different concepts. Bandwidth
describes simply the breadth of frequencies comprising a spectrum. Tis is often expressed in terms of octaves. Commonly referred to in music, an octave is the interval
between one frequency and another with half or double its frequency. As an example, the frequency range from f1 f2
to > f1 represents one octave if f2 = f1 /2. = 2f1
Hz represents one octave of bandwidth, as do ranges 8–16 Hz, and 16–32 Hz. Also, the range from f1 octave if f0
. Te range from 4 to 8 to f0
< f1 represents one In a classic empirical study, Kallweit and Wood (1982) found
a useful relationship between bandwidth and resolution. For a zero-phase wavelet with at least two octaves of bandwidth, they showed that the temporal resolution TR case could be expressed as TR
= 1⁄(1.5fmax ), where fmax
in the noise-free is the
maximum frequency in the wavelet. Other definitions are possible, but for two octaves or more of bandwidth the clue is that one can approximately relate temporal resolution to the highest, and only the highest, frequency of a wavelet. Tis leads to some very useful and quite accurate predictions. Examples are that wavelet breadth is TB TPT
= 1⁄(0.7fmax = 1⁄(1.4fmax ).
Figure 5.2 demonstrates the expected improvement in resolution associated with increasing fmax
. We see that when
the maximum frequency value is increased while holding the minimum frequency fmin
obtained. Meanwhile, the right side of the figure shows three more temporal wavelets where the fmin holding fmax
any change while the side lobes diminish as fmin
fixed. We see that the main lobe shows hardly is lowered.
Tus, filling in the low frequencies gives wavelets with less pronounced side lobe amplitudes. It is attractive because it is smoother, and with less side lobe energy in the wavelet it is unlikely that a small-amplitude event will be lost amid the side lobes from neighbouring, large-amplitude reflections.
195 fixed, sharper temporal wavelets are value is changed while
) and peak-to-trough is
Lasse Amundsen
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