Apparent Wavelength

Key Takeaways

• The apparent wavelength determines the maximum receiver spacing that avoids spatial aliasing of coherent seismic events: Spatial aliasing occurs when the receiver spacing dx exceeds half the apparent wavelength of the wavefield component being sampled: dx greater than lambda_a / 2. Once aliased, the component wraps to an apparent velocity on the opposite side of the wavenumber spectrum, making it indistinguishable from energy arriving from a different direction and impossible to remove by conventional f-k filtering without also damaging signal. For a reflection arriving with apparent velocity V_a = 3,000 m/s at dominant frequency f = 50 Hz, the apparent wavelength is 60 metres and the maximum receiver spacing is 30 metres. For ground roll with apparent velocity 350 m/s at 20 Hz, the apparent wavelength is 17.5 metres and the maximum spacing to avoid aliasing the noise is 8.75 metres. Since a spacing of 30 metres (needed to avoid aliasing the signal) will alias the ground roll (which requires 8.75-metre spacing), the field acquisition strategy must either use dense single-receiver spacing (8.75 metres or less) and rely on processing to separate signal from noise, or use group arrays of geophones to attenuate the aliased ground roll in the field before recording. In practice, most WCSB 3D surveys use 50-metre receiver spacings, which aliases all ground-roll components with apparent wavelengths below 100 metres; f-k filtering and other techniques handle the aliased noise in processing.
• Receiver group arrays attenuate noise by summing over a length comparable to the apparent wavelength of the noise: A linear array of n geophones at spacing d metres, summed before recording, has a spatial response (array response function) that passes waves with apparent wavelengths much longer than the array length (nd) while attenuating waves with apparent wavelengths near or shorter than the array length. The first null of the array response occurs when the apparent wavelength equals the array length: at this wavelength, the contributions from successive geophones in the array are spread exactly one wavelength apart in phase and cancel by destructive interference. For ground roll with apparent wavelength 18 metres, an 18-metre group array (e.g., 6 geophones at 3-metre spacing, summed) produces a null response, giving maximum attenuation of the noise. Signal reflections with apparent wavelengths of 60 to 200 metres are attenuated by only 5 to 15 percent, preserving most of the signal amplitude. In northeast British Columbia Montney surveys, a typical surface-array design uses 6 geophones per group at 5-metre spacing (30-metre array length) to attenuate ground roll with apparent wavelengths of 20 to 40 metres while passing Montney reflections with apparent wavelengths above 80 metres in the 35 to 80 Hz signal band.
• Apparent wavelength changes with frequency for dispersive surface waves, requiring broadband consideration in array design: Ground roll and other surface waves are dispersive: different frequencies travel at different phase velocities, so the apparent wavelength at each frequency is frequency-dependent. Low-frequency ground roll (8 to 15 Hz) typically has apparent wavelengths of 30 to 60 metres in soft near-surface conditions, while high-frequency ground roll (25 to 50 Hz) may have apparent wavelengths of only 5 to 12 metres. An array designed to null the dominant 15 Hz ground roll (apparent wavelength 45 metres) will pass the high-frequency 40 Hz ground roll (apparent wavelength 8 metres) because the array length (45 metres) is much longer than the noise apparent wavelength at 40 Hz and the response wraps through multiple aliased cycles. This is one reason that f-k filtering in the processing domain complements but does not replace field array design: the field array attenuates the dominant low-frequency ground roll whose apparent wavelength is comparable to the array length, while the f-k filter handles the residual broadband noise that the array cannot attenuate without aliasing the high-frequency signal components.
• In 3D seismic acquisition, apparent wavelength governs the maximum receiver line spacing in the cross-line direction to avoid cross-dip aliasing: A 3D seismic survey samples the wavefield on a grid of receiver locations, and spatial aliasing can occur independently in the in-line direction (along receiver lines) and in the cross-line direction (across receiver lines). Cross-line aliasing occurs when the receiver line spacing exceeds half the apparent wavelength of dipping reflectors as seen across the lines. For a reflector dipping at 15 degrees at right angles to the receiver lines, the cross-dip apparent velocity is V / sin(15) = approximately 3.9 V (where V is the P-wave velocity at the reflector depth); at 50 Hz and V = 2,000 m/s, the cross-line apparent wavelength is 3.9 times 2000 / 50 = 156 metres, requiring receiver lines spaced at most 78 metres apart to avoid aliasing the 15-degree dip at 50 Hz. In the Alberta Foothills, where structural dips of 20 to 45 degrees in the thrust sheets produce cross-line apparent wavelengths as short as 40 to 80 metres for 50 Hz energy, receiver line spacings of 20 to 40 metres are used in high-quality 3D surveys over these areas to capture the steeply dipping reflections without aliasing, significantly increasing the acquisition cost compared to 100-metre line spacings acceptable in the flat Deep Basin.
• Apparent wavelength analysis in microseismic monitoring determines the frequency content and wave type of recorded arrivals from hydraulic fracture events: Microseismic events from hydraulic fracturing operations generate both compressional (P) and shear (S) wave arrivals that travel through the formation and arrive at the monitor well or surface array with apparent wavelengths determined by their true wavelengths and the angle of approach. The true P wavelength at 200 Hz and V_P = 4,000 m/s is 20 metres; at an arrival angle of 45 degrees (for a source 400 metres horizontally from the monitor well at equivalent depth), the apparent P wavelength along a vertical monitor well array is 20 / sin(45) = 28 metres, requiring element spacing of less than 14 metres to avoid aliasing the P arrivals. The S arrival at the same frequency and arrival angle has a true wavelength of approximately 12 metres (V_S = 2,400 m/s) and apparent wavelength of 17 metres, requiring element spacing of less than 8.5 metres. Standard downhole geophone arrays for microseismic monitoring use element spacings of 10 to 15 metres, which provides adequate apparent-wavelength sampling for P arrivals but may alias S arrivals at the highest frequencies; this is generally acceptable because P arrivals are used for primary event location and S arrivals provide supplementary polarity information rather than the primary location constraint.