Apparent Wavelength: Definition, Spatial Aliasing, and Arrays

What Is Apparent Wavelength?

Apparent wavelength is the distance between successive zero-crossings, or equivalently between successive peaks, of a wave as measured along a receiver line when that wavefront arrives at an oblique angle to the line rather than perpendicular to it. It differs from true wavelength because the receiver array measures the spatial trace of the wavefront projected along the line direction, not the true perpendicular distance between successive wavefronts. The relationship is governed by the angle of approach: apparent wavelength equals the true wavelength divided by the sine of the angle of approach, where the angle of approach is measured between the incoming wavefront direction and the receiver line axis. Because the sine of any angle less than 90 degrees is less than one, the apparent wavelength is always equal to or greater than the true wavelength, approaching infinity as the wave travels nearly parallel to the receiver line and equaling the true wavelength only when the wave travels exactly perpendicular to the receiver array. This concept is fundamental to seismic acquisition design, because the apparent wavelength of noise sources such as ground roll determines the receiver spacing required to avoid spatial aliasing, and the apparent wavelength of signal sources determines the array length needed to attenuate noise without distorting signal.

How Apparent Wavelength Is Defined and Derived

Consider a planar wavefront advancing through the earth at velocity V (the true phase velocity of the wave type) and carrying a temporal frequency f. The true wavelength in the direction of propagation is lambda = V/f. Now suppose this wavefront reaches a horizontal line of receivers oriented along the x-axis, and the wavefront is tilted so that it arrives at an angle theta to the receiver line (equivalently, the wavefront normal makes an angle of 90 minus theta with the x-axis). As the wavefront crosses the receiver line, successive receivers encounter the peak of the wave in sequence. The time delay between adjacent receivers spaced a distance dx apart is dt = dx × cos(90 - theta) / V = dx × sin(theta_i) / V, where theta_i is the angle of incidence measured from the vertical (the complement of theta). But the spatial period observed along the receiver line is the distance between successive wave peaks measured along the line, which is lambda_a = V_a / f, where V_a is the apparent velocity of the wave along the receiver line. By Snell's Law and the geometry of the wavefront crossing, V_a = V / sin(theta_i) (for a horizontally traveling wave, theta_i is the angle from vertical, and sin(theta_i) = horizontal component of the slowness vector). The apparent wavelength follows immediately as lambda_a = V_a / f = V / (f × sin(theta_i)) = lambda / sin(theta_i).

In the common field geometry for land seismic acquisition, the receiver line runs horizontally along the surface, and waves of interest arrive from below at various angles. A direct P-wave arriving nearly vertically (theta_i close to 0) has sin(theta_i) nearly equal to theta_i in radians, and the apparent wavelength is very large: the wavefront is nearly flat as seen by the receiver array, and successive receivers experience almost simultaneous wave arrivals. A refracted wave arriving at critical angle, or ground roll arriving nearly horizontally (theta_i close to 90 degrees), has sin(theta_i) close to 1 and an apparent wavelength nearly equal to the true wavelength. The problematic case for spatial aliasing is noise waves with low apparent velocity (large sin(theta_i) and short apparent wavelength), which require fine receiver spacing to sample adequately without aliasing.

The relationship to the angle of incidence and to Snell's Law is direct. Snell's Law states that the horizontal component of the slowness vector (1/velocity) is conserved across a horizontal interface: p = sin(theta) / V = constant for a ray family, where p is the ray parameter (also called the horizontal slowness). The apparent velocity V_a = 1/p = V/sin(theta). The apparent wavelength lambda_a = V_a/f = V/(f × sin(theta)) = lambda/sin(theta). Thus apparent wavelength is the spatial-domain equivalent of the ray parameter concept: high-apparent-velocity arrivals (small sin(theta), nearly vertical) have long apparent wavelengths and low wavenumbers; low-apparent-velocity arrivals (large sin(theta), nearly horizontal, or surface waves traveling horizontally) have short apparent wavelengths and high wavenumbers. This wavenumber representation is the basis for frequency-wavenumber (f-k) filtering, where signal and noise are separated in the f-k domain based on their apparent velocity (slope in f-k space) rather than in the time-offset domain.

Spatial Aliasing and the Nyquist Wavenumber

Spatial aliasing is the seismic analog of temporal aliasing in digital signal processing. In temporal sampling, the Nyquist theorem states that a signal of frequency f must be sampled at a rate of at least 2f samples per second (the Nyquist rate) to prevent aliasing of high-frequency energy into lower-frequency bands. The spatial equivalent applies to receiver arrays: a wavefield with apparent wavelength lambda_a must be sampled at a spatial interval delta_x no greater than lambda_a/2 to prevent spatial aliasing. In terms of the wavenumber k_a = 1/lambda_a (in cycles per metre), the Nyquist wavenumber is k_N = 1/(2 × delta_x), and spatial aliasing occurs for any wavefield component with k_a greater than k_N, i.e., with apparent wavelength less than 2 × delta_x.

The practical consequence of spatial aliasing in seismic data is severe. Aliased noise from coherent arrivals such as ground roll wraps into the data at apparent wavenumbers reflected about the Nyquist wavenumber. If delta_x = 25 m (82 ft), the Nyquist wavenumber is 1/(2 × 25) = 0.020 cycles/m (the Nyquist spatial frequency), and any wavefield with apparent wavelength less than 50 m (164 ft) will be aliased. Ground roll at 300 m/s (984 ft/s) apparent velocity and 10 Hz frequency has apparent wavelength 30 m (98 ft), which is less than 50 m: at 25 m receiver spacing, this ground roll is aliased. In the f-k domain, the aliased ground roll appears as a band of energy with reversed apparent velocity slope, overlapping the signal cone and making it impossible to design an f-k reject filter that removes the ground roll without also removing the primary reflection signal. This aliasing contamination propagates through all subsequent processing steps including velocity analysis, multiple attenuation, and migration, degrading data quality in ways that cannot be fully corrected without the original, properly sampled field records.

The anti-alias design criterion for receiver spacing is therefore: delta_x must satisfy delta_x less than or equal to V_noise_min / (2 × f_noise_max), where V_noise_min is the minimum apparent velocity of the coherent noise to be properly sampled and f_noise_max is the maximum frequency of that noise. For ground roll at a site with minimum apparent velocity 350 m/s (1,148 ft/s) and maximum frequency 18 Hz, the required receiver spacing is at most 350/(2 × 18) = 9.7 m (32 ft). If the survey budget constrains receiver spacing to 25 m (82 ft), then ground roll above 7 Hz will be spatially aliased at 25 m spacing, and the acquisition team must rely on analog geophone array design to attenuate the ground roll in the field before digital recording, rather than relying on digital processing to remove it afterward.