Apparent Velocity: Definition, Seismic Noise, and F-K Filtering

Apparent velocity is the speed at which a seismic wavefront appears to travel along a surface or along a seismic recording line, as opposed to the true velocity at which the wave travels through the subsurface medium. Because seismic waves typically arrive at the surface at an angle rather than straight down, the wavefront sweeps laterally across the receiver array at a rate that exceeds the true propagation velocity through the rock. This distinction between apparent and true velocity is fundamental to seismic data acquisition design, noise identification and suppression, normal moveout (NMO) correction, array beam steering, and frequency-wavenumber (f-k) filtering across every major hydrocarbon-producing basin in the world.

Key Takeaways

• Apparent velocity (Va) is the speed of a wavefront measured along the Earth's surface or along a seismic line; for a wave with true velocity V arriving at angle of incidence θ from vertical, Va = V / sin(θ).
• Different wave types have characteristic apparent velocities on shot records: refracted P-waves arrive at apparent velocities at or above the refractor velocity, the direct wave travels at the near-surface layer velocity, surface waves (ground roll) arrive at 150-400 m/s (490-1,310 ft/s), and the air wave travels at approximately 330 m/s (1,083 ft/s).
• Apparent velocity is the primary parameter used in f-k (frequency-wavenumber) filtering to separate coherent noise from reflection signal in seismic processing.
• Array design for geophones and hydrophones is based on the expected apparent velocities of desired signal and coherent noise, allowing arrays to act as spatial filters.
• In borehole seismic (VSP), apparent velocity along the receiver array distinguishes upgoing reflections from downgoing direct and multiply reflected waves.

Definition and the Apparent Velocity Formula

Consider a planar seismic wavefront traveling through a homogeneous medium at velocity V. If this wavefront strikes the Earth's surface (or a horizontal receiver array) at an angle of incidence θ measured from the vertical (equivalently, at an angle of 90° - θ from horizontal), the point of intersection of the wavefront with the surface moves laterally at a velocity greater than V. The apparent velocity along the surface is:

Va = V / sin(θ)

where θ is the angle of incidence from the vertical (the angle between the ray path and the surface normal). When θ approaches 90 degrees (a nearly horizontal wave arriving nearly parallel to the surface), the denominator sin(θ) approaches 1, and apparent velocity approaches true velocity. When θ is small (a wave arriving nearly vertically), sin(θ) is small, and apparent velocity becomes very large. In the limiting case of a perfectly vertical ray (θ = 0), the apparent velocity is theoretically infinite: the wavefront hits all surface points simultaneously and there is no apparent lateral motion. This is the geometry of a primary reflection from a horizontal reflector at zero offset, and NMO correction transforms the hyperbolic moveout of such reflections toward this vertical-incidence geometry.

The formula is a direct consequence of Snell's Law. For a wave in a layer with velocity V1 refracting along an interface with velocity V2 (V2 greater than V1), the critical angle is θc = arcsin(V1/V2). At and beyond the critical angle, refracted energy travels along the interface and returns to the surface as head waves with apparent velocity equal to V2. This is the basis of seismic refraction surveying, which has been used since the earliest days of geophysical exploration to map near-surface velocity layers and, in exploration contexts, to determine depths to basement and to major velocity contrasts. See the Oil Authority article on seismic acquisition for context on how surface seismic data are collected.

How Apparent Velocity Governs Shot Record Interpretation

On a raw seismic shot record, multiple wave types are recorded simultaneously across the receiver array. Each wave type has a characteristic apparent velocity that governs its slope (moveout) on the shot record's time-distance display. Understanding these apparent velocities is the first step in designing noise suppression strategies and in quality-controlling field data. The principal wave types and their characteristic apparent velocities are as follows.

The direct wave travels from the shotpoint through the near-surface layer directly to each receiver without reflection or refraction. Its apparent velocity along the surface equals the true P-wave velocity of the near-surface layer (typically 400 to 1,800 m/s, or 1,300 to 5,900 ft/s, depending on whether the surface is unconsolidated sediment, weathered rock, or competent bedrock). On a shot record, the direct wave appears as a linear event with a slope equal to the reciprocal of the near-surface layer velocity.

Refracted waves (head waves) arrive at apparent velocities equal to the velocity of the refractor from which they return. In a two-layer case, first arrivals beyond the crossover distance travel at apparent velocity V2 (the sub-weathering velocity or basement velocity), typically 1,800 to 6,000 m/s (5,900 to 19,700 ft/s). In seismic refraction surveys, these apparent velocities are measured directly on the travel-time versus offset plot and used to compute refractor depths and velocities. In reflection seismic acquisition, the refraction arrivals are muted during processing, but their apparent velocities are recorded and used to build the near-surface velocity model required for static corrections.

Surface waves (ground roll) are low-frequency, high-amplitude waves that travel along the Earth's surface. They are dispersive (different frequencies travel at different velocities) with phase velocities typically ranging from 150 to 400 m/s (490 to 1,310 ft/s), depending on the shear-wave velocity of the near surface. On a shot record, ground roll appears as a cone of energy with low apparent velocity (steep moveout slope) and low frequency (typically less than 20 Hz). Ground roll is the dominant coherent noise problem in land seismic surveys worldwide. Because its apparent velocity is much lower than that of primary reflections (which may have apparent velocities of 2,000 to 10,000 m/s or more at typical recording offsets), it is separable from signal in the f-k domain.

The air wave travels through air at approximately 330 m/s (1,083 ft/s) at sea level and standard temperature, slightly less at high elevation or in extreme cold. It appears on shot records as a linear event with a distinctive slope corresponding to sonic velocity in air. The air wave is energetic in shallow surveys and in explosive-source land surveys, and its low apparent velocity puts it close to ground roll in the f-k domain. Notch filters in receiver arrays are sometimes designed to attenuate the air wave, since its apparent velocity is well-defined.

Primary reflections arrive with apparent velocities that depend on offset and the true stacking velocity of the reflector. At near offset, where reflection angles are small, apparent velocity is high (approaching infinity for a flat reflector at zero offset). At far offsets, where ray paths are more oblique, apparent velocity is lower. The hyperbolic moveout of a primary reflection on a common-midpoint (CMP) gather represents the variation of apparent velocity with offset, and NMO correction using the stacking velocity flattens this moveout, effectively making all offsets appear to have infinite apparent velocity (simultaneous arrival).

Multiples have apparent velocities that are generally lower than those of primaries at the same time, because multiples travel longer ray paths at shallower angles. Interbed multiples and water-bottom multiples are distinguished from primaries partly on the basis of their apparent velocity behavior in CDP gathers and in the f-k domain.

Fast Facts: Apparent Velocity at a Glance

Frequency-Wavenumber (F-K) Filtering and Apparent Velocity

The frequency-wavenumber (f-k) transform converts a seismic record from the time-distance (t-x) domain to the frequency-wavenumber domain. In the f-k domain, a linear event with a specific apparent velocity Va maps to a line through the origin with slope f/k = Va. Events with different apparent velocities map to different slopes in the f-k plane, allowing them to be separated by applying a fan-shaped or polygonal mask in the f-k domain before transforming back to t-x.

The f-k filter is the primary tool for attenuating coherent noise with low apparent velocity (ground roll, air waves, and direct arrivals) while preserving primary reflections with higher apparent velocity. The design of an effective f-k filter requires accurate knowledge of the apparent velocities of both the desired signal and the noise to be rejected. If the noise apparent velocity overlaps significantly with the signal apparent velocity (which can happen at far offsets where primary reflection moveout is large), f-k filtering will damage signal along with noise. In this case, alternative methods such as surface-consistent noise attenuation, high-resolution radon transforms, or singular value decomposition (SVD) filters are preferred.

A complication in f-k filtering is spatial aliasing. The Nyquist wavenumber for a receiver array with group interval dx is kmax = 1/(2 dx). If a noise event has apparent velocity Va and dominant frequency f, its wavenumber is k = f/Va. If k exceeds kmax (i.e., the wavelength of the noise is shorter than two group intervals), the noise aliases to a different apparent velocity in the f-k domain and cannot be cleanly separated from signal. This is why seismic survey design balances receiver spacing against the expected apparent velocities of signal and noise, a process governed by the same apparent velocity concept. See the Oil Authority article on seismic acquisition for survey geometry parameters.

In marine seismic surveys, the equivalent low-apparent-velocity noise includes cable noise (vibrations traveling along the hydrophone streamer at speeds of 1,400 to 1,500 m/s, close to the water velocity), swell noise (irregular energy from ocean waves), and direct-wave energy from the air gun array. Because the water velocity of approximately 1,500 m/s (4,920 ft/s) is higher than typical land near-surface velocities, marine direct-wave and cable noise appear at higher apparent velocities than land ground roll, reducing (but not eliminating) the overlap with primary reflections in the f-k domain.