Band-Reject Filter: Definition, Notch Filter, and Noise Removal

A band-reject filter is a signal-processing operator that attenuates a defined range of frequencies, called the stopband, and transmits all frequencies outside that range with minimal change. It is the frequency-selective complement of the band-pass filter: where a band-pass filter keeps only the frequencies inside a window, a band-reject filter removes only the frequencies inside a specific window and passes everything else. In seismic data acquisition and processing, band-reject filters are used to eliminate discrete-frequency interference that would otherwise contaminate the recorded signal, including power-line hum, harmonic distortion from vibroseis sweeps, cable strum vibration on marine streamers, and drill-string resonance noise in vertical seismic profile (VSP) surveys. When the stopband is very narrow, targeting a single interference frequency, the filter is commonly called a notch filter.

How a Band-Reject Filter Works

Unlike the attenuation of broadband noise, which is best handled by a band-pass filter that limits the total frequency range of the data, many interference sources in seismic and well-logging environments generate energy concentrated at one or a few specific frequencies. Power transmission lines radiate electromagnetic fields that induce currents in recording cables, geophone strings, and sensor electronics at exactly the power-line frequency and its harmonics. Vibroseis trucks generate harmonic distortion at integer multiples of the fundamental sweep frequency. Ocean currents cause marine streamers to vibrate at their mechanical resonance frequency, generating so-called cable strum noise. Drilling equipment generates bit bounce and drill-string resonance energy at predictable frequencies related to rotary speed and bit design. In all of these cases, the interference is spectrally localized and can in principle be removed by attenuating only the affected frequencies, leaving the rest of the spectrum intact.

A band-reject filter achieves this by defining a stopband: the range of frequencies to be attenuated. The simplest specification is a two-parameter notch design, defined by a center frequency f_c and a bandwidth BW. Frequencies within approximately f_c +/- BW/2 are attenuated; frequencies outside this range are passed. The depth of the notch, measured in decibels, indicates how completely the interference is removed: a 40 dB notch reduces the interference amplitude by a factor of 100, while a 60 dB notch reduces it by a factor of 1,000. In practice, notch depths of 30 to 60 dB are achievable with standard digital filter designs, which is typically sufficient to reduce the interference below the noise floor of the processed data.

The shape of the band-reject response in the transition zones between the passband and stopband follows the same design options available for band-pass filters. A Butterworth band-reject design produces smooth, monotonically varying transition zones with no ripple. A Chebyshev band-reject design achieves a steeper rolloff at the cost of ripple in either the passband or stopband. A windowed sinc notch (FIR design) applies a rectangular or tapered window to a sinc-function operator in the time domain to achieve controlled transition and minimal side lobes. For most seismic applications, the Butterworth FIR design offers the best balance of stopband depth, transition width, and minimal wavelet distortion of the passband signal.

Narrow Notch Filters: Power-Line Noise Removal

The most common use of a narrow notch filter in seismic and geophysical well-logging is the removal of power-line electromagnetic interference. In countries operating 50 Hz electrical grids (including the United Kingdom, most of Europe, Australia, the Middle East, South Africa, India, and China), the primary interference frequency is 50 Hz. In countries operating 60 Hz grids (including the United States, Canada, Mexico, most of Central America, and parts of Japan), the primary frequency is 60 Hz. The interference appears in seismic recordings as a monofrequency sinusoidal component superimposed on the seismic trace; in wireline log data it appears as a 50 Hz or 60 Hz sinusoidal oscillation riding on the log curve.

A narrow notch centered at 50 Hz with a bandwidth of 2 to 4 Hz (Q of 12.5 to 25) is typically sufficient to reduce 50 Hz power-line noise in well-log data. For seismic field records, where the interference may also include harmonics at 100 Hz, 150 Hz, and 200 Hz (for a 50 Hz grid), multiple notch filters may be applied simultaneously or sequentially, each centered on one of the harmonic frequencies. The design challenge is that the power-line frequency is not perfectly stable: it drifts slowly as load on the grid changes. In North American 60 Hz systems, the frequency may vary between approximately 59.7 and 60.3 Hz over the course of a recording day. A fixed notch at exactly 60.0 Hz may fail to capture the full interference energy when the frequency drifts to 59.8 Hz, leaving residual contamination. Two solutions are used in practice: slightly widening the notch stopband (for example, 58.5 to 61.5 Hz, accepting some loss of signal) or applying an adaptive notch filter that estimates the instantaneous interference frequency from the data and updates the notch center in real time.

Adaptive notch filtering uses an algorithm that continuously estimates the frequency, amplitude, and phase of the interference signal from the recorded data, typically using a least-mean-squares (LMS) or recursive least-squares (RLS) adaptive algorithm. The estimated interference waveform is then subtracted from the input trace, leaving the residual seismic signal. This approach is more effective than a fixed notch when the interference frequency drifts and is also preferable when the seismic signal of interest contains energy near the interference frequency, because it subtracts the specific interference rather than blanket-attenuating the entire frequency band around 60 Hz.

Wide Band-Reject Filters and Marine Cable Strum

Not all band-reject applications target a single narrow frequency. When an interference source generates energy across a relatively wide frequency range or produces a harmonic series extending over tens of hertz, a wider stopband may be needed. Marine seismic streamers, which are towed through ocean water at depths of 5 to 15 m (16 to 49 ft), vibrate in the current in a manner analogous to a plucked string. This cable strum generates narrowband noise at the mechanical resonance frequency of the streamer section and its harmonics. The fundamental resonance frequency depends on the streamer tension, the linear mass density of the cable, and the span length between the coupling points; typical values are in the range of 5 to 30 Hz for ocean-bottom cable and towed-streamer configurations. The strum energy can be intense enough to dominate the low-frequency part of the spectrum near the resonance frequency, interfering with the low-frequency signal sought in broadband acquisition programs.

Removing cable strum noise by band-reject filtering requires identifying the strum fundamental and its harmonics from the data itself, typically by examining frequency-wavenumber (f-k) spectra or amplitude-frequency spectra of noise records recorded during non-shooting intervals. Once the resonance frequencies are identified, a series of narrow notch filters is applied at each harmonic. Because the strum frequency may vary slowly along the streamer (as tension varies) and over time (as current speed and tow depth change), adaptive methods similar to those used for power-line removal are increasingly used. Alternatively, in multi-sensor streamers equipped with both hydrophones and micro-electromechanical system (MEMS) accelerometers, the cable vibration signal is recorded directly by the accelerometers and can be subtracted from the hydrophone signal, eliminating the need for post-processing notch filtering.