Band-Reject Filter: Notch Filter, Seismic Noise Removal, and Signal Processing
A band-reject filter is a signal-processing operator that attenuates a defined range of frequencies (the stopband) while transmitting all frequencies outside that range with minimal change in amplitude or phase. 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 and its immediate sidebands, the filter is commonly called a notch filter, and this specific implementation is the most frequently encountered form of band-reject filtering in oilfield signal processing applications.
The technical implementation of a band-reject filter is straightforward in the frequency domain: the Fourier transform of the input signal is computed, the amplitude of frequency components within the stopband is multiplied by a gain function that ranges from 1.0 (full transmission) outside the stopband to 0.0 (complete attenuation) within the stopband, and the inverse Fourier transform is applied to produce the filtered output in the time domain. The shape of the gain function within the stopband transition region determines whether the filter is a Butterworth band-reject (maximally flat passband with monotonic rolloff into the stopband), a Chebyshev band-reject (equiripple in the stopband, steeper rolloff for the same filter order), or a Gaussian band-reject (smooth transition, zero side lobes in the time-domain response). In seismic processing, the Butterworth and zero-phase implementations are preferred because they introduce minimal phase distortion in the passband frequencies that carry the reflection signal, preserving the wavelet shape at the reflection events that bracket the stopband frequency.
Key Takeaways
- Power-line notch filtering in land seismic: In North American land seismic surveys, power lines running across or near the survey area generate electromagnetic interference at 60 Hz (the power distribution frequency) and at harmonics of 60 Hz (120, 180, 240 Hz). This interference is coupled into the recording system through the seismic cables and receiver connections, appearing as a continuous sinusoidal tone at 60 Hz in the time domain superimposed on the seismic signal. A notch filter at 60 Hz with a stopband width of 2 to 4 Hz (58 to 62 Hz attenuated, all other frequencies passed at full amplitude) removes the power-line hum while preserving the reflection signal at 58 Hz and below and 62 Hz and above. Since 60 Hz falls within the reflection signal band of most WCSB surveys (typically 10 to 80 Hz), a simple low-cut or high-cut filter cannot remove the power-line hum without also removing reflection signal on either side of 60 Hz; only the notch filter can selectively suppress the interference within the signal band. In Alberta and BC, power-line routes across active seismic survey areas are mapped during acquisition planning, and receiver lines parallel to or under power lines are flagged for notch filter application during processing.
- Vibroseis harmonic notch filtering: Vibroseis seismic sources generate ground-force waveforms that deviate from the theoretical linear sweep due to mechanical non-linearities in the vibrator's baseplate hydraulic system. These non-linearities produce harmonic distortion at integer multiples of the instantaneous sweep frequency, creating harmonic energy (second harmonic, third harmonic, etc.) that arrives at the receiver as coherent noise offset in time from the fundamental sweep correlation. In correlating vibroseis data, the second harmonic of a 6 to 80 Hz sweep arrives as broadband noise with a time offset of approximately one octave of the sweep rate, contaminating the correlated section. Band-reject filters applied to the uncorrelated sweep record before correlation, targeting the 12 to 160 Hz second-harmonic band, attenuate the harmonic distortion before it is folded into the signal band by the correlation process. This pre-correlation harmonic notch filtering is now a standard step in WCSB vibroseis processing workflows, applied with stopband parameters derived from the sweep parameters (start frequency, end frequency, sweep length, and taper design) of each vibroseis source configuration used in the acquisition.
- VSP drill-string notch filtering: In vertical seismic profile surveys acquired while the drill string is rotating or reciprocating in the borehole (active drilling VSP), the drill string transmits mechanical vibration from the rig surface directly to the downhole geophone tool, contaminating the seismic record with drill-string resonance energy at the rotary table RPM frequency and its harmonics. For a rotary table turning at 80 RPM (1.33 Hz), drill-string resonance appears at 1.33, 2.67, 3.99 Hz and upward through the sonic frequency range. Band-reject filters with stopband centres at each harmonic frequency, applied to the VSP record before deconvolution and stacking, remove the drill-string tones while preserving the reflection signal between the tones. In LWD (logging while drilling) seismic operations where the drill string is always rotating, drill-noise notch filtering is essential for recovering any usable signal; without it, the drill-string harmonics can exceed the seismic signal amplitude by 10 to 30 dB, making the raw record appear as pure noise to a processor unfamiliar with the spectral signatures of drill-string energy.
- Marine seismic cable strum and notch filtering: Marine seismic streamers towed at 4 to 6 knots in ocean currents produce vortex-induced vibration (cable strum) when the cable diameter interacts with the flow to produce von Karman vortex shedding at a frequency determined by the Strouhal relationship: f = St x V / D, where St is approximately 0.2 (the Strouhal number for a cylinder in flow), V is the current speed, and D is the cable diameter. For a 50 mm diameter marine streamer in a 0.5 m/s crosscurrent, the Strouhal frequency is approximately 2 Hz; for higher flow speeds or smaller diameter cables, the frequency can reach 20 to 30 Hz and fall within the reflection signal band. Band-reject notch filters applied at the Strouhal frequency and its harmonics suppress the cable strum without removing reflection signal at adjacent frequencies. Modern solid marine streamers with integrated fluid buoyancy and Kevlar strength members have largely replaced liquid-filled streamers in premium WCSB marine surveys (Gulf of Mexico operations by Alberta-headquartered producers such as Canadian Natural Resources and Suncor), and the solid streamers have different cable strum spectra that require notch filter parameters different from the traditional liquid-filled cable specifications.
- Interaction with the reflection signal band: The critical limitation of band-reject filtering is that any reflection signal energy at the stopband frequencies is removed along with the noise, creating a spectral gap in the filtered data at the notch frequency. For a narrow notch of 2 to 4 Hz width, this gap removes only a small fraction of the total signal bandwidth (for example, 2 Hz out of a 70 Hz wide signal band, or 2.9% of the bandwidth). The wavelet distortion caused by this spectral gap is typically small: a 60 Hz notch applied to a 10 to 80 Hz signal produces a filtered wavelet that differs from the unnotched wavelet by approximately 3% in peak amplitude and by a small ripple in the side lobes, both of which are negligible for structural interpretation but may affect quantitative AVO and impedance inversion if not accounted for in the wavelet extraction and forward-modelling steps. For this reason, quantitative seismic interpretation workflows on notch-filtered data should use a wavelet extracted from the notch-filtered data (not from the unfiltered data) to ensure that the inversion forward model correctly accounts for the spectral hole in the data.
Time-Domain Response of Band-Reject Filters
The time-domain response of a band-reject filter to a seismic impulse (a zero-duration spike) is a waveform consisting of a main spike at time zero with a long oscillatory tail at the notch frequency. This ringing tail, which decays exponentially with a time constant inversely proportional to the notch width, is a fundamental property of the band-reject filter in the time domain and must be considered when applying notch filters to seismic data that contains strong reflectors near the target zone. For example, a 60 Hz notch filter with 2 Hz stopband width applied to a stacked seismic trace containing a high-amplitude reflection event will produce an approximately 100 ms ringing tail at 60 Hz on the filtered trace, which can interfere with weaker adjacent reflection events within 50 ms of the strong reflector. In the WCSB context, this is most problematic when a hard reflector such as the base of salt in the Devonian carbonate section or a hard carbonate cap above a reef produces a strong reflection that is spatially and temporally close to the Montney or Duvernay target horizon.
The ringing problem is mitigated by using a wide notch (5 to 10 Hz stopband width instead of 1 to 2 Hz), which reduces the time constant of the ringing at the cost of removing more signal bandwidth near the notch frequency. Alternatively, the notch filter can be applied only to those traces and time windows where the interference frequency is confirmed to be present (using a time-frequency spectrogram of the pre-stack data), rather than globally to all traces, preserving full signal bandwidth on traces that are not contaminated by power-line or other discrete-frequency noise. Modern machine-learning-based seismic noise attenuation algorithms can identify and remove coherent discrete-frequency interference adaptively without applying a fixed notch filter, preserving the signal bandwidth at the interference frequency more effectively than traditional spectral-domain notch filters; however, these approaches require careful quality control to ensure that the algorithm has not learned to attenuate genuine signal that coincidentally has a near-sinusoidal character, which can happen with high-amplitude monochromatic noise superimposed on a reflection signal at the same frequency.
Band-Reject Filtering in Production Data Time-Series Analysis
Beyond seismic data, band-reject filtering appears in oilfield production engineering and reservoir monitoring contexts where time-series signals must be cleaned of periodic interference. In WCSB gas field bottomhole pressure monitoring using permanent downhole gauges (typically quartz crystal or strain gauge gauges deployed in Montney wells on fiber or electrical cables), the pressure record may be contaminated by diurnal temperature cycles (24-hour period) that affect the gauge electronics temperature and create a periodic pressure drift of 0.005 to 0.02 MPa amplitude. A band-reject filter with a stopband centred at a period of 24 hours (frequency 0.0000116 Hz, or 1/86,400 Hz) removes the diurnal cycle from the pressure record while preserving the pressure transient signal from production changes, shutdowns, and reservoir depletion trends that occur on different time scales. This application of the notch filter concept in the very low frequency domain (sub-0.01 Hz) uses the same mathematical principle as the seismic 60 Hz notch but at frequencies six orders of magnitude lower, demonstrating that the band-reject filter is a universal signal processing tool applicable across the full frequency range of oilfield measurements.
Production decline curve analysis using time-series band-reject filtering is also applied to remove the periodic signature of seasonal demand-driven production rate cycles from Montney and Duvernay gas wells operated under nomination-constrained production profiles: WCSB gas producers in northeast BC typically reduce output in summer when natural gas demand is low and increase it in winter when heating demand peaks, creating an approximately annual (1/8760 Hz) production cycle in the rate data. Removing this seasonal production cycle from the rate-time record using a band-reject filter with a 0.8 to 1.2 year period stopband allows the underlying Arps decline-curve model to be fitted to the intrinsic formation deliverability trend rather than the demand-driven production schedule, producing more accurate EUR (estimated ultimate recovery) estimates for reserves reporting under NI 51-101 (Canada's National Instrument governing oil and gas disclosure) standards.