Frequency Domain

The frequency domain is a mathematical representation of a signal in which the independent variable is frequency (measured in hertz) rather than time. Any time-domain signal (such as a seismic trace showing amplitude versus time) can be transformed into the frequency domain using the Fourier transform, which decomposes the signal into its component sine waves of different frequencies, amplitudes, and phases. The frequency-domain representation shows which frequencies are present in the signal, how strong each frequency is (the amplitude spectrum), and how the phase of each frequency relates to the others (the phase spectrum). Many seismic processing operations are applied in the frequency domain because they are simpler or more computationally efficient there than in the time domain.

Key Takeaways

The Fourier transform converts a time-domain signal into its frequency-domain equivalent. The inverse Fourier transform converts it back. In practice, the discrete Fourier transform (and its fast algorithm, the FFT) is applied to digitized seismic data sampled at regular time intervals (typically every 1 to 4 milliseconds).
In the frequency domain, a seismic trace appears as an amplitude spectrum (amplitude as a function of frequency, showing the energy distribution across the bandwidth) and a phase spectrum (phase shift at each frequency). Filters applied in the frequency domain multiply each frequency component by a gain factor, which is much simpler than the convolution operation required in the time domain.
Spectral whitening (or whitening deconvolution) is applied in the frequency domain to balance the amplitude spectrum, boosting weaker frequencies and attenuating dominant ones to produce a more uniform response across the bandwidth. This improves the seismic resolution by preventing a narrow band of dominant frequencies from masking the finer detail in the data.
The frequency domain has a direct spatial equivalent: the wavenumber domain (or f-k domain), in which the x-axis is wavenumber (cycles per metre) rather than time. Dip filters and noise suppression filters that target noise propagating at specific angles can be applied efficiently in the f-k domain.
Seismic data is typically acquired with a bandwidth of 5 to 100 Hz on land and 2 to 150 Hz in marine surveys. The frequency content directly determines the vertical resolution of the seismic data: a dominant frequency of 40 Hz with a velocity of 3,000 m/s produces a dominant wavelength of 75 metres and a resolution limit (quarter wavelength) of about 19 metres.

What Is the Frequency Domain?

Play a chord on a piano. Three keys pressed together produce three notes simultaneously. Your ear hears the combined sound. An audio spectrum analyzer attached to a microphone shows the sound as three peaks on a graph, each peak at the frequency of one of the piano keys. The spectrum analyzer is showing you the frequency-domain representation of the chord: not how the sound varies over time, but which frequencies make it up.

A seismic trace is a complex waveform that contains many different frequencies mixed together. The time-domain seismic trace shows you the combined waveform. The Fourier transform of the trace shows you the frequency-domain version: which frequencies are present, how much energy is at each frequency, and how the phases of the different frequencies relate to each other. The three-peak piano chord becomes a seismic spectrum with a roughly bell-shaped curve that peaks at the dominant frequency and falls off toward zero at the low and high ends of the bandwidth.

What makes the frequency domain useful for processing is that many operations that are complicated in time (such as filtering out a specific frequency band or equalizing the response across the bandwidth) are simple multiplications in frequency. Instead of applying a complex filter operation sample by sample through the full length of the trace, you can transform to frequency, multiply by the filter gain at each frequency, and transform back. The result is identical but the computation is faster.

Fast Facts

The Fast Fourier Transform (FFT) algorithm, first published in its modern form by Cooley and Tukey in 1965, made frequency-domain processing practical for large seismic datasets. Before the FFT, computing the Fourier transform of a seismic trace required N² operations for a trace of N samples. The FFT reduces this to N log₂(N) operations, a speedup that is enormous for long traces. A trace of 4,096 samples requires 16,777,216 operations using the direct approach but only 49,152 using the FFT, a factor of 341 reduction. This efficiency gain is what allowed frequency-domain processing to become the standard rather than a computational curiosity.

Frequency-Domain Operations in Seismic Processing

Band-pass filtering is the most common frequency-domain operation. The seismic trace is transformed to the frequency domain, the amplitude of all frequencies below a low cutoff (say 5 Hz) and above a high cutoff (say 90 Hz) is set to zero, and the trace is transformed back to time. The result removes very low-frequency noise (ground roll, swell noise) and very high-frequency noise (ambient noise, cable strum) while preserving the signal bandwidth of 5 to 90 Hz.

Spectral shaping (or spectral balancing) equalizes the amplitude spectrum so that all frequencies in the pass band have roughly the same amplitude. This is important when the source generates more energy at some frequencies than others, or when the earth preferentially attenuates certain frequencies during wave propagation. An equalized spectrum produces a wavelet that is more compact in time (better resolution) and more consistent across the survey area.

Phase rotation converts the seismic wavelet from one phase character to another (for example, from minimum phase to zero phase) by multiplying each frequency component by the appropriate phase shift in the frequency domain. Most seismic interpretation is done on zero-phase data, where the peak of the wavelet is centered on the reflection event, because zero-phase wavelets are easier to pick and correlate to well logs.

The Frequency Domain in Electromagnetic Surveys

In controlled-source electromagnetic (CSEM) surveys, the source signal is typically a square wave or a multi-frequency sine wave transmitted at a fundamental frequency (0.1 to 4 Hz) and its harmonics. The receiver data is analyzed in the frequency domain: the amplitude and phase of the fundamental frequency and harmonics at each receiver offset are the data points used to map subsurface resistivity. This frequency-domain CSEM approach is used by EMGS and CGG Electromagnetic in their marine surveys on the Norwegian Continental Shelf and other deepwater areas.

The frequency domain is sometimes called the spectral domain or the Fourier domain. FD is the abbreviation used in seismic processing documentation. Related terms include Fourier transform (the mathematical operation that converts a time-domain signal to its frequency-domain representation; the inverse Fourier transform converts back; the FFT is the computationally efficient algorithm used in practice), amplitude spectrum (the frequency-domain plot showing the amplitude of a signal as a function of frequency; shows the bandwidth and spectral shape of a seismic trace or wavelet), band-pass filter (a filter that passes frequencies within a specified range and attenuates frequencies outside it; most commonly applied in the frequency domain by multiplying the amplitude spectrum by zero outside the pass band), wavenumber (the spatial frequency of a wave, measured in cycles per metre; the spatial counterpart of temporal frequency; the f-k (frequency-wavenumber) domain is a two-dimensional frequency domain used for dip filtering and noise suppression in seismic processing), and spectral whitening (a frequency-domain processing step that equalizes the amplitude spectrum of a seismic trace to produce a uniform response across the bandwidth; improves temporal resolution by preventing dominant frequencies from masking finer structure).

How Frequency-Domain Filtering Recovered a Montney Seismic Survey from Noise

An operator had acquired a 2D seismic survey over a Montney tight gas prospect in northeast British Columbia, shooting with an explosive source along a grid of lines crossing the Dawson Creek area. The raw seismic data was of poor quality in several areas, contaminated by 60 Hz power-line interference from a transmission corridor that ran parallel to some of the receiver lines. The 60 Hz noise appeared as a coherent hum overlaid on the reflection signal.

The processing contractor transformed the affected traces to the frequency domain and applied a notch filter centered at 60 Hz, setting the amplitude to zero in a narrow band from 57 to 63 Hz. The inverse Fourier transform produced clean traces with the 60 Hz hum removed. Because the seismic signal bandwidth for this survey was 8 to 80 Hz, the 57 to 63 Hz notch removed only a small portion of the signal bandwidth along with the noise. The improvement in signal quality in the notch-filtered data was dramatic: reflection events that were masked by the hum became visible and correlatable across the affected lines.

The notch filter processing was applied during the early quality control pass and added approximately CAD 8,000 to the processing budget for the additional run. The alternative, re-acquiring the affected lines away from the power-line corridor, would have required a renegotiation of land-access and safety permits and an additional field crew mobilization estimated at CAD 180,000. The frequency-domain solution was available because the noise was spectrally distinct from the signal: 60 Hz is a single frequency, while the seismic signal is broadband. When noise and signal occupy the same frequency range, time-domain or frequency-domain filtering cannot separate them, and different acquisition geometry is the only solution.