Questions: Time-Series and Frequency-Domain Analysis in Seismology

Question 1

A seismologist wants to remove 60 Hz power-line noise from a seismic recording. In which domain is this most efficiently accomplished?

• A. The time domain, by subtracting a 60 Hz sine wave from the raw seismogram
• B. The frequency domain, by computing the Fourier transform, zeroing the 60 Hz component, then inverse transforming
• C. The spatial domain, by combining recordings from multiple seismometers to cancel the noise
• D. The wavelet domain, which is required for any frequency-based operation

Answer: (B)

The Fourier transform converts the time-domain signal to a frequency spectrum where different frequency components are separated and identifiable. Suppressing 60 Hz noise is trivial in the frequency domain — multiply the spectrum by a notch filter at 60 Hz, then inverse transform back. Doing this in the time domain requires convolving with a complex filter. The efficiency of the frequency domain for filtering operations is the central practical motivation for using the Fourier transform.

Question 2

Stacking multiple seismic recordings of the same subsurface point improves signal-to-noise ratio primarily because:

• A. Coherent signal adds linearly while random noise adds in quadrature, improving SNR by √N for N stacked traces
• B. Each additional trace adds more signal power, making the total signal N times stronger while noise stays constant
• C. Stacking applies an implicit bandpass filter above 40 Hz, removing high-frequency noise
• D. Multiple recordings cover a wider area of the subsurface, providing more spatial information

Answer: (A)

The signal is coherent across traces — same arrival time and waveform shape. Random noise is incoherent — different in each trace. When N traces are summed, the coherent signal adds linearly (amplitude scales as N) while random noise adds in quadrature (amplitude scales as √N), so signal-to-noise ratio improves by N/√N = √N. This is why seismic surveys acquire many redundant recordings — each additional trace provides diminishing but real improvement.

Question 3

Deconvolution improves the temporal resolution of a seismic record by removing the effect of the source wavelet from the recorded signal.

• T. True
• F. False

Answer: True

A seismogram is the convolution of the source wavelet, the Earth's reflectivity series (what you want), and the instrument response. Deconvolution inverts this: because convolution in time equals multiplication in frequency, deconvolution is spectral division by the source wavelet. The result compresses the broad, ringy wavelet into a sharp spike, improving the ability to resolve closely spaced reflectors. Deconvolution is a standard step in seismic processing pipelines for this reason.

Question 4

A seismic bandpass filter passing 10–40 Hz both removes low-frequency microseismic ocean noise and high-frequency power-line noise while preserving the geologically relevant signal.

• T. True
• F. False

Answer: True

Ocean-generated microseismic noise dominates below ~1 Hz; power-line and cultural noise concentrate at 50–60 Hz. Crustal reflection signals often fall in the 5–40 Hz range. A 10–40 Hz bandpass filter suppresses both noise bands while passing the geologic signal — dramatically improving signal-to-noise ratio. The Fourier transform makes this selective frequency removal straightforward via multiplication by a window function in the frequency domain.

Question 5

Why does the fact that convolution in the time domain equals multiplication in the frequency domain make deconvolution practical?

Model answer: A seismogram is the convolution of wavelet, Earth reflectivity, and instrument response. Undoing convolution (deconvolution) is a difficult inverse problem in the time domain. But in the frequency domain, convolution becomes multiplication, so deconvolution becomes simple division: divide the recorded spectrum by the wavelet spectrum. This transforms a computationally hard operation into straightforward pointwise arithmetic, making it practical for large seismic datasets.

The convolution theorem is what makes frequency-domain processing powerful throughout signal processing and geophysics. Operations that are difficult or expensive in the time domain (deconvolution, arbitrary filtering) become trivial in the frequency domain. The tradeoff is that a forward and inverse Fourier transform must be applied, but FFT algorithms make this efficient even for large datasets.