Questions: Seismic Data Processing and Noise Filtering

Question 1

In a common midpoint (CMP) gather, traces recorded at different source-receiver offsets show the same reflection arriving at different times. What is the shape of the arrival time curve plotted against offset?

• A. A straight line with positive slope — the further the offset, the later and more uniformly delayed the arrival
• B. A hyperbola — the travel time increases with offset in a curved relationship governed by the seismic velocity and reflector depth
• C. A flat line — because all traces share the same reflection point, arrivals are simultaneous regardless of offset
• D. An exponential curve — because energy loss increases non-linearly with distance traveled

Answer: (B)

The travel time t for a reflection at depth z with velocity v and offset x follows t² = t₀² + x²/v² (the NMO equation), which is the equation of a hyperbola. The zero-offset two-way time t₀ forms the apex of the hyperbola, and traces at larger offsets record the same reflection progressively later. This hyperbolic shape is not arbitrary — it is a direct geometric consequence of the extra path length traveled by off-center rays. Recognizing this shape is the foundation for velocity analysis: the correct seismic velocity is the one that, when applied in the NMO correction, flattens the hyperbola into a horizontal alignment across all offsets.

Question 2

After applying NMO correction and stacking 50 traces from a CMP gather, by approximately what factor does the signal-to-noise ratio improve?

• A. 50 — because there are 50 times as many traces contributing to the signal
• B. 25 — because stacking averages out half the noise from 50 traces
• C. 7 — because signal adds coherently while random noise averages down as the square root of the number of traces
• D. 2 — because stacking mainly removes the two largest noise spikes

Answer: (C)

When n traces are stacked, coherent reflections add in amplitude proportional to n, while random (incoherent) noise adds in amplitude proportional to √n (since random noise has zero mean and its amplitude sums as the root of the number of samples). The signal-to-noise ratio therefore improves by n/√n = √n. For n = 50, this is √50 ≈ 7. This √n improvement is why industry seismic surveys invest heavily in recording many traces per CMP (high fold): each doubling of fold provides roughly a 40% SNR improvement. Option A would only be correct if noise were coherent and added like signal, which random noise does not.

Question 3

NMO correction changes the physical content of the seismic data by adding new geological information to the traces.

• T. True
• F. False

Answer: False

False. NMO correction is a purely geometric operation — it time-shifts each trace to remove the offset-dependent travel-time delay, making all traces in a CMP gather look as though they were recorded at zero offset. It does not add information; it removes a geometric artifact of the acquisition geometry. The geological information (the reflection events) was already in the original traces — NMO correction simply aligns that information so that stacking can be applied. One side effect is 'NMO stretch' at large offsets and shallow times, where the time-shifting distorts the waveform shape, but this is an artifact of the correction, not new geological data.

Question 4

Velocity analysis must be performed before NMO correction can be applied, because the correct stacking velocity determines which hyperbola to flatten.

• T. True
• F. False

Answer: True

True. The NMO equation requires knowledge of the seismic velocity at each depth (or equivalently, a stacking velocity function with time). Without the correct velocity, you either over-correct (flattening too much, causing NMO stretch artifacts) or under-correct (leaving residual moveout, so stacking does not optimally align reflections). Velocity analysis is done by testing a range of velocities and finding the one that produces the flattest gather — the velocity that maximizes the coherent stack amplitude is the best stacking velocity. This is why velocity analysis is a distinct processing step that precedes NMO correction in the standard processing sequence.

Question 5

Explain why stacking improves the signal-to-noise ratio. What happens to the seismic reflection signal and what happens to random noise when multiple traces are averaged together?

Model answer: Seismic reflections are coherent signals — after NMO correction, they align horizontally across the CMP gather, so they add constructively when stacked (amplitudes sum). Random noise is incoherent — it has random polarity and timing, so it partially cancels when summed (amplitudes add as √n for n traces). The result is that the signal amplitude grows as n while the noise amplitude grows as √n, improving the signal-to-noise ratio by n/√n = √n. Stacking 50 traces therefore improves SNR by approximately 7.

This is the fundamental principle behind all coherency-based enhancement methods in signal processing. The key prerequisite is that NMO correction has already aligned the reflections — if reflections are not aligned before stacking, they too will partially cancel, and the SNR improvement is lost. This is why accurate velocity analysis is so critical: poor velocities mean poor NMO correction, which means poor stacking, which means poor SNR.