Seismic Methods II: Processing, Deconvolution, Migration, Interferometry, Attributes, AVO, VSP and Multicomponent Seismics
2. Seismic deconvolution
Deconvolution undoes the convolution of the wavelet with the reflectivity, to sharpen the trace and bring it closer to the earth's reflectivity series. In the Wiener approach a filter f of n points is found that makes the filtered trace as close as possible, in the least-squares sense, to a desired output d. The normal equations involve the trace's autocorrelation, which is Toeplitz, and the cross-correlation of the desired output with the trace. For spike deconvolution the desired output is a spike, so the output is the flattened (whitened) spectrum within the signal band. A small percentage of white noise is added to the zero lag of the autocorrelation (prewhitening) so that spectral holes are not amplified without limit.
Predictive deconvolution predicts the trace a distance α (the prediction lag) ahead from its past n samples, and the prediction error is the output. A lag of one sample is spike deconvolution; a longer lag keeps the first α of each wavelet and removes only repetitions longer than that. To attack short-period multiples the lag is set to the multiple's period: a water layer of depth z and speed Vw gives a repeat period 2z/Vw, so z = 150 m with Vw = 1500 m/s gives 0.2 s. Long-period multiples are left alone by this method and need Radon or surface-related multiple elimination.
3. Migration: dip relations and Kirchhoff migration
A stacked section plots each event at the midpoint, as if the reflection came from straight below. A dipping reflector is therefore drawn too far down-dip, at too gentle a dip, a point scatterer or fault edge makes a diffraction hyperbola, and a tight syncline gives a bow-tie. Migration moves energy to its true position, collapses diffractions and steepens dips. In the zero-offset (exploding-reflector) picture, with constant speed V, a true dip ξ and the time slope dT/dx on the section obey sin ξ = (V/2)·(dT/dx), and the event moves up-dip by Δx = (V²·T/4)·(dT/dx). Worked example: V = 2000 m/s, T = 1.0 s, dT/dx = 0.4 ms/m give sinξ = 1000 × 0.0004 = 0.4 (ξ ≈ 23.6°), Δx = 400 m, and the migrated depth is (VT/2)·cosξ = 1000 × 0.9165 ≈ 917 m, less than the unmigrated 1000 m.
Kirchhoff migration treats each output point as the apex of a diffraction hyperbola and sums the input amplitudes along that curve, with weights for obliquity and spherical spreading and a wavelet-shaping filter. It needs the velocity model, handles steep dips within the chosen aperture (the width of data summed), accepts irregular geometry, and works pre-stack or post-stack. Too wide an aperture admits noise, too narrow an aperture loses steep dips. Other methods are phase-shift and Stolt f-k (fast, for constant or depth-only velocity) and finite-difference wave-equation migration (dip accuracy limited by the approximation, such as 15° or 45°).
- Time versus depth migration: time migration suits mild lateral velocity change; strong lateral variation (salt, thrust belts) needs depth migration with an accurate velocity-depth model, usually pre-stack.
- Wrong velocity: a velocity too low under-migrates (diffractions not collapsed), too high over-migrates and draws upturned “smiles”. Post-stack migration also assumes the stack is a zero-offset section, which fails for conflicting dips, so such data go to pre-stack migration.
4. Reverse-time migration, time reversal and time-reversal imaging
Reverse-time migration (RTM) solves the full two-way wave equation numerically, so it has no dip limit and can image overturning, prismatic and turning waves, which one-way methods cannot. The steps:
(1) propagate the source wavefield S(x, t) forward through the velocity model;
(2) inject the recorded traces at the receivers and propagate them backward in time, from the last sample to the first, giving R(x, t);
(3) apply the imaging condition, the zero-lag cross-correlation I(x) = Σt S(x, t)·R(x, t), which is large where the downgoing and back-propagated wavefields coincide in time, at a reflector. It is costly, sensitive to the velocity model, and needs a Laplacian filter against low-frequency backscatter artefacts. RTM is the preferred tool near salt flanks.
Time reversal rests on a symmetry: in a lossless medium the wave equation contains only second time derivatives, so if u(x, t) is a solution then so is u(x, −t). A time-reversal mirror records the field on a surface, then re-emits the records in reversed order, and the wave retraces its path and refocuses at the original source, even through a complex scattering medium, with a focus about half a wavelength wide. Absorption adds a first-derivative term, breaks the symmetry and blurs the focus. Time-reversal imaging uses this to locate sources, for example microseismic events or earthquakes, by back-propagating their recordings and looking for the focus at t = 0; RTM is the same operation applied to reflection data with a source wavefield as the reference.
5. Seismic interferometry and the representation theorems
Seismic interferometry builds the response between two receivers, as if one were a source, from recordings that never had a source there. Take two receivers A and B. Cross-correlating their recordings of the same wavefield gives a trace whose peak lies at the difference of the two travel times from the source, which is the travel time between A and B for sources lying on the line through them. Summing the correlations over many sources, or over long stretches of random ambient noise, then yields an estimate of the Green's function between A and B, with both a causal part (positive lag) and an acausal part (negative lag). Example: two stations 30 km apart, with a surface-wave group speed of 3 km/s, show a peak at ±30/3 = ±10 s.
The theory is the pair of representation theorems. Both take two wave states in one medium, with Green's functions G(x, xA) and G(x, xB) for sources at xA and xB, and relate a field inside a closed surface S to integrals over S. The correlation type brings in the complex conjugate (the time-reversed field) of one state: the causal Green's function plus its time-reverse, G(xB, xA, t) + G(xB, xA, −t), equals a surface integral of the cross-correlations of G(x, xA) and G(x, xB) and of their normal derivatives, up to factors of density and frequency. It needs a lossless medium, because only then is time-reversal a symmetry. The convolution type has no conjugate, convolves the two fields, holds also in lossy media, underlies wavefield extrapolation (the Kirchhoff-Helmholtz and Rayleigh integrals), and in interferometry adds travel times where correlation subtracts them. Source-receiver reciprocity, G(xB, xA) = G(xA, xB), is what lets receivers play the part of sources.
- Practical limits: the theorem assumes sources all round the receivers. With one-sided or uneven noise the arrival times remain good, but amplitudes are biased and spurious events appear. The main contribution comes from sources near the line through the two receivers (the stationary-phase region).
- Uses: ambient-noise surface-wave tomography of the crust, the virtual-source method (borehole receivers acting as sources below a complex overburden) and passive imaging.
6. Attributes, bright and dim spots, and AVO
A seismic attribute is a quantity derived from the data to bring out a property. The complex trace z(t) = x(t) + i·y(t), with y the Hilbert transform of x, gives three: the envelope (reflection strength) A = √(x² + y²), insensitive to phase and tied to impedance contrast; the instantaneous phase φ, which displays continuity and sequence boundaries; and the instantaneous frequency (1/2π)·dφ/dt, which drops in the shadow of gas through absorption and is erratic at thin beds. Geometric attributes such as coherence (similarity of neighbouring traces) map faults and channels, and curvature maps structure. An attribute is only as good as the processing behind it, so calibrate it against wells; a correlation is not a cause.
A bright spot is a local amplitude anomaly, classically from a gas sand under shale, where impedance falls and R is large and negative, so the event may also reverse polarity relative to the seabed or to a water-sand top. Supporting signs are a flat event (a gas-water contact, which is horizontal on a depth section), amplitude that rises with offset, a low-frequency shadow below and a velocity pull-down. A dim spot is an amplitude decrease, for example a high-impedance sand whose positive R is reduced when gas lowers its impedance towards that of the shale above. Bright spots are not proof of gas: coal, igneous sills, tuned thin beds and hard carbonate streaks also give them.
AVO (amplitude variation with offset) reads fluids and lithology from the change of reflection amplitude with angle, using the Zoeppritz equations or their approximations. The two-term Shuey form, valid to about 30°, is R(θ) ≈ A + B·sin²θ, with intercept A the normal-incidence R and gradient B. Worked example: A = −0.12, B = −0.20 give R(30°) = −0.12 − 0.20 × 0.25 = −0.17, a bright, increasing-amplitude event. Gas sands are sorted into classes by their impedance relative to the shale above.
| Class | Sand impedance against shale | Normal-incidence R | Change with offset |
|---|---|---|---|
| I | Higher | Large positive | Amplitude falls, may reverse polarity |
| II | About equal | Near zero | Amplitude grows from a small value; polarity may flip |
| III | Lower | Large negative | Becomes more negative, a bright spot |
| IV | Lower, with a hard shale above | Negative | Magnitude falls with offset |
7. Seismic stratigraphy, resolution, high-resolution seismics, VSP and multicomponent seismics
Seismic stratigraphy reads depositional sequences, successions bounded by unconformities and their correlative conformities, from reflection terminations: onlap and downlap (baselap) at the base, toplap and erosional truncation at the top. Reflections follow bedding surfaces, which are time lines, rather than facies boundaries. Seismic facies are described by reflection configuration (parallel, divergent, clinoform, hummocky, chaotic, free), continuity, amplitude, frequency and interval velocity, and are interpreted as environment and relative sea-level change.
Resolution has two parts. Vertically, with dominant wavelength λ = V/f, two interfaces are resolved when the bed is thicker than about λ/4 (the tuning thickness, where the top and base reflections of a typical thin bed interfere constructively); below it thickness is inferred from amplitude, not from separation. Horizontally, unmigrated resolution is the first Fresnel zone, of radius r = √(z·λ/2) = (V/2)·√(T0/f), which migration reduces towards about half a wavelength at best. Worked example: V = 3000 m/s, f = 30 Hz, T0 = 1.2 s give λ = 100 m, z = 1800 m, r = √(1800 × 100/2) = 300 m. Absorption, A = A0·exp(−π·f·t/Q), cuts high frequencies, so resolution is traded against penetration. High-resolution seismics works at the shallow end of that trade: high-frequency sources (sparker, boomer, chirp, hammer, mini-vibrator), closely spaced groups, sub-millisecond sampling and small bins, for engineering, groundwater, coal and shallow-gas targets, with careful statics and positioning.
VSP places receivers at many depths in a well with the source at the surface. In a zero-offset VSP the first-arrival times give true vertical travel times, so the interval velocity is Δz/Δt between two depths: first arrivals at 0.200 s (400 m) and 0.260 s (550 m) give 150/0.060 = 2500 m/s. The record holds a downgoing field (direct wave and multiples) and an upgoing field (reflections), separated by f-k or median filtering; a reflector lies at the depth where its upgoing event meets the first-arrival line, and a corridor stack of the upgoing field ties the surface seismic to the well. Offset and walkaway VSPs image away from the well and ahead of the bit.
Multicomponent seismics records the full particle-motion vector with three-component receivers (and four-component ocean-bottom nodes at sea, since shear waves do not travel in water). A P wave converted to S at a reflector gives the PS (C) wave, with its reflection point nearer the receiver, so common-conversion-point binning is needed. S waves feel mainly the rock frame and not the pore fluid, so they image through gas clouds, and shear-wave splitting into fast and slow components shows fracture-induced anisotropy. With vertical-incidence two-way times T_PP and T_PS to the same reflector, Vp/Vs = 2·T_PS/T_PP − 1: T_PP = 1.0 s and T_PS = 1.4 s give 1.8.
Key takeaways
- Nyquist frequency = 1/(2Δt). Processing undoes the convolutional model: gain, deconvolution, velocity analysis (semblance), NMO, stack, migration. In the f-k plane an event of apparent velocity V lies on f = V·k.
- Spike deconvolution whitens the spectrum; predictive deconvolution with lag equal to the multiple's period (2z/Vw for a water layer) removes short-period multiples. It needs a stationary wavelet and white reflectivity.
- Migration moves dipping events up-dip and steeper (sinξ = (V/2)·dT/dx, Δx = (V²T/4)·dT/dx). Kirchhoff sums along diffraction hyperbolae; RTM solves the two-way wave equation and images by zero-lag correlation of source and back-propagated fields.
- Time reversal needs a lossless medium and refocuses to about half a wavelength. Correlation-type interferometry needs a lossless medium; the convolution type holds with loss and adds times.
- A bright spot is a large negative R (gas under shale) but coal and sills mimic it; AVO uses R ≈ A + B·sin²θ and classes I to IV. Envelope, phase and frequency come from the complex trace.
- Vertical resolution is λ/4 and unmigrated horizontal resolution is √(zλ/2). A VSP gives velocity as Δz/Δt, converted waves give Vp/Vs = 2·T_PS/T_PP − 1, and sequences are read from onlap, downlap, toplap and truncation.
Practice questions (17)
Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.
Which of the following is NOT an assumption of conventional Wiener spiking deconvolution applied to a stacked or pre-stack trace?
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Answer: D — The source signature was measured in the field and is supplied as input
Spiking deconvolution is statistical: it infers the wavelet from the autocorrelation of the trace itself, which is why it needs a white reflectivity, a stationary wavelet and, for a causal stable inverse, minimum phase. A measured signature is used by deterministic deconvolution (signature deconvolution), not by the statistical method, so it is the one item that is not an assumption.In a marine survey the water depth is 120 m and the speed of sound in water is 1500 m/s. A predictive deconvolution is to be designed against the water-bottom reverberation, with the prediction lag set equal to its repetition period. The lag is ____ ms.
Numerical answer — type the value.
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Answer: 160
The water-bottom multiple repeats with the two-way time through the water layer: 2 × 120/1500 = 0.16 s = 160 ms. Using the one-way time (80 ms) is the usual slip.Ground roll with an apparent velocity of 300 m/s and a frequency of 30 Hz is recorded with a group interval of 10 m. Which statement is correct?
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Answer: A — Its wavenumber is 0.1 cycles/m, above the spatial Nyquist value of 0.05 cycles/m, so it is aliased
On the line f = V·k the wavenumber is k = f/V = 30/300 = 0.1 cycles/m. The spatial Nyquist wavenumber is 1/(2Δx) = 1/20 = 0.05 cycles/m, so the event is aliased and wraps to a false wavenumber, which is why an f-k filter cannot remove it cleanly and field arrays are needed. The 0.2 cycles/m value would be the Nyquist wavenumber for a 2.5 m interval, 0.01 cycles/m inverts V and f, and 1/Δx is twice the true Nyquist wavenumber.Which of the following statements about deconvolution are correct?
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Answer: A — Predictive deconvolution with a prediction lag of one sample is equivalent to spike deconvolution; B — Prewhitening adds a small amount of white noise to the zero lag of the autocorrelation so that the filter stays stable
With a one-sample lag the prediction error is what cannot be predicted from the past, which is the spike, so this is spike deconvolution, and prewhitening is the standard stabiliser. Deconvolution can only rebalance frequencies that carry signal, and boosting empty parts of the spectrum raises noise. A lag matched to one multiple period targets repetitions at that period and leaves long-period multiples, which need Radon or surface-related methods.What does migration do to a dipping reflector on a zero-offset stacked section?
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Answer: C — It moves the event up-dip and steepens it; true dip exceeds apparent dip
Because sin ξ = tan ζ in the constant-speed case, the true dip ξ exceeds the apparent dip ζ on the section, and the event moves laterally up-dip while the depth to it decreases. Moving it down-dip reverses the direction: the stack plots the energy below the midpoint, which lies down-dip of the true reflecting point.On a constant-velocity (V = 3000 m/s) stacked section, an event at two-way time T = 1.0 s has a time slope dT/dx = 0.2 ms/m. Using sin ξ = (V/2)·(dT/dx) and the geometry of the exploding-reflector model, the lateral up-dip displacement of the event on migration is ____ m.
Numerical answer — type the value.
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Answer: 450
sin ξ = (3000/2) × 0.0002 = 0.3. The unmigrated normal distance is VT/2 = 1500 m, and the lateral displacement is (VT/2)·sinξ = 1500 × 0.3 = 450 m (equivalently (V²T/4)·dT/dx = 2.25 × 10⁶ × 0.0002). Entering 0.2 ms/m as 0.2 s/m would make sin ξ exceed 1, which is impossible.How does Kirchhoff migration form the image at a chosen output point?
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Answer: B — It sums the input amplitudes along the diffraction curve of that point, with obliquity and spreading weights, and places the sum at the point
Kirchhoff migration is a weighted summation along the diffraction hyperbola whose apex is the output point; it needs a velocity model to define the curve. Back-propagation with a correlation imaging condition describes reverse-time migration, and the stepwise phase shift describes the phase-shift method for depth-only velocity.Which of the following statements about reverse-time migration (RTM) and time reversal are correct?
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Answer: A — RTM can image steep dips and overturning waves because it solves the two-way wave equation; C — The wave equation of a lossless medium is unchanged when t is replaced by −t
The lossless wave equation has only second time derivatives, so it is symmetric in time, and RTM, which propagates the full field, is free of the dip limit of one-way methods. Absorption introduces a first time derivative and breaks the symmetry, so refocusing degrades. RTM propagates waves through an assumed velocity model and is sensitive to its errors.Two broadband stations 40 km apart record ambient noise from all around. The cross-correlation of their records has its main peak at a lag of +12.5 s, and a mirror peak at −12.5 s. What is the group velocity of the surface wave between them?
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Answer: A — 3.2 km/s, from the inter-station distance divided by the lag
The correlation peak at lag τ is the travel time of the wave between the two stations, as if one were a source, so V = 40 km/12.5 s = 3.2 km/s. The peaks at ±τ are the causal and acausal parts of the Green's function (waves going each way), not a two-way trip. Doubling the speed or halving it comes from misreading the lag.Which statement correctly contrasts the correlation-type and convolution-type representation theorems?
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Answer: D — The correlation type needs a lossless medium, and the convolution type holds also with losses
The correlation-type theorem uses the complex conjugate (time-reversed) field, and time-reversal is a symmetry only when there is no loss; the convolution-type theorem has no conjugate and is valid in dissipative media too. Retrieving the Green's function from noise by cross-correlation is the correlation type. The two do not differ merely in the source, and the loss requirement is not reversed.A strong, laterally limited amplitude anomaly lies over a structural closure. One segment of it is horizontal, cuts across the dip of the neighbouring reflections and ends abruptly at the closure outline. What is the best explanation?
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Answer: B — A fluid contact such as gas over water, because a contact is horizontal while the bedding is not
A gas-water (or oil-water) contact is a gravity-controlled horizontal surface, so a flat event that cuts dipping bedding and terminates on the structural outline is a flat spot, a direct hydrocarbon indicator. Multiples repeat the shape of the primaries and do not end at a structural contour, sills need not be horizontal (many are saucer-shaped) and have no reason to stop at a contour, and migration smiles are curved noise artefacts.A gas sand beneath shale gives a large negative normal-incidence reflection coefficient whose magnitude increases with offset, so a bright event brightens further on far-offset traces. Which AVO class is this?
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Answer: C — Class III, as the sand impedance is below the shale's and R grows more negative with offset
A large negative intercept and a negative gradient (more negative with offset) is Class III, the classic bright-spot gas sand. Class I has a positive intercept, Class II a near-zero one, and Class IV has a negative intercept whose magnitude decreases with offset, so the stated increase rules it out.Using the two-term Shuey form R(θ) = A + B·sin²θ with intercept A = 0.08 and gradient B = −0.40, the reflection coefficient is zero at an angle of incidence of ____ degrees (give the answer to one decimal place).
Numerical answer — type the value.
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Answer: 26.6
R = 0 when sin²θ = −A/B = 0.08/0.40 = 0.2, so sinθ = 0.4472 and θ = 26.57°, which is 26.6° to one decimal place. At larger angles R becomes negative, so the polarity reverses, as for a Class I or II sand.In a zero-offset VSP the first arrivals are 0.300 s at 600 m depth and 0.375 s at 780 m depth. What is the interval velocity between the two receivers?
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Answer: B — 2400 m/s, the depth difference divided by the time difference
A VSP first arrival is a one-way vertical time, so Vint = Δz/Δt = (780 − 600)/(0.375 − 0.300) = 180/0.075 = 2400 m/s. The 2080 m/s choice is the average velocity to 780 m (780/0.375), not the interval value. Halving or doubling treats the times as two-way or the interval as doubled, neither of which applies.A zero-offset reflection arrives at a two-way time of 1.6 s. The average velocity is 2000 m/s and the dominant frequency is 40 Hz. Using r = √(z·λ/2) with λ = V/f, the radius of the first Fresnel zone on the unmigrated section is ____ m.
Numerical answer — type the value.
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Answer: 200
Depth z = V·T/2 = 2000 × 1.6/2 = 1600 m and λ = V/f = 2000/40 = 50 m, so r = √(1600 × 50/2) = √40000 = 200 m. Using the full two-way time as the depth (3200 m) gives 283 m, the usual slip.From a multicomponent survey the two-way vertical time to a reflector is 1.6 s on the PP section and 2.2 s on the PS (converted-wave) section. Using Vp/Vs = 2·T_PS/T_PP − 1, the average Vp/Vs down to the reflector, to two decimal places, is ____.
Numerical answer — type the value.
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Answer: 1.75
T_PS/T_PP = 2.2/1.6 = 1.375, so Vp/Vs = 2 × 1.375 − 1 = 1.75. Taking the ratio of the times directly (1.375) forgets that the PS time is the average of the P and S times.Which of the following statements about seismic stratigraphy and multicomponent seismics are correct?
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Answer: A — Onlap and downlap are baselap terminations at the base of a depositional sequence; B — Toplap and erosional truncation are terminations seen at the top of a sequence
Baselap (onlap, downlap) marks the base of a sequence and toplap with erosional truncation marks its top. Reflections follow bedding, which is a time line, so they are chronostratigraphic. The S leg of a converted wave feels the rock frame and is hardly affected by gas in the pores, which is why PS data can image through gas clouds that scramble P waves.