Stacked data

4.1 Stacked data

4.1.1 Bandlimited reflectivity

The ultimate goal of a migration is to transform the seismic data into bandlimited reflectivity. In the simplest context, reflectivity means the normal-incidence reflection coefficient of P-waves. Potentially, every point in the subsurface has a reflectivity value associated with it. In a one dimensional layered medium, P-wave reflectivity is given by r k = 2(I k −I k−1 )/(I k +I k−1 ) where

I k =ρ k v k is the P-wave impedance of the k th layer. In the continuous case, this can be written as r δz (z) = .5∂ z ln(I(z))δz where δz is a small increment of depth. In a 3D context, reflectivity of a small earth element, δvol, is conveniently described by

(4.1) At best, equation (4.1) can only be an expression for normal incidence reflectivity. More generally,

r(x, y, z) = .5 ∇(log(I(x, y, z))) #

reflectivity must be acknowledged to be a function of the angle of incidence as well as spatial position.

79 For the case of plane elastic waves in layered media, the Zoeppritz equations (Aki and Richards,

4.1. STACKED DATA

1980) are an exact prescription of this variation. The Zoeppritz equations are highly complex and nonlinear and the estimation of their approximate parameters from seismic data is called the study of amplitude variation with offset or AVO. (A nearly synonymous term is amplitude variation with angle or AVA.) Such complex reflectivity estimates are key to true lithology estimation and are a subject of much current research. Ideally, AVO analysis should be conducted simultaneously with migration or after migration. For the current discussion, it is assumed that only the normal-incidence reflectivity, r(x, y, z), is of interest and that a stacked seismic section provides a bandlimited estimate of r(x, y, z).

The estimate of reflectivity must always be bandlimited to some signal band of temporal fre- quencies. This is the frequency range over which signal dominates over noise. Unless some sort of nonlinear or model-based inversion is done, this signal band is determined by the power spectrum of the source, the data fold, the types of coherent and random noise, and the degree of anelastic attenuation (Q loss). The optimal stacked section will have a zero-phase, white, embedded wavelet. This means that it should obey the simple convolutional model

(4.2) where w(t) is a zero phase wavelet whose amplitude spectrum is white over some limited passband.

s(t) = r(t) • w(t)

If w(t) has residual phase or spectral color then these will adversely affect the resolution of the final migrated image. They should be dealt with before migration.

4.1.2 The zero offset section

A discussion of post-stack migration greatly benefits from a firm theoretical model of the CMP (common midpoint) stack. A simple model of stacked seismic data is the zero-offset section or “ZOS” model. This model asserts that the pre-stack processing and CMP stack estimates a signal-enhanced version of what we would have recorded had there been a single, coincident soure/receiver pair at each CMP. Each stacked trace represents a separate physical experiment that can be approximately described by the scalar wave equation (assuming acoustic energy only). Taken together, the ensemble of stacked traces has a more complicated basis in physical theory.

The major steps the in the estimation of the signal enhanced ZOS are Spherical spreading correction: A time dependent amplitude scaling that approximately corrects

for spherical divergence. Deconvolution: The source signature and the average Q effect is estimated and removed. Sort to (x, h) coordinates: The data is considered to be gathered in (s, g) (source,geophone) coor-

dinates and then sorted to midpoint, x, and half-offset, h (Figure 4.1). Statics correction: Near surface static time delays are estimated and removed. NMO removal: The data is taken through stacking velocity analysis and the resulting velocities are

used to remove normal moveout time delays. Residual statics correction: Residual time delays (small) are sought and removed. Usually this is

a surface-consistent step. Trace balancing: This might be considered optional but some step is needed to correct for source

strength and geophone coupling variations.

80 CHAPTER 4. ELEMENTARY MIGRATION METHODS Common

14 midpoint coordinate 12

VelocityWlens

ce 10 Sour 8

Receiver coordinate Figure 4.1: A 2D seismic line is depicted in the

Figure 4.2: Most, but not all, zero-offset ray- (s, g) plane. Each shot is represented as having

paths are also normal-incidence raypaths. six receivers with a central gap. Common mid- point, x, and common offset, h coordinates are depicted.

CMP stacking: All traces with a common midpoint coordinate are summed. Events which have been flattened on CMP gathers are enhanced. Other events and random noise are reduced.

The ZOS is said to be signal enhanced because the stacking process has been designed to select against multiples. The simplest (and dominant) raypath which returns energy to its source is called the normal-incidence raypath. Quite obviously, the normal-incidence reflection will send energy back up along the path that it traveled down on. However, it is possible to have zero-offset paths that are not normal-incidence paths (Figure 4.2).

4.1.3 The spectral content of the stack

The sort from (s, g) to (x, h) marks the transition from single channel to surface consistent processing. Where it actually takes place is somewhat arbitrary because a seismic processing system can always resort the data whenever required. However, it is a logical transition that must occur somewhere before stack. (x, h) coordinates are defined by

of which the inverse transformation is

(4.4) The relationship between (s, g) and (x, h) is depicted graphically in Figure 4.1.

s = x − h and g = x + h.

Mathematically, a 2D pre-stack seismic dataset is a 3D function, ψ(s, g, t). This dataset can be written as an inverse Fourier transform of its spectrum through

ψ(s, g, t) =

φ(k s ,k g , f )e 2πi(k s s+k g g−ft) dk s dk g df (4.5)

4.1. STACKED DATA

g (meters)

2500 m/s 5 2000 m/s 1500 m/s 1000 m/s

maximum frequency (Hertz)

Figure 4.3: These curves show the maximum spatial sample rate required to avoid aliasing of k g on shot records (assuming point receivers) as a function of the maximum signal frequency. A curve is drawn for each of five near surface velocities. This is based on expression (4.15).

where the notation V ∞ indicates the integration covers the entire relevant, possibly infinite, portion of (k s ,k g , f ) space. Imposing the coordinate transform of equations 4.4 gives

ψ(x, h, t) = φ(k s ,k g , f )e 2πi(k s (x−h)+k g (x+h)−ft) dk s dk g df (4.6)

or ψ(x, h, t) =

φ(k s ,k g , f )e 2πi((k s +k g )x+(k s −k g )h−ft) dk s dk g df. (4.7)

This motivates the definitions k x =k s +k g and k h =k s −k g (4.8) or the inverse relationship

This allows equation 4.7 to be written

ψ(x, h, t) =

φ(k ,k

h , f )e

2πi(k x+k

h h−ft) dk x dk h df. (4.10)

where the factor of 1/2 comes from the Jacobian of the coordinate transformation given in equation

4.9. Both of equations 4.5 and 4.10 are proper inverse Fourier transforms which shows that the wavenumber relationships given in equations 4.8 and 4.9 are the correct ones corresponding to the (s, g) to (x, h) coordinate transformation.

Quite a bit can be learned from equations 4.9 about how the spectral content of the pre-stack data gets mapped into the post-stack data. The maximum midpoint wavenumber, k xmax comes

82 CHAPTER 4. ELEMENTARY MIGRATION METHODS

from the sum of k smax and k gmax . As will be shown formally later, the maximum wavenumber is directly proportional to horizontal resolution (i.e. the greater the wavenumber, the better the resolution). The meaning of the maximum wavenumbers is that they are the largest wavenumbers that contain signal. They typically are less than the Nyquist wavenumbers but they cannot be greater. Thus k smax ≤k sN yq = 1/(2∆s) and k gmax ≤k gN yq = 1/(2∆g). It is customary to sample the x coordinate at ∆x = .5∆g so that k xN yq = 1/(2∆x) = 2k gN yq which means that the stacking

process will generate k x wavenumbers up to this value from combinations of k s and k g according to equation 4.8. However, because spatial antialias filters (source and receiver arrays) are not very effective, it must be expected that wavenumbers higher than Nyquist will be present in both the k s

and k g spectra. The only way to generate unaliased wavenumbers up to k xN yq in the stacking process is if ∆s = ∆g so that k xN yq =k sN yq +k gN yq . This means that there must be a shotpoint for every receiver station which is very expensive acquisition. Normal 2D land shooting puts a shotpoint for every n receiver stations where n ≥ 3. This means that k x wavenumbers greater than a certain limit

will be formed from completely aliased k s and k g wavenumbers. This limiting unaliased wavenumber is

1 1 1 1 n+1

k xlim =

For n=3, k = xlim 2 3 k xN yq so that shooting every third group will result in a conventional stack with the upper third of the k x spectrum being completely useless.

Even if n = 1, there will still be aliased contributions to k x if the k s and k g spectra are aliased as they normally are. For example, k xN yq can be formed with unaliased data from k sN yq +k gN yq but it can also be formed from the sum of k s = 0 and k g = 2k gN yq and there are many more such aliased modes. Similarly, wavenumbers less that k xlim can be formed by a wide variety of aliased combinations. Further analysis is helped by having a more physical interpretation for k s and k g so that their potential spectral bandwidth can be estimated.

As explained in section 2.9, the apparent velocity of a wave as it moves across a receiver array is given by the ratio of frequency to wavenumber. For a source record (common source gather ), this has the easy interpretation that

k g sin θ 0

which means that

f sin θ 0

v 0 So, if f max is the maximum temporal frequency and (sin θ 0 ) max = 1 then

f max

k gmax =

v 0min

where v 0min is the slowest near surface velocity found along the receiver spread. So, to avoid any aliasing of k g , assuming point receivers, the receiver sampling must be such that k gN yq ≥k gmax which leads to the antialiasing condition

v 0min

Assuming the equality in expression (4.15) allows the maximal sampling curves in Figure 4.3 to

be calculated. Alternatively, if coarser sampling than suggested in these curves is planned, then

a portion of the k g spectrum will be aliased. An array can be designed to suppress the aliased

4.1. STACKED DATA

source receivers

sources

receiver

Figure 4.4: (A) Wavenumber k g is measured on a common source gather and estimates the emergence angle, θ 0 , of a monochromatic wave. (B) Wavenumber k s is measured on a common receiver gather and estimates the takeoff angle α 0 of the ray that arrives at the common receiver.

wavenumbers though this is not as effective as temporal antialias filtering. The wavenumbers k s are those which would be measured by an f -k analysis on a common

receiver gather. Reciprocity suggests that a common receiver gather is similar to a common source gather with the source and receiver positions interchanged (Figure 4.4). It cannot be expected that the amplitudes will be completely reciprocal on such gathers but the traveltimes should be. Since the relationship between apparent velocity and frequency-wavenumber ratios is a based on

traveltimes, not amplitudes, the arguments just made for the interpretation of k g also apply to k s . The interpretation is that f /k s gives the apparent velocity of a monochromatic wave as it leaves a source such that it will arrive at the common receiver. Thus

sin α 0

where α 0 is the take-off angle required to reach the common receiver and v 0 is as before. Comparing equations 4.12 and 4.16 leads to the conclusion that the potential bandwidth of k s is just as broad as that of k g .

For the k h wavenumber, the stacking process rejects all but k h = 0. Stacking can be modelled by an integral over h as

ψ 0 (x, t) =

ψ(x, h, t)dh.

all h

If equation (4.10) is substituted into equation (4.17) and the order of integration reversed, it results that

0 (x, t) =

e 2πik h h dh  φ(k x ,k h , f )e

2πi(k

x x−ft) dk x dk h df. (4.18)

all h

The integral in square brackets is δ(k h ) which, in turn, collapses the k h integral by evaluating it at

84 CHAPTER 4. ELEMENTARY MIGRATION METHODS R

depth=z

Figure 4.5: The zero-offset Fresnel zone is defined as the width of a disk on a subsurface reflector from which scattered energy arrives at R with no more than a λ/2 phase difference.

k h = 0 to give

2πi(k

x x−ft) dk x df. (4.19)

So the CMP stacking process passes only the k h = 0 wavenumber which is a very severe f -kfilter applied to a CMP gather. It is for this reason that f -kfiltering applied to CMP gathers does not typically improve the stack. However, f -k filtering of source and receiver gathers can have a

dramatic effect. The k h = 0 component is the average value over the CMP gather and corresponds to horizontal events. Of course, this is done after normal moveout removal that is designed to flatten those events deemed to be primaries.

So far, the discussion has been about wavenumber spectra with nothing being said about the effect of stacking on temporal frequencies. Bearing in mind the goal to estimate bandlimited reflec- tivity, the ideal stacked section should have all spectral color removed except that attributable to the reflectivity. Most pre-stack processing flows rely on one or more applications of statistical decon- volution to achieve this. Typically these deconvolution techniques are not capable of distinguishing signal from noise and so whiten (and phase rotate) both. If the pre-stack data input into the stacking process has been spectrally whitened, it will always suffer some attenuation of high frequencies due to the stacking out of noise. This is because the stacking process improves the signal-to-noise ratio √ by

f old. Moreover, this effect will be time-variant in a complicated way because of two competing effects. First, the fold in any stacked section is actually time-variant because of the front-end mute. Until the time at which nominal fold is reached, the stacking effect is therefore time variant. Second, the signal-to-noise ratio in any single pre-stack trace must decrease with both increasing time and increasing frequency due to attenuation (i.e. Q effects). As a result, it is virtually guaranteed that the post-stack data will need further spectral whitening. It is also highly likely that some sort of wavelet processing will be required to remove residual phase rotations.

4.1.4 The Fresnel zone

The zero-offset rays shown in Figure 4.2 are only a high frequency approximation to what really happens. The zero-offset recording at a point on the earth’s surface actually contains scattered waves from a very broad zone on each subsurface reflector. This contrasts with the high-frequency raytracing concepts that suggest the reflection occurs at a point. Figure 4.5 illustrates the concept of the zero-offset Fresnel zone. The width, w of the Fresnel zone is defined as the diameter of a

4.1. STACKED DATA

Original Fresnel zone

60Hz.

λ/2 400 Fresnel zone (meters)

100Hz.

180Hz. 140Hz.

200 after 3-D migration

after 2-D migration

depth (meters)

Figure 4.6: The width of the Fresnel zone (based Figure 4.7: A 3D migration collapses the Fres- on equation (4.21)) is shown versus depth for a

nel zone to a disk of diameter λ/2 while a 2D variety of frequencies and an assumed velocity

migration only collapses the Fresnel zone in the of 3000 m/s.

inline direction

disk whose center is marked by the ray P and edge by the ray Q. Ray P is a simple zero-offset ray while ray Q is defined to be λ/4 longer than ray P. Assuming constant velocity, this means that scattered energy that travels down-and-back along path Q will be λ/2 out of phase at the receiver, R, compared with that which travels along path P. All paths intermediate to P and Q will show a lesser phase difference. Therefore, it is expected that path Q will interfere destructively with P and is taken to define the diameter of the central disk from which scattered energy shows constructive interference at R. The equation defining the Fresnel zone diameter is therefore

+ (w/2) 2 −z=

which is easily solved for w to give

√ w=2 (λ/4 + z) 2 −z 2 = 2zλ 1 + λ/(8z) = 2zv/f 1 + v/(8fz)

(4.21) where, in the third expression, λf = v, has been used. If z ≫ λ, then the approximate form √

w∼ 2zλ is often used. The size of a Fresnel zone can be very significant in exploration as it is often larger than the target (Figure 4.6).

Migration can be conceptualized as lowering the source and receiver to the reflector and so shrinks the Fresnel zone to its theoretical minimum of λ/2. That the Fresnel zone does not shrink to zero is another way of understanding why reflectivity estimates are always bandlimited. However, seismic data is a bit more complicated than this in that it possesses signal over a bandwidth while the Fresnel zone concept is monochromatic. Roughly speaking, the effective Fresnel zone for broadband data can be taken as the average of the individual Fresnel zones for each frequency.

The 3D Fresnel disk collapses, under 3D migration, from a diameter of w to a diameter of λ/2. However, much seismic data is collected along lines so that only 2D migration is possible. In this case, The Fresnel zone only collapses in the inline direction and remains at its unmigrated size in the crossline direction (Figure 4.7). Thus the reflectivity shown on a 2D migrated seismic line must

be viewed as a spatial average in the crossline direction over the width of the Fresnel zone. This is one of the most compelling justifications for 3D imaging techniques, even in the case of sedimentary

86 CHAPTER 4. ELEMENTARY MIGRATION METHODS

basins with flat-lying structures.