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4.4.1
The reflection method
Although earthquake seismology and refraction seismology enable scientists to determine gross Earth structures and crustal and upper-mantle structures, reflec- tion seismology is the method used to determine fine details of the shallow structures, usually over small areas. The resolution obtainable with reflection seismology makes it the main method used by oil-exploration companies to map subsurface sedimentary structures. The method has also increasingly been used to obtain new information on the fine structures within the crust and at the crust– mantle boundary.
For land profiles, explosives can be used as a source. Other sources include the gas exploder, in which a gas mixture is exploded in a chamber that has a movable bottom plate resting on the ground, and the vibrator, in which a steel plate pressed against the ground is vibrated at increasing frequency (in the range 5–60 Hz) for several seconds (up to 30 s for deep crustal reflection profiling). Vibrators require an additional step in the data processing to extract the reflections from the recordings: the cross-correlation of the recordings with the source signal.
Of the many marine sources, the two most frequently used for deep reflection profiling are the air gun, in which a bubble of very-high-pressure air is released into the water, and the explosive cord. Many air guns are usually used in an array towed behind the shooting ship.
Deconvolution is the process which removes the effects of the source and receiver from the recorded seismograms and allows direct comparison of data recorded with different sources and/or receivers. For the details of the methods of obtaining and correcting seismic-reflection profiles, the reader is again referred to the textbooks on exploration geophysics (e.g., Telford et al. 1990; Dobrin and Savit 1988; Yilmaz 2001; Claerbout 1985).
The basic assumption of seismic reflection is that there is a stack of horizon- tal layers in the crust and mantle, each with a distinct seismic P-wave velocity. Dipping layers, faults and so forth can be included in the method (see Section 4.4.4). P-waves from a surface energy source, which are almost normally inci- dent on the interfaces between these layers, are reflected and can be recorded by geophones (vertical-component seismometers) close to the source. Because the rays are close to normal incidence, effectively no S-waves are generated (Fig. 4.39). The P-waves reflected at almost normal incidence are very much smaller in amplitude than the wide-angle reflections near to, and beyond, the criti- cal distance. This fact means that normal-incidence reflections are less easy to rec- ognize than wide-angle reflections and more likely to be obscured by background
158 Seismology
noise, and that sophisticated averaging and enhancement techniques must be used to detect reflecting horizons.
4.4.2
A two-layer model
Consider the two-layer model in Fig. 4.34. By application of Pythagoras’ theorem, the travel time t for the reflection path SCR is given by Eqs. (4.29) and (4.30) as
t = SC α1 +CRα 1 = 2 α1 z2 1+ x2 4 or t2= 4z 2 1 α2 1 + x2 α2 1 (4.65)
which is the equation of a hyperbola. At normal incidence (x= 0), the travel time is t= t0, where
t0=
2z1
α1
(4.66)
This is the two-way normal-incidence time. At large distances (x z1) the travel time (Eq. (4.65)) can be approximated by
t≈ x α1
(4.67)
This means that, at large distances, the travel-time curve is asymptotic to the travel time for the direct wave, as illustrated in Fig. 4.32(b). In reflection profiling, since we are dealing with distances much shorter than the critical distance, the travel-time–distance plot is still curved. Notice that, with increasing values of α1, the hyperbola (Eq. (4.65)) becomes flatter. If travel-time–distance data were obtained from a reflection profile shot over such a model, one way to determine α1 and z1would be to plot not t against x, but t2against x2. Equation (4.65) is then the equation of a straight line with slope 1/α21and an intercept on the t
2 axis of t2
0 = 4z21/α21.
The normal-incidence reflection coefficient for P-waves is given by Eq. (4.62). Since normal-incidence reflections have small amplitudes, it is advantageous to average the signals from nearby receivers to enhance the reflections and reduce the background noise. This averaging process is called stacking. Common-depth- point (CDP) stacking, which combines all the recordings of reflections from each subsurface point, is the method usually used. Common-offset stacking, which combines all the recordings with a common offset distance, is less popular. Figure 4.40 shows the layout of shots (or vibrators) and receivers used in CDP reflection profiling. The coverage obtained by any profile is
coverage= number of receivers
P P P P Figure 4.40. Common-depth-point (CDP) reflection profiling. In this example, eight geophones () record each shot (
). In (a), shot A is fired and a reflection from a particular point P on the reflector (the interface between the two layers) is recorded by geophone 8. In (b), all the geophones and the shotpoint have been moved one step to the right, and shot B is fired; the reflection from point P is recorded by geophone 6. Similarly, a reflection from P is recorded (c) by geophone 4 when shot C is fired and (d) by geophone 2 when shot D is fired. The four reflections from point P can be stacked (added together after time corrections have been made). In this example, because there are four reflections from each reflecting point on the interface (fewer at the two ends of the profile), there is said to be four-fold coverage. Alternatively, the reflection profile can be described as a four-fold CDP profile.
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where the shot spacing is in units of receiver spacing. In the example of Fig. 4.40, the number of receivers is eight and the shot spacing is one. This results in a four-fold coverage. Reflection-profiling systems usually have 48 or 96 recording channels (and hence receivers), which means that 24-, 48-, or 96-fold coverage is possible. The greater the multiplicity of coverage, the better the system is for imaging weak and deep reflectors and the better the final quality of the record section. In practice, receiver spacing of a few tens to hundreds of metres is used, in contrast to the kilometre spacing of refraction surveys.
In order to be able to add all these recordings together to produce a signal reflected from the common depth point, one must first correct them for their different travel times, which are due to their different offset distances. This cor- rection to the travel times is called the normal-moveout (NMO) correction.
The travel time for the reflected ray in the simple two-layer model of Fig. 4.34 is given by Eq. (4.65). The difference between values of the travel time t at two distances is called the moveout, t. The moveout can be written
t = α2 1 z2 1+ x2 a 4 − 2 α1 z2 1+ x2 b 4 (4.69)
where xaand xb(xa> xb) are the distances of the two geophones a and b from the shotpoint. The normal moveout tNMO is the moveout for the special case when geophone b is at the shotpoint (i.e., xb= 0). In this case, and dropping the subscript a, Eq. (4.69) becomes
tNMO= 2 α1 z2 1+ x2 4 − 2z1 α1 (4.70)
If we make the assumption that 2z1 x, which is generally appropriate for reflection profiling, we can use a binomial expansion for
z2 1+ x2/4: z2 1+ x2 4 = z1 1+ x 2 4z2 1 = z1 1+ x 2z1 21/2 = z1 1+1 2 x 2z1 2 − 1 8 x 2z1 4 + 1 16 x 2z1 6 + · · · (4.71)
To a first approximation, therefore,
z2 1+ x2 4 = z1 1+1 2 x 2z1 2 (4.72)
Substituting this value into Eq. (4.70) gives a first approximation for the normal moveout tNMO. tNMO = 2z1 α1 1+1 2 x 2z1 2 − 2z1 α1 = x2 4α1z1 (4.73)
Figure 4.41. (a) Reflections from an interface, recorded at distances xa, xb, xc, xd, xe and xf. Three travel-time curves (1, 2 and 3) are shown for two-way
normal-incidence time t0and increasing values of velocity. Clearly, curve 2 is the best fit to the reflections. To stack these traces, the NMO correction (Eq. (4.74)) for curve 2 is subtracted from each trace so that the reflections line up with a constant arrival time of t0. Then the traces can be added to yield a final trace with increased signal-to-noise ratio. (b) The power in the stacked signal is calculated for each value of the stacking velocity and displayed on a time–velocity plot. The velocity which gives the peak value for the power in the stacked signal is then the best stacking velocity for that particular value of t0. For (a), velocity 2 is best; velocity 1 is too low and velocity 3 too high. (After Taner and Koehler (1969).)
Using Eq. (4.66) for t0, the two-way normal-incidence time, we obtain an alter- native expression for tNMO:
tNMO=
x2
2α2 1t0
(4.74)
This illustrates again the fact that the reflection time–distance curve is flatter ( tNMO is smaller) for large velocities and large normal-incidence times. This NMO correction must be subtracted from the travel times for the common-depth- point recordings. The effect of this correction is to line up all the reflections from each point P with the same arrival time t0 so that they can be stacked (added together) to produce one trace. This procedure works well when we are using a model for whichα1and z1are known, but in practice we do not know them: they are precisely the unknowns which we would like to determine from the reflections! This difficulty is overcome by the bootstrap technique illustrated in Fig. 4.41. A set of arrivals is identified as reflections from point P if their travel times fall on a hyperbola. Successive values ofα1and t0are tried until a combination defining a hyperbola that gives a good fit to the travel times is found. These valuesα1and
162 Seismology
t0then define the model. In the example of Fig. 4.41, it is obvious that curve 2 is correct, but with real data it is not sufficient to rely on the eye alone to determine velocities – a numerical criterion must be used. In practice, the reflections are stacked using a range of values forα1and t0. The power (or some similar entity) in the stacked signal is calculated for each value ofα1and t0. For each value of
t0the maximum value of the power is then used to determine the best velocity. A plot of power against both velocity and time, as in Fig. 4.41(b), is usually called a velocity-spectrum display.
Once the records have been stacked, the common-depth-point record section shows the travel times as if shots and receivers were coincident. The stacking process necessarily involves some averaging over fairly short horizontal distances, but it has the considerable advantage that the signal-to-noise ratio of the stacked traces is increased by a factor of √n over the signal-to-noise ratio of the n individual traces.
4.4.3
A multilayered model
A two-layer model is obviously not a realistic approximation to a pile of sediments or to the Earth’s crust, but it serves to illustrate the principle of the reflection method. A multiple stack of layers is a much better model than a single layer. Travel times through a stack of multiple layers are calculated in the same way as for two layers, with the additional constraint that Snell’s law (sin i/α = p, a constant for each ray) must be applied at each interface (Eq. (4.55)). The travel times and distances for a model with n layers, each with thickness zjand velocity αj, are best expressed in the parametric form
x= 2 n j=1 zjpαj 1− ρ2α2 j (4.75) t= 2 n j=1 zj αj 1− ρ2α2 j
where p= sin ij/αj. It is unfortunately not usually possible to eliminate p from these equations in order to express the distance curve as one equation. In this multilayered case, the time–distance curve is not a hyperbola as it is when n= 1 (Eq. (4.65)). However, it can be shown that the square of the travel time, t2, can be expressed as an infinite series in x2:
t2= c
0+ c1x2+ c2x4+ c3x6+ · · · (4.76) where the coefficients c0, c1, c2, . . . are constants dependent on the layer thick- nesses zjand velocitiesαj. In practice, it has been shown that use of just the first two terms of Eq. (4.76) (c0 and c1) gives travel times to an accuracy of within about 2%, which is good enough for most seismic-reflection work. This means that Eq. (4.76) can be simplified to
which is, after all, like Eq. (4.65), a hyperbola. The value of the constant c0 is given by c0= n m=1 tm 2 (4.78)
where tm= 2zm/αmis the two-way vertical travel time for a ray in the mth layer. The two-way normal-incidence travel time from the nth interface, t0,n, is the sum of all the tm: t0,n= n m=1 tm = n m=1 2zm αm (4.79)
Equation (4.78) can therefore be more simply written as
c0= (t0,n)2 (4.80)
The second constant of Eq. (4.77), c1, is given by
c1= n m=1 tm n m=1 α2 mtm (4.81)
We can define a time-weighted root-mean-square (RMS) velocityα2nas
α2 n= n m=1α 2 mtm n m=1 tm (4.82)
With these expressions for c0and c1, Eq. (4.77) becomes
t2= (t0,n)2+
x2
α2
n
(4.83)
This now has exactly the same form as the equation for the two-layer case (Eq. (4.65)), but instead of t0we have t0,n, and instead of the constant velocity above the reflecting interfaceα1, we now haveαn, the time-weighted RMS velocity above the nth interface. This means that NMO corrections can be calculated and traces stacked as described in Section 4.4.2. The only difference is that, in this multilayered case, the velocities determined are not the velocity above the interface but the RMS velocity above the interface.
Figure 4.42 shows a typical velocity-spectrum display for real data. By stacking reflection records and using such a velocity-spectrum display, one can estimate both t0,nandαnfor each reflector. However, determining t0,nandαnis not enough; to relate these values to the rock structure over which the reflection line was shot, we have to be able to calculate the thicknesses and seismic velocities for each layer and to obtain an estimate of the accuracy of such values.
Let us suppose that the RMS velocity and normal-incidence times have been determined for each of two successive parallel interfaces (i.e., t0,n−1, t0,n,αn−1
164
RMS VELOCITY (km s−1)
Figure 4.42. Steps in the computation of a velocity analysis. (a) The twenty-four individual reflection records used in the velocity analysis. (b) The maximum amplitude of the stacked trace shown as a function of t0, two-way time along the trace. Notice that the reflections are enhanced compared with the original traces. Although the main reflections at t0= 1.2 and 1.8 s do stand out on the original traces, these intercept times are clearly defined by the stacked trace, and
subsequent deeper reflections that were not clear on the original traces can now be identified with some confidence on the stacked trace. (c) The velocity spectrum for the traces in (a). The peak power at each time served to identify the velocity which would best stack the data. The stacking velocity clearly increases steadily with depth down to about 3 s. After this, some strong multiples (rays that have bounced twice or more in the upper layers and therefore need a smaller stacking velocity) confuse the velocity display. (From Taner and Koehler (1969).)
andαnare known). Then, by using Eqs. (4.79) and (4.82), we can determine the velocity of the nth layerαn:
α2 n n m=1 tm = n m=1 α2 mtm = n−1 m=1 α2 mtm+ α2ntn (4.84) and α2 n−1 n−1 m=1 tm = n−1 m=1 α2 mtm (4.85)
Subtracting Eq. (4.85) from Eq. (4.84) gives
α2 n n m=1 tm− α2n−1 n−1 m=1 tm= αn2tm (4.86) or α2 nt0,n− α2n−1t0,n−1= αn2(t0,n− t0,n−1) (4.87)
Rearranging this equation givesαn, the velocity of the nth layer (also known as the interval velocity), in terms of the RMS velocities:
αn =
α2
nt0,n− α2n−1t0,n−1
t0,n− t0,n−1 (4.88)
Afterαnhas been determined, zncan be calculated from Eq. (4.79):
t0,n = n m=1 2zm αm = n−1 m=1 2zm αm +2zn αn = t0,n−1+ 2zn αn (4.89) Thus, zn = αn 2(t0,n− t0,n−1) (4.90)
Therefore, given the two-way normal-intercept times and corresponding stack- ing (RMS) velocities from a velocity analysis, the velocity–depth model can be determined layer by layer, starting at the top and working downwards.
Multiples are rays that have been reflected more than once at an interface. The most common multiple is the surface multiple, which corresponds to a ray that travels down and up through the layers twice. Reflections with multiple ray paths in one or more layers also occur. In marine work, the multiple which is reflected at the sea surface and seabed is very strong (Figs. 9.23 and 10.10). The periodicity of multiple reflections enables us to filter them out of the recorded data by deconvolution.
166 Seismology
Example: calculation of layer thickness and seismic velocity from a normal-incidence reflection line
A velocity analysis of reflection data has
t0,1= 1.0 s, α1= 3.6 km s−1
t0,2= 1.5 s, α2= 4.0 km s−1
Calculate a velocity–depth model from these values, assuming that there is a constant velocity in each layer. Sinceα1must be equal toα1, the velocity of the top
layer, z1, can be calculated from Eq. (4.90):
z1=
3.6 × 1.0
2.0 = 1.8 km Nowα2can be calculated from Eq. 4.88:
α2=
(4.02× 1.5) − (3.62× 1.0)
1.5 − 1.0 =√22.08 = 4.7 km s−1 Finally, z2is then calculated from Eq. (4.90):
z2=
4.7 × 0.5
2.0 = 1.175 km
Notice that the velocityα2is larger than the RMS velocityα2in this example.
Unfortunately, the interval velocities and depths frequently cannot be determined accurately by these methods (see Problem 21). In exploration work the inaccuracy of velocity–depth information can usually be made up for by detailed measurements from drill holes. However, when reflection profiling is used to investigate structures deep in the crust and upper mantle, such direct velocity information is not available. There is an added problem in this situation: when the depths of horizons are much larger than the maximum offset, the NMO correction is insensitive to velocity. This means that any velocity that is approximately correct will stack the reflections adequately, and so the interval velocities calculated from the stacking velocities will not be very accurate. As an example, consider a two-layer model:
α1= 6 km s−1, z1= 20 km, α2= 8 km s−1
The reflection hyperbola (Eq. (4.65)) for this interface is t= 1 3 400+x 2 4
and the two-way normal incidence time t0is 6.667 s. The NMO correction is
(Eq. (4.73))
tNMO=
x2
480
The maximum offset used in deep reflection profiling is rarely greater than about 5 km and often less. At this offset, the correct NMO correction for the reflection is 0.052 s. However, this correction has to be estimated from the data. Using the
correct value for t0, tNMO= 0.062 s when α1= 5.5 km s−1and tNMO= 0.044 s
whenα1= 6.5 km s−1. Clearly, a signal with predominant frequency of 20 Hz or
less (i.e., time for one cycle>0.050 s) cannot give an accurate value for α1.
Higher-frequency signals give more accurate velocity analyses, as discussed in Section 4.4.4. The best way to obtain reliable interval-velocity measurements in such cases is to supplement the reflection profiles with wide-angle-reflection profiles. Such profiles, perhaps 50–80 km in length, allow deep wide-angle reflections to be recorded. Wide-angle reflections have larger amplitudes than those of normal-incidence reflections (Fig. 4.39), and their travel times (Eqs. (4.65) and (4.67)) give more reliable values for the interval velocities.
Example: amplitude of reflections
For normal-incidence reflections the P-wave reflection coefficient is given by Eq. (4.62). An increase in impedance will therefore give a positive reflection coefficient and a decrease in impedance will give a negative reflection coefficient. As a simple example, imagine a layer of sediment with an impedance of 4× 106kg
m−2s−1sandwiched in sediment with an impedance of 3× 106kg m−2s−1. The
reflection coefficient at the top of the layer is 4− 3 4+ 3 =
1 7= 0.143 while the reflection coefficient at the base of the layer is
3− 4 3+ 4 =
−1
7 = −0.143
To see a real example of the use of reflection coefficients, consider the bottom-simulating reflector (BSR) which is a common feature of marine seismic-reflection lines. The BSR arrives about 200–300 ms after the seafloor reflection, has the opposite polarity, follows the seabed reflection (hence its name) and so frequently cuts across reflections from any sedimentary stratigraphy. The BSR is a consequence of the presence of gas hydrates within the sediments. A gas hydrate is a rigid water-molecule cage that encloses and is stabilized by methane or other hydrocarbons. Hydrates are stable at temperatures over 0◦C at the elevated pressures reached in water over 300 m deep – effectively an ‘ice’ that is stable above 0◦C. These hydrates have a narrow stability field, the base (controlled by
temperature) being only a few hundred metres below the seafloor. The base of the stability field marks the boundary between high-velocity hydrated sediments above and normal or gas-filled sediments below. Since there is a change in physical properties of the sediment at this boundary, it will yield a seismic reflection. Figure 4.43 shows a seismic-reflection line across the Cascadia margin where the Juan de Fuca plate is subducting beneath North America. The BSR is clearly visible, as is the deformation of the sediments filling this oceanic trench (see also Section 9.2.2). Figure 4.44 shows the detail of the seafloor and BSR reflections – the