Lateral resolution refers to how close two reflecting points can be situated horizontally, yet be recognized as two separate points rather than one. Consider the spherical wavefront that impinges on the horizontal planar reflector AA′ in Figure 11.1-3. This reflector can be visualized as a continuum of point diffractors. For a coincident source and receiver at the earth’s surface (location S), the energy from the subsurface point (0) arrives at t0 = 2z0/v. Now let the incident wavefront advance in depth by the amount λ/4. Energy from subsurface location A, or A′, will reach the receiver at time t1 = 2(z0 + λ/4)/v. The energy from all the points within the reflecting disk with radius OA′ will arrive sometime between t0 and t1. The total energy arriving within the time interval (t1 − t0), which equals half the dominant period (T/2), interferes constructively. The reflecting disk AA′ is called a half-wavelength Fresnel zone  or the first Fresnel zone . Two reflecting points that fall within this zone generally are considered indistinguishable as observed from the earth’s surface.
The Fresnel zone
Since the Fresnel zone depends on wavelength, it also depends on frequency. For example, if the seismic signal riding along the wavefront is at a relatively high frequency, then the Fresnel zone is relatively narrow. The smaller the Fresnel zones, the easier it is to differentiate between two reflecting points. Hence, the Fresnel-zone width is a measure of lateral resolution. Besides frequency, lateral resolution also depends on velocity and the depth of the reflecting interface — the radius of the wavefront is approximated by (Exercise 11-1)
In terms of dominant frequency f (equation 1), the Fresnel-zone width is
Figure 11.1-4 A constant-velocity zero-offset section from an earth model that consists of four reflectors, each with four nonreflecting segments A, B, C, and D. Lateral resolution is governed by the size of the Fresnel zone. The lateral extent of each gap is indicated by the solid bars on top. Note that A is hardly recognizable on any of the four horizons; B can be inferred on the shallow horizon at 0.5 s; C is difficult to infer after 2 s, while D is recognizable at all depths. All of these observations depend on noise level and how easily the diffractions can be recognized.
|t0 (s)||v (m/s)||f (Hz)||r (m)|
Table 11-2 shows the Fresnel zone radius, where r = OA′ in Figure 11.1-3 for a range of frequency and velocity combinations at various depths t0 = 2z/v. From Table 11-2, note that the shallower the event (and the higher the dominant frequency), the smaller the Fresnel zone. Since the Fresnel zone generally increases with depth, spatial resolution also deteriorates with depth.
Figure 11.1-4 shows reflections from four interfaces, each with four nonreflecting segments. The actual sizes of these segments are indicated by the solid bars on top. On the seismic section, the reflections appear to be continuous across some of these segments. This is because the size of some of the nonreflecting segments is much less than the width of the Fresnel zone; they are beyond the lateral resolution limit.
Spatial resolution is better understood in terms of diffractions. Note that in Figure 11.1-4, the diffraction energy is smeared across the nonreflecting segments on the deeper reflectors. Since migration is the process that collapses diffractions, it is reasonable to think that migration increases spatial resolution. Remember that migration can be achieved by downward continuation of receivers from the surface to the reflecting horizons. As a result of downward continuation, the observation points get closer and closer to the reflection points and, therefore, the Fresnel zone gets smaller and smaller. A smaller Fresnel zone means a higher spatial resolution (equation 2).
Migration and the Fresnel zone
Migration tends to collapse the Fresnel zone to approximately the dominant wavelength (equation 1) . Therefore, we anticipate that migration will not resolve the horizontal limits of some of the nonreflecting segments along the deeper reflectors in Figure 11.1-4. Tables 11-1 and 11-2 can be used to estimate the potential resolution improvement that may result from migration. Unless three-dimensional (3-D) migration (3-D poststack migration) is performed, the actual resolution will be less than that indicated. Two-dimensional (2-D) migration only shortens the Fresnel zone in the direction parallel to the line orientation. Resolution in the perpendicular direction is not affected by 2-D migration.
Figure 11.1-5 indicates how vertical and lateral resolution problems are inter-related. We want to determine the edge of the pinchout. The basis of the pinchout model is a wedge of material represented at a given midpoint location by a two-term reflectivity sequence, one term associated with the top and one with the bottom of the wedge. The true thickness of the wedge at various locations is indicated on top of Figure 11.1-5a. The velocity within the wedge is 2500 m/s.
Edge of the pinchout
We first consider the reflectivity sequence with two spikes of equal amplitude and identical polarity. The vertical-incidence seismic response (Figure 11.1-5a) is obtained by convolving the sequences with a zero-phase wavelet with a 20-Hz dominant frequency. (The zero-phase response simplifies event tracking from the top and bottom of the wedge.) Based on this response, the edge of the wedge can be inferred as left of location B, where the waveform reduces to a single wavelet (Figure 11.1-5a). From the resolution threshold criterion, the smallest thickness that can be resolved is (2500 m/s)/(4Hz) = 31.25 m. Figures 11.1-5a, b, and c show the same pinchout modeled using three different zero-phase wavelets with increasing dominant frequency (20, 30, and 40 Hz). Separation between the true location of pinchout A and the position of the minimum resolvable wedge thickness B decreases with increasing wavelet bandwidth.
While the resolution threshold criterion allows us to say only that the thickness of the wedge is less than 31.25 m left of location B, an amplitude-based criterion can provide a substantially more accurate location for the edge of the wedge. Again, refer to Figure 11.1-5a and observe the sudden change in amplitude at location A where the true edge of the pinchout is located. Hence, the edge still can be reliably detected, even though it may not be resolved, provided the signal-to-noise ratio is favorable. Assuming that the relative size of the top and bottom reflection coefficients is known, amplitudes also can be used to estimate the wedge thickness between locations B and A.
Figures 11.1-5a, b, and c show an apparent lateral variation in layer thickness. To see the difference between the true thickness and the apparent thickness (peak-to-peak time), refer to Figure 11.1-5d. This figure shows the data of Figure 11.1-5b with the actual geometry of the wedge superimposed on the seismic response. Since the composite wavelet has only one positive peak, the apparent thickness between locations A and B is nearly zero. At location B, the composite wavelet has a flat top. Immediately to the right of location B, the flat top disappears and the composite wavelet splits. The flat-top character can be identified as the limit of vertical resolution . A short distance to the right of the point at which the composite wavelet first splits into two peaks, the apparent thickness becomes equal to the true thickness. This thickness is called the tuning thickness and is equal to peak-to-trough separation (one half the dominant period) of the convolving wavelet . Beyond the point of tuning thickness, note the apparent thickening of the layer between locations B and C. To the right of location C, the apparent and true thicknesses become equal.
Besides apparent thickness, the maximum absolute amplitude of the composite wavelet along the pinchout also changes . To the left of location A in Figure 11.1-5b, note the single isolated zero-phase wavelet. Immediately to the right of location A, the response of the two closely spaced spikes with identical polarity results in the maximum absolute amplitude. This amplitude gradually decreases to a minimum exactly at the tuning thickness. It then increases and reaches the amplitude value of the original single wavelet to the right of location C.
Maximum amplitude and apparent thickness change in reverse when the reflectivity model consists of reflection coefficients with equal amplitude and opposite polarity (Figure 11.1-5e). The composite waveform resulting from this reflectivity model is discussed by Widess . Two spikes of opposite polarity with a small separation between them act as a derivative operator. When applied to a zero-phase wavelet, this operator causes a 90-degree phase shift. This phase shift can be seen in Figure 11.1-5e on the wavelets between locations A and B. Widess  observed that the composite wavelet within this zone basically retains its shape while its amplitude changes.
Figure 11.1-5f shows data of Figure 11.1-5e with the actual geometry of the wedge superimposed on the seismic response. Note that the wedge appears thicker than it actually is between locations A and B. Also note the apparent thinning of the layer between locations B and C. Beyond location C, the apparent and true thicknesses become equal. Immediately to the right of location A in Figure 11.1-5e, the response of the two closely spaced spikes with opposite polarity results in the cancellation of the amplitudes. The largest absolute amplitude of the composite wavelet gradually increases to a maximum immediately to the right of location B. It gradually decreases and reaches the amplitude value of the original single wavelet to the right of location C.
From the above discussion, we see that peak-to-peak time measurements and amplitude information can aid in detecting pinchouts that may otherwise be unresolvable. If the size of the reflection coefficients were known, then the amplitudes could be used to map the thickness below the resolution limit.
Nevertheless, the reliability of the analysis depends to some extent on the signal-to-noise ratio. Finally, the deceptive changes in amplitude and apparent thickness must be noted during the mapping of the top and bottom of the pinchout.
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