Prestack time migration

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

As stated early in the chapter, the rigorous solution to the problem of conflicting dips with different stacking velocities is prestack time migration. The robust alternative in practice is to apply NMO and DMO corrections followed by poststack time migration. The migration step can, however, be moved up to precede CMP stacking. Specifically, by migrating each of the NMO- and DMO-corrected common-offset sections, one has the opportunity to update the velocity field and generate CMP gathers which can be used for analysis of amplitude variation with offset as well as to obtain an improved migrated stack.

Figure E-2  The prestack time migration ellipse. See Section E.5 for details.

We now review the kinematics of prestack time migration. The theory of extrapolation of nonzero-offset wavefields is provided in wavefield extrapolation and migration, while the stationary-phase traveltime trajectory associated with nonzero-offset wave propagation is given in nonzero-offset traveltime equation. Figure E-2 shows a sketch of a nonzero-offset raypath in a constant-velocity medium from a source location S to a reflection point R and to a receiver location G. The traveltime equation associated with the raypath SRG is given by

 ${\displaystyle vt={\sqrt {z^{2}+(y+h)^{2}}}+{\sqrt {z^{2}+(y-h)^{2}}},}$ (32)

where v is the medium velocity, t is the total traveltime from S to R to G. The medium is represented by the midpoint y and depth z coordinates.

Equation (32) can be put into the following alternative form:

 ${\displaystyle {\frac {y^{2}}{(vt/2)^{2}}}+{\frac {z^{2}}{(vt/2)^{2}-h^{2}}}=1,}$ (33)

which represents an ellipse in the y − z plane for a constant t with the following parameters:

1. Semi-major axis in midpoint y direction: a = vt/2.
2. Semi-minor axis in depth z direction: ${\displaystyle b={\sqrt {(vt\!/\!2)^{2}-h^{2}}}.}$
3. Distance from center to either focus: ${\displaystyle {\sqrt {a^{2}-b^{2}}}=h.}$
4. Distance from one focus to a point on the ellipse to the other focus: vt.

From its properties, note that this ellipse in the y − z plane can be associated with a recording geometry for a source-receiver pair which is situated at the foci with a separation equal to the offset 2h (Figure E-2). The ellipse itself describes the geometry of a reflecting interface such that the reflections recorded by a source-receiver pair situated at the foci have the same arrival time t. This means that the nonzero-offset traveltime section from such a recording would contain traces with zero amplitudes, except for one sample at time t on the trace at a midpoint location that coincides with the center of the ellipse. Hence, the ellipse of equation (33) in the y − z plane describes the impulse response of a prestack migration operator applied to nonzero-offset data.

Figure D-5  The response characteristics of the DSR operator (equation D-23) [1][1] (a) Real part of the y − z plane at 16 Hz and h = 400 m. Note the semielliptical wavefronts. (b) Real part of the y − z plane at t = 1024 ms, h = 400 m. Because of the wraparound in h, we observe two wavefronts, one for h = 400 m and one for h = 0. (c) Real part of the y − t plane at z = 200, 400, 600, and 800 m superimposed. These are the table-top trajectories for h = 400 m. The loci of the arrival times are determined by a stationary-phase approximation to DSR (see Section D.2) [2]. Periodicity in y and t result from approximating Fourier integrals by sums.

Equation (32) describes the nonzero-offset traveltime trajectory in the y − t plane for a constant z associated with a point scatterer. Figure D-5 (Section D.1) shows the elliptic wavefront in the y − z plane equivalent to the elliptic reflector geometry described here, and the table-top traveltime trajectory in the y − t plane.

When equation (33) is specialized to the zero-offset case, h = 0, we obtain

 ${\displaystyle {\frac {y^{2}}{(vt/2)^{2}}}+{\frac {z^{2}}{(vt/2)^{2}}}=1,}$ (34a)

which describes a circle in the y − z plane for a constant t with a radius vt/2. This circle represents the impulse response of a poststack migration operator applied to zero-offset data.

When equation (32) is specialized to the zero-offset case, h = 0, we obtain

 ${\displaystyle vt=2{\sqrt {y^{2}+z^{2}}},}$ (34b)
Figure D-6  The response characteristics of the exploding reflectors operator ER(Y) (equation D-29) [1]. (a) Real part of the y − z plane at 16 Hz. Note the circular wavefronts. (b) Real part of the y − z plane at t = 1024 ms. (c) Real part of the y − t plane at z = 200, 400, 600, and 800 m superimposed. These are the hyperbolic trajectories. The loci of the arrival times are determined via the stationary-phase approximation to ER(Y) (see section D.2) [2]. Periodicity in y and t result from approximating Fourier integrals by sums.

which describes the well-known diffraction hyperbola in the y − t plane for a constant z. Figure D-6 (mathematical foundation of migration) shows the circular wavefront in the y − z plane equivalent to the circular reflector geometry described here, and the zero-offset hyperbolic traveltime trajectory in the y − t plane.

In migration principles, we discussed semicircle superposition and diffraction summation concepts for zero-offset migration using equations (34a) and (34b), respectively. Specifically, zero-offset migration can be conceptualized as spreading of amplitudes on each input stacked trace on the y − t plane along semicircular trajectories on the y − z plane of the migrated section. Alternatively, for a given output sample of a trace on the z − t plane of the migrated section, amplitudes along the hyperbolic trajectory on the y − t plane of the input stacked section can be summed and placed on that output sample location. The Kirchhoff summation technique for migration incorporates to the process of diffraction summation the amplitude and phase factors described in migration principles.

Similarly, prestack time migration can be conceptualized either by way of semi-elliptical superposition using equation (33) or diffraction summation over the traveltime surface described by equation (32). The traveltime surface, which is shown in Figure 5.3-1a, is known as Cheops’ pyramid [3]. The result of summation of amplitudes over the pyramidal surface is placed at its apex. The question that is of practical importance is how to define the summation paths over this surface.

To consider alternative methods of summation, refer to the traveltime equation (32) that describes the pyramidal surface in Figure 5.3-1a, and, first, make the change of variables from depth z to event time τ in the migrated position by using the relation z = vt/2, then rewrite this equation in terms of the event position after migration, which is the lateral coordinate ym of the apex of the pyramid

 ${\displaystyle t={\sqrt {{\frac {\tau ^{2}}{4}}+{\frac {(y-y_{m}+h)^{2}}{v^{2}}}}}+{\sqrt {{\frac {\tau ^{2}}{4}}+{\frac {(y-y_{m}-h)^{2}}{v^{2}}}}}.}$ (35)

Hence, within the context of equation (35), the summation involves mapping amplitude at a point on the pyramidal surface with coordinates (y, h, t) to the apex with coordinates (ym, h = 0, τ).

Whatever the summation strategy, it may not be desirable to map the amplitudes on the pyramidal surface directly onto the apex of the pyramid. Instead, it is desirable first to collapse the pyramidal surface described by equation (35) to a traveltime curve that passes through the apex of the pyramid at y = ym described by

 ${\displaystyle t={\sqrt {\tau ^{2}+{\frac {4h^{2}}{v^{2}}}}}.}$ (36)

As a result, amplitude at a point on the pyramidal surface with coordinates (y, h, t) is mapped onto a point with coordinates ${\displaystyle \left(y_{m},\ h,\ {\sqrt {\tau ^{2}+4h^{2}\!/\!v^{2}}}\right)}$ on the hyperbolic curve of equation (36). Now, you have the opportunity to perform velocity analysis using equation (36) and refine the velocity field used in the first summation step. The second step in the summation involves applying NMO correction using equation (36), stacking the amplitudes along the offset axis, and placing the result at the apex of the hyperbola of equation (36) at time τ and offset h = 0. This apex coincides with the apex of the pyramidal surface with coordinates (ym, h = 0, τ).

The two most obvious choices of summation paths to collapse the pyramidal surface of equation (35) to the hyperbolic curve of equation (36) are described below:

1. Summation curves of constant offset: Consider a set of vertical cross-sections of the traveltime pyramid illustrated in Figure 5.3-1a parallel to the midpoint axis as illustrated in Figure 5.3-1b. Sum the amplitudes along each of the constant-offset tabletop traveltime curves, independently, and place the result for each at the apex Ah of the summation curve with coordinates ${\displaystyle \left(y_{m},\ h,\ {\sqrt {\tau ^{2}+4h^{2}\!/\!v^{2}}}\right)}$. The summation collapses the pyramidal surface onto the hyperbolic traveltime curve of equation (36), which is formed by combining the apex points Ah of the constant-offset curves. This hyperbolic traveltime curve is orthogonal to the constant-offset summation curves.
2. Summation curves of constant time: Consider a set of horizontal cross-sections of the traveltime pyramid as illustrated in Figure 5.3-1c [4][5]. Sum the amplitudes along each of the constant-time curves, independently, and place the result for each at Ah on the summation curve with coordinates ${\displaystyle \left(y_{m},\ h,\ {\sqrt {\tau ^{2}+4h^{2}\!/\!v^{2}}}\right)}$. The summation collapses the pyramidal surface onto the hyperbolic traveltime curve of equation (36), which is formed by combining the points Ah of the constant-time curves. This hyperbolic traveltime curve is orthogonal to the constant-time summation curves.
Figure 5.3-1  Left column adapted from Fowler [6]: The nonzero-offset traveltime surface associated with a point scatterer and the various summation trajectories for prestack time migration. Right column: The nonzero-offset traveltime surfaces as in the left column after DMO correction. See text for details.

The pyramidal surface described by equation (35) is converted by DMO correction to a hyperboloid of revolution as illustrated in Figure 5.3-1d and described by the following equation [7]:

 ${\displaystyle t={\sqrt {\tau ^{2}+{\frac {4(y-y_{m})^{2}}{v^{2}}}+{\frac {4h^{2}}{v^{2}}}}}.}$ (37)

As for equation (35), within the context of equation (37), the summation required by prestack time migration involves mapping amplitude at a point on the hyperboloidal surface with coordinates (y, h, t) to the apex with coordinates (ym, h = 0, τ).

Again, it is desirable first to collapse the hyperboloidal surface described by equation (37) to a traveltime curve that passes through the apex of the hyperboloid at y = ym described by equation (36). As a result, amplitude at a point on the hyperboloidal surface with coordinates (y, h, t) is mapped onto a point with coordinates ${\displaystyle \left(y_{m},\ h,\ {\sqrt {\tau ^{2}+4h^{2}\!/\!v^{2}}}\right)}$ on the hyperbolic curve of equation (36). As for the pyramidal surface, the second step in the summation involves applying NMO correction, stacking the amplitudes along the offset axis, and placing the result at the apex of the hyperbola of equation (36) at time τ and offset h = 0. This apex coincides with the apex of the hyperboloidal surface with coordinates (ym, h = 0, τ).

For constant offset h, rewrite equation (37) in the following form:

 ${\displaystyle {\frac {t^{2}}{\tau ^{2}+4h^{2}\!/\!v^{2}}}-{\frac {(y-y_{m})^{2}}{h^{2}+v^{2}\tau ^{2}\!/\!4}}=1.}$ (38)

Note that, as a result of the transformation from the pyramidal surface to the hyperboloidal surface, the table-top summation curves at constant offset are transformed to hyperbolic curves described by equation (38).

For constant time t, rewrite equation (37) in the following form:

 ${\displaystyle (y-y_{m})^{2}+h^{2}={\frac {v^{2}}{4}}(t^{2}-\tau ^{2}).}$ (39)

Note that the constant-time curves of the pyramidal surface are transformed to circles described by equation (39).

So, for the constant-offset and constant-time summation techniques described above, we may consider substituting the pyramidal surface with the hyperboloidal surface in the following manner:

1. Summation curves of constant offset: Consider a set of vertical cross-sections of the traveltime hyperboloid parallel to the midpoint axis. Sum the amplitudes along each of the constant-offset hyperbolic traveltime curves of equation (38) as illustrated in Figure 5.3-1e, independently, and place the result for each at the apex Ah of the summation curve with coordinates ${\displaystyle \left(y_{m},\ h,\ {\sqrt {\tau ^{2}+4h^{2}\!/\!v^{2}}}\right)}$. The summation collapses the hyperboloidal surface onto the hyperbolic traveltime curve of equation (36), which is formed by combining the apex points Ah of the constant-offset curves. This hyperbolic traveltime curve is orthogonal to the constant-offset summation curves.
2. Summation curves of constant time: Consider a set of horizontal cross-sections of the traveltime hyperboloid [7]. Sum the amplitudes along each of the constant-time circles of equation (39) as illustrated in Figure 5.3-1f, independently, and place the result for each at Ah on the summation curve with coordinates ${\displaystyle \left(y_{m},\ h,\ (v\!/\!2){\sqrt {t^{2}-\tau ^{2}}}\right)}$. The summation collapses the hyperboloidal surface onto the hyperbolic traveltime curve of equation (36), which is formed by combining the points Ah of the constant-time circles. This hyperbolic traveltime curve is orthogonal to the constant-time summation curves.

References

1. Yilmaz, 1979, Yilmaz, O., 1979, Prestack partial migration: Ph.D. thesis, Stanford University.
2. Clayton, 1978, Clayton, R., 1978, Common midpoint migration: Stanford Expl. Proj., Rep. No. 14, Stanford University.
3. Claerbout (1985), Claerbout, J.F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications.
4. Bancroft and Geiger (1994), Bancroft, J. C. and Geiger, H. D., 1994, Equivalent-offset CRP gathers: 64th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 672–675.
5. Bancroft et al., 1997, Bancroft, J. C., Margrave, G., and Geiger, H. D., 1997, A kinematic comparison of DMO-PSI and equivalent offset migration (EOM): 67th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1575–1578.
6. Fowler, 1997, Fowler, P., 1997, A comparative overview of prestack time migration methods: 67th Ann. Int. Soc. Explor. Geophys. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1571–1574.
7. Gardner et al. (1986), Gardner, G. F. H., Wang, S. Y., Pan, N. D., and Zhang, Z., 1986, Dip moveout and prestack imaging: Expanded Abstracts, 75–84, 18th Ann. Offshore Tech. Conf.