# Wave-equation datuming

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

Figure 8.1-2 shows a simple case of datuming. A zero-offset section is computed over three point scatterers buried beneath midpoint location A in a medium with the layered velocity structure as denoted between Figures 8.1-2a and b. The point diffractors are situated at the layer interfaces at 800, 1300, and 1900 m depths. The traveltime trajectory associated with the shallow point diffractor is a hyperbola. The traveltimes associated with the deeper diffractors are nearly hyperbolic (normal moveout). Figure 8.1-2  Layer replacement by Kirchhoff summation. (a) Input zero-offset section (at datum level z = 0) based on a velocity-depth model consisting of three point scatterers buried at depths of 800, 1300, and 1900 m. Interval velocities are indicated on the right side of the section. (b) Step 1: Downward continuation of the wavefield from z = 0 to datum level z = 800 m using a velocity of 2000 m/s. (c) Step 2: Upward continuation of (b) back to z = 0 using a velocity of 2500 m/s. (d) The zero-offset section derived independently using interval velocities that are indicated on the left side of the section. This section should be compared to (c).

Extrapolate the zero-offset wavefield at z = 0 (Figure 8.1-2a) using the velocity of the first layer (2000 m/s) and compute the wavefield at the first interface, z = 800 m. The result is shown in Figure 8.1-2b. As expected, the hyperbola associated with the shallow point scatterer largely collapses to its apex since this scatterer is located at the first interface. Because the receivers now are closer to the other two deeper scatterers, events associated with them also are compressed. Figure 8.1-2b shows the zero-offset section that would have been recorded if the receivers were placed along the first interface. The energy from the shallow point scatterer on this section now arrives at t = 0 because the datum for this section is the interface at which the scatterer is located.

Now, extrapolate the wavefield at the first interface (z = 800 m) shown in Figure 8.1-2b back up to the surface (z = 0) using the velocity of the second layer (2500 m/s). Figure 8.1-2c shows the result, which is compared to the zero-offset section in Figure 8.1-2d. The latter was derived independently from the velocity-depth model denoted between Figures 8.1-2c and d. With this two-step wave extrapolation, the first layer with the 2000-m/s velocity was replaced with the second layer with the 2500-m/s velocity.

In principle, wave-equation datuming can be performed by any extrapolation method based on phase-shift, finite-difference, or Kirchhoff summation techniques. Nevertheless, the Kirchhoff summation is more convenient in handling datum surfaces with arbitrary shapes.

It is important to distinguish between datuming and migration. Datuming produces an unmigrated time section at a specified datum z(x), which can be arbitrary in shape. Migration involves computing the wave-field at all depths from the wavefield at the surface. In this respect, datuming is an ingredient of migration, when migration is done as a downward-continuation process. In addition to downward continuation, migration, of course, requires invoking the imaging principle (t = 0) (migration principles).

Wave-equation datuming has several practical applications — horizon flattening, forward modeling of seismic wavefields, and layer replacement. These are performed in either the prestack or poststack mode. The main difference between the two implementations is that the velocity must be halved when doing poststack datuming to conform to the exploding reflectors model (introduction to migration). Horizon flattening involves downward continuing from one marker horizon to another. When this is done successively for all marker horizons in a section, the technique is useful in reconstructing the past structural history in a given survey area . Seismic modeling involves a series of upward continuations through a specified set of velocity interfaces starting at the bottom of the model and ending at the top. An example is provided in Section K.1.