Page 242 - Acquisition and Processing of Marine Seismic Data

4.9 τ-P TRANSFORM (SLANT STACK) 233

Eq. (4.34) is constant along the ray path in a new line, indicated by 2, is tangential to the
layered media. reflection hyperbola at point B, which is trans-
For a plane wave decomposition of a shot or formed to point B in the τ-p gather. The process
0
CDP gather, the amplitudes of all traces in the continues in this manner; each time after
gather must be summed up along several increasing the inclination of the line, the ampli-
slanted lines, each having a time lag of the sam- tudes on the CDP gather along the line are
pling rate (Δt). Fig. 4.19 schematically shows summed up, and the obtained sum is assigned
plane wave decomposition of a reflection hyper- to the corresponding point on the τ-p gather.
bola on a CDP gather. Let’s assume that the CDP The line with the p ¼ 1/V dip has the largest
contains only one reflection hyperbola. A single inclination, indicated by number 5 in Fig. 4.19,
reflection hyperbola on shot or CDP gathers and is asymptotical to the reflection hyperbola
becomes an elliptical hyperbola on τ-p gathers. on the CDP gather. This line corresponds to a
First, the amplitudes on the CDP gather are plane wave propagating horizontally (e.g., at a
summed along the horizontal line (p ¼ 0) indi- 90-degree angle from vertical), and the sum of
cated by 1 in Fig. 4.19. This horizontal line is tan- the amplitudes along this line is assigned to
0
gential to the reflection hyperbola around the point E on the τ-p gather. The process of sum-
0
point A, and point A is moved to a point A in mation of the amplitudes along straight lines is
the τ-p gather. Then the line is inclined, and this known as a linear τ-p transform. Amplitudes,

FIG. 4.19 Schematic representation of plane wave decomposition of a single reflection hyperbola on a CDP gather by τ-p
transform.