Theory of 3-D angle gathers in wave-equation seismic imaging

Next: Common-azimuth approximation Up: Fomel: 3-D angle gathers Previous: Introduction

# Traveltime derivatives and dispersion relationships for a 3-D dipping reflector

Theoretical analysis of angle gathers in downward continuation methods can be reduced to analyzing the geometry of reflection in the simple case of a dipping reflector in a locally homogeneous medium. Considering the reflection geometry in the case of a plane reflector is sufficient for deriving relationships for local reflection traveltime derivatives in the vicinity of a reflection point (Goldin, 2002). Let the local reflection plane be described in coordinates by the general equation

 (1)

where the normal angles , , and satisfy

 (2)

The geometry of the reflection ray paths is depicted in Figure 1. The reflection traveltime measured on a horizontal surface above the reflector is given by the known expression (Slotnick, 1959; Levin, 1971)

 (3)

where is the length of the normal to the reflector from the midpoint (distance in Figure 2)

 (4)

and are the midpoint coordinates, and are the half-offset coordinates, and is the local propagation velocity.

plane3b
Figure 1.
Reflection geometry in 3-D (a scheme). and and the source and the receiver positions at the surface. is the reflection point. is the normal projection of the source to the reflector. is the mirror'' source. The cumulative length of the incident and reflected rays is equal to the distance from to .

According to elementary geometrical considerations (Figures 1 and 2), the reflection angle is related to the previously introduced quantities by the equation

 (5)

plane2b
Figure 2.
Reflection geometry in the reflection plane (a scheme). is the midpoint. As follows from the similarity of triangles and , the distance from to is twice smaller than the distance from to .

Explicitly differentiating equation (3) with respect to the midpoint and offset coordinates and utilizing equation (5) leads to the equations

 (6) (7) (8) (9)

Additionally, the traveltime derivative with respect to the depth of the observation surface is given by

 (10)

and is related to the previously defined derivatives by the double-square-root equation
 (11)

In the frequency-wavenumber domain, equation (11) serves as the basis for 3-D shot-geophone downward-continuation imaging. In the Fourier domain, each derivative translates into ratio, where is the wavenumber corresponding to and is the temporal frequency.

Equations (6), (7), and (10) immediately produce the first important 3-D relationship for angle gathers

 (12)

Expressing the depth derivative with the help of the double-square-root equation (11) and applying a number of algebraic transformations, one can turn equation (12) into the dispersion relationship

 (13)

For each reflection angle and each frequency , equation (13) specifies the locations on the four-dimensional ( , , , ) wavenumber hyperplane that contribute to the common-angle gather. In the 2-D case, equation (13) simplifies by setting and to zero. Using the notation and , the 2-D equation takes the form

 (14)

and can be explicitly solved for resulting in the convenient 2-D dispersion relationship

 (15)

In the next section, I show that a similar simplification is also valid under the common-azimuth approximation. Equations (13) and (15) describe an effective migration of the downward-continued data to the appropriate positions on midpoint-offset planes to remove the structural dependence from the local image gathers.

Another important relationship follows from eliminating the local velocity from equations (11) and (12). Expressing from equation (12) and substituting the result in equation (11), we arrive (after a number of algebraical transformations) to the frequency-independent equation

 (16)

Equation (16) can be expressed in terms of ratios and , which correspond at the zero local offset to local structural dips ( and partial derivatives), and ratios and , which correspond to local offset slopes. As shown by Sava and Fomel (2005), it can be also expressed as

 (17)

where refers to the vertical offset between source and receiver wavefields (Biondi and Shan, 2002).

In the 2-D case, equation (16) simplifies to the form, independent of the structural dip:

 (18)

which is the equation suggested by Sava and Fomel (2003). Equation (18) appeared previously in the theory of migration-inversion (Stolt and Weglein, 1985).

 Theory of 3-D angle gathers in wave-equation seismic imaging

Next: Common-azimuth approximation Up: Fomel: 3-D angle gathers Previous: Introduction

2013-03-02