Next: Amplitude considerations Up: Fomel & Prucha: Angle-gather Previous: Introduction

# Traveltime considerations

 rays Figure 1 Reflection rays in a constant-velocity medium: a scheme.

Let us consider a simple reflection experiment in an effectively constant-velocity medium, as depicted in Figure 1. The pair of incident and reflected rays and the line between the source s and the receiver r form a triangle in space. From the trigonometry of that triangle we can derive simple relationships among all the variables of the experiment Fomel (1995, 1996a, 1997).

Introducing the dip angle and the reflection angle ,the total reflection traveltime t can be expressed from the law of sines as
 (1)
where v is the medium velocity, and h is the half-offset between the source and the receiver.

Additionally, by following simple trigonometry, we can connect the half-offset h with the depth of the reflection point z, as follows:
 (2)

Finally, the horizontal distance between the midpoint x and the reflection point is
 (3)

Equations (1-3) completely define the kinematics of angle-gather migration. Regrouping the terms, we can rewrite the three equations in a more symmetric form:
 (4) (5) (6)
For completeness, here is the inverse transformation from t, h, and to z, , and :
 (7) (8) (9)
The inverse transformation (7-9) can be found by formally solving system (4-6).

The lines of constant reflection angle and variable dip angle for a given position of a reflection (diffraction) point have the meaning of summation curves for angle-gather Kirchhoff migration. The whole range of such curves for all possible values of covers the diffraction traveltime surface - Cheops' pyramid'' Claerbout (1985) in the space of seismic reflection data. As pointed out by Fowler (1997), this condition is sufficient for proving the kinematic validity of the angle-gather approach. For comparison, Figure 2 shows the diffraction traveltime pyramid from a diffractor at 0.5 km depth. The pyramid is composed of common-offset summation curves of the conventional time migration. Figure 3 shows the same pyramid composed of constant- curves of the angle-gather migration.

 coffset Figure 2 Traveltime pyramid, composed of common-offset summation curves.

 cangle Figure 3 Traveltime pyramid, composed of common-reflection-angle summation curves.

The most straightforward Kirchhoff algorithm of angle-gather migration can be formulated as follows:

• For each reflection angle and each dip angle ,
• For each output location ,
1.
Find the traveltime t, half-offset h, and midpoint x from formulas (4), (5), and (6) respectively.
2.
Stack the input data values into the output.
As follows from equations (4-6), the range of possible 's should satisfy the condition
 (10)
The described algorithm is not the most optimal in terms of the input/output organization, but it can serve as a basic implementation of the angle-gather idea. The stacking step requires an appropriate weighting. We discuss the weighting issues in the next section.

Next: Amplitude considerations Up: Fomel & Prucha: Angle-gather Previous: Introduction
Stanford Exploration Project
4/20/1999