Kirchhoff Migration:

Double Square Root Vs. Zero Offset

In the highly competitive market of seismic data processing, particularly in the area of Prestack 3D Time Migration, there is often very little consideration placed on what type of algorithm is being used. A 3D volume can be sorted into equivalent offset volumes and each volume can be migrated with a Kirchhoff zero offset migration, and this can be (and commonly is) called Kirchhoff Pre-Stack 3D Time Migration.

In order to achieve the best time imaging potential and provide the most accurate gathers for AVO analysis it is recommended that the Kirchhoff Double Square Root equation be used for Prestack 3D and 2D Time Migration. Equations displayed in Figures 1 and 2 show the differences in the kinematics associated with the two migration principles.

In Figure 1 the double square root algorithm sums data into an output trace by adding the two one way times from both the shot and receiver paths. The zero offset algorithm in Figure 2 only uses the distance from the input trace to the output location in its computation of travel times.

Figure 1: Kirkhoff Double Square Root Basic Equation

Figure 2: Kirkhoff Zero Offset Migration Basic Equation

The Kirchhoff Double Square Root equation allows complex diffraction patterns to be imaged by using each trace’s unique geometry in association with structure and velocity. Amplitude relationships associated with angle of incidence can be better preserved by integrating higher order terms in the accumulation of migrated gathers by using the Double Square Root Equation.

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