Downward continuation

PHASE-SHIFT MIGRATION

The phase-shift method of migration begins with a two-dimensional Fourier transform (2D-FT) of the dataset. (See chapter .) This transformed data is downward continued with $\exp(ik_z z)$ and subsequently evaluated at $t=0$ (where the reflectors explode). Of all migration methods, the phase-shift method most easily incorporates depth variation in velocity. The phase angle and obliquity function are correctly included, automatically. Unlike Kirchhoff methods, with the phase-shift method there is no danger of aliasing the operator. (Aliasing the data, however, remains a danger.)

Equation (7.14) referred to upcoming waves. However in the reflection experiment, we also need to think about downgoing waves. With the exploding-reflector concept of a zero-offset section, the downgoing ray goes along the same path as the upgoing ray, so both suffer the same delay. The most straightforward way of converting one-way propagation to two-way propagation is to multiply time everywhere by two. Instead, it is customary to divide velocity everywhere by two. Thus the Fourier transformed data values, are downward continued to a depth $\Delta z$ by multiplying by

$$e^{ i k_z \Delta z } \quad =\quad \exp \left( - i {... ...{ 1 - { v^2 k_x^2 \over 4 \omega^2 } } \Delta z \right)$$ (15)

Ordinarily the time-sample interval $\Delta \tau$ for the output-migrated section is chosen equal to the time-sample rate of the input data (often 4 milliseconds). Thus, choosing the depth $\Delta z = (v/2) \Delta \tau$, the downward-extrapolation operator for a single time step $\Delta \tau$ is

$$C\quad =\quad \exp \left( - i \omega \Delta \tau \sqrt{ 1 - { v^2 k_x^2 \over 4 \omega^2 } } \right)$$ (16)

Data will be multiplied many times by $C$, thereby downward continuing it by many steps of $\Delta \tau$.

Downward continuation