Phase Encoding

My frequency encoding scheme will now be developed in the same framework of phase-encoding (Romero et al., 2000), which typically consists of the following three steps. 1) Different shot gathers are uniquely phase encoded. 2) They are summed together to form supergathers, which are then 3) migrated all at once. The first step amounts to multiplying the $s^{\textrm{th}}$ shot gather with a unique phase encoding function Ns, a step expressed as

$$\tilde{{\bf {d}}}_s = N_s{\bf {d}}_s.$$ (2.5)

In this dissertation, tilde notes an encoded version of $\circ$ . Then $\tilde{{\bf {d}}}_s$ are summed over all sources to give the encoded supergather $\tilde{\bf {d}}$ :

$$\tilde{\bf {d}} = \sum_{s=1}^S \tilde{{\bf {d}}}_s = \sum_{s=1}^S N_s{\bf {d}}_s,$$ (2.6)

$$= \tilde{{\bf {L}}}{\bf {m}},$$ (2.7)

where the multisource phase-encoded prestack modeling operator is defined as

$$\tilde{{\bf {L}}} \stackrel{\mathrm{def}}{=}\sum_{s=1}^S N_s {\bf {L}}_s.$$ (2.8)

Finally, the third step involves applying the adjoint operator $\tilde{{\bf {L}}}^{\dagger}$ to the encoded supergather $\tilde{\bf {d}}$ in equation 2.6, before applying the imaging condition, to get the migrated image $\tilde{{\bf {m}}}$ as

$$\tilde{{\bf {m}}} = \sum_{\omega}\tilde{{\bf {L}}}^{\dagger}\tilde{\bf {d}}$$ (2.9)

$$= \sum_{\omega} \sum_{s=1}^S\sum_{q=1}^S N_s^{*} N_q {\bf {L}}_s^{\dagger}{\bf {L}}_q {\bf {m}}$$ (2.10)

$$= \hat{{\bf {m}}} + {\bf {c}},$$ (2.11)

$$\noalign{where} \hat{{\bf {m}}} \stackrel{\mathrm{def}}{=}\sum_{\omega} \sum_{s=1}^S \vert N_s\vert^2 {\bf {L}}_s^{\dagger} {\bf {L}}_s {\bf {m}}$$ (2.12)

$$=\:\sum_{\omega}\sum_{s=1}^S {\bf {L}}_s^{\dagger} {\bf {L}}_s {\bf {m}},$$ (2.13)

$$\noalign{and} {\bf {c}} \stackrel{\mathrm{def}}{=}\sum_{\omega}\sum_{s=1}^S\sum_{q\neq s}^S N_s^{*} N_q {\bf {L}}_s^{\dagger} {\bf {L}}_q {\bf {m}}$$ (2.14)

$$=\sum_{\omega}\sum_{s=1}^S\sum_{q\neq s}^S N_s^{*} N_q W^{*}_s(\o... ...{q}(\omega) \underline{{\bf {L}}}_s^{\dagger}\underline{{\bf {L}}}_q {\bf {m}}.$$ (2.15)

Here, $\hat{{\bf {m}}}$ is the sequential shot-gather migration and ${\bf {c}}$ is crosstalk noise. Equation 2.13 follows assuming the phase encoding function $N_s$ is of pure phase so that $N_s^{*} N_s=1$ , and equation 2.15 follows from equation 2.4.

Note the crosstalk noise, ${\bf {c}}$ , is the only part of $\tilde{{\bf {m}}}$ that depends on the random phase encoding function, over which an ensemble average, denoted by $\langle\:\rangle$ , is taken to produce

$$\langle{\bf {c}}\rangle = \sum_{\omega}\sum_{s=1}^S\sum_{q\neq s}^S \langle N_s^{*} N_q\rangle {\bf {L}}_s^{\dagger} {\bf {L}}_q {\bf {m}}.$$ (2.16)