Phase-encoded Reverse-time Migration

The phase-encoded multisource traces are first formed by summing the encoded traces from all mono-frequency shot gathers. Here I assume a fixed spread geometry. Then the phase-encoded RTM method back-projects every multisource trace and migrating with the following formula

$$m({\bf {x}}) = \sum_r \sum_s G({\bf {x}}\vert{\bf {r}})^* \tilde{D}({\bf {r}}) [N({\bf {s}})G({\bf {x}}\vert{\bf {s}})]^* ,$$

$$= \sum_r \overbrace{G({\bf {x}}\vert{\bf {r}})^*\tilde{D}({\bf {r}}... ...{s} \overbrace{[N({\bf {s}})G({\bf {x}}\vert{\bf {s}})]^*}^{forward~propagate},$$

$$= B({\bf {x}}) F({\bf {x}}),$$ (21)

where ${\bf {x}}~\epsilon~V$ , $B({\bf {x}}) = \sum_{g} G({\bf {x}}\vert{\bf {r}})^*\tilde{D}({\bf {r}})$ is the back-projected reflection wavefield and $F({\bf {x}})=\sum_{s} N({\bf {s}})G({\bf {x}}\vert{\bf {s}})$ is the forward-modeled multisource wavefield at the trial image point ${\bf {x}}$ . Also, the $gth$ component of the encoded multisource data $\tilde{D}$ is denoted by $\tilde{D}({\bf {r}})=\omega^2 \sum_{s} N({\bf {s}}) D({\bf {r}}\vert{\bf {s}})$ , where $D({\bf {r}}\vert{\bf {s}})$ represents the recorded single-source shot gather, and ${\bf {r}}$ indicates the receiver position on the surface. The data and migration operators are in the frequency domain at angular frequency $\omega$ , and $N_i$ is mapped to the function $N({\bf {x}}_i)$ .