Theory

I now present the spectral encoding strategy for removing crosstalk artifacts in multisource imaging. First, I identify the source spectrum in the forward modeling equation. Then, I outline a typical phase-encoded multisource procedure, before developing the proposed frequency encoding method.

In the frequency domain a seismic trace with a source at ${\bf {x}}_s$ and a receiver at ${\bf {x}}$ can be expressed (Stolt and Benson, 1986), based on the Born approximation to the Lippman-Schwinger equation, as

$$d({\bf {x}}\vert{\bf {x}}_s) = \int G({\bf {x}}\vert{\bf {x}}')m_o({\bf {x}}')G({\bf {x}}'\vert{\bf {x}}_s)W_{s}(\omega)\,\mathrm{d}{\bf {x}}'.$$ (2.1)

Here, $G({\bf {b}}\vert{\bf {a}})$ denotes the Green's function from ${\bf {a}}$ to ${\bf {b}}$ ; $m_o({\bf {x}}') \stackrel{\mathrm{def}}{=}s({\bf {x}}')\delta s({\bf {x}}')$ is the reflection coefficient-like term at ${\bf {x}}'$ , where $\delta s({\bf {x}}')$ is the slowness perturbation from an assumed background slowness $s({\bf {x}}')$ ; and Ws is the spectrum of the $s^{\textrm{th}}$ source weighted by $-2\omega^2$ and can be pulled outside the integral since it is independent of ${\bf {x}}'$ . For conciseness $W_{s}(\omega)$ is hereafter referred to as `source spectrum' or simply `spectrum' for short. As the earth model is discretized into $M$ grid points, equation 2.1 can be recast in matrix-vector form as

$${\bf {d}}_s = W_{s}(\omega) \underline{{\bf {L}}}_s {\bf {m}}, \quad \forall s=1,\ldots,S$$ (2.2)

$$\noalign{which is conventionally expressed as} {\bf {d}}_s = {\bf {L}}_s {\bf {m}}, \quad \forall s=1,\ldots,S$$ (2.3)

$$\textrm{where~~~~~~~~~}{\bf {L}}_s = W_{s}(\omega) \underline{{\bf {L}}}_s.$$ (2.4)

Here, $\gls{m}\in\mathbb{R}^M$ is the reflectivity model; $\gls{ds}\in\mathbb{C}^{n_h}$ represents the $s^{\textrm{th}}$ shot gather; S is the number of shots; nh is the number of receivers per shot; $\gls{Ls}\in\mathbb{C}^{n_h\times M}$ represents the prestack modeling operator for the $s^{\textrm{th}}$ shot gather, and $\underline{{\bf {L}}}_s$ is Ls deprived of Ws. Equations 2.2 to 2.4 are in the frequency domain and recognize that quantities such as ${\bf {d}}_s$ , $\underline{{\bf {L}}}_s$ , and ${\bf {L}}_s$ all depend on $\omega$ , which is silent to reduce notational clutter; however, $\omega$ is explicitly retained in $W_{s}(\omega)$ , because $W_{s}(\omega)$ represents the proposed frequency encoding function. Note also the subscript in $W_{s}(\omega)$ , implying that different sources may have different spectrums.