Operator overview

In migration by downward-continuation, the wavefield at depth $z+\Delta z$ is obtained by phase-shift from the wavefield at depth z Claerbout (1985)  

$$\mathcal W\left({z+\Delta z} \right)= \mathcal W\left(z \right)e^{-i k_z\Delta z}.$$ (1)

where the depth wavenumber k_z depends linearly through a Taylor series expansion on its value in the reference medium (k_zo) and the slowness difference in the depth interval from z to $z+\Delta z$, $s\left({\bf x},z \right)- s_o\left(z \right)$: 

$$k_z\approx {k_z}_o+ \left. \frac{\partial k_z}{\partial s} \right\vert _{s=s_o}\left(s- s_o\right),$$ (2)

where, by definition, $\Delta s=s\left({\bf x},z \right)-s_o\left(z\right)$, and ${\bf x}$ denotes spatial position at depth z. The expression for $\left. \frac{\partial k_z}{\partial s} \right\vert _{s=s_o}$ can take many different forms, summarized in Sava (2000).

From Equations (1) and (2), we can write that

$$\begin{eqnarray} \mathcal W\left({z+\Delta z} \right)&=& \mathcal W\left(z \righ... ...c{\partial k_z}{\partial s} \right\vert _{s=s_o}\Delta s\Delta z}.\end{eqnarray}$$ (3) (4)

Equation (3) represents a general form of the main mixed-domain downward-continuation operator. This operator can be broken up into a group of functional operators as follows:

• Wavefield continuation using mixed-domain phase-shift:

  $$\begin{eqnarray} \mathcal W\left({z+\Delta z} \right)&=& \mathcal W\left(z \righ... ...}^{\texttt{SL op}} \Delta z}}_{\texttt{FX op}} }_{\texttt{WC op}};\end{eqnarray}$$ (5)

• Imaging condition which transforms the wavefield into an image at any given depth level:

  $$\begin{eqnarray} \mathcal W\left({z+\Delta z} \right)\overbrace{\to}^{\texttt{IG op}} R\left(z+\Delta z\right).\end{eqnarray}$$ (6)

$\mathcal W\left({z+\Delta z} \right)$ and $\mathcal W\left(z \right)$ are the wavefields at depths $z+\Delta z$ and z respectively, $\Delta z$ is the depth step, s_o is the constant reference slowness in the slab from z to $\Delta z$,$s\left(\bf{x} \right)$ is the variable slowness in the same depth slab, and $\texttt{DSR}\left(s_o\right)$ represents the depth wavenumber expressed using the double-square root equation, and which is a function of the reference slowness (s_o).

For Equations (5) and (6), we can distinguish 5 functional operators. Each operator is initialized with a call to a function (XXin) and executed with a call to another function (XXop). In a typical example, the functional operators perform the following tasks:

1.

Wavefield continuation operator (WCin & WCop)

Continues the wavefield between two depth levels, using one or more reference slownesses.

Interface: integer function WCop(wfld,iws,izs,ith,FKop,FXop) result(st)

• complex, dimension(:,:,:,:,:), pointer :: wfld (wavefield slice)
• integer :: iws (index of the frequency slice)
• integer :: izs (index of the depth slice)
• integer :: ith (thread number)
• FKop :: f-k operator
• FXop :: f-x operator

Implemented examples:

• weimwc1 (mixed-domain wavefield continuation operator for 1 reference slowness)
• weimwcN mixed-domain wavefield continuation operator for N reference slownesses)

2.

Slowness operator (SLin & SLop)

Selects the number and values of the reference slownesses (s_o), and sets-up the interpolation map between the wavefields continued using the various reference slownesses.

Interface: integer function SLop() result(st)

Implemented examples:

• weislo1 (slowness selector for 1 reference slowness)
• weisloN (slowness selector for N reference slownesses)

3.

f-k operator (FKin & FKop)

Performs phase-shift using the full 3-D DSR equation Claerbout (1985), the common-azimuth equation Biondi and Palacharla (1996), or the offset plane-waves equation Mosher and Foster (2000).

Interface: integer function FXop(iws,izs,ifk,ith,wfld) result(st)

• integer :: iws (index of the frequency slice)
• integer :: izs (index of the depth slice)
• integer :: ifk (index of the reference velocity)
• integer :: ith (thread number)
• complex, dimension(:,:,:,:,:), pointer :: wfld (wavefield slice)

Implemented examples:

• weiwem (3-D prestack or 3-D offset plane-waves phase-shift)
• weicam (3-D common-azimuth phase-shift)

4.

f-x operator (FXin & FXop)

Performs phase shift that accounts for lateral slowness variation. Examples of (f-x) operators include but are not limited to split-step Fourier Stoffa et al. (1990), local Born Fourier or local Rytov Fourier Huang et al. (1999), Fourier Finite-Difference Ristow and Ruhl (1994), generalized screen propagators Le Rousseau and de Hoop (1998), etc.

Interface: integer function FXop(iws,izs,ifk,ith,wfld) result(st)

• integer :: iws (index of the frequency slice)
• integer :: izs (index of the depth slice)
• integer :: ifk (index of the reference velocity)
• integer :: ith (thread number)
• complex, dimension(:,:,:,:,:), pointer :: wfld (wavefield slice)

Implemented example:

• weissf (Split-step correction)

5.

Imaging operator (IGin & IGop)

Performs imaging in the offset-domain or the offset ray-parameter domain. This operator can also incorporate amplitude-preserving corrections.

Interface: integer function IGop(wfld,iws,ith) result(st)

• complex, dimension(:,:,:,:,:), pointer :: wfld (wavefield slice)
• integer :: iws (index of the frequency slice)
• integer :: ith (thread number)

Implemented examples:

• weihcig (Offset-domain common image-gathers)
• weipcig (Offset ray-parameter p_h common image-gathers)