Wave-equation time migration

Wave-equation and image rays

The phase information contained in seismic waves is governed by the eikonal equation, which takes the form (Chapman, 2004)

$$\nabla T \cdot \nabla T = \frac{1}{V^2(\mathbf{x})}\;,$$ (1)

where $\mathbf{x}$ is a point in space, $T(\mathbf{x})$ is traveltime from the source to $\mathbf{x}$, and $V(\mathbf{x})$ is the phase velocity. In an anisotropic medium, the phase velocity depends additionally on the direction of traveltime gradient $\nabla T$. For simplicity, we limit the following discussion to the isotropic case and two dimensions. Extensions to anisotropy and 3-D are possible but would complicate the discussion. In 2-D, $\mathbf{x}=\{x,z\}$, and the isotropic eikonal equation 1 can be written as

$$\left(\frac{\partial T}{\partial x}\right)^2 + \left(\frac{\partial T}{\partial z}\right)^2 = \frac{1}{V^2(x,z)}\;.$$ (2)

The simplest wave equation that corresponds to eikonal equation 1 is the acoustic wave equation

$$\frac{\partial^2 W}{\partial x^2} + \frac{\partial^2 W}{\par... ...^2} = \frac{1}{V^2(x,z)}\,\frac{\partial^2 W}{\partial t^2}\;,$$ (3)

where $W(x,z,t)$ is a wavefield propagating with velocity $V$. Equation 3 provides the basis for a variety of wave-equation migration algorithms, from one-way wave extrapolation in depth to two-way reverse-time migration (RTM) (Biondi, 2006; Etgen et al., 2009).

Consider a family of image rays (Hubral, 1977), traced orthogonal to the surface. Image-ray coordinates $x_0$ and $t_0$ as functions of the Cartesian coordinates $x$ and $z$ satisfy the following system of partial differential equations (Cameron et al., 2007):

$$\vert\nabla x_0\vert^2 = \left(\frac{\partial x_0}{\partial x}\right)^2+\left(\frac{\partial x_0}{\partial z}\right)^2 =\frac{1}{J^2(x,z)}\;,$$ (4)

$$\nabla x_0\cdot\nabla t_0 = \frac{\partial x_0}{\partial x}\frac{\partial t_0}{\partial x}+ \frac{\partial x_0}{\partial z}\frac{\partial t_0}{\partial z}=0\;,$$ (5)

$$\vert\nabla t_0\vert^2 = \left(\frac{\partial t_0}{\partial x}\right)^2+\left(\frac{\partial t_0}{\partial z}\right)^2 =\frac{1}{V^2(x,z)}\;,$$ (6)

with boundary conditions $x_0(x,0)=x$ and $t_0(x,0)=0$. Equation 6 is the familiar eikonal equation 2. Equation 5 expresses the orthogonality of rays and wavefronts in an isotropic medium. Equation 4 defines the quantity $J(x,z)$ as the geometrical spreading of image rays.

Applying a change of variables from $\{x,z\}$ to $\{x_0,t_0\}$ transforms eikonal equation 2 to the coordinate system of image rays and leads to an elliptically anisotropic eikonal equation which is

$$\left(\frac{\partial T}{\partial x_0}\right)^2\,\frac{1}{J^2... ...al T}{\partial t_0}\right)^2\,\frac{1}{V^2} = \frac{1}{V^2}\;.$$ (7)

The corresponding wave equation is given by

$$\frac{\partial^2 W}{\partial x_0^2}\,V_d^2 + \frac{\partial^2 W}{\partial t_0^2} = \frac{\partial^2 W}{\partial t^2}\;,$$ (8)

where $V_d=V/J$. Equation 8 is a particular version of the more general Riemannian coordinate transformation analyzed by Sava and Fomel (2005). Some other versions of Riemannian coordinate transformations are analysed by Shragge (2008); Shragge and Shan (2008). Note that at the surface of observation $z=0$, the solution of equation 8 coincides geometrically with the solution of the original Cartesian equation 3.

The significance of equation 8 lies in the following fact established by Cameron et al. (2007). When time migration is performed using coordinates $x_0$ and $t_0$ and the conventional traveltime approximation based on the Taylor expansion of diffraction traveltime around the image ray, such as the classic hyperbolic approximation

$$t^2(x) \approx t_0^2+\frac{(x-x_0)^2}{V_0^2(x_0,t_0)}\;,$$ (9)

the coefficient $V_d$ appearing in equation 8 is simply related to time-migration velocity $V_0$ appearing in equation 9. More specifically,

$$V_d^2(x_0,t_0) = \frac{V^2}{J^2} = \frac{\partial \left[t_0\,V_0^2(x_0,t_0)\right]}{\partial t_0}\;.$$ (10)

Following Cameron et al. (2008a), we call $V_d$ Dix velocity because it corresponds to the classic Dix inversion applied to the time-migration velocity (Dix, 1955). In the absence of lateral velocity variations (a $V(z)$ medium), image rays do not spread; therefore, $J=1$, $V_d=V$ and corresponds to true velocity in the medium, and $V_0$ corresponds to root-mean-square (RMS) velocity. In the presence of lateral velocity variations, the conversion between time-migration coordinates and true Cartesian coordinates is more complicated (Cameron et al., 2008a; Li and Fomel, 2015; Bevc et al., 1995). However, to extrapolate seismic wavefields in image-ray coordinates using equation 8, it is sufficient to use an estimate of the Dix velocity $V_d$, which is readily available from conventional time-domain processing and equation 10.

Wave-equation time migration