Theory of 3-D angle gathers in wave-equation seismic imaging

Traveltime derivatives and dispersion relationships for a 3-D dipping reflector

Theoretical analysis of angle gathers in downward continuation methods can be reduced to analyzing the geometry of reflection in the simple case of a dipping reflector in a locally homogeneous medium. Considering the reflection geometry in the case of a plane reflector is sufficient for deriving relationships for local reflection traveltime derivatives in the vicinity of a reflection point (Goldin, 2002). Let the local reflection plane be described in $\{x,y,z\}$ coordinates by the general equation

$$x\,\cos{\alpha} + y\,\cos{\beta} + z\,\cos{\gamma} = d\;,$$ (1)

where the normal angles $\alpha$ , $\beta$ , and $\gamma$ satisfy

$$\cos^2{\alpha} + \cos^2{\beta} + \cos^2{\gamma} = 1\;,$$ (2)

The geometry of the reflection ray paths is depicted in Figure 1. The reflection traveltime measured on a horizontal surface above the reflector is given by the known expression (Slotnick, 1959; Levin, 1971)

$$t(h_x,h_y) = \frac{2}{v}\,\sqrt{D^2+h_x^2+h_y^2- \left(h_x\,\cos{\alpha} + h_y\,\cos{\beta}\right)^2}\;,$$ (3)

where $D$ is the length of the normal to the reflector from the midpoint (distance $MM'$ in Figure 2)

$$D = d - m_x\,\cos{\alpha} - m_y\,\cos{\beta}\;,$$ (4)

$m_x$ and $m_y$ are the midpoint coordinates, $h_x$ and $h_y$ are the half-offset coordinates, and $v$ is the local propagation velocity.

plane3b
Figure 1. Reflection geometry in 3-D (a scheme). $S$ and $R$ and the source and the receiver positions at the surface. $O$ is the reflection point. $S'$ is the normal projection of the source to the reflector. $S''$ is the ``mirror'' source. The cumulative length of the incident and reflected rays is equal to the distance from $S''$ to $R$ .

According to elementary geometrical considerations (Figures 1 and 2), the reflection angle $\theta$ is related to the previously introduced quantities by the equation

$$\cos{\theta} = \frac{D}{\sqrt{D^2+h_x^2+h_y^2- \left(h_x\,\cos{\alpha} + h_y\,\cos{\beta}\right)^2}}\;.$$ (5)

plane2b
Figure 2. Reflection geometry in the reflection plane (a scheme). $M$ is the midpoint. As follows from the similarity of triangles $S''SR$ and $S'SM$ , the distance from $M$ to $S'$ is twice smaller than the distance from $S''$ to $R$ .

Explicitly differentiating equation (3) with respect to the midpoint and offset coordinates and utilizing equation (5) leads to the equations

$$t_{m_x} \equiv \frac{\partial t}{\partial m_x} = -\frac{2}{v}\,\cos{\theta}\,\cos{\alpha}\;,$$ (6)

$$t_{m_y} \equiv \frac{\partial t}{\partial m_y} = -\frac{2}{v}\,\cos{\theta}\,\cos{\beta}\;,$$ (7)

$$t_{h_x} \equiv \frac{\partial t}{\partial h_x} = \frac{4}{v^2\,t}\,\left(h_x\,\sin^2{\alpha} - h_y\,\cos{\alpha}\,\cos{\beta}\right)\;,$$ (8)

$$t_{h_y} \equiv \frac{\partial t}{\partial h_y} = \frac{4}{v^2\,t}\,\left(h_y\,\sin^2{\beta} - h_x\,\cos{\alpha}\,\cos{\beta}\right)\;.$$ (9)

Additionally, the traveltime derivative with respect to the depth of the observation surface is given by

$$t_z \equiv \frac{\partial t}{\partial z} = -\frac{2}{v}\,\cos{\theta}\,\cos{\gamma}$$ (10)

and is related to the previously defined derivatives by the double-square-root equation

$$- v\,t_z = \sqrt{1- \frac{v^2}{4}\,\left(t_{m_x} - t_{h_x}\right)^2 - \frac{v^2}{4}\,\left(t_{m_y} - t_{h_y}\right)^2}$$

$$+ \sqrt{1- \frac{v^2}{4}\,\left(t_{m_x} + t_{h_x}\right)^2 - \frac{v^2}{4}\,\left(t_{m_y} + t_{h_y}\right)^2}\;.$$ (11)

In the frequency-wavenumber domain, equation (11) serves as the basis for 3-D shot-geophone downward-continuation imaging. In the Fourier domain, each $t_x$ derivative translates into $-k_x/\omega$ ratio, where $k_x$ is the wavenumber corresponding to $x$ and $\omega$ is the temporal frequency.

Equations (6), (7), and (10) immediately produce the first important 3-D relationship for angle gathers

$$\cos{\theta} = \frac{v}{2\,\omega}\,\sqrt{ k_{m_x}^2 + k_{m_y}^2 + k_z^2}\;.$$ (12)

Expressing the depth derivative with the help of the double-square-root equation (11) and applying a number of algebraic transformations, one can turn equation (12) into the dispersion relationship

$$\boxed{ \begin{gathered}\left(k_{m_x}^2 + k_{m_y}^2\right)\,\fr... ...s^2{\theta}}{v^2}\,\frac{\sin^2{\theta}}{v^2}\;. \end{gathered} }$$ (13)

For each reflection angle $\theta$ and each frequency $\omega$ , equation (13) specifies the locations on the four-dimensional ($k_{m_x}$ , $k_{m_y}$ , $k_{h_x}$ , $k_{h_y}$ ) wavenumber hyperplane that contribute to the common-angle gather. In the 2-D case, equation (13) simplifies by setting $k_{h_y}$ and $k_y$ to zero. Using the notation $k_{m_x}=k_m$ and $k_{h_x}=k_h$ , the 2-D equation takes the form

$$k_m^2\,\sin^2{\theta} + k_h^2\,\cos^2{\theta} = \frac{4\,\omega^2}{v^2}\,\cos^2{\theta}\,\sin^2{\theta}$$ (14)

and can be explicitly solved for $k_h$ resulting in the convenient 2-D dispersion relationship

$$k_h = \frac{2\,\omega\,\sin{\theta}}{v}\, \sqrt{1-\frac{4\,k_m^2\,v^2}{\omega^2\,\cos^2{\theta}}}\;.$$ (15)

In the next section, I show that a similar simplification is also valid under the common-azimuth approximation. Equations (13) and (15) describe an effective migration of the downward-continued data to the appropriate positions on midpoint-offset planes to remove the structural dependence from the local image gathers.

Another important relationship follows from eliminating the local velocity $v$ from equations (11) and (12). Expressing $v^2$ from equation (12) and substituting the result in equation (11), we arrive (after a number of algebraical transformations) to the frequency-independent equation

$$\boxed{ \tan^2{\theta} = \frac{ k_z^2\,\left(k_{h_x}^2 + k_{h_y}^... ..._y}\,k_{m_y}\right)^2} {k_z^2\,\left(k_{m_x}^2 + k_{m_y}^2 + k_z^2\right)}\;. }$$ (16)

Equation (16) can be expressed in terms of ratios $k_{m_x}/k_z$ and $k_{m_y}/k_z$ , which correspond at the zero local offset to local structural dips ($z_{m_x}$ and $z_{m_y}$ partial derivatives), and ratios $k_{h_x}/k_z$ and $k_{h_y}/k_z$ , which correspond to local offset slopes. As shown by Sava and Fomel (2005), it can be also expressed as

$$\tan^2{\theta} = \frac{k_{h_x}^2 + k_{h_y}^2 + k_{h_z}^2}{k_{m_x}^2 + k_{m_y}^2 + k_z^2}\;,$$ (17)

where $k_{h_z}$ refers to the vertical offset between source and receiver wavefields (Biondi and Shan, 2002).

In the 2-D case, equation (16) simplifies to the form, independent of the structural dip:

$$\tan{\theta} = \frac{k_h}{k_z}\;,$$ (18)

which is the equation suggested by Sava and Fomel (2003). Equation (18) appeared previously in the theory of migration-inversion (Stolt and Weglein, 1985).

Theory of 3-D angle gathers in wave-equation seismic imaging