next up previous [pdf]

Next: Common-azimuth approximation Up: Fomel: 3-D angle gathers Previous: Introduction

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

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

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

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

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

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

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

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$ .
[pdf] [png] [xfig]

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

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

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$ .
[pdf] [png] [xfig]

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

$\displaystyle t_{m_x}$ $\displaystyle \equiv$ $\displaystyle \frac{\partial t}{\partial m_x} =
-\frac{2}{v}\,\cos{\theta}\,\cos{\alpha}\;,$ (6)
$\displaystyle t_{m_y}$ $\displaystyle \equiv$ $\displaystyle \frac{\partial t}{\partial m_y} =
-\frac{2}{v}\,\cos{\theta}\,\cos{\beta}\;,$ (7)
$\displaystyle t_{h_x}$ $\displaystyle \equiv$ $\displaystyle \frac{\partial t}{\partial h_x} =
\frac{4}{v^2\,t}\,\left(h_x\,\sin^2{\alpha} -
h_y\,\cos{\alpha}\,\cos{\beta}\right)\;,$ (8)
$\displaystyle t_{h_y}$ $\displaystyle \equiv$ $\displaystyle \frac{\partial t}{\partial h_y} =
\frac{4}{v^2\,t}\,\left(h_y\,\sin^2{\beta} -
h_x\,\cos{\alpha}\,\cos{\beta}\right)\;.$ (9)

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

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

and is related to the previously defined derivatives by the double-square-root equation
$\displaystyle - v\,t_z$ $\displaystyle =$ $\displaystyle \sqrt{1-
\frac{v^2}{4}\,\left(t_{m_x} - t_{h_x}\right)^2 -
\frac{v^2}{4}\,\left(t_{m_y} - t_{h_y}\right)^2}$  
  $\displaystyle +$ $\displaystyle \sqrt{1-
\frac{v^2}{4}\,\left(t_{m_x} + t_{h_x}\right)^2 -
\frac{v^2}{4}\,\left(t_{m_y} + t_{h_y}\right)^2}\;.$ (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

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

\begin{equation*}\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} }\end{equation*} (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

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

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

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

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

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

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

next up previous [pdf]

Next: Common-azimuth approximation Up: Fomel: 3-D angle gathers Previous: Introduction