Basic Principles of Reflection Tomography

For reflection data, there are two things that can cause traveltime perturbation: slowness perturbation $\Delta s$ and reflector movement $\Delta r$. Figure demonstrates the basic geometry for the reflection tomography problem. Here, l_n is the normal ray, l_o is the offset ray with aperture angle $\theta$, and $\Delta r$ is the normal shift between exact reflector and apparent reflector.

According to Fermat's principle, the traveltime perturbation caused by slowness perturbation, $\Delta t_o$, can be mapped approximately to slowness perturbation by the following linear relationship:

$$\begin{eqnarray} \Delta t_o \approx \int_{l_o} \Delta s dl_o.\end{eqnarray}$$ (1)

According to van Trier 1990, the reflector movement $\Delta r$ can be assumed equal to the residual zero-offset migration of the reflector. Consequently, $\Delta r$ can be mapped to the slowness perturbation along the normal (zero-offset) ray, which can be expressed by the following equation

$$\begin{eqnarray} \Delta r \approx -\frac{1}{s_0}\int_{l_n} \Delta s dl_n,\end{eqnarray}$$ (2)

where s₀ is the local slowness at the reflection point. According to Fermat's principle, the reflector movement $\Delta r$ causes $-2\Delta r cos\theta$ change in ray length. As a result, the traveltime perturbation caused by reflector movement is

$$\begin{eqnarray} \Delta t_n \approx 2s_o\Delta r cos\theta \approx -2cos\theta \int_{l_n} \Delta s dl_n.\end{eqnarray}$$ (3)

 

ref Figure 1 Geometry for reflection wave propagation. lo is the offset ray. ln is the normal ray. $\theta$ is the aperture angle of the offset ray. $\Delta r$ is the normal shift between apparent reflector and correct reflector.

By summing $\Delta t_o$ and $\Delta t_n$, we can obtain the total traveltime perturbation:

$$\begin{eqnarray} \Delta t = \Delta t_o + \Delta t_n \approx \int_{l_o} \Delta s dl_o -2cos\theta \int_{l_n} \Delta s dl_n.\end{eqnarray}$$ (4)

Equation (4) provides a linear relationship between reflection traveltime perturbation $\Delta t$ and slowness perturbation $\Delta s$ which can be used for backpropagation.

For migration velocity analysis, reflection traveltime perturbation, $\Delta t$, can be effectively obtained from angle-domain common-image-gathers (ADCIG) Clapp (2001). Figure is a sketch of ADCIG. Here, $\Delta r$ is the normal shift between correct reflection position and apparent reflection position; $\Delta r_2$ is the residual moveout; and $\Delta r_1$ is the total normal shift. According to Biondi and Symes 2003, the traveltime perturbation $\Delta t_o$ can be calculated from total normal shift by following equation:

$$\begin{eqnarray} \Delta t_o \approx 2 s_ocos\theta \Delta r_1.\end{eqnarray}$$ (5)

Combining equation (5) and (3), we can obtain reflection traveltime perturbation from residual moveout by following equation:

$$\begin{eqnarray} \Delta t \approx 2 s_ocos\theta \Delta r_2.\end{eqnarray}$$ (6)

As we can see, $\Delta t_o$ and $\Delta t_n$ can provide independent data information for velocity inversion. However, from reflection data, we can not obtain them separately since the reflection data alone can not provide the exact reflector position. Instead, we can only obtain $\Delta t$ which is the sum of $\Delta t_o$ and $\Delta t_n$ for reflection tomography.

 

expl_data_adcig Figure 2 Illustration of calculating $\bf \Delta t$ for reflection tomography from angle-domain CIGs