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Theoretical part

You can either write your answers on paper or edit them in the file hw2/paper.tex. Please show all the mathematical derivations that you perform.

  1. In class, we derived analytical solutions for one-point and two-point ray tracing problems for the special case of a constant gradient of slowness squared
S^2(\mathbf{x}) =
S^2(\mathbf{x_0})+2 \mathbf{g} \cdot (\mathbf{x}-\mathbf{x_0})\;.
\end{displaymath} (1)

    In this homework, you will consider another special case, that of a constant gradient of velocity
V(\mathbf{x}) = {\frac{1}{S(\mathbf{x})}} =
{V(\mathbf{x_0})+\mathbf{G} \cdot (\mathbf{x}-\mathbf{x_0})}\;.
\end{displaymath} (2)

    Recall the ray tracing system from Homework 1
    $\displaystyle \frac{d \mathbf{p}}{d \xi}$ $\textstyle =$ $\displaystyle -\nabla V\;,$ (3)
    $\displaystyle \frac{d \mathbf{x}}{d \xi}$ $\textstyle =$ $\displaystyle \mathbf{p} V^3(\mathbf{x})\;,$ (4)
    $\displaystyle \frac{d T}{d \xi}$ $\textstyle =$ $\displaystyle V(\mathbf{x})\;.$ (5)

    and consider the one-point ray tracing problem with the initial conditions $\mathbf{x}(0)= \mathbf{x}_0$ and $\mathbf{p}(0)=\mathbf{p}_0$.
    1. Show that the solution of equation (3) for the constant gradient of velocity is
\mathbf{p}(\xi) = \mathbf{p}_0 - \mathbf{G} \xi\;.
\end{displaymath} (6)

      and express velocity along the ray as a function of $\mathbf{p}_0$, $\mathbf{G}$, and $\xi$:
V(\mathbf{x}) = \frac{1}{\sqrt{\mathbf{p} \cdot \mathbf{p}}} =
\end{displaymath} (7)

    2. Let $a = \mathbf{p} \cdot (\mathbf{x} - \mathbf{x}_0)$. Using the chain rule, find the expression for
\frac{d a}{d \xi} =
\end{displaymath} (8)

      and solve it to show it that
a(\xi) = V(\mathbf{x}_0) \xi
\end{displaymath} (9)

a_0(\xi) = \mathbf{p}_0 \cdot (\mathbf{x} - \mathbf{x}_0) = V(\mathbf{x}) \xi
\end{displaymath} (10)

    3. One way to seek the solution for the one-point ray tracing problem is to look for scalars $\alpha$ and $\beta$ in the representation
\mathbf{x}(\xi) = \mathbf{x}_0 + \alpha(\xi) \mathbf{p}_0 + \beta(\xi) \mathbf{G}\;.
\end{displaymath} (11)

      Under what condition does the linear system of equations
      $\displaystyle V(\mathbf{x}) \xi = \mathbf{p}_0 \cdot (\mathbf{x} - \mathbf{x}_0)$ $\textstyle =$ $\displaystyle \alpha \mathbf{p}_0 \cdot \mathbf{p}_0 + \beta \mathbf{p}_0 \cdot \mathbf{G}$ (12)
      $\displaystyle V(\mathbf{x}) - V(\mathbf{x}_0) = \mathbf{G} \cdot (\mathbf{x} - \mathbf{x}_0)$ $\textstyle =$ $\displaystyle \alpha \mathbf{p}_0 \cdot \mathbf{G} + \beta \mathbf{G} \cdot \mathbf{G}$ (13)

      have a unique solution for $\alpha$ and $\beta$? Solve the system to find $\alpha$ and $\beta$ and obtain an analytical expression for the ray trajectory $\mathbf{x}(\xi)$.
    4. Express the squared distance between the ray end points
      $\displaystyle (\mathbf{x} - \mathbf{x}_0) \cdot (\mathbf{x} - \mathbf{x}_0)$ $\textstyle =$ $\displaystyle \alpha^2 \mathbf{p}_0 \cdot \mathbf{p}_0 + 2 \alpha \beta \mathbf{p}_0 \cdot \mathbf{G} +
\beta^2 \mathbf{G} \cdot \mathbf{G}$  
        $\textstyle =$   (14)

      in terms of $\mathbf{G}$, $\mathbf{p}_0$, and $\xi$.
    5. In the two-point problem, the unknown parameters are $(\mathbf{p}_0 \cdot \mathbf{G})$ and $\xi$. Express $(\mathbf{p}_0 \cdot \mathbf{G})$ from your equation (7) and substitute it into your equation (14). Solve for $\xi$.
    6. Finally, use $\xi$ and $(\mathbf{p}_0 \cdot \mathbf{G})$ expressed in terms of $\vert\mathbf{x} - \mathbf{x}_0\vert$, $\mathbf{G}$, $V(\mathbf{x}_0)$, and $V(\mathbf{x})$ and substitute them into the one-point traveltime solution obtained by integrating equation (5)[*]
T(\xi) =
\frac{1}{\vert\mathbf{G}\vert} \mbox{arccosh}\l...
...\mathbf{G}) V(\mathbf{x}) V^2(\mathbf{x}_0) \xi}\right)\;.
\end{displaymath} (15)

      Your result will be the analytical two-point traveltime
      $\displaystyle \widehat{T}(\mathbf{x}_0,\mathbf{x}) =$   $\displaystyle \hfill  $ (16)

  2. In class, we discussed the hyperbolic traveltime approximation for normal moveout
T(h) \approx \sqrt{T_0^2 + \frac{h^2}{V_0^2}}\;.
\end{displaymath} (17)

    More accurate approximations, involving additional parameters, are possible.
    1. Consider the following three-parameter approximation
T(h) \approx T_0 \left(1-\frac{1}{S}\right) +
\frac{1}{S} \sqrt{T_0^2+S \frac{h^2}{V_0^2}}\;,
\end{displaymath} (18)

      where $S$ is the so-called ``heterogeneity'' parameter.

      Evaluate parameter $S$ in terms of the velocity $V(z)$ and the reflector depth $z_0$.

S =
\end{displaymath} (19)

      by expanding equation (18) in a Taylor series around the zero offset $h=0$ and comparing it with the corresponding Taylor series of the exact traveltime. The exact traveltime is given by the parametric equations
      $\displaystyle h$ $\textstyle =$ $\displaystyle \int\limits_0^{z_0} \frac{p V(z) dz}{\sqrt{1-p^2 V^2(z)}}\;,$ (20)
      $\displaystyle T$ $\textstyle =$ $\displaystyle \int\limits_0^{z_0} \frac{dz}{V(z) \sqrt{1-p^2 V^2(z)}}\;.$ (21)

    2. Let $\tau = T-p h$. Show that $\tau$ can be approximated to the same accuracy by
\tau(p) \approx \displaystyle \tau_0 \left(1-\frac{1}{S_...{\tau_0}{S_{\tau}} \sqrt{1-S_{\tau} {V_{\tau}^2} p^2}\;.
\end{displaymath} (22)

      Find $\tau_0$, $V_{\tau}$, and $S_{\tau}$.

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