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Theoretical part

You can either write your answers on paper or edit them in the file hw1/paper.tex. Please show all the mathematical derivations that you perform.

  1. In class, we used a mysterious parameter $\sigma$ to represent a variable continuously increasing along a ray. There are other variables that can play a similar role.
    1. Transform the isotropic ray tracing system
      $\displaystyle \frac{d \mathbf{p}}{d \sigma}$ $\textstyle =$ $\displaystyle S(\mathbf{x}) \nabla S$ (1)
      $\displaystyle \frac{d \mathbf{x}}{d \sigma}$ $\textstyle =$ $\displaystyle \mathbf{p}$ (2)
      $\displaystyle \frac{d T}{d \sigma}$ $\textstyle =$ $\displaystyle S^2(\mathbf{x})$ (3)

      into an equivalent system that uses $\xi$ instead of $\sigma$, where $\xi$ is constrained by equation (4):
      $\displaystyle \frac{d \mathbf{p}}{d \xi}$ $\textstyle =$ $\displaystyle \frac{\nabla S}{S^2(\mathbf{x})}$ (4)
      $\displaystyle \frac{d \mathbf{x}}{d \xi}$ $\textstyle =$   (5)
      $\displaystyle \frac{d T}{d \xi}$ $\textstyle =$   (6)

      What are the physical units of $\xi$?
    2. Suppose you are given $T(\mathbf{x})$ - the traveltime from the source to all points $\mathbf{x}$ in the domain of interest. Your task is to find $\xi(\mathbf{x})$ for all $\mathbf{x}$. Derive a first-order partial differential equation that connects $\nabla \xi$ and $\nabla T$.

  2. The so-called ``parabolic'' or $15^{\circ}$ eikonal equation (Bamberger et al., 1988; Claerbout, 1985; Tappert, 1977) has the form
    \begin{displaymath}
{\frac{\partial T}{\partial x_1}} +
{\frac{1}{2 S(\mathb...
...frac{\partial T}{\partial x_2}\right)^2} =
{S(\mathbf{x})}
\end{displaymath} (7)

    where $\mathbf{x}=\{x_1,x_2\}$ is a point in space, $T(\mathbf{x})$ is the traveltime, and $S(\mathbf{x})$ is slowness.
    1. Derive the ray tracing system for equation (7)
      $\displaystyle \frac{d x_2}{d x_1}$ $\textstyle =$ $\displaystyle \hspace{5in}$ (8)
      $\displaystyle \frac{d p_1}{d x_1}$ $\textstyle =$   (9)
      $\displaystyle \frac{d p_2}{d x_1}$ $\textstyle =$   (10)
      $\displaystyle \frac{d T}{d x_1}$ $\textstyle =$   (11)

      where $p_1$ represents $\partial T/\partial x_1$ and $p_2$ represents $\partial T/\partial x_2$.

    2. Assuming constant slowness $S(\mathbf{x}) \equiv S_0$, solve the ray tracing system for a point source at the origin $\{x_1,x_2\} = \{0,0\}$.

    3. Using the ray tracing solution, find the shape of the wavefronts defined by equation (7 in the case of a constant slowness.

    4. The isotropic eikonal equation
      \begin{displaymath}
\left(\frac{\partial T}{\partial x_1}\right)^2 +
\left(\frac{\partial T}{\partial x_2}\right)^2 = S^2(\mathbf{x})
\end{displaymath} (12)

      describes wavefronts of the wave equation
      \begin{displaymath}
S^2(\mathbf{x}) \frac{\partial^2 P}{\partial t^2} =
\nabla^2 P + \cdots
\end{displaymath} (13)

      with omitted possible first- and zero-order terms. What wave equation corresponds to equation (7)?
      \begin{displaymath}
S(\mathbf{x}) {\frac{\partial^2 P}{\partial t^2}} =
\end{displaymath} (14)


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Next: Computational part Up: Homework 1 Previous: Prerequisites

2013-09-17