Homework 1

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
      

      $$\frac{d \mathbf{x}}{d \sigma} = \mathbf{p}$$ (1)

      $$\frac{d \mathbf{p}}{d \sigma} = S(\mathbf{x})\,\nabla S$$ (2)

      $$\frac{d T}{d \sigma} = S^2(\mathbf{x})$$ (3)

      
      
      into an equivalent system that uses $\lambda$ instead of $\sigma$, where $\lambda$ represents the length of the ray trajectory:
      

      $$\frac{d \mathbf{x}}{d \lambda} =$$ (4)

      $$\frac{d \mathbf{p}}{d \lambda} =$$ (5)

      $$\frac{d T}{d \lambda} = S(\mathbf{x})\;.$$ (6)

      
      
      Remember to check physical dimensions.
   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 $\lambda(\mathbf{x})$ - the length of the ray trajectory at all $\mathbf{x}$. Derive a first-order partial differential equation that connects $\nabla \lambda$ and $\nabla T$.

2. The so-called ``parabolic'' or $15^{\circ}$ eikonal equation (Bamberger et al., 1988; Claerbout, 1985; Tappert, 1977) has the form
   

   $${\frac{\partial T}{\partial x_1}} + {\frac{1}{2\,S(\mathb... ...frac{\partial T}{\partial x_2}\right)^2} = {S(\mathbf{x})}$$ (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)
      

      $$\frac{d x_2}{d x_1} = \hspace{5in}$$ (8)

      $$\frac{d p_1}{d x_1} =$$ (9)

      $$\frac{d p_2}{d x_1} =$$ (10)

      $$\frac{d T}{d x_1} =$$ (11)

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

   2. Assuming a 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
      

      $$\left(\frac{\partial T}{\partial x_1}\right)^2 + \left(\frac{\partial T}{\partial x_2}\right)^2 = S^2(\mathbf{x})$$ (12)

      
      
      describes wavefronts of the wave equation
      

      $$\nabla^2 P = S^2(\mathbf{x})\,\frac{\partial^2 P}{\partial t^2} + \cdots$$ (13)

      
      
      with omitted possible first- and zero-order terms. What wave equation corresponds to equation (7)?
      

      $$S(\mathbf{x})\,{\frac{\partial^2 P}{\partial t^2}} =$$ (14)

      
      

Homework 1