Homework 2

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
   

   $$S^2(\mathbf{x}) = S^2(\mathbf{x_0})+2\,\mathbf{g} \cdot (\mathbf{x}-\mathbf{x_0})\;.$$ (1)

   
   
   In this homework, you will consider another special case, that of a constant gradient of velocity
   

   $$V(\mathbf{x}) = {\frac{1}{S(\mathbf{x})}} = {V(\mathbf{x_0})+\mathbf{G} \cdot (\mathbf{x}-\mathbf{x_0})}\;.$$ (2)

   
   
   It is convenient to change the parameterization of the ray tracing system, with parameter $\xi$ defined by equation (3) below:
   

   $$\frac{d\,\mathbf{p}}{d\,\xi} = -\nabla V\;,$$ (3)

   $$\frac{d\,\mathbf{x}}{d\,\xi} = \mathbf{p}\,V^3(\mathbf{x})\;,$$ (4)

   $$\frac{d\,T}{d\,\xi} = V(\mathbf{x})\;.$$ (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
      

      $$\mathbf{p}(\xi) = \mathbf{p}_0 - \mathbf{G}\,\xi\;.$$ (6)

      
      
      and express velocity along the ray as a function of $\mathbf{p}_0$, $\mathbf{G}$, and $\xi$:
      

      $$V(\mathbf{x}) = \frac{1}{\sqrt{\mathbf{p} \cdot \mathbf{p}}} =$$ (7)

      
      

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

      $$\frac{d\,a}{d\,\xi} =$$ (8)

      
      
      and solve it to show it that
      

      $$a(\xi) = V(\mathbf{x}_0)\,\xi$$ (9)

      
      
      and
      

      $$a_0(\xi) = \mathbf{p}_0 \cdot (\mathbf{x} - \mathbf{x}_0) = V(\mathbf{x})\,\xi$$ (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
      

      $$\mathbf{x}(\xi) = \mathbf{x}_0 + \alpha(\xi)\,\mathbf{p}_0 + \beta(\xi)\,\mathbf{G}\;.$$ (11)

      
      
      Under what condition does the linear system of equations
      

      $$V(\mathbf{x})\,\xi = \mathbf{p}_0 \cdot (\mathbf{x} - \mathbf{x}_0) = \alpha\,\mathbf{p}_0 \cdot \mathbf{p}_0 + \beta\,\mathbf{p}_0 \cdot \mathbf{G}$$ (12)

      $$V(\mathbf{x}) - V(\mathbf{x}_0) = \mathbf{G} \cdot (\mathbf{x} - \mathbf{x}_0) = \alpha\,\mathbf{p}_0 \cdot \mathbf{G} + \beta\,\mathbf{G} \cdot \mathbf{G}$$ (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
      

      $$(\mathbf{x} - \mathbf{x}_0) \cdot (\mathbf{x} - \mathbf{x}_0) = \alpha^2\,\mathbf{p}_0 \cdot \mathbf{p}_0 + 2 \alpha\,\beta\,\mathbf{p}_0 \cdot \mathbf{G} + \beta^2\,\mathbf{G} \cdot \mathbf{G}$$

      $$=$$ (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)
      

      $$T(\xi) = \frac{1}{\vert\mathbf{G}\vert}\,\mbox{arccosh}\l... ...\mathbf{G})\,V(\mathbf{x})\,V^2(\mathbf{x}_0)\,\xi}\right)\;.$$ (15)

      
      
      Your result will be the analytical two-point traveltime
      

      $$\widehat{T}(\mathbf{x}_0,\mathbf{x}) = \hfill \$$ (16)

      
      

2. In class, we discussed the hyperbolic traveltime approximation for normal moveout
   

   $$T(h) \approx \sqrt{T_0^2 + \frac{h^2}{V_0^2}}\;.$$ (17)

   
   
   More accurate approximations, involving additional parameters, are possible.
   1. Consider the following three-parameter approximation
      

      $$T(h) \approx T_0\,\left(1-\frac{1}{S}\right) + \frac{1}{S}\,\sqrt{T_0^2+S\,\frac{h^2}{V_0^2}}\;,$$ (18)

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

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

      $$S =$$ (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
      

      $$h(p) = \int\limits_0^{z_0} \frac{p\,V(z)\,dz}{\sqrt{1-p^2 V^2(z)}}\;,$$ (20)

      $$T(p) = \int\limits_0^{z_0} \frac{dz}{V(z)\,\sqrt{1-p^2 V^2(z)}}\;.$$ (21)

      
      

   2. Let $\tau = T-p\,h$. Show that the function $\tau(p)$ can be approximated to the same accuracy by
      

      $$\tau(p) \approx \displaystyle \tau_0\,\left(1-\frac{1}{S_... ...ac{\tau_0}{S_{\tau}}\,\sqrt{1-S_{\tau}\,{V_{\tau}^2}\,p^2}\;.$$ (22)

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

Homework 2