Homework 4

Theoretical part

1. Consider the elastic wave equation
   

   $$\rho \ddot{u}_i = C_{ijkl,j} u_{k,l} + C_{ijkl} u_{k,lj}\;$$ (1)

   
   
   in the case of an isotropic elasticity
   

   $$C_{ijkl} = \lambda \delta_{ij} \delta_{kl} + \mu (\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk})\;.$$ (2)

   
   
   Using the geometric representation
   

   $$u_i(\mathbf{x},t) = a_i(\mathbf{x}) f\left(t-T(\mathbf{x})\right)$$ (3)

   
   
   and assuming the P-wave polarization in the direction of the gradient of $T$, derive the elastic P-wave amplitude transport equation and show its similarity to the corresponding equation for the case of acoustic variable-density wave propagation.
2. Consider a 2-D common-midpoint gather $G(t,x)$, which contains a geometric event $A_0 f\left(t-T(x)\right)$ with a constant amplitude $A_0$ along a hyperbolic shape
   

   $$T(x) = \sqrt{t_0^2+\frac{x^2}{v^2}}\;.$$ (4)

   
   
   The gather gets transformed by the slant-stack (Radon transform) operator
   

   $$R(\tau,p) = \mathbf{D}_\tau^{1/2} \int G(\tau + p x, x) d x\;.$$ (5)

   
   
   where $\mathbf{D}_\tau^{1/2}$ is a waveform-correcting half-order derivative operator.

   Using the theory of geometric integration, show that $R(\tau,p)$ will contain a geometric event $A_1(p) f\left(\tau-T_1(p)\right)$. Find $T_1(p)$ and $A_1(p)$.

3. Using the hyperbolic traveltime approximation
   

   $$T = \sqrt{\left(T_0-\frac{z}{V_0}\right)^2 + \frac{(x_0-x)^2}{V_0^2}}$$ (6)

   
   
   makes the geometric imaging analysis equivalent to analyzing wave propagation in a constant-velocity medium. In particular, we can easily verify that the traveltime satisfies the isotropic eikonal equation
   

   $$\left(\frac{\partial T}{\partial x}\right)^2 + \left(\frac{\partial T}{\partial z}\right)^2 = \frac{1}{V_0^2}\;.$$ (7)

   
   

   Suppose that you switch to the more accurate shifted-hyperbola approximation
   

   $$T = \left(T_0-\frac{z}{V_0}\right) (1-\frac{1}{S}) + \frac{... ...\left(T_0-\frac{z}{V_0}\right)^2 + S \frac{(x_0-x)^2}{V_0^2}}$$ (8)

   
   
   1. How would you need to modify the eikonal equation?
   2. How would you need to modify the following expressions for the escape time and location for use in the angle-domain Kirchhoff time migration?
      

      $$\hat{T}(theta) = \frac{T_0-z/V_0}{\cos{\theta}}$$ (9)

      $$\hat{x}(theta) = x_0 + V_0 \left(T_0-\frac{z}{V_0}\right) \tan{\theta}$$ (10)

      
      

Homework 4