GP 605 -- Midterm 2001

due November 15

• Implement Tikhonov regularized least squares in Scilab. This involves solving problems of the form
  
  $$\min _x \left( \Vert Ax -d\Vert^2 + \lambda \Vert Rx\Vert^2\right).$$
  
  
  Incorporate two different regularization operators $R$, the identity, and a discrete second difference.
• Implement the L-curve method. This involves solving the above equation for a range of $\lambda$ values and plotting the logarithm of $\Vert Ax -d\Vert$ against the logarithm of $\Vert Rx\Vert$. This curve often takes the shape of an ``L'', in which case the optimal value of $\lambda$ is the knee in the curve.
• For a data/jacobian matrix that I will make available, pick (manually) the knee of the L-curve in order to obtain the optimal value of the regularization parameter $\lambda$.
• With this optimal value of $\lambda$, solve the regularized optimization problem to obtain the regularized model $x_\lambda$.
• Your first estimate of the data uncertainties $\sigma$ is then $\sigma^2 = {\rm variance}(A x_\lambda - d)$.
• Now estimate the data variance without computing a model by smoothing the data. You can do this using the same tools you developed above by solving
  
  $$\min _\mu \left( \Vert\mu - d\Vert^2 + \lambda \Vert R\mu\Vert^2\right)$$
  
  
  where $\mu = A x_T$. In this formulation, your estimate of the data errors $\sigma^2$ is the variance of $\mu _\lambda -d$.

Click here for an ascii file containing your data. These are in milliseconds. The geometry of the model is as follows. The top of the model is at z=0. The bottom is at z=40 meters. There are 100 receivers, evenly spaced, starting at 0.4 meters. Take the source to be at (0,0) so the rays propagate vertically. It's up to you to figure out how many layers to use.

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