Background

You are in charge of a top-security prison where scores of bad guys are kept.  Years ago some tunnels were dug but they were never filled in.  You've received information to the effect that the prisoners are trying to build new tunnels, or possibly connect the old ones.  In order to find out what's going on you talked CSM into making a cross-hole tomography survey under the prison.  You job is to analyze the data and draw quantitative conclusions as to how many tunnels there are and whether they are connected.

Cross Hole Tomography

In practice, we partition the space between the boreholes in cells. Assuming the rays don't bend (i.e.  linearized the problem) the forward operator is a matrix where the ij-th component is the length of the i-th ray in the j-th cell.

Geometry

20 receivers down a 10 meter deep borehole record a perfect pulse triggered by 20 sources in another borehole 10 meters further. The sources and receivers are equally spaced. For more information  see figure 1:
 

There are 13 cells in each direction, so 13 x 13 total cells.  Each shot is recorded at all receivers in the file tttrue.

Forward Operator

The forward operator is very simple: the travel time to each receiver is the
slownesses integrated over the path from source to receiver, which is a straight line in this case! There is no ray-bending. However, to make life even easier, the jacobian matrix associated with forward operator is already given as vector. You can partition the vector into a matrix of dimensions that agree with the geometry of the problem using partition.m. Partition takes a vector of length n and returns a matrix with n/m rows and m columns.

MATHELB

A mini matlab tutorial  can be downloaded to find all the functions in matlab you need for this assignment. Also, in m:\handouts\gp605\kvanwijk\hansen_toolbox you can find the toolbox by Per Christian Hansen with functions like L-curve.m and an svd function that projects out the null space: csvd.m.

• Estimate the variance in the data.  You can do this in several ways.  First you could use the l_curve.m function to estimate the optimal regularization parameter and then use the least squares model that results from this regularization parameter to predict times. Comparing the variance of these predicted times and data will give you an estimate of the variance of the noise.  However, this doesn't always work, so you should also look at the Chi-squared curve as a function of the number of singular values used.  If you find that this curve is very flat near the Chi-squared = 1 point, you may need to reduce the number of singular values used so that you don't overfit the data.   Finally, you can try smoothing the data in small windows.  By comparing the smoothed data to the raw data, you might be able to estimate the local noise variance.  Throughout, you'll want to use Hansen's compact SVD csvd.m.  This will project out the null space for you.

• Once you have an estimate of the data variance,  find the smoothest model that fits the data, according to the recipe for Optimally Truncated SVD (i.e.  Chi-squared =1 or the data fit does not improve). Use your estimate of the variance  in the definition of  Chi-squared.  You don't have to use truncated SVD.  You could also use second difference regularization if you prefer.   But the goal is the find the smoothest model that explains the data using your own estimate of the data uncertainties.

• To help you understand what features are well and poorly resolved, plot one of the well resolved model singular vectors and one of the poorly resolved singular vectors.  All models should be plotted as either 2D contour plots (13 x 13) or as 3D perspective plots.  Be sure to explain what you see in these eigenvector plots.  Also be sure to label what you infer to be the orientation of the sources and receivers:  you can figure this out from your eigenvector plots.

• Show an illumination plot.  This is the total length of rays in each cell.  Again this should be either a contour plot or a perspective plot, not a 1D plot.

• Show plots of the resolution matrix and the covariance matrix associated with your best estimate of the true model.

• Try to say something quantitative about the range of models that fit the data.

• Finally, don't forget your prime objective, which is to make a quantitative interpretation on the tunnels using the data.

• When you've done all this, you'll need to write up your results in a coherent fashion.