% Lab #2 for Geophysical Inverse Theory ESS523, Fall 2005, Univ of WA
% by Andy Ganse, course TA
% course professor:  Ken Creager
% 11 Oct 2005
% -------------------------------------------------------------------------

This week's lab is an extension of the parameter estimation we did at the
very end of reviewtut.m, where we fit a parabola to our synthetically
homemade noisy data points for the ballistic problem (e.g. Example 1.1 in
the Aster et al book).

Now that we have that fit we'd like to quantitatively describe the quality
of that fit, using the statistical tools we learned about like p-values and
covariance.

This lab HW basically works thru the book's section 2.2, concentrating on
examples 2.1 and 2.2, which are both contained within the example 2.1 Matlab
script on the CD accompanying the book.

TO DO:
------
Adapt the Matlab script from the CD for the book's example 2.1 to your
  script for last week's lab.  Ie, last week you chose your own "true" model
  params and your own constant stdev for the noise you added to your synth data.
  So each of your data plots and model estimates from last week will look
  different from each other's and from the book's.  But otherwise the approach
  is the same.  You're mainly just cutting and pasting chunks of the book's
  Matlab script into your own.

That example 2.1 script has a lot in it.  Don't worry about the Monte Carlo
  parts for now.  But do add the following to your own problem from the end
  of reviewtut.m last week:
    - the model estimates (already done last time)
    - the covariance matrix of your model estimates
    - the p-value for your data misfit, and interpret it in terms of one of
      the three general cases on pp19-20.
      (don't forget that your own # deg freedom here (=N-M) will differ from
      the book's and possibly from each other depending how many data points
      you made)
    - calc and plot the 95% confidence ellipses for your estimated model params
      The Matlab script can plot all three, and that's fine if you like, but
      you only need to turn in the one for m2,m3.  (init vel & gravity)
      Add to that plot (by hand is fine) the location of your model estimate
      and the true one.
    - mark (just by hand is fine) the extent of the ellipse as projected down
      onto the m2 and and m3 axes.  Then compute and mark (again by hand is
      fine) the 95% conf interval for m2 & m3 individually by ignoring the
      covariances (off-diag components of the cov matrix), as per eqn 2.30 in
      the book.  Which interval is larger?  So if you ignore the off-diagonal
      info in the covariance matrix, are you over- or under-estimating the
      uncertainty in (say) the gravity parameter?
    - so you're turning in printouts of those numbers and plots.

Lastly, two extra credit questions :
    - The book's script calculates:   Gw \ yw
      That "\" is a handy LS operator in Matlab once you see what it does:
      verify that the Gw\yw = ginv*yw
    - The script used a Matlab function called "norm()".  What does it do?

Okay, the above looks like a lot to do, but hopefully it's actually not too
bad since most of it is already in the book's Matlab script, and you're just
cutting and pasting...

Cheers!