TODO


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- write cross section section in paper
- explain integral in paper
- develop signal generator
- develop high energy and multi-ring fitter
- write talk for Monday
- write a section on atmospheric backgrounds in paper
- check index of refraction calculation against tables in the paper
- write more tests for the solid angle calculation
    - use table from paper
- make sure normal direction is normalized
- speed up likelihood calculation
    - rotate and translate path points in path_init
    - create fast interp1d for equally spaced points
    - optimize with -O2
    - only calculate average time if mu_direct[i] is not too small and PMT is
      hit
    - add likelihood function which calculates total expected charge and only
      loops over hit PMTs
    - add fast likelihood which doesn't use a 2D lookup table
- add function to compute the Hessian via finite differences
- add function to calculate covariance matrix
- add function to compute the approximate volume in the likelihood space where
  the likelihood is greater than some value
- add output to likelihood
- add a set of seed points to start the likelihood
- minimize the likelihood in a do while loop until it converges to a single
  point
- once the first position is found, compute the Hough transform to look for
  other rings
- add rayleigh scattering to likelihood function
- add muon shower PDF? delta rays?
- calculate path length through acrylic?
- update estar table with energies above 1 GeV
- speed up normalize() and interp1d()
- add shower PDF to get_total_charge_approx()
- compute mean of starting points
- speed up time PDF calculation for fast likelihood