[Ifeffit] background subtractions

[Matt Newville]

Hi, 
 
I figured I should comment on some of the issues on background
subtraction.  I agree with 99.9% of what Bruce said. This probably
represents the UWXAFS/Ifeffit party line, and so it's definitely
worth questioning.  Much about this topic is not completely known,
and I'm certainly open for suggestions...

Carlo mentioned an algorithm:
> I know that Steve Wasserman and Jeff Terry have a
> Mathematica background subtraction algorithm which allows
> you to back FT the region below Rbkg and subtract it from
> chi.  This seems to be roughly the same idea as the
> background fit but is less biased by the fitting.  I would
> appreciate your comments.

I don't know the details of Steve and Jeff's method, but it sounds
close to Autobk/spline().  Two other methods are also similar: 1)
the method of Cook and Sayers, which took mu(E) and smoothed it to
give the background function, and 2) the method of Li, Bridges, and
Booth (in their work on Atomic XAFS), where they subtracted a first
shell Feff calculation for chi(k) from the mu(k) and then filtered
out the high frequency parts to reveal structure in the background.
There may be other similar methods or variations as well.

These all share the idea that mu0(E) is defined by the lowest
frequency components of mu(E).  The subtleties are then 1) whether
you use the higher-frequency components in the fit for the
background (as most 'classic' background subtractions and Cook and
Sayers do), 2) whether you use an estimate of the first shell EXAFS,
and 3) how you control the flexibility of the spline.

Of course, the main issue with background subtraction in EXAFS is
that the background extends into the first shell, and may affect the
results for the first shell parameters.  This is a "well-known"
problem in EXAFS, though perhaps more widely assumed and accepted
than actually studied.  There's also the related idea that
background subtraction should leave a "pretty" chi(k) and chi(R).  
Again, I think this not as well-established as we might hope, but
is probably reasonable.  For now, I'll assume it's right....

If the background "leaks into" the first shell (as everyone knows it
does), then the first shell probably also extends to the low-R
"background region".  Certainly, if there were no overlap of the
first shell and background, then any old background subtraction
method will work just fine.

This suggests that we *should* take the first shell into account
when determining the background.  I heartedly agree with Bruce on
this (and most other things).  How this happens is not necessarily
obvious.  In Autobk/spline(), the first shell leakage to low-R can
be taken into account crudely by supplying a 'standard': a chi(k)
spectra with approximately the right first shell.  The idea is that
the first shell of the standard can be scaled in amplitude to match
the first shell of the data and then predict the leakage of the
first shell to low-R. Without a 'standard', Autobk/spline() simply
minimizes the low-R components, which ignores the first shell
leakage.  Li, Bridges, and Booth sort of boot-strapped the solution
(first using a simple background to get a fitted chi, then
subtracted that from mu to get a refined background).

Refining the background in Feffit/feffit() is roughly the same
concept, but allows a better mode for the first shell.  It also
allows us to assess correlations between background and structural
parameters, which seems useful.  Another important difference is
that Autobk/spline() use a very different FT range than
Feffit/feffit(), so low-R 'ugly bumps' that were reduced in
Autobk/spline() might appear with other FT settings.

The ability to assess the correlations of the background and
structural parameters was a main motivation for putting this feature
in Feffit.  Everyone "knew" that you can get "wrong answers" for the
first shell parameters because the background was "wrong", but it
was rare to measure how wrong the first shell parameters could be.  
By refining background parameters with structural parameters, we can
now measure it easily.  In general, the background parameters and
the structural parameters seem to be slightly correlated, with E0
and R1 influenced more than N and sigma2.

Some people are uncomfortable with the idea of measuring background
parameters twice, which is what running Autobk/spline() and then
refining the background with Feffit/feffit().  I don't see a problem
with this from an 'information theory' point of view.  There are a
fixed set of parameters set aside to account for the background.
Autobk/spline() measures them and the Feffit/feffit() refines that
measurement.  More simply, if a parameter can be measured, it is
allowed to be measured.

But that presents a good opportunity to bring up the bkg_cl()
command in Ifeffit.  By itself, bkg_cl() gives a pretty bad
background, though it does give a reasonably good and stable
normalization.  Importantly, it does no spline fit at all, and so
"uses up" no independent points (if you're hung up on counting such
things).  Doing this, and then including the background in feffit()
really does put the refinement of all variables in one fit.  Many
(possibly most) people greatly prefer refining everything at once.
It's easy to do (athena even has a pull-down menu and guesses the
central atom for you!!), and a good thing to try if you're worried
about the relation between background subtraction and fitted
parameters.

--Matt