[Ifeffit] Fit in R and k space

Matt Newville newville at cars.uchicago.edu
Tue Sep 20 11:01:31 CDT 2005

Hi Michel,

On Tue, 20 Sep 2005, Michel Schlegel wrote:

> Pardon my poor knowledge of EXAFS fitting and reliability of
> the results, but more generally, I wondered if EXAFS fitting
> in raw unfiltered k-space is always relevant.

I would not call your knowledge of EXAFS fitting "poor".  I 
think you've pretty much got every point exactly right.

For the original question (why prefer R-space v. k-space or even
Q-space for fitting), the difference is only important when
there are spectral content (k- and R-range) that you want to
ignore.  The content you'd want to ignore is most commonly the
high-R shells that you don't (yet?) have a model for. That is,
if your model could account for all spectral features, fitting
in k- and R-space would be equivalent.

For the more common case in which the model is more limited than
the data, using R-space makes it very easy to specify which
components to ignore.  Fitting in Q-space is nearly identical to
R-space, expect for an issue you point out later....

Being able to limit the spectral content in this way is entirely
to get a good measure of the fit statistics.  When fitting in
original k-space, you cannot say "fit the first shell and ignore
the fact that I'm not modelling the second shell".

As Michel points out, the limited k-range and the physics of
EXAFS does mean there is 'spectral bleeding', so that the
frequencies (R-values) for a single 'shell are not perfectly
sharp.  Looking at any plot of |chi(R)| it is pretty obvious
that the peaks are not delta functions, or even particularly
sharp.  As Michel put it "the contribution from non-modelled
distant shells would affect the structural parameters from the
modelled shells".  This is unavoidable, but is also generally a
small effect.  The best things to do are to be aware of this
possibility and try to model any significant further shells than
those you're really willing to say you've got right.

This is why we prefer R-space to Q-space and the older approach
of 'Fourier Filtering'.  It is often difficult and sometimes
impossible to really isolate the 'First Shell' from the 'Second
Shell', which can make it dangerous to compare isolated 'First
Shells' from systems with different 'bleeding' of higher shells.

Michel also wrote:

> Finally, another issue - for which I would not lay out my
> neck, though - is the noise. In EXAFS, the signal to noise
> increases with k, and of course fitting in the raw k space is
> another way of dealing conveniently with the noise -or
> relative uncertainty, as we may name it. However, the FT is a
> way of filtering out some of this noise, and so maximise the
> signal from a shell. Likewise, fitting in filtered q-space may
> yield more accurate results, because some of the noise is
> filtered out. However, I concur that somehow the fact that the
> uncertainty on the high-q part of the filtered contribution os
> greater than in the low q-part should be implemented somehow.

The FT does not actually filter out noise.  It can be used to
filter out the highest frequencies, and so the sharp spikey
spectra that oftens shows up in k-weighted chi(k) at high k
(where the noise is larger than the signal).  Sadly, this is the
part of the noise we care the least about: there is also 
noise with the same frequency (hopefully lower amplitude!) as
the signals we're analyziing.  That's not removed by ignoring 
the high frequency components.

In general, I would say that fitting in R-space is slightly
preferred over fitting in Q-space and much preferred over
fitting in k-space.  The only real reason it is much preferred
to k-space fitting is that you can systematically ignore shells
that you are not modelling.



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