[ RadSafe ] Convolution, etc.

[Bill Prestwich]

Well, just for the record the general situation is as follows. When measuring a distribution with an instrument, the result obtained is a distorted
representation because of the instrumental response. Mathematically, the measured distribution is related to the physical distribution by an integral transform,
provided the instrument responds linearly. The kernel of the transform is the instrumental response function. Convolution is a special case when the response
function shape is invariant, so this does not strictly apply in the case of gamma-ray spectrometry. Unfolding is the process of inverting the integral transform
to obtain the physical distribution. In the special case of convolution, the inversion is deconvolution.
Chher,
Bill

Dale Boyce wrote:

> See "Radiation Detection and Measurement" Second Edition by Glenn Knoll page
> 672
>
> V. Computerized Spectrum Analysis
>     A. Deconvolution or Unfolding...
>
> The terms are used interchangeably, and technically are correct, although it
> can be confusing.  For instance when you convolve two Poisson distributions
> you wind up with another Poisson. Each point in one distribution is the
> amplitude of of an admixture of the other distribution centered at that
> point.
>
> Clear as mud, probably.  In gamma spectroscopy the peaks are not convolved
> in the same sense, but the admixture of different gamma lines result in a
> distribution of pulse heights while not a convolution of  intrinsic line
> widths it is a convolution of the system response.
>
> In neutron spectroscopy, especially in sorting out something like Bonner
> sphere data it is somewhat more closely analagous to the convolution of
> intrinsic response.
>
> I'm sure that I haven't gone into enough detail to clear anything up for
> people.  I'd recommend looking at Knoll's book. If anyone is interested, I
> can give some more detailed description of the convolution of functions.  A
> good example is the calculation of the activity of the daughter of a
> radioactive product of an irradiation.  One can either directly solve the
> differential equation, or one can convolve the ingrowth of the daughter with
> the ingrowth of the parent.
>
> This last example is probably a good example of why it is appropriate to use
> the terms deconvolution and unfolding.  If you were to take measurements of
> EOB activity of a daughter of a direct product of an irradiation as a
> function of length of irradiation, you would mathematically solve for the
> amplitudes of the various exponential terms by linear or non-linear least
> squares. The same technique you would use for gamma or neutron spectroscopy.
>
> Dale
>
> Dale E. Boyce
> daleboyce at charter.net
>
> ----- Original Message -----
> From: "John Jacobus" <crispy_bird at yahoo.com>
> To: <JPreisig at aol.com>; <radsafe at radlab.nl>
> Sent: Monday, March 20, 2006 8:19 AM
> Subject: Re: [ RadSafe ] A neutron spectrometry note
>
> > Dr. Preisig,
> > I have also heard the terms used interchangeably in
> > the past with regard to work that was done by the
> > Naval Research Laboratory on light emissions from
> > TLDs.  The signal is deconvoluted.  The spectrum is
> > unfolded.
> >
> > http://en.wikipedia.org/wiki/Deconvolution
> > and http://mathworld.wolfram.com/Deconvolution.html
> >
> > However, as noted in
> > http://rkb.home.cern.ch/rkb/AN16pp/node38.html
> > "If f(u) and fy (y) are known it may be possible to
> > solve the above equation for fx (x) analytically (
> > deconvolution or unfolding)."
> >
> > --- JPreisig at aol.com wrote:
> >
> >> Hmmmmmm,
> >>
> >>      This is from:   jpreisig at aol.com   .
> >>
> >>       Hey all of you,
> >>
> >>             Hope all is well where you are.
> >>
> >>       Lately, I've noticed some researchers/workers
> >> in neutron spectrometry
> >>       using the terms unfolding and deconvolution
> >> interchangably.  This is
> >>       not a correct thing to do.
> >>
> >>             Unfolding (and iterative unfolding) are
> >> described in the book
> >>       Accelerator Health Physics by Patterson and
> >> Thomas.  It has been
> >>       commonly used to analyze neutron spectrometry
> >> data.  There are
> >>       recent improvements in this technique, I
> >> think.
> >>
> >>             Deconvolution (and convolution) are
> >> described in the Mathematical
> >>       Physics book by Matthews and Walker.
> >> Deconvolution is regularly used in
> >>       the fields of geophysics and seismology.
> >> That's it.
> >>
> >>              Alpha creep, huh?  Wow.  Creep is also
> >> used in seismology to
> >>       describe rather slow movements of (earthquake)
> >> faults.
> >>
> >>               Take care.        Regards,   Joseph R.
> >> (Joe) Preisig, Ph.D.
> >>
> >>
> >
> > +++++++++++++++++++
> > "Those who corrupt the public mind are just as evil as those who steal
> > from the public purse."
> > Adlai Stevenson
> >
> > -- John
> > John Jacobus, MS
> > Certified Health Physicist
> > e-mail:  crispy_bird at yahoo.com
> >
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