An Algorithm for Forward Fitting
The intent of the Forward-Fitting process is to provide a parametric
model which is a best fit to the observed count rate
profiles. Ideally, this can be done without making maps of any kind.
The idea is to find a model with the fewest parameters for a given
c2 or C-statistic giving the "goodness-of-fit" of the computed
model modulation profile to the observed profile. The model can be a
superposition of circular Gaussians or a superposition of more
complicated basis functions.
We restrict ourselves initially to models with two circular Gaussians.
Such a model has 8 free parameters:
- (x1,y1),(x2,y2) for the
centroids of the sources,
- F1, F2, for their fluxes,
- and w1, w2
for their widths.
It is easy to show that the signal from a single
sub-collimator cannot distinguish the difference between a point
source and a circular Gaussian, so with little loss of generality,
we can start with the case of two point sources. (Later, the
algorithm distinguishes between Gaussians and point sources by
comparing successive detector profiles.)
Our strategy proceeds as follows:
- Find the centroid of the source. This can be done in various ways:
- Find the best correlation between the single-revolution
signal f(roll_angle) from the coarsest collimator and a function
of the form:
This is very fast and the values of A and B lead to the azimuth
and phase of the flare source.
- OR: Make a coarse back-projection map using the two coarsest
subcollimators, and find the peak of the map.
- Make calibrated event lists for all collimators using the flare
centroid for map center.
- Using the profiles of the phase_map_ctr for each collimator,
regularize the count rate and make corrections for gridtran,
livetime and modamp variations. (Regularization is required for
successful interpretation of fourier spectra and for
( hsi_rebin.pro, hsi_regularize.pro)
- Compute the power spectrum AS(i)( n), i=1-9
of each of the 9 regularized profiles.
- Determine the location of significant peaks in the appropriate
ranges of each AS(i)(
Look for double peaks, starting with the
coarsest collimators and working towards the finest. At a given
pitch, source separations greater than that pitch will produce a
double peak. This will usually be seen in signals from 2 or 3
with pitches spanning the separation size. The
separation of the sources can then be directly determined from the
power-spectral maxima ( n1,
- For collimators with | n1
significantly greater than 0.
there will be a slow amplitude modulation ("beating") of the signal.
The beat signal is best understood as a long-wavelength waveform in
the U-V plane, which modulates the visibilities. (See Figure)
The beat waveform, like the parent visibility, has amplitude, phase
and direction. The direction of this beat waveform
is the same as the direction of the separation vector for the
source pair. In general this direction is not readily found
from the power spectra AS(i)
( n), because the signal has been
regularized from the point of view of the flare centroid, which is
almost always a different direction than the separation vector,
Given the period | n1- n2| and the mean frequency
n2)/2 one may
find the separation vector by using the Hilbert transform and
phase shifting. But
first one must filter the spectrum.
For two circular Gaussian sources with a radial
separation vector, g(k) will be a low-frequency sinusoid.
- Fourier filter the signal by using only spatial
frequencies whose spectral power exceeds
- Compute the complex signal
fh(k) = f(k) + i HILBERT(f), ()
where HILBERT is a function in the IDL 5.x library.
- Then eliminate the high frequencies by a phase shift:
g(k) = fh(k) X ei n0k
where the shift is given by the peak frequency of the
If the beat amplitude is large (i.e. nearly equal sources), and
the separation happens to be radial, the amplitude and phase can
be found immediately from g(k).
Since in general the separation is nonradial, which causes the
the beat signal to be irregularly periodic, it
must be found for a complete set of regularizations in different
- Reduce the size of the beat amplitude arrays by rebinning
by a factor of 2-4.
- Find the phase of the "beat" signal by re-regularizing
g(k) in a range of directions (every 10°
or so). The signal with the
highest "spectral purity" gives the direction of
the separation vector.
( uv_project.pro, spectral_width.pro)
- Alternatively, exploit the fact that the beat amplitude
is regularly sampled in one dimension of the UV
plane and irregularly sampled in the perpendicular
dimension: Compute a 2-D Fourier transform of g(k)
and direct FT's, and use 2-D power spectra to find the
beat amplitude and phase.
- The separation, direction and centroid give
- The total flux on the coarsest scales provides the sum F1+F2.
- Determine the amplitude of each source distinuished in the
Fourier power spectrum.
- The collimator-to-collimator ratios of the spectral power in
each source give the ratios of a Gaussian function of the
widths wi and fluxes Fi, from which the
final fluxes and widths may be determined.
Once the parameters have been found, the model count rate can be compared
the observed one, and the c2 or C-statistic can be computed.
are sufficiently small, we are done. If not, small adjustments of the
parameters can be made in a search for a better fit. If this does not
lead to a satisfactory statistic, it will be necessary to seach for a third
source or more complicated sources.