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Cosmic Signal Simulations
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Fast-Spherical Harmonic Transform:
We need to generate a large number of full sky realizations
starting from a known spherical harmonic coefficient spectrum.
T(pixel) = Sum on L, Sum on M (-L, L) a_{LM} Y_{LM}(pixel)
We have fast code that calculates the a_{LM} for each theory
and set of input parameters. The problem is to generate maps.
This requires calculating the (real) spherical harmonic functions'
numerical value at each pixel. We have been able to do this readily
out to L = 40 for a map with 6144 pixels at a reasonable speed.
Using this technique to get L = 360 and 100,000 pixels takes
days to compute.
Our goal is to generate a 4 million pixel maps
from a power spectrum coefficients covering L =1,2000
(roughly 8 million functions to be calculated, since M goes -L,L)
using a fast algorithm so that we can do this for thousands of
realizations.
This can be related to the inverse of the problem -
discussed below as spectral analysis -
of estimating the power spectrum from the map
except in spectral analysis we plan to average over M.
A recemt paper by Muciaccia, Natoli, and Vittorio
(
(astro-ph/9703084) )
develops a fast algorithm breaking the issue into two one dimensional
problems: (1) a fast Fourier transform over longitude and
(2) Legendre polynomial evaluation in latitude. The paper claims
success with the algorithm and indicate that the next step
is to implement it on parallel processors.
Another approach to fast spherical harmonic transform,
similar in approach to the fast Fourier transform
has been given by Healy, Rockmore, and Moore.
We have copies of this paper for student use.
"FFTs for the 2-Sphere - Improvements and Variations" (1996) &
"An FFT for the 2-sphere and Applications", Proc of ICASSP-96
Volume 3, pp. 1323-1326.
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Fast Pixelization Routines:
We have a set of pixelization routines for finding the pixel number
for a given direction and direction for a given
pixel number. We also have routines for finding nearest neighbors pixels.
The spherical harmonics have a high degree of symmetry and this
could be exploited with a suitable symmetry in the pixel scheme
and the basic routine is sufficiently simple for parallelization.
Sa pixelization scheme and making
those routines fast is a potential project.
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Sky Signal Simulation Framework:
This effort is to provide a simulated map of the sky for a given
input frequency and instrument antenna response.
It is the sum of the following seven signals extrapolated/interpolated
to the given frequency and the convolved with the instrument antenna
response.
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Cosmic Signal
- the problem just above
- the need for a fast spherical harmonic evaluation
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Foregrounds 1: Clusters of Galaxies
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Foregrounds 2: Extragalactic Sources
We have generated lists of extragalactic sources containing
their angular position and signal flux level. The issue is to
be able to convolve with antenna response function and place
signal level in the map.
S = Integral of G(theta,phi) Signal
where theta and phi are the angles of the pixel from the source
and G(theta,phi) is the antenna/instrument response.
There are 10,000 sources to be convolved and added into 4 million pixels
for each of the various antenna gain functions.
One approach past brute force is to search for only those
sources near to the pixel and thus of significant signal level.
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Foregrounds 3: Galactic Synchrotron Emission
We have template maps at a given frequency and resolution.
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Foregrounds 4: Galactic Free-Free Emission
We have only a model of this emission.
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Foregrounds 5: Galactic Dust Emission
Doug Finkbeiner has developed a high-quality, high-resolution
dust map based upon two satellite missions (IRAS & COBE).
This map exists at one wavelength and must be extrapolated.
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Foregrounds 6: Planets and Solar System Artifacts
Similar problem to the extragalactic sources but the planets and
large asteroids of interest move noticeably. There is the issue of
parallax and distance to the object of interest
Thus one needs to set up an approach
to handling the ephemeris for their positions.
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Measured Signal Simulation Framework:
Take the sky signal and simulate the experiment, i.e.
the response of the instrument during the data taking.
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Instrument Noise:
relatively easy Gaussian and 1/f
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Scan pattern on the sky
- need to generate quickly hundreds of
millions of observations - simulate scan and determine pizel number
and add in instrument performance. This is relative good candidate
for parallelizing. Some instruments have multiple independent elements
in the focal plane and the scan several hours apart is basically
independent of the previous scanning.
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Data Processing of Observations:
Take the (simulated or actual) signal and generate time-ordered,
cleaned and calibrated data.
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Large Data Set Handling
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Unpack data stream, check for errors
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Merge Pointing and relevant information to create complete data
stream
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Remove baseline and calibrate for amplitude (gain)
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Data Quality Checking and Flagging
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Map Generation from Observations:
Take the time-ordered, clean calibrated signal and generate a sky map.
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Large sparse matrix inversion or iteration procedure
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Maximum entropy approach
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Algorithm Development and Implementation
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Separate and recover cosmological signal, foregrounds, and
instrumental effects
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Fast-Spherical Harmonic Transform:
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Maxtix Manipulation and Eigenvalue/Eigenvector Problem
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Spectral Analysis
- e.g. direct spherical Fourier Transform:
one example is estimating spherical power spectrum from spherical map.
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11-dimensional parameter space searching : parallelization
