Transcript 621 Noise Propagator for Laser Tomography AO

The Noise Propagator for Laser
Tomography Adaptive Optics
Don Gavel
NGAO Telecon
October 8, 2008
Noise Propagator Issue
• Simulations are showing that the “law of averages” is not working as
expected with multiple laser guidestars
• Dividing a fixed amount of laser power over a larger number of
guidestars results in an increase in the noise in the solution
• Increasing number of guidestars with fixed laser power per guidestar
results in no decrease in noise in the solution
• Law of averages (sqrt(n) noise reduction) only holds if guidestars
are overlapping or very close to overlapping
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Analysis of Noise Propagator
• An analytic approach was taken to understand this problem
independently of the numerical simulations
• The answer is a basic consequence of linear algebra
Definition
• Noise propagator is the ratio of the standard deviation of the noise
in the estimate of wavefront along the direction of a guidestar to the
standard deviation in the noise of the measurement.
Determination
• One way to determine the noise propagator is to form the rssdifference of an estimate to the zero noise case
• Another way, for linear systems, is to simply have zero atmospheric
index fluctuation (r0=infinity) and assess the response to
measurement noise.
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LTAO is a linear system of equations
• We (LAOS, TSW, etc.) model LTAO as a linear system
y  Ax  n
• Where x is a vector of all the delta-indices of the “voxels” in the
atmospheric volume (n_subaps x n_layers)
• y is the vector of all the phase measurements (n_subaps x
n_guidestars)
• A is the linear relation between them – representing the
accumulation of indices times dz to get accumulated optical path
distance
• n is the noise in the measurement.
• Note: I’m skipping the phase-to-slope and slope-to-phase
operations. These operations are also assumed linear and don’t
change the nature of the argument. (for example 50 mas noiseequivalent-angle equals ~35 nm phase error)
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Minimum variance solution and
noise propagator
• The LTAO problem is underdetermined because n, the number of
unknowns (voxels), exceeds m, the number of measurements
• Aside: When n>m, the difference n-m is the number of blind modes
• The minimum variance solution is

xˆ  PA T APA T  N 0

1
y
Where P is the a-priori covariance of the solutions (e.g Kolmogorov
spectrum and Cn2 profile), and N0 is the assumed covariance of the
measurement noise.
• The noise propagator is found by setting y = n, N=Imxm and solving
for the covariance of Axˆ
1
1
T
T
T
T
T
~
~
Axx A  APA APA  N 0  APA  N 0  APA T
• In the case where the “signal” APAT is much greater than the
assumed noise N0 , the noise propagator is nearly identity Imxm !
• This is the case for LTAO: sqrt(N0) is 35 nm compared to sqrt(APAT)
of several microns in a typical atmosphere.
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Noise propagator and the law of
averages
• Why doesn’t the noise propagator follow a law of averages when
more guidestars are added?
• Because although more equations are added, more unknowns are
added too. As long as n m and the equations are non-redundant
and the solution is unconstrained, the noise propagator is identity.
• Overlapping guidestars introduces redundant equations: i.e. more
equations without adding more unknowns.
• Law of averages starts to apply when n’<m, where n’ = the number
of degrees of freedom you can measure = m – the redundancy of
the measurements – the number of observable a-priori constraints
on the solution = rank(APAT). Then, noise propagator goes as
sqrt(n’/m)
• This is consistent with what was observed in the LAOS runs done by
Chris Neyman (e-mail of Aug 26) and consistent with subsequent
example runs explained in the next few slides.
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About redundancy in measurements
• Redundant equations result in the matrix APA T becoming singular.
The matrix APA T  N 0  is nearly singular if N0 is relatively small.
• Numerical inversion generates large gains in the reconstructor in its
valiant attempt to remain consistent with all measurements. It is better
to use a pseudo-inverse using the singular value decomposition with
thresholds on singular values (“regularization”). This allows
redundancies to be suppressed without causing large gains – and
brings back the law of averages!
• Increasing N0 to keep the matrix full rank (also a form of
regularization) has roughly the same effect.
• Redundancy in LTAO happens when a newly added guidestar does
not improve the resolution of layers: (q1-q2)zmax < d. Or, somewhat
equivalently, nGS > nlayers
• We can force the law of averages through choices in the model: e.g.
limiting number of layers or assuming a “severe” Cn2 profile limiting apriori uncertainty to just a few layers (and using regularization).
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Examples
Noise Propagator
• I ran a couple cases independent of LAOS and TSW, just using
matrix arithmetic in IDL to illustrate these points
• Case runs: 2 dimensional (x and z): 3 to 5 guidestars over 30
arcsec, 32 subapertures across x,3 to 32 layers over z, various Cn2
profiles. Regularization is SVD pseudo inverse thresholded at 1%
(ratio of singular value to largest singular value).
• Case 1: 32 layers,3 guidestars: -30, 0, +30 arcsec Cn2 uniform over altitude
Altitude
z
Telescope pupil
x
Telescope pupil position x
Distribution of estimate covariance over volume Noise propagator back
In response to unit measurement noise
to WFS
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More examples
• Case 2: 5 guidestar: -30,-15,0,15,30 arcsec, 32 layer uniform Cn2
Distribution of estimate covariance over volume
In response to unit measurement noise
Noise propagator back to WFS
• Case 3: 5 guidestars and 3 layers
3 5  0.775
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Last example:
forcing ground-layer averaging
Noise Propagator
• Case 4: 3 guidestars -10,0,10 arcsec, 3 layers, Cn2=exp{-z/0.73km)
Altitude
z
Telescope pupil
x
Distribution of estimate covariance over volume
In response to unit measurement noise
1 3  0.577
Telescope pupil position x
Noise propagator back to
WFS
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Conclusion
• Adding more guidestars can serve one of two purposes
1. Introducing denser sampling of the atmosphere
In which case, noise propagator remains unity (increasing noise with
decreasing power per guidestar)
2. Introducing measurement redundancy
In which case, noise propagator follows law of averages
• But it can’t do both
• Noise regularization is essential
– To prevent high reconstructor gains
– To recognize and apply law of averages to redundancy
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