1. The Noise Propagator for Laser Tomography Adaptive Optics Don Gavel NGAO Telecon October 8, 2008

2. 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 2

3. 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 rss- difference of an estimate to the zero noise case Another way, for linear systems, is to simply have zero atmospheric index fluctuation (r 0 =infinity) and assess the response to measurement noise. 3

4. LTAO is a linear system of equations We (LAOS, TSW, etc.) model LTAO as a linear system 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 noise- equivalent-angle equals ~35 nm phase error) 4

5. 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 Where P is the a-priori covariance of the solutions (e.g Kolmogorov spectrum and Cn2 profile), and N 0 is the assumed covariance of the measurement noise. The noise propagator is found by setting y = n, N = I mxm and solving for the covariance of In the case where the “signal” APA T is much greater than the assumed noise N 0, the noise propagator is nearly identity I mxm ! This is the case for LTAO: sqrt( N 0 ) is 35 nm compared to sqrt( APA T ) of several microns in a typical atmosphere. 5

6. 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(APA T ). 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. 6

7. About redundancy in measurements Redundant equations result in the matrix becoming singular. The matrix is nearly singular if N 0 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 N 0 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: (  1 -  2 )z max n layers 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 a- priori uncertainty to just a few layers (and using regularization). 7

8. Examples 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: 8 z x Telescope pupil Altitude Distribution of estimate covariance over volume In response to unit measurement noise 32 layers,3 guidestars: -30, 0, +30 arcsec Cn2 uniform over altitude Noise propagator back to WFS Telescope pupil position x Noise Propagator

9. More examples Case 2: 5 guidestar: -30,-15,0,15,30 arcsec, 32 layer uniform Cn2 Case 3: 5 guidestars and 3 layers 9 Distribution of estimate covariance over volume In response to unit measurement noise Noise propagator back to WFS

10. Last example: forcing ground-layer averaging Case 4: 3 guidestars -10,0,10 arcsec, 3 layers, Cn2=exp{-z/0.73km) 10 Distribution of estimate covariance over volume In response to unit measurement noise Noise propagator back to WFS x Telescope pupil Telescope pupil position x z Altitude Noise Propagator

11. 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 11