Adaptive Optics for Extremely Large Telescopes III

ANALYSIS OF THE FROZEN FLOW ASSUMPTION USING GEMS TELEMETRY DATA

Angela Cortés1,a

, Alexander Rudy2,b

, Benoit Neichel3, Lisa Poyneer

4, Mark Ammons

4, and Andrés

Guesalaga1

1Pontificia Universidad Católica de Chile, Santiago, Chile

2University of California Santa Cruz

3Gemini Observatory Southern Operations Center, La Serena, Chile

4Lawrence Livermore National Laboratory

Abstract.We use telemetry data from the Gemini south multi-conjugate adaptive optics system (GeMS) to

study the validity of the frozen Flow hypothesis using two types of algorithms: i) the spatio-temporal cross-

correlations of the wave-front sensor (WFS) measurements; and ii) the Predictive Fourier Control (PFC)

framework. The pros and cons of each technique are identified as well as their ability to determine the number of

layers present and the associated velocities. Their potential use to determine the altitude, i.e. turbulence profiler, is

also addressed. Examples derived from simulations and on-sky data are presented.

1 Introduction

In the past decade, Adaptive Optics (AO) instruments have greatly improved the performance and

utility of ground based telescopes [1]. As telescope aperture size has increased, the effect of

atmospheric turbulence quickly becomes the limiting factor for telescope resolution and sensitivity. AO

systems correct for the deviations in the optical path due to atmospheric turbulence, by applying a

phase compensator in real time. The required compensation speed is set by the Greenwood frequency,

the rate at which phase becomes uncorrelated, . As AO systems become larger, the time-delay

between the measurement of the uncorrected phase and the application of a phase compensation

becomes the dominant source of error. Predictive Control can eliminate this “servo-lag” error, and PSF

reconstruction can compensate for this error through post-processing. Eliminating the time-lag error

requires an understanding of the way that atmospheric turbulence crosses a telescope aperture.

One of the main hypotheses used to study turbulence for adaptive optics systems is Taylor’s frozen

flow hypothesis. The frozen flow hypothesis has three main assumptions [2]:

1. The atmospheric turbulence is located in horizontal layers, independent of each other.

2. Each layer moves with a constant velocity.

3. The time required for a layer to move across the telescope aperture is too short to permit significant

changes of the turbulent pattern of the layer.

This work is focused on the evaluation of these assumptions, using two different methods to determine

the wind velocity and direction. If the frozen flow hypothesis holds, a predictive AO system must

a e-mail: ajcortes@uc.cl

b e-mail: arrudy@ucsc.edu

Third AO4ELT Conference - Adaptive Optics for Extremely Large TelescopesFlorence, Italy. May 2013ISBN: 978-88-908876-0-4DOI: 10.12839/AO4ELT3.13364

Adaptive Optics for Extremely Large Telescopes III

simply translate the observed turbulence across the telescope aperture to compensate for the phase error

at the next time step.

2 GeMS

GeMS is the Gemini South AO MCAO facility instrument. It provides a uniform, diffraction-limited

image over a field of view of 60", using five laser guide stars, observed by five Shack Hartmann (SH)

WFS, with 16x16 subapertures each, for a total of 204 valid subapertures [3]. The corrections are done

by three deformable mirrors (DMs), conjugated to different altitudes, one at the ground layer, and the

other two at 4.5 and 9 km. The three DMs have 917 actuators total, however, only 684 are active, the

rest are shadowed by the geometry of the system.

The system can operate at up to 800 Hz. However, the control rate is usually limited by the power of

the sodium lasers, and the conditions of the atmospheric sodium layer, which determine the brightness

of the artificial guide stars.

To examine the temporal behavior of the atmosphere, and to test the wind identification methods

described here, telemetry data from GeMS was saved during 30-120s intervals on several different

nights.

GeMS operates in closed loop. However, both methods require open-loop wavefront measurements,

and so use “pseudo-open-loop data” (POL), phase aberrations that have been reconstructed using the

GeMS interaction matrix, the shape of the DMs, and the residual errors measured with the WFS. GeMS

data shows “fratricide” effects, where the Rayleigh scattering of each laser guide star is visible in the

other wavefront sensors. This effect was masked out in the POL WFS telemetry.

In the following sections, several different telemetry cases were examined. The telemetry was taken on

different nights, in different seasons, and shows winds at a variety of velocities and altitudes.

Representative examples were chosen to show the power of these wind identification methods, they do

however not represent the full range of conditions that were observed.

3 Wind Profiler

The Wind Profiler uses a time-delayed cross-correlation, presented in a paper before [4]. The technique

was extended from Wang [5] and Schoeck [6]. Wind Profiling was developed from the SLODAR

method [7], which uses wavefront distortions measured with the same telescope on two close-by stars.

Due to the geometry of the two stars relative to the telescope (optical triangulation), the correlation

between the slopes measured on the two stars, decreases with increasing altitude. Different layers of the

turbulence distribution can then be identified. With GeMS, the Wind Profiler determines the altitudes

of the layers using the laser guide stars. The Wind Profiler is using the idea of the original SLODAR

for the optical triangulation, using laser guide stars instead of the natural and temporal cross correlation

rather than the spatial.

3.1 Theory

Temporal cross-correlation identifies the translation of turbulence across the aperture. At t=0 the

peaks gives the altitude of the turbulence, the center corresponds to the ground layer, and moving away

from the center it will give the altitude of higher layers. Performing it on time-delayed telemetry will

give us information about the turbulence over time. Tracking correlation peaks will allow us to get the

speed and the direction of the wind.

The time-delayed cross correlation used is the combination of two WFSs, A and B:

(1)

where contains the X and Y slopes of the WFS-A in subaperture at time , and are

relative subaperture displacements in the WFS grid. The time delay of the measurement, , is a

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multiple of the acquisition time that in our case ranges from 1.25ms to 0.4 s. is the number

of overlapping illuminated subapertures for offset , represents the average over the time

series, and denotes summation over all valid overlapping illuminated subapertures.

We then apply a 2D deconvolution to each time delayed cross-correlation of the WFS, using the

autcorrelation of each WFS applying Fast Fourier Transform (FFT), that is:

(2)

where A is the average of the autocorrelations of WFS A and B.

The Frozen Flow Hypothesis (FFH) was examined before by Schoeck [6] using only one WFS, and

therefore the autocorrelation of the WFS data. Performing the autocorrelation for one or for multilayers

will give a value that is constant with the time. In this work, the cross-correlation between two different

WFS can be used. As GeMS has 5 WFSs, we can get different baselines, and add common baselines

(the ones that have the same orientation) to increase the measured signal. Performing this process with

data coming from 2 different WFSs will provide information of the velocity and direction of wind at

different altitudes, and is not assuming Kolmogorov or anything on the structure of the atmosphere.

3.2 Results from sky

The analysed data was a selection of the most representative cases, and the one that also have better

correlation peaks to perform the tracking.

The data can be divided into 4 important cases: data with strong dome turbulence, data with mainly

ground-layer turbulence, data with mid-altitude turbulence and data with high-altitude turbulence. Here

we are going to display one example of each case to be analysed.

3.2.1 Dome and Ground Layer Turbulence

For this case, we have two very good examples that illustrate how the turbulence behaves at lower

altitudes. The first case is when the turbulence is located inside the dome, when it is stuck over the

telescope and the peak of the correlation it will be at the central point. For the second case, when we

have turbulence at a ground layer, but mainly outside the dome, the turbulence will move with the

wind. In this case, the peak will follow the translation of the turbulence and match the wind direction.

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Figure 1. Left column: correlation signal at t=0, middle: t=12 s, right: decay of the central correlation peak,

dashed line: decay of the correlation strength and the continuous line is the real measurement. (Top) Dome

Turbulence: the correlation peak remains in the center with the time. (Bottom) Ground Layer Turbulence: the

correlation peak moves in time. The decay rate corresponds to a wind speed of 8.8 m/s.

Both scenarios are shown in Figure1, where the correlation peak starts in the central point, which

suggests low-altitude turbulence. The case that is displayed at the top shows how the main turbulence

remains in the central point with the time, as opposed to the case displayed at the bottom, which moves

down. This case is one of the most important ones, as we found that the decay rate of the correlation

peak is constant. For the case at the top, we can use this fact. Measuring the decay of the central point,

will give us the proportion of turbulence that is located inside the dome (see top right of Fig.1).

3.2.2 Mid and High Altitude Turbulence

When the peak of the correlation is not at the center, implies that the turbulence is at higher altitude, the

distance of the center in pixels will give us the altitude of the layer.

Figure 2: Left column: correlation signal at t=0, middle: t=12 s, right: decay of the central correlation peak,

dashed line: decay of the correlation strength and the continuous line is the real measurement. (Top) Mid altitude

turbulence. The peak starts three pixels away from the center, and moves down-right, the blue line shows the

alignment direction of the two WFS, the decay rate gives a wind speed of 10 m/s. (Bottom) High altitude

turbulence. Here one peak starts at 5 pixels from the center, and another at the center, the decay rate gives a wind

speed of 17.7 m/s.

Figure 2, shows cases when we have turbulence at higher altitudes, the peaks starts away from the

centre and moves given us the direction of the wind. As was mentioned before, the decay rate of the

correlation intensity is constant and will help to estimate the dome turbulence as was mentioned before.

4 Fourier Wind Identification

We use the Fourier Wind Identification (FWI) technique applied to GeMS telemetry data in order to

measure atmospheric frozen flow. FWI was developed as part of a Predictive Fourier Control

framework for Adaptive Optics Systems [8] that aims to minimize temporal errors, including servo-lag,

which manifest as a mis-estimation of the current atmospheric phase.

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4.1 Description of the Fourier Wind Identification Technique

The Fourier Wind Identification technique uses the Fourier basis set. Although the GeMS reconstructor

does not use the Fourier basis set, we have applied our technique here to reconstructed open loop phase

measurements from GeMS. These measurements are constructed using the telemetry streams from the

GeMS instrument in real time.

The process used to construct the open loop phase is documented in Poyneer, van Dam and Véran

2009. First, the open-loop phase is converted to the Fourier basis set in its spatial dimensions. The

Fourier basis set provides a convenient way to examine translating frozen-flow, as the individual

Fourier modes translate quite simply across an aperture.

The temporal power spectral density (PSD) of each Fourier mode is then estimated from the data.

Individual Fourier modes are split into segments of length S, which are windowed to emphasize the

middle of the segment, and which overlap with neighboring segments. The overall frame rate, sets

the maximum estimated temporal frequency at . The length of the segments sets the frequency

sampling spacing at . For the GeMS telemetry data, we found that an interval length of 2048

worked well. We used the full length of each telemetry set (between 2-5 seconds), using half

overlapped segments, to estimate the PSD. The half overlapped segments increase the signal-to-noise

in the resulting PSD (for a more detailed discussion of this method, see [9]).

1 0

0 5

0 0

0 5

Fit

Pea

ks

1 0

0 5

0 0

0 5

Found

Pea

ks

1 0 0 5 0 0 0 5

fx m 1

1 0

0 5

0 0

0 5

Theo

ry

[-25.2,0.3], 63.2%

1 0 0 5 0 0 0 5

fx m 1

[-17.8,7.7], 51.3%

252015105

0510152025

f tH

z

Not Possible

Possible

Match

252015105

0510152025

f tH

z

1 0 0 5 0 0 0 5

fx m 1

[-4.8,9.7], 84.1%

GeMS during 11109092452_pol _wf s2 Analysis on 2013-10-01 with d68e8c07

Figure 4: The fit of peaks identified in PSDs to frozen

flow layers. Each column shows a different layer. The

top panel shows the identified peaks. The bottom

panel shows the theoretical peaks that would exist in a

perfect detection. The middle panel shows which

peaks from the data (green) match the theoretical

values, and which ones weren’t found (red). Some

peaks cannot be fit, as they are too close to 0 Hz, and

are shown in white.

Figure 3: A power spectral density for a single Fourier

mode found in GeMS telemetry. The PSD is on a log

scale, and is for the k=13, l=5 Fourier mode. The PSD

shows a clear peak at 0 Hz, corresponding to the

steady-state errors in the system, and a peak at 5Hz,

indicating that this Fourier mode translates with a

frequency of 5 Hz. There is also a peak further out at

12Hz. These two peaks correspond to translations of

this Fourier mode at two different velocities.

40 30 20 10 0 10 20 30 40

vx (m s)

40

30

20

10

0

10

20

30

40

v y(m

s)

Identified Wind Layers for 6 8s bins

0%

20%

40%

60%

80%

100%

Wind Liklihood

Figure 5: A wind likelihood map produced

from GeMS telemetry data. A single layer of

wind with a likelihood > 80% is apparent at 7

m/s and 10 m/s.

Figure 6: The time-evolution of the wind-layer

detection in GeMS telemetry using 6.8s bins. The

size of the circle represents the strength of the metric.

The wind layer detection is stable for the duration of

the telemetry.

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Once we have created a PSD for each Fourier mode, we can look for temporal peaks which are

indicative of frozen flow. In Fourier space, temporal peaks will appear with frequencies given by

(3)

We model each peak as an oscillation at the temporal frequency , with an added white noise

broadening term [1]. A sample PSD, with fitted peaks is shown in Figure 3. Each peak corresponds to a

potential match to Equation. The peaks that appear close to are eliminated as they correspond to

the slowly varying steady-state errors found in every system. Peaks are fit for each spatial Fourier

mode separately.

Using the identified peaks from all the Fourier modes, the FWI algorithm works backwards through

Equation 3. The frequencies in Equation 3, when shown on an grid, appear as a plane in

frequency space, with the (piston) term always at . Figure 4 shows the process of

matching found peaks in a PSD to a theoretical plane in Fourier space. FWI then produces a metric in

velocity space that shows the percentage of matched peaks at each velocity. Areas with high metric

scores are velocities at which frozen flow has been detected.

4.2 FWI Performance on GeMS Telemetry Data

The FWI method was applied to the four test cases described in Section 2. In each case, frozen flow

layers were identified.

Figure 6 shows the results from a simple case that demonstrates the identification of a single layer. The

single layer is travelling at and is easily identified in the wind velocity metric. In

order to test the extent to which this wind velocity vector was constant for the duration of the telemetry

data, we analyzed the data using only single intervals of 2048 time steps (~ 6 seconds each) and looked

at the progression of the wind vector. As shown in Figure 6, the strongest identified layer does not

appear to shift appreciably. The weaker layers show a much noisier behavior.

Figures 7 and 8 show two more complex cases. Figure 7 shows a case where two distinct layers are

identified, with almost a 90º angle between their wind vectors. In this case, it is clear that the FWI

algorithm has identified two independent layers. However, since the current implementation of the

algorithm does not do any tomography, the frozen flow layers cannot be physically separated from this

information alone.

Figure 7: A wind likelihood map constructed from

GeMS telemetry showing two frozen flow layers with an

almost 90º offset, both traveling near 9 m/s. The two

layers are likely at different altitudes, but are both

detected strongly, suggesting that two frozen-flow layers

were present at GeMS during this data set.

Figure 8: A wind likelihood map constructed from GeMS

telemetry showing one strong detection of a frozen flow

layer at 11 m/s and a weaker detection at 25 m/s. It is

important to note that the weaker detection does not

necessarily imply a weaker contribution to the total

atmospheric turbulence.

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Figure 8 also shows a complex case, where a single strong layer is identified at 11 m/s and a secondary,

weaker layer is shown travelling at 25 m/s. In this case, it is interesting to note that the FWI method

only detects layer strength in terms of the number of peaks identified in the PSD, suggesting that the

weaker layer is a poor match to the PSDs, and not that it contains less atmospheric power.

5 Comparison of the FWI method and the Cross-Correlation Method

Figure 9 shows the same set of telemetry analyzed by the cross-correlation method described in Section

B and the FWI method described in Section 4. The two methods agree, in that that they both measure a

frozen flow layer travelling at 17.5 m/s, and a slower moving ground layer, traveling at 6 m/s.

The Cross-Correlation method does a good job of identifying layer altitudes, and easily separates the

dome seeing layer (represented by the peak at the center of the left and center panels in Figure 9) from

the faster high altitude layer. However, the method used to track the peaks in the cross-correlation data

has trouble identifying them.

The FWI method easily identifies the ground layer with a very strong match, and identifies a weakly

localized high altitude layer, travelling at 17.5 m/s–a very good match to the cross-correlation data.

It is clear that the FWI method can identify layers which don’t appear to have much correlated

turbulence strength, and that the wind layer likelihood metric used by FWI does not correspond to

turbulence strength. In contrast, the cross-correlation method identifies the layers with the most

turbulent strength readily, but has trouble tracking consistent, but weaker layers.

6 Analysis and Conclusions

The Wind Profiler method, using spatial-temporal correlations, and the Fourier Wind Identification

method are able to detect Frozen Flow in telemetry data from GeMS. Wind identification was

compared on the same telemetry data, and found to be consistent between the two methods. Neither

method assumes a Kolmogorov Turbulence power spectrum.

Using the Fourier technique, non-Frozen flow turbulence was automatically rejected suggesting that

frozen flow turbulence was easily detected in all telemetry cases. The FWI method made no attempt to

estimate the altitude of identified turbulence.

The Wind Profiler method found that frozen-flow turbulence had a melting rate that is proportional to

the wind speed. As well, the Wind Profiling method can separate turbulence by altitude, and can detect

Figure 9: Comparison of the cross-correlation measurement of the wind velocity and the FWI analysis,

showing a wind layer likelihood map. (Left and center) Cross-correlation frames showing a fast moving peak

and a slow moving peak. (Right) A FWI likelihood map which shows a fast moving peak and a slow moving

peak. Both methods show a fast moving component at 17.5 m/s and a slower moving component at 7 m/s. The

methods detect wind velocities consistently, although each method detects wind velocities with a different

degree of certainty.

17.7 m/s peak 7 m/s peak

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dome seeing (static terms). The method is weakened by difficulty of automatically tracking correlation

peaks.

The two wind identification methods here proved to be self-consistent and demonstrate a few of the

trade-offs between a correlation method and a Fourier-based method.

7 Acknowledgments

This work has been supported by the Chilean Research Council (CONICYT) through scholarship for

first author and research grant Fondecyt 1120626. And thanks to the Gemini people for providing the

data and support.

8 References

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Constant derived from DIMM Data”. Beyond conventional adaptive optics: a conference devoted

to the development of adaptive optics for extremely large telescopes. Proceedings of the Topical

Meeting held May 7-10, (2001), Venice, Italy.

2. M. Schöck, ‘‘An analysis of turbulent layers with a wavefront sensor,’’ Ph.D. dissertation

(University of Wyoming, Laramie, Wyoming, 1998).

3. B. Neichel et al., "The Gemini MCAO System GeMS: nearing the end of a lab-story", Proc. SPIE

7736, Adaptive Optics Systems II, 773606 (July 28, 2010).

4. A. Cortés, B. Neichel, A. Guesalaga, and et al., "Atmospheric turbulence profiling using multiple

laser star wavefront sensors", MNRAS, (2012)

5. Wang, L., Schöck, M. and Chanan, G. “Atmospheric turbulence profiling with SLODAR using

multiple adaptive optics wavefront sensors”. Appl. Opt. 47, 1880-92. (2008)

6. Matthias Schoeck and Earl J. Spillar "Analysis of turbulent atmospheric layers with a wavefront

sensor: testing the frozen flow hypothesis", Proc. SPIE 3762, Adaptive Optics Systems and

Technology, 225 (September 27, 1999).

7. R. Wilson, “SLODAR: measuring optical turbulence altitude with a Shack–Hartmann wavefront

sensor,” Monthly Notices of the Royal Astronomical Society 337, 103–108, Wiley Online Library

(2002)

8. Poyneer, L. A., van Dam, M. A. & Veran, J.-P. “Experimental verification of the frozen flow

atmospheric turbulence assumption with use of astronomical adaptive optics telemetry”. Journal of

the Optical Society of America A 26, 833 (2009).

9. Poyneer, L. A. & Veran, J.-P. “Predictive wavefront control for adaptive optics with arbitrary

control loop delays”. Journal of the Optical Society of America A 25, 1486 (2008).

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