analysis of the frozen flow assumption...
TRANSCRIPT
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: [email protected]
b e-mail: [email protected]
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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