Testing turbulence model at metric scales with mid-infrared VISIR images
  at the VLT
Mon. Not. R. Astron. Soc. 000, 1–8 (2005) Printed 4 November 2018 (MN LATEX style file v2.2)
Testing turbulence model at metric scales with mid-infrared VISIR
images at the VLT
A. Tokovinin1⋆, M. Sarazin2, and A.Smette3
1Cerro-Tololo Inter American Observatory, Casilla 603, La Serena, Chile
2European Southern Observatory, Karl-Schwarzschild-Strasse, 2 D-85748 Garching bei München, Germany
3European Southern Observatory, Alonso de Cordova 3107, Casilla 19001, Vitacura, Santiago, Chile
ABSTRACT
We probe turbulence structure from centimetric to metric scales by simultaneous imagery
at mid-infrared and visible wavelengths at the VLT telescope and show that it departs signifi-
cantly from the commonly used Kolmogorov model. The data can be fitted by the von Kármán
model with an outer scale of the order of 30 m and we see clear signs of the phase structure
function saturation across the 8-m VLT aperture. The image quality improves in the infrared
faster than the standard λ−1/5 scaling and may be diffraction-limited at 30-m apertures even
without adaptive optics at wavelengths longer than 8 micron.
Key words: site testing – atmospheric effects
1 INTRODUCTION
As the ground-based telescopes become bigger, more emphasis is
made on studying and modeling atmospheric optical distortions at
large spatial scales. This knowledge is required, e.g. for specifying
the stroke of deformable mirrors in adaptive optics or the range of
fringe-trackers in interferometers. Moreover, the size of the atmo-
spheric coherence length exceeds 1 m at infrared (IR) wavelengths.
Thus, even classical long-exposure imagery in the IR is affected
by the departures of turbulence statistics from the standard Kol-
mogorov model (variance proportional to the 5/3 power of the base-
line) which was so successfully used in the visible range.
Optical path-length fluctuations were measured with long-
baseline interferometers, but the published results are controver-
sial. Saturation of the fringe motions at baselines from 8 m to
16 m was first observed by Mariotti & Di Benedetto (1984). Later,
Davis et al. (1995) found a strong departure from the Kolmogorov
law and a saturation of the path-length fluctuations at the 80-
m baseline at a level of about 10µm rms. On the other hand,
Colavita et al. (1987) have not found any departure from the Kol-
mogorov law at baselines up to 12 m by doing a temporal analysis
of the fringe motion, and Nightingale & Buscher (1991) reached
the same conclusion from interferometric measurements of the
fringe motion in a 4-m telescope aperture. The controversy could
be caused by the use of the frozen-flow hypothesis in the inter-
pretation of fringe motion on a single baseline. Another reason is
the difficulty in separating fringe motion caused by the atmosphere
from the mechanical noise due to instrument instabilities, tracking,
⋆ E-mail: atokovinin@ctio.noao.edu
Turbulence outer scale L0 can be evaluated from the covari-
ances of the image motion in small telescopes, as implemented
in the GSM instrument (Ziad et al. 2000), or from the analysis of
adaptive-optics data, e.g. by Fusco et al. (2004). These methods use
the von Kármán (VK) turbulence spectrum (cf. Appendix A) and
adjust its parameters (r0, L0) to fit the data. Measurements with
the GSM at different sites show that typical L0 values are of the
order of 20 m. Maire et al. (2006) compared directly fringe motion
in a long-baseline interferometer with the L0 measured by GSM
and found a good agreement.
This short and non-exhaustive review demonstrates that fur-
ther work on characterizing large-scale turbulence structure is
needed. A direct comparison of mid-IR and optical images at a
large telescope offers a new, independent way to probe turbulence
models at spatial scales from centimeters to meters. To our knowl-
edge, no such work has been done previously, so we made an exper-
iment at the Very Large Telescope (VLT) located at the ESO obser-
vatory Cerro Paranal in Chile. The main purpose of this experiment
is to evaluate the atmospheric phase structure function (SF) and to
check if the VK model is adequate. So far, the mathematically con-
venient VK model has been used without being actually tested.
The experiment is described in Sect. 2. The results are pre-
sented in Sect. 3 and the conclusions are given in Sect. 4. Appen-
dices contain some technical material. For reader’s convenience we
reproduce the formulation of atmospheric models in Appendix A.
c© 2005 RAS
http://arxiv.org/abs/0704.0470v1
2 A. Tokovinin, M. Sarazin, & A. Smette
Subtract backg.
FFT −−> OTF
Correct OTF(0)
Compute SF
Divide by T0* VISIR
Guider
S−H sensor
Guide star
Object
Tilt errors
Good spots
Fit r0
Average, OTF
Average A−B
Sel. long axis
Re−center
DIMM, MASS,
meteo
Bright star
Divide by Ref
Baseline
3cm−15cm
von Karman (r0,L0)
0.4m−4m
Figure 1. Overview of the experiment and data interpretation.
2 EXPERIMENT DESCRIPTION
2.1 From PSF to structure function
It is well known that the modulus of the long-exposure optical
transfer function (OTF) T (f) in a perfect telescope is related to
the atmospheric SF Dϕ(r) as (Tatarskii 1961; Roddier 1981)
T (f) = T0(f) exp[−0.5Dϕ(λf)], (1)
where T0(f) is the diffraction-limited OTF, f is the vector of spa-
tial frequency on the sky in rad−1, f = |f |, λ is the imaging wave-
length and r is the baseline vector. This relation can be inverted
to reconstruct Dϕ(r) from the known image point spread function
(PSF). However, it is only feasible at the baselines r = λf where
Dϕ(r) is not very large or small, otherwise T (f) is close to either 1
or 0 and the sensitivity to the atmospheric turbulence is lost. In the
visible range, the method probes centimetric and decametric scales,
in the mid-IR it is sensitive to the metric scales because λ is much
larger.
The 8-m VLT telescope offers a unique platform for our exper-
iment with access to metric baselines. The aberrations are removed
by active optics and the turbulence inside the dome is low, ensuring
that the image blur is dominated by the atmospheric seeing. Optical
images as small as 0.18′′ FWHM have been recorded under excep-
tional conditions, proving that the VLT intrinsic quality is nearly
ideal even in the visible range.1 For our purpose, the mid-IR im-
ager VISIR installed at the Cassegrain focus of the UT3 telescope
is the best choice.
A deeper analysis of the mid-IR images shows that some de-
partures from the ideal-telescope model (1) are inevitable. Resid-
ual image motion (tilt) is the largest source of uncertainty, as it can
cause additional blur (e.g. wind shake), while, on the other hand,
part of the atmospheric tilt is removed by guiding. Residual aber-
rations in the VLT optics and instrument can add something to the
atmospheric PSF, too. Therefore, opto-mechanical wave-front dis-
tortions of instrumental nature cannot be separated cleanly from the
large-scale atmospheric distortions. In this respect our new exper-
iment is not fundamentally different from long-baseline interfer-
ometers, but it was worth trying nevertheless because instrumental
effects in both cases are different.
1 ESO Press Release, July 21, 2000.
http://www.eso.org/outreach/press-rel/pr-2000/pr-16-00.html
Table 1. Instrument parameters
Instrument VISIR VISIR SH
Filter PAH1 Q2 None
Aperture diam., m 8.115 8.115 0.34 square
λ/∆λ, µm 8.6/0.42 18.7/0.88 0.6/∼0.3
Pixel scale, arcsec 0.075 0.075 0.280
Detector format 256×256 256×256 592×573
Exposure time, s 30×2 90×1 ∼45
Chopping period, s 4 2 None
2.2 Overview
The overall scheme of the experiment is presented in Fig. 1.
Two different stars are observed through the VLT: the object with
VISIR, the guide star with the Shack-Hartmann (SH) sensor of the
VLT active optics (AO) system. The same guide star is used for the
guiding, called field stabilization. The angular distance between the
object and the guide star was from 3′ to 5′. Details of the optical
and IR imagery and data reduction are given below and in Table 1.
Data on the seeing and turbulence profile at the Paranal
observatory are collected by the dedicated site monitor
equipped with the Differential Image Motion Monitor (DIMM)
(Sarazin & Roddier 1990) and Multi-Aperture Scintillation Sensor
(MASS) (Kornilov et al. 2003). The monitor points to a bright
star near zenith and measures the total seeing ǫ and the seeing in
the free atmosphere ǫFA produced by turbulence above 500 m.
Naturally, ǫFA < ǫ unless most turbulence is above 500 m. The
VLT dome is higher than the DIMM tower, hence the seeing at
VLT can be better than that measured by DIMM, but still worse
or equal to ǫFA. The MASS also measures the adaptive-optics
time constant τ0 and a crude turbulence profile. The effective wind
speed in the free atmosphere V was evaluated from the relation
V = 0.31r0,FA/τ0. The speed of the ground wind Vgr was taken
from the Paranal ambient conditions database.
2.3 Conditions of the experiment
The data for this study have been obtained by A.S. on June 19, 21,
and 22, 2006. Table 2 lists relevant average parameters for each
data set. On all nights the sky was clear, with stable air temperature
and very low humidity. Individual (non-averaged) data from DIMM
and MASS are plotted in Fig. 2 to characterize the variability of the
turbulence. The conditions were rather stable on all 3 nights, with
c© 2005 RAS, MNRAS 000, 1–8
http://www.eso.org/outreach/press-rel/pr-2000/pr-16-00.html
Turbulence model with mid-IR images 3
Figure 2. Temporal evolution of the seeing ǫ and the seeing in the free atmosphere ǫFA as measured by DIMM and MASS respectively during the data
acquisition periods on 3 nights. The seeing derived from the SH spots and reduced to zenith is over-plotted as large squares.
Table 2. Night log
Date Time Air ǫ, ǫFA, V Vgr
Jun 2006 UT mass ′′ ′′ m/s m/s
19/20 23:38–23:44 1.7 1.38 0.63 30 5.6
21/22 1:40–1:49 1.5 1.21 0.68 13 2.5
22/23 23:07–23:16 1.1 0.54 0.27 16 6.4
A B BA
Figure 3. The pairs of positive and negative PSFs in the PAH1 filter regis-
tered with VISIR on June 21/22 (left, file 5) and June 22/23 (right, file 1).
Only the central 20x20 pixels (1.5′′) of each image are displayed with a
square-root intensity stretch.
the DIMM seeing always dominated by the ground layer. However,
the seeing derived from the SH spots indicates that the ground see-
ing contribution at VLT was less than at DIMM on June 19/20 and
21/22.
2.4 VISIR data
Mid-IR images of bright stars were obtained with VISIR in two
filters called PAH1 (8.6µm) and Q2 (18.7µm), cf. Table 1. A stan-
dard chopping-nodding technique was used. At each telescope po-
sition (nod), the image was shifted on the detector back and forth by
modulation (chop) of the VLT secondary mirror M2 with a period
of 2-4 s. The chop throw was 10′′ in the North-South direction, with
a little pause to stabilize the image after each chop. For the PAH1
filter, for example, a total of 30 images with 2 s exposure in each
chop position are taken to produce the data cube with cumulative
exposure of 60 s. Then the telescope is moved by 10′′ to the East
and a second data cube is taken. Here we do not take advantage
of the nodding and analyze only the average difference between
images in two chopping positions A and B. This A−B difference
suppresses the background and its slow drifts. It is averaged over
all image pairs in the cube and contains positive (A) and negative
(B) images of the same star, considered here separately as two in-
dependent realizations of the PSF.
The positive and negative PSFs are extracted as two 64x64
Figure 4. The cuts of the OTFs along fx axis for the VISIR images dis-
played in Fig. 3 (top – June 21/22, bottom – June 22/23). Asterisks –
positive images, plusses – negative images. The line shows the diffraction-
limited OTF T0(λf).
pixel (4.8′′) subsections and processed in parallel. The background
is computed in the corners of these images (outside the radius of 32
pixels) as a median and subtracted. For reference only, these PSFs
are approximated with 2-dimensional Gaussians to determine the
Full Width at Half Maximum (FWHM) ǫl and ǫs in the long and
short axes. The ellipticity is computed as e = (ǫl − ǫs)/(ǫl + ǫs)
and the position angle θ of the long axis (counted from the x-axis
counter-clockwise) is recorded as well.
Figure 3 shows typical examples of the individual PSFs in
the PAH1 filter on two nights. On June 21/22, the PSF is blurred
c© 2005 RAS, MNRAS 000, 1–8
4 A. Tokovinin, M. Sarazin, & A. Smette
Figure 5. The atmospheric OTF in the VISIR PAH1 filter obtained after
correcting the observed OTFs for diffraction on June 22/23, file 1. All points
are plotted without radial averaging (asterisks – image A, triangles – image
B) to show the asymmetry. Compare with Fig. 4.
by the turbulence. It is slightly elongated (e = 0.1...0.2) and has
some structure, different between positive and negative images. On
June 22/23, under a better seeing, the PSF is closer to a diffraction-
limited one, but still elongated (e = 0.08...0.12). The elongation
can be caused by a combination of several factors including some
residual low-order aberrations, vibrations, etc. However, the domi-
nant contribution to the image elongation is likely related to the tilt
anisoplanatism between the guide star and the object, estimated in
Appendix C. The direction of the elongation points approximately
to the guide star.
Each PSF is Fourier-transformed and the normalized modulus
of the FT is identified with the observed T (f) (Fig. 4). The spa-
tial frequencies f are translated to the baselines r = λf . A small
correction is needed for the normalization (cf. Appendix B). The
experimental OTF T (f) is divided by the calculated T0(f) to ob-
tain the atmospheric OTF and then the SF. In calculating T0(f),
we use the pupil diameter D = 8.115 m because it is not vignetted
by the cold stop inside the VISIR instrument. Figure 5 shows the
results of this division for the sharpest PAH1 image (compare with
Fig. 4, bottom). The saturation of the OTF (hence SF) is obvious.
The vertical spread is caused by the asymmetry and the differences
between the images A and B.
Note that at small baselines the derived atmospheric OTFs and
SFs are sensitive to the normalization errors, while at large base-
lines the experimental T (f) becomes noisy and its modulus is bi-
ased by any image defects such as noise and bad pixels. For these
reasons, we use for further analysis the baseline range from 0.4 m
to 4 m, where T (f) is most reliable. In the following, the asym-
metry is neglected and the atmospheric SFs are calculated from the
radially-averaged OTFs.
The data in the Q2 filter are rather noisy compared to the
PAH1 filter. The PSFs are nearly diffraction-limited. At some nod-
ding positions, the image is affected by bad detector pixels and/or
horizontal stripes in the background. The OTFs registered on June
22/23 do not differ from the diffraction-limited ones within the er-
rors. Therefore, no reliable estimates of the atmospheric SF are de-
rived from the Q2 images on this night and we can only affirm that
the phase fluctuations were much less than 1 rad at 18.7µm.
Figure 6. The atmospheric OTF derived from the average SH spot de-
convolved by the average reference spot (June 22/23, 23:07). The fitted
Kolmogorov model is plotted in dashed line.
2.5 Shack-Hartmann data
Long-exposure images from the Shack-Hartmann (SH) sensor were
recorded quasi-simultaneously with the data. A typical exposure
time is 45 s, but the SH “sees” the star only during 1/2 of the chop-
ping cycle. On the other hand, the guiding was done continuously
because the guiding “box” moved to compensate for the chopping.
The SH images are “raw”, i.e. not corrected for bias, hot pix-
els, flat field, etc. The geometry of the lenslet array is square, with
24 spots across pupil diameter, hence sub-aperture size d = 0.34m.
The pixel scale 0.30′′ was calculated from the opto-mechanical
data of the SH sensor and telescope. However, the real pixel scale
depends, among other things, on the actual distance between the
lenslet array and the detector. So we obtained images of the known
double star HIP 73246 with a separation of 3.512′′ measured by
Hipparcos and derived the pixel scale 0.280′′ ± 0.004′′ for the SH
sensor at the Cassegrain focus of UT3.
The images of individual spots show various distortions and
are all elongated in one direction due to the un-corrected atmo-
spheric dispersion (except for the data taken near zenith). Initially,
we selected for the analysis only the sharpest spots in each frame.
The spots were extracted, over-sampled, re-centered and averaged.
The influence of local CCD defects is reduced by the averag-
ing. The modulus of the Fourier Transform T (f) of the average
spot was calculated and normalized so that T (0) = 1. Then it
was divided by the diffraction-limited OTF for the square aperture
T0(f) = (1− |λfx/d|) (1− |λfy/d|), assuming λ = 0.6µm.
We presume that the elongation of the spots is caused primar-
ily by the atmospheric dispersion. The long axis of T (f) is found by
fitting a 2-dimensional Gaussian. The cut along this axis is fitted to
the Kolmogorov atmospheric OTF (eq. A2), giving an estimate of
the Fried’s parameter r0,Kolm and seeing ǫSH = 0.98λ/r0,Kolm
(Fig. 6). The atmospheric SF at short baselines is derived from the
same cut.
We found that even the sharpest spots were distorted by resid-
ual aberrations in the lenslets. A mosaic of selected sharp spots
recorded under good seeing near zenith (Fig. 7) shows various de-
grees of distortion and elongation. This became apparent on June
22/23, under excellent conditions, when the SH measured a seeing
of about 0.75′′ , worse than the DIMM seeing. To overcome this
problem, we used the image of the reference point source. A total
of 306 sharpest reference spots were selected, re-centered and aver-
aged in the same way as the star images. Then we re-processed the
c© 2005 RAS, MNRAS 000, 1–8
Turbulence model with mid-IR images 5
Figure 7. Mosaic of 8 individual good SH spots (June 22/23, 23:15), each in a 3′′ × 3′′ field. The data are taken at air mass 1.1, with negligible atmospheric
dispersion. The ellipticity of the average spot is e = 0.08.
Figure 8. Structure functions on June 21/22, Q2 filter (files 1, top, and 3,
bottom). The SFs derived from the VISIR images A and B are plotted as
plusses and crosses, the SFs from the SH spots – as asterisks. Full lines –
VK models, dashed lines – subtraction and addition of Dtilt, dotted lines –
Kolmogorov SFs.
data by selecting the same spots and de-convolving them by the av-
erage reference spot instead of T0(f). A good agreement between
DIMM and SH was reached (Fig. 2).
3 RESULTS
The results of image processing are gathered in the Table 3. The
time (to 1 min) refers to the end of each acquisition. The seeing
at 0.6µm estimated from the SH spots ǫSH is listed as well (it
is not reduced to zenith as in Fig. 2). The next columns give the
parameters of the elliptical Gaussians approximating the positive
(A) and negative (B) mid-IR images: the FWHM ǫs of the short
axis (in arcseconds), the ellipticity e, and the position angle θ of
the long axis in degrees.
The SFs derived from the visible and mid-IR PSFs are con-
verted to linear units (µm2) by multiplying them with (λ/2π)2 and
combined on the same plots (Figs. 8,9,10). They are compared to
Figure 9. Structure functions as in Fig. 8 for the PAH1 filter (June 21/22,
files 5, top, and 6, bottom).
the VK models (Appendix A). The model parameters r0 are derived
from the SH spots, and the outer scale L0 is selected to match the
data qualitatively. These parameters are also listed in Table 3.
The exact degree of tip-tilt compensation by the field stabi-
lization servo cannot be evaluated. A large part of the atmospheric
tilt produced by the ground layer is compensated, but tilts from
high layers are actually amplified (Appendix C). However, the total
effect of the tilt compensation is not large and cannot explain the
deviation of the SFs from the Kolmogorov model. In the figures,
the dashed lines show the VK models with complete tilt compen-
sation and tilt doubling, thus bracketing possible effects of the field
stabilization system.
Figure 8 shows the data from two consecutive acquisitions
made with an interval of only 4 min. We see that L0 increased from
50 m to 200 m. Further data show that it decreased again in the next
6 min. (Fig. 9). Such “bursts” of L0 are typical (Ziad et al. 2000).
On June 22/23, the images in the Q2 filter are so close to the
diffraction limit that the SFs derived from them are uncertain (L0
values marked by colons in Table 3). The SFs derived from the
PAH1 image shows saturation (Fig. 10).
c© 2005 RAS, MNRAS 000, 1–8
6 A. Tokovinin, M. Sarazin, & A. Smette
Table 3. Data log
Time File Filt ǫSH , Image A Image B r0, L0,
UT ′′ ǫs e θ ǫs e θ m m
June 19/20
23:38 5 Q2 1.348 0.573 0.09 22 0.582 0.08 23 0.082 40
23:40 6 Q2 - 0.674 0.07 38 0.663 0.08 43 0.086 200
23:42 7 Q2 - 0.643 0.03 30 0.624 0.03 9 0.086 200
23:44 8 Q2 - 0.585 0.05 22 0.578 0.08 17 0.082 50
June 21/22
1:40 1 Q2 0.980 0.506 0.10 -25 0.513 0.09 -30 0.112 50
1:42 2 Q2 0.867 0.512 0.08 -19 0.530 0.08 -30 0.115 100
1:44 3 Q2 0.895 0.515 0.08 -19 0.529 0.06 -28 0.117 200
1:46 4 Q2 0.861 0.493 0.03 -20 0.505 0.02 -24 0.113 60
1:48 5 PAH1 0.719 0.379 0.10 -1 0.388 0.08 -9 0.110 30
1:49 6 PAH1 0.989 0.373 0.17 -7 0.400 0.19 3 0.111 40
June 22/23
23:07 1 PAH1 0.583 0.239 0.08 23 0.247 0.09 21 0.181 35
23:08 2 PAH1 0.575 0.245 0.12 32 0.251 0.09 22 0.182 40:
23:11 3 Q2 0.580 0.457 0.02 14 0.456 0.01 27 0.186 60:
23:12 4 Q2 0.614 0.455 0.02 20 0.454 0.02 8 0.186 60:
23:14 5 Q2 0.584 0.448 0.04 20 0.459 0.02 17 0.194 200:
23:16 6 Q2 0.584 0.451 0.02 4 0.448 0.02 14 0.194 200:
Figure 10. Structure function as in Fig. 8 for June 22/23, PAH1 filter, file
4 CONCLUSIONS
We were able to measure directly the structure function of atmo-
spheric wave-front distortions at the VLT up to metric scales. The
results show a broad agreement with the VK turbulence model.
Hence, we can use this model with an increased degree of confi-
dence for predicting the long-exposure PSF in the infrared or eval-
uating the deformable-mirror stroke.
Interpretation of the measured SFs in terms of atmospheric
turbulence model cannot be done without reservations, however.
Several instrumental effects bias these SFs. Instead of uncertain
modeling of these effects, we simply present the results “as they
are” and hope that new, deeper studies will be prompted by this
work. It is preferable to use a good-quality optical imager rather
than SH for continuing this study.
The standard theory predicts an improvement of the image
size in a very large telescope (neglecting diffraction) as λ−1/5, e.g.
by 1.70 times between 0.6µm and 8.6µm. In fact, on a good night
the VLT image quality at 8.6µm is limited by diffraction, and we
observe a clear saturation of the SF. It means that the λ−1/5 scaling
does not work. In a larger, 30-m telescope, the FWHM resolution at
8.6µm will be 0.067′′ (diffraction-limited) for the VK model with
decametric outer scales because the SF saturates at large baselines.
This example illustrates a dramatic effect of turbulence model for
predicting the long-exposure image quality in the IR, demonstrated
here experimentally.
ACKNOWLEDGMENTS
We thank Stephane Guisard for obtaining reference SH images on
our request. A suggestion by anonymous Referee to de-convolve
the SH spots with images of the reference source helped us to re-
solve the problem of lenslet aberrations.
APPENDIX A: MODELS OF TURBULENCE
The von Kármán (VK) turbulence model describes the spatial
power spectrum of atmospheric wave-front phase by the formula
(Tatarskii 1961; Sasiela 1994; Ziad et al. 2000)
Wϕ(κ) = 0.0229r
−11/6
, (A1)
where κ is the modulus of the spatial frequency (one over period).
The model has two parameters, the coherence radius r0 (also called
Fried radius) and the outer scale L0. The Kolmogorov turbulence
model is a specific case of (A1) with L0 = ∞.
The phase structure function (SF) Dϕ(r) is obtained from
Wϕ(κ). An analytic expression for the SF in the VK model can
be found in (Tatarskii 1961) and in (Tokovinin 2002). In the limit
L0 = ∞ (Kolmogorov model) the SF is
Dϕ(r) = 6.8839(r/r0)
. (A2)
If we try to approximate a VK SF (or PSF) with a simpler Kol-
mogorov model, the derived r0,Kolm will be larger than the true r0.
c© 2005 RAS, MNRAS 000, 1–8
Turbulence model with mid-IR images 7
An approximate relation between these parameters can be estab-
lished numerically. Here we use a formula adapted from (Tokovinin
2002),
r0 ≈ r0,Kolm
1− 1.5(r0,Kolm/L0)0.356. (A3)
Thus we translate the r0,Kolm obtained by fitting the SH spots to
the r0 parameter appropriate for the VK model by means of (A3).
This ensures a good match between the Kolmogorov and VK SFs
at short baselines (e.g. Fig. 8).
APPENDIX B: CORRECTION FOR THE MISSING FLUX
The wings of the PSF (essentially caused by diffraction) outside the
selected field are cut off, hence T (0) (integral of the PSF) is under-
estimated. The fraction of energy in the Airy PSF contained in the
circle of angular radius a is (Born & Wolf 1965):
E(a) = 1− J
0 (πad/λ)− J
1 (πad/λ). (B1)
We take a = pNgrid = 2.4
′′ , where p = 0.075′′ is the pixel size
and Ngrid = 32 is the half-size of the PSF frame.
A second correction is needed because we calculate the back-
ground as the average intensity at the distance ≈ a from the center.
Together with the background, we subtract some fraction of the
PSF. This fraction ∆E, integrated over the field, can be estimated
by differentiating (B1) as
[E(a+∆a)− E(a)]. (B2)
We take ∆a = λ/D to average out the “wiggles” of E(a). For
PAH1 images, E(a) ≈ 0.98 and ∆E ≈ 0.01. We divide T (0)
by E(a) − ∆E before the normalization, thus accounting for the
missing flux. The correction in the Q2 filter is larger, reaching 5%.
Without such correction, we over-estimate T (f) at small baselines
and hence under-estimate the SF. We see in Fig. 9 that the first
points of mid-IR SFs are still below the model curves, indicating
that a larger correction for missing flux was probably needed for
the PAH1 images.
APPENDIX C: EFFECT OF THE TIP-TILT SERVO ON
THE SF
The variance of the image centroid motion in one coordinate σ2α (in
square radians) can be computed by the known formula
α = Ktilt(λ/D)
(D/r0)
, (C1)
where Ktilt = 0.170 for the Kolmogorov turbulence, e.g. (Sasiela
1994). The image motion is achromatic because the right hand of
Eq. C1 does not depend on λ. For the VK model, our numerical
calculation leads to the approximation
Ktilt ≈ −2.672 + 2.308x − 0.898x
, (C2)
where x = log
(L0/D). The approximation is valid for L0/D <
300 with an accuracy of better than 1%. For a typical situation at
VLT, L0 = 20m, Ktilt = 0.013, i.e. an order of magnitude smaller
than for the Kolmogorov model.
The field stabilization servo measures the tilt of the guide star
α2, filters it with the closed-loop response h(t) and applies to com-
pensate for the object tilt α1. The residual tilt error is then
∆α = α1 − α2 ⊙ h. (C3)
It follows that the power spectrum of the residual tilt variance is
Figure C1. Temporal power spectra of the tilt (full line) and residual tilts
in longitudinal and transverse directions after compensation with a 1-Hz
servo. Parameters: D = 8.115m, L0 = 20m, V = 10m/s, s = 0.72D,
η = 0, ν0 = 1Hz.
W∆α(ν) = Wα(ν)[1 + |h̃(ν)|
]− 2Re[W12(ν)h̃
(ν)], (C4)
where Wα(ν) is the power spectrum of the tilt, W12(ν) is the cross-
power spectrum between guide star and object, and Re stands for
the real part. The residual tilt variance σ2∆α can be conveniently
expressed as a fraction r of the un-corrected variance,
W∆α(ν)dν = rσ
α. (C5)
The servo response is modeled here by a simple integrator with a
3-db cutoff frequency ν0,
h̃(ν) =
1 + i(ν/ν0)
. (C6)
The power spectrum and cross-spectrum of tilt for VK turbu-
lence model is computed by Avila et al. (1997). For a single layer
moving with the speed V at direction η (η = 0 for the wind blow-
ing from the object to the guide star), the cross-spectrum depends
on the separation s of the beam footprints. For example, a layer
at H = 4 km and a guide star at θ = 5′ from the object lead to
s = θH = 5.8m. In these conditions, the 1-Hz servo actually in-
creases the tilt variance: σ2∆α = [2.36, 1.03]×σ
α for V = 10m/s,
where the coefficients [rl, rt] = [2.36, 1.03] refer to the longitudi-
nal and transverse directions. This situation is illustrated in Fig. C1.
On the other hand, for the ground layer (s = 0) and V = 5.5m/s,
we obtain rl = rt ≈ 0.1, i.e. a good tilt correction.
Residual tilt errors of the VLT field stabilization system can
be modeled. However, the input information for such a model must
include the closed-loop servo response h̃(ν), altitude profiles of
turbulence C2n(h) and L0(h), profiles of the wind speed and di-
rection, and the geometry of the object and guide star. We do not
have all this information for the data at hand. Instead, we evaluate
the effects of the servo by assuming either rl = rt = 0 (perfect
compensation) or rl = rt = 2 (un-correlated tilts). The first case is
closer to reality when a large part of turbulence is near the ground.
The wave-front structure function D(x, y) can be represented
by the combination of the quadratic part caused by tilts Dtilt(x, y)
and the tilt-removed part D0,
D(x, y) = D
(x, y)+D
(x, y) = D
(x, y)+σ
).(C7)
When the field-stabilization servo is at work, the resulting SF
Dg is
c© 2005 RAS, MNRAS 000, 1–8
8 A. Tokovinin, M. Sarazin, & A. Smette
Dg(x, y) = D0(x, y) + σ2α(rlx
2 + rty
= D(x, y) + σ2α(rl − 1)x
2 + σ2α(rt − 1)y
where it is assumed that the x-axis is directed to the guide star.
With such orientation of the coordinates, we can neglect the cross-
term proportional to xy, which otherwise would be required in
the Eq. C8. Our two options (complete compensation, r = 0, or
tilt doubling, r = 2) correspond to the subtraction or addition of
Dtilt(x, y) from the VK model. The tilt variance σ2α is calculated
with eqs. C1,C2.
As an example, consider the case of file 5 (PAH1) on June
21/22. A good match between MASS and SH seeing (Fig. 2) and
MASS profiles indicate that most of turbulence was located in a
strong layer at 4 km. Our model leads to [rl, rt] = [2.4, 1.0] (as-
sumed parameters: s = 5.8m, V = 10m/s, η = 0, L0 = 30m,
seeing 1.0′′, ν0 = 1Hz). We compute the SF according to Eq. C8
and translate it to the PSF using Eq. 1. The resulting PSF at 8.6µm
has a minimum FWHM of 0.37′′ and an ellipticity of 0.08, elon-
gated towards the guide star at position angle −39◦. The actual
FWHM was 0.38′′ with e = 0.1 elongated at −5◦. Thus, tilt aniso-
plamatism can qualitatively explain the image elongation.
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This paper has been typeset from a TEX/ LATEX file prepared by the
author.
c© 2005 RAS, MNRAS 000, 1–8
	Introduction
	Experiment description
	From PSF to structure function
	Overview
	Conditions of the experiment
	VISIR data
	Shack-Hartmann data
	Results
	Conclusions
	Models of turbulence
	Correction for the missing flux
	Effect of the tip-tilt servo on the SF
