Atmospheric image blur with finite outer scale or partial adaptive correction
P. Martinez^{1}  J. Kolb^{1}  A. Tokovinin^{2}  M. Sarazin^{1}
1  European Southern Observatory, KarlSchwarzschildStrasse 2, 85748 Garching, Germany
2  CerroTololo Inter American Observatory, Casilla 603, La Serena, Chile
Received 12 March 2010 / Accepted 22 March 2010
Abstract
Context. Seeinglimited resolution in large telescopes
operating over wide wavelength ranges depends substantially on the
turbulence outer scale and cannot be adequately described by one seeing
value.
Aims. This study attempts to clarify frequent confusions between
seeing and the fullwidth at halfmaximum of longexposure images in
large telescopes, also often called delivered image quality.
Methods. We study the effects at the focus of a telescope of
finite turbulence outer scale and partial adaptive corrections, both
corresponding to a reduction in the lowfrequency content of the phase
perturbation spectrum, by means of analytical calculations and
numerical simulations.
Results. If a von Kàrmàn turbulence model is adopted, a simple
approximate formula predicts the dependence of atmospheric
longexposure resolution on the outer scale over the entire practically
interesting range of telescope diameters and wavelengths. In the
infrared (IR), the difference from the standard Kolmogorov seeing
formula can exceed a factor of two. We find that a loworder adaptive
turbulence correction produces residual wavefronts effectively of small
outer scale, so even a very low compensation order leads to a
substantial improvement in resolution over seeing, compared to the
standard theory.
Conclusions. Seeinglimited resolution of large telescopes,
especially in the IR, is currently under estimated by not accounting
for the outer scale. On the other hand, adaptiveoptics systems
optimized for diffractionlimited imaging in the IR can improve the
resolution in the visible by as much as a factor of two.
Key words: techniques: high angular resolution  instrumentation: high angular resolution  telescopes
1 Introduction
Image blur of astronomical objects caused by terrestrial atmosphere is
traditionally called ``seeing''. In the 2nd half of the 20th century
this phenomenon was understood and quantified (Young 1974). This
understanding was based on considering the distorted wavefronts as a
random stationary process with a powerlaw spectrum  the
KolmogorovObukhov model (Roddier 1981; Tatarskii 1961). This theory describes the shape of
the atmospheric longexposure point spread function (PSF) and many
other phenomena by a single parameter, e.g; the Fried's coherence
radius r_{0} (Fried 1966). The theory predicts the dependence of the PSF
fullwidth at half maximum (FWHM)
on wavelength
and r_{0} to be
In this paper, we assume that r_{0} and refer to observations at zenith. By adopting a standard wavelength nm, we can replace r_{0} with and this single parameter is nowadays usually called ``seeing''. Here we use the term seeing in this precise sense, meaning at 500 nm at zenith.
The success of this theory led most people to believe that the atmospheric parameters r_{0} or actually exist and can be measured to high accuracy, given adequate means. The match between real physical quantities such as PSF or various statistical estimates of distorted wavefronts to the KolmogorovObukhov theory varies from very good to poor, but it is never perfect. The concept of seeing becomes questionable if we push it too far.
The physics of turbulence implies that the spatial power spectral
density (PSD) of phase distortions
(
is
the spatial frequency in m^{1}) deviates from the pure power law at
low frequencies. A popular von Kàrmàn (vK) turbulence model
(Ziad et al. 2000; Conan 2000; Tatarskii 1961) introduces an additional
parameter, the outer scale L_{0}
Equation (2) is the definition of L_{0}. The Kolmogorov model corresponds to . In the vK model, r_{0} describes the highfrequency asymptotic behavior of the spectrum, and thus loses its sense of an equivalent wavefront coherence diameter as defined originally by Fried (1966). Obviously, Eq. (1) is no longer valid as well.
It remains an open question whether wavefront statistics actually correspond to Eq. (2). Proving the vK model experimentally would be a difficult and eventually futile goal because largescale wavefront perturbations are anything but stationary. However, it is firmly established that the phase spectrum does deviate from a power law (Ziad et al. 2000; Tokovinin et al. 2007). Equation (2) with an additional parameter L_{0} provides a useful firstorder description of this behavior. Existing experimental data on L_{0} are interpreted here in this sense.
In this paper, we study the modifications of Eq. (1) implied by the finite outer scale. Our analytical calculations are confirmed by extensive numerical simulations. We show that for finite L_{0} the atmospheric FWHM becomes smaller than predicted by Eq. (1), and that this difference can be substantial. The practical consequences for operation of large telescopes are discussed. The lack of lowfrequency power is typical not only for the vK turbulence, but also for partially corrected wavefronts resulting, e.g., from tiptilt compensation (fast guiding) or loworder adaptiveoptics (AO) correction. This correction leads to a small effective L_{0}. We apply the same analytical treatment to this case and study the shrinking of the PSF halo under partial AO compensation.
2 Analytical treatment
The calculation of the longexposure PSF is performed by multiplying the
telescope optical transfer function (OTF) by an additional term, the
atmospheric OTF
where is the angular spatial frequency (in inverse radians), is the imaging wavelength, and is the phase structure function (SF) (Goodman 1985; Roddier 1981). This expression is in general, applicable to any turbulence spectrum and any telescope diameter. In the case of a large ideal telescope with diameter , the diffraction can be neglected and the longexposure OTF and PSF are accurately described by Eq. (3).
The analytic expression for the phase structure function in the von Kàrmàn model can be found in Conan (2000), Consortini et al. (1972), and Tokovinin (2002). For infinite L_{0}, it transforms into .
Figure 1 (top) plots the SFs for Kolmogorov and vK models with the same r_{0}. In the latter case, the SF saturates at r > L_{0}, reaching asymptotically the level 0.17 (L_{0}/r_{0})^{5/3}. It reaches halfsaturation at r = 0.17 L_{0}. The Kolmogorov SF with the same r_{0} crosses the vK saturation level at r = 0.109 L_{0}. This tells us that the effect of a finite outer scale is strong at distances much shorter than L_{0}, and that it would be misleading to compare L_{0}directly with the telescope diameter.
Putting the vK SF into Eq. (3), we find that for finite L_{0}, does not go to zero at large arguments, therefore its inverse Fourier transform (the PSF) formally does not exist. However, when this level is small and can be neglected. In Fig. 1 (bottom), we compare the PSF profiles for different values of L_{0}/r_{0}, including .
A firstorder approximation of the FWHM of atmospheric PSFs (
) under vK turbulence was
proposed by Tokovinin (2002) to be
This formula is valid for L_{0}/r_{0} > 20 to an accuracy of 1%. We recall the reader that while r_{0} depends on the wavelength, L_{0} does not. At smaller L_{0}/r_{0} values, the atmospheric PSF develops a strong corehalo structure, and its FWHM becomes less and less meaningful. The actual PSF in a telescope is a convolution of the atmospheric blur with diffraction, aberrations, guiding errors, etc. As neither of these factors is described by a Gaussian, calculation of the combined FWHM as a quadratic sum of individual contributions is not accurate.
A formula for the FWHE, fullwidth at halfenergy (
), can be derived with the same accuracy (Tokovinin 2002)
Figure 1: Top panel: comparison of the von Kàrmàn (full line) and Kolmogorov (dashed line) phase structure functions with the same r_{0}. Bottom panel: normalized atmospheric PSFs for different L_{0}/r_{0} ratios. 

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The following section gathers results obtained with extensive numerical simulations to confirm the reliability and validity domain of Eq. (4), and thereby Eq. (5).
3 Numerical simulations
3.1 Random wavefronts
The atmospheric turbulence was generated with 1000 uncorrelated phase screens on a array equivalent to a 100 m width physical size (pixel size 12.2 mm). The principle of the generation of a phase screen is based on the Fourier approach (McGlamery 1976): randomized white noise maps are colored in the Fourier space by the turbulence PSD (Eq. (2)), and the inverse Fourier transform of an outcome corresponds to a phase screen realization. The large size of the simulated phase screens is mandatory for correctly sampling the L_{0} and computing PSF for large telescopes. The simulations considered several L_{0} cases (10, 22, 32.5, 50, 65 m, and ).
Several investigations were carried out on the phase screens to ascertain that their statistics indeed correspond to the input parameters r_{0} and L_{0}. For example, we compared the phase variance, and the variances in the first 100 Zernike coefficients for D = 42 m (r_{0}=12.12 cm and L_{0}=65 m) with their expected values given by Conan (2000) and found good agreement. The phase variance matches expectations to within 1.7, while the variance in the tip and tilt components agrees with their theoretical values to within 1.3 and 0.7, respectively. The case of is particular: the variance in the tip and tilt coefficients does not fit their theoretical values, corresponding instead to a finite outer scale in the range of 200 m to 500 m. This is a consequence of the finite size of the simulated phase screens.
Figure 2: The atmospheric FWHM of simulated longexposure PSFs versus telescope diameter for several L_{0} values ( m, ). The blue curves trace the diffraction FWHM . 

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Several telescope diameters were considered ranging from 10 cm to 42 m. The wavelength domain ranges from the Uband to Mband, while the seeing ranges from 0.1 to 1.8 .
All simulations adopt fast Fourier transforms (FFT) of arrays to generate the longexposure PSFs (over 1000 realizations). The same set of phase screens is used for all telescope diameters. As a result of the very large arrays involved in handling both phase screen statistics and aliasing effect (e.g., largeaperture cases), small telescope diameters (<1 m) may be affected by coarse pupil sampling. The effect of speckle structure is also stronger for small D, causing a larger random scatter in the results.
3.2 Measurement of the FWHM
We determined the FWHM of the simulated longexposure PSFs in the following way. The PSFs were first azimuthally averaged. The 10th order polynomial was then fitted to this curve, and the radius where it crosses the 1/2 of the maximum intensity was determined. The results of this routine were then compared with those of another algorithm (Kolb 2005) applied to the same set of PSFs, and both gave similar values (e.g., 1 pixel for D = 8 m and L_{0} = 22 m; i.e., 0.006 ).
The simulated PSFs are broadened by diffraction and are thus not
directly comparable to Eq. (4). We approximately account for
this by subtracting quadratically the diffraction FWHM
evaluating the true width to be
(6) 
This provides a good approximation of as long as the diffraction blur is small, , but cannot be applied to small diameters, as mentioned above, because the individual broadening factors are not Gaussian. This explains why our results for small Dare inaccurate.
3.3 Outer scale and telescope diameter
The first series of simulations aims at defining the general trend of atmospheric FWHM in large telescopes in the presence of finite outer scale. We compare to Eq. (4) and to the seeing , fixed to in this case. Some results are presented in Fig. 2.
From Fig. 2, it is straightforward to see that in all cases, even for because all simulated wavefronts have finite outer scale. As expected, the validity of Eq. (4) is confirmed, except for the small D where our treatment of diffraction is too crude. All these cases correspond to L_{0}/r_{0} > 80 where the effect of the finite L_{0} is still mild.
Figure 3: Dependence of on wavelength ( top, ) and seeing ( bottom, m ). Other parameters: L_{0} = 22 m, D = 8 m. 

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3.4 Wavelength and seeing dependence
For the second series of simulations, we considered an 8m telescope, a fixed outer scale L_{0} = 22 m, and 0.83 seeing at 0.5 m, while the imaging wavelength was allowed to vary from the Uband to the Mband (from 0.365 to 4.67 m). The results are presented in the top panel of Fig. 3. We note the stronger dependence of on wavelength, compared to the Kolmogorov case. The third series of simulation considered the same L_{0} and D, as the seeing conditions were allowed to vary (Fig. 3, bottom). The agreement with Eq. (4) is demonstrated for both wavelength ( L_{0}/r_{0} > 10) and seeing dependence ( L_{0}/r_{0} > 20).
3.5 Discussion
The previous subsections have provided general results about the atmospheric FWHM in the presence of a finite outer scale. To relate these results to true situations, we discuss the particular case of the 8m Very Large Telescope at Paranal (Chile) assuming standard seeing conditions (0.83 at 0.5 m), and median outer scale value (L_{0}=22 m, i.e. L_{0}/r_{0} = 180). Results shown in Fig. 3 indicate that the FWHM of the von Kármán PSF is lower by than for standard theory ( ) in the visible. This difference is even more dramatic in the nearIR, where the FWHM ( ) is lower by 29.7 (Hband) and 36.3(Kband).
Figure 4: Top: ratio of seeing to FWHM ( ) as a function of the wavelength for several L_{0} cases. Bottom: similarly, ratio of to FWHE ( ). For both plots, seeing is 0.83 at 0.5 m. 

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In the same way, Fig. 4 (top) quantifies the ratio of to (i.e., Eq. (1) to (4)) but for several L_{0} values ranging from the Paranal median value (including the 1 outer scale values, 13 and 37 m) to 150 m. The difference with the standard Kolmogorov seeing formula is substantial and can exceed a factor of two in the IR. Likewise, Fig. 4 (bottom) compares the ratio of to .
4 Resolution under partial compensation
In analogy with the finite outer scale impact, we discuss here the consequences of a reduction in the lowfrequency content of the phase perturbation spectrum generated by AO partial correction or tiptilt compensation.
The purpose of adaptive optics (AO) systems is to compensate for the effect of atmospheric wavefront distortions and to reach a diffractionlimited resolution. To do this, the actuator spacing d (or an equivalent measure of AO compensation order) must be on the order of 2 r_{0} or smaller (Roddier 1998). However, AO systems that do not fulfill this condition may still improve the resolution. A good example is the use of the loworder AO system PUEO for observations at visible wavelengths (Rigaut et al. 1998). A resolution gain of up to a factor of two has been reported. To our knowledge, the reduction of the atmospheric PSF under partial compensation has not been explored in a systematic way.
Residual wavefronts after AO compensation contain highfrequency
ripple, whereas the perturbations at spatial frequencies lower than
f_{c} = 1/(2 d) are corrected. This can be modeled by performing highpass
filtering of the atmospheric PSD. The form of this filter varies,
depending on the AO system. The calculations here only illustrate the
principle and should be adapted to each AO system if an exact result
is sought. We model the AO compensation with a multiplicative factor
F
where m = 6. The PSD (Eq. (2)) is multiplied by F, the SF is calculated and used to compute the residual PSF in the same way as for the vK spectrum. The SF saturates at , reaching the value . The shape of the SF and the saturation value depend on the filter F(x). We experimented with several filters and chose Eq. (7) with m=6 because it matches approximately the known formula of Roddier (1998). Comparing this to the saturation level of the vK SF, given by 0.17 (L_{0}/r_{0})^{5/3}, we find that the effective outer scale of the residual wavefront is 2 d.
Once the SF saturates at , the atmospheric OTF reaches a constant level . We can represent such an OTF as a sum of the constant term and a decreasing part. This corresponds to the sum of a diffractionlimited PSF scaled by and a wide residual halo. The shape of the halo can therefore be calculated by replacing the atmospheric OTF with out to the distance where the minimum is reached, and setting it to zero for higher frequencies. We note that we renormalize the halo OTF to one at the coordinate origin.
The FWHM and the diameter of a circle containing half the energy (FWHE) were computed for the halo of partially compensated PSFs. We compare these parameters to the noncompensated (Kolmogorov) PSFs in Fig. 5. Even when the actuator spacing d is much larger than r_{0} and the AO system does not perform well in the classical sense ( ), the gains in FWHM and FWHE are already substantial. Maximum resolution gain is reached at , when the coherent PSF core is still very weak. As the compensation order increases further, an increasing fraction of energy goes into the core, and the PSF halo becomes weaker and wider. At small d and high S, the halo becomes even wider than the uncompensated atmospheric PSF, being produced by residual phase errors on spatial scales smaller than r_{0}.
Tiptilt correction is a particular case of loworder AO compensation. It is well known that maximum resolution gain is achieved at (Fried 1966). The gain studied here does not depend on the telescope diameter D, but rather on the dimensionless parameter d/r_{0}, in full analogy with the effect of the outer scale.
All three effects  outer scale, partial AO correction and tiptilt compensation  reduce the lowfrequency content of the phase perturbation spectrum. When they act together, the gain in resolution over Kolmogorov turbulence is not cumulative. For example, at finite outer scale the tiptilt fluctuations become smaller and their correction achieves a smaller resolution gain. The resolution gain from partial AO correction (Fig. 5) is also smaller because of the finite L_{0}.
Figure 5: Gain in the FWHM (full line) and FWHE (dotted line) diameter of the PSF halo (compared to the Kolmogorov PSF) resulting from partial AO compensation (outer scale not included). The dotted line shows the coherent energy S. 

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5 Conclusions and discussion
This study has been largely motivated by the confusion between seeing and
the FWHM of longexposure images in large telescopes, also often
called delivered image quality (DIQ). In an ideal large telescope
(no aberrations, internal turbulence, and wind shake), the DIQ is
always lower than predicted by the standard theory, owing
to the finite turbulence outer scale.
The seeing is usually measured by the Differential Image Motion Monitors (Martin 1987; Sarazin & Roddier 1990, DIMMs). This method is sensitive to smallscale wavefront distortions and provides estimates of r_{0} that are almost independent of L_{0}(Ziad et al. 1994)^{}. Using the standard theory, we overestimate the FWHM expected for a large telescope. As the PSF is broadened by nonatmospheric factors, this mismatch can hide telescope defects. This is particularly a problem in the IR, where the difference from the standard theory is large. Therefore, before achieving a truly seeinglimited telescope performance in the IR, we must take account of the finite L_{0}. Stated in other words, our telescopes could perform better than we predict them to based on the standard theory and DIMM measurements.
On the other hand, If we wish to deduce atmospheric seeing from the width of the longexposure PSF, the situation is reversed. The true seeing is poorer than we predict. The effect of finite L_{0} is apparent for all telescope diameters. A simultaneous measurement of L_{0} is thus required to be accurate. Estimating seeing from the width of the spots in activeoptics ShackHartmann sensor (long exposures) should be performed with these considerations in mind.
Since internal telescope defects and outer scale act in opposite directions, they can partially compensate for each other. An agreement between DIMM measurements and DIQ can thus be found where it should not be (Sarazin & Roddier 1990). By comparing simultaneous PSFs at visible and midIR wavelengths, it is possible to extract two parameters, and L_{0}, assuming that the telescope's contribution to the image degradation can be neglected (Tokovinin et al. 2007).
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Footnotes
 ...(Ziad et al. 1994)^{}
 Seeing monitors based on the absolute image motion are affected by finite L_{0}, giving biased, larger r_{0} values.
All Figures
Figure 1: Top panel: comparison of the von Kàrmàn (full line) and Kolmogorov (dashed line) phase structure functions with the same r_{0}. Bottom panel: normalized atmospheric PSFs for different L_{0}/r_{0} ratios. 

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In the text 
Figure 2: The atmospheric FWHM of simulated longexposure PSFs versus telescope diameter for several L_{0} values ( m, ). The blue curves trace the diffraction FWHM . 

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In the text 
Figure 3: Dependence of on wavelength ( top, ) and seeing ( bottom, m ). Other parameters: L_{0} = 22 m, D = 8 m. 

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In the text 
Figure 4: Top: ratio of seeing to FWHM ( ) as a function of the wavelength for several L_{0} cases. Bottom: similarly, ratio of to FWHE ( ). For both plots, seeing is 0.83 at 0.5 m. 

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In the text 
Figure 5: Gain in the FWHM (full line) and FWHE (dotted line) diameter of the PSF halo (compared to the Kolmogorov PSF) resulting from partial AO compensation (outer scale not included). The dotted line shows the coherent energy S. 

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In the text 
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