Issue 
A&A
Volume 530, June 2011



Article Number  A56  
Number of page(s)  7  
Section  Astronomical instrumentation  
DOI  https://doi.org/10.1051/00046361/201116553  
Published online  06 May 2011 
Stellar scintillation in the short exposure regime and atmospheric coherence time evaluation
Sternberg Astronomical Institute,
Universitetskij prt, 13,
119992
Moscow,
Russia
email: victor@sai.msu.ru
Received:
20
January
2011
Accepted:
24
March
2011
Aims. Accurately measuring the atmospheric coherence time is still a significant problem despite a variety of applicable methods. The MultiAperture Scintillation Sensor (MASS) designed for the vertical profiling of optical turbulence also provides a measurements of coherence time, but its results were found to be biased. Hence there is a need for a more robust method to determine τ_{0}.
Methods. The effect of smoothing the stellar scintillation by a finite exposure of the detector is considered. The short exposure regime is described and its limits are defined. The reanalysis of previous measurements with the MASS is performed to test the applicability of this approach in real data processing. It is shown that most of the actual measurements satisfy the criteria of short exposures.
Results. The expressions for the mean wind speeds in the free atmosphere from the measurement of the scintillation indices are derived for this regime. These values provide an estimate of the atmospheric coherence time τ_{0} without the need of empirical calibration. The verification of the method based on real measurements of the resulting τ_{0} agree well with independent methods.
Key words: site testing / instrumentation: adaptive optics / atmospheric effects
© ESO, 2011
1. Introduction
In recent decades the adequacy of optical turbulence (OT) measurements in the Earth’s atmosphere above potential astronomical sites and operating observatories has become more and more important. Developments in astronomy demand more in the way we determine OT. One such requirement is the ability to use cheap tools to monitor the OT automatically and on long time scales.
For these methods, which use modest feeding optics, the analysis of the spatial wavefront distortion is replaced by the analysis of temporal variations of the wavefront inside a limited area. The widely used Differential Image Motion Monitor DIMM (Sarazin & Roddier 1990), which measures the turbulence integrated over the whole atmosphere, and the MultiAperture Scintillation Sensor MASS (Kornilov et al. 2003; Tokovinin et al. 2003), which determines the altitude distribution of the OT, belong to these methods.
The transition from a spatial to a spatialtemporal description of the OT is based on the hypothesis of frozen turbulence (Taylor 1938) and was analyzed by many authors (see e.g. Roddier 1981; Martin 1987). A necessary component of this analysis is the knowledge of wind velocities in the atmosphere.
The MASS instrument was developed to measure the vertical distribution of the OT. The MASS data contain information about the temporal properties of stellar scintillation that is necessary to estimate the atmospheric coherence time τ_{0} with a method that involves the differential exposure scintillation index DESI as described in Tokovinin (2002).
After the actual measurements of OT with the MASS, it became clear that the τ_{0} values derived in this way were significantly underestimated and therefore required additional calibrations and corrections during analysis (Travouillon et al. 2009). Empirical recipes of correction do not give confidence results because this systematic underestimation may be caused by several factors (Tokovinin 2011). On the other hand, a large and permanently increasing volume of collected data requires uniform treatment for an objective comparison of OT characteristics between various sites.
In this paper we analyze the effect of temporal averaging that affects the scintillation indices measured by the MASS instrument and their role in the evaluation of τ_{0}. In addition we propose a modification of the DESI method that does not depend on an empirical calibration. The presentation of the modified method is preceded by theoretical description of scintillation in the regime of short exposures.
The regime of short exposures allows us to establish a simple relationship between the measured scintillation statistics and the characteristic wind speed that enters the definition of τ_{0}. In the last section, the method is tested with MASS data taken at Mt. Shadzhatmaz (Kornilov et al. 2010) and Mt. Maidanak (Kornilov et al. 2009).
We found that the main cause of the mentioned underestimations of τ_{0} was the wrong interpretation of the formula for DESI. These discrepancies virtually disappear after the correction in the data processing software.
The modified method does not require any empirical calibration. It is theoretically clearer, and gives an estimate of the mean wind in the free atmosphere as well as a more accurate atmospheric coherence time.
2. Temporal averaging of the scintillation
The theory of weak perturbations implies that the scintillation index s^{2} – the variance of the relative fluctuations of light intensity – is described by the sum of the scintillation indices produced by individual turbulent layers: (1)where W(h) is the weighting function (WF), which depends on the size and shape of the receiving aperture and does not depend on the altitude distribution of the structural coefficient of the refractive index . The WF represents a power of the scintillation generated by a layer of unit intensity located at a height h.
The MASS method involves simultaneous measurements of the scintillation indices in four concentric apertures of different diameters (hereafter A, B, C and D), which lead to 10 independent scintillation indices. The vertical OT profile is restored from the measured indices and the theoretically calculated WFs (Tokovinin et al. 2003; Kornilov et al. 2007). The calculation of a set of functions W(h) is based on the assumption that light intensity measurement has a zero exposure, i.e. the averaging factor is related only to the receiving aperture.
The scintillation index s^{2} measured with finite exposure time is determined by the integrated effect of all turbulent layers along the line of sight as in the case of zero exposure (1) but with another WFs.
The expression for the new WFs W′(w,τ,h) depends not only on the layer altitude h but also on the wind speed w = w(h) and averaging time τ (Tokovinin 2002; Kornilov 2011). It differs from the expression for W(h) with an additional multiplicand A_{s}(w,τ,f) in the integrand: (2)Here, f is the modulus of the spatial frequency, A(f) is spatial aperture filter (axisymmetric for MASS apertures), r_{F} is the Fresnel radius .
Multiplicand A_{s}(wτf) = A_{s}(ξ) is the spectral filter averaging with the wind shear wτ and can be expressed through the Bessel functions J_{0} and J_{1} and Struve functions H_{0} and H_{1} (Kornilov 2011): (3)It has a simple asymptotic behavior for low values of argument ξ = wτf: A_{s}(ξ) ≈ 1 − π^{2}ξ^{2}/6, which follows from its series expansion in the neighborhood of 0. The approximation provides an accuracy better than 0.02 until πξ < 1. When ξ → ∞, the function A_{s}(ξ) ≈ 1/πξ, and starting from ξ ≈ 1, the relative difference is less than 0.04.
These asymptotes correspond to the two extreme cases: the short and the long exposure regimes. The regime of short exposures (SE) considered earlier by Tokovinin (2002) will be the subject of this paper. The regime of long exposures (LE) was studied recently by Kornilov (2011) in the application of the potential impact of stellar scintillation on the accuracy of photometric measurements.
The main feature of these regimes is that the weighting function W′(w,τ,h) can be represented as the product of a function depending on wind shear and the function that independs on winds. It allows one to separate the wind effect and the geometry of light propagation.
3. Short exposures
In Eq. (2) the integrand is significantly different from 0 in the region of intersection of the aperture (f ≲ 1/D) and Fresnel (f ≲ 1/r_{F}) filters. Outside this domain, i.e. when f > min { 1/D,1/r_{F} } , the integrand rapidly tends to 0. If the wind shear τw ≪ max { D,r_{F} } then fτw ≲ 1 and hence a quadratic approximation 1 − π^{2}(wτf)^{2}/6 of the spatial filter A_{s}(w,τ,f) can be applied.
Let us replace A_{s} for a certain aperture with the quadratic approximation in Eq. (2) and take w and τ outside the integral over frequency. Then (4)In this difference the first integral is the scintillation power for zero exposure. The second integral (we denote it as ) is the atmospheric second moment of wind additionally weighted with U(h). The weighting function U(h) is obtained by multiplying the initial spatial spectrum by π^{2}f^{2} and subsequently integrating over f. Because the effect of highfrequency spectral components increases after this multiplication, the functions U(h) become essentially different from those usually used W(h) and have dimensions of m^{−7/3}. The set of U(h) functions is shown in Fig. 1.
Fig. 1 Weighting functions U(h) for a set of MASS apertures computed for a typical spectral response of its detectors and the light source of spectral class A0 V. The solid curves depict the WFs for normal indices, the dashed curves those for crossindices. The thin lines denote the functions for a number of smaller apertures, their diameters are marked in centimeters. 
Additional features of these WFs are as follows:

the spectral band width effect is more important than beforeowing to the highfrequency spectral components, which aremore intense. The difference of WFs for white and red starsreaches 10% in aperture A;

for an infinitely small aperture an asymptote does not exist because at D = 0 the integral diverges. However, there is an envelope of ~h^{−1/6} to which the curves converge as D → 0 (see Fig. 1);

the asymptote for the aperture D ≫ r_{F} is also very interesting: U(h) ≈ 17.22 λ^{−2/3} D^{3} h^{4/3}. Compared to the asymptotic behavior of the normal scintillation index, it is chromatic and has a stronger dependence on D.
Note that the WFs shown in the figure by the dashed lines are calculated for crossindices, i.e. the values obtained directly from the MASS instrument measurements rather than differential indices. Differential indices were always calculated from the normal indices and covariances in the form of linear combinations, but from the viewpoint of the OT restoration it is a superfluous intermediate step, which we have dismissed.
Fig. 2 Indices normalized to as function of an exposure τ. The bold curve corresponds to the median of γ_{12} = 0.938, the dashed line to the limit of the SE regime γ_{12} = 0.870. Other curves are marked by appropriate γ_{12}. The upper horizontal line represents the lower boundary of the SE regime, the lower horizontal line the upper limit of the LE regime. 
The SE regime and the LE regime are selfreproducible in the sense that if they occur in each turbulent layer they will be observed for the whole atmosphere. To determine the limits of the SE regime, one can use the condition of the applicability of the quadratic approximation of the wind shear filter A_{s}(ξ), namely that at its border the function is equal to 1 − π^{2}ξ^{2}/6 = 5/6.
Because for small wτ this function varies slowly with frequency f within the main spectral peak (its maximum is located at frequency ≲ 0.7/r_{F}) of the integrand, the value of will also slightly vary with wτ. Consequently we can assume that the quadratic approximation of the form (4) for a given aperture and atmospheric conditions is applicable while or (5)Of course, this inequality defines the SE regime only within some level of accuracy. One should take into account the error of approximation of the function A_{s}(ξ) (≈ 0.02) and the contribution of the secondary peaks of the scintillation spectrum. A rough estimate of the accuracy of the numerical factor in (5) is ≈ 0.05.
Similarly, it is possible to write the limit of the LE regime in the form of . Both boundaries are indicated in Fig. 2 with dashed lines. The figure demonstrates examples of the typical behavior of the measured indices with exposures ranging from 1 to 64 ms in different wind conditions. Note that the left and right branch of each curve can be generated by different turbulent layers because the contribution of an individual layer is ~ in the SE regime while it is ~ in the LE regime.
4. Verification of the applicability of the SE regime in the MASS measurements
Since the value is calculated and not measured the restriction (5) is not directly applicable. However instead of one can use an additional index measured with a shorter exposure τ′ < τ. Simple algebraic manipulations lead to the condition (6)It follows that for exposures τ = 2 ms and τ′ = 1 ms should be more than , for 3 and 1 ms and for 4 and 1 ms . Condition (6) cannot be used when exposure τ′ itself is too long for the SE regime.
The analysis of actual MASS data obtained at Mt. Shatdzhatmaz in the period May 2009 to June 2010 was performed with the usual exposure τ = 1 ms. The special version of the MASS software recorded additional data in output files to later calculate the indices with exposures 2, 4, 8, 16, 32 and 64 ms (see Fig. 2).
Cumulative distributions of ratios and averaged over a minute are shown in Fig. 3. As might be expected from the form of the WFs (Fig. 1), smaller ratios are observed for smaller apertures, which are more sensitive to the OT motion, and the distribution for the AB crossindex is very close to the curve for the aperture B. Numerical characteristics of the distributions are given in Table 1 and show that the 2ms measurement is almost always within the SE regime. Only 18% of the measurements in aperture A fail to meet the γ_{21} > 0.870 condition.
The characteristics of the distributions can vary greatly from season to season. For instance, during the February–March 2010 period, which was characterized by strong winds, the median γ_{21} decreased to 0.90, while the fraction of measurements that do not satisfy the criterion of SE increased to 38%. On the other hand, all measurements fell into the SE regime in October 2009, which was notable for its stable calm weather.
Characteristics of the γ_{21} and γ_{41} distributions for measurements at Mt. Shadzhatmaz in May 2009 − June 2010.
Fig. 3 Cumulative distributions of γ_{21} (black curves) and γ_{41} (gray curves) for the apertures A, B, C, D and AB crossindex based on measurements at Mt. Shatdzhatmaz. Vertical lines cut off the points that do not satisfy the SE regime. 
Characteristics of the γ_{21} and γ_{31} distributions for measurements at Mt. Maidanak in August 2005–December 2007.
For measurements with 4 ms exposure (value γ_{41}), the situation is radically different. In this case, the vast number of measurements does not satisfy the SE condition and consequently the quadratic approximation can be used only in rare cases and with caution.
A similar analysis was performed for measurements at Mt. Maidanak. The standard MASS data were used, from which indices and were calculated to construct the cumulative distribution of γ_{21} and . General properties of these distributions do not differ from those of Mt. Shatdzhatmaz. Some differences are caused by the significantly larger apertures C and D of the first generation device (Kornilov et al. 2003). Characteristics of the distributions are given in Table 2. The fraction of measurements dropping out of SE regime in the A aperture is 19.4% for exposure 2ms and 64% for 3ms exposure.
In the situation where γ_{21} approaches unity, a very high precision of the ratio is required because the curves for γ_{21} = 0.995 and γ_{21} = 0.99 differ very much. For these curves even 8 ms measurements are in the SE regime.
5. Reduction to zero exposure
Typical exposures of 1 ms are taken with the MASS. This leads to a wind shear on the order of 3 cm assuming a wind speed in the tropopause ~30ms^{1}. The value is comparable with the size of the device apertures and with typical Fresnel radius, so a 1 ms exposure cannot be considered to be an infinitely small one. The procedure used to correct indices to zero exposure is provided during the MASS data processing. The algorithm is based on numerical simulations (Tokovinin 2002).
However, for the measurements in the SE regime, the required correction can be calculated with the help of a direct method using the two indices and obtained with different exposures. Although the ratio of exposures may be arbitrary, it is convenient to consider the case of a single τ and a double 2τ exposures. Then the expression (4) becomes (7)After solving this system for , we obtain (8)One can express in terms of and covariance ρ_{1} of adjacent counts as . The formula can be rewritten as (9)The resulting correction is somewhat smaller than the one adopted in Kornilov et al. (2007): . Of course, formulas (8) and (9) are exact only if the condition (5) of the SE regime is satisfied for .
The statistical error of ρ_{1} is very close to that of . Consequently, the correction of the scintillation index increases its standard error by a factor of 1.18, which is not significant in practice.
6. Atmospheric second moment of wind
We can estimate the integral from the system of Eqs. (7). Solving the system for the indices measured in the japerture (j = 1,...,10) with exposures τ_{1} and τ_{2} relative to this unknown, we obtain (10)where Δ_{j} denotes the measured quantity. The integral is of little interest by itself because it includes an additional factor U_{j}(h), which distorts the contribution of different altitudinal layers. Real estimation of the second atmospheric moment of the wind can be obtained if we find a close to unity linear combination A_{U}(h) of the functions U_{j}(h).
This type of combination is shown in Fig. 4. Good results can even be obtained when only using functions corresponding to normal indices. The method and results of the decomposition A_{U}(h) ≈ 1 are described in detail in Appendix A.
Fig. 4 Approximation A_{U}(h) of a constant level with help of the set of U(h). Black curves (1) show the approximation with 10 indices, gray (2) with 4 indices. For clarity the gray graphs are shifted to the level of 0.8. Thick lines correspond to the weight of ~h, thin lines to the other power of altitude. Curve 3 corresponds to the weighting of ~h^{7/5}. Gray dashed 4 is the WF for the aperture A. 
Expansion coefficients of A_{U}(h) = ∑ _{j}c_{j}U_{j}(h) are on the order of 10^{15}m^{7/3}. Summing Eq. (10) for the indices with these coefficients, we obtain (11)Clearly the boundary layer (h < 0.5 km) wind will not be fully accounted for in the integral and the surface wind will be completely excluded despite the strong surface turbulence. The behavior of A_{U}(h) is similar to the approximating function under the integral defining the intensity of turbulence in the free atmosphere J_{free}. Therefore this quantity can be used for further normalization. The contribution of the surface layer can be accounted for later on base of the DIMM data.
Dividing both sides of the expression (11) by J_{free}, we obtain the mean square of the wind speed ⟨ w^{2} ⟩ in the free atmosphere and its expression in terms of measurable quantities: (12)
7. Atmospheric coherence time
It was shown by Kellerer & Tokovinin (2007) that the mean wind can be a good estimator of , which is used in the definition of the atmospheric coherence time τ_{0}. We recall that the instrument FADE (Fast Defocusing of stellar image, Tokovinin et al. 2008) measures the time constant (interferometric coherence time) also using . Although the authors indicate that on average , we will assume that both values are equivalent in the following argumentation.
Evaluation of (in the free atmosphere) from the measurements is obtained directly from the formula for ⟨ w^{2} ⟩ : (13)Substituting this value in the definition of the atmospheric time constant τ_{0}, we obtain the following expression: (14)where we substituted the Fried parameter r_{0} with OT intensity J_{tot} in the whole atmosphere. For the wavelength λ = 500 nm: (15)If there is information about the OT power in the boundary layer and surface wind speed V_{0}, its contribution can be accounted for at the stage of calculating the mean wind . In order to do this, we correct the mean square of the wind speed by adding in proportion of the surface layer intensity J_{GL}: (16)or (17)Substituting the corrected value in formula (15), we obtain an estimation of the atmospheric coherence time for the whole atmosphere: (18)Note that the usual method of adding coherence time is not correct if used to calculate , which has square metric.
The relative accuracy of τ_{0} does not practically depend on the accuracy of the J_{tot}, and is twice as good as the accuracy of the measured Δ_{j}.
8. Verification of the method of evaluation τ_{0}
The method presented here was tested with data obtained at Mt. Shadzhatmaz and Mt. Maidanak, which were analyzed in Sect. 4. To construct ∑ _{j}c_{j}Δ_{j}, we used the coefficients from the Table A.1 defined for stars of spectral class A0 V. This simplification introduces a systematic error when processing the measurements of stars of other spectral classes, which is why further results should not be regarded as final (less than 5%, see Sect. 3). We also computed the coherence time for free atmosphere, i.e. the J_{free} value was used instead of J_{tot} in formula (15).
For the measurements at Mt. Shadzhatmaz, the differences Δ_{j} were computed from indices and . The results of the evaluation are shown in Fig. 5, where the cumulative distributions of τ_{0} obtained with different decompositions are presented. The method results in 6.69 ms median when using 4 normal indices and 6.53 ms using all 10 indices. The difference between these distributions is completely explained by the fact that usage of all indices involves an additional contribution of moving turbulence between 0.5 and 1 km.
Fig. 5 Cumulative distribution of free atmosphere τ_{0} from measurements at Mt. Shatdzhatmaz obtained using differences with decomposition in 4 (solid line) and 10 (dashed line) indices. 
Fig. 6 Cumulative distribution of free atmosphere τ_{0} from measurements at Mt. Maidanak obtained using DESI (grey dashed – before software correction, grey solid – after) and using and differences (black solid and dashed curves respectively). 
For the standard MASS output (Maidanak campaign), the normal indices , , and were used to study biasing of τ_{0} estimations with a different temporal span. At first we investigated the effect of the DESI formula correction, i.e. a replacement of the index with the index . As can be seen from Fig. 6, the correction increases the τ_{0} estimations by a factor ≈ 1.4. The cumulative distributions of the coherence time calculated from and are also presented in the figure. Evidently that the median differs only slightly for the corrected DESI algorithm (5.82 ms) and the 2 ms median (5.98 ms).
The τ_{0} values obtained with are systematically higher by ≈ 0.6 ms than τ_{0} calculated with . This shift is caused by the 3 ms exposure beyond the SE regime. When the τ_{0} increases, the greater number of the 3 ms measurements satisfies the criterion of SE and because of that, the relative difference between the curves for 2 ms and 3 ms is reduced, becoming less than 4% at τ_{0} ≈ 15 ms. To obtain the systematic shift of the 2 ms curve, additional measurements with a shorter exposure are required. However, taking into account the quadratic dependence of indices on exposure, we can estimate that its median is not overstated by more than 5%.
Random errors of τ_{0} were computed during the averaging of over 1 min as the standard error of the mean. The relative error of is approximately constant in the entire range of values. At very low winds (less than 5ms^{1}) the errors increase. The median of the relative error is ≈ 0.02, only 1% of the measurements have errors greater than 0.1.
9. Discussion and conclusion
The analysis of actual MASS measurements described in Sect. 4 has shown that for the commonly used 1 ms exposure, 80% of the cases are within the SE regime and that the correction to zero exposure (9) is suitable. Despite some conventionality of the regime boundary definition, it cannot be significantly weakened. On the contrary, the study of real data requires some tightening of the limit to guarantee that all turbulent layers are within the SE regime. Our statistics shows that exposures of 3 ms rarely satisfy the criterion of SE.
Such a control is important because measurements of τ_{0} in large wind shear will be affected by systematic errors. Fortunately these cases are not as interesting as the opposite ones.
The reaction of the DESI method on fast turbulence motion is not so evident. One can derive from the definition of the differential exposure signal (Tokovinin 2011) and formula (10): (19)This means that the DESI method estimates the coherence time not by using the atmospheric wind moment, but its derivative with exposure. The DESI differences are smaller by an order of magnitude than the values themselves and therefore the DESI method is noisier. Indeed, for Mt. Maidanak data the median of relative random errors is ≈ 0.03 using the difference , while using DESI the median of relative errors amounts ≈ 0.1.
The case of a calm atmosphere is most interesting from the viewpoint of adaptive optics and interferometry. In this situation the differences of the indices are small and poorly defined owing to an increase of the relative error. The most dangerous is a systematic error due to an incorrect accounting for photon noise. To reduce the impact of these errors, an adaptable choice of exposure may be realized (Tokovinin 2011).
Fortunately, in real situations this effect is not too large. For example, measurements at Mt. Maidanak show that an error Δp = 0.01 in the parameter p for photon noise correction changes the τ_{0} median by 0.05 ms. At τ_{0} ≈ 15 ms (95% level of cumulative distribution) this error results in a shift of the curve by 0.7 ms when used and 0.3 ms when .
Generally speaking, the SE regime is not very suitable to determine wind conditions. Much more accurate results can be obtained by the determination of the averaging time t for which the index becomes equal to . This method is similar to the calculation of correlation peak width, see e.g., Caccia et al. (1987). However, in the middle domain (see Fig. 2) the behavior of indicates that similar integral characteristics are closer to the mean wind speed ⟨ w ⟩ than to the mean square ⟨ w^{2} ⟩ . The mean speed may be used for some applications, but for the τ_{0} evaluation the value ⟨ w^{2} ⟩ is more suitable.
The analyzed data were obtained at sites with rare highaltitude jet streams. For other observatories, especially those closer to the equator, the situation could be worse. There it will make sense to reduce the exposure to 0.5−0.7 ms, which is equivalent to increasing the wind range by 1.5−2 times. The main objective of a correct choice of exposure is to ensure that we are within the SE regime for a statistically overwhelming fraction of cases.
The main advantage of making measurements in the SE regime is the simplicity of the theoretical description and its practical usage. The described method for a τ_{0} evaluation does not contain any empirical calibration, it gives more accurate estimations and additional information about the mean wind in the free atmosphere.
All quantities on the right hand side of Eq. (15) are measured with MASS/DIMM simultaneously. However, the joint MASS/DIMM data processing was realized only in the recent version of the program atmos (Kornilov et al. 2010) and the algorithm described in Sect. 7 will be implemented in the next release of this version. An alternative way is to use a fairly simple postprocessing of existing MASS output, but in both cases the meteo data on the surface wind are additionally needed to take the ground layer into account. Free atmosphere τ_{0} evaluation using both methods is implemented starting with the atmos2.97.3 version.
Acknowledgments
The author thanks his colleagues who took part in campaigns of optical turbulence measurements at Mt. Maidanak and Mt. Shatdzhatmaz, by whose efforts the data set used in the analysis was collected. Concerned discussion on the atmospheric coherence time from MASS data with participants of the conference “Comprehensive characterization of astronomical sites” also stimulated the implementation of this work. The author is particularly grateful to A. Tokovinin and N. Shatsky for useful discussions, T. Travouillon for his efforts to make the paper clearer and M. Sarazin for his interest in the appearance of this work.
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Appendix A: Decomposition in the weighting functions
The decomposition of the atmospheric wind moment is done in the same way as an expansion of altitude atmospheric moments with sets of measured indices (Tokovinin et al. 2003) in the program atmos. In this case we solve the system of linear equations Uc = 1, where U is the WFs matrix with the dimension k × n, c is the vector of coefficients, 1 is the unit vector corresponding to the altitude grid h. The number of nodes n = 50 is significantly greater than the maximal number of indices k = 10. As usual a loguniform grid is used, which has more nodes at low altitudes.
Coefficients c_{j} (10^{15}m^{7/3}) of the h^{0} decomposition in WFs U(h) when we used all indices, 4 normal indices, 4 normal indices for the original MASS device – Cols. 2, 3, 4 respectively.
The system is solved by SVD with a regularization of the solution by discarding low singular values s_{i} < 10^{3} s_{0}. The quality of the solution was controlled by two parameters: the maximum deviation from 1 (not accounting for the initial part of the curve) and the noise enhancement factor . The solution depends on how the system was weighted. We used an implicit weight by grid density and some explicit weight.
Without the explicit weight, the solution has the first maximum at the minimum altitude but strongly oscillates and has large NF > 20. The study of additional weight in the form of ~h^{p} has shown that if p ≈ 1 the solution is good enough: it is close to 1 and has a moderate NF value.
The approximating curves A_{U}(h) calculated with all 10 indices and only 4 normal indices are shown in Fig. 4. The curve for aperture A, for which an index and temporal covariances are used in the DESI method to assess the atmospheric coherence time, is also shown. The coefficients c_{j} are presented in Table A.1 for all variants. They were calculated for a typical set of MASS apertures, its detector response and A0 V spectral class of light source.
The approximation that uses all indices is close to 1 for all nodes of the typical grid for restoration of the OT profile. There is a slight (≈ 0.06) excess at the altitude of 0.7 km and the difference for other nodes is lower. The approximation with only 4 normal index looses 25% of the contribution of 1 km layer and almost 75% of the contribution of 0.5 km layer. It is expected that the decomposition in the complete set of indices will give an estimation of τ_{0} with smaller systematic errors because the noise properties of crossindices are not worse than those of normal indices and the factor NF in the first variant is only twice as large.
All Tables
Characteristics of the γ_{21} and γ_{41} distributions for measurements at Mt. Shadzhatmaz in May 2009 − June 2010.
Characteristics of the γ_{21} and γ_{31} distributions for measurements at Mt. Maidanak in August 2005–December 2007.
Coefficients c_{j} (10^{15}m^{7/3}) of the h^{0} decomposition in WFs U(h) when we used all indices, 4 normal indices, 4 normal indices for the original MASS device – Cols. 2, 3, 4 respectively.
All Figures
Fig. 1 Weighting functions U(h) for a set of MASS apertures computed for a typical spectral response of its detectors and the light source of spectral class A0 V. The solid curves depict the WFs for normal indices, the dashed curves those for crossindices. The thin lines denote the functions for a number of smaller apertures, their diameters are marked in centimeters. 

In the text 
Fig. 2 Indices normalized to as function of an exposure τ. The bold curve corresponds to the median of γ_{12} = 0.938, the dashed line to the limit of the SE regime γ_{12} = 0.870. Other curves are marked by appropriate γ_{12}. The upper horizontal line represents the lower boundary of the SE regime, the lower horizontal line the upper limit of the LE regime. 

In the text 
Fig. 3 Cumulative distributions of γ_{21} (black curves) and γ_{41} (gray curves) for the apertures A, B, C, D and AB crossindex based on measurements at Mt. Shatdzhatmaz. Vertical lines cut off the points that do not satisfy the SE regime. 

In the text 
Fig. 4 Approximation A_{U}(h) of a constant level with help of the set of U(h). Black curves (1) show the approximation with 10 indices, gray (2) with 4 indices. For clarity the gray graphs are shifted to the level of 0.8. Thick lines correspond to the weight of ~h, thin lines to the other power of altitude. Curve 3 corresponds to the weighting of ~h^{7/5}. Gray dashed 4 is the WF for the aperture A. 

In the text 
Fig. 5 Cumulative distribution of free atmosphere τ_{0} from measurements at Mt. Shatdzhatmaz obtained using differences with decomposition in 4 (solid line) and 10 (dashed line) indices. 

In the text 
Fig. 6 Cumulative distribution of free atmosphere τ_{0} from measurements at Mt. Maidanak obtained using DESI (grey dashed – before software correction, grey solid – after) and using and differences (black solid and dashed curves respectively). 

In the text 
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