## On the normalization of scintillation autocovariance for generalized SCIDAR

Optics Express, Vol. 17, Issue 13, pp. 10926-10938 (2009)

http://dx.doi.org/10.1364/OE.17.010926

Acrobat PDF (214 KB)

### Abstract

The Generalized SCIDAR (Scintillation Detection and Ranging) technique consists in the computation of the mean autocorrelation of double-star scintillation images taken on a virtual plane located a few kilometers below the telescope pupil. This autocorrelation is normalized by the autocorrelation of the mean image. Johnston *et al*. in 2002 [*C*^{2}_{N}. Those authors restricted their analysis to turbulence at ground level. Here we generalize that study by calculating analytically the error induced by that normalization, for a turbulent layer at any altitude. An exact expression is given for any telescope–pupil shape and an approximate simple formula is provided for a full circular pupil. We show that the effect of the inexact normalization is to overestimate the *C*^{2}*
_{N}
* values. The error is larger for higher turbulent layers, smaller telescopes, longer distances of the analysis plane from the pupil, wider double-star separations, and larger differences of stellar magnitudes. Depending on the observational parameters and the turbulence altitude, the relative error can take values from zero up to a factor of 4, in which case the real

*C*

^{2}

*value is only 0.2 times the erroneous one. Our results can be applied to correct the*

_{N}*C*

^{2}

*profiles that have been measured using the Generalized SCIDAR technique.*

_{N}© 2009 Optical Society of America

## 1. Introduction

*C*

^{2}

_{N}(

*h*), is a key parameter in the field of high angular-resolution imaging through the atmosphere. For example, in optical astronomy, measurements of

*C*

^{2}

_{N}(

*h*) are required to develop adaptive optical systems for ground-based telescopes and to evaluate existing or potential astronomical sites.

*h*, strength

*C*

^{2}

_{N}and thickness

*δh*is given by

*r*stands for the modulus of the position vector

**r**and

*K*(

*r,h*) is given by the Fourier transform of the power spectrum

*W*of the irradiance fluctuations. An expression of

_{I}*W*can be found in Reference [8

_{I}8. F. Roddier, “The Effect of Atmospheric Turbulence in Optical Astronomy,” Progress in Optics **XIX**, 281–376 (1981).
[CrossRef]

*K*(

*r,h*) [7

7. J.-L. Prieur, G. Daigne, and R. Avila, “SCIDAR measurements at Pic du Midi,” Astron. Astrophys. **371**, 366–377 (2001).
[CrossRef]

*k*=2

*π/λ*and

*u*is the spatial frequency. It is worth noting that the scintillation variance,

*𝓒*(0), is proportional to

*h*

^{5/6}, as can be easily deduced from Eqs. 1 and 2 by changing the integration variable to

*ξ*=

*h*

^{1/2}

*f*.

*θ*and crossing a turbulent layer at an altitude

*h*casts on the ground two identical scintillation patterns shifted from one another by a distance

*θh*. The spatial autocovariance of the compound scintillation exhibits peaks at positions

**r**=±

*θh*with an amplitude proportional to the

*C*

^{2}

_{N}value associated to that layer. The determination of the position and amplitude of those peaks leads to

*C*

^{2}

_{N}(h). This is the principle of the so-called Classical SCIDAR (CS), in which the scintillation is recorded at ground level by taking images of the telescope pupil while pointing a double star. As the scintillation variance produced by a turbulent layer at an altitude

*h*is proportional to

*h*

^{5/6}, the CS is blind to turbulence close to the ground, which constitutes a major disadvantage because the most intense turbulence is often located at ground level [9

9. R. Avila, E. Masciadri, J. Vernin, and L. Sánchez, “Generalized SCIDAR measurements at San Pedro Mártir: I. Turbulence profile statistics,” Publ. Astron. Soc. Pac. **116**, 682–692 (2004).
[CrossRef]

10. M. Chun, R. Wilson, R. Avila, T. Butterley, J.-L. Aviles, D. Wier, and S. Benigni, “Mauna Kea ground-layer characterization campaign,” Mon. Not. R. Astron. Soc. **394**, 1121–1130 (2009).
[CrossRef]

*et al*. in 1994 [11] proposed to optically shift the measurement plane a distance

*d*below the pupil. For the scintillation variance to be significant,

*d*must be of the order of 1 km or larger. This is the principle of the Generalized SCIDAR (GS) which was first implemented by Avila

*et al*. in 1997 [12

12. R. Avila, J. Vernin, and E. Masciadri, “Whole atmospheric-turbulence profiling with generalized Scidar,” Appl. Opt. **36(30)**, 7898–7905 (1997).
[CrossRef]

*h*produces autocovariance peaks at positions

**r**=±

*θ*(

*h*+

*d*), with an amplitude proportional to (

*h*+

*d*)

^{5/6}. The cut of the peak centered at

**r**=

*θ*(

*h*+

*d*), along the direction of the double-star separation is given by

*h*between −

*d*and 0,

*C*

^{2}

_{N}(

*h*)=0 because that space is virtual. To invert Eq. 4 and determine

*C*

^{2}

_{N}(

*h*), a number of methods have been used like Maximum Entropy [12

12. R. Avila, J. Vernin, and E. Masciadri, “Whole atmospheric-turbulence profiling with generalized Scidar,” Appl. Opt. **36(30)**, 7898–7905 (1997).
[CrossRef]

6. V. A. Klückers, N. J. Wooder, T. W. Nicholls, M. J. Adcock, I. Munro, and J. C. Dainty, “Profiling of Atmospheric Turbulence Strength and Velocity Using a Generalised SCIDAR Technique,” Astron. Astrophys. Suppl. Ser. **130**, 141–155 (1998).
[CrossRef]

1. R. A. Johnston, C. Dainty, N. J. Wooder, and R. G. Lane, “Generalized scintillation detection and ranging results obtained by use of a modified inversion technique,” Appl. Opt. **41(32)**, 6768–6772 (2002).
[CrossRef]

7. J.-L. Prieur, G. Daigne, and R. Avila, “SCIDAR measurements at Pic du Midi,” Astron. Astrophys. **371**, 366–377 (2001).
[CrossRef]

13. R. Avila, J. L. Avilés, R. Wilson, M. Chun, T. Butterley, and E. Carrasco, “LOLAS: an optical turbulence profiler in the atmospheric boundary layer with extreme altitude-resolution,” Mon. Not. R. Astron. Soc. **387**, 1511–1516 (2008).
[CrossRef]

*λ*is the wavelength, and the maximum altitude for which

*C*

^{2}

_{N}values can be measured [13

13. R. Avila, J. L. Avilés, R. Wilson, M. Chun, T. Butterley, and E. Carrasco, “LOLAS: an optical turbulence profiler in the atmospheric boundary layer with extreme altitude-resolution,” Mon. Not. R. Astron. Soc. **387**, 1511–1516 (2008).
[CrossRef]

*D*is the telescope pupil diameter.

*h*by

*h*sec(

*z*).

## 2. Analytical development

### 2.1. Autocovariance of double-star scintillation images

*h*above the telescope pupil plane. The detector plane is made the conjugate of a plane at a distance

*d*below the pupil plane. The instantaneous image produced by a single star on the detector can be written as:

*A*represents the mean irradiance,

*f*(

**r**) is the random spatial modulation factor produced by the scintillation,

**r**is the position vector on the detection plane and

*P*(

**r**) stands for the pupil irradiance transmittance. The diffraction of the pupil due to the propagation distance

*d*is neglected, to disentangle this effect from the problem investigated here. A brief discussion of the effect of pupil diffraction in GS-kind measurements can be found in reference [13

13. R. Avila, J. L. Avilés, R. Wilson, M. Chun, T. Butterley, and E. Carrasco, “LOLAS: an optical turbulence profiler in the atmospheric boundary layer with extreme altitude-resolution,” Mon. Not. R. Astron. Soc. **387**, 1511–1516 (2008).
[CrossRef]

*A*is considered to be constant,

*f*is a dimensionless, real, stationary, homogeneous, isotropic and ergodic random function with mean value equal to 1, and

*P*is equal to 1 inside the pupil and 0 outside.

*and the individual stellar irradiances are*

**θ***A*

_{1}and

*A*

_{2}, the image can be written as:

*f*is shifted due to the image plane displacement (

*d*) and the layer altitude (

*h*), whereas the pupil

*P*is shifted only due to

*d*. This feature is illustrated in Fig. 1. The particular case of

*h*= 0 – which leads to identical shifts for the off-axis terms

*f*and

*P*– was analyzed by Johnston

*et al*. [1

1. R. A. Johnston, C. Dainty, N. J. Wooder, and R. G. Lane, “Generalized scintillation detection and ranging results obtained by use of a modified inversion technique,” Appl. Opt. **41(32)**, 6768–6772 (2002).
[CrossRef]

*A*

_{1}and

*P*are constants, we have:

*f*, 〈

*f*(

*)*

**ρ***f*(

*+*

**ρ***)〉 is only a function of*

**r****r**and can therefore be out of the integral over

*ρ*. Defining

*f*〉=1.

*C*(

**r**) is given by Eq. 5 and the footprint

*S*(

**r**) is equal to twice the telescope pupil diameter 2

*D*. Thus

*C*(

**r**) is much narrower than

*S*(

**r**).

### 2.2. Autocorrelation of the average of the double-star scintillation images

*f*〉=1 and

*A*

_{1},

*A*

_{2}and

*P*are constant. Applying similar considerations as those who led to Eq. 16, the autocorrelation of the mean image is written as

_{〈I〉}(

**r**) along the double star separation, for three ranges of values of

*dθ*that will be of interest below.

### 2.3. Normalization

*C*(

**r**−

*(*

**θ***d*+

*h*))-1 or

*C*(

**r**+

*(*

**θ***d*+

*h*))-1 (cf. Eq. 15), from which

*C*

^{2}

_{N}(

*h*) can be retrieved. For that purpose, the procedure generally used in scidar-like experiments consists of the following computation:

*dθ*. In the first two, the operation performed with Eq. 19 indeed cancels out the terms corresponding to the pupil autocorrelation, leaving the scintillation autocovariance term isolated:

**r**), one can determine either of the lateral autocovariance peaks, as long as they are not superimposed, i.e. as long as

*θh*is larger than the peaks width

*L*(see Eq. 5). This case is represented in Fig. 2(a). It is useful to express

*a*and

*b*in terms of the difference of the magnitudes of the two stars Δ

*m*≡|

*m*

_{1}−

*m*

_{2}|. Knowing that

*m*

_{1,2}≡-2.5log

_{10}(

*A*

_{1,2}) and defining

*α*≡

*A*

_{1}/

*A*

_{2}=10

^{−0.4Δm}, one can show that

**• Case 2.**For

*dθ*≥

*D*, like in the Low Layer SCIDAR [13

**387**, 1511–1516 (2008).
[CrossRef]

*C*(

**r**+

*(*

**θ***d*+

*h*)) and

*C*(

**r**−

*(*

**θ***d*+

*h*)) lie further apart from the origin than the region where

*S*(

**r**) is non-zero. This can be seen in Fig. 2(b) where

*S*(

**r**) is represented by the magenta dashed line and the scintillation covariance peaks are the red curves. The lateral peaks lie in zones where the black line is overlapped with the green and yellow lines, i.e., where only

*S*(

**r**+

*) or*

**θ**d*S*(

**r**−

*) contribute to Γ*

**θ**d_{〈I〉}(

**r**). In this case the three additive terms of Eqs. 16 and 18 can be treated separately in Eq. 19, yielding:

_{c}(

**r**), Λ

_{l}(

**r**) and Λ

_{r}(

**r**), whereas if

*d*=0 they do appear in the expression of Λ(

**r**).

**• Case 3.**For 0<

*dθ*<

*D*, like in most of the GS measurements, the lateral scintillation covariance peaks lie in zones where the central pupil autocorrelation

*S*(

**r**) overlap the lateral pupil autocorrelations

*S*(

**r**+

*) and*

**θ**d*S*(

**r**−

*) (see Fig. 2c). Within those zones of overlapping, Γ*

**θ**d_{〈I〉}(

**r**) (black line) is larger than

*S*(

**r**+

*) and*

**θ**d*S*(

**r**−

*). Thereby, the normalization performed using Eq. 19 no longer results in the cancellation of those terms. To prove that analytically, without loss of generality one can focus on the zone of influence of the right-hand-side covariance peak, namely,*

**θ**d*r*>

*θ d*. Assuming that

*θ d*is much larger than the width

*L*of the covariance peaks (Eq. 5), in this zone, the other two covariance peaks

*C*(

**r**) and

*C*(

**r**+

*(*

**θ***d*+

*h*)) are equal to 1 (see Eq. 12 and remember that 〈

*f*〉=1). Hence, when substituting Eqs. 16 and 18 into the numerator of Eq. 19, the terms containing

*S*(

**r**) and

*S*(

**r**+

*) cancel out. Equation 19 can then be written as:*

**θ**d*S*(

**r**) and

*S*(

**r**+

*), the term*

**θ**d*S*(

**r**−

*) in the numerator cannot be canceled. The effect of this inexact normalization is calculated in § 3.*

**θ**d1. R. A. Johnston, C. Dainty, N. J. Wooder, and R. G. Lane, “Generalized scintillation detection and ranging results obtained by use of a modified inversion technique,” Appl. Opt. **41(32)**, 6768–6772 (2002).
[CrossRef]

*h*=0) and that the condition they impose for the case corresponding to our case 2 is stronger:

*dθ*>2

*D*.

*x*)=Λ

_{‖}(

*x*)-Λ

_{‖}(

*x*), where

*x*is the abscissa of the axis aligned along the direction of

*, Λ*

**θ**_{‖}(

*x*)=Λ(

*x*,0) and Λ

_{⊥}(

*x*)=Λ(0,

*x*). This procedure was introduced for the CS [3

3. A. Rocca, F. Roddier, and J. Vernin, “Detection of atmospheric turbulent layers by spatiotemporal and spatioangular correlation measurements of stellar-light scintillation,” J. Opt. Soc. Am. A **64**, 1000–1004 (1974).
[CrossRef]

*θ*(

*d*+

*h*)>2

*L*), which is usually the case, there is no need of performing that operation. Nevertheless, it is often done. It can easily be shown that in that case, Λ

_{⊥}(

*x*)=0 and Λ

_{‖}(

*x*) is the one-dimensional expression of Eq. 29. So, our analysis is valid for the GS data reductions based on the function Ψ(

*x*).

*x*) can also be obtained by calculating the difference of the parallel and perpendicular cross-sections of Γ

*and divide the result by the parallel cross-section of Γ*

_{I}_{〈I〉}. However, if this procedure is used in the GS, the result cannot be reduced to Eq. 29. Therefore, our analysis does not account for that data reduction procedure.

## 3. Error induced by the inexact normalization in GS

**r**), defined in Eq. 19, has been thought to be equal to

*θd*.

*F*(

**r**) is indeed what one wants to estimate experimentally in GS measurements. However, the quantity that has actually been measured is Eq. 29. The corresponding relative error is:

*a*and

*b*(Eqs. 21 and 22), it is straight forward to show that

*r*>

*θd*). For the left-hand-side peak, one interchanges −

*and +*

**θ**d*, yielding the same result because*

**θ**d*S*(

**r**) is an even function.

*F*(

**r**) can be estimated from the measurement of Λ(

**r**) and the calculation of

*ε*(

**r**), which involves either analytical or experimental determination of the pupil autocorrelation

*S*(

**r**) and the knowledge of the conjugation altitude

*d*and the double star parameters

*θ*,

*a*and

*b*.

*S*(

**r**) can be calculated for any pupil function

*P*(

**r**) using the Wiener-Khinchin theorem [25] which states that

*𝓕*and

*𝓕*

^{−1}are the Fourier transform and its inverse operators.

*ε*is calculated for the particular case of a circular pupil without central obscuration. The interest of having such an expression is to understand the influence of the observational parameters on ε and to be able to make a rapid estimate of its value. In § 3.2, exact values of ε are computed for pupils obscured by a secondary mirror and for different values of the observational parameters. The reader interested only in the exact values of

*ε*can skip § 3.1.

### 3.1. Approximate expression for a full circular pupil

*D*as:

*P*(

**r**) equal to

*𝓓*(

**r**,

*D*). The autocorrelation of

*P*(

**r**) is computed using Eq. 33. Let

*S*

_{𝓓}(

**r**) be the autocorrelation for the case of

*P*(

**r**)=

*𝓓*(

**r**,

*D*). As

*P*(

**r**) is centro-symmetric, so is

*S*(

_{𝓓}**r**). A diametral cut of

*S*(

_{𝓓}**r**) is shown in Fig. 3(a). To simplify Eq. 32 we can make the following approximation:

*r*>

*θd*, which is the case for Eq. 32, all the arguments of

*S*in that equation are positive. We can then ignore the absolute value in Eq. 35. In the substitution of

*S*(

*r*) by

*S*(

_{𝓓}*r*) in Eq. 32, attention must be paid for the cases where

*SD*vanishes. Three different situations can occur:

*S*(

_{𝓓}*r*+

*θd*) nor

*S*(

_{𝓓}*r*) nor

*S*(

_{𝓓}*r-θd*) are null. This occurs for

*r*+

*θd*<

*D*.

*S*(

_{𝓓}*r*+

*θd*)=0 but

*S*(

_{𝓓}*r*)≠0 and

*S*(

_{𝓓}*r*−

*θd*)≠0, which takes place when

*r*+

*θd*≥0 but

*r*<

*D*.

*S*(

_{𝓓}*r*+

*θd*)=

*S*(

_{𝓓}*r*)=0 and

*S*(

_{𝓓}**r**−

*θd*)≠0, which happens if

*r*+

*θd*≥0,

*r*≥

*D*but

*r*−

*θd*<

*D*.

*a*+2

*b*=1, simple algebraic manipulations lead to:

*r*=

*θ*(

*d*+

*h*) (see Eq. 16 and Fig. 2). Replacing r by this expression in Eq. 36 gives

*h*, as defined in Eq. 6, has been introduced.

_{max}*θd*increases then the relative error increases. Note From Eq. 24 that the maximum value of

*b*is 1/4, which occurs when Δ

*m*=0. Therefore 1-2

*b*>0. Moreover, 1−

*b*>

*b*. Thus, in the second case, if

*θ h*increases, the denominator decreases faster than the numerator, implying that

*ε*increases. In the first case too, if

_{𝓓}*θ h*increases

*ε*increases. Concerning the dependence on

_{𝓓}*D*, similar arguments lead to the conclusion that if

*D*increases, the error decreases. In the third case, the relative error is independent of

*θ, d, h*and

*D*. From Eq. 36 it can be deduced that in the three cases,

*ε𝓓*>0. This means, from Eq. 31, that Λ(

**r**)>

*F*(

*r*), resulting in an overestimation of the turbulence intensity.

*ε*and

_{𝓓}*ε*as a function of

*h*, for

*D*=1 m,

*d*=4 km,

*b*=1/4 and different values of

*θ*. The altitude takes values within the height-range where the optical turbulence strength can be significant in the atmosphere. The exact values of the relative error ε are computed following Eqs. 32 and 33 and the relation

*r*=

*θ*(

*d*+

*h*). Comparing the solid and dashed lines for a given value of

*θ*, it can be seen that the exact and approximate expressions of the relative error follow the same trend and give similar values. The mean relative difference between

*ε*and

*ε*for the values plotted in Fig. 4 is 23%. The curves saturate at (

_{𝓓}*b*−1)/

*b*and are plotted only for

*h*<

*h*

_{max}for each value of

*θ*.

### 3.2. Exact values for a circular pupil with central obscuration

*S*of such a pupil is different from that of a non-obscured pupil, as shown in Fig. 3. This affects the value of

*ε*. In Fig. 4b, solid lines represent

*ε*for the same conditions as in Fig. 4a but with a central obscuration of diameter 0.3

*D*, which is a common value for optical telescopes. Like in Fig. 4a, dashed lines represent

*ε*. The mean and maximum difference between the exact and the approximate computations – for the parameter ranges considered in Fig. 4b – are 36% and 77% respectively, which are non-negligible. Nevertheless, if one wishes a rapid estimate of ε within a factor of 1.77 with respect to the exact value, Eq. 37 is a useful expression. An interesting feature is that

_{𝓓}*ε*does not increase monotonically as

*h*increases, but has a local minimum.

*ε*(

*h*). In each panel all the parameters but one remain constant. One of those parameters,

*e*, is defined as follows:

*eD*is the diameter of the secondary mirror. By examining Figs. 4b, 5a, 5b, 5c and 5d one can understand the effect of each parameter:

*h, θ, D, d, e*and

*b*. There appears to be two characteristic altitudes in each plot: the error remains mostly constant up to a given altitude

*h*

_{1}, then it increases rapidly as

*h*increases up to another altitude

*h*

_{2}from which

*ε*remains strictly equal to (1−

*b*)/

*b*until

*h*reaches

*h*

_{max}. For example, in Fig 5b, for

*d*=10 km (red curve),

*h*

_{1}~16 km and

*h*

_{2}~30 km. Those “cut-off” altitudes

*h*

_{1}and

*h*

_{2}depend strongly on

*D*and

*θ*, less strongly on

*d*and seem to be independent of

*e*and

*b*. For a given altitude of the turbulence, closer double stars and/or larger telescopes and/or analysis plane closer to the pupil give smaller errors, unless the saturation regime is reached. Larger secondary shadows produce larger oscillations of

*ε*(

*h*) for

*h*<

*h*

_{1}.

*h*=2 km, using a double star of separation

*θ*=5″, equal-magnitude components (

*b*=0.25) on a 2-m telescope with a secondary mirror of diameter 0.6 m and the analysis plane at

*d*=2 km, the relative error is

*ε*=0.048, whereas for

*h*=22 km,

*θ*=8″, stars with 1 magnitude difference (b=0.203),

*D*=1m,

*e*=0.3 and

*d*=4

*km*, the relative error is

*ε*=3.86. In such a case, considering Eq. 31 and the proportionality between

*C*

^{2}

_{N}and the scintillation autocovariance (Eq. 1), to obtain the correct

*C*

^{2}

_{N}value one would have to multiply the erroneous value by 1/(1+3.86)~0.21.

## 4. Conclusion

*et al*. in 2002 [1

**41(32)**, 6768–6772 (2002).
[CrossRef]

*h*. We provide a procedure to calculate and correct the error induced by the inaccurate normalization for pupils of any shape and give an approximate analytical expression of that error in the case of a circular pupil with no central obscuration. The approximate expression is useful to make a rapid estimate of the error and to understand the influence of the observational parameters. For an accurate correction of

*C*

^{2}

_{N}(

*h*) measurements the exact expression of the error (Eq. 32) should be used. It has been shown that the error is more significant for higher turbulent layers, smaller telescopes, longer distances of the analysis plane from the pupil, wider double-star separations, and larger differences of stellar magnitudes.

## Acknowledgments

## References and links

1. | R. A. Johnston, C. Dainty, N. J. Wooder, and R. G. Lane, “Generalized scintillation detection and ranging results obtained by use of a modified inversion technique,” Appl. Opt. |

2. | J. Vernin and F. Roddier, “Experimental determination of two-dimensional spatiotemporal power spectra of stellar light scintillation. Evidence for a multilayer structure of the air turbulence in the upper troposphere,” J. Opt. Soc. Am. A |

3. | A. Rocca, F. Roddier, and J. Vernin, “Detection of atmospheric turbulent layers by spatiotemporal and spatioangular correlation measurements of stellar-light scintillation,” J. Opt. Soc. Am. A |

4. | J. Vernin and M. Azouit, “Traitement d’image adapté au speckle atmosphérique. I-Formation du speckle en atmosphère turbulente. Propriétés statistiques,” J. Opt. (Paris) |

5. | J. Vernin and M. Azouit, “Traitement d’image adapté au speckle atmosphérique. II-analyse multidimensionnelle appliquée au diagnostic à distance de la turbulence,” J. Opt. (Paris) |

6. | V. A. Klückers, N. J. Wooder, T. W. Nicholls, M. J. Adcock, I. Munro, and J. C. Dainty, “Profiling of Atmospheric Turbulence Strength and Velocity Using a Generalised SCIDAR Technique,” Astron. Astrophys. Suppl. Ser. |

7. | J.-L. Prieur, G. Daigne, and R. Avila, “SCIDAR measurements at Pic du Midi,” Astron. Astrophys. |

8. | F. Roddier, “The Effect of Atmospheric Turbulence in Optical Astronomy,” Progress in Optics |

9. | R. Avila, E. Masciadri, J. Vernin, and L. Sánchez, “Generalized SCIDAR measurements at San Pedro Mártir: I. Turbulence profile statistics,” Publ. Astron. Soc. Pac. |

10. | M. Chun, R. Wilson, R. Avila, T. Butterley, J.-L. Aviles, D. Wier, and S. Benigni, “Mauna Kea ground-layer characterization campaign,” Mon. Not. R. Astron. Soc. |

11. | A. Fuchs, M. Tallon, and J. Vernin, “Folding of the vertical atmospheric turbulence profile using an optical technique of movable observing plane,” in |

12. | R. Avila, J. Vernin, and E. Masciadri, “Whole atmospheric-turbulence profiling with generalized Scidar,” Appl. Opt. |

13. | R. Avila, J. L. Avilés, R. Wilson, M. Chun, T. Butterley, and E. Carrasco, “LOLAS: an optical turbulence profiler in the atmospheric boundary layer with extreme altitude-resolution,” Mon. Not. R. Astron. Soc. |

14. | A. Fuchs, M. Tallon, and J. Vernin, “Focussiong on a turbulent layer: Principle of the Generalized SCIDAR,” Publ. Astron. Soc. Pac. |

15. | R. Avila, J. Vernin, and S. Cuevas, “Turbulence Profiles with Generalized Scidar at San Pedro Mártir Observatory and Isoplanatism Studies,” Publ. Astron. Soc. Pac. |

16. | B. Kern, T. A. Laurence, C. Martin, and P. E. Dimotakis, “Temporal coherence of individual turbulent patterns in atmospheric seeing,” Appl. Opt. |

17. | R. Avila, J. Vernin, and L. J. Sánchez, “Atmospheric turbulence and wind profiles monitoring with generalized scidar,” Astron. Astrophys. |

18. | R. W. Wilson, N. J. Wooder, F. Rigal, and J. C. Dainty, “Estimation of anisoplanatism in adaptive optics by generalized SCIDAR profiling,” Mon. Not. R. Astron. Soc. |

19. | A. Tokovinin, J. Vernin, A. Ziad, and M. Chun, “Optical Turbulence Profiles at Mauna Kea Measured by MASS and SCIDAR,” Publ. Astron. Soc. Pac. |

20. | B. García-Lorenzo and J. J. Fuensalida, “Processing of turbulent-layer wind speed with Generalized SCIDAR through wavelet analysis,” Mon. Not. R. Astron. Soc. |

21. | S. E. Egner, E. Masciadri, and D. McKenna, “Generalized SCIDAR Measurements at Mount Graham,” Publ. Astron. Soc. Pac. |

22. | J. Vernin, H. Trinquet, G. Jumper, E. Murphy, and A. Ratkowski, “OHP02 gravity wave campaign in relation to optical turbulence,” Environmental Fluid Mechanics7, 371-+ (2007). [CrossRef] |

23. | S. E. Egner and E. Masciadri, “A G-SCIDAR for Ground-Layer Turbulence Measurements at High Vertical Resolution,” Publ. Astron. Soc. Pac. |

24. | J. J. Fuensalida, B. García-Lorenzo, and C. Hoegemann, “Correction of the dome seeing contribution from generalized-SCIDAR data using evenness properties with Fourier analysis,” Mon. Not. R. Astron. Soc. |

25. | J. W. Goodman, |

**OCIS Codes**

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(280.7060) Remote sensing and sensors : Turbulence

(290.5930) Scattering : Scintillation

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: April 15, 2009

Revised Manuscript: May 20, 2009

Manuscript Accepted: June 9, 2009

Published: June 16, 2009

**Citation**

Remy Avila and Salvador Cuevas, "On the normalization of scintillation
autocovariance for generalized SCIDAR," Opt. Express **17**, 10926-10938 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-13-10926

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### References

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- B. Kern, T. A. Laurence, C. Martin, and P. E. Dimotakis, "Temporal coherence of individual turbulent patterns in atmospheric seeing," Appl. Opt. 39, 4879-4885 (2000). [CrossRef]
- R. Avila, J. Vernin, and L. J. S’anchez, "Atmospheric turbulence and wind profiles monitoring with generalized scidar," Astron. Astrophys. 369, 364 (2001). [CrossRef]
- R. W. Wilson, N. J. Wooder, F. Rigal, and J. C. Dainty, "Estimation of anisoplanatism in adaptive optics by generalized SCIDAR profiling," Mon. Not. R. Astron. Soc. 339, 491-494 (2003). [CrossRef]
- A. Tokovinin, J. Vernin, A. Ziad, and M. Chun, "Optical Turbulence Profiles at Mauna Kea Measured by MASS and SCIDAR," Publ. Astron. Soc. Pac. 117, 395-400 (2005). [CrossRef]
- B. Garcıa-Lorenzo and J. J. Fuensalida, "Processing of turbulent-layer wind speed with Generalized SCIDAR through wavelet analysis," Mon. Not. R. Astron. Soc. 372, 1483-1495 (2006). arXiv:astro-ph/0608595. [CrossRef]
- S. E. Egner, E. Masciadri, and D. McKenna, "Generalized SCIDAR Measurements at Mount Graham," Publ. Astron. Soc. Pac. 119, 669-686 (2007). [CrossRef]
- J. Vernin, H. Trinquet, G. Jumper, E. Murphy, and A. Ratkowski, "OHP02 gravity wave campaign in relation to optical turbulence," Environmental Fluid Mechanics 7, 371-+ (2007). [CrossRef]
- S. E. Egner and E. Masciadri, "A G-SCIDAR for Ground-Layer Turbulence Measurements at High Vertical Resolution," Publ. Astron. Soc. Pac. 119, 1441-1448 (2007). [CrossRef]
- J. J. Fuensalida, B. Garcıa-Lorenzo, and C. Hoegemann, "Correction of the dome seeing contribution from generalized-SCIDAR data using evenness properties with Fourier analysis," Mon. Not. R. Astron. Soc. 389, 731-740 (2008). [CrossRef]
- J. W. Goodman, Statistical Optics (Wiley-Interscience, New-York, 1985).

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