## Improved detection of atmospheric turbulence with SLODAR

Optics Express, Vol. 15, Issue 22, pp. 14844-14860 (2007)

http://dx.doi.org/10.1364/OE.15.014844

Acrobat PDF (2021 KB)

### Abstract

We discuss several improvements in the detection of atmospheric turbulence using SLOpe Detection And Ranging (SLODAR). Frequently, SLODAR observations have shown strong ground-layer turbulence, which is beneficial to adaptive optics. We show that current methods which neglect atmospheric propagation effects can underestimate the strength of high altitude turbulence by up to ~ 30%. We show that mirror and dome seeing turbulence can be a significant fraction of measured ground-layer turbulence, some cases up to ~ 50%. We also demonstrate a novel technique to improve the nominal height resolution, by a factor of 3, called Generalized SLODAR. This can be applied when sampling high-altitude turbulence, where the nominal height resolution is the poorest, or for resolving details in the important ground-layer.

© 2007 Optical Society of America

## 1. Introduction

1. M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D Campbell, J. C. Y Chin, S. K. Hartman, R. E. Lafon Jr., P. J. Stomski, and D. M. Summers, “The W. M. Keck Observatory Laser Guide Star Adaptive Optics System: Performance Characterization,” Publ. Astron. Soc. Pac., **118**, 310–318 (2006). [CrossRef]

2. G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A.-M. Lagrange, P.-Y. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in *Adaptive Optical System Technologies II*,
P. L. Wizinowich and B. Domenico, eds., *Proc. SPIE*4839, 140–149 (2003).

3. M. Iye, H. Takami, N. Takato, S. Oya, Y. Hayano, O. Guyon, S. A. Colley, M. Hattori, M. Watanabe, M. Eldred, Y. Saito, N. Saito, K. Akagawa, and S. Wada, “Cassegrain and Nasmyth adaptive optics systems of 8.2-m Subaru telescope,” in *Adaptive Optics and Applications III*,
W. Jiang and Y. Suzuki, eds., Proc. SPIE **5639**, 1–10 (2004). [CrossRef]

4. J. A. Stoesz, J.-P. Veran, F. J. Rigaut, G. Herriot, L. Jolissaint, D. Frenette, J. Dunn, and M. Smith, “Evaluation of the on-sky performance of Altair,” in *Advancements in Adaptive Optics*,
D. B. Calia, B. L. Ellerbroek, and R. Ragazzoni., eds., Proc. SPIE **5490**, 67–78 (2004). [CrossRef]

*r*

_{0}, the coherence time,

*τ*

_{0}, and the anisoplantanic angle,

*θ*

_{0}. Other useful parameters include the outer scale,

*L*

_{0}, and the power law,

*β*, of the power spectrum of spatial phase fluctuations (for Kolmogorov,

*L*

_{0}= ∞ and

*β*= 11/3). Measurement of these parameters has been emphasized with the planned construction of Extremely Large Telescopes (ELT) and new adaptive optic technologies, such as Ground Layer Adaptive Optics (GLAO). This has led to numerous campaigns to characterize the atmospheric turbulence profile at current or proposed observatory sites, for example, the Cerro Tololo campaign [6

6. A. Tokovinin and T. Travouillon, “Model of optical turbulence profile at Cerro Pachón,” Mon. Not. R. Astron. Soc. **365**, 1235–1242 (2006). [CrossRef]

10. P. Pant, C. S. Stalin, and R. Sagar, “Microthermal measurements of surface layer seeing at Devasthal site,” Astron. Astrophys. Suppl. Ser. **136**, 19–25 (1999). [CrossRef]

11. M. Azouit and J. Vernin, “Optical Turbulence Profiling with Balloons Relevant to Astronomy and Atmospheric Physics,” Publ. Astron. Soc. Pac. **117**, 536–543 (2005). [CrossRef]

12. T. Travouillon, “SODAR calibration for turbulence profiling in TMT site testing,” in *Ground-based and Airborne Telescopes*,
L. M. Stepp, ed., Proc. SPIE **6267**, 626720 (2006). [CrossRef]

13. 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. **63**, 270–273 (1973). [CrossRef]

14. A. Fuchs, M. Tallon, and J. Vernin, “Folding-up of the vertical atmospheric turbulence profile using an optical technique of movable observing plane,” in *Atmospheric Propagation and Remote Sensing III*,
W. A. Flood and W. B. Miller, eds., Proc. SPIE **2222**, 682–692 (1994). [CrossRef]

15. R. W. Wilson, “SLODAR: measuring optical turbulence altitude with a Shack-Hartmann wavefront sensor,” Mon. Not. R. Astron. Soc. **337**, 103–108 (2002). [CrossRef]

16. T. Butterley, R. W. Wilson, and M. Sarazin, “Determination of the profile of atmospheric optical turbulence strength from SLODAR data,” Mon. Not. R. Astron. Soc. **369**, 835–845 (2006). [CrossRef]

6. A. Tokovinin and T. Travouillon, “Model of optical turbulence profile at Cerro Pachón,” Mon. Not. R. Astron. Soc. **365**, 1235–1242 (2006). [CrossRef]

*N*×

*N*), of square sub-apertures or lenslets. The 2-D spatial cross-covariance of the sub-aperture spot motions (or Z-tilts, as sub-aperture image data are thresholded prior to centroid calculation, see [16

16. T. Butterley, R. W. Wilson, and M. Sarazin, “Determination of the profile of atmospheric optical turbulence strength from SLODAR data,” Mon. Not. R. Astron. Soc. **369**, 835–845 (2006). [CrossRef]

*C*

_{N}^{2}(

*h*)

*dh*[16

16. T. Butterley, R. W. Wilson, and M. Sarazin, “Determination of the profile of atmospheric optical turbulence strength from SLODAR data,” Mon. Not. R. Astron. Soc. **369**, 835–845 (2006). [CrossRef]

*L*

_{0}, and the power law,

*β*, of the power spectrum of spatial phase fluctuations. The process of obtaining

*C*

_{N}^{2}(

*h*)

*dh*information from the observational data is further explained in Section 4. The vertical resolution is uniform, given by

*δh*=

*w*/

*θ*where

*w*is sub-aperture size, or lenslet size mapping to the telescope pupil. The highest sampled layer,

*h*

_{N-1}= (

*N*- 1)

*δh*≈

*H*=

_{max}*D*/

*θ*, where N is number of sub-apertures across the telescope pupil, with the ground layer denoted as

*h*

_{0}= 0. The vertical resolution and maximum sample height are scaled by inverse of the air mass,

*χ*, or cos(

*ζ*), where

*ζ*is the zenith distance.

*δh*= 75 m and H

*= 1200 m when observing*

_{max}*δ*Apodis having separation,

*θ*= 102.9″ with a zenith distance,

*ζ*= 50°. For high altitude sampling, we have obtained

*δh*= 2400 m and

*H*= 40800 m when observing

_{max}*α*Crucis having separation,

*θ*= 4.1″ with a zenith distance,

*ζ*= 35°.

**369**, 835–845 (2006). [CrossRef]

*h*

_{0}= 0, found in part to be caused by strong mirror and dome seeing turbulence. By applying a high pass filter with cut-off of 1-2 Hz to the temporal centroid data streams, it was possible to remove the mirror and dome seeing turbulence from the ground-layer measurement. However, at this stage we point out that the ground-layer at Siding Spring dominates the seeing, particularly so on nights with poor seeing. We describe the process of removing dome and mirror seeing turbulence from SLODAR data in Section 3.

*δh*. We call this technique Generalized SLODAR and we report on the methodology, numerical simulations and observational results in Section 4.

## 2. Propagation effects of high-altitude layers

15. R. W. Wilson, “SLODAR: measuring optical turbulence altitude with a Shack-Hartmann wavefront sensor,” Mon. Not. R. Astron. Soc. **337**, 103–108 (2002). [CrossRef]

15. R. W. Wilson, “SLODAR: measuring optical turbulence altitude with a Shack-Hartmann wavefront sensor,” Mon. Not. R. Astron. Soc. **337**, 103–108 (2002). [CrossRef]

**369**, 835–845 (2006). [CrossRef]

**369**, 835–845 (2006). [CrossRef]

**369**, 835–845 (2006). [CrossRef]

17. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in *Progress in optics*, 19, (North-Holland Publishing Co., Amsterdam, 1981), pp. 281–376. [CrossRef]

*P*(

_{ϕ}*f*) is the phase power spectrum with no propagation, in other words as the wavefront leaves the layer at height

*h*.

*P*

_{ϕ}^{0}(

*f*) is then the power spectrum at the ground,

*h*= 0. The nulls of the propagated power spectrum,

*P*

_{ϕ}^{0}(

*f*) = 0 occurs at spatial frequencies,

_{n}*f*= [(

_{n}*n*+ 0.5)

^{1/2}]/

*r*, for integer

_{f}*n*= 0,1,..., and where

*r*= (

_{f}*λh*)

^{1/2}denotes the layer’s Fresnel length. This results in a measurable decrease, in the variance of the image motion across sub-aperture with size comparable to the Fresnel length. The effect is increased by the removal of global tilt in the reduction process, as this eliminates low spatial frequencies.

*h*= 15 km has a Fresnel length of 8.6 cm at a wavelength of 0.5 microns. In our site testing observations at Siding Spring, we used small (5.8 cm on the 40″ and 8.5 cm on the 24″ telescopes) sub-apertures for SLODAR because the seeing is often poor, with

*r*

_{0}about 8 cm in median seeing. These sub-apertures sizes are similar to the 5 cm sub-apertures used by the European Southern Observatory (ESO) portable SLODAR system using a 40 cm telescope [15

**337**, 103–108 (2002). [CrossRef]

**369**, 835–845 (2006). [CrossRef]

*H*= 7709 m, or pupil separation, Δ = 6, projected onto H, where Δ =

*Hθ*/

*w*. The target double star referenced in calculations is

*α*Cen with separation,

*θ*= 9.44″. The plots show the corresponding covariance impulse function Δ = 6 calculated with numerical simulations involving 300 phase screens using Fresnel propagation (only for propagation effects) and the calculated theoretical covariance impulse response using the methodology outlined by Butterley et al. [16

**369**, 835–845 (2006). [CrossRef]

*H*, and for sub-aperture sizes,

*w*= 5.8 cm and

*w*= 11.6 cm. The covariance impulse functions are modelled for the ANU 17×17 SLODAR instrument using the methodology outlined by Butterley et al. [16

**369**, 835–845 (2006). [CrossRef]

*w*= 11.6 cm size sub-apertures are modelled using a telescope pupil with twice the diameter (

*D*= 2 m) compared to the

*w*= 5.8 cm size sub-apertures (

*D*= 1 m), but impulse functions are identical as pupil geometry is unchanged [16

**369**, 835–845 (2006). [CrossRef]

*H*= Δ

*δh*, where in Fig. 3 have the values

*δh*= 1.29 km and

*H*= 20.6 km (Δ = 16). The propagation effects are lessened for the larger sized sub-aperture,

_{max}*w*= 11.6 cm, but are still significant.

*w*= 5.8 cm, and therefore susceptible to propagation effects in the underestimation of high altitude turbulence. The observational data represents an individual run of the double star

*α*Cen, with angular separation,

*θ*= 9.44″. The dataset consists of 2000 frames at 200 fps and exposure of 2 ms under excellent seeing,

*r*

_{0}= 18.2 cm, for Siding Spring. The dataset is selected as an example as it exhibits significant high-altitude turbulence, subsequently verified in the temporal-spatial cross-covariance data, moving at speeds ~ 20 ms

^{-1}. The centroid data was filtered with a 1 Hz high-pass FIR (Finite Impulse Response) filter to remove static mirror and dome seeing contributions from the ground-layer Δ = 0 measurement (see Section 3).The estimation of the

*C*

_{N}^{2}(

*h*)

*dh*profile was implemented by fitting of the transverse (T) theoretical covariance functions for a Kolmogorov turbulence power spectrum,

*β*= 11/3. From Fig. 4 the inclusion of propagation effects increase the strength of the high-altitude turbulence

*H*> 15 km by ~ 25%.

## 3. Removal of mirror and dome turbulence

## 4. Improving the vertical resolution

*δh*=

*w*/

*θ*, where

*w*is the sub-aperture width and

*θ*is the angular separation of the observed double star (see Fig. 1). To improve the height resolution,

*δh*, assuming fixed

*θ*and fixed exposure time,

*τ*, is to reduce the size of the sub-apertures,

*w*. The number of signal photons per

*τ*is directly proportional to

*w*

^{2}so reducing

*w*will cause photon starvation and hence restrict observations to the brighter double stars (few in number). The minimum useable sizes of

*w*range from

*w*= 5 cm for the portable ESO 8×8 SLODAR system [15

**337**, 103–108 (2002). [CrossRef]

*w*= 5.8 cm for our ANU 17×17 SLODAR system. Reducing

*w*will also present a number of second order effects, including less sensitivity to high altitude layers due to typical high wind speeds and propagation effects. The high wind speeds,

*v*will reduce the variance of the observed tilts contributed by increasing the effective sampling distance of wavefront tilts from

*w*to

*τv*when

*τv*>

*w*[19

19. H. M. Martin, “Image motion as a measure of seeing quality,” Publ. Astron. Soc. Pac., **99**, 1360–1370 (1987) [CrossRef]

*w*, resulting in reduction in the variance of the observed tilts contributed by the layer. Hence a physical limit exists for minimum size of

*w*and therefore a minimum height resolution,

*δh*, to ensure a satisfactory performance of SLODAR.

*δh*. The SHWFS conjugation heights are a fractional amount of the nominal height resolution,

*δh*. By combining

*N*datasets at regularly spaced SHWFS conjugation heights, new information is provided about the atmospheric turbulence, and is possible to achieve a Generalized SLODAR height resolution,

_{G}*δh*

^{*}=

*δh*/

*N*.

_{G}**369**, 835–845 (2006). [CrossRef]

*δh*. This model assigns equal a priori probability to all heights (unbiased model). The fitting procedure can be modelled as system of linear equations in matrix form,

**Ax**=

**b**.

**A**is the kernel matrix with column vectors corresponding the theoretical covariance impulse response functions.

**b**is an ensemble average of the observed atmospheric turbulence covariance profile (that also includes systematic and statistical noise), represented as a column vector.

**x**is the quantity that we seek, an estimate of the atmospheric turbulence profile, represented as a column vector of strengths of each thin layer.

**x**, is based on the input data,

**b**, assumptions made by the model,

**A**, and the process to recover

**x**, (inversion models). The system is over-determined as there are more equations than variables so matrix

**A**cannot be directly inverted and a least squares solution is sought. The system solution,

**x**, can be found by least squares inversion,

**x**=

**A**

^{+}

**b**, where

**A**

^{+}is the pseudo inverse of

**A**. However, we note as

**b**contains unwanted noise so the system may be un-stable and hence the solution,

**x**, invalid.

**x**>

**0**and that

**b**is possibly corrupted with Gaussian noise. Such an inversion model is the Non-Negative Least Squares (NNLS) algorithm. We have found through simulation that the NNLS algorithm recovers the input atmosphere model more accurately than other regularization algorithms, such as MAXENT and Tikhonov regularization. The simulation utilized RegTools (Regularization Tools) [20

20. P. C. Hansen, “Regularization Tools,” http://www2.imm.dtu.dk/%7Epch/Regutools/.

*lsqnonneg*and performs well on compact sources (minimal smoothing). Hence the NNLS algorithm suitable with the thin-layer model assumption of the atmosphere, as verified with typical high resolution measurements of atmospheric turbulence [11

11. M. Azouit and J. Vernin, “Optical Turbulence Profiling with Balloons Relevant to Astronomy and Atmospheric Physics,” Publ. Astron. Soc. Pac. **117**, 536–543 (2005). [CrossRef]

**369**, 835–845 (2006). [CrossRef]

*C*(Δ) describes the theoretical covariance of x-directional slopes for a cross-pair of sub-apertures with lateral pupil spatial offset (

^{′x}_{i,j,i′,j′}*δi*,

*δj*) and layer height,

*H*= Δ

*δh*, after global tilt subtraction. The number of cross-pair lenslets having the same lateral pupil spatial offset (

*δi*,

*δj*) for a given layer height,

*H*,is denoted by

*N*. The notation used to describe the theoretical covariance impulse response function,

_{cross}*X*(Δ,

_{L}*δi*,

*δj*), is shown in Fig. 6. The indices [

*i*,

*j*] refer to the lenslet index for star A and [

*i*

^{′},

*j*

^{′}] for star B. A cross-pair of lenslets has a lateral pupil spatial offset defined by (

*δi*,

*δj*)=(

*i*

^{′}-

*i*,

*j*

^{′}-

*j*), specified in units of the sub-aperture width,

*w*. The lenslet index,

*i*, takes on integer values

*i*= {1,2, ...,

*N*}, where

*N*is the number of lenslets mapped across the diameter of the telescope pupil. Likewise for indices

*j*,

*i*

^{′}and

*j*

^{′}. The lateral pupil spatial offset,

*δi*, takes on integer values

*δi*= {1 -

*N*, 2 -

*N*,...,0,1,...

*N*- 2,

*N*-1}, specified in units of sub-aperture width,

*w*. Likewise for

*δj*.

*δi*,

*δj*), (units of

*w*). A turbulent layer height at

*H*corresponds to a lateral pupil spatial separation, Δ =

*Hθ*/

*w*of telescope pupils, specified in units of the sub-aperture width,

*w*. The lateral pupil spatial separation, Δ, is an offset of the projected telescope pupils along the x-direction at the layer altitude,

*H*, and takes on integer values Δ = {0,1,2, ...,

*N*- 1}. The physical separation of a pair of sub-apertures with a spatial offset (

*δi*,

*δj*) projected on a layer at height,

*H*, is then (

*u*,

_{x}*u*) where

_{y}*u*= |

_{x}*δi*+ Δ|

*w*and

*u*= |

_{y}*δj*|

*w*, is used by

*C*(Δ) function. Hence the estimated strengths of the layers are defined with height bins of widths

^{′x}_{i,j,i′x,j′x}*δh*and centered at Δ

*δh*. However, the practical height resolution,

*δh*, may be poorer depending on the signal-to-noise ratio of observational data and the inversion model implemented to recover the estimated strengths.

*X*(Δ,

_{L}*δi*,

*δj*), is an accurate model, and takes into consideration the pupil geometry (mapping of circular or square sub-apertures on the annular telescope), turbulence power spectrum (

*β*,

*L*) and effects of ‘global’ tilt subtraction (tilt anisoplanatism) required to remove telescope tracking errors. The parameters Δ and (

_{o}*δi*,

*δj*) are integer valued and hence

*X*(Δ

_{L}*δi*,

*δj*) is a discrete function that models the impulse response of equally spaced thin layers with height, Δ

*δh*. The discrete impulse response function

*X*(Δ,

_{L}*δi*,

*δj*) is in a format that is compatible with the discrete observational covariance profile

**C**

*(*

^{′x,obs}_{L,k}*δi*,

*δj*). Hence the discrete function

*X*(Δ,

_{L}*δi*,

*δj*) can be specified in matrix form,

**A**, to model the system as a set of linear equations,

**Ax**=

**b**, and then inverted to solve for layer strengths,

**x**=

**A**

^{+}

**b**. To further explain the process the discrete observational covariance profile

**C**

*(*

^{′x,obs}_{L,k}*δi*,

*δj*) can be modelled as a linear equation in the form

*δi*and

*δj*maps the theoretical covariance impulse response,

*X*(Δ,

_{L}*δi*,

*δj*), of a particular height,

*H*= Δ

*w*/

*θ*. Expressing as a set of linear equations,

**Ax**=

**b**, where

**x**is a column vector of layer strengths:

*col*{

**x**} denotes the process that serializes the 2-D data,

**x**, into a 1-D column vector by stacking columns of

**x**with increasing

*δi*.

*y*= 0 of the 2-D theoretical covariance function,

*X*(Δ,

_{L}*δi*,

*δj*), or by setting

*j*

^{′}=

*j*or

*δj*= 0.

*X*(Δ,

_{L}*δi*) , is calculated for integer valued lateral pupil spatial separations, Δ = {0,1,2, ...,

*N*- 1} and the condition Δ = 0 corresponds to completely overlapped telescope pupils projected on the SHWFS. This configuration is when the SHWFS is conjugated to the telescope pupil (

*h*

_{0}=0 km), refer Fig. 6.

*δh*, can be found at non-integer lateral pupil spatial separations, Ω

_{k}= Δ+

*η*= {0 +

_{k}*η*, 1 +

_{k}*η*, 2 +

_{k}*η*,...,

_{k}*N*- 1 +

*η*}, where

_{k}*η*takes values between 0 and 1, where

_{k}*k*is the index of the group of

*N*Generalized SLODAR datasets,

_{G}*k*= {0,1, ...,

*N*- 1}. The value Ω

_{G}_{k}can be obtained by moving the SHWFS conjugation height,

*h*

_{0}, upwards by fractional amounts of the height resolution,

*h*

_{0}

^{*}=

*η*, and is illustrated in Fig. 6. Moving the conjugation height,

_{k}δh*h*

_{0}

^{*}, results in a lateral pupil spatial offsets,

*δm*=

_{k}*δi*+

*η*and

_{k}*δm*=

_{k}*η*for

_{k}*δi*= 0, corresponding to a lateral pupil spatial separations, Ω

_{k}=

*η*. Hence telescope pupils are no longer completely overlapped at the SHWFS but separated by a fractional amount of a lenslet. The non-integer lateral pupil spatial separations, Ω

_{k}_{k}, can be thought of sampling new and unique spatial offsets,

*δi*+

*η*, in the telescope pupil.

_{k}*k*, having unique lateral pupil spatial separations, Ω

_{k}, and with equal height resolutions,

*δh*. The methodology for Generalized SLODAR is shown in Fig. 7.

_{k}, must be equally spaced and hence require

*η*to also be equally spaced. The fractional spacings,

_{k}*η*are then given by

_{k}*η*=

_{k}*k*/

*N*and therefore

_{G}*η*= (1/

_{k}*N*){0, 1,2, ...,

_{G}*N*- 1}.

_{G}*k*, to be

**C**

*(*

^{′x,obs}_{L,k}*δi*). Note that the symbol defined for the observed covariance profile

**C**

*(*

^{′x,obs}_{L,k}*δi*) should be clearly distinguished from the theoretical covariance function for a cross-pair of lenslets,

*C*(Δ). We now need to transform the observed covariance profile from a local lenslet-based coordinate system to a global coordinate system,

^{′x,obs}_{i,j,i′,j′}**C**

*(*

^{′x,obs}_{L,k}*δm*), referenced to

_{k}*η*= 0, or lateral spatial offsets in the telescope pupil at

_{k}*h*

_{0}= 0. The global coordinate system of an individual Generalized SLODAR dataset,

*k*, is defined as

*δm*=

_{k}*δi*+

*η*, specified in units of the sub-aperture width,

_{k}*w*. To construct the observed super-resolution covariance profile,

**C**

*(*

^{′x,obs}_{L,k}*δm*

^{*}), requires the

**C**

*(*

^{′x,obs}_{L,k}*δm*) profiles to be first scaled to normalize fluctuations in seeing and then interleaved. The scaling parameter,

_{k}*a*, normalizes

_{k}**C**

*(*

^{′x,obs}_{L,k}*δm*) to have equal seeing and hence remove any bias effects, and defined as

_{k}**A**

*(*

^{′x,obs}_{L,k}*δi*= 0) refers to the peak of the centroid-noise removed auto-covariance function for dataset

*k*, and proportional to the total atmospheric seeing. For most cases the scaling parameter,

*a*, is close to unity,

_{k}*a*≈ 1. The observed super-resolution covariance profile,

_{k}**C**

*(*

^{*′x,obs}_{L,k}*δm*

^{*}), is then

*m*and

*l*are now indices that reference a higher sampled SHWFS at fractional spacings of a sub-aperture,

*w*

^{*}=

*w*/

*N*with total samples, of

_{G}*N*

^{*}=

*N*. Due to the complexity and time required to compute

_{G}N*X*(Ω

_{L}^{*},

*δm*

^{*}) it is best to approximate with interpolation methods. Through numerical simulations involving phase screens, it found that cubic interpolation method is suitable for the

*X*(Ω

_{L}^{*},

*δm*

^{*}) function and spline interpolation for

*X*

_{T}^{*}(Ω

^{*},

*δm*

^{*}) function.

*X*(Ω

_{L}^{*},

*δm*

^{*}), can now be constructed in matrix form,

**A**, to model the system as a set of linear equations,

**Ax**=

**b**, and then inverted to solve for layer strengths,

**x**=

**A**

^{+}

**b**. Due to the larger size of the matrix

**A**, it best to use a positively constrained,

**x**> 0 inversion method for compact sources (minimal smoothing to

**x**), such as the Non-Negative Least Squares (NNLS) algorithm implemented as the MATLAB iterative routine

*lsqnonneg*.

*N*= 3. For

_{G}*N*= 6 the results are progressively poorer due to a larger matrix being increasingly sensitive to noise.

_{G}*N*= 3 resulting in an effective height resolution,

_{G}*δh*

^{*}=

*δh*/3. We confirm this by clearly separating two phase screens separated in height by

*h*= 2

*δh*

^{*}.

*α*Cen and the ANU 17×17 SLODAR instrument on the ANU 40″ telescope. The parameters of the simulation are listed in Tab. 1. The Generalized SLODAR is simulated by sequentially moving the layers down in vertical height by

*δh*

^{*}or 400m for each fractional generalized pupil offsets,

*η*. A lateral pixel offset of 1cm corresponds to vertical height of 200m. Therefore, for each dataset, decreasing the separation of telescope pupils of star A and star B as projected onto the phase screens

_{k}*H*

_{1}and

*H*

_{2}by two pixels (2×200m) achieved Generalized SLODAR. The wavefronts for each star at the SHWFS is calculated by extracting the part of the phase screen that the pupils project on and then adding together for each layer

*H*

_{1}and

*H*

_{2}.

*α*Cen, angular separation of 9.44″, at Siding Spring Observatory. The first dataset was captured at 10:04 21 June 2006 (UTC) with SHWFS conjugation height

*h*

_{0}

^{*}= 550 m (

*η*

_{1}= 0.5) and second dataset was captured at 10:59 21 June 2006 (UTC) with SHWFS conjugation height

*h*

_{0}

^{*}= 990 m (

*η*

_{2}= 0.9). The third dataset having SHWFS conjugation height

*h*

_{0}= 0 m (

*η*

_{0}= 0) was taken 12:43 21 June 2006 (UTC) and excluded in the analysis as the atmospheric seeing changed significantly (poor seeing) during the 1hr45mins of observing downtime. Note the fractional spacings of

*η*

_{1}= 0.5 and

*η*

_{2}= 0.9 are not regularly spaced but the methodology and results remain valid. Each dataset consists of 4000 frames captured at 200 fps using a fixed exposure of 2 ms with centroid sequences from each lenslet pre-processed by 1 Hz high pass FIR filter to remove mirror and dome seeing contributions from the ground-layer turbulence measurement bin. The results are shown in Fig. 9 and clearly demonstrate an improvement in height resolution by a factor two over the nominal resolution of 1100 m, providing an ‘effective’ resolution of 550 m. The error bars are one standard deviation calculated by dividing the dataset into 10 segments of 400 frames. As the SHWFS conjugation height

*h*

_{0}= 0 m (

*η*

_{0}= 0) was excluded from the analysis we added a single impulse response function for Ω

^{*}= 0 to model the ground-layer. We note that the turbulence bin for height 550 m does not register any strength. The error bar for this bin constrains the lowest turbulence to be below ~ 50 m, as otherwise the finite width of the covariance impulse response for layers at ~ 50 m would cause spill-over exceeding the error bar.

## 5. Conclusions

## Acknowledgements

## References and links

1. | M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D Campbell, J. C. Y Chin, S. K. Hartman, R. E. Lafon Jr., P. J. Stomski, and D. M. Summers, “The W. M. Keck Observatory Laser Guide Star Adaptive Optics System: Performance Characterization,” Publ. Astron. Soc. Pac., |

2. | G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A.-M. Lagrange, P.-Y. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler, and G. Zins, “NAOS, the first AO system of the VLT: on-sky performance,” in |

3. | M. Iye, H. Takami, N. Takato, S. Oya, Y. Hayano, O. Guyon, S. A. Colley, M. Hattori, M. Watanabe, M. Eldred, Y. Saito, N. Saito, K. Akagawa, and S. Wada, “Cassegrain and Nasmyth adaptive optics systems of 8.2-m Subaru telescope,” in |

4. | J. A. Stoesz, J.-P. Veran, F. J. Rigaut, G. Herriot, L. Jolissaint, D. Frenette, J. Dunn, and M. Smith, “Evaluation of the on-sky performance of Altair,” in |

5. | J. W. Hardy, |

6. | A. Tokovinin and T. Travouillon, “Model of optical turbulence profile at Cerro Pachón,” Mon. Not. R. Astron. Soc. |

7. | F. Rigaut, “Ground Conjugate Wide Field Adaptive Optics for the ELTs,” in |

8. | A. Tokovinin, “Seeing Improvement with Ground-Layer Adaptive Optics,” Publ. Astron. Soc. Pac., |

9. | D. R. Andersen, J. Stoesz, S. Morris, M. Lloyd-Hart, D. Crampton, T. Butterley, B. Ellerbroek, L. Jollissaint, N. M. Milton, R. Myers, K. Szeto, A. Tokovinin, J.-P. Véran, and R. Wilson, “Performance Modeling of a Wide Field Ground Layer Adaptive Optics System,” Publ. Astron. Soc. Pac., |

10. | P. Pant, C. S. Stalin, and R. Sagar, “Microthermal measurements of surface layer seeing at Devasthal site,” Astron. Astrophys. Suppl. Ser. |

11. | M. Azouit and J. Vernin, “Optical Turbulence Profiling with Balloons Relevant to Astronomy and Atmospheric Physics,” Publ. Astron. Soc. Pac. |

12. | T. Travouillon, “SODAR calibration for turbulence profiling in TMT site testing,” in |

13. | 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. |

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

15. | R. W. Wilson, “SLODAR: measuring optical turbulence altitude with a Shack-Hartmann wavefront sensor,” Mon. Not. R. Astron. Soc. |

16. | T. Butterley, R. W. Wilson, and M. Sarazin, “Determination of the profile of atmospheric optical turbulence strength from SLODAR data,” Mon. Not. R. Astron. Soc. |

17. | F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in |

18. | Richard W. Wilson, Durham University, Centre for Advanced Instrumentation, Department of Physics, South Road, DH1 3LE, Washington, United Kingdom, (personal communication, 2006). |

19. | H. M. Martin, “Image motion as a measure of seeing quality,” Publ. Astron. Soc. Pac., |

20. | P. C. Hansen, “Regularization Tools,” http://www2.imm.dtu.dk/%7Epch/Regutools/. |

21. | C. M. Harding, R. A. Johnston, and R. G. Lane, “Fast Simulation of a Kolmogorov Phase Screen,” Appl. Opt., |

22. | M. C. Britton, “Arroyo,” in |

**OCIS Codes**

(000.2170) General : Equipment and techniques

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(110.6770) Imaging systems : Telescopes

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: June 15, 2007

Revised Manuscript: August 30, 2007

Manuscript Accepted: September 9, 2007

Published: October 25, 2007

**Citation**

Michael Goodwin, Charles Jenkins, and Andrew Lambert, "Improved detection of atmospheric turbulence with SLODAR," Opt. Express **15**, 14844-14860 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-22-14844

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### References

- M. A. van Dam, A. H. Bouchez, D. Le Mignant, E. M. Johansson, P. L. Wizinowich, R. D Campbell, J. C. Y Chin, S. K. Hartman, R. E. Lafon, Jr., P. J. Stomski and D. M. Summers, "The W. M. Keck Observatory Laser Guide Star Adaptive Optics System: Performance Characterization," Publ. Astron. Soc. Pac., 118, 310-318 (2006). [CrossRef]
- G. Rousset, F. Lacombe, P. Puget, N. N. Hubin, E. Gendron, T. Fusco, R. Arsenault, J. Charton, P. Feautrier, P. Gigan, P. Y. Kern, A.-M. Lagrange, P.-Y. Madec, D. Mouillet, D. Rabaud, P. Rabou, E. Stadler and G. Zins, "NAOS, the first AO system of the VLT: on-sky performance," Proc. SPIE 4839, 140-149 (2003).
- M. Iye, H. Takami, N. Takato, S. Oya, Y. Hayano, O. Guyon, S. A. Colley, M. Hattori, M. Watanabe, M. Eldred, Y. Saito, N. Saito, K. Akagawa, and S. Wada, "Cassegrain and Nasmyth adaptive optics systems of 8.2-m Subaru telescope," Proc. SPIE 5639, 1-10 (2004). [CrossRef]
- J. A. Stoesz, J.-P. Veran, F. J. Rigaut, G. Herriot, L. Jolissaint, D. Frenette, J. Dunn, and M. Smith, "Evaluation of the on-sky performance of Altair," Proc. SPIE 5490, 67-78 (2004). [CrossRef]
- J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, 1998).
- A. Tokovinin and T. Travouillon, "Model of optical turbulence profile at Cerro Pach´on," Mon. Not. R. Astron. Soc. 365, 1235-1242 (2006). [CrossRef]
- F. Rigaut, "Ground Conjugate Wide Field Adaptive Optics for the ELTs," in ESO Conference and Workshop Proceedings, 58, E. Vernet, R. Ragazzoni, S. Esposito, and N. Hubin, eds., (Garching, Germany: ESO, 2002), p. 11.
- A. Tokovinin, "Seeing Improvement with Ground-Layer Adaptive Optics," Publ. Astron. Soc. Pac. 116, 941-951 (2004). [CrossRef]

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