Active optics Shack-Hartmann sensor: using spot sizes to measure the seeing at the focal plane of a telescope

0.61 MB·English

by  P. Martinez

Mon.Not.R.Astron.Soc.000,1–??(2011) Printed12January2012 (MNLaTEXstylefilev2.2)
Active optics Shack-Hartmann sensor: using spot sizes to measure
the seeing at the focal plane of a telescope
P. Martinez,1 ⋆ J. Kolb,2 M. Sarazin,2 J. Navarrete 3
1UJF-Grenoble1/CNRS-INSU,InstitutdePlane´tologieetd’AstrophysiquedeGrenoble(IPAG)UMR5274,Grenoble,F-38041,France
2EuropeanSouthernObservatory,Karl-Schwarzschild-Strasse2,D-85748,Garching,Germany
3EuropeanSouthernObservatory,AlonsodeCordova3107,Casilla19001,Vitacura,Santiago,Chile
2
1
0
2
Released2011XxxxxXX
n
a
J
ABSTRACT
1 Real-time seeing estimation at the focus of a telescope is nowadays strongly emphasized
1
as this knowledge virtually drives the dimensioning of adaptive optics systems and instru-

mentoperationalaspects. In this contextwe study the interestof using active opticsShack-
]
M Hartmann(AOSH)sensorimagestoprovideaccurateestimateoftheseeing.TheAOSHprac-
tically delivers long exposure spot PSFs – at the critical location of the telescope focus –
I beingdirectlyrelatedtotheatmosphericseeinginthelineofsight.AlthoughAOSHsensors
.
h are not specified to measure spot sizes but slopes, we show that accurate seeing estimation
p fromAOSHimagescanbeobtainedwithadedicatedalgorithm.Thesensitivityandcompar-
-
o isonoftwoalgorithmstovariousparametersisanalyzedinasystematicway,demonstrating
r thatefficientestimationoftheseeingcanbeobtainedbyadequatemeans.
t
s
Keywords: Sitetesting–atmosphericeffects–methods:numerical–methods:dataanalysis
a
[

1
v
0 1 INTRODUCTION atzenith.Forthispurpose,severalflavorsofimagescanbeusedat
3 thecriticallocationofthetelescopefocii:(1/)scientificinstrument
Theevaluationoftheseeingisparamountforselectingastronomi-
3 images,(2/)guideprobeimages,(3/)activeopticsShack-Hartmann
2 calsitesandfollowingtheirtemporalevolution.Likewise,itsesti- images.AttheVLT,focalplanesareequippedwithanarmusedfor
. mationisfundamentalforthedimensioningofadaptiveopticssys-
1 acquisitionofanaturalguidestar.Thelightfromthisstaristhen
temsandtheirperformancepredications.Thisknowledgevirtually
0 splitbetweenaguideprobeforanaccuratetrackingofthesky,and
drivesinstrumentsoperationalaspectsatatelescope,andmoreem-
2 a Shack-Hartmann wavefront sensor used by the active optics to
phasisismadetodevelopanduseaccuratereal-timeseeingestima-
1 controltheshapeoftheprimarymirror.
toratthefocusofatelescope.
:
v The atmospheric seeing is commonly measured by the dif- However seeingestimationfromthefull-widthathalfmaxi-
Xi ferential image motion monitor (DIMM, Sarazinetal., 1990) in mum(FWHM)ofaPSFstronglyreliesontheexposuretimethat
most observatories, or by means of alternative seeing monitors, mustbelongenoughsothattheturbulencehasbeenaveraged;en-
r
a e.g.,GeneralizedSeeingMonitor(GSM,Ziadetal.,2000),Multi- suringthatallrepresentationsofthewavefrontspatialscaleshave
Aperture Scintillation Sensor (MASS, KornilovandTokovinin passed through the pupil. This is thus dependent on pupil size
2001). Being localized away from the telescope platform, such a and turbulence velocity, though it is commonly admitted that 30
devicedeliversseeingestimatethatcansignificantlydifferfromthe seconds average properly the turbulence, introducing significant
effectiveseeingseenatatelescopefociibecauseofpointingorien- FWHMbiasesotherwise.Thisalreadydiscardstheuseoftheguide
tationand/or height above theground differences, or local seeing probeasexposuretimesarenotlongerthan50milli-second,while
bias(dome contribution). Theeffect of the twolatter willlargely scientificinstrumentimagesdonotentirelycomplywiththereal-
expandinthecontextofthenextgenerationoftelescopes:theex- timeaspectoftheseeingestimation.Instrumentsareaffectedbyan
tremelylargetelescopes. observationalbias:unavailabilityforalargerangeofseeingcondi-
Ontheotherhand,theatmosphericseeingcanbededucedat tions,andareadditionallyaffectedbythetelescopefieldstabiliza-
thefocusofalargetelescopefromthewidthofthelong-exposure tion.
pointspreadfunction(PSF)assumingsuccessivecorrectionstoap- ActiveOpticsShack-Hartmann(AOSH)deliverscontinuously
ply(turbulenceouterscale,wavelength,andairmass).Weremind real-timeimagesoflongexposurespotPSFs(typically45seconds)
thereaderthattheseeingisdefinedatλ=500nmforobservations atthesamelocationof scientificinstruments.AOSHimagespro-
videsimultaneouslyvariousdata:slopes,intensities,andspotsizes.
Whenshortexposuresareused,theinformationprovidedbyboth
⋆ [email protected] slopesandintensities(i.e.,scintillation)canbeusedtoretrieveCn2
(cid:13)c 2011RAS
2 P.Martinezet al.
profile using correlations of these data from two separated stars
(Robertetal.,2011).Inthispaper,weproposetousethespotsizes
inthesub-apertures toretrievetheatmospheric seeingintheline
ofsight.Inotherwords,weproposetousetheactiveopticsAOSH
sensorsystemofthetelescopeasaturbulencemonitortoprovide
accurateseeingestimationdirectlyatthetelescopefocus.Forthis
purpose, careful assumption of the long-exposure PSF profile(to
ensurepreciseFWHMevaluation), andaccuratederivation ofthe
seeingfromtheestimatedFWHMarerequired.
Inthiscontext,weinvestigatetheperformanceandlimitations
oftwodifferentmethodstoestimatetheseeingfromlong-exposure
AOSHspotPSFs.Thecomparisonandselectionoftheuppermost
modus operandi isabyproduct ofthiswork. Ourstudy iscarried
out by means of extensive simulations, and using real data from
theAOSHimagesobtainedattheVLT.InSect.2wedescribetwo
differentalgorithmsdevelopedtoretrievetheseeingfromlongex-
posureimages,andinSect.3wepresentoursimulationhypothe-
Figure1.SimulatedAOSHimagebasedontheVLTAOSHgeometry.
sis,anddiscussourresults.InSect.4real-datafromtheVLTAOSH
databasearere-analyzedandcomparedtosynchronousimagemea-
surementsobtainedattheVLT(UT4)withtheinstrumentFORS2 theouterscaleL .Theexpressionforthephasestructurefunction
0
(Appenzeller,etal.1998).FinallyinSect.5wedrawconclusions. (D )withfiniteouterscaleL canbefoundin(e.g.,Tatarskii1961;
φ 0
Tokovinin 2002). As large-scale wavefronts are anything but sta-
tionary,AOSHlong-exposurespotPSFscanbeinafirstorderde-
2 EXTRACTINGSEEINGFROMSHACK-HARTMANN scribedbyEq.2,whileaposterioricorrectionbyL0 experimental
SPOTSIZES estimateismandatory.TheexaminationoftheL0influencewillbe
treatedinthefollowing.
Accurate seeing estimation from the full-width at half maximum
(FWHM)ofaturbulence-limitedlongexposurePSFrequirestwo
conditions:(1/)carefulassumptionofthelong-exposurePSFpro- 2.2 SeeingandFWHM
filetoensurepreciseFWHMevaluation;(2/)accuratederivationof
TheKolmogorov-ObukhovmodelpredictsdependenceofthePSF
the seeing from the estimated FWHM.In the following, we treat
FWHMε onwavelengthλandr :
these two aspects, and present in details the two dedicated algo- 0 0
rithms. ε =0.976λ/r . (3)
0 0
Equation3isthedefinitionoftheseeing(assumedatλ=500nm,
2.1 Long-exposurePSFprofile andforobservationsatzenith).However,thephysicsofturbulence
impliesthatthespatialpowerspectraldensity(PSD)ofphasedis-
Thetheoreticalexpressionofalong-exposurePSFcanbedescribed
tortions W (v) deviates from the pure power law at low frequen-
through the expression of its optical transfer function (OTF) ob- φ
cies.ApopularvonKa`rma`n(vK)turbulencemodel(e.g.,Tatarskii
tainedbymultiplyingthetelescopeOTF,denotedT (f),bytheat-
0 1961; Ziadetal., 2000) introduces an additional parameter, the
mosphericOTF:
outerscaleL thatdescribesthelow-frequencybehaviorofthetur-
0
T (f)=exp[−0.5D (λf)], (1) bulencemodelphasespectrum:
a φ
where f is the angular spatial frequency, λ is the imaging wave- Wφ(v)=0.0229r0−5/3(|v|2+L−02)−11/6. (4)
length,andD (r)isthephasestructurefunction(Goodman1985;
φ Equation4isthedefinitionofL ,wherevisthespatialfrequencyin
Roddier,1981).Thisexpressionismiscellaneousandapplicableto 0
m−1.TheKolmogorov-ObukhovmodelcorrespondstoL =∞.In
anyturbulencespectrumandanytelescopediameter. 0
thevKmodel,r describesthehigh-frequencyasymptoticbehavior
The analytic expression for the phase structure function in the 0
of the spectrum. In thiscontext, theouter scaleof theturbulence
Kolmogorov-ObukhovmodelcanbefoundinTatarskii(1961)and
(L ) plays a significant rolein theimprovement of image quality
isexpressedby D (r) = 6.88(r/r )5/3,wherer istheFried’sco- 0
φ 0 0 (i.e.,FWHM)atthefocusofatelescope.Theimagequalityisdif-
herenceradius(Fried1966).Finally,thelong-exposureOTFcanbe
ferent(andinsomecasesbyalargefactor,e.g.,30%to40%inthe
expressedas:
near-infrared)fromtheatmosphericseeingthatcanbemeasuredby
T(f)=T (f)×exp[−3.44(λf/r )5/3], (2) dedicatedseeingmonitors,suchastheDIMM.
0 0
ThedependenceofatmosphericlongexposureresolutiononL is
0
andlong-exposure PSFsareaccuratelydescribedbyEq.2.Inthe efficientlypredictedbyasimpleapproximateformula(Eq.5)intro-
caseofalargeidealtelescopewithdiameterD≫r thediffraction
0 ducedbyTokovinin(2002),andconfirmedbymeansofextensive
termT canbeneglected,whileinthecaseofAOSHsub-apertures
0 simulations (Martinezetal., 2010a,b), where we emphasized that
ofsizeditcannot(d≈r0). theeffectoffiniteL isindependentofthetelescopediameter.The
0
WenotethatEq.2assumesnonrealisticbehaviorofthelow- validityofEq.5hasbeenestablishedinaL /r >20andL /D6500
0 0 0
frequencyoftheturbulencemodelphasespectrum.Itisfirmlyes- domain,whereourtreatmentofthediffractionfailedforsmalltele-
tablished that thephase spectrum deviates from thepower law at
scopediameters(D<1m, Martinezetal.,2010a).
lowfrequencies(Ziadetal.,2000;Tokovininetal.,2007),andthis
behavior is described in a first order by an additional parameter, FWHM≈ε 1−2.183(r /L )0.356, (5)
0 p 0 0
(cid:13)c 2011RAS,MNRAS000,1–??
Seeing estimationfromShack-Hartmannimages 3
As a consequence, to deduce atmospheric seeing ε (at 500nm)
0
from the FWHM of a long-exposure PSF the correction implied
byEq.5ismandatorypriortoairmassandwavelengthcorrection.
Wewillshowthatthisdoesconcerneventhecaseofsmallsize(d)
AOSHsub-apertures,whereL ≫d.
0
2.3 Algorithmsprinciple
We consider two different algorithms to extract the FWHM of a
longexposureAOSHimage.Thesealgorithmsdifferfromtheiras-
sumptionsonthesub-aperturespotprofileandthewaysub-aperture
diffractionisaccountedfor,whiletheyrelyonacommonprelimi-
narystepfortheselectionofspotsretainedfortheanalysis.
2.3.1 AOSHspotsselection
Figure2.FWHMestimationfromsimulatedAOSHimagesasfunctionof
Forbothalgorithms,backgroundestimationisperformedonacor-
theseeingobtainedwithA1andA2.
neroftheimagewithoutspots,andhotpixelsaresettotheback-
ground. The cleanest and un-vignetted spots are selected in each
frame for the analysis. These extracted spots are oversampled by This quadratic subtraction defined in Eq. 8 is justified by the as-
afactortwo,re-centeredandaveraged.Theaveragingreducesthe sumptionthatthetwotermsofEq.7canbeapproximatedtoGaus-
influenceofpotentiallocalCCDdefects(e.g,badpixels,biasstruc- sianfunctions.
tures,etc...).Inpractice,theaveragedspotisbasedonhundredsof
selectedspots.
3 CALIBRATIONANDRESULTS
2.3.2 OTF-basedalgorithm
In order to calibrate and compare these two algorithms, we test
The first algorithm, hereafter A1, has been proposed by them with simulated AOSH images. To match our simulations
Tokovininetal.,(2007)andisbasedonthelong-exposurespotPSF to real situation, we assume identical conditions as the ones en-
profile defined in Eq. 2. The modulus of the long-exposure opti- countered with the AOSH system of the VLT. Assumptions for
cal transfer function of the averaged spot is calculated and nor- the AOSH model and atmospheric turbulence are described bel-
malized. Itisthen dividedby thesquaresub-aperture diffraction- low.Thewavelengthconsideredthroughthestudyis500nmexcept
limitedtransferfunctionT (f): whentheeffectofthewavelengthisanalyzed.
0
T (f)=(1−[λf /d])×(1−[λf /d]) (6)
0 x y
3.1 Simulationhypothesis
wheredisthesizeoftheAOSHsub-aperture.Atthisstagethecut
oftheT(f)alongeachaxiscanbeextractedandfittedtotheexpo- 3.1.1 Shack-Hartmannmodel
nentialpartofEq.2toderiveasingleparameterr ,orequivalently
0 OursimulationsarebasedonadiffractiveShack-Hartmannmodel
FouriertransformedtoderivetheFWHMoftheresultingspotPSF
thatreproducestheVLTAOSHgeometry:24sub-aperturesacross
profile using a 2-dimensional elliptical Gaussian fit. The orienta-
thepupildiameter,22pixelspersub-aperture,0.305′′ pixelscale,
tionofthelongandsmallaxisofT(f)isfoundbyfittingitwitha
d = D/24 = 0.338m.ThevalidityoftheAOSHmodelhasbeen
2-dimensionalellipticalGaussian.
verifiedthroughseveralaspectssuchastheplatescale,spotsizes,
slopemeasurements,andphasereconstruction.
2.3.3 PSF-basedalgorithm
The second algorithm, hereafter A2, has been proposed by 3.1.2 Atmosphericturbulence
Noetheetal.,(2006),andisinuseasadiagnostictoolattheVLT
The atmospheric turbulence is simulated with 300 uncorrelated
sinceMay2010.A2reliesontheassumptionthatAOSHspotscan
phase screens of dimension 3072 × 3072 pixels (i.e. 45 meters
bedescribedbythefollowingrotationallysymmetricPSFprofile:
width).Theprincipleofthegenerationofaphasescreenisbasedon
F(u)∝F0(u)⊗exp[−(u/r0)5/3] (7) theFourierapproach:randomizedwhitenoisemapsarecoloredin
where u is the spatial coordinate, F the diffraction limited PSF theFourierspacebytheturbulencepower spectral density(PSD)
0
oftheAOSHsub-aperture.Thesymbol⊗denotestheconvolution function, and the inverse Fourier transform of an outcome corre-
product.A2reliesonanapproximatePSFprofilebasedonthean- spondtoaphasescreenrealization.
alyticalexpressionoftheOTFdefinedinEq.2.ThealgorithmA2 Thevalidity of theatmospheric turbulence statistichasbeen
worksasfollow:aninitialestimationoftheFWHM(hereafter,θ)is carriedout onthesimulatedphasescreens: thevalueoftheouter
derivedbyfittingtheaveragedspotprofiletotheexponentialpart scale(L0),Friedparameter(r0)andseeingofthephasescreenshave
of Eq.7. Then A2 approximately accounts for F (u) by quadrat- beenconfirmedbydecompositionontheZernikepolynomialsand
0
ically subtracting the sub-aperture diffraction θ = λ/d from the variancemeasurementsoverthe300uncorrelatedphasescreens.In
0
estimatedFWHM: addition thevalidity of thelong-exposure AOSHimage has been
verified.InFig.1,weshowanexampleofasimulatedAOSHim-
FWHM ≈ qθ2−θ02. (8) age.
(cid:13)c 2011RAS,MNRAS000,1–??
4 P.Martinezet al.
Figure3.Sub-aperturediffractionremovalasfunctionoftheimagingwavelengthappliedonAOSHimagesgeneratedat500nmforA1(left)andA2(right).
Figure4.A1characterization:impactoftheturbulenceouterscaleL0(left)andtheAOSHimagingwavelength(right).
WehavegeneratedAOSHimagesthroughatmosphericturbu- similar although A2 constantly provides smaller FWHM estima-
lencewithseeingconditionsfrom0.1to1.8′′with0.1′′increment. tion than A1. In addition, A1 provides a smoother response than
Exceptwhentheturbulenceouterscaleisanopen-parameter,L is A2totheseeing;A2exhibitsirregularitiesinitsbehavior.
0
alwaysdefinedat22meters–VLTParanalmedianvalue,seefor Two regimes are observable: (1/) when the seeing is better than
instanceDali,etal.(2010). 0.6′′(A1)or0.3′′(A2)theFWHMishigherthantheseeing,which
islikelyduetotheunder-samplingoftheSH(pixelscaleis0.305′′),
(2/)whentheseeingdegradesfurtherthan0.6′′ (A1)or0.3′′ (A2)
3.2 Results the FWHM is smaller and asymptotically converges towards Eq.
5 (L =22m). This last result is important as it demonstrates that
InthissectionweprovideresultsfromthecomparisonofA1and 0
although d ≪ L , the FWHM of a long-exposure PSF obtained
A2,whilethebestalgorithmrevealedwillbefurthercharacterized 0
with very small telescope diameters (here AOSH sub-apertures
againstseveralimportantparameters.Thiscalibrationprocessisa
of ∼30cm) does follow Eq. 5. We note that A2 convergence to
necessary step prior to on-sky implementation of such numerical
Eq.5(L =22m)isinaccurateforworstseeingconditionsthan1.5′′.
tool,andwillprovideinsightsforthere-analysisoftheVLTAOSH 0
databasepresentedinSect.4.
–Sub-aperturediffractionremoval:
A1andA2differfromthewaytheyaccountforthethesub-aperture
3.2.1 Algorithmscomparisonandselection
diffraction. A1 is based on the deconvolution of a square sub-
–Responsetotheseeing: apertureOTF(Eq.6),whileA2makesuseofaquadraticsubtrac-
Figure 2 shows the FWHM extracted with both A1 and A2 as tion of θ (Eq. 8). To compare these two approaches, we assess
0
a function of the seeing. The black dotted line corresponds to a thesensitivityofbothalgorithmsonthesub-aperturediffraction.In
pureequivalencebetweenseeingandFWHM(i.e.Eq.5assuming practice,A1andA2havebeentestedonthesamesetofAOSHim-
L =∞), while the black dashed line corresponds to Eq. 5 with agessimulatedat500nm,whiletheinputimagingwavelengthused
0
L =22m being in agreement with the statistic of the simulated tofeedthealgorithmsforremovingthediffractionλ/dvariesfrom
0
atmospheric turbulence. The general trend of both algorithms is 400to800nm.Sincethebandwidthcenterofthewavefrontsensor
(cid:13)c 2011RAS,MNRAS000,1–??
Seeing estimationfromShack-Hartmannimages 5
pathcanvarywiththeguidestartype,pushingthistestfurtherwith 3.2.4 Spotsampling
suchalargerangeofimagingwavelengths(i.e.,wideramount of
Tounderstandtheasymptoticaltrendofthealgorithmresponseto
diffraction)isherejustified.
theseeingpresentedintheprevioussubsections,weanalyzetheef-
Results are presented in Fig. 3 and show that A1 and A2
fectofthespotsamplingontheFWHMmeasurement.Resultsare
behave similarity for seeing conditions > 1.0′′, while for lower
presentedinFig.5(left)whereitisshownthatforagivenseeing,
seeing by contrast to A1, A2 exhibits strong non reliable irreg-
theestimationgetsbetterwhenthesamplingimproves.FWHMat
ularities. Subtracting quadratically the diffraction FWHM θ as
0 0′′/pixelsamplingcorrespondstotheoreticalvaluesobtainedwith
carriedoutbyA2isaccurateaslongastheeffectofthediffraction
Eq.5.Aroughestimationindicatesthat5pixelsperFWHMisre-
is small relatively compared to the turbulence (d ≫ r ), and
0 quiredfor accurate measurement, though itcan becalibratedand
therefore fails at large r (good seeing conditions). In fact, the
0 correctedincaseofpoorsampling.
FWHMθ variesfrom0.24′′ to0.49′′ whenthewavelengthvaries
0
from400to800nm. TheactualAOSHPSFsareaconvolutionof
the atmospheric blur and diffraction, and since neither of these
3.2.5 Fieldstabilization
individuals broadening factors are Gaussian, calculation of the
combinedFWHMasaquadraticsumofindividualcontributionsis ThetelescopefieldstabilizationremovesthelowfrequencyTip-Tilt
notaccurate. componentsgeneratedby,amongotherthings,windshacking.As
aconsequencethefieldstabilizationalsoremovestheslowturbu-
–Algorithmselection: lence,whichmayreducetheFWHMofaPSFimage.Toassessthe
Considering all the aspects treated previously, it appears that A1 impactofthefieldstabilizationwecomparetheFWHMmeasure-
ismore appropriate than A2. Therefore, inthe following wewill mentsonsimulatedAOSHimageswheretheTip-Tiltcontribution
considerA1onlyforfurthercharacterizationforthesakeofclarity (inthefulltelescopepupil)hasbeencompletelyremoved(i.e.,per-
untilSect.4,whererealdataobtainedattheVLTwithA2willbe fect field stabilization with infinite bandwidth) to that of AOSH
re-analyzed. Thenext subsections concentrate onthedependency imagesleftunmodified. Thistest hasbeen carriedout for several
ofA1tovariousparameterswiththeobjectiveofallowingproper seeingconditions(0.1,0.9,1.8′′)andouterscalevalues(10m,22m,
calibrationofthealgorithm.Nonetheless,itshouldbenotedthatA2 andinfinitecase).Noimpactatallhasbeenrevealedbutata0.01′′
demonstratesidenticaldependencytotheseparameterssuchthata level, which is negligible. The estimation of the seeing from the
calibrationofA1orA2canbecarriedoutsimilarly. widthofAOSHspotsisthereforenotsensitivetothefieldstabiliza-
tion,whichrisesasarelevant advantage. TheTip-Tiltinthesub-
aperturecomesfromthecontributionofseverallow-ordermodes,
whichexplainswhytheAOSHspotPSFFWHMissensitivetothe
3.2.2 Turbulenceouterscale turbulenceouterscalebutnottofieldstabilization.
In Fig. 4 (left) the extracted FWHM with A1 as function of the
seeing isshown for several outer scale (L ) values: 10, 22m, and
0 3.2.6 Atmosphericdispersion
infinite outer scale (although likely about 200 meters due to the
physical finitesize of the phase screens used in simulation). It is Except for data taken at zenith, individual spot inthe AOSHim-
observablethatA1startstobesensitivetotheouterscaleforseeing agecanexhibitelongationinonedirectionduetouncorrectedat-
higherthan0.5′′.FromresultspresentedinFig.4(left)itisfurther mosphericdispersion.A1allowstheestimationofFWHMintwo
established(seeSect.3.2.1)thatAOSHsub-aperturesarenotsmall orientations:thesmallandthelargeaxes.Thealgorithmhasbeen
enoughsothataspotFWHMcanbeapproximatedtoε ,i.e.,the tested on 200 real AOSH images obtained at the VLT-UT3 Nas-
0
AOSHspot FWHMmeasurement doesdependontheouterscale mythfocus(notequippedwithanatmosphericdispersioncompen-
andthereforefollowsEq.5. sator, ADC) on May 12th 2010. Results of the FWHM extracted
alongthesmallandlargeaxisarepresentedinFig.5(right)where
theairmass isover-plotted. Resultsshow that A1 does differenti-
atetheelongatedaxis,andtheratiobetweenbothaxisfollowsthe
3.2.3 Imagingwavelength evolutionoftheairmass.Thedifferencebetweenthetwoaxiscan
beaslargeas0.3′′ whichissubstantial.Theabilitytodistinguish
ToanalyzehowtheAOSHimagingwavelengthimpactstheestima-
smallandlargeaxisoftheFWHMisthereforemandatory.Wenote
tionoftheseeing,wegenerateAOSHimagesforvariousimaging
thatA2,byassumingarotationallysymmetrictheoreticalspotPSF
wavelengthcoveringthevisiblespectrumfrom400to800nm.Re-
profile(Eq.7)doesnotallowtodifferentiatethesetwoorientations.
sultsarepresentedinFig.4(right)andindicatethataccurateknowl-
edgeoftheSHimagingwavelengthiscritical.Forallwavelengths,
A1responseisconstantlysmoothandcanthereforebeefficiently
3.2.7 DetectorPSF
calibrated. A1response alwaysasymptotically converges towards
Eq.5(L =22m),whereweremindthereaderthatr iswavelength Thediffusionofchargesinthedetectormaterialbeforetheyareal-
0 0
dependent.Theshorterthewavelength,thefastertheconvergence locatedtoonepixelisobservableasanartificialenlargementofthe
to Eq. 5. This behavior was already reported in Martinezetal., PSF.Inmostcases,thespatialresponseofthedetectorisnottriv-
(2010a),whereweexaminedthevalidityofEq.5withwavelength ialtodetermine.Figure6(left)presentstheimpactofthedetector
inthecontext oftelescope images. Asnoticed inSect.3.2.1, and PSFonthemeasuredFWHMinthecaseof1.0′′seeing.Wefound
generalizedhereforallwavelength,bellow0.6′′ theimpactofthe thatthedetectorPSFdoesenlargetheAOSHspotFWHMandthat
AOSHpixel scaleisobservable. The reliabilityof seeingestima- theeffectcanbesignificant;itstartstobesubstantialfrom1pixel,
tionfails at very good seeing conditions when seeing < 2× pixel whichislargerthanwhatisusuallyencounteredinscientificdetec-
scale(here0.61′′). tors,inparticularstheonesattheVLT.
(cid:13)c 2011RAS,MNRAS000,1–??
6 P.Martinezet al.
Figure5.Left:ImpactoftheAOSHspotsampling.Right:Impactoftheuncorrectedatmosphericdispersionontheseeingestimation(realdata).
Figure6.Top:ImpactofthedetectorPSFontheseeingestimation.Bottom:Impactofthesignal-to-noise(RON=15ADUs)ontheseeingestimation.
3.2.8 Signal-to-noise takingintoaccount someof thecriticalparametersstudiedinthe
previoussections.Accountingforthemedianouterscalevalueof
AttheVLTtheread-out-noise(RON)leveloftheAOSHisabove
Paranal(22m, Dali,etal.2010),theunder-samplingoftheAOSH
15 ADUs, while typical signal is about 20000 ADUs (per sub-
attheVLT,andtheimagingwavelength,wefoundthatthefollow-
aperture).Infigure6(right)wepresentthemeasuredFWHMinthe
ing equation provides accurate seeing estimate from A2 FWHM
caseof1.0′′seeingasafunctionofthesignallevel(inADUs)fora
measurements:
RONof15ADUs.Itisshownthatpoorsignal-to-noise(SNR)ratio
imageenlargestheFWHMwhichisevidentforsignallowerthan
ǫ =1.18×(FWHM1.84−0.15)1/2 (9)
10000ADUs,whileforsignalhigherthan10000ADUstheFWHM 0
measurements areroughly stable.Thestandard signal ADUlevel
Fromthe sensitivityanalysis of A1(likewiseA2) tovarious
obtainedattheVLTisthereforehighenoughtoavoidanyimpact
parameters,weproposetodrawhereaclassificationofthecritical
ontheFWHMestimation,althoughitcanbecalibratedotherwise
parameters associated with an error budget. Based on the previ-
knowingtheflux.
oussystematicexaminationofallparametersimpactontheseeing
estimation, we can conclude that L , λ, and the detector PSFare
0
themajorparameterstoconsider.Allotherparameterscanbeex-
4 APPLICATIONTOTHEVLTAOSHDATABASE cludedfromtheerror budget, beingeither withnegligibleimpact
(fieldstabilization,SNR),withaknownandconstantvalueandthus
4.1 Correctionlawanderrorbudget
calibrated (the spot sampling), or apart of the algorithm analysis
Torelateoursimulationstorealsituation,were-analyzedtheVLT (atmospheric dispersion). TheParanal median value for the outer
AOSHdatabaseobtained over thepast year.At theVLT,FWHM scaleis22m. Fortheerror budget ofthistermweconsider the±
estimatesarerecordedasdiagnosticinformationsincethecommis- onesigmadeviationaroundthemedianvalue22m(11mand42m).
sioningofthetelescope,whiletheA2algorithmisusedsinceMay AttheVLT,thebandwidthofthewavefrontsensorpathiscentered
2010.Asshownintheprevioussections,A2(similarlyA1)requires around550nminaverage.Asitcanvarywiththeguidestartype,
tobecalibrated.FromFig.2itisstraightforwardtofitthedataof we consider a central wavelength of 550 nm ± 50nm. Hence we
A2,andtoderiveacorrectionlawtotransformFWHMintoseeing assumehereadeviationbetween500and600nm.Forthedetector
(cid:13)c 2011RAS,MNRAS000,1–??
Seeing estimationfromShack-Hartmannimages 7
PSFterm,sincethevalueisunknownattheVLT,weconsiderthat
itisconfinedbetweenourreferenceandideal0valueand1pixel.
Wefoundthattheerrorterms(σ)canbeexpressedas:
σ =0.033×ǫ , (10)
L0 0
σ =−0.009+0.096×ǫ (11)
λ 0
σ =0.011×ǫ (12)
detector 0
Assumingthatalltermsareindependent,wecanthereforewritethe
errorbudgetofEq.9suchas:
σ2 =σ2 +σ2+σ2 (13)
ǫ0 L0 λ detector
WenotethatforA2,bycontrastwithA1,anadditionaltermshould
be considered: σ , standing for the atmospheric dispersion im-
AD
pact. Fromthis budget error, we can estimatethat all these inde- Figure7.SimultaneousseeingestimationwithFORS2andtheVLTAOSH
pendenterrorsourcesaffecttheseeingderivedfromAOSHimages database,whileAOSHdataarecorrectedbyEq.9(bluedots),andleftun-
at∼10%level.Ideally,adrasticreductionofthiserrorlevelispos- corrected (reddots).FORS2dataarecorrected byEq.5(L0).Bestfitof
siblethroughsimultaneousandinstantaneousmeasurementof L , each setofdata is over-plotted (full redand blue lines), while theblack
0
fulllinerepresentsthereference,i.e.,theidealperfectmatchofFORS2and
andaccurateknowledgeoftheguidestartype.
AOSHseeingestimates.
4.2 ApplicationtotheVLTdatabase
described by a Gaussian, and thus the calculation of the FWHM
WeappliedEq.9retroactivelyontheVLTAOSHdatabaseobtained based onaGaussian fitisnot accurate. Thistypically induces an
at UT4. Inparticular, weuse aset of 6500 simultaneous FWHM over-estimationoftheseeingat∼10%level.
measurements obtained since May 2010 with the AOSH (using WeinformthereaderthatEq.9canbeusedontheVLTAOSH
A2)andtheFORS2instrumentimages.Weremindthereaderthat database,beingtemporallyvalidfromMay2010uptonow.From
FWHM estimation on long-exposure PSF from any scientific in- thecommissioningoftheVLTtoMay2010,anothercorrectionlaw
strumentfollowsEq.5.WenotethatFORS2datamightbebiased mustbederivedsinceanotheralgorithmwasinuse.Inaddition,in
byseveralparameters,amongofthemthetelescopefieldstabiliza- thefuture, A1should replaceA2at theVLT,and thusEq. 9will
tion, and the accuracy of the FWHM extraction which is based requiretobere-adaptedaccordingly.
ontheSExtractorsoftware(SourceExtractor,Bertin,etal.1996).
Briefly,SExtractorisrunonthereducedFORS2images,andstars
areidentifiedbasedonSextractorparameters,whiletheirFWHMs
5 CONCLUSION
aremeasuredbytheprogramthroughtheuseofaGaussianfitap-
plied on the data. The FWHM is afterwards extracted from the ActiveOpticsShack-Hartmann(AOSH)sensoroffersanadvanta-
estimated Gaussian profile. The examination of the reliability of geouspossibilitytoestimatetheseeingatthefocusofatelescope:
FORS2 data is beyond the scope of this paper, and therefore the (1/) it delivers long exposure PSFs, (2/) it is not affected by any
comparison between AOSHand FORS2datamust be here, care- observationalbias(continuousreal-timeseeingestimation)bycon-
fullyconsidered. trasttoscientificinstruments,(3/)weshowthatitisnotsensitiveto
The correction to apply to the data is carried out as follow: thetelescopefieldstabilization.
(1)FORS2FWHMdataarecorrectedbyEq.5assumingL =22m Thecomparisonofthetwoproposedalgorithmsgaveadvan-
0
(Paranalmedianvalue),whileA2FWHMareleftuncorrected by tagetotheoneproposedbyTokovininetal.,(2007)whichisbased
Eq.9,(2)FORS2FWHMdataarecorrectedbyEq.5(L =22m)and on the theoretical OTF expression of the long-exposure PSF. Al-
0
A2FWHMdataarecorrectedbyEq.9.ResultsareshowninFig.7, thoughPSF-basedandOTF-basedalgorithmsexhibitsimilargen-
wherethereddotscorrespondto(1)andthebluedotsto(2).Fig- eralbehavior,wefoundthatA1providesasmootherresponsethan
ure7demonstratesthatFORS2correctionfor L andAOSHcor- A2totheseeingandbetteraccountforthesub-aperturediffraction,
0
rectionwithEq.9bring seeingcorrespondence inabetter agree- andatmosphericdispersion.
ment. This is obviously expected as both FORS2 and AOSH de- We show that the estimation of the seeing from AOSH im-
liver FWHMestimateat thesamelocation: the focusof the UT4 ages is sensitive to several parameters but can be calibrated. We
telescope. This result further confirm the reliabilityof seeing ex- establish that even considering the small size of the AOSH sub-
tractionfromAOSHdata. apertures (∼30cm), it does depend on the turbulence outer scale
The power-laws of both set of data are provided in the leg- L andthereforefollowsEq.5.WeshowthatAOSHcanbeused
0
endofFig.7andareover-plottedforthesakeofclarity.Whenthe tobuildstatisticsusingmedianvalueof L ,inareal-timefashion
0
AOSHdataarecorrectedby Eq.9,itisobservable thattheslope but relying on median L value, or requiring instantaneous mea-
0
ofthepower-lawthroughthedataislinearandequal tounity, by surement of L . We note that in this context several independent
0
contrast to the situation where AOSH data are left uncorrected. campaigns have converged to the median value of L of 22 m at
0
Onlyaconstantandhomogeneousover-estimationofFORS2see- Paranal(e.g.,Dali,etal.2010).
ingremains.Suchanoverestimationreflectsthepresenceofabias, The VLT AOSH database available since the commission-
which likelyfindsitsorigininthe FWHMestimation carriedout ingofthetelescopedoes notprovide seeinginformationbut spot
bySExtractor.Forinstance,long-exposurePSFprofilescannotbe FWHMs.Inthiscontext,Eq.9isrequiredtoderivetheseeingin-
(cid:13)c 2011RAS,MNRAS000,1–??
8 P.Martinezet al.
formation.Weremindthereader,thatEq.9istemporallyvaliduntil Dalietal.,2010,A&A,524,A73+
May2010.Withtheseconsiderationsinmind,there-analysisofthe Fried,D.L.,1966,JournaloftheOpticalSocietyofAmerica,56,
pastyearsoftheVLTAOSHdatabasedemonstratesaseeingcor- 1372+
respondence in better agreement between AOSH and the FORS2 Goodman,J.W.,1985,StatisticalOpticsbyJosephW.Goddman
imager.Thisresultfurtherconfirmsthereliabilityofseeingextrac- NewYork,NY:JohnWileyandSons,1985
tionfromAOSHimages. KornilovV.G.andTokovinin,A.A.,Measurementoftheturbu-
Finally,thequalificationandcalibrationofthealgorithmA1 lenceinthefreeatmosphere above Mt.Maoverset, Astronomy
are nearly completed and clear the path to its operational imple- Reports
mentationattheVLTforthreemonthsoftestperiodstartinginfall Martinez,P.,Kolb,J.,Tokovinin,A.,andSarazin,M.,2010,A&A,
2011.Finally,wenotethatmoreemphasisisgiventouseclosed- 516,A90+
loop real-time AO instruments data in the near future to get the Martinez,P.,Kolb,J.,Sarazin,M.,andTokovinin,A.,2010,The
estimateoftheseeingatthecriticallocationofthetelescopefocus. Messenger,141,5-8
Noethe,L.,Sarazin,M.,2006,ESOinternaldocument
Robert,C.,Voyez,J.,Ve´drenne,N.,andMugnier,L.,2011,Pro-
ACKNOWLEDGMENTS ceedinginComprehensivecharacterizationofastronomicalsites
Roddier, F., 1981, Progress in optics. Volume 19. Amsterdam,
The activity outlined in the paper was supported by the Euro- North-HollandPublishingCo.,1981,p.281-376
peanCommission,SeventhFrameworkProgramme(FP7),Capaci- Sarazin,M.,andRoddier,F.,1990,A&A,227,294-300
tiesSpecificProgramme,ResearchInfrastructures;specificallythe Tokovinin,2002,PASP,114,1156-1166
FP7, Preparing for the construction of the European Extremely Tokovinin,A.,Sarazin,M.,andSmette,A.,2007,MNRAS,378,
LargeTelescopeGrantAgreement,ContractnumberINFRA-2007- 701-708
2.2.1.28.WeacknowledgePhilippeDuhouxfromESOforhishelp Tatarskii, 1961, V.I. Wave propagation in a turbulent medium,
withtheVLTcontrolsoftware. DoverPubl.,Inc.,NewYork
Ziad,A.,Conan,R.,Tokovinin,A.,Martin,F.,andBorgnino,J.,
2000,AppliedOptics,39,5415-5425
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