High-Resolution Altitude Profiles of the Atmospheric Turbulence with PML at the Sutherland Observatory

3 MB·

by  L. Catala

MNRAS000,1–14(2016) CompiledusingMNRASLATEXstylefilev3.0
High-Resolution Altitude Profiles of the Atmospheric
Turbulence with PML at the Sutherland Observatory
L. Catala1,2?, A. Ziad3, Y. Fante¨ı-Caujolle3, S.M. Crawford1, D.A.H. Buckley4,1, J.
Borgnino3, F. Blary3, M. Nickola5 and T. Pickering6
1South African Astronomical Observatory, Observatory Road, Observatory 7935, South Africa
2University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa
3Universit´e Cˆote d’Azur, OCA, CNRS, Laboratoire J.L. Lagrange UMR 7293, Parc Valrose F-06108 Nice Cedex 2, France
7 4Southern African Large Telescope, P.O. Box 9,Observatory 7935, South Africa
1 5Space Geodesy Program Hartebeesthoeck Radio Astronomy Observatory, PO Box 443, Krugersdorp 1740, South Africa
0 6Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
2

n
a AcceptedXXX.ReceivedYYY;inoriginalformZZZ
J

8
2 ABSTRACT
With the prospect of the next generation of ground-based telescopes, the extremely

] large telescopes (ELTs), increasingly complex and demanding adaptive optics (AO)
M systemsareneeded.Thisistocompensateforimagedistortioncausedbyatmospheric
turbulence and fully take advantage of mirrors with diameters of 30 to 40 m. This
I
. requiresamoreprecisecharacterizationoftheturbulence.ThePML(ProfilerofMoon
h
Limb) was developed within this context. The PML aims to provide high-resolution
p
- altitude profiles of the turbulence using differential measurements of the Moon limb
o position to calculate the transverse spatio-angular covariance of the Angle of Arrival
r fluctuations.Thecovarianceofdifferentialimagemotionfordifferentseparationangles
t
s is sensitive to the altitude distribution of the seeing. The use of the continuous Moon
a
limb provides a large number of separation angles allowing for the high-resolution
[
altitude of the profiles. The method is presented and tested with simulated data.

1 Moreover a PML instrument was deployed at the Sutherland Observatory in South
v Africa in August 2011. We present here the results of this measurement campaign.
5
3 Key words: turbulence – atmospheric effects – site testing.
2
8
0
.
1 1 INTRODUCTION profile via the turbulence structure function (Cn2(h)), which
0
gives a measure of the turbulence strength of a layer at an
7 Over the past decades a number of instruments have been
altitude h. However, all profilers have limitations: either a
1 developed in order to measure the atmospheric turbulence,
low-resolution altitude profile of the whole atmosphere or a
: whichaffectthequalityofimagesfromgroundbasedoptical
v high-resolutionaltitudeprofileofonlyasectionoftheatmo-
i telescopes. sphere at ground layer(GL) or in the free atmosphere(FA).
X Thedifferentialimagemotionmonitor(DIMM,Sarazin
Despite their limitations these instruments have provided
r & Roddier (1990)) and the multi-aperture scintillation sen-
a very useful information for site selection, continuous seeing
sor (MASS, Kornilov et al. (2002)) are the most commonly
monitoring, and the determination of the essential parame-
usedinstrumentsforcontinuousmonitoringatobservatories
ters needed for the design of adaptive optics (AO) systems.
aroundtheworld.Otherinstruments,suchasthegeneralized
seeingmonitor(GSM,Martinetal.(1994)),theslopedetec-
With the advent of the extremely large telescopes
tionandranging(SLODAR,Wilson(2002))andthescintil-
(ELTs)withdiametersgreaterthan30m,theconstraintson
lationdetectionandranging(SCIDAR,Fuchsetal.(1998)),
thedesignofanAOsystemarebecomingmoredemanding,
have been extensively used during site testing campaigns.
requiring the development of new atmospheric turbulence
Those instruments can be classified in 2 main categories:
monitoring instruments that provide more accurate mea-
the instruments that only measure atmospheric turbulence
sures of atmospheric profiles with high resolution through
parametersvaluesintegratedthroughtheentireatmosphere
theentireatmosphere.ThequalityoftheAOcorrectionover
andtheprofilersprovidinganestimationsoftheturbulence
alargefieldofviewforthisnextgenerationofground-based
telescopesreliesontheaccuratedeterminationoftheoptical
(cid:63) E-mail:[email protected] parametersoftheatmosphericturbulence(Costille&Fusco
©2016TheAuthors
2 L. Catala et al.
2011).Inparticular,thedistributionoftheturbulenceinthe
different layers of the atmosphere is a critical parameter.
A new instrument, Profiler of Moon Limb (PML, Ziad
etal.(2013)),hasbeendevelopedinordertoprovideCn2(h)
profiles from differential measurement of the wavefront An-
gle of Arrival (AA, Borgnino (1990)) fluctuations along the
lunarlimb.Thedifferentialmeasurementismadepossibleby
the use of 2 sub-apertures, similar to the DIMM technique.
Directmeasurementmethodsuseseriesofsingleimagesthat
areaffectedbytelescopevibrationsandwindshake.Weget
rid of these effects thanks to the differential method. Mea-
surements are done from the difference between the mea-
sured Moon edge position in one sub-aperture with that
measured in the other. Since both apertures are similarly
affected by telescope vibrations and wind shake, those are
suppressed by the differential measurement. Moreover, the
useoftheMoonlimbofferstheadvantageofprovidingavery
high-resolution altitude profile of the turbulence (Cn2(h)) in
addition to all the integrated atmospheric parameters that Figure 1.PrincipleofPMLmeasurement.Whenmeasuringthe
otherinstrumentsprovide:thecoherencelength(r ),thesee- angular covariance of a system with a fixed base, B, the contri-
0
ing (ε ), the coherence time (τ ) and the isoplanatic angle bution of a layer at an altitude h peaks for an angular value of
0 0 θ= B.
(θ0). The PML instrument is an expansion of the monitor h
ofouterscale(MOSP,Maireetal.(2007))conceptbasedon
a direct measurement method using series of single images
campaign, we chose to resolve using a simulated annealing
of the Moon limb to retrieve the outer scale profile (L0(h)) method (Maire et al. 2007).
from the structure function of transverse AA fluctuations.
Here we recall the theoretical expression of the spatio-
Similarly, the PML provides a measure of the turbulence
angular covariance of the transverse AA fluctuations in the
profile(Cn2(h))fromthedifferentialcovarianceofthesefluc- case of the Von Karman turbulence model (Von Karman
tuations.
1948), detail how we compute it, and test the response of
In the context of the Sutherland site characterization
our inversion grid.
andinordertoprovideinformationforasimulationstudyon
potential AO performances on the Southern African Large
Telescope (SALT), a PML observing campaign was car- 2.1 Theoretical Angle of Arrival covariance
ried out at Sutherland. This campaign was primarily used
The optical atmospheric turbulence is commonly described
to work on the instrument data processing and inversion
by the spectrum of its index of refraction (n) fluctuations
mPMetLhomdedaseuvreelompemnetsnta,tbtuhte Salustoheprrloavniddesditefi.rst results from that follows a Kolmogorov’s law: Φn(k) = 0.033Cn2 k−11/3,
where k is the wave-number. However, the Kolmogorov’s
In this paper we present the theoretical background
model assumes an infinite outer scale ( ) value. In order
of the PML working principle in the second section. An L0
to take into account the finite size of the outer scale, other
overview of the optical layout of the instrument is given
models were developed. In the case of the PML instrument
insection3.Thefourthsectionisdedicatedtothemeasure-
mentWaendusdeadtasipmrouclaetsesidngdatetachtnoiqupero.be the validity of the w[is2Leπg0ui]v2see]−nt1h1b/ey6V.(BBonaosrKegdnarionmnoatenhtmisalom.d1oe9ld:9eΦ2l,;ntA(hkve)ilA=aA0e.t0sp3a3la.t[1i2a9πl9]c73o)vC:an2ri[akn2c+e
method and test our data analysis process. Those simula-
tPioMnLs awraespdreepsleonyteeddainttthheeSfiuftthhesrelacntidonS.AIAnOAuOgbussetrv2a0t1o1rythine Cα(B,D)=1.19sec(z) dhCn2(h)S(B,D,L0(h)), (1)
Z
SouthAfrica.Theresultsofthisobservingcampaign,aswell
with
as comparison with ancillary instruments, are presented in
1
tinhgerseixmtharskesctaiboonu.tInthtehePsMevLenmthethseocdt.ionweprofferconclud- S(B,D,L0(h))=Z df f3(f2+ L0(h)2)−1(cid:34)1/6 (cid:35)
[J0(2πfB)+J2(2πfB)] 2J1π(πDDff) 2, (2)
2 THEORATICAL BACKGROUND AND where z isthezenithangle, B istheseparationbetweentwo
RECONSTRUCTION METHOD sub-apertures of diameter D, L0(h) is the wavefront outer
The PML uses Moon images from two sub-apertures in or- scale at the altitude h, f the spatial frequency and Jm are
Bessel function of order m.
der to measure the profile of the atmospheric turbulence as
Inthecase ofdifferentialmeasurements,andforobser-
illustrated in Figure 1. In order to recover the turbulence
vations in two directions separated by an angle θ (Figure
profile(Cn2(h))wecompareourdatatoatheoreticalmodel. 1) the differential angular covariance can be expressed as
The method uses the covariance of the transverse AA fluc-
follows (Ziad et al. 2013):
tuations. Comparing the theoretical and measured covari-
anceisanon-linearinverseproblemthat,fortheSutherland C∆α(B,D,θ)=2Cα(θh,D) Cα(B θh,D) Cα(B+θh,D). (3)
− − −
MNRAS000,1–14(2016)
PML - Turbulence Profile 3
Figure2.Theoretical”S”functions,givenbyeq.6.Top:S (θ,h),S0(θ,h)andS+(θ,h)fromlefttoright.Bottom:KS matrix.Inallfive
figures h is increasing from bottom to top. Values are give−n for the 33 single layers of the reconstruction grid. θ goes from -356 to 356
arcsecondsfromlefttorightforthethreetopfiguresandthebottomleft.ThebottomrightfigureshowsthepositivevaluesofKS that
wewillusefortheinversion,asweonlymeasurepositiveθ.
θh is the spatial distance of the perturbed wavefront inter- KCn(h)=1.19sec(z)∆hCn2(h).
cepted by an angle θ at an altitude h (Figure 1). -Theshapeterm,containingtheouterscalevalueinforma-
Using eq. 1 and 2, this gives: tion:
C∆α(B,D,θ)=1.19sec(z) dhCn2(h)[2S0h−S−h−S+h], (4) -KTL(hhe, ffi)lt=erifn3g(ft2e+rmLs0,1(hli)n2k)−ed11/t6o. the system sub-pupils and
Z base: (cid:102) (cid:103)
wCS+hhone=rseiSd,(eBSr0ih+ng=θht,ShD(eθ,hLo,v0De(,rhaL)l)l0.(aht)m),oSs−phh=ereS(aBs−aθshu,pDe,rLp0o(shit)i)onanodf KK−0JJ((hh,,θθ,, ff))==[[JJ00((22ππffθ(Bh)−+θhJ2))(2+πJf2θ(h2)π]f(2BJ1−π(πDθDhf)f))](cid:102)22,J1π(πDDff)(cid:103)2
tehxipnre∆sshiiondiascsraetseulmay:ers at altitudes hi we can rewrite this aKn+Jd(h,θ, f)=[J0(2πf(B+θh))+J2(2πf(B+θh))](cid:102)2J1π(πDDff)(cid:103)2.
This allows us to rewrite the S integrals in the following
C∆α(B,D,θ)=1.19sec(z) i ∆hiCn2(hi)[2S0hi −S−hi −S+hi], (5) form:
X
For easier calculation, we split the components solely de-
pendentonpredefinedparameters(altitudegridandsystem S0, ,+(h,θ)= dfKL(h, f).K0J, ,+(h,θ, f), (6)
parameters)fromthosedependentonparametersthatneed − Zf −
to be determined(Cn2(h), L0(h)), as follows:
-Theenergyterm,containingtheturbulencestrengthinfor-
mation: The S0, ,+(h,θ) functions can be determined for each indi-
−
MNRAS000,1–14(2016)
4 L. Catala et al.
Fdiiffgeurernet3l.ayTehreaolrteittuicdael:Sh0=,−,+50fmor,s2i5n0gmle,la5y0e0rms,w1ikthmL,20k(hm),=52k0mma.nLdef1t:0kSm−..CEeanctherl:inSe0s.oRnigthhte:sSe+fi.gEuarcehsccoorlorersrpeopnrdesetontssintghleefruonwcstiofrnosmfotrhae
topthreeimagesofFigure2.
vidual layer and hence, summing over all altitudes, gives:
C∆α(θ)= KCn(h).[2 dfK0J(h,θ, f).KL(h, f)
f
h Z
X
− f dfK+J(h,θ, f).KL(h, f)− f dfK−J(h,θ, f).KL(h, f)].
Z Z
(7)
If we consider the case of a fixed , the three dfKJ.KL
L0 f
components can be precalculated and stored in a matrix
R
KS =2S0−S−−S+ (Figure 2). We can then write:
C=KCn.KS,
where KCn is a 1xN matrix and KS is a NxM matrix, with
N the number of layers of the reconstruction grid and M
the number of separation angles (θ) along the Lunar limb.
Usingachosenaltitudegridandtheθ valuessetbythe Figure4.Covarianceforasinglelayerofunitystrength(KCn=
system configuration, we can compute all the KJ functions 1) at h=350 m and with L0 = 20m. Blue line: 2S0. Blue dotted
and,inturn,theSfunctionsatfixed andthecorrespond- lines:S+, .Notethepositionofthepeakofthelateralcomponent
iSn0g, ,K+Sfumnacttriioxn.sTwhehitloepthroewboofttFoimgurgerLa2p0shhsowresptrheesetnhteotrheetiKcaSl Nlfooorctasetiemtdhpeal−ithc+iitg/yh−avBhnadl.uBbeleoatfctkeCrl∆ivnαiesd:uuCael∆izαtao(thtioh=ne1fokafmcat)ltl=hcauKtrCvwenes.[tt2ooSog0ke−tKhSe−Cr−.nS=+]1.
ma−trix that will be used for the inversion. The bottom left
shows the full KS matrix, including negative θ values. As
we only perform measurements for positive θ, we will use resultinginthetheoreticaldifferentialcovariance(blackline)
the positive side of the matrix for the inversion shown on forasinglelayerat350m.Here,forsimplicityweconsidera
the bottom right side figure. The number of separation an- layer of unity strength, KCn =1, and L0 =20m. Note that
gles available (x-axis) is determined by the system layout weonlymeasurepositiveseparationangles,therefore,forthe
and is given by the number of pixels along the Lunar limb. reconstruction we only consider the positive components of
Herewehavesetthenumberoflayersto33witharangeof the covariance.
altitudes going from 10 m to 24 km above the telescope en- In the more realistic case of an atmosphere made up
trancepupil. issetto20m.Wealsoshowthe2Dcurveof ofmultiplelayersofvariablethicknessatdifferentaltitudes,
L0
thetheoreticalS0, ,+ functionsfor7individuallayersinFig- withdifferentturbulencestrengthandouterscalevalue,one
ure 3. From this r−epresentation, we can clearly see that the willinputeachofthelayersparameters(hi,∆hi,Cn2(hi) and
positionofthepeakofcovarianceinthelateralcomponents L0(hi)), before adding up all the layers contribution to get
(left and right figures) is dependent on the layer altitude h: the equivalent covariance for the overall atmosphere. From
θpeak = Bh, with B the base between the 2 sub-apertures at those parameters, we chose two beforehand (hi, ∆hi). The
the telescope entrance pupil. Hence, for lower altitudes the other two (L0(hi), Cn2(hi)), can be retrieved by minimiz-
peak of covariance is located at larger separation angles θ. ing the difference between theoretical and measured values.
Figure 4 shows the combined S0,S− and S+ (blue lines) However, while the different components (S0h,S−h,S+h) of the
MNRAS000,1–14(2016)
PML - Turbulence Profile 5
Table 1.Reconstructionaltitudegrid.
GL h[m] 10 150 250 350 450 550 650 750 850 950
dh=100m
FA h[km] 1.25 1.75 2.25 2.75 3.25 3.75 4.25 4.75
dh=500m
h[km] 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 16 18 20 22 24
dh=1km dh=2km
differentialangularcovariancehaveastrongdependenceon expectsthatforaturbulentlayerlocatedatoneoftherecon-
theouterscale,itsimpactonthedifferentialcovarianceitself struction grid altitudes, all the turbulence will be reflected
is mitigated by the fact that it is given by the combination in that layer after the inversion. In the case of a turbulent
oftwicethecentralcovarianceminusthetwolateralcovari- layer located in between two altitudes of the reconstruction
ance. It is only when the outer scale is small (in the metric grid, one will expect the reconstruction to spread the tur-
range) that its impact on the differential covariance cannot bulencebetweentheadjacentlayers.Ifwetaketheinput10
be neglected anymore (Borgnino et al. 1992). In the partic- km layer (on the x-axis), located between the 9.5 km and
ular case of astronomical observatory sites the outer scale 11 km layers of the reconstruction grid (on the x-axis), the
is known to be in the decametric range. Hence we can sim- turbulence is redistributed with 63.7% in the 9.5 km layer
plifytheinversionproblembytakingafixedtheouterscale and 36.3% in the 11 km one. Similarly the 21 km layer is
valueandreducethereconstructiontotheturbulenceprofile split with 48.15% in the 20 km layer and 51.85% in the 22
alone.SimilarlytotheworkdoneontheMOSPinstrument km layer. The redistribution agrees with the theoretical ex-
(Maire et al. 2007), we will use a simulated annealing algo- pectationsandvalidatesboththechoiceofouraltitudegrid
rithmfortheminimizationprocessleadingtothereconstruc- and inversion method.
tionoftheturbulenceprofile,Cn2(h).dh.Otherminimization
technique were also tested and presented in (Blary et al.
2014).
3 OPTICAL LAYOUT AND MEASUREMENT
METHOD
2.2 Altitude grid and Inversion Response
3.1 PML optical layout
In order to cover both the GL and the FA part of the at-
mospheric turbulence, we chose a 33 layers grid. There are The PML (Ziad et al. 2010, 2013) was designed to provide
10 layers for the GL below 1 km and 23 layers for the FA high-resolution altitude profiles of the atmospheric turbu-
between1and25km.Thedetailofthealtitudegridisgiven lence. Similar to the DIMM technique, it uses a differential
in Table 1. The number of layers was chosen in agreement method via a two sub-aperture mask mounted at the en-
withpreviousfindingsthatshowedthenecessitytoknow30 trance pupil of the telescope, allowing telescope vibration
to40layersinordertofeedthelateradaptiveopticssystems andwindshakeeffectstobeignored.Theprofilesarerecon-
using tomography (Costille & Fusco 2012). structed from the differential covariance functions. The use
We tested the response of the reconstruction grid to ofthecontinuousLunarlimb,ascomparedtoadoublestar
89individualturbulentlayerswithaltitudesrangingfrom5 with SLODAR, provides a large number of separation an-
m to 30 km. For simplicity all layers were of unit strength gles,allowingforthehigh-resolutionofthealtitudeprofiles.
(KCn = 1) and with an outer scale value (L0) fixed to 20 ThePMLconsistsofa16-inchMEADEtelescopetube
m.WeshowtheresponseresultsinFigure5.Inthetopfig- mounted on an Astro-Physics AP3600 equatorial mount
ure, the x-axis shows the altitudes of the input turbulent with a mask made of 2 holes with separation B = 0.267m,
layer,whilethey-axisshowsthealtitudesofthereconstruc- anddiameter D=0.06m (Figure6,left).Whenpointingthe
tion grid. The pink ellipses represent the relative amount telescopeattheMoontwoimagesofthelimbareproduced,
of turbulence in each layer of the reconstruction grid. The correspondingtothetwosub-apertures.Inordertoseparate
bottomgraphofFigure5showshoweachofthereconstruc- the 2 images, a Dove prism (D) is introduced in the optical
tion grid altitude is sensitive to turbulence in the adjacent path (Figure 6,right). The Dove prismflipsover one of the
layers.Thebaseofeachtrianglegivestherangeofaltitudes images and avoids overlap of the images. The image acqui-
forwhichtheindividuallayersofthereconstructiongridcan sition is performed by a PCO Pixelfly CCD operating at a
partiallysenseturbulence.Theheightateachaltitudegives frame rate of 33 Hz. The CCD, with a pixel size of 9.9 mi-
the sensitivity strength from 0 to 1, the latter being 100% crons,producesimagesof640x480pixels.Theimagescaleis
sensitive. 0.594arcsecperpixel.Theexposuretimeneedstobeshort
Foreachofthe89singleturbulentlayersthereconstruc- enough to ”freeze”the turbulence, typically of the order of
tion process should apportion the turbulence of the input a few ms (i.e. τ ). Here it was set to 5 ms. The number of
0
layer between the 33 layers of the reconstruction grid. One images used for each measurement was set to a thousand
MNRAS000,1–14(2016)
6 L. Catala et al.
Figure 5. Response of the inversion algorithm to single turbulent layers. Top: the single turbulent layer input altitude is given on
the x-axis while the y-axis represents the altitudes of the reconstruction grid. For each input layer (x) the relative distribution of the
turbulencethroughoutthereconstructiongridlayers(y)isrepresentedbythepinkellipses.Bottom:Thecolortriangle-likeshapedcurves
representthesensitivityofeachaltitudegridheighttoturbulenceinall89inputlayers.
images per data set. For each acquisition we used the sta- The eventual residual rotation is measured by stacking
tistical properties of the atmospheric turbulence to retrieve all images of an acquisition. This is equivalent to a long-
its parameters. exposureimageandsuppressestheseeingeffecttoonlykeep
the static optical misalignment. The difference between the
topandbottomedgepositionsgivesustheresidualrotation
angle. The x-drift is measured for each image as compared
3.2 Image pre-processing and ”cleaning”
to a reference image chosen to be the first of an acquisi-
Prior to the data analysis that will lead to the profile re- tion.Finallythedriftparalleltothey-axisduetoimperfect
construction, there are a number of steps that need to be polaralignmentismeasuredbyfittingalinetothedatarep-
followed to make sure that we removed any instrumental resentingthemeanedgepositionthroughoutanacquisition
bias due to optical misalignment and imperfect tracking. as shown in the image 4 of Figure 7.
This will ensure that when performing the differential mea- After applying all corrections, we can measure the po-
surements we properly match the same point on the Moon sition of the Moon edges on the images that we will use for
edge from both images, hence only measuring the edge mo- the profile reconstruction.
tionduetoatmosphericturbulence.Thefullpre-processing,
summarized in Figure 7, involves the following:
1. Flat fielding and Dark frame subtraction.
3.3 Angle of arrival covariance - Experimental
2. Measuring image rotation, should any remain after the
measurements and profile reconstruction
optical alignment of the Dove prism.
3. Measuring shift in the x-direction, if any. Foreachimage,wedeterminetheedgepositionbytakingthe
4. Measuring image drift due to telescope pointing inaccu- image derivative before using a barycenter method for the
racy. if any. detection (Maire et al. 2007). The method is illustrated in
5. Applying, rotation, shift and drift correction to the im- Figure8.Oncetheedgepositionshavebeendeterminedfor
ages. allsetsoftwoimagesofanacquisiton,wecancalculatethe
experimentaldifferentialcovarianceoftheAAasillustrated
MNRAS000,1–14(2016)
PML - Turbulence Profile 7
Figure 6. PML optical layout. Left: Schematic of the overall instrument setup with the 2 sub-aperture mask at the entrance pupil of
thetelescope.Right:SchematicoftheopticalpathfromthetelescopeentrancepupiltotheimagingCCD.L1isacollimatinglens,DP
representstheDoveprismandL2refocussesthecollimatedbeamontotheimagingCCD.
Figure 7.Fullalignmentprocesssummary.1:FlatfieldingandDarkframesubstraction.2:Measuretheresidualrotationbetweenthe
Top and Bottom images. 3: Find the x-displacement of each image (dxi) with respect to the reference image. 4: Find the amount of
driftingbetweenimages(k).Thefigureinstep4showstheaveragedpositionofthelimb(y)forall1000imagesofanacquisition(x)and
theslopeoflinefits(cyanandyellow)givesustheamountofdriftingbetweentwoconsecutiveimages(darkbluedotsandcyanlineare
fromtheTopimagewhilereddotsandyellowlinearefromtheBottomimage).5:Applyingtheshift(dxi)anddrift(k.i)correctionto
eachimage,aftercorrectionfortherotation.Thentheimagesarereadytobeusedforedgedetectionanddataextraction.
MNRAS000,1–14(2016)
8 L. Catala et al.
Figure 8. Edge Detection. Top left: original image and an ex-
ampleofaverticalcutshowingaHeavysidestepfunctionatthe
edgeoftheLunarlimb.BottomLeft:Derivativeoftheimageand
the peak function of its vertical cut. Right: Zoom in around the
peak of the image derivative and windowing used to perform a
barycentermeasurementofthepeakposition. Figure 9.Experimentalcovariance.Detailofthedifferentialco-
variancemeasurement.Wecalculatetheproductofthedifference
betweentop(blue)andbottom(red)positionsat xi and (xi+θ)
in Figure 9: forall xi positionsalongtheedge.Theaverageofallproductsis
thedifferentialcovariance(Cmeas(θ))valueataseparationθ
C∆mαeas(θ)=h[αT(xi)−αB(xi)][αT(xi+θ)−αB(xi+θ)]i, ∆α
where αT(xi) and αB(xi) aretheverticalcoordinatesofthe
limbataninitial xi coordinatefor,respectively,thetopand ically reject the solution but accept it with a probability
bottom images. αT(xi+θ) and αB(xi+θ) are the positions p = e(−∆E/T). This cost-increasing acceptance probability
at the xi coordinate along the limb separated by an angle allowsforexploringthefullparameterspaceandavoidsbe-
θ fromtheinitialposition.Thebracketssigns( )represent coming trapped in a local minimum. This acceptance prob-
hi
theaveragevalueforalltheproductscorrespondingtoaspe- ability is set by the ”temperature”parameterT, in analogy
cificseparationangleθ alongtheedge.Onecanseethatthe withthermodynamics.TheSAalgorithmstartswithahigh
largerθ,thefewernumberofmeasurementsalongthefinite initial temperature to explore a wide area of the parameter
length of the Lunar limb. After the image pre-processing, space and a ”cooling”schedule slowly lowers the ”tempera-
the”cleaned”imagesaregenerallybetween550and600pix- ture”towards the reduction of the search around the global
els wide. This gives a maximum of 599 measurements for minimum. We will stop the search, and keep the current
the smallest separation angle (θ1 = 1 pixel = 0.594 arcsec- best set of Cn2(hi) values as our best fit result, when, at a
onds) and a single measurement for the largest separation fixed ”temperature”, no improvement to the cost function
angle (θmax 599 pixels 356 arcseconds). The measure- canbemade.AsimilartechniquewasalsousedinMaireet
∼ ’
menterroristhereforemuchlargerforlargerthansmallerθ. al. (2007).
Intheinversionprocesswewillweighthefitsbythenumber
of data points for each θ. For one acquisition, we calculate
C∆α(θ) foreachofthe1000images.Thefinaldifferentialco- 3.4 PML Fried parameter extraction
variance function for the acquisition is obtained by taking
the average of all thousandsC∆α(θ). In addition to the turbulence profile the PML data can be
Using both, measured and theoretical, covariances one used to measure the integrated seeing by determining the
canreconstructtheturbulence.Wegenerateanatmospheric Friedparameter.Foreachacquisition,wehavethetemporal
ppurotfieleth(ehic,or∆rehsip,oCnn2d(ihnig) tahnedoreLt0ic(ahli)c)ovwairtihanwcehifcuhncwtieoncobme-- vpaorsiiattioionnaolofntghethpeoesditgioe.nTohviesrptrhoevi1d0e0s06im00aDgeIsMaMndmfoeraseuarceh-
fore comparing it to the measured one. mentsperacquisition.InthecaseofthePMLthemotionis
We use a simulated annealing (SA) algorithm (Kirk- only measured in the direction perpendicular to the Lunar
patrick et al. 1983) to find the best fit value. The SA al- limb that corresponds to the direction perpendicular to the
gorithm is a random search technique, which exploits an sub-aperturesseparationbase,hencethetransversemotion.
analogy with thermodynamics and the way in which a TheclassicalrelationbetweentheFriedparameter(r0)and
metal cools and freezes into a minimum energy, assimi- thevarianceoftheimageposition(σ2)canbefoundinFried
lated here to our global minimum. Starting from an ini- (1966);Tatarskii(1971).Inourcasewewillusetheabsolute
tial set of Cn2(hi) values we compute the initial cost func- variance (σ2abs) which is calculate from the absolute posi-
tainocne:(EEnn==00)=betθw[Ce∆ethnαeot(hθe)−thCeo∆mrαeeatsic(aθl)]a2.ndThmene,asfuorreedacchovsaurbi-- tawiloeonnusgse(tdyh−ethe<degyeex>wp)rereacsstoihmoenprudttheeraiσnve2atdbhseboyrvaeZwriatodhneees1t.0aF0l0o.r(im1a9lal9g4pe)os,.sbiTtaihoseennds
sequentiteration(n),weapplyasmallvariationtotheprevi-
ousCn2(hi) vaPlues,calculatethenewcostfunction,andthen on σ2abs and including the outer scale (L0):
tchoemcpousttedthecerceoasstesdiaffnedrewnceek∆eeEp=thEen+n1e−wEsne.tIfofitpisarnaemgaetteivres., r5/3=0.179sec(z)λ2[D−1/3−1.52L0−1/3]
If the cost increases, ∆E is positive, we do not systemat- 0 σ2
abs
MNRAS000,1–14(2016)
PML - Turbulence Profile 9
Table 2.ReconstructionProfilefromcovariancesimulations.
Relative strength of the
turbulence layers [%]
Altitude 33simulatedlayers
[km] simu recon recon
(NOISY)
0.01 31 30.6 31
0.15 6 6 6.1
0.25 4 4 3.8
0.35 3 2.9 3.3
0.45 2 2 1.6
0.55 3 3 3.4
0.65 2 2 1.5
0.75 5 5 5.5
0.85 5 5 4.6
0.95 2 2 2.3
1.2 2 2 1.9
1.7 2.5 2.5 2.5
Figure10.Simulatedcovariancefor33layerswithaprofilegiven
2.25 2 2 2
inthesecondcolumnofTable2(simu).Theblacklineshowsthe
2.75 1.5 1.5 1.6
perfectcovarianceoverlappedtothenoisycovariance(redcrosses)
3.25 1 1 0.6
producedbyaddingGaussiannoisetotheperfectcurve.
3.75 5 5 6
4.25 3 3 1.6
4.75 1.6 1.6 2.9
5.5 1.4 1.4 0.6
We can apply this method to either the top or bottom 6.5 0.5 0.5 1.1
images independently. Note that when tends towards 7.5 1.4 1.4 0.9
L0
infinity, the expression simplifies to the more general 8.5 2.2 2.2 2.7
Kolmogorov’s case. 9.5 2.3 2.3 2
11 2 2 2.2
12 1.5 1.5 1.6
13 2.1 2 1.7
Similarly, a method that uses the differential motion
14 0.3 0.3 0.5
have been developed for both the transverse motion, per-
15 0.5 0.5 0.8
pendicular to the direction of the sub-apertures separation, 16 0.9 0.9 0.7
andthelongitudinalmotion,paralleltothedirectionofthe 18 0.7 0.7 0.6
sub-apertures separation. It can also be used to determine 20 0.3 0.9 0.4
the Fried parameter. In the PML case we are only looking 22 1.2 1.2 1.7
atthetransversedifferentialvariance(σ2).Theoriginalfor- 24 1.1 1.1 0.4
t
mula was derived by Sarazin & Roddier (1990):
r05/3=λ2∗sec(z)∗D−1/3∗ σK2t, 4 SIMULATIONS
t In order to probe our reconstruction method we simulated
differentialcovariancesforaprofilewithaltitudesmatching
with,
our reconstruction grid. We looked at two cases, one with
Kt =0.358 (1 0.811 S−1/3); a perfect covariance curve and one with a noisy covariance
∗ − ∗ curve. The noisy data were produced from a perfect covari-
where S = B, D is the apertures diameter and B the sepa- ancetowhichweaddedGaussiannoise(Figure10).Thead-
D
ration between the two apertures and z is the zenith angle. ditionalnoiseiswithin5percentofthevalueofthe”clean”
An updated value of the constant Kt is given in Tokovinin simulated data.
(2002): We show the reconstruction results in Figure 11, with
the input simulated data in red and reconstruction in blue.
Kt =0.364 (1 0.798 S−1/3 0.018 S−7/3). On the left we show the covariances while on the right we
∗ − ∗ − ∗
havethecorrespondingturbulenceprofiles.Forbothgraphs
Themethodusingtoporbottomimagesindependently wegivethemeanrelativeerrorbetweentheinputdataand
was useful during preliminary tests to make sure that the the output reconstruction. The top two panels correspond
resultsfromtopimageswereconsistentwiththosefromthe totheperfectcovariancecase,whilethebottompanelscor-
bottomimages.Howeverthosearestronglyaffectedbytele- respond to the noisy data case. In addition, the relative
scope vibrations and wind shake. The measurements of the strengthofthelayersfromthesimulatedandreconstructed
Fried parameter presented in section 6 were extracted with profiles are reported in Table 2.
the more reliable differential method implemented with the The relative error of the reconstruction from a perfect
later Kt value. data set seems negligible on the covariance, with a value
MNRAS000,1–14(2016)
10 L. Catala et al.
Figure11.ExampleofsimulatedCovarianceandthebestfitfromourreconstructionmethodinthecaseofa33layersmodelmatching
thereconstructiongrid.Left:Covariancefitshowingthesimulatedcovariance(redcrosses)overlappedwiththereconstructedones(blue
line).Right:Correspondingprofiles,forthesimulatedprofile(redline)andthereconstructedone(bluestars).Top:Reconstructionfrom
a perfect covariance curve (black line in Figure 10). Bottom: Reconstruction from a noisy covariance curve (red crosses in Figure 10).
ThesimulatedandreconstructedprofilesrelativeturbulencestrengthsaregiveninTable2.Foreachgraphweprovidethevalueofthe
relativeerrorbetweenthesimulatedandreconstructeddata.
close to zero. However, this still reflects as a 1.57% relative large part of the error originates from an incorrect redistri-
error on the reconstructed profile. In the case of the noisy bution between adjacent layers. Also, in some cases, poorer
dataset,therelativeerroronthereconstructedcovarianceis optimization of the algorithm could generate convergence
0.45%, which reflects as 13.5% on the reconstructed profile. issues and additional error in the redistribution.
The error on the profile is more important for the higher Overall, when running a set of 100 noisy data simula-
layersoftheatmosphere.Athigheraltitudesthecovariance tions, the mean relative error on the profile reconstruction
peaks from different layers get closer to each other (Fig- is14%forthefullrangeofaltitude,25%forthe5to24km
ure 3) and hence the response of the reconstruction is more range and 5% for altitudes below 5 km.
sensitive to turbulence in adjacent layers, seen as wider tri-
anglesinthebottomgraphofFigure5.Asaresultweshould
see some error coming from an incorrect redistribution be-
tween adjacent layers. In order to evaluate this effect, we 5 FIRST MEASUREMENTS AND RESULTS
compared the reconstructed and original simulated profiles AT THE SUTHERLAND SITE
after applying a smoothing over three consecutive layers:
The PML was deployed at the SAAO Sutherland observa-
i+1
Cn2(hi) = 13 Cn2(hk). After smoothing, the relative error tory in South Africa during August 2011. During the PML
k=i 1 observingcampaignwealsohadaMASS-DIMMandaGSM
between thePp−rofiles goes down to 4.3%, confirming that a running alongside it. On all nights that the PML was oper-
MNRAS000,1–14(2016)