The Use of the ARM WSI to Estimate the Atmospheric Optical Depth atNight

Contents

1. Introduction

2. Measured Parameter, Instrument Description and Data Characteristics2.1 Radiation field2.2. Instrument Description2.3. Data Characteristics

3. Image Processing and Optical Thickness Determination3.1. Preliminaries3.2. Primary Image Processing3.3. The Signal — to — Noise and the Standard Error of a Measurement3.4. Results

4. Future Work: Optical Thickness from a radiative transport model

5. References

List of Tables

Ileana Cristina MusatDecember 2000

Advisor:Professor Robert G. Ellingson

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The Use of the ARM WSI to Estimate the Atmospheric Optical Depth at Night

Abstract

Atmospheric constituents can be studied during the night using the light of stars as radiation source. The Whole SkyImager, an instrument that shows high accuracy at low levels of illumination and has a wide dynamic range, is used.The digital images obtained are processed to infer the atmospheric extinction coefficient and the star radiance. Astudy of the accuracy of the star s observations is undertaken. Subsequently, the broadband visible extinction will becalculated using the SBDART atmospheric radiation model, in which the Sun s brightness (visual magnitude, --26)is scaled to match the brightest stars in the image (from Sirius, --1.46, to Polaris, +2.02 visual magnitude). Agoodness of fit of the model with observations will be assessed, taking into account various profiles of temperature,pressure, columnar densities of gases, aerosol provenience types etc. Finally, a study of the accuracy of the starsobservations with the instrument was undertaken.

1. Introduction

Since the first pointing of a telescope towards the sky by Galileo Galilei, the extinctionproperties of the atmosphere have hampered the astronomical observations made with theinstruments placed on the Earth’s surface. Even during the clearest nights, the inhomogeneouscomposition and the random movements of the turbulent atmosphere have distorted the imagesof the distant objects. Performing precise extinction measurements is essential for absolutephotometry measurements. In this regard, atmosphere researchers and astronomers pursue acommon goal: better knowledge of the atmospheric transmission. While astronomers endeavor toprecisely measure the celestial objects radiances as if these would be seen from the top of theatmosphere, thus deducing the atmospheric extinction from their data, for the atmosphericscientists the attenuation of light in the atmosphere is a valuable mean to study its composition.

Differential photometry and adaptive optics techniques are methods to avoid an exactknowledge of the extinction while observing from the ground. Measuring the absolute fluxes ofstars, i.e. absolute photometry, is more delicate, and the measurements imply a very goodknowledge of the atmospheric extinction. The standards for absolute photometry measurementsare known non variable stars, which must be quasi-evenly spread over the sky, for at least one ofthem to enter in the field of view of the telescope together with the currently observed star.Absolute standard measurements with photo-multipliers have been done, for example, by Hayesand Latham (1975), and by Tueg, White and Lockwood (1977), while the standards for theCharged Coupled Device (CCD) detectors are currently set by the Sloan Digital Survey Project(Kent, 1994). In fact, the CCD detection of fainter stars produced the need to replacing the oldsystem of measuring absolute fluxes in logarithmic magnitudes (Megessier, 1995; Lupton et al.,1999)

The Whole Sky Imager (WSI) is an instrument, which uses a CCD as a sensor. TheAtmospheric Radiation Measurements (ARM) program (Stokes and Schwartz, 1996) hasdeployed WSI instruments at the observation sites in the Oklahoma, Alaska and the TropicalWestern Pacific. The instrument acquires data 24 hours a day, but this study will be concernedwith nighttime observations. When used during the night, it measures silicon magnitudes ofstars (Scholl, 1994).

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During the night, the main sources emitting or reflecting the radiation captured by theCCD device are:

• celestial objects (stars, planets, nebulae, the Milky Way etc.);• cosmic rays;• city lamps (Na or Hg vapor lamps);• airglow (a thermosphere emission of excited atoms and ions: Na, OH-1, etc.).

Some of the sources exhibit line emission while other have a blackbody continuum emission andother just reflect the sunlight.

During the night, the principal extinction (absorption and scattering) sources are:• the atmosphere: gases ( H2O vapor, N2, O2, CO2, CO etc.); solid particles (aerosols);

liquid droplets (in clouds; this component is eliminated by choosing cloudlessobservation nights).

• the magnetosphere: plasma (ions, free electrons), less important for visible and nearinfrared observations.

• the interstellar medium (ionized H clouds; dust, made of silicates).

The extraterrestrial extinction is not taken into account in this study, because it is not importantfor atmospheric attenuation and is not discussed herein.

In the material that follows, Section 2 describes the WSI instrument from the point ofview of the hardware and the incorporated CCD electronic characteristics, as well as the dataacquired with it. Section 3 contains the methods chosen for processing the digital images, thetotal atmospheric optical depth results obtained to date and an error analysis of thisdetermination. Section 4 is dedicated to outlining the manner for using the SBDARTatmospheric radiation model to calculate an equivalent star s signal at the top of the atmosphere.This last section describes future work to be accomplished, and it exposes the principal ideasregarding the method to be used for determining the nightly columnar aerosol load from totalopacity of the atmosphere and the possibility of sensitivity studies within the model. Theultimate objective of this study is to assess the potential of the WSI instrument for an accuratedetermination of aerosol optical depth. The present paper is the first part in a morecomprehensive effort.

2. Measured Parameter, Instrument Description and Data Characteristics

2.1 Radiation field

The monochromatic radiation with the intensity ITOA at the top of the atmosphere isattenuated exponentially down to a value I (z, θ), at the height z, according to the Bouguer -Lambert-Beer s law (Rohlfs, 1996; Cox, 2000):

I (z, ) I e- (z, ) TOAθ τ θ= (2.1)

where _(z, _) is the optical path through the atmosphere at the height z, which depends on theinclination of the radiation beam relative to the vertical (θ).

Consider the Earth s mean radius as RE, r (z) the atmosphere s index of refraction and ρ(z) the density of the atmosphere at the height z above the Earth s surface. The optical path along

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the line of sight at a zenith angle θ is related to the vertical optical path, called optical thickness(or opacity, in astronomy), τ (0) through a linear relation:

τ θ τ θ( ) ( ) ( )= ⋅0 Χ , (2.2)

where X(θ) is the relative air mass, which depends on the spherical shape of the atmosphere andon the variation with height of the density and the refractive index of the air:

Xz dz

RE

RE

zrr

z dz( )( )

( ) ( ) sin ( )

( )θ ρ

θ

ρ=

−+

∞∫

∞∫

1 2 2 20 00

, (2.3)

In the case of the plane—parallel atmosphere, or at zenith angles smaller than 800 (McClatchey etal., 1978), the relative air mass value becomes:

X( ) sec( )θ θ= . (2.4)

If studying the atmosphere s components separately, the density ρ(z) should be replacedwith the density of the respective component, τ(θ) and τ(0) will represent the component soptical path along the line of sight, and along vertical, respectively. The integrated verticaloptical thickness, or extinction, of the monochromatic radiation due to a certain atmosphericcomponent (molecules, aerosols), is a summation of the absorption and scattering along thevertical:

τ( ) ( ) ( )00 0

=∞∫ +

∞∫k z dz s z dz, (2.5)

where k and s are the monochromatic absorption and scattering coefficients, respectively.

In the visible, the major molecular scattering (Rayleigh-type) is due to the molecules ofN2 and O2 and to aerosol, and the absorption takes place by O2, O3 and aerosol.

The integrated optical thickness at a wavelength λ in the visible, and for a zenith angle θ, isthen a summation of the diverse contributions mentioned and may be written:

τ( )00 0 0111

=∞∫ +

∞∫ +

∞∫

=∑

=∑

=∑ kidz sidz s jdz

j

N

i

M

i

M, (2.6)

where M is the total number of types of molecular components and N is the total number of typesof aerosols (which are considered here for simplicity, non-absorbent scatterers, i.e. non-sulfateand non-carbonaceous).

The interaction coefficients are proportional to the cross sections per particle and to thenumber density of molecules, for each level z in the atmosphere:

τ σ σ σ( ) , ,001 0 011

=∞∫ +

=∑

∞∫ +

∞∫

=∑

=∑abs inidz

i

M

scatt inidz jaerosolsn j

j

Ndz

i

M, (2.7)

where σ -s are the scattering cross sections: σ scatt, i , for molecules, and σ jaerosols

, for aerosols,and the absorption cross section for molecules, σ abs, i , respectively.

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Equation (2.7) may be rewritten taking into account the mean cross sections of interaction(absorption or scattering) per particle, which do not depend on z anymore, and the numberdensity at level z of each species, integrated to form the vertical column abundance:

τ σ σ σ( ) , ,001 0 011

=∞∫ +

=∑

∞∫ +

∞∫

=∑

=∑abs i

mean nidzi

M

R i nidz jaerosols n jdz

j

N

i

M, (2.8)

If NA is the Avogadro s number and m _ i, the mass of a mole of the respective substance, theabsorption term in equation (2.8) becomes:

τ σ λµι

abs abs iNA

g mwi

pp

i

Mdp( ) , ( ) ( )0

01=

⋅∫ ′

=∑ ′ , (2.9)

where wi is the (mass) mixing ratio of the absorbent species of molecules i at level withpressure p′, while p is the total atmospheric pressure at the level for which the optical thicknessis calculated, in this case, the Earth s surface.

The molecular scattering term in equation (2.8) is also known. The Rayleigh scatteringcross section per molecule σ R (z′) depends on n (z′), the number concentration at level z′, or,equivalently, on the pressure p(z′) at that level, such that it is appropriate to express it as afunction of pressure and wavelength (Cox, 2000):

σ λλ λ λ

R zA A A p z

p( , ) [ ]

( )′ = + + ′14

22

34 0

1 , (2.10)

where p0 is a fixed constant pressure, and the A coefficients are constants.

The aerosol scattering term in the equation (2.8) is usually modeled knowing that thescattering cross section, s, for aerosol at the wavelength λ varies as λ-1. The optical thickness atevery λ is obtained by comparison with the known aerosol optical depth at the wavelength of550nm (Oikarinen et al., 1996), or is taken as the product of a specific aerosol extinction crosssection (B) at 550nm with the vertical column mass load m, in [Kg/m2], (Tegen et al., 1999):

τ aerosol (λ) = τ aerosol (550nm) s (λ) / s (550nm), (2.11)

τ aerosol (550nm) = B m. (2.12)

The knowledge of diverse air masses X (θ) and of the vertical column mass abundances orvolume abundances (in [Kg/m2] and [number molecules/m2], respectively) makes possible thecomputation of the total clear sky thickness, and reciprocally.

2.2 Instrument Description

The WSI measures the radiance, or intensity (I), of the radiation field. The relation of themeasured parameter to the optical thickness is given by:

ln I = ln ITOA — τ X(θ), (2.13)

Fig. 1. Schematic diagram of the Whole Sky Imager (WSI), a cooled CCD.

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where I is the intensity at the observation place on the Earth s surface, ITOA is the same intensityat the top of the atmosphere, and τ is the total optical thickness.

The (2.13) relation is actually used for hetero-chromatic case, for a [400; 900] nmbroadband and a band centered at 800nm, having a full width at half maximum (FWHM) of 70nm, in NIR, i.e. [710; 890] nm).

More specifically, a pencil of energy (or the number of photons which are carrying it)coming from a given direction in the hemisphere above the instrument, passes through the entireoptical depth of the atmosphere, encounters a specific pupil of the fisheye lens, is bent through it,and reaches the horizontal surface on which the detector (CCD) is mounted. An image of theinstrument s measuring unit is given in Fig.1.

The wavelength band falling on chip is determined by the color filter used. The range ofthe intensities can be adjusted by a neutral density filter, which follows a logarithmic function oftransfer, whose task is to allow a wide range (day and night) response.

An image of the optical system (the fish-eye lens) is given in Fig.2 (Smith, 1991). Theaperture pupil for a ray of light is specific for the beam inclination relative to the optical systemaxis, and the subsequent lens system, added behind the fish-eye lens, is designed to compensatefor the distortions. Because the lens produces a shift and a rotation (Smith, 1991) of the ray,the object imaged is appears mirrored, in the detector plane.

Fig.2 Fish-eye lens design: optical axis, entrance pupils and aperture stops (Smith, 1991).

The optical system as a whole has a 8mm effective focal length at f/2.8 speed, such thatits effective aperture (or diameter) is 2.8mm. The low speed should be combined with longerexposure times, mainly when the object brightness is faint, as is the case during the night.

Table 1 summarizes the characteristics of the CCD camera (Shields, 2000):

The field of view is determined by the chip s dimensions and by the effective focal lengthof the optical system.

Each element of the chip s 512 x 512 matrix represents the intensity of the radiationreceived from the directions in space, which are within a specific cone (solid angle) with the topplaced in the center of the respective pixel point. The width of this cone is dictated by the size ofthe square pixel. For a (19 m) 2 pixel size, and a 2„ solid angle field-of-view, the value of thesolid angle seen by a pixel is (0.351ß)2 / (pixel)2 or (21arcmin)2 / (pixel)2.

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The CCD detection is currently widely used in imaging and spectroscopy for astronomy(Zandik and Beletic, 1998; Farnham et al., 2000) and atmospheric science (Swenson and Mende,1994; Bennet, 1994; Haque and Swenson, 1999; Hielkopf and Graham, 2000).

Table 1 The characteristics of the CCD device mounted on WSI Instrument.

Dimension 9.7 mm x 9.7mmPixel size 19 _m x 19 _mNumber of pixels 512 x 512Well Depth 315,000 electronsCharge Transfer Efficiency (CTE) 0.999998Readout Noise 6.09electrons/pixel/readDark Count 1.08 electrons/second/pixel at —30 ßCDynamic Range 94 dBDigital resolution 16bit at 40,000pixels/second =0.4MHzCCD Gain 4.37 electrons/ADU, ADU (or count) is

the Analog-to-Digital conversion unit

2.3 Data Characteristics

The WSI data used in this study were acquired at the South Great Plains (SGP) locationof the Cloud and Radiation Test-bed (CART) site, pertaining to the Atmospheric RadiationMeasurements (ARM) program. . The images of the whole sky are generally taken with anexposure time of 60 seconds, at 6 minutes intervals, and more rarely at 12 minutes. This studyuses images obtained during 33 (partially) clear-sky nights during 1999, information for which issummarized in Table 2.

Table2. Clear-sky images, specified as month, day, hour, minute:

April 7:0506...0554

May 3:0336...0512

June 14:0300...0718

July 2:0330...0342

July 8:0400...0418

April 9:0330...0400

May 5:0254...0330

June 15:0306...0712

July 4:0342...0448

October 1:0130...0448

April 10:0200...0618

May 11:0312...0724

June 17:0524...0606

July 5:0342...0530

October 2:0130...0336

April 16:0742...0830

May 15:0324...0954

June 19:0800...0818

July 6:0336...0548

October 5:0354...1118

April 18:0418.0348...0806

June 2:0318...0354

June 26:0306...0318

July 7:0336...0348

October 6:0100...0730

October 11:0400...0718

October 12:0130...0506

October 14:0130...1124

October 15:0254...1118

October 16:0354...0936

October 21:0912...1130

Sept. 29:0418

Sept.30:0418

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During each exposure, two images are acquired. One in "open hole" mode, which meansthat no colored filter is used and the band wavelength is "roughly [400; 900] nm" (WSI web site,2000). The other image is registered through a 70nm FWMH, Gaussian shaped filter, centeredon 800nm wavelength (in near infrared).

The raw images are not currently available to the user. The images available to the userare already calibrated, with the calibration coefficient, for each combination of neural and colorfilters, provided.

The calibration is an essential part of the image validation, and the quality of the datadepends crucially on it. The final goal is to remove from the instrumental output all the fakesignals due to the CCD electronics, to the light scattering within the cooling recipient, etc. Thecalibration encompasses:

• flat-fielding (correction of the irregularities of the CCD chip and fiber optic);

• substracting the dark image (correction for electrons not generated by photons, butthermally generated inside the half-conducting lattice);

• correcting for the roll-off (variation of the optical efficiency and the filter efficiency withthe zenithal angle, i.e. correcting for the longer path through the optical system of theheavily refracted off-zenith rays);

• multiplying by the calibration constant, i.e. converting the (photo)electron counts intocalibrated radiances(intensities).

In Fig. 3, an image of 72 averaged frames, taken during one night, shows that the opticaleffect of vignetting, which causes a reduction of illumination in some part of the image (Bornand Wolf, 1965), is quasi - non-existent, and that the flat-fielding has been carefully done. In thesame figure, some saturated pixels (over a bright planet image), non-discharged by the readout,remain constantly bright after the operation of averaging. In the opposite part of the image, someobvious defect pixels, dead pixels (Howell, 1992), are to be excluded from the processingprocedure, too.

For the user s convenience, a calibration file is available, containing a flat-field matrix, aroll-off matrix, general astrometric data for the sun (for daylight observations) and for the stars(equatorial coordinates relative to the mean Equinox 2000.0 for the brightest stars), geographiclocation of the instrument, as well as the functions necessary to transform image coordinates(pixel location in the frame) into local astronomical coordinates (azimuth and zenithal distance).

A typical nighttime calibrated image contains all the sky objects visible, rendered with a21 arc-minutes per pixel resolution, and - near the edges - the Moon and Sun s occultor,different nearby buildings, distant city lights, planes trajectories etc. A mask for the data,containing edge pixels and suspect pixels is provided, such that the corner pixels of the matrix,which do not pertain to the circular projection of the hemispheric field-of-view, are easilyeliminated.

The histogram of the intensity points in an image is very skewed towards low values. Outof a total 512 x 512 points, some 100 points are very bright, with values from 100 to 8000(107mW/m2/sr/ m) units, while the overwhelming majority are smaller or zero.

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A pixel with the index (i, j), imaging the night sky and pertaining to a luminous object(star, planet, etc.) contains the sum of the intensities from the object, from the neighboring sky,and a noise term, due to CCD electronics:

I ( i, j ) = I object + I sky + Noise. (2.14)

The process implying photon detection is statistically a Poisson process. The arrivingphotons generate photo- electrons in the silicon chip with a high efficiency compared to, say,photographic plate. The electrons are stored in (quantic) wells of energy having a well depthwhich upper limits the number of accumulated electrons. After the exposure time is over, theelectrons are transported to the edge of each column, during the readout process, and from hereto the edge of the chip, towards an amplifier, which transforms the charge accumulated in tinyelectrical potential differences. The charge is conducted to the edge of the chip, driven by theelectric field produced by the clocking voltages (Howell, 2000). An Analog to Digital Converterdigitizes the analog values (real numbers, in Volts) into natural numbers (gray values).

The 16-bit digitization specific for this project s camera implies that the maximumnumber of gray levels in an image is 216. The calibrated images at the user s disposal are scaleddown to an 8-bit digitization.

3. Image Processing and Optical Thickness Determination

3.1 Preliminaries

Image Processing takes into account the character of the detector and the objects detected(stars). The instrument is not a telescope, such that the resolution is not very good. The goal is todiscriminate the flux from different stars, in a crowded whole sky image.

Each bright star is tracked during a clear-sky night. In the astronomical sense, not all thestars are constant emitters; some of them are variables, with a period of several days. For ourpurposes, this inconstancy of the flux at the top of the atmosphere will be neglected and the ITOA

is assumed constant.

Suppose a star is at the zenith point of the observing place at some time. Its light arrivesat the instrument after passing through the shortest path through the atmosphere, the verticalpath. During the night, as it moves to a different azimuth and zenith position, the radiancereceived from the star will follow the slanted path described in Section 2.1.

In astronomy, the images taken with CCD devices are in the focal plane of a telescopesuch that the field of view encompasses only the principal object (i.e. a star), and the nearbyenvironment. The separation object vs. background is almost always neat, or can be made so byaperture photometry techniques, i.e., adding the light of the star from the pixels which contain it,if a certain model shape of the star is assumed to be true (McLean, 1996, Gilliland, 1992).

In the case of the WSI, the field of view is extremely large, thus crowded. The stars -with the exception of the brightest and closest (i.e. brightest stars, with magnitude under 6) - arenot well separated, they are unfocussed and sometimes spatially under-sampled.

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Fig. 3. Above, an image obtained out of 72 frames during the same night. The center of rotation isnear Pollaris, and the traces in the lower half of the image are due to two bright planets. Thespherical symmetry of the airglow is a proof of the good calibration of the instrument. Below, ascan along the row passing through the center of the image, showing no vignetting effect.

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First of all, the stars are not point sources, because of the spreading introduced by theoptical system of the instrument. The "point-spread-function" (PSF) is assumed most commonlyto have a Gaussian shape; in this study the FWHM is assumed to be 1.18 pixels. Other PSFshapes are currently fitted for Hubble Space Telescope images (Anderson and King, 2000).

Secondly, the lack of a tracking mechanism gives birth actually to a "smear" of object’slight to other pixels, as it changes his position. Thus, the variation of zenith and azimuth of thestar’s center during the given exposure time (60s) produces within the image a displacement,which does not surpass one to two pixels. The area swept by a moving star image duringexposure time is maximum 2 adjacent pixels. This depends also on whether the star and thepixel centers are aligned or not.

3.2 Primary Image Processing

The digital image processing determines the bright objects in the image, calculates thebackground intensity, I sky, for each image, then substracts the background intensity and the noisefrom the bright stars images and determines the radiance value attenuated through theatmosphere, and transforms the object s image coordinates (pixel number) to local astronomicaland to equatorial astronomical coordinates and searches the Yale Bright Stars Catalogue formatching objects.

The bright objects finding uses the DAOPHOT FIND program (Stetson, 1979, 1987,1990 and 2000) to search for positive anomalies (deviations) of intensity in an image, verifiesthem for roundness and sharpness and declares them as potential candidates to be stars or defect(bright) pixel.

The object s light is superimposed over the sky brightness. The astronomers recommendcalculating a level of background for every region of the sky in which the object of interest lies,and not using an average over whole frame sky. For this study, the sky is obtained from the mode(most frequent value) of the distribution of the intensities of the points situated in a circularwindow with the radius 10 pixels around the star. The star itself is searched in a circular windowwith the radius 2 - 3 pixels. This software operation is identical to placing a hardware aperturecentered over the star s image, and thus is called software aperture technique (Howell, 2000).This aperture can be a circular annulus, centered on the source, and with a certain width, or canbe a square window. The mode value is replaced with a linear combination of the mean andmedian values in the case when the histogram of the intensity values is very skewed (DaCosta,1992).

The value of the sky background is then substracted from every pixel pertaining to thesource-star, or equivalently, the source pixels summation is diminished by an amount equal tothe background times the number of source’s pixels. Thus the source s flux has been estimated.

The astronomical algorithm provides the coordinate transformation between horizontalcoordinates (azimuth and zenithal distance), which are dependent on the observer s place andtime, and equatorial coordinates relative to the 2000.0 mean Equinox (right ascension and

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declination), which are listed in the stellar catalogues (Scholl, 1993). The pixel coordinateswithin the image are related to the horizontal coordinates through a function given within thecalibration data. The function includes the correction for the radial variation in the angularresolution, because as the zenith angle becomes larger, the area of the sky projected on thesurface of the corresponding ex-centric pixel becomes larger, while more objects willagglomerate within the same solid angle (Garcia et al., 1997).

As it is known, the observed position of a celestial body is affected by the atmosphere srefraction. The correction from observed to true zenith distance is proportional to thetangent of the zenithal angle and to the real part of the refraction index (r-1), which depends onthe total air pressure (p), the water vapor partial pressure (p w) and the temperature (T), each ofthem functions of observer’s altitude. The algorithm designed to compensate for the refractionand to calculate the true zenith of a star uses an empirical formula (Meeus, 1991). Because thecorrection for the object at horizon is 35 arc-min, and only 1 arc-min for the object at 45ß zenithangle, and taking into account the under-sampling characteristic for a wide field of viewinstrument, the star position, if affected only by refraction, would almost surely lie in the samepixel or, at horizon, in a direct neighbor pixel. In general, the star s movement during theexposure time is more important than the variation of position due to the refraction.

3.3 The Signal - to - Noise and the Standard Error of a Measurement

The quality of the measured star s radiance is assessed by computing the signal to noiseratio (S/N) for a CCD-type detection. The CCD equation (Howell, 2000) written specificallyfor a stellar object is:

( )

( )( exp Re )

S

N starNstar

Nstar npixnb

npixNsky NDark t N ad g f

=

+ + + ⋅ + +1 2 2 2σ

, (3.1)

where

• N star, in electrons, is the total number of photoelectrons produced in the CCD by the sourceextended over n pix pixels.

• N sky is the number of photoelectrons produced in semiconductor by the background (orsky ) diffuse sources, and calculated as the mode of the distribution of n b pixels

surrounding the object.

• N Dark (in electrons/pixels/second) is the dark current, composed out of thermally generatedelectrons within the semi-conductor.

• t exp is the exposure time.

• N Read (in electrons/pixel/read) is the readout noise, i.e. the number of electrons retrieved inthe electric signal, and which are due to the random fluctuations introduced by the on-chipamplifier, the A/D circuit and from the output electronics.

• g (in electrons/ADU) is the gain of the analog-to-digital converter, which transformsvoltages (real numbers) into digits (natural numbers).

• (σ f) 2 = 0.289 and is a factor which gives the error introduced by the A/D converter.

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The signal measured in radiance units [mW/m2/sr/nm] may be converted back tophotoelectrons, N star, through the equation:

N star = I star t exp / C, (3.2)

where C is the calibration constant specific for the combination of neutral and density filters usedin measuring I star .

The denominator (the noise) is a combination of Poisson and shot noises. It is known thatthe photons arrival is a process, which obeys the Poisson distribution law, such that the standarddeviation is given by the square root of the number of events:

σ (N star) = (N star) 1/2 (3.3)

σ (N sky) = (N sky) 1/2 (3.4)

σ (N Dark t exp) = (N Dark t exp) 1/2 (3.5)

The last two terms in the denominator (N Read and the last term) are shot noises, or whitenoises, which are pure noises, and will not appear as square roots in the equation.

The standard deviation of the measurement of the star:

σ = 1 / (S/ N) star, (3.6)

is further used in the error analysis as the root mean square (rms) error of one radiancemeasurement.

When the errors due to the CCD detection process are much smaller than the rms error ofthe source s photons fluctuation, then the equation (3.1) yields the maximum signal to noiseratio:

(S / N) max = (N star) 1/2, (3.7)

which is also true for a very bright source.

Source flux is obtained with small error if the source is bright compared to the sky. Faintsources are more prone to be erroneous. In the case of the WSI, the sources are not very largespatially, such that the error in determining the flux should be not high, if the instrumental noiseis stable and low.

3.4 Results

The actual variation of the radiance with the sec (θ) is presented in Figs. 4-8 for the nightof October 15, and in Figs. 9-12 for the night of October 14, for a number of stars brighter thanthe visual magnitude +2.02. The linear fit of the ln (I) against sec ( θ) is performed, and theresults are given in the Table 3. The total optical thickness is given by the slope of the linear fit,while the intercept value represents an estimate of the source s intensity at the top of theatmosphere, ln (I TOA).

If the atmospheric composition and density profile were constant during one night, or oneseason, then an extinction coefficient would be more accurately determined by averaging an

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increasing number of measured values during that time span. The extinction coefficient would bea constant, expressing the vertical optical thickness of an unchanging atmosphere. But becausethe atmosphere is neither static, nor well mixed, the extinction coefficient could vary from nightto night or even during the same night.

Fig. 16 presents the variation of the (S/N) star, in percent, normalized to (S/N) max, for starssituated at different air masses, which were observed during the nights of October 14 and 15,1999. It can be seen that the S/N ratio decreases for larger values of the zenith angle, because thesignal itself is decreasing, due to the increasing optical path.

Due to the fact that the stars are spectrally different, for an imager working in narrowband, it would be possible to measure different extinction values using stars situated at the samezenith angle, but emitting differently at the observing wavelength.

In astronomy, the extinction coefficient is measured throughout the observation night,and it is not uncommon to consider it a variable of the observing night. It is also true that forlocation of the astronomical observatory, a place often situated at high altitude and with a largenumber of clear sky nights per year, there are some mean extinction tables that can be usedinstead.

Table 3.a Extinction measurements of the brightest stars (14 October, 1999). Theparameters within the table are A, the ln (I TOA), and B, the optical thickness of the atmosphere,for every star. N is the number of the measurements, while chisquare and chi/(N-2) are thegoodness of fit parameter, and the chisquare parameter scaled to the number of the degrees offreedom, respectively. The probability P should be closer to 1 for a good fit. The standard errorsof A and B are given by sigA and sigB entries.

A = ln (I) B=tau(0) Star chisquare N chi/(N-2) P sig A sigB7.35 -0.19 Adara 0.093 12 0.01 1 0.31 0.108.13 -0.11 Aldebaran 8.763 59 0.15 1 0.02 0.016.88 -0.11 Alioth 0.615 12 0.06 1 0.17 0.047.13 -0.12 Alnath 4.335 61 0.07 1 0.03 0.028.07 -0.15 Altair 16.586 36 0.49 0.995 0.02 0.017.10 -0.12 Bellatrix 2.568 47 0.06 1 0.05 0.038.47 -0.07 Betelgeuse 21.994 43 0.54 0.993 0.02 0.017.19 -0.19 CMaDelta-25 0.454 12 0.05 1 0.33 0.118.70 -0.13 Capella 69.498 72 0.99 0.494 0.01 0.017.28 -0.15 Castor 3.625 41 0.09 1 0.05 0.037.55 -0.14 Deneb 47.389 69 0.71 0.967 0.02 0.017.04 -0.10 Dubhe 2.322 45 0.05 1 0.06 0.027.58 -0.12 Fomalhaut 18.752 45 0.44 1 0.03 0.016.88 -0.12 GemGamma-24 0.809 42 0.02 1 0.06 0.046.98 -0.13 Hamal 9.522 72 0.14 1 0.10 0.096.94 -0.11 Menkalinan 7.674 66 0.12 1 0.04 0.027.32 -0.17 Mirfak 9.656 72 0.14 1 0.06 0.066.80 -0.14 Mirzam 0.757 25 0.03 1 0.16 0.077.76 -0.12 Pollux 2.691 37 0.08 1 0.03 0.028.30 -0.09 Procyon 5.293 25 0.23 1 0.04 0.026.95 0.08 Regulus 66.398 8 11.07 0 0.26 0.088.43 -0.10 Rigel 32.151 45 0.75 0.887 0.02 0.019.91 -0.14 Sirius 712.088 22 35.60 0 0.01 0.018.77 -0.18 Vega 29.675 41 0.76 0.859 0.01 0.00

15

In Table 3, A and B are the parameters of the linear fit of the linear equation given in(2.13), and N is the number of measurements of the radiance (I) and varies largely, because starsrise or set at different moments during the night. The fit is obtained by minimizing the sum in theexponential of a Gaussian probability, which represents the distribution of the measurements(Bevington, 1992).

Table 3.b As in Table 3.a, the extinction measurements of the brightest stars, but in 15October, 1999. A = ln (I) B=tau(0) Star chisquare N chi/(N-2) P sig A sigB

7.30 -0.20 Adara 1.24 22 0.06 1 0.18 0.068.14 -0.15 Aldebaran 15.74 72 0.22 1 0.02 0.016.97 -0.17 Alioth 0.45 20 0.03 1 0.18 0.067.17 -0.19 Alnath 2.83 70 0.04 1 0.04 0.038.11 -0.20 Altair 5.73 26 0.24 1 0.03 0.017.16 -0.19 Bellatrix 3.36 56 0.06 1 0.05 0.048.50 -0.11 Betelgeuse 33.31 55 0.63 0.984 0.01 0.017.11 -0.19 CMaDelta-25 0.93 22 0.05 1 0.19 0.078.75 -0.19 Capella 54.34 72 0.78 0.916 0.01 0.017.32 -0.20 Castor 7.57 51 0.15 1 0.04 0.027.53 -0.17 Deneb 23.23 57 0.42 1 0.02 0.017.14 -0.16 Dubhe 5.06 52 0.10 1 0.05 0.027.62 -0.17 Fomalhaut 5.85 33 0.19 1 0.05 0.026.74 -0.05 GemGamma-24 30.58 53 0.60 0.99 0.05 0.036.96 -0.12 Hamal 3.53 72 0.05 1 0.08 0.077.00 -0.15 HyaAlpha-30 0.10 10 0.01 1 0.31 0.126.93 -0.15 Menkalinan 5.33 72 0.08 1 0.04 0.037.41 -0.27 Mirfak 14.49 72 0.21 1 0.11 0.106.86 -0.19 Mirzam 1.00 36 0.03 1 0.12 0.067.74 -0.14 Pollux 7.99 47 0.18 1 0.03 0.028.33 -0.12 Procyon 11.73 35 0.36 1 0.03 0.017.65 -0.24 Regulus 3.32 13 0.30 0.986 0.17 0.088.51 -0.17 Rigel 27.97 55 0.53 0.998 0.02 0.019.70 -0.09 Sirius 1309.49 24 59.52 0 0.02 0.018.78 -0.23 Vega 12.04 32 0.40 0.999 0.02 0.01

The linear fit has the goodness of fit parameter, χ2, given by the sum of squared errorsbetween ln(I star ) and (A+B sec(θ)) and weighted by the standard deviation of the ln(I star ), whichis equal to the ratio of the individual measurement errors of star intensity, σi , to the I value:

χσ

2 2

1=

− +

=∑ [

( )]

yi A Bxi

ii

N, (3.8)

where σ i = σ [ln (I star )]= σ ( I star )/ I star .

If we assume that the star s intensity expressed in photoelectrons, N star , is distributed as aPoisson process, (identical, to a scalar, with I star measured radiance), then the standard deviationof the intensity is the squared root of the mean. This is the way the fitting presented in the Table3 has been done.

16

Alternatively, one can choose to estimate the standard deviations from measurements. Theintensity I in the standard deviation σi formula is I star, i.e. the intensity measured in the n pix

pixels of a star, as the difference of the signal and background:

I star = n pix (I — I sky). (3.9)

Thus the standard deviation of the radiance measurement of a star is given by the formula oferror propagation:

σ 2 (I star) = npix2 [σ 2 (I pix) + σ 2 (I sky)]. (3.10)

The background intensity has been determined as the mode of n b pixels, and the mode wascalculated (DaCosta, 1992) sometimes as the difference 3 median — 2 mean and sometimes as amean. Thus the standard error of I sky is equal to standard error of the mean (Bevington, 1992):

σ 2 (I sky) = σ 2 (I pix) / n sky. (3.11)

The χ2 /(N-2) entry in the Table 3 represents the reduced chi-square, where N-2 is the number ofdegrees of freedom, an amount that helps to assess whether the fitting function is appropriate todescribe the data. If it is close to 1, then the estimated variance of the sample distribution and thevariance of the parent distribution are close, and the sample is indeed drawn from a statisticalpopulation having the assumed distribution function. P entry in the table represents theprobability that, continuing to measure, to obtain a fraction P of measurements equal orexceeding χ2.

The last two entries in the table, σA and σB, are the uncertainties in the parameters A andB of the fit. Overall, the chi-square test shows that the fitting functions are good. Thedetermined optical thickness is reported as: τ(0) – σ B, and the I TOA – σA.

An average of the parameters over more clear sky nights could be obtained, with thesupposition that the vertical distribution of the constituents and of the thermodynamic parametersremained unchanged. It is not the goal of this study to obtain mean extinction coefficients for aparticular location or case.

In sum, the results shown in Table 3 accurately indicate that it is indeed possible toestimate the atmospheric total optical depth from WSI measurements.

4. Future work: Optical Thickness from a radiative transfer model

If the extinction due to atmospheric gases were known, it would be possible to inferrelatively accurate the effective aerosol optical depth from the total optical depth (see Section 2).Models for calculating atmospheric extinction exist (e.g. SBDART, LBLRTM, etc.) and sinceARM instruments measure the vertical distribution of temperature, water vapor and ozone, itshould be possible to calculate the extinction due to atmospheric gases. Appropriatemethodology for error analysis should be developed, too.

The data obtained in this manner might be used to track aerosol during the night and togive continuity to similar data obtained from sun photometers during daylight hours.

Acknowledgements. This research was sponsored in part by the ARM program, under the DOEgrant DEFG0294ER61746. The author wishes to express her gratitude for the scientific guidingand encouragements received from Dr. Robert G. Ellingson.

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5. References

WSI, 2000: http://www.arm.gov/docs/instruments/static/wsi.html

Stokes, G.M. and S. E. Schwartz, 1994: The Atmospheric Radiation Measurement (ARM)program: Programmatic background and design of the cloud and radiation test bed, Bull.Amer. Meteor. Soc., 75, 1201-1221

Ricchiazzi, P., S. Yang, C. Gautier, D. Sowle, 1998: SBDART: A research and teaching softwaretool for plane-parallel radiative transfer in the Earth s atmosphere, Bull. Amer. Meteor.Soc., 79, 2101-2114.

Smith, W.J., 1992: Modern lens design: a resource manual, McGraw-Hill, 471pp.

Swenson, G., and S. Mende, 1994: Atmospheric airglow imagined with Charge CoupledDevices, Proc. SPIE, 2266, 109-121.

Garcia, F.J., Taylor, M.J., Kelley, M.C., 1997: Two-dimensional spectral analysis ofmesospheric airglow image data, Applied Optics, 36 (29), 7374-7385.

McLean, I., 1996: Electronic Imaging in Astronomy, Wiley-Praxis Series in Astronomy andAstrophysics, 470 pp.

Hayes, D.S., and D.W. Latham, 1975: A rediscussion of the atmospheric extinction and theabsolute spectral-energy distribution of Vega, Ap. J., 197, 593-601.

Tueg, H., N. M. White, G. W. Lockwood, 1977: Absolute Energy Distributions of Alpha Lyraeand 109 Virginis from 3295 A to 9040 A, Astron. Astrophys., 61, 679-684.

Farnham, T.L., D.G. Schleicher, M.F. A’Hearn, 2000: The HB Narrowband Comet Filters:Standard Stars and Calibrations, Icarus, 147, 180-204.

Zadnik, J., and J.W.Beletic, 1998: Effect of CCD readout noise in astronomical speckleimaging, Applied Optics, 37(2), 361-368.

Oikarinen, L., H. Saari, K. Rainio, J. Graeffe, H. Astola, 1996: Star-pointing spectrometer formeasurements of atmospheric ozone, Proc. SPIE, 2830, 224-235.

Gilliland, R.L., 1992: Details of noise sources and reduction process, in Astronomical CCDObserving and Reduction Techniques, ASP Conference Series, 23, 68-89.

Stetson, P.B., 1979: Photographic Stellar Photometry with the PDS Microdensitometer,Astronomical Journal, 84(7), 1056 -1066.

Stetson, P.B., 1987: Daophot: A computer program for crowded-field stellar photometry,Publications of the Astronomical Society of the Pacific, 99, 191-222.

Stetson, P.B., 1990: On the Growth-curve method for calibrating stellar photometry with CCD s,Publications of the Astronomical Society of the Pacific, 102, 932-948.

Stetson, P.B., 2000: Homogeneous photometry for star clusters and resolved galaxies. II.Photometric standard stars, Publications of the Astronomical Society of the Pacific, 112,925-931.

Howell, S.B., 2000: Handbook of CCD Astronomy, Springer, 165 pp.

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DaCosta, G.S., 1992: Basic Photometry Techniques, in Astronomical CCD Observing andReduction Techniques, ASP Conference Series, 23, 90-104.

Rohlfs, Kristen, 1996: Tools of Radio Astronomy, 2nd edition, Springer, 423 pp.

McClatchey, R.A., R. W. Fenn, J. Selby, F. Volz, J. Garing, 1978: Optical Properties of theAtmosphere, Chapter14 in Handbook of Optics, McGraw-Hill, 1600 pp.

Bennett, C.L., 1994: FTIR measurements of thermal infrared sky radiance and transmission,Proc.SPIE, 2266, 25-35.

Shields, J., 2000: personal communication.

Scholl, M, 1994: Six-feature star-pattern identification algorithm, Applied Optics, 33(20), 4459-4464.

Scholl, M, 1993: Experimental demonstration of a star-field identification algorithm, OpticsLetter, 18(6), 402-404.

Haque, R., and G. R. Swenson, 1999: Extraction of motion parameters of gravity-wavestructures from all-sky OH image sequences, Applied Optics, 38(21), 4433-4442.

Hielkopf, J. and P. M. Graham, 2000: A wide-field spectral imager, Publications of theAstronomical Society of the Pacific, 112, 801-808.

Lupton, R.H., J. E. Gunn, A. S. Szalay, 1999: A modified magnitude system that produces well-behaved magnitudes, colors, and errors even for low signal-to-noise ratio measurements,Astronomical Journal, 118, 1406-1410.

Kent, S.M., 1994: Sloan Digital Sky Survey, in Science with astronomical Near-infrared skysurveys, Astrophysics and Space Science, 217, Kulwer A.P., 27-30.

Anderson, J. and I. R. King, 2000: Toward high-precission astrometry with WFPC2. I. Derivingan accurate Point-Spread Function, Publications of the Astronomical Society of thePacific, 112, 1360-1382.

Megessier, C., 1995: Accuracy of the astrophysical absolute flux calibrations: visible and near-infrared, Astron.Astrophys., 296, 771-778.

Born, Max and Emil Wolf, 1965: Principles of optics, Pergamon, 808pp.

Tegen, I., P.Hollrig, M.Chin, I.Fung, D.Jacob, J.Penner, 1997, Contribution of different aerosolspecies to the global aerosol extinction optical thickness: Estimates from model results,J.Geopgys.Res., 102, 23895-23915.

Warner, J.X., and R.G.Ellingson, 2000: A new narrowband radiation model for water

vapor absorption, J.Atmos.Res., 57, 1481-1496.

Bevington, P. and K. Robinson, 1992: Data reduction and error analysis for the physicalsciences, 2nd edition, McGraw-Hill, 330pp.

Fig. 4. Variation of ln(Intensity) as a function of air mass, for the brightest stars visible during the night of15 October, 1999.

Fig. 5. As in Fig.4, but for other stars.

Fig. 6. As in Fig.4, but for other stars.

Fig. 7. As in Fig.4, but for other stars.

Fig. 8. Variation of ln(Intensity) as a function of air mass, for the brightest stars visible during the night of14 October, 1999.

Fig. 9. As in the Fig.8 but for other stars.

Fig. 10. As in Fig.8 but for other stars.

Fig.11. As in Fig.8 but for the star Vega.

Fig. 12. Results obtained using a variable software aperture, averaged, for the night of 15 October 1999.

Fig. 13. As in Fig.12, but for other stars.

Fig. 14. As in Fig.12, but for other stars.

Fig. 15. As in Fig.12, but for other stars.

Fig. 16. The Signal to Noise ratio for some of the stars (scaled to maximum S/N), shows the decreasing ofthe source s signal, as detected by the instrument, as the star is moving towards low altitudes.