GAP-2015-XXX 1

Effect of Multiple Scattering on the AerosolAnalysis

Max Malacari, Bruce DawsonSchool of Physical Sciences, University of Adelaide

GAP-2015-XXX, October 2015

Abstract

In this note we show using a Monte Carlo photon-by-photon simulation ofthe CLF that aerosol profiles reconstructed using observations of the CLF laserare sensitive to multiple scattering of the laser, with typical VAODs being un-derestimated by a few percent. We then use this simulation to develop a param-eterisation of the multiple scattered fraction of the CLF laser for use in aerosolprofile reconstruction. Finally, we demonstrate how this parameterisation can beapplied to the existing Data Normalised and Laser Simulation aerosol analysesin order to correct aerosol profiles for multiple scattering of the CLF laser.

1 Introduction

Measurements of laser tracks from the Central Laser Facility and eXtreme Laser Fa-cility (CLF and XLF) are used to infer the vertical distribution of aerosols abovethe array. Sets of 50 vertical depolarized shots are measured every 15 minutes byeach of the 4 FD stations during each night of observation. As the aerosol concentra-tion in the atmosphere varies over short time-scales, each set of shots within an hourblock is averaged and used to calculate the Vertical Aerosol Optical Depth (VAOD).Two techniques are currently employed for the reconstruction of the VAOD: the DataNormalized (DN) method in which the hourly averaged track profiles are comparedto averages collected under clear conditions (called Rayleigh nights), and the LaserSimulation (LS) method in which these hourly averages are compared to simulationsgenerated with different aerosol attenuation conditions.

In both of these VAOD reconstruction procedures we assume that any light thatis not initially scattered out of the laser beam and towards the detector is lost. Thisassumption ignores the possibility that some photons will reach the detector after oneor more additional scatterings. As multiple scattering serves to make the laser appearslightly brighter than it would under the assumption of single scattering only, both ofthese VAOD reconstruction methods will compensate for this effect by reconstructingan aerosol atmosphere that is marginally cleaner than reality. While the effect ofmultiple scattering on reconstructed aerosol profiles is not expected to be large, weshow in this GAP note through simulations that the systematic it introduces to theaerosol transmission factor is non-negligible.

GAP-2015-XXX 2

2 Photon-by-photon simulation

Until now, multiple scattering studies have focussed on the multiple scattered signalreceived at a detector due to an isotropic light source, such as fluorescence light emittedfrom air showers [4, 2, 3, 1]. In this section we discuss in some detail a new simulationwhich has been written to study the multiple scattered signal received at a detectorfrom a source who’s source function is determined by the combination of the Rayleighand aerosol phase functions.

For this work we have developed a new photon-by-photon multiple scattering simu-lation of the CLF laser called MSSim. The simulation exploits the cylindrical symmetryof a vertically moving light source in order to greatly increase the number of collectedphotons and decrease the simulation time by many orders of magnitude. This tech-nique has been used previously in similar studies such as Mike Roberts’ isotropicmultiple scattering parameterization [4].

In the simulation, the light source is located at ground level and emits photonsone at a time vertically upwards through a parametrized atmosphere of height Hmax.A cylindrical detector ring of height Hring and radius R surrounds the light sourceand photons propagate until they either pass through this ring or leave the simulationvolume. The detection volume is slightly larger in radius than the detector ring toallow for the detection of photons that might multiple scatter from behind the detectorback into the field of view. Figure 1 gives a graphical representation of this detectionvolume. Photons are traced on their way through the atmosphere as they scatter onmolecules and aerosols until they either pass through this ring or leave the detectionvolume. Information about photons that pass through this ring, including their arrivaldirection, number of interactions in the atmosphere, and height of first interaction arerecorded. We discuss the atmospheric description and simulation procedure in moredetail below.

R

Rout

R

Rout Hmax

Hring

Figure 1: Top (left) and side (right) view of the detection volume. The laser axis isshown in green.

GAP-2015-XXX 3

2.1 Atmosphere description

The molecular and aerosol atmospheres are described using simple exponential models,under the assumption that the atmosphere is horizontally uniform. The molecularscattering coefficient at height h above ground level (per unit distance) is fully definedby its horizontal attenuation length at sea level (for a 355 nm wavelength) LM, andits scale height HM

αM(h) = (1/LM) · exp(−(h+Hg)/HM) (1)

where Hg is the prevailing ground height of the detector. Upon interacting with amolecule in the air, the probability of scattering into a solid angle dΩ at an angle θ tothe photon’s direction of travel is described by the analytically determined Rayleighphase function (

1

σ

dσ

dΩ

)M

=3

16π(1 + cos2 θ) (2)

The aerosol atmosphere is described in a similar way, with an added provision fora mixing layer of uniform density

αA(h) =

{1/LA : h < Hmix(1/LA) · exp(−(h−Hmix)/HA) : h ≥ Hmix

The probability of scattering into a solid angle dΩ after interacting with an aerosolparticle is described by the modified Henyey-Greenstein function(

1

σ

dσ

dΩ

)A

=1− g2

4π

(1

(1 + g2 − 2g cos θ)3/2 + f3 cos2 θ − 12(1 + g2)3/2

)(3)

where f and g are backscattering and asymmetry parameters respectively. While afully analytical description of the Mie phase function is not possible, the modifiedHenyey-Greenstein function has been shown to be a reasonable approximation.

2.2 Propagating a photon

Given a photon’s current position and direction of travel, ~P1 and p̂1, we must determineits new position and direction of travel, ~P2 and p̂2, after it has interacted with amolecule or aerosol particle in the atmosphere. This is shown graphically in Figure 2.Given the new position of the photon we can then determine if the line joining ~P1 and~P2 intersects the boundary of the detection volume.

Given a random number P on the interval [0, 1), the optical depth traversed by aphoton before interacting is given by

τ = − ln(1− P ) = 1cos θ

∫ h2h1

α(h′) dh′ (4)

where θ is the zenith angle of the photon’s current trajectory and h1 and h2 correspondto the current height of the photon and the height of next interaction respectively.

The procedure for propagating a photon through the atmosphere is as follows:

1. Roll a random optical depth for both the aerosol and molecular atmosphere usingEquation 4, τA and τM.

GAP-2015-XXX 4

θ1

θ2

p̂1

p̂2

~P1

~P2

Hmix

Ground

h

αA

Figure 2: Geometry of the current photon position and direction (~P1 and p̂1) and

the new photon position and direction (~P2 and p̂2). The aerosol atmosphere has auniform density below the mixing layer height Hmix and then decays exponentiallyabove that height with a scale height of HA. Note that the molecular atmosphere (notshown) decays exponentially with height starting from ground level and dominates thescattering processes in the air.

2. Invert the optical depths in order to solve for h2, the height of next interaction.See below.

3. Choose the interaction corresponding to the smallest h2 in the case of an upward-going photon, or the largest h2 in the case of a downward-going photon.

4. Find the photon’s new position ~P2 by translating the photon along its directionof travel to height h2

~P2 = ~P1 + (|h2 − h1|/ cos θ) · p̂1

5. Check whether this trajectory passes through the boundary of the detectionvolume (r ≥ R and/or h ≥ Hmax), and if so whether or not it passes throughthe detector ring.

6. If the photon is still within the detection volume, given the interaction typesample the corresponding phase function to determine the photon’s new directionof travel.

GAP-2015-XXX 5

7. Transform the photon’s new direction of travel into the global coordinate systemin order to get p̂2.

In the case of the molecular atmosphere it is straightforward to invert the rolledoptical depth in order to determine the height of next interaction. Using Equations 1and 4 we can show that

h2 = −HM · ln(− LMHM· τM · cos θ + exp

(− (h1 +Hg)/HM

))−Hg

As the aerosol atmosphere is piecewise in nature, the calculation of the height ofnext interaction depends on whether the photon’s current position is above or belowthe mixing layer, and whether or not the photon is upward or downward travelling.

2.2.1 Condition 1: h1 ≥ Hmix and θ < π/2In this case both h1 and h2 will be above the mixing layer. Solving for h2 we find

h2 = −HA · ln(− LAHA· τA · cos θ + exp

(− (h1 −Hmix)/HA

))+Hmix

2.2.2 Condition 2: h1 ≥ Hmix and θ > π/2In this case h1 is above the mixing layer and, depending on the rolled value of τA,h2 could be above or below the mixing layer. First we use Equation 5 to check if anoptical path length of τA is reached before the top of the mixing layer. If this is notthe case then we compute the remaining optical path length that must be traversedbelow the mixing layer, τnew

τnew = τA −1

cos θ· HALA

(exp

(− (h1 −Hmix)/HA

)− 1)

Given τnew we can then calculate h2

h2 = τnew · LA · cos θ +Hmix

2.2.3 Condition 3: h1 < Hmix and θ < π/2

In this case h1 is below the mixing layer and, depending on the rolled value of τA, h2could be above or below the mixing layer. Similarly to Condition 2 we first calculateh′2 assuming a mixing layer of infinite height

h′2 = τA · LA · cos θ + h1If h′2 falls above Hmix then we calculate the remaining optical depth above the mixinglayer, τnew

τnew = τA −1

cos θ· 1LA

(Hmix − h1)

and then solve for h2

h2 = −HA · ln(− LAHA· τnew · cos θ + 1

)+Hmix

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2.2.4 Condition 4: h1 < Hmix and θ > π/2

In this case both h1 and h2 are below the mixing layer. h2 is simply given by

h2 = τA · LA · cos θ + h1

If the photon’s new position lies outside the detection volume, the point of inter-section ~Pint of the photon’s trajectory with the detection volume must be calculated.If the height of this position is less than the height of the detector ring the photon’spropagation information is recorded in the output file. The point of intersection canbe found from the intersection of the line joining ~P1 and ~P2 with a cylinder of radiusR, and can be expressed in terms of a parameter t between 0 and 1

~Pint = ~P1 + t · (~P2 − ~P1)

3 Simulation parameters

Fixed parameters used for the simulation are shown in Table 1. The contribution tothe multiple scattered fraction due to photons that scatter behind the detector ringwas determined by changing Rout and observing the effect on the multiple scatteredfraction. The contribution was found to be negligible, and increasing Rout beyond thedetector-laser distance only served to increase the simulation time by many orders ofmagnitude. The detection volume radius is therefore set to the detector ring distancefor all subsequent simulations.

The simulation parameters were chosen to encompass a reasonable range of differentaerosol attenuation conditions, ranging from completely aerosol free (Rayleigh nights)to very ”dirty”. The most aerosol contaminated atmosphere simulated correspondsto a VAOD at infinity of 0.2, which is well beyond the maximum VAOD quality cutused in data analysis of 0.1. The laser-detector distance was varied over a rangecorresponding to the approximate distances between individual FD stations and theCLF and XLF (∼26-39 km).

Parameter Value

Hmax 30 kmHring 100 mR 20, 26, 30 kmRout Set to RLM 14.2 km (suitable for a 355 nm laser)HM 8 kmHg 1.4 km (observatory’s prevailing ground height)HA 1.2 kmHmix 1 kmLA 11, 22, 55, 110 kmf 0.4g 0.6

Table 1: Summary of the fixed parameters used for the multiple scattering simulations.

GAP-2015-XXX 7

4 Simulation results

The simulation outputs a number of different parameters for each photon that hitsthe detector ring including: the coordinates of the position at which the photon hitsthe detector, the direction of travel, the total path length traversed since leaving theground, the first interaction height, the total number of interactions, the nature of thelast interaction (molecular or aerosol) before the photon is incident on the detector.

5 Parameterizing the MS fraction

Existing multiple scattering parameterisations describe the quantity of multiple scat-tered light received at a fluorescence detector telescope from a shower-like isotropicsource in the atmosphere. As light is assumed to be “lost” when it is attenuated asit traverses the atmosphere between its emission point and the fluorescence detector,these paramaterisations provide an important correction to the received light flux.

Based on previous analyses (e.g., we tried a linear parameterization of the form

fζ(α, τ, R) = k · αa · τ b ·Rc (5)

where fζ gives the fraction of multiple scattered light within a single 100 ns time bincompared to the total signal for an integration angle of ζ. The terms α, τ , and Rcorrespond to the total scattering cross section (aerosol and molecular) at the lasertrack (at the centre of the time bin) in m−1, the total optical depth (aerosol andmolecular) between the laser track and the detector, and the distance between thelaser track and the detector respectively in metres. The parameters a, b, and c weredetermined using a grid search for the minimum χ2/ndf, with k being a free parameterin the fit. The parameter space was searched with a step size of 0.01 in each dimension.

We found the multiple scattered fraction to have a negligible dependence on thedistance between the detector and the laser track. This is due to the fact that thisdistance is somewhat incorporated into the optical depth term?

Figures 4a and 4b show an example of the fitted parameterization (multiple scat-tered signal relative to the total signal) for ζ = 1.6◦ along with the fit residuals. Thereduced χ2 for this integration angle is 1.25 and the residuals are typically less than∼ 1% of the total integrated light. Overplotted in red is the multiple scattered fractionpredicted by the Roberts model, which was developed for an isotropic light source. Inthe case of of a laser source, the angular distribution of the emitted light is determinedby the combination of the Rayleigh and aerosol phase functions. As the laser beam isviewed approximately perpendicular to its direction of travel through the atmosphere,

6 Effect of the optical spot

7 Pixelisation of the FD camera

8 Data Normalized analysis

In order to quantify the effect of multiple scattering on reconstructed aerosol profileswe have developed a toy model of the Data Normalized (DN) analysis that has the

GAP-2015-XXX 8

x

-20-10

010

20

-20

-10

0

10

20

0

5

10

15

20

25

30

(a)

Azimuth [deg]-15 -10 -5 0 5 10 15

Ele

vatio

n [d

eg]

0

5

10

15

20

25

30

210

310

410

(b)

Figure 3: (a) Example of the trajectories of the first 1000 photons in a simulation run.Photons that have been singly scattered out of the laser beam are plotted in blue,and multiply scattered photons are plotted in red. The laser beam itself is plottedin green. (b) The camera view for this same simulation (very long integration time),where the laser axis is centred in the middle of the camera. This shows the arrivaldirection of the photons that pass through the detector ring. The high intensity stripeshows detected photons that have been singly scattered out of the laser beam. Allphotons arriving off-axis have been multiply scattered.

GAP-2015-XXX 9

0.99τ 0.62α0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

1.6

f

0

0.05

0.1

0.15

0.2

0.25

0.3

VAOD = 0.2

VAOD = 0.1

VAOD = 0.04

VAOD = 0.02

Pure Rayleigh

Parameterization

Roberts isotropic param.

/ndf = 1.252χ

(a)

Entries 6969

Mean -0.0006573

RMS 0.005015

MC-fparf-0.03 -0.02 -0.01 0 0.01 0.02 0.03

N

0

50

100

150

200

250

300

350

400 Entries 6969

Mean -0.0006573

RMS 0.005015

Entries 6969

Mean -0.0006573

RMS 0.005015

(b)

Figure 4: (a) The multiple scattered signal relative to the total signal as a function ofthe total scattering coefficient at the laser track and the total optical depth betweenthe detector and the laser track, shown for a light integration angle of ζ = 1.6◦. Shownin blue is the parameterized prediction, and shown in red is the prediction from MikeRoberts’ parameterization for an isotropic emitter. (b) Distribution of the residualsbetween the Monte Carlo simulation and the parameterized prediction for ζ = 1.6◦.

GAP-2015-XXX 10

capability of taking the output of the photon-by-photon simulation as its input. TheDN method deduces the aerosol concentration in the atmosphere under the assumptionof horizontal uniformity by comparing the observed traces from vertical lasers at anygiven observation time to traces recorded on clear, aerosol free nights (Rayleigh nights).The comparison of laser profiles to those recorded on Rayleigh nights allows the laserand fluorescence detector absolute calibration, laser polarization state, and molecularmultiple scattering to be normalized out, thereby simplifying the situation (CRAP!).The toy DN analysis uses the same underlying method as the production DN code.Here we give a brief description of the procedure.

SA + SM

TA1 · TM1 TA2 · TM2

DetectorLaser

φ1 φ2

θ

h

Figure 5: The laser-detector geometry. Laser light is scattered out of the beam atheight h and angle θ.

Laser light is scattered out of the beam as it travels up through the atmosphere.The number of photons remaining in the beam at height h is proportional to theproduct of the aerosol and molecular transmission factors (TA1 and TM1) between theground and that height. The number of photons scattered out of the beam at anangle θ towards the detector is given by the sum of the aerosol and molecular cross-sections (SA and SM), which depend on the local density of aerosol and molecularscatterers as well as their scattering phase functions. This light is then attenuated onits way to the detector according to the aerosol and molecular transmission factorsbetween the laser beam and the detector (TA2 and TM2).

The number of photons received at the detector originating from height h on anight when aerosols are present can be expressed as

Naer(h) = N0(E0, λ) · TA1 · TM1 · (SA + SM) · TA2 · TM2 (6)

where N0(E0, λ) is the initial number of photons in a beam of energy E0 and wave-length λ. Likewise, the number of photons received at the detector on a Rayleigh nightcan be expressed as

NRay(h) = N0(E0, λ) · TM1 · SM · TM2 (7)assuming that the molecular atmosphere remains approximately constant and thatthe relative calibration of the laser and the fluorescence detector between the time of

GAP-2015-XXX 11

observation and the Rayleigh night is known. Molecular atmosphere is constant intime..... SOMETHING HERE.

The aerosol transmission between the ground and a point at height h at an elevationangle of φ is given by

TA = exp(−VAOD/ sinφ) (8)

If the laser is at 90 degrees (φ1 = 90◦) then Equations 7, 8, and 9 can be combined

and rearranged to give an expression for the VAOD at height h

VAOD(h) =−1

1 + 1/ sinφ2ln

(NaerNRay

1

1 + SA/SM

)(9)

In order to reconstruct the VAOD at any given time we need the observed lasertrace at that time, as well as the laser trace on the Rayleigh night corresponding tothat CLF epoch. In addition we need to know the fraction of light that is scatteredout of the beam by aerosols and molecules as a function of height. These quantitiesdepend on the local aerosol concentration at height h, αA(h), which is not knowna-priori.

SA(h) =αA

αA + αM· fA(θ) (10)

SM(h) =αM

αA + αM· fM(θ) (11)

where the functions fA and fM describe the fraction of light scattered into the solid-angle subtended by the detector by aerosols and molecules respectively. These quan-tities can be determined from Equations 2 and 3.

In order to get around the fact that we don’t initially know αA we exploit the factthat scattering in the atmosphere is dominated by molecular scattering, and that theaerosol is strongly forward peaked and therefore has a small contribution to scatteringin the direction of the detector (θ ∼ 90◦). This allows us to make the approxima-tion that the scattering of laser light out of the beam is almost purely molecular, orSA � SM. The first guess VAOD is therefore given by

VAODfirst guess =−1

1 + 1/ sinφ2ln

(NaerNRay

)(12)

As aerosols are strongly concentrated near the ground this approximation will havethe effect of lowering the reconstructed VAOD lower in the atmosphere where aerosolscattering out of the beam is non-negligible. We can correct for this via an iterativeprocedure. Given the first guess VAOD we can perform a fit to determine αA, and aftermaking some educated assumptions about the shape of the aerosol phase function cancalculate SA and SM using Equations 11 and 12. This procedure can then be repeateda handful of times until the true VAOD is reached, although 2 or 3 iterations is oftensufficient.

It should also be noted that this aerosol reconstruction method assumes that laserlight is only singly scattered on its way from the laser beam to the detector.

GAP-2015-XXX 12

Height [km a.g.l]0 2 4 6 8 10 12 14

VA

OD

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Figure 6: Example of a DN VAOD that has been reconstructed using lasers producedin this simulation (with multiple scattering turned off). The black curve shows the firstguess VAOD, assuming that aerosol scattering out of the laser beam is negligible. Thered curve (and the grey points) shows the resultant VAOD after aerosol scatteringout of the beam has been iteratively corrected for. The reconstructed VAOD is ingood agreement with the 3-parameter input VAOD, shown in blue, which was used tosimulate the laser.

9 Effect of multiple scattering on DN reconstructed

VAODs

9.1 Effect on shower profiles

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ζ (deg) a b k χ2/ndf

0.5 0.86 0.89 77.73 1.420.6 0.83 0.90 70.11 1.430.7 0.80 0.91 61.10 1.400.8 0.77 0.93 51.72 1.400.9 0.75 0.94 47.73 1.391.0 0.73 0.95 43.50 1.361.1 0.71 0.95 39.34 1.331.2 0.69 0.96 35.11 1.321.3 0.67 0.97 31.08 1.301.4 0.65 0.98 27.33 1.281.5 0.64 0.98 26.58 1.261.6 0.62 0.99 23.13 1.251.7 0.60 1.00 20.04 1.231.8 0.59 1.00 19.23 1.211.9 0.57 1.01 16.53 1.192.0 0.56 1.01 15.76 1.182.1 0.55 1.02 14.92 1.172.2 0.54 1.02 14.15 1.152.3 0.52 1.03 12.03 1.132.4 0.51 1.03 11.35 1.122.5 0.50 1.03 10.70 1.112.6 0.49 1.03 10.06 1.092.7 0.48 1.03 9.44 1.082.8 0.47 1.03 8.85 1.082.9 0.46 1.04 8.25 1.073.0 0.45 1.04 7.72 1.053.1 0.44 1.04 7.20 1.063.2 0.43 1.04 6.71 1.043.3 0.42 1.04 6.25 1.033.4 0.41 1.04 5.81 1.033.5 0.40 1.05 5.39 1.023.6 0.39 1.05 5.00 1.023.7 0.38 1.05 4.64 1.013.8 0.38 1.04 4.78 0.993.9 0.37 1.04 4.43 0.98

Table 2: Fitted parameters for the function fζ(α, τ) obtained via a grid search of theparameter space.

GAP-2015-XXX 14

References

[1] Maria Giller and Andrzej Śmiakowski. An analytical approach to the multiplyscattered light in the optical images of the extensive air showers of ultra-highenergies. Astroparticle Physics, 36(1):166–182, aug 2012.

[2] J P\c ekala, P Homola, B Wilczy\’nska, and H Wilczy\’nski. Atmospheric multiplescattering of fluorescence and Cherenkov light emitted by extensive air showers.\nimpra, 605:388–398, jul 2009.

[3] J. Pkala and H. Wilczyski. Point spread function due to multiple scattering of lightin the atmosphere. Nuclear Instruments and Methods in Physics Research SectionA: Accelerators, Spectrometers, Detectors and Associated Equipment, 729:296–301,nov 2013.

[4] M D Roberts. The role of atmospheric multiple scattering in the transmission offluorescence light from extensive air showers. Journal of Physics G: Nuclear andParticle Physics, 31(11):1291, 2005.