Monte-Carlomodelingusedtosimulatepropagationof

photonsinamediumNilsHaëntjens– OceanOpticsClass2017

basedonlecturesfromEmmanuelBoss andEdouardLeymarie

WhatisMonteCarloModeling?

• MonteCarlomodelingconsistinrepeatedrandomsamplingtoobtainnumericalresults.• Usedtobuildasolutiontotheradiativetransferequationbysimulatingpropagationofphotonsinamedium.• Increasingcomplexitycanbeaddedasthemodelisdeveloped.

ImplementingStepbyStepStepsandconceptsneededtoimplementaMonte-Carlomodelusedtosimulatethepropagationofphotonsinamedium:• Randomnumbergenerator• Absorption• Scattering• VariableWeightofPhotons• Indexofrefraction

RandomNumberGeneration• FirststeprequiredforaMonteCarlomodelistohaveamethodtogeneraterandomnumbers• Randomnumbergeneratorsproduce”pseudo”randomnumbers• Essentialpropertiesofarandomnumbergenerator:• repeatability:usingseeds• randomness:produceindependentuniformlydistributedrandomnumbers• longperiod:sequenceusedtoproducetherandomnumberuseafiniteperiod• insensitivetoseeds:periodandrandomnesspropertiesarenotaffectedbytheinitialseeds

Exercise1:RandomNumberGenerator

• Generate10000randomnumbersX in[0 1]• Matlab:rand.m

• CheckthattheX areuniformlydistributed• dividethe[0 1] axisinto20equalintervals• frequencyoccurrenceofX is500ideally• standarddeviation< 36

• Howsensitiveistherandomnumbergeneratortochangeinseeds?

Exercise1:ResultsMatlab2017a:rand.m; seed=1;n=10000;

Attenuationofacollimatedbeam

• Howphotonsareabsorbedinthemedium?• AbsorptionobeysBeer’sLaw:

𝐸 = 𝐸#𝑒%&'withz thedepthwithinthemedium(m)

a Absorptioncoefficient(m-1)

ØTheprobabilityofabsorptioninthemediumwithin[z z+δz] isa δz with δz << 1/a

Exercise2:Attenuationofacollimatedbeam

• Assumephotoncanbe:• absorbed• transmitted• NOTscatteredØc = a = 1 m-1

• Noboundaries• norefraction

• FixedstepΔz = 20 cm• absorptioncanonlyoccurattheendofastep

New photon

Move photon Dz

New Position z=z+ Dz

Absorb?Increase

absorptionevents

Yes

n=10,000Yes

No

Stop

Exercise2:Results

Exercice 2b:Improvedalgorithm

• Asinglecalculationperphoton(faster)

• Removeassumptionoffixedstep

• Assumehomogeneouswater

New photon

Random number

Increase absorptionevents

N=10,000 ? Yes Stop

Absorption depth

𝑧 = 𝑙𝑐 =

−ln(𝑋)𝑎

𝑝 𝑙 = 𝑒%4, 𝑙 ≥ 0

P 𝑙 = 9𝑒%4:

#

𝑑𝑙 = 1 − 𝑒%4

𝑙 = − ln 1 − 𝑋 = − ln 𝑋 , 0 ≤ 𝑋 ≤ 1

Fromthedefinitionoftheopticaldistanceltheprobabilitydensityfunctionforattenuationoflightis

Thecumulativedistributionfunctionis

Todeterminel inMCSimulations,𝑃 𝑙 = 𝑋

Thegeometricpathlengthz (inmeters)canbecomputedwith

assumenoscattering

Absorption

Reflection

AddingScattering

• Keeptrackofeachphoton:• position,direction,andterminationpoint

Ørequiredforscattering

• Terminationofphoton:• absorbedbymedium• reflected(z <0)

• Assumeboundarieshavesameindexofrefraction• Variablesteplength

𝑧 =−ln(𝑋)𝑐

ScatteringProbabilities

Theprobabilitythataphoton,whenscattered,willscatteratpolarangle𝜓andazimuthalangleΦ awayfromtheincidentdirectionisgivenbythescatteringphasefunction𝛽B(𝜓,Φ) ofthemedium

𝑝(Ψ)and𝑝(Φ) areindependentofoneanotherForseawaterandforair,theazimuthalangleΦ withrespecttotheincidentdirectionisuniformlydistributedover[0 2π].

The polarangle𝜓 cumulativedistributionfunctionisΦ = 2πX

2𝜋9 𝛽B 𝜓 sin 𝜓 𝑑𝜓J

#

= 𝑋

Exercise3:AddingScattering

initialize array

new photon

move photonincrement reflected count

absorbed? calculate new direction

yes

yes

no

noIn medium

VariableWeightPhotons

• Increasecomputationalspeed• Biasingthedistributionfunction• tracemorephotonsthatarelikelytofindtheirwaytotheareaofinterestwithoutchangingthefinalcomputedresult

• Decreasephotons’weightalongtheirpath• Athresholdissettodeterminewhenthephoton’sweightisnotsignificantanymore.

SpecularReflection

Fromtheairsidewhenaphotonreachesanair-waterinterfaceIncident,reflectedandtransmittedangles

Fractionofreflectedlight

Air

Water

1n

2n

θi

θt

θr

DiffuseReflection

Onthewaterside,travelingfromwatertoair,thereisacriticalincidentangle𝜃𝑐 abovewhichthereisa100%reflection

Fractionofreflectedlight

),,( zyx

),,( zyx µµµ

),,( zyx -

),,( zyx µµµ -1n

2n

iq

Exercise4:SurfaceinteractionsStart

Initializing Photon

Update Reflectance and Photon Weight

Move Photon at a Variable Step

Update Photon Weight Due to Absorption

Photon in Water?

Weight Too Small?

Change Photon Direction

Last Photon?

No

Yes

Yes

Yes

No

No

NoYes

Internally Reflected?

Get Photon Position and Direction

Update Reflection

Keepaddingfeatures

• Keeptrackofphotons• Adddetectors• Addsourcesoflight• Seafloorinteractions• Non-homogenousmedium• Polarization• ....

SimulO – “SimulationOptique”• “Userfriendly”3DMonteCarlo

• photonsarefollowedfromthesourcetothepointwheretheyareabsorbed• Buildanydeviceassemblingandsizingelementaryobjects• SetOpticalproperties

• HomogenousVolumesproperties• refractiveindexofthematerial• absorptionandscatteringcoefficients• scatteringphasefunction

• uploadyours• built-in:purewater,isotropic,Henvey-Greenstein,Fournier-Forand

• HomogenousSurfaceproperties• transparent,specularorLambertian reflection

• Photonemissionlightsource• Photoncountingtools

• numberofcollisionsontheelementaryobjects• averageofphotonpathlength,averagenumberofscatteringeventsper

photons,numberofphotonsabsorbed

byEdouardLeymarie (Laboratoire d’Océanographie deVillefranche)

SimulO Applications• Self-ShadingsimulationsforBoussole

• 1billionphotons/wavelength/IOP• Doxaran D.,Leymarie E.,Nechad B.Dogliotti A.,RuddickK.,Gernez P.andE.Knaeps (2016).Improvedcorrectionmethodsforfieldmeasurementsofparticulatelightbackscatteringinturbidwaters.OpticsExpress,24(4),3615-3637.

• Song,G.,Xie,H.,Bélanger,S.,Leymarie,E.andM.Babin (2013).Spectrallyresolvedefficienciesofcarbonmonoxide(CO)photoproduction inthewesternCanadianArctic:particlesversussolutes.Biogeosciences,10,3731–3748

• Babin,M.,D.Stramski,R.A.Reynolds,V.M.Wright,andE.Leymarie (2012)Inpress.Determinationofthevolumescatteringfunctionofnaturalwatersamples.AppliedOptics,51,17,3853-3873

• Leymarie,E.,D.Doxaran,andM.Babin (2010).Uncertaintiesassociatedtomeasurementsofinherentopticalpropertiesinnaturalwaters,AppliedOptics,49,5415-5436

Surface

LimitationsofSimulO

• Ramanscatteringisnotimplemented• Nopolarization• Assumeperfectlyflatsurface(nowind)• Assumeblacksky(correctionwillbesubmitted)

ReflectiveTubeAbsorptionMeter(RTAM)Ratiobetweenmeasuredandthetrueabsorptioncoefficientsasafunctionofthereflectivityofthetube.

Kirk1992

Self-ShadingestimatedbybackwardMC

Sea surface

Lu Sensor

Forward representation

Sea surface

Lu Sensor

Self-ShadingestimatedbybackwardMC

Simulation1:Luinfinitely

smallNotshaded

LutrueNij

Simulation2:Lusensor+structureShaded

LumeasuredMij

30*120 hemisphere matrixAtm : a=0, b=0, n=1

water : a, b, bb/b, λ, n=1.34

Lu Sensor

Shading =𝑀𝑖𝑗

𝑁𝑖𝑗

LuShadingMatrix

notshaded→1shaded→0

• Assumptions• homogeneouswater• atmosphereisnotsimulated• flatseasurface(nowaves)• blacksky

Self-shadingwithablackskyHyperNav onfloat