Monte-Carlo modeling used to simulate propagation of photons in a medium

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1 Monte-Carlo modeling used to simulate propagation of photons in a medium Nils Haëntjens Ocean Optics Class 2017 based on lectures from Emmanuel Boss and Edouard Leymarie

2 What is Monte Carlo Modeling? Monte Carlo modeling consist in repeated random sampling to obtain numerical results. Used to build a solution to the radiative transfer equation by simulating propagation of photons in a medium. Increasing complexity can be added as the model is developed.

3 Implementing Step by Step Steps and concepts needed to implement a Monte- Carlo model used to simulate the propagation of photons in a medium: Random number generator Absorption Scattering Variable Weight of Photons Index of refraction

4 Random Number Generation First step required for a Monte Carlo model is to have a method to generate random numbers Random number generators produce pseudo random numbers Essential properties of a random number generator: repeatability: using seeds randomness: produce independent uniformly distributed random numbers long period: sequence used to produce the random number use a finite period insensitive to seeds: period and randomness properties are not affected by the initial seeds

5 Exercise 1: Random Number Generator Generate random numbers X in [0 1] Matlab: rand.m Check that the X are uniformly distributed divide the [0 1] axis into 20 equal intervals frequency occurrence of X is 500 ideally standard deviation < 36 How sensitive is the random number generator to change in seeds?

6 Exercise 1: Results Matlab 2017a: rand.m; seed = 1; n = 10000;

7 Attenuation of a collimated beam How photons are absorbed in the medium? Absorption obeys Beer s Law: E = E # e %&' with z the depth within the medium (m) a Absorption coefficient (m -1 ) ØThe probability of absorption in the medium within [z z+δz] is a δz with δz << 1/a

8 Exercise 2: Attenuation of a collimated beam Assume photon can be: absorbed transmitted NOT scattered Øc = a = 1 m -1 No boundaries no refraction New photon Move photon Dz New Position z=z+ Dz n=10,000 Yes Stop Fixed step Δz = 20 cm absorption can only occur at the end of a step Absorb? No Yes Increase absorption events

9 Exercise 2: Results

10 Exercice 2b: Improved algorithm From the definition of the optical distance l the probability density function for attenuation of light is p l = e %4, l 0 The cumulative distribution function is P l = 9 e %4 dl = 1 e %4 z = l c # : To determine l in MC Simulations, P l = X l = ln 1 X = ln X, 0 X 1 The geometric path length z (in meters) can be computed with ln (X) = a assume no scattering New photon Random number Absorption depth N=10,000? Increase absorption events A single calculation per photon (faster) Remove assumption of fixed step Assume homogeneous water Yes Stop

11 Adding Scattering Keep track of each photon: position, direction, and termination point Ørequired for scattering Termination of photon: absorbed by medium reflected (z < 0) Assume boundaries have same index of refraction Variable step length z = ln (X) c Reflection Absorption

12 Scattering Probabilities The probability that a photon, when scattered, will scatter at polar angle ψ and azimuthal angle Φ away from the incident direction is given by the scattering phase function βb(ψ, Φ) of the medium p(ψ) and p(φ) are independent of one another For seawater and for air, the azimuthal angle Φ with respect to the incident direction is uniformly distributed over [0 2π]. Φ = 2πX The polar angle ψ cumulative distribution function is J 2π 9 βb ψ sin ψ dψ = X #

13 Exercise 3: Adding Scattering initialize array new photon increment reflected count move photon no In medium yes absorbed? no calculate new direction yes

14 Variable Weight Photons Increase computational speed Biasing the distribution function trace more photons that are likely to find their way to the area of interest without changing the final computed result Decrease photons weight along their path A threshold is set to determine when the photon s weight is not significant anymore.

15 Specular Reflection From the air side when a photon reaches an airwater interface Incident, reflected and transmitted angles n 1 Air θ i θ r Fraction of reflected light n 2 Water θ t

16 Diffuse Reflection On the water side, traveling from water to air, there is a critical incident angle θ c above which there is a 100 % reflection ( x, y, z) Fraction of reflected light n 2 ( µ x, µ y, µ z ) n 1 q i ( µ x, µ y,-µ z ) ( x, y, -z)

17 Exercise 4: Surface interactions Start Initializing Photon Update Reflectance and Photon Weight Move Photon at a Variable Step No Get Photon Position and Direction Yes Internally Reflected? No Photon in Water? Yes Update Photon Weight Due to Absorption Weight Too Small? Yes No Change Photon Direction Update Reflection No Last Photon? Yes

18 Keep adding features Keep track of photons Add detectors Add sources of light Seafloor interactions Non-homogenous medium Polarization...

19 SimulO Simulation Optique by Edouard Leymarie (Laboratoire d Océanographie de Villefranche) User friendly 3D Monte Carlo photons are followed from the source to the point where they are absorbed Build any device assembling and sizing elementary objects Set Optical properties Homogenous Volumes properties refractive index of the material absorption and scattering coefficients scattering phase function upload yours built-in: pure water, isotropic, Henvey-Greenstein, Fournier-Forand Homogenous Surface properties transparent, specular or Lambertian reflection Photon emission light source Photon counting tools number of collisions on the elementary objects average of photon pathlength, average number of scattering events per photons, number of photons absorbed

20 SimulO Applications Surface Self-Shading simulations for Boussole 1 billion photons / wavelength / IOP Doxaran D., Leymarie E., Nechad B. Dogliotti A., Ruddick K., Gernez P. and E. Knaeps (2016). Improved correction methods for field measurements of particulate light backscattering in turbid waters. Optics Express, 24(4), Song, G., Xie, H., Bélanger, S., Leymarie, E. and M. Babin (2013). Spectrally resolved efficiencies of carbon monoxide (CO) photoproduction in the western Canadian Arctic: particles versus solutes. Biogeosciences, 10, Babin, M., D. Stramski, R. A. Reynolds, V. M. Wright, and E. Leymarie (2012) In press. Determination of the volume scattering function of natural water samples. Applied Optics, 51, 17, Leymarie, E., D. Doxaran, and M. Babin (2010). Uncertainties associated to measurements of inherent optical properties in natural waters, Applied Optics, 49,

21 Limitations of SimulO Raman scattering is not implemented No polarization Assume perfectly flat surface (no wind) Assume black sky (correction will be submitted)

22 Reflective Tube Absorption Meter (RTAM) Ratio between measured and the true absorption coefficients as a function of the reflectivity of the tube. Kirk 1992

23 Self-Shading estimated by backward MC Sea surface Lu Sensor Sea surface Lu Sensor Forward representation

24 Self-Shading estimated by backward MC Assumptions homogeneous water atmosphere is not simulated flat sea surface (no waves) black sky Simulation 1 : Lu infinitely small Not shaded Lu true N ij Simulation 2 : Lu sensor + structure Shaded Lu measured M ij Lu Shading Matrix 30*120 hemisphere matrix Lu Sensor Atm : a=0, b=0, n=1 water : a, b, bb/b, λ, n=1.34 Shading = M ij N ij not shaded 1 shaded 0

25 Self-shading with a black sky HyperNav on float

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