MCRT: L6. Initial weight of packet: W = L / N MC At each interaction multiply weight by probability of scattering: W = a W

Transcription

1 MCRT: L6 Variance reduction techniques improve signal-to-noise of simulation using same number of MC packets Examples of where MCRT is inefficient optically thin (need lots of photons) Weights keep packet alive but reduce weight by albedo on every scattering, kill when below some weight Russian roulette: decide whether to keep following low weight packets Forced first scattering use for optically thin simulations Next-event estimator force packets in a particular direction Initial weight of packet: W = L / N MC At each interaction multiply weight by probability of scattering: W = a W After N scatterings, the packet s weight will be W a N Test if packet s weight below a minimum value, then decide whether to terminate or keep random walk going: if (W.lt. W_min) then if (ran.lt. p) then W = W/p else terminate photon packet endif end if

2 Weighting Regular or analog Mont Carlo: L-packets have equal weight W 0 = L/N MC where L is the source luminosity (power) Packets random walk with scattering events until absorbed (ξ > albedo) and then terminated Alternative: at each scattering, multiply packet s weight by albedo (probability L-packet would have scattered) Keep track of photons until they exit simulation or if weight falls below some value use Russian Roulette to decide fate of packet Run-time of simulation is like running regular MC with albedo = 1 Russian Roulette Follow weighted photon packet until weight less than some value user-defined value, e.g., 1.E-3 W 0 Give packet a probability for surviving, say p = 0.1 Decide whether packet lives or dies If packet lives, then keep the Monte Carlo simulation going with a new weight W/p Typically W_min = 1E-3, p = 0.1

3 Forced first interaction If region optically thin, regular MC will lead to very few photons being scattered and most randomly chosen optical depths are larger than τ e from source to edge of grid Emit photon, compute τ e to edge of grid along direction of travel, then choose a random τ in range (0, τ e ) using: τ = ln(1 ξ[1 e τ e ]) Have forced photon to interact between source and τ e to edge of grid, so need to multiply photon weight by (1 e τ e ) Images courtesy of Tom Robitaille Constant density sphere, central point source, 10 7 photons

4 Constant density sphere, central point source, 10 7 photons Constant density sphere, central point source, 10 7 photons

5 Constant density sphere, central point source, 10 7 photons Constant density sphere, central point source, 10 7 photons

6 Most photons scatter Most photons do not scatter

7 Algorithm: The first time a photon is emitted, we force it to scatter, and we weigh the energy of the photon according to the probability of scattering before escaping the grid. We call this forced first scattering See Appendix in Wood & Reynolds (1999, ApJ 525, 799) 1. Compute escape escape 2. Sample between 0 and escape 3. Weight photon energy by: W =1 4. Travel then scatter e escape All photons scatter!

8 Regular MC Force First Scattering Constant density sphere, central point source, 107 photons Regular MC Force First Scattering Constant density sphere, central point source, 107 photons

9 Regular MC Force First Scattering Constant density sphere, central point source, 107 photons Regular MC Force First Scattering Constant density sphere, central point source, 107 photons

10 Regular MC Force First Scattering Constant density sphere, central point source, 107 photons Peeling Off or Next Event Estimator When photons exit from (x, y, z), regular MC bins according to the exit angles (θ, φ) and then projects photons onto an image plane into (xim,yim) location bins to produce images x im = y cos φ x sin φ y im = z sin θ y cosθ sin φ x cos θ cos φ This can require many MC photons to get good signal-to-noise especially for 3D systems Alternatively, at each interaction, compute the probability for a photon to emerge in a chosen direction and project a weighted MC photon into the image (or detector)

11 Photon binning

12

13 Peeling-off Peeling-off

14 Peeling-off W = P (, ) e escape Peeling-off W = P (, ) e escape

15 Peeling-off W = P (, ) e escape Peeling-off W = P (, ) e escape

16 Peeling-off W = P (, ) e escape Peeling-off

17 Peeling-off Peeling-off Every photon contributes multiple times to all viewing angles

18 Formulae for peeling off for direct and scattered photons: e τ e W direct = 4 π d 2 W scattered = Φ(θ)e τ e where τ e is the optical depth to the observer/detector. For external detectors τ e is optical depth to edge of the grid. d is the distance from the photon location to the observer. Φ(θ) is phase function for scattering the photon through the angle θ towards the observer, cos θ = n photon. n observer. In astronomy we can generally ignore distances within the medium if the source-observer distance is large, so assume d is the same for all photons Detectors in medicine or internal viewing of a galaxy d 2 Source Detector W scattered = Φ(θ)e τ e d 2