MCmodel

A 3D Monte Carlo scattering code to keep track of individual parcels in a plane-parallel homogenous slab

The theory behind the model, many of its parameters, and model validation have been described by Petrich et al. (2012), https://doi.org/10.1016/j.coldregions.2011.12.004

The Monte Carlo code itself is implemented in C for speed and integrated with a Python interface for convenience.

License: Apache 2.0.

Installation

Requirements

• Python 2.7, 3.5, 3.6, 3.7. Should work on Linux, MacOS, and Windows.
• Numpy
• When compiling from source, a C-compiler, and typically setuptools and wheel.

From PyPI

Run pip install mcmodel. Done. Later, upgrade with pip install -U mcmodel, or uninstall with pip uninstall mcmodel.

Use without installation

Get the latest source code from https://github.com/cpetrich/MCmodel Run python setup.py build and copy the contents of the built/lib.[...] directory into the same directory as your own modeling code.

Installation with pip

Get the latest source code from https://github.com/cpetrich/MCmodel A wheel will be built locally and installed with pip. Build the module and create a wheel file with python setup.py bdist_wheel, cd dist, pip install mcmodel-[...].whl. To uninstall the package later run pip uninstall mcmodel.

Test

To test the result of the build process, a rather incomplete test suite can be run with python setup.py test.

Limitations

The code is currently not reentrant since it uses global state for the random number generator, the phase function, and buffers that hold detailed information about the most recent particle track. I.e., only one instance of the model per process.

The special case - in a scattering model - of "no scattering" (w0 == 0) is implemented only for k == 0 (i.e. no absorption).

Usage

The usual mode of operation is:

0. seed the random number generator
1. define a scattering phase function
2. call the scattering code with slab optical properties, and position and direction of one parcel of light
3. store entrance and exit positions and angles of the parcel and information about plane crossings inside the medium
4. if desired, store complete track of the light parcel
5. continue with (2) a couple of thousand or million times.
6. read the simulation results from disk, and determine parcel intensity due to absorption and Fresnel reflection
7. analyze result as desired

An example of seeding the random number generator of the Monte Carlo model is

import os
import mcmodel
mcmmodel.set_seed(int.from_bytes(os.urandom(4),'little'))

A helper function exists to generate common scattering phase functions, e.g.

import mcmodel_util
pf = mcmodel_util.make_phase_function('Henyey-Greenstein', (0.98, 0))

The phase function is transfered to the model code like

mcmodel.define_phase_function(pf['lookup_cdf'],pf['lookup_phi'])

A random number generator is usually used to generate an incoming distribution (that random number generator is separate from the one used by the Monte Carlo model and should also be seeded).

import random
random.seed()

A sample from the angular distribution of a Lambert emitter (i.e., overcast over snow-covered ground) is

import math

elevation_angle = math.arcsin(random.uniform(-1., 0))
azimuth = random.uniform(-math.pi, math.pi)

Note that the Monte Carlo code works in elevation angles. A normally incident beam at the upper surface of the slab has an elevation angle of -pi/2.

The domain parameters and incident angle are passed on to the Monte Carlo code in a dictionary, e.g.

in = {'angle': elevation_angle,
      'azimuth': azimuth,
      'thickness': 1, # slab thickness
      'k': 1}         # densiy of microscopic interactions used by the model

with additional optional input parameters described below. Many of the output data are obtained through updates to a dictionary that is passed to the scattering simulation. To calculate the track of a single parcel call

out = {}
mcmodel.simulate(in, out)

where exit position and angles are

exit_position = (out['x'], out['y'], out['z'])
exit_elevation_angle = out['angle']
exit_azimuth = out['azimuth']
path_length = out['path_length']

Attenuation of the parcel inside the medium is accounted for by decreasing its amplitude. The precise method of this depends on the definition and units of thickness and microscopic extinction coefficient passed into the model above. However, the intensity of the emerging parcel will be of a form similar to

exit_intensity = math.exp(-kappa * path_length)

where kappa is the absorption coefficient.

Intensity reduction due to Fresnel reflection (other than total reflection) would be accounted for based on entrance and exit angles and relative refractive indices. To do this in hindsight is not trivial and requires the simulations to be run with a relative refractive index of 1 and appropriate scaling of the input angular distribution to account for non-trivial refractive indices. This is outside the scope of this overview. [TODO: in a future version, add an option to have Monte Carlo model take care of this by selecting a path stochastically.]

API

The C code exports currently four functions (see Usage above):

• mcmodel.set_seed(...)
• mcmodel.define_phase_function(..., ...)
• mcmodel.simulate(..., ...)
• mcmodel.get_last_particle_track()

In addion, there is one helper function defined in Python

• mcmodel_util.make_phase_function(...)

Parameters

The model domain is a horizontal slab of thickness thickness, extending from z=0 at the upper surface to z=-thickness at the bottom. Parcles are injected at depth z=z0 (-thickness <= z0 <= 0) and x=0 and y=0. Injection at z0 = -thickness and z0 = 0 are on the outside of the slab at the lower and upper interface, respectively (i.e. still subject to refraction).

If an anisotropic scattering coefficient is used, i.e. sigma_anisotropy != 0, then the vertical and horizontal scattering coefficients are sigma_v = k * w0 and sigma_h = k * w0 * (sigma_anisotropy+1), respectively.

Input Dictionary of mcmodel.simulate()

Name	Default	Meaning
angle	none	elevation angle of incident parcel [-pi; pi]
azimuth	none	azimuth angle of incident parcel [-pi; pi)
z0	0	vertial coordinate of parcel injection
thickness	none	slab thickness
k	none	microscopic extinction coefficient
w0	1	single scattering albedo
n_surface	1	refractive index n_slab / n(z>0)
n_bottom	1	refractive index n_slab / n(z<-thickness)
sigma_anisotropy	0	anisotropy of scattering coefficient (cf. Trodahl et al.), sigma_anisotropy = sigma_h/sigma_v - 1
do_record_track	False	log coordinates of scattering the parcel. Retrieve with mcmodel.get_last_particle_track()
do_record_plane_crossings	False	log position and angle of parcel every time it crosses specified planes
record_planes	[]	list of z coordinates to record parcel crossings at

Output Dictionary of mcmodel.simulate()

Name	Meaning
angle_in	copy of angle in input dictionary
azimuth_in	copy of azimuth in input dictionary
angle	exit eleation angle of parcel
azimuth	exit azimuth of parcel
x	lateral exit coordinate, same unit as thickness in input
y	lateral exit coordinate, same unit as thickness in input
z	vertical exit coordinate, same unit as thickness in input
path_length	total path length of parcel in the medium, same unit as thickness in input
exit_direction	+1: exit at z=0, -1: exit at z=-thickness
min_z	lowest z coordinate along the track
max_R	largest value of sqrt( x^2 + y^2 ) along the track
n_collisions	number of scattering events along the path (NB: not total reflection)
n_missed_collisions	number of absorption events along the path, all of which have been ignored (count is slightly too high if w0>0)
n_refractions	number of refractions experienced along the path
n_total_reflections	number of total reflections along the path
n_plane_crossings	number of plane crossing events along the path
plane_crossings	2D numpy array containing 6 entries (columns) per crossing event (see below)

The six column entries for each plane crossing are:

1. elevation angle
2. azimuth angle
3. x-coordinate at the point of crossing
4. y-coordinate at the point of crossing
5. z-coordinate at the point of crossing
6. parcel path length so far