3. Model descriptions

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General

MCARaTS numerical world is composed of several modules. Some concepts behind them are described below.

Geometry

The position of photon is defined in the 3D Cartesian coordinate system. The direction of photon transport is defined by zenith (polar) angle (theta) and azimuth angle (phi) from the X-axis, as in the following figure.

The incident and emergent directions for pixel radiances can be specified by the user. The zenith and azimuth angles of source light-transport direction are defined as theta0 and phi0, respectively. Similarly, emergent zenith and azimuth angles are defined as theta1 and phi1, respectively. The horizontal boundary condition is cyclic. The vertical boundaries z=zmin and zmax correspond to the bottom and top, respectively, of the atmosphere (BOA and TOA, respectively).

Surface

The surface is modeled as one of the following models

• black (no reflection)

• Lambertian model

• Diffuse-specular mixture model: diffuse reflection is Lambertian, and specular reflection is Fresnellian for rough surface facets.

• Rahman-Pinty-Verstraete BRDF model

• Li-Sparse-Ross-Thick BRDF model

Surface model and its parameters can vary two-dimensionally.

Atmosphere

The model atmosphere is divided into cell volumes three-dimensionally. The number of layer is defined as nz.

The user can specify cloud layers ("3D layers") arbitrarily with horizontal inhomogeneity. Other layers are assumed to be horizontally homogeneous. The 3D layers are divided into cells in the vertical and horizontal directions. The numbers of 3D cells in x-, y-, and z-directions are defined as nx, ny, and nz3, respectively.

Modeled atmospheric media include the followings:

• Scattering constituents: For each one constituent, extinction coefficient, single scattering albedo and a specification parameter for phase function are arbitrarily specified. The user defines multiple kinds of phase function, which can be Rayleigh scattering phase function, Henyey-Greenstein function, or a function interpolated from user-specified database.

• Gaseous absorbing constituents: Modeled by correlated k-distribution.

All properties can be specified in 1D (for layers) and/or in 3D (for 3D cells). The atmosphere and surface system is schematically illustrated in the figure below.

Phase functions

One method for phase function data representation is to prepare the phase functions for

• multiple kinds of aerosol polydispersions,

• multiple types of hydrometeors (e.g., cloud water and ice, drizzle droplets, rain water, snow, and graupel)

  • different shapes,

  • different size distributions, and/or

  • different effective radii and effective variances.

Total number of phase functions could be 100 to 10,000, which is significantly smaller than the number (1,000,000 or larger) of atmospheric cells in usual cases. To obtain highly accurate radiance estimates, every phase functions should be tabulated for 200 or more points of scattering angles.

In the mcarats, the phase functions should be given in a database file, and 3D distribution of a phase function specification parameter should be given. The latter data should be specified cell-by-cell, so that used computer memory should be a word length multipled by the number of atmospheric cells.

Thus, the strategy to prepare the phase function database and 3D distribution of phase function specification parameter is economical for limited computer memory!

In addition, the phase function specification parameter is not digitalized in the latest version. The phase functions used in the code are interpolated from the database table!

Integrators

Various radiative quantities can be computed by the code:

• Upward/downward/direct fluxes at layer interfaces, averaged over finite areas

• Voxel-averaged 3-D heating rates

• Radiances

  • Radiances averaged over finite horizontal areas at arbitrary layer interfaces

  • Local radiances averaged over finite solid angles, looking from arbitrary points

  • Radiances averaged over finite horizontal areas and for planes at arbitrary layer interfaces

• Pathlength statistics for the radiances

  • Layer air mass factors

  • Air mass factors for user-specified weighting functions

  • Time-resolved radiance histogram

For radiances, the user should specify the location of view point, field of view (FOV), and FOV-center direction with zenith and azimuth angles, a method of angular pixel mapping, and numbers of angular pixels.

Algorithms

Used algorithms are described in Iwabuchi (2006, J. Atmos. Sci.). The model uses Monte Carlo methods for simulating photon trajectories and samples fluxes, heating rates, radiances, and other radiometric quantities. The local estimation method is used for radiance averaged over some specific area or over specific solid angle. The maximum cross section method is used for acceleration of photon tracing in inhomogeneous media. Other methods useful for variance reduction are also used and documented in the scientific paper. The following figure is an algorithm flow chart.