In addition to the single scattering properties in COP, this version contains:

- the complete 4x4 scattering matrix of hexagonal ice crystals.
- the scattering matrix of imperfect hexagonal ice crystals.

### 1. The Scattering Matrix

The state of polarization of electromagnetic radiation can completely be described by the 4 elements of the Stokes vector. The Stokes vector of scattered radiation is related to that of the incoming radiation by a 4x4 scattering matrix:

For the present data base, only particles in random orientation are considered. In this case, the scattering matrix contains only six independent elements which are not dependent on the scattering azimuth angle:

These six elements are available in COPE.

### 2. Imperfect hexagonal ice crystals

Imperfect hexagons are modelled using statistical deviations of ray paths during the ray tracing (Figure 1) in hexagonal ice crystals. Each time a ray hits a crystal surface, the normal to this surface is tilted against its original direction by a certain angle, and it is also rotated around it (Figure 2). This tilt angle is chosen by use of random numbers, which are uniformly distributed between 0 and a given maximum value. The azimuth angle is randomly distributed between 0 and 2*Pi. The directions of the refracted and reflected ray are calculated according to this tilted surface element.

Figure 1: Ray tracing in a hexagonal column. |
Figure 2: Tilting of plane normals during the ray tracing. |

A maximum tilt angle of 1° only affects the direct forward and backward scattering. This scattering matrix may better be suited for lidar applications than the original scattering matrix without any tilting because the extremely strong increase in the direct backscattering at 180° does not occur here. A maximum tilt angle of 30° removes all halo features in the scattering function and may therefore be a more realistic model of ice crystals in many cirrus clouds which don't show halos.

### 3. The data files

The scattering matrices of randomly oriented hexagonal columnar ice crystals are stored in the directory **dat/corr**. There is one file for each scattering matrix. The file names reflect size and shape of the crystal, the wavelength, and the maximum tilt angle. They consist of 9 letters (e.g.**c40550r01.cs**) with the following meaning:

**c4**means**c**olumn No. 4. For the sizes and numbers of the particles, have a look at the table.**0550**means a wavelength of 550 nm.Currently available are the wavelengths:

0.550 µm, 0.659 µm, 0.865 µm, 1.6 µm, 3.7 µm.**r**means**r**andom orientation.**01**is a maximum tilt angle of 1°.Currently available are the maximum tilt angles:

1°, 30°, 90°

These data files contain the following information:

- INPUT parameters for ray tracing:
- length of a-axes and c-axes of the ice crystal
- wavelength
- refractive index
- maximum tilt angle

- OUTPUT of ray tracing:
- extinction, scattering, and absorption cross sections
- single scattering albedo
- asymmetry parameter
- relative amount of unscattered rays (going through two planparallel faces of the crystal)
- mean geometrical cross section of the crystal
- the scattering matrix elements F
_{11},F_{22}/F_{11}, F_{33}/F_{11},F_{44}/F_{11},-F_{12}/F_{11}, F_{34}/F_{11}

(**The phase function F**! )_{11}is normalized to 1 ! This may differ from the convention used by other authors

There are two additional columns after the matrix elements, which are related to a scattering matrix test and correction algorithm after *Hovenier and van der Mee (1996)*.

### 4. The program **cope.f**

This program uses results of the ray tracing program for particles of different sizes to calculate size distribution weighted averages of the single scattering properties. **cope.f** relieson certain conventions about the number of size intervals and the sizes of the particles (see table). For each calculation of size distribution weighted scattering properties, the scattering matrices of the 8 columns must exist for the desired wavelength. There is also a convention about the composition of the names of the files (see above).

The currently available contents of the data base is indicated in the file **cope.cfg**. This file is read by **cope.f** to fill in its selection menus.