The Discretized Spatial Grid

Discretizing the spatial grid is accomplished by dividing the model atmosphere into N_i layers. This is done by specifying N_i+1 altitudes, not necessarily equally spaced, at which the solution will be determined. Each altitude level, or grid point, is assigned a value for all relevant quantities of the model atmosphere.

The extinction coefficient is calculated at each point on the vertical grid using,

$$k_{e,i} = \sum_{l=1}^{N_l} n_l(z_i) \sigma_{e,l}$$ (7.32)

where $k_{e,i} \equiv k_e(z_i)$ and $\sigma_{e,l}$ is the extinction cross-section for species l. The surface is at level i=1 and the top of the atmosphere corresponds to level i=N+1. To calculate the optical depth at each grid point, the extinction coefficient is taken as varying linearly between adjacent levels, i.e. k(z)=a+bz. Substituting into the definition for optical depth, equation (2.73), and summing down to level i gives,

$$\tau_i = \sum_{i'=i}^{N_i-1} \left[ \frac{ k_{e,i'} + k_{e,i'+1} } {2} \right] (z_{i'+1} - z_{i'})$$ (7.33)

where $\tau_i \equiv \tau(z_i)$.

The source function is also taken as varying linearly through a layer, i.e. ${\bf J}={\bf a} + {\bf b}z$. Substituting this into equations (4.26) and (4.27) gives an expression for the Stokes vector coefficient at i in terms of that at i+1 and a combination of the source function coefficients at i and i+1 as follows,

  

$${\bf I}_n^{cm}(\tau_{i+1},\mu) {\bf I}_n^{cm}(\tau_i,\mu) e^{-\Delta\tau_{i,i+1}} + \alpha_{i+1} {\bf J}_n^{cm}(\tau_i,\mu) + \beta_{i+1} {\bf J}_n^{cm}(\tau_{i+1},\mu)$$ = (7.34)

$${\bf I}_n^{cm}(\tau_i,-\mu) {\bf I}_n^{cm}(\tau_{i+1},-\mu) e^{-\Delta\tau_{i,i+1}} + \alpha_i {\bf J}_n^{cm}(\tau_{i+1},\mu) + \beta_i {\bf J}_n^{cm}(\tau_i,\mu)$$ = (7.35)

where,

$$\alpha_i \frac{1-e^{-\Delta\tau_{i,i+1}}}{\Delta\tau_{i,i+1}} - e^{-\Delta\tau_{i,i+1}}$$ = (7.36)

$$\beta_i 1 - \frac{1-e^{-\Delta\tau_{i,i+1}}}{\Delta\tau_{i,i+1}}$$ = (7.37)

and

$$\Delta\tau_{i,i+1} = \frac{\tau_i - \tau_{i+1}}{\mu}.$$ (7.38)

The solution throughout the atmosphere is computed by first considering the level at the top of the atmosphere. The downward Stokes vector at the level immediately below it is calculated using equation (4.35) with i=N and the appropriate boundary condition ${\bf I}_n^{cm}(\tau_{N_i+1},-\mu)={\bf0}$ (for a given $\mu$). Once known, the Stokes vector at i=N-1 can be calculated in a similar manner. This is continued until the surface, i=1, is reached. Once the surface reflected Stokes vector is determined (see section 2.7), the upward Stokes vectors are calculated using equation (4.34). Iterations are performed from i=1 to N. At this point, calculations for one scattering order have been completed. The convergence criterion is checked and, if not met, a new source function is determined and the above procedure is repeated.

The vertical resolution required is mainly a function of optical depth and inhomogeneity of the atmosphere. Tests using different vertical grids have revealed that a vertical spacing of 1 km is more than adequate for most simulations. Above the top of the ozone layer ($\sim$45 km), where the pressure is very small, a larger grid spacing may be used as there is little scattering. An exception to this includes modeling the limb. After corrections for the sphericity of the Earth have been made (see Section 4.6), where a spacing of 0.5 km may be necessary due to the longpath length between layers. Another exception is when an optically thick aerosol layer is present, such as tropospheric dust or marine aerosols in the boundary layer.

To determine the validity of taking the source function as varying linearly between levels, another method was tested. The average source function of two adjacent levels was assumed to be representative of the layer between them. This simpler treatment gives the following constants,

$$\alpha_i = \beta_i = \frac{ 1-e^{-\Delta\tau_{i,i+1}}} {2}.$$ (7.39)

This and the linear approximation yielded very similar solutions, usually within one part in 10³, which implies that the linear approximation is valid. For thicker atmospheres (or a courser grid), quadratic interpolation may be necessary.