The Discretized Angular Grid

Integration of all quantities over $\mu$ is also done numerically. Gaussian-Quadrature is chosen as it has been proven to be superior to other quadrature integration techniques (Liou, 1980). In Gaussian-Quadrature, the discrete values of $\mu$ are determined by the roots of the Legendre polynomial of order 2N_j. The weights, w_j, have a more complicated origin but are always normalized such that,

$$\sum_{j=-N_j}^{N_j} w_j = 2$$ (7.40)

and where $j\neq 0$. The integral is replaced with a summation as follows,

$$\int_{-1}^{1} f(\mu) d\mu \longrightarrow \sum_{j=-N_j}^{N_j} f(\mu_j) w_j$$ (7.41)

again where $j\neq 0$.

 

Table 4.3: Pivots and weights for Gaussian-Quadrature of order eight and twenty (Abramowitz and Stegun, 1964).

 

Nj	$\pm \mu_j$	wj
4	0.18343	0.36268
	0.52553	0.31370
	0.79667	0.22238
	0.96028	0.10123
10	0.07652	0.15275
	0.22779	0.14917
	0.37371	0.14210
	0.51087	0.13169
	0.63605	0.11819
	0.74633	0.10193
	0.83912	0.08328
	0.91223	0.06267
	0.96397	0.04060
	0.99313	0.01761

For example, the source function of equation (4.28) becomes,

$${\bf J}_{n+1}^{cm}(\tau_i;\mu_j) = \frac{\tilde{\omega}(\tau_... ..._i;\mu_j;\mu_{j'}) {\bf I}_n^{sm}(\tau_i;\mu_{j'})] \, w_{j'}.$$ (7.42)

The pivots (roots) and weights for Gaussian-Quadrature orders eight and twenty are given in Table 4.3.

In the calculation of irradiance, the integration is only performed over one hemisphere. This presents no real problem as the summation can run from 1 to N_j. However, care must be taken to ensure that the sum of the weights is exactly unity. To ensure this in the code, a renormalization is performed.

One drawback of using Gaussian-Quadrature is that, as the pivots are prescribed, they rarely correspond to angles desired. One way around this is to add extra pivots at any desired angles. These will have a zero weighting associated with them as they should not influence the integration. One example of when this is useful is in the calculation of the Stokes vector near the limb.