Spherical Correction

For many applications, calculations using plane-parallel geometry is adequate. However, for solar zenith angles larger than 75$^{\circ }$ the attenuation of the direct solar beam is overestimated in a plane-parallel atmosphere. This is because as $\mu$ tends towards zero, the plane parallel pathlength enhancement, $1/\mu$, becomes infinite. Also, for solar zenith angles larger than 90$^{\circ }$, some parts of the atmosphere will still be directly illuminated which is not possible using plane parallel geometry. Similar problems arise in the calculation of scattered light near the limb, or horizon. Pathlengths can be both overestimated or underestimated, depending on whether the streams are upward or downward. In a plane-parallel atmosphere, any viewing direction below the horizon will have a direct surface component. In a spherical atmosphere, for an observation made at 20 km, there is no direct surface component until 4.5$^{\circ }$ below the local horizon.

These deficiencies are illustrated in Figure 4.2. The dashed lines represent a plane-parallel atmosphere superimposed on the spherical one (solid lines). Light scattered downward is over-attenuated in a plane-parallel atmosphere while light scattered upward is both under-attenuated and possesses a direct surface component.

  

Figure 4.2: Comparison of plane-parallel and spherical atmospheres.

To remedy these inadequacies a number of options are available. One possibility is to solve the radiative transfer equation in three-dimensional spherical coordinates. The drawback of this is the huge increase in both complexity of the solution and computation time required. The alternative adopted herein is to integrate the source function through a series of spherical shells similar to that depicted in Figure 4.2. In this formulation, the physical distance between layers is no longer simply $\Delta z/\mu$, but a more complicated function of $\theta$ and altitude.

  

Figure 4.3: Pathlength between spherical shells.

In practice, equations (4.34) and (4.35) are still used. However, the definition of $\Delta \tau_{i,i+1}$ is modified to,

$$\Delta \tau_{i,i+1} = \frac{k_{e,i} + k_{e,i+1}}{2} s_{i,i+1}$$ (7.43)

where s_i,i+1 is the physical pathlength between shell layers i and i+1, also called the slant pathlength. This is shown in Figure 4.3 and is calculated as follows. Consider two levels at heights z_i and z_i+1. The local zenith angle at height z_i+1 is $\theta_{i+1}$ and it has a tangent height given by,

$$z_{t,i+1}=(R_e + z_{i+1})\sin{\theta_{i+1}} - R_e$$ (7.44)

where z_t,i+1 is the tangent height for the zenith angle at height z_i+1 and R_e=6378.1 km is the mean radius of the Earth and is taken as a constant. An ellipsoidal Earth is not considered. From this the slant pathlength can be determined,

$$s_{i,i+1} = \vert\sqrt{(R_e+z_{i+1})^2 - (R_e+z_{t,i+1})^2} - \sqrt{(R_e+z_{i})^2 - (R_e+z_{t,i+1})^2}\vert$$ (7.45)

where the absolute values ensure a positive length if z_i+1<z_i. Note that the local zenith angle at height z_i will differ from that at z_i+1. They are related by,

$$\sin{\theta_i} = \left( \frac{R_e + z_{i+1}}{R_e + z_i} \right) \sin{\theta_{i+1}}.$$ (7.46)

This further complicates the solution process as this variation of local zenith angle along the line-of-sight necessitates interpolation of the source function which has been calculated at discrete angles. This is done using quadratic interpolation.

In addition, a distinction must be made between upward radiance streams which originate at the surface and upward radiance streams which originate at the top of the atmosphere (originally as downward streams). Radiance streams will originate from the surface as long as $z_{t,i} \leq 0$ and streams will originate from the top of the atmosphere if z_t,i > 0. If the latter case is true then at some point in the integration along the path, the tangent height will be reached. If it does not correspond exactly to a vertical grid point then a dummy grid point is created between levels. Source function and extinction values are linearly interpolated onto it. Similarly, the solar zenith angle will vary along the line-of-sight. This correction has not been performed but it should not matter as long as the sun is fairly high in the sky. Only if the line-of-sight passes through the terminator is this correction important.

Two checks are available to ensure the spherical and plane-parallel versions are consistent: for high sun and viewing away from the limb, the two should yield the same solution; and for $R_e \rightarrow \infty$, the two must also yield the same solution. These tests have been performed and both plane-parallel and spherical versions give nearly identical radiances.