Refraction

In constructing the mathematical model to describe the transfer of radiation in Earth's atmosphere, one potentially important effect has been neglected: refraction. As light passes between mediums possessing different optical properties, it is refracted or bent. Refraction has been shown to be important for both occultation (Smith and Hunten, 1990) and in the calculation of J-values near sunset (DeMajistre et al., 1995; Balluch and Lary, 1997) and so its effect on limb radiances is likely to be important. The relationship between initial and refracted zenith angles is given by Snell's law,

$$m_i \sin\theta_i = m_{i-1} \sin\theta_i'$$ (7.47)

where the m_i is the refractive index of an atmospheric layer between z_i and z_i+1, $\theta_i$ is the incident zenith angle and $\theta_i'$ is the refracted zenith angle. Snell's law can be derived from the Fresnel equations (e.g.: Jackson, 1960). For light propagating through Earth's atmosphere and passing from lower density to higher density air, by Snell's law, the path the light traverses will be bent towards the surface. As the density of the air is continually changing, light is continually being refracted. This effect is shown schematically for the ER-2 in Figure 4.4.

  

Figure 4.4: The difference between apparent and actual paths through the limb of the atmosphere as a result of refraction.

The continuous nature of atmospheric refraction makes implementing it in radiative transfer models difficult and no attempt will be made to do so here. Instead, a correction term will be utilized which will shift the elevation angle of the limb radiance (and derived quantities) to one which is more representative of the atmospheric region sensed.

  

Figure 4.5: The refraction of light at discrete layers and its impact on tangent height.

The effects of refraction can be quantified using Snell's law and trigonometry and an expression which describes the change in tangent height due to refraction can be derived. Consider a pencil of light at height z_i, making an angle $\theta_i$ with the local surface normal, and propagating downward with initial tangent height z_t,i. Upon refraction, it has a local zenith angle of $\theta_i'$, as shown in Figure 4.5 . At the next layer interface, z_i-1, the local zenith angle is $\theta_{i-1}$. They can be related using the law of sines,

$$\frac{\sin(\pi-\theta_{i-1})}{R_e + z_i} = \frac{\sin \theta_{i-1}}{R_e + z_i} = \frac{\sin\theta_i'}{R_e+z_{i-1}}$$ (7.48)

and upon substitution of Snell's law,

$$\frac{\sin \theta_{i-1}}{R_e + z_i} = \left( \frac{m_i}{m_{i-1}} \right) \left( \frac{\sin\theta_i}{R_e+z_{i-1}} \right)$$ (7.49)

the local zenith at z_i-1 can be expressed in terms of the local zenith angle at z_i. Carrying out one more iteration,

$$\frac{\sin \theta_{i-2}}{R_e + z_{i-1}} = \left( \frac{m_i}{m... ...{i-1}} \right) \left( \frac{\sin\theta_i}{R_e+z_{i-2}} \right)$$ (7.50)

and upon canceling rearranging,

$$(R_e+z_{i-2}) \sin\theta_{i-2} = \frac{m_i}{m_{i-2}} (R_e+z_i) \sin\theta_i$$ (7.51)

the pattern becomes clear. Using the definition for tangent height,

$$\sin\theta_i = \frac{R_e + z_{t,i}}{R_e + z_i}$$ (7.52)

and generalizing the result,

$$R_e + z_{t,j} = \frac{m_i}{m_j}(R_e + z_{t,i})$$ (7.53)

an expression relating the tangent height of the refracted pencil at height z_j to the apparent (unrefracted) tangent height is arrived at. This is called the formula of Bouguer which can also be arrived by considering how the optical path behaves in a medium with spherical symmetry (Born and Wolf, 1975).

The change in tangent height as a result of refraction is,

 

$$\Delta z_{t} = z_{t,a} - z_{t,c} z_{t,a} - \left( \frac{m_a}{m_{c}} \right) (R_e + z_{t,a}) + R_e$$ =

$$(R_e + z_{t,a}) \left( 1- \frac{m_a}{m_{c}} \right)$$ = (7.54)

where z_t,a is the apparent tangent height as seen by the CPFM and z_t,c is the actual, or corrected, tangent height. The refractive indices are calculated at the altitude of the ER-2, m_a, and the altitude of the corrected tangent height, m_c. As the corrected tangent height is an unknown, m_c will be initially evaluated at the apparent tangent height, and upon calculating $\Delta z_{t}$, a second iteration can be performed evaluating m_c at $z_{t,a}-\Delta z_{t}$.

An expression for refractive index of air (or any ideal gas) can be derived by considering the molecules comprising air to be simple dipoles under the influence of an electric field. This treatment yields the general result,

$$m^2-1= k N \frac{1}{\lambda-\lambda_o}$$ (7.55)

where N is the number density, $\lambda_o$ a resonance wavelength (in general, there will be many resonance wavelengths), and k a constant. The dispersive term can be expanded in a Laurent series and for air from the near-UV to the near-IR, only the $\lambda^{-2}$ term will be important. Using, $m^2-1 \approx 2(m-1)$, Cauchy's formula is arrived at (Born and Wolf, 1959),

$$m - 1 = A \left(1 + \frac{B}{\lambda^2} \right)$$ (7.56)

where $A=2.879\times 10^{-4}$ and $B=5.67\times 10^{-3}~\mu$m² at surface pressure. If B is taken as pressure independent and utilizing the fact that $m-1\propto N$, the refractive index at any height can be obtained by multiplying the left hand side of equation (4.56) by e^-z/H, where H is some suitable scale height obtained by assuming an isothermal atmosphere. This allows equation (4.54) to be written as,

$$\Delta z_t = (R_e+z_a) A \left( 1 + \frac{B}{\lambda^2} \right) (e^{-z_c/H} - e^{-z_a/H})$$ (7.57)

where R_eA=1.84 km which roughly represents the maximum change in tangent height resulting from refraction. Equation (4.57) has been plotted for wavelengths of 300 nm and 800 nm in Figure 4.6 for the ER-2 at 20 km. Note that for a satellite, the equation is nearly identical except that z_a is replaced with the orbital altitude. For ODIN, as an example, z_a=600 km which would increase the values of $\Delta z_{t}$ from Figure 4.6 by about 17%.

  

Figure 4.6: Effect of refraction on tangent height as determined using equation (4.57) for the ER-2 at 20 km.

Clearly, from Figure 4.6, refraction can become important for tangent heights near the surface. If the corrected tangent height is $z_{t,a}-\Delta z_{t}$ then the corrected elevation angle, EA_c, is given by,

$$\cos{\rm EA}_c=\frac{R_e + z_{t,a}-\Delta z_{t}}{R_e + z_a} = \cos{\rm EA}_a - \frac{\Delta z_{t}}{R_e + z_a}.$$ (7.58)

where EA_a is the CPFM apparent elevation angle. For $\vert{\rm EA}\vert<15^{\circ}$, a small argument expansion for cosine ( $\cos x\approx 1-x^2/2$) may be used,

$${\rm EA}_c^2 - {\rm EA}_a^2 = ({\rm EA}_c - {\rm EA}_a)({\rm EA}_c + {\rm EA}_a) = \frac{2 \Delta z_{tan}}{R_e + z_a}$$ (7.59)

and using ${\rm EA}_a + {\rm EA}_c \approx 2{\rm EA}_a$, the corrected elevation angle is,

$${\rm EA}_c = {\rm EA}_a + \frac{\Delta z_t}{\vert{\rm EA}_a\v... ... \lambda^{-2})(e^{-z_c/H}-e^{-z_a/H})} {\vert{\rm EA}_a\vert}.$$ (7.60)

Note the correction term is proportional to (EA_a)^-1 which means for larger elevation angles, the effects of refraction decrease (although for EA $>15^{\circ}$, the small argument expansion cannot be used). Also, for EA $\rightarrow 0$, $\Delta z_t \rightarrow 0$, so that the correction term goes to zero. Further, using z_c assumes that the majority of the signal is originating at the tangent point which, depending primarily on wavelength, may not be the case. As will be discussed further in Chapter 5, most of the information reaching the instrument is from light scattering into the line of sight before the slant optical depth, $\tau_s$, reaches unity. At 500 nm, equation (4.60) becomes,

$${\rm EA}_c = {\rm EA}_a + \frac{0.95}{\vert{\rm EA}_a\vert}(e^{-z_c/H}-e^{-z_a/H})$$ (7.61)

 where EA_a and EA_c are in degrees.

The corrections, which will decrease the elevation angle, will be applied to the elevation angle of the measurements. The three different cases are:

1.

${\rm EA}\geq0$: No corrections as refraction effects small above 20 km

2.

$-4.5^{\circ}<{\rm EA}<0$: Equation (4.60) used with z_c taken as altitude at which slant optical depth equals unity (if $\tau_s=1$ is reached above z_t,c) or z_t,c (otherwise).

3.

${\rm EA}<-4.5^{\circ}$: Equation (4.60) used with z_c taken as altitude at which slant optical depth equals unity (if $\tau_s=1$ is reached above 0 km) or 0 km (otherwise).

Equation (4.61) predicts a maximum change of about 0.2$^{\circ }$ near an elevation angle of $-4.5^{\circ}$.