3.2 Atmospheric optical propagation

Ground-based optical observations of sprites and elves are often carried out at viewing elevation angles near horizontal and as a result are subject to particularly strong effects of Rayleigh-scattering and refraction along the line of sight. These effects and some other causes of extinction are investigated below.

3.2.1 Cloud and Rayleigh-scattering: temporal considerations

Figure 3.1: Factors affecting intensity and timing of Rayleigh-scattered lightning flashes. The observer is located at coordinates (0,0) and the observer's horizontal is aligned with the abscissa. The white lines show the viewing direction used for the cases in Figures 3.2 and 3.3. Only single-scattering is considered. (a) Scatter coefficient and calculations for red light at 700 nm wavelength. (b) Calculations for blue light at 400 nm.

Figure 3.1 shows calculations of Rayleigh-scattering between a source at 5 km altitude and 675 km range (great circle distance) and a receiver on the ground, located at the origin of the plots. The axes are aligned with the observer's local horizontal and vertical and show values of the various quantities as a function of position in the vertical plane that includes the source and the observer. Because the source (lightning) and receiver are beyond each other's horizons, no direct light can be observed and any light originating at the source can reach the receiver only after Rayleigh-scattering from the sky. We only consider single-scattering for reasons that will be clear in the following discussion. Thus the region in which received light is scattered is bounded by the lines of local horizontals of the source and observer. The top panel in (a) shows with a logarithmic color scale the Rayleigh-scatter coefficient^3.1 for light with 700 nm wavelength, which is proportional to the atmospheric density. The curvature of the atmosphere is evident. The second plot shows the intensity of the direct flash, which is diminished as $r^{-2}$ and attenuated by Rayleigh-scattering.

The third plot shows the analogous quantity as seen from the observer's point of view -- that is, the fraction of light which, if scattered towards the observer at each given point, would reach the observer. In this case $r^{-2}$ does not play a role for an optical detector with a fixed solid angle of acceptance.

The fourth plot shows the propagation time for singly-scattered light. The ground-path (that taken by a lightning sferic) propagation time for this distance is 2.25 ms. The white line indicates a typical photometer line of sight elevation of 6$^\circ$. The fifth plot shows the product of the values in the second and third plots, illustrating that a Rayleigh-scattered lightning flash seen at all elevation angles comes primarily from light scattering relatively near the observer, and thus from the lower ($<$30 km altitude) atmosphere. As a result, the signal seen by a photometer arrives with minimal delay with respect to the causative flash (time of sferic). This observation also justifies the single-scattering assumption for red light. Figure 3.2 shows the signal which would be seen by a photometer pointed along the white line (6$^\circ$ elevation). There is an inevitable onset delay as compared with the radio pulse, but nevertheless an instantaneous lightning flash results in an optical signal which peaks within no more than $\sim$20 $\mu$s of the arrival of the sferic.

Figure 3.2: Predicted photometric signatures of a Rayleigh-scattered lightning flash. The model flash is perfectly impulsive in time.

Figure 3.1(b) shows the analogous plots for light with 400 nm wavelength, for which the Rayleigh-scatter coefficient is $\sim$9.9 times that for 700 nm. This case shows a qualitatively different outcome. The signal contribution shows scattering primarily near the source, and up to mesospheric altitudes, suggesting that for blue light single-scattering is likely a poor assumption, and that, therefore, optical flashes seen in blue from over the horizon should appear especially diffuse. Another way to state this difference is to say that the optical depth of the entire atmosphere looking out at 6$^\circ$ elevation is $\sim$0.35 for 700 nm light and $\sim$3.5 for 400 nm light. Figure 3.2 shows the modeled photometer response for a blue photometer. The delay due to scattering is at least 30 $\mu$s, and the impulsive flash is seen to be spread out considerably in time ($\sim$70 $\mu$s).

Figure 3.3: Flash onset delay variation with viewing azimuth. Values shown are for 700 nm wavelength, 675 km range, and 6.5$^\circ$ degree viewing elevation.

Figure 3.3 shows the results of a similar model for singly-scattered light at 700 nm but where the third dimension is included, and the dependence on the photometer viewing azimuth is investigated. This result shows that the effect of Rayleigh-scattering does add some azimuthal dispersion to an observed scattered lightning flash, but only amounting to about 10 $\mu$s for a photon directed 10$^\circ$ (in azimuth) away from the source lightning flash.

These simple calculations show that for red light (700 nm wavelength) the photometric onset delay with respect to the arrival of a lightning sferic is only on the order of 10 $\mu$s. Similarly, the delay between photometers viewing different azimuths up to 10$^\circ$ away from the lightning flash is only on the order of 10 $\mu$s. These two results are important considerations for Section 4.1, where the larger delay and azimuthal dispersion seen by a photometric array is shown to be a unique signature of elves.

The modeled time-broadening resulting from long-distance atmospheric propagation is small compared with the $\sim$1 ms duration often seen for scattered lightning flashes when viewed towards their source azimuth. The duration and shape of such optical pulses is instead dominated by multiple elastic scattering within the clouds of the thunderstorm [Thomason and Krider, 1982; Guo and Krider, 1982].

3.2.2 Scattering and absorption as a function of wavelength

A full account of the extinction processes important at visible and near-visible wavelengths includes Mie scattering due to small aerosol particles and spectral absorption by O$_2$ and other species [e.g., Erlick and Frederick, 1998]. These complex effects are quantifiable with available models but the importance of aerosols can vary greatly with atmospheric conditions. Aerosol content cannot be easily recorded along each viewing path and at each time, leading to significant uncertainties in optical transmission properties for low viewing elevation angles.

Figure 3.4: Atmospheric transmission calculated with MODTRAN3. (a) Variation with viewing elevation angle. (b) Variation with observer altitude.

In addition, attenuation at blue wavelengths is highly sensitive to the viewing elevation angle and to the observer altitude. Figure 3.4 shows examples of calculations using the Plexus interface to the MODTRAN3 model for atmospheric extinction.^3.2 Most of the observations reported in this dissertation were realized from Langmuir Laboratory, situated at $\sim$3.2 km altitude, providing a considerable advantage over lower sites, as shown in panel (b) of Figure 3.4.

3.2.3 Atmospheric refraction

The net effect of atmospheric refraction is to elevate the apparent position of distant objects. As viewed from a given altitude, this effect varies with temperature profiles and weather conditions in the lower atmosphere, but may typically be as large as $\sim$0.6$^\circ$ for objects above the atmosphere seen at the horizon from sea level. For example, this effect causes stars to set later than would be expected based on the geometry of the solid Earth.

At low viewing elevation angles, atmospheric refraction is significant for interpreting sprite and elve features, especially in video observations. In the interpretation of video images used in this dissertation, a simple empirical form for the dependence of refraction on viewing elevation angle adapted from Montenbruck and Pfleger [1998, p. 46] is used. A more detailed discussion of this subject may be found in the work of Stanley [2000, p. 141].