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Contrast and signal-to-noise ratio in long-distance starlight imaging|
William C. Priedhorsky
The author is with the Los Alamos National Laboratory, Mail Stop D436, Los Alamos, New Mexico 87545.
Received 24 July 1995; revised manuscript received 29 January 1996.
A small telescope on an airplane or in low Earth orbit can, in principle, resolve ground objects under starlight with useful resolution. For an ~50-cm aperture and ~100-s exposure, one can obtain a resolution of tens of centimeters from an aircraft and a few meters from orbit. Such starlight images are photon poor, and feature detection depends on photon statistics. Scattered light, atmospheric absorption, and foreground airglow all degrade image contrast. I report an investigation into image signal-to-noise ratio using first-order analytical approximations. We find that, for a given angular resolution, the signal-to-noise ratio for spaceborne images is degraded approximately a factor of 1.7, compared with airborne images, by foreground airglow. Image signal-to-noise ratio improves as the passband moves to the red and as skies become brighter from artificial illumination.
Key words: Photon counting, remote sensing, night vision, starlight imaging, microchannel plates. © 1996 Optical Society of America
20 July 1996
With a broadband photon-counting detector, one can form images under starlight conditions at long range. The key is long exposures to build up photon statistics, and photon counting makes these possible. Because each detected photon is time tagged with high precision, images can be corrected for slew, jitter, and changing target perspective. We therefore need a high-speed photon-counting imager that can be used to report, for each photon, not just the two dimensions of position but a third dimension of time, and does this at count rates sufficient for optical imaging. Film or CCD’s are not suitable for this task; however, new fast schemes for microchannel plate readout meet this need.1 In principle, the three-dimensional information allows us to restore the original resolution of the telescope. But no matter how well a restoration algorithm works, long-range starlight images will be limited by photon statistics.
I have investigated the contrast and signal-to-noise ratio of long-range starlight images. The model includes the major sources of scattering and absorption, to first order, as summed over broad wavelength bands. Ground covers of various albedos are considered. We have concentrated on the light levels of a moonless night. The light from any appreciable Moon would increase contrast and flux and introduce shadows to the scene; a moonless night is therefore the worst case for nighttime imaging. In Section 2 I discuss the sources of light in a starlight image; Section 3 quantitatively analyzes illumination levels; Section 4 presents calculations of contrast levels; Section 5 considers measurements in the red and near IR; Section 6 analyzes the effects of artificially illuminated bright skies; and Section 7 summarizes the conclusions.
2. Sources of Light, Scattering, and Absorption
The 0.3–1.0-μm light of the night sky comes from three roughly equal sources: starlight (both individual stars and the diffuse galactic light, scattered by interstellar dust), sunlight scattered from interplanetary dust (zodiacal light), and airglow.2 The first two come from large distances, but the airglow comes from a layer at approximately 100 km that lies below orbital altitude. Once it strikes the atmosphere, light is absorbed by ozone in the stratosphere, scattered by molecules and aerosols, and absorbed at longer wavelengths by water and oxygen.
These contributions are summarized in Fig. 1. An object on the ground will be illuminated by (1) starlight, direct and scattered, (2) zodiacal light, direct and scattered, and (3) airglow, direct and scattered. Starlight imaging depends on the detection of this ground reflection and discrimination of objects by their varying albedos. Unfortunately, the ground reflection must be detected in the presence of a diffuse, uniform background. The nature of this background depends on altitude. From airplane altitude, above most atmospheric scattering but below the ozone layer, the background includes (1) upward scattered starlight, (2) upward scattered zodiacal light, and (3) upward scattered airglow. Scattered ground reflection is of a lesser order and is not included in the calculations below. From a spacecraft, one must add (4) the direct flux from the airglow. The situation is more difficult than at aircraft altitude, because the airglow layer is in the field of view and becomes an additional source of background. It is therefore critical to understand the breakdown of radiation sources in the night sky; I analyze these in Section 3. When the Moon is half-full or more, it becomes the largest source of illumination, and foreground airglow is not significant. Contrasts will be similar to daytime, although at reduced illumination levels.
Fig. 1. Sources of illumination for starlight imaging. The reflected ground signal must compete with scattered light. From spacecraft altitude, it must also compete with foreground airglow.
3. Quantitative Analysis of Illumination
A. Sky Brightness
To calculate image contrast, we must quantitatively divide the light of the night sky into its respective components. In this analysis, we follow and expand the calculations of O’Connell,3 who compared sky backgrounds in space and on the ground. Starting with absolute measurements of the sky background at good sites, one can subtract the astronomical and scattered light contributions and deduce the airglow fraction.
We use the clear night sky spectrum of Turnrose,4 who analyzed a large quantity of background data taken from Palomar Observatory. This spectrum is based on small aperture measurements that exclude bright stars. In the astronomical visual (V) band, Turnrose reports a typical sky brightness of V = 21.42 magnitudes per square arcsecond at the zenith. Walker,5 surveying the best astronomical sites in California, found the darkest sky to be 21.9 magnitudes per square arcsecond at Junipero Serra Peak. This is an exceptionally dark sky, and only 35% darker than Palomar. Höhn and Büchtemann6 measured the sky brightness from 400 to 800 nm in a range of weather conditions. Their clear sky brightness is consistent with Turnrose’s. We chose the Turnrose spectrum because of the large size and excellent calibration of its database.
Figure 2 shows sky brightness found by Turnrose4 in units of tenth magnitude stars per square degree. The magnitude scale is that used by O’Connell,3 where magnitude μλ = −2.5 log Sλ − 21.1, and Sλ is in ergs s−1 cm−2 Å−1, with μ5500 corresponding to the standard astronomical V. As with all magnitude scales, 1 magnitude step is a factor of approximately 2.5 in brightness, with brighter objects more negative in magnitude. O’Connell’s is an energy flux scale, in which an object with Sλ = constant radiates equal energy in each wavelength band.
Walker5 gave the dependence of sky brightness with elevation. Let a be a constant such that the flux falling on a horizontal surface (ergs cm−2 s−1) is
where I(t|cos Θ) is the intensity at optical depth t, measured from the top of the atmosphere, and Θ is the zenith angle. The total atmospheric optical depth is τ. For example, I(τ|−1) is the intensity at the ground looking toward the zenith in ergs cm−2 s−1 sr−1. For uniform illumination, with I(τ|cos Θ) constant over Θ, the factor a in Eq. (1) is 1. Walker’s data show that the sky brightens at low altitudes; integrating over elevation, his data imply a = 1.22.
Fig. 2. Spectrum of the night sky at Palomar from Turnrose.4 Prominent features in the spectrum include airglow emission at 557.7 nm, artificial mercury lines at 365.0/365.5 nm, and the rise in airglow intensity toward long wavelengths because of the OH bands.
We can thus compare starlight illumination with full moonlight or sunlight. From Turnrose4 the zenith sky at 5500 Å is the equivalent of 350 tenth magnitude stars per square degree. From Eq. (1) we found that the horizontal illumination is the equivalent of one star of magnitude −6.6 at the zenith. This can be compared to the full Moon (magnitude −12.7 before atmospheric absorption) or the Sun (−26.74 before atmospheric absorption). One finds that night skylight is 200 times fainter than full moonlight and 2 × 108 times fainter than sunlight.
B. Atmospheric Effects
In the optical 300–700-nm band, the atmosphere can be approximated by an absorbing layer of ozone atop a scattering layer in the lower atmosphere. The optical depths for the two components in clear conditions are shown in Table 1.7 These values do not include O2 and H2O absorption, which are small in this region. The u, b, υ, r bands selected for the table correspond roughly to the astronomical U, B, V, and R but have been adjusted to be contiguous. These bands were added to obtain broadband fluxes in the analysis below.
Table 1. Atmospheric Effects for the Visible Band
The diffuse flux at the zenith, I(τ|−1), can be decomposed into several components:
The right-hand terms on the first line are direct airglow, as attenuated by the atmosphere, and airglow scattered in the atmosphere. The terms on the second line are direct zodiacal light, and zodiacal light scattered in the atmosphere. On the third line are direct diffuse galactic light (dgl), and diffuse galactic light scattered in the atmosphere. The diffuse galactic light is starlight scattered by interstellar dust, and comprises the diffuse part of the starlight signal. Finally, the fourth line is starlight scattered in the atmosphere (direct starlight is not measured in a small aperture experiment and is therefore considered separately in Table 2).
Table 2. Calculated Breakdown of Zenith Sky Fluxa
aFluxes in units of μλ = 10 stars per square degree. Totals may disagree because of rounding.
C. Incident Fluxes
We extracted average incident fluxes for zodiacal light, diffuse galactic light, and starlight from the literature. We assumed the star numbers cited by Allen8 in calculating the stellar flux at 5500 Å and assumed a mean stellar spectrum as calculated by Sternberg and Ingham.9 Both the absolute numbers and spectrum of the stellar flux are consistent with spot measurements analyzed by Toller et al.10 These measurements, taken from the Pioneer probe at large heliocentric distance, sampled the stellar flux free from airglow or zodiacal light. The small flux from the diffuse galactic light was derived from data quoted by Lillie and Witt,11 and the spectrum was assumed to be the same as starlight. Zodiacal light was taken from tables of Levasseur-Regourd and Dumont,12 which are consistent to within ±10% with the partial sky maps of Kwon et al.,13 and was assumed to have the same spectrum as the Sun. To obtain a typical nighttime value, we averaged the zodiacal flux along the circle 90° from the Sun.
D. Scattered Contribution
Chamberlain14 gives an excellent discussion of the scattering of diffuse sources of sky light. The zenithal scattered flux at the ground depends on the angular distribution of the incident flux. We approximate the distribution of starlight, diffuse galactic light, and zodiacal light as isotropic, whereas the airglow brightens toward the horizon as the path length through the airglow layer increases. For a plane-parallel geometry, the airglow intensity incident on the atmosphere would scale as 1/(cos, Θ). For an atmosphere at finite height, the spherical geometry of the Earth comes into play. Tables giving the scattered flux from an isotropic source, and a layer at 100 km, can be found in Ashburn15 for the case of Rayleigh scattering. For moderate optical depths (less than 0.50), the approximations below are adequate:
if the illumination is constant over zenith angle Θ;
if the illumination varies as 1/cos Θ.
Here the incident flux Iinc(τabs|−1) is that incident on the lower atmosphere after ozone absorption at 20–30-km altitude. Iscat,diffuse is the scattered light, including Iscat,ag, Iscat,zl, and Iscat,dgl, arising from initially diffuse light. τscat is the total optical depth for scattering, and τabs is the total optical depth for absorption (so that τ = τscat + τabs). We assume that the ground albedo is a negligible source of scattered light [the zenith scattered light increases by approximately 20% for a ground albedo of 0.25 (Ref. 15)]; this approximation is reasonable because ground albedos are typically small at wavelengths for which scattering is important.
The scattered flux seen at the zenith from the ground is the same as the scattered flux seen at nadir above the scattering atmosphere. This conclusion can be derived from Chamberlain14 and the X and Y function tables of Chandrasekhar et al.16 These tables apply for isotropic scattering, but, for an azimuthally symmetric source, they are appropriate for Rayleigh scattering because it too is symmetric between the up and down hemispheres. In other words,
to a good approximation for moderate optical depth. Iscat includes both Iscat,diffuse and atmospherically scattered starlight Iscat,star.
E. Decomposition of the Night Sky
Applying the approximations above to data from Refs. 8 and 9 and 11–13, we can resolve Turnrose’s4 observed sky flux into its constituents. To obtain the breakdown of Table 2, the calculated quantities (c) were subtracted from the observed intensity (o) to obtain the derived airglow (d), then added to the calculated starlight (c) to obtain the total flux and the fraction of sky light contributed by airglow.
We conclude that airglow is the major contributor in these bands, yielding more than half of the light of the night sky. This differs from older analyses such as that of Roach and Gordon,2 because of their overestimate of zodiacal light compared with later investigators.12,13 Although some of the airglow is in discrete lines, much is in continuum and bands and cannot be spectrally segregated from ground reflection. For example, the airglow contribution in the υ band would drop only from 53% to 45% if the strong 557.7-nm O i line were filtered out.
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