11/5/12 Richard R. Auelmann. Low Light Imaging

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1 Low Light Imaging Nearly all Earth imaging from space is performed in daylight using reflected sunlight illumination in the visible near infrared (VNIR) band. Night imaging, if at all available, is performed using the thermal emission in either the mid wave infrared (MWIR) or long wave infrared (LWIR) bands. The MWIR is preferred when resolution is critical because of roughly two and a half times lower diffraction blur than LWIR for the same size aperture. But even for MWIR, the diffraction blur is roughly six times larger than passive VNIR with the same aperture. Here we consider the use of low light imaging devices, operating in a reflective band (visible, VNIR or SWIR), to image from space under conditions of twilight, moonlight, starlight and artificial illumination from building lights and street lamps. Such imaging devices have been widely used in tactical applications usually operating at a very low optical Q to achieve adequate signal to noise. But to use low light devices to image from space, one favors larger Q so as to maintain adequate spatial resolution with a limited aperture diameter. Also, haze radiance, which increases with sensor-to-target path length though the atmosphere has a larger relative noise impact than in tactical applications. While SWIR has coarser resolution than VNIR for a given Q and aperture diameter, it has lower haze radiance. Integration (exposure) times as long as one second may be necessary to achieve a minimum signal to noise ratio (SNR) under low ambient light conditions. This contrasts to a few milliseconds required for daylight imaging. Such long integration times rule out the use of line scan detectors with 64 or even 128 channels of time-delayed integration (TDI) because it would take too long to image even a small point target. So one must use a frame camera. A camera (with a 4K by 4K array) could be added to a commercial imaging satellite without too much difficulty. Long integration times pose two critical problems: image smear due to the orbital motion of the satellite and smear due to aim point drift. With one-second of integration there might be up to 20 pixels of image smear depending on the orbit and viewing geometry. Such smear, if uncompensated would completely blur the image. Use of multiple frames offer a possible solution to both smear problems, but at the expense of increased image processing and higher data transmission rates. Low Light Detector Technology Charged Couple Device (CCD) and Complimentary Metal Oxide Semiconductor (CMOS) are the two competing technologies. CMOS detector arrays cost less, use less power, support higher frame rates, are less susceptible to radiation, and offer greater functional flexibility. But CCDs offer lower noise, higher quantum efficiency and higher fill factors. So for low light level sensing from space, CCD technology is the first choice (Reference 1). At low signal levels, the readout noise associated charge couple devices (CCDs) tends to be a limiting factor. The tactical world has used intensified charged couple devices (ICCDs) for low light level imaging. They employ high intensifier gains (up to a million) that minimize the impact of readout noise and thereby allow low light imaging with reasonable integration times. ICCDs do however have drawbacks. Because of the high gains, image blooming due to local bright lights can be severe. Also the intensifier introduces additional image blur, which reduces the effective angular resolution. And finally they are complex devices as illustrated in Figure 1. 1 R. R. Auelmann

2 Figure 1. Intensified Charged Couple Device (ICCD) A new low light level imaging technology has emerged. This is the Electron Multiplying Charged Couple Device (EMCCD), which dispenses with the image intensifier (Reference 2). Unlike the ICCD, signal amplification takes place after the photons are sensed, but before readout noise is added. Such a device is depicted in Figure 2. It consists of an area array CCD, which collects the image. The image is then transferred to an identical covered CCD, which serves as the storage area while the next image is collected. Successive lines are shifted from the storage area to the shift register, which then moves the electrons to the gain register where the electrons are multiplied. The gain register uses voltages much higher than the normal clock voltages. Gains up to a thousand can be achieved. EMCCDs are much less prone to blooming under bright lights than ICCDs, and resolution is unaffected by the amplification process. And they are less complex. Figure 2. Typical EMCCD Configuration 2 R. R. Auelmann

3 Current EMCCD arrays include: Texas Instruments TC 253 SPD: 656 x 496, 7.4-µm pixels, Front-side illuminated Texas Instruments TC 285 SPD: 1004 x 1002, 8-µm pixels, Front-side illuminated Princeton Instruments PhotonMAX: 1024 x 1024, 13-µm pixels, Backside illuminated In what follows we quantify the levels of illumination required to provide a modest signal to noise ratios (SNR 3) using CCDs and EMCCDs as a function of scene illumination levels. The challenge for low light level imaging from space is two-fold: space telescopes are aperture limited, and the integration times are limited due to orbital motion constraints. Both factors work to limit the SNR. Models to compute the SNR for these devices are as follows: SNR = SNR = K Δρ R S t int 2 [ K( ρ R S + R H ) + Φ D ] t int + σ readout M K Δρ R S t int η 2 F M 2 2 [ K( ρ R S + R H ) + Φ D ] t int + σ readout CCD ICCD SNR = M K Δρ R S t int η 2 F M 2 2 [ K( ρ R S + R H ) + Φ D ] t int + σ readout K = detector responsivity Δρ = reflectance difference (for computing signal) ρ = reflectance for computing noise R S = scene radiance based on ρ =1 R H = haze radiance ( ) Φ D = dark noise electron flux σ readout = readout noise η F = excess noise factor (1.3 for EMCCD, 1.6 for ICCD) M = gain factor t int = integration time EMCCD Both the ICCD and the EMCCD employ a gain factor, but they are applied differently. For the ICCD the gain is applied in front of the detectors so it only amplifies the scene and background electrons. For the EMCCD gain is applied in the gain register so it also amplifies the dark noise. Thus detector cooling (to reduce the dark noise) is more important for an EMCCD than for either a CCD or ICCD. Also neither Clocking Induced Charge (CIC) nor Charge Transfer Efficiency (CTE) is included in this model even though these effects are more important than for a CCD especially when operated at high gains. So it makes sense to operate at a relatively low gain (M = 50). Both η F and M are equal to 1 for a CCD. 3 R. R. Auelmann

4 Ambient Low Light Conditions Most low light measurements are expressed in terms of photometric rather than radiometric units where the correspondence is given in Table 1. Photographers use light meters to measure illuminance (lux), which is the photometric unit (defined over the visible spectrum) that corresponds to the radiometric unit of irradiance (W/m 2 ). Pyranometers, on the other hand provide irradiance (W/m 2 ) measurements over a broad spectral band (typically 0.3µm to 3 µm) and are widely used by organizations such as the National Renewable Energy Laboratory for measuring solar collection efficiency. However, they are accurate to ~ 1 W/m 2, which is too coarse for even twilight illumination. TABLE 1. Photometric and Radiometric Units Radiometric Power, Watt (W) Photometric Luminous flux, lumen (lm) Power/unit area onto a surface Irradiance, W/m 2 Illuminance, lux=lm/m 2 Power/unit area from a surface Radiant exitance, W/m 2 Luminous emittance, lux=lm/m 2 Power/unit area/solid angle Radiance, W/m 2 /sr Luminance, lux/sr=lm/m 2 /sr Quantum efficiency, η QE Luminosity function, η eye Figure 1 is a plot of illuminance (lux) on the Earth surface as a function of the Sun elevation angle (reproduced from the RCA Electro-Optics Handbook) with values for twilight conditions, moonlight and starlight. Twilight illumination is due entirely to the diffuse reflection of the Sun off the atmosphere. There is no direct solar illumination because the Sun is below the horizon and is divided into three zones depending on the Sun angle: Civil twilight: between Sun at horizon (sunrise & sunset) and Sun 6 below horizon Nautical twilight: Sun between 6 and 12 below horizon Astronomical twilight: Sun between 12 and 18 (dawn & dusk) below horizon Twilight illuminance ranges from a high of roughly 800 lux at sunrise and sunset to between 1 and 10 lux with the Sun 6 below the horizon (civil twilight limit) to as low as lux when the Sun is 12 below the horizon. Lunar illuminance has a peak of 0.2 lux (for a full moon). By comparison average outdoor sunlight illuminance is on the order of 50,000 lux. 4 R. R. Auelmann

5 Figure 3. Illuminance on Under Various Conditions (Left-hand scale has units of lux) Conversion of Photometric to Radiometric Units Our interest is to assess the performance of satellite imaging systems operating in the visible (0.4 µm 0.7 µm), VNIR (0.4 µm 0.9 µm) and SWIR (0.9 µm 1.7 µm) bands under low light conditions. To accomplish this we require estimates of spectral irradiance (W/m 2 /µm). Computer codes, such as MODTRAN and MOSART, normally used to provide such information, appear incapable of providing meaningful values when the Sun is below the horizon. So we must rely on photometric data such as given in Figure 3. The challenge then is to convert such data to the radiometric units of interest (W/m 2 /µm). One problem is that most of the available data is in terms of illuminance rather than spectral illuminance. A further difficulty is that illuminace is weighted by the luminosity function, which accounts for the sensitivity of the human eye to visible light. The luminosity function V λ (Figure 4) is analogous to the quantum efficiency of a detector, except that it is set equal to unity at the most sensitive eye wavelength of µm (in the green band). At µm there are 683 lux per W/m 2. (conversely 1 lux = W/m 2 ). 5 R. R. Auelmann

6 Figure 4. Luminosity Function L F Depicts Response of the Human Eye If one had values for the spectral irradiance (over the visible spectrum) they could compute both the spectral illuminance and the illuminance using the following steps: (1) weight the spectral irradiances by the luminosity function, (2) multiply the weighted results by 683 to obtain the spectral illuminance, and (3) integrate over the visible band to obtain the illuminance. In our case we have values of illuminance (not spectral illuminance) and want to deduce the corresponding irradiance not just for the visible band, but also for the VNIR and SWIR bands. To accomplish this we also need estimates of the relative spectral irradiance for the bands of interest. Relative means we know the shape of the spectral irradiance but not absolute values. Since twilight illumination is due to diffuse solar illumination, I propose to use a code such as MOSART to compute the shape of the spectral irradiance when the Sun is at the horizon (the apparent limit of such codes) and normalize it at µm wavelength. The key assumption we make is that while the absolute values of the spectral irradiance change dramatically as the Sun dips below the horizon, its profile remains essentially unchanged. Let E λ denote the spectral irradiance and E Nλ denote the normalized value set equal to one at µm. The E λ (W /m 2 /µm) = C SF ( W /m 2 ) E Nλ ( µm 1 ) where C SF is the to be determined scale factor. Figure 5 shows E Nλ as a function of wavelength obtained using the MOSART code. It is normalized at the peak wavelength of the illuminosity function V λ (shown in red). The dashed line is the product V λ E Nλ. Let I be the measured illuminance (say from Figure 1) related to the spectral irradiance: I( lux) = 683 " lux % $ ' C # W /m 2 & SF 0.7 V λ 0.4 E Nλ dλ 6 R. R. Auelmann

7 Figure 5. Relative Irradiance for the Visible Band (MOSART Results) Values for the non-dimensional integrals (for the case shown in Figure 5) are 0.7 E Nλ dλ = V λ E Nλ dλ = It follows that C SF ( W /m 2 ) = I( lux) ( 683) ( ) = I( lux) which is the key result. With it we can compute the irradiance using 0.7 E( W /m 2 ) = C SF ( W /m 2 ) E Nλ 0.4 ( ) = I lux This establishes the relationship between luminance and irradiance for the visible spectrum. Photon Detection at all Three Reflective Bands. To extend these results to the VNIR and SWIR bands we employ the expanded relative spectral irradiance values in Figure 6. These dλ 7 R. R. Auelmann

8 are from the same MOSART run as Figure 5. And they are also normalized to one at µm. The blue curve (an extension of the scale from Figure 4) is the normalized value for units of W/m 2, while the green curve expresses the irradiance in terms of photon flux (ph/s) rather than Watts. The conversion factor is λ /( hc) where h is Planck s constant ( W s 2 ) and c is the speed of light ( m /s). For photon detectors, the green curve is more relevant. The main difference between the two curves is the strong bias to longer wavelengths for photon detection. So at µm there are ph /s per Watt. So 1 lux corresponds roughly to ph /s/m 2 for the visible spectrum (or equivalently lux per ph/s/m 2 ) Figure 6. Relative Irradiance (0.4 µm to 1.7 µm) Normalized at µm (MOSART result for diffuse solar radiation with Sun on horizon) Analogous to the previous results, we denote the equations using photons/sec with a star () superscript. The E λ ( ph /s/m 2 /µm) = C SF ( ph /s/m 2 ) E Nλ ( µm 1 ) ( ) = lux I lux $ ' & % ph /s/m 2 ) C SF ( 0.7 V λ 0.4 E Nλ dλ 8 R. R. Auelmann

9 Values for the non-dimensional integrals (the green curve) are 0.7 E Nλ dλ = V λ E Nλ dλ = It follows that and that C SF ( ph /s/m 2 ) = ( ) ( ) = I( lux) I lux ( ) E ( ph /s/m 2 ) = C SF ( ph /s/m 2 ) E Nλ = I( lux) dλ These results apply to the visible spectrum. For the VNIR and SWIR bands we need only compute the integrals 0.9 E Nλ dλ = for VNIR E Nλ dλ = for SWIR 0.9 It follows that for a given illuminance (lux), the VNIR and SWIR band values for E are 2.1 times and 1.8 times larger than for the visible band, respectively. Specifically, VNIR : E ph /s/m 2 SWIR : E ph /s/m 2 ( ) = I(lux) ( ) = I(lux) The remaining task of determining the radiance and signal to noise values is relatively straightforward. ( ) Scene Radiance. To evaluate sensor signal we require the radiance R S ph /s/m 2 /sr reflected off a horizontal scene and measured at the top of the atmosphere in the direction of the satellite sensor. For diffuse solar illumination the radiance is given by R S = C SF π λ 2 λ 1 E Nλ ( τ atm ) secψ dλ where τ atm is the atmospheric transmission along the local vertical (Figure 7) and ψ is the sensor zenith angle. At this point the scene reflectance is set equal to 1. (This is so values for reflectance 9 R. R. Auelmann

10 can be assigned later to compute the signal and the noise). As an aside, if the reflected scene radiance were due to direct (rather than the diffuse) solar radiance, it would be multiplied by the cosine of the zenith angle ψ sun. 1 Atmospheric Transmission Wavelength, λ (µm) Figure 7. Vertical Atmospheric Transmission Assume the sensor zenith angle is 40. The integrand E Nλ shown in Figure 8. The corresponding integrals are ( τ atm ) secψ for that case is λ 2 λ 1 E Nλ ( τ atm ) secψ dλ = Visible = VNIR = SWIR and with C SF ( ph /s/m 2 ) = I( lux)we have R S ( ph /s/m 2 /sr) = I(lux) Visible = I(lux) VNIR = I lux ( ) SWIR 10 R. R. Auelmann

11 Integrand Wavelength (µm) Figure 8. Integrand E Nλ ( τ atm ) secψ with 40 Sensor Zenith Angle Haze Radiance. We have no independent measurements for the haze radiance when the Sun is below the horizon. But we do have MOSART code comparisons between the haze radiance and diffuse solar radiance for each of the three bands when the Sun is at the horizon and the sensor zenith angle is 40. These ratios are: Visible : R H /R S = VNIR : R H /R S = SWIR : R H /R S = In lieu of any better information we assume that these ratios hold for twilight conditions as well. Scene and Haze Electron Flux. The signal electron flux (electrons/s) output of the detector is simply K R S where K is the sensor responsivity given by K = π $ λ mean & 4 % Q 2 ' ) η fill η QE τ opt ( (electrons photons 1 m 2 sr) where η fill is the aperture area fill factor, η QE is the quantum efficiency of the detector, and τ opt is the optical train transmission. Here we assume that both η QE and τ opt are constants. 11 R. R. Auelmann

12 Twilight Imaging. As seen from Figure 2, imaging under twilight conditions is more promising than imaging under moonlight or starlight conditions. Illuminance at civil twilight (Sun 6 below the horizon) is on the order of 1 lux. Here we compare the sensor performance in the visible, VNIR and SWIR bands. Signal to Noise Ratio. Table 2 list the integration times to achieve a SNR = 5 for the three bands using a CCD (and an EMCCD except for SWIR) at an illuminance of one lux. Note that a larger readout noise was assumed for the EMCCD than for the CCD. Somewhat surprisingly there is no advantage in using the EMCCD at this SNR level. Any advantage for an EMCCD shows up at significantly lower SNR levels. In all cases with the same Q the SWIR outperforms the VNIR, and the VNIR outperforms the visible. Indeed even a Q = 2 SWIR has a shorter integration time than a Q = 1 VNIR. The advantages of the shorter integration times must be weighed against the coarser angular resolution of the SWIR for the same aperture diameter, where effective angular resolution is defined by: α eff = λ /D mean Q RER λ mean D & ( 1+ 1 ' Q 1.35 Values for this parameter using a 1-m diameter aperture are also listed in the table. ) + 1/1.35 TABLE 2. Integration Times to Achieve SNR = 5 Spectral band visible VNIR SWIR Type Detector CCD EMCCD CCD EMCCD CCD CCD λ mean µm Q η QE τ opt K (e/ph)m 2 sr 1.16E E E E E E-13 Scene radiance, R S 7.68E E E E E E+16 Haze radiance, R H 2.72E E E E E E+15 Δρ ρ K ΔρR S elect/s K ρr S elect/s K R H elect/s M η F Dark noise flux, N D elect/s Readout noise, σ read Integration time, t int sec SNR Aperture diameter, D m Angular resolution, α eff µr R. R. Auelmann

13 Alternately we can compare the SNR achieved for a given integration time (say 1 sec) as shown in Figure 9, Except for the one case noted the results are for a CCD. The results are for the same input parameters as listed in Table 2. Possibly the most significant point is that a Q = 1 SWIR sensor may be able to provide useful imagery at illuminance levels as low a 0.1 lux, which levels are reached in the region of nautical twilight (Sun between 6 and 12 degrees below the horizon). Note the VNIR EMCCD has a small advantage over the VNIR CCD at SNR less than about 3. Figure 9. SNR as a Function of Illuminance for 1-sec Integration Time for a CCD. Orbital Smear and Aim Point Drift. A 1-sec integration time poses a problem with smear due to the orbital motion of the satellite, and random drift of the LOS off the aim point. The first effect is entirely predictable from the viewing geometry. The second effect is not. It is most efficient to image point targets at that the point of closest approach to the satellite ground track. Then for a frame camera there is only the azimuth smear component given by: Azimuth smear (pixels) = n 2 vt r tanψ where n is the number of detectors in the cross track direction r is the slant range, v is the satellite velocity, t is the integration time and ψ is the zenith angle. The geometry is shown in Figure 10. The LOS is maintained pointed at target aim point by rotating the LOS about ζ. The smear increases linearly as one departs from the line perpendicular to the LOS and passing through the aim point. 13 R. R. Auelmann

14 Figure 10. Image Shear Due to Azimuth Smear The orbit parameters for three commercial imaging satellites are listed in Table 3. Figure 11 shows the smear magnitudes for a 4 K by 4K detector array in these orbits as a function of the zenith angle for a 1-sec integration time. TABLE 3. Commercial Satellite Orbits Orbit Parameters units GEOEYE 1 WorldView 1 WorldView 2 Complete orbits per day, I Coverage repeat period, N solar days Residual, K Orbits per solar day, Q = I + (K/N) ` Orbits per repeat period, R = NQ Orbit period min Mean orbit rate, n rad/sec Semi-major axis, a km Orbit altitude km Sun synchronous inclination deg North-to-South Equatorial crossing hour 10:30AM 10:30AM 10:30AM Orbit speed km/sec Ground speed km/sec Earth rotation per orbit period (deg) Distance between passes km Daily step interval km R. R. Auelmann

15 Figure 11. Azimuth Smear As a Function of Zenith Angle for Three Orbits Multiple Frame Imaging. Here we consider the use of N sub frames (taken one after the other) in order to reduce the smear, incurred over a total 1-sec interval, by 1/N due to the orbital motion of the satellite and aim point drift. Multiple sub-frames can be effective in reducing both smear effects. A three sub-frame example is shown in Figure 12 (though 10 sub frames is more likely). Because each subframe is viewed from a slightly different angle the scenes skewed differently for each sub frame. So the raw sub frame data must be shifted to compensate for the skewing. Fortunately, the required shifts are predictable. Only after the skew corrections are made can the sub frame signals be summed. Preferably, the sub frame skew-shift operations and signal summation would be done on board (rather than on the ground) so as to minimize the amount data that has to be transmitted to the ground. 15 R. R. Auelmann

16 Figure 12. Multiple Sub Frames Shifted and Summed The low SNR of an individual sub frame is a concern, because readout noise is introduced when each sub frame signal is output, instead of just once when a single frame is used. At best, sub frame integration time varies inversely as the number of sub frames N. The aggregate SNR is equal to the N times the sub frame SNR because the noise adds randomly. Sub-frame and full frame SNRs are shown in Figure 13 for CCD and EMCCD VNIR detectors as a function of the N for a scene illuminance of 1 lux and a total integration time of 1 sec. The EMCCD dramatically outperforms the CCD primarily because it is far less sensitive to readout noise. Multiple frames can also reduce the impact of aim point drift. First recognize that the aim point smear in each sub frame is reduced by 1/N. But this benefit can only be realized if the frames are registered to one another. Because the aim point drift is random, the translations required to bring the sub frames into coincidence are not known a priori. The required translations are found by matching high intensity points or lines among sub frames. This would appear to be a problem because of the very low SNR for the individual sub-frames. However, these low values correspond to average scene reflectance. For a one or two kilometer point target area one would expect for find numerous scene artifacts with much higher reflectance difference values, so the prospects for frame co-registration are 16 R. R. Auelmann

17 much better than implied by the sub frame SNR values in Figure 13. One drawback is that such coregistration is probably best be done on the ground, which not only requires that all the sub frame date be transmitted to the ground sub frame data to the ground, but that the ground station be prepared to perform the co-registration SNR Number of Subframes Figure 13. SNR Dependence on Number of Sub Frames (1 Lux Illuminance) Use of a Terminator Orbit. There is minimal payoff in adding a low light level imaging capability to a satellite in a conventional (~10:30 AM) Sun synchronous orbit. Any payoff would be limited to winter months at latitudes above 60 N and below 60 S which normally are of low interest. And chances of adding coverage on the northbound leg would only be possible for scenarios where there is significant artificial illumination, because the required integration times become excessive for illumination levels less than one lux. Adding a low light level imaging capability to a satellite in a terminator Sun synchronous orbit with a 6 AM north to south crossing of the equator makes more sense. The Sun elevation angle for such an orbit is presented in Section 6. Presumably the conventional line scanner would be used for most daylight imaging, but would operate at somewhat lower SNR levels and with longer shadows because of the lower Sun elevation angles. Daylight imaging would be limited to a little over seven months (late spring, summer and early fall) of the northern hemisphere and only for a short period near Winter Solstice for the southern hemisphere. But during the imagery would be available on both southbound and northbound orbit legs. 17 R. R. Auelmann

18 An added low light level imager would provide a point target imaging capability during those months not covered by the conventional imager over the same regions and limited to the northern hemisphere and some southern latitudes that are in the twilight zone. This applies to both northbound and southbound legs. The relative robustness of the low light level imager in the terminator orbit stems for the fact that the coverage areas of most interest are always in twilight where the illumination is between1 to 10 lux. The relative merits of low light imaging for a conventional 10:30 AM and 6AM Sun synchronous orbits are summarized in Table 4. TABLE 4. Relative Merits of Conventional and Terminator Sun Synchronous Orbits 18 R. R. Auelmann

19 Imaging in Moonlight Here we develop a model for estimating the radiance of the Earth due to the reflected sunlight off the Moon (moonlight). One of the difficulties is that it is hard to estimate the solar reflectance off the Moon because it does not follow Lambert s law due to the large amount of dust on the Moon. The viewing geometry is depicted in Figure 14 where the lunar phase angle Φ is the angle between Sun and the Earth as viewed from the Moon. A full moon occurs when Φ = 0 and a quarter Moon occurs when Φ = 90. Let T denote the point on the Earth to be imaged. ψ moon is the zenith angle of the Moon as measured at T and ψ is the zenith angle of the imager. Note that imager need not be the Sun-Earth-Moon plane, though it is shown that way in Figure 14. Figure 14. Sun-Earth-Moon-Imager Geometry Spectral Irradiance. Fortunately there are radiometric (rather than photometric) measurements (Figure 15) of the lunar spectral irradiance at the surface of the Earth for Φ = 0 when transmitted through 1.5 masses of atmosphere (corresponding to ψ moon = The reference is: Per Knudsson and Mette Qwner-Petersen, Real-Time Rea-Sky Dual-Conjugate Adaptive Optics Experiment, SPIE Proc. Vol 6272 (2006). To apply these results to values of ψ moon, we back-out the atmospheric transmission loss, using transmission values listed in the Infrared Handbook. The results are listed in the last column of Table R. R. Auelmann

20 Figure 15. Lunar Spectral Irradiance for Lunar Zenith Angle (cited reference) TABLE 5. Scene Spectral Irradiance E(λ) λ E (ψ moon = 48 ) τ atm τ atm E(no Atm.) µm 0.001W/m 2 /µm ψ = 0 ψ = W/m 2 /µm R. R. Auelmann

21 Scene Radiance. Given the scene spectral irradiances E( λ) we can compute the scene radiance R T for different viewing geometries using the equation where R T = cosψ moon π λ 2 λ E(λ) τ atm ψ moon,λ hc λ 1 τ atm ( ψ moon,λ) τ atm ( ψ,λ) = τ atmo ( λ) ( ) τ atm ( ψ,λ) dλ [ ] secψ +secψ moon The scene radiance for the spectral band is 0.45 µm to 0.90 µm is listed in Table 6 for two sensor zenith angles and a range of lunar zenith angles. From previous daylight imaging analysis we have the ratios R H /R T for this spectral band and zenith angles computed from direct solar illumination. So to a first approximation, it seems reasonable to use this ratio to estimate the haze radiance R H for reflected sunlight off the Moon and on the Earth. TABLE 6. Scene and Haze Radiances for Selected Geometries (Full Moon) ψ ψ moon R H /R T R T R H deg deg photons/m 2 /sr E E E E E E E E E E E E E E E E E E E E E E E E E E E E+14 Signal to Noise Ratio. The general expression for SNR is SNR = Δρ K R T t int 2 2 [ K( ρ R T + R H ) + N D ] t int + σ readout + σ quantization where K is the sensor responsivity # K electrons m2 sr& % ( = π 2 λ m $ photons ' 4 Q η η τ 2 fill QE opt For the specified spectral band, and idealized conditions η fill, η QE, τ opt and Q all equal to one: 21 R. R. Auelmann

22 K = m 2 sr Then for the range of values in Table 6, K R H far exceeds the expected value of the dark noise N D. So for an integration time of 1 sec, the radiance noise dominates the readout and quantization noise, and the SNR can be approximated by SNR = Δρ R T ρ R T + R H K t int Results are plotted in Figure 16 for the radiance values in Table 6. Note the SNR scales as the square root of the responsivity K. Figure 16. Approximate SNR for a Full Moon ( Φ = 0) Figure 17 shows the reduction in scene irradiance as a function of the phase angle Φ. The brightness for a quarter moon ( Φ = 90 o ) is only 8.5% that of a full moon (Φ = 0 o ). The larger than expected falloff with lunar phase angle is attributed to the fact that the solar reflectance off the moon is not Lambertian. So to achieve the same SNR as shown in Figure 16, but for a quarter moon, the integration time would have to be almost 12 seconds. 22 R. R. Auelmann

23 Figure 17. Brightness Dependence Upon Phase Angle (from previous cited reference) References 1. Janesick, James, Dueling Detectors, Optical Engineering Magazine, February Denvir, D. J., and Conroy, E., Electron Multiplying CCD Technology: The new ICCD, Andor Technology Ltd, UK 23 R. R. Auelmann