Predicting Moonlight Brightness for Night Landscape Photography

The brightness of moonlight illumination is quantified and a method presented for calculating exposure times required for use of moonlight as a photographic light source. Results are compared against a previous method and are found superior. Test data validating the results are presented. The method is recommended for night photography under moonlight conditions.

Keywords: moonlight brightness, luminance value, exposure value, night photography.

Landscape photography by moonlight is a growing field of interest, as it results in unusual images that often have a surreal quality. The techniques of photography by moonlight are usually characterized by a trial-and-error approach, since the amount of moonlight illumination is poorly understood, if at all, by most practitioners. This paper brings together knowledge of moonlight obtained from the scientific literature, as well as the authorís experience of photographing by moonlight for over a period of 30 years. The method proposed provides accurate predictions of moonlight illumination that can be used to predict appropriate camera settings for achieving desired photographic results.

The moon reflects sunlight onto the earth. The amount of this reflected light varies over time due to a number of factors, including: lunar phase; the distances between the sun, moon and earth; and whether the moon is itself illuminated by grazing light or light that comes from a direction more parallel to the line of view from the earth. Other factors include the height of the moon in the sky as seen by an observer on the earth, and attenuation effects from scattering of in-falling moonlight by water vapor, aerosols, and dust in the atmosphere.

Light in-falling onto an object is termed illuminance. Krisciunas and Schaefer 1991 gives the illuminance of the Moon at the outer edge of the earthís atmosphere as

where I* is in units of foot-candles, and m is the astronomical V (visual) magnitude[1] of the moon.

From Allen 1976, p. 144, we have the Lunar Phase Law (V magnitude brightness vs. phase):

where a is the lunar phase angle, defined as the angular distance (in degrees) between the Earth and the Sun as seen from the Moon. The sign convention is taken where a has positive values during the moonís waxing phases, and negative values during the waning phases. It is noted the linear term of this equation as provided in Allen 1976 does not include the absolute value function, and the present work adopts the form of equation (2) per Krisciunas and Schaefer 1991. This formula closely approximates the lunar illumination except right around Full Moon, where the Moon brightens significantly due to the ìopposition effectî. For |a|<7ƒ, the combined effects of collimated inter-particle backscatter and the lack of cross-light shadows in depressions and behind rocks on the lunar surface increase the brightness by about 35% above that predicted by equation (2), according to Krisciunas and Schaefer 1991. Palmer indicates that greater amounts of brightening have been observed under conditions of very small phase angles, and attributes this to retro-reflections in small vitreous particles present in meteor crater ejecta. However, brightening nearest opposition is limited by the earthís shadow: penumbral eclipse begins with phase angles of about 1.5 degrees, and the umbra is reached at about 0.9 degrees. Total eclipse occurs when the moon is at phase angle of less than about 0.4 degrees. For the current work, correction for opposition effect is made by increasing the predicted lunar illumination using a factor that varies linearly from 1.00 at |a|=7 to 1.35 at a=0, with a test for the presence of eclipse.

Following the techniques available in Meeus 1991 (and updated in 1998), the phase of the moon is calculated in terms of the fractional illumination of the lunar disk, P. This task is greatly eased by making use of the BASIC code provided in Duffett-Smith 1990. The lunar phase angle, a, can then be obtained using the formula from Meeus 1998, page 345:

Combining equations (1) and (2) gives the atmospheric edge lunar illumination (foot-candles) in terms of lunar phase angle (degrees)

Moonlight is scattered somewhat as it transits the atmosphere, which reduces the illumination measurably. Krisciunas and Schaefer 1991 gives reduced illumination as

where k is the ìextinction coefficientî in units of magnitudes per air mass, and X_m is the optical path length traversed by the in-falling Moonlight, measured in air masses. One unit air mass is traversed on a path from the earth surface to a point on the outer edge of the atmosphere at the zenith. Green 1992 states the air mass is often approximated as the secant of the zenith distance (angular distance between the moon and the point in the sky directly above the observer) for any direction not near the horizon, and recommends Rozenbergís (1966) formula for air mass:

Equation (6) goes as sec(Z) when far from the horizon, but is limited to 40 air masses on the horizon.

The value of k, the ìextinction coefficientî, depends on the transparency of the atmosphere, which varies according to existing conditions. It is inferred from Krisciunas and Schaefer 1991 that the value of k can vary from 0.15 with very clear air to values at least as large as 0.24 when the air is hazy. A measured value of 0.172 mag/air mass is given for a particular night at the 2800-m level of Mauna Kea on the Island of Hawaii. Allen 1976, page 131, provides a table of the fractional transmission of the atmosphere to total radiant energy through dust-free air, with cases based on water vapor content of the atmosphere. The values in this table are approximated using equation (5), with a specific k value assigned to each case. The value of k for each case is selected to minimize the RSS deviation between the table values and those obtained using equation (5). When this is performed, the following values of k are obtained:

Water vapor in cm of precipitable water per unit air mass	0.0	0.5	1.0	2.0	3.0	4.0
Atmospheric Extinction Coefficient, k (mag/air mass)	0.109	0.161	0.176	0.183	0.190	0.198

This gives confidence that the value of k due to the presence of water vapor alone is in the range from about 0.11 to 0.20.

Green 1992 cites Hayes and Latham 1975 as a source for understanding the ìthree sources of extinction in the earth's atmosphere that must be considered when dealing with ground-based astronomical photometry: molecular absorption, Rayleigh scattering by molecules, and aerosol scattering.î

At wavelength lambda = 510 nm, which is the peak spectral response for the rods of the human eye used in night vision, molecular absorption (which occurs in spectral lines and bands) is rather negligible, although for altitudes under 10 deg, ozone can cause extinction > 0.01 magnitude per air mass. We adopt Schaefer's (1992) value A_oz = 0.016 magnitudes per air mass for the small ozone component contributing to atmospheric extinction.

Rayleigh scattering by air molecules can be represented by the following equation (after Hayes and Latham 1975 for lambda = 510 nm = 0.51 micron):

Aerosol scattering is due to particulates including dust, water droplets, and manmade pollutants, and the extinction due to this is generally given by the formula

where the scale height, H, is usually taken as 1.5 km (Hayes and Latham 1975; however, this may vary by a factor of 2 on any given night) and lambda is the observed wavelength (in microns). The quantity alpha_o varies from site to site; Tueg et al. (1977) and Hayes and Latham 1975 find typical values near alpha_o = 0.9, but we adopt alpha_o= 1.3 after Angstroem (1961) and Schaefer (1992). Schaefer remarks that the variation in A_o îis rabid . . . because the aerosol component varies greatly on all time scales". Volcanic aerosols, in particular, are highly variable from site to site and year to year. . . . I adopt A_o = 0.05 as an average value. Thus, we will take the extinction due to aerosols for the human eye as

so that for elevations near sea level, A_aer is about 0.12 magnitudes per air mass.

Assuming an elevation of 0.1 km (the elevation of the authorís home), and applying the formulas provided in Green 1992, a composite atmospheric extinction coefficient, k, is obtained:

Assuming a value of 0.20 for the extinction coefficient, k, and applying equation (6) to equation (5) gives the atmospheric attenuated lunar illumination (foot-candles) as

The following figure presents a family of curves representing lunar illumination at the earthís surface calculated using equations (4) and (10), k=0.20, and zenith angle increments of 15 degrees. For simplicity, the effect of eclipse on illumination at small phase angles has been ignored. With this consideration, the peak illumination at a=0, k=0.2, and z=0 is 0.0327 foot-candles.

As a check on the programming of the equations into the spreadsheet used to generate the plotted values, and on selection of value for the extinction coefficient, the value of I was calculated for a full moon on the zenith, which was then compared against a value obtained from Schlyter. The values (taken prior to correction for the opposition effect) correlate at 0.0248 foot-candles at k=0.173 magnitudes per air mass.

One final correction to apply accounts for the variation in illumination arising from changing distance between the sun, moon, and earth. Illumination from a source decreases with the inverse square of the distance from that source. Thus the illumination from the sun is greatest at perihelion and least at aphelion. Similarly, the illumination from the moon (assuming constant solar illumination) is greatest at perigee and least at apogee. The discussion of illumination thus far has assumed the earth at 1.00 Astronomical Unit (AU) distance from the sun, and the moon at the mean distance from the earth of 384401 kilometers (km). The distance from the sun to the earth varies from 0.9833 AU at perihelion to 1.0167 AU at aphelion (Allen 1976 p.162). The distance from the moon to the earth varies widely due to the complex lunar orbit, with the extremes of the range being 356400 km at perigee and 406700 km at apogee (Allen 1976 p.147). The variations in brightness due to these effects are

The range of variation in lunar illumination is thus 6.9% for variation in sun distance and 30% for variation in moon distance. While the sun distance variation results in a photographically negligible change in illumination, the moon distance variation results in a change that is slightly more than one-third stop of light, which has a noticeable effect when slide films are used.

The illumination from a ìnominalî full moon, under assumed conditions of a=3 degrees, k=0.2, z=15 degrees, and with the sun-earth and earth-moon distances at average values, is I=0.0269 foot-candles. Under conditions favorable for maximum brightness, assuming a=0 degrees, k=0.11, z=0 degrees, and with the earth at perihelion and moon at perigee, and neglecting the presence of eclipse and retro-reflective backscatter, a peak value of I=0.0426 foot-candles is calculated. Palmer has measured a peak value at the edge of penumbral eclipse, of 9.2 lux, or 0.855 foot-candles, which gives a measure of the effect of the retro-reflective back scatter (approximately 20 times, or 4.33 stops brighter!).

Having the incident lunar illumination in units of foot-candles is a good start, however for the purposes of photography we need to know the luminance of the surface being photographed, in units of candles per square foot (or candelas per square meter, if we are referring to SI units). We assume that incident illumination is diffusely reflected by the surface it falls upon, and it is that reflected light that is captured in the camera, either on film or on a CCD.

From Adams 1981 (page 12 footnote)[2], diffuse reflected luminance is obtained from incident luminance using

where E is the surface brightness in candles-per-square-foot, I is the incident illuminance in foot-candles, and r is the surface reflectance.

A good approach to selecting camera settings is provided in Gordon 1985. Image brightness, or the quantity of light falling on the focal plane of a camera, is related to subject brightness, relative lens aperture (or f-stop), and in-path optical density by

where B is the focal plane image brightness, E is the subject surface brightness (candles per square foot), r is the lens focal ratio (focal length divided by aperture), and D is the in-line density of the optical path. In the language of photography, the argument 10^D is often called ìfilter factorî.

Photographic emulsion responds to total exposure, or intensity times time. The relative response of the film to light is termed the film speed. The exposure that results in a film optical density of 1.375 density units on transparency film, or 1.00 density units on monochrome negative film, is assumed to be the desired quantity. Gordon gives the relation that provides this desired film density as

where S is the ISO rating of the film speed (e.g. 64 for Kodachrome), and T is the exposure time in seconds. Thus, under the ìnominalî full moon conditions discussed above (I=0.0269 foot-candles) and an 18% reflectance surface, applying equation (12) gives E=0.00154 candles per square foot. Assuming a film speed of ISO 100, a lens aperture of f/1.4, and no significant attenuators in the light path (i.e. no ëfilter factorî), we can apply equation (14) to obtain the required exposure time of 12.7 seconds. In practical application, a correction factor for film reciprocity failure would be applied since the required exposure is longer than 0.1 second. If we call this correction factor c, the final formula for exposure time fully corrected for film reciprocity effects is

The exact value of c to apply depends on the response of individual film emulsion types, and sometimes differs between different batches of the same emulsion. In most cases, tests must be performed on a given film to determine appropriate values of c for that film, as film manufacturers seldom make such information available. Testing of Fuji Provia 100F performed by the author indicates the correction factor for this film is

Combined with equations (3), (4), (10), and (11), equation (15) gives us the full set of information needed to calculate the required exposure to obtain a daylight-like rendering of a landscape scene under moonlight conditions, given the lunar phase in terms of fractional illumination. Experience indicates that artistic renderings that provide ìthe feeling of moonlightî typically require an under-exposure by one-half to one stop, with a one-and-one-half stop underexposure being the useful limit with transparency films.

Another approach to finding exposure time uses a pair of photographic tools known as the Luminance Value and Exposure Value scales. These scales provide linear functions for the logarithmic relationships between light exposure and response of photographic emulsions. Values of the Luminance Value scale, also called Light Values, or LV, increases by one unit for every doubling of the surface brightness of an object being photographed. The conventional calibration of the LV scale defines LV=10 units as equivalent to a surface brightness of 10 candles per square foot. On the other hand, the EV scale provides sets of camera settings (shutter speed and lens relative aperture, or f-stop) that give equivalent exposure at the focal plane. For example, exposures of 1/30 at f/16, 1/15 at f/22, 1/125 at f/8, etc, all give equivalent exposure, and correspond to EV 13. The base of the EV scale is defined at EV 0, which corresponds to a 1 second exposure at f/1.0. The relationship between the EV and LV scales is given by

In practice, tables of LV, EV, and the adjustment for film speed (EIA) are consulted to avoid having to work the logarithms at the point of application.

Using this method, the EV value required to photograph a given scene is the LV value of an 18% reflectance surface brightness in that scene when a film speed of ISO 100 is employed. A lower film speed requires a lower EV value, and a higher film speed requires a higher EV value, for the same scene surface brightness. All factors of two involved in any of the parameters are converted to a one-unit increment in either LV or EV, thus simplifying the complex logarithmic relationships into simple linear terms.

The relationship between surface brightness, in candles per square foot, and LV units is:

Combining equations (12) and (18) gives the photographic Light Value index in terms of incident light, in units of foot-candles, and surface reflectance:

As an example of the application of equations (16) and (19), return to the ìnominal full moonî situation with I=0.0269 foot-candles. With a Kodak 18% reflectance standard gray card positioned facing the moon, the value of LV becomes

Thus, using ISO 100 speed film, an EV value of -2.66 is obtained. An f-stop of f/1.4 has an EV_A of 1.0, and the required EV_T is EV_T =-2.66-1.0=-3.66. This corresponds to an exposure time of 12.7 seconds, neglecting any reciprocity failure effects of the photographic emulsion. A similar calculation for f/5.6 gives 204 seconds.

Until quite recently, LunarLight Photography used a method of obtaining subject surface brightness under moonlight conditions that has been based on test data accumulated over a period in excess of 30 years. This data was obtained from making trial exposures under field conditions, and suffered from lack of rigor or control. Lacking anything better, the data served as the basis for an exposure-prediction program called NightLandscape, which has evolved over the years. The most recent version of this program, NightLandscape v4.0d, calculates lunar position and fraction illumination using the BASIC routines provided in Duffett-Smith 1990, and then calculates lunar illumination using data table lookups. Results from this paper confirms that substantial error existed in the subject brightness obtained by the old method. The following figure illustrates the predicted surface brightness of a Kodak 18% gray card facing in-falling moonlight, as determined by both methods.

The old method had been calibrated around full moon, and was known to be less accurate beyond a few days on either side of full moon. The loss of accuracy results from lack of opportunities to make trial exposures beyond the few days during each lunation when there is sufficient light to take pictures with reasonably short exposure times (less than ten minutes at moderate lens apertures). The pronounced peak in the new prediction curve due to the opposition effect was not accounted for in the old method. In fact, attempts to fit an interpolation curve to points very close to full moon produced significant underexposure of the film on days either side of full moon. Improved understanding of the opposition effect has provided insight into this problem, which provided no end of frustration to any practitioner who attempted to use the old predictive interpolation curve, only to end up with underexposed film. The new prediction method is being incorporated into a revision of the NightLandscape program, which will be called NightLandscape v5.0.

Testing of the new method began in early October 2003 on LunarLight Photography Film Roll Number 2001-10-02. The material used was Fuji Provia 100F transparency film, type RDP III. Exposures were made of a Kodak 18% gray card under moonlight and daylight conditions on two separate days. The daylight exposures were made using a Minolta SRT 102 camera equipped with a Minolta MC-series 135mm lens. Subject luminance was measured using the cameraís internal light meter. The moonlight exposures were made using the same equipment, and the exposure times were calculated using the new prediction method. Resulting optical density of the film was measured using a Pentax spot meter and a light table. All density measurements were made to provide relative densities, so as to eliminate any absolute bias in the test measurement.

Normal daylight exposure of an 18% gray card, using well-calibrated equipment, should result in a film optical density of 1.375 over the film base density for transparency films. The daylight exposures that served as controls in this test showed optical densities within 0.022 of the desired nominal value. (Typical film sensitivity is about 0.5 density per stop of exposure). For all tests, the air was clear, and a k value of 0.2 was assumed. Results from the moonlight tests were as follows:

Parameter	Test 1	Test 2	Test 3	Test 4
Date	4 Oct 2003	12 Oct 2003
Time (UT)	0400	0800
% Illumination	.748	.963
Phase Angle, a	60.3	22.3
R_E, AU	1.0001	0.9981
R_M, km	383375	404346
z, degrees	56.1	24.8
I, foot-candles	0.00473	0.0125
LV	-5.17	-3.76
Measured Density, D_meas	1.178	1.233
Departure from optimal, D_meas-D₀	-0.197	-0.142
Error, f-stops	-1/3 stop	-1/4 stop

A fair measure of the departure from nominal film density found in this test may be the result of assuming a value of k, the atmospheric extinction coefficient, of 0.20 at an atmosphere near sea level, where Green 1992 indicates a value of 0.27 is more appropriate. This experiment will be continued using this consideration in the near future. But even so, an error of ±1/3 stop is far better than the error thought to exist in the body of test data obtained by trial-and-error that served as the basis of the old method. This range of error is typical for camera light meters and film density variations arising from calibration tolerance in film processing labs. Thus the new method provides results well within what is considered acceptable for general photography, and is recommended to practitioners of moonlight photography.

Duffett-Smith, Peter, ìAstronomy with your personal computerî 2nd. Ed., Cambridge University Press, 1990

Green, Daniel, ìMagnitude Corrections for Atmospheric Extinctionî, International Comet Quarterly, Vol. 14, July 1992, pages 55-59.

Krisciunas, K. and Schaefer B.E., ìA model of the brightness of moonlightî, Publ. Astron. Soc. Pacif. 103 (667), 1033-1039 (1991)

Meeus, Jean, ìAstronomical Formulae for Calculatorsî, William-Bell, Richmond, Virginia, 1979.

Meeus, Jean, ìAstronomical Algorithmsî 2nd. Ed., William-Bell, Richmond, Virginia, 1998.

Palmer, J, ìMy Opposition to Lunacyî, private web page with URL http://www.optics.arizona,edu/Palmer/moon/lunacy.htm. Dr Palmer, of University of Arizona, Tucson, indicates he has measured lunar illuminance on the earth surface as high as 9.2 lux at the edge of penumbral eclipse under very favorable conditions of low humidity and atmospheric clarity.

P. Schlyter, ìRadiometry and photometry in astronomyî, private web page with URL http://www.stjarnhimlen.se/comp/radfaq.html. Mr. Schlyter apparently obtained the value of I=0.267 lux from one of the references cited at the end of his web page, however he does not indicate which one.

[1] It is customary in astronomical photometry to break the spectrum of light into the frequency bands UBVRI, or Ultraviolet, Blue, Visual, Red, Infrared, using band-pass filters. Landscape photography is usually concerned only with the Visual band

LunarLight Photography Technical Information

Predicting Moonlight Brightness for Night Landscape Photography