Mon. Not. R. Astron. Soc. 365, 1295–1299 (2006)
doi:10.1111/j.1365-2966.2005.09821.x
Modelling artificial night-sky brightness with a polarized multiple
scattering radiative transfer computer code
Dana Xavier Kerola †
Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA
Accepted 2005 November 11. Received 2005 October 19; in original form 2005 May 17
ABSTRACT
As part of an ongoing investigation of radiative effects produced by hazy atmospheres, computational procedures have been developed for use in determining the brightening of the night
sky as a result of urban illumination. The downwardly and upwardly directed radiances of
multiply scattered light from an offending metropolitan source are computed by a straightforward Gauss–Seidel (G–S) iterative technique applied directly to the integrated form of
Chandrasekhar’s vectorized radiative transfer equation. Initial benchmark night-sky brightness tests of the present G–S model using fully consistent optical emission and extinction
input parameters yield very encouraging results when compared with the double scattering
treatment of Garstang, the only full-fledged previously available model.
Key words: radiative transfer – scattering – light pollution.
1 INTRODUCTION
A computer program originally developed by Kerola for use in determining the polarization and intensity of the sunlit sky (Kerola
1996), and later adapted to analyse scattered light from the coma
of Comet Hale–Bopp (Kerola & Larson 1999, 2001), has now been
modified for use in evaluating night-sky brightness enhancements
as a result of urban light pollution. The intensity and polarization
of the scattered light can be determined as functions of the chemical composition of the atmosphere (e.g. gas, dust, haze) and the
source-to-observer viewing geometry.
There has been heightened awareness recently of the accelerating
decrease in the darkness of the night sky. This current attempt to
construct a reliable methodology for a complete calculation of nightsky brightening from anthropogenic sources should be of interest
not only to workers in observational astronomy making photometric
gauges on their own of the impact of urban growth on ground-based
telescopic research campaigns, but to demographers and other environmental scientists who use satellite remote sensing information
in urban studies.
There have been very few previous efforts to compute urban sky
glow using detailed radiative transfer formulations to account for
the effects of atmospheric scattering. The most notable, and the one
that is an underpinning for all later reports, is the work of Garstang
(principally Garstang 1986, 1989). Garstang laid the foundation
by developing a tractable expression for use as a starting point in
quantifying the upward propagation of light from municipal sources.
His expression of the so-called ‘optical emission function’ is the
starting point for so many subsequent studies studies. Garstang’s
model treats in detail the geometry involved in calculating single
and double scattering by air molecules and atmospheric particulates.
There has been one other attempt by Yocke, Hogo & Henderson
(1986) to construct a far less elaborate single scattering model to
be used in a limited application to study the night-sky brightening
impact of a proposed nuclear waste repository near a national park.
In contrast to the scant availability of rigorous radiative transfer
models devoted to computing night-sky brightness, there have been a
considerable number of attempts, dating back to the 1970s, often regionally oriented, to assess artificial sky glow from very simple models, or from photometric measurements. Walker (1970, 1973) studied observatory sites across California and Arizona. Berry (1976)
and Pike (1976) each looked at Ontario, Canada, while Bertiau, de
Graeve & Treanor (1973) examined areas in Italy. All of these prior
works relied on population data to estimate urban optical emission.
Within the past few years there has been a flurry of activity, also from
Italy, by Cinzano et al. (2000) to resurrect and extend the model of
Garstang to generate satellite-calibrated nadir-viewing synthetic images to represent the impact of urban lighting. Cinzano (1994) has
also compiled an exhaustive bibliography of light pollution studies,
which will give the reader a complete glimpse of the heritage of
work in this field.
2 C R E AT I O N O F A P O L A R I Z E D M U LT I P L E
S C AT T E R I N G R A D I AT I V E T R A N S F E R M O D E L
FOR LIGHT POLLUTION STUDIES
E-mail: [email protected]
2.1 Radiative transfer equation including polarization
†Present affiliation: Northrop Grumman Electronic Systems, Space Sensors
Division, Azusa, CA 91702, USA.
Although it might be argued that it is overkill to include the polarization of light in a model of artificial night-sky brightness, where
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D. X. Kerola
the particular constituent (molecule or particle) under consideration
by
we are dealing with such intrinsically low illumination levels, and
where there is such a lack of highly accurate spectropolarimetric
measurements characterizing all contributions to diffuse night-sky
brightness, the present effort to solve the vectorized radiative transfer equation (RTE) is undertaken in the spirit of computational completeness.
Our starting point is the formulation of the RTE by Chandrasekhar
(1960). In its integro-differential form, it is
µdI(τ, µ, ϕ) = I(τ, µ, ϕ) − J(τ, µ, ϕ).
N=
n(z) = n 0 e−z/H .
(1)
where Il and Ir are the components of the polarized intensities oriented parallel and perpendicular to the scattering plane. The parameter U is associated with the angle between the maximal electric
field vibration direction and the plane of scattering, while V provides a measure of the degree of elliptical polarization of the wave.
Equation (1) quantifies the radiant energy transport in a specified
observation direction given by the pair (µ, ϕ), where µ is the cosine
of the zenith angle and ϕ is the azimuth angle. The change in the
intensity of the electromagnetic radiation along the view direction
caused by energy scattered into the observed beam is given by the
source term:
(σλ )AERO = Qπa 2 ,
0
−τ/µ0
+ ( /4)e
P(µ, µ0 , ϕ, ϕ0 )F 0 .
(3)
Here, P is a 4 × 4 phase matrix given by the sum of the separate
Rayleigh (molecular) and aerosol (particulate) matrices, is the
single scattering albedo, τ is the optical depth of the medium, µ 0 is
the cosine of the zenith angle of the source, and F 0 is the intensity
of the incident source (which, in our nighttime adaptation, will be
set to zero). The first term on the right-hand side of equation (3),
where the integration is carried out over all scattering directions,
is the contribution from multiple scattering. All of the information
regarding the net variation of the observed sky light with view direction is contained in P. It is not the intent here to recapitulate the
explicit details of the phase matrix elements. For that, the interested
reader needs to refer expressly to the work of Chandrasekhar (1960)
for the Rayleigh phase matrix, and to the book by Coulson (1988),
among others, for the scattering matrix for small particles. However,
some discussion ought to be made here concerning the physical importance of τ and the means by which we gauge its numerical value.
The total extinction optical depth given by the sum of absorption
and scattering can be written as
(τλ )EXT ≡ (τλ )ABS + (τλ )SCA .
(σλ )RAYL = (128π a /3λ )
5 6
4
m2 − 1
m2 + 1
2
.
(9)
With a limited elementary review now complete on how optical
depth is computed, let us proceed to show exactly where such an
important quantity appears in our light pollution model.
2.2 Description of the Gauss–Seidel iteration technique
for solving the radiative transfer equation
For an arbitrary set of linear equations, Gauss–Seidel (G–S) iteration
provides a powerful and relatively rapid means of obtaining an accurate solution to the algebraic system represented by equation (1).
Its main distinguishing feature is that the most recently calculated
value of an unknown from a previous step is then used to update
the value of the variable in the next step. In any application of G–S
iteration, an initial estimate of the unknowns must be made. One
notable example of the earliest use of G–S iteration in radiative
transfer problems is the work of Herman & Browning (1965). The
procedure followed here is patterned directly after their treatment.
The atmosphere is divided into a number of layers, each of optical
depth τ . Working with the integrated RTE written in central differencing notation, we begin to march downward from the top of
the atmosphere (TOA) where τ = 0, to the bottom, where τ = τ tot ,
the total optical depth through the entire atmosphere. At each level
(L) then, in our downward march, the individual Stokes parameters
are computed for a pre-selected set of observation directions (µ, ϕ).
The governing equation for the downward traverse is
(4)
Unless we are dealing with monochromatic light whose wavelength (λ) coincides with the spectral absorption of a particular
atmospheric gas, the total optical depth will be a result mainly from
scattering, for which we can then write
(τλ )SCA ≡ (τλ )RAYL + (τλ )AERO
= Nmolecules (σλ )RAYL + NAERO (σλ )AERO .
(8)
where Q is the efficiency factor for scattering. As a function of
the size parameter (x = 2πa/λ) and the particle’s real refractive
index (m), Q oscillates tremendously, reaching a value of Q = 2 for
large values of x. When looking at the other extreme, where x 1, we enter the Rayleigh scattering regime, for which the scattering
cross-section, derived by Stratton (1941) is
2π
P(µ, µ , ϕ, ϕ )I dµ dϕ J(τ, µ, ϕ) = (1/4π)−1
(7)
In equation (7), n 0 is the concentration at sea level (≈2.55 ×
1019 molecules cm−3 for the Rayleigh atmosphere) and H is the
scaleheight (≈8.4 km for dry air). Scaleheight is a measure of how
distended the air, or a layer of particles, is. The aerosol component
of the atmosphere would exhibit a similar behaviour, except that no
single values can be assigned to n 0 and H for particles, obviously
because of their high degree of variability, both spatially and temporally. Representative urban tropospheric particulate concentrations
would lie in the range ∼102 –104 cm−3 . In addition, aerosols are
polydisperse, i.e. there is a particle size distribution that complicates calculation of the aerosol scattering cross-section, and hence
the aerosol optical depth. For an ensemble of polydisperse particles,
with a particle effective radius, a, the scattering cross-section can
be written as
(2)
1
(6)
with the integral evaluated over the appropriate altitude range z.
An exponential decrease of the atmospheric constituent concentration with altitude is expected, giving
The specific intensity I is a four-element vector, the Stokes vector,
defined as
I = (Il + Ir , Il − Ir , U , V ),
n(z) dz
(5)
Here, N molecules (N AERO ) is the total number of molecules (aerosols)
along a line of sight contained in a 1-cm2 cross-sectional area. The
Rayleigh scattering cross-section and the scattering cross-section
for particles are (σ λ ) RAYL and (σ λ ) AERO , respectively. The column
number density (N) is related to the volumetric concentration (n) of
I L+1 (τ, µ, ϕ) = I L−1 (τ, µ, ϕ) e−2τ/µ
+ (1 − e−2τ/µ )J i (τ, µ, ϕ),
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(10)
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where J i is the average source term within layer i. When the bottom
of the atmosphere is reached, a completely analagous equation, with
µ changed to −µ, is written for the upward traverse:
to apply the inverse square law scaling to arrive at the sky brightness
for the remote observing site.
I L−1 (τ, µ, ϕ) = I L+1 (τ, µ, ϕ) e−2τ/µ
2.3 Execution of the Gauss–Seidel code: summary
of adjustable input parameters
+ (1 − e−2τ/µ )J i (τ, µ, ϕ).
(11)
The last computed downward intensity is used in the calculation
of the first upward intensity. In the course of development and testing
of the FORTRAN program, it became apparent that virtually constant
values of a sufficient set of diagnostic intensities were achieved
quite quickly for small optical depths, even though no convergence
criteria were specified per se.
With our procedures in place, we merely need to shut off any incoming sunlight (the F 0 term in equation 2), and instead trigger an
upwelling initial intensity generated by the municipality under consideration. We draw upon the formulation given by Garstang (1986,
1989) for the net ‘upwardly directed’ urban beams. He expresses
this emission function (lumens sterad−1 ) as
I (ψ) = (L P/2π)[2G(1 − F) cos ψ + 0.554Fψ 4 ].
(12)
Here, L is the per capita visible light output in lumens, P is the
population of the city or town being considered, G is the fraction
of the light that is isotropically reflected from the ground, F is the
fraction radiated directly into the upward hemisphere, and ψ is the
zenith angle of the up-bound rays. As depicted in Fig. 1, a fair
approximation to the brightness (b) perceived by an observer at
location x, situated on the ground receiving illumination from the
sky from a direction −ψ (antiparallel to direction ψ), will be
b(−ψ) = πI (−ψ)/x 2 .
(13)
The full multiple scattering calculations are performed twice:
first to generate the set of city-centre, ground-level downwardly
directed intensities, I (−ψ), and simultaneously to produce the set
of converged downwardly directed intensities for the very TOA. This
second set of values – call them I TOA (−ψ) – is now used for the
new initial input as the code is exercised again, this time to generate
the final set of downwardly directed intensities pertinent to what an
observer would detect at a remote site, a distance x from the centre of
the illuminating city source. During this second, complete multiple
scattering calculation, the elevation of the remote site is input as
well to adjust for the height dependency of the molecular optical
depth. Aerosol scattering existent over the immediate environs of the
remote site can now be included as well. Upon reaching convergence
of the intensities after this second cycle of execution, it is reasonable
Built into our radiative transfer model is the versatility to accommodate varied sets of fundamental input quantities characterizing the
meteorological conditions prevailing at the time of calculation of a
core city’s sky glow. The parametrization of the ambient weather
conditions (i.e. cloud cover, haze and dust content) and the topographic adjustments for the central city and observing site are
achieved quite concisely by means of simple input files. Furthermore, for a given case study, the primary invariant quantities the
program needs are the spectrally averaged wavelength of the urban
light sources, and, associated with that, the number of layers into
which to subdivide the model atmosphere. For a typical V-band situation (i.e. λ ≈ 0.55 µm), with a modest amount of aerosol content
(τ aerosol ≈ 0.05), it is sufficient to divide the atmosphere into 10
layers, and to perform the G–S iterations a total of eight times to
obtain a good solution. In addition, hardwired into the calculations
is the brightness of the natural sky-background; expressed in terms
of apparent visual magnitude (m), the model is currently operative
with a value m natural = 21.9 arcsec−2 . The natural sky brightness
expressed in nanolamberts (cf. Garstang 1986) becomes
bnatural = 34.08 exp{20.7233 − 0.92104m natural }.
With these essential quantities set, the code is executed to produce a tabulation of the sky brightness (expressed in mag arcsec−2 )
as a function of the view zenith angle for an observer situated either within the local environs of the extended metropolitan emitting
region, or at some distance x away. The model is presently constructed to ascertain the net brightness produced from two separate
cities. Work is underway to automate the procedures for summing
the apparent brightness as a function of view azimuth for sky glow
produced by an arbitrary number of offending municipal regions.
2.4 Initial benchmark comparison of the Gauss–Seidel model
against the Garstang model
As mentioned early on, the heritage of full-fledged radiative transfer
modelling of night-sky brightness as a result of urban light pollution
is firmly rooted in the work over a decade and a half ago of, principally, Garstang (1986, 1989). As alluded to earlier, new models are
now emerging which make use of Garstang’s formulation for the
Figure 1. Brightness (b) at remote site located distance X from offending urban nighttime light source of radius r, emitting with upward intensity.
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D. X. Kerola
upward emission function (as does the present G–S model). The
in-depth studies presented by Cinzano et al. have resulted in detailed comparisons of calculated upwardly directed optical radiances with calibrated Defense Meteorological Satellite Program
(DMSP) Operational Line-scan System (OLS) measurements. It
is a long-range goal to adapt our procedures to accomplish objectives similar to those of Cinzano and his collaborators. However, first it is important to emphasize some of the unique aspects already imbued in our operational G–S code, which to my
knowledge none of the previous models has been able to handle.
Foremost, the capacity of our model to treat the effects of multiple scattering and polarization should help advance the field. Furthermore, as a natural consequence of determining the source term
J per atmospheric layer as we march from the TOA to the ground in
the course of the G–S iterations, the overall characterization of the
absorption plus scattering, or extinction, produced by aerosols and
air molecules becomes more simplified than in the Garstang models.
One case in point is how the Garstang (1986) parameter ‘K’, which
measures the relative importance of particles versus molecules, is
expressed readily in the G–S model as the ratio of two optical depths
(i.e. K = τ aerosols /τ Rayleigh ). We then are able to specify the mixing
ratio fractions of particulates and molecules per layer in order to
suitably weight the contributions from aerosols versus the clear air
component.
Because of the differences in formulation and parametrization
between the G–S model and the previous models, direct one-to-one
comparisons of results are exceedingly difficult to make. However,
by way of performing an initial test of the soundness of our approach,
we try as closely as possible to use the same input parameters as
Garstang (1986) employed in his first model application, wherein
he calculated the brightness of the sky as a function of view zenith
angle as observed from Boulder, Colorado due solely to the lights
from Denver. Hence, we set P = 1.3 × 106 , L = 1000, x = 40 km,
F = 0.15 and G = 0.15, identical to what Garstang used. We adopt
τ AERO = 0.05 for the total particle vertical optical depth, which corresponds closely to his quantity K = 0.5. We have also replicated
fairly well Garstang’s adopted natural sky brightness variation with
view zenith angle, although ours is brighter at the larger angles,
explaining partly why our net observed sky glow is larger at the
steeper views. Fig. 2 shows the generally acceptable agreement using the two distinct approaches. The G–S model results are still
somewhat preliminary, but the underlying calculation of the phase
matrices and Stokes vector components has been extensively verified in the course of the previous adaptations of the code in the
Hale–Bopp coma studies (Kerola & Larson 1999, 2001) so that enhancements and refinements to the night-sky code can be made as
needed. One such necessary extension will be to account for changes
to the amount of scattering produced by a spherical atmosphere, a
condition which we obtain when considering large separation distances between observing site and urban light source.
Figure 2. Comparison of the DXK G–S model versus the Garstang (1986)
model for a test case of the view over Boulder, CO (negative view angle
looking away from source) due to the lights of Denver, 40 km distant. The
upper curve (solid) is the result of the G–S model; approximate Garstang
values are given as asterisks. The lower curve (dashed) is the natural skybackground calculated in the G–S model, with the Garstang approximate
values shown as small diamonds.
G–S iterations; this is a bonus. The reader must be aware however
that in the current code’s configuration, the degree of polarization
is only valid for the case of an observer at or near the centre of the
radiating city, looking upward. In continuing to enhance the usefulness of the program, it will be necessary to make a number of
non-trivial transformations for calculating the viewing angles and
scattering angles (thereby affecting polarization) appropriate for remote observatory sites.
None the less, some discussion right now is warranted regarding the potential use of polarization (quantified either via models
or observations) as a diagnostic of atmospheric particulate properties. Any modelling to date has dealt only with the contribution to
night-sky polarization from natural background sources. A major
reference compendium by Leinert et al. (1998) examines in detail
the compensation which has to be made to calibrate astronomical
observations of extraterrestrial objects as a consequence of how the
Earth’s lower atmosphere scatters light, e.g. from the air glow, the
Zodiacal light, the diffuse galactic light, and the integrated light from
stars. As cited in the work of Leinert et al. (1998), Staude (1975)
examined the effects of first-order Rayleigh and Mie scattering in
a spherical atmosphere by a uniform, unpolarized source (the air
glow), as well as for the Zodiacal light and Milky Way for different
viewing geometries.
Now, to give indications of what the present G–S model is predicting for polarization produced solely from ‘man-made’ sources,
we exercised the code for two contrasting situations of atmospheric
transparency: one with quite clear skies (τ MIE = 0.03), and a second
case for τ MIE = 0.20, signifying rather turbid air quality. The municipal beams are taken to be unpolarized, both in their direct upward
propagation, and the fractional parts (F and G) reflected from the
ground. Fig. 3 shows the degree of polarization (DP = −Q/I ) in
the vertical plane for an observer at the city centre at ground level as
a function of view zenith angle. Arbitrarily we chose the location to
be Tucson, Arizona, where the median elevation is about 2500 feet
above mean sea level, implying a Rayleigh optical depth for visible
3 I M P O RTA N C E O F I N C O R P O R AT I N G
P O L A R I Z AT I O N I N T O T H E M O D E L
The primary purpose of the present work is to describe a new method
to compute all orders of scattering of man-made light propagating
from a localized municipal source so that the resultant brightening of the sky can be ascertained. In adapting the original code,
it was judged easier to preserve the vectorized formulation of the
RTE as the nighttime problem was being contemplated. For the
nocturnal problem at hand, the calculation of the polarization at any
given level in the atmosphere occurs automatically in performing the
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Figure 3. Degree of polarization predicted by the G–S model for realistic
atmosphere conditions, each case having a Rayleigh optical depth τ RAYL =
0.088. The tropospheric aerosol ensemble has particle effective radius a =
1.0 µm and variance b = 0.25.
light τ RAYL = 0.088. A conservatively scattering urban aerosol with
a particle effective radius a = 1.0 µm and variance b = 0.25 was
selected. The results should be similar for any town or city, small or
large, possessing those particular optical depths and particle optical
characteristics. Not surprisingly, the trend is toward depolarization
as the haziness of the atmosphere increases. The fact that all of the
DP values are negative is also no surprise, because we are dealing
with an opposition effect (i.e. exact backscattering).
Until the code can be extended to perform the geometric transformations required to compute the polarized angular scattering for
an observer external to the emitting city, it would be premature to
prognosticate what the polarization ought to be at a remote observing site. One could surmise that the ‘anthropogenic’ DP values for
a distant observatory should still be relatively small, and perhaps
again negative (only in the vertical plane), because we might be
restricted this time to mainly small scattering angles (i.e. nearly
forward scattering), depending what the effective scattering layer
height is.
At this juncture, given the incomplete state of the art in trying to
completely model the polarization and intensity of the diffuse nighttime sky, it would be especially laudable to begin carefully coupling
the various model results for the natural plus artificial night-sky polarization. Certainly such an effort will be formidable, and undoubtedly require collaboration amongst many workers. Obviously there
are many variable, uncertain parameters to be taken into account.
In concert with such an analytical enterprise, if simultaneous highly
sensitive photopolarimetric observations from within cities, and in
deserts and on mountain tops can be had, it might just be possible
to retrieve some basic physical properties of the small particulates
which comprise the nocturnal lower atmosphere environment. In
that spirit, let me transition to the concluding section to suggest a
few additional areas where attention needs to be paid.
1299
parent is the need for a consistent of ‘observables’ with which to
begin correlating past and present light pollution investigations. The
recent studies of Cinzano et al. and Elvidge et al. (1997), which use
the calibrated DMSP data, are vital and encouraging. Without such
important validation via satellite observations of the visible radiance measured looking down at a given locale, on a given night,
it is almost pointless for atmospheric radiative transfer modellers,
regardless of the level of sophistication of the techniques they are to
employ, to attempt any kind of meaningful sensitivity study of the
effect of various model parameters on night-sky brightness. However, with reliable radiance measurements from the DMSP OLS,
a re-examination can be made, for example, of Garstang’s optical
emission parameters L, F and G.
Although the main goal in this paper has been the communication
of a methodology for studying artificial night-sky glow using the
complete treatment of polarized multiple scattering, an immediate
next step is the use of the G–S model for prediction of at-sensor
radiances. Such a project to apply the ‘forward’ radiative transfer
model for purposes of performing atmospheric correction of the
DMSP OLS measurements will require further enhancements to
the existing G–S code in order to treat, among other things, the
instrumental spectral response. There will also have to be some
means developed (perhaps through ground-level measurements) for
obtaining a more consistent, credible set of radiative transfer model
input parameters. If these sorts of considerations can be adequately
addressed, significant progress can be made in further stimulating
the field of light pollution modelling.
AC K N OW L E D G M E N T S
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4 N E C E S S A RY D I R E C T I O N S F O R F U T U R E
LIGHT POLLUTION STUDIES
Like the inordinately strong optical flux density produced by the
greater Las Vegas, Nevada metropolitan area, what is glaringly ap-
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2005 The Author. Journal compilation This paper has been typeset from a TEX/LATEX file prepared by the author.