Abstract

The radiant-intensity function of a ground-based light source, otherwise called here the city emission function (CEF), is a pivotal modulator of night-sky radiance and is one of the key factors affecting light pollution propagation in a nighttime environment. It is difficult or rather impossible to compute a CEF from databases that are usually incomplete in their description of artificial sources other than public lighting. However, we have developed an indirect remote-sensing method to retrieve the CEF from sky-brightness measurements made at a local meridian that intersects a horizontal circle at the azimuthal position of a city or town. The inversion algorithm is validated for sensitivity and specificity in the reproduction of the initial emission functions, and demonstrates that the solution model succeeds in reconstructing different types of CEFs. The numerical inversion of the integral equation was successful even under extreme conditions, including elevated ground reflectance or zero uplight cases. The good numerical convergence and applicability of the model to experimental data is demonstrated by processing the radiance patterns obtained under natural conditions. The model provides a great opportunity for systematic retrievals of CEF needed by astronomers, physicists, environmental designers, city and suburban planners, as well as biologists and illumination engineers who deal with public lighting systems and must recognize the potentially adverse effects of night-lighting characteristics on the environment.

© 2017 Optical Society of America

1. INTRODUCTION

Light pollution, unwanted light, has proven to be one of the most critical environmental concerns in recent decades due to increasing nighttime lighting levels, especially in urban commercial development globally. The beams emitted from artificial light sources employed to illuminate urban spaces and structures inherently produce diffuse light, which spreads in all directions over a larger area and causes unwanted illumination of desired dark space. This diffuse light is alternatively referred to as a secondary radiation which occurs due to scattering processes in the Earth’s atmosphere [1,2].

The transportation of photons from a light source to a distant observer (or measuring device) is not straightforward; the signal recorded is an amalgam of many beams traveling along different optical paths (see, e.g., Fig. 1 in [3]). For instance, two beams emitted at low and high elevation angles can both contribute to the signal entering a sensor within a small solid angle, but these beams travel along different optical paths. In particular, the scattering efficiency in a turbid atmosphere might depend strongly on the scattering angle [4]. However, the intensity of light would also drop rapidly as the optical path length increases because the probability of absorption (or scattering) increases with the distance the photons travel in the atmosphere. In addition, the flux density of electromagnetic radiation emitted from a ground-based light source typically depends on the luminaire (light emitter) properties [5], lighting design (most typically shielding, [6,7]), and also ambient environment—for instance, the optical properties of reflecting surfaces and obstacles such as buildings and trees. Therefore, the total angular emission function from whole-city light sources is difficult to determine even if the inventory of light sources and their parameterization are generally known.

Certainly, the diffuse light field in a nocturnal environment cannot be predicted without knowing the bulk radiant intensity function [8] and delineating its exact shape or at least its nature [9,10]. It is well established that the radiant intensity for diffusely reflected light is a decreasing function of zenith angle, while the emission from poorly shielded lights can show a dominant peak at low elevation angles. The form of emission function makes the diffuse light of a night sky more or less intense far from a city or town. For instance, cosine radiators with no direct uplight efficiently illuminate the city surroundings, while the distant places would remain dark. In contrast, light emission above the horizon can cause light pollution to propagate for long distances. Consequently, there is no doubt that the intensity distribution of upwardly directed urban beams is a crucial part of any light pollution model. In analogy with our previous papers (e.g., [3,10,11]) we will use the term city emission function (CEF) throughout this paper. Emissions from a finite dimensional ground surface (bulk CEF) are obtained here as a cosine projected ground radiance. CEF can rarely be obtained directly because of many limitations—for instance, satellite data provided by Visible Infrared Imaging Radiometer Suite (VIIRS) are usually available only for a closed, bounded set of emission angles and should be subject to corrections due to adverse atmospheric effects such as intensity decay along the beam direction or signal distortion in haze or thin clouds [12]. The correction functions depend on an overall delineation of the geographical distribution of the atmospheric optical thickness, spatial distribution of atmospheric constituents, and clouds in the Earth’s atmosphere, which are all often unknown or uncertain at the given discrete points in time. Some effort has also been expended toward aerial surveys of CEF [13,14] that are, however, expensive when carried out routinely and are designed for position-specific monitoring rather than for obtaining bulk CEF.

Most of the methods of measurements are indirect, as experimental data normally involves properties pertaining to the secondary optical effects. The method of retrieval of an approximate radiant intensity function of ground-based light sources was developed only recently [15], while it works with a parameterized empirical formula. The retrieval model is rather about obtaining a scaling parameter that simulates an uplight fraction. Concurrently, a generalized inversion of multi-angle and multi-spectral radiance data was applied to the spectral sky radiance data [11] in order to determine the intensity pattern of upwardly directed beams accounting for the cumulative effect of all city lights. The method is advantageous as it uses a stabilizer in solving the inverse problem numerically, and supplementary data are collected at a set of discrete wavelengths and distances from the monitored light source. However, all sky data are needed, including those from dark regions of the sky where the intensity of a diffuse light decreases by several orders of magnitude below the levels typical for sky elements aside the light source. This imposes quite strong requirements on the quality of clear-sky radiance data. Also, the mathematical formulation of the kernel function is very complex, thus imposing an additional burden on a numerical solution and its stabilization and relatively large errors when processing low intensities at the side of sky opposite the relative azimuthal position of the city or town.

Thus, we have developed a new approach to retrieval of an area-averaged CEF using night-sky brightness measurements made only on a local meridian from zenith to horizon. The method benefits from a simplified analytical formulation of the kernel of integral equation, fast algorithm, and lowered experimental errors because the data are to be exclusively taken at the brightest part of the sky. Such data can be gathered by traditional devices, e.g., CCD sensors which operate linearly within several orders of magnitude and do not need extremely high dynamic range. The method is specifically designed for meaningful, routinely easily measured data.

2. FORMULATION OF THE THEORETICAL PROBLEM SOLVED

It has been shown in [11] that extraction of radiance I(zE) of a round-based light source from observed radiance data is a discrete linear inverse problem, where the mapping between the unknown radiance I(zE) and measured radiance of a night sky Lλ(z,a) can be formulated as follows:

zE=0π/2K(zE;z,a)I(zE)dzE=L(z,a).

In Eq. (1), z, a are the observational zenith and azimuth angles, and zE is the zenith angle for beams emitted from an artificial light source. K(zE;z,a) is the so-called kernel of the integral equation and contains the basic physics of the radiative transfer in the Earth’s atmosphere for beams traveling from a light source to a detector. In general, K(zE;z,a) accounts for the superposition of optical signals from all light-emitting pixels with a total surface area of S:

Kλ(zE;z,a)=M(z)St[h(zE),zE]t[h(zE),z]h2(zE)×dh(zE)dzEΓ[h(zE),zE,z,a]cos3zEds,
where h is the altitude where the scattering event occurs for light beams that are emitted from an elementary surface area ds toward zE and observed at zenith angle z. The transmission function t is to characterize the beam attenuation when passing through an atmospheric layer between surface and the altitude h, taking into account that the angle of propagation is either zE or z. Additional information on the scattering and transmission functions Γ and t can be found in [16], while M(z)cos1z for zenith angles below 80°–85°. For long optical paths, the refraction by Earth’s atmosphere might be important, especially if z approaches 90°. In general, the relation between zE and z depends on the position of a light-emitting pixel
tan2(zE)=tan2(z)+Di2h22Dihtan(z)cos(acityaipixel),
where Di is the distance to the ith pixel. In Eq. (3), the azimuth angle aipixel of the ith pixel is measured relatively to acity that is the azimuth angle of the geographical average location for all of the city pixels as seen from the position of an observer. In fact, the midpoint is calculated as the center of mass for the 2D physical shape of a city. Note that Eq. (3) is an improvement to formula (10) introduced in [16]. However, the notation used here is different from what we have introduced in [16]; e.g., zE has been called z0 in earlier work.

It is a common practice in solving light-scattering problems to use two coordinate systems: the first one is centered in the position of an observer, while the second one is used to characterize the positions of light-emitting pixels independent of the measuring point. These two systems are preferred because the azimuth and zenith angles of the measured sky elements are obtained relative to the position of an observer, but the coordinates of light-emitting pixels are fixed and should not be redefined with each change of the observer’s position. It is therefore convenient to express aipixel in the form

aipixel=acityarccos(D2+Di2Ri22DiD),
with D being the distance from observer to the geographical midpoint of a city/town and Ri being the radial distance from the city center to the ith pixel. The positions of city pixels relative to the observer can be expressed in the Cartesian coordinate system
xi=Dcos(acity)+Ricos(acityφi),yi=Dsin(acity)+Risin(acityφi),
where φi is the azimuth of the ith pixel as seen from the midpoint. The above polar to Cartesian transform might appear useful in presenting the model graphically.

If observations are made along a local meridian from zenith to horizon and the city is not too irregular in its shape (or too asymmetric relative to the weighted center), or a measuring device is not too close to the city edges, then a=acity and formula (2) can be approximated as follows:

Kλ(zE;z,a)=M(z)Stλ[h(zE),zE]tλ[h(zE),z]h2(zE)×dh(zE)dzEΓ[h(zE),zE,z,a]cos3zE,
where integration over all city pixels has been replaced by a total surface area S, taking into account the mean value theorem for integrals. After a bit of mathematical manipulations, we can obtain
d[tan2(zE)]dh=2tanzEcos2zEdzEdh=2D2h3+2Dh2tanz,
so
dhdzE=D1+tan2zE(tanztanzE)2,
and thus we can immediately find the altitude of a scattering event as a function of the emission and observational zenith angles
h(z)=Dtanz+tanzE.
Under clear-sky conditions, the function Γ() is dominated by aerosol and Rayleigh scattering, meaning that we can use a simple concept of weighted contribution of both scattering functions
Γλ(h,θ)=τmHmexp{h(z)Hm}3(1+cos2θ)16π+ΩaτaHaexp{h(z)Ha}1ga24π(1+ga22gacosθ)3/2,
where τm and τa are the total molecular and aerosol optical thicknesses of the atmosphere, Hm and Ha are the corresponding scale heights, ga is asymmetry parameter of aerosol particles, and Ωa is the single scattering albedo of aerosols. The scattering angle θ for radiance collected at local meridian has a simple form: θ=πzzE.

Asymmetry parameter ga is a scalar quantity that characterizes angular redistribution of photons scattered by the aerosol particles and usually ranges from 0.5 to 0.8 (see, e.g., [17]). The lowest possible value of ga is 1, meaning that all photons that interact with an aerosol particles are exclusively scattered backward, which is almost unrealistic. The perfect forward scatter with ga=+1 is also rare, while isotropic scattering with ga=0 is well known in light-scattering theories and used as the equivalent to diffuse reflection. Single scattering albedo Ωa can vary over a closed interval from 0 to 1, where Ωa=0 is for total absorption (all photons are removed) and Ωa=1 is for conservative scattering (all photons are scattered). However, most typically Ωa is as large as 0.9–0.95 [18]. It is well established that both aerosol and Rayleigh optical depths depend on the wavelength λ. The Rayleigh component can be approximated analytically as τm0.00879λ4.09 [19,20], but the aerosol optical depth is a complex function of aerosol microphysics that in turn makes τa a non-trivial function of λ. Nevertheless, a set of approximate formulae for aerosol optical depth are in use, while most consider τa to be inversely proportional to the wavelength (e.g., [21,22]). The typical values of τa for the visible spectrum are 0.1–0.6. The remaining two parameters, Hm and Ha, are called molecular- and aerosol-scale heights and can be obtained under assumption of an exponentially stratified atmosphere. Here, Hm (or Ha) is basically the altitude up to which a homogeneous molecular (or aerosol) atmosphere would extend. The values usually adopted in numerical modeling are 8 km for Hm and 1–2 km for Ha [23,24]. Thanks to the simple analytical form of the transmission coefficients, we can find

t[h(zE),zE]t[h(zE),z]=exp{[M(zE)+M(z)]×[τm+τaτmexp(h(z)Hm)τaexp(h(z)Ha)]},
with M(zE)cos1zE and M(z)cos1z. In conclusion, the kernel of the integral equation acquires the form
K(zE;z)=SDcos3zEcosz(1+tan2zE)4π×exp{coszE+coszcoszcoszE×[τm+τaτmexp(h(z)Hm)τaexp(h(z)Ha)]}×{τmHmexp[h(z)Hm]3[1+cos2(z+zE)]4+τaHaexp[h(z)Ha]Ωa(1ga2)[1+ga2+2gacos(z+zE)]3/2},
where the dependence on the azimuth angle has been excluded because the observations are made for a=acity. The retrieval of I(zE) is only possible if the kernel functions are linearly independent; otherwise the conversion of Eq. (1) to algebraic form would lead to a system of redundant equations. An example for the inverse problem that suffers from linearly dependent equations is retrieval of particle size distribution from extinction measurements at wavelengths much larger than the size of particles (see Fig. 1(b) in [25]).

We have computed K(zE;z)/S for a set of discrete distances showing that kernel functions vary over several orders of magnitude (Fig. 1). Although the overall angular behaviors are similar at large observational zenith angles, the values of K(zE;z) decrease steeply as z approaches zero. This makes the model sensitive to observations taken at different zenith angles.

 figure: Fig. 1.

Fig. 1. Decimal logarithms of the kernel functions K(zE;z)/S computed after Eq. (12) for τm=0.15, τa=0.2, ga=0.85, and Ωa=0.9. (a) D=1km, (b) D=10km, (c) D=50km. The remaining parameters are the same as in the text (Section 2).

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3. INVERSION OF SKY RADIANCE MEASUREMENTS

Equation (1) represents a mapping from I(zE) data region to sky radiance data region using an integral transform operator K(zE;z). However, most often I(zE) is an unknown function that has to be determined from the experimental data of L(z,a). The inversion of radiance data is a typically ill-posed problem, because the algebraized form of Eq. (1),

KI=L,
has an unstable solution if sought as
I=K1L.
The matrix K is obtained by algebraization of the integral operator K(zE;z); K1 is the inverse matrix of K, I is the solution vector for ground radiance I constructed from discrete points of zE, and L is the data vector containing experimental radiance measured at discrete zenith angles z. The vectors I and L need not necessarily have the same number of elements; e.g., we may use I=[I(zE,1),I(zE,2),,I(zE,M)] and L=[L(z1),L(z2),,L(zN)], where M can differ from N. It is, however, required that both zE and z range from 0 to π/2 and z is measured along the local meridian that intersects the horizon at the position of a light source.

It is not surprising that some elements of the vector I obtained from Eq. (14) can appear negative because a small eigenvalue of K produces large elements in K1, and thus any small experimental error of L can be extremely amplified. This is why many solutions satisfying the measure of “goodness” |KIL|2ε2 may exist, thus implying there is a great difficulty in finding a correct value even if the oscillatory forms of I are completely filtered out. However, it is well established that the true solution must be smooth and this can be controlled by quadratic forms of I, such as ITJI [26], where J is usually a near-diagonal matrix and IT is the transposed matrix of I. The solution vector I is then found by minimizing ITJI subject to the constraint |KIL|2ε2. Considering that K is an element of Hilbert space of quadratically integrable functions (a compact operator from I space to radiance space), then a unique solution to vector I must exist so that I=(KTK+αJ)1KTL, where α is an undetermined Lagrangian multiplier (i.e., the parameter of regularization; see pages 36–43 in [27]). The solution of Eq. (13) is difficult even for over-determined systems wherein NM because K1 is very often large, so a simple inversion in the form of Eq. (14) tends to be unstable. The instability is not eliminated if a least squares method is used with (KTK)1. Therefore, it is generally very useful to introduce a stabilizing (smoothing) functional and follow with the method described below.

Equation (1) is the Fredholm integral equation of the first kind for which the regularization methods are well suited [28]. We proceed here with the Tikhonov inversion adapted by Kabanov et al. [29] for use in optical particle sizing because his approach appeared to be well applicable to many related problems, including that solved in this paper. We summarize the main solution steps here to make the algorithm complete. Kabanov has discretized the solution using the stepwise kernel functions

Kmn=Km(zn)=Gm1,m(zn)+Gm,m(zn)+Gm+1,m(zn),m1,mMK1n=K1(zn)=G1,1(zn)+G2,1(zn),KMn=KM(zn)=GM1,M(zn)+GM,M(zn),
with
Gm,p(zn)=(zE,m1+zE,m)/2(zE,m+zE,m+1)/2K(zE;zn)ωm(zE)(zEzE,p)ωm(zE,p)dzE,
where
ωm(zE)=(zEzE,m1)(zEzE,m)(zEzE,m+1),
and ωm(zE,p) is the first derivative of ωm in zE=zE,p. The near-diagonal matrix J is chosen in accordance with [26,29] in the form J=(zE,M+1zE,0)D+(zE,M+1zE,0)B, D=diag{δ1,δ2,,δM}, δm=(zE,m+1zE,m1)/2, and the elements of the matrix B are as follows:
Bmn=(zE,mzE,m1)1,n=m1,m=2M,Bmn=1zE,mzE,m1+1zE,m+1zE,m,n=m,m=1M,Bmn=(zE,nzE,n1)1,n=m+1,m=1M1,Bmn=0,|nm|>1,m,n=1M.

The nature of diagonal and near-diagonal elements of the symmetric matrix J is explained in detail in [30] (see Section 19.5), where the symbol H is used instead of J. In our convention H denotes an altitude, so we avoided a use of the same symbol for two different quantities.

Further, we apply the following substitution in Eq. (19):

Kcos(zE;z)=K(zE;z)coszECEF(zE)=I(zE)coszE,
where CEF(zE) is cosine-projected radiance, otherwise called radiant intensity function. Experimentation with several distinct kernels is not as rare, as it may appear useful in more accurate retrievals [31].

Applying the mirror symmetry to the observed radiance data, we have generated synthetic data for the meridian opposite to the azimuthal position of a light source. This approach allows for extension of zE to the interval of values from zE,0=π/2 to zE,M+1=π/2, thus ensuring that CEF and its first derivative will both be smooth and continuous functions of zE near zenith. More specifically, we require that d(CEF)/d(zE) exists at zE=0, while the light emissions to the zenith are not necessarily zero. Extension of the input data function as just described has proved to be very useful in numerical tests performed on synthetic data. Doubling the interval for zE (and also for z) results in the fact that CEF(zE), found in the process of minimization, has to be multiplied by the factor of 2, or the elements of the solution vector are obtained as the sum of mirror elements from both meridians. The retrieval process is completed by obtaining the city emission function CEF(zE) or ground radiance I(zE)=CEF(zE)/coszE.

Finally, we have made a sensitivity study to evaluate the effect of τa, ga, and Ωa on K(zE;z). Basically, we alter one of the just-mentioned aerosol properties while keeping all remaining parameters unchanged in Eq. (12). It is demonstrated in Fig. 2 that asymmetry parameter ga has a small impact on the unmodified (original) kernel K(zE;z) except for situations when observations are made in the near-horizon zone. However, aerosol optical thickness (τa) as well as single scattering albedo (Ωa) can both influence K(zE;z) considerably at intermediate zenith angles also, meaning that retrieval of CEF would be sensitive to both aerosol properties analyzed here.

 figure: Fig. 2.

Fig. 2. Same as Fig. 1, but for K(zE;z)/S as a function of τa, ga, and Ωa. (a) τa=0.5 (solid line), τa=0.1 (short dashes), τa=0.03 (long dashes); (b) ga=0.5 (solid line), ga=0.7 (short dashes), ga=0.95 (long dashes); (c) Ωa=0.7 (solid line), Ωa=0.5 (short dashes), Ωa=0.3 (long dashes).

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4. NUMERICAL EXPERIMENTS ON SYNTHETIC AND REAL DATA

The model developed in this paper can be advantageously and easily applied to routinely measured data. Undoubtedly, any new solution to the inverse problem should be validated for sensitivity and specificity in reproducing the initial functions. This is the only way to assess the performance of the inversion method developed here and to remedy the algorithm if the results prove to be unsatisfactory. In our case, an analytical model for CEF is ideally suited for testing the inversion routines as it allows modeling of different types of emission patterns that can be representative of distinct light sources.

For synthetic reconstruction purposes, we have used Garstang’s model [32],

CEF(zE)=GEF(zE)[2G(1F)coszE+0.554FzE4],
which is useful as a zero-order approximation to imitate light sources with and/or without direct uplight. Here, F is the fraction of the light emitted from a light source directly upward and G characterizes the fraction of the light that is isotropically reflected from the ground. The coefficient of proportionality PL/(2π) was omitted for brevity (P is the population of a city that produces an output L lumens per head of the population).

We have computed the radiance for N=35 zenith angles using τm=0.15, τa=0.2, ga=0.85, Ωa=0.9 as input data to the model. The radiance data [L(z1),L(z2),,L(zN)] generated for a local meridian forms a basis for reconstructing the city emission function by applying the minimization routines introduced in previous sections. The solution vector C (in fact the cosine-projected radiance vector) was found by minimizing Tikhonov’s functional while penalizing the large norms that normally tend to produce bad solutions. The radiance data displayed in Fig. 3(a) (triangles) was computed from the synthetic CEF that is depicted in Fig. 3(b) (solid line). Consequently, the regularization parameter α that controls solution Cα=(KcosTKcos+αJ)1KcosTL was determined using a rapidly convergent iteration method. Basically, the procedure we have used is straightforward because KcosCαLε has a strictly monotonous and continuous form when evaluated as a function of α. It is demonstrated in Fig. 3(b) that a regularized solution matches the synthetic CEF very well (compare dashed line against solid line) and re-generated radiance data are perfectly accurate [see dashed line in Fig. 3(a)]. The systematic tests have been carried out on a set of emission functions that characterize various types of light sources—from cosine radiator to an unshielded emitter and various types of surfaces—from very dark to extremely reflective ones. The synthetic CEFs as well as the results of numerical inversions are brought together in Fig. 4 and prove that all emission functions were reproduced successfully with an overall discrepancy being smaller than 3%. All results are consistent, and the tests confirmed a high sensitivity and specificity of the inversion scheme.

 figure: Fig. 3.

Fig. 3. (a) Synthetic radiance data (black triangles) and radiance function generated using CEF found in the process of inversion. (b) CEF used as an input to the regularization algorithm (solid line) and CEF obtained in the process of minimization.

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 figure: Fig. 4.

Fig. 4. Synthetic CEFs (solid lines) of different types computed in accordance with Eq. (20) and the corresponding regularized solutions (dashed lines).

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A controlled experiment has been performed by Žwak [33] in an open atmosphere which provides a unique opportunity to validate our model on real sky-radiance data. The measurements were conducted shortly before midnight, 3–4 km away from an electric substation illuminated by intense badly shielded luminaires that were controlled during the field experiment. Although the zenith radiance has only been lowered to the range of 40%–50% after turning off all the lights, a much higher horizon darkening was manifested for the quadrant facing toward the electric substation. A decrease of zenith radiance by only about 50% when turning off the local lights is mainly due to parasitic light from other distant cities and towns that were, of course, out of control of this targeted experiment and have contributed to a background luminance of a night sky. Other quadrants remain dark because the light from houses was largely absent at midnight, while the effect of street lights from distant cities was significantly suppressed thanks to new technologies that reduce the amount of light emitted to low elevation angles. This is the reason we expect that the targeted field experiment should report an approximate cosine-like CEF for a dark environment when the lighting of the electric substation is turned off, but a gradual increase of CEF near the horizon occurs after the local lights are turned on again. The results of the numerical inversion are in accord with our anticipation, as is evident from Fig. 5.

 figure: Fig. 5.

Fig. 5. CEF (in arbitrary units) obtained from a field experiment with local lights turned off (blue short dashed line) and turned on (black long dashed line). Triangles are used for CEF determined from background-compensated radiance data, while Garstang’s fit is demonstrated as the red solid line.

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Figure 5 indicates that the radiant intensity with all the lights switched off is an even more steep function of the zenith angle than predicted by the cosine law. However, this is not unexpected as the observations were made under dark sky conditions with all light coming from nearby cities and a town situated mostly at the side opposite to the position of the electric substation. The emission function changed markedly after the local lights were switched on, as documented by the long dashed line. It is therefore obvious that local lights from the electric substation contribute appreciably to the upward emissions, which can be verified by using background-compensated radiance data in retrieval of the CEF. To verify this, we have subtracted the radiances obtained under dark conditions from the radiance data obtained when local lights were turned on. In this way, the CEF was also adjusted for the effect of city lights. The CEF related to the electric substation is depicted in Fig. 5 (see triangles) and approximated by Garstang’s function (solid line) with the best fit parameters Ffit=15% and Gfit=2.5%. Note that Gfit has no relation to the ground albedo. Instead, Ffit and Gfit are to self-characterize the artificial lights of the electric substation.

In order to complete our analysis, we investigated the applicability of the inversion model to data obtained under limit conditions, specifically when the radiance pattern is taken inside an urban area with artificial lights distributed everywhere around. One can assume that no dominant radiance peak appears at a specific azimuthal position, so there is no preferred direction of observation. We have made such an experiment during the field campaign in the city of Žilina (Slovak Republic) with poor radiance gradation from zenith toward horizon [dashed line in Fig. 6(a)]. In fact, the retrieval method is not well designed to determine CEF from data collected in an urban zone or its close surroundings due to uncertainty in the value of D that influences Eq. (12) though Eq. (9) in a non-trivial way. However, it is suggested in this case to use the distance to the midpoint of a city. For the city of Žilina we have chosen D as small as 0.5 km, but we do not recommend use of D below 0.1 km for numerical convergence reasons. The reproduced data of sky radiance are found to be consistent with those observed at z>50° [see Fig. 6(a)], but the solution to the inverse problem underestimates the radiance at low zenith angles. The measure in which the reconstructed radiance-vector (LR) differs from that of the vector for measured radiance (LM) was expressed here by means of overall error {(LMLR)sinzdz}/{LMsinzdz}. Although this error was found to be below 10%, the solution function for CEF [Fig. 6(b)] bears a striking similarity to what we have found in Fig. 5 for a dark landscape or to what we know as the cosine-law. This is because the radiance data were taken inside an urban area and such data do not contain much information on emissions to low elevation angles. In other words, the reflected light is important at distances that are small compared to the size of the city/town, while light emissions to low-elevation angles are the most important modulators of sky radiance far from a light-emitting source. The best conditions for retrieval of CEF are therefore at intermediate distances due to the highest information content of radiance data. It is difficult to specify the optimum distance, as it depends on many factors such as atmospheric turbidity, size of a city or town, and lighting technology. Roughly, the interval for optimum values of D can be estimated from the product of Kcos(zE;z) and CEF(zE) when drawn for different input parameters as a function of zE. The information content is generally expected to be low, when D has almost no effect on Kcos·CEF near its peak values or when it influences only the lowest values of Kcos·CEF.

 figure: Fig. 6.

Fig. 6. (a) Sky radiance (in arbitrary units) taken in the city of Žilina and the reconstructed one. (b) Solution to the inverse problem.

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An increasing number of negative values in the solution vector or CEF producing a bad fit to the radiance data are both indicators that the inverse algorithm fails to find a solution within the required error bounds. There could be several reasons for such a result, including large experimental errors, inappropriately chosen distance to the light source (a city or town), optical instability of atmospheric environment, presence of invisible clouds, phantom effects due to parasitic light, or a theoretical model that might be inadequate to describe the mechanism of formation of sky glow in a few situations, e.g., when beams from two or more dominant light sources are superimposed on (or intermingled in) the measured signal without a possibility of subtracting one from the other.

Sometimes these situations can be avoided by changing the position of the measuring point relative to the light sources; then one light source might become dominant at the expense of others. If a separation of the optical signals from two or more light sources is difficult or rather unrealizable, then a transform to a single integral equation is impossible and one should follow with the iterative solution to the problem. In fact, it would mean that two or more emission functions are retrieved concurrently using all-sky photometry made at a set of discrete measuring points. The solution concept should start with an initial “zero approximation” to the emission function CEF1i=0 for city 1. This kind of generating function allows for obtaining a 0th solution to the emission function for city 2 (CEF2j=0) using the radiance data measured on a local meridian of city 2. Retrieval of CEF2j=0 is possible because the product of CEF1i=0 and Kcos can be computed numerically, meaning that the theoretical contribution of city 1 to the sky radiance is known, and then CEF2j=0 is the only function to be determined. In the next round, the product of CEF2j=0 and a respective kernel function Kcos will model the contribution of city 2 to the night-sky radiance that is consequently subtracted from the radiance data acquired on a local meridian of city 1, thus resulting in a successful solution of the inverse problem for CEF1i=1. Integrating the product of CEF1i=1 and Kcos, the radiance due to emissions from city 1 can be predicted and subtracted from the data measured on a local meridian of city 2. The resulting function allows us to solve the inverse problem for CEF2j=1 analogously to what we have made before. If convergent, the scheme just described guarantees the solutions for CEF1 and CEF2 would improve with increasing the number of iterations. The procedure is terminated if the conditions CEF1iCEF1i1<ε1 and CEF2jCEF2j1<ε2 are both satisfied simultaneously. Here, ε1 and ε2 are the error margins defined depending on the accuracy requirements. It is essential to highlight that the solution scheme for a combined inverse problem might suffer from lowered stability, especially when the measurement errors are large. It is expected that the better the match found for generating function CEF1i=0, the fewer the iterations necessary to obtain the final solutions for city emission functions. Among many others, the cosine function seems to be a good trial function for CEF1i=0. Garstang’s model [Eq. (20)] is also a good alternative for modeling the zero-order approximation of the CEF1.

5. CONCLUSIONS

The radiant intensity function is one of the key inputs to modeling sky glow under arbitrary meteorological conditions. The photons emitted from artificial light sources produce a veiling luminance that makes nights essentially brighter and can preclude traditional astronomical observations in an optical window from roughly 400 nm up to the red edge at about 700 nm. Therefore, there is a need to build an overall picture of CEFs and establish a database of CEFs for cities worldwide. In this paper, we have developed and validated a simple and cost-effective method for retrieval of CEFs using ground-based radiance observations. The night-sky brightness measurements have to be made on a local meridian from zenith to horizon for the quadrant facing toward the light-emitting source (a city or town).

The method is based on a Tikhonov regularization that penalizes large norms with the aim of finding a stable and smooth solution. The inversion technique we have developed has been verified on synthetically generated radiance data showing that reproduced CEF is consistent with that of initial CEF. A very good match between input and reconstructed radiance patterns has proven the sensitivity and specificity of the solution technique and its capability in retrieval of arbitrary CEFs, specifically those that normally characterize urban zones. It is required that the radiance data should be taken under clear sky conditions; otherwise, the retrieval algorithm might not succeed or can produce inaccurate or incorrect results. The radiance measurements should primarily be made at intermediate distances from a city due to the highest information content of the experimental data. In other words, the distance should not be smaller than 1–2 city diameters from the average location of all light-emitting pixels (most typically from the geographical midpoint of a city) and should not exceed several tens of city diameters.

The remote-sensing technique we have introduced requires a priori information on the optical properties of the atmosphere, especially the aerosols that are important modulators of ground-reaching diffuse radiation in a cloud-free nighttime atmosphere. It is strongly recommended that an optically stable day be selected and that measurements be avoided during the air mass transformation, as this can have a potential effect on a local aerosol system. Diligence in fulfilling these requirements can minimize the sensitivity of the proposed approach to incorrect estimations of the main atmospheric parameters, as these can be determined separately, e.g., from daylight radiance or spectral irradiance experiments [34] using nephelometers or lidars [35].

Generally, it is difficult to estimate a sensitivity of CEF retrievals to the radiance measurement errors because inversion errors usually depend on many factors concurrently. For instance, a success rate of an inversion technique can depend on instrument field of view, interval of scattering angles, and grid density of radiance data, as well as the “dynamic range” of the unknown function, i.e., on how steeply CEF changes with the emission zenith angle. Some of analyses made on the error components in related linear inversions have shown that inversion accuracy is rarely better than 10%–20% even if experimental errors are kept as low as 5% [36,37]. Optimally, the optical measurements should be accurate within 1% to make retrievals successful [38], but this accuracy is difficult to achieve. Therefore, we have accepted a 5% error margin to be a reasonable limit for experimental error.

This paper presents the core principles of the inversion of radiance data and provides a great opportunity for systematic retrievals of CEF. In cooperation with the light-pollution community, it can motivate researchers to complete the database of CEFs for as many cities as possible. We therefore encourage experimentalists to share their data and put measured radiance/luminance patterns at our disposal for further analysis.

Funding

Agentúra na Podporu Výskumu a Vývoja (APVV) (APVV-14-0017); Agentúra Ministerstva Školstva, Vedy, Výskumu a Športu SR (Agency of the Ministry of Education, Science, Research and Sport of the Slovak Republic for the Structural Funds of EU) (2/0016/16).

REFERENCES

1. A. A. Kokhanovsky, Light Scattering Media Optics: Problems and Solutions, 3rd ed. (Praxis, 2004).

2. G. E. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, 2006).

3. M. Kocifaj, “Night sky luminance under clear sky conditions: theory vs. experiment,” J. Quant. Spectrosc. Radiat. Transfer 139, 43–51 (2014). [CrossRef]  

4. M. I. Mishchenko, L. D. Travis, R. A. Kahn, and R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997). [CrossRef]  

5. C. D. Elvidge, D. M. Keith, B. T. Tuttle, and K. E. Baugh, “Spectral identification of lighting type and character,” Sensors 10, 3961–3988 (2010). [CrossRef]  

6. D. M. Duriscoe, C. B. Luginbuhl, and C. D. Elvidge, “The relation of outdoor lighting characteristics to sky glow from distant cities,” Light. Res. Technol. 46, 35–49 (2014). [CrossRef]  

7. C. B. Luginbuhl, G. W. Lockwood, D. R. Davis, K. Pick, and J. Selders, “From the ground up I: light pollution sources in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 185–203 (2009). [CrossRef]  

8. C. Luginbuhl, D. Duriscoe, C. Moore, A. Richman, W. Lockwood, and R. Davis, “From the ground up II: sky glow and near-ground artificial light propagation in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 204–212 (2009). [CrossRef]  

9. M. Aubé, “Physical behavior of anthropogenic light propagation into the nocturnal environment,” Philos. Trans. R. Soc. London B 370, 20140117 (2015). [CrossRef]  

10. M. Kocifaj and H. A. Solano-Lamphar, “Angular emission function of a city and skyglow modeling: a critical perspective,” Publ. Astron. Soc. Pac. 128, 124001 (2016). [CrossRef]  

11. M. Kocifaj, “Modeling the night-sky radiances and inversion of multi-angle and multi-spectral radiance data,” J. Quant. Spectrosc. Radiat. Transfer 139, 35–42 (2014). [CrossRef]  

12. F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016). [CrossRef]  

13. H. U. Kuechly, C. C. M. Kyba, T. Ruhtz, C. Lindemann, C. Wolter, J. Fischer, and F. Hölker, “Aerial survey and spatial analysis of sources of light pollution in Berlin, Germany,” Remote Sens. Environ. 126, 39–50 (2012). [CrossRef]  

14. C. C. M. Kyba, T. Ruhtz, C. Lindemann, J. Fischer, and F. Hölker, “Two camera system for measurement of urban uplight angular distribution,” AIP Conf. Proc. 1531, 568–571 (2013). [CrossRef]  

15. M. Kocifaj and H. A. Solano-Lamphar, “Skyglow: a retrieval of the approximate radiant intensity function of ground-based light sources,” Mon. Not. R. Astron. Soc. 439, 3405–3413 (2014). [CrossRef]  

16. M. Kocifaj, “Light-pollution model for cloudy and cloudless night skies with ground-based light sources,” Appl. Opt. 46, 3013 (2007). [CrossRef]  

17. R. C. Levy, L. A. Remer, and O. Dubovik, “Global aerosol optical properties and application to moderate resolution imaging spectroradiometer aerosol retrieval over land,” J. Geophys. Res. 112, D13210 (2007).

18. H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002). [CrossRef]  

19. V. E. Zuev and G. M. Krekov, Optical Models of the Atmosphere (Gidrometeoizdat, 1986).

20. P. M. Teillet, “Rayleigh optical depth comparisons from various sources,” Appl. Opt. 29, 1897–1900 (1990). [CrossRef]  

21. P. Pesava, H. Horvath, and M. Kasahara, “A local optical closure experiment in Vienna,” J. Aerosol Sci. 32, 1249–1267 (2001). [CrossRef]  

22. G. P. Gushchin, The Methods, Instrumentation and Results of Atmospheric Spectral Measurements (Gidrometeoizdat, 1988).

23. F. Waquet, P. Goloub, J. L. Deuzé, J. F. Léon, F. Auriol, C. Verwaerde, J. Y. Balois, and P. Francçois, “Aerosol retrieval over land using a multiband polarimeter and comparison with path radiance method,” J. Geophys. Res. 112, D11214 (2007). [CrossRef]  

24. N. G. Loeb and W. Su, “Direct aerosol radiative forcing uncertainty based on a radiative perturbation analysis,” J. Climate 23, 5288–5293 (2010). [CrossRef]  

25. H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

26. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Dover, 2002).

27. F. Cakoni and D. Colton, Quantitative Methods in Inverse Scattering Theory: An Introduction (Springer, 2006).

28. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

29. M. V. Kabanov, M. V. Panchenko, Y. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Y. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988).

30. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

31. G. P. Box, K. M. Sealey, and M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992). [CrossRef]  

32. R. H. Garstang, “Model for artificial night-sky illumination,” Publ. Astron. Soc. Pac. 98, 364–375 (1986). [CrossRef]  

33. Z. Žwak, data provided (personal communication, 2012).

34. M. Kocifaj, H. Horvath, O. Jovanović, and M. Gangl, “Optical properties of urban aerosols in the region Bratislava-Vienna I, methods and tests,” Atmos. Environ. 40, 1922–1934 (2006). [CrossRef]  

35. M. Adam, M. Pahlow, V. A. Kovalev, J. M. Ondov, M. B. Parlange, and N. Nair, “Aerosol optical characterization by nephelometer and lidar: the Baltimore Supersite experiment during the Canadian forest fire smoke intrusion,” J. Geophys. Res. 109, D16S02 (2004). [CrossRef]  

36. C. Dellago and H. Horvath, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—I, basic considerations and models,” J. Aerosol Sci. 24, 129–141 (1993). [CrossRef]  

37. H. Horvath and C. Dellago, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements–II, case studies,” J. Aerosol Sci. 24, 143–154 (1993). [CrossRef]  

38. K. S. Shifrin, “Study of the properties of matter from single scattering,” in Theoretical and Applied Problems in the Scattering of Light (Nauka i Tekhnika, 1971) (in Russian).

References

  • View by:

  1. A. A. Kokhanovsky, Light Scattering Media Optics: Problems and Solutions, 3rd ed. (Praxis, 2004).
  2. G. E. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, 2006).
  3. M. Kocifaj, “Night sky luminance under clear sky conditions: theory vs. experiment,” J. Quant. Spectrosc. Radiat. Transfer 139, 43–51 (2014).
    [Crossref]
  4. M. I. Mishchenko, L. D. Travis, R. A. Kahn, and R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
    [Crossref]
  5. C. D. Elvidge, D. M. Keith, B. T. Tuttle, and K. E. Baugh, “Spectral identification of lighting type and character,” Sensors 10, 3961–3988 (2010).
    [Crossref]
  6. D. M. Duriscoe, C. B. Luginbuhl, and C. D. Elvidge, “The relation of outdoor lighting characteristics to sky glow from distant cities,” Light. Res. Technol. 46, 35–49 (2014).
    [Crossref]
  7. C. B. Luginbuhl, G. W. Lockwood, D. R. Davis, K. Pick, and J. Selders, “From the ground up I: light pollution sources in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 185–203 (2009).
    [Crossref]
  8. C. Luginbuhl, D. Duriscoe, C. Moore, A. Richman, W. Lockwood, and R. Davis, “From the ground up II: sky glow and near-ground artificial light propagation in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 204–212 (2009).
    [Crossref]
  9. M. Aubé, “Physical behavior of anthropogenic light propagation into the nocturnal environment,” Philos. Trans. R. Soc. London B 370, 20140117 (2015).
    [Crossref]
  10. M. Kocifaj and H. A. Solano-Lamphar, “Angular emission function of a city and skyglow modeling: a critical perspective,” Publ. Astron. Soc. Pac. 128, 124001 (2016).
    [Crossref]
  11. M. Kocifaj, “Modeling the night-sky radiances and inversion of multi-angle and multi-spectral radiance data,” J. Quant. Spectrosc. Radiat. Transfer 139, 35–42 (2014).
    [Crossref]
  12. F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
    [Crossref]
  13. H. U. Kuechly, C. C. M. Kyba, T. Ruhtz, C. Lindemann, C. Wolter, J. Fischer, and F. Hölker, “Aerial survey and spatial analysis of sources of light pollution in Berlin, Germany,” Remote Sens. Environ. 126, 39–50 (2012).
    [Crossref]
  14. C. C. M. Kyba, T. Ruhtz, C. Lindemann, J. Fischer, and F. Hölker, “Two camera system for measurement of urban uplight angular distribution,” AIP Conf. Proc. 1531, 568–571 (2013).
    [Crossref]
  15. M. Kocifaj and H. A. Solano-Lamphar, “Skyglow: a retrieval of the approximate radiant intensity function of ground-based light sources,” Mon. Not. R. Astron. Soc. 439, 3405–3413 (2014).
    [Crossref]
  16. M. Kocifaj, “Light-pollution model for cloudy and cloudless night skies with ground-based light sources,” Appl. Opt. 46, 3013 (2007).
    [Crossref]
  17. R. C. Levy, L. A. Remer, and O. Dubovik, “Global aerosol optical properties and application to moderate resolution imaging spectroradiometer aerosol retrieval over land,” J. Geophys. Res. 112, D13210 (2007).
  18. H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
    [Crossref]
  19. V. E. Zuev and G. M. Krekov, Optical Models of the Atmosphere (Gidrometeoizdat, 1986).
  20. P. M. Teillet, “Rayleigh optical depth comparisons from various sources,” Appl. Opt. 29, 1897–1900 (1990).
    [Crossref]
  21. P. Pesava, H. Horvath, and M. Kasahara, “A local optical closure experiment in Vienna,” J. Aerosol Sci. 32, 1249–1267 (2001).
    [Crossref]
  22. G. P. Gushchin, The Methods, Instrumentation and Results of Atmospheric Spectral Measurements (Gidrometeoizdat, 1988).
  23. F. Waquet, P. Goloub, J. L. Deuzé, J. F. Léon, F. Auriol, C. Verwaerde, J. Y. Balois, and P. Francçois, “Aerosol retrieval over land using a multiband polarimeter and comparison with path radiance method,” J. Geophys. Res. 112, D11214 (2007).
    [Crossref]
  24. N. G. Loeb and W. Su, “Direct aerosol radiative forcing uncertainty based on a radiative perturbation analysis,” J. Climate 23, 5288–5293 (2010).
    [Crossref]
  25. H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.
  26. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Dover, 2002).
  27. F. Cakoni and D. Colton, Quantitative Methods in Inverse Scattering Theory: An Introduction (Springer, 2006).
  28. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).
  29. M. V. Kabanov, M. V. Panchenko, Y. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Y. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988).
  30. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).
  31. G. P. Box, K. M. Sealey, and M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
    [Crossref]
  32. R. H. Garstang, “Model for artificial night-sky illumination,” Publ. Astron. Soc. Pac. 98, 364–375 (1986).
    [Crossref]
  33. Z. Žwak, data provided (personal communication, 2012).
  34. M. Kocifaj, H. Horvath, O. Jovanović, and M. Gangl, “Optical properties of urban aerosols in the region Bratislava-Vienna I, methods and tests,” Atmos. Environ. 40, 1922–1934 (2006).
    [Crossref]
  35. M. Adam, M. Pahlow, V. A. Kovalev, J. M. Ondov, M. B. Parlange, and N. Nair, “Aerosol optical characterization by nephelometer and lidar: the Baltimore Supersite experiment during the Canadian forest fire smoke intrusion,” J. Geophys. Res. 109, D16S02 (2004).
    [Crossref]
  36. C. Dellago and H. Horvath, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—I, basic considerations and models,” J. Aerosol Sci. 24, 129–141 (1993).
    [Crossref]
  37. H. Horvath and C. Dellago, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements–II, case studies,” J. Aerosol Sci. 24, 143–154 (1993).
    [Crossref]
  38. K. S. Shifrin, “Study of the properties of matter from single scattering,” in Theoretical and Applied Problems in the Scattering of Light (Nauka i Tekhnika, 1971) (in Russian).

2016 (2)

M. Kocifaj and H. A. Solano-Lamphar, “Angular emission function of a city and skyglow modeling: a critical perspective,” Publ. Astron. Soc. Pac. 128, 124001 (2016).
[Crossref]

F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
[Crossref]

2015 (1)

M. Aubé, “Physical behavior of anthropogenic light propagation into the nocturnal environment,” Philos. Trans. R. Soc. London B 370, 20140117 (2015).
[Crossref]

2014 (4)

M. Kocifaj, “Modeling the night-sky radiances and inversion of multi-angle and multi-spectral radiance data,” J. Quant. Spectrosc. Radiat. Transfer 139, 35–42 (2014).
[Crossref]

M. Kocifaj, “Night sky luminance under clear sky conditions: theory vs. experiment,” J. Quant. Spectrosc. Radiat. Transfer 139, 43–51 (2014).
[Crossref]

D. M. Duriscoe, C. B. Luginbuhl, and C. D. Elvidge, “The relation of outdoor lighting characteristics to sky glow from distant cities,” Light. Res. Technol. 46, 35–49 (2014).
[Crossref]

M. Kocifaj and H. A. Solano-Lamphar, “Skyglow: a retrieval of the approximate radiant intensity function of ground-based light sources,” Mon. Not. R. Astron. Soc. 439, 3405–3413 (2014).
[Crossref]

2013 (1)

C. C. M. Kyba, T. Ruhtz, C. Lindemann, J. Fischer, and F. Hölker, “Two camera system for measurement of urban uplight angular distribution,” AIP Conf. Proc. 1531, 568–571 (2013).
[Crossref]

2012 (1)

H. U. Kuechly, C. C. M. Kyba, T. Ruhtz, C. Lindemann, C. Wolter, J. Fischer, and F. Hölker, “Aerial survey and spatial analysis of sources of light pollution in Berlin, Germany,” Remote Sens. Environ. 126, 39–50 (2012).
[Crossref]

2010 (2)

C. D. Elvidge, D. M. Keith, B. T. Tuttle, and K. E. Baugh, “Spectral identification of lighting type and character,” Sensors 10, 3961–3988 (2010).
[Crossref]

N. G. Loeb and W. Su, “Direct aerosol radiative forcing uncertainty based on a radiative perturbation analysis,” J. Climate 23, 5288–5293 (2010).
[Crossref]

2009 (2)

C. B. Luginbuhl, G. W. Lockwood, D. R. Davis, K. Pick, and J. Selders, “From the ground up I: light pollution sources in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 185–203 (2009).
[Crossref]

C. Luginbuhl, D. Duriscoe, C. Moore, A. Richman, W. Lockwood, and R. Davis, “From the ground up II: sky glow and near-ground artificial light propagation in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 204–212 (2009).
[Crossref]

2007 (3)

M. Kocifaj, “Light-pollution model for cloudy and cloudless night skies with ground-based light sources,” Appl. Opt. 46, 3013 (2007).
[Crossref]

R. C. Levy, L. A. Remer, and O. Dubovik, “Global aerosol optical properties and application to moderate resolution imaging spectroradiometer aerosol retrieval over land,” J. Geophys. Res. 112, D13210 (2007).

F. Waquet, P. Goloub, J. L. Deuzé, J. F. Léon, F. Auriol, C. Verwaerde, J. Y. Balois, and P. Francçois, “Aerosol retrieval over land using a multiband polarimeter and comparison with path radiance method,” J. Geophys. Res. 112, D11214 (2007).
[Crossref]

2006 (1)

M. Kocifaj, H. Horvath, O. Jovanović, and M. Gangl, “Optical properties of urban aerosols in the region Bratislava-Vienna I, methods and tests,” Atmos. Environ. 40, 1922–1934 (2006).
[Crossref]

2004 (1)

M. Adam, M. Pahlow, V. A. Kovalev, J. M. Ondov, M. B. Parlange, and N. Nair, “Aerosol optical characterization by nephelometer and lidar: the Baltimore Supersite experiment during the Canadian forest fire smoke intrusion,” J. Geophys. Res. 109, D16S02 (2004).
[Crossref]

2002 (1)

H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
[Crossref]

2001 (1)

P. Pesava, H. Horvath, and M. Kasahara, “A local optical closure experiment in Vienna,” J. Aerosol Sci. 32, 1249–1267 (2001).
[Crossref]

1997 (1)

M. I. Mishchenko, L. D. Travis, R. A. Kahn, and R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[Crossref]

1993 (2)

C. Dellago and H. Horvath, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—I, basic considerations and models,” J. Aerosol Sci. 24, 129–141 (1993).
[Crossref]

H. Horvath and C. Dellago, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements–II, case studies,” J. Aerosol Sci. 24, 143–154 (1993).
[Crossref]

1992 (1)

G. P. Box, K. M. Sealey, and M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
[Crossref]

1990 (1)

1986 (1)

R. H. Garstang, “Model for artificial night-sky illumination,” Publ. Astron. Soc. Pac. 98, 364–375 (1986).
[Crossref]

Adam, M.

M. Adam, M. Pahlow, V. A. Kovalev, J. M. Ondov, M. B. Parlange, and N. Nair, “Aerosol optical characterization by nephelometer and lidar: the Baltimore Supersite experiment during the Canadian forest fire smoke intrusion,” J. Geophys. Res. 109, D16S02 (2004).
[Crossref]

Arboledas, L. A.

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F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
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F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
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F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
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F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
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H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

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H. U. Kuechly, C. C. M. Kyba, T. Ruhtz, C. Lindemann, C. Wolter, J. Fischer, and F. Hölker, “Aerial survey and spatial analysis of sources of light pollution in Berlin, Germany,” Remote Sens. Environ. 126, 39–50 (2012).
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M. Kocifaj, H. Horvath, O. Jovanović, and M. Gangl, “Optical properties of urban aerosols in the region Bratislava-Vienna I, methods and tests,” Atmos. Environ. 40, 1922–1934 (2006).
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H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
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H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

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M. Kocifaj, H. Horvath, O. Jovanović, and M. Gangl, “Optical properties of urban aerosols in the region Bratislava-Vienna I, methods and tests,” Atmos. Environ. 40, 1922–1934 (2006).
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H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
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H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

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M. V. Kabanov, M. V. Panchenko, Y. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Y. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988).

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H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
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H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

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C. D. Elvidge, D. M. Keith, B. T. Tuttle, and K. E. Baugh, “Spectral identification of lighting type and character,” Sensors 10, 3961–3988 (2010).
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F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
[Crossref]

C. C. M. Kyba, T. Ruhtz, C. Lindemann, J. Fischer, and F. Hölker, “Two camera system for measurement of urban uplight angular distribution,” AIP Conf. Proc. 1531, 568–571 (2013).
[Crossref]

H. U. Kuechly, C. C. M. Kyba, T. Ruhtz, C. Lindemann, C. Wolter, J. Fischer, and F. Hölker, “Aerial survey and spatial analysis of sources of light pollution in Berlin, Germany,” Remote Sens. Environ. 126, 39–50 (2012).
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F. Waquet, P. Goloub, J. L. Deuzé, J. F. Léon, F. Auriol, C. Verwaerde, J. Y. Balois, and P. Francçois, “Aerosol retrieval over land using a multiband polarimeter and comparison with path radiance method,” J. Geophys. Res. 112, D11214 (2007).
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R. C. Levy, L. A. Remer, and O. Dubovik, “Global aerosol optical properties and application to moderate resolution imaging spectroradiometer aerosol retrieval over land,” J. Geophys. Res. 112, D13210 (2007).

Lindemann, C.

C. C. M. Kyba, T. Ruhtz, C. Lindemann, J. Fischer, and F. Hölker, “Two camera system for measurement of urban uplight angular distribution,” AIP Conf. Proc. 1531, 568–571 (2013).
[Crossref]

H. U. Kuechly, C. C. M. Kyba, T. Ruhtz, C. Lindemann, C. Wolter, J. Fischer, and F. Hölker, “Aerial survey and spatial analysis of sources of light pollution in Berlin, Germany,” Remote Sens. Environ. 126, 39–50 (2012).
[Crossref]

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C. B. Luginbuhl, G. W. Lockwood, D. R. Davis, K. Pick, and J. Selders, “From the ground up I: light pollution sources in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 185–203 (2009).
[Crossref]

Lockwood, W.

C. Luginbuhl, D. Duriscoe, C. Moore, A. Richman, W. Lockwood, and R. Davis, “From the ground up II: sky glow and near-ground artificial light propagation in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 204–212 (2009).
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C. Luginbuhl, D. Duriscoe, C. Moore, A. Richman, W. Lockwood, and R. Davis, “From the ground up II: sky glow and near-ground artificial light propagation in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 204–212 (2009).
[Crossref]

Luginbuhl, C. B.

D. M. Duriscoe, C. B. Luginbuhl, and C. D. Elvidge, “The relation of outdoor lighting characteristics to sky glow from distant cities,” Light. Res. Technol. 46, 35–49 (2014).
[Crossref]

C. B. Luginbuhl, G. W. Lockwood, D. R. Davis, K. Pick, and J. Selders, “From the ground up I: light pollution sources in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 185–203 (2009).
[Crossref]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, and R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[Crossref]

Moore, C.

C. Luginbuhl, D. Duriscoe, C. Moore, A. Richman, W. Lockwood, and R. Davis, “From the ground up II: sky glow and near-ground artificial light propagation in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 204–212 (2009).
[Crossref]

Nair, N.

M. Adam, M. Pahlow, V. A. Kovalev, J. M. Ondov, M. B. Parlange, and N. Nair, “Aerosol optical characterization by nephelometer and lidar: the Baltimore Supersite experiment during the Canadian forest fire smoke intrusion,” J. Geophys. Res. 109, D16S02 (2004).
[Crossref]

Olmo, F. J.

H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
[Crossref]

H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

Ondov, J. M.

M. Adam, M. Pahlow, V. A. Kovalev, J. M. Ondov, M. B. Parlange, and N. Nair, “Aerosol optical characterization by nephelometer and lidar: the Baltimore Supersite experiment during the Canadian forest fire smoke intrusion,” J. Geophys. Res. 109, D16S02 (2004).
[Crossref]

Pahlow, M.

M. Adam, M. Pahlow, V. A. Kovalev, J. M. Ondov, M. B. Parlange, and N. Nair, “Aerosol optical characterization by nephelometer and lidar: the Baltimore Supersite experiment during the Canadian forest fire smoke intrusion,” J. Geophys. Res. 109, D16S02 (2004).
[Crossref]

Panchenko, M. V.

M. V. Kabanov, M. V. Panchenko, Y. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Y. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988).

Parlange, M. B.

M. Adam, M. Pahlow, V. A. Kovalev, J. M. Ondov, M. B. Parlange, and N. Nair, “Aerosol optical characterization by nephelometer and lidar: the Baltimore Supersite experiment during the Canadian forest fire smoke intrusion,” J. Geophys. Res. 109, D16S02 (2004).
[Crossref]

Pesava, P.

P. Pesava, H. Horvath, and M. Kasahara, “A local optical closure experiment in Vienna,” J. Aerosol Sci. 32, 1249–1267 (2001).
[Crossref]

Pick, K.

C. B. Luginbuhl, G. W. Lockwood, D. R. Davis, K. Pick, and J. Selders, “From the ground up I: light pollution sources in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 185–203 (2009).
[Crossref]

Pkhalagov, Y. A.

M. V. Kabanov, M. V. Panchenko, Y. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Y. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988).

Portnov, B. A.

F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
[Crossref]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Remer, L. A.

R. C. Levy, L. A. Remer, and O. Dubovik, “Global aerosol optical properties and application to moderate resolution imaging spectroradiometer aerosol retrieval over land,” J. Geophys. Res. 112, D13210 (2007).

Richman, A.

C. Luginbuhl, D. Duriscoe, C. Moore, A. Richman, W. Lockwood, and R. Davis, “From the ground up II: sky glow and near-ground artificial light propagation in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 204–212 (2009).
[Crossref]

Ruhtz, T.

C. C. M. Kyba, T. Ruhtz, C. Lindemann, J. Fischer, and F. Hölker, “Two camera system for measurement of urban uplight angular distribution,” AIP Conf. Proc. 1531, 568–571 (2013).
[Crossref]

H. U. Kuechly, C. C. M. Kyba, T. Ruhtz, C. Lindemann, C. Wolter, J. Fischer, and F. Hölker, “Aerial survey and spatial analysis of sources of light pollution in Berlin, Germany,” Remote Sens. Environ. 126, 39–50 (2012).
[Crossref]

Rybnikova, N. A.

F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
[Crossref]

Sánchez, C.

H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
[Crossref]

H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

Sauerzopf, H.

H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
[Crossref]

H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

Sealey, K. M.

G. P. Box, K. M. Sealey, and M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
[Crossref]

Seidl, S.

H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
[Crossref]

H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

Selders, J.

C. B. Luginbuhl, G. W. Lockwood, D. R. Davis, K. Pick, and J. Selders, “From the ground up I: light pollution sources in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 185–203 (2009).
[Crossref]

Shifrin, K. S.

K. S. Shifrin, “Study of the properties of matter from single scattering,” in Theoretical and Applied Problems in the Scattering of Light (Nauka i Tekhnika, 1971) (in Russian).

Solano-Lamphar, H. A.

M. Kocifaj and H. A. Solano-Lamphar, “Angular emission function of a city and skyglow modeling: a critical perspective,” Publ. Astron. Soc. Pac. 128, 124001 (2016).
[Crossref]

M. Kocifaj and H. A. Solano-Lamphar, “Skyglow: a retrieval of the approximate radiant intensity function of ground-based light sources,” Mon. Not. R. Astron. Soc. 439, 3405–3413 (2014).
[Crossref]

Stamnes, K.

G. E. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, 2006).

Su, W.

N. G. Loeb and W. Su, “Direct aerosol radiative forcing uncertainty based on a radiative perturbation analysis,” J. Climate 23, 5288–5293 (2010).
[Crossref]

Teillet, P. M.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Thomas, G. E.

G. E. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, 2006).

Tikhonov, A. N.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

Travis, L. D.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, and R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[Crossref]

Tuttle, B. T.

C. D. Elvidge, D. M. Keith, B. T. Tuttle, and K. E. Baugh, “Spectral identification of lighting type and character,” Sensors 10, 3961–3988 (2010).
[Crossref]

Twomey, S.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Dover, 2002).

Uzhegov, V. N.

M. V. Kabanov, M. V. Panchenko, Y. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Y. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988).

Veretennikov, V. V.

M. V. Kabanov, M. V. Panchenko, Y. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Y. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988).

Verwaerde, C.

F. Waquet, P. Goloub, J. L. Deuzé, J. F. Léon, F. Auriol, C. Verwaerde, J. Y. Balois, and P. Francçois, “Aerosol retrieval over land using a multiband polarimeter and comparison with path radiance method,” J. Geophys. Res. 112, D11214 (2007).
[Crossref]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

Waquet, F.

F. Waquet, P. Goloub, J. L. Deuzé, J. F. Léon, F. Auriol, C. Verwaerde, J. Y. Balois, and P. Francçois, “Aerosol retrieval over land using a multiband polarimeter and comparison with path radiance method,” J. Geophys. Res. 112, D11214 (2007).
[Crossref]

West, R. A.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, and R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[Crossref]

Wolter, C.

H. U. Kuechly, C. C. M. Kyba, T. Ruhtz, C. Lindemann, C. Wolter, J. Fischer, and F. Hölker, “Aerial survey and spatial analysis of sources of light pollution in Berlin, Germany,” Remote Sens. Environ. 126, 39–50 (2012).
[Crossref]

Zuev, V. E.

V. E. Zuev and G. M. Krekov, Optical Models of the Atmosphere (Gidrometeoizdat, 1986).

Žwak, Z.

Z. Žwak, data provided (personal communication, 2012).

AIP Conf. Proc. (1)

C. C. M. Kyba, T. Ruhtz, C. Lindemann, J. Fischer, and F. Hölker, “Two camera system for measurement of urban uplight angular distribution,” AIP Conf. Proc. 1531, 568–571 (2013).
[Crossref]

Appl. Opt. (2)

Atmos. Environ. (1)

M. Kocifaj, H. Horvath, O. Jovanović, and M. Gangl, “Optical properties of urban aerosols in the region Bratislava-Vienna I, methods and tests,” Atmos. Environ. 40, 1922–1934 (2006).
[Crossref]

J. Aerosol Sci. (3)

C. Dellago and H. Horvath, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements—I, basic considerations and models,” J. Aerosol Sci. 24, 129–141 (1993).
[Crossref]

H. Horvath and C. Dellago, “On the accuracy of the size distribution information obtained from light extinction and scattering measurements–II, case studies,” J. Aerosol Sci. 24, 143–154 (1993).
[Crossref]

P. Pesava, H. Horvath, and M. Kasahara, “A local optical closure experiment in Vienna,” J. Aerosol Sci. 32, 1249–1267 (2001).
[Crossref]

J. Atmos. Sci. (1)

G. P. Box, K. M. Sealey, and M. A. Box, “Inversion of Mie extinction measurements using analytic eigenfunction theory,” J. Atmos. Sci. 49, 2074–2081 (1992).
[Crossref]

J. Climate (1)

N. G. Loeb and W. Su, “Direct aerosol radiative forcing uncertainty based on a radiative perturbation analysis,” J. Climate 23, 5288–5293 (2010).
[Crossref]

J. Geophys. Res. (5)

M. Adam, M. Pahlow, V. A. Kovalev, J. M. Ondov, M. B. Parlange, and N. Nair, “Aerosol optical characterization by nephelometer and lidar: the Baltimore Supersite experiment during the Canadian forest fire smoke intrusion,” J. Geophys. Res. 109, D16S02 (2004).
[Crossref]

F. Waquet, P. Goloub, J. L. Deuzé, J. F. Léon, F. Auriol, C. Verwaerde, J. Y. Balois, and P. Francçois, “Aerosol retrieval over land using a multiband polarimeter and comparison with path radiance method,” J. Geophys. Res. 112, D11214 (2007).
[Crossref]

R. C. Levy, L. A. Remer, and O. Dubovik, “Global aerosol optical properties and application to moderate resolution imaging spectroradiometer aerosol retrieval over land,” J. Geophys. Res. 112, D13210 (2007).

H. Horvath, L. A. Arboledas, F. J. Olmo, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Optical characteristics of the aerosol in Spain and Austria and its effect on radiative forcing,” J. Geophys. Res. 107, D194386 (2002).
[Crossref]

M. I. Mishchenko, L. D. Travis, R. A. Kahn, and R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (2)

M. Kocifaj, “Modeling the night-sky radiances and inversion of multi-angle and multi-spectral radiance data,” J. Quant. Spectrosc. Radiat. Transfer 139, 35–42 (2014).
[Crossref]

M. Kocifaj, “Night sky luminance under clear sky conditions: theory vs. experiment,” J. Quant. Spectrosc. Radiat. Transfer 139, 43–51 (2014).
[Crossref]

Light. Res. Technol. (1)

D. M. Duriscoe, C. B. Luginbuhl, and C. D. Elvidge, “The relation of outdoor lighting characteristics to sky glow from distant cities,” Light. Res. Technol. 46, 35–49 (2014).
[Crossref]

Mon. Not. R. Astron. Soc. (1)

M. Kocifaj and H. A. Solano-Lamphar, “Skyglow: a retrieval of the approximate radiant intensity function of ground-based light sources,” Mon. Not. R. Astron. Soc. 439, 3405–3413 (2014).
[Crossref]

Philos. Trans. R. Soc. London B (1)

M. Aubé, “Physical behavior of anthropogenic light propagation into the nocturnal environment,” Philos. Trans. R. Soc. London B 370, 20140117 (2015).
[Crossref]

Publ. Astron. Soc. Pac. (4)

M. Kocifaj and H. A. Solano-Lamphar, “Angular emission function of a city and skyglow modeling: a critical perspective,” Publ. Astron. Soc. Pac. 128, 124001 (2016).
[Crossref]

C. B. Luginbuhl, G. W. Lockwood, D. R. Davis, K. Pick, and J. Selders, “From the ground up I: light pollution sources in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 185–203 (2009).
[Crossref]

C. Luginbuhl, D. Duriscoe, C. Moore, A. Richman, W. Lockwood, and R. Davis, “From the ground up II: sky glow and near-ground artificial light propagation in Flagstaff, Arizona,” Publ. Astron. Soc. Pac. 121, 204–212 (2009).
[Crossref]

R. H. Garstang, “Model for artificial night-sky illumination,” Publ. Astron. Soc. Pac. 98, 364–375 (1986).
[Crossref]

Remote Sens. Environ. (1)

H. U. Kuechly, C. C. M. Kyba, T. Ruhtz, C. Lindemann, C. Wolter, J. Fischer, and F. Hölker, “Aerial survey and spatial analysis of sources of light pollution in Berlin, Germany,” Remote Sens. Environ. 126, 39–50 (2012).
[Crossref]

Sci. Adv. (1)

F. Falchi, P. Cinzano, D. Duriscoe, C. C. M. Kyba, C. D. Elvidge, K. Baugh, B. A. Portnov, N. A. Rybnikova, and R. Furgoni, “The new world atlas of artificial night sky brightness,” Sci. Adv. 2, e1600377 (2016).
[Crossref]

Sensors (1)

C. D. Elvidge, D. M. Keith, B. T. Tuttle, and K. E. Baugh, “Spectral identification of lighting type and character,” Sensors 10, 3961–3988 (2010).
[Crossref]

Other (12)

A. A. Kokhanovsky, Light Scattering Media Optics: Problems and Solutions, 3rd ed. (Praxis, 2004).

G. E. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, 2006).

V. E. Zuev and G. M. Krekov, Optical Models of the Atmosphere (Gidrometeoizdat, 1986).

Z. Žwak, data provided (personal communication, 2012).

K. S. Shifrin, “Study of the properties of matter from single scattering,” in Theoretical and Applied Problems in the Scattering of Light (Nauka i Tekhnika, 1971) (in Russian).

G. P. Gushchin, The Methods, Instrumentation and Results of Atmospheric Spectral Measurements (Gidrometeoizdat, 1988).

H. Horvath, F. J. Olmo, L. A. Arboledas, O. Jovanović, M. Gangl, W. Kaller, C. Sánchez, H. Sauerzopf, and S. Seidl, “Size distributions of particles obtained by inversion of spectral extinction and scattering measurements,” in Optics of Cosmic Dust, G. Videen and M. Kocifaj, eds., NATO Science Series (Springer, 2002), Vol. 79, p. 143.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Dover, 2002).

F. Cakoni and D. Colton, Quantitative Methods in Inverse Scattering Theory: An Introduction (Springer, 2006).

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

M. V. Kabanov, M. V. Panchenko, Y. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Y. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University, 2007).

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Figures (6)

Fig. 1.
Fig. 1. Decimal logarithms of the kernel functions K(zE;z)/S computed after Eq. (12) for τm=0.15, τa=0.2, ga=0.85, and Ωa=0.9. (a) D=1km, (b) D=10km, (c) D=50km. The remaining parameters are the same as in the text (Section 2).
Fig. 2.
Fig. 2. Same as Fig. 1, but for K(zE;z)/S as a function of τa, ga, and Ωa. (a) τa=0.5 (solid line), τa=0.1 (short dashes), τa=0.03 (long dashes); (b) ga=0.5 (solid line), ga=0.7 (short dashes), ga=0.95 (long dashes); (c) Ωa=0.7 (solid line), Ωa=0.5 (short dashes), Ωa=0.3 (long dashes).
Fig. 3.
Fig. 3. (a) Synthetic radiance data (black triangles) and radiance function generated using CEF found in the process of inversion. (b) CEF used as an input to the regularization algorithm (solid line) and CEF obtained in the process of minimization.
Fig. 4.
Fig. 4. Synthetic CEFs (solid lines) of different types computed in accordance with Eq. (20) and the corresponding regularized solutions (dashed lines).
Fig. 5.
Fig. 5. CEF (in arbitrary units) obtained from a field experiment with local lights turned off (blue short dashed line) and turned on (black long dashed line). Triangles are used for CEF determined from background-compensated radiance data, while Garstang’s fit is demonstrated as the red solid line.
Fig. 6.
Fig. 6. (a) Sky radiance (in arbitrary units) taken in the city of Žilina and the reconstructed one. (b) Solution to the inverse problem.

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

zE=0π/2K(zE;z,a)I(zE)dzE=L(z,a).
Kλ(zE;z,a)=M(z)St[h(zE),zE]t[h(zE),z]h2(zE)×dh(zE)dzEΓ[h(zE),zE,z,a]cos3zEds,
tan2(zE)=tan2(z)+Di2h22Dihtan(z)cos(acityaipixel),
aipixel=acityarccos(D2+Di2Ri22DiD),
xi=Dcos(acity)+Ricos(acityφi),yi=Dsin(acity)+Risin(acityφi),
Kλ(zE;z,a)=M(z)Stλ[h(zE),zE]tλ[h(zE),z]h2(zE)×dh(zE)dzEΓ[h(zE),zE,z,a]cos3zE,
d[tan2(zE)]dh=2tanzEcos2zEdzEdh=2D2h3+2Dh2tanz,
dhdzE=D1+tan2zE(tanztanzE)2,
h(z)=Dtanz+tanzE.
Γλ(h,θ)=τmHmexp{h(z)Hm}3(1+cos2θ)16π+ΩaτaHaexp{h(z)Ha}1ga24π(1+ga22gacosθ)3/2,
t[h(zE),zE]t[h(zE),z]=exp{[M(zE)+M(z)]×[τm+τaτmexp(h(z)Hm)τaexp(h(z)Ha)]},
K(zE;z)=SDcos3zEcosz(1+tan2zE)4π×exp{coszE+coszcoszcoszE×[τm+τaτmexp(h(z)Hm)τaexp(h(z)Ha)]}×{τmHmexp[h(z)Hm]3[1+cos2(z+zE)]4+τaHaexp[h(z)Ha]Ωa(1ga2)[1+ga2+2gacos(z+zE)]3/2},
KI=L,
I=K1L.
Kmn=Km(zn)=Gm1,m(zn)+Gm,m(zn)+Gm+1,m(zn),m1,mMK1n=K1(zn)=G1,1(zn)+G2,1(zn),KMn=KM(zn)=GM1,M(zn)+GM,M(zn),
Gm,p(zn)=(zE,m1+zE,m)/2(zE,m+zE,m+1)/2K(zE;zn)ωm(zE)(zEzE,p)ωm(zE,p)dzE,
ωm(zE)=(zEzE,m1)(zEzE,m)(zEzE,m+1),
Bmn=(zE,mzE,m1)1,n=m1,m=2M,Bmn=1zE,mzE,m1+1zE,m+1zE,m,n=m,m=1M,Bmn=(zE,nzE,n1)1,n=m+1,m=1M1,Bmn=0,|nm|>1,m,n=1M.
Kcos(zE;z)=K(zE;z)coszECEF(zE)=I(zE)coszE,
CEF(zE)=GEF(zE)[2G(1F)coszE+0.554FzE4],

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