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
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 ). 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 . 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 , 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  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 . 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 , 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  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  that extraction of radiance of a round-based light source from observed radiance data is a discrete linear inverse problem, where the mapping between the unknown radiance and measured radiance of a night sky can be formulated as follows:
In Eq. (1), , are the observational zenith and azimuth angles, and is the zenith angle for beams emitted from an artificial light source. 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, accounts for the superposition of optical signals from all light-emitting pixels with a total surface area of :16], while for zenith angles below 80°–85°. For long optical paths, the refraction by Earth’s atmosphere might be important, especially if approaches 90°. In general, the relation between and depends on the position of a light-emitting pixel 3), the azimuth angle of the th pixel is measured relatively to 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 . However, the notation used here is different from what we have introduced in ; e.g., has been called 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 in the form
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 and formula (2) can be approximated as follows:
Asymmetry parameter 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., ). The lowest possible value of is , meaning that all photons that interact with an aerosol particles are exclusively scattered backward, which is almost unrealistic. The perfect forward scatter with is also rare, while isotropic scattering with is well known in light-scattering theories and used as the equivalent to diffuse reflection. Single scattering albedo can vary over a closed interval from 0 to 1, where is for total absorption (all photons are removed) and is for conservative scattering (all photons are scattered). However, most typically is as large as 0.9–0.95 . It is well established that both aerosol and Rayleigh optical depths depend on the wavelength . The Rayleigh component can be approximated analytically as [19,20], but the aerosol optical depth is a complex function of aerosol microphysics that in turn makes a non-trivial function of . Nevertheless, a set of approximate formulae for aerosol optical depth are in use, while most consider to be inversely proportional to the wavelength (e.g., [21,22]). The typical values of for the visible spectrum are 0.1–0.6. The remaining two parameters, and , are called molecular- and aerosol-scale heights and can be obtained under assumption of an exponentially stratified atmosphere. Here, (or ) 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 and 1–2 km for [23,24]. Thanks to the simple analytical form of the transmission coefficients, we can find1) 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 ).
We have computed 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 decrease steeply as approaches zero. This makes the model sensitive to observations taken at different zenith angles.
3. INVERSION OF SKY RADIANCE MEASUREMENTS
Equation (1) represents a mapping from data region to sky radiance data region using an integral transform operator . However, most often is an unknown function that has to be determined from the experimental data of . The inversion of radiance data is a typically ill-posed problem, because the algebraized form of Eq. (1),
It is not surprising that some elements of the vector obtained from Eq. (14) can appear negative because a small eigenvalue of produces large elements in , and thus any small experimental error of can be extremely amplified. This is why many solutions satisfying the measure of “goodness” may exist, thus implying there is a great difficulty in finding a correct value even if the oscillatory forms of 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 , such as , where is usually a near-diagonal matrix and is the transposed matrix of . The solution vector is then found by minimizing subject to the constraint . Considering that is an element of Hilbert space of quadratically integrable functions (a compact operator from space to radiance space), then a unique solution to vector must exist so that , where is an undetermined Lagrangian multiplier (i.e., the parameter of regularization; see pages 36–43 in ). The solution of Eq. (13) is difficult even for over-determined systems wherein because 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 . 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 . We proceed here with the Tikhonov inversion adapted by Kabanov et al.  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 functions26,29] in the form , , , and the elements of the matrix are as follows:
The nature of diagonal and near-diagonal elements of the symmetric matrix is explained in detail in  (see Section 19.5), where the symbol is used instead of . In our convention 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):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 to the interval of values from to , thus ensuring that CEF and its first derivative will both be smooth and continuous functions of near zenith. More specifically, we require that exists at , 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 (and also for ) results in the fact that , 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 or ground radiance .
Finally, we have made a sensitivity study to evaluate the effect of , , and on . 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 has a small impact on the unmodified (original) kernel except for situations when observations are made in the near-horizon zone. However, aerosol optical thickness () as well as single scattering albedo () can both influence considerably at intermediate zenith angles also, meaning that retrieval of CEF would be sensitive to both aerosol properties analyzed here.
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 ,
We have computed the radiance for zenith angles using , , , as input data to the model. The radiance data 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 (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 that controls solution was determined using a rapidly convergent iteration method. Basically, the procedure we have used is straightforward because has a strictly monotonous and continuous form when evaluated as a function of . 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.
A controlled experiment has been performed by Žwak  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 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 and . Note that has no relation to the ground albedo. Instead, and 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 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 as small as 0.5 km, but we do not recommend use of below 0.1 km for numerical convergence reasons. The reproduced data of sky radiance are found to be consistent with those observed at [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 () differs from that of the vector for measured radiance () was expressed here by means of overall error . 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 can be estimated from the product of and when drawn for different input parameters as a function of . The information content is generally expected to be low, when has almost no effect on near its peak values or when it influences only the lowest values of .
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 for city 1. This kind of generating function allows for obtaining a 0th solution to the emission function for city 2 () using the radiance data measured on a local meridian of city 2. Retrieval of is possible because the product of and can be computed numerically, meaning that the theoretical contribution of city 1 to the sky radiance is known, and then is the only function to be determined. In the next round, the product of and a respective kernel function 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 . Integrating the product of and , 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 analogously to what we have made before. If convergent, the scheme just described guarantees the solutions for and would improve with increasing the number of iterations. The procedure is terminated if the conditions and are both satisfied simultaneously. Here, and 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 , 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 . Garstang’s model [Eq. (20)] is also a good alternative for modeling the zero-order approximation of the .
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  using nephelometers or lidars .
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 , 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.
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).
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