scattering radiance in limb-viewing geometry

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 104, NO. D24, PAGES 31,261-31,274, DECEMBER 27, 1999 Multiple scattering radiance in limb-viewing geometry Liisa Oikarinen, Elina Sihvola, and Erkki Kyr61i Geophysical Research Division, Finnish Meteorological Institute, Helsinki, Finland Abstract. At present, satellite-based limb-viewing measurements in near-uv, visible, and near-ir wavelength range are based on the attenuation of direct solar light (the Stratospheric Aerosol and Gas Experiment instruments). This paper studies a new technique: the measurement of backscattered solar radiance spectrum in limb-viewing geometry. A multiple-scattering backward Monte Carlo algorithm "Siro" has been constructed for realistic radiative transfer modeling of these measurements. By Monte Carlo simulation the difficult spherical geometry of limb-viewing can be accurately modeled, as can constituent densities and boundary conditions that vary in three dimensions. Previous multiple-scattering models applicable to limb-viewing all assume a spherical shell atmosphere. The backward technique is very efficient for simulating a receiver that has a narrow field of view. In this paper the role of multiple scattering is studied by the Siro model in an atmosphere including scattering by molecules and aerosols and absorption by 03. Simulations show that the multiple to total scattering ratio increases from almost zero at 300 nm to % at visible and near-ir wavelengths (depending on solar geometry and albedo of Earth's surface). A single-scattering model is not sufficient for the analysis of limb radiance measurements. When the solar zenith angle is small, limb radiance is very sensitive to the surface albedo. A bright spot of diameter 50 km on an otherwise dark surface already causes a noticeable increase of intensity. 1. Introduction The near-uv, visible, or near-ir spectrum of sunlight scattered by the atmosphere contains a great deal of information on the composition of the atmosphere. Several new remote sensing instruments measure a continuous radiance spectrum in this wavelength range. Presently, satellite-based measurements are made in nadir-viewing geometry (e.g., the Global Ozone Monitoring Experiment (GOME) instrument onboard ERS- 2). With this technique, total ozone columns and some other trace gases as well as vertical concentration profiles with a low vertical resolution (7-10 kin) can be obtained [Burrows et al., 1999]. This vertical resolution is not sufficient for all scientific applications. For example, the understanding of stratospheric chemistry and dynamics would benefit from a better altitude resolution of measurements. Altitude resolution in the stratosphere can be improved by limb-viewing measurement geometry. The line of sight (LOS) of a limb-viewing detector is a tangential path through the atmosphere (Figure 1). The Now at Department of Physics, University of Helsinki, Finland. Copyright 1999 by the American Geophysical Union. Paper number 1999JD / 99 / 1999 JD ,261 limb is scanned vertically, and spectra are measured at different tangent altitudes. This means that a good vertical resolution (1-3 kin) is inherent to these measurements. The stratospheric optical depth of the LOS is longer than in nadir viewing so a greater sensitivity to small amounts of stratospheric absorbers is to be expected. Limb-viewing geometry is already employed by the Stratospheric Aerosol and Gas Experiment (SAGE) solar occultation instruments [Mauldin et al., 1985], which measure the attenuation of direct solar UV, visible, and near-ir radiation. With the solar occultation tech- nique, only one sunrise or sunset can be measured during one satellite orbit; that is, only one or at most two profile soundings can be made per orbit. Also, the measurements only occur at twilight, when illumination and photochemistry in the atmosphere are changing rapidly. The geographical coverage of measurements is improved by stellar occultation, a technique to be used by the Global Ozone Monitoring by Occultation of Stars (GO- MOS) instrument [Bertaux et al., 1991]. Stellar occultation is a good technique at the nightside of the Earth, but daytime measurements suffer from a high noise level caused by backscattered solar background. Typically, the brightness of Earth's limb increases exponentially toward lower altitudes, and the accuracy of bright limb stellar occultation measurements decreases rapidly toward lower altitudes.

2 31,262 OIKARINEN ET AL.: MULTIPLE SCATTERING IN LIMB-VIEWING GEOMETRY SCIAMACHY ] GOME Sun OSIRIS SCIAMACHY [ GOMOS background 1 Figure 1. The geometry of limb-viewing and nadir-viewing measurements of backscattered solar light. Limb-viewing instruments include the Optical Spectrograph and InfraRed Imager System (OSIRIS), scanning imaging absorption spectrometer for atmospheric cartography (SCIA- MACHY), and Global Ozone Monitoring by Occultation of Stars (GOMOS). Nadir-viewing instruments include the Global Ozone Monitoring Experiment (GOME) and SCIAMACHY. This paper studies a new limb-viewing technique: measurement of backscattered solar radiance spectrum in limb-viewing geometry at UV, visible, and near-ir wavelengths. Because radiance measurements can be carried out always when the LOS of the instrument is illuminated, this technique can fill the dayside gap of occultation measurements. Three European limb-viewing instruments that measure backscattered solar radiance will be operating in next few years. The Optical Spectrograph and InfraRed Imager System (OSIRIS) instrument [Payne et al., 1995] on the Swedish satellite Odin will be launched in May Scanning imaging absorption spectrometer for atmospheric cartography (SCIAMACHY) [Bovensmann et al., 1999] and GO- MOS will be launched on the Envisat-1 satellite of the European Space Agency in the year This measurement technique has already been used in UV between 88 and 340 nm by the UV spectrometer of the Solar Mesosphere Explorer (SME) to measure ozone profiles in the mesosphere [Rusch et al., 1984]. Accurate radiative transfer modeling is necessary for the development of retrieval algorithms for the limbviewing radiance measurements. Because of the essentially different geometry, radiative transfer models used with ground-based measurements or nadir-viewing instruments like GOME [Rozanov et al., 1997] cannot directly be applied to limb viewing. Development of retrieval algorithms for limb radiance measurements in the near-uv, visible, and near-ir wavelength range is at an early state. Accurate and well-validated radiative transfer models for this geometry are not yet publicly available. A radiative transfer (RT) model for simulating limbviewing measurements must be able to model spherical geometry and multiple-scattering radiance. A very accurate model also takes into account refractive bending and polarization of light. Lenoble [1985] and Thorne [1990] review RT models for spherical atmospheres in general. Marchuk et al. [1980] review Monte Carlo models for spherical atmospheres published in Russian journals. Limb-viewing geometry has been modeled in spherical shell atmospheres by Monte Carlo simulation [Collins et al., 1972; Adams and Kattawar, 1978; Kattawar and Adams, 1978; Marchuk et al., 1980] and by a model that uses a Gauss-Seidel iteration scheme [ Thorne, 1990; Herman et al., 1994, 1995]. The widely used and publicly distributed moderate resolution transmittance code (MODTRAN) developed by the U.S. Air Force Research Laboratory (AFRL) [Berk et al., 1989] fulfils most of the requirements of limb geometry. Single-scatter (SS) solar radiance and all layer transmittances in the spherical shell atmosphere are calculated using full spherical refractive geometry. Multiple-scattering (MS) source functions are supplied by the plane parallel discrete ordinates radiative transfer method (DISORT) [$tamnes et al., 1988]. MS results for near-horizon Sun are inaccurate, and the model can only be used for solar zenith angles smaller than 90 ø (i.e., no modeling of twilight conditions). Unfortunately, the current implementation of DISORT to MODTRAN can also be inaccurate for treatment of limb geometry. The calculation of layer source functions is performed for the observer viewing angle, instead of the correct viewing angle, which varies along the LOS. Also, the MS source functions are averages over solar azimuth (A. Berk, Spectral Sciences, Inc., personal communication, 1998). The MODTRAN model includes a variety of spherical model atmospheres and is able to simulate radiance at all wavelengths above 200 nm (including band models for absorption in the IR). Although some suitable RT models exist, the limbviewing measurement technique has been studied only

3 OIKARINEN ET AL.: MULTIPLE SCATTERING IN LIMB-VIEWING GEOMETRY 31,263 a little. Marchuk et al. [1980](p. 123) have investigated the angular and spectral distribution of intensity in limb-viewing geometry by a multiple-scattering Monte Carlo model. Their results show that the MS term is important: multiple scattering was found to constitute 30-60% of the total intensity when solar zenith angle is small. The atmospheric model included scattering by aerosols and molecules and absorption by aerosols and ozone. Only five wavelengths between 400 and 800 nm were studied, and the statistical accuracy of the Monte Carlo calculations was not better than 10%. Thorne [1990] has investigated the sensitivity of limb radiance to the amount of ozone and volcanic dust in the strato- sphere. The role of multiple scattering is important because it reflects the complexity of radiative transfer modeling required in the analysis of limb measurements. In this paper we extend the earlier studies of the amount of multiple scattering and light reflected from ground to cover all possible solar positions and also the UV wavelength range. Two other topics important for the analysis of limb scan data are also studied shortly. We investigate from what altitudes of the atmosphere measurements at a given tangent altitude and wavelength actually carry information. The last study examines the effect of a nonuniform surface albedo. We have developed a Monte Carlo algorithm "Siro" (" sirota" is the Finnish word for "to scatter") especially for multiple-scattering simulations in limb-viewing geometry with a realistic model atmosphere. The Monte Carlo technique was selected because of its conceptual simplicity and minimal need of geometrical approximation. In a Monte Carlo simulation the radiance eters and boundary conditions are allowed to vary with latitude and longitude in addition to altitude. This is accomplished by introducing constituent profiles into the code by continuous (stepwise analytical) functions. Siro models refractive bending, but polarization of light is not considered in the initial version. Section 2 of this paper gives a short introduction to the OSIRIS, SCIAMACHY, and GOMOS (GOMOS background term measurement) instruments. Sections 3 and 4 introduce the Siro model. Section 5 presents new results by Siro. 2. Some Future Limb-Viewing Instruments Among the main applications of this paper are the SCIAMACHY [Bovensmann et al., 1999] and GOMOS [Bertaux et al., 1991] instruments on Envisat-1 of European Space Agency and the OSIRIS instrument [Payne et al., 1995] on the Swedish Odin satellite. SCIA- MACHY will measure scattered radiance alternately with nadir viewing and limb viewing in (or near) the plane of the satellite trajectory so that the same volume of atmosphere will be observed by both methods. The stellar occultation instrument GOMOS will mea- sure limb radiance spectra as a background term, which is measured independently of the stellar signal. Table 1 summarizes some characteristics of these three instru- ments. Vertical profiles of aerosol extinction and concentration of 03, NO2, and some other gases absorbing in the spectral range of the instruments are to be retrieved from the measurements. The direction from which the solar beam enters the is determined statistically by following a large number of individual photon trajectories through the atmosphere. Siro uses the backward (or adjoint) Monte Carlo method, which was first introduced to atmospheric opatmosphere relative to the instrument's LOS has a large effect to the radiance observed by the detector. Figure 2 explains the coordinates that are used in this work to specify the direction of Sun (zenith OT and azimuth bt tics by Collins et al. [1972], Adams and Kattawar relative to the tangent point of LOS). The relative di- [1978], and Kattawar and Adams [1978]. The same idea rection of Sun depends on the satellite orbit and the was also presented by Marchuk et al. [1980]. Lenoble azimuth range in which the instrument's FOV is aland Chen [1992] and Perliski and Solomon [1993] have lowed to vary. Figure 3, which has been produced by used the backward Monte Carlo method to compute air a simplified orbit propagator, gives an idea of the pomass factors for ground-based zenith sky measurements sition of Sun in the measurements by OSIRIS, SCIAduring twilight conditions. The word "backward" refers to time reversal: the photons are started from the detector, and their path is followed backward to the point where they leave the atmosphere toward the source, the Sun. This technique is very effective when the receiver MACHY, and GOMOS. In the nominal viewing direction of OSIRIS and SCIAMACHY (LOS in orbit plane) the possible directions of Sun (OT, qst) form approximately a circle, the radius of which depends on the orbit (the descending node of Odin is 0600 LT, and Envisat's has a narrow field of view (FOV). Now all simulated is 1000 LT) and the month of the year. The maximum photons contribute to the signal, whereas in a forward simulation, only a small fraction of the photons coming from the Sun would eventually reach the instrument. radius occurs at summer solstice, and the minimum occurs in February. GOMOS is pointed at stars of varying azimuths relative to the orbit plane, and the relative di- Siro differs from the spherical models presented by other authors by being fully three dimensional. Previrection of the Sun is more scattered. ous algorithms used a spherical shell model atmosphere, where the reflective properties of the ground do not vary with geographical location and the density profiles are only functions of altitude. In Siro, atmospheric param- 3. Radiative Transfer Modeling The radiance spectrum Ix(rdet, - det) detected by an instrument at location rdet from direction det (the

4 31,264 OIKARINEN ET AL.' MULTIPLE SCATTERING IN LIMB-VIEWING GEOMETRY Table 1. Some Characteristics of Future Satellite-Based Spectrographs OSIRIS, SCIAMACHY, and GOMOS for Measuring Limb Radiance at Near-UV, Visible, and Near-IR Wavelengths. OSIRIS SCIAMACHY GOMOS Background Term Plat form Odin Envisat- 1 Envisat- 1 Launch date May 2000 in 2001 in 2001 Spectral range nm nm, nm, nm, and nm, nm nm, and nm Spectral resolution I nm (UV) to nm _>5 nm (vis) and 2 nm (vis) _ 0.7 nm (IR) Instantaneous FOV vertical I km vertical 3 km minimum vertical I km horizontal 23 km horizontal 100 km minimum horizontal 35 km b Note that the Global Ozone Monitoring by Occultation of Stars (GOMOS) parameters are for the background term measurement; they are not all valid for the occultation measurement. The Optical Spectrograph and InfraRed Imager System (OSIRIS) values are from technical reports of the Odin group; the scanning imaging absorption spectrometer for atmospheric cartography (SCIAMACHY) and GOMOS values are from relatedocuments by the European Space Agency. qnstantaneous field of view (FOV) gives roughly the spatial resolution for small integration time (0.1 s). bfov for a nominal 0.5 s exposure depends on the scewness of the occultation. The worst vertical resolution is - 3 km (occultation in orbit plane). negative of LOS direction) is modeled by solving the radiative transfer equation in a model atmosphere. Vlx(r, ) -- -k Xt(r)[Ix(r, )- Jx(r, )]. (1) An upper boundary condition is set by the incoming solar beam at the top of the atmosphere, and a lower boundary condition is determined by the reflective properties of the ground or a cloud top surface. Detector I I Tangent altitude Sun Line of sight q0r (LOS) ' Tangent point Figure 2. Coordinates for characterizing the geometry of a limb measurement. The direction of the Sun relative to the detector line of sight (LOS) is defined by the zenith angle 0T and azimuth angle bt at the LOS tangent point. The zenith angle is defined as the angle between the z axis and the direction of the Sun. The azimuth is defined as the angle between the LOS and the projection of the Sun's direction on the plane perpendicular to the z axis. In (1), Ix(r, ) is the radiance at point r in the atmosphere propagating to direction. The first term in the right-hand side of (1) presents loss of radiation. The total volum extinction coefficient k Xt(r) includes absorption and scattering by molecules and particles. The source function Jx(r, ) gives the gain of radiation. At the wavelengths and atmospheric region studied in this work the source function is only due to elastic scattering by molecules and particles: Jx (r, ) = coo(r) / Px (r,, ')I s+ms (r, ')d '. 4 r (2) Single-scattering albedo coo(r) at location r is the ratio of the scattering coefficient to total extinction coefficient k[catt(r)/k}xt(r). The scattering phase function Px(r,, ) is a weighted sum of phase functions for molecular and particle scattering: P (r,, ') +P (r,, ') k[ - i ) ). Function k scatt (r) is the coefficient for scattering from molecules (equal to the product of the spectral scattering crossection and the local density) and k scatt(r) is the coefficient for scattering from aerosols. The total scattering coefficient k Catt(r) at wavelength A and point r is k}catt(r) - k SCatt(r) + k SCatt(r). (4) We call SS that part of radiation where a photon reaching the detector has undergone only one scattering event on its way through the atmosphere. All other

5 OIKARINEN ET AL.: MULTIPLE SCATTERING IN LIMB-VIEWING GEOMETRY 31,265 Coordinates of Sun at tangent point [ sciamach Y 30...! 'za:"-: "-... -,".'.x. :... :... _ / / ß ß 90[... ::::::: ':::; i:::'..i::::::i: :i::][::!!f...' detector, T} S(s) is the transmittance of the path from :':":':::':: :':":':-.::'::., ß : -:..:-'.' '.. the Sun to the scattering point r(s), and T P(s) is the ::::::::. :.,. :::::::::::::::::::::::::::::.',,:::::i.:.. ß... : ::::::::::::::::::::::::::::::::::::::::::::::::::::::: 90 1 Azimuth (o)(measured from line of sight) Figure 3. The direction (OT, CfT) of the incident solar beam relative to the LOS of OSIRIS, SCIAMACHY, and GOMOS. For OSIRIS the possible directions are inside the innermost circle, and for SCIAMACHY the directions are between the two outermost circles. The dots present the direction of the Sun for GOMOS (all possible occultations with stars brighter than magnitude two and one orbit/month). For GOMOS, azimuths 180ø< bt <360øhave been mapped to the mirror image in 0 ø < bt < 180 ø. The shaded area gives approximately the dark limb where no scattered solar light is measured. photons belong to the MS term. Those photons scattered only once by an atmospheric molecule or particle but also reflected by the ground are considered as multiply scattered (we assume that the FOV of the detector is so small that ground-reflected light can not be seen directly in limb-viewing geometry). The SS component Ix ss can be expressed as an integral over the LOS of light scattering toward the detector ixss(rdet, det)_ i, un / T s($)t) p($) LOS ß P$ [ride t ($), n0($), r(s)] k} cart [r(s)]ds. (s) Here s is a distance along the LOS measured from the transmittance of the path from the scattering point to the detector. The solar irradiance incident on the at- mosphere is denoted by I un. It arrives to the LOS with direction f 0(s). Because of refraction, f 0(s) and the apparent direction of the detector f det(s) vary slightly along the LOS. The SS term can easily be solved by numerical integration of (5) and is therefore a commonly used approximation for the scattered radiance. The atmospheric model used in the Monte Carlo simulations of this work includes scattering by molecules and aerosols and absorption by ozone. The vertical distribution of absorption and scattering coefficients is shown in Figure 4 for 250 nm, where Oa absorption is the dominating process, and 500 nm, where the absorption coefficient of Oa is of the same order of magnitude or smaller than the scattering coefficients. The profiles present the U.S. standard atmosphere, rural boundary layer aerosols with surface visibility 23 km, summer tropospheric aerosols, and background strato- spheric aerosol conditions (stepwise analytical functions fit to the corresponding profiles in the MODTRAN model). The Rayleigh scattering cross section is taken from the MODTRAN model. For ozone UV absorption cros sections by Paur and Bass [1985] and visible and near-ir cross sections from a compilation by E. P. loo Wavelength 250 nm \ I,! -'q I : I ' Wavelength 500 nm 100 :,,,, 80 60!... ;...?::,,,i... i i... ";.. xi '. : ; : 60...:... ;-'::",.:... ;... :... i... i ø Extinction coefficient (km- 1) 10 's 10 '6 10 '4 10 '2 Extinction coefficient (km- 1) Figure 4. Extinction coefficients used in the simulations at 250 and 500 nm. Rayleigh extinction is shown by a solid line, aerosol extinction is shown by a dashed line, and ozone extinction is shown by dotted line.

6 31,266 OIKARINEN ET AL.' MULTIPLE SCATTERING IN LIMB-VIEWING GEOMETRY Rayleigh!!!... Aerosol _._ ;... ß ::::::::::::::::::::::::::::::::::: i......, ß, ¾--4...,... :::::::::::::::::::::::::::::::::::::::::::::::::::: Scattering angle (ø) Figure 5. Phase functions for Rayleigh scattering and aerosol scattering (Henyey-Greenstein with asymmetry factor!7-0.75). solar beam illuminating the whole dayside of the atmosphere [Collins et al., 1972]. In a typical limb case the backward method automatically concentrates the first scattering event close to the tangent point of the LOS, which is exactly the area where the largest contribution to the scattered radiance can be expected to originate. The basic idea of Siro is similar to the "Flash" algorithm presented by Collins et al. [1972]. The essential difference of the two methods is in the presentation of atmospheric and surface parameters and the calculation of optical depths for the photon paths. These are accomplished in Siro in such a way that spherical symmetry of the model atmosphere is not required, whereas Collins et al. [1972] assume a spherical shell atmosphere. Siro starts with a computation of the LOS of the instrument. We assume the instrument FOV is very narrow and shoot photons to one discrete direction only. The LOS is traced with short steps of length As. The Shettle (Naval Research Laboratory, Washington, D.C., total optical depth ri (measured along the LOS) from personal communication, 1994) are used. the detector to each cell i of the LOS is then calculated. The phase functions P and P are taken to be in- Within each small cell the absorption and scattering codependent of r. A Rayleigh scattering phase function efficients are assumed to be constant. The first (or last with alepolarization equal to 0 is used [see, e.g., Lenoble, 1993, Equation (12.14)]. P is modeled by the if counting in the forward direction) scattering cell for a photon is determined by finding the largest ri that Henyey-Greenstein phase function with asymmetry facsatisfies tor = 0.75 [Lenoble, 1993, Equation (13.20)]. The phase functions have been plotted on Figure 5 as a function ri _<-ln[1- RN(1 - TLøS)] (6) of the scattering angle (the angle between f and where RN is a uniformly distributed random number Reflection from the ground is assumed to be Lamberbetween 0 and 1. Transmittance to the last point of tian, and a surface albedo of 0.3 is used unless otherwise the LOS is denoted by T Løs The scaling factor 1 - stated. The Siro algorithm can easily be modified to T Løs forces a scattering to occur at some point along include other absorbers, more complicated phase functhe LOS. The photon weight, initially equal to 1.0, is tions (e.g., Mie aerosol phase functions), and different multiplied by the single-scattering albedo co0(rl) at the types of surface reflection (which can all vary with altiselected first scattering point rl to prevent the bias from tude, latitude, and longitude), but for the purposes of requiring the interaction to be a scattering event and by this paper the presented model is sufficient. 1 - T Løs to preventhe bias from forcing a scattering 4. Backward Monte Carlo Algorithm to occur along the LOS. Siro If the solar zenith angle is so large that the LOS tangent point is in shadow (only one or both ends of LOS In a Monte Carlo computation one photon at a time is followed along its three-dimensional path through the atmosphere. The photon's probability to be absorbed or scattered at a given point in the atmosphere is calculated, and random numbers are used to select the results of collisions. Because the propagation of light illuminated), instead of (6), a different sampling scheme suggested by Marchuk et al. [1980, p. 89] is used. The first scattering cell i is sampled from the uniform distribution 1/N LOs, 1 _ i _ N LOs, where N LOs is the number of simulation steps along the detector LOS. To eliminate bias, the photon weight is multiplied by is reversible, the probability of a photon going through ;½ s (- )W the atmosphere along a particular path is independent of the direction in which we follow the path. The back- Next the SS contribution from this photon is calculated by multiplying the photon weight by the probaward Monte Carlo method uses this idea. We shoot bility that the photon would come directly from Sun to photons from the detector and trace their paths backward. When the receiver FOV is small, the backward technique increases the speed of the simulation drastically because those photons, which in the forward method would never reach the FOV of the detector, will now not be simulated at all. Another advantage over a forward method is a simpler description of the photon source, which is now the detector instead of the point rl. This probability is the product of the transmittance of path r 1-Sun multiplied by the scattering probability to the direction of Sun. The resulting SS photon weight is added to the cumulative SS weight. Several different solar positions can be simulated together, in which case the transmittance of rl-sun and the scattering probability are calculated separately for each solar position.

7 OIKARINEN ET AL.: MULTIPLE SCATTERING IN LIMB-VIEWING GEOMETRY 31,267 We then continue tracing the photon path from r l to simulate higher scattering orders. A random number is used to decide whether the first scattering occurs from a molecule or an aerosol particle. A new propagation direction for the photon is obtained by sampling the scattering angle from the phase function and selecting the rotation angle 5 of the plane of scattering by 5-2 rrn. The photon path is then traced step by step in the new direction until the path intersects the upper boundary of the atmosphere or the ground. In the former case we proceed to select a scattering point and calculate second-order scattering contributions, etc., as in the case of the LOS and SS component. In the latter case, if the selected RN is larger than the optical depth to the ground, the photon hits the lower boundary, and a Lambertian reflection is modeled. The photon weight is multiplied by the albedo of the surface at the point of reflection, and a new direction is sampled from a uniform distribution. The photon's history is followed until its weight becomes smaller than a given minimum value. After the simulation of a sufficiently large set of photons, the cumulative weight of each solar position and scattering order is divided by the total number of photons shot from the detector, which gives the ratio of the detected radiance to incident solar irradiance. A minimum weight of 10-6 and step size of I km are normally sufficient. Refraction of the photon trajectories is taken into account by bending the ray path at the middle point of each short step As. The bending angle for a step is calculated by Snell's law from the values of the refractive index at the end points of the step. The refracted path from a scattering point to the Sun is found by iteration. The statistical error in the retrieved radiance is es- timated from the standard deviation of intensities ob- tained from 10 independent simulations. For example, when 10,000 photon histories are simulated, the results from each subset of 1000 photons are recorded for the estimation of statistical error. Usually, simulation of 10,000 photon histories gives a statistical accuracy better than or equal to 1%. With the atmospheric model presented in section 3 the simulation time for one solar position, one tangent altitude, and one wavelength varies from I min to over an hour in a Silicon Graphics Origin 2000 computer. The required number of photon histories and computer time vary with solar position (small OT requires fewer photons than large 0T), surface albedo, and SS albedo. As a routine correctness check, the SS component of the Monte Carlo simulation is checked against the value obtained by nonstatistical numerical integration of (5). To validate the Siro model, limb-viewing simu- lations presented by Adams and Kattawar [1978], Kattawar and Adams [1978], and Collins et al. [1972] for homogeneous Rayleigh or aerosol scattering atmospheres have been repeated by Siro. Results by Siro agree very well with the SS values and SS/total ratios of Adams and Kattawar [1978] and Kattawar and Adams [1978]. In most cases, results agreed to the last significant digit given. The largest differences in SS/total ratio were 2-5%. The relative difference between Siro and the multiple scattering radiance (SS+MS) of the Flash model [Collins et al., 1972] in a pure Rayleigh scattering atmosphere was on the average 5% and at worst 9% in the 36 cases compared. This is considered acceptable, because the statistical error of the Flash results is large (only 200 photon histories per detector position were simulated), because Flash includes polarization and Siro does not, and because the intensities and solar zenith angles of Flash simulations were estimated from the plots by Collins et al. [1972] and may not be very accurate. 5. Simulation Results In a fixed static spherically symmetric atmosphere the scattered solar radiance is only a function of wavelength, tangent altitude of the LOS, and the relative direction of the Sun. The SS radiance spectra of Figure 6 give a general idea of the spectral and spatial distribution of limb intensity. The radiance spectrum at a given tangent altitude reflects the wavelength dependence of molecular and aerosol scattering and absorption. Radiance is largest at visible or longest UV wavelengths. It decreases rapidly toward the UV because of increasing absorption by ozone. The slow decrease of radiance toward the IR is caused by decreasing molecular and aerosol scattering coefficients. In the optically thin limit the radiance measured along a fully illuminated LOS increases approximately exponentially as the tangent altitude decreases ffi: Wavelength (nm) Figure 6. Single scattering radiance simulated by the moderate resolution transmittance code MODTRAN 3.5 with the U.S. standard atmospheric model and the aerosol model described in section 3. The tangent altitude of detector's LOS is 10, 20, 30, and 50 km; the zenith angle of the Sun 0T=80øand azimuth 0T=90 ø. Spectra have been smoothed to I nm spectral resolution.

8 31,268 OIKARINEN ET AL' MULTIPLE SCATTERING IN LIMB-VIEWING GEOMETRY (a) Total (single + multiple scattering) radiance ø I... o.0 ;, 20 o i... i "'.i " ' ',, ":}' '... }... ' 'x [ : :: 6o..... :......?'.../'......',....':' / loo, 0 2'0 0 6'0 8 ' Azimuth (o) Single scattering radiance 0 :,... o.o2s :.., '01o... '... ; "... k o.o35., ':""""--,..i... "5:>",.q,.'..,,,. i : :..;.:..;...,:i..].: '... (b) loo o 2'0 o i i,, ' ' Azimuth (o) Multiple scattering radiance 20. (c) ß ' o., ""-.,.. - -'-, J ' ' F-' _Z :... i... ' ::::::::::::::::::::::::::::::::::::::::::::::::::::::: i... ::::i ' i,,, Azimuth (o) Figure 7. (a) Total limb radiance in units of incident solar irradiance I,x Sun at / nm as a function of the Sun's direction (0T, OT). The tangent altitude of LOS is 20 km. (b) and (c) The radiance has been resolved to single scattering (SS) and multiple scattering (MS) (second-order and higher) terms. The statistical accuracy of single and total scattering results is better or equal to 1% for OT _<92.5øand 1-40% for 0T=95ø--100ø(worst for large OT and qst=90ø). The accuracy of MS is 1-3% at OT <_80 ø, % at OT = ø and 5-50% for 0T= ø.

9 OIKARINEN ET AL' MULTIPLE SCATTERING IN LIMB-VIEWING GEOMETRY 31,269 We present here three examples of simulation results from the Siro model. The first one investigates the proportion of MS in limb-viewing measurements. A similar study has been presented by Marchuk et al. [1980] for a smaller range of wavelengths and solar positions. In the second example we study what altitude regions of the atmosphere are actually seen by limb soundings at different UV-visible and near-ir wavelengths and LOS tangent altitudes. The last example studies the sensitivity of limb radiance to a bright surface of restricted area under the LOS tangent point Second and Higher Orders of Scattering The amount of MS and the effect of surface albedo are interesting for the analysis of limb measurements. A full MS calculation is very time consuming and would not be performed in a retrieval algorithm unless absolutely necessary. Knowing the magnitude of the reflected component helps to asses the accuracy at which surface albedo should be known to model correctly limb radiance. We study the dependence of limb radiance, and the proportion of multiple scattering especially, on the direction of the Sun (OT, q T) in an average case: tangent altitude of LOS 20 km and wavelength of 500 nm. In this case most of the radiation originates near the tangent point of the LOS (except for cases where the Sun is very close to or below the horizon), and the absorption and scattering coefficients are of the same magnitude (Figure 4). Figures 7, 8, and 9 show results obtained by simulating 10,000 photons per solar position for OT < 95 ø and 20,000 photons for OT = ø, where the uniform sampling scheme was used instead of (6). The Rayleigh scattering phase function peaks for forward and backward scattering, and the aerosol phase function has a strong forward maximum (Figure 5). The limb radiance as a function of solar position reflects the shape of the phase functions clearly (Figure 7). In the MS component the dependence is weaker, partly because the multiple independent scattering events smear out the angular shape of the phase functions and partly because the intensity component that has been reflected from ground by Lambert reflection shows no dependence on 0T. However, even photons that have undergone three collisions (third-order scattering) show small maxima for forward and backward Sun. Extinction along the Sun-LOS path increases as 0T increases; that is, the Sun gets closer to the horizon, and the amount of light reaching the detector decreases. This is why the forward scattering maximum (at 0T=0 ø) does not appear exactly in the direction of scattering angle 0 ø, but at a zenith 0T slightly smaller than 90 ø, and the backward maximum (0T=180 ø) appears around OT=70 ø. The proportion of multiple scattering to total radiance in the same simulation is shown in Figure 8. MS is most important, constituting about half of the total radiance when the solar zenith angle OT is large. In the region of strongestotal radiance (detector looking almost directly at the Sun) the relative contribution of MS is smaller. The ratio MS/total radiance decreases as OT increases, but at very large OT, when the Sun is almost behind the horizon, the proportion of MS starts to increase again. Comparing Figure 8 with Figure 3, which shows the range of (OT, qst) for OSIRIS, SCIA- MACHY, and GOMOS, we see that MS is important for measurements by all the three instruments. Figure 9 shows the proportion of photons whose trajectory includes a reflection from ground. The proportion of reflected light is largest for small solar zenith angles and decreases as 0T increases. As a result, the sensitivity of limb radiance to surface albedo is largest Multiple scattering/total radiance (%) 20 60' " '-" '" - f'".."' --.x... i"...4{ i: ,,i."-"-,,,, '---: '--- _ O0...,,,,.,...i... :::::::::-.-.i-,.... i O Azimuth (ø) Figure 8. Proportion of MS radiance to total radiance as a function of the Sun's direction (OT, 0T) at 500 nm and tangent altitude 20 km simulated by Siro. Statistical error of the results is _ 1% for OT _ 92.5øand 1-80% for other OT (absolut error of MS/total ratio in percent).

10 31,270 OIKARINEN ET AL.: MULTIPLE SCATTERING IN LIMB-VIEWING GEOMETRY Reflected from ground/total radiance (%) :...?'7... 2'...'2 ' 2C'- 2 z:2...-i- "i -': 57œ " 15.. I... ;222.;?..;:: '.? Azimuth (o) Figure 9. Proportion of light that has undergone at least one reflection from ground (Siro simulation for wavelength 500 nm and tangent altitude 20 km). Statistical error is _<1% for 0T _<92.5 øand up to 100% for other 0T (absolute error of reflected/total ratio in percent). at small solar zenith angles and decreases when the Sun gets close to the horizon. For example, when the surface albedo is 0.3, light reflected from the ground constitutes 20% of the total signal at small solar zenith angles (wavelength 500 nm and tangent altitude 20 km). In the extreme case of albedo 1.0 the ratio reflected/total radiance would be 40% at small The ratio MS/total scattering varies significantly with wavelength. Figure 10 shows the wavelength dependence of this ratio for LOS tangent altitudes 20 and 50 km and solar position 0 -=80 ø, 0r=90ø(a typical OSIRIS measurement geometry). Because of large absorption by ozone the proportion of MS is very small at near UV but increases quickly toward longer wavelengths. Between 300 and 350 nm the ratio of multiple scattering to total scattering reflects clearly the features of ozone Huggins band absorption and between 500 and 700 nm it reflects Chappuis bands absorption. The MS/total ratio is also expected to have sharp minima at the wavelengths of water vapor and molecular oxygen absorption bands at long visible and near-ir wavelengths, not simulated in this model (these absorp- 20 lo g o! ]i' Tangent altitude 20 km Tangent altitude 50 km Wavelength (nm) Figure 10. Wavelength dependence of the ratio multiple/total scattering for solar position 0 =80 ø, b =90øand tangent altitudes 20 and 50 km. The UV spectrum ( nm) was simulated at I nm wavelength increments, and the rest of the spectrum was simulated at 5 nm steps. The ratio at UV shows sharp ozone absorption features, whereas most of the fine structure between 350 and 700 nm is due to the statistical inaccuracy of the simulation.

11 OIKARINEN ET AL.: MULTIPLE SCATTERING IN LIMB-VIEWING GEOMETRY 31,271, 100!!, 1 -B-High Sunl visible, albedo \ Low Sun, visible, albedo \ ---I---High Sun, visible, albedo Ii' - -High Sun, UV, albedo ;;,2... : Scattering order Figure 11. The proportion of different orders of scattering to total radiance in four different simulation cases at tangent altitude 20 km. High Sun corresponds to (OT, qbt)=(20 ø, 90ø), and low Sun corresponds to (OT, ½T)=(80 ø, 90ø). The visible wavelength of the simulation was 500 nm, and the UV wavelength was 300 nm. tion bands are visible in the spectra of Figure 6 at low tangent altitudes). In Figure 11 the proportion of different scattering orders to total radiance is shown for four cases: high Sun, visible wavelength, albedos 1 and 0; high Sun, UV, albedo 1; and low Sun visible wavelength, albedo 0. Again, we see that in the UV, SS dominates, and at visible wavelengths the effect of surface albedo is largest for high Sun. The contribution of third and higher scattering orders is significant in the visible. Our results on ratio MS/total scattering and the effect of surface albedo are in agreement with those of Marchuk et al. [1980, p. 123] for five wavelengths between 400 and 800 nm, a set of high and low Sun geometries, and surface albedos 0 and 0.8. Only a qualitative comparison of the results is possible because the atmospheric model used by Marchuk et al. [1980] is not available to us. Probably because of the poor statistical accuracy (10%) of their simulation, Marchuk et al. [1980] failed to see the wavelength dependence of ratio MS/total scattering. look at mean photon path lengths Asi at different layers i (at altitudes zi). More exactly, the path length Asi is a weighted sum of the path lengths ASin of the n- 1,..., N individual photons simulated: N - (7) n=l where w, is the weight of photon n. The altitude distribution of Asi is interesting for the vertical inversion of constituent concentration profiles. The shape of the distribution reflects the true altitude resolution of the measurement technique. Figure 12 shows mean path lengths for wavelength 500 nm and different tangent altitudes. Down to 20 km they have a strong peak at the tangent altitude, but then the altitude distribution becomes flatter. Limb ra- diance at this wavelength will also carry some information about tropospheric species, mainly through light reflected from ground. In Figure 13, mean photon path lengths for a measurement at tangent altitude 10 km have been plotted for different wavelengths. This plot shows clearly that because of strong absorption by ozone in the UV, the photons at wavelength 300 nm or below received by a limb-viewing detector have visited only upper parts of the stratosphere. The lowest altitude from which radiation is received depends on wavelength. Therefore the UV spectrum measured at just one (low) tangent altitude already includes plenty of altitude-resolved information. This is not a surprising result, because the retrieval of ozone profiles by nadir-viewing instruments is based on the same fact. On the other hand, if only the UV region of the limb spectra were used for constituent retrieval, the true altitude resolution of the measure- Wavelength 500 nm 5.2. Altitude Distribution of Mean Photon Path Lengths When the transmittance to the tangent point of the LOS is above 0.1 (approximately), most of the radiation observed by a limb-viewing instrument originates near the tangent point of the LOS. This applies for visible and near-ir wavelengths at tangent altitudes higher than - 15 km. To study more closely the altitude region relevant for the radiance measurement at a given wavelength and tangent altitude, we divide the atmosphere into 1 km thick concentric altitude layers and Average photon path length (km) Figure 12. Mean photon path lengths (7) for wavelength 500 nm and different tangent altitudes of LOS (0r=80 ø, ½r=90ø).

12 31,272 OIKARINEN ET AL' MULTIPLE SCATTERING IN LIMB-VIEWING GEOMETRY 8O! 60 < o Tangent altitude 10 km......! 300 nm 350 nm 500 nm nm _L-L=--. '.:'-: ':-, i, i... I Average photon path length (km) Figure 13. Mean photon path lengths (7) for different wavelengths (tangent altitude 10 km, and a =80 ø,,=9oø). ments would not be determined so much by the FOV of the instrument but more by the wavelength dependence of UV absorption A 3-D Example: Nonuniform Surface Albedo We now study how a completely reflecting surface (albedo 1) of circular area under the tangent point of the LOS affects the radiance measured at 500 nm, tangent altitude 20 kin, and at=30 ø, bt=90 ø. The diameter ds of the bright spot is varied, and the surface albedo outside the spot is 0. The simulation presents a situa- Table 2. Sensitivity of Limb Radiance I:x/I u' (in Units of Incident Solar Irradiance) to a Bright Circular Spot on the Earth's Surface Under the Line of Sight (LOS) Tangent Point. oz, deg do,, km zo,, km I,X/I Sun (I- Io)/Io, % I tion where the land under the LOS is partly covered by snow or low-altitude clouds. From the results one can estimate the spatial resolution at which surface albedo should be known to simulate realistically limb radiance over a given geographicalocation. Table 2 gives the received radiance as a function of the (half) central angle a spanned by the bright spot (Figure 14). Also, the diameter ds of the spot and the altitude z at which the LOS of the detector crosses the boundary of the two surface types are given. In this case of quite extreme albedos the surface properties at several hundreds of kilometers away from the tangent point (up to 500 kin) have some effect on the detected radiance. A' small bright spot of diameter 50 km just under the tangent point has a noticeable effect in this simulation of high Sun (small at). However, if at is large or if albedo variations of the surface are less extreme than in this example, surface albedo variations of diameter 100 km or smaller can safely be neglected in a model. 6. Conclusions In the model atmosphere described in section 3, MS constitutes 10-50% of the total radiance at visible wavelengths in limb-viewing mee surements. The ratio MS/total scattering dependstrongly on surface albedo. The ratio can be even above 50% if there is a snow or a cloud cover under the tangent point of the LOS. The proportion of MS and the effect of surface albedo depend on the solar position relative to the detector LOS; the effects decrease as at increases. In near UV the MS/total scattering ratio drops fast, and below 310 nm the SS model is quite accurate. The large proportion of MS and light reflected from ground makes retrieval of constituent density profiles from limb radiance measurements a challenging problem. ART model used by a constituent retrieval algorithm must be both fast and accurate. The simulations of section 5 show that a SS model is not sufficient for the ,030 inf 8.31e le e e e e e e e-02 0 LOS Albedo on the spot is 1, and albedo outside the spot is 0. The simulation is for tangent altitude 20 km, wavelength 500 nm, and OT ø, bt ø. Angle r, diameter and altitude z are explained in Figure 14. The right column gives the increase of intensity compared to a completely black surface (albedo 0 everywhere). Figure 14. Explanation of angle c, altitude za, and diameter ds used to describe the dimensions of a bright spot on the Earth's surface. The bright spot is indicated by a thick line on the plot. ( q :re> qo e, qto' too o

13 OIKARINEN ET AL.: MULTIPLE SCATTERING IN LIMB-VIEWING GEOMETRY 31,273 analysis of the whole spectral range and all solar posi- Berk, A., L. S. Bernstein, and D.C. Robertson, MODtions of the measurements by OSIRIS, SCIAMACHY, TRAN: A moderate resolution model for LOWTRAN 7, and GOMOS. Because the relative amount of MS ra- AFGL Tech. Rep. F C-0079, 38 pp., Air Force Geophys. Lab., Hanscom Air Force Base, Mass., diance depends on wavelength (Figure 10) and, most Bertaux, J. L., G. Megie, T. Widemann, E. Chassefiere, R. importantly, reflects atmospheric absorption features, Pellinen, E. Kyr61 i, S. Korpela, and P. Simon, Monitor- MS effects cannot be corrected for by a simple scal- ing of ozone trend by stellar occultations: The GOMOS ing factor (total radiance is not of the form constxss). instrument, Adv. Space Res., (3), , Finding a suitable MS approximation will be an impor- Bovensmann, H., J.P. Burrows, M. Buchwitz, J. Frerick, S. Noel, V. V. Rozanov, K. V. Chance, and A. P. H. tant task in the development of retrieval algorithms for Goede, SCIAMACHY: Mission objectives and measurethe limb-viewing measurements. ment modes, J. Atmos. $ci., 56, , One also has to choose how to treat surface albedo Burrows, J.P., M. Weber, M. Buchwitz, V. V. Rozanov, A. in constituent retrieval algorithms. One possibility is Ladstttermann, M. Eisinger, and D. Perner, The Global to try to retrieve the albedo value from the limb scan Ozone Monitoring Experiment (GOME): Mission concept data. Another possibility is to model surface albedo and first scientific results, J. Atmos. $ci., 56, , (and tropospheric extinction) by a priori data. If the Collins, D. G., W. G. Blttner, M. B. Wells, and H. G. Horak, latter solution is selected, the simulation results of sec- Backward Monte Carlo calculations of the polarization tion 5 suggest that the surface albedo should be known characteristics of the radiation emerging from sphericalat 10% accuracy level. Especially, cloud conditions dur- shell atmospheres, Appl. Opt., 11, , ing the measurements should be known. Graigner, J., and J. Ring, Anomalous Fraunhofer line profiles, Nature, 193, 762, The mean photon path lengths at different altitude Herman, B. M., A. Ben-David, and K. J. Thome, Numerical layers depend strongly on wavelength, especially at technique for solving the radiative transfer equation for a near-uv wavelengths (Figure 13). A common retrieval spherical shell atmosphere, Appl. Opt., $$, , approach in occultation measurementseparates the inversion into two parts: a spectral inversion and a ver- Herman, B. M., T. R. Caudill, D. E. Flittner, K. J. Thome, tical inversion (e.g., onion peeling). For limb radiances this approach must be used with caution: the mean path length distributions of Figure 13 show that spectral and geometrical effects are not decoupled. We have presented a backward Monte Carlo algorithm Siro that has proven to be a versatile tool for studying the limb-viewing measurementechnique. Also, Siro can be used as a reference against which to validate approximations adopted by other faster methods. Siro itself is too slow to be used as part of an operational retrieval algorithm. A very realistic model would perform the radiative transfer calculation separately for different polarization directions of light. Marchuk et al. [1980] have simulated the intensity error due to neglect of polarization in some limb-viewing geometries. In their model the error was 2-6% at 400 nm and below 1% at 550, 600, 700, and 800 nm. A more realistic radiance simulation would also take into account the Ring effect, i.e., the filling of solar Fraunhofer lines by inelastic scattering processes in the atmosphere [Graigner and Ring, 1962]. The next version of Siro will model polarization of light. More studies on three-dimensional effects are also foreseen. Acknowledgments. This work was supported by the Academy of Finland. The authors acknowledge A. Berk for providing valuable suggestions for improving the Monte Carlo algorithm and the text. The authors also thank G. Leppelmeier for checking the manuscript and the anonymous referees for their comments that greatly helped to improve the paper. References Adams, C. N., and G. W. Kattawar, Radiative transfer in spherical shell atmospheres, I, Rayleigh scattering, Icarus, 35, , and A. Ben-David, Comparison of the Gauss-Seidel spherical polarized radiative transfer code with other radiative transfer codes, Appl. Opt., $, , Kattawar, G. W., and C. N. Adams, Radiative transfer in spherical shell atmospheres, II, asymmetric phase functions, Icarus, 35, , Lenoble, J., Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures, 300 pp., A. Deepak, Hampton, Va., Lenoble, J., Atmospheric Radiative Transfer, 532 pp., A. Deepak, Hampton, Va., Lenoble, J., and H. B. Chen, Monte Carlo study of the effects of stratospheric aerosols and clouds on zenith sky absorption measurements, the International Radiation Symposium, Int. Assoc. of Meteorol. and Atmos. Phys., Tallin, Estonia, August Marchuk, G.I., G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics, 208 pp., Springer, New York, Mauldin, L. E., III, N.H. Zaun, M.P. McCormick, J. H. Guy, and W. R. Vaughn, Stratospheric Aerosol and Gas Experiment II instrument: A functional description, Opt. Eng., œ, , Paur, R. J., and A.M. Bass, The ultraviolet cross-sections of ozone, II, Results and temperature dependence, in Atmospheric Ozone: Proceedings of the XV Quadrennial Ozone Symposium, edited by C. S. Zerefos and A. Ghati, pp , D. Reidel, Norwell, Mass., Payne, W. F., E. J. Llewellyn, and J. S. Matsushita, OSIRIS: An imaging spectrograph for Odin, paper presented at the 46th International Astronautical Congress, International Astronautical Federation, Oslo, Norway, October Perliski, L. M., and S. Solomon, On the evaluation of air mass factors for atmospheric near-ultraviolet and visible absorption spectroscopy, J. Geophys. Res., 98, 10,363-10,374, Rozanov, V. V., D. Diebel, R. J. Spurr, and J.P. Burrows, GOMETRAN: A radiative transfer model for the satellite project GOME, the plane parallel version, J. Geophys. Res., 102, 16,683-16,695, 1997.

14 31,274 OIKARINEN ET AL.: MULTIPLE SCATTERING IN LIMB-VIEWING GEOMETRY Rusch, D. W., G. H. Mount, C. A. Barth, R. J. Thomas, and M. T. Callan, Solar mesosphere explorer ultraviolet spectrometer: Measurements of ozone in the mbar region, J. Geophys. Res., 89, 11,677-11,687, Stamnes, K., S.-C. Tsay, W. Wiscombe, anal K. Jayaweera, Numerically stabile algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media, Appl. Opt., 27, , Thorne, K. J., Radiative transfer model for a spherical atmosphere, Ph.D. thesis, Univ. of Ariz., Tuscon, E. Kyr61 i and L. Oikarinen, Finnish Meteorological Institute, Geophysical Research Division, P.O. Box 503, FIN Helsinki, Finland. ( erkki.kyrola@fmi.fi; liisa.oikarinen@fmi.fi) E. Sihvola, University of Helsinki, Department of Physics, P.O. Box 9, FIN UNIVERSITY OF HELSINKI, Finland. ( elina.sihvola@helsinki.fi) (Received October 5, 1998; revised June 22, 1999; accepted September 8, 1999.)