# Effect of surface albedo variations on UV-visible limb-scattering measurements of the atmosphere

## Abstract

[1] In a new technique to measure stratospheric ozone profiles a near-UV and visible spectrometer is used to detect backscattered solar light in limb-viewing geometry. Several satellite-based instruments using this technique will be launched in 2001–2002. A limb-viewing instrument does not see Earth's surface or tropospheric clouds directly. However, indirect light reflected from the surface or low-altitude clouds (which have an effect similar to a highly reflective surface) can make up tens of percents of limb radiance. Our simulations by a three-dimensional Monte Carlo radiative transfer model show that reflection from an area at Earth's surface that extends over 1000 km along the instrument line of sight and 200 km across it contributes to the signal. The exact area depends on tangent altitude of the measurement, wavelength, surface elevation, and atmospheric conditions. Over this distance the reflectivity of the surface can vary by almost 100%, which should be taken into account in measurement data analysis. Normally, radiative transfer models that are used to analyze scattering measurements assume that the atmosphere and the surface are spherically symmetric. In such a model the surface albedo is a constant, and it is not possible to include a spatially varying surface into the model. We present an approximative method to average the reflectivity of Lambert surfaces, which can be used to estimate limb radiance over an inhomogeneous surface by a spherically symmetric model.

## 1. Introduction

[2] In a new technique to measure stratospheric ozone, a spectrometer for near-UV and visible wavelength range (in some cases also near-IR) is used to detect backscattered solar light at the limb of the Earth's atmosphere. The instrument, located on a satellite, records intensity spectra at different tangent altitudes of the limb. Typically, altitudes between 10 and 60 km are sampled at 1–2 km tangent altitude intervals. The spectra measured at different altitudes are inverted to density profiles of atmospheric trace gases and particles by fitting modeled spectra to the data. This measurement technique has been demonstrated recently with the Shuttle Ozone Limb Sounding Experiment (SOLSE) and the Limb Ozone Retrieval Experiment (LORE) flown on shuttle flight STS-87 [*McPeters et al.*, 2000; *Flittner et al.*, 2000]. Limb scatter measurements are made in a continuous manner by the Optical Spectrograph and Infrared Imager System (OSIRIS) [*Llewellyn et al.*, 1997], launched aboard the Odin satellite in February 2001, and scanning imaging absorption spectrometer for atmospheric cartography (SCIAMACHY) [*Bovensmann et al.*, 1999], which will start its operation in 2002.

[3] Traditionally, atmospheric measurements using scattered solar UV and visible radiation have been conducted in nadir-viewing geometry. In the analysis of nadir-viewing measurements, reflection from the surface (bare surface or clouds) has an important role. A correction for surface reflection is made by means of a cloud fraction and a cloud top height, which are deduced from the measurement with the help of a cloud climatology or data from other measurements [*Kuze and Chance*, 1994; *Hsu et al.*, 1997]. In limb-viewing geometry the instrument is not pointed directly at Earth's surface. However, a substantial amount of light reflected from Earth's surface or clouds can be received through scattering [*Oikarinen et al.*, 1999].

[4] Data inversion methods for limb-scattering measurements are yet under development, and the role of surface reflection has not been discussed much. In the inversion scheme used by *Flittner et al.* [2000] to analyze SOLSE and LORE measurements, effects of surface reflection are canceled by scaling the data by a reference spectrum measured at a high tangent altitude (where absorption of radiation is small). This approach may not work for OSIRIS and SCIAMACHY, which will be scanning the limb vertically while moving along a polar orbit. Measurements at different tangent altitudes are made over a different surface, and the relative amount of reflected light can vary within a scan. In this case it may become necessary to model the reflected intensity with the help of surface albedo data from concurrent measurements by other instruments.

[5] Modeling of radiative transfer in limb-viewing geometry is computationally very demanding [*Griffioen and Oikarinen*, 2000; *Rozanov et al.*, 2001]. Models that do not require spherical symmetry of the atmosphere or the reflecting lower surface are even more time-consuming than spherical shell models [e.g., *Oikarinen et al.*, 1999]. In practice, to analyze measurement data in a reasonable time, one has to assume that the surface has a constant albedo. However, the instantaneous line of sight (LOS) of a limb measurement extends from several hundreds to over a thousand kilometers in the atmosphere. Over such a distance the reflectivity of the underlying surface can vary a lot.

[6] We have used the backward Monte Carlo model Siro [*Oikarinen et al.*, 1999] to simulate limb intensity measurements over an inhomogeneous surface. In 2 of this paper we look at the composition of the reflected part of limb intensity. In 3 we study how reflection at different points of the surface under the measurement LOS contribute to limb intensity. A simple approximative scheme for spatial averaging of the surface albedo is presented in 4. This scheme can be used in data retrieval for estimating reflected intensity with a spherical shell model without having to employ a time-consuming three-dimensional radiative transfer model. Alternatively, if an effective (constant) albedo can be retrieved from the limb measurements, the results of 3 and 4 help to interpret the retrieved albedo. In 10 the results of 4 are applied to simulated measurements by OSIRIS over a realistically varying surface.

[7] The present study is limited to surfaces that reflect radiation like a Lambert surface [e.g., *Born and Wolf*, 1980]. Simulations are done using three different aerosol models, which correspond to different tropospheric and stratospheric aerosol conditions.

## 2. Reflected Intensity Component

*R*that has been reflected from Earth's surface at least once (either from the direct solar beam or from scattered diffuse light) and a component

*S*that has not reached Earth's surface,

*S*equals the radiance that would be measured if the surface was completely nonreflecting, that is, surface albedo

*A*= 0 everywhere. In addition to the reflectivity of the surface the magnitudes of

*R*and

*S*depend strongly on atmospheric conditions, most importantly, the amount of aerosol both in the troposphere and in the stratosphere. The radiances

*R*and

*S*also depend on wavelength λ and measurement geometry.

[9] Three different aerosol models have been used in the simulations of this work (Figure 1): (1) a background stratospheric aerosol model combined with a tropospheric aerosol model, which corresponds to surface visible range (vis) of 23 km (average aerosol conditions), (2) background stratospheric aerosols and tropospheric aerosols of vis = 2 km (thick troposphere case), and (3) an extreme volcanic stratospheric model combined with a troposphere of vis = 23 km (aerosol conditions after a major volcanic eruption). The aerosol extinction profiles were taken from the MODTRAN model [*Berk et al.*, 1989]. A single scattering albedo of 0.9 is used at 0–10 km altitudes and of 1.0 at other altitudes. The Henyey-Greenstein phase function [e.g., *Lenoble*, 1993, equation (13.20)] with asymmetry factor *g* = 0.75 was used for modeling the angular dependence of scattering by all aerosol types. The atmospheric model also includes Rayleigh scattering and absorption by ozone. For these constituents we used the same model as in [*Oikarinen et al.*, 1999] (U.S. Standard 1976 density and ozone concentration profiles).

[10] Figure 2 shows how the reflected intensity *R* varies as a function of λ in the three model atmospheres. In Figure 2a, *R*(λ) is shown relative to the incident solar irradiance at the top of the atmosphere, *I*^{0}(λ), and in Figure 2b it is shown relative to total limb intensity *I*(λ). A measurement at tangent altitude *z*_{T} = 20 km and solar illumination directly from above the tangent point of the LOS has been simulated (solar zenith angle θ_{Sun} = 0°). The background stratospheric aerosol case with vis = 23 km is shown also for θ_{Sun} = 80° for comparison (Sun 10° above the horizon at the LOS tangent point, solar azimuth angle 90°). In the simulations of Figure 2 the surface was assumed to have a constant diffuse albedo of *A* = 1 to show the reflected intensity at its largest. For most real surfaces, *A* < 1, and *A* varies somewhat as a function of wavelength.

[11] In Figure 2 we see that reflected light is important in the UV at λ > 300 nm and in the visible. When *A* is large, *R*(λ) can constitute more than half of the total intensity *I*(λ). Below 300 nm, stratospheric ozone blocks transmission of light to Earth's surface. The shape and intensity of *R*(λ) depend strongly on the aerosol load in both the troposphere and the stratosphere. In the UV between ∼300 and ∼350 nm the ratio *R*(λ)/*I*(λ) reflects the ozone absorption spectrum. The fine structure of ozone Huggins band is lost in Figure 2, where the spectra have been simulated at 10 nm wavelength intervals. In the visible the ratio *R*(λ)/*I*(λ) mostly reflects the smooth decrease of Rayleigh and aerosol extinction as a function of λ.

[12] In case of a Lambert surface, *R* decreases as the solar zenith angle increases approximately as cos(θ_{Sun})exp[−*L*(θ_{Sun})τ], where τ is the total optical depth of the atmosphere for θ_{Sun} = 0° and *L* is the air mass factor (*L*(θ_{Sun}) ∼ [cos(θ_{Sun})]^{−1} when θ_{Sun} is less than ∼70°). A more detailed simulation of *R* as a function of solar illumination geometry has been shown by *Oikarinen et al.* [1999].

*I*as an ensemble of photons, we can further divide the reflected intensity

*R*into parts according to the number of reflections the photons have undergone,

*R*

_{1}is the intensity of photons that have hit the lower surface only once,

*R*

_{2}is the intensity of photons reflected twice, etc. (Figure 3). A photon that belongs to

*R*

_{2}has seen at least two scattering events in addition to the two reflections. Some (single-scattering) radiative transfer models approximate

*R*by

*R*

_{SS}, the part of

*R*

_{1}that consists of photons reflected from the direct solar beam and scattered once in the atmosphere at the LOS toward the detector (see Figure 3).

*A*, we can write (a similar approach has been used by

*Lenoble*[1998])

*R**

_{1}(

*R**

_{2}) is

*R*

_{1}(

*R*

_{2}) for

*A*≡ 1. (In the following we will denote by an asterisk all quantities obtained from a simulation where

*A*= 1 everywhere on the surface.)

*x*,

*y*) of the surface by

*A*(

*x*,

*y*). In this case all terms in equation (3) become integrals over the surface. For example, term

*A*

^{2}

*R**

_{2}is replaced by

*x*

_{1},

*y*

_{1}) is the location of the first collision with the surface and (

*x*

_{2},

*y*

_{2}) is the location of the second collision.

## 3. Contribution of Different Points of the Surface

[16] The LOS of a limb-viewing instrument travels in the atmosphere a distance of 1000–2000 km. To get an idea on how different points of the surface both below the LOS and across it contribute to the intensity at the detector, we simulate reflected intensity *R* as a function of position (*x*, *y*) of the surface. More precisely, we define *R*(*x*, *y*) as the intensity for photons that have their first collision with the surface at point (*x*, *y*). The origin of the surface coordinate system is assigned to the point that lies directly below the tangent point of the LOS. The *x* coordinate runs along the LOS, and the *y* coordinate runs across it. The detector is located at the −*x* side, looking toward +*x*. In practice, we have to divide the surface below the LOS into small cells and calculate the number of photons that hit each cell. As a trade-off between the statistical precision of the Monte Carlo simulation and the spatial resolution of the simulation, the cell dimensions were chosen to vary from 20 to 100 km in the *x* direction and from 5 to 50 km in the *y* direction.

[17] Figure 4 shows *R**(*x*, *y*) (i.e., *R*(*x*, *y*) for *A* ≡ 1) for background stratospheric aerosols, vis = 23 km, λ = 500 nm, *z*_{T} = 20 km, and θ_{Sun} = 0°. Reflection from a region about ±75 km in the *y* direction and ±500 km in the *x* direction around the tangent point constitutes ∼95% of the total reflected intensity. In Figure 4 we have assumed that the horizontal field of view (FOV) of the instrument in the *y* direction is infinitesimally small. The horizontal FOV of true instruments varies typically from tens of kilometers to a hundred kilometers at the tangent point, and the contribution function in *y* direction is wider correspondingly.

[18] In a geometry where θ = 0° the distribution *R**(*x*, *y*) is symmetric with respect to the LOS. It also remains quite symmetric in cases where solar illumination is from the side (shown later in Figure 5). On the contrary, function *R**(*x*, *y*) is not symmetric with respect to the line *x* = 0, i.e., the tangent point. Instead, the distribution is approximately symmetric with respect to the maximum point of the distribution, *x*_{max}, which lies slightly closer to the detector than the tangent point.

*R**(

*x*,

*y*) on tangent altitude, we first define two projections of the distribution:

*R**

_{y}(

*y*) and

*R**

_{x}(

*x*) for the simulation case of Figure 4 as a function of distance

*y*or

*x*and tangent altitude

*z*

_{T}. The distribution at each tangent altitude has been scaled by dividing by total limb intensity

*I*at that altitude. The width of

*R*

_{y}*(

*y*) increases with tangent altitude (Figure 5, left). This is due to the fact that when the tangent point (or any point of the LOS) is moved upwards in the atmosphere, the surface area seen at a given solid angle from the tangent point increases. As the tangent altitude increases, the optical depth of the LOS decreases. This makes the maximum point of

*R**

_{x}(

*x*) (Figure 5, right) to move toward

*x*

_{max}= 0. When the optical depth to the tangent point is less than ∼0.1,

*x*

_{max}= 0. At the same time,

*R**

_{x}(

*x*) becomes slightly broader in shape. Note that the LOS path inside the atmosphere is shorter for high

*z*

_{T}than for low

*z*

_{T}. This makes

*R**

_{x}(

*x*) become slightly sharper again at tangent altitudes higher than ∼55 km.

[20] The shapes of *R**_{y}(*y*) and *R**_{x}(*x*) depend on the optical properties of the atmosphere and thus on the wavelength of radiation. At near-UV wavelengths the distribution *R**(*x*,*y*) becomes slightly narrower than at λ = 500 nm, and *x*_{max} moves toward the detector at low *z*_{T}. This is seen in Figure 5b, where the distributions are shown for the same case as in Figure 5a but for λ = 337 nm.

[21] Figure 5c shows results from a simulation where all other parameters are as in Figure 5a but θ_{Sun} = 80° instead of θ_{Sun} = 0°. When the surface is a Lambert reflector, the solar illumination geometry affects the shape of *R**_{y}(*y*) (and *R**_{x}(*x*) if the solar azimuth is not 90°) only through small effects resulting from the curvature of Earth's surface. The angle of incidence and atmospheric transmission of the direct solar beam become somewhat dependent on the location (*x*, *y*). The distribution *R**_{y}(*y*) in Figure 5c is slightly weighted to the solar side. Compared with a fully symmetric distribution, *R**_{y}(*y*) gradually decreases by 1% per 10 km in *y* from the solar to the antisolar side. It would be natural to assume that the effects of the curvature of Earth's surface increase when θ_{Sun} or the optical depth of the atmosphere τ increases. However, *R**_{y}(*y*) does not become significantly more asymmetric than in Figure 5c for a larger θ_{Sun} or τ because at the same time the proportion of diffuse light increases a lot, which partly smoothes out the asymmetry effect.

[22] The tropospheric aerosol model (or vis) does not affect the optical depth of the LOS and, consequently, has very little effect on the shape of *R**_{x}(*x*) (Figure 6a), although it has a large effect on the magnitude of the total reflected intensity. The stratospheric aerosol model, to the contrary, affects the optical depth of the LOS and the shape of *R**_{y}(*y*) and *R**_{x}(*x*) a lot. For volcanic stratospheric aerosols (Figure 6b) the distribution *R**_{x}(*x*) is very narrow when *z*_{T} is below the dense stratospheric aerosol layer located at 20 km altitude (compare Figure 6b with Figure 1d) but very flat for *z*_{T} > 20 km. For volcanic aerosols the distribution declines more rapidly at the +*x* side of *x*_{max} than at the −*x* side at low tangent altitudes (below ∼25 km), whereas for background stratospheric aerosols, *R**_{x}(*x*) is almost symmetric with respect to the maximum point of the distribution, *x*_{max}. The distribution for background stratospheric aerosols, λ = 337 nm, behaves similarly (Figure 6b), but the effect is not at all as strong as for volcanic aerosols. At high tangent altitudes, *R**_{x}(*x*) for volcanic aerosols reaches longer to the +*x* side of *x*_{max} than to the −*x* side.

[23] Surface elevation affects the distribution of *R**(*x*, *y*) in *y* direction. The distribution becomes narrower when the surface elevation *z*_{S} increases. The width of the resulting distribution is close to that obtained for a sea level surface at tangent altitude *z*_{T} − *z*_{S}.

## 4. Approximation of Reflected Intensity

[24] The intensity reflected from a surface whose albedo varies as a function of (*x*, *y*) can be modeled accurately only by a three-dimensional radiative transfer model. We now look into some approximations that would make it possible to model reflection from an inhomogeneous surface by a model that accepts only a constant surface albedo.

### 4.1. Approximation of Second- and Higher-Order Reflection Terms

[25] Let us first examine the proportion of different reflection orders (equation (2)) in the case *A*(*x*, *y*) = 1. Table 1 shows the proportion of *R** and the components *R*_{1}* and *R*_{2}* to the total limb intensity *I* for all three model atmospheres at λ = 500 nm, θ_{Sun} = 0°. For background stratospheric aerosols the same has also been shown for θ_{Sun} = 80°, and to present a case where the LOS optical depth is larger than at 500 nm, for λ = 337 nm, θ_{Sun} = 0°. At λ = 500 nm, *R*_{1}* constitutes more than 80% or the reflected component, and *R*_{1}* and *R*_{2}* together make up almost all of the reflected light. At λ = 337 nm, on the contrary, even *R*_{3}* is significant. In Table 1 we also show the proportion of *R*_{SS}* to total intensity. Taking into account only *R*_{SS}* will greatly underestimate the amount of reflected light in all the cases studied here. Particularly in the thick troposphere case (vis = 2), this approximation fails to see all the reflected radiation because it all results from reflection of diffuse light. The data in Table 1 are for *z*_{T} = 20 km. Tangent altitude does not affect the relative proportion of different reflection orders much.

Aerosol Model^{a} |
θ_{Sun}, degrees |
λ, nm | R*, % |
R_{1}*, % |
R_{2}*, % |
R_{SS}*, % |
l, km |
---|---|---|---|---|---|---|---|

Background, vis 23 | 0 | 500 | 59 | 51 | 7 | 18 | 15 |

Background, vis 23 | 0 | 337 | 39 | 26 | 9 | 2 | 11 |

Background, vis 23 | 80 | 500 | 13 | 11 | 2 | 1 | 15 |

Background, vis 2 | 0 | 500 | 26 | 21 | 4 | 0 | 4 |

Volcanic, vis 23 | 0 | 500 | 64 | 52 | 9 | 6 | 26 |

- a Here vis is the surface visible range, given in kilometers.

[26] The rightmost column of Table 1 gives the average distance *l* between reflections for photons that have been reflected more than once. As expected, this distance is shortest in those cases where the optical depth of the lower atmosphere is largest (vis = 2 km or λ = 337 nm). The volcanic stratospheric aerosol layer reflects a larger amount of upwelling radiation back toward the surface than a background stratospheric aerosol layer and increases *l*.

*R*

_{2}or higher orders,

*A*(

*x*

_{1},

*y*

_{1}) =

*A*(

*x*

_{2},

*y*

_{2}). In other words, assume

*R*

_{2}**(

*x*,

*y*) = ∫ ∫

*R*

_{2}*(

*x*,

*y*,

*x*

_{2},

*y*

_{2})

*dx*

_{2}

*dy*

_{2}. The approximation in equation (7) is good when the spatial variation of

*A*(

*x*,

*y*) is slow and the variation has a scale that is large compared with the distance between collisions

*l*. The error of the approximation in equation (7) is largest in cases where

*A*is changing quickly from ≈0 to ≈1. In reality this occurs in measurements over an ocean which is partly covered by thick clouds. The validity of equation (7) was tested by simulations in the five cases of Table 1 with three types of surfaces which include

*A*= 0 →

*A*= 1 boundaries: (1) a sharp boundary across the LOS, (2) a sharp boundary parallel to the LOS, and (3) a chessboard-like surface consisting of quadrangles where

*A*= 0 or

*A*= 1. In the simulations with the first two surface types the boundary was “moved” along or across the LOS to see the effect of boundaries lying at different distances from the tangent point. For both case 1 and case 2 the error of the approximation in equation (7) was always below 1%, which was approximately the statistical accuracy of the Monte Carlo simulation. In the chessboard-surface case the dimensions of the quadrangles were varied so that dimensions both smaller and larger than

*l*(see Table 1) were tested. In this case the approximation in equation (7) resulted in an overestimation of the total limb intensity by at most 2%. The error was largest for small quadrangles with dimensions less than ∼

*l*.

*R**

_{1}= ∫ ∫

*R**

_{1}(

*x*,

*y*)

*dxdy*and

*R**

_{2}= ∫ ∫

*R***

_{2}(

*x*,

*y*)

*dxdy*. In 5 we develop a method to estimate

*K*(

*x*,

*y*) without a three-dimensional radiative transfer model. Equation (9) with the

*K*(

*x*,

*y*) estimate is useful if we have a model which can resolve the total reflected intensity to components

*R*

_{1}* and

*R*

_{2}* (and

*R*

_{3}* in some cases).

*R** to

*R*

_{1}* and

*R*

_{2}*, we have to neglect effects of multiple reflections altogether.

*R** = ∫ ∫

*R**(

*x*,

*y*)

*dxdy*, overestimates the true

*I*by

*A*is a constant. The error is largest for

*A*≈ 0.5, and it decreases toward

*A*= 0 and

*A*= 1. For the background stratospheric cases of Table 1 at λ = 500 nm (regardless of vis), equation (10) results in a total intensity value that is larger than the true intensity by, at most, 3% for θ

_{Sun}= 0° and by, at most, 0.6% for θ

_{Sun}= 80°. For background stratospheric aerosols, λ = 337 nm, and for volcanic aerosols, λ = 500 nm, the error of equation (11) is ≤5% of the total intensity. The error of equation (10) in cases where

*A*(

*x*,

*y*) is not constant will be studied in 8 and 10.

### 4.2. Approximation of Distribution *K*(*x*, *y*)

*K*(

*x*,

*y*) (equation (8)) can be approximated on the basis of some physical considerations without three-dimensional radiative transfer modeling. Let us look separately at the dependence of

*K*(

*x*,

*y*) on

*x*and

*y*and assume that

*K*(

*x*,

*y*) can be separated into

#### 4.2.1. Approximation in *y*-dimension

*Y*(

*y*) in equation (12) by

*z*

_{max}, the altitude of the LOS point where

*x*=

*x*

_{max}(we now define

*x*

_{max}as the maximum point of distribution

*X*

^{appr}(

*x*) defined in 6). Often

*z*

_{max}≈

*z*

_{T}. Parameter

*C*

_{1}is a scaling factor selected so that ∫

_{−∞}

^{+∞}

*Y*

^{appr}(

*y*)

*dy*= 1. In equation (13) we assume that the shape of

*Y*(

*y*) is defined by reflection from points that lie on a line which is perpendicular to the LOS and crosses the LOS at

*x*=

*x*

_{max}. In addition, equation (13) neglects all multiple scattering effects, e.g., multiple reflections and scattering from atmospheric particles after the reflection, which are difficult to take into account by a simple formula.

[32] The shapes of distributions *R*_{y1}*(*y*) and *R*_{y2}*(*y*) from simulations of 3 have been compared with *Y*^{appr}(*y*) from equation (13) on the plots on the left-hand sides of Figures 7 and 8. The distributions from Figures 5 and 6 at *z*_{T} = 20 km are compared, except for the λ = 337 nm case, where distributions for *z*_{T} = 15 km are shown to exhance the effects of the LOS optical depth. Here *R*_{y1(2)}*(*y*) is defined like *R**_{y} in equation (5) and is scaled to total area 1.0. The distribution *R*_{y1}*(*y*), shown by a dashed line, is slightly narrower than *R*_{y2}*(*y*) (dotted line). In case of background stratospheric aerosols, λ = 500 nm (Figure 7a), the first-order distribution *R*_{y1}*(*y*) has a sharper shape than *Y*^{appr}(*y*), suggesting that equation (13) underestimates the average optical depth τ by ignoring multiple scattering. At λ = 337 nm, on the contrary, the distributions for both first- and second-order reflection are wider than *Y*^{appr}(*y*) (Figure 7b). In this case the probability of multiple scattering in the vicinity of the LOS is higher than at λ = 500 nm, which changes the distribution *Y*(*y*).

[33] The amount of boundary layer aerosol (vis) does not affect *Y*(*y*) as much as extinction of the upper troposphere and stratosphere. Some trial and error reveals that in all the cases studied here, equation (13) will actually be improved (or at least will not become worse) if a value of τ ≈ 0.8 is used instead of the optical depth of a single vertical path between *z* = 0 km and *z* = *z*_{max}. In the following, however, we will use the original definition of τ. Finally, equation (13) does not take effects of the solar zenith angle into account, so the approximate distributions *Y*(*y*) for the case in Figure 7a (θ_{Sun} = 0°) and Figure 7c (θ_{Sun} = 80°) are equal.

#### 4.2.2. Approximation in *x*-dimension

*X*(

*x*) in equation (12) is to take the distribution of multiple scattering source function, or even more simply, the single scattering source function along the LOS. This function gives the relative contribution of different points of the LOS to total intensity. The single scattering distribution gives

*T*

^{ps}(

*x*) is the transmittance of the path from the Sun to the LOS point corresponding to

*x*, and

*T*

^{op}(

*x*) is the transmittance of the path from this point to the detector. Function

*P*(

*x*) is the scattering phase function, which is a weighted sum of phase functions for molecular and particle scattering, and

*k*(

*x*) is the total scattering coefficient at point

*x*on the LOS. Finally,

*C*

_{2}is a constant that scales the total area of

*X*

^{appr}(

*x*) to 1. If a multiple scattering source function is used instead of single scattering, the distribution in equation (14) includes an integration of radiation coming from all directions to point

*x*instead of just the contribution of direct radiation from the Sun.

[35] The distributions *R**_{1}(*x*) and *R**_{2}(*x*), defined analogously to *R**_{y1(2)}, are almost identical in shape in the case of background stratospheric aerosols (Figures 7 and 8a, right). In the case of volcanic aerosols, *R**_{2}(*x*) becomes somewhat wider than *R*_{1}*(*x*) (Figure 8b). At high tangent altitudes (*z*_{T} > 30 km), approximation *X*^{appr}(*x*) from equation (14) agrees very well with *R*_{1}*(*x*) and *R*_{2}*(*x*).

[36] In the case of background stratospheric aerosols the form of equation (14) is also quite correct at low tangent altitudes, but it predicts the center point of the distribution, *x*_{max}, to lie slightly closer to the detector than it really does. The error in the location of *x*_{max} is ∼10 km for λ = 500 nm, *z*_{T} = 20 km, and ∼20 km for λ = 337 nm, *z*_{T} = 15 km (Figures 7a–7b and Figure 8a). The shift disappears at *z*_{T} ≈ 25 km.

[37] For volcanic stratospheric aerosols at *z*_{T} = 20 km, the altitude of maximum aerosol extinction, the shift between *R*_{x}*(*x*) and *X*^{appr}(*x*) is as large as 40 km (Figure 8b). In this case the shapes of the true and approximated distributions are also different: *X*^{appr}(*x*) rises more slowly and declines more rapidly than *R*_{1}*(*x*) and *R*_{2}*(*x*). At *z*_{T} = 30 km, equation (14) already agrees well with *R*_{1}*(*x*) and *R*_{2}*(*x*). Calculation of *X*^{appr}(*x*) from the source function of both single and multiple scattering (with *A*(*x*, *y*) = 1) does not correct the shift of *x*_{max} at low tangent altitudes.

### 4.3. Accuracy of the Albedo-Averaging Scheme

*K*(

*x*,

*y*) approximated by -–(14),

*R*for any

*A*(

*x*,

*y*) with radiative transfer simulations only for constant surface albedos

*A*= 0 and

*A*= 1. If our radiative transfer model can resolve

*R** to

*R*

_{1}*,

*R*

_{2}*, etc., we can estimate the reflected intensity by equations (9) and (15). If we only know

*R** (e.g.,

*R** =

*I*

_{A=1}−

*I*

_{A=0}), we can estimate reflected intensity by equations (10) and (15). To evaluate equation (15), we have to calculate the optical depth of a model atmosphere from 0 km to

*z*

_{max}in order to use equation (13) (alternatively, the value τ = 0.8 can be used), and we have to output the LOS source function from a radiative transfer model or calculate equation (14).

[39] The error of these approximations was studied in the five example cases of Figures 7 and 8 by simulating a measurement over the edge of two regions of *A* = 1 and *A* = 0. A sharp edge across the LOS presents the worst case for equation (14). On the contrary, the error of equation (13) is at its largest when the measurement is done along an underlying belt-shaped cloud that has a width about equal to the half width of the distribution *Y*^{appr}(*y*). However, in all cases studied here, such a belt-shaped albedo variation causes a smaller error than an edge across the LOS situated at a suitable distance from the tangent point. For these surfaces, where the albedo only gets values *A* = 0 and *A* = 1, the difference of equations (9) and (10) is ≤1% in all cases of Figures 7 and 8.

[40] In all cases with background stratospheric aerosols the error of equation (9) or (10) and equation (15) was below ±3% of total intensity for every surface edge scenario. At λ = 337 nm a third-order reflection term has to be included in equation (9) to achieve this accuracy; in the other cases the first- and second-order terms are sufficient. The asymmetry of *R*_{y}*(*y*) in the case with θ_{T} = 80° caused an error smaller than 2% when the surface had an edge parallel to the LOS. In the volcanic aerosol case the error for an edge across the LOS was as large as 30% (25% with *X*^{appr}(*x*) from a multiple scattering source function) when the edge was positioned at the point where the distribution *X*^{appr}(*x*) has its maximum. This case, which presents the worst possible case for the approximations, is illustrated in Figure 9. The plots show a simulation of a fictive measurement where the instrument is staring at *z*_{T} = 20 km and an *A* = 1 → *A* = 0 edge is moved along the LOS direction.

*A*

_{ave}is a simple area average of

*A*(

*x*,

*y*) over −600 km ≤

*x*≤ 600 km, −50 km ≤

*y*≤ 50 km, an area selected as the average important region on the basis of Figures 5 and 6. In equation (17),

*A*

_{tp}=

*A*(0,0), the albedo value just below the tangent point. For the most part of the simulation of Figure 9, approximation of

*K*(

*x*,

*y*) by equation (15) results in clearly better estimation of reflected intensity than equations (16) and (17).

## 5. A Simulated Measurement Over a Realistic Surface

[42] In the UV and in the visible the reflectivity of Earth's surface is small except for areas covered by snow, where reflectivity can be near 1.0, and barren regions like the Sahara Desert, where the reflectivity is ∼0.3. The most important small-scale spatial variation of surface albedo is caused by tropospheric clouds, which cover ∼60% of the Earth's surface [e.g., *Seinfeld and Pandis*, 1998] and can have a reflectivity close to 1.

[43] In 4 we studied the variations of limb intensity and the accuracy of the various approximations when the instrument travels over a sharp boundary of a completely reflecting and completely absorbing surface. We next simulate limb measurements over more complicated scenes of varying cloudiness and surface albedo. To imitate realistic spatial variation of *A*(*x*, *y*), we take one orbit of reflectivity data (from day 104 of year 1999) measured by the National Atmospheric and Oceanic Administration (NOAA) Advanced Very High Resolution Radiometer (AVHRR) [*Kidwell*, 1995] at channel 1 (0.58–0.68 μm). The data set gives the top of the atmosphere reflectivity measured in nadir-viewing geometry. The data were not corrected for atmospheric effects, so they does not give the absolute value of *A*(*x*, *y*) accurately, but they reflect the variation of *A* versus *x* and *y* quite realistically. The effects of surface elevation were neglected in this simulation; clouds were assumed to be just highly reflective areas on Earth's surface. The AVHRR data set has a resolution of 3 km along track and 5 km across track. The NOAA AVHRR instrument is flying on a polar orbit like OSIRIS and SCIAMACHY will do.

[44] Figure 10 shows the intensity measured by OSIRIS at λ = 500 nm in a simulated flight over the AVHRR reflectivity scene. The distribution *R*(*x*, *y*) was averaged over FOV = 40 km in the *y* direction, which corresponds to the FOV of the OSIRIS instrument. For simplicity we assume background stratospheric aerosols, vis = 23 km, and θ_{Sun} = 60° throughout the flight. On a true satellite orbit the solar illumination geometry varies along the orbit, and it also depends on the time of the year. For OSIRIS θ_{Sun} is always >58°. Furthermore, we assume the instrument is continuously staring attangent altitude 20 km.

[45] In addition to the “true” simulated intensity, Figure 10 again shows the intensity obtained from equations (9) and (10), with *K*(*x*, *y*) from equation (15), and the intensity obtained by equations (16) and (17). Figure 10a shows the local surface albedo just below the LOS tangent point (on Figure 10a the value has been averaged over 30 km in the *x* dimension to make the plot readable). Figure 10b shows the intensity relative to the exoatmospheric solar irradiance, and Figure 10c shows the relative difference between the first three approximations and true intensity. Only that portion of the orbit where the AVHRR instrument was well in daylight has been shown. The instrument is flying from northern Canada over the Arctic (reaching the northernmost point of the orbit at 1500 km), then south from Siberia toward a mountainous area in China (at 6500 km), and then over the Indian Ocean, where it is flying over a very cloudy region (at 9000–11,000 km on the plot).

[46] In the measurement configuration of OSIRIS the satellite's projection on the ground moves 400–1300 km during one vertical scan of the limb. When the satellite is flying over the boundary of two large homogeneous areas which have very different albedos (for example at distances 6000 km and 9000 k min Figure 10), the reflected intensity can change rapidly even during one scan. Most of the fine structure in *A*(*x*, *y*) is smoothed out, however.

[47] In the simulation of Figure 10, equation (10) is somewhat worse than equation (9). In regions where *A*(*x*, *y*) ≈ 0.5, equation (10) gives a total intensity value which is larger by 2% than the more correct value obtained from equation (9). The error of equation (9) (first- and second-order term) is below 2%, except for five points where it is ≤3%. The error of equation (10) is mostly below 3%. On the average, all approximationsslightly overestimate the true intensity. Estimation of *K*(*x*, *y*) by equation (15) gives better results than simple area averaging equation (16) at locations where *A*(*x*, *y*) is changing rapidly; the error of equation (16) is ≈6% at two locations. Using the tangent point albedo, equation (17) results in an even worse estimation of reflected intensity. The error reaches 15% at several points in the simulation of Figure 10 (not shown in Figure 10c). Equation (9) or (10) with equation (15) resulted in an intensity error ≤10% when the simulation of Figure 10 was repeated using the volcanic aerosol model.

## 6. Conclusions

[48] A limb-viewing UV-visible sensor will receive reflected light from a surface area that extends over ≂1000 km along the LOS and ≂200 km across the LOS. The exact area depends on tangent altitude of the measurement, elevation of the lower boundary, atmospheric conditions(aerosol conditions in particular), and wavelength. The tropospheric contamination in a limb intensity spectrum also originates from about the same area.

[49] In a measurement where the instrument is moving during a limb scan (e.g., along a polar orbit) the relative amount of reflected intensity varies from one exposure to another. The variation is large, often tens of percents of total limb intensity, when the instrument is traveling over the edge of a clear region and a cloudy or snowy region. Even if the instrument records a full vertical image of the limb at once, surface reflectivity appears slightly different at different wavelengths and tangent altitudes. Ignoring these variations of surface albedo results in errors in the retrieval of absorber and scatterer optical depths.

[50] Many data inversion methods make use of “differentiation” of the spectra to eliminate unknown low-frequency spectral structure from the data. This does not eliminate the effects of surface variations completely because the reflected intensity component *R*(λ) includes sharp spectral absorption features from the troposphere and stratosphere in addition to the spectrum of *A*(λ) itself (*R*(λ)/*I*(λ) in Figure 2b is notconstant).

[51] Surface albedo effects can be taken into account in measurement data analysis by modeling the reflected intensity component with the help of concurrent reflectivity and cloud data measured by other satellite instruments. We have presented an approximative method to calculate the reflected intensity over a surface that has an arbitrary albedo distribution *A*(*x*, *y*) without having to deploy a three-dimensional radiative transfer model. The surface albedo *A*(*x*, *y*) at different points below the measurement is weighted by function forms given in equations (13) and (14), which have to be evaluated using an appropriate model atmosphere.

[52] The presented approximation scheme works quite well under background stratospheric aerosol conditions for measurements at all tangent altitudes and for volcanic aerosols at high tangent altitudes (*z*_{T} ≥ 30 km). Under these conditions, reflected intensity for OSIRIS (θ_{Sun} always large) can be approximated at an accuracy better than 3% of total limb intensity. The error is comparable in size to other multiple-scattering modeling errors [*Griffioen and Oikarinen*, 2000]. The error increases a little as the solar zenith angle decreases; at θ_{Sun} = 0° the error is ≤5%. Equation (14) is not very accurate under volcanic stratospheric aerosol conditions and measurements at low tangent altitudes (*z*_{T} ≤ 20 km). In this case the presented averaging scheme predicts a total limb intensity that is wrong by 30% in the worst case (sharp boundary across the LOS, θ_{Sun} = 0). Even in this case, the presented averaging method results in better approximation of *R*(*x*, *y*) than the use of a simple area average of the reflectivity of equation (16) or the local reflectivity below the tangent point of equation (17). The sensitivity of the data inversion methods to these approximation errors should be studied further.

[53] The results of this paper apply to surfaces that reflect approximately according to the Lambert law. A strong specular component changes the shape of *R**(*x*, *y*) from that obtained for a Lambert surface (especially in the y direction). In case of specular reflection the shape of *R**(*x*, *y*) also becomes more dependent on the solar zenith angle.

## Acknowledgments

[54] This work was partly supported by the Academy of Finland. The author thanks E. Kyrölä and the anonymous reviewers for useful comments on the paper and L. Flynn for supplying the AVHRR data.