## Abstract

Multiangle approaches for radiance-to-flux conversion require accurate coregistration between the observations from nadir- and oblique-viewing directions. The along-track mode of Earth Radiation Budget (ERB) scanning instruments, such as the Clouds and the Earth’s Radiant Energy System (CERES), provides some multiangular observations with almost the same target observed from nadir, aft, and fore directions. To improve the overlaps of multiangle observations, this study explains how to introduce a yaw steering angle in the along-track scan mode so as to reduce the residual collocations errors. The implementation of this correction to the CERES/*Terra* along-track mode shows that the distances between the nadir and the oblique (55°) observations are reduced from about 40 to 2 km. Both oblique radiances are shown to be equal with small rms differences: 3.9% (all scenes) and 1.8% (homogeneous scenes), compared, respectively, to 7.0% and 3.5% before the scan adjustment.

## 1. Introduction

With scanning radiometers, satellites generally observe the earth from only one viewing direction. However, from a single observation and because of the anisotropy of the radiation field, the estimation of the hemispherical flux is a critical issue. The angular dependence of the solar-reflected radiances [in the shortwave (SW) spectral domain] and of the terrestrial emitted radiances [longwave (LW) domain] is complex and results from many surface and atmospheric effects. Some recent missions with multiangular capabilities, such as the Polarization and Directionality of Earth Reflectances (POLDER; Buriez et al. 2005) and the Multiangle Imaging Spectroradiometer (MISR; Diner et al. 1999), allow the capture of angular signatures but they do not measure the radiation within the entire range of the SW and LW spectral domains. In a different direction, the Clouds and the Earth’s Radiant Energy System (CERES; Wielicki et al. 1996) and the Geostationary Earth Radiation Budget (GERB; Harries et al. 2005) are measuring the earth radiation components with SW and LW broadband channels. The CERES algorithms compute fluxes by using a statistical Angular Distribution Model (ADM). Considerable efforts have been devoted for improving this ADM (Loeb et al. 2003a, 2005). The method consists in classifying the terrestrial scenes and clouds according to their angular behavior, using the dataset of observations acquired over a large range of angles and provided by the rotating azimuth plane scan mode of CERES. In the future, the broadband radiometer (BBR) of the mission Earth-CARE [European Space Agency (ESA) 2001] should perform three successive observations in the along-track plane, at nadir and 55° fore and aft directions, in order to improve the flux estimates.

Although the construction of the statistical CERES ADM is robust, the question of its validation is raised and has been largely studied by Loeb et al. (2003b) for CERES on the Tropical Rainfall Measuring Mission (TRMM) satellite. Further analyses with different methods remain useful, as well as the study of possible future improvements using instruments with several simultaneous multiangular observations. The CERES along-track (AT) mode partly serves this goal by providing almost simultaneous observations from different directions. The ERBE instruments also had this capability. Smith et al. (1994) have used about 25 days of along-track data to check the limb-darkening models for the 12 ERBE scene types. The analyses are, however, limited by collocation errors between the nadir and oblique observations. These collocation errors occur because the field-of-view (FOV) size and shape changes with the scan angle, but mainly because the oblique and nadir observations are displaced by the lateral distance corresponding to the earth’s rotation between the observations.

This paper explains how the collocation errors between observations from an along-track scanner can be reduced. An “instrument coverage request” was made to the National Aeronautics and Space Administration (NASA) and implemented for eight days during February and March 2005. We have analyzed the observations of this CERES “true” along-track (TAT) mode. Such an experiment helps also in the preparation of the Earth-CARE mission. With radar, cloud profiling radar, imager, and BBR, the Earth-CARE mission is defined for improving the understanding of the cloud–aerosol–radiation interactions [European Space Agency (ESA) 2001], and requires an accuracy of 10 W m^{−2} for the flux retrieval. The future Earth-CARE BBR instrument will use three telescopes for each along-track view. The telescope assembly is moved in the across-track direction by a mechanism to compensate the rotation of the earth. To our knowledge, such yaw steering mechanisms have never been implemented for earth observation purposes. The analysis of the CERES TAT mode then should provide a useful simulation dataset for the three-view BBR.

To get collocated observations, the satellite axis or the azimuth of the instrument has to move as a function of latitude. This rotation angle (or yaw angle) depends on the orbit characteristics. In this paper, the second section discusses the mathematical description of satellite orbits around the earth. The yaw angle required for the true along-track is introduced in section 3. The section 4 compares the theoretical model to real observations and discusses the consistency of the multiangular radiometric observations.

## 2. Angle between the ground track and a meridian

### a. Motion of the satellite orbital plane

Let us denote *O* as the center of the earth (center of the attractive body) and *Oz* the polar axis. We consider two referential frames, ℜ = (*O*; *x*, *y*, *z*), inertial (Galilean) and ℜ* _{E}* = (

*O*;

*x*,

_{E}*y*,

_{E}*z*), terrestrial (non-Galilean). The terrestrial equator plane contains (

*O*;

*x*,

*y*,

*x*,

_{E}*y*) and ℜ

_{E}*rotates in ℜ with angular speed Ω̇*

_{E}_{E}:

corresponding to one round trip in one sidereal day.

We consider the periodic motion of a body, the artificial satellite *S*, in the gravitational field of the earth, with the mean motion *n*. Only circular orbits (altitude *h*) are studied here. The inclination, angle of the orbital plane of the satellite with respect to the equatorial plane of the planet, is noted *i*. The intersection of these two planes defines two particular points, the nodes (ascending and descending) of the orbit.

The perturbation theory, initialized by Lagrange, shows that the orbital plane (defined by *i*, inclination, and by Ω, right ascension of the ascending node, in a Galilean referential) is submitted to a motion, with the pole axis as the rotation axis (see, e.g., Brouwer and Clemence 1961; King-Hele 1964; Kaula 1966). This motion, called *precession motion*, is secular (i.e., proportional to time). It is due to the nonsphericity of the planet and to the action of other celestial bodies. However, the principal cause of this movement is the oblateness of the earth, characterized by the *J*_{2} term of the geopotential. Considering only this term, the velocity Ω̇ of the nodal precession is given by

with *K*_{0} a coefficient that only depends on *J*_{2}, on the planet mass *M*, and equatorial radius *R* (Capderou 2005). For earth,

corresponding to *K*_{0} = 9.964° day^{−1}.

The other zonal harmonics, although only the even ones (i.e., *J*_{4}, *J*_{6}, . . .), also contribute to secular deviation. If the development of the terrestrial geopotental is continued at order *m*, the expression of Ω̇ is more complex, with terms in *J*^{2}_{2}, *J*_{4}, *J*_{6}, . . . , *J _{m}* (Capderou 2005). The relative difference, between the Ω̇ values at order 2 and 4, is about 0.3%, between the values at order 4 and 6, and following, is negligible here. In this work, Ω̇ is computed at order 4.

### b. Calculation of the angle between the ground track and a meridian

The ground track of the satellite *S* is defined as the intersection of the straight line segment *OS* with the earth’s surface (for this application, we may treat the earth as a sphere of radius *R*). We calculate the angle between the satellite ground track and a meridian for an arbitrary point on the ground track. In the frame ℜ, the satellite orbit cuts the meridian at an angle *j*. Referring to Fig. 1, *P* is the subsatellite point (latitude *ϕ*); *N* is the point on the ground track corresponding to the ascending node (the dihedral angle at *N* gives the inclination *i*); and *PQ* is the meridian through *P*, where *Q* is on the equator. The dihedral angle at *P* is the angle *j* that we wish to determine. Using the relation of spherical trigonometry in the spherical triangle *PQN* (right in *Q*), we obtain

We consider the plane tangent to the sphere of radius *R* at the relevant point *P* (latitude *ϕ*), and orthogonal unit vectors * e_{λ}* and

*, respectively, along the parallel and the meridian related to this point. In this orthonormal basis (*

**e**_{ϕ}*P*,

*,*

**e**_{λ}*), the angular velocity vector of the satellite ground track,*

**e**_{ϕ}**, in the Galilean referential frame ℜ, is**

*V*where the term with Ω̇, [see Eq. (2)], represents the angular velocity of the orbital plane, in ℜ at this latitude.

The contribution of the angular velicity of the earth motion, in ℜ, at this latitude, is

with Ω̇_{E} given by Eq. (1).

We deduce the components of the angular velocity of the satellite ground track, denoted ** V**′, in the terrestrial referential frame ℜ

*, by vector composition:*

_{E}The value of *j*′, angle between the ground track and the meridian, in ℜ* _{E}*, is deduced from the Cartesian coordinates of

**′.**

*V*We use *κ* to denote the *daily recurrence frequency*. This quantity, which is important in the study of recurrent orbits (Capderou 2005), is defined by

Using *κ*, we thus obtain

and rewriting *j* with Eq. (4) results in

When the latitude of the point *P* equals the maximal attained latitude, noted *ϕ _{m}*, one can check that the ground track is in fact normal to the meridian. We recall that

*ϕ*is given by

_{m}### c. Angular adjustment

We define the *angular adjustement,* denoted *δ*, as the difference between the two angles *j* and *j*′. With Eqs. (4) and (10), writing tan(*j* − *j*′), we obtain

For a given satellite (*i* and *κ* fixed), *δ* is the function of the sole variable *ϕ*. These functions are plotted for the satellites TRMM (non–sun synchronous) and *Terra* (sun synchonous), Figs. 2 and 3, respectively.

The maximum of the function *δ*(*ϕ*) is attained at the equator, for *ϕ* = 0. This value, *δ _{M}*, is given by

For example, for TRMM, with *i* = 35.00° and *κ* = 15.4361, we find *δ _{M}* = 2.25°; for

*Terra*, with

*i*= 98.21° and

*κ*= 233/16 (recurrent satellite),

*δ*= 3.85°.

_{M}The minimum of the function, noted *δ*_{0}, is obtained for *ϕ* = ±*ϕ _{m}:*

With the expression of the derivative (*dδ*/*dϕ*), one shows that the tangent to the curve *δ* (*ϕ*) is horizontal for *ϕ* = 0 and vertical for *ϕ* = ±*ϕ _{m}*.

## 3. True along track

### a. Local orbital frame

Up to now, the satellite has been treated as a point, or at least, we have considered only the motion of its center of gravity. But as a vehicle, the satellite can also move about its center of inertia. Although this kind of motion is largely irrelevant for the purposes of calculating its trajectory, it is of course crucial when we come to ask what the instruments aboard will be able to view.

Manipulation of the angular orientation of the satellite is referred to as attitude control. The attitude of the satellite tends to vary under the action of couples, which may be external, due to radiation pressure or atmospheric drag on solar panels, or internal, due to mechanical motion of the instrument motors. A stabilization system is thus required to maintain the satellite in the right position relative to the local orbital frame. For any point *S* on the orbit, this frame is defined by the following three axes, illustrated in Fig. 4:

the yaw axis

*SZ*, directed toward the center of the earth, also called the nadir axis;_{c}the pitch axis

*SY*, directed normally to the orbital plane; and_{c}the roll axis

*SX*, lying in the orbital plane and completing an orthogonal right-handed system of axes. This axis lies along the velocity vector of the satellite when the eccentricity is zero._{c}

We shall refer to the axes of the local orbital frame as the Cardan axes, with the appropriate subscript. The angles obtained by rotation relative to these axes are the Cardan angles.

### b. Scanning modes

There are various ways for an instrument to look at the earth:

With a rotation about

*SX*_{c}, the instrument scans perpendicularly to its displacement. This is orthogonal or across-track scanning (i.e., the so-called XT mode).With a rotation about

*SY*_{c}, the instrument scans along the ground track. This is along-track scanning (i.e., the so-called AT mode).

An instrument like CERES can alternate between across-track and along-track scanning, or scan obliquely by a rotation of the instrument about an axis in the plane *SX _{c}Y_{c}* [named in this case the Programmed Azimuth Plane (PAP) scanning mode]. The angle between this rotation axis and

*SX*, denoted yaw angle

_{c}*α*, may theoretically vary on an entire circle. The yaw angle is measured in a plane perpendicular to the yaw axis.

### c. Adjusted along-track mode

The across-track swath corresponds to a yaw angle of 0° (or 180°), The along-track swath corresponds to a yaw angle of 90° (or 270°). In this case, the swath does not exactly cover the ground track (see Fig. 5) for the same reasons that the across-track swath is not exactly perpendicular to the ground track. By adjusting the yaw angle as a function of the latitude overflown, the ground track can be covered by the swath. This adjustment, resulting from the rotation of the earth, is equal to the angular adjustment calculated above [see Eq. (11)]. With this correction, the track covers the ground track (see Fig. 6). This scanning mode is called *adjusted along-track* or *the* TAT mode. The yaw angle is *α* = 90° + *δ*.

## 4. Comparisons with observations

### a. Collocation efficiency

As seen before, the CERES radiometer can be programmed to rotate the scan plane in azimuth. This ability first benefits the comparisons between radiances from two satellites by aligning their respective scan planes. Haeffelin et al. (2001) and Szewczyk et al. (2005) apply this technique for the intercomparison of CERES/Scanner for Radiation Budget (ScaRab) and of the different CERES flight models on *Terra* and *Aqua*. Following the former theoretical discussion, we have asked for a specific CERES instrument coverage request, giving the instrument rotation and function of the latitude. It was successfully programmed by NASA (thanks to Z. Peter Szewczyk) and tested on the FM2 and FM3 instruments, respectively, on *Terra* and *Aqua*, during eight days in February and March 2005, over about 120 revolutions (see example in Fig. 7). Since the ultimate objective is to study the radiation anisotropy mainly in the SW domain, for which the errors in the flux estimate are the largest, this true along-track mode worked only for the daylight side of the orbits (cross track for the nighttime side). The corresponding CERES SSF (edition 2) products have been analyzed by looking for the best collocations between the nadir and the 55° oblique fore and aft observations. The 55° value for the oblique angle is chosen according to the recommendations of Davies (1984) and Bodas-Salcedo et al. (2003). From radiation transfer modeling, they found that viewing zenith angles (VZAs) around 55° minimize errors when deriving fluxes from along-track scanners.

The minimum distances between each pair of observations (nadir aft, nadir fore, aft fore) have been sought for. A triplet distance has been defined from the standard deviation of the distances between the three observations and their center of gravity of the triangle. This distance is shown in Fig. 8 for the along-track data of *Terra* and TRMM and Fig. 9 shows the dramatic improvement of the modified along-track scanning mode of *Terra*. For the old configuration, the triplet distance is as large as 40 km for *Terra* and 10 km for TRMM, and it cancels at a latitude of 82° and 35°, respectively [for *ϕ* = *ϕ _{m}*, see Eq. (13)]. Figures 10 and 11 focus on the TAT for the six days of

*Terra*data and the three days of

*Aqua*data for 2005, respectively. The jumps in the series of points are presumably caused by the stepwise changes of the azimuth (constant within 0.6°). The figures are slightly different from

*Terra*to

*Aqua*. But in each case, they are almost identical for all the orbits (several dozens are plotted). One-third of the distances (the triplet distances as defined above) is between 2 and 3 km, two-thirds less than 1 km (from which, one-tenth less than 0.3 km). The distances increase with latitude, most probably due to the corresponding variation of sensibility (linked to the derivative of the function) shown in Fig. 3.

### b. Angular consistency of radiances

A way to evaluate the triplet collocation is to compare the two oblique LW radiances. Except for small possible azimuthal anisotropy (shadowing effect) for the daytime data (Minnis et al. 2004), these radiances should be equal. Figure 12a shows the comparison between the LW radiances from the 55° aft and fore directions for the TAT data of CERES/*Terra* in February 2005. The rms difference is small but significant (3.9%) and in all probability due to coregistration errors: though the distances between the FOV are small (less than 2 km) the differences in the surface and shape of the FOV and the time lag between the two observations (4 mn) is not negligible and may produce large radiance differences due to the high space and time variability of clouds. Since the pixel length at oblique is approximately twice as the nadir footprint (which is 20 km), we have also tested the comparisons keeping only homogeneous areas. The corresponding homogeneity requirement is based on the radiance scatter over a distance of 50 km in the along-track direction, and is selected if the standard deviation is less than 5% of the mean radiance. When only homogeneous areas are selected (Fig. 12b), the rms differences are noticeably reduced (1.8%). Compared to the rms differences of 7% and 3.5% found with the normal AT mode, respectively, for all scenes and homogeneous scenes, the modification of the AT mode yields definitive improvements. In both cases, for the TAT mode, the mean difference is close to zero (−0.3% and −0.2%) showing the equality of both oblique radiances.

For the TRMM data, homogeneous targets, where the collocation errors are minimal, were also selected to create triplets. These targets are located around 35°S and 35°N, where the satellite motion accompanies the earth rotation—and in fact, where AT and TAT modes are equivalent (see Figs. 2 and 8). For the nighttime data (Fig. 13a), the rms differences between both oblique data are close to zero (0.7%) as expected. For the daytime (Fig. 13b), the same differences are somewhere significant (1.4%). A step ahead is to separate ocean from continent. For daytime data, we found rms differences of 1.0% and 2.3%, and mean differences of 0.0% and −1.30%, respectively, for the ocean and continent. This is probably related to the LW azimuthal anisotropy (Minnis et al. 2004) since the departure from zero is more manifest for the continent than for the ocean. For comparison, the mean differences between radiances at nadir and at 55° fore is up to 6% (Fig. 14) in agreement with the well-known limb-darkening effect.

## 5. Conclusions

The CERES *true along-track mode* was successfully implemented on *Terra* and *Aqua* during eight days in February and March 2005. The present results show that the TAT mode reduces most collocation errors (i.e., the distance between the FOV centers) to less than 2 km and a few less than 0.5 km. The residual collocation errors vary according to the latitude because the azimuth rotation of the instrument is not continuous but changes in steps of 0.6°. The best collocations (better than 0.3 km) were obtained around the equator (5°N and 5°S), where the latitude dependence of the correction is the lowest.

The goal of this operation is to observe different earth targets from three directions (nadir, and 55° fore and aft) in nearly simultaneous conditions. A simple test of angular consistency is the comparison of the two oblique LW radiances, which are theoretically equal (except possible small shadowing effects). Due to large scatters in the intercomparisons, the test fails in most cases except for homogeneous areas (over 50 km). This is almost certainly due to residual coregistration errors that occur even when the centers of the FOV are close each other (the size and the shape of FOV vary according to the scan angle in the CERES design). Furthermore, there are about 4 min between the fore and aft observations. Considering the high space and time variability of the cloud field, small coregistration errors over heterogeneous areas can generate radiance differences greater than the angular effects. Over homogeneous areas, the tests have provided confidence that both the angular behavior is correct and that the coregistration errors have negligible effects.

Despite the restriction to homogeneous areas, the selected dataset from this study is unique with regards to broadband channels, and would be useful for additional checks of the CERES ADM and for the future algorithms of flux retrieval for the Earth-CARE BBR. When the comparison test of the LW oblique radiances is successful, the selected triplet is assumed free from significant coregistration errors and useful to study the SW angular effects. Methods based on MISR and POLDER should lead to more extended simulation dataset of SW radiances but require one to properly take into account the narrow-to-broadband conversions. The CERES TAT data favorably provide direct broadband SW and LW radiances.

## Acknowledgments

The CERES data were obtained from the Atmospheric Sciences Data Center at the NASA Langley Research Center. The authors thank N. G. Loeb and Z. P. Szewczyk for helpful discussions and for the implementation of the true along-track data, R. Kandel and E. Lopez-Baeza for their devotion to the BBR and EarthCARE projects, and three anomymous reviewer for their constructive comments. This work was funded by ESA Contract ESTEC 17772/04/NL/GS.

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## Footnotes

*Corresponding author address:* Michel Capderou, Laboratoire de Météorologie Dynamique (IPSL), Ecole Polytechnique, F-91128 Palaiseau, France. Email: capderou@lmd.polytechnique.fr