## Abstract

This paper presents a simple approach to adjust microwave brightness temperature distributions obtained from slant-path measurements for projection effects. Horizontal displacement in the direction of sight is caused by signal contributions from other than near-surface layers that are projected to the footpoint of observation. In particular at frequencies sensitive to ice particle scattering the horizontal projection effect can amount to values as big as the vertical cloud extent. Based on cloud model–generating, three-dimensional hydrometeor distributions at subsequent model time steps and a modified one-dimensional radiative transfer model, the high correlation of effective radiance contribution altitudes and brightness temperatures at 37.0 and 85.5 GHz is demonstrated. For these altitudes, described by the centers of gravity of the spectral weighting functions, regression equations are derived with standard errors below 0.61 km at 85.5 GHz and 0.22 km at 37.0 GHz for both the Special Sensor Microwave/Imager (SSM/I) and Tropical Rainfall Measurement Mission Microwave Imager. Once the centers of gravity are retrieved a simple geometry correction can be applied to the measurements.

Application to model cloud fields at various time steps and different oberservation geometries shows a significantly improved correspondence of brightness temperature and hydrometeor distributions. This method is also applied to SSM/I observations during the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment in the equatorial Pacific. Considerable improvements of single-channel rain retrievals based on 85.5-GHz measurements compared to shipborne radar data were achieved, which suggests that a major uncertainty of so-called scattering algorithms can be explained by geometry effects that can be easily corrected. Multichannel algorithms, however, require a more elaborate integration scheme to allow for both frequency and scene-dependent adjustments.

## Introduction

After almost 20 years of passive microwave remote sensing of precipitation, which became operational with the launch of the first Special Sensor Microwave/Imager (SSM/I) in June 1987, those parts of the inversion problem introduced by the coarse spatial resolution of the sensors have received great attention and are currently regarded as the most severe ones. Superimposed are the uncertainties of the vertical hydrometeor distribution, surface and atmospheric background emission, and comparably minor problems such as effects of aspherical particles, etc. The payload of the forthcoming Tropical Rainfall Measurement Mission (TRMM) includes an updated version of the SSM/I, the TRMM Microwave Imager (TMI), which provides a better spatial resolution and an additional low-frequency channel at 10.7 GHz (see Table 1). The enhanced footprint size will contribute to a better assessment of beam-filling effects; however, those channels mainly sensitive to surface rainfall still suffer from coarse resolution.

If temporally accumulated products are required, the above error sources are even exceeded by temporal sampling errors. Bell et al. (1990) derived a theroretical estimate for this error that strongly depended on averaging scales and reached only 10% assuming a 10-km satellite field of view over (500 km)^{2} and 1-month averaging scales. However, for instantaneous rain retrievals, the spatial sampling effects dominate the error budget. Part of this is the so-called beam-filling problem, which originates from the nonlinear and nonunique relation of area-averaged surface rain rates and area-averaged brightness temperatures (e.g., Chiu et al. 1990). Note that temporal sampling errors are random in nature and decrease with an increasing averaging period, while errors induced by coarse sensor resolution bias the rainfall estimate. Their correction requires introduction of advanced information that may again require averaging in space and time, for example, the assumption of rain intensity distributions over sufficiently large areas and time periods.

When the imaging capabilities of passive microwave sensors are analyzed in more detail, another aspect comes into play. In case of vertically extended rain clouds with large horizontal inhomogeneities, the received signal cannot be described by plane-parallel radiative transfer considerations. The strong horizontal gradients of hydrometeor extinction and scattering properties require the inclusion of radiation emerging along the horizontal, that is, crossing vertical boundaries (i.e., the so-called leakage). Special cases are the high emission by warm cloud sides compared to lower emission from cloud tops, which is either directly emitted toward the sensor or emitted downward and further reflected by the surface into the direction of sight. The relative contribution of such components to the total signal was demonstrated by several researchers using simple box-type cloud models (Weinman and Davies 1978; Kummerow and Weinman 1988; Haferman et al. 1993; Petty 1994; Liu et al. 1996). Two main conclusions can be drawn: 1) if 3D effects are neglected, simulated brightness temperatures become too high, which leads to a general underestimation of surface rain rates. The amount of underestimation depends on fractional cloud amount, which in turn depends on the assumed sensor field of view. 2) At those frequencies where both scattering and emission contribute to the upwelling radiation, the net effect of warm cloud-side emission, cloud-side reflection at the surface, and scattering–emission by the cloud body itself can approach values of the cloud-free ocean (Petty 1994), so that a single-channel rain detection would not even identify the rain cloud.

The general underestimation of rainfall introduced by plane-parallel modeling is tackled by beam-filling corrections, which may consist of a static factor (e.g., Hong 1994), a dynamic adjustment (Wang 1996), or by minimizing the beam-filling problem by using the channel with the highest available resolution to drive the rain retrieval (Petty 1994). In all cases, the beam-filling correction addresses the above-mentioned nonlinearity between area-averaged rain rates and brightness temperatures and their variation with channel frequency.

The 3D problem, however, includes another aspect that reaches beyond the beam-filling issue, although it is sometimes classed with the beam-filling problem (Wang 1996). First, the actual geometry of the cloud, that is, extent and aspect ratios in three dimensions, and the clustering of single clouds or organized cells have to be addressed. Second, the projection of scattering and emission contributions at different altitudes into the horizontal plane may lead to significant shifts of the location of radiance sources toward the surface point of the beam. Roberti et al. (1994) have analyzed this shift effect and attempted to correct for it, which was then implemented into the rain retrieval of Haferman et al. (1996). However, the relative improvement of the results was not explicitly proven. Petty (1994) and Petty et al. (1994) have demonstrated that the issues of beam filling and effects caused by the 3D cloud geometry are strongly connected, which suggests that a proper beam-filling correction must include some sort of assessment of actual cloud geometry factors and their effect on brightness temperatures as observed from space. Our study makes an attempt to provide a rather sophisticated method for the correction of 3D effects as a basis for improved beam-filling corrections. This procedure is demonstrated using cloud model and 3D radiative transfer simulations. In section 2, the simulation framework is described and hydrometeor fields and associated brightness temperatures at TMI and SSM/I frequencies for several time steps of an oceanic squall line are presented. This section also introduces a modification of a one-dimensional (1D) radiative transfer model that provides very similar results compared to a 3D model but with much less computational effort. In section 3 the basis for the correction method is founded; the method is then developed in section 4 and an error analysis is presented. Finally, case studies of SSM/I data from the Global Precipitation Climatology Project Algorithm Intercomparison Project (AIP-3) are used to demonstrate the improvement of a single-channel rainfall retrieval. The paper is completed by a discussion and conclusions.

## Simulations over ocean

A dynamic cloud model simulation was taken from an experiment performed in the framework of the Tropical Ocean Global Atmosphere (TOGA) Coupled Ocean–Atmosphere Response Experiment (COARE) based on the model of Tao et al. (1987). The case represents an organized line of convective cells from the growing until the mature stage, that is, time steps between 180 and 360 min with 60-min intervals. Figures 1 and 2 show the distributions of liquid water and ice paths at each stage. The surface rainfall intensities mainly follow the liquid water patterns with heaviest showers at the right cloud edge of up to 80 mm h^{−1}.

The model output provides profiles of temperature, pressure, and relative humidity as well as hydrometeor liquid water and ice concentrations for nonprecipitating droplets and crystals, rain, snow, and graupel. A gamma-particle size distribution was assumed for the precipitating particles, while the nonprecipitating particles were described by a modified gamma distribution. The precipitation rates, that is, size distributions, were derived from liquid water and ice densities by fitting nonlinear regressions. For snow and graupel, a size-dependent density variation was introduced according to Magono and Nakamura (1965) and Klaassen (1988). All hydrometeors were considered spherical and homogeneous. Thus all variables driving passive microwave emission and scattering are available and represent a consistent environment.

Two microwave radiative transfer models were implemented to account for 3D effects: 1) a backward Monte Carlo model (MCM) (Roberti et al. 1994) and 2) a plane-parallel model using Eddington’s second approximation of radiative transfer (e.g., Weinman and Davies 1978). The latter was developed and compared to a quasi-exact matrix–operator model (MOM) (Bauer and Schluessel 1993) that finds standard errors below 1.5 K. The Eddington model was adjusted to provide approximated 3D conditions by slant-path simulations for both the downwelling and upwelling directions but to allow for computer time and memory resource savings. This approach was also chosen by Liu et al. (1996;his Fig. 6) for a simplified cloud geometry.

For completeness, the differences of vertically polarized brightness temperatures between our two models at 10.7, 37.0, and 85.5 GHz at cloud model simulation time step *t* = 360 min were calculated. Figures 3a,c,e show the results at the original cloud model resolution, while Figs. 3b,d,f show the differences after application of antenna patterns that were moved stepwise across the scene. These were approximated by Gaussian functions determined by the cross-track and along-track footprint sizes associated with 3-dB sensitivity limits as given in Table 1.

The general behavior shows an increase of brightness temperature differences with frequency due to the increase of the scattering contribution to the signal, is commonly overexpressed by the Eddington method. However, the average errors are in the range of a few degrees and are even more reduced when antenna patterns are applied. Most importantly, the differences between MCM and the modified Eddington model (MEM) do not show any features associated with displacements or misalignments associated with 3D effects not captured by the MEM approach. Thus MEM can be considered as a quasi-3D radiative transfer model, which requires much less computer time than any other model. The accuracy of MEM compared to MCM is similar to that of the 1D Eddington model compared to MOM or other techniques (e.g., Kummerow 1993).

## Weighting functions and centers of gravity

To determine the contribution of single atmospheric layers, *C*_{i}, to the upwelling brightness temperature *T*_{B}, the radiance source term *J,* as calculated for an optically thin layer (e.g., Kummerow 1993), has to be multiplied by the integral transmission of the overlying layers:

where *z*_{i} and *z*_{i+1} are the lower- and upper-altitude boundaries of the actual layer, *σ*_{j} denotes the *j*th volume extinction coefficient, and *μ* = cos(*θ*) with zenith angle *θ.* Following Mugnai et al. (1993) and Muller et al. (1994), these functions can be normalized by the upwelling brightness temperature at the top of the atmosphere (*z* = *z*_{T}) and the layer thickness at the altitude where the radiation originates:

so that integration over the atmospheric column including the surface contribution (not shown here) gives a value of 1. Note that in contrast to Mugnai et al. (1993) and Muller et al. (1994) the approximation of 3D radiative transfer by the two-path 1D radiative transfer by MEM requires the calculation of the layer contributions to the upwelling radiances along both paths. Thus layer indices between *i* = −*n* + 1 and *i* = *n* − 1 occur in (1), and layer boundary altitudes are defined as negative for the downward-directed radiation path.

In general, when cloud, rainwater, and ice are present, the weighting functions show highly variable distributions. The maximum contributions to the upwelling radiances are determined by the change of optical depth and emission or scattering efficiency with altitude. These are driven by many free parameters such as temperature, water vapor, liquid water, ice concentrations, particle type, and particle size spectra. Inserting the emission and scattering parts of the source term into (1) provides expressions for the contributions to *T*_{B} by emission and scattering from each layer in the column after application of (2). Figures 4a,b give examples for the resulting weighting functions at one surface location but two different viweing angles, that is, *x* = 83 km, *y* = 70 km, *ϕ* = 90°, and *ϕ* = 270°, where *ϕ* is counted positive in counterclockwise direction with a positive *y* direction set to *ϕ* = 0°. The distribution in Fig. 4a is mainly determined by large amounts of cloud ice, snow, and graupel in the line of sight (upward-directed path) with a very high surface rain rate of 56.6 mm h^{−1}. At those frequencies, which are mainly sensitive to ice particle scattering, the weighting function peaks are shifted toward high altitudes. At 85.5 GHz the weighting function peaks at about 6.5 km, while at 37.0 GHz the superimposed effects of emission by liquid water below the freezing level (at about 4.5-km altitude) and ice above the freezing level produce two distinct peaks at different altitudes, that is, 2 and 5 km. The lower-frequency channels get some contribution from higher layers due to significant ice amounts, but the liquid rain still dominates the signal, particularly at 10.7 GHz. The resulting vertically polarized brightness temperatures are 250.6 K (10.7 GHz), 273.6 K (19.35 GHz), 248.8 K (37.0 GHz), and 218.6 K (85.5 GHz), respectively. Including the layer contributions of the downward-directed path (negative altitude range) shows no significant contribution from either the surface (*z* = 0) or other layers. The second profile (Fig. 4b) shows a different picture since its footpoint is located at the same position and the same amount of surface rainfall. However, in the slant path, there is almost no cloud ice. Thus all weighting functions peak at almost the same altitude well below the freezing level, while the contributions above are mainly caused by some nonprecipitating cloud water and water vapor emission. For this profile, the corresponding vertically polarized brightness temperatures are 254.7 K (10.7 GHz), 283.3 K (19.35 GHz), 272.1 K (37.0 GHz), and 277.6 K (85.5 GHz), respectively. Obviously, the contribution from the surface and the downward-directed path are much higher than viewing the scene from the other direction. A large fraction of the downward-emitted radiation is reflected at the surface and not obscured by cloud layers in the upward direction. This example illustrates the ambiguity of brightness temperature–surface rainfall relations as a consequence of inclined view through a well-structured cloud.

The projection effect that causes brightness temperatures, which originate from radiance emission and scattering at high altitudes, becomes associated with (*x, y*) coordinates at the surface, which are located at the footpoint of the line of sight off the location of the bulk emission scattering. The amount of displacement depends on the altitude of the bulk radiance contribution. This in turn is related to the weighting functions and is strongly dependent on frequency and the hydrometeor profiles. Roberti et al. (1994) attempted a geometry correction, that is, the calculation of the displacement between footpoint and approximate location of maximum radiance contribution, by the formula

where *d*_{M} denotes the displacement and *z*_{M} denotes the altitude of the scene-averaged weighting function maximum with local zenith angle *θ.* This correction provided only moderately improved results that are caused by both the scene averaging and the selection of *z*_{M} as the variable parameter. As noted earlier, the weighting functions may be highly variable and comparably smooth so that the location of their maximum becomes a very unstable parameter to be chosen for correction. A much better estimate is provided by the center of gravity because it includes an integration over the entire altitude range accounting for all contributions instead of selecting one particular altitude:

Here, *z*_{CG} denotes the altitude and *C*_{CG} is the normalized contribution to *T*_{B} between the upmost layer of the downward-directed beam and the top of the atmosphere (*z* = *z*_{T}) toward the satellite, as obtained from (2). The location of the centers of gravity at all six frequencies are also presented in Figs. 4a,b. In this case, the *z*_{CG}’s are found at different altitudes than the weighting function peaks, which makes sense physically due to the integrated contribution of all the layers to the signal at the top of the atmosphere. Thus the effective altitude of the radiation origin does not correlate well with the weighting function peak unless it is located at high altitudes near the cloud top. The same is true for the brightness temperatures at 85.5 GHz, which are better correlated to the *z*_{CG}’s than to the *z*_{M}’s. The lower correlation for *z*_{M} is explained by the highly variable emission and scattering phenomena contributing to the brightness temperatures.

A cross section parallel to the *x* axis at model time step *t* = 360 min at *y* = 70 km was selected. Corresponding brightness temperatures at three window frequencies, that is, 10.7, 37.0, and 85.5 GHz, and associated *z*_{CG} values were computed for the original model resolution (Figs. 5 and 7) and after application of TMI antenna patterns (Figs. 6 and 8) at azimuth angles *ϕ* = 90° (view from the left) and *ϕ* = 270° (view from the right), respectively.

The cross section represents a small area of large surface rain rates reaching 60 mm h^{−1} at *x* = 82 km with moderate ice particle concentrations located in multiple cells at *x* = 50 km (snow, graupel) and 75 km (graupel). The minimum brightness temperatures at 85.5 GHz indicate the response to ice scattering. However, comparing the two viewing directions (Figs. 5a vs 7a, Figs. 6a vs 8a), these features are found at different positions separated by approximately 15 km. Depending on the viewing direction, each minimum is shifted by 7.5 km with respect to the convective cell. This behavior is still visible at 37.0 and 85.5 GHz when antenna patterns are applied, while the dynamic range of brightness temperatures at 10.7 GHz is considerably reduced by the smoothing effects of its coarse resolution (Figs. 6a, 8a). Comparing the locations of the minimum values of *T*_{B} and the maximum values of *z*_{CG}, only minor differences appear, that is, the higher the value of *z*_{CG}, the more response in *T*_{B} at the respective frequency either by scattering or emission (Figs. 5a versus 5c, Figs. 7a versus 7c). This is different at 10.7 GHz due the strong emission contribution by the cloud sides and its reflection at the sea surface, which leads to negative *z*_{CG}, indicating the stronger contribution to the upwelling radiance from the downwelling path. Then the location of the maximum of *T*_{B} corresponds to the inflection point of the *z*_{CG} curve.

As a result of antenna pattern application, this local mismatch effect is reduced, and the gross locations of the *T*_{B} and *z*_{CG} extrema coincide well, as expressed by the very high correlation coefficient of −0.97 between *T*_{B} and *z*_{CG} at 85.5 GHz (Figs. 6a versus 6c, Figs. 8a versus 8c). This is, however, a function of emission versus scattering contribution since mismatch effects are mainly caused by the above-described phenomenon and thus affect mainly lower-frequency channels. Another very interesting feature is the much weaker expressed sensitivity of the polarization differences to viewing direction before and after application of antenna patterns. Comparing Fig. 5b with Fig. 7b and Fig. 6b with Fig. 8b reveals only minor differences for both amounts and positions of polarization difference minima, which are associated with large cloud optical depths. Even though the dynamic range of polarization differences is smaller and the features are broader, the presented results indicate a much smaller degree of affect by 3D effects than for brightness temperatures that have been proposed by Petty et al. (1994). Finally, the assessment of *z*_{CG} allows an estimate of the most probable altitude at which the geophysical parameter, for example, rain rate, is actually observed. Based on Figs. 6 and 8 a single-channel algorithm using only 10.7-GHz measurements to retrieve rainfall would actually be most sensitive to the rainfall variations close to the surface, while a similar approach employing the 19.35-GHz channel would rather be sensitive to rainfall at 1–2 km. For higher frequencies, not only the altitude range increases, which contributes most significantly to the upwelling radiance, but also its variability and therefore its connection to surface rainfall becomes more unstable.

## Projection correction

After calculating the radiative transfer for all model time steps and maximizing the projection effect by choosing two distinct viewing directions orthogonal to the squall line axis, retrieval equations for *z*_{CG} were derived. To avoid overexpression of those *T*_{B} − *z*_{CG} pairs for which more data points were available over the dynamic range, the *z*_{CG} range was subdivided into intervals that were then equally filled from the original dataset by random selection. Linear regressions were computed for those channels of the SSM/I and TMI where the projection effect is most significant—37.0 and 85.5 GHz. Only those data to which the antenna patterns were applied were used to address realistic observation conditions. The resulting retrieval equations are of the form

The coefficients *a*_{i} and required channel measurements *T*^{i}_{B} are given in Table 2. Once *z*_{CG} as a function of frequency is determined from the measurements, a correction can be applied as follows:

which translates into the (*x, y*) coordinates of the employed model simulation as

with the original coordinates of the brightness temperature measurement (*x, y*), the corrected location (*x**, *y**), and the azimuth angle of the observation *ϕ.* For real SSM/I or TMI measurements the correction has to be applied backward in the line of sight, that is, toward the satellite in radial direction.

Figure 9 shows the results of the corrections when applied to the simulated brightness temperatures at 85.5 GHz at model time step *t* = 360 min and azimuth angle *ϕ* = 90°, while Fig. 10 gives the corresponding result for an azimuth angle, *ϕ* = 270°. Thus the correction is applied parallel to the *x* axis since the viewing angle was set constant over the scene. The upper panels are included to illustrate the extent and amounts of vertically integrated cloud ice and liquid portions to be better compared to the brightness temperature distributions. The middle panels show the uncorrected (left) and corrected (right) TMI simulations, while the lower panels give similar results for the SSM/I. When the correction is applied to the 37-GHz simulations the geometric shift is small and the correspondence between liquid water and ice paths and brightness temperatures is not obvious (not shown here), due to the comparably narrow dynamic range in the brightness temperatures. This becomes even more evident at the lower SSM/I spatial resolution, which causes comparably flat brightness temperature patterns. Figures 9 and 10 show the reflection of radiation emitted by warm cloud sides into the line of sight at the right edge of the cloud.

Most obvious is the good coincidence of the ice extent (solid line) with the corrected 85.0-GHz patterns, which are shifted on average by at least 10 km to the right at *ϕ* = 270° and to the left at *ϕ* = 90° for both sensors (Figs. 9d,f; 10d,f). After correction, the local brightness temperature minima match the maxima of the ice paths very well. For the SSM/I the coarse spatial resolution smooths this effect; however, the gross correction improvement is still visible (Figs. 9e,f; 10e,f). The correction procedure has the advantage that the backshift is applied pixelwise and thus accounts for the local projection effect in contrast to scene averaged procedures. Additionally, the stretching of the cloud signature by the projection effect, as already noted by other authors (Roberti et al. 1994; Haferman et al. 1996), is corrected by the pixelwise retrieval of *z*_{CG} and subsequent shift correction.

Another important issue is that comparing the corrections applied to the same simulations and various azimuth angles should produce azimuth-independent results. While the original simulations, for example, for 85 GHz (Figs. 9c, 10c), show a relative displacement of more than 20 km locally dependent on the intensity of the convection, thus increasing *z*_{CGz}, this difference has disappeared after correction (Figs. 9d, 10d). The same is observed at SSM/I resolution (Figs. 9e, 10e versus Figs. 9f, 10f).

## SSM/I data

To validate the correction procedure, case studies were analyzed using SSM/I satellite data that were collected during the period between November 1992 and February 1993. This data were processed for purposes of the AIP-3 over the TOGA COARE region. Collocated ship radar measurements were recorded and regarded as ground validation for AIP-3. For details of ship radar specification, data processing, and correction, please refer to Ebert et al. (1996). Since the projection correction procedure was developed from a fixed model environment, case studies had to be chosen that represent similar conditions, that is, they were measured over the same area and climatological regime. However, during TOGA COARE a large variety of cloud types and precipitation systems was observed (Ebert 1996), which is not entirely covered by our cloud model study.

All available 178 cases for SSM/I overpasses were included for further analysis. The radar data were averaged to the 85.5-GHz footprint using 3-dB footprint sizes as given in Table 1. This was done twice, that is, for the uncorrected footprint and corrected footprint locations. Only those radar–satellite data pairs were used in which both gave positive rainfall amounts to avoid any influence of rain detection problems. Since the correction is to be applied for each frequency, single-channel algorithm had to be chosen that uses only the 85.5-GHz *T*_{B}. Thus the algorithm developed by Adler et al. (1991) from a combined cloud–radiative transfer model was taken, which relates surface rainfall, RR (mm h^{−1}) to the 85.5-GHz horizontally polarized *T*_{B}, that is, *T*_{BH} (85.5 GHz):

Only A-scan satellite pixels with odd pixel numberings were included to ensure a centered overlap of all channel footprints since basically all are required for the SSM/I projection correction. Two different criteria for a satellite-derived low rain-rate cutoff were applied to demonstrate the influence of low versus high rain rates on the statistics. When a cutoff value of 1 mm h^{−1} was used, 45 cases remained, providing at least five pixels for intercomparison, while 15 cases remained when a cutoff for rain rates below 5 mm h^{−1} was employed. Figure 11 provides the corresponding error statistics, shown here as differences between the respective value after correction and before correction for these two setups. The cases were ordered so that the correlation differences decreased with case number (given here as percentage of total case number). The biases and root-mean-square errors (rmse) correspond to the order of cases in the correlation graph.

Using a cutoff rain rate of 1 mm h^{−1} (Fig. 11a), it becomes obvious that in 70% of the 45 cases an improvement of the correlation is obtained. The biases and rmse’s remain relatively unchanged with the exception of a few individual cases. When rain rates below 5 mm h^{−1} are excluded, the improvement by the projection correction becomes even more obvious. Now, 80% of the 15 cases have better correlations with 40% by more than 0.5. With two exceptions, the biases become significantly better, as do the rmse’s. This shows that in those cases in which higher rainfall intensities associated with stronger convection are observed the projection effect obviosuly distorts the viewing geometry, which can be corrected again by our approach.

The two arrows in the right panel of Fig. 11b denote one case in which the correction retrieved worse results than before (upper arrow) and another case in which significant improvements resulted in all error categories (lower arrow). In the first case (*F-10* on 1101 UTC 20 December 1992) the rainfall patterns derived from the satellite data did not match the radar measurements at all with respect to location and intensities. In the other case (*F-11* on 1851 UTC 27 December 1992) convective cells aligned parallel to the satellite scan were observed for which the satellite estimates, that is, the relation between low brightness temperatures at 85 GHz and surface rainfall, were already good except for the misplacement by the projection effect. In general, the projection correction seemed to perform best when rather isolated features, that is, cells or squall lines, were present, while in case of more homogeneous and dense cloud features no improvement was noted.

## Conclusions

The objective of the correction of 3D effects is the improvement of the geometrical relation between hydrometeor and corresponding brightness temperature distributions that are distorted by the satellite-viewing geometry. This paper presents a simple method to correct for the horizontal displacement and stretching of brightness temperature distributions as a results of the inclined view of current and future passive microwave sensors with a conical scan geometry. A model cloud obtained from a dynamic cloud model at different time steps was taken as input for 3D radiative transfer simulations. A modified 1D model was developed for purposes of computation efficieny, which proved to give very similar results compared to the 3D computations.

The centers of gravity of the brightness temperature weighting functions were then used to determine the position of effective radiance contribution along the line of sight. This provides a measure for the displacement of radiation origin with respect to the footprint location, which in turn can be easily used for correction. Regression formulas for the altitudes of the centers of gravity were obtained after applying spatial weights to the simulations according to realistic antenna patterns of SSM/I and TMI. Retrieval errors were found to be below 0.61 km at 85.5 GHz and 0.22 km at 37.0 GHz. Application to the model cloud itself and to independent measurements of the SSM/I during AIP-3 showed considerable improvements of hydrometeor versus 37.0 and 85.5-GHz *T*_{B} distributions. The systematic improvement for both *F-10* and *F-11* as well as ascending and descending overpasses also shows that the displacement of cloud structures as observed by the satellite compared to ground radar measurements is not explained by other means, for example, satellite navigation errors.

Effects of sensor-viewing geometry and their correction become particularly important when hydrometeor profiling algorithms are developed. In this case, databases are constructed that relate *vertical* hydrometeor profiles to brightness temperature vectors, which makes sense from a microphysical point of view due to the dominance of vertical cloud formation processes in convective situations that provide unique radiation signatures measurable by satellite sensors. However, SSM/I and TMI satellite retrievals are always applied to measurements originating from slant paths, that is, cloud columns in which the microphysical relation between neighboring layers is not similar to that driven by vertical processes. Generally spoken, if profiling algorithms use only vertical hydrometeor distributions for database creation, the actual measurements of inclined profile signatures may not be covered by the database. Thus either the radiative transfer simulations have to be carried out for real 3D geometries or 3D corrections are to be applied to the measurements, which improve the geometrical correspondence between hydrometeor and brightness temperature distributions. In the case of rain-mapping algorithms based on single- or multiple-channel regressions as well as rather empirical techniques, the geometry correction provides a simple opportunity for improvements as demonstrated by this study.

More effort has to be spent on the generalization of this procedure for two reasons. 1) The correction presented in this paper was derived from a single event in tropical oceans. Thus an extended version is required that can be applied to various regimes of surface temperature, water vapor paths, near-surface wind speeds, and cloud types. 2) Since the projection effect is a strong function of frequency, as is the single-channel correction, the combination of brightness temperatures for multichannel algorithms requires a sophisticated scheme for geometric adjustment of pixel positions and footprint overlaps. The results of this study show, however, that the projection correction is primarily required for measurements at higher frequencies, for example, at 85 GHz, so that by introducing a high-resolution grid by oversampling all measurements, the above-presented correction procedure can be applied without additional effort of footpoint location unification.

## Acknowledgments

The authors wish to thank Drs. W.-K. Tao and C. Kummerow of the NASA/Goddard Space Flight Center for the permission to use the simulations of the Goddard Cumulus Ensemble Model for this study. We also thank C. Kummerow and G. Petty for their valuable comments. Finally, we gratefully acknowledge the support by P. Huebl for the analysis of AIP-3 satellite data. This work was partly funded by the Deutsche Agentur für Raumfahrtangelegenheiten (DARA), Kennz. AN-50 EE 9501-ZK, European Community ENV4-CT96-0281, and ESA-ESTEC 11947/96/NL/CN.

## REFERENCES

## Footnotes

*Corresponding author address:* Mr. Peter Bauer, DLR, Space System Analysis Division, Linder Höhe, 51170 Köln, Germany.