## 1. Introduction

Most instantaneous passive-microwave rain-retrieval algorithms currently implemented use a cloud database constructed offline (Kummerow et al. 1996; Kummerow and Giglio 1994a,b; Smith et al. 1994; Tesmer and Wilheit 1998). The databases associate calculated microwave brightness temperatures to sample rain events representing those that are expected to produce the eventual measurements, namely, in our case, precipitation over the tropical ocean. Once a representative database is constructed, one processes each set of instantaneous measurements by searching the database for those scenarios whose associated radiances are closest to the measurements. The details of the search and eventual estimation procedures differ from one retrieval algorithm to the other, but the general principle is the same. In the case of the Tropical Rainfall Measuring Mission’s (TRMM) Microwave Imager (TMI), the passive-microwave instantaneous retrieval algorithm uses a large database (Kummerow et al. 1996), which was constructed using various cloud model simulations (Soong and Tao 1984; Tripoli 1992). Radiative transfer calculations followed by the appropriate filters were used to associate to each simulated rain event (itself consisting of surface wind and temperature and relative humidity and hydrometeor profiles, all area-averaged to match the TMI resolution) the brightness temperatures that one would expect the TMI’s 10.7 GHz H- and V-pol, 19.3 GHz H- and V-pol, 21.3 GHz V-pol, 37 GHz H- and V-pol, and 85.5 GHz H- and V-pol channels to measure.

*p*(

**R**|

**T**

_{b}) for the rain

**R**given brightness temperatures

**T**

_{b}in terms of the more readily computable probability

*p*(

**T**

_{b}|

**R**) for the brightness temperatures given the rain:

*p*

**R**

**T**

_{b}

*p*

**T**

_{b}

**R**

*p*

_{prior}

**R**

*p*

_{prior}(

**R**)” represents any a priori knowledge about the rain; in practice,

*p*

_{prior}would be an archival lognormal distribution if no independent information about

**R**is available].

The main obstacle to conducting these studies is the large number of variables for which one has to account. In section 2, we begin by studying the vertically stratified rain by itself in order to derive an economical representation of the rain profiles. In section 3, we study the brightness temperatures and derive expressions for the conditional covariances on both sides of (1). In section 4, we further use our results to derive first-order parametrized retrieval formulas that estimate rain rates and their uncertainties from measured brightness temperatures.

## 2. Principal component analysis of the vertical rainfall *R*

*R*

_{1}, . . . ,

*R*

_{8}the rain in the first eight 0.5-km layers above the surface (numbered from the surface up), one can compute the covariance matrix of these variables from one’s database and diagonalize it. The matrix of change of basis specifies which eight (new) linear combinations of the

*R*

_{i}’s are mutually uncorrelated, and the associated eigenvalues determine the amount of information carried by each of the new variables. Large eigenvalues indicate a correspondingly large variation in the associated variable while smaller eigenvalues indicate that the value of the corresponding variable changes relatively little over the whole data. In practice, we calculated the covariance matrix of log(

*R*

_{i}) for the TRMM cloud-simulations database and then diagonalized it. The main result is that the largest eigenvalue is significantly larger than the remaining seven, and its eigenvector is very close to

*R*

^{′}

_{1}

^{8}

_{1}

*R*

_{i}. The eigenvalues were

^{−6}

*r*stands for log(

*R*)];

*R*

^{′}

_{1}

*R*

^{′}

_{2}

*R*

^{′}

_{8}

Indeed, when reconstructed using *R*^{′}_{1}*R*^{′}_{2}*R*^{′}_{3}*R*^{′}_{4}*R*^{′}_{5}*R*^{′}_{6}*R*^{′}_{7}*R*^{′}_{8}

In order to verify that this high correlation between the rain in the various layers is not due to an artifact of the cloud models used to generate the TRMM passive-microwave database in the first place, a similar analysis was applied to actual data from the TRMM radar. We analyzed the data from 60 orbits completed in Sept 1998. The natural logarithms of the rain-rate estimates of the TRMM-combined algorithm (Haddad et al. 1997) for the fourteen 250-m layers between 750 m and 4 km were used. The first three altitude bins near the surface were ignored to avoid surface clutter problems. The covariance matrix was calculated and diagonalized. The results obtained are quite similar to the ones found for the passive microwave database rain rates. For convective events, the eigenvalues were 12.46 > 5.1 > 0.94 > 0.3 > · · · > 1 × 10^{−2}. The coefficients of the eigenvector Σ_{i} *a*_{i} log(*R*_{i}) for the first eigenvalue 12.46 all verified 0.17 < *a*_{i} < 0.3, quite close to the value 1/*R*^{′}_{1}^{−3}. The coefficients of the eigenvector Σ_{i} *a*_{i} log(*R*_{i}) corresponding to the first eigenvalue were in the range 0.21 < *a*_{i} < 0.29. Merging all cases together, the eigenvalues were 11.8 > 0.43 > 0.14 > · · · > 9 × 10^{−3}, with an eigenvector Σ_{i} *a*_{i} log(*R*_{i}) for the first eigenvalue satisfying 0.21 < *a*_{i} < 0.29. As in the case of the passive-microwave database, the first eigenvalue is far bigger than the remaining ones, although since we do have 14 layers, the second eigenvalue could not be negligible. It is particularly interesting to note that for the convective, stratiform, or all merged events, the eigenvector Σ_{i} *b*_{i} log(*R*_{i}) for this second eigenvalue always had the form (*b*_{1}, . . . , *b*_{7}, −*b*_{8}, −*b*_{9}, . . . , −*b*_{14}), with 0.13 < *b*_{1}, . . . , *b*_{6}, *b*_{9}, . . . , *b*_{14} < 0.34 and *b*_{7} ≅ *b*_{8} ≅ 0.05. In other words, the second eigenvector quantifies the difference between the rain below 2.25 km and the rain above 2.75 km. This is remarkably similar to the case of the TRMM cloud-simulations database; indeed, (2) specifies that the second eigenvector for the rain described in the database is the difference between the rain below 2 km and the rain above 2.5 km. Figure 2 shows the scatter diagram of the first two eigenvectors in the cases of the TRMM passive-microwave database and of some TRMM radar data obtained during 10 orbits in Jan 1999. In both cases, the second eigenvector varies most (and is therefore most descriptive) for the moderate rain rates. This is consistent with the fact that near-surface evaporation can be significant, especially under stratiform rain, a process that is best quantified by the difference between the rain aloft and the rain near the surface. In summary, our principal component analysis confirms that the TRMM passive-microwave database is consistent with measurements in the Tropics and suggests that in the Tropics, one should be able to describe the vertical distribution of rain using two variables only: the mean rain rate (the first eigenvector) and the difference between the rain in upper and lower layers.

## 3. Conditional covariances of *R* and *T*_{b}

^{−1}anywhere in the rainy column. This explains the small positive bias of the clear-air database histogram compared with the measurements. The rainy samples, however, show remarkable large-scale agreement with the data, although there is a notably larger incidence of high brightness temperatures in the database. This is most likely due to the database’s emphasis on substantial convective precipitation and a corresponding probable underrepresentation of stratiform rain. The same features are evident in the histograms for the 10.7-H, 19.3-H, 19.3-V and 21.3-V channels. At 37 GHz (Fig. 4), the database contains a relatively small but significant number of rain samples with low associated brightness temperatures extending well below the clear-air values, while no measurements fell in this region. This feature is more pronounced in the 85.5 GHz (Fig. 5), where the database contains no rain samples with brightnesses exceeding the clear-air cases, while the measurements actually peak in that region. This is an indication that the database overrepresents high-scattering events, as noted earlier, and that scattering is overestimated in the radiative transfer calculations (because of an overproduction of ice, nonrepresentative frozen hydrometeor size distributions, or unrealistically slow fall velocities; see, e.g., Panegrossi et al. 1998 for a discussion of some of these effects). A test for goodness of fit for the 37-V data reveals that the (approximately

*χ*

^{2}) statistic

*χ*

^{2}variable with 13 degrees of freedom, is at the 99.91st percentile. This value is not as small as one may have liked; while it may have been affected by the very large sample size, which greatly increases the penalty for a mismatch between observations and simulated temperatures, the mismatch seems mostly due to the questionable representativity of the observations themselves. Thus on the whole, while the database temperatures are not totally inconsistent with the observations, the evident mismatches indicate that the current database overemphasizes scattering and/or over-represents high-scattering cases and that it would be prudent to make the database more representative in order to improve our interpretation of the TMI’s observations. The “too-much-scattering” problem could be addressed by tuning the frozen hydrometeor size distributions and fall speeds in the cloud models themselves, and the general representativity of the database can be improved by resampling it. These possible remedies, however, require much effort and are quite beyond the scope of the current study.

Figure 6 shows the conditional means *T*_{b} | *R*^{′}_{1}*R*^{′}_{1}^{−1}. For large rain rates, the 85.5-GHz uncertainties climb beyond 80 followed closely by the 37-GHz *σ*’s. This is due to the fact that the effect of scattering from ice increases as the frequency increases, while the amount of ice is not closely related to the amount of rain. The uncertainties in the other channels remain below 40 K. Table 2 and Fig. 7, showing the correlation coefficients of the various channels given *R*^{′}_{1}^{−1} and approaches +1 for high rain rates. More significant is the consistently positive correlation between 19.3 GHz V-pol and the higher-frequency channels, and its relatively weaker correlation with the low-frequency channels. This suggests that in the TRMM database, both 19.3-GHz channels are affected by scattering in the cloud model used even at very low rain rates, the effect being more significant at V-polarization than at H.

*T*

^{′}

_{1}

*T*

^{′}

_{2}

*T*

^{′}

_{i}

*R*

_{i}’s given

*T*

^{′}

_{1}

^{−1}and continue rising as the rain drops, exceeding 100% as soon as the rain rates drop below 20 mm h

^{−1}and remaining near 15 mm h

^{−1}for the lowest rain rates. Since the rain rate remains positive, much of this deviation must necessarily lie above the mean, making an overestimation of the rain likely if the joint behavior of the database radiances is not consistent with that of the measurements.

## 4. Estimation of *R* using microwave brightness temperatures *T*_{b}

The principal component analyses allowed us to reduce the number of variables required to describe the rain as well as those required to describe the measured brightness temperatures. It is therefore natural to investigate the possibility of estimating the rain from the measured radiances directly using the reduced set of variables without having to consult a database in real time. Since the vertical distribution of rain can be adequately described using a single variable *R*^{′}_{1}*R*^{′}_{1}

In practice, it would be simplest to use a subset of all available microwave channels to estimate *R*^{′}_{1}*R*^{′}_{1}*T*′ of brightness temperatures *T*_{i} among 10.7, 19.3, 21.3, 37, and 85.5 GHz is a priori nonlinear, we modify it slightly by trying to maximize the correlation’s numerator *E*{*R*^{′}_{1}*T*′} keeping *E*{*T*′^{2}} constant. This, in effect, minimizes the scatter between *T*′ and *R*^{′}_{1}*T*′ are found, one can easily compute the mean and variances of *R*′ given *T*′ and thus determine the inverse relation and its uncertainty.

*T*

^{′}

_{opt}

*R*

^{′}

_{1}

*R*

_{average}),

*T*

^{′}

_{opt}=

*T*

_{10.7H}

*T*

_{19.3H}

*T*

_{21.3V}

*T*

_{37H}

*T*

_{85.5H}

*T*′’s. The rms uncertainty corresponding to

*T*

^{′}

_{opt}

*σ*of more than 50%. The result without the 19.3-GHz channels is not very good either.

Using *T*^{′}_{opt}*T*^{′}_{opt}*R*^{′}_{1}*R*^{′}_{1}*R*^{′}_{1}*T*^{′}_{opt}*R*_{1} plotted against the original *R*_{1}, and Fig. 13 shows the results of the reconstruction for each of the remaining four layers *R*_{2}, . . . , *R*_{5}. Table 3 gives the precisions of the retrieved *R*_{i} in each layer. While the figures suggest that the estimates of the rain in each layer are reasonably close to their actual values, the table confirms that the relative error in the retrieval remains below 55%. These results are quite encouraging.

## 5. Conclusions

Our study of the joint behavior of the rain in a horizontally stratified atmosphere and the associated microwave radiances shows that the single, most crucial variable characterizing the rain profile is the vertically averaged rain rate, followed by the difference between the high-altitude subfreezing-level rain and the precipitation closer to the surface, the remaining rain eigenvariables having negligibly small variances, implying that they can safely be considered constant (equal to their respective means). The measurements of the passive microwave channels can similarly be described using two linear combinations of the brightness temperatures. The conditional standard deviation of the rain rates given these eigenradiances is a nearly linear function of the conditional mean rain rate when the latter is high, equal to about 55% of the rain rate, but the proportion rises to 65% when the rain is around 30 mm h^{−1} and exceeds 100% when the rain drops below 20 mm h^{−1}. The study also shows that for a higher-resolution situation, such as the case of an airborne sensor, the vertical rain rates can be adequately estimated using five of the TRMM passive-microwave channels and an associated database similar to that used for TRMM, with an rms uncertainty (due to the variations accounted for in the model database) below 55%.

## Acknowledgments

Svetla Veleva is gratefully acknowledged for several helpful discussions. This work was performed at the Jet Propulsion Laboratory, California Institute of Technology, Pasedena, California, under contract with the National Aeronautics and Space Administration.

## REFERENCES

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*J. Appl. Meteor.,***33,**3–18.——, and ——, 1994b: A passive microwave technique for estimating rainfall and vertical structure information from space. Part II: Applications to SSM/I data.

*J. Appl. Meteor.,***33,**19–34.——, W. S. Olson, and L. Giglio, 1996: A simplified scheme for obtaining precipitation and vertical hydrometeor profiles from passive microwave sensors.

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The term *R*^{′}_{2}*R*^{′}_{1}*N**N* = 8 for the database (×) and *N* = 14 for the TRMM radar (•)

Citation: Journal of Atmospheric and Oceanic Technology 17, 12; 10.1175/1520-0426(2000)017<1618:ETUIPM>2.0.CO;2

The term *R*^{′}_{2}*R*^{′}_{1}*N**N* = 8 for the database (×) and *N* = 14 for the TRMM radar (•)

Citation: Journal of Atmospheric and Oceanic Technology 17, 12; 10.1175/1520-0426(2000)017<1618:ETUIPM>2.0.CO;2

The term *R*^{′}_{2}*R*^{′}_{1}*N**N* = 8 for the database (×) and *N* = 14 for the TRMM radar (•)

Citation: Journal of Atmospheric and Oceanic Technology 17, 12; 10.1175/1520-0426(2000)017<1618:ETUIPM>2.0.CO;2

Comparison of the 10.7-GHz *T*_{b} histograms

Comparison of the 10.7-GHz *T*_{b} histograms

Comparison of the 10.7-GHz *T*_{b} histograms

Comparison of the 37-GHz *T*_{b} histograms

Comparison of the 37-GHz *T*_{b} histograms

Comparison of the 37-GHz *T*_{b} histograms

Comparison of the 85.5-GHz *T*_{b} histograms

Comparison of the 85.5-GHz *T*_{b} histograms

Comparison of the 85.5-GHz *T*_{b} histograms

Means and standard deviations of *T*_{b} given an “average” *R* = exp(*R*^{′}_{1}

Means and standard deviations of *T*_{b} given an “average” *R* = exp(*R*^{′}_{1}

Means and standard deviations of *T*_{b} given an “average” *R* = exp(*R*^{′}_{1}

Correlation coefficients for the passive channels given *R*^{′}_{1}

Correlation coefficients for the passive channels given *R*^{′}_{1}

Correlation coefficients for the passive channels given *R*^{′}_{1}

Means of *R* given *T*^{′}_{1}

Means of *R* given *T*^{′}_{1}

Means of *R* given *T*^{′}_{1}

Root-mean-square variation of *R* given *T*^{′}_{1}

Root-mean-square variation of *R* given *T*^{′}_{1}

Root-mean-square variation of *R* given *T*^{′}_{1}

Selected candidate *T*′’s: *T*^{′}_{opt}

Selected candidate *T*′’s: *T*^{′}_{opt}

Selected candidate *T*′’s: *T*^{′}_{opt}

The quantity *R*^{′}_{1}*T*^{′}_{opt}*R*^{′}_{1}

The quantity *R*^{′}_{1}*T*^{′}_{opt}*R*^{′}_{1}

The quantity *R*^{′}_{1}*T*^{′}_{opt}*R*^{′}_{1}

The quantity *R*_{1}, retrieved from *T*^{′}_{opt}*E*{*R*^{′}_{2}*E*{*R*^{′}_{5}*R*_{1}

The quantity *R*_{1}, retrieved from *T*^{′}_{opt}*E*{*R*^{′}_{2}*E*{*R*^{′}_{5}*R*_{1}

The quantity *R*_{1}, retrieved from *T*^{′}_{opt}*E*{*R*^{′}_{2}*E*{*R*^{′}_{5}*R*_{1}

The quantity *R*_{i}, retrieved from *T*^{′}_{opt}*E*{*R*^{′}_{2}*E* {*R*^{′}_{5}*R*_{i}’s

The quantity *R*_{i}, retrieved from *T*^{′}_{opt}*E*{*R*^{′}_{2}*E* {*R*^{′}_{5}*R*_{i}’s

The quantity *R*_{i}, retrieved from *T*^{′}_{opt}*E*{*R*^{′}_{2}*E* {*R*^{′}_{5}*R*_{i}’s

The rms error on the rain rate estimated for each layer from the mean rain rate *R*^{′}_{1} and *R*^{′}_{2}}, . . . , *R*^{′}_{8}

Correlation coefficients of *T _{b}* given

*R*; most coefficients do not vary significantly with

*R.*

Error in the rain rate calculated for each layer using the mean rain rate *R*^{′}_{1} estimated from *T*^{′}_{opt} and using *E*{*R*^{′}_{2}}, . . . , *E*{*R*^{′}_{5}