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  • View in gallery

    Map of oceanic regions used in this study including the northwestern Pacific (NWP), eastern Pacific (EP), southwestern Pacific, and tropical Atlantic (TA) Oceans.

  • View in gallery

    The first (solid lines) and second (dotted lines) EOFs of radiative index vectors for globally observed raining TMI pixels for the categories defined in Eq. (3).

  • View in gallery

    A general guide for the physical interpretation of EOF space, as based on the Biggerstaff and Seo (2010) mapping of individual profiles from simulated storms in two-dimensional EOF space.

  • View in gallery

    (left) The distribution of frequency in the two-dimensional EOF parameter space for the 5-yr dataset of raining pixels over NWP for categories (a) 1, (b) 2, and (c) 3. The CFAD for PR-derived radar reflectivity over regions (center left) A, (center right) B, and (right) C for the three cloud classifications as noted by the boxes drawn in (a)–(c).

  • View in gallery

    The distribution of (top) frequency, (middle top) PR rain rate, (middle bottom) TMI rain rate, and (bottom) storm height in the two-dimensional EOF parameter space for the 5-yr global tropical ocean dataset of raining pixels for categories (a)–(d) 1, (e)–(h) 2, and (i)–(l) 3.

  • View in gallery

    The distribution of (left) frequency, (center left) PR rain rate, (center right) TMI rain rate, and (right) storm height in the two-dimensional EOF parameter space for category 1 of the 5-yr tropical ocean dataset of raining pixels over (a)–(d) NWP, (e)–(h) EP, (i)–(l) SWP, and (m)–(p) TA.

  • View in gallery

    As in Fig. 6, but for category 2.

  • View in gallery

    As in Fig. 6, but for category 3.

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Regional Variability within Global-Scale Relations between Passive Microwave Signatures and Raining Clouds over the Tropical Oceans

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  • 1 Department of Earth Science Education, Kongju National University, Kongju, South Korea
  • | 2 School of Meteorology, University of Oklahoma, Norman, Oklahoma
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Abstract

Empirical orthogonal function (EOF) analysis of radiance vectors associated with emission and scattering indices for the Tropical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI) has been performed to examine the regional variability in relations between brightness temperature and rain rate over portions of the tropical oceans known to exhibit regional differences due to different thermodynamic environments and different large-scale forcing. The TMI indices and rain rates used in this study are the products of the Goddard profiling algorithm (GPROF), version 6. The EOF framework reduces the nine-dimensional space of the brightness temperatures and their polarizations to just two dimensions associated with the EOF coefficients. Vertical profiles of reflectivity from the TRMM precipitation radar (PR) are used to show that the statistically obtained EOFs represent bulk physical characteristics of raining clouds. Hence, EOF analysis provides an efficient framework for diagnosing regional differences in cloud structures that affect brightness temperature–rain-rate relations. The EOF framework revealed fundamental differences in the behavior of TMI surface rain-rate retrievals versus retrievals that are based on the PR aboard the TRMM satellite. In EOF space, TMI rain rates were bimodally distributed, with one mode indicating higher rain rates with greater high-density ice and rainwater content in the cloud and the other mode being consistent with moderately heavy warm rain from shallow convection. In contrast, the PR rain-rate distribution showed high rain rates being assigned over a much greater diversity of cloud structures. The manifold of EOF space constructively shows that, of the regions examined here, the tropical northwestern Pacific Ocean region produces the greatest occurrence of particularly strong cumulonimbus clouds.

Corresponding author address: Eun-Kyoung Seo, Dept. of Earth Science Education, Kongju National University, Kongju 314-701, South Korea. E-mail: ekseo@kongju.ac.kr

Abstract

Empirical orthogonal function (EOF) analysis of radiance vectors associated with emission and scattering indices for the Tropical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI) has been performed to examine the regional variability in relations between brightness temperature and rain rate over portions of the tropical oceans known to exhibit regional differences due to different thermodynamic environments and different large-scale forcing. The TMI indices and rain rates used in this study are the products of the Goddard profiling algorithm (GPROF), version 6. The EOF framework reduces the nine-dimensional space of the brightness temperatures and their polarizations to just two dimensions associated with the EOF coefficients. Vertical profiles of reflectivity from the TRMM precipitation radar (PR) are used to show that the statistically obtained EOFs represent bulk physical characteristics of raining clouds. Hence, EOF analysis provides an efficient framework for diagnosing regional differences in cloud structures that affect brightness temperature–rain-rate relations. The EOF framework revealed fundamental differences in the behavior of TMI surface rain-rate retrievals versus retrievals that are based on the PR aboard the TRMM satellite. In EOF space, TMI rain rates were bimodally distributed, with one mode indicating higher rain rates with greater high-density ice and rainwater content in the cloud and the other mode being consistent with moderately heavy warm rain from shallow convection. In contrast, the PR rain-rate distribution showed high rain rates being assigned over a much greater diversity of cloud structures. The manifold of EOF space constructively shows that, of the regions examined here, the tropical northwestern Pacific Ocean region produces the greatest occurrence of particularly strong cumulonimbus clouds.

Corresponding author address: Eun-Kyoung Seo, Dept. of Earth Science Education, Kongju National University, Kongju 314-701, South Korea. E-mail: ekseo@kongju.ac.kr

1. Introduction

Satellite passive microwave observations have been used to retrieve rainfall over the tropics and parts of the middle latitudes for more than two decades (Wilheit et al. 1977, 1982; Spencer et al. 1983; Wu and Weinman 1984; Simpson et al. 1988; Kummerow et al. 2000, 2001). One of the approaches for the retrieval is to apply the Bayesian algorithm to invert brightness temperatures TB into rain rates by using a predefined database consisting of the relations between cloud/rain structures and microwave radiances (Evans and Stephens 1993; Kummerow et al. 1996; Olson et al. 1996; Evans et al. 2002; Seo and Liu 2005). The database is often constructed from simulations of precipitation systems using cloud-resolving models (Tao and Simpson 1993; Mugnai et al. 1993). Accordingly, microwave-based rainfall retrievals have been validated on both a point-by-point basis and over time- or area-averaged domains using various ground and airborne observations (Heymsfield et al. 2000; Houze et al. 2004; Wolff et al. 2005; Wolff and Fisher 2008; Lin and Hou 2008). In most cases, retrieved rain itself has been a target of assessment (e.g., Kummerow et al. 2000, 2001; Bowman 2005; Yuter et al. 2005).

Rather than focus on validating accumulated rainfall totals retrieved from passive microwave observations, Panegrossi et al. (1998) examined the consistency between simulated and observed manifolds of TB. Their point was that satellite retrievals of rainfall required similarity between the multidimensional manifolds of observed and simulated storms. Without this similarity, the retrieval databases would not properly reflect the diversity of naturally occurring cloud systems and rain retrievals would be somewhat uncertain.

As a further step, Seo and Liu (2005) and Biggerstaff and Seo (2010) used empirical orthogonal function (EOF) analysis to effectively compare the manifolds of TBs in predefined simulation databases with those from observations by reducing the number of dimensions associated with the microwave frequencies and their polarizations. On the basis of earlier work (Biggerstaff et al. 2006; Seo et al. 2007a) that examined the impact of different hydrometeors on microwave emission and scattering measurements, Biggerstaff and Seo (2010) related different parts of the EOF manifold back to characteristic hydrometeor profiles and illustrated how the EOF framework provides a convenient method for examining multidimensional relations in two-dimensional EOF space without significant loss of physical interpretation.

The EOF method has also been used to quantify the uncertainty in retrievals of cloud properties, including rainfall. Seo and Biggerstaff (2006) noted that the frequencies used in the Tropical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI) provided good correlations with rain and high-density precipitation-sized ice content but were insensitive to cloud ice and low-density snow. As a consequence, retrievals of latent heating profiles were not as robust as retrievals of rain. EOF methods have also been used to adjust model database hydrometeor profiles to provide a better match between observed and simulated brightness temperatures in an effort to improve rain retrievals (Seo et al. 2007a).

Still, the fundamental nature of relations between TB and rain rate (RR) has not been well examined in EOF space. Such information may be necessary to provide additional guidance for physical improvements in the retrieval process. Indeed, one of the fundamental validation criteria should be that the manifold of TB–RR relations in simulated databases matches that observed in nature.

It is well known, however, that latitudinal variability in the dominant type of cloud and in the structure of precipitating storms from the tropics to middle latitudes and even the polar regions exists (Houze 1981). In fact, significant variability is even evident across a narrow band of latitude, such as across the tropics. Petty (1995) found a strong regional dependence from shipboard present-weather reports in the frequency of thunderstorms in association with the variability of tropical cloud systems. Large discrepancies in rainfall retrievals that were based on satellite infrared and passive microwave were found between eastern and western Pacific Ocean rainfall systems (Berg et al. 2002). Petersen and Rutledge (2001) found using TRMM precipitation radar (PR) and Lightning Imaging Sensor data that precipitation systems in the eastern Pacific are shallower, especially for small storm systems, and have a higher percentage of stratiform rain. In the western Pacific warm pool, storm systems contain more ice and are slightly more intense in terms of radar reflectivity structure than those sampled over the other tropical oceanic regions. Given these observations, it is clear that there is significant variability in cloud systems across the tropics.

Because microwave TBs are a function of hydrometeor profiles, such variability in cloud structure and rain should be reflected in their radiative characteristics. Therefore, another validation criterion should be that passive microwave retrievals of rain exhibit the same degree of regional and seasonal variability that has been found in natural precipitation systems. Such variability in cloud structure and rain should be reflected in their radiative characteristics since microwave TBs are a function of hydrometeor profiles. Biggerstaff and Seo (2010) investigated the regional dependencies in rainy clouds by obtaining the TB manifolds of TMI observations. In their study, four different tropical regions show unique cloud structures. For example, the tropical northwestern Pacific had the strongest convective clouds and most diverse cloud structures while the tropical eastern Pacific exhibited a narrow distribution of convective cloud structures. Hence, they showed that the EOF framework was suitable for diagnosing regional variability.

As a further step in the evaluation of microwave rain retrievals, this study makes use of the EOF analysis method to examine naturally occurring TB–RR relations over the TRMM coverage domain and the existence of regional dependencies in the relations over the four tropical oceanic regions examined in Biggerstaff and Seo (2010). In particular, we will show how the frequency distribution of cloud structures in EOF space provides a fingerprint that can be used to diagnose regional variability in TB–RR relations. The EOF manifold fingerprints can be used as a first-order validation target for retrieving cloud properties from passive microwave observations since they effectively reduce the nine-dimensional space associated with the TMI observations to just two dimensions that can be readily visualized.

2. Data and analysis methods

To demonstrate the regional dependencies in TB–RR relations and to investigate the difference in characteristic behavior for TMI-derived rain rates versus those retrieved from the PR, a large number of observations of both TMI TBs and rain-rate estimates are needed over regions that are known to exhibit regional variability. To provide context for how this regional variability compares to the global-scale TB–RR relations that exist over the tropical oceans, a similar dataset is needed for the entire tropical ocean area sampled by TRMM.

a. Observational data

To determine the global-scale TB–RR relations that exist over the tropical oceans, TMI TBs (TRMM product 1B11) were collected at 10.65, 19.35, 37.0, and 85.5 GHz with both horizontal and vertical polarizations during a 5-yr period, from December 2001 to November 2006, along with the corresponding TMI-retrieved rain rates at the surface (TRMM product 2A12) and PR-derived rain rates near the surface (TRMM product 2A25). All products were version 6. To evaluate the regional dependency, this dataset was partitioned into four separate regions across the tropical ocean (Fig. 1). The regions chosen are from the tropical northwestern Pacific (NWP), the tropical southwestern Pacific (SWP), the tropical eastern Pacific (EP), and the tropical Atlantic Ocean (TA). These four regions represent a broad range of cloud regimes and are expected to exhibit regional dependencies in cloud structure that is due to differences in thermodynamic environments and large-scale forcing (e.g., Berg et al. 2002, 2006). Because this study is focused on regional dependency, the data were not partitioned by season. A future study will examine seasonal variability to compare with the regional variability examined here.

Fig. 1.
Fig. 1.

Map of oceanic regions used in this study including the northwestern Pacific (NWP), eastern Pacific (EP), southwestern Pacific, and tropical Atlantic (TA) Oceans.

Citation: Journal of Applied Meteorology and Climatology 51, 5; 10.1175/JAMC-D-11-055.1

b. Generation of radiance indices

Except for the 85.5-GHz frequency that is sampled every 4.5 km, the TMI samples the atmosphere every 9 km along the scanning direction. Because the actual fields of view (FOV) of an individual measurement vary with frequency (Kummerow et al. 1998), the 9-km pixels of data represent various degrees of oversampling of the atmosphere, especially for the lower frequencies. To provide a common scale for observations, a nominal TMI footprint size of 14 km × 14 km was selected in the Goddard profiling algorithm (GPROF), version 6 (Olson et al. 2006). The horizontally averaged surface rainfall rates over these nominal pixels were used.

All collocated TMI TBs at the four frequencies were converted to attenuation and scattering indices (Petty 1994). The attenuation index P is a normalized polarization difference, ranging from 0 and 1, with a small value of P denoting opaque liquid cloud and values near 1 indicating a cloud-free FOV. The index P depends on the emission of microwave radiation by liquid hydrometeors and is insensitive to scattering by ice particles (Petty 1994). The scattering index S, on the other hand, represents volume scattering associated with frozen precipitation aloft. A large value of S corresponds to strong ice scattering. To aid intuitive interpretation of the radiance indices in terms of the behavior of TBs, the radiance indices of Petty (1994) are modified as in Biggerstaff and Seo (2010) such that
e1
Defined in this manner, Pm corresponds to larger liquid hydrometeor volume and higher emission, just like greater values of the lower-frequency TMI TBs. In a similar way, larger negative values of Sm correspond to high volumes of high-density ice and greater scattering signatures, just like the higher-frequency TMI TBs. Here, we make use of the modified attenuation indices at 10.65, 19.35, 37.0, and 85.5 GHz and the modified scattering index at 85.5 GHz. This set of modified radiance indices becomes the components of a characteristic radiative index vector I, where I contains the four attenuation indices and the 85.5-GHz scattering index and is defined as
e2
The subscripts in the components of I denote the frequencies of the TMI channels. Using the observed TMI TBs, a database of Is was constructed over the four tropical oceanic regions in Fig. 1 and for the overall global tropical ocean area sampled by TRMM.

c. Classification of raining pixels

To match the TMI observational scale, the rain intensities from the PR, whose horizontal resolution is about 4.3 km at nadir, were averaged over the nominal TMI pixels (14 km × 14 km area). Only those pixels that had nonzero average PR rain rates were retained for further analysis. To classify the type of precipitating cloud over the nominal pixels, the convective areal fraction (convF) was used. The convF is defined as the ratio of the number of convective PR pixels (provided by TRMM product 2A23) to the total number of collocated PR pixels within the nominal footprint. On the basis of convF, rain pixels are classified into one of the following three categories:
e3
Thus, each nominal footprint has a radiation index vector I, a PR-derived rain rate, a TMI-derived rain rate, and classification based on convF.

d. EOF analysis

In the past, TMI TB–RR relations have been examined for individual frequencies (e.g., Mugnai et al. 1993). Since there are nine possible TMI measurements that arise from the five microwave frequencies, with four having both horizontal and vertical polarizations, such an approach makes it difficult to visualize the multidimensional relations between the TMI observations and rain intensity. Moreover, Biggerstaff et al. (2006) and Seo and Biggerstaff (2006) showed that retrievals of cloud properties from TMI measurements were better represented when correlations between TMI frequencies were taken into account. Using the radiative index vector I in place of the TMI channels reduces the parameter space from nine to five dimensions, but further reduction can be accomplished using EOF analysis applied to the radiative index vectors.

The EOF analysis defines new axes that explain the largest variance in the dataset. The EOFs show radiative characteristics, that is, simultaneous correlation structures among components of the anomalies of Is. If the dataset is well represented in two or three dimensions in the EOF framework, it is possible to demonstrate effectively the multiple I–RR relations. EOF analysis of the radiative index vectors was performed following the method described in Biggerstaff and Seo (2010).

1) Global tropical ocean EOFs

The two major EOFs, which are found in the TMI TBs, can explain about 88% of the total variance in globally observed Is (Fig. 2 and Table 1). For all three cloud classifications, the first EOF shows positive amplitudes at all of the modified attenuation indices (Pm10, Pm19, Pm37, and Pm85), which are simultaneously linked with a negative amplitude at the modified scattering index (Sm85). The primary difference between the cloud types, for the first EOF, is the degree to which the scattering index is negative. Category-1 (convective) clouds have a strongly negative scattering signature while category-3 (nonconvective) clouds have a much weaker negative scattering signature. Category-2 (mixed) clouds fall between the other two classifications. Hence, category-1 clouds have a tendency for increases in liquid hydrometeor content to occur jointly with increases in high-density ice content aloft that results in strong scattering at 85.5 GHz, or vice versa (Seo et al. 2007b). In contrast, category-3 clouds have a much weaker relation between liquid content at low levels and high-density ice content aloft. These EOF patterns are consistent with deep, mature-phase convection dominating category 1 and stratiform precipitation dominating category 3 (Biggerstaff and Seo 2010).

Fig. 2.
Fig. 2.

The first (solid lines) and second (dotted lines) EOFs of radiative index vectors for globally observed raining TMI pixels for the categories defined in Eq. (3).

Citation: Journal of Applied Meteorology and Climatology 51, 5; 10.1175/JAMC-D-11-055.1

Table 1.

The major eigenvalues for Is in each category for the 5-yr global dataset of raining pixels.

Table 1.

The second EOF explains about 20% of the variance of the dataset for all three cloud classifications (Table 1). Unlike the first EOF that had similar structure across the different cloud types, the second EOF exhibits a fundamental difference between category-1 cloud and other classifications. For convective classification, all of the indices are positive (or negative). On the other hand, for categories 2 and 3 the second EOF shows all positive amplitudes except for the modified 85.5-GHz attenuation index. Note that the negative amplitude for Pm85 represents the deviation from the mean Pm85. Therefore, the small negative amplitude for Pm85 indicates saturation of radiation by liquid hydrometeors. It is apparent that in the mixed and nonconvective rainy pixels there are several occurrences that have high enough liquid water to saturate the 85.5-GHz channel without there being enough high-density ice aloft to decrease the 85.5-GHz emission through scattering.

Another important difference in the second EOF across the cloud types is the amplitude of the correlation between the attenuation indices and the scattering index. Category-1 clouds have a much weaker positive correlation than do the category-3 clouds. Hence, increased liquid water content in category-3 clouds is correlated with less high-density ice aloft as compared with the high-density ice content for category-1 clouds with similar liquid water contents, resulting in positive amplitudes in Sm85 and the attenuation indices. Note also that the second EOF for the modified 10.6-GHz attenuation index in category 3 is near zero. This implies that there is little variance in rain rate from the mean value of the distribution for nonconvective clouds, which is typical of stratiform rain.

2) Regional EOFs

The EOFs of the four selected regions are similar to those shown in Fig. 5 of Biggerstaff and Seo (2010) since they were taken over the same areas. Hence, we do not repeat the figure here. The only difference is that Biggerstaff and Seo (2010) used a dataset collected over a 1-yr period (December 1999–November 2000), whereas this study makes use of a 5-yr dataset. As before, the first EOFs explain about 77% of the respective total variances across the ocean domains. The differences in the first EOF across the four oceanic regions for the different cloud classifications are similar to those observed for the global database discussed above. This implies that, from a passive microwave viewpoint, there is a similar magnitude of variance in the structure of deep, mature-phase convective clouds across the four tropical oceanic regions. On the other hand, the second-EOF patterns have relatively large regional variability, consistent with differences in the number of occurrences or radiative structure of shallow rain and/or tall ice clouds over the four oceanic regions (Biggerstaff and Seo 2010).

3) EOF manifolds and their physical interpretation

The EOF patterns in Fig. 2 represent the simultaneous correlations among radiative indices for the entire oceanic area sampled by TRMM. That is, they reflect global-scale oceanic correlation patterns. We make use of the global tropical ocean EOFs to generate two-dimensional manifolds in which the global TB–RR relations can be described and regional differences in the TB–RR relations can be determined. This reduction in dimensionality is obtained by projecting the ith radiative index vector I from a particular database onto the first and second global EOFs such that
e4
where and denote the first and second EOFs, respectively, for the global dataset of raining pixels and ai and bi represent the first- and second-EOF coefficients, respectively. When all of the radiative index vectors are projected, the global TB–RR relations can be displayed in two dimensions with one axis (the horizontal) representing the first EOF and the orthogonal (vertical) axis representing the second EOF. Likewise, differences in regional TB–RR relations can be determined by projecting the radiative index vectors from each region onto the global EOFs and comparing the results.

The two-dimensional EOF parameter space corresponds reasonably well to different bulk vertical structures of precipitating clouds. Figure 3 summarizes the bulk physical interpretation of the EOF space from Biggerstaff and Seo (2010) that was based on the mapping of individual profiles from simulated storms (see their Fig. A1). The dashed boundaries in Fig. 3 indicate that the different regions are not sharply delineated. Rather, the descriptions within each area identify typical characteristics in a continuous spectrum of raining cloud structures over EOF space. It should be understood that the EOFs represent perturbations from the mean profile. Hence, the descriptions are for hydrometeor concentrations relative to the mean values. Furthermore, large scattering signals are attributed to graupel, as Biggerstaff et al. (2006) showed that the 85.5-GHz scattering was dominated by high-density ice particles in simulated storms. Smaller scattering signatures can be associated with snow (Seo and Liu 2005), however.

Fig. 3.
Fig. 3.

A general guide for the physical interpretation of EOF space, as based on the Biggerstaff and Seo (2010) mapping of individual profiles from simulated storms in two-dimensional EOF space.

Citation: Journal of Applied Meteorology and Climatology 51, 5; 10.1175/JAMC-D-11-055.1

To elucidate further the physical interpretation of EOF space, the manifold of EOF coefficients by cloud classification over the NWP region is plotted along with a contoured-frequency-by-altitude-diagram (CFAD) analysis of the vertical profiles of radar reflectivity from the PR for subdomains within EOF space (Fig. 4). Profiles of PR reflectivity illustrate the differences in vertical structure for the three cloud classifications as well as differences in cloud structure within a given EOF manifold.

Fig. 4.
Fig. 4.

(left) The distribution of frequency in the two-dimensional EOF parameter space for the 5-yr dataset of raining pixels over NWP for categories (a) 1, (b) 2, and (c) 3. The CFAD for PR-derived radar reflectivity over regions (center left) A, (center right) B, and (right) C for the three cloud classifications as noted by the boxes drawn in (a)–(c).

Citation: Journal of Applied Meteorology and Climatology 51, 5; 10.1175/JAMC-D-11-055.1

According to the general guide given in Fig. 3, region C would contain the clouds with the strongest rain rates and greatest amounts of graupel aloft. Region B would consist of clouds that had moderate values of both rain and graupel, and region A would be associated with moderate-to–moderately high graupel aloft and low-to-modest surface rain rates. Indeed, the PR profiles for these different regions validate the general bulk physical interpretation of EOF space.

CFADs of reflectivity over region C show that this area contains clouds with high reflectivity near the surface (likely associated with heavy rain) and strong reflectivity aloft that would be associated with high graupel content. In contrast, region B exhibited mean reflectivity that was about 3–5 dB less near the surface and around 10 dB less at 8-km altitude. The weaker reflectivity in region B would be associated with both more-modest surface rain and more-modest graupel aloft than would the clouds in region C. In a similar way, region A had the lowest value of mean reflectivity at the surface and had reflectivity aloft that was nearly as strong as that of region C. Clouds in region A, therefore, would most likely have moderately high graupel content and low surface rain rates.

Biggerstaff and Houze (1993) showed that the transition zones between convective areas and well-defined stratiform regions with clear radar bright bands can contain relatively high radar reflectivity aloft, likely associated with graupel, and weak radar reflectivity at low levels, associated with low rain rates. The CFADs of radar reflectivity over region A are consistent with transition-zone structures. Another explanation is that this part of the parameter space represents weak-to-modest isolated deep convection. As noted by Biggerstaff and Seo (2010), 17% of the pixels in their category-3 data came from isolated convection. Even with the reduction of the TMI observations to nominal footprint scales, the inherent variability in measurements as a function of microwave frequency results in isolated rain emission signatures being averaged over nonraining areas while the scattering signature is somewhat preserved.

In terms of cloud classification, category-1 clouds have the highest mean radar reflectivity at low levels and the slowest decrease in mean reflectivity aloft across all portions of EOF space. In contrast, category-3 clouds have the weakest low-level radar reflectivity and the greatest decrease in mean reflectivity aloft. Furthermore, category-3 clouds tend to show evidence of a marked radar bright band associated with melting. The bright band is even evident in region C where deep convection would likely be found. Because radar bright bands associated with melting are disrupted by strong vertical motions (Biggerstaff and Listemaa 2000), the differences in the brightband signature suggest that convection in category 3 is weaker, on average, than convection in category 1. The weak nature of convection in category 3 is reflected in EOF space by a limited manifold in the direction of positive first-EOF coefficient.

3. Global TB–RR relations

a. Hydrometeor frequency distribution in EOF space

To assess the importance of regional variability in TB–RR relations, we need first to illustrate the global-scale TB–RR relations across the TRMM satellite–covered tropical oceans. Here, the radiative index vectors for the entire database for each cloud type are projected onto the global EOFs. The frequency distribution of the scatterplots of first- versus second-EOF coefficients defines the boundary of the EOF manifold as well as the relative number of occurrences of different types of hydrometeor profiles (Fig. 5).

Fig. 5.
Fig. 5.

The distribution of (top) frequency, (middle top) PR rain rate, (middle bottom) TMI rain rate, and (bottom) storm height in the two-dimensional EOF parameter space for the 5-yr global tropical ocean dataset of raining pixels for categories (a)–(d) 1, (e)–(h) 2, and (i)–(l) 3.

Citation: Journal of Applied Meteorology and Climatology 51, 5; 10.1175/JAMC-D-11-055.1

In each cloud type, the highest frequency of EOF pairs is shifted slightly to the left of the origin, indicating that the mean is biased by extreme hydrometeor concentrations so that a number of reconstructed individual hydrometeor profiles would require a reduction in both liquid and ice concentrations from the mean profile. In other words, clouds having a small amount of rain at low levels and a small amount of high-density ice at upper levels frequently occur for all categories over the selected regions. After the exclusion of the outermost frequency contour, convective pixels exhibit greater diversity than nonconvective pixels as the frequency distribution covers a broader area in EOF space for category-1 cloud classification.

b. Rain-rate and echo-top distributions in EOF space

The projection of an individual radiative index vector onto the global EOFs creates a pair of EOF coefficients that have a given rain rate from both the PR and TMI products that were associated with that radiative index vector. To display the rain intensity within the EOF parameter space, all of the rain rates from the projected Is were averaged over 10 unit × 10 unit EOF cells (Fig. 5). The second row in Fig. 5 shows the average PR rain rates in the EOF space, and the average TMI rain rates are displayed in the third row in Fig. 5. Average echo tops derived from the PR reflectivity (TRMM product 2A23) are shown in the last row of Fig. 5. The PR has a sensitivity of about 17 dBZ, which is insufficient to detect the top of low-density snow and small cloud ice layers. Hence, the PR echo tops might underestimate the actual cloud tops.

1) PR–RR relations vs TMI–RR relations in EOF space

The most striking difference in TB–RR relations is between the character of the PR and TMI rain-rate distributions in EOF space. In all three cloud classifications, the PR rain rates have a gradient that is directed roughly parallel to the line y = x, where the magnitude of the first and second EOFs is equal. In contrast, the TMI rain-rate distribution appears to be bimodal, with one mode that exhibits increasing rain rate with increasing first EOF (rain-rate gradient parallel to the horizontal axis) and another mode that reflects warm rain from shallow convection in the upper portion of the EOF manifold, consistent with the low echo tops in that part of the manifold (last row in Fig. 5). In addition, the PR rain-rate distribution shows high rain rates over a much wider diversity of clouds while TMI retrievals of high rain rates are primarily restricted to that part of the EOF manifold that corresponds to deep, mature-phase convection that has high emission and large scattering signatures (i.e., the lower-right portion of the EOF manifold where the average echo tops are the highest).

Another important distinction between PR and TMI rain-rate distributions is that part of the EOF manifold has no TMI rain estimates. The area along the left edge of the EOF manifold, associated with low rain content and various degrees of scattering signature, has average PR rain rates of less than 2–4 mm h−1, depending on the cloud type. According to the echo-top distribution, this part of the parameter space is also populated with relatively tall clouds. Given the lack of TMI rain along with the reflectivity structure for that part of EOF space (Fig. 4), it is likely that this part of the EOF space is best associated with isolated deep convection where the inherent emission signature is averaged over a mostly nonraining field of view leading to a near-zero TMI rain rate.

2) Global-scale TB–RR relations by cloud type

Comparing the area with EOF coefficient frequency greater than 0.03% (first green contour in the first row in Fig. 5) with the PR rain-rate distribution in EOF space (second row in Fig. 5), it is clear that most of the convectively classified footprints have rain rates that are between 4 and 40 mm h−1. The same region of EOF space for nonconvective classifications has PR rain rates between 1 and 18 mm h−1. The separation between convective and nonconvective rain rates is less pronounced in the TMI retrievals. The echo-top distribution (last row in Fig. 5) suggests that most of the convective footprints are composed of clouds with echo tops ranging from 3- to 9-km altitude, whereas the category-3 footprints are mostly below 6 km. In addition, there is a much broader area of low echo tops (<4 km) for the nonconvective classification than for the convective footprints. Even taking the low sensitivity of the PR into account, echo tops of less than 6 km suggest that the cloud systems consist of weak convection as compared with most mesoscale convective systems (MCSs). Using a shipborne radar, Cifelli et al. (2007) noted a significantly weaker reflectivity profile for sub-MCS convection versus MCS convection over two regions of the eastern Pacific. Thus, much of the precipitation sampled across the tropical oceans is from small cloud systems.

The highest rain rates and tallest clouds in all three classifications are found outside the 0.2% EOF coefficient frequency contour. The convective classification has the greatest rain rates and tallest clouds, with the nonconvective classification having the lowest rain rates, as would be expected from the differences in updraft speeds and precipitation growth mechanisms in convective versus stratiform clouds (Biggerstaff and Listemaa 2000). High PR rain-rate intensity is not constrained to the deepest convective clouds where the greatest emission and scattering occur. Instead, the PR yields similar magnitudes of rain intensity for shallow (warm rain) clouds and deep, mature-phase clouds. There is more of a separation in extreme rain rates for the nonconvective classification, with the highest values being found in moderately to relatively deep clouds.

4. Regional variability in TB–RR relations

The radiative index vectors for each of the four tropical ocean regions are projected onto the global-scale EOFs and partitioned by cloud classification (Figs. 68). Both the NWP and SWP regions contain slightly more than one million rainy footprints, and EP and TA contain about 0.7 million rainy footprints (Table 2). The convective cloud classification accounts for about 10%–15% of the observations over all regions. The mixed-cloud classification accounts for about 20%, and the nonconvective category held the remaining 60%–70% of the rainy footprints (Table 2). The EP region has the lowest percentage of convective classification and the highest nonconvective classification. NWP and SWP have about 1.5 times as frequent rainy clouds than have EP and TA over the same horizontal area. Thus, the tropical Pacific Ocean regions are more productive in generating rainy clouds than are the other tropical oceans.

Fig. 6.
Fig. 6.

The distribution of (left) frequency, (center left) PR rain rate, (center right) TMI rain rate, and (right) storm height in the two-dimensional EOF parameter space for category 1 of the 5-yr tropical ocean dataset of raining pixels over (a)–(d) NWP, (e)–(h) EP, (i)–(l) SWP, and (m)–(p) TA.

Citation: Journal of Applied Meteorology and Climatology 51, 5; 10.1175/JAMC-D-11-055.1

Fig. 7.
Fig. 7.

As in Fig. 6, but for category 2.

Citation: Journal of Applied Meteorology and Climatology 51, 5; 10.1175/JAMC-D-11-055.1

Fig. 8.
Fig. 8.

As in Fig. 6, but for category 3.

Citation: Journal of Applied Meteorology and Climatology 51, 5; 10.1175/JAMC-D-11-055.1

Table 2.

Number and percentages (in parentheses) of occurrences of rainy pixels for each cloud classification for the 5-yr dataset over the four tropical oceanic regions.

Table 2.

a. Regional variability for convective cloud classification

For category-1 clouds, the manifold of EOF space occupies about the same range in the first-EOF magnitude except for NWP, which extends farther along the horizontal axis than do the other distributions. Hence, the NWP region contains some convective elements that produce the largest departure from the mean tropical oceanic convective system. This part of the parameter space is where deep, mature-phase clouds with relatively large quantities of both rain and graupel would be prevalent. Indeed, the distribution of echo tops (last column of Fig. 6) shows that this part of the NWP manifold has echo tops exceeding 12-km altitude. The NWP region exhibits the largest area of such tall clouds in EOF space, which is consistent with the greater emission and scattering signature found in strong convective clouds (Biggerstaff and Seo 2010). Thus, the manifold of EOF space constructively shows that the NWP region contains the most frequent occurrence of the strongest cumulonimbus clouds in terms of the amount of rain, high-density ice, and echo-top heights. Relative to the other oceans, EA and TA are likely to generate a greater number of raining clouds with very low echo tops (~2-km altitude) and fewer clouds with echo tops higher than ~13-km altitude.

In terms of the second-EOF magnitude, the EP region exhibits the smallest range of negative values for first-EOF coefficients between 0 and 100. This suggests that convective clouds over EP would have slightly fewer scattering signatures than would convective clouds over the other tropical oceanic regions. This is consistent with the lower sea surface temperature over EP relative to the western Pacific and likely less vigorous convection over EP. The range of negative second EOFs for convective clouds over TA falls between NWP and EP. The smaller overall fingerprint of the EOF manifolds for EP and TA suggests that those regions produce a smaller variety of convective clouds in terms of vertical structure of hydrometeors, consistent with the smaller range in echo tops as compared with the NWP and SWP regions.

In terms of TMI rain rates, the TB–RR relations are remarkably similar across all four tropical ocean regions. The only feature that is significantly different is for the far-right side of the NWP manifold where the tallest echo tops are observed. Here, the TMI rain rates decrease, on average, relative to slightly more shallow convection that has lower magnitude of the first-EOF coefficient. None of the other tropical manifolds extend to this part of the parameter space. The PR rain rates for this part of the NWP manifold also suggest that the tallest clouds are not associated with the highest rain rates. It is possible that more-vigorous updrafts in the tallest clouds lead to lower precipitation efficiency with reduced surface rain rates despite having a higher integrated water and precipitation ice content (Lau and Wu 2003).

The convective TMI TB–RR relations are similar across the four oceanic regions within common EOF space, but the PR-based TB–RR relations are not. In the top portion of the EOF manifold, where warm rain from shallow convective clouds is found, the NWP exhibits average rain rates in excess of 50 mm h−1 while for the same EOF space the EP exhibits rain rates closer to 30 mm h−1. The echo tops in these two areas are about the same. Both SWP and EP have their highest rain rates in the EOF space that contains the tallest convective clouds, with SWP having the highest rain rates of all tropical oceanic regions in this part of EOF space. In that regard, both SWP and EP exhibit a dependence on the magnitude of the first-EOF coefficient that is close to the TMI TB–RR relations.

In contrast, the TA region has its highest PR rain rates where moderately strong clouds with slightly above-average rain for a given precipitation ice content would be found. Echo tops in this area range from 4 to 10 km in altitude. The 40–50 mm h−1 rain rates found there are nearly double the 20 mm h−1 rain rates associated with the tallest clouds over TA. In general, for positive first-EOF coefficients, the echo tops over TA are less than those over other tropical oceans for the same EOF space. Thus, the same passive microwave signatures are associated with more shallow clouds over TA than they would be over the other tropical oceanic regions. Indeed, PR reflectivity profiles (not shown) revealed that the TA region generates shallower deep convective clouds than do the other oceans.

b. Regional variability for mixed and nonconvective cloud classification

The EOF coefficient frequency contours show the influence of more-uniform stratiform precipitation structures in the mixed and nonconvective cloud classifications. If one neglects the outermost contour, the distributions become centered closer to the origin of the EOF manifold and exhibit less dependence on the second EOF. As the structures become more stratiform, there is less variability and smaller deviation from the mean hydrometeor profile. Despite the similar overall pattern, there is a difference in the location of the peak frequency distribution between the western Pacific and other tropical ocean regions. For category-3 clouds, both NWP and SWP have a relative maximum in frequency near the horizontal axis with first-EOF values around 20 (Fig. 8). Regions EP and TA, on the other hand, have their maxima shifted toward the left, where the first-EOF magnitude is around −30. This difference in the location of the peak frequencies suggests that EP and TA have many clouds with little rain and little precipitation ice while NWP and SWP have many clouds that contain moderate concentrations of both rain and high-density ice.

The mixed and nonconvective classifications also exhibit a smaller range in first-EOF magnitude than does the convective classification, with TA generally having the smallest manifold and NWP the largest. As in the convective classification, the far-right edge of the NWP manifolds in the mixed and nonconvective classifications is associated with clouds that have higher echo tops than any part of the same EOF space for TA.

The range of negative second-EOF magnitude increases in all of the tropical ocean regions as the cloud classification becomes more stratiform. As in the global-scale manifolds, this part of the EOF manifold is associated with relatively tall clouds with low rain rates and moderately strong scattering signatures, suggesting that this area is dominated by isolated convection or by stratiform.

Rain-rate distributions for mixed and nonconvective clouds exhibit less structural variability than does the convective cloud classification. The four tropical ocean regions have their maxima in about the same part of the EOF space. The amplitude of the maxima varies regionally, however. For the mixed-cloud classification, NWP and SWP exhibit the highest maximum PR and TMI rain rates while TA exhibits the lowest maximum rain rate. Peak PR rain rates are around 49 mm h−1 in NWP as compared with about 29 mm h−1 in TA for the same part of the EOF space. A similar trend occurs in the nonconvective cloud classification, only with EP having a higher PR maximum rain rate than NWP and SWP. In terms of maximum TMI rain rates, however, TA is weaker than NWP, EP, and SWP for the same EOF space.

5. Discussion and conclusions

EOF analysis of the radiative index vectors I formed from TMI attenuation and scattering indices provides an efficient method for visualizing in two dimensions the multidimensional relations between passive microwave observations and cloud structures, including rain rates. It was shown that, although the TB EOFs are obtained statistically, different portions of the EOF manifold can be physically related to different vertical structures of raining clouds. Indeed, the frequency distribution in the EOF manifold provides a fingerprint that can be used to match observations of naturally occurring clouds to simulated clouds. For example, Biggerstaff and Seo (2010) illustrated how EOF manifold and frequency distributions obtained from observations can be used to determine how well numerical simulations of cloud systems produce hydrometeor profiles that resemble those in nature. In a more general sense, the EOF framework could be used for quantitative comparisons, in terms of the vertical structures of clouds, between any set of simulations or observations.

In this study, EOF analysis was used to project I from different tropical oceanic regions onto the global-scale EOFs for investigation of regional differences in relations between TMI brightness temperatures TB and rain rates. The TMI TBs and rain rates used in this study are the products of GPROF, version 6.

The EOF framework underlines a fundamental difference between TB–RR relations for TMI and PR rain retrievals. The TMI RRs are bimodal, with one mode related to shallow, warm rain, and the other mode associated with increasing rain rates with increasing cloud depth in the direction of cloud with greater rain and high-density ice concentrations. In contrast, the PR RRs exhibit high values distributed across a spectrum of cloud depths that have EOF coefficients associated with various amounts of rain emission and ice-scattering signatures. Hence, the same PR rain rate is produced for a wide variety of radiative signatures. This difference in behavior with respect to echo-top height could be important for rain retrievals using infrared cloud-top temperatures. The PR RR distributions indicate that the same echo-top height can be associated with a wide range of rain rates.

The PR-based TB–RR relations also show that the gradient in rain rate increases sharply toward higher values of the first EOF coefficient, where the highest rain rates are found. As a consequence, microwave rainfall retrievals likely have larger uncertainty in high-rain-rate regimes than in lower-rain-rate regimes. To reduce the uncertainty in Bayesian retrieval methods, it might be useful to adopt a flexible searching range (or error variance). That is, a large (small) searching range can be applied to presumably low (high) rain rates. Furthermore, the shape of the searching range does not need to be isotropic. The PR-based TB–RR distributions suggest that the search shape could be elongated in the y = −x direction (i.e., in the direction parallel to the contours of rain rate in EOF space).

The regional variability in TB–RR relations is different in the TMI and PR rain retrievals. Within the same EOF space, the TMI retrievals show very little regional dependence across the four tropical ocean regions examined here. The PR, on the other hand, exhibits differences in the EOF location of peak convective rain intensities across the different tropical ocean areas. In other words, the maximum rain rates are associated with different types of clouds in terms of their radiative signature and hydrometeor profile. The PR rain rates also have significant differences in the amplitude of rain intensity within the same EOF space for all cloud types. Hence, for the same I, meaning similar precipitation structure, clouds produce different rain rates depending on over which tropical oceanic region they were. The rain-rate differences approach a factor of 2, with TA generally having lower rain rates than either NWP or SWP for the same EOF space.

The four tropical ocean regions examined here were chosen because regional dependency is expected to occur as a result of differences in their environmental thermodynamics and large-scale forcing. Indeed, the regional variability in the PR rain rates is consistent with those expectations. Given the diagnosed regional variability in TB–RR relations, it appears that unique databases need to be developed for specific subsets of the tropical oceans. The analysis presented here illustrates how EOF fingerprints over different parts of the tropical oceans can be used to aid statistical validation of rain retrievals. The databases used in retrievals of cloud properties need to have the same EOF frequency distribution fingerprint as the observations over those regions.

Acknowledgments

TRMM data were provided by the Goddard Distributed Active Archive Center. The first author was funded by the Korea Meteorological Administration Research and Development Program under Grant CATER 2009-2108. The second author was supported by the U.S. National Science Foundation under Grants ATM-0619715 and ATM-0802717.

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