W EI-K UO T AO

D ANIEL E. J OHNSON

ABSTRACT

1. Introduction
Tropical precipitating cloud systems are one of the most important elements of the global climate system. The latent heat released by precipitating cloud systems is the primary agent of the vertical energy transport from the surface to the atmosphere. The excess of radiative energy gained at the lower-latitude surface, thus, par- ticipates in the atmospheric general circulation (Simp- son 1992). A great many studies have examined the roles of trop- ical precipitation in maintaining atmospheric circula- tions on many scales, ranging from mesoscale to syn- optic to planetary (e.g., Matsuno 1966; Gill 1980; Hart- mann et al. 1984; Yoshizaki 1991; Mapes 1993; Shige and Satomura 2000). Although there have been disputes on the interactions between tropical large-scale atmo- spheric circulations and convection (Emanuel et al. 1994; Stevens et al. 1997; Mapes 1997), early studies in the Marshall Island areas (Nitta 1970, 1972) and re- cent Tropical Ocean and Global Atmosphere (TOGA) Coupled Ocean–Atmosphere Response Experiment (COARE)¹ studies over the warm-pool region have shown that convection plays an important role in the large-scale dynamics (see a review of Stevens et al. 1997). Yanai et al. (2000) have shown that during TOGA COARE the conversion of perturbation available potential energy generated by convective heating main- tains the perturbation kinetic energy of the intraseasonal oscillations (ISO; Madden and Julian 1971, 1972, 1994), which occasionally influence large-scale climate shifts, such as El Niño (Takayabu et al. 1999). Houze (1982) and Johnson and Young (1983) em- phasized the differences of diabatic heating profiles that exist between convective and stratiform regions of me- soscale convective systems (MCSs). They found that for convective regions of MCSs the heating profile has warming at all levels with a maximum at midlevels, whereas in stratiform regions there is a warming peak in the upper troposphere and a cooling peak at low levels. The resulting MCS heating profile is positive at all levels, but with a maximum value in the upper tro¬ posphere. Hartmann et al. (1984) demonstrated that the heat source resembling a heating profile of MCSs with a maximum value in the upper troposphere produces a Walker circulation, which is in much better agreement with observations than those that are produced with a more conventional heat source (resembling a convective heating profile) with a simple linear global model. The Tropical Rainfall Measuring Mission (TRMM), a joint Japanese–U.S. cooperative Earth probe satellite, was successfully launched in 1997 to advance under- standing of the global energy and water cycle (Kum- merow et al. 2000). The precipitation radar (PR) and the TRMM Microwave Imager (TMI) have been pro¬ viding distributions of rainfall throughout the Tropics and have contributed significantly toward reducing un- certainty in satellite estimates of rainfall. The primary goal of TRMM is to use this precipitation information to determine the four-dimensional (i.e., temporal and spatial) patterns of latent heating over the whole tropical region. Tao et al. (2001) used TRMM precipitation infor- mation to estimate the four-dimensional latent heating structure over the global Tropics for 1 month (February 1998). Three different latent heating algorithms—the hydrometor heating (HH; Yang and Smith 1999a,b), the convective–stratiform heating (CSH; Tao et al. 1993a), and the Goddard profiling (GPROF) heating (Olson et al. 1999) algorithms—were used and their results were intercompared. The HH and GPROF algorithms are mi- crowave radiometer based for the TMI. Only for one of the three algorithms, the CSH algorithm, can PR prod- ucts be used as input, as well as TMI products. This is because the CSH algorithm utilizes only the surface rain rate and an estimate of the fractions of rainfall produced by convective and stratiform processes. The CSH al- gorithm assumes that the shape of the overall MCS heat- ing profile is determined by the relative amounts of convective and stratiform heating, which are propor- tional to the relative amounts of convective and strati- form precipitation. The first spaceborne precipitation radar, PR provides height information based upon the time delay of the precipitation-backscattered return power and has en- abled us to directly obtain vertical profiles of precipi- tation over the global Tropics (Kozu et al. 2001; Oka- moto 2003). The classification between convective and stratiform rain became more straightforward utilizing observed precipitation profiles (Awaka et al. 1998). The accuracy of this classification is very important for es- timating latent heating (Houze 1997). Moreover, the ob- served precipitation profiles should be related to those of latent heating (Tao et al. 1990). Thus, the accuracy of latent heating estimates from the PR should be su- perior to that from the TMI, which measures radiances that are the end product of the integrated effects of electromagnetic absorption/emission and scattering through a precipitating cloud along the sensor view path. The concept of a spectral approach originates from Austin and Houze (1973) and Houze (1973) in which the precipitation-top height (PTH) observed by surface- based radar data was utilized in estimating the vertical transports by cumulus-scale convection. This spectral approach has been extended by Houze and Leary (1976), Leary and Houze (1980), Houze et al. (1980), and Cheng and Houze (1980). Takayabu (2002) used a similar concept and obtained a spectral expression of precipitation profiles to examine convective and strat- iform rain characteristics statistically over the equatorial area (10°N–10°S) observed by the TRMM PR. In her study, all nadir data of PR2A25 version 5 (Iguchi et al. 2000) for the period of 1998–99 were utilized. Precip- itation profiles with a 0.3 mm h − 1 precipitation-top threshold were accumulated and stratified by PTHs. The threshold of 0.3 mm h − ¹ corresponds to 17.2 (stratiform) and 15.5 (convective) dBZ above the 0°C height and 17.0 (stratiform) and 14.2 (convective) dBZ just below the 0°C height in typical initial equivalent reflectivity factor–rainfall rate (Z – R ) relations used in the PR2A25 version-5 algorithm (Iguchi et al. 2000). Properties of convective rain profiles show near-monotonic change with cumulative frequency. Stratiform rain profiles con- sist of two groups. One group consists of shallow strat- iform rain profiles, which are very weak and increase downward. The other group consists of anvil rain pro- files, characterized by maximum intensity around the melting level, much less intensity above, and a down- ward decrease below as indicated in traditional radar observations. Based on the results of spectral precipitation statistics of Takayabu (2002), we introduce a new retrieval al- gorithm for latent heating estimates utilizing the TRMM PR data. It is named the spectral latent heating (SLH) algorithm and utilizes a cloud-resolving model (CRM) to produce its spectral lookup tables. The primary dif- ference from the CSH algorithm is that the SLH algo- rithm requires information about not only convective/ stratiform characteristics and surface precipitation rate, but also about the precipitation profile. The total heating profile shape is affected basically by two factors: the fraction of stratiform rainfall and the shape of the con- vective heating profile. The importance of the first factor has been shown by Johnson (1984) and has been taken into account by the CSH algorithm. By utilizing the information about precipitation profiles, the second fac- tor is also taken into account by the SLH algorithm. This paper will describe the procedure of the SLH algorithm. A set of heating-profile lookup tables for convective, shallow stratiform, and deep stratiform (an- vil) regions with melting levels are produced with nu- merical simulations of tropical cloud systems in TOGA COARE. A consistency check of the SLH algorithm will also be done with the CRM-simulated outputs in this paper.
2. Approach
Figure 1 shows the procedure for deriving the SLH algorithm. As well as for the CSH algorithm, a set of lookup tables is produced from CRM-simulated heating profiles. The primary difference from the CSH algo- rithm’s lookup table is that the SLH algorithm’s lookup table refers to the PTH of corresponding CRM-simu- lated precipitation profiles. Because of the scarcity of reliable validation data and difficulties associated with the collocation of validation data and satellite measure- ments, a consistency check of the SLH algorithm is performed using CRM-simulated precipitation profiles as a proxy for the PR data. The algorithm-reconstructed heating profiles from CRM-simulated precipitation pro- files are compared with CRM-simulated ''true'' heating profiles, which are computed directly from the model thermodynamic equation. The consistency check is a useful and necessary precondition for the application of the algorithm to actual TRMM PR data.
a. CRM simulations
The CRM used in this study is the two-dimensional version of the Goddard Cumulus Ensemble (GCE) mod- el, and is primarily documented in Tao and Simpson (1993). Recent improvements were presented in Tao (2003) and Tao et al. (2003). The GCE model has been used to develop the other TRMM heating algorithms, namely, the CSH and GPROF algorithms. Here, we introduce a brief summary of the model configuration. The model includes solar and infrared radiative transfer processes and explicit cloud–radiation interactive processes (Tao et al. 1996). Simulations pre- sented in this study employ a parameterized Kessler- type two-category liquid water scheme (cloud water and rain) and a three-category ice-phase scheme (cloud ice, snow, and graupel) by Rutledge and Hobbs (1984). Sub- grid-scale (turbulent) processes in the GCE model are parameterized using a scheme based on Klemp and Wil- helmson (1978) and Soong and Ogura (1980). The ef- fects of both dry and moist processes on the generation of subgrid-scale kinetic energy have been incorporated in the model. The model domain is 1024 km in the x direction (horizontal) and 22.4 km in the z direction (vertical). The horizontal resolution is 1000 m. The ver- tical resolution varies from 100 m at the lower boundary to 1000 m at the top of the domain. In this study, tropical convective systems in TOGA COARE are simulated with the so-called cloud ensemble modeling approach. In this approach, many clouds of different sizes in various stages of their life cycles can be present at any model simulation time. Observed large-scale advective tendencies of temperature, mois- ture, and horizontal momentum are used as the main large-scale forcings that govern the GCE model in a semiprognostic manner (Soong and Ogura 1980). These are applied uniformly over the model domain with the assumption that the model domain is much smaller than the large-scale disturbance. Large-scale advective ten- dencies for potential temperature θ and specific humid- ity q are defined as and
and were derived every 6 h from the version-2.1 TOGA COARE intensive flux array (IFA)-averaged soundings of Ciesielski et al. (2003). Here, v is the horizontal wind vector, ω is vertical pressure velocity, α is the specific volume, and C ′ , = C pd (1 − q obs ) + C pv q obs is the mois- ture-weighted heat capacity at constant pressure, where C ρq and C ρν are the heat capacity at constant pressure for dry air and water vapor, respectively. The version- 2.1 analysis uses the National Center for Atmospheric Research  (NCAR)  Atmospheric  Technology  Division (ATD) humidity-corrected sonde data (Wang et al. 2002). Because accurate calculations of the large-scale hor- izontal momentum forcing terms are difficult to obtain from observations in the Tropics, the terms are instead replaced by a nudging term
where V is the model domain–averaged horizontal ve- locity, v obs is the observed large-scale horizontal vector over the IFA region, and τ is the specified adjustment time scale of approximately 1 h (Grabowski et al. 1996; Xu and Randall 1996). This method constrains the do- main-averaged horizontal velocities to follow the ob- served values and thereby provides a simple means of controlling the cloud system dynamics by the large-scale momentum and shear. Cyclic lateral boundary condi- tions are incorporated to ensure that there is no addi- tional heat, moisture, or momentum forcing. For the simulations in this paper, four subperiods with 9-day durations (10–18 December 1992, 27 December 1992–4 January 1993, 9–17 February 1993, and 18–26 February 1993), and one subperiod with an 8-day du- ration (19–26 December 1992), during the TOGA COARE intensive observation period (IOP) are chosen. The simulations have been done with a data sampling frequency of 5 min.
b. Latent heating
In diagnostic studies (e.g., Yanai et al. 1973; Yanai and Johnson 1993), it is customary to define the ap- parent heat source Q 1 of a large-scale system by av- eraging horizontally the thermodynamic equation as
where π = (p/P 00 ) 8 ' C p is the nondimensional pressure, P 00 is the reference pressure (1000 hPa), C p is the spe- cific heat of dry air at constant pressure, and R is the gas constant for dry air. The second term in the paren- theses in Eq. (4) is the large-scale advection term given by Eq. (3). The Q 1 can be directly related to the con- tributions of cloud effects, which can be explicitly es- timated by the cloud-resolving model:
The overbars denote horizontal averages, the primes in- dicate deviations from the horizontal averages, ρ is the air density, Q R is the cooling/heating rate associated with radiative processes, and D g is the subgrid-scale (smaller than the cloud scale) diffusion, which is usually small in comparison with other terms above the boundary lay- er (Soong and Tao 1980). The first term on the right- hand side of Eq. (5) is the vertical eddy heat flux con- vergence from upward and downward cloud motions. The term LH is the net latent heating due to the phase change of water:
where L v, L r, and L s are the latent heats of vaporization, fusion, and sublimation, respectively. Variables c, e, f, m, d, and s stand for the rates of condensation, evap- oration, freezing of raindrops, melting of snow and grau- pel, deposition of ice particles, and sublimation of ice particles, respectively. The vertical integration of Q 1 − Q R (hereinafter represented as Q 1 R ) from the surface to the tropopause z, yields (Yanai et al. 1973)
where P o and S o are the surface precipitation rate and surface sensible heat fluxes. The surface sensible heat fluxes are generally smaller than the surface precipita- tion rate and can be neglected. The precipitation falling at a given time is not related to latent heating that is occurring at that instant, but rather to the accumulated latent heating that led up to the precipitation over a finite time period. Therefore, LH and Q 1 R should be basically integrated over the time periods encompassing the life cycles of cloud processes producing the precipitation. However, it is extremely difficult to tabulate, for example, the effect of individual mesoscale systems. Instead, we here depend on the sta- tistics. In the CRM simulation, as well as in the real world, cloud systems develop and decay. Although in- stantaneous matching between a certain rainfall profile and a heating profile is an ill-posed concept, statistical tabulation still could be done if the life cycle of cloud systems is realistically reproduced in the CRM. For ex- ample, a shallow convective rain profile may be at a developing stage of a mesoscale system with a certain probability A, and may be just an isolated convection with a probability of 1 − A. The latent heating asso- ciated with the two cases should be different. However, if the CRM reproduces the statistics of the rain systems well enough, CRM-based tables can statistically rep- resent an average heating profile for a certain rain pro- file. Besides, we accumulate LH and Q 1R over a period of 5 min for each data sampling, because accumulation over long periods is inadequate for moving convective systems. Additional sensitivity tests with periods of 1 and 2 min do not alter the overall results of our study. Hereinafter, heating (LH and Q 1 R ) accumulated over a period of 5 min for each data sampling is represented as instantaneous heating. The SLH algorithm aims to estimate LH and Q 1 e for each precipitation profile. Although the PR footprint- scale (4 km) estimates of Q 1 R are less meaningful be- cause Q 1 R is a large-scale variable, we still need to derive the value Q 1R in order to compare with Q 1 derived from the diagnostic studies for validation. On the other hand, for the footprint-scale estimates, LH is more meaningful than Q 1 − Q R . Therefore, we will show the results main- ly with LH in this paper.
3. Algorithm development
In this section, we first discuss the classification of rain types, which is essential to the algorithm. Second, we show how lookup tables are constructed. Section 3c describes how these lookup tables are applied to pre- cipitation data in order to retrieve heating profiles.
a. Classification of rain types
Takayabu (2002) separated convective and stratiform precipitation based on the TRMM PR version-5 2A23 convective–stratiform separation algorithm, which clas- sified shallow, isolated precipitation elements into the stratiform category. Recently, Schumacher and Houze (2003) suggested that because the shallow, isolated ech- oes represent warm-rain processes, they should be clas- sified as convective. After their suggestion we revised spectral plots of Takayabu (2002, her Fig. 1) and present them as Fig. 2, with a reclassification of shallow, iso- lated rain (rain type 15 in product 2A23) into convective rain. The abscissa is cumulative frequency. Precipitation profiles with a 0.3 mm h − ¹ precipitation-top threshold were accumulated and stratified with PTHs. All nadir precipitation profiles over the ocean in the 10°N–10°S belt for the 3 yr of 1998–2000 from PR2A25 version 5 are used. A shallow stratiform category still exists in Fig. 2. This is because when a bright band exists or the convective reflectivity threshold is not met, shallow and nonisolated rain pixels are classified into stratiform rain (J. Awaka 2003, personal communication). The TRMM PR rain-type classifications, in which brightband identification is very important, cannot be directly applied to GCE outputs (Awaka et al. 1996). This is because the microphysical schemes utilized in CRMs (e.g., Lin et al. 1983; Rutledge and Hobbs 1984) typically do not contain an explicit description of par- tially melted precipitation particles that leads to a bright band of enhanced radar reflectivity. Thus, the GCE con- vective and stratiform separation method (Lang et al. 2003) is used with some modifications to produce the lookup tables from the CRM simulation data in this study. The GCE method begins with the method of Churchill and Houze (1984) applied to surface rainfall. Model grid points that have a precipitation rate 2 times as large as the value of the background average are identified as convective cores. The points surrounding the core point are also convective, as is any point with a precipitation rate above 20 mm h − ¹ . The background average is taken from a box that has five grid points on a side. Following this, two additional criteria are applied to identify active convection aloft with little or no surface rainfall, such as tilted updrafts and new cells ahead of a convective line (Tao et al. 1993b). Nonsurface precipitation regions are considered to be convective if cloud water exceeds a certain threshold (i.e., the minimum of either 0.5 g kg − ¹ or one-half of the maximum cloud water at that specific simulation time), or if the updraft exceeds a certain threshold (i.e., the minimum value between 3 m s−¹ and one-half of the maximum updraft at that specific simulation time) below the melting level. Although the TRMM PR rain-type classification can- not be directly applied to them, we need to maintain the consistency. As shown in Fig. 3, the GCE method has a stratiform category that exists below the 0°C level (grid points with rain index 1 and 4; see Table 1 for the definition of rain index). These shallow rain grid points, in which surface rainfall, cloud water, or updraft do not exceed the above thresholds, are classified into the strat- iform category. However, in the same way that rain- type 15 in product 2A23 is reclassified from stratiform to convective, we would like to reclassify the shallow, isolated rain grids (e.g., x = 264 km to x = 267 km in Figs. 3a–c) as being convective. As for shallow and nonisolated rain grids, we also reclassified them based on their adjacent rain types. The grid points of shallow precipitation (e.g., x = 249 km to x = 251 km in Figs. 3a–c) adjacent to or surrounded by the convective grid points are classified as being convective. As a result, the grid points with rain index 4 in Figs. 3a–c are re- classified from stratiform into convective. Their heating profiles are characterized by lower-level warming topped by cooling, such as shallow convection in the “episodic trade wind regimes” observed during TOGA COARE (Johnson and Lin 1997). On the other hand, the shallow rain grid points (e.g., x = 549 km to x = 551 km in Figs. 3d–f) adjacent to or surrounded by deep stratiform precipitation with PTH higher than the melting level (hereinafter the term ''anvil'' represents deep stratiform regions both with and without surface precipitation) are classified as being stratiform. As a
result, the grid points with rain index 1 in Figs. 3d–f remain in a stratiform category. Their heating profiles are characterized by lower-level cooling and upper-level heating. All GCE-simulated precipitation profiles with a 0.3 mm h − ¹ precipitation-top threshold are accumulated and stratified with PTH (Fig. 4), as well as the spectral plot of TRMM PR–observed precipitation profiles in Fig. 2. The abscissa is cumulative frequency. Actual numbers of sampled profiles for convective (stratiform) precip- itation are 304 205 (2 528 451) or (11:89). Spectral properties of GCE-simulated convective precipitation profiles in Fig. 4 correspond well to those of TRMM PR–observed ones in Fig. 2, having a smooth increase of precipitation intensity with an increase of PTH. Pre- cipitation rates of the GCE-simulated convective pre- cipitation profiles are stronger than those of the PR- observed ones, which could be due to regional differ- ences. The GCE-simulated precipitation profiles are from the TOGA COARE simulations, while the PR- observed ones are from all nadir data over ocean areas in the 10°N–10°S belt. On the other hand, properties of the GCE-simulated stratiform precipitation profiles show less similarity to those of the PR-observed ones. While the downward decrease of the precipitation rates below the melting layer is indicated in the GCE-simulated stratiform pre- cipitation, as well as the PR-observed ones, the upward rapid decrease of those above the melting layer in the PR-observed precipitation is not indicated in the GCE- simulated stratiform precipitation. This is mainly be- cause the PR can neither retrieve the small ice-phase precipitation nor observe the PTH accurately enough in the upper-level regions of the anvils where small ice- phase hydrometors dominate. Thus, insensitivity of the PR to the small ice-phase precipitation should be taken into account by the algorithm.
b.  Construction of lookup tables
Heating-profile lookup tables have been constructed for the three rain types—convective, shallow stratiform, and deep stratiform both with and without surface rain (Figs. 5 and 6). For the construction of lookup tables, the GCE-simulated precipitation profiles and corre- sponding heating profiles from the four subperiods with 9-day durations (10–18 December 1992, 27 December 1992–4 January 1993, 9–17 February 1993, and 18–26 February 1993) are used. Another simulation for the one subperiod with an 8-day duration (19–26 December 1992) is not used for the construction of lookup tables, but it is later used for the consistency check of the algorithm. The PTH is used as an index for the table for con- vective rain (Figs. 5a and 6a). The GCE-simulated pre- cipitation profiles with a 0.3 mm h − 1 precipitation-top threshold and corresponding heating profiles are accu- mulated and averaged for each PTH with model grid F IG . 5. Ensemble mean, GCE-simulated latent heating profiles, plotted as functions of PTH from (a) convective and (b) stratiform regions, and (c) precipitation rates at the melting level from anvil regions. Contours indicate values of the confidence interval for the mean at the 95% level with Student's t test. Contour interval is 2.0 K h − 1 for convective and 1.0 K h − 1 for stratiform and anvil intervals. Properties of convective heating profiles show near-monotonic changes with PTH, so that it is appro- priate to use the PTH as an index for tables for con- vective rain. The levels of Q 1 R heating maximums are higher than those of the LH heating maximums because of the eddy heat flux convergence [the first term on the right-hand side of Eq. (5)], but the difference between LH and Q 1g is not so large. This is because LH is a major contributer to Q 1 R (Tao et al. 1993b). The LH heating is largely due to condensation and deposition in the convective updrafts, and LH cooling is largely due to rain evaporation found in the lowest levels just above the surface. The level of the LH peak shifts up- ward until the PTH reaches 6 km and then reaches its plateau around 4 km, just below the melting level (4.4 km). These results indicate that liquid water processes (i.e., condensation) dominate. This is consistent with the results from videosonde observations during TOGA COARE (Takahashi et al. 1995). Notice that the shallow convective heating profiles (PTH < 6 km) are charac- terized by a cooling aloft, such as the episodic trade wind regimes observed during TOGA COARE (Johnson and Lin 1997). We use these heating profiles (shown in Figs. 5a and 6a) as the convective heating profile lookup table for the SLH algorithm, except for heating profiles with a PTH higher than 16.5 km, whose confidence level for the mean is low because of the small number of profiles. As for the stratiform precipitation, the GCE-simulated precipitation profiles with a 0.3 mm h − 1 precipitation- top threshold and corresponding heating profiles are also averaged for each PTH (Figs. 5b and 6b). However, as mentioned earlier, we cannot expect the PR to retrieve the small ice-phase precipitation and observe the PTH accurately enough in the upper-level regions of the an- vils where small ice-phase hydrometors dominate. Therefore, only the stratiform heating profiles with a PTH lower than 4.4 km (shown in Figs. 5b and 6b), characterized by lower-level cooling, are used as the shallow stratiform heating profile lookup table for the SLH algorithm. As shown in Figs. 5b and 6b, the anvil profiles with a PTH higher than the melting level are characterized by upper-level warming and lower-level cooling, which is also found in observations (e.g., Johnson and Young 1983). The upper-level warming in these anvil regions is largely due to condensation and deposition, while the lower-level cooling is largely due to the evaporation of raindrops and melting of ice particles (Tao et al. 1990). As an analogy to Eq. (7), the vertically integrated heat- ing Q = (LH, Q 1R ), from the melting level z m to the tropopause z,, may be related to the precipitation rate at the melting level P ,,, by
where f is the fraction of the precipitation rate at the melting level P m , which is carried over from the con- vective region. The increases of the upper-level heating amplitude with P nn in Figs. 5c and 6c are consistent with Eq. (8). The PR can measure the precipitation rate at the melting level as can a surface-based radar (e.g., Leary and Houze 1979), although it cannot observe the PTH accurately enough in the upper-level regions of the anvils. Thus, we replace PTH with the precipitation rate at the melting level P m as the index in the lookup table for anvil regions (Figs. 5c and 6c). It is noted that the Q 1R heating is larger than the LH heating above the 12- km level because of the eddy heat flux convergence, which is induced by the radiative cooling at the top of anvil clouds (Sui et al. 1994). The fraction f to be used in Eq. (8) is not negligible and is examined here. Houze et al. (1980), Gamache and Houze (1983), and Chong and Hauser (1989) showed that the stratiform precipitation falling into the melting layer from the anvil cloud above is a combi- nation of condensate generated in and carried over from the convective region plus condensate that is produced by the anvil region's own upward motion, analyzing the water budgets of a precipitating tropical mesoscale con- vective system. In order to evaluate this advection ef- fect, we calculated the following values for the con- vective and anvil tables (Figs. 5a and c):
Here, P s is the precipitation rate at the observable lowest level and tildes denote the variables in the lookup table. Although not shown here, the simulated precipitation profiles are also accumulated and averaged as well as the simulated heating profiles. As shown in Fig. 7, R conv is larger than unity and R anvil is smaller than unity, which indicate the contribution to the anvil water budget made by the horizontal transfer of condensate from the con- vective region. It is noted that R conv decreases with PTH. Convection with low PTH includes shallow convection that is growing into deep convection. Generally, in such growing shallow convection an updraft carries the grow- ing particles upward, because these particles are not yet heavy enough to overcome the updraft (see Fig. 2 of Houze 1981). Thus, total condensates generated in shal- low convection are much larger than surface precipi- tation, which results in large R conv . On the other hand, R anvil decreases with ¯ P ww . The index of ¯ P w may be in- terpreted as the distance from the convective region: large ¯ F m indicates a small distance from the convective region, while small to the convective region, because condensate carried over from the convective region is large, P m is large and R anvil is small. Condensate carried over from the convective region continues to grow, mainly because of condensation and deposition, and begins to fall relative to the ground. Thus, condensate carried over from the convective region decreases with the distance from the convective region. This accounts for small R enxil far from the convective region. The height of max- imum heating decreases slightly with ¯ ,,, in Figs. 5c and 6c, which may be a manifestation of ascending front- to-rear flow and consistent with the above interpretation.
c. Procedure of heating retrieval
In order to retrieve heating profiles, the SLH algo- rithm utilizes the observed information on precipitation type (convective or stratiform), PTH, P x , and P m . The stratiform region is first separated into a shallow strat- iform region and an anvil region based on the PTH in comparison with the melting level. The algorithm then derives heating profiles based on the following proce- dure for each of the three regions (Fig. 8). For convective regions, as well as shallow stratiform regions, a heating profile corresponding to the PTH is selected in the convective heating-profile (Figs. 5a and 6a) and PTH-based shallow stratiform heating-profile (Figs. 5b and 6b) lookup tables, respectively. Based on Eq. (7), the amplitude is determined by
For anvil regions with the melting level (PTH > 4.4 km), on the other hand, the heating profile correspond- ing to P m is selected in the anvil heating-profile lookup table (Figs. 5c and 6c). The upper-level warming am- plitude is then determined by
It should be noted that the contribution by the hori- zontal transfer of condensate from the convective region to the anvil region (i.e., R cony, R envil ) is already included in the lookup tables. This is because Õ(z) and Õbigh(z) are normalized by respectively. Because R conv is larger than unity and R anvil is smaller than unity, as shown in Fig. 7, the vertically integrated amount of Q ( z ) is estimated to be larger than P s in the convective regions, while that of Q bigh ( z ) is estimated to be smaller than P w in the anvil regions. The evaporative cooling rate below the melting level in the anvil regions is proportional to the reduction of the precipitation profile toward the surface from the melting level (Tao et al. 1990). The downward decrease of the intensity of anvil rain below the melting layer has been statistically indicated in the PR observation (see Fig. 2 of Takayabu 2002), as well as in case studies of traditional radar observations (e.g., Leary and Houze 1979). Thus, the algorithm computes the lower-level cooling amplitude Q 10w as a function of the difference of the rainfall rate between the surface and the melting level P rrr − P r :
The above procedure allows us to estimate the heating profiles in the anvil regions both with and without sur- face precipitation.
4. Consistency check
a. Comparison with the GCE model
For a consistency check of the SLH algorithm, we reconstructed heating profiles averaged over the TOGA COARE IFA region for the 19–26 December 1992 pe- riod using the simulated parameters (i.e., PTH, convec- tive/stratiform characteristics, P s , P m ) as inputs. The simulated data during 19–26 December 1992 are in- dependent from other simulated data that are used for the construction of lookup tables. This period is char- acterized by the transitions among cloud regimes and has also been used by the Global Energy and Water Cycle Experiment (GEWEX) Cloud System Study (GCSS) working group 4 (WG4) model intercomparison project for CRMs and single-column models (SCMs, Moncrieff et al. 1997). Tao et al. (2000) showed that there was excellent agreement between heating profiles derived from the sounding and simulated by the GCE model. Then, the algorithm-reconstructed heating pro- files from GCE-simulated precipitation profiles are com- pared with GCE-simulated true heating profiles, which are computed directly from the model thermodynamic equation (Fig. 9). Figures 9a and 9d show the time series of the sim- ulated heating and reconstructed heating, respectively. Here we show only the results for LH. The results of Q 1R are similar to those of LH. The pattern in temporal variations of the SLH algorithm-reconstructed heating profile agrees well with that of the GCE-simulated heat- ing profile. In particular, the SLH algorithm-recon- structed heating profiles captured the evolution of a qua- si-2-day oscillation (Takayabu et al. 1996) during the period of 1800 UTC 23 December–1800 UTC 25 De- cember 1992. During the period 1800 UTC 23 Decem- ber–1200 UTC 24 December, the transition from the shallow convective stage to the deep convective stage can be seen in the GCE-simulated as well as in the reconstructed LH profiles. During 0600–1500 UTC 25 December, both simulated and reconstructed LH profiles show upper-level heating and lower-level cooling. It is noted that the reconstructed results show more smoothed features. This is because the lookup tables represented the averaged profiles for each index, either PTH or P .... Figures 9b and 9e show the time series of the sim- ulated heating and reconstructed heating in the convec- tive regions, respectively. The pattern in temporal var- iations of the reconstructed heating profile agrees very well with that of the simulated heating profile. The dif- ferences between the shallow convective stage and the deep convective stage are realistically reconstructed. This is a result of the utilization of observed informa- tion, not only on precipitation type and intensity, but also on the precipitation depth (i.e., PTH). Figures 9c and 9f show the time series of the simu- lated heating and reconstructed heating in the stratiform regions (shallow stratiform plus anvil), respectively. Al- though the reconstructed time series are smoother than the simulated ones, the pattern in temporal variations of the reconstructed heating profile agrees well with that of the simulated heating profile. For example, there is no surface rain in the stratiform region around 1200 UTC 25 December, as shown later (Fig. 14), because hydrometers generated at the middle and upper levels evaporate completely before reaching the surface. Still, the SLH algorithm reconstructs the heating profiles characterized by upper-level heating and lower-level cooling around 1200 UTC, consistent with the GCE simulations. This comes from utilizing the precipitation rate at the melting level ( P w ) for anvil rain. The SLH algorithm gives smoother features of the reconstructed heating time series for the stratiform regions than those for the convective regions. This is because in the lookup table the heating profile does not vary much with P m for anvil rain (Figs. 5c and 6c). Eight-day-averaged profiles of the simulated LH and the reconstructed LH are shown in Figs. 10a and 10b. These profiles are simply the time averages of the pro- files shown in Fig. 9. The 8-day-averaged profiles of the reconstructed LH agree very well with the simulated ones in total, convective, and stratiform regions, al- though the reconstructed convective heating above the melting level is a little bit stronger than the simulated one. Also, the 8-day-averaged profiles of the recon- structed Q 1 R (Fig. 10d) agree very well with the simu- lated ones in the total, convective, and stratiform regions (Fig. 10c). Both simulated and reconstructed LH profiles have a distinct cooling near 4.4 km, which is due to the melting processes. On the other hand, neither the sim- ulated nor reconstructed Q 18 profile has such a distinct cooling near 4.4 km. This is because the eddy heat flux convergence compensates for the distinct cooling due to the melting (Sui et al. 1994; Shie et al. 2003).
b. Estimation errors
Based on the assumption that heating profiles cor- respond statistically to precipitation profiles or precip- itation parameters (i.e., PTH, P n ), we construct lookup tables. However, instantaneous heating structures are very fluctuating. Here, we reconstructed the heating pro- files grid by grid for the four 9-day subperiods that were used for the lookup tables, using the simulated param- eters as inputs. Then, the differences between recon- structed heating profiles and simulated ones are exam- ined statistically to see the errors of instantaneous and grid-by-grid estimates with this table method. It serves as a guide in the application of the algorithm and in- terpretation of the results. The discussion of errors due to regional or seasonal change is beyond the scope of this study and will be discussed in the forthcoming Part II. Figure 11 shows histograms of grid numbers per χ (reconstructed LH − simulated LH) for the lower ( z = 3.5 km) and upper ( z = 7.6 km) levels, in which con- vective heating and stratiform heating have their max- imum, respectively. Grid points where nonzero heating is simulated by the GCE model or reconstructed by the SLH algorithm are used for the plots. Figure 11 indicates that for grid-by-grid estimation in the convective re- gions, there are errors that are quite large at 3.5 km, while those at 7.6 km considered small. It is noted that at 7.6 km in convective regions, the number of grid points with χ > 0 K h − 1 (overestimation) is larger than those grid points with χ < 0 K h − ¹ (underestimation). This is consistent with the results of the consistency check shown in Fig. 10 where the reconstructed con- vective heating above the melting level is a little stron- ger than the simulated one. On the other hand, at 7.6 km in the stratiform region, the number of grid points with χ < −10 K h−¹ is larger than those with χ > 10 K h − ¹ , while the number of grid points with 0 K h − ¹ < χ < 5 K h − ¹ is larger than those of grid points with −5 K h−¹ < χ < 0 K h−¹. This is because the SLH algorithm cannot capture the large heating associated with old convective cells remaining in the upper levels of the trailing stratiform regions. These results suggest that the SLH algorithm estimates should be averaged for quantitative use. A preliminary evaluation of the horizontally averaged estimates is also performed using a GCE-simulated syn- thetic observation (Fig. 12). Horizontal averaging re- duces the root-mean-square (rms) error, and for aver- aging over an ~50 km width the rms error is reduced to about 1 K ² h −². From this result, an averaging over an ~50 km width is recommended when using the SLH algorithm estimates quantitatively. Large rms errors can be found at about 9 km. Again, this may be because the SLH algorithm cannot capture large heating associated with old convective cells remaining in the upper levels of the trailing stratiform regions.
c.  Comparison with the CSH algorithm
Tao et al. (2001) represented the first attempt at using TRMM rainfall information to estimate the latent heat- ing structure over the global Tropics for February 1998. Three different latent heating algorithms—the HH al- gorithm, the CSH algorithm, and the GPROF heating algorithm—were used and their results were intercom- pared. The CSH algorithm-estimated heating profiles only showed one maximum heating level and were in better agreement in diagnostic studies than the HH al- gorithm- or GPROF algorithm-estimated heating pro- files. Only for one of the three algorithms, the CSH algorithm, can PR product be used as input, as well as the TMI products. Here, we compare the SLH algorithm performance with the CSH algorithm performance, us- ing the simulated parameters as inputs. Because the re- sults of both methods are derived from the same model, the comparison of SLH and CSH is not a rigorous test, but rather a consistency check. Its purpose is to see their advantages and disadvantages purely originating from the different retrieval procedures between SLH and CSH. The CSH algorithm has been developed based on the assumption that the shape of the overall MCS heating profile is determined by the relative amounts of con- vective and mesoscale heating, which are proportional to the relative amounts of convective and stratiform pre- cipitation:
Here, P conv and P stra are observed surface precipitation rates in the convective and stratiform regions, and Q(z)conv and Q(z)stra are model-generated convective and stratiform heating profiles, normalized by the convective and stratiform rainfall, respectively. An appropriate se- lection of latent heating profiles from the lookup table is very important for the CSH algorithm (Tao et al. 2000). Figures 13a–c show the time series of the LH profiles simulated by the GCE model and reconstructed by the SLH algorithm, and the CSH algorithm during 19–26 December 1992, respectively. Eight-day-averaged con- vective and stratiform LH profiles during 19–26 De- cember 1992 (Fig. 10a), normalized by the convective and stratiform rainfall, respectively, are used for the CSH algorithm to reconstruct LH profiles from simu- lated P conv and P stra . Clearly this is the most adequate set of heating profiles. Figure 13, however, indicates that the SLH algorithm reconstructs the time series of the heating profile better than the CSH algorithm, es- pecially for the period of 1800 UTC 23 December–1800 UTC 25 December 1992 when the evolution of a quasi- 2-day oscillation can be seen. During the period 1800 UTC 23 December–1200 UTC 24 December, the tran- sition from the shallow convective stage to the deep convective stage can not be seen in the CSH algorithm reconstructed LH profiles. In addition, during 0600– 1500 UTC 25 December, the CSH algorithm-recon- structed LH profiles is negligible, while the GCE-sim- ulated and the SLH algorithm-reconstructed LH profiles show upper-level heating and lower-level cooling. Figure 14 shows the time series of precipitation rates during the period of 1800 UTC 23 December–1800 UTC 25 December when the evolution of a quasi-2-day os- cillation can be seen. The subperiods in Figs. 14a–d indicate shallow convective, developing, mature, and decaying stages of the quasi-2-day oscillation, respec- tively. Figures 15a–d show LH profiles for the subpe- riods of Figs. 14a–d, respectively. During the shallow convective stage the cloud en- semble with low PTH dominated (see Figs. 3a–c). The SLH-reconstructed LH profile with maximum below the melting level is in good agreement with that simulated by the GCE model. In contrast, the CSH algorithm re- constructed LH profile was at a maximum above the melting level (Fig. 15a). The underestimation of low- level heating made by the CSH algorithm is caused by two reasons. First, the CSH algorithm fixes the con- vective heating profile in the 8-day-averaged profile (Fig. 10a), and, thus, cannot retrieve a shallow convec- tive profile. Second, the ratio of total convective heating to surface precipitation (i.e., tion is much larger than that for deep convection, as is shown in Fig. 7. In the CSH algorithm, because the convective heating profile is normalized by the averaged convective rainfall, the ratio of total convective heating to surface precipitation is also fixed in the 8-day-av- eraged profile, which is smaller than that of shallow convection, and the heating amplitude only varies be- cause of the convective component of the surface pre- cipitation rate ( P conv ). These account for the underesti- mation of low-level heating. During the developing stage, both the SLH and CSH algorithms reconstructed the LH profile in good agree- ment with that simulated by the GCE model (Fig. 15b). It is noted that both the SLH and CSH algorithms re- constructed somewhat stronger upper-level heating and weaker lower-level heating than those simulated by the GCE model because of the underestimation of convec- tive heating and overestimation of anvil heating. This may be because the contribution to the anvil water bud- get made by the horizontal transfer of condensate from the convective region simulated by the GCE model is more than that assumed by both algorithms in their ta- bles, which results in an underestimation of convective heating and an overestimation of anvil heating. During the mature stage, the GCE model simulated LH profile (Fig. 15c) is similar to that at the developing stage (Fig. 15b), but the lower-level amplitude at the mature stage should be smaller than that at the devel- oping stage. This is due to lower-level cooling resulting from increasing anvil rainfall. The SLH algorithm-re- constructed LH profile with a maximum at 7 km is in good agreement with the GCE model simulation. In con- trast, the CSH algorithm-reconstructed LH profile has a tendency to underestimate in the upper levels and to overestimate in the lower levels. It is because the rel- ative contribution of P stra to total surface precipitation at the mature stage is smaller than at the developing stage, as shown in Fig. 14. The decaying stage includes a period when P ssa is zero. The CSH algorithm reconstructed the LH profile with a very small amplitude. This is because the CSH algorithm estimates no heating profiles from the no- surface rain [see Eq. (14)]. On the other hand, the SLH algorithm-reconstructed LH profile characterizes upper- level heating and lower-level cooling in good agreement with the GCE model simulation (Fig. 15d). This is be- cause the SLH algorithm can compute heating profiles from the precipitation rate at the melting level in the stratiform region ( P ,,, ). It is noted that the melting-level cooling in this decaying stage is stronger in the recon- structed profile than in the simulated profile. These results show that the utilization of PTH and P … provides two distinct advantages for the SLH al- gorithm. First, the differences in heating profiles be- tween the shallow convective stage and the deep con- vective stage can be realistically retrieved. Second, heat- ing profiles in the decaying stage with no surface rain can also be retrieved. The CSH algorithm takes into account the fact that when cloud latent heating is separated into convective and stratiform regions, the profiles for each region take on a characteristic shape, even for systems from dif- ferent geographic locations (Johnson 1984; and a review by Houze 1997). Therefore, the CSH algorithm is best used for temporal scales over the lifetime of a system. For example, the 2-day-averaged profile of LH recon- structed by the CSH algorithm for the period covering the entire life time of the quasi-2-day oscillation (1800 UTC 23 December–1800 UTC 25 December) is in good agreement with that simulated by the GCE model, as well as that reconstructed by the SLH algorithm (Fig. 16). It is, however, inferred from the disagreement be- tween the CSH algorithm-reconstructed and the GCE- simulated LH profiles during the shallow convective stage shown in Fig. 15a, that even for monthly averaged heating profiles the CSH algorithm may not retrieve the heating features unless it used the ''representative'' heating profiles normalized by surface rain rate. Tao et al. (2001) showed that the CSH algorithm-retrieved heating profile over the Atlantic Ocean for 1 month has an upper-level (about 8 km) heating maximum and is not in agreement with a distinct, single midlevel (about 4–5  km) heating maximum seen in the mean heating profile determined from a diagnostic budget study dur- ing the Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE)² (Nitta 1978; Thompson et al. 1979). The reasons for the disagree- ment between the CSH algorithm-reconstructed and the GCE-simulated LH profiles during the shallow convec- tive stage shown in Fig. 15a explain the disagreement between the CSH algorithm-retrieved and the diagnos- tically determined profiles over the Atlantic Ocean. Tao et al. (2000, their Fig. 16) showed that the CSH algorithm-retrieved heating profiles for the 19–26 De- cember 1992 period, computed using the convective and stratiform heating profile associated with the February convective period in TOGA COARE, results in more cooling at the lower levels and larger heating at the upper levels. They suggested that this is because the cooling below and heating aloft are greater in the strat- iform region for February than for December. For a preliminary evaluation of the SLH algorithm's sensitiv- ity to the selection of the lookup tables, we reconstructed the heating profiles for 19–26 December 1992 using the lookup table produced only from the February simu- lations (i.e., 9–17 February 1993 and 18–26 February 1993). Figure 17 shows the GCE-simulated and the SLH algorithm-reconstructed heating profiles. The 8-day-av- eraged SLH algorithm-reconstructed profiles are in good agreement with the GCE-simulated ones in total, con- vective, and stratiform regions, although the recon- structed convective heating profile is somewhat higher than the simulated one. These results indicate that the SLH algorithm is less sensitive to the selection of the lookup table than is the CSH algorithm. This is primarily because the lookup table of the SLH algorithm prepares multiple profiles of latent heat release for various PTHs, while the CSH algorithm depends on two averaged ver- tical profiles of latent heat release for convective and stratiform regions. Second, it is because the SLH al- gorithm uses not only P s , but also P m . However, it is necessary to examine the universality of the lookup table by simulations of other field experiments [e.g., GATE, the South China Sea Monsoon Experiment (SCSMEX), 3 and the Kwajalein Experiment (KWAJEX) 4 ] for global application of the SLH algorithm to the TRMM PR data.
5. Summary and future work
The spectral representation of rain profiles observed by the precipitation radar of the Tropical Rainfall Mea- suring Mission satellite provides details about convec- tive and stratiform rain characteristics over the equa- torial area (Takayabu 2002). Based on these spectral statistics of TRMM PR data, we introduce a new re- trieval algorithm (the spectral latent heating algorithm) for the PR data. Numerical simulations of cloud systems in the TOGA COARE IFA were conducted using observed large-scale advective temperature and moisture forcings. To con- struct the lookup tables, GCE-simulated precipitation profiles and corresponding heating profiles from the four subperiods with 9-day durations (10–18 December 1992, 27 December 1992–4 January 1993, 9–17 Feb- ruary 1993, and 18–26 February 1993) are used. Those from the one subperiod with an 8-day duration (19–26 December 1992) are set aside to serve as ''observed'' fields for the consistency check of the algorithm. The heating profiles are sorted according to the pre- cipitation-top height of convective and stratiform re- gions. Considering the sensitivity of the PR, we used a threshold of 0.3 mm h − 1 to determine the PTH. Prop- erties of the convective heating profiles show near- monotonic change with PTH, so that it is appropriate to use PTH as an index for tables for convective rain. Stratiform heating profiles consist of two groups. One group includes shallow stratiform heating profiles (PTH < 4.4 km), characterized by lower-level cooling. The other group consists of anvil heating profiles (PTH > 4.4 km), characterized by upper-level warming and low- er-level cooling. We cannot obtain the PTH accurately enough for the anvil regions because of the insensitivity of PR to the small ice-phase hydrometors. Therefore, we selected the precipitation rate at the melting level instead of PTH as a parameter for the lookup table for the anvil regions. For a consistency check of the SLH algorithm, we reconstructed latent heating profiles for 19–26 Decem- ber 1992 with the lookup tables. Patterns in temporal variations of simulated heating profiles and reconstruct- ed ones agree very well. Differences between the shal- low convective stage and the deep convective stage are reconstructed more realistically than with other algo- rithms currently used. This is because we prepare spec- tral vertical profiles of latent heat release for various PTHs. The heating profile in the decaying stage with no surface precipitation can also be retrieved. This comes from utilizing the precipitation intensity at the melting level for anvil rain. One might infer that the SLH algorithm (i.e., lookup tables) is tied to a specific CRM—the GCE model used in this study. Krueger and Lazarus (1999) showed the similarities in the bulk characteristics between the ob- servations and the results from the several CRMs that participated in the GCSS WG4 case-2 study. This is because the same observed forcing data (time varying), large-scale advective tendencies of potential tempera- ture, water vapor mixing ratio, and horizontal momen¬ tum were used as the main forcing as well as in this study. However, even if the same forcing data are used, we might find the differences among lookup tables from the CRMs. For example, the differences in ice micro- physics among the CRMs might affect the use of the 0.3 mm h − 1 threshold. Comparisons of lookup tables from the CRMs using the same forcing data might be interesting not only for the SLH algorithm, but also for the CRMs intercomparison. For global application of the SLH algorithm to TRMM PR data, it is necessary to examine the univer- sality or regionality of the lookup table. As suggested by Houze (1989), the vertical profile of heating for con- vection of a given depth may vary from location to location. For example, convective cells with enhanced liquid water processes (i.e., condensation) have latent heating concentrated below the freezing level, whereas convective cells with significant ice processes (i.e., rim- ing and/or depositional growth) should provide stronger latent heating above the freezing level. We are now extending our study to simulations of other field ex- periments (e.g., GATE, SCSMEX, and KWAJEX) in order to examine the universality of the lookup table and will report our results in the forthcoming Part II of this study. We will also perform simulations with At- mospheric Radiation Measurement Program (ARM) 5 data in order to produce lookup tables for precipitation over land. For a true test, the SLH algorithm will be applied to PR data and the results will be compared with heating profiles derived diagnostically from sounding data of TRMM field experiments (e.g., SCSMEX; Johnson and Ciesielski 2002). Comparisons of the SLH algorithm estimates from TOGA COARE shipborne radar data (Short et al. 1997), with heating profiles derived diag- nostically from sounding data during TOGA COARE (Ciesielski et al. 2003), may also be a good test but are left to future studies. It is also planned to compare the SLH algorithm retrieved latent heating profiles with those derived from other methods (Tao et al. 1993a; Olson et al. 1999; Yang and Smith 1999a,b) using TRMM data. Acknowledgments. The first author expresses special thanks to Chung-Lin Shie for his kind guidance on using the GCE model. He also appreciates discussions with Bill Olson, Song Yang, and Steve Lang during his short- term visit to NASA GSFC, partly supported by a re- search fellowship from the Japan Society for the Pro- motion of Science for Young Scientists. This study is supported by the JAXA/EORC Tropical Rainfall Mea- suring Mission (TRMM) project. Yukari N. Takayabu would like to express her heartfelt gratitude to the late Prof. Tsuyoshi Nitta for motivating her to develop the latent heating algorithm utilizing TRMM PR data. Au- thors W.-K. Tao and D. E. Johnson are mainly supported by the NASA headquarters Atmospheric Dynamics and Thermodynamics Program and the NASA TRMM. They thank Dr. R. Kakar at NASA headquarters for his sup- port. The authors thank three anonymous reviewers for their constructive comments that improved the clarity of the presentation in this paper. We also thank Paul Ciesielski (CSU) for providing us with the large-scale forcing datasets for the model simulations. The Grid Analysis and Display System (GrADS) package was utilized for the figures. Acknowledgments are also made to JAXA/EORC for computer time used in this research.