Spectral Retrieval of Latent Heating Profiles from TRMM PR Data. Part I: Development of a Model-Based Algorithm

CRM simulations

The CRM used in this study is the two-dimensional version of the Goddard Cumulus Ensemble (GCE) model, 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 presented 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). Subgrid-scale (turbulent) processes in the GCE model are parameterized using a scheme based on Klemp and Wilhelmson (1978) and Soong and Ogura (1980). The effects 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 vertical 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, moisture, 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 tendencies for potential temperature θ and specific humidity q are defined as

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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^′_p = C_pd(1 − q_obs) + C_pvq_obs is the moisture-weighted heat capacity at constant pressure, where C_pq and C_pv 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 horizontal momentum forcing terms are difficult to obtain from observations in the Tropics, the terms are instead replaced by a nudging term

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where v is the model domain–averaged horizontal velocity, 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 domain-averaged horizontal velocities to follow the observed values and thereby provides a simple means of controlling the cloud system dynamics by the large-scale momentum and shear. Cyclic lateral boundary conditions are incorporated to ensure that there is no additional 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 duration (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.

Latent heating

In diagnostic studies (e.g., Yanai et al. 1973; Yanai and Johnson 1993), it is customary to define the apparent heat source Q₁ of a large-scale system by averaging horizontally the thermodynamic equation as

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where π = (p/P₀₀)^R/C_p is the nondimensional pressure, P₀₀ is the reference pressure (1000 hPa), C_p is the specific heat of dry air at constant pressure, and R is the gas constant for dry air. The second term in the parentheses in Eq. (4) is the large-scale advection term given by Eq. (3). The Q₁ can be directly related to the contributions of cloud effects, which can be explicitly estimated by the cloud-resolving model:

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The overbars denote horizontal averages, the primes indicate deviations from the horizontal averages, ρ is the air density, Q_R is the cooling/heating rate associated with radiative processes, and D_θ is the subgrid-scale (smaller than the cloud scale) diffusion, which is usually small in comparison with other terms above the boundary layer (Soong and Tao 1980). The first term on the right-hand side of Eq. (5) is the vertical eddy heat flux convergence from upward and downward cloud motions. The term LH is the net latent heating due to the phase change of water:

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where L_υ, L_f, 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, evaporation, freezing of raindrops, melting of snow and graupel, deposition of ice particles, and sublimation of ice particles, respectively. The vertical integration of Q₁ − Q_R (hereinafter represented as Q_1R) from the surface to the tropopause z_t yields (Yanai et al. 1973)

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where P₀ and S₀ are the surface precipitation rate and surface sensible heat fluxes. The surface sensible heat fluxes are generally smaller than the surface precipitation 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_1R 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 statistics. In the CRM simulation, as well as in the real world, cloud systems develop and decay. Although instantaneous 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 example, 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 associated 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 represent an average heating profile for a certain rain profile. 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_1R) 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_1R for each precipitation profile. Although the PR footprint-scale (4 km) estimates of Q_1R are less meaningful because Q_1R is a large-scale variable, we still need to derive the value Q_1R in order to compare with Q₁ derived from the diagnostic studies for validation. On the other hand, for the footprint-scale estimates, LH is more meaningful than Q₁ − Q_R. Therefore, we will show the results mainly with LH in this paper.

Classification of rain types

Takayabu (2002) separated convective and stratiform precipitation based on the TRMM PR version-5 2A23 convective–stratiform separation algorithm, which classified shallow, isolated precipitation elements into the stratiform category. Recently, Schumacher and Houze (2003) suggested that because the shallow, isolated echoes represent warm-rain processes, they should be classified 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, isolated 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 partially melted precipitation particles that leads to a bright band of enhanced radar reflectivity. Thus, the GCE convective 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 cannot 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 stratiform 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 reclassified 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) precipitation 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. Precipitation rates of the GCE-simulated convective precipitation profiles are stronger than those of the PR-observed ones, which could be due to regional differences. 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 precipitation, 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 because 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.

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 corresponding 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 convective rain (Figs. 5a and 6a). The GCE-simulated precipitation profiles with a 0.3 mm h⁻¹ precipitation-top threshold and corresponding heating profiles are accumulated and averaged for each PTH with model grid intervals. Properties of convective heating profiles show near-monotonic changes with PTH, so that it is appropriate to use the PTH as an index for tables for convective rain. The levels of Q_1R 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_1R is not so large. This is because LH is a major contributer to Q_1R (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 upward 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 characterized 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 anvils 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 heating Q = (LH, Q_1R), from the melting level z_m to the tropopause z_t, may be related to the precipitation rate at the melting level P_m by

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where f is the fraction of the precipitation rate at the melting level P_m, which is carried over from the convective region. The increases of the upper-level heating amplitude with P_m 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 combination 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 convective system. In order to evaluate this advection effect, we calculated the following values for the convective and anvil tables (Figs. 5a and c):

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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 convective 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 growing 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 shallow convection are much larger than surface precipitation, which results in large R_conv. On the other hand, R_anvil decreases with P̃_m. The index of P̃_m may be interpreted as the distance from the convective region: large P̃_m indicates a small distance from the convective region, while small P̃_m indicates a large distance. Close to the convective region, because condensate carried over from the convective region is large, P̃_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 P̃_m and large R_anvil far from the convective region. The height of maximum heating decreases slightly with P̃_m in Figs. 5c and 6c, which may be a manifestation of ascending front-to-rear flow and consistent with the above interpretation.

Procedure of heating retrieval

In order to retrieve heating profiles, the SLH algorithm utilizes the observed information on precipitation type (convective or stratiform), PTH, P_s, and P_m. The stratiform region is first separated into a shallow stratiform 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 procedure 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

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For anvil regions with the melting level (PTH > 4.4 km), on the other hand, the heating profile corresponding to P_m is selected in the anvil heating-profile lookup table (Figs. 5c and 6c). The upper-level warming amplitude is then determined by

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It should be noted that the contribution by the horizontal transfer of condensate from the convective region to the anvil region (i.e., R_conv, R_anvil) is already included in the lookup tables. This is because Q̃(z) and Q̃_high(z) are normalized by P̃_s and P̃_m in Eqs. (11) and (12), 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_high(z) is estimated to be smaller than P_m 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_low as a function of the difference of the rainfall rate between the surface and the melting level P_m − P_s:

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The above procedure allows us to estimate the heating profiles in the anvil regions both with and without surface precipitation.

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 period using the simulated parameters (i.e., PTH, convective/stratiform characteristics, P_s, P_m) as inputs. The simulated data during 19–26 December 1992 are independent from other simulated data that are used for the construction of lookup tables. This period is characterized 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 profiles from GCE-simulated precipitation profiles are compared 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 simulated 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 heating profile. In particular, the SLH algorithm-reconstructed heating profiles captured the evolution of a quasi-2-day oscillation (Takayabu et al. 1996) during the period of 1800 UTC 23 December–1800 UTC 25 December 1992. During the period 1800 UTC 23 December–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 P̃_m.

Figures 9b and 9e show the time series of the simulated heating and reconstructed heating in the convective regions, respectively. The pattern in temporal variations of the reconstructed heating profile agrees very well with that of the simulated heating profile. The differences between the shallow convective stage and the deep convective stage are realistically reconstructed. This is a result of the utilization of observed information, 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 simulated heating and reconstructed heating in the stratiform regions (shallow stratiform plus anvil), respectively. Although 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_m) 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 profiles 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, although 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 reconstructed Q_1R (Fig. 10d) agree very well with the simulated 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 simulated nor reconstructed Q_1R 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).

Estimation errors

Based on the assumption that heating profiles correspond statistically to precipitation profiles or precipitation parameters (i.e., PTH, P_m), we construct lookup tables. However, instantaneous heating structures are very fluctuating. Here, we reconstructed the heating profiles grid by grid for the four 9-day subperiods that were used for the lookup tables, using the simulated parameters as inputs. Then, the differences between reconstructed heating profiles and simulated ones are examined 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 interpretation 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 convective heating and stratiform heating have their maximum, 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 regions, 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 convective heating above the melting level is a little stronger 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 synthetic observation (Fig. 12). Horizontal averaging reduces the root-mean-square (rms) error, and for averaging over an ∼50 km width the rms error is reduced to about 1 K² 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.

Comparison with the CSH algorithm

Tao et al. (2001) represented the first attempt at using TRMM rainfall information to estimate the latent heating structure over the global Tropics for February 1998. Three different latent heating algorithms—the HH algorithm, the CSH algorithm, and the GPROF heating algorithm—were used and their results were intercompared. The CSH algorithm-estimated heating profiles only showed one maximum heating level and were in better agreement in diagnostic studies than the HH algorithm- or GPROF algorithm-estimated heating profiles. 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, using the simulated parameters as inputs. Because the results 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.

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 convective and stratiform LH profiles during 19–26 December 1992 (Fig. 10a), normalized by the convective and stratiform rainfall, respectively, are used for the CSH algorithm to reconstruct LH profiles from simulated 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, especially 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 transition 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-reconstructed LH profiles is negligible, while the GCE-simulated 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 oscillation can be seen. The subperiods in Figs. 14a–d indicate shallow convective, developing, mature, and decaying stages of the quasi-2-day oscillation, respectively. Figures 15a–d show LH profiles for the subperiods of Figs. 14a–d, respectively.

During the shallow convective stage the cloud ensemble 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 reconstructed 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 convective heating profile in the 8-day-averaged profile (Fig. 10a), and, thus, cannot retrieve a shallow convective profile. Second, the ratio of total convective heating to surface precipitation (i.e., R̃_conv) for shallow convection 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-averaged profile, which is smaller than that of shallow convection, and the heating amplitude only varies because of the convective component of the surface precipitation rate (P_conv). These account for the underestimation of low-level heating.

During the developing stage, both the SLH and CSH algorithms reconstructed the LH profile in good agreement with that simulated by the GCE model (Fig. 15b). It is noted that both the SLH and CSH algorithms reconstructed somewhat stronger upper-level heating and weaker lower-level heating than those simulated by the GCE model because of the underestimation of convective heating and overestimation of anvil heating. This may be because the contribution to the anvil water budget 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 tables, 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 developing stage. This is due to lower-level cooling resulting from increasing anvil rainfall. The SLH algorithm-reconstructed LH profile with a maximum at 7 km is in good agreement with the GCE model simulation. In contrast, 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 relative 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_stra 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 because the SLH algorithm can compute heating profiles from the precipitation rate at the melting level in the stratiform region (P_m). It is noted that the melting-level cooling in this decaying stage is stronger in the reconstructed profile than in the simulated profile.

These results show that the utilization of PTH and P_m provides two distinct advantages for the SLH algorithm. First, the differences in heating profiles between the shallow convective stage and the deep convective stage can be realistically retrieved. Second, heating 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 different 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 reconstructed 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 between 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 during the Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE)² (Nitta 1978; Thompson et al. 1979). The reasons for the disagreement between the CSH algorithm-reconstructed and the GCE-simulated LH profiles during the shallow convective stage shown in Fig. 15a explain the disagreement between the CSH algorithm-retrieved and the diagnostically 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 December 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 stratiform region for February than for December. For a preliminary evaluation of the SLH algorithm's sensitivity 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 simulations (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-averaged SLH algorithm-reconstructed profiles are in good agreement with the GCE-simulated ones in total, convective, and stratiform regions, although the reconstructed 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 vertical profiles of latent heat release for convective and stratiform regions. Second, it is because the SLH algorithm 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.