Choice of the DSD shape and coefficients

Of the different kinds of drop size distributions that could be used we tested two different shapes: the exponential type N(D) = N₀ exp(−ΛD) and the gamma type N(D) = N₀D^μ exp(−ΛD), where D (cm) is the drop diameter, N(D) is the number density of drops with diameters between D and D + dD, and N₀ (m⁻⁴), Λ (cm⁻¹), and μ are the shape parameters for the distribution. We will hereinafter consider the exponential type as a special case of gamma-type distribution when μ is equal to 0.

Over the past few years, an approach has been developed to compare the measured DSDs arising from different sources. This approach is known as the “normalized DSD” approach and was first proposed by Willis (1984). The form we are using here is described in detail in Dou et al. (1999a,b), and consists of rewriting the gamma-type equation as

View Expanded

where N^*₀ is the normalized N₀ and is equivalent to N₀ when μ is equal to 0. D₀ = (3.67 + μ)/Λ is the median volume diameter, and κ(D, μ) is a normalization correction factor,

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where Γ is the complete Gamma function. The DSD is then normalized with respect to the water content because introducing (3.2) into the definition of W leads to the following form, which is independent of μ:

W⁻⁴πρ_wN^*₀D⁴₀(3.3)

Here, ρ_w is the density of liquid water.

Using this normalized approach, it is possible to write the relationship between two integrated rainfall parameters X and Y as X = mN^*(1−n)₀Yⁿ, where n is independent of μ, and m is weakly dependent on μ. For instance, the fitted a coefficient of W = aN^*(1−b)₀Z^b varies from 3.68 × 10⁻⁶ for μ = 0 (exponential case) to 3.56 × 10⁻⁶ for μ = 4, and the b coefficient varies from 0.56 to 0.54, respectively. The coefficients given here are obtained following Ferreira and Amayenc (1999) and Ferreira et al. (2000, unpublished manuscript, hereinafter FAOT), where N^*₀ is computed by fitting the initial rain relations of 2A25 version 4 for temperatures between 273 and 293 K under the Mie assumption at 13.85 GHz. The inferred N^*₀ from 2A25 for convective rain is 17.0 × 10⁶ m⁻⁴, and the inferred N^*₀ for stratiform rain is 4.8 × 10⁶ m⁻⁴ (hereinafter these values will be referred as N^*_0orig). These values for N^*₀ are close to those found by Dou et al. (1999a), derived from airborne microphysics probe measurements made during the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE): N^*₀ equal to 19.2 × 10⁶ m⁻⁴ for convective rain and 3.9 × 10⁶ m⁻⁴ for stratiform rain, with μ varying from 0.35 to 4.05 as a function of the rain rate (and equal to 0.99 for 10 < R < 30 mm h⁻¹).

Database construction and profiles computation

The retrieval method is based on the Bayesian technique (or the estimated expected value method) described in Kummerow et al. (1996) and Olson et al. (1996). A database of cloud/precipitation profiles is generated from numerical cloud-resolving model simulations. In the current study, the database’s cloud and precipitation profiles are generated using simulations from two different models: the Goddard Cumulus Ensemble (Tao and Simpson 1993) and the University of Wisconsin Nonhydrostatic Modeling System (Panegrossi et al. 1998). Each of the four simulations utilized has different characteristics that represent different meteorological situations. The first two simulations represent an oceanic mesoscale convective system observed on 22 February 1993 (Jorgensen et al. 1994, 1995; Olson et al. 1996, among others) during TOGA COARE. The simulation is performed at 1-km (128 km × 128 km domain) and 3-km (384 km × 384 km domain) horizontal spacings, respectively. The third simulation, performed using the University of Wisconsin model, is a thunderstorm complex observed during the Cooperative Huntsville Meteorological Experiment that is performed at 1-km spacing on a 50 km × 50 km domain. The last simulation is of Hurricane Gilbert (1988), also performed using the University of Wisconsin model at 3.3-km spacing on a domain of 205 km × 205 km. These last two simulations are described in greater detail in Panegrossi et al. (1998). The profiles are extracted at different time intervals from the simulations to sample a greater variety of situations.

In these cloud-resolving model simulations, the cloud liquid and cloud ice particles are assumed to be monodisperse (20 μm); rain, snow, and graupel have exponential-type particle size distributions based on Marshall and Palmer (1948) (MP) for rain and Rutledge and Hobbs (1984) for snow and graupel. The density of liquid water is set to 1.0 g cm⁻³, and the densities of snow and graupel are set to 0.1 and 0.4 g cm⁻³, respectively.

Given a measured profile of water content retrieved from 2A25, the associated profile of atmospheric variables (vector P) is found in the database by

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where x is the vector representing the set of variables considered in the retrieval, x_k is the vector of model-simulated atmospheric variables, σ is the estimated uncertainty in the 2A25 water contents, and η_i is a weight depending upon the considered atmospheric layer. The η_i are set to 0 in the ice layer (i.e., above the assumed altitude of the 0°C isotherm), and they are a maximum below the melting layer. The η_i then decrease linearly with decreasing altitude to take into account the decreasing confidence in the attenuation-corrected reflectivity and also the lack of data at the edges of the swath near the surface. The W_Obsi and W_Simik are the water content observed and obtained from the simulations, respectively, in layer i. The summation over k pertaining to dbase (dbase is the database of model simulations) represents the summation over the large ensemble of model-simulated atmospheric profiles x_k. Here M is defined as M = Σ¹⁴_i=1 η²_i, and N is also a normalization factor defined as

View Expanded

The solution profile, expressed by (3.6), is a weighted average of all possible profiles in the model dbase: the better the agreement between W_Obsi and W_Simik, the greater the weighting of profile x_k. In the next section the quality of this adjustment is discussed.

Because the PR is able to provide information only where a rain signal exists, the problem of filling the nonprecipitating regions (i.e., below the PR detection threshold) has been solved using the TMI-observed T_B. Some description of the characteristics of the nonrainy regions is needed when taking into account the areally extensive TMI antenna patterns in the forward radiative transfer calculation. Using the TMI-observed T_B, we attempt to minimize the possible brightness temperature bias that the clear-air regions might introduce in the study. Thus, a small database of nonprecipitating atmospheric profiles with only cloud, water vapor, and temperature is generated. When a PR ray is detected as nonprecipitating (standard 2A25: rain/no-rain flag) the nonprecipitating profile that minimizes Σ (ΔT_B)² is selected from the nonprecipitating database. The ΔT_B represents the difference between observed and simulated brightness temperatures, and the sum Σ is made over all the frequencies.

As a standard product, the PR reflectivity is given at a range resolution of 250 m along each ray. To reduce the number of levels to the 14 levels corresponding to the forward radiative transfer model calculation, a vertical filter is applied based on the Cressman (1959) technique. This filtering process allows us to diminish possible errors resulting from uncertainties in the altitude of the radar range gates and to ensure the consistency of the vertical spatial resolution between the model-simulated profiles and the radiative transfer model forward calculation.

Characteristics of the 7 August 1998 scene

Figure 1 illustrates the characteristics of this scene. The domain extends approximately from 4° to 13°N and 162° to 173°W. This case presents an interesting structure extending from northeast to southwest directions, with some farther extent to the southwest to another precipitating region. The PR convective/stratiform classification shows mostly stratiform (4067 pixels) rain with spots of convection (938 pixels) embedded in the stratiform (Fig. 1b). The maximum integrated precipitating water is about 29.76 kg m⁻², attained in a convective region. The horizontal distribution of liquid cloud water (Fig. 1a) is well correlated with the horizontal distributions of rain and ice precipitation (Figs. 1c and 1d, respectively). The maximum precipitating water content in the first layer (0–0.5-km altitude) is 4.55 g m⁻³, corresponding to a convective pixel. The maximum of ice is 2.03 g m⁻³ in another convective pixel, just above the 0°C isotherm. It should be noted that there is no a priori reason for the maximum ice and the maximum rain to be collocated within the same profile or pixel.

It may also be noticed on Figs. 1b–d that convective pixels generally correspond to the highest precipitation. Nevertheless, local maxima of rain may correspond to stratiform regions. This result is consistent from the physics point of view but may also be an artifact from the methodology, because the differentiation between convective and stratiform profiles in the database (based on dynamics of the model) is different from the PR classification (based on both horizontal and vertical patterns of reflectivity). Nevertheless, the horizontal cross sections of the different species show good horizontal consistency despite the fact that each of the profiles has been retrieved independently from the others. Table 1 shows, for the two types of precipitation, the mean difference between the PR-calculated water content and the database-adjusted water content in grams per cubic meter (see section 3b). This table demonstrates that the observed W matches correctly the W of the profiles in the database. It is important to remember that we are using the PR-retrieved water contents to select database profiles that are actually used to set the ice-phase precipitation and other variables not directly measured by the PR. The differences between the two Ws are not negligible, but they are small enough to ensure the consistency of the ice and cloud profiles with the liquid precipitation profiles derived directly from the radar. We can also see from Table 1 that the stratiform precipitation profiles are better matched than the convective profiles because of their smoother shapes, which are better represented in the database. The convective profiles exhibit more irregular shapes with a higher variability and thus are not represented as well in the database. Table 1 gives also the distribution of the layers used in this study.

Brightness temperature simulations without antenna pattern effects

A retrieval of the rain profile structure is performed with the inferred values from 2A25, that is, N^*₀ = 17 × 10⁶ m⁻⁴ for convective rain and N^*₀ = 4.8 × 10⁶ m⁻⁴ for stratiform rain, using a gamma-type distribution with μ = 1. The brightness temperature forward calculation is performed using the corresponding Mie coefficients computed for the two different N^*₀. This sensitivity study allows us to understand the variations of the computed T_B for individual profiles. Figure 2 illustrates the differences of radiometric signatures between the two rain regimes. It can be seen that, depending on the frequency, the result varies. First, at 10 GHz the saturation regime (defined as the regime in which scattering balances emission, leading to a quasi-constant T_B as W increases) is not attained (Figs. 2a,b). Because of the difference in N^*₀, the convective rain profiles always yield lower brightness temperatures than do the stratiform rain profiles having the same total integrated water content. This result is from the poor absorption efficiency of the smaller drops for which the proportion increases dramatically as N^*₀ increases. Second, for intermediate frequencies such as 19, 21, and 37 GHz (Figs. 2d–g), similar behavior is observed as long as the total water content remains below the saturation value. The saturation value depends on the considered frequency and on the chosen N^*₀: approximately 7 kg m⁻² (10 kg m⁻²) at 19 GHz, 5 kg m⁻² (7 kg m⁻²) at 21 GHz, and 4 kg m⁻² (5 kg m⁻²) at 37 GHz for stratiform (convective) rain. On the other hand, above the saturation value, the brightness temperature tends to increase as N^*₀ increases. This result is due to the greater proportion of larger drops as the rain rate increases that compensate for the effect of the N^*₀ increase. The total absorption efficiency is higher, even with the contribution from scattering. Third, the highest frequency (Figs. 2h,i) is always saturated, and its behavior is similar to the saturated regime of the intermediate frequencies. However, this reason is not so obvious from Fig. 2, because the DSD is set to a constant value for the ice phase in this study.

Simulated TB for the original N^*₀

Figure 3 illustrates the brightness temperature simulations that the antenna pattern affects. The T_B computation is performed using an original N^*₀ equal to N^*_0orig (see section 3a), that is, a convective N^*₀ approximately 3.5 times the stratiform N^*₀. One may notice that the agreement is generally good between the observed and simulated brightness temperatures. The general spatial patterns are very well reproduced at each frequency; the main differences arise when considering the intensities of the computed T_B as compared with the observed. The first obvious feature is the lack of emission at 10 GHz-V (Figs. 3a,b). The background brightness temperature is correctly simulated, but the maximum in rain is 233.8 K in the observations and only 231.5 K in the simulation. At 10 GHz-H (not shown), the respective maxima are 200.6 K in the observations but only 196.0 K in the simulation. One can see that this difference is less important at 19 GHz-V (Figs. 3c,d) for which the regions of maximum emission are similar in shape. The observed maximum is 272.1 K, and the computed maximum is only 269.7 K (for 19 GHz-H, the maxima are respectively 268.5 and 266.4 K). Both simulated and observed 21-GHz-V results exhibit the same kinds of features. Their respective maxima are 274.8 K for the observation and 272.7 K for the simulation. Note that in the region of strong convection some saturation may occur (scan numbers 60–63; ray numbers 82–98). The comparison between the observed and computed 37-GHz results is more critical, mostly because of the lack of information regarding ice precipitation provided by the radar. The PR appears to be poorly sensitive to ice, which makes the restitution of the profiles above the 0°C isotherm somehow harder to validate. We also know that one of the weaknesses of our database of model-simulated profiles is the proper description of the ice-phase microphysics. This weakness is why, even if the main emission patterns are still reproduced, the 37-GHz-V field (Figs. 3g,h) exhibits strong evidence of scattering by ice in the simulated T_B that does not appear in the observed T_B. These effects are primarily confined to regions classified as convective by the radar. In the vertical polarization, the observed maximum is 274.5 K as compared with the computed maximum of 268.5 K. The observed maximum is 269.1 K and the computed maximum is 263.0 K at 37 GHz-H. Last, the 85-GHz T_B simulation is even more affected by the ice-scattering problem. It is clear that the simulated 3D domain has too much ice in its upper layers, leading to a general low bias in 85-GHz T_B for both polarizations. The minimum observed T_B at 85 GHz-V is 286.4 K, and the minimum observed T_B at 85 GHz-H is 284.3 K; the minimum computed T_Bs are 288.5 and 286.4 K, respectively, for V and H polarizations.

Simulation with different N^*₀

In this section, we examine the effect of the DSD on the simulated brightness temperatures, given a domain of water profiles derived from the radar. It should be emphasized that, even if we vary the N^*₀ in the forward radiative transfer calculation, the three-dimensional domains of hydrometeor water contents described above are unchanged. Thus, interpretation of the results should be always considered with respect to the T_Bs simulated and described in the previous section that were based on N^*_0orig.

Table 2 presents the average brightness temperatures obtained for the stratiform type of rain and for the different DSDs considered. The “MP” refers to an exponential-type distribution with N^*₀ = 8 × 10⁶ m⁻⁴ for both convective and stratiform pixels (for exponential distribution, the N^*₀ identifies with the classic N₀). The other distributions are the gamma type with μ equal to 1 and the N^*₀ eventually scaled but keeping the ratio between the N^*₀ convective and N^*₀ stratiform (e.g., 0.5N^*_0orig has an N^*₀ = 2.4 × 10⁶ m⁻⁴ for stratiform rain and 8.5 × 10⁶ m⁻⁴ for convective rain). The first thing to note from Table 2 is the decreasing sensitivity of T_B to changes in N^*₀ as the frequency increases. The 21-GHz case is special, because this channel is mostly sensitive to the water vapor and cloud water and thus is not affected by DSD changes. It is also clear that the average brightness temperature for the lower channels decreases as N^*₀ increases. These results are consistent with the test presented in section 4b. We can see how the variation of the number of the biggest drops, with high-absorption efficiencies, affects the so-called emissivity channels. The trend at 85 GHz is reverse, because this channel’s radiance is mostly due to ice scattering. It is important to note also that the MP case is very close to the N^*_0orig case. This suggests that the MP case could be a good approximation of the DSD at low spatial resolution (1ow-frequency channels) and for stratiform rain.

Table 3 is the same as Table 2, but for convective rain. The same general trends appear for convective rain and stratiform rain; nevertheless, it may be noted that the average T_Bs for convective rain (both simulated and observed) are smaller than the average T_Bs for stratiform rain. For this particular scene, the difference appears to be small, but tests performed on other scenes confirm this tendency. This result is consistent with the remark made in section 4b. On the other hand, the average observed 85-GHz T_Bs (both polarizations) are clearly lower in convective rain than in stratiform rain, which seems to be consistent with the idea of higher ice density or concentration in regions of strong updrafts. Greater scattering in the convective regions appears to be represented in the cloud-model simulations, because the simulated 85-GHz T_Bs exhibit the same feature as the observed T_Bs. It may be noted that, in a rain retrieval algorithm, the convective and stratiform rain might be difficult to distinguish because of the combined effects of both high spatial heterogeneity of convection and the possibly higher N^*₀ of the convective rain DSD.

Presented in Table 4 are the root-mean-square (rms) differences between the observed and simulated T_Bs; in Table 5 the variances of the rms differences are listed. The variances are on the same order of magnitude as the rms difference itself for each channel, which indicates the high variability of the quality of the simulation from one pixel to another. On the other hand, the rms difference is less than 8 K for all of the emissivity channels and about 15 K at 85 GHz. This range represents an error of less than 10% between the simulated and the observed T_B.

Another interesting feature is the apparent frequency dependence of the best fit between the observed and simulated T_Bs. At 10 GHz, it seems that the best fit is obtained using the MP DSD or the N^*_0orig. However, at the higher-frequency channels, a higher N^*₀ is required. One may note also that the best fit is obtained using 2N^*_0orig for the horizontal polarization of 19 and 37 GHz, but the best fit is obtained using 3N^*_0orig for the vertical polarization of 19, 21, and 37 GHz. This result might be an indication of biased surface modeling in the radiative transfer simulations of this study. The variances do not show this polarization-dependent feature, but it is clear that the smallest variances (except for the 10-GHz channel) are always obtained using the 2- or 3N^*_0orig case. This apparent need for different N^*₀ for different channels could also be due to the fact that different N^*₀ were assumed in 2A25 and the forward radiative transfer calculations. Future studies will investigate this particular point.