1. Introduction

Estimation of raindrop size distribution (DSD) parameters is important in obtaining accurate rain-rate estimates from radar measurements. In theory, a multiparameter radar rainfall measurement such as a dual-polarization radar (Bringi and Chandrasekar 2001) or a dual-frequency radar (e.g., Meneghini et al. 1992) can provide the information about DSD, which in turn can be utilized for a better rain-rate estimate. In reality, however, it is generally difficult to extract such information for each radar resolution volume because of the difficulty in satisfying measurement accuracy requirements needed to make the DSD estimation.

The DSD is often approximated with a gamma model:

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where D is the drop diameter, {N₀, μ, Λ} or {N_T, μ, Λ} are parameters of the gamma model, and Γ() is the complete gamma function (Ulbrich 1983). Although N_T is the zeroth moment of the DSD, it has been treated also as a DSD parameter (Chandrasekar and Bringi 1987). The parameter μ is often fixed for simplicity and to make it possible to estimate DSD from a dual-parameter radar measurement.

It appears that two DSD parameters (e.g., N₀ and Λ with a fixed μ) are not fully independent from one resolution volume to another; the spatial variability of DSD parameters can be represented by a “two scale” model (Kozu and Nakamura 1991). The two-scale model is generally defined as a model in which one DSD parameter is allowed to vary rapidly (e.g., for each range bin or from moment to moment) and the other is constant over a certain space or time domain (e.g., one radar beam or one rain event). This model has been used for extracting DSD-related information from multiparameter measurements combining data having different spatial resolutions in the Tropical Rainfall Measuring Mission (TRMM) (Iguchi et al. 2000; Haddad et al. 1997b). In such a scheme, an appropriate two-scale DSD model is important to achieve reasonable results when the estimated DSD is used to obtain “final goal” rainfall parameters (in the above cases, the rainfall rate and the attenuation coefficient at each TRMM radar resolution volume).

We can study an appropriate two-scale model if we have actual DSD data, but such a situation is only available at a limited number of experimental sites. Therefore, we often have to rely on indirect DSD information such as the relation between radar reflectivity factor Z and rain-rate R (Z–R relation) obtained from simultaneous radar and rain gauge observations. Ulbrich (1983) attempted to relate N₀ with μ in the three-parameter gamma distribution model using a set of Z–R relations listed by Battan (1973) and instantaneous disdrometer-measured DSD data. However, the relationship thus obtained may not reflect the microphysics of precipitation but rather results from a correlation between N₀ and μ because of the functional form of the DSD model (Chandrasekar and Bringi 1987). Haddad et al. (1997a) proposed a unique parameterization of DSD in which rain rate is treated as a DSD parameter and the other two parameters are defined so that they are mutually uncorrelated with rain rate. A feature of their parameterization is to treat rain rate as one of the three DSD parameters. They also discussed spatial variability of the other two DSD parameters, suggesting that they are less variable over a several-kilometer range based on a Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) Particle Measuring Systems, Inc., probe dataset. Assuming that rain rate is only a variable DSD parameter over a several-kilometer domain is a type of the two-scale DSD model. However, their parameterization appears to be too specifically tuned to the TOGA COARE data, leading to a complicated representation of the DSD model.

In this paper, we propose a DSD model that is much simpler than Haddad et al. (1997a) but is a versatile method to derive a two-scale DSD model over a space (or time) domain in which certain statistical DSD properties exist. Such properties are estimated from statistical relations among measurable remote sensing or in situ data. The existence of a rain parameter relation, such as a Z–R relation, over a spatial or temporal domain is a representation of the two-scale DSD model, if we assume a DSD model described by two parameters. We note here that Z and R can also be recognized as DSD parameters. Therefore, the existence of a Z–R relation within the domain means that the DSD model can be described only with one parameter within the domain. Because the Z–R relation has been widely obtained and used for radar rainfall remote sensing, our approach could be used not only in specific locations but on wider occasions. The general estimation method is first discussed. An example that uses a Z–R relation to obtain a relation between gamma DSD model parameters is presented. The DSD parameters used are the same as those in Ulbrich (1983), but we will relate N₀ and Λ instead of relating N₀ and μ as Ulbrich (1983) did. The derived N₀–Λ relation essentially represents the DSD model that is fully consistent with the measured Z–R relation with a fixed μ assumption. This model is suitable for relating various rain parameters such as Z, effective radar reflectivity factor Z_e, R, and the attenuation coefficient k, as we will see later in this paper. As an application, we use this model to establish the rain parameter relations for the TRMM precipitation radar (PR) rain retrieval algorithm 2A25 (Iguchi et al. 2000). Iguchi et al. (2000) described the outline of the DSD model for 2A25 as a part of their algorithm description. In the last part of this paper, we focus our attention on the description of the DSD model for 2A25. A part of this paper is also recognized as a detailed follow-up of Iguchi et al. (2000) on the aspect of the DSD model.

2. Basic formulation of the problem

a. Measurable rainfall parameters and relation to DSD parameters

Although natural DSDs are highly variable, three-parameter models can approximate the natural DSDs well for the purpose of relating higher-order moments of the DSD (Kozu and Nakamura 1991). Two-parameter models are less flexible but still provide a good fit to the natural DSDs in limited applications (Kozu and Nakamura 1991). In this paper, we use the gamma DSD model given by Eq. (1) in which the shape parameter μ is fixed.

For the DSD modeling proposed in this paper, we assume that an integral rain parameter x_i measurable with remote sensing is proportional to the ith-order moment of DSD. Those include Z (6th moment), R (approximately 3.67th moment), and rain attenuation coefficients (approximately 3rd–4.5th moments depending on frequency) (Kozu 1991). Those quantities are assumed to be measured without any attenuation effects or to be already attenuation corrected. Let x_i be such a quantity with the multiplicative coefficient c_xi:

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where M_i is the ith-order moment of DSD. In a similar way, we assume that another integral rain parameter x_j is also measured:

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By using Eqs. (2) and (3), we recognize that the (x_i, x_j) pair can be converted to the pair (N₀, Λ). This is the basic concept of a dual-parameter radar measurement proposed by Ulbrich and Atlas (1978) and others. This type of “point” dual-parameter measurement has been extended to “areawide” dual-parameter measurements in which a certain rain parameter is assumed to be constant over the area. The two-scale DSD model is directly applicable to the dual-parameter measurements for such areawide dual-parameter measurements. Kozu and Nakamura (1991) assumed N₀ or N_T as constant over a certain time domain while Λ was allowed to vary. This is a simple case of the two-scale DSD model.

In this paper, we discuss a general approach to find a proper two-scale model in the case in which a statistical rain parameter relationship between x_i and x_j is obtained by some method such as regression and probability density matching. The existence of such an x_i– x_j relation can be recognized as the situation in which the variation of DSD can be parameterized with a single parameter for the domain in which the x_i–x_j relation holds. Note that this situation means a two-scale DSD model is applicable for the domain, even though the DSD parameter constant in the domain is not N₀ or Λ, but an x_i–x_j relation that can be converted to an N₀–Λ relation, as we will see later. For simplicity, we first convert the integral rain parameters x_i and x_j to their natural logarithms X_i and X_j:

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Equations (2) and (3) are then converted to the following forms:

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Equations (5) and (6) indicate that the measurable integral rain parameters X_i and X_j are related to the conventional DSD parameters lnN₀ and lnΛ with simple linear transformations. Therefore, any relation between X_i and X_j can be converted to an N₀–Λ relation. The N₀–Λ relation derived in this way can be interpreted as the one that represents the DSD variation consistent with the measured X_i–X_j relation. If we have another measurable rain parameter in addition to X_i and X_j, we could estimate the relations among {N₀, μ, Λ} or we could improve the accuracy in estimating the N₀–Λ relation, depending on the properties of the third rain parameter. In this paper, we stay with the case in which we have two measurable rain parameters.

b. Finding the rain parameter relation

The basic concept of the approach we propose here is to find the “best” X_i–X_j relation that can be converted to the corresponding N₀–Λ relation. A linear regression between X_i and X_j is commonly used for this purpose. However, linear regression has a disadvantage because of the asymmetric nature between X_i and X_j. For example, a regression line from R to Z is not necessarily the best one for estimating R from Z. A better choice to find the statistical relation between X_i and X_j, having similar statistical properties, could be obtained from principal component (PC) analysis, in which the first principal component is the new rain parameter representing the most variable rain parameter on the X_i–X_j plane. For example, if we use the normalized X_i and X_j (X_ni and X_nj, respectively),

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where and are mean values of X_i and X_j and σ_Xi and σ_Xj are their standard deviations, respectively, we obtain the relation

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that corresponds to the first principal component of X_ni and X_nj. By going back to the original variables x_i and x_j, we have the power law

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Using Eq. (8) or Eq. (9) with Eqs. (4)–(6), we can obtain a relation between N₀ and Λ. Note that the principal component analysis is not the only way to derive the linear relation between X_i and X_j. In many cases, rain parameter relations are obtained from regressions, and there may be cases in which we have to utilize them to derive the DSD model.

3. Use of the X_i –X_j relation

If we have a statistical relation that has the form of Eq. (8), then we can convert it to the corresponding N₀–Λ relation. From Eqs. (4), (5), (6), and (8),

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with

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Equations (10) and (11) represent the relationships between the conventional DSD parameters that are consistent with the integral rain parameter relations [Eqs. (8) and (9)]. We note here the reason why the assumption of an x_i–x_j relation is a two-scale DSD model. The existence of an x_i–x_j relation means that the DSD can be described by just one variable parameter, either x_i or x_j. In other words, the x_i–x_j relation itself is a “slowly variable” parameter and either x_i or x_j is a “rapidly changing” parameter. In a similar way, if we convert the DSD parameter to N₀ and Λ, we can say that the N₀–Λ relation, such as Eq. (10), is a slowly variable parameter and either N₀ or Λ is a rapidly changing parameter. A confusing point in this case is that, for example, if we assume that Eq. (10) is slowly variable, we have to fix two parameters, a and b, instead of just one. However, because the DSD is still described with only one variable parameter in this case, we treat this as one of the two-scale DSD models.

Using the above result, let us try to find the DSD model matched to a measured Z–R relation. Hereinafter, we use the following units for N(D), D, Z, R, N₀, and Λ: mm⁻¹ m⁻³, mm, mm⁶ m⁻³, mm h⁻¹, mm^−μ−1 m⁻³, and mm⁻¹, respectively. In the case of the Z–R relation (Z = αR^β), we note that i = 6, j = 3.67, c_xi = 1, and c_xj = 6π(3.778 × 10⁻⁴). Note also that we assume that the Atlas and Ulbrich (1977) terminal velocity formula

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applies, where the unit of υ_t is meters per second. Therefore, a and b are expressed as

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Ulbrich’s (1983) approach was that he used two integral rain parameters proportional to moments [Eqs. (2) and (3)] and eliminated Λ to obtain a power law between the two integral rain parameters [Eq. (9)]. As a result, α became a function of N₀ and μ and β became a function of μ only. His expression of β was equivalent to setting b = 0 in Eq. (13). Using these results, he directly related μ with the exponent β in the Z–R relation. However, such an assumption is not necessary and may not be valid in general because N₀ may be a function of rain rate or Λ. In our approach, we select a fixed μ, but if we have another rain parameter relationship or any other DSD-related information in the area of interest then we adjust μ based on such additional information, making the DSD model more flexible and reliable.

4. Example of DSD modeling with Z–R relation

As an example, we use DSD data obtained by a disdrometer of the Joss–Waldvogel type (Joss and Waldvogel 1967) in April and May of 2004 at Kototabang, West Sumatra, Indonesia (Fukao 2006). The disdrometer data were integrated for 3 min to reduce errors due to small sample size and were used to obtain Z–R relations. During this period, a clear transition from an inactive phase to an active phase of the Madden–Julian oscillation (MJO; Madden and Julian 1994) was observed. In response to this transition, the DSD shape and Z–R relations changed significantly (Kozu et al. 2005). Therefore, the data should be suitable to test the DSD modeling for two different Z–R relations. We show two cases for Z–R-relation-based N₀–Λ relations assuming μ = 3. Figure 1 shows scattergrams between dBR and dBZ for 14–17 April (MJO inactive phase) and 2–3 May (active phase) directly calculated from the disdrometer data, where DSDs in the former period were broader than in the latter period. The corresponding Z–R relations from the principal component analysis are Z = 346.65R^1.468 and Z = 101.9R^1.667, respectively, as shown by solid lines in Fig. 1. Using Eqs. (10) and (13), we can convert them to the corresponding N₀–Λ relations: N₀ = 295.9Λ^2.691 and N₀ = 142.0Λ^4.177, respectively. For the calculation of R we are using Eq. (12) to keep consistency with the moment approximation to rain rate.

These statistical N₀–Λ relations derived from the measured Z–R relations can be compared with instantaneous values of N₀ and Λ calculated from individual disdrometer samples using the following equation with the 3.67th and 6th moments of DSD:

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Figure 2 shows the comparison of instantaneous values of N₀ and Λ with the statistical N₀–Λ relations converted from the Z–R relations obtained with the principal component analysis. Note that the Z–R-relation-based statistical N₀–Λ relations fit the instantaneous values based on Eq. (14) well, although they do not necessarily match the least squares fit of instantaneous values of N₀ and Λ. Such a discrepancy may be a limitation of this DSD modeling method, but, as will be shown later, this does not cause much of a problem in the application to estimate other integral rain parameter relations, such as between k and Z_e and between R and Z_e.

The above result indicates that once we have a statistical relation between Z and R, we can translate it to that between N₀ and Λ with a fixed μ. Using the Z–R-relation-based two-scale model, we can obtain any rain parameter relations for the period of interest. Note that once the DSD model is established, we do not need to assume that an integral rain parameter is proportional to a moment of DSD, as we needed to in Eqs. (2) and (3). For example, Fig. 3 shows the scattergram between k (dB km⁻¹) and Z_e (mm⁶ m⁻³) for 13.8 GHz and 20°C. The solid lines represent the results calculated from the N₀–Λ relations (two-scale models) mentioned above using the Mie theory. Furthermore, the open squares are instantaneous values directly calculated from the disdrometer data also using the Mie theory. The results indicate that the k–Z_e relations from the two-scale model fit the instantaneous values well despite the slight discrepancy in N₀–Λ relations between the statistical relation and instantaneous values mentioned above.

A question we have here is about the effect of fixing μ on other rain parameter relations, such as the k–Z_e relations, since the N₀–Λ relation is sensitive to changes in μ. In the above model of 14–17 April 2004, for example, N₀–Λ relations for μ = 0, 3, 6, and 10 are N₀ = 5560.8Λ^−0.3068, N₀ = 295.9Λ^2.691, N₀ = 2.140Λ^5.691, and N₀ = 0.000 448Λ^9.691, respectively, given that Z = 346.65R^1.468. The effect of μ variation can be evaluated with integral rain parameter relations relevant to microwave propagation and meteorology, such as k–Z_e and W–Z_e relations, where W represents the liquid water content (g m⁻³).

Figure 4 shows the k–Z_e and W–Z_e relations for 14–17 April calculated from the N₀–Λ relations mentioned above using the Mie theory assuming μ = 0, 3, 6, and 10. It is found that the variation of these relations with changing μ is very small. The variations for 2–3 May have been found to be similar or smaller than those of the 14–17 April case.

We note that the natural variation of the μ value ranges between 0 and 20 (Kozu and Nakamura 1991; Tokay and Short 1996; Illingworth and Blackman 2002), with the mode around 4–6. However, we should also note the shortcomings of the Joss–Waldvogel disdrometer. It tends to underestimate the number of small diameter drops, especially in heavy rain, and cannot resolve large drop sizes of 5.0–5.5 mm (Tokay et al. 2002). Considering this fact, we believe that a μ value slightly smaller than those obtained from the past Joss–Waldvogel disdrometer data (i.e., μ = 3) would be a reasonable choice as a nominal value of μ. We also note that the change in the above-mentioned k–Z_e and W–Z_e relations assuming μ = 10 or larger is very close to the results for μ = 6. Therefore, the present evaluation should be sufficient in safely concluding that the effect of μ variation on the rain parameter relations is minor. For these reasons, we assume μ = 3 for all calculations of rain parameter relations for TRMM PR rain retrieval.

As an overall validation, we show the result using longer-period disdrometer data for which the correlation coefficient between dBR and dBZ is lower than the shorter events shown above. Figure 5a shows the Z–R relations (disdrometer-measured values and their principal component analysis fitting of Z = 192.8R^1.596). From the fitting line, we have obtained N₀–Λ relations for different μ values. Using those results, we calculate k–Z_e relations for 13.8 GHz and 20°C, assuming μ = 0 and 10, which are shown in Fig. 5b along with instantaneous values directly calculated from disdrometer data. As shown in this figure, the Z–R-relation-based DSD model fits the disdrometer data very well.

5. Application to TRMM PR rain retrieval algorithm

The above Z–R-relation-based DSD model is applied to versions 5 and 6 of the TRMM PR 2A25 rain retrieval algorithm. In this section, we describe a detailed procedure of making the DSD model for the 2A25 algorithm. The “default” or “standard” Z–R relations for the DSD model were derived for stratiform and convective rains, based on several Z–R relations obtained in oceanic and semioceanic tropics in the past (see Table 1). The “past” does not mean the PR-measured Z_e but Z–R relations obtained in past studies using ground-based radars and rain gauges or disdrometers. By utilizing those past study results (power-law Z–R relations), we try to obtain a DSD model applicable to the TRMM PR measurement that mainly aims at the measurement of tropical to subtropical rainfall. The term “semioceanic” is somewhat ambiguous. We treat it as meaning an observation site within about 100 km from the coastline, such as Florida; Darwin, Australia; and Ile-Ife, Nigeria (in Table 1).

The “global average” Z–R relation was obtained as follows:

1. Choose a set of Zs (Z₁, Z₂, … , Z_n between 14 and 50 dBZ) and calculate corresponding rain rates (R_1i, R_2i, … , R_ni), where i denotes the Z–R relation number, that is, a specific Z–R relation listed in Table 1.

2. Average all R_jis [i.e., calculate R_{j_av} = (R_j1 + R_j2 + … + R_jm)/m, where m is the number of relevant Z–R relations] for j = 1, 2, … , n. A set of pairs, (Z₁, R_{1_av}), (Z₂, R_{2_av}), … , (Z_n, R_{n_av}), is then obtained.

3. By making a principal component analysis on a log–log space, a linear relation between dBZ and dBR, which is converted to a power law Z = aR^b, is obtained.

We used Z–R relations from rainfall data in tropical oceanic and semioceanic climate considering that the main target of TRMM observation is tropical rain. Although there might be some systematic difference in DSD between land and ocean regions that was also suggested by TRMM PR observation (Iguchi et al. 2000), it was decided that introducing such a land/ocean separation in the DSD model was too premature in producing the “operational” standard products. The resulting global average Z–R relations are Z = 300R^1.38 (stratiform) and Z = 185R^1.43 (convective). Using these default Z–R relations for the 2A25 algorithm, we generated the corresponding N₀–Λ relations for μ = 3, N₀ = 3175Λ^1.54 (stratiform), and N₀ = 2724Λ^2.25 (convective), and a set of rain parameter relations, such as k–Z_e relations, that was summarized in Table 1 of Iguchi et al. (2000) as initial values.

6. Summary

We have proposed a method to derive a two-scale DSD model over a spatial or temporal domain in which statistical rain parameter relations exist. The two-scale DSD model is generally defined as a model in which one DSD parameter is allowed to vary rapidly (e.g., for each range gate or from moment to moment) and the other is constant over a certain space or time domain (e.g., one radar beam or one rain event). The existence of an integral rain parameter relation such as a Z–R relation over a spatial or temporal domain is considered to be an example of the two-scale DSD model. We described the procedure of employing a rain parameter relation derived from standard statistical data processing with an assumption of the gamma DSD model. The test results using Z–R relations obtained at Kototabang, West Sumatra, showed that the resulting two-scale DSD model expressed by the N₀–Λ relation varies depending on the assumption of μ but that higher-order rain parameter relations such as k–Z_e relations from those different models are very close to each other, indicating the stability of the model against the change of μ in the application to rain parameter relations. This result was applied to the DSD model for the TRMM PR 2A25 (versions 5 and 6) algorithm. A detailed procedure of the model construction was described.