Feasibility of Raindrop Size Distribution Parameter ... - J-Stage

Journal of the Meteorological Society of Japan, Vol. 87A, pp. 53–66, 2009 DOI:10.2151/jmsj.87A.53

Feasibility of Raindrop Size Distribution Parameter Estimation with TRMM Precipitation Radar Toshiaki KOZU Shimane University, Matsue, Japan

Toshio IGUCHI National Institute of Information and Communications Technology, Koganei, Japan

Takuji KUBOTA Japan Aerospace Exploration Agency (JAXA), Tsukuba, Japan

Naofumi YOSHIDA Remote Sensing Technology Center of Japan, Tsukuba, Japan

Shinta SETO Institute of Industrial Science, The University of Tokyo, Tokyo, Japan

John KWIATKOWSKI George Mason University, Fairfax, Virginia, USA and

Yukari N. TAKAYABU Center for Climate System Research, The University of Tokyo, Kashiwa, Japan (Manuscript received 30 June 2008, in final form 7 November 2008)

Abstract This paper studies the feasibility of estimating raindrop size distribution (DSD) parameters from Tropical Rainfall Measuring Mission (TRMM) Precipitation Radar (PR) measurements. A methodology is described for DSD estimation with PR, in which parameter “e ” or “a ” in the Z-R relation Z = aRb is used as a DSD parameter. The e parameter is an adjustment factor for a in the relation between the attenuation coefficient k and the effective radar reflectivity factor Ze (k = a Zeb ) that makes the attenuation correction stable by using the path-integrated attenuation estimated from the surface echo as a reference. e is also recognized as a path-averaged DSD parameter. Large (small) e corresponds to small (large) a, i.e., to small (large) median volume diameters (D0s) with the assumption of the gamma DSD model. e exhibits a clear diurnal variation over land suggesting that afternoon convection causes DSDs with large D0s. In contrast, there is no significant diurnal variation over the ocean. e also exhibits a clear negative correlations with the storm-top height deduced from the PR and with the lightning flash rate, both of which again suggest that deep convections over land produce large D0s. There are several error sources that may produce bias errors in the DSD estimates: non-uniform beam filling (NUBF) within the PR antenna beam, non-liquid hydrometeors aloft (such as hail), and variation in the Normalized Radar Cross Section (NRCS) under rain as compared with no-rain conditions. Preliminary evaluations are performed on these error sources, which generally cause negative errors in e (i.e., overestimation of raindrop size). Nevertheless, comparisons of PR- and disdrometer-estimated a (in Z=aRb) generally are in agreement at various locations over both land and oceanic sites. This result suggests the feasibility of PR estimation of DSD. It is concluded that,

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at the present stage, PR estimates of global DSD distribution should be considered to be “qualitative.” Nevertheless, it would be useful to study the tuning of spaceborne radar algorithms and climatological studies of cloud microphysics.

1. Introduction It is crucial for global mapping of rainfall with a spaceborne rain radar to use appropriate relations between the effective radar reflectivity factor (Z e) and the rain rate (R), and between the rain attenuation coefficient (k) and Z e, i.e., to know the information of raindrop size distribution (DSD). In the course of developing a TRMM Precipitation Radar (PR) algorithm to estimate rain rate profiles, we planned to estimate a DSD parameter from the PR observation itself, at least in part (Iguchi et al. 2000; Meneghini et al. 2000). It is obvious that the global map of the DSD or Z e -R relation is novel and would provide important information for the climatology of cloud microphysics and radar rainfall remote sensing. External DSD information is also necessary to provide the Z e -R and k-Z e relations for the PR when the DSD estimation by PR is not reliable, and to validate the PR-based DSD estimation results. Until now, there have been a number of DSD measurements over the world related to TRMM or spaceborne radar validation (Tokay and Short 1996; Yuter and Houze 1997; Tokay et al. 2001; Tokay et al. 2002; Kozu et al. 2006) . However, there has been little effort to directly validate the PR-based DSD information, which is essential to assure the validity of the PR 2A25 algorithm. Chandrasekar et al. (2005) compared the PR-estimated N0* (the intercept parameter of a normalized DSD) and the median volume diameter D 0 with those from groundbased polarimetric radar. However, the comparison was limited to two sites (in Florida and Brazil) and covered only two rain scenes. In this paper, we study the feasibility of using PR for estimating DSD parameters. Our interest is in whether or not the PR can reasonably estimate the DSD. The answer may be part of the validation of the 2A25 algorithm since it is based on this very capability. We mainly use DSD parameter “e ” (defined in the next section), or “a,” a coefficient in the relation between the radar reflectivity factor (Z) and R (Z-R relation) Z = aRb, Corresponding author: Toshiaki Kozu, 1060 Nishikawatsu, Matsue, Shimane 690-8504, Japan E-mail: [email protected] ©2009, Meteorological Society of Japan

as a DSD parameter for discussion. In Sections 2 and 3, we describe the basic principles of DSD estimation, and the general characteristics of the DSD parameter estimated. In Sections 4 and 5, we discuss possible er ror sources that can cause bias errors in the statistics of the PR-estimated DSD parameter, and provide the results of validation against a set of disdrometer-based DSD parameters. Based on these error considerations and validation results, we present global distributions of the DSD parameters in Section 6 to demonstrate the climatology of DSD.

2. Description of DSD parameter 2.1 Z-R and N0 -L relations as a DSD model For the 2A25 version 5 and 6 algorithms, we prepared two sets of DSD models: one model for stratiform and the other for convective. Each model is described as several members of Z-R relations. The nominal Z-R relations among the members are Z = 185R1.43 (convective) and Z = 300R1.38 (stratiform). These nominal relations were derived from several Z-R relations at various places, mainly tropical oceanic sites, measured in the past. The idea that a Z-R relation is a DSD model may seem somewhat strange. However, the Z-R relation is equivalent to relating two DSD parameters such as the amplitude and scaling parameters of the exponential distribution. To obtain other members of the Z-R relation, we change the coefficients 185 or 300 within ±4dB while keeping the exponent fixed. For details of the DSD model description, see Kozu et al. (2009). A flowchart of the DSD model and rain parameter relations is given in Fig. 1. By assuming a power law in the relation between raindrop terminal velocity v(D) and D (diameter), we can convert these Z-R relations to N0 -L relations, assuming a modified gamma DSD model N(D) with a fixed m (Kozu et al. 2009): N(D) = N0D m e-LD.

(1)

For the TRMM PR DSD model, we assume m = 3, considering the range of natural m variation (Kozu and Nakamura 1991; Tokay and Short 1996). Note that these Z-R relations can be recognized as the relation between the 6th and 3.67th moments and as two types of DSD parameter sets, a convective model and a stratiform model.

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Fig. 1. Outline flow of TRMM PR DSD estimation.

2.2 Estimation of DSD parameters e and a The set of N0 -L relations that can be derived from the Z-R relation set is used to calculate k-Z e and R-Z e relations for 13.8 GHz using the Mie theory for various raindrop temperatures and a bright-band model (Awaka et al. 1985). To apply the Hitschfeld-Bordan (H-B) method (Iguchi et al. 2000), the exponent b in the k = a Z eb relation is fixed throughout the profile and in all members of the DSD model set for each convective and stratiform model; b = 0.7923 (convective) and b = 0.7713 (stratiform). In other words, all regressions to obtain k-Z e relations are made with fixed b values. In this way, a DSD model (a Z-R relation in either the convective or stratiform model) is now converted to an a value in k = a Z eb relation. Note that the corres ponding R-Z e relation has variable exponents. Note also that Z in the DSD model is just a 6th moment of DSD and is different from Z e. By combining the surface-reference technique (SRT), which is the estimation of two-way path-in tegrated attenuation (PIA) by subtracting an “appar

ent” (attenuated) surface normalized radar crosssection (NRCS) from that of the non-rainy area, and the Hitschfeld-Bordan method (Iguchi et al. 2000), the k-Z e relation is adjusted so that a stable attenuation correction of the measured (attenuated) radar reflec tivity factor profile is obtained. This is equivalent to estimating the e parameter in the k = ea 0Z eb relation, where a 0 is the a value for the nominal k-Z e relation corresponding to the nominal Z-R relation. That is, the a coefficient used for the attenuation correction is ea 0, and e is just a non-dimensional factor. Once a 0 is fixed, however, e can be recognized as a parameter determining the k-Z e relation. The nominal a (a 0) values used for the calculation of e are averaged values for raindrop temperatures of 0°C, 10°C and 20°C, and are 0.0002836 for stratiform and 0.0004139 for convective models. Note that the units of k and Z e are dB km-1 and mm6m-3. As a result, the adjusting parameter e is directly related to a in the original Z-R relation as a member of the DSD model. In other words, estimating e is equivalent to estimating DSD. Note, however, that e (or a) is not directly converted to a single DSD but rather determines a line (on the logR-logZ plane) on which DSD should move. Consider a two-parameter DSD model (in this case, a modified gamma model with m fixed). A single DSD is determined by a point on a two-dimensional plane defined by the two axes of DSD parameters. Specifying e or a does not determine a “point” but determines a “line” in the DSD parameter plane. Nevertheless, this would be very beneficial because we can specify, for example, the Z-R relation, the most valuable information for quanti tative radar rainfall remote sensing. Figure 2a depicts the relation between e and a for TRMM PR convective and stratiform DSD models. An e larger (smaller) than unity corresponds to an a smaller (larger) than that of the nominal value. Specifically, e is approximated to the a coefficients with the following relations: Convective:

a = 1671.7x2 - 851.79x + 184.20,

(2) Stratiform:

a = 2473.8x2 - 1327.3x + 298.84, (3)

where x = log10 e. In the following, we discuss the properties of e or a for the convective model because DSD estimation in stratiform rain is not reliable owing to the low two-way path-integrated attenuation (PIA).

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2.3 Some comments on e and related DSD parameters As noted above, the actual coefficient used for the rain retrieval is not e but ea 0. Therefore, the relations between log10(ea 0) and a for convective and stratiform rains should be closer to one another. This result is seen in Fig. 2b. By fixing R, Z, or Z e, we can specify a single DSD. Relations connecting N0 and L with e and R are given in Appendix. Before discussing the properties of e , we should note the actual e parameter used in the 2A25 algorithm. In this algorithm, the e parameter directly obtained by comparing the H-B solution with the SRT is called e 0. Depending on the accuracy of the SRT, e 0 is de- weighted so that e takes a value between e 0 and unity. This is called e and used for attenuation correction in the 2A25 algorithm (Iguchi et al. 2000). In this paper, we use e because of its higher stability than e 0, in spite of the fact that it may be somewhat influenced by the

Fig. 3. Diurnal variation of e over land and the ocean. Data source: 20–35° north and south, June 1998 – February 2001, summer hemisphere (JJA or DJF) only. Solid curves with diamonds and “+” symbols indicate mean and mean±standard deviation, res pectively.

Fig. 2. Relations between (a) log10e and a in the Z-R relation, Z = aZ b, for the TRMM PR DSD model, and (b) between log10(ea 0) and a.

nominal value (e =1 corresponding to a 0). Using the gamma DSD model assumed for the 2A25 algorithm, an e larger (smaller) than unity means that a is smaller (larger) than the nominal value (a = 185 for convective and a = 300 for stratiform) and that the median volume diameters (D 0s) are smaller (larger) than the corresponding nominal values. Since L and D 0 are approximately related with L = ( m+3.67)/D 0 (Ulbrich 1983), large (small) D 0 corresponds to small (large) L for a fixed m . Qualitatively, we can also say that an e larger (smaller) than unity suggests that the DSDs are narrower (broader) than those derived from the nominal DSD models, i.e., the nominal Z-R relations.

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3. General properties of e In this paper, we mainly discuss the statistics of e and a converted with the relation given by Eq. (2). To obtain the statistics, we use TRMM 2A25 and 3A25 version 6 products (JAXA and NASA 2005). A quality check of e is performed in the statistical processing in the 3A25 algorithm. Actually, statistical processing is performed only when the difference in the PIA estimates from the PR 2A21 algorithm is judged to be reliable or marginally reliable and the PIA from SRT and that from the radar reflectivity profile are consistent with one another (JAXA and NASA 2005). 3.1 Diurnal variation Convective rainfall over land generally increases in the early afternoon (1200 to 1800 local time), while the diurnal variation of convective rain over the ocean is very small (Takayabu 2002). To examine the corresponding diurnal variation of DSD, we show the diurnal variation of e in the form of two-dimensional histograms for summer (JJA in 20°N-35°N and DJF in 20°S-35°S) from June 1998 to February 2001 over land and the ocean. To emphasize the effect of afternoon convection in summer, the period and location are limited to the summer time in sub-tropical to mid-latitude regions. As seen in Fig. 3, the e value over land clearly decreases by about 0.15 (the mean value changes from 0.85 to 0.7), corresponding to an increase in a in Z = aRb from 253 to 356. In contrast, there is little diurnal variation over the ocean. These results are in accordance with past findings (Joss et al. 1968; Ajayi and Owolabi 1987; Kozu et al. 2006) in that convective rain over land has larger a values. 3.2 Relation to storm-top height and lightning fre quency The result in Section 3.1 suggests that there should be some correlation between DSD and the storm-top height and lightning activity, since deep convection is generally associated with high storm-top height and active lightning. Using a TRMM 3A25 product (stormHtMean) and a TRMM Lightning Imaging Sen sor (LIS)-derived 10-degree grid lightning-flash rate, we obtain Figs. 4 and 5, describing the correlations be tween e and the storm-top height and between e and the lightning-flash rate. In Fig. 4, we can see a significant negative correlation between e and the storm-top height, especially when the storm-top height exceeds about 5 km. However, the correlation is not significant when the storm-top height is below 5 km. In Fig. 5, we

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can also see a clear negative correlation between e and the lightning-flash rate. Note that no lightning cases are omitted to make the figure clearly portray the relation between lightning and e. The results in Figs. 4 and 5 again suggest that deep convections having lightning activity, mainly occurring over land, are associated with large a and D 0 values. This is generally consistent with past findings in mid-latitude regions and in the tropics. However, it should be noted that this result may be partly due to the error sources, which will be discussed in the next section.

4. Error considerations Major error sources in the PR observation include fading error and radar calibration error. Fading error is related to both rain and surface measurements but is random in nature. Thus, it does not have much effect on the average values of the DSD parameters used for climatological studies. Radar calibration error causes a bias error in the effective radar reflectivity factor and in the NRCS measurements. As for PIA estimation from the SRT, however, calibration error is not a problem since the PIA is derived by subtracting two NRCSs (in decibels). Since the PR is expected to be well calibrated within 0.8 dB (3s) (Kozu et al. 2001), we neglect the calibration error in the consideration of the DSD estimation presented here. Other major error sources in DSD estimations from the PR are (1) the non-uniformity of rain within a radar beam (NUBF), (2) non-liquid hydrometeors aloft (such

Fig. 4. Correlation between storm-top height and e over the land and ocean. Data source: 3A25 1998-2000, 3 years, 5-deg. grid.

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Fig. 5. Correlation between lightning flash rate and e over the land and ocean. Data source: 3A25 and LIS data, 1998–2000, 3 years, 10-deg. grid.

as hail, graupel, or sleet), and (3) variation of NRCSs between no-rain and rainy conditions, which affects the basic assumptions underlying the surface reference method. (1) NUBF causes errors in the R-Z e relation, but they are relatively minor in comparison with the NUBF errors in PIA estimation based on NRCS measurements. The DSD will be well estimated if there is negligible NUBF and a small non-liquid hydrometeor concentration aloft, and if the SRT is accurate. When there is a nonnegligible NUBF, complicated errors occur (Iguchi et al. 2009). In general, PIA estimates from the SRT are underestimated in comparison with those from the effective radar reflectivity profile, although the effec tive radar reflectivity profile is modified depending on the 3-D structure of the NUBF. As a result, e tends to become smaller (i.e., the estimate of a in the Z = aRb relation becomes larger) than the “true” a that we obtain in a no-NUBF case. Version 5 of the 2A25 algorithm employs an NUBF correction scheme; however, “overcorrection” of the NUBF, leading to large surface rain rate, was sometimes found. In the version 6 algorithm, therefore, NUBF correction is not adopted. Thus, it would overestimate the a value to some extent due to the NUBF. An improved NUBF correction scheme will be employed for the version 7 algorithm, with tests already underway. According to one preliminary test, e values are underestimated by 0.1 to 0.15 in the version 6 products corresponding to a

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20 to 30% overestimation of a. Interestingly, the e error due to NUBF does not depend on whether it is over land or the ocean, according to the test. However, the NUBF correction method is still under development, so we can only say that NUBF will cause some underestimation of e (or overestimation of a). We have to wait for the development of the version 7 algorithm for a more quantitative evaluation of the NUBF effect. (2) The effect of non-liquid hydrometeors aloft has not been well quantified. This problem is equivalent to the fact that the hydrometeor and DSD models along the path above the freezing level are different from the real situation; in particular, the difference in the k-Z e relation is critical. In the version 5 and 6 2A25 models (Iguchi et al. 2000), a hydrometeor at the top of convective rain still contains one percent water. This may not be a valid assumption. In most cases, however, this should not cause too much of a problem since the radar reflectivity above the freezing level is generally much smaller than in the rain region and thus the attenuation due to non-liquid hydrometeors is not significant. In some cases, especially over land, hail may produce Z es greater than 50 dBZ (Witt et al. 1998), in which case the attenuation may not be negligible. If we were to erroneously overestimate the attenuation aloft, a smaller attenuation would be allocated in the rain region, leading to a smaller e (i.e., a larger estimate of a). A preliminary test allocating no attenuation above the freezing level suggests that this effect is statistically minor (the change in e is less than 0.04, or less than 8% error in a). There is a possibility, however, that there is a large bias error in some cases of strong hail storms. (3) The SRT assumes that the NRCS in a non-rainy area is the same (either the NRCS close to a rainy area spatially or that in the same area under no-rain conditions) as that in a rainy area (Meneghini et al. 2000). Strictly speaking, however, this is not true. Over land, soil moisture and surface roughness may change, causing variations in NRCS. Over the ocean, there may be a difference in wind fields between rainy and nonrain areas, and raindrop striking may cause differences in ocean surface roughness and thus in the NRCS. Seto and Iguchi (2007) made a detailed study of the error of SRT depending on the method used to obtain the reference value of NRCS for both over land and the ocean. Based on their results, we have conducted a follow-up analysis to evaluate the degree of DSD estimation error due to the SRT-based PIA estimation error. Although the number of samples available is relatively small, taking “NRCS under

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weak rain” (“weak” meaning that rain attenuation is negligible) as the reference for the SRT (called the WRR method, in Seto and Iguchi 2007) is considered to be the most reasonable NRCS reference over land, since it can account for the soil-wetness and roughness change due to rain. The difference between the two types of e values has been obtained; one from [the standard version 6 2A25 algorithm using the standard 2A21-derived PIA] (e s) and the other from [the standard version 6 2A25 algorithm using the WRR-derived PIA] (ew). The result indicates the interesting feature that (ew - e s) generally becomes slightly positive over land. Since the WRR would give a more accurate PIA (Seto and Iguchi 2007) than the standard 2A21 algorithm, to some extent, this result suggests that the standard product underestimates e (and overestimates a) in general. The e errors of 0.02 to 0.04 (corresponding to an error in a of 4 to 8%) occur mainly in arid regions such as the Sahel and the Arabian Peninsula, suggesting that the NRCS change due to soil moisture and roughness variations is more significant in arid regions. Over the ocean, the bias error has been shown to depend on the incidence angle, so statistically no significant bias error in e is anticipated. In summary, all three “bias” error sources mentioned above cause the underestimation of e. NUBF can cause errors both over land and the ocean; the amount of error is not well quantified but may reach 0.1 or so. In contrast, the errors related to non-liquid hydrometers and the SRT are anticipated to occur mainly over land. The magnitude of such errors, however, is expected to be relatively small, at least statistically. From these error considerations, we can say that the DSD estimates from PR should exhibit some overestimation in a in the Z-R relation, especially over arid regions, but at least qualitatively the PR estimation of DSD can provide information on DSD climatology (as a map of DSD parameter e or a).

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PR-derived DSD validation with disdrometer data

Although there are several error sources, TRMM PR is expected to provide a statistical overview of DSD globally. In general, event-scale validation (direct comparison) is difficult because of the various error sources involved in such comparisons. Therefore, we make a validation using the Joss-Waldvogel dis drometer (Joss and Waldvogel 1967) data in a statisti cal sense. The procedure is as follows. (1) We focus the comparison on seasonal variations of

DSD (specifically a in the Z-R relation). (2) To obtain many independent samples and to utilize existing disdrometer data, we neglect annual variations of DSD. (3) The seasonal variation of e, converted to a with Eq. (2), is obtained by using 0.5° grid 3A25 data around the disdrometer site, applying four-grid averaging (to make a 1°×1° box), and determining the eight-year average from 1998 to 2005. The number of independent e samples for the average is then 32. (4) Three-minute integrated disdrometer samples taken over a month are used to obtain a statistical relation between Z and R using a fixed exponent (= 1.43). Note that the year in which the disdrometer data were obtained depends on the location and does not necessarily match the PR observation period. Table 1 lists the periods of the disdrometer data used for the present analysis. If there are multiple years for the same month, those a values are averaged. To obtain “convective” Z-R coefficients only from disdrometer data, the following simple convective rain extraction scheme, which basically regards moderate-to-intense and variable rainfall as convective, is used. (i) Let Rd and Zd be the respective disdrometer-derived rain rate and radar reflectivity factor. (ii) Calculate Y = -(1/1.5)(log10Rd - log107) + log104000. _ Y, then the rain is judged “convective.” (iii) If log10Zd > (iv) If log10Zd < Y, calculate the magnitude of change Table 1. Periods of disdrometer data collection used for validation. Location a Gadanki

Period

May − Dec. National Atmos. Res. Lab. (NARL) / National 1999

b Kototabang

Aug. 2001 − July 2003

c Singapore

Jan. − Oct. 1998 (Jan. − May 1995) July 2003 − June 2004 May 1979 − July 1981 July 1992 − Feb. 1993 July 1999 − Nov. 1999

d Palau e Kashima f Kapingamarangi g Kwajalein

Provider

Inst. of Information and Comm. Tech. (NICT) Res. Inst. for Sustainable Humanosphere (RISH), Kyoto University Nanyang Tech. Univ. (NTU) Institute of Obs. Res. for Global Change (IORGC) NICT National Aeronautics and Space Admin. (NASA) National Oceanic and Atmos. Admin. (NOAA) / NASA

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rate of Rd. If this is equal to or greater than 1.2 mm h-1 min-1, then the rain is judged “convective.” Threshold (ii) is based on the Z-R relation Z = 216R1.5 calculated for the Marshall-Palmer DSD (Kozu 1991). The line in (ii) is perpendicular to Z = 216R1.5 in the logR-logZ plane and crosses the

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point (R, Z) = (7 mm h-1, 4000 mm6m-3 ). The above classification method has been visually verified using Singapore disdrometer data. (5) The coefficient a values in the Z-R relation derived from (3) and (4) are compared as a seasonal variation. Figures 6a through g compare PR-derived and

Fig. 6. Comparison of a (in Z = aRb ) derived from PR and from disdrometer data at several locations. PR-derived a is from 0.5-deg. grid data averaged to 1-deg. grid for 3 months, for 1998–2005. For the meaning of open triangles in (c), (e), and (g), refer to the text.

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disdrometer-derived a values at (a) Gadanki, India (79°E, 13°N), (b) Kototabang, West Sumatra (100°E, 0°S), (c) Singapore (104°E, 1°N), (d) Palau (134°E, 7°N), (e) Kashima (141°E, 36°N), (f) Kapingamarangi Atoll (155°E, 1°N), and (g) Kwajalein Atoll (167°E, 9°N). The agreement depends on location as described below. (a) At Gadanki, the seasonal variation of DSD is very distinctive. In the southwest monsoon season (MayOct.), the DSD is broad; in particular, the DSD in the dry season (May-June) is very broad due to high temperatures and intensive local convection (Reddy and Kozu 2003; Kozu et al. 2006). In contrast, the DSD in the northeast monsoon season (Nov.-Dec.) is very narrow. Due to this, a tends to decrease from May to December as clearly indicated in both the PRand disdrometer-derived a values. The noisy feature of the disdrometer-derived a values for May to July may be due to the small sample size. Since there is little rainfall at Gadanki between January and April, no dis drometer data is obtained in this season. (b) Kototabang is located just on the equator at a mountainous site in West Sumatra. There is a clear diurnal variation in rainfall due to local convection. Rain falls throughout the year with relatively mild rainy seasons in November-December and March-April (Kozu et al. 2006). Although the disdrometer-derived a exhibits somewhat noisy features, good agreement is generally obtained between the disdrometer- and PR-derived a values. (c) The climate of Singapore is hot and humid, and is classified as a mild Asian monsoon climate, with a prenortheast monsoon season from October to November, a northeast monsoon season from December to March, a pre-southwest monsoon season from April to May, and a southwest monsoon season from June to September. Seasonal and diurnal variations in rainfall are relatively mild, and there is rainfall throughout the year. From the PR-derived a, there seems to be a slight tendency whereby a is larger in the presouthwest monsoon season and smaller in the middle of the northeast monsoon season, but the disdrometer data using 1998 has greater a values for January to May than those from the PR. This discrepancy may be due to the intense El Niño that occurred between August 1997 and May 1998. Hamid et al. (2001) found that during the El Niño period, lightning was more active in the area around Indonesia, suggesting that convection is more “intense” and deep in an El Niño year. This fact may be related to the development of

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drop sizes with intense and deep convection-associated lightning. In fact, if we use the 1995 disdrometer data for comparison, the similarity between the disdrometer- and PR-derived a values becomes much closer, as indicated with open triangles in Fig. 6c. (d) The Palau disdrometer is located on Peleliu Island, which is an oceanic site at the southern edge of the Palau Islands that belongs to the Republic of Palau. The comparison of PR- and disdrometer-derived a values agree well, suggesting the validity of the PR estimation of DSD. (e) The Kashima disdrometer data were collected about 30 years ago. The data were mainly used for satellite-Earth communication link analyses. In Ka shima, a increases slightly in the winter season, as seen in the disdrometer-derived a values. However, this trend is not so clear in the PR-derived values. This may be due to the small number of samples near the Kashima area in winter. If we use 5-degree grid PR data, a seasonal variation similar to the disdrometer data is obtained (see open triangles). (f) Kapingamarangi Atoll disdrometer data were extensively analyzed by Tokay and Short (1996) and Atlas et al. (1999). The same dataset is used for this analysis. The PR- and disdrometer derived a values generally agree well. For September, however, there is significant discord between the two. This may be due to the small number of disdrometer samples for September (only 20 samples of 3-minute averaged convective DSDs). (g) There are two Joss-Waldvogel disdrometers at Kwajalein Atoll. We use the Legan site disdrometer (Gage et al. 2002). The PR- and disdrometer-derived a values do not agree well, with the disdrometerderived a values being systematically higher. The reason for this offset is not clear, but one possibility is that the disdrometer in this site experienced noisy conditions and small drop counts (less than about 0.8 mm) had to be cut-off to reduce the noise component (C. R. Williams, private communication 2004), causing inaccurate raindrop size distributions. Nevertheless, small seasonal variations from July to November are seen in both the disdrometer and PR results, which are consistent with one another. To assess the effect of the low-channel cut-off, we tried to make extrapolated DSDs assuming an exponential distribution in the small drop-size regions (less than 1 mm). For this extrapolation, we used the negative slope obtained from a linear regression of the logarithm of DSD versus diameter between 1 and 2 mm. We have calculated the a coefficient from the artificially modified DSDs,

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resulting in a values somewhat closer to the PR result, as indicated by the open triangles in the figure.

6. Climatology of DSD From the above discussion of PR and disdrometer comparisons, we can conclude that the PR- and dis drometer-derived a values are generally consistent with each other in terms of their seasonal variations and absolute values. This is an encouraging result

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justifying the validity of PR capability of DSD esti mation for convective rainfall, in spite of the fact that there are several unknown error sources as discussed in Section 4. We believe, therefore, that PR-derived DSD parameters (e or a) can represent the global distribution of DSD, at least qualitatively. From this viewpoint, in Figs. 7a–d, we provide three-month and 10-year (March 1998 to February 2008) averaged global a distributions to demonstrate

Fig. 7. Seasonal variation of global a distribution as a representation of DSD climatology from TRMM PR.

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the climatology of DSD within the TRMM coverage as a seasonal variation. We note that e is related to a as seen in Fig. 2a, and that e values of 0.8, 1.0, and 1.1 correspond to respective Z-R relations of Z = 282R1.43, Z = 185R1.43 and Z = 152R1.43. From Figs. 7a–d, we can recognize the following characteristics of DSD. (1) In general, a values over land are larger than over the ocean. This should be true; however, we have to wait for the next study on the error sources discussed above to quantify the degree of the land-ocean con trast of DSD. (2) In general, tropical oceanic rainfall has smaller a values, especially in the tropical central to east Pa cific, than other ocean regions. This characteristic was also suggested by Shige et al. (2008). (3) In the mid-latitude ocean, larger a values are observed in the winter season than in the summer season. This is probably related to the dominancy of the mid-latitude frontal systems that prevail in winter (Takayabu 2008). (4) In the Amazon, there appears to be some seasonal variation; in the latter half of the wet season, MAM, a values appear to be smaller, and in the dry season to pre-monsoon season, SON, they appear to become larger. This would also be related to easterly- and westerly regimes in the Amazon basin (Williams et al. 2002) and would reflect the difference in microphysics of these regimes depending on the season. A similar property of DSD was also observed in Sumatra in relation to the active and inactive phases of the Madden-Julian Oscillation (MJO) (Kozu et al. 2005).

7. Conclusion We have described the methodology of DSD estimation with PR, where the parameter “e ” or “a” in the Z-R relation, Z = aRb, is used as a DSD parameter that can be estimated by the PR. e exhibits a clear diurnal variation over land, suggesting that afternoon convections cause large raindrops. In contrast, there is no significant diurnal variation over the ocean. e also exhibits a high negative correlation with the stormtop height deduced from the PR and with the lightning flash rate derived from the LIS, both of which again suggest that deep convections over land produce large raindrops. There are, however, several error sources that may produce bias errors in the DSD estimates. These include non-uniform beam filling within the PR antenna beam (NUBF), non-liquid hydrometeors aloft (such as hail), and the variation in NRCS under rain compared with no-rain conditions. These errors have

to be studied in more detail, but some preliminary evaluations have been performed. The variation in NRCS due to soil moisture and roughness variations appears to cause errors in e and is more significant in arid regions, whereas over the ocean the bias errors depend on the incidence angle. Therefore, no sta tistically significant bias error in e is anticipated. These error sources generally cause negative bias er rors in e (i.e., overestimation of a). Nevertheless, com parisons of PR- and disdrometer-derived a (in Z=aRb) generally show good agreement at various places over both land and oceanic sites. This result suggests the validity of the PR estimation of DSD. In summary, we conclude that at the present stage, PR estimation of global DSD characteristics should be considered to be “qualitative.” Nevertheless, it should be useful to study the tuning of spaceborne radar algorithms and the regional and seasonal variation of cloud microphysics. Finally, global distributions of the DSD parameter a have been presented as seasonal variations showing some climatological features of DSD. The clear contrast between land and ocean DSDs, i.e., larger a values over land, may be due to the much more intense convection over land than the ocean, although the error sources discussed in Sec tion 4 should enhance the contrast to some extent. In particular, low sea-surface temperatures in the tropical east Pacific would cause shallow and weak convection, leading to smaller a values.

Acknowledgments Disdrometer data were provided by the National Atmospheric Research Laboratory (NARL), India, Drs. J. T. Ong (NTU, Singapore), C. R. Williams (NOAA, USA), K. K. Reddy (formerly IORGC, Japan), S. Mori (IORGC, Japan), H. Hashiguchi (RISH, Kyoto University), and A. Tokay (NASA/GSFC, U.S.A.). We thank Drs. K. Okamoto (Tottori Univ. of Env.), R. Meneghini (NASA/GSFC), N. Takahashi (NICT), S. Shimizu (JAXA), and T. Shimomai (Shimane University) for their valuable discussions and support in this research. We also thank Mr. A. Numata (RESTEC) for his support in data processing. This work has been partly supported by the TRMM project at JAXA, by CREST of JST (Japan Science and Technology Corporation) and by a Grant-in-Aid for Scientific Research funded by the Ministry of Education, Culture, Sports, Science and Technology (MEXT).

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and L-R relations as follows.

Appendix Relations between e and N0 and between e and L for the PR DSD model 1. Formulation of rain rate dependence of N0 and L From the DSD model, starting with a basic Z-R relation (actually the relation between the 6th and 3.67th moments of DSD, the M6 -M3.67 relation, with the assumption of the gamma DSD model with m = 3), it follows that the logarithms of N0 and L are related with a linear equation (Kozu et al. 2009). If we use cR M3.67 as an approximation of the rain rate, it can be shown that the logarithms of N0 and R and the logarithms of L and R are also related with linear equations. (Note that cR = 0.0006×3.78p, with an approximate terminal velocity expression, v(D) = 3.78D 0.67 (m s-1) with D in mm) (Atlas and Ulbrich 1977). In this appendix, we assume that the units of N0, L, M3.67, M6, and R are mm-4m-3, mm-1, mm3.67m-3, mm6m-3, and mm h-1, respectively. Note that M6 and M3.67 can be expressed as follows with the assumption of the gamma DSD model (Kozu and Nakamura 1991): lnM 3.67 = lnR − lncR = lnN0 + lnΓ (µ + 4.67 )− (µ + 4.67 ) lnΛ , (A1)

lnM 6 = lnN0 + lnΓ (µ + 7 ) − (µ + 7 ) lnΛ .

(A2)

From Eq. (A1), lnL (lnN0) can be expressed as a function of lnR and lnN0 (lnL) as follows:

ln Λ = − lnR + lncR + lnN0 + lnΓ (µ + 4.67 ) (µ + 4.67 ),

lnN0 = lnR − lncR − lnΓ (µ + 4.67 ) + (µ + 4.67 )ln Λ , (A1")

(A1')

In addition, we assume that a Z-R relation (the M6 -cR M3.67 relation) is given by the following natural logarithmic expression (note that Z = M6 ): lnM6 = lna +blnR.

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(A3)

By substituting lnM6 in Eq. (A3) with Eq. (A2), then using either Eq. (A1') or Eq. (A1"), and performing some rearrangement, we can express either lnN0 or lnL as a function of lnR. In summary, we can analytically express the N0 -R

lnN0 = A N + BN ×lnR,

(A4)

lnL = AL + BL ×lnR,

(A5)

where the coefficients in Eqs. (A4) and (A5) are given by AN =

µ + 4.67 ln Γ (µ + 7 ) − lna 2.33 −

BN =

AL =

BL =

µ+7 (lncR + lnΓ (µ + 4.67 )) , µ + 4.67

µ + 4.67 µ + 7 −b , µ + 4.67 2.33

(A6)

(A7)

1 ln Γ (µ + 7 ) − ln Γ (µ + 4.67 )- lna − lncR , 2.33 (A8)

1− b . 2.33

(A9)

Note that the R used here represents the rain rate approximated with M3.67, which is very close to the rain rate obtained by using the Gunn-Kinzer terminal velocity (Kozu 1991), and is not corrected for the terminal velocity difference caused by the air density decrease aloft.

2. Dependence of A N , BN , AL , and BL on e The correction factor e, i.e., the ratio of a to a 0 (a 0 indicating the a value for the nominal DSD model, i.e., the average of a values for 20°C, 10°C and 0°C raindrop temperatures) modifies the N0 -R and L-R relations. It has been empirically found that Eqs. (A10) and (A11) are very good approximations of the coefficients of the N0 -R and L-R relations. A N = aAN + bAN×log10e ,

(A10)

AL = aAL + bAL×log10e.

(A11)

By performing regression analyses between log10 e and A N and between log10 e and A L , the coefficients listed in Table A1 are obtained. Note that BN and BL are constant (not dependent on e ). Note also that this

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Table A1. Coefficients to obtain DSD parameters N0 and L from rain rate and e . The units of N0 and L are mm-4m-3 and mm-1, respectively. Rain type

Convective Stratiform

aAN

12.443

10.862

bAN

13.874

13.387

aAL

2.004

1.797

bAL

1.809

1.746

BN

-0.4155

-0.2509

BL

-0.1845

-0.1631

procedure is applicable only to the version 5 and 6 products of PR 2A25, and applies mainly to convective rains where the reliability of SRT-based path attenu ation is generally high.

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