A variety of major sources of rainrate retrieval errors in a conventional dual-wavelength radar technique (DWRT) as well as Ze-R method are analyzed based on simulations upon utilizing a disdrometer-measured raindrop size distribution (DSD) as well as vertical rain field structure (VRS) data collected by Tropical Rainfall Measuring Mission’s (TRMM) precipitation radar (PR). A spaceborne dual-wavelength radar geometry comprising 13.6 and 35 GHz operating frequencies are considered in the simulations. Through first fold of the simulation, we statistically examined the significance of the VRS effect in the DWRT, which is found to be negligible. Next, we attempted to gauge relative sensitivities of the DWRT and Ze14 (at 13.6 GHz)-R method to the natural fluctuation of DSD. Statistical error analyses suggest some distinct lower bounds of rainrate retrieval accuracies of the two estimates. For instance, if the minimum sensitivity of 35-GHz radar is equivalent to about 10 dBZ and the rainrate is about 10 mm h−1, DWRT shows ∼51% of improvement in the accuracy for 3-km range resolution, while it has ∼44% improvement for 1-km range resolution compared to the Ze14-R method. Finally, the effects of nonuniform rain field (NUR) and mismatching in the observed field-of-views (FOV) of the radars are analyzed. It is noticed that the NUR effect introduces small amount of enhancement in the errors comparing to that due to the effect of the DSD variation and/or the coupling of the Mie scattering effect, while mismatching in the FOV significantly enhances errors as well as biases in the DWRT estimates.
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 The Tropical Rainfall Measuring Mission (TRMM) satellite is now operating in non-Sun synchronous orbit enabling its onboard instruments to capture tropical and part of subtropical rainfall distributions. The advantageous position of the TRMM satellite and its sophisticated instruments, like precipitation radar (PR), enable the mission to successfully monitor rainfall distributions/structures in the tropical regions. The TRMM data satisfied its objectives to capture tropical surface rainfall distribution [Kummerow et al., 2000].
 However, in addition to the restrictions on the observations of precipitation or other phenomena due to midlatitude or high-latitude disturbances, the TRMM’s observation is limited by its lifetime. A TRMM follow-on mission is now under discussions in order to give continuity to TRMM’s observation with larger coverage. The objectives of the future mission are not only to extend/expand its observations but also to enhance TRMM’s capability by improving its inherent limitations in the accuracy of the measurements. The future mission is expected, for example, to provide additional microphysical information by discriminating storm into regions of solid, liquid and/or mixed hydrometeors [Meneghini et al., 1998; Nakamura et al., 1998]. For these purposes, a dual-wavelength radar is preferable. The dual-wavelength radar data will allow the use of algorithms that are less sensitive to fluctuations in the drop size distribution (DSD) and independent of the absolute calibration of the radars [Atlas and Ulbrich, 1977; Meneghini et al., 1998]. Among those algorithms, a conventional dual-wavelength radar technique (DWRT) is a simple one. The conventional DWRT is expected to improve rainrate retrieval accuracy by providing independent rainrate estimation from rain attenuation and/or to give better rain attenuation correction [Fujita, 1983; Meneghini and Kozu, 1990].
 Each rainrate retrieval algorithm has its own response to different error sources. The estimation of rainrate from a radar based on a conventional, so-called radar reflectivity (Z)-rainrate (R) method, often plagued by large uncertainties. The conventional Z-R method, for example, has large error due to DSD variation [Wexler and Atlas, 1963; Battan, 1973]. Unlike this, rainrate retrieval from the microwave attenuation by the dual-wavelength radar [Fujita, 1983] assures better accuracy, since rain attenuation has good correlation with rainrate and less sensitive to DSD variation [Eccles and Mueller, 1991; Atlas and Ulbrich, 1977]. Analyses of the errors in the dual-wavelength radar algorithm (namely, DWRT), however, become complicate since the DWRT utilizes two radar signatures at two different frequencies and at two range gates. The naturally fluctuated DSDs and their scattering and attenuation effects as well as their distributions along and/or across the radar beam play a primary role to infer radar integral parameters within radar beam heights. Any fluctuation in the radar integrals and/or their measurement uncertainties finally couple errors in the radar retrievals.
 In this work, we attempt to examine relative sensitivities of DWRT to the variety of error sources, as described previously, based on simulations. Taking into account of the fact that the majority of radar algorithms to estimate rainrate are primarily influenced by the DSD variability, this study primarily focus to gauge relative sensitivities of the DWRT as well as the Ze (namely, at 13.6-GHz; hereafter, referred as Ze14)-R algorithm to the DSD variability and try to establish lower bounds of their accuracies. For a definite evaluation of error statistics in the radar retrievals due to natural fluctuations of the DSD combined with the radar beam height, however, a complete data set with measured vertical profiles of raindrop spectra is necessary. At present, we are limited with such data set. We, therefore, attempt to determine roughly the error statistics in the DWRT as well as Ze14-R estimates based on simple simulation frameworks.
 Through the first fold of the simulation scheme, we attempt to generate and determine the significance of the vertical structure of rainstorm (VSR) in the DWRT upon statistically relating a disdrometer-measured DSD at the ground level with vertical profiles of radar reflectivity measured by TRMM PR. Section 2 outlines the TRMM PR measurements of the vertical profiles of radar reflectivity.
 Through another fold of the simulation, we made an attempt to evaluate the effect of natural variation of the DSD in the DWRT as well as Ze14-R estimates based on a simple (namely, slant-path) structure of rain field upon utilizing the disdrometer-measured DSD data. The simulation is tried to make as simple as possible so that we can analyze DSD variability induced errors in the retrievals. The simulation utilizes a large set of disdrometer-measured (for ∼819 days) DSD data. Section 2 outlines the disdrometer measurement and outputs radar integral parameters from the measured DSD. This fold of simulation disregards any effects of the VSR. Also, any evolutions or breakups of the raindrops and/or other plausible error sources are disregarded in the simulation. By interpreting the DSD-derived radar integrals measured in time space into height space, we generate a slanted (which is assumed to be uniform within a specified FOV of the radars), rain field structure, which evolve with time (namely, 1-min averaging) series of the DSD measurement. The simulation also addresses the issues of the effects of the raindrop temperatures as well as statistical fluctuations in the radar signals and noises. Also, an approach to evaluate the effect of the Mie scattering is considered in this simulation study.
 Lastly, we addressed the effect of the nonuniformity of rain in DWRT estimates upon simulating horizontally nonuniform rain fields within the radar FOVs by envisioning a running linear average of the measured radar reflectivity factors generated by a slanted uniform rain field structure scheme. The effect of the mismatched FOVs of the two radars in the horizontal plane is also analyzed upon generating mismatched FOVs by shifting one of the radar’s horizontal FOV-structures in time series.
Section 3 explains the simulation methods. Taking into account of the objectives and the allocated frequencies of TRMM follow-on mission’s radar, a dual-wavelength radar observation operating at the frequencies of 13.6 and 35 GHz in a downward looking geometry is considered in the simulations. Results of the simulations are presented and analyzed in section 4. Finally, section 5 summarizes and discusses the results.
2.1. Disdrometer Measurement and Radar Integrals from DSD Data
 Raindrop size distribution data measured by a disdrometer at Nagoya University (35.1°N, 137.0°E) for the years of 1998, 1999 and 2000 (∼819 days) are utilized in this study to extract vertical structures of radar integrals. Table 1 outlines the measurement period of the utilized DSD data. The DSD derived radar integral parameters used throughout this paper are defined as, Rdis (mm h−1) as true rainrate, Zdis (dB) as true radar reflectivity factor, Zedis (dB) as effective radar reflectivity factor computed based on Mie model and measured radar reflectivity factor, Zmdis (dB) as Zedis with rain attenuation.
Table 1. Measurement Period of DSD Data Utilized in the Simulationa
Here ✓, all used; *, partly used; x, not used.
 The DSD data used in this study were collected by RD-69 disdrometer developed by Joss and Waldvogel . The disdrometer has a sampling cross-sectional area of 50 cm2and sorts raindrops into 20 size intervals ranging from 0.035 to 0.52 cm.
 Integral rainfall parameters are determined as a function of the raindrop size distribution measured by the disdrometer (with the sensor surface area (F) of 50 cm2and integration time step (Δt) of 60 s) as follows:
where i is the disdrometer channel, Di (cm) the center diameter of the channel, X(Di) the drop count at the i-th channel, v(Di) (cm s−1) the fall velocity of raindrops following Gunn and Kinzer's  interpolation.
Figure 1 depicts the scattergram of the measured rainrate with radar reflectivity factor that obtained from the 3-year DSD data set. The solid line in the figure gives the regression for Zdis-Rdis (in dB). In the figure, we notice that there are cut off lines in the lower part. One is the lower end and another is right of the first one and is steeper. The former is because of the smallest channel. The latter one is found to be due to a dead time of sensor, that is, maximum count in a specified period exists. Also, considering the small sampling area of this instrument, Poisson random errors can largely affect the 1 min. integrated disdrometer data [e.g., Kozu, 1991; after Cornford, 1967]. Such possible random errors are tried to minimize by utilizing only the channel data having more or equal to 4 numbers of drops. However, our primary interest is to focus on the higher order moments such as Rdis and Zdis, these thresholds are thought to be sufficient, and utilized the data as they are.
 The integrals, Zedis and specific attenuation (kdis) (dB km−1) in rain at 13.6 and 35 GHz radio waves (hereafter, Zedis14, Zedis35 (Zes) and kdis14, kdis35 (ks), respectively) can be determined as
where m is the complex refractive index of water, which is a function of rain temperature (T) and radar frequency, σb is the radar backscattering cross section and σe the extinction cross section. The complex refractive indices for water are taken from Ray  at 0 as well as 20°C raindrop temperatures. Since we are concerned with downward looking dual-wavelength (which is nonpolarimetric) radar measurement, the shape of the raindrop is assumed to be spherical considering that the symmetry axes of falling raindrops are aligned along the vertical direction [after Kozu, 1991]. The computation of the radar backscattering cross section and the extinction cross section is based on the Mie model.
 The kdis-Rdis relations assessed from the Mie model at the raindrop temperature of 0oC for 13.6 and 35 GHz radar frequencies are fitted in Figures 2a and 2b, respectively. The solid lines in the figures give the best-fit regressions. As can be noticed in the figure, the kdis-Rdis relationship, namely at 35 GHz, is appeared to be almost linear being less dependent on DSD [Wexler and Atlas, 1963; Atlas and Ulbrich, 1977]. Unlike to the kdis-Rdis relationships, the Zedis-Rdis relations vary largely with the DSD variation. The variation depends on the frequency, the scattering properties and little on the temperature of the raindrops. Figures 2c and 2d give Zes, which incorporate the Mie backscattering effects at 0°C raindrop temperature, comparing with the true Zdis. In some regions, Zes are deviated from 1:1 reference line showing enhancement and/or reduction. In the following sections, we will discuss and address these effects upon simulating the vertical profiles of the DSD derived radar quantities.
2.2. Vertical Storm Structure Data From TRMM PR
 Sufficiently small field of view of 4.3 km at nadir angle, antenna scanning capability in the cross track geometry over ±17 degrees resulting 220 km swath width with a nominal altitude of 350 km and a vertical resolution of 250 meters are some of the observational features of the TRMM PR [Kummerow et al., 1998]. In the region of light precipitation, PR suffers from a sensitivity limitation (having equivalent minimum detectable rain of ∼18 dBZ with signal-to-noise ratio of 1), which is one of the limitations of the PR [Short, 1998]. The PR applies 13.8 GHz radio wave, which suffers from rain attenuation in strong rain and rain attenuation correction is required. The rain attenuation correction is done by iteration type Hitschfeld-Bordan correction [Hitschfeld and Bordan, 1954; Iguchi and Meneghini, 1994] or by referencing surface signature reduction due to rain attenuation [Meneghini and Nakamura, 1990]. This method is a modified one of the Z-R method, in which radar signature is converted to rainrate using a typical or empirical radar reflectivity (Z)-rainrate (R) relationship. One of the standard algorithms of the TRMM PR, so called 2A25 retrieves the vertical profiles of the radar reflectivity factor with rain attenuation correction [Iguchi et al., 2000]. In the first fold of our simulation scheme, we utilized the PR data on the vertical reflectivity profile, i.e., a product of 2A25, and it is assumed to be true.
 In order to determine quantitatively the significances of errors in the downward looking dual-wavelength radar rain estimates, however, we are limited by measured data on the vertical profiles of DSD. Also, currently we left with any downward looking dual-wavelength radar data that match the proposed frequency (13.6, 35 GHz) pair with sufficient validation (though some downward looking dual-wavelength radar observations, for example, by Japan’s Communications Research Laboratory’s dual-wavelength (10, 35 GHz) radar, are realized so far, we have not yet had plenty of data with sufficient validation). In order to partially deal with this problem and determine the relative significance of variety of error sources in the dual-wavelength radar estimates, vertical rain field structure is simulated and the same is outlined as below.
3.1. Statistical Scheme
 Throughout this simulation fold, we attempt to examine qualitatively the significance of vertical structure of rainstorm in the DWRT estimates. Taking into account the TRMM PR’s valuable data on the vertical structure of the storm observed from space platform, we preferred to utilize these core data set in our simulation. The expected vertical profiles of radar integral parameters valid for dual-wavelength (i.e., 13.6 and 35 GHz) radar observation is then generated “statistically” by coupling the DSD derived radar integrals with the PR-observed vertical reflectivity profiles. This “statistical scheme” basically assures consistency between radar reflectivity and DSD; and links them statistically by assuming that DSD aloft remains basically the same as that measured in the ground levels. The simulation flow is outlined as below:
 1. Obtain radar integral parameters (namely, Rdis, Zdis, Zes and ks for 13.6 and 35 GHz radar frequencies) through (1)–(5) from 3-year DSD data set.
 2. Sort the individual radar integral parameters into radar reflectivity (Zdis) channels of 0.2 dBZ width (as shown in Figure 3a). At this point, it is important to make a note that we slotted the integrals into Zdis classes so that they can be statistically related with the vertical profiles of PR-observed reflectivity (hereafter, Zpr(j), where j is the PR’s range resolution, which is equivalent to 250 m). The 0.2 dBZ is fixed taking into account of the availability of Zpr(j) reflectivity data. Taking into account of the PR’s sensitivity limitation, we utilized the Zdis data equal to or above 18 dBZ.
 3. Relate the slotted radar Zdis-dependence parameters in accordance to Zpr(j), which is also slotted into the channels having the same width, i.e., 0.2 dBZ. It is noted here that slotted Zdis values are not always coincide with its counterpart, that is, Zpr(j). Hence, uniformly distributed random numbers with 0 mean and standard deviation of 1 were introduced. The product of the random numbers with the slotted Zdis provides us a large distribution of the slotted Zdis (as shown in Figure 3a). Next, upon sampling the DSD derived maximum distributed slotted Zdis by the profiled Zpr(j), which pick ups the first nearest value from the distributions, we approximated vertical profiles of the Zdis [hereafter, Zdis(j)] and other Zdis-dependent integral parameters (hereafter, Rdis(j) and Ze14dis(j), Ze35dis(j), k14dis(j) and k35dis(j), respectively, at 13.6 and 35 GHz radar frequencies). As a result, the approximated radar integrals fluctuate with the profiles of Zpr(j) along the radar beam heights.
 It is worth mentioning here that the simulation was limited only in the liquid rain layer (i.e., below melting layer of the precipitation) taking into account of DSD measurement, which is valid for raindrop. To do so, first, the PR-detected bright band height was located. Then, the top range bin of the rainstorm at the third range bin (750 m) below the bottom of the bright band is determined which is approximated in the standard algorithm. Though, the criterion is deterministic, it gives us quite safe margin to limit the simulation almost in the rain layer.
 The profiles of Zedis14(j) and Zedis35(j) at 13.6 and 35 GHz radar frequencies, respectively, after including the downward path-accumulated attenuation (PAA) from the respective kdis(j)-profiles over the rainstorm provide us the profiles of measured radar reflectivity factors (Zmdis14 and Zmdis35). Figure 3b displays an example of the simulated Zmdis14 and Zmdis35 along with the attenuation corrected reflectivity observed by TRMM PR. The profiles of the simulated Zmdis(j) at 13.6 and 35 GHz radar frequencies provide the estimate of differential attenuation (DA), i.e., the difference of the differences of Zms at two frequencies at two range bins given as
where Zms depend on radar frequency (f), range (j) and rain attenuation. Δj is the path length between the range bins. The suffixes 1 and 2 denote 13.6 and 35 GHz radar frequencies, respectively.
 Rainrate over different range bins can be estimated (hereafter, DWRT estimate) using the differential attenuation-rainrate relationship in the form of
The coefficients adf and bdf are further defined from our own derived (assessed in the framework of Mie model at the raindrop temperature of 0°C by employing 3-year DSD data set) best fit coefficients of the power law relationship between specific attenuation (kdis) and rainrate (Rdis) for 13.6- and 35-GHz radars (as given in Table 2) in the form of
Table 2. Zdis-Rdis, Zedis-Rdis and kdis-Rdis Relationships Derived From Disdrometer-Measured DSD
f, GHz T, °C
kdis = aRdisb
Zdis = aRdisβ
Zedis = aRdisβ
 Needless to say that the radar signals from the space platform suffer from atmospheric attenuation, attenuation in dry snow as well as in melting layer of precipitation before encountering the rain layer. However, the attenuation due to dry air and clouds is almost negligible compared to the rain attenuation. Note that attenuation by supercooled cloud water or atmospheric water and oxygen, along the way of radar signals (especially at Ka-band), however, should be taken into account for longer ranges. Attenuation of radar signal in dry snow is also small (e.g., for 0.86 cm wavelength radar at −10°C with typical bulk density of 0.04 g cm−3 is about 0.03 dB km−1 for ∼2 mm h−1 rainrate) and can be ignored [Battan, 1973; Matrosov, 1998]. The specific attenuation in the melting layer of the precipitation is strong but the layer is thin. Also, at this point, we have to mention that the differences in the measured reflectivity factors at these radar frequencies couple non-Rayleigh backscattering effect with attenuation through solid or the melting precipitation particles. Only with the use of PR data, however, it is insufficient to address quantitatively these complex effects in the regions of dry or melting snow isolating from rain layer.
3.2. Slant-Path Scheme of Uniform Rain Field
 As it is mentioned earlier, the effects of the DSD variability and/or their vertical distribution along the radar beam height are the primary sources of errors in radar retrievals. For a definite evaluation of the error statistics in the radar retrievals linked with the natural fluctuation of the DSD in the radar beam height, at present, this study is limited with such data set. In this context, the effect of the natural variation of DSD in radar retrievals is supposed to be roughly estimated, if the DSD derived radar integrals are utilized to approximate their vertical structures. Hence, in this fold of the simulation framework, an attempt has been made here to determine a rough statistics of errors in the DWRT due to the fluctuation in the measured DSD with an approach to generate a simple (namely, “slanted”) structure of rain field by interpreting the DSD derived radar integrals measured in time space into height space. In this way, the generated vertical structure of the rain field evolves with the time (namely, 1-min averaging) series of the measured DSD aligning the generated rain field structures as in “slanted” manner along the radar beam height. The generated rain field structure is assumed to be “uniform” and “scanned instantaneously” by the radars with ideal beams. By assuming a typical fall velocity of raindrops of ∼8 m/s, assuming that the 1-min-averaged DSD data approximately correspond to 500 m height (hereafter, range bin). A rain depth of 5–km (corresponding to ten range bin numbers) is assumed to be observed by the radars in a downward looking configuration with matched FOVs.
Figure 4a explains the concept of the simulation. In this figure the values i correspond to the number of 1-min-averaged DSD data, whereas j corresponds to the range bin numbers. The effective radar reflectivity factors (Zedis14[i + j] and Zedis35[i + j]), the specific attenuation (kdis14[i + j] and kdis35[i + j]), the disdrometer-measured true rainrate (Rdis[i + j]) and the true Zdis[i + j], hence are obtained at each range bin. The kdis14[i + j] and kdis35[i + j] are then accumulated separately to calculate total downward path-accumulated attenuation, PAA (accAdis14[j] and accAdis35[j]) through the rainstorm. In the simulation, the bottom and top of the rainstorm are assumed to be the center of the 0th and 10th range bins. The measured radar reflectivity factors (Zmdis[j]) at each range bin for 13.6- and 35-GHz radars (hereafter, Zms[j]) are then obtained after including path-accumulated attenuations to the corresponding Zes [i + j]. The effect of rain attenuation increases as increase the rain depth, rainfall intensity as well as the operating frequency. Figure 5 evidences these facts comparing the simulated true Zdis with Zms (i.e., Zedis with PAA), for example, at the rain bottom for 13.6- (Figure 5a) and 35- (Figure 5b) GHz radars. As depicted in the figures, the 35-GHz radar suffers largely from the path accumulated rain attenuation than the 13.6-GHz radar. The profiles of the Zms, which provide us the differential attenuation, hence the DWRT estimates through (5) to (8).
 Another consideration of this part of the simulation is to examine the effect of the statistical fluctuation in the radar signal and noises. Needless to say that radar echo suffers from so-called intrinsic radar signal fluctuation. Hence, averaging over many returns is required to obtain the rain echo intensity [Marshall and Hitschfeld, 1953]. In the case of spaceborne dual-wavelength radar retrievals, the inconsistency in the numbers of independent samples, which are possibly different for 13.6- and 35-GHz radars due to differences in dwelling time over different swath widths, may serve as additional error sources.
 In the case of TRMM PR, a frequency agility technique is applied to obtain 64 independent samples using 32 pairs of pulses of 1.6 μs differing with a frequency by 6 MHz with a fixed pulse repetition frequency (PRF) of 2776 Hz [Kummerow et al., 1998]. Thirty-two pairs of pulses are transmitted for each of 49 angle bins with an angle bin interval of 0.7° over the 215 km PR swath. Assuming the swath of 40 km for the future spaceborne 35-GHz radar which means it will have longer dwelling time for each pixel, the number of independent samples can be taken as 350. For data processing simplicity the numbers could be 256.
 Also, the actual radar signal is always contaminated by the external as well as internal receiver noises. The independent measurement of the system noise level in TRMM PR is performed for each angle bin in order to estimate the echo intensity from the total measured power, resulting 256 independent samples for system noise. By assuming the same multiplication, the independent samples of averaging for the noise level measurement from the 35-GHz radar could be 1024.
 Taking into account of the different number of independent samples, also, we attempt to examine their random error. The numbers of 64 (256) and 256 (1024) samples of the 13.6- and 35-GHz radar signal (noise), respectively, are simulated using exponentially distributed random numbers having mean values of Zes and noise power. The equivalent mean noise power of 18 and 14 dBZ (in one pulse base in order to express better sensitivity) are considered for the 13.6- and 35-GHz radar, respectively. In the computation, a noise correction procedure is applied by linearly subtracting the averaged fluctuated noise from the averaged fluctuated signal with noises.
3.3. Slant-Path Structure of Nonuniform Rain Field
 In the previous simulation of the slanted uniform rain field structure, one of the basic assumptions was that each 1-min-averaged DSD data are assumed to recover uniform profiles of the radar quantities both along (slanted-path) and across (horizontally) the hypothetical antenna beam-patterns giving point-structure of the instantaneous FOV (IFOV). In other words, implicit in the above simulation is the uniformity of rain across the antenna beams. However, the uniformity of rainrate across the radar antenna beam is not always guaranteed and the retrieval of averaged rainrate across the beam is desired [Nakamura, 1991; Amayenc et al., 1993; Durden et al., 1998; Kozu and Iguchi, 1999]. The assumption of horizontally uniform rain holds no longer, if one considers a wider beam illumination, and the FOV fills various rain fields that vary with time series. Simply envisioning a running linear average of the measured radar reflectivity factors from several point-IFOV-structures could provide the nonuniform rain field situation. In this simulation part, we averaged the point-IFOV-scale structure of the Zms in the previous simulation that varies along with time series. The averaging is quite similar to that of the “block averaging of scanned sub-IFOVs-structures” introduced by Jameson and Kostinski  in their work to investigate non-Rayleigh signal statistics caused by relative motion of the scatterers. Figure 4b gives the conceptual illustration of a wider beam, which linearly averages four variables corresponding to, for example, four point-IFOV-structures (averaged of the 1st IFOV to 4th IFOV) for first observation. Hence, we construct the first nonuniform-FOV, which essentially fills with horizontally nonuniform Zms. In the same way, we applied a recursive-averaging scheme for the 3-year data for both radars.
3.4. Spatially Mismatched-FOV-Structure
 In the above simulations, one of the assumptions is that both radars illuminate identically the same rain volume (namely, matched-FOVs) within a storm by hypothetical beam patterns. Indeed, these conditions may not always be guaranteed for actual dual-wavelength radar observations from the moving platforms. If there exists mismatching in the scanned FOVs of the two radars, the accuracy of DWRT will be affected seriously [Rinehart and Tuttle, 1982].
 In this simulation, we roughly examined the effect of mismatched FOVs in the DWRT by manually arranging the sampling volume of the radars so that they will be matched partially in the horizontal plane. To do so, first it is assumed that the radars scan nonuniform rain field within the horizontal plane of the FOVs, which averages four point-IFOV-structures (as of Figure 4b). Then, a forward recursion is applied to scan the spatially distributed averaged-FOVs from the 3-year data. Figure 4c explains the concept of the simulation. While both radars follow the same recursion sequence, i.e., without leading or lagging in point-IFOV-structure, we considered the situation as 0%-mismatched FOVs. This condition is equivalent to the slant-path structure of nonuniform rain field. Next mismatching in the averaged-FOVs of the two radars is introduced by allowing one of the radar (e.g., 35 GHz) to lead or lag in the point-IFOV-structure. For example, in Figure 4c, the first averaged FOV of 13.6-GHz radar contains first four IFOVs-structures (i.e., 1st, 2nd, 3rd and 4th), while the 35-GHz radar partially views that rain volume by leading in one IFOV-structure (i.e., the average of the 2nd, 3rd, 4th and 5th IFOV-structures). This situation is assumed to be the equivalent of 25%-mismatching. In the same manner, by leading in two IFOV-structures by the 35-GHz radar gives 50%-mismatching, while leading in three IFOV-structures results in 75%-mismatching. The above procedure is recursively applied with the whole DSD data set. The simulation with such arrangement of the observed rain fields of the two radars, translates the effects of mismatched-FOVs incorporating the effects of the NUR in the DWRT retrievals.
4. Result Analysis and Discussions
4.1. Effects of the Vertical Storm Structure, the DSD Variability, the Variation in Raindrop Temperatures, and the Statistical Fluctuations in Radar Signals and Noises
 The first attempt of this analysis is set mainly to examine significances of the vertical structure of the rainstorm and the effect of natural variability of DSD in DWRT retrievals. Figure 6 reveals the DWRT estimated rainrates that assessed through the statistical and slanted structure of uniform rain field schemes, for example, over 3-km range resolution. The results are based on 3-year DSD data and a large set of vertical reflectivity profile data collected by the PR. The estimated rainrates are compared with the DSD derived ‘true’ rainrate upon averaging over the respective range resolution (hereafter, path-averaged true rainrate). Figures 6a and 6b, for example, give the scatterplots of true versus DWRT retrieved rainrates that assessed through the statistical scheme and slanted structure of uniform rain field, respectively. As noticed, the two estimates reveal almost similar behavior. Both scatterplots show that the estimated rainrates have fairly good correlations with true one. For a precise comparison, below we examine the error statistics of these two estimates in terms of means and standard deviations. The means and standard deviation are calculated as follows: (1) specified true rainrate channels with a width (namely, 1 mm h−1), (2) took mean of the estimated rainrates in each channel, and (3) calculated the standard deviation of the estimates in each channel of the true rainrate.
 The standard deviations (shown in Figure 6c) of the estimates assessed through the statistical scheme in most of the rainrate regions appear slightly larger than that from the slanted structure. However, in some regions, the statistical scheme exhibits less standard deviation. The differences in the values of the standard deviations of the two estimates, however, are considerably small. This scenario manifests the fact that the effect of the vertical structure of the storm in the DWRT appeared to be negligible compared to the effect of the DSD variability.
 In addition to the above analyses, we approach to examine maximum allowable errors in the DWRT retrievals while to neglect the effect of DSD variability and the vertical structure of the rainstorm. In order to assess this effect, we assume an extremely simplified (namely, a slab-like) uniform rain field structure having a depth of 5 km. Then, the rain slab is assumed to be scanned by 13.6- and 35-GHz radar. The antenna gain is assumed to be uniform within the footprint. The assumed vertical structure of the slab-like rainstorm along the radar beam heights is same as that of Figure 4a except the radar beam fills uniformly with radar integrals that is, effective reflectivity factor (Zeslab), attenuation coefficient (kslab), which are the derived one from Ze-R and k-R relationships (given in Table 2, from Mie model, with explicit assumption of mean rainrate of 0.1–30 mm h−1).
 Then, the random error in reflectivity measurements is considered at low signal-to-noise ratios. The noise-equivalent effective reflectivity [Zeslab(N)] of 18 and 14 dBZ (for a single pulse) for 13.6- and 35-GHz radar are taken into account, respectively. Then, with the given Zeslab(N) that averaged over S(f) independent pulses, standard error in the reflectivity measurement is approximated [e.g., Hogan and Illingworth, 1998] at the top bin of the rainstorm (jtop) as
where S(f) is selected typically as 64 and 256, respectively, for 13.6- and 35-GHz radars. In the case of the bottom range bin ((jbot) of the rain slab, in addition to the above effect, some possible fluctuations are considered in the radar signals and noises. The fluctuations in the Zeslab and Zeslab(N) at the bottom bin of the rain slab are incorporated by introducing negative exponentially distributed random numbers. As outlined previously, averaging over the numbers of 64 (256) and 256 (1024) independent samples for Zeslab (Zeslab(N)), respectively, for the 13.6- and 35-GHz radar, we generated the fluctuated- Zeslab and Zeslab(N), fZeslab and fZeslab(N).
 Then, the standard error of a reflectivity measurement with a given fZeslab(N) averaged over S(f) (64 and 256, respectively, for 13.6- and 35-GHz radar) at the bottom bin of the rainstorm (jbot) is analytically approximated as
 Then the standard error in the DA-measurement (ΔDAslab) can be expressed analytically from (5) as
where the subscripts 1 and 2 represent 13.6 and 35 GHz radar frequencies, respectively. Finally, by employing the DA-Rdis relationship (as given in equation (6)), the standard error in the DWRT retrieved rainrate is approximated.
Figures 6a and 6b include the calculated standard error bar (i.e., ± standard deviations fluctuating around the mean values) of the DWRT estimate over 3-km range through this extremely simplified model. The calculated amount of error through the slab-like model is almost theoretically expected while disregarding any effects of the DSD fluctuations and or the effect of the vertical structure of the storm. Figure 6d compares the standard error bars in the DWRT estimates accessed through those three schemes over 3-km range. As notice in the figure, the errors are considerably large, while considering the effects of the DSD variability (slant-path scheme). The vertical structure of the storm (statistical scheme) introduces negligible amount of error in the DWRT compared to the effect of the DSD variability.
 Taking into account of these facts, an attempt has been made to examine relative sensitivities of DWRT retrieval mainly to the natural fluctuation of the DSD by comparing it with the results from different approaches. A typical example of the result is displayed in Figure 7, for example, over 1-km range resolution, where the left-hand panel gives scattergrams of the estimated rainrate with true rainrate averaged over the 1-km range resolution and the right-hand panel shows fluctuation in the standard error bars (calculated as ± standard deviations from respective mean distributions, as shown in left-hand panel figures with thick lines, of the estimates along each 1 mm h−1 channels of the true rainrate). The simulation results presented in Figure 7 are drawn without adding any signal and noise fluctuation effects, and based on 3-year database.
 The DWRT retrieved rainrate that is based on the slanted structure of the uniform rain field is presented in Figures 7a and 7b. This retrieval is same as Figure 6b (with dots), except range resolution, i.e., 1-km. As in the case of 3-km range resolution (Figure 6b), the scattergram of the DWRT estimated rainrate shown in Figure 7a1 has fairly good correlation with the true rainrate. However, in contrast to Figure 6b, the fluctuations of the scatters appeared quite larger revealing poor performance of the DWRT with poor range resolution. The mean distribution of the DWRT estimate in strong rain region appears to be slightly deviated from the 1:1 reference line (thin solid line). The deviation increases gradually with increasing rainrate intensity. This is only due to the imperfect fittings of the Ze14dis-Rdis and kdis-Rdis relationships mainly in strong rain region (as shown in Figure 2). The standard error fluctuations (Figure 7a2) in the DWRT retrieved rainrate, however, are considerably small over whole rainrate regions. This result is compared with the results from other approaches (namely, Ze14-R method, DWRT with NUR, FOV-MM, Rayleigh approximation and truncated drop size distributions) and discussed below.
 The second row of Figure 7 gives estimated rainrates from a conventional, so called Ze-R (namely, Zedis14 = 206Rdis1.36, as fitted in Figure 1) method, which is based on the slant-path scheme of uniform rain field. In contrast to the DWRT retrieval, the Ze14-R method exhibits larger variation in the retrieved rainrate. At the region of weak rainrates, however, the variation appears considerably small. The variation increases as increases the rainfall intensity. As a result, the standard deviation, thus, the standard error in the Ze14-R estimates (as shown in Figure 7b2) increases with increasing rainfall intensity. By comparing the standard deviations of the DWRT and Ze14-R estimates over the 1-km range resolution, we found that the two standard deviations crossover at the rainrate of about 6 mm h−1. Below the crossover-point the standard deviation of the Ze14-R estimate is lower than that of the DWRT, whereas above ∼6 mm h−1 rainrate, DWRT gives lower standard deviation (better accuracy). From the similar comparison over 3-km range resolution, it is found that standard deviations of the Ze14-R and the DWRT estimates crossover at ∼2 mm h−1 rainrate.
 In another consideration, in addition to the DSD fluctuation effects in the DWRT retrievals, additional effects, as outlined previously, such as radar signal and noise fluctuation and noise correction procedure are examined. As a result, it is found that the statistical fluctuation has almost negligible effects except in the region of very weak rain, where the noise is dominant (not shown). The effect of raindrop temperatures (namely, at 0 and 20°C) in the DWRT and Ze14-R retrievals (not shown) is also examined and found almost identical behavior of estimates at both temperatures, as it is expected.
 Taking into account of the crossovers of the standard errors in the Ze14-R and DWRT retrievals, one can expect that a combination of the Ze14-R and DWRT improves rainrate retrieval accuracy. However, one should take care of the effect of strong rain attenuation in the heavy rain region for 35-GHz radar (as depicted, for example, in Figure 5b), which severely limits the applicable upper range of DWRT. The strong attenuation causes fractional detection of heavy rain while to consider applicable minimum detection thresholds of 35-GHz radar from spaceborne platform [Adhikari and Nakamura, 2002]. To avoid the fractional detection of heavy rain, the Ze14-R method is supposed to be useful again in the strong rain region instead of the DWRT. The upper range of the DWRT can be determined, once the 35-GHz radar signal falls below the minimum sensitivity threshold. As a result, a combined (i.e., Ze14-R; DWRT and again Ze14-R) form of rainrate retrieval technique can be considered to improve the rainrate retrieval accuracy.
Figure 8, for example, displays the scatterplots of the estimated rainrate by the combined technique versus the measured rainrate over 1- (left-hand panel) and 3- (right-hand panel) km range resolutions. The equivalent noise thresholds are typically specified as of 18 (upper panel) and 10 (lower panel) dBZ for 35-GHz radar. The standard deviations of the combined estimates are given in Figures 9a and 9b, respectively, over 3- and 1-km range resolutions. The standard deviations of the estimates by the DWRT and Ze14-R method alone are also depicted in the figures. As a comparison, it is clearly revealed that the combined technique largely enhances the retrieval accuracy. For instance, at 10 mm h−1 rainrate with the assumption of 10 dBZ detection threshold of the 35-GHz radar, the combined estimate shows ∼51% improvement in the accuracy over 3-km range, while over 1-km range it has ∼44% improvement compared to the Ze14-R method. The results also demonstrate some improvement in the accuracy of the retrievals while increasing the minimum sensitivity for 35-GHz radar.
4.2. Effect of Nonuniform Rain Field
 Another focus of this study is to examine the effect of the nonuniformity of rain (NUR) in the DWRT retrievals. Based on the simulation illustrated in Figure 4b, which outputs the slanted vertical structures of the nonuniform rain field, the effect of the NUR is examined. A typical example of the simulation result is displayed in Figures 7c1 and 7c2, representing the DWRT estimates over 1-km range resolution. Figure 7c1 is basically same as Figure 7a1, except it incorporates additional effect of NUR. Qualitatively, Figures 7a1 and 7c1 reveal quite similar features. For example, the mean distribution of the DWRT estimated rainrate with nonuniform rain field (Figure 7c1) are more deviated from 1:1 slope line than the mean estimates from uniform rain field (Figure 7a1), which is mainly because of averaging of linear Zms in addition to the effect of imperfectly fitted kdis-Rdis relationships (consisting with Figures 2a and 2b). We examined this fact by averaging more horizontal nonuniform rain fields, for example, eight point-IFOV-structures. As a result, the means of the DWRT-estimates found to be deviated largely, i.e., more negatively biased than in the case of four point-IFOV-structure averaging.
 The error statistics of the DWRT estimates without and with the additional effect of NUR are compared in Figure 7c2. As a result, it can be noticed that the two estimates are not in good agreement. The effect of nonuniformity resulted in noticeably larger standard deviations than that in the case of uniform rain fields. The error increases with the increase in the intensity of the rain. Nevertheless, the additional amounts of errors due to the nonuniformity of rain are not much significant comparing to the primary effects of DSD variation and/or the coupling of Mie scattering effects. From the similar analysis with larger range resolutions (namely, 3-and 5 km, not shown here), the similar behavior is observed. The standard error fluctuation, as expected, found to be smaller with larger range resolutions.
4.3. Effect of Horizontally Mismatched-FOVS
 A typical example of the effect of the horizontally mismatched-FOVs of the two radars (outlined as in Figure 4c) in DWRT estimates over 1-km range resolution is shown in Figures 7d1 and 7d2. The result is typical with an apparent mismatching of 25%. As can be noticed in the figure, the additional effect of the explicit mismatching resulted in a large variation in the DWRT retrieval. Also, the mean values deviate remarkably from 1:1 reference. Here, it is worth mentioning here that along with the effects of mismatching, the simulation incorporates the effect of the NUR, which originally couples the effects of the natural fluctuation of DSDs and the effect of the Mie scattering. This is because the simulation begins by averaging the measured radar reflectivity factors (i.e., four point-IFOV-structures), which varies along with the time series. Thus, the result, here, accumulates the effect of the nonuniformity of rain in addition to the effects of the mismatched FOVs. The mean estimates with mismatched-FOVs are appeared to be more deviated (namely, more negatively biased) than that with matched FOVs, where the deviations increase with the increase in the mismatching as well as with the increase in the rainrates. The bias is originally due to the combined effects averaging and imperfectly fitted kdis-Rdis relationships. In addition, in the case of the apparent mismatching, the simulation procedure added more biases. This is because, while simulating the mismatched-FOVs, we assumed that the 13.6-GHz radar first “scans” the averaged-FOV-structure (after averaging of four point-IFOV-structures), where the vertical structures of the true rainrates is consistently aliened. Unlike to these, the four-point averaged-IFOV-structures of the 35-GHZ radar shift gradually by leading one point measurement (equivalent to 1 min) so that the nonuniform-FOV-structures of the radar beams are mismatched. Such simulation arrangement gradually introduces inconsistency in the generated averaged-structures of the Zm35 with true rainrates as well as with Zm14, which finally results in the additional biases in the DWRT-estimates. The inconsistency increases with increasing mismatching; hence increases the biases in the estimated rainrates. Figure 7d2 gives an example of the standard error statistics of the DWRT retrievals with (25%) and without the mismatching. The retrievals labeled as matched FOV is redundantly drawn from Figure 7a2, which couples the effect of the Mie scattering through natural variability in DSD. The result depicts clearly that the mismatching introduces large errors. The error increases as increases the mismatching of the FOVs as well as with the increase in the rainrates. As noticed in the cases of the NUR, the fluctuations are larger over shorter-range resolution. Also, it is noticed that the standard fluctuation increases largely as increases the mismatching (e.g., 50% and 75%, not shown) in the field-of-views of the two radars as well as with increasing rainrates intensity.
 From the above comparisons, it is revealed that the DWRT estimates incorporating the effects of the NUR do not differ considerably from those statistics drawn due to the effects of the primary error sources. The statistics (namely, bias in the mean estimates) due to the effect of nonuniform rain field derived in this work are not in exact agreement with the results reported by Nakamura . Unlike Nakamura's  findings, i.e., the bias due to the nonuniformity of rain in the dual-frequency method varies largely from overestimation to underestimation, our analysis shows that the bias is almost always negative (underestimated), especially in the strong rain region. This difference is likely due to the differences in the use of data, applied model, and the computation framework. Differing from Nakamura's  analysis method that was based on simple rain models and explicit assumption of Rayleigh approximation, our work is based on the more practical framework of the Mie model to obtain the radar cross-sections from the measured DSD. In our simulation, the effect of the fluctuated DSD primarily couples Mie scattering effect and their combination emanates errors/biases in the DWRT having mixed probability. We, therefore, try to verify this disagreement by approaching with an explicit assumption of Rayleigh approximation to obtain the radar cross sections from the measured DSD for the given radar frequencies.
 In Rayleigh approximation, in fact, the Zes are equivalent to the Z, as σb is being proportional to D6, while ks differ from that of the outputs of the Mie model because of the D3 proportionality of absorption cross-section in the Rayleigh approximation. Table 2 includes the fitted parameters of the power law models of the kdis-Rdis relations that obtained from the Rayleigh approximation. By explicitly assuming that the Zes are equivalent to Z and based on these relations, we generated the profiles of the radar integral parameters through the vertical structure of the slanted rain field (as outlined in Section 3). Rainrates from the DWRT (5) are then retrieved based on the DA-R relationship (6) upon utilizing the approximated kdis-Rdis relations obtained from Rayleigh approximation for 13.6 and 35 GHz radar frequencies. Figure 7e1 gives a typical example of the DWRT estimates from Rayleigh approximation over 1-km range resolution. Figure 7e1 is same as Figure 7a1, except the computation approximations. If the scatterplots of these two estimates are compared with the 1-km path averaged true rainrates, it can be noticed that different features of the estimates at different rain intensities. For instance, in contrast to Figure 7a1, in Figure 7e1 the estimated rainrate at low rain region has much less variations, and the variation increases gradually as increases the rainrates. The mean distribution of the estimated rainrates with Rayleigh’s approximation is found to be varied largely with the rain intensities from underestimation to overestimation, which is in agreement with Nakamura's  result. The mean estimate that assessed from the Mie model, however, is either underestimated or in agreement with the true rainrates. Figure 7e2 compares the error statistics, which reveals that, in contrast to the Rayleigh’s approximation, the Mie scattering couples considerably larger amount of errors in the weak rain region. This comparison evidences that the Mie scattering effects which inevitably couple with fluctuation of the DSDs is the dominant and responsible factor to introduce errors in the DWRT estimates. In addition, the systematic underestimation or overestimation of Zes, which are slightly inconsistent at two radar frequencies, as well as the slightly nonlinear kdis-Rdis relationships are responsible for the biases/errors.
 Since the radar cross-sections assessed through Mie model mainly has a function of the sizes of the raindrops and the propagating frequencies, the above effect in a typical spaceborne dual-wavelength radar is, theoretically, quite understandable. Here, we examine the size dependence of the biases/errors in the DWRT by rerunning the simulation (based on Mie model and slanted structure of uniform rain field case) upon gradually truncating the upper size ranges (namely, from D = 0.045–0.255 cm, and up to 0.045–0.52 cm, which correspond to channel: 2–13, and up to 2–20) of the measured DSD. Figures 7f1 and 7f2 give a typical example of the simulation result, for example, with DSD that regards larger diameters of the raindrops (D = 0.045–0.255 cm). As a result, it is revealed that the estimated rainrate gives fairly less fluctuations in the weak rain region. In the region of the strong rainrates, the mean estimate is slightly over estimated. These features are being changed gradually as gradual increasing the contribution of the bigger raindrops. More clearly, the fluctuations of the estimates increase gradually in whole rain region. It means that the bigger raindrops exhibiting larger Mie scattering effect are responsible for the errors in the DWRT retrievals associated even in lower rain regions. On the other hand, the increased Mie scattering effects associated with larger raindrops contribute to the gradual underestimation of the DWRT retrievals in the higher rain region. These behaviors are strongly coupled in the DWRT retrievals that incorporate the effect the NUR and mismatched-FOVs as well. It means, the Mie scattering effect that coupled with the naturally fluctuated DSD is primarily responsible for the errors in the DWRT (13.6 and 35 GHz) estimates.
 A variety of major sources of errors in the retrievals of a conventional dual-wavelength radar technique as well as Ze14 (namely, Ze at 13.6 GHz)-R method are analyzed based on simulations utilizing disdrometer-measured raindrop size distribution data as well as vertical storm structure data collected by Tropical Rainfall Measuring Mission’s precipitation radar. A spaceborne dual-wavelength radar geometry having a combination of 13.6 and 35 GHz radio frequencies, that is, a candidate frequency pair for TRMM follow-on mission’s dual-wavelength radar, is considered in the simulations.
 At first, the simulation studied the significance of the vertical structure of rainstorm in the DWRT upon statistically relating DSD spectra with the vertical profiles of radar reflectivity factor measured by TRMM PR. The simulation analyses reveal that the vertical structure of the storm introduces almost negligible amount of error in the DWRT.
 Next, we attempted to examine relative sensitivities of DWRT as well as Ze14-R retrievals to the natural fluctuation of the DSD based on a simple (namely, slant-path) structure of rainstorm. The effect of the natural variation of the DSD and/or the couplings of the Mie scattering effect are “primary and inevitable” (hereafter, primary error sources) for radar observations from the space. It is found that these effects introduce considerable amount of errors in the DWRT estimates, mainly in the weak rain region. The error is larger, while the range resolution is shorter. Also, the error increases as increases rainfall intensity. The estimated rainrates exhibit fairly good correlation with the measured rainrates over any range resolution. In addition, the simulation also addresses the effects of the variation in raindrop temperatures and statistical fluctuations in the radar signals and noises, which are found to be almost negligible.
 Statistical error analysis suggested some distinct merits/drawbacks of the DWRT and Ze14-R methods. For example, the result suggested that below ∼6 and ∼2 mm h−1 rainrates, respectively, over 1- and 3-km range resolutions the Ze14-R method could retrieve more accurate rainrate than the DWRT. Taking into account of these merits/drawbacks and upon considering detection thresholds typical for 35 GHz spaceborne radar, a combined rainrate retrieval technique in the form of Ze14-R, and DWRT is proposed, which significantly improves the rainrate retrieval accuracy. The results suggest, for instance, if the minimum sensitivity of 35-GHz radar is equivalent to be about 10 dBZ, and if rainrate is about 10 mm h−1, the combined estimates give ∼51% of improvement in the accuracy for 3-km range resolution, while it has ∼44% improvement for 1-km range resolution compared to Ze14-R method.
 Another focus of this study was to analyze the effects of some additional sources of biases/errors that sometime may encounter as inevitable in the DWRT. Nonuniform rain field and mismatching in the observed field-of-views of the radars are major among them. Through an art of simulation, we introduced the effect of the nonuniform rain fields in the observation of the 13.6- and 35-GHz radar by envisioning a running linear average of the Zms at the point-scale-pixels that vary along with the time series of DSD data. As a result, it is found that nonuniform rain field within the observed FOVs introduces noticeable amount of errors. However, the additional nonuniformity effect appears negligible while comparing with the effect of the primary error sources, especially in the lower rain region. The magnitude of the error, due to the nonuniform rain field, increases with the increase in the rainrates. Also, the nonuniformity introduces bias in the mean of the DWRT-estimated rainrates. The bias increases with increasing rainrate intensity. The bias is due to the averaging effect of the Zms which is nonlinearly related with rainrates giving larger weight to the higher rainrates as well as to the larger nonuniformity.
 Finally, we simulated the effect of mismatching in the observed field-of-views of the 13.6- and 35-GHz radars in the estimates of the DWRT by explicitly arranging the sampling volume of the radars that fills by the nonuniform rain fields within the horizontal plane of the FOVs. The simulated results display that the mismatching effect introduces a significantly large amount of errors as well as biases in the DWRT estimates. This is due to the larger nonuniformity in the rain fields (and their averaging effects) that introduced by the simulation, resulting in the increase in the errors (and biases) with the increases in the mismatching.
 Overall, we evaluated relative significances of a variety of error sources mainly in the retrievals of a conventional dual-wavelength radar technique comprising of 13.6- and 35-GHz spaceborne radars based on simulations. The simulated statistics presented in this paper, however, may deviate in some extend, if a more realistic beam illumination model incorporating evolutions/breakups in the falling hydrometeors that linked to different phases of storm height or any other plausible errors are taken into account.
 We are thankful to Mr. H. Minda, at the Hydrospheric Atmospheric Research Center, Nagoya University, for arranging disdrometer data. Thanks are due to Dr. D. A. Short, ENSCO, Inc., and Prof. D. Narayana Rao, Sri Venkateswara University, for their helpful discussions. We extend our thanks to the National Space Development Agency of Japan for providing TRMM PR data.