THE DUAL-FREQUENCY (13.6/35.5 GHz) Precipitation

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1 2612 IEEE TRANSACTIONS ON GEOSCIENCE AN REMOTE SENSING, VOL. 45, NO. 8, AUGUST 2007 Rain Retrieval Performance of a ual-frequency Precipitation Radar Technique With ifferential-attenuation Constraint Nanda B. Adhikari, Toshio Iguchi, Member, IEEE, Shinta Seto, and Nobuhiro Takahashi Abstract Assessments on the performance of dual-frequency (13.6/35.5 GHz) Precipitation Radar (PR) rain retrieval techniques are performed by means other than relying on the surface reference technique. A PR inversion technique (PR-IT) with an independent estimate of the differential attenuation, i.e., the difference of attenuation differences between two frequencies over a certain range, is introduced as an alternative to surface reference or iterative methods for resolving the path-integrated attenuation (PIA) information. Retrieval performance of the proposed method is tested to some vertical rain profiles synthesized with arbitrarily defined and disdrometer-measured raindrop-size distribution data. Retrievals of two other PR-ITtype methods, namely the PR-IT with sets of preselected surface PIAs at two frequencies from wide ranges and the PR-IT iterative algorithm, are also considered for the analysis. The comparison of the simulated results obtained from these three methods is presented and discussed. Index Terms ifferential attenuation, dual-frequency rain retrieval, Global Precipitation Measurement (GPM). I. INTROUCTION THE UAL-FREQUENCY (13.6/35.5 GHz) Precipitation Radar (PR) aboard the Global Precipitation Measurement (GPM) satellite is expected to provide some information on the raindrop-size distribution (S) of precipitation. However, since at the GPM radar frequencies, the radar returns suffer from attenuation while propagating through the rain, cloud, and mixed-phase precipitation, retrievals of S parameters will not be reliable in many cases unless we correct the attenuation effect in the PR measurements. Past studies of airborne or spaceborne dual-frequency radar techniques utilized the surface reference technique (SRT) to estimate the path-integrated attenuation (PIA), which in turn was used to correct the attenuation effect in the measured reflectivity. The attenuation-corrected PR reflectivities are then used to retrieve some S parameters. Among such techniques, the PR inversion technique (PR-IT) described by [1] is a basic Manuscript received June 2, 2006; revised November 17, N. B. Adhikari, T. Iguchi, and N. Takahashi are with the Environment Sensing and Network Group, Applied Electromagnetic Research Center, National Institute of Information and Communications Technology (NICT), Tokyo , Japan ( adhikari@nict.go.jp). S. Seto was with the Environment Sensing and Network Group, Applied Electromagnetic Research Center, National Institute of Information and Communications Technology (NICT), Tokyo , Japan. He is now with the Institute of Industrial Science, University of Tokyo, Tokyo , Japan. igital Object Identifier /TGRS one, in which the attenuation-corrected reflectivities are linked to the S parameters by solving PR backward-recursion equations. In practice, however, the SRT-based PR-IT estimates could be unreliable in a number of cases, depending on the statistical properties of the surface echoes as well as on the rainfall intensities [2]. To help bypass the SRT-induced uncertainties, recent studies by Mardiana et al. [3] and Rose and Chandrasekar [4] propose an iterative means of PIA adjustment in the PR-IT retrieval. The non-srt iterative method as well has some limitations in its application. For instance, the estimates of such iterative method would be unrealistic if the radars are not well calibrated [3]. Also, as it is reported by Rose and Chandrasekar [4] and Liao and Meneghini [5], the iterative solutions are rather unrealistic for certain combinations of S parameters of moderately large rainfall rates or with large PIA. Taking note of these limitations, this paper studies a possible alternative way to resolve the PIA information in the PR-IT by means other than relying on the SRT or iterative methods. In particular, this paper takes into account an independent estimate of the differential attenuation (A) as a new constraint to the PR-IT by considering the fact that the A estimates is generally independent of radar calibration error and can be obtained directly from the PR measurements by taking the difference of the differences of the measured reflectivity factors (in decibels) at two radar frequencies over certain range intervals [6], [7]. In order to link the A estimate to the PR- IT algorithm, this paper outlines a simple iterative procedure; its retrieval performance is evaluated with both synthesized arbitrarily defined rain profiles and a profile based on a large set of disdrometer-measured S data. In addition, this paper considers the retrievals of two other PR-IT-type methods, namely the PR-IT with sets of predefined-surface PIA at two frequencies from wide ranges and the PR-IT iterative algorithm, as described by [3], and compares the retrieval performance of these three methods. II. PR-IT RETRIEVAL In the PR-IT, the measured reflectivity factors (Z m )(mm 6 m 3 ) are first corrected for attenuation, and the dualfrequency ratio (FR) of the effective reflectivity factors (Z e )(mm 6 m 3 ) is then used to estimate the S parameters [1]. The Z m, for the given wavelengths λ i (e.g., i =1 and /$ IEEE

2 AHIKARI et al.: RAIN RETRIEVAL PERFORMANCE for Ka- and Ku-bands, respectively) at a specific range r, can be expressed as Z m (r, λ i )=Z e (r, λ i )exp 0.2ln(10) r 0 k(s, λ i )ds = Z e (r, λ i )A(r, λ i ) (1) where k (in decibels per kilometer) is the specific attenuation and A is the attenuation factor. We assume that the S follows a gamma distribution in the form of N(, r) =N 0 (r) µ exp [ Λ(r)] (m 3 mm 1 ) (2) where Λ(r) (= ( µ)/ 0 (r) [8]) is the slope parameter, µ is the shape parameter, (in millimeters) is the drop diameter, 0 (in millimeters) is the median volume diameter, and N 0 (per cubic meter per millimeter) is the intercept parameter. The effective reflectivity factor Z e (r, λ i ) at the given wavelength can be written as a function of the S parameters and the backscattering cross section (σ b ) (in square millimeters) Z e (r, λ i )=C z (λ i )N 0 (r, λ i )I b [λ i, 0 (r)] (3) where C z (λ i )=λ 4 i /[π5 (m 2 1)/(m 2 +2) 2 ] (in quadratic millimeters) with the complex refractive index m of the particle [9], and I b [λ i, 0 (r)] = σ b (λ i,) µ exp [ Λ(r)] d. (4) Similarly, the specific attenuation k, in terms of the extinction cross section (σ t ) (in square millimeters) and the S parameters, can be given as k(r, λ i )=0.01 log 10 (e)n 0 (r, λ i )I t [λ i, 0 (r)] (5) where I t [λ i, 0 (r)] = σ t (λ i,) µ exp [ Λ(r)] d. (6) Using (1), the ratio of the dual-frequency Z e s can be expressed as f(r) = Z m(r, λ 1 )A(r, λ 2 ) Z m (r, λ 2 )A(r, λ 1 ) where, from (3), the same ratio can be given as (7) g [ 0 (r)] = C z(λ 1 )I b [λ 1, 0 (r)] C z (λ 2 )I b [λ 2, 0 (r)]. (8) The ratio in (7) or (8) is known as the FR. For a given λ 1, λ 2, µ, and the temperature (T ), the FR in (8), as a function of 0, can be computed using Mie s theory, and the result can be stored as a lookup table. The 0 and N 0 parameters at the jth range bin (1 j s), then, can be obtained by solving the following set of PR recursion equations [1] and equating (7) and (8) as f(r j )= Z m(r j,λ 1 )A(r j,λ 2 ) Z m (r j,λ 2 )A(r j,λ 1 ) (9) 0 (r j )=g 1 [f(r j )] (10) N 0 (r j,λ i )=Z m (r j,λ i ) A(r j,λ i )C z (λ i )I b [λ i, 0 (r j )]} 1. (11) The recursion process starts at the bottom range gate (r s ) and continues backward until r j = r 1 (rain top), where the attenuation factor at the (j 1)th range bin can be given as s A(r j 1,λ i )=A(r s,λ i )exp ah N 0 (r p,λ i )I t [λ i, 0 (r p )] p=j (12) where a =0.2 ln(10), h (in kilometers) is the range resolution and the subscript s represents the bottom range gate. In order to start the recursion, we need to give the PIA estimate at the bottom range gate, that is, A(r s,λ i ) in (12). Note here that the PIA throughout this paper is defined as 10 log 10 (A) and is measured in decibels. With the given PIA information, the recursion subsequently retrieves the 0 and N 0 profiles. The retrieved 0 and N 0 profiles, then, can be used to retrieve Z m in a discrete form as } j Z m (r j,λ i )= Z e (r j,λ i )exp ah N 0 (r p,λ i )I t [λ i, 0 (r p )] p=1 (13) Z m(r j,λ i ) A(r j,λ i ) Ã(r j,λ i ) (14) where Ã(r j,λ i ) is the retrieved attenuation factor given as } j Ã(r j,λ i )= exp ah N 0 (r p,λ i )I t [λ i, 0 (r p )]. (15) p=1 The retrieved Z m by (14) and the measured reflectivity factors Z m (1), then, can be compared to examine the retrieval performance of this method as Z m (r s,λ i ) Z m (r s,λ i ) = Ã(r s,λ i ) A(r s,λ i ) = C(λ i) (16) where C(λ i ) is a factor. Solving (9) (15) with C(λ i ) in (16), being close to unity, defines the proper solution of the PR-IT method. Note that different adjustment of the initial A(r s,λ i ) give different values of C(λ i ). This paper considers following three different approaches to adjust A(r s,λ i ) in (12) and compares their retrieval performance using simulated data sets.

3 2614 IEEE TRANSACTIONS ON GEOSCIENCE AN REMOTE SENSING, VOL. 45, NO. 8, AUGUST 2007 A. PR-IT With Predefined PIAs This part of the analysis examines the retrieval performance of the PR-IT method by executing it with preselected surface PIAs [= 10 log 10 A(r s,λ i )] in (12) from wide ranges (namely, PIA(r s,λ 1 )=0 40 (db) and PIA(r s,λ 2 )=0 10 (db) spaced with 0.1-dB step size, respectively, for Ka- and Kubands). With these initial PIA values, the backward-recursion process (9) (12) retrieves subsequent values of C(λ i ) in (16). Among them, only the proper one shows the unity values. By examining the unity value of both C(λ 1 ) and C(λ 2 ) in the 2- field of the PIAs, we evaluate the retrieval performance of the PR-IT method. Error analysis with ±1% of deviations in C(λ 1 ) and C(λ 2 ) from the unity value is also considered in the analysis. B. PR-IT Iterative In this part of analysis, we consider Mardiana et al. [3] iterative method, mainly for the reason of comparison of the estimates derived from other two methods. Note that this iterative method tries to adjust A(r s,λ i ) in (12) iteratively until C(λ i ) in (16) becomes close to unity (with an allowable tolerance of 0.001). Note that the recursion process (9) (12) in this method starts with an initial assumption of A(r s,λ i )=1.0 at the surface range bin, and the recursion procedure continues by updating the A(r s,λ i ) estimates in each iteration step. For more details on this method, readers are referred to see [3]. C. PR-IT With A Constraint In this approach, we consider the estimate of the A (that is, the difference of the attenuation differences between two frequencies over a certain range) as a new independent constraint to PR-IT method for adjusting A(r s,λ i ) in (12). The A can be approximated by taking the difference of the differences of the measured reflectivity factors (in decibels) at two frequency channels at two range gates [e.g., between the top (r 1 ) and the bottom (r s ) range gates] as Ã= [10 log 10 Z m (r 1,λ 1 ) 10 log 10 Z m (r s,λ 1 )] [10 log 10 Z m (r 1,λ 2 ) 10 log 10 Z m (r s,λ 2 )] (17) [ Ze (r 1,λ 1 )A(r 1,λ 1 ) = 10 log 10 Z e (r 1,λ 2 )A(r 1,λ 2 ) Ze(r ] s,λ 2 )A(r s,λ 2 ). Z e (r s,λ 1 )A(r s,λ 1 ) (18) Using (3), (18) can be expressed as à = 10 log Ib [λ 1, 0 (r 1 )] I b [λ 2, 0 (r s )] 10 I b [λ 1, 0 (r s )] I b [λ 2, 0 (r 1 )] A(r } 1,λ 1 )A(r s,λ 2 ). (19) A(r 1,λ 2 )A(r s,λ 1 ) Therefore, if I b [λ 1, 0 (r 1 )] I b [λ 2, 0 (r s )] =1 (20) I b [λ 1, 0 (r s )] I b [λ 2, 0 (r 1 )] (19) represents the difference of the attenuation difference [ ] A(r1,λ 1 )A(r s,λ 2 ) A = 10 log 10. (21) A(r 1,λ 2 )A(r s,λ 1 ) The condition (20) is of course satisfied if scattering is Rayleigh scattering, in which case I b (λ 1 )=I b (λ 2 ). It is, therefore, often believed that the A information is reliable only when the Rayleigh approximation for scattering is valid at both frequencies. However, the validity of the A approximation is wider than the Rayleigh scattering case. Notice that the condition (20) does not necessarily require the Rayleigh scattering. The condition can be meet if 0 (r 1 )= 0 (r s ).Evenifthis condition is not meet, the ratio I b (λ 1, 0 )/I b (λ 2, 0 ) does not change any more than a factor of two in the usual rainfall, in which the dual-frequency method is applicable in a vertical observation with the Ka- and Ku-band combination [10]. Taking note of these facts, this paper considers the A information approximated by (17) as a constraint to PR-IT method. To couple the approximated A value to the PR-IT algorithm, a simple iterative procedure is considered; detail of which is outlined as follows: 1) approximate A value between two range intervals (e.g., between the top and the bottom range gates) by (17); 2) define the minimum and maximum initial boundaries of the Ku-band PIA values [in decibels, i.e., 10 log 10 (A)] at the surface range gate [e.g., PIA n=0 min (r s,λ 2 )= 0 and PIA n=0 max(r s,λ 2 )=5dB, respectively], then find the middle value [e.g., PIA n=0 mid (r s,λ 2 )=2.5 db] as the initial guess of the Ku-band PIA; 3) with PIA n=0 mid (r s,λ 2 ) and A estimates, find the respective Ka-band initial PIA as PIA n=0 mid (r s,λ 1 )= PIA n=0 mid (r s,λ 2 )+A; 4) solve (9) (12) using these PIAs [after converting to the respective A(r s,λ i ) values] and, then, retrieve the first set of estimates of 0 n=0 (r j ) and N0 n=0 (r j ) by (10) and (11), hence, à n=0 (r s,λ i ) by (15); 5) convert à n=0 (r s,λ i ) to PIÃn=0 (r s,λ 2 ) and define it as one of the new Ku-band PIA boundaries [either as lower PIA n=1 min (r s,λ 2 ) if PIÃn=0 (r s,λ 2 ) > PIA n=0 mid (r s,λ 2 ) or upper PIA n=1 max(r s,λ 2 ) if PIÃn=0 (r s,λ 2 ) < PIA n=0 mid (r s,λ 2 )]; 6) locate the next Ku-band PIA middle point PIÃn+1 mid (r s,λ 2 ) [from either PIA n min(r s,λ 2 ) and PIA n+1 max(r s,λ 2 ) or PIA n+1 min (r s,λ 2 ) and PIA n max(r s,λ 2 )]; then find the respective Ka-band PIÃn+1 mid (r s,λ 1 ) as in 3) and repeat the process for (n +1)th successive times, where n is the number of iterations. In this analysis, we consider the tenth successive iteration as the final solution by allowing a tolerance of about ±0.01% in the PIA estimates. Once the S parameters are retrieved, the rainfall rate can be calculated as R(r j )= π N(, r j )ν() 3 d (mm h 1 ) (22) where ν() =4.854 exp( 0.195)(m s 1 ) is the terminal fall velocity given by the study in [11]. III. SIMULATIONS Simulations of several vertical rain profiles with different sets of 0 and N 0 parameters are considered to examine the

4 AHIKARI et al.: RAIN RETRIEVAL PERFORMANCE 2615 retrieval performance of those three methods described above. In addition, a simulation with a large set of disdrometermeasured S data at the ground level is considered to examine the sensitivity of the PR-IT with A method to the effect of S variability. Assuming that the scatterers to be composed of only raindrops, a rain layer of 4.25 km is considered in the simulation. The rain particles are assumed to be spherical. The temperature of 0 C is assumed to be constant throughout the rain layer. Calculation of the backscatter and extinction cross sections is done based on Mie s formula. The complex refractive indexes of the rain particle are taken from [9]. The PR measurement is considered at and 13.6-GHz frequencies and the radars observe the rainstorm from the top with a range resolution of 250 m. Both radars observe the rainstorm with ideal beams. The number of range bins is 17, ranging between the surface and the 4.25-km altitude. A. Uniform Vertical Rain Profiles With the assumption of constant 0, N 0, and µ parameters between the surface and the 4.25-km altitude, the rain model is considered as uniform rain layer. With the given S parameters and the described PR measurement assumptions, the profiles of the true or the baseline-measured reflectivity factors Z m (r j,λ i ) can be generated in a discrete form from (1) with (2) (5). Several uniform vertical rain profiles are generated with different discrete combinations of 0 and N 0 values. B. Nonuniform Vertical Rain Profiles By assuming different values of 0 at different altitudes, a rain model is defined as nonuniform vertical rain layer. The other parameters, that is, N 0 = 8000 mm 1 m 3 and µ =0 are taken as constant over the whole rain layer. The PR simulations (assumed as in the case of the uniform vertical rain profiles) with these S parameters provide us with the profiles of the baseline Z m (r j,λ i ) from (1) with (2) (5) for further analysis. C. Simulation With isdrometer-measured S In this simulation, we utilize a large set of S data collected by an impact-type R69-disdrometer at Nagoya University (35.1 N, E) for three years (from 1998 to 2000, 819 precipitating days) to simulate the vertical rain-field structure. The measured Ss are first used to transform Z e (λ i ), k(λ i ), and R and, then, to simulate the vertical profiles of these radar rainfall parameters following a simulation scheme, as described in [7]. According to this scheme, the simulated vertical structure of the rainstorm is assumed to evolve with the disdrometermeasured S along 1-min time series. In particular, with this simulation scheme, 17 1-min-averaged Ss are used to construct a vertical path of 4.25 km, where each S is used to represent the rain in a 0.25-km layer. These Ss then subsequently provides us with the profiles of the known rainfall rate Z e s and ks at both radar frequencies. The simulated vertical rain profiles are assumed to be observed by the PR with the ideal beams and matched FOVs. The simulation, then, Fig. 1. Contour diagram showing the ratios of the estimated to the simulated measured reflectivity factors [i.e., C(λ 1 ) and C(λ 2 ) in (16), respectively, at Ka- and Ku-bands, having values of 0.99, 1.00, and 1.01] obtained from the PR-IT method with predefined PIA values at the surface. The simulation is considered with a uniform rain model (4.25 km in height) with µ =0, N 0 = 8000 mm 1 m 3,and(a) 0 =1.1 mm and (b) 0 =1.5 mm for a gamma-shaped S model. Open squares indicate the modeled PIA values. ark filled-squares are from the PR-IT iterative method. The lines indicated by arrows are the simulated PIAs using the PR-IT A constraint, while the filled-circles are the respective final solutions obtained from the PR-IT with A method. calculates the PIA profiles at both radar frequencies, and the simulated PIAs with Z e profiles provide the profiles of Z m s, as in (1) (for more details on the simulation procedure, readers are referred Fig. 2 of [7]). A huge number of the simulated Z m profiles at the two radar frequencies are then used to examine the retrieval performance of the PR-IT with A method. IV. RESULTS AN ISCUSSIONS A. PR-IT With Predefined PIA Fig. 1 shows an example of the simulation results obtained from this method that performed with the uniform vertical rain profiles generated with the known S parameters of µ =0, N 0 = 8000 mm 1 m 3, 0 =1.1 mm [hereafter, referred to as case A, Fig. 1(a)] and 0 =1.5 mm [hereafter, referred to as case B, Fig. 1(b)]. In the figures, the ratios of the retrieved to the simulated measured reflectivity factors at Ka- and Ku-bands [having values of C(λ 1 )=C(λ 2 )=0.99, 1.00, and 1.01] are shown by contour lines in the 2- field of the PIAs at rain bottom. For clarity, the figures focus only those regions where the lines of C(λ 1 )=C(λ 2 )=1.0 are closer. Note here that to avoid the double-valued solution for 0, as described in [1], the results include only those data that fall in the region of 0 > 0.8 mm. For the reason of comparison, the figures also include the PIA estimates obtained from the PR-IT iterative and the PR-IT with A methods. The results of these two methods are discussed separately in the following sections. As depicted in Fig. 1(a) for case A (a case with relatively small PIA), the lines of C(λ 1 )=1and C(λ 2 )=1, obtained with the predefined PIAs, pass through the true PIA value with an intersection. Unlike this, for case B (a case with relatively large PIA), the lines of C(λ 1 )=1and C(λ 2 )=1start getting closer at the region of smaller PIA values rather than at the true PIA values, and they remained closer in the region the larger PIA and even beyond the true value. This leads the solution to a large uncertainty. We have performed similar analyses to several other uniform vertical rain profiles simulated with different discrete set of 0 and N 0 parameters (not shown

5 2616 IEEE TRANSACTIONS ON GEOSCIENCE AN REMOTE SENSING, VOL. 45, NO. 8, AUGUST 2007 here). Overall results of such simulations, which are typical with a finite step size of 0.25-km-range resolution and the rain-depth of 4.25 km, show that the PR-IT method with the predefined PIA values works only in the restricted range of 0 from mm with N 0 = 8000 mm 1 m 3. B. PR-IT Iterative The estimates of the PR-IT iterative method for cases A and B are compared with other two methods in Fig. 1(a) and (b), respectively. Notice in the figure that, starting with the 0-dB initial PIAs, the iterative retrieval continues until C(λ 1 )= C(λ 2 )=1 (with the tolerance of ±0.001). As a result, it appears that this method behaves similarly as the PR-IT with the predefined PIAs in that, for case A, the iterative method converges correctly to the true value, while for case B, it converges to the incorrect PIA values. We have examined the PR-IT iterative method with other different initial values of A(r s,λ i ) in (12) (not shown here). As a result, for case A, each trial brings the convergence to the true value, while for case B, none of them converges to the correct value. The simulated results obtained from this and the PR-IT method with the predefined PIAs agree with the findings in the study in [4] in that for certain combination of 0 and N 0 parameters of weak to moderate rainfall intensities, these methods work well, while in the cases of moderately strong rainfall, the solutions differ largely from the true value. C. PR-IT With A Constraint The estimates of the PR-IT with A constraint are displayed in Fig. 1(a) and (b), respectively, for cases A and B by comparing them with those retrievals obtained from the above two methods. In the figures, the simulated PIA values using the A constraint are displayed with bold lines (indicated by arrows), while the respective final solutions obtained from the PR-IT with A constraint with the tenth iteration are shown by the filled-circle. As expected, the simulated PIA values with PR-IT with A constraint pass through the true value and the final estimates agree with the respective true PIA values in both cases. As mentioned previously, with the assumption of the uniformity in the rain model with identical size distributions at the end points of the processing interval, the non-rayleigh scattering at both two range gates become the same; hence, condition (20) meets and the A approximation by (17) becomes exact. As a result, in the cases of such uniform rainstorm, the PR-IT method with A constraint properly retrieves the expected PIA values. Once the final estimates of the PIA values at both frequencies are obtained from the PR-IT with A constraint, the profiles of the S and rainfall parameters then can be estimated by the PR-IT method. Fig. 2 gives an example of such estimated vertical profiles of 0, N 0, the rainfall rate, and the corresponding PIAs [Fig. 2(a) (d), respectively]. The results shown in Fig. 2 are simulated with a uniform vertical rain column, 4.25 km in height, based on 0 =1.8 mm, N 0 = 4000 mm 1 m 3, and µ =0. Note that this combination represents the incorrect convergence region of the PR-IT iterative method, as specified Fig. 2. Vertical profiles of (a) 0,(b)N 0, (c) rainfall rate, and (d) Kuand Ka-band PIA of the model (solid lines) and those retrieved from the PR-IT with A (dark points) and PR-IT iterative (dotted dashed lines) methods. A uniform rain model, 4.25 km in height with 0 =1.8 mm and N 0 = 4000 mm 1 m 3, is considered for this simulation. The shown final solutions of the PR-IT with A and the iterative methods are drawn with 10 and 559 iterations, respectively. by the study in [4]. Final estimates of both PR-IT with A and the iterative methods are shown in the figures for the reason of comparison. Note that for the PR-IT with A method, the final solution is drawn from the tenth iteration, while for PR-IT iterative method, the iteration is converged with 559 iterations. As depicted in Fig. 2(a) (d), depending on the rain-column height, as discussed in [4], the PR-IT iterative method in most of the lower range bins fails to retrieve the correct PIA and S values. Unlike the PR-IT iterative method, the PR-IT with A method retrieves properly the expected profiles of the PIAs, 0, N 0, and the rainfall rate. Unlike the cases of uniform rainstorm, for cases of nonuniform vertical rain columns or cases where the non-rayleigh scattering is not the same at both ranges, the ratio given in the left-hand side of (20) deviates from the unity value, hence, expression (17) becomes inexact. In such cases, since the approximated A value by (17) is not changed in the iteration, the errors in the A estimate persist so that even if the correct Ku-band PIA is found, the Ka-band PIA would be offset by the error in the A. However, unless the 0 (or rainfall rate) changes substantially at the two ranges r 1 and r s, the deviations of this ratio (20) from unity at two range gates, generally, cancel to result in a value close to unity. We have examined this fact by using several nonuniform vertical rain profiles generated with different 0 values varying as a function of altitude, where the 0 values at the bottom-range gates are assumed larger than that in the top-range gates. An example of the test result obtained from the PR-IT with A method using 175 such simulated nonuniform vertical rain profiles is depicted in Fig. 3 in terms of mean-range-profiled 0 [Fig. 3(a)] and rainfall rate [Fig. 3(b)], where the modeled (or true) values are shown by dashed lines and the estimated values are shown by solid

6 AHIKARI et al.: RAIN RETRIEVAL PERFORMANCE 2617 Fig. 3. (a) and (b) Plots of mean range-profiled 0 and rainfall rate. ashed lines indicate the true values, while the solid lines with standard error bars (mean with ± standard deviations) show the PR-IT with A estimates simulated for 175 nonuniform vertical rain columns using µ =0, N 0 = 8000 mm 1 m 3, and different 0 values varying as a function of altitude. Fig. 4. Scatter plots of the PIA estimates obtained from the PR-IT with A method versus true surface PIAs simulated at (a) Ku- and (b) Ka-bands using disdrometer-measured three-year S data set. (c) Estimated rainfall rate based on the derived S parameters from the PR-IT with A method versus the disdrometer-derived true rainfall rate averaged over the simulated 4.25-km vertical rain column. lines with standard error bars (mean with ± standard deviations calculated at each range gate). As expected, the estimated mean values of the 0 and rainfall rate deviate slightly from the respective modeled values. The bias and the root-mean-squared (rms) deviation of 0, which are defined as s j=1 bias = rms-deviation = 1 s [ retrieve 0 (r j ) 0 true (r j ) ] s j=1 true 0 (r j ) s j=1 [ retrieve 0 (r j ) true 0 (r j ) ] 2 1 s s j=1 true 0 (r j ) (23) } 1/2 (24) averaged over those retrieved profiles are found as 0.9% and 4.4%, respectively. Similarly, the calculated bias and the rms deviation of the rainfall rate are 1.3% and 13.9%, respectively. The bias is negative for 0 and positive for the rainfall rate. Note that depending on the relative size of 0 at the processing-range gates, the estimates of the PIAs, hence, 0 and the rainfall rate, of the nonuniform vertical rain profiles would be either positively or negatively biased. For instance, from a simple simulation test using a rain model of decreasing 0 with increasing altitude (namely, with 0 =1.6 and 1.4 mm at the bottom- and top-range gates, respectively, and constant N 0 of 8000 mm 1 m 3 ), it appears that the surface PIA values would be negatively offset by about 2.1 db for Ka-band and by about 0.23 db for Ku-band. While similar analysis with another different nonuniform rain model of increasing 0 with increasing altitude (for instance, with 0 = 1.5 and 1.3 mm at the top- and bottom-range gates, respectively, and constant N 0 of 8000 mm 1 m 3 ) shows an overestimation of the surface PIAs offset by about 1.9 and about 0.19 db at Ka- and Ku-band channels, respectively (not shown here). More sensitivity tests of this method are further considered by using a sufficiently large number of nonlinear vertical rain profiles simulated with 3-year measured-s data sets described as in the previous section. Note here that unlike in the simulation of the uniform or nonuniform vertical rain profiles, the shape parameter µ for this analysis is considered to be 7.7 (which is the approximated mean value obtained from the threeyear S data derived through the moment (third, fourth, and sixth) method [12]). Also, as we are concerned here with the measured S, we consider the mass-weighted mean diameter ( m, which can be obtained from the ratio between the fourth and third moments) instead of 0. In order to minimize the possible Poisson random errors in the 1-min-sampled disdrometermeasured S data, as discussed in [12], this analysis utilizes

7 2618 IEEE TRANSACTIONS ON GEOSCIENCE AN REMOTE SENSING, VOL. 45, NO. 8, AUGUST 2007 only those data that have the number of drops at each specific drop channel equal to or more than four and for the disdrometerderived rainfall rate equals to or more than 1 mm h 1.Fig.4 shows the final solutions obtained from this method using these data sets, in terms of the Ku- [Fig. 4(a)] and Ka-band [Fig. 4(b)] PIA at the surface, and the path-averaged rainfall rates over the 4.25-km rain-depth [Fig. 4(c)]. As a result, the estimates obtained from those huge numbers of rain profiles correlate fairly well with the measured values. However, depending on the rainfall intensities or PIA values the estimates suffers from considerable amount of errors. As shown in the figures, at the low rain rate and PIA values the estimates are suffered from the large amount of errors, while at the moderate to strong rainfall rate and larger PIA values, the errors are relatively smaller compared to that at the low rainfall rate. V. S UMMARY In this paper, we have performed some assessments on the rain retrieval by PR (13.6/35.5 GHz, as will be in GPM) methods by means other than referencing the surface echoes. The primary focus of this paper is given to introduce a new PR retrieval technique, a variation of the PR iterative algorithm described in the study in [3] and [4], with A constraint obtained independently from the profiles of the PR measured reflectivity factors. The algorithm is based on the PR-IT originally described by the study in [1]. The algorithm is then tested using some simulated vertical rain profiles synthesized with arbitrarily defined S parameters and disdrometer-measured S data sets. For the reason of comparison, this paper also considers the retrievals of two other PR-IT-type methods namely the PR-IT with sets of predefined-surface PIA and the PR-IT iterative algorithm. Based on the uniform rain model, this paper first assesses the PR-IT retrievals by executing it with the preselected surface PIA values at two frequencies from the wide ranges then evaluates how well the estimated S parameters give back the assumed PIAs. As a result, the retrievals of this and the PR-IT iterative methods show similar strengths and limitations. In particular, as described by [4], these two methods work well in the regions of weak to moderate rainfall intensities but limited to moderately strong rainfall rates or larger PIAs. Unlike in these methods, in the cases of uniform rainstorm, the estimates of the PIAs, hence, the S parameters and the rainfall rate obtained from the PR-IT with A method agree fairly well with the expected values in both weak and strong rain regions. In the cases of nonuniform rainstorm, however, the estimates of the proposed method deviate from the expected values depending on the variation in the 0 or rainfall rate at the two range gates of the processing interval of the algorithm. However, unless the 0 or rainfall rate changes substantially at the processing two range gates, the deviations are relatively small. ACKNOWLEGMENT The authors would like to thank K. Nakamura for his encouragements and insights on the disdrometer simulation. They would also like to thank the reviewers for helpful comments. REFERENCES [1] R. Meneghini, T. Kozu, H. Kumagai, and W. C. Boncyk, A study of rain estimation methods from space using dual-wavelength radar measurements at near nadir incidence over ocean, J. Atmos. Ocean. Technol., vol. 9, no. 4, pp , Aug [2] R. Meneghini, T. Iguchi, T. Kozu, L. Liao, K. Okamoto, J. A. Jones, and J. Kwiatkowski, Use of the surface reference technique for pathattenuation estimates from TRMM Precipitation Radar, J. Appl. Meteorol., vol. 39, no. 12, pp , ec [3] R. Mardiana, T. Iguchi, and N. Takahashi, ual-frequency rain profiling method without the use of surface reference technique, IEEE Trans. Geosci. Remote Sens., vol. 42, no. 10, pp , Oct [4] C. R. Rose and V. Chandrasekar, A systems approach to GPM dualfrequency retrievals, IEEE Trans. Geosci. Remote Sens., vol. 43, no. 8, pp , Aug [5] L. Liao and R. Meneghini, A study of air/space-borne dual-wavelength radar for estimates of rain profiles, Adv. Atmos. Sci., vol. 22, no. 6, pp , [6] N. B. Adhikari and K. Nakamura, Simulation based analysis of rainrate estimation errors in dual-wavelength precipitation radar from space, Radio Sci., vol. 38, no. 4, p. 1066, OI: 1029/2002RS [7] N. B. Adhikari and T. Iguchi, Effect of raindrop size distribution variability in dual-frequency radar rain retrieval algorithms assessed from disdrometer measurements, IEEE Geosci. Remote Sens. Lett., vol. 3, no. 2, pp , Apr [8] C. W. Ulbrich, Natural variations in the analytic form of the raindrop size distribution, J. Clim. Appl. Meteorol., vol. 22, no. 10, pp , [9] P. S. Ray, Broadband complex refractive indices of ice and water, Appl. Opt., vol. 11, no. 8, pp , Aug [10] T. Iguchi, Possible algorithms for the dual-frequency Precipitation Radar (PR) on the GPM core satellite, in Proc. 32nd Int. Conf. Radar Meteorol., Albuquerque, NM, Oct. 2005, vol. 5R4. [11] R. Gunn and G. G. Kinzer, The terminal velocity of fall for water droplets in stagnant air, J. Meteorol., vol. 6, no. 4, pp , [12] T. Kozu, Estimation of raindrop size distribution from spaceborne radar measurement, Ph.. thesis, Kyoto Univ., Kyoto, Japan, Nanda B. Adhikari received the M.Eng. degree from the State Engineering University of Armenia, Yerevan, Armenia, in 1994 and the Ph.. degree from Nagoya University, Nagoya, Japan, in He is currently a Research Fellow with the National Institute of Information and Communications Technology, Tokyo, Japan, working on retrieval algorithms for dual-frequency spaceborne radar for the Global Precipitation Measurement mission. Toshio Iguchi (M 95), photograph and biography not available at the time of publication. Shinta Seto received the Bachelor s, Master s, and octoral s degrees in engineering from the University of Tokyo, Tokyo, Japan, in 1998, 2000, and 2003, respectively. From 2003 to 2006, he was with the National Institute of Information and Communications Technology (formerly known as Communications Research Laboratory), Tokyo, as a Postdoctoral Researcher and he worked for the development of spaceborne dual-frequency Precipitation Radar. Since 2006, he has been with the Institute of Industrial Science, University of Tokyo. His current research interests are precipitation retrieval using microwave remote sensing and its application to water-cycle studies. Nobuhiro Takahashi, photograph and biography not available at the time of publication.