THE CORE satellite of the Global Precipitation Measurement

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1 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 6, JUNE Applicability of the Iterative Backward Retrieval Method for the GPM Dual-Frequency Precipitation Radar Shinta Seto, Member, IEEE, and Toshio Iguchi, Member, IEEE Abstract The Dual-frequency Precipitation Radar (DPR) on the core satellite of the Global Precipitation Measurement mission will measure the radar reflectivity factor in the Ku- and Ka-bands. A rain-rate retrieval algorithm that does not require a surface reference was developed (called the MA04 method). However, MA04 cannot give the true solution in some cases of heavy rainfall. MA04 is a simplified version of the iterative backward retrieval method (IBRM), and the IBRM is equivalent to the forward retrieval method with a constraint. The purpose of this study is to clarify the essential conditions under which the IBRM and MA04 can give the true solution (the conditions are referred to as the applicability as in the title). For the purpose, DPR measurements are simulated under simplified assumptions. The applicability of the IBRM and MA04 is closely related to the magnitude of internal attenuation. The upper limit of rain rate for which the IBRM can obtain a true solutionis10to20mm h 1 if the internal attenuation occurs between the top and middle of the target range bin and the vertical resolution is 0.25 km. The upper limit of rain rate for which MA04 can obtain a true solution is dependent on the number (n) of range bins, and it is 24 to 36 mm h 1 in the case of n =12.MA04 can apply a wider range of rainfall than the IBRM because MA04 tends to select the solution with the smallest attenuation among possible solutions. Index Terms Attenuation, radar, rain. I. INTRODUCTION THE CORE satellite of the Global Precipitation Measurement mission will be launched in 2013 with a Dual-frequency Precipitation Radar (PR) (DPR). The DPR, a successor of the PR on the Tropical Rainfall Measuring Mission satellite, consists of a Ku-band (13.6-GHz) radar and a Ka-band (35.5-GHz) radar. The former radar is similar to the PR. The dual-frequency measurements of the DPR are expected to allow for accurate estimates of drop-size distribution (DSD) and rain rate. Methods to retrieve DSD and rain rate from dual-frequency measurements have been studied for more than 30 years [1] [4]. Echo power measured by a spaceborne PR Manuscript received January 19, 2010; revised June 28, 2010 and October 14, 2010; accepted December 15, Date of publication February 6, 2011; date of current version May 20, This work was supported by the Japan Aerospace Exploration Agency under the 5th and 6th Precipitation Measuring Mission Research Announcements. S. Seto is with the Institute of Industrial Science, The University of Tokyo, Tokyo , Japan ( seto@iis.u-tokyo.ac.jp). T. Iguchi is with the National Institute of Information and Communications Technology, Koganei , Japan ( iguchi@nict.go.jp). Digital Object Identifier /TGRS is converted to a radar reflectivity factor (Z m ) for some aerial range bins and to a surface backscattering cross section (σ 0 ) for the range bin located at the Earth s surface. The number of measurements by the DPR in a matched beam is, at most, 2n +2(2n from Z m and 2 from σ 0 ), where n is the number of aerial range bins. Meneghini et al. [5], [6] developed a retrieval method for an airborne DPR by assuming that the DSD obeys a two-parameter gamma distribution function. Using this method (referred to as ME92 hereafter), they retrieved 2n DSD parameters from 2n +2 measurements. ME92 is categorized as a backward retrieval method (BRM). BRM requires a surface reference technique (SRT) to estimate the path-integrated attenuation (PIA) by comparing σ 0 at the target pixel and σ 0 at nearby no-rain pixels (called reference pixels). In the SRT, the surface conditions between the target pixel and the reference pixels are generally assumed to be the same in terms of σ 0 ; however, this assumption is not guaranteed, and actual differences in surface conditions often cause biases in the PIA. An SRT was applied in the standard algorithm of the PR [7], but it was found that rainfall-induced changes in surface conditions affect σ 0 and yield biases in the PIA and rain rate [8]. Moreover, instantaneous rain-rate estimates over land are sometimes largely biased if the land surface type of the target pixel is drastically different from that of the reference pixels. In the case of airborne measurements, the unstable attitude of the aircraft can cause biases in the PIA and rain rate [9]. Over ocean, σ 0 at a Ku-band radar is affected by the presence of rain through modification of the sea surface roughness by rain impacts [10]. Generally, we cannot expect very accurate estimates from BRM. Mardiana et al. [9] proposed a new retrieval method which dispenses with the SRT and applied it to airborne dualfrequency measurements. This method is called MA04 hereafter. Whereas the PIA is estimated using the SRT in ME92, the PIA is assumed before the retrieval in MA04. After the retrieval, the PIA is calculated from the retrieved DSD, and the calculated PIA is compared with the assumed PIA. Until the calculated and assumed PIAs are judged to be the same, the assumed PIA is modified, and the retrieval is iterated. As will be explained in Section III, MA04 is a simplified version of the iterative BRM (IBRM). MA04 showed good performance for airborne measurements in [9], but Rose and Chandrasekar [11] and Liao and Meneghini [12] reported that MA04 failed for heavier rainfall cases. Adhikari et al. [13] also reported /$ IEEE

2 1828 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 6, JUNE 2011 the DSD of rainfall follows a gamma distribution function as shown in N(D) =N 0 D μ exp [ ( μ)d/d 0 ] (1) where D (in millimeters) is the drop size (diameter), N (in mm 1 m 3 ) is the number density, and N 0 (in mm 1 μ m 3 ), D 0 (in millimeters), and μ are the DSD parameters. μ is a known parameter and is fixed to one throughout this paper, and N 0 and D 0 are unknown parameters to be retrieved. Within any range bin, horizontal and vertical uniformities of rainfall are assumed. Fig. 1. Schematic figure of the simulated DPR measurements of rainfall. cases of MA04 failure and developed a modified method using the difference of attenuation (DA) between two frequencies. However, because DA can be estimated accurately only when the rain rate is vertically constant or the Mie effect is negligible, this modified method cannot be widely used. The purpose of this study is to objectively clarify the essential conditions under which MA04 and the IBRM can give the true solution (the conditions can be referred to as the applicability of MA04 and the IBRM in this paper). For the purpose, DPR measurements are simulated, and the settings will be given in Section II. To capture the essence, simplified assumptions are employed in this paper, although more realistic assumptions should be employed to discuss the applicability in some actual cases. In Section III, methods to retrieve DSD from the simulated DPR measurements are explained. Each method can be decomposed into subproblems, and the solutions of the subproblems are investigated in Section IV. Section V discusses the applicability of MA04 and the IBRM. As the applicability is closely related to the definition of internal attenuation, major results are presented for different settings of internal attenuation in Section VI. Finally, a summary and conclusions of this paper are given in Section VII. II. SIMULATION OF DPR MEASUREMENTS A. Assumptions Precipitation measurements by the DPR were simulated with the following assumptions. The DPR measures precipitation in the nadir direction (with the incident angle of 0 ) to provide Z m at n range bins and σ 0 at the Earth s surface. The width of each range bin is L (in kilometers), and L is fixed to 0.25 km. As is shown in Fig. 1, numbers 1 to n are assigned to the range bins from the top to the bottom, and an r-axis (in kilometers) is set vertically downward such that r = r i 1 at the top of range bin i, r = r i at the bottom of range bin i, and r = r n corresponds to the Earth s surface. Above the top of range bin 1 (r <r 0 ), there are no precipitation and no source of attenuation. Between r 0 and r n, all the precipitation is in liquid phase as rainfall, and B. Theory Rain rate [denoted by R (in millimeters per hour)] can be calculated from the DSD as R = C R D=0 v(d)d 3 N(D)dD (2) where C R is a constant for the unit conversion (= 0.6π 10 3 ) and v(d) (in meters per second) is the falling velocity of a raindrop with the drop size of D. Following Gunn and Kinzer [14], we employ the following equation to calculate v(d): v(d) =4.854 D exp( 0.195D). (3) The real (attenuation-free) effective radar reflectivity factor [denoted by Z e (in mm 6 m 3 )] and attenuation coefficient [denoted by k (in decibels per kilometer)] are given as Z e = C Z k = C k D=0 D=0 σ b (D)N(D)dD (4) σ t (D)N(D)dD (5) where σ b (D) (in square millimeters) and σ t (D) (in square millimeters) are the backscattering cross section and total extinction cross section, respectively, calculated based on Mie theory, C k is equal to 0.01 log 10 (e), and C Z is given in C Z = λ4 π 5 n 2 w +2 n 2 w 1 where λ (in centimeters) is the wavelength of microwave and n w is the complex refractivity index of water, which depends on the physical temperature of raindrops [denoted by T (in kelvins); T is fixed to 300 K for simplification]. Z e and k can be regarded as functions of the two unknown DSD parameters, namely, N 0 and D 0, as follows: 2 (6) dbz e (N 0,D 0 )=10log 10 N 0 + F (D 0 ) (7) k(n 0,D 0 )=N 0 G(D 0 ) (8)

3 SETO AND IGUCHI: APPLICABILITY OF THE IBRM FOR THE GPM DUAL-FREQUENCY PRECIPITATION RADAR 1829 where dbz e =10log 10 (Z e ) and F and G are functions of D 0 given as F (D 0 )=10log 10 {C Z G(D 0 )=C k D=0 D=0 σ b (D)D μ exp [ ( μ)d/d 0 ] dd } σ t (D)D μ exp [ ( μ)d/d 0 ] dd. (9) (10) The measured radar reflectivity factor [denoted by Z m (in mm 6 m 3 )] is given in r Z m (r) =Z e (r)exp 0.2 ln 10 k(s)ds (11) s=r 0 where s is a dummy parameter of r. Equation (11) can be rewritten in decibel units as dbz m (r) =dbz e (r) 2 r s=r 0 k(s)ds (12) where dbz m =10log 10 (Z m ). The second term of the righthand side of (12) is denoted by dba(r) dba(r) 2 r s=r 0 k(s)ds. (13) The PIA is given as follows as r = r n corresponds to the Earth s surface: PIA = dba(r n )=2 r n s=r 0 k(s)ds. (14) C. Internal Attenuation The DPR does not provide dbz m (r) as a continuous function of r but provides values at discrete range bins. The measured value of dbz m at discrete range bin i is denoted by [dbz m ] i. (In the same way, hereafter, [X] i denotes the representative value of X at discrete range bin i.) In each range bin, as rainfall is assumed to be homogeneous, dbz e (r) and k(r) are constants, and they can be regarded as [dbz e ] i and [k] i, respectively. It is reasonable that [dba] i is defined as [dba] i [dbz e ] i [dbz m ] i. The following inequalities should be always satisfied (please refer to Fig. 2): dba(r i 1 )<[dba] i <dba(r i ) (15) [dbz e ] i dba(r i 1 )>[dbz m ] i >[dbz e ] i dba(r i ). (16) As dba(r i ) dba(r i 1 )=2 [k] i L, [dba] i and [dbz m ] i can be written as follows with parameter Fig. 2. Schematic figure to explain the internal attenuation for range bin i. μ is set to one. α (0 <α<1): [dba] i = dba(r i 1 )+α 2 [k] i L (17) [dbz m ] i =[dbz e ] i dba(r i 1 ) α 2 [k] i L. (18) Here, the term α 2 [k] i L indicates the attenuation occurring inside range bin i. This is called the internal attenuation in this study. Parameter α represents the strength of the internal attenuation and should be larger than zero and smaller than one. However, in many studies (e.g., [5], [9], and [11]), the internal attenuation was calculated with α =1, whereas some few studies (e.g., [15]) assumed α =0.5. In the main part of this study, α was set to one so that our results could be directly compared with those of the previous studies. The effect of different choices of α on our results will be discussed in Section VI. Some new notations will be introduced in Section III and later. Equation (18) is rewritten as [dbz m ] i + dba(r i 1 )=[dbz e ] i α 2 [k] i L. (19) The right-hand side of (19) can be determined only by the variables of range bin i, and the value of this equation is denoted by [dbz f ] i as written in [dbz f ] i [dbz m ] i +dba(r i 1 )=[dbz e ] i α 2 [k] i L. (20) By adding 2 [k] i L to both sides of (19), (21) is given as [dbz m ] i + dba(r i )=[dbz e ] i +(1 α) 2 [k] i L. (21) The right-hand side of (21) can be determined only by the variables of range bin i, and the value of this equation is denoted by [dbz b ] i as written in [dbz b ] i [dbz m ] i + dba(r i ) =[dbz e ] i +(1 α) 2 [k] i L. (22) When α =1, [dbz b ] i is eventually equal to [dbz e ] i.

4 1830 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 6, JUNE 2011 III. RETRIEVAL METHODS The problem of retrieving (N 0,D 0 ) of n range bins from [dbz m ] of the n range bins at the two frequencies is referred to as the main problem in this study. In this section, three kinds of retrieval methods to solve the main problem are reviewed: the forward retrieval method (FRM), the BRM, and the IBRM. A. FRM In the FRM, the DSD parameters are sequentially determined from the top (range bin 1) to the bottom (range bin n) without the use of the SRT. At range bin i, asdba(r i 1 ) is known, [dbz f ] i can be calculated by (20); then, we must solve a subproblem to retrieve (N 0,D 0 ) from [dbz f ] i at the two frequencies. This subproblem is a set of two nonlinear equations with two unknowns and is denoted by SP-F. Once SP-F is solved for range bin i, the DSD and its related variables at range bin i (including [k] i ) are given, and dba(r i ) can be calculated as dba(r i 1 )+2 [k] i L. Then, the same process can be applied for range bin (i +1).AsdBA(r 0 ) is known to be zero, the aforementioned process can start at range bin 1. B. BRM In the BRM, the DSD parameters are sequentially determined from the bottom (range bin n) to the top (range bin 1) using the SRT. At range bin i,ifdba(r i ) is known, [dbz b ] i can be calculated by (22); then, we must solve a subproblem to retrieve (N 0,D 0 ) from [dbz b ] i at the two frequencies. This subproblem is a set of two nonlinear equations with two unknowns and is denoted by SP-B. Once SP-B is solved for range bin i,thedsd and its related variables for range bin i become known, and dba(r i 1 ) can be calculated as dba(r i ) 2 [k] i L. Then, the same process can be applied for range bin (i 1). Inthe BRM, as dba(r n ) is estimated by the SRT, the aforementioned process can start at range bin n. ME92 is categorized as BRM. C. IBRM The IBRM applies the same procedure as in the BRM but without the use of the SRT. Instead of estimating the PIA by the SRT, the PIA can be assumed arbitrarily. Once the PIA is assumed, the same procedure as in the BRM is applied to retrieve (N 0,D 0 ) and related variables of n range bins. The PIA can be calculated from the retrieved (N 0,D 0 ), and it is compared with the assumed PIA. This comparison is equivalent to checking whether dba(r 0 ) is retrieved to be zero. If the calculated PIA and the assumed PIA are not the same (if dba(r 0 ) is retrieved not to be zero), the assumed PIA is considered wrong. If the two PIAs are the same (if dba(r 0 ) is retrieved to be zero), the retrieved (N 0,D 0 ) is a possible solution, but it is not necessarily the true solution. Therefore, an infinite number of candidates of PIA should be tested in the IBRM, although it is impossible. MA04 is a simplified version of the IBRM, and it tests a finite number of candidates of PIA. In MA04, the PIA is assumed to be zero at the first iteration. Then, at the second iteration and later, the PIA is assumed to be the same as the PIA calculated in the previous iteration. The iteration stops once a possible solution is found. D. Comparison of Methods Both in the FRM and the IBRM, the solution should satisfy the condition that dba(r 0 ) is zero. Hereafter, this condition is called the upper boundary condition of dba. In the BRM, there is no checking of whether the upper boundary condition of dba is satisfied or not. On the other hand, the condition that the PIA (= dba(r n )) is true is called the lower boundary condition of dba. The lower boundary condition is satisfied in the BRM if the SRT is perfect. In terms of the boundary conditions of dba, the FRM and IBRM are equivalent, whereas the BRM differs from the FRM and IBRM. According to the categorization of retrieval methods for a single-frequency radar [16], the FRM and the IBRM are kinds of initial value methods, whereas the BRM is a kind of final value method. IV. SOLUTIONS OF THE SUBPROBLEMS The essence of the three kinds of retrieval methods can be reduced to either SP-F or SP-B. Except for solving the subproblems, no difficulties are involved in the retrieval methods. Therefore, in this section, solutions of SP-B and SP-F are investigated. A. SP-B Solving SP-B is relatively easy. As shown in (22), [dbz b ] is equal to [dbz e ] when α =1; therefore, [dbz b ] is simply written as follows for the two frequencies: [dbz b ] u =10log 10 (N 0 )+F u (D 0 ) (23) [dbz b ] a =10log 10 (N 0 )+F a (D 0 ) (24) where superscript u (a) indicates the Ku-band (Ka-band). By subtracting (23) from (24), (25) is given [dbz b ] δ = F δ (D 0 ) (25) where superscript δ indicates the difference between the Ka-band and the Ku-band so that X δ =(X a X u ) holds for any frequency-dependent variable and function X. Equation (25) has only one unknown parameter, which is D 0.Fig.3 shows the function of F δ (D 0 ). As shown in the figure, F δ (D 0 ) takes the maximum where D 0 is 0.82 mm. Hereafter, the D 0 that maximizes F δ (D 0 ) is denoted by D 0s. In this case (μ =1), D 0s =0.82 mm. When [dbz b ] δ is negative, D 0 is uniquely determined by (25). However, when [dbz b ] δ is positive, two different D 0 s can satisfy (25). One of the two D 0 s is larger than D 0s, and the other is smaller than D 0s. The unique solution is usually constrained such that D 0 is larger than or equal to D 0s. Later, this constraint is called the D 0 constraint. Under the D 0 constraint, D 0 is uniquely determined. Once D 0 is determined, N 0 can be calculated by substituting D 0 into (23) or (24). As shown previously, SP-B has only one solution under the D 0 constraint unless [dbz b ] is biased. When a [dbz b ] δ that

5 SETO AND IGUCHI: APPLICABILITY OF THE IBRM FOR THE GPM DUAL-FREQUENCY PRECIPITATION RADAR 1831 Fig. 3. Function of F δ (D 0 ). is larger than F δ (D 0s ) is given due to some errors (such as biases in the PIA), SP-B has no solutions. B. SP-F By (20), [dbz f ] is written as follows for the two frequencies when α =1: [dbz f ] u =10log 10 (N 0 )+F u (D 0 ) 2 N 0 G u (D 0 ) L (26) [dbz f ] a =10log 10 (N 0 )+F a (D 0 ) 2 N 0 G a (D 0 ) L. (27) The difference in [dbz f ] between the two frequencies is given in [dbz f ] δ = F δ (D 0 ) 2 N 0 G δ (D 0 ) L. (28) As the two unknown parameters remain in (28), it is more difficult to solve SP-F than SP-B. The solutions of SP-F are investigated as shown in Fig. 4. In Fig. 4, the horizontal axis is D 0, and the vertical axis is [dbz f ] δ. When [dbz f ] δ and D 0 are known, N 0 can be calculated by (28). Once N 0 is known, [dbz f ] u and [dbz f ] a can be calculated by substituting N 0 and D 0 into (26) and (27). Fig. 4(a) (d) shows dbn 0 (= 10 log 10 N 0 ), [dbz f ] u, [dbz f ] a, and the rain rate, respectively. In these figures, the gray shading indicates the region where N 0 is negative. On the borderline of the gray-shaded region, N 0 = 0. The borderline is the same as the line of F δ (D 0 ) shown in Fig. 3, as (28) becomes [dbz f ] δ = F δ (D 0 ) when N 0 =0.The dotted line in Fig. 4(b) corresponds to Δ[dBZ f ] u /ΔD 0 =0. On the plane of (D 0, [dbz f ] δ ), Δ[dBZ f ] u /ΔD 0 is defined by Δ[dBZ f ] u /ΔD 0 = { [dbz f ] u ( D 0 +ΔD 0, [dbz f ] δ) [dbz f ] u ( D 0, [dbz f ] δ)} /ΔD 0. (29) The solutions of SP-F are given as the crossing points of the two lines in Fig. 4(b): the line corresponding to a given [dbz f ] δ, which is parallel to the horizontal axis, and the contour for a given [dbz f ] u. For example, let us consider the case that [dbz f ] u =42dB and [dbz f ] a =37dB. The contour of [dbz f ] u =42dB has two crossing points with the horizontal line of [dbz f ] δ = 5 db. Then, two solutions are found at D 0 =1.08 mm and D 0 =1.78 mm. In this example, SP-F has two solutions. Moreover, Fig. 4(b) suggests that SP-F always has two solutions and that one solution is located left of the dotted line and the other solution is located right of the dotted line. In the aforementioned example, both solutions have D 0 larger than D 0s, so there are two solutions even with the D 0 constraint. Generally, it can be concluded that the number of solutions of SP-F without the D 0 constraint is always two and it is one or two with the D 0 constraint, unless [dbz f ] is biased. If a biased [dbz f ] is given because of some errors, SP-F may have no solutions. Fig. 4(c) can also be used to find the solutions of SP-F, and it can be confirmed that Fig. 4(c) gives the same solution as Fig. 4(b). In the aforementioned example ([dbz f ] u = 42 db and [dbz f ] a =37dB), the contour of [dbz f ] a =37dB crosses the line of [dbz f ] δ = 5 db at the two points: D 0 = 1.08 mm and D 0 =1.78 mm. In Fig. 4(c), the dotted line corresponds to Δ[dBZ f ] a /ΔD 0 =0, and it is the same as the line of Δ[dBZ f ] u /ΔD 0 =0because it always holds that Δ[dBZ f ] δ /ΔD 0 =0anywhere on the (D 0, [dbz f ] δ ) plane. V. S OLUTIONS OF THE MAIN PROBLEM The solutions of the main problem are discussed schematically as shown in Fig. 5. In the FRM without the D 0 constraint [Fig. 5(a)], the main problem for n range bins has, at most, 2 n solutions, as SP-F without the D 0 constraint has two solutions at each range bin. However, if the false solution is selected for SP-F at a range bin, SP-F at the following range bins may have no solutions because [dbz f ] is biased. Therefore, the main problem may not have 2 n solutions. In this example, the number of solutions is ten when n =4. In the FRM with the D 0 constraint [Fig. 5(b)], the main problem may have multiple solutions because SP-F with the D 0 constraint has one or two solutions. Among the ten solutions in Fig. 5(a), three solutions that do not satisfy the D 0 constraint are excluded, leaving seven candidate solutions. In the BRM without the D 0 constraint [Fig. 5(c)], there may be multiple solutions because SP-B with the D 0 constraint has one or two solutions. In the BRM with the D 0 constraint [Fig. 5(d)], there is always a unique solution because SP-B with the D 0 constraint has a unique solution, and the unique solution is the true solution only when the PIA by the SRT is true. As pointed out in Section III, the IBRM is equivalent to the FRM in terms of the boundary conditions of dba. Moreover, in the IBRM, the D 0 constraint needs to be applied to obtain the unique solution of SP-B. Therefore, the IBRM is equivalent to the FRM with the D 0 constraint. For example, in the case of Fig. 5, the IBRM has seven solutions. Based on this equivalence, the solutions and the applicability of the IBRM will be further discussed in the following.

6 1832 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 6, JUNE 2011 Fig. 4. Graphical investigation of the solutions of SP-F (α =1). In each figure, the horizontal axis is D 0, and the vertical axis is [dbz f ] δ. Contours are (a) dbn 0,(b)[dBZ f ] u,(c)[dbz f ] a, (d) the rain rate, (e) D 0 of the false solution, and (f) the rain rate. The dotted line indicates that [dbz f ] u / D 0 = [dbz f ] a / D 0 =0. The dashed line indicates that D 0 = D 0s. The red solid line indicates (b) the contour of [dbz f ] u =42dB and (c) the contour of [dbz f ] a =37dB. The blue solid line indicates [dbz f ] δ = 5 db. The green solid lines in (e) and (f) indicate that the D 0 of the false solution is equal to D 0s. The gray region indicates that N 0 is calculated to be negative, and no solutions can exist. In (f), the region with positive N 0 is divided into four regions: Region 0 to region 3. μ is set to one. Fig. 5. Examples of the solutions of the main problem: (a) In the FRM without the D 0 constraint, (b) in the FRM with the D 0 constraint, (c) in the BRM without the D 0 constraint, and (d) in the BRM with the D 0 constraint. Closed circles indicate the true solution, open circles indicate false solutions, and grayshaded circles indicate a solution with D 0 smaller than D 0s. The cross symbols indicate that the next range bin has no solutions. A. Applicability of the IBRM for General Cases With n =1 At first, the applicability of the IBRM is investigated for general cases with n =1. For example, let us consider the case where the true solution corresponds to (D 0, [dbz f ] δ )= (1.78 mm, 5 db). This point is on the contour of [dbz f ] u = 42 db, as shown in Fig. 4(b). In this case, another solution corresponds to the other crossing point of the contour of [dbz f ] u =42 db and the line of [dbz f ] δ = 5 db. This solution is called a false solution. The D 0 of the false solution is 1.08 mm. Fig. 4(e) shows the D 0 of the false solution. On the dotted line, the true and false solutions coincide to give the same D 0. Under the D 0 constraint, if the D 0 of a true solution is smaller than D 0s, the true solution is never selected. The case is categorized as type 0, and the region in Fig. 4(f) indicated by white background (region 0) corresponds to type-0 solutions. If D 0 of a true solution is larger than D 0s and if D 0 of the corresponding false solution is smaller than D 0s, the true solution is always selected. The case is categorized as type 1, and the region in Fig. 4(f) indicated by blue background (region 1) corresponds to type 1. The contours in Fig. 4(f) indicate the rain rate, as well as those in Fig. 4(d). In the case of type 0, the false solution exists in region 1. Moreover, in Fig. 4(f), region 2 is defined as the region right of the dotted line except for region 1, and region 3 is defined as the region left of the dotted line except for region 0. If a true

7 SETO AND IGUCHI: APPLICABILITY OF THE IBRM FOR THE GPM DUAL-FREQUENCY PRECIPITATION RADAR 1833 TABLE I CATEGORIZATION OF SP-F TABLE II SETTINGS OF THE CASE STUDIES solution exists in region 2 (region 3), the case is categorized as type 2 (type 3), and the corresponding false solution exists in region 3 (region 2). Both in the cases of type 2 and type 3, the IBRM has multiple solutions. The categorization is summarized in Table I. Only in the case of type 1, the IBRM certainly gives the true solution. As is shown in Fig. 4(f), the rain rate is less than 10 mm h 1 for type 1. The applicability of the IBRM is limited to relatively weak rainfall. B. Applicability of the IBRM for Some Cases Including n>1 Next, the applicability of the IBRM is investigated for some specific cases including n>1. The settings of the cases are given in Table II. When n>1, the DSD parameters are set to be the same for all the range bins, but it is not assumed in the retrieval algorithm. As has been noted, the essence of the main problem lies in subproblems for a single layer. The treatment of multiple range bins is straightforward; the fact that the DSD parameters are taken to be constant does not restrict the generality of the results. N 0 = in all the cases, and D 0 is set in the following manner. D 0 =1.4mm (R =5.7mm h 1 ) in cases a to c. D 0 =1.7 mm (R =16.9 mm h 1 ) in cases d to i. D 0 =2.0mm (R =41.9mm h 1 ) in cases j to l. In order to find solutions, the IBRM is applied in the following manner; the PIA at the Ku-band and the PIA at the Ka-band are independently assumed to be the true PIAs plus biases between 5 and 5 db (with a step of 0.1 db). In total, = candidates are tested. In this way of proceeding, false solutions with a bias of larger than 5 db or smaller than 5 db do not appear, but the primary solutions of the IBRM occur. The results are shown in Fig. 6. In each subfigure in Fig. 6, the horizontal (vertical) axis gives the biases in the PIA at the Ku-band (Ka-band), and a solid (dotted) line is drawn where dba(r 0 )=0holds at the Ku-band (Ka-band). Only at the origin, the lower boundary condition of dba is satisfied, implying the existence of the true solution. Crossing points of the solid line and the dotted line, where the upper boundary condition of dba is satisfied, correspond to possible solutions in the IBRM. The light gray shading indicates that an assumed PIA is negative, and the dark gray shading indicates that no solutions exist for the assumed PIA. 1) Type 1: Cases a to c are categorized into type 1. When n =1(case a), as shown in Fig. 6(a), the solid line and the dotted line cross only at the origin. This means that, whenever the upper boundary condition of dba is satisfied, the lower boundary condition of dba is satisfied. Therefore, the true solution can be given by the IBRM for case a. When n =2 [case b; shown in Fig. 6(b)] and when n =10[case c; shown in Fig. 6(c)], the solid line and the dotted line cross only at the origin. Hence, the true solution can be given by the IBRM for the two cases, as well as for case a. The aforementioned examples strongly suggest that the IBRM can always give the true solution as long as all the range bins are categorized as type 1. 2) Type 2: Cases d to i are categorized into type 2. When n =1(case d), as shown in Fig. 6(d), the solid line and the dotted line cross not only at the origin but also at another point (named point B). The solution corresponding to point B is a false solution, but it can be selected by the IBRM as it satisfies the upper boundary condition of dba. Therefore, the true solution cannot be always selected by the IBRM for case d. When n =2[case e; shown in Fig. 6(e)], the solid and dotted lines cross at the origin and at point B, which is exactly the same with point B of case d. The same PIA bias for cases d and e can be explained with Fig. 7, which is a schematic figure similar to Fig. 5, but the horizontal axis of Fig. 7 corresponds to the bias in dba(r i ). When the true solution is selected at range bin 1 and the false solution is selected at range bin 2, the bias in dba(r 2 ) or the PIA corresponds to point B because SP-F at range bin 2 is the same as SP-F at range bin 1. If the false solution is selected at range bin 1, [dbz f ] at range bin 2 is biased, and SP-F at range bin 2 has no solutions. Therefore, the number of solutions for case e is limited to two. As long as (N 0,D 0 ) = (10 000, 1.7 mm) in all the range bins, regardless of n, there will be two solutions: the true solution and the false solution which has the bias corresponding to point B. The latter solution is given when the true solution is selected at range bins 1 to (n 1), and the false solution is selected at range bin n. If the false solution is selected at any range bin between 1 and (n 1), SP-F at the next range bin has no solutions. Therefore, the number of solutions is limited to two for any n (Fig. 7). In Fig. 6(f) (i), we can see that the solid line crosses the dotted line at the origin and at point B when n =4, 6, 8, and 10, respectively. Furthermore, these figures show that the solid line and the dotted line become close to each other when n increases. When n =10 [Fig. 6(i)], the two lines appear to overlap. However, according to the investigation described earlier, there should be only two solutions for case i. Hence, the lines do not exactly overlap with each other in Fig. 6(i).

8 1834 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 6, JUNE 2011 Fig. 6. Results of the IBRM. In each figure, the horizontal (vertical) axis is the bias in the assumed PIA at the Ku-band (Ka-band), and a solid (dotted) line indicates that dba(r 0 ) becomes equal to zero at the Ku-band (Ka-band). The light gray shading indicates that the assumed PIA is negative, and the dark gray shading indicates that no solutions exist in the BRM with a given PIA. (a) to (l) correspond to cases a to l, respectively. (a) D 0 =1.4 mm, n =1; (b) D 0 =1.4 mm, n =2;(c)D 0 =1.4 mm, n =10;(d)D 0 =1.7 mm, n =1;(e)D 0 =1.7 mm, n =2; (f) D 0 =1.7 mm, n =4;(g)D 0 =1.7 mm, n =6;(h)D 0 =1.7mm, n =8;(i)D 0 =1.7mm, n =10;(j)D 0 =2.0mm, n =1;(k)D 0 =2.0mm, n =2;(l)D 0 =2.0mm, n =10. When the upper boundary condition of dba is exactly satisfied, the corresponding solution is called an exact solution. The true solution and the solution corresponding to point B are exact solutions. When the upper boundary condition of dba is almost but not exactly satisfied, the corresponding solution is called an approximate solution. Unless the numerical

9 SETO AND IGUCHI: APPLICABILITY OF THE IBRM FOR THE GPM DUAL-FREQUENCY PRECIPITATION RADAR 1835 Fig. 7. Structure of the solutions of the main problem in cases d to i. The horizontal axis corresponds to the bias of dba. The symbols have the same meaning as in Fig. 5. calculation is executed very precisely, it is difficult to distinguish approximate solutions from exact solutions, and thus, approximate solutions become possible solutions in the IBRM. In case i, there seem to be infinite possible solutions: Two of these are exact solutions, and the others are approximate solutions. Accumulation of numerical error can give rise to multiple approximate solutions when n is large. The aforementioned examples suggest that the IBRM cannot always give the true solution when some range bins are categorized into type 2. 3) Type 3: Cases j to l are categorized into type 3. When n =1[case j; shown in Fig. 6(j)], the solid line and the dotted line cross at the origin and another point (named point C). This means that the true solution cannot always be selected by the IBRM for case j. Point C has a negative bias in the PIA. When n =2[case k; shown in Fig. 6(k)], four crossing points can be seen: at the origin, point C, and other two points (named point D and point E). The origin and point C correspond to solutions when the true solution is selected at range bin 1. Points D and E correspond to solutions when the false solution is selected at range bin 1. When n increases, the solid line and the dotted line approach each other (figures not shown). When n =10[case l; shown in Fig. 6(l)], the two lines almost overlap. (In Fig. 6(l), part of the dotted line is drawn on the right-hand side of the figure, but the other part of the dotted line almost overlaps the solid line.) On the overlapped line, there are multiple exact solutions (the true solution and false solutions including those corresponding to points C, D, and E) and an infinite number of approximate solutions. The aforementioned case studies suggest that the true solution cannot always be selected when some range bins are categorized into type 3. As a conclusion, the IBRM can be applied only when all the ranges are categorized into type 1. C. Applicability of MA04 Next, the applicability of MA04 is discussed. MA04 is a simplified version of the IBRM, as explained in Section III. In MA04, once the upper boundary condition is satisfied (within a small error), the retrieved (N 0,D 0 ) is regarded as the unique Fig. 8. Portion of Fig. 4(f) is enlarged. Categorization into regions 0 to 3 is shown by different background colors, and the rain rate is indicated by the contours. The red/purple/blue lines indicate the boundary lines of the regions for which MA04 can give the true solution in the cases of n =6, n =9,and n =12according to Rose and Chandrasekar [8]. μ is set to one. solution, and the search is terminated. Rose and Chandrasekar [11] empirically derived the conditions in which MA04 can give the true solution under the same assumptions of this study (μ =1and α =1). The upper limit is given by the following equation: ( D 0 = a + b ) 2 Nw 0.5 (30) where a and b are constants dependent on n and N w is a normalized N 0 defined as follows: N w = N 0 D μ Γ(μ +4) 0. (31) 6 ( μ) μ+4 The lines of (30) are shown on the (D 0, [dbz f ] δ ) plane in Fig. 8 for the cases of n =6, 9, and 12. The threshold lines are drawn only where N w < 8000 and D 0 < 2.5 mm as Rose and Chandrasekar [11] assumed. Basically, Fig. 8 shows an enlarged portion of Fig. 4(f), so the categorization and the rain rate are shown as background colors and the contours, respectively. If the true solution is located to the right of the line (the rain rate is lighter than the upper limit), MA04 provides the true solution. The lines exist in region 2, indicating that MA04 can give the true solution for part of type 2. The reason why MA04 can give the true solution not only for type 1 but also for part of type 2 is explained as follows. MA04 assumes a PIA of zero in the first iteration and stops the iteration once a possible solution is given. Consequently, this method usually selects the solution with the smallest PIA among multiple possible solutions. In type 2, as the true solution has smaller PIA than the other exact solution(s), the true solution can be selected by MA04 when n

10 1836 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 6, JUNE 2011 Fig. 9. Same as in Fig. 4 but for α =0.5. μ is set to one. (a) dbn 0 ;(b)[dbz f ] u ;(c)[dbz f ] a ;(d)r (in millimeters per hour); (e) false D 0 (in millimeters); (f) region. is small and when rain rate is weak. However, when n becomes large or when rain rate becomes strong, approximate solutions with a smaller PIA than that of the true solution emerge and prevent MA04 from selecting the true solution. The upper limit of the rain rate for which MA04 can give the true solution is 12 to 18 mm h 1 in the case of n =12, according to Rose and Chandrasekar [11]. MA04 can be applied for heavier rainfall than the IBRM as it selects the solutions with the smallest PIA among multiple possible solutions. [dbz f ] a =10log 10 (N 0 )+F a (D 0 ) α 2 N 0 G a (D 0 ) L (33) [dbz f ] δ = F δ (D 0 ) α 2 N 0 G δ (D 0 ) L. (34) A variable X calculated with α is denoted by X{α}.In(34),as F δ (D 0 ) is independent of α, the following equations hold: N 0 {α} =(1/α) N 0 {1} (35) dbn 0 {α} = dbn 0 {1} +10log 10 (1/α). (36) VI. EFFECTS OF INTERNAL ATTENUATION Up to the previous section, the characteristics of the solutions of the main problem and the applicability of the IBRM and MA04 were investigated. In reality, the results are strongly dependent on the internal attenuation, which was explained in Section II-C. In the previous sections, α was set to one for consistency with previous studies. However, in nature, α should be larger than zero and smaller than one; hence, α =1overestimates the internal attenuation. In this section, investigation is conducted with different values of α (0 <α<1). First, the dependence of the results in Fig. 4 on α is examined. For a general α, (26) (28) are rewritten respectively as [dbz f ] u =10log 10 (N 0 )+F u (D 0 ) α 2 N 0 G u (D 0 ) L (32) Once N 0 is known, [dbz f ] u can be calculated by (32). As the third term of the right-hand side of (32) is equal to [dbz f ] δ F δ (D 0 ), it is independent of α. In the right-hand side of (32), only the first term is dependent of α. Therefore, [dbz f ] u {α} is given as follows: [dbz f ] u {α} =[dbz f ] u {1} +10log 10 (1/α). (37) In the same way, [dbz f ] a {α} is given as follows: [dbz f ] a {α} =[dbz f ] a {1} +10log 10 (1/α). (38) As R is proportional to N 0, R{α} is given as R{α} =(1/α) R{1}. (39)

11 SETO AND IGUCHI: APPLICABILITY OF THE IBRM FOR THE GPM DUAL-FREQUENCY PRECIPITATION RADAR 1837 Fig. 9 is the same as Fig. 4 except for α =0.5. In Fig. 9(a) (d), the α dependence of the variables shown in (36) (39) can be confirmed. Comparison of Figs. 4(f) and 9(f) suggests that any point on the (D 0, [dbz f ] δ ) plane belongs to the same region irrespective of α. This can be proved as follows. First, the borderline between region 0 and region 3 does not change with α because D 0s is independent of α. Next, the borderline of regions 3 and 2 (the line of Δ[dBZ f ] u /ΔD 0 =0) does not change with α because [dbz f ] u changes by the same amount with a change in α at any point on the (D 0, [dbz f ] δ ) plane. The reason why the borderline between region 2 and region 1 does not change with α can be explained as follows. For example, let us consider the case of (D 0, [dbz f ] δ )= (1.78 mm, 5 db). In Fig. 9(b), this point is on the contour of [dbz f ] u 45 db, which is the same as the contour of [dbz f ] u =42 db in Fig. 4(b). Therefore, D 0 of the false solution in the case of α =0.5 [Fig. 9(e)] is the same as that in the case of α =1.0 [Fig. 4(e)] and is equal to 1.08 mm. At any point on the (D 0, [dbz f ] δ ) plane, D 0 of the false solution does not change with α. Therefore, the borderline of region 1 and region 2, where D 0 of the false solution is equal to D 0s, does not change with α. Hence, it is proved that any point on the (D 0, [dbz f ] δ ) plane belongs to the same region, irrespective of α. As is given in (39), the rain rate is dependent on α. Inthe case of α =0.5, the rain rate corresponding to the borderline of region 1 and region 2 (as the upper limit for which the IBRM can be applied) is twice as large as that of the case of α =1 and is 10 to 20 mm h 1 when α =0.5. Inasimilarway,we can guess that the upper limit of rain rate that MA04 can be applied in the case of n =12should be around 24 to 36 mm h 1 if α =0.5 by doubling the estimates in [11]. In previous studies and in this study up to Section V, the applicability of the IBRM and that of MA04 were underestimated because the internal attenuation was overestimated. We can see that, in (32) (34), α and L always appear as their product (α L). Hence, the effects of L on the investigation are the same as those of α. For example, if L is halved to km, the upper limit of the rain rate for which MA04 can be applied is doubled compared with the case of L =0.25 km. This resolution (L =0.125 km) will be realized by the DPR with oversampling in most range bins. For example, the upper limit for which the IBRM can be applied is 20 to 40 mm h 1 when α =0.5 and L =0.125 km. As an extreme case, if L becomes zero (if Z m (r) is measured as a continuous function), the upper limit becomes infinite, which means that the IBRM and MA04 can always give the true solution as long as D 0 is larger than D 0s. VII. SUMMARY AND CONCLUSION Previous studies [11] [13] reported that MA04 cannot give the true solution for heavier rainfall. This study shows the essential conditions under which MA04 and the IBRM can give the true solution. MA04 is a simplified version of the IBRM, and the IBRM is equivalent to the FRM with the D 0 constraint. The IBRM can always give the true solution only when the true solution belongs to region 1 at all the ranges. Otherwise, the IBRM has multiple solutions and cannot necessarily give the true solution. The rain rate corresponding to region 1 is no larger than 20 mm h 1 (in the case of α =0.5 and L =0.25 km). The applicability of MA04 is wider than that of the IBRM as MA04 tends to select the solution with the smallest PIA among multiple possible solutions by the IBRM. When n increases, the upper limit decreases because of the advent of approximate solutions. Rose and Chandrasekar [11] showed that the upper limit was around 12 to 18 mm h 1 when n =12, but the result was underestimated because they overestimated the internal attenuation by setting α =1. In the case of α =0.5 and L = 0.25 km, the value should be doubled at 24 to 36 mm h 1. In this paper, to show the essence of the applicability of MA04 and the IBRM, the following simplified assumptions are applied: 1) vertically constant profile; 2) liquid precipitation only; and 3) no observation errors in Z m. Although these assumptions are not realistic, the basic findings of this paper are not affected. 1) The main problem can be decomposed into subproblems, and the details of the vertical profile are not crucial to the basic results. 2) As solid and melting particles are not involved in this paper, we should investigate also for the layers with solid and melting particles in near-future studies. For the liquid precipitation layer, the analysis of this study is useful even if solid and melting precipitation layers exist over liquid precipitation layers. Once the attenuation by solid and melting precipitation layers is retrieved, by correcting Z m for the attenuation in advance, Z m can be assumed to have no attenuation by nonliquid precipitation particles. 3) The inclusion of observational errors in Z m into the analysis obscures the basic characteristics of the findings; since this is the main objective of this paper, this error source has been excluded from consideration. To develop a robust algorithm, observation errors in Z m should be considered in the near future. REFERENCES [1] J. Goldhirsh and I. Katz, Estimation of rain drop size distribution using multiple wavelength radar systems, Radio Sci.,vol.9,no.4,pp , [2] M. Fujita, An algorithm for estimating rain rate by a dual-frequency radar, Radio Sci., vol. 18, no. 5, pp , [3] J. Testud, P. Amayenc, and M. Marzoug, Rainfall-rate retrieval from a spaceborne radar: Comparison between single frequency, dual frequency, dual-beam techniques, J. Atmos. Ocean. Technol., vol. 9, no. 5, pp , Oct [4] L. Liao, R. Meneghini, L. Tian, and G. M. Heymsfield, Retrieval of snow and rain from combined X- and W-band airborne radar measurements, IEEE Trans. Geosci. Remote Sens.,vol.46,no.5,pp , May [5] 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 [6] R. Meneghini, H. Kumagai, J. R. Wang, T. Iguchi, and T. Kozu, Microphysical retrievals over stratiform rain using measurements from an airborne dual-wavelength radar-radiometer, IEEE Trans. Geosci. Remote Sens., vol. 35, no. 3, pp , May [7] R. Meneghini, J. A. Jones, T. Iguchi, K. Okamoto, and J. Kwiatkowski, A hybrid surface reference technique and its application to the TRMM Precipitation Radar, J. Atmos. Ocean. Technol.,vol.21,no.11,pp , Nov

12 1838 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 6, JUNE 2011 [8] S. Seto and T. Iguchi, Rainfall-induced changes in actual surface backscattering cross sections and effects on rain-rate estimates by spaceborne precipitation radar, J. Atmos. Ocean. Technol., vol. 24, no. 10, pp , Oct [9] R. Mardiana, T. Iguchi, and N. Takahashi, A dual-frequency rain profiling method without the use of a surface reference technique, IEEE Trans. Geosci. Remote Sens., vol. 42, no. 10, pp , Oct [10] D. E. Weissman and M. A. Bourassa, Measurements of the effect of rain-induced sea surface roughness on the QuickSCAT scatterometer radar cross section, IEEE Trans. Geosci. Remote Sens., vol. 46, no. 10, pp , Oct [11] C. R. Rose and V. Chandrasekar, A systems approach to GPM dualfrequency retrieval, IEEE Trans. Geosci. Remote Sens., vol. 43, no. 8, pp , Aug [12] 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 , Nov [13] N. B. Adhikari, T. Iguchi, S. Seto, and N. Takahashi, Rain retrieval performance of a dual-frequency precipitation radar technique with differential-attenuation constraint, IEEE Trans. Geosci. Remote Sens., vol. 45, no. 8, pp , Aug [14] R. Gunn and G. G. Kinzer, The terminal velocity of fall for water droplets in stagnant air, J. Meteorol., vol. 6, pp , [15] T. Kozu, K. Nakamura, R. Meneghini, and W. C. Boncyk, Dualparameter radar rainfall measurement from space: A test result from an aircraft experiment, IEEE Trans. Geosci. Remote Sens., vol. 29, no. 5, pp , Sep [16] T. Iguchi and R. Meneghini, Intercomparison of single-frequency methods for retrieving a vertical rain profile from airborne or spaceborne radar data, J. Atmos. Ocean. Technol., vol. 11, no. 6, pp , Dec Shinta Seto (M 10) received the B.E., M.E., and Ph.D. degrees from The University of Tokyo, Tokyo, Japan, in 1998, 2000, and 2003, respectively. From 2003 to 2006, he was a Postdoctoral Researcher with the National Institute of Information and Communications Technology (formerly the Communications Research Laboratory), Koganei, Japan, where he worked on the development of spaceborne dual-frequency precipitation radar. Since 2006, he has been with the Institute of Industrial Science, The University of Tokyo. His current research interests include precipitation retrieval using microwave remote sensing and its application to water cycle studies. Toshio Iguchi (M 97) received the B.Sc. degree from Hokkaido University, Sapporo, Japan, in 1976, the M.Sc. degree from The University of Tokyo, Tokyo, Japan, in 1978, and the Ph.D. degree from York University, Toronto, ON, Canada, in Since 1985, he has been with the National Institute of Information and Communications Technology (formerly the Communications Research Laboratory), Koganei, Japan. From 1991 to 1994, he visited the Goddard Space Flight Center, National Aeronautics and Space Administration, Greenbelt, MD, performing the U.S. Japan collaborative experiment for measuring rain using airborne radar. Since then, he has focused primarily on issues related to remote sensing of precipitation from space.