APPLICABILITY OF THE ITERATIVE BACKWARD RETRIEVAL METHODFOR THE GPM DUAL-FREQUENCY PRECIPITATION RADAR
Shinta Seto
Institute of Industrial Science, the University of Tokyo
Toshio Iguchi
National Institute of Information and Communications Technology (NICT), Japan

1. INTRODUCTION
The Dual-frequency Precipitation Radar (DPR) on the core satellite of the Global Precipitation
Measurement (GPM) mission will measure radar reflectivity factor (Zm) in the Ku-band (13.6 GHz) and Ka-band(35.5GHz). There have been several methods to retrieve two parameters (N0, D0) of the drop size distribution(DSD) function from the Zm in the two frequencies assuming that the DSD function is modeled by a gamma
distribution function as shown in Eq. (1).

N ( D)

N 0 D P exp> (3.67  P ) D / D0 @,(1)

where P is the third parameter of the DSD function, which is a known parameter in this study.
While Meneghini et al. [1] developed a retrieval method with the use of surface reference technique (SRT),
Mardiana et al. [2] proposed another retrieval method without the use of SRT. However, it was pointed out that
Mardiana’s method (referred to as MA04 hereafter) cannot give the true solution for heavier rainfall cases [3].
The purpose of this study is to show the conditions that MA04 can give the true solution.
2. APPLICABILITY OF THE IBRM AND MA04
MA04 is a special case of the iterative backward retrieval method (IBRM), and the IBRM is equivalent to
the forward retrieval method (FRM) with a constraint on D0 (D0-constraint) [D0 is larger than 0.82mm in the caseof P=1]. In the FRM, a problem to retrieve (N0, D0) from Zm at each range always has two solutions: the truesolution and a false solution. When the true solution does not satisfy the D0-constraint (type-0), the IBRM cannever give the true solution. On the other hand, when the true solution satisfies the D0-constraint and the falsesolution does not (type-1), the IBRM can always give the true solution. In other cases (type-2 and type-3), as
both the true solution and the false solution satisfy the D0-constraint, the IBRM cannot always give the truesolution.

Fig. 1. Categorization into the types 0 to 3 in thecase of P=1, D=1, and L=0.25[km]. The contourshows the rain rate [mm h-1]

Fig. 2. The part of Fig. 1 is enlarged. The threelines indicate the upper limit of the rain rate forwhich Mardiana’s method can be applied.

Figure 1 shows categorization into the four types on (D0, [dBZf]G) plane. [dBZf]G is the difference of [dBZf]
between the Ka-band and the Ku-band, and [dBZf] at a discrete range bin is defined as Eq. (2).
[dBZf]҂[dBZe]íD×2×[k]×L,(2)

where [dBZe] is the effective radar reflectivity factor (in decibel), [k] is the attenuation coefficient, L is the widthof the range bin (=0.25km), and D is a parameter. The second term of the right-hand side of Eq. (2) indicates theattenuation occurred inside the range bin (called “internal attenuation” in this study). The value of D should belarger than 0 and smaller than 1, but D is set to be 1 in this study as well as in the previous studies such as [1], [2],and [3]. Therefore, [dBZf] of a range bin can be determined only by (N0, D0) of the range bin. In Fig. 1, thecontour indicates rain rate of the range. The rain rate for which the IBRM can always give the true solution is lessthan 10 mm h-1 (depending on D0).
Figure 2 enlarges a portion of Fig. 1. The three lines show the upper limit of the rain rate for which MA04
can give the true solution according to [3]. It is shown that MA04 can give the true solution not only for type-1
but for part of type-2, where path integrated attenuation (PIA) of the true solution is smaller than that of the falsesolution. As MA04 tends to select the solution with the smallest PIA when the IBRM has multiple solutions, it
can give the true solution for part of type-2. However, if the number of range bins (n) is large, false solutions witha smaller PIA than that of the true solution emerge because of the accumulation of numerical error, and it
becomes difficult for MA04 to select the true solution.

3. APPLICATION OF MA04 TO SIMULATED DATASETTo check the applicability of MA04 under realistic conditions, a simulated DPR observation dataset wasproduced based on the TRMM standard product 2A25 (version 6) as follows.
(1) In 2A25, range bins with liquid precipitation (from node-D and clutter free bottom) are taken.
(2) k can be calculated by k=aZeb (a, b, Ze are given in 2A25)(3) As k/Ze is a function of D0, D0 can be retrieved, and N0 is also retrieved by substituting D0 into k or Ze.
According to the 2A25 standard algorithm, P=3 is assumed. The physical temperature of rain drops (T) is set as0oC at node-D and increases by 1.5oC per range bin.
(4) From D0, N0, and P(=3), Ze and k in the two frequencies of DPR are calculated.
(5) Zm is calculated by assuming D=1. No attenuation above the top of node-D and no attenuation by nonprecipitating particles are considered.
(6) Where Zm in the Ka-band is less than 12dB, the range bin and lower range bins are excluded from simulationdata.
The simulated dataset was produced for one month (July 2001). The categorization explained in Sec. 2 isapplied to the dataset. If all the range bins in a pixel is categorized into the same type, the pixel is called auniform type pixel, otherwise it is called a mixed type pixel. Almost half of the rain pixels are uniform typepixels. Among uniform type pixels, type-1 is dominant, and type-0 is second dominant in terms of the number ofpixels, but type-2 and type-3 are more dominant than type-0 in terms of total amount of rainfall.
MA04 is applied and the retrieved rain rate at the lowest range bin is evaluated. Figure 3 shows the results
for uniform type pixels. In most cases of type-0, retrieved rain rate is underestimated. On the other hand, in mostcases of type-1, retrieved rain rate is very close to the true value. In type-2, there is severe underestimation
particularly when the number of range bins is large. When the number of range bins is 1 to 4, the results arepretty good. In type-3, regardless of the number of range bins, the retrieved rain rate is mostly underestimated.
These results correspond well to the discussion in sec. 2; MA04 can give the true solution in type-1 and par t of
type-2 (when the number of range bins is small).
4. SUMMARY AND FUTURE STUDIESThe limitation of applicability of MA04 has been pointed out by previous studies, and the conditions on
which MA04 can give the true solution is discussed objectively and is tested by using a simulated DPR
observation dataset in this study. In near future studies, more realistic simulated dataset including solid andmelting layers and with observation errors should be used to test actual applicability of MA04, and MA04 should
be modified to have no underestimation.

Fig. 3 The evaluation of MA04 for type-0 through type-3 (uniform pixels). The horizontal (vertical) axis isthe true (retrieved) rain rate at the lowest range bin. Dark (light) shade indicates the large (small) populationof pixels. Colored lines show the average of retrieved rain rate (the number of range bins is 1-4 for blue, 5-8
for light blue, 9-12 for green, 13-16 for yellow, and 17-20 for red).

REFERENCES[1] Meneghini, R., T. Kozu, H. Kumagai, and W. C. Boncyk, “A study of rain estimation methods from space using dual-wavelength radarmeasurements at near-nadir incidence over ocean.”, J. Atmos. Oceanic Technol., Vol. 9, pp364-382, 1992.
[2] Mardiana, R., T. Iguchi, and N. Takahashi, “A dualfrequency rain profiling method without the use of a surface reference technique”,
IEEE Trans. Geosci. Remote Sens., Vol. 42, pp2214-2225, 2004.
[3] Rose, C. R., and V. Chandrasekar, “A systems approach to GPM dual-frequency retrieval”, IEEE Trans. Geosci. Remote Sens., Vol. 43,
pp1816-1826, 2005.