## 1. Introduction

Dual-frequency precipitation radar (DPR) on the core satellite of the Global Precipitation Measurement (GPM) mission consists of two radars: a Ku-band precipitation radar (KuPR) and a Ka-band precipitation radar (KaPR). KuPR uses the frequency of 13.6 GHz and is similar to the Tropical Rainfall Measurement Mission (TRMM)’s precipitation radar (PR) with a frequency of 13.8 GHz. KaPR uses a higher frequency of 35.5 GHz. The scan width of KuPR is about 250 km, and each scan has 49 pixels; this is regarded as a normal scan. For each pixel of a normal scan, KuPR measures the vertical profile of precipitation echoes with an interval of 125 m and a resolution of 250 m. When KuPR measures the inner half of a normal scan, KaPR examines the same location, so that dual-frequency measurements are available. However, when KuPR measures the outer half of a normal scan, KaPR examines the area interleaved by two normal scans (called an interleaved scan) and measures the vertical profile of precipitation echoes with an interval of 250 m and a resolution of 500 m to detect weaker precipitation. In summary, DPR produces three types of measurements: 1) dual-frequency measurement (for the inner half of normal scans); 2) KuPR’s single-frequency measurement (for the outer half of normal scans); and 3) KaPR’s single-frequency measurement (for interleaved scans). The DPR level-2 standard algorithm was developed for all three measurement types.

In our previous dual-frequency precipitation retrieval algorithm (Seto et al. 2013), the attenuation correction method of Hitschfeld and Bordan (1954; HB method) was used in combination with the dual-frequency ratio (DFR) of the effective radar reflectivity factor [denoted by *Z*_{e} (mm^{6} m^{−3})]. Therefore, this algorithm is called the HB-DFR method (H-D method). The HB method assumes the relationship between specific attenuation [denoted by *k* (dB km^{−1})] and *Z*_{e} to be *k* = *αZ*_{e}^{β} [*α* (dB km^{−1} mm^{−6β} m^{3β}) and *β* are predetermined parameters], whereas the H-D method modifies the *k*–*Z*_{e} relationship to *k* = *εαZ*_{e}^{β} (*ε* is a correction factor) by means of DFR and uses this for the attenuation correction. The H-D method is applicable to the vertical profile, where single-frequency measurements and dual-frequency measurements coexist. This is called a partial-dual-frequency measurement, and it can occur often for type 1 measurements due to differences in the minimum detection level and attenuation between the two radars. If the H-D method is applied to single-frequency measurements, then *ε* is set to 1, so that the H-D and H-B methods are identical.

The H-D method does not work well for heavy precipitation. Similar results can be obtained by the iterative backward retrieval method (Mardiana et al. 2004; Rose and Chandrasekar 2005, 2006a,b; Adhikari et al. 2007; Seto and Iguchi 2011). The deficiency can be attributed to the fact that two drop size distribution (DSD) parameters are generally not uniquely determined by dual-frequency-measured radar reflectivity factors (denoted by *Z*_{m}). To solve this problem, additional information is necessary. In this study, the surface reference technique (SRT) is introduced to improve the H-D method.

The new attenuation correction method is called the HB-DFR-SRT method (H-D-S method). The method uses DFR and the SRT to modify the *k*–*Z*_{e} relationship. The details of the H-D-S method and other attenuation correction methods are given in section 2. The synthetic dataset of DPR used for the evaluation is explained in section 3. In section 4, the methods are tested under an idealized condition with no vertical variation in *ε* and no SRT errors, but the effects of vertical variation in *ε* and SRT errors on the performance of attenuation correction are discussed in sections 5 and 6, respectively. Section 7 is for a summary and conclusions.

## 2. Attenuation correction methods

In this section, the H-D-S method is explained after some previous attenuation correction methods are reviewed.

### a. HB method

*k*–

*Z*

_{e}relationship shown below:

*r*(km) is the range or distance from the radar,

*α*is a parameter dependent on the range, and

*β*is a parameter independent of the range. With Eq. (1), the vertical profile of

*Z*

_{e}is analytically derived from that of

*Z*

_{m}(mm

^{6}m

^{−3}) by,

*ζ*is defined by the following:

*s*(km) is a dummy parameter of

*r*.

### b. H-S method

*k*–

*Z*

_{e}relationship. The SRT is used to estimate path-integrated attenuation [PIA (dB)] by comparing the measured surface backscattering cross section [

*σ*

^{0}

_{m}(dB)] and the reference surface backscattering cross section [

*σ*

^{0}

_{ref}(dB)] below (Meneghini et al. 2000, 2004),

*k*–

*Z*

_{e}relationship of Eq. (1), the PIA is calculated by

*r*

_{s}(km) is the distance from the radar to the earth’s surface. PIAs calculated by Eqs. (4) and (5) are generally not the same. The

*k*–

*Z*

_{e}relationship is then modified to

*ε*

_{S}is a constant independent of

*r*. With this

*k*–

*Z*

_{e}relationship,

*Z*

_{e}(

*r*) and PIA are calculated below:

*ε*

_{S}so that the two PIAs calculated by Eqs. (4) and (8) are equal. If we consider the errors in the PIA estimated by the SRT,

*ε*

_{S}is determined in the same way as in the TRMM PR standard algorithm (Iguchi et al. 2000, 2009). The latter case is explained in more detail in section 6.

### c. H-D method

The H-D method is an attenuation correction method for dual-frequency measurements. Here, DFR is defined as the ratio of KaPR’s *Z*_{e} to KuPR’s *Z*_{e}. The relationship between DFR and DSD is reviewed before the procedure of the H-D method is explained.

*N*

_{w}(m

^{−3}mm

^{−1}) and

*D*

_{m}(mm):

*D*(mm) is drop size,

*N*(m

^{−3}mm

^{−1}) is the number density of rain drops, and

*n*is defined below:

*μ*is the third DSD parameter but is fixed to 3 in this study, and Γ is the complete gamma function. When the DSD is given by Eq. (9),

*Z*

_{e}is calculated by the following:

*f*(s

^{−1}) is frequency and

*I*

_{b}is defined by

*c*(mm s

^{−1}) is the speed of an electromagnetic wave,

*σ*

_{b}(mm

^{2}) is the backscattering cross section, and

*K*is a function of the refractivity factor. DFR is a function of

*D*

_{m}and is independent of

*N*

_{w}, as follows:

*f*

_{1}= 13.6 GHz and

*f*

_{2}= 35.5 GHz.

The procedure for the H-D method is as follows:

At each frequency, the HB method is applied with the initial

*k*–*Z*_{e}relationship of Eq. (1).At each range bin, DFR is calculated from the dual-frequency

*Z*_{e}*s*. Term*D*_{m}is derived from DFR by Eq. (13), and*N*_{w}is calculated from*D*_{m}and*Z*_{e}by Eq. (11). DSD parameters can be obtained at this point, but they are tentative estimates.- At each frequency, from the tentative DSD estimates, the specific attenuation is calculated bywhere
*I*_{e}is defined bywhere*σ*_{e}(mm^{2}) is the extinction cross section. Generally,*k*calculated from the DSD by Eq. (14) and*Z*_{e}calculated by the HB method contradict the assumed*k*–*Z*_{e}relationship. This implies that the assumed*k*–*Z*_{e}relationship is not appropriate. Then, the*k*–*Z*_{e}relationship is modified to Eq. (16) so that it agrees with the calculated*k*and*Z*_{e}:where*ε*_{D}, unlike*ε*_{S}, is range dependent. Since*ε*_{D}is adjusted at each range bin in the H-D method, it has a greater number of degrees of freedom than the H-S method. - The HB method is applied with the modified
*k*–*Z*_{e}relationship of Eq. (16). Factor*Z*_{e}is given bywhere*ζ*_{D}is defined by

*ε*

_{D}converges. In practice, the maximum number of iterations is 100.

### d. H-D-S method

*k*–

*Z*

_{e}relationship to

*ε*

_{S}and

*ε*

_{D}are determined sequentially. First,

*ε*

_{S}is set to 1,

*ε*

_{D}is determined using the H-D method, and then

*ε*

_{S}is determined. If we assume that the SRT is perfectly correct, then

*ε*

_{S}can be determined, so that the two PIA estimates calculated by Eqs. (4) and (20) are equal. As is the case for the H-D method, the H-D-S method can be applied to partial-dual-frequency measurements. If the H-D-S method is applied for a single-frequency measurement, it works as the H-S method. Therefore, we believe that the H-D-S method is suitable for DPR where single-frequency and dual-frequency measurements coexist.

*r*

_{t}(km) denotes the distance from the radar to the top of precipitation, and PIA

_{0}(dB) denotes a given PIA (e.g., estimated by the SRT). The H-S method is a single-frequency attenuation correction method, which satisfies both initial and PIA conditions. However, as far as we know, no dual-frequency attenuation correction methods other than the H-D-S method always satisfy both the initial and PIA conditions. For example, the backward retrieval method (Meneghini et al. 1992, 1997) always satisfies the PIA condition, but it generally does not satisfy the initial condition. Conversely, forward retrieval methods, such as the H-D method and the iterative backward retrieval method, always satisfy the initial condition but generally do not satisfy the PIA condition.

## 3. Evaluation with synthetic dataset

In this section, a simple synthetic dataset of DPR measurements is prepared to evaluate the attenuation correction methods.

### a. Synthetic dataset

A synthetic dataset of DPR measurements was produced as described in Seto et al. (2013). The process used to produce the synthetic dataset is briefly explained here, but please see Seto et al. (2013) for the full details. The dataset contains the vertical profile of *Z*_{m} at KuPR and KaPR with an interval of 250 m and a resolution of 250 m (oversampling is not represented).

*Z*

_{e}and the parameters for the

*k*–

*Z*

_{e}relationship (

*ε*,

*α*, and

*β*) were taken from the TRMM PR standard product, 2A25. For each range bin,

*k*was calculated. According to Eqs. (11) and (14), the ratio of

*k*to

*Z*

_{e}is a function of

*D*

_{m}, not

*N*

_{w}, as shown below:

*D*

_{m}from

*k*/

*Z*

_{e}, and then

*N*

_{w}is calculated from

*D*

_{m}and

*Z*

_{e}by Eq. (11). Precipitation rate [denoted by

*R*(mm h

^{−1})] is calculated by Eq. (24). This value is used as the truth for evaluation:

*C*

_{R}= 0.6π × 10

^{−3}and

*V*(m s

^{−1}) is the falling velocity, which is given below for liquid precipitation (Gunn and Kinzer 1949):

Neglecting the difference in frequency between TRMM PR and KuPR, the *k* and *Z*_{e} of KuPR are assumed to be the same as those of TRMM PR. For KaPR, from the DSD parameters, *k* and *Z*_{e} are calculated by Eqs. (11) and (14) with *f* = 35.5 GHz.

*Z*

_{m}is calculated by

*Z*

_{m}at each range bin is given approximately by the equation below:

*i*) denotes the

*i*th range bin from the top of precipitation,

*j*is a dummy parameter of

*i*, and

*L*(km) is the vertical width of the range bin (=0.25 km). The third term on the right-hand side of Eq. (27) is attenuation attributed to the precipitation particles in the

*i*th range bin.

No measurement errors were added to *Z*_{m}. No noise effects were considered, so *Z*_{m} was used for calculation even if it was lower than the actual minimum detection level of around 18 dB*Z*. No clutter effects were simulated. Multiple scattering, nonuniform beamfilling, and high-density ice particles (graupel and hail) may affect real measurements of DPR, but none of them was considered. Conditions were assumed so that the basic performance of the attenuation correction methods could be evaluated and compared. An evaluation of the DPR level 2 standard algorithm, in which the H-D-S method is used under realistic conditions including the effects of noise, cloud liquid water, and surface clutter, is given in Kubota et al. (2014).

In this study, 125 orbits of the TRMM PR standard product (orbit numbers 20675–20799, observed in July 2001) were used to make the synthetic dataset. Profiles with a bright band were used because some profiles without a bright band were not of very good quality.

### b. Setting

*k*–

*Z*

_{e}relationships of the H-S, H-D, and H-D-S methods can be written in the following common form:

*ε*(

*r*) =

*ε*

_{S}in the H-S method,

*ε*(

*r*) =

*ε*

_{D}(

*r*) in the H-D method, and

*ε*(

*r*) =

*ε*

_{S}

*ε*

_{D}(

*r*) in the H-D-S method. The parameters of

*α*and

*β*are common to all methods. In section 4, the

*α*and

*β*of KuPR are the same as those of TRMM PR. This makes the

*ε*of KuPR the same as that of TRMM PR, which is vertically constant. However, in reality, we cannot set

*α*and

*β*so that

*ε*becomes vertically constant. In sections 5 and 6, the

*α*of KuPR is intentionally made different from that of TRMM PR. In all sections, the

*α*of KaPR is 8 times as large as the

*α*of KuPR at each range bin, and the

*β*of KaPR is the same as that of KuPR.

### c. Precipitation rates

For quantitative evaluation, the precipitation rates were calculated in the following common way after attenuation correction. From the *k* and *Z*_{e} estimates of KuPR, the *D*_{m} and *N*_{w} are calculated by Eqs. (23) and (11), and *R* is calculated from the DSD by Eq. (24). This value is compared with the true *R*. If *Z*_{e} is overestimated and *ε* is correct, then *R* is always overestimated. If *ε* is overestimated and *Z*_{e} is correct, then *R* is generally overestimated. Therefore, the evaluation of *R* is not only an evaluation of *Z*_{e} or the result of the attenuation correction, but it is also an evaluation of *ε*, or the *k*–*Z*_{e} relationship used for attenuation correction.

## 4. Results under ideal conditions

In this section, the attenuation correction methods are evaluated under the condition where PIA_SRT is perfect, and the parameter *ε* does not change vertically.

### a. Surface precipitation rate

For each pixel, the precipitation rate at the lowest range bin is called the surface precipitation rate (denoted by *R*_{s}). Figure 1 shows a comparison of *R*_{s} estimated by the H-S method and the true *R*_{s}. Figure 1a extends for a range of 0.1–100 mm h^{−1} on a log scale, and Fig. 1b extends for a range of 0–5 mm h^{−1} on a linear scale. The H-S method estimates *R*_{s} almost perfectly.

In Fig. 2, *R*_{s} estimated by the H-D method is compared with the true *R*_{s}. When the true *R*_{s} is 2 mm h^{−1} or less, the estimates are generally good, but when the true *R*_{s} is larger than 2 mm h^{−1}, *R*_{s} is underestimated. According to Seto and Iguchi (2011), when dual-frequency *Z*_{m}*s* are given, two DSD parameters are not uniquely determined. The H-D method tends to select solutions with lower *R*_{s} values; hence, it does not work well for nonweak precipitation. In Fig. 2a, an underestimation is also apparent when the true *R*_{s} is less than 0.5 mm h^{−1}. This is because *D*_{m} is not uniquely determined, and a larger *D*_{m} is selected when DFR is larger than 1. However, this error is not significant on a linear scale, as shown in Fig. 2b.

As in Fig. 1, but for the H-D method.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

As in Fig. 1, but for the H-D method.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

As in Fig. 1, but for the H-D method.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

An evaluation for the H-D-S method is shown in Fig. 3. By using the SRT, the H-D-S method partly mitigates the negative bias found in the H-D method. However, an underestimation is apparent when the true *R*_{s} is higher than 2 mm h^{−1}. The reason for this will be examined in the next subsection.

As in Fig. 1, but for the H-D-S method.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

As in Fig. 1, but for the H-D-S method.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

As in Fig. 1, but for the H-D-S method.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

### b. Vertical profile

Figure 4 shows the averaged vertical profile of estimates of some variables when the true *R*_{s} is between 2 and 10 mm h^{−1}, or when the estimates of *R*_{s} differ substantially among the methods. The vertical axis shows the position relative to bright band. At each relative position, the variables are averaged. Because the number of liquid-precipitation range bins is fixed to 12 (including the bottom of the bright band), the number of samples is constant from the top of the bright band to the surface. This number (12) is determined to maximize the number of samples (10 858).

Comparison of the averaged vertical profile of (a) *ε* of KuPR, (b) deviation of *Z*_{e} of KuPR from the truth, and (c) precipitation rates under the condition that the SRT has no error and *ε* is vertically constant. A bright band exists in the area with a darker background.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

Comparison of the averaged vertical profile of (a) *ε* of KuPR, (b) deviation of *Z*_{e} of KuPR from the truth, and (c) precipitation rates under the condition that the SRT has no error and *ε* is vertically constant. A bright band exists in the area with a darker background.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

Comparison of the averaged vertical profile of (a) *ε* of KuPR, (b) deviation of *Z*_{e} of KuPR from the truth, and (c) precipitation rates under the condition that the SRT has no error and *ε* is vertically constant. A bright band exists in the area with a darker background.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

Figure 4a shows the vertical profile of the *ε* of KuPR. The truth is vertically constant and is slightly larger than 1. The H-S method gives almost perfect estimates. As long as *ε* is vertically constant, *ε* can be represented only by *ε*_{S}. The H-D method gives correct estimates in upper-range bins, but it shows a negative bias at lower-range bins. The negative bias expands downward. Because the H-D-S method attempts to correct the negative bias by using a vertically constant *ε*_{S}, *ε* cannot be effectively corrected. A positive bias in *ε* emerges in upper-range bins and the negative bias in *ε* does not disappear in lower-range bins.

Figure 4b shows the vertical profile of *Z*_{e} of KuPR, but the deviation of *Z*_{e} from the truth on the decibel scale is shown for a better representation. In the H-S method, because the *k*–*Z*_{e} relationship is estimated perfectly, the estimated *Z*_{e} is also perfect. In the H-D method, because *ε* is negatively biased in lower-range bins, *Z*_{e} is also underestimated. In the H-D-S method, because *ε* is overestimated in upper-range bins and is underestimated in lower-range bins, the positive bias in *Z*_{e} first increases but later decreases downward. Term *Z*_{e} at the surface is estimated almost perfectly because the PIA is given correctly.

Figure 4c shows the vertical profile of *R*. In the H-S method, the vertical profiles of *ε* and *Z*_{e} are perfectly estimated, and the vertical profile of estimated *R* is also perfect. In the H-D method, both *ε* and *Z*_{e} are underestimated, and *R* is negatively biased in lower-range bins. In the H-D-S method, in upper-range bins, *ε* and *Z*_{e} are overestimated and, consequently, *R* is overestimated. In lower-range bins, *R* is underestimated, probably because the effect of the negative bias in *ε* is stronger than the effect of the positive bias in *Z*_{e}.

### c. Improvement of the H-D-S method

The H-D-S method fails to completely modify the bias in the H-D method. Because the negative bias in the H-D method expands downward, a vertically constant *ε*_{S} is not sufficient. Here, we propose an improved H-D-S method by using a vertically variable factor denoted by *ε*_{S2}. This method is called the H-D-S2 method.

*k*–

*Z*

_{e}relationship:

*k*–

*Z*

_{e}relationship, PIA is calculated by

*ε*

_{S2}. The first is that PIA calculated by Eq. (31) becomes the same as PIA_SRT. The second is to minimize the value of the equation below:

*ε*

_{S2}is normally 1, but it follows a lognormal distribution as

*ε*

_{S}(explained in section 6). Here, the term

*ε*

_{S2}(

*s*) on PIA, and

*W*

^{2}(

*s*) represents the error variance of

*ε*

_{S2}(

*s*). Considering that the error in the H-D method expands in lower-range bins,

*W*

^{2}(

*s*) is empirically set to be equal to

*ζ*

_{D}(

*s*). The second equation is to try to maximize the likelihood of

*ε*

_{S2}(

*s*).

In Fig. 4, the vertical profiles of variables estimated by the H-D-S2 method are shown alongside other methods. In Fig. 4a, compared with the H-D-S method, the vertical profile of *ε* becomes closer to the truth. Compared with the H-D method, *ε* becomes slightly larger in upper-range bins and is modified well in lower-range bins. This leads to an improvement in the vertical profiles of *Z*_{e} and *R*, as shown in Figs. 4b and 4c, respectively. In Fig. 5, values of *R*_{s} estimated by the H-D-S2 method are evaluated. When the true *R*_{s} is less than 20 mm h^{−1}, the estimates are not significantly biased, but a severe overestimation is apparent when the true *R*_{s} is higher than 20 mm h^{−1}.

As in Fig. 3, but for the H-D-S2 method.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

As in Fig. 3, but for the H-D-S2 method.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

As in Fig. 3, but for the H-D-S2 method.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

Figure 6 summarizes the evaluation of *R*_{s} by the four methods (H-S, H-D, H-D-S, and H-D-S2). Figure 6a shows the bias ratio, which is defined as the ratio of the bias to the true *R*_{s}. Figure 6b shows the RMSE. Under the condition where the SRT is perfect and *ε* does not change vertically, the H-S method is best in terms of the bias ratio and RMSE, and dual-frequency methods are of no use for attenuation correction. However, this condition is not realistic. In the following sections, the effects of SRT error and vertical changes in *ε* are discussed.

Summary of the evaluation of *R*_{s} by the four methods under the condition where the SRT has no error and *ε* is vertically constant. (a) The bias ratio and (b) RMSE are shown.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

Summary of the evaluation of *R*_{s} by the four methods under the condition where the SRT has no error and *ε* is vertically constant. (a) The bias ratio and (b) RMSE are shown.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

Summary of the evaluation of *R*_{s} by the four methods under the condition where the SRT has no error and *ε* is vertically constant. (a) The bias ratio and (b) RMSE are shown.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

## 5. The effect of vertical change of *ε*

To produce a realistic situation where *ε* changes vertically, TRMM PR’s *α* multiplied by an artificial factor (0.8–1.2) is given as KuPR’s *α* at liquid-precipitation range bins where attenuation becomes large and estimates differ according to the method used. The artificial factor is dependent only on the distance from the bright band. The same factors are used for different pixels if the distance from the bright band is the same, so that they are not cancelled out in the averaged vertical profile.

Figure 7 shows the vertical profile in the same format as Fig. 4. As shown in Fig. 7a, the true *ε* changes vertically. In the H-S method, because *ε* is always vertically constant, it contains random errors. The H-D method can simulate the shape of the vertical profile of *ε*, but it is negatively biased in lower-range bins. The H-D-S method partly mitigates the negative biases, but it degrades the estimates in upper-range bins compared with the H-D method. The H-D-S2 method is better than the H-D-S method because the shape of the vertical profile of *ε* is estimated well and the bias is not very large.

As in Fig. 4, but under the condition where the SRT has no error and *ε* is vertically variable.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

As in Fig. 4, but under the condition where the SRT has no error and *ε* is vertically variable.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

As in Fig. 4, but under the condition where the SRT has no error and *ε* is vertically variable.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

The vertical profile of *Z*_{e} shown in Fig. 7b is not very different from that shown in Fig. 4b. In the H-S method, *ε* has random errors, but the errors are largely cancelled out, so *Z*_{e} is not significantly affected. In the other methods, because *ε* is biased, the vertical profile of *Z*_{e} is also biased. Figure 7c shows the vertical profile of *R*. In the H-S method, random errors in *ε* cause the estimated *R* to have unnatural variation. In the other methods, the vertical profile of *R* is smooth but biased. The H-D-S2 method produces better estimates than both the H-D-S and H-D methods.

Figure 8 shows the evaluation of *R*_{s} in the same format as Fig. 6. A comparison of the bias ratio shown in Fig. 8a with that shown in Fig. 6a reveals no significant effects of the variation in *ε*. However, a comparison of the RMSE shown in Fig. 8b with that shown in Fig. 6b indicates that the H-S method becomes worse. Under the condition where *ε* is vertically variable, there are clear advantages of dual-frequency methods. Among the four methods, the H-D-S2 method has the smallest RMSE when the true *R*_{s} is between 0.5 and 4 mm h^{−1}. The H-D-S method has a smaller RMSE than the H-S method when the true *R*_{s} is between 0.5 and 2 mm h^{−1}. The RMSE of the H-D method is comparable to that for the H-S method when the true *R*_{s} is around 1 mm h^{−1}. However, for heavy precipitation (when the truth is larger than 5 mm h^{−1}), the H-S method is best.

As in Fig. 6, but under the condition where the SRT has no error and *ε* is vertically variable.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

As in Fig. 6, but under the condition where the SRT has no error and *ε* is vertically variable.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

As in Fig. 6, but under the condition where the SRT has no error and *ε* is vertically variable.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

## 6. The effect of errors in the SRT

In this section, the error of PIA_SRT is considered in addition to the vertical variation in *ε*. PIA_SRT deviates from the truth by random error of −1 to 1 dB. The error characteristic is realistic in the sense that the error of PIA_SRT is generally caused by spatial and temporal variations in surface conditions and is not related to precipitation intensity (Seto and Iguchi 2007). If the error of PIA_SRT is the same, then estimates for weak precipitation are more severely affected than those for heavy precipitation.

*ε*

_{S}to minimize the value of the equation below:

*σ*

_{S}

^{2}. Also,

*ε*

_{S}is assumed to follow a lognormal distribution, and the variance of log(

*ε*

_{S}) is denoted by

*σ*

_{ε}

^{2}. Here,

*σ*

_{S}is set to 1.0 dB and

*σ*

_{ε}is set to 1.0.

In the H-D-S method, *ε*_{S} is determined in the same way as in the H-S method, but PIA in Eq. (33) is calculated by Eq. (20). The estimated PIA in the H-D-S method is denoted by PIA_{fin}. In the H-D-S2 method, *ε*_{S2} is selected under two conditions. PIA calculated by Eq. (31) becomes equal to PIA_{fin}, and the value of Eq. (32) is minimized.

Figure 9 shows the evaluation of *R*_{s} in the same format as in Figs. 6 and 8. As the H-D method does not use the SRT, the result is the same as that shown in Fig. 8. In the other methods, because of the SRT errors, the bias ratio becomes larger when the true *R*_{S} is between 0.5 and 10 mm h^{−1} and RMSE increases.

As in Fig. 6, but under the condition where the SRT has random errors and *ε* is vertically variable.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

As in Fig. 6, but under the condition where the SRT has random errors and *ε* is vertically variable.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

As in Fig. 6, but under the condition where the SRT has random errors and *ε* is vertically variable.

Citation: Journal of Atmospheric and Oceanic Technology 32, 5; 10.1175/JTECH-D-14-00065.1

The H-D method has a smaller RMSE than the H-S method has when the true *R*_{S} is between 0.5 and 5 mm h^{−1}. For weak precipitation, the SRT error is severe, but DFR works well. When the true *R*_{s} is larger than 5 mm h^{−1}, the H-S method has a smaller RMSE than the H-D method has. For heavy precipitation, the SRT is relatively reliable, but DFR is not reliable because of heavy attenuation. The H-D-S method produces intermediate results between the H-D and H-S methods in terms of RMSE. Although the H-D method produces an underestimation and the H-S method produces an overestimation, the bias in the H-D-S method is not very large when the true *R*_{s} is between 0.5 and 5 mm h^{−1}. The H-D-S2 method does not work well because it produces a larger RMSE than the H-D-S method does and has a large bias even when the true *R*_{s} is less than 20 mm h^{−1}.

## 7. Summary and conclusions

The H-D-S method, a new dual-frequency attenuation correction method, was developed in this study. The method uses both DFR and the SRT to modify the *k*–*Z*_{e} relationship used in attenuation correction. The H-D-S method can always satisfy both the initial condition and the PIA condition as well as the H-S method can. The H-D-S method can be applied for partial-dual-frequency measurements and automatically switches to the H-S method for single-frequency measurements. Therefore, the H-D-S method is suitable for DPR measurement where single-frequency and dual-frequency measurements coexist.

The H-D-S method and other attenuation correction methods were tested with a simple synthetic DPR dataset. The H-D method that we had previously developed tends to have a larger bias in *ε* and *R* in lower-range bins. The H-D-S method partially mitigates this bias. Under the condition where there is error in the SRT and *ε* changes vertically, the H-D method works well for weak precipitation, whereas the H-S method works well for heavy precipitation. The H-D-S method produces stable estimates. The H-D-S2 method works better than the H-D-S method when the true *R*_{s} is less than 20 mm h^{−1} and when the SRT is perfect, but it needs to be improved when the SRT has errors. Quantitative evaluation should be undertaken with a dataset of real DPR measurements.

## Acknowledgments

This study is supported by the Japan Aerospace Exploration Agency under the 7th Precipitation Measurement Mission Research Announcement.

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