## 1. Introduction

The C- and X-band radars are preferred choices for mobile ground-based and airborne deployments because the radar antenna is much smaller in size than the S-band antenna for a specified beamwidth. The C- and X-band radars are more sensitive when detecting lighter precipitation than is the S-band radar. However, measurements with the higher-frequency radars are more susceptible to attenuation. The amount of attenuation is proportional to the intensity of the precipitation. For example, horizontal X-band copolarization reflectivity *Z*_{HH} and differential reflectivity *Z*_{DR} are usually underestimated due to attenuation of the radar signal as it propagates through precipitation. Therefore, attenuated X-band radar measurements must be corrected for attenuation effects before retrieving rain-rate and microphysical information from them.

In the case of single-polarization radar, an empirical relation between the one-way specific attenuation of the horizontally polarized signal *A*_{H} and the reflectivity is used for attenuation correction (Hitschfeld and Bordan 1954). The attenuation correction based on the *A*_{H}–*Z*_{HH} empirical relation is unstable and sensitive to calibration error in *Z*_{HH}. On the other hand, the accuracy and stability of the attenuation correction scheme is vastly improved when dual-polarization observations are used (Bringi et al. 1990).

*K*

_{DP}; ° km

^{−1}) derived from the gradient of differential phase shift (Φ

_{DP}; °) along the beam is proportional to rain intensity (Sachidananda and Zrnić 1986; Ryzhkov and Zrnić 1996; Scarchilli et al. 1993). One-way specific attenuation (

*A*

_{H}; dB km

^{−1}) and differential attenuation (

*A*

_{DP}; dB km

^{−1}) are nearly linearly proportional to

*K*

_{DP}(Bringi et al. 1990; Jameson 1991). As

*K*

_{DP}is unaffected by attenuation, radar system bias due to changes in transmit power, and antenna and receiver gain factors, it is more commonly used for estimating attenuation. Noting that ΔΦ

_{DP}is twice the sum of

*K*

_{DP}over a specified range, the corresponding two-way total attenuation (

*A*

_{h}; dB) and differential attenuation (

*A*

_{hv}; dB) can be estimated in the Φ

_{DP}-based method (Bringi et al. 1990; Anagnostou et al. 2006) to be approximatelyThe coefficients

*α*and

*β*relate ΔΦ

_{DP}to

*A*

_{h}and

*A*

_{hv}, respectively. However, the dynamic range of coefficients of

*α*and

*β*is too large to be accurately derived (Bringi and Chandrasekar 2001). Variations in

*α*and

*β*are caused by natural variations in the drop size distribution (DSD), drop shape oscillation, and temperature, and also they are insensitive to the Φ

_{DP}of drizzle and cloud droplets in the rain volume at these wavelengths.

Polarization radar measurements, namely differential reflectivity, provide additional information about precipitation and allow for better microphysical characterization of hydrometeors (Vivekanandan et al. 1999; Chandrasekar et al. 2006). The *Z*_{DR} provides information about the mean drop shape and mean size of DSD. Therefore, the inclusion of *Z*_{DR} in the attenuation correction scheme has the potential to reduce the uncertainty in estimated attenuation.

The majority of attenuation correction techniques use the *A*_{H}–*K*_{DP} or *A*_{h}–ΔΦ_{DP} relations, and *Z*_{DR} is used as a constraint for estimating the differential attenuation. Testud et al. (2000) modified the rain-profiling algorithms of the spaceborne Tropical Rainfall Measuring Mission rain radar and proposed the ZPHI algorithm, using Φ_{DP} as an external constraint. Bringi et al. (2001) further improved the ZPHI algorithm and demonstrated its applicability to C-band polarimetric radar attenuation correction. This technique effectively recalibrates the attenuation-corrected *Z*_{DR} to the expected 0-dB value in the regions of drizzle (Smyth and Illingworth 1998). It uses an optimization approach to determine linear coefficients that maximize the consistency between estimated path-integrated attenuation and ΔΦ_{DP} profiles. Gorgucci and Chandrasekar (2005) adopted the method for the X band and evaluated the error structure of the attenuation correction methodologies using the differential phase measurement. Lim et al. (2011) implemented the algorithm to a network of X-band radars.

Park et al. (2005a,b) adopted the algorithm from Bringi et al. (2001) for X-band radar measurements and provided a detailed description of the *A*_{H}–*K*_{DP} and *A*_{H}–*A*_{DP} nonlinear relations for X band based on T-matrix scattering calculations (Barber and Yeh 1975). The relations are derived as a power-law expression where the proportionality constant is sensitive to DSD, temperature, and drop shape. The exponent in the relation is mostly sensitive to the temperature of the raindrop. The proportionality constant in the *A*_{H}–*A*_{DP} relation is less sensitive to DSD and the exponent in the relation is sensitive to temperature, as in the *A*_{H}–*K*_{DP} relation. Snyder et al. (2010) demonstrated that the attenuation correction based on Park et al. (2005b) had better levels of agreement with S-band radar in a tornadic supercell case. Gorgucci et al. (2006) developed an alternative method based on the self-consistency principle for estimating both attenuation and differential attenuation with less than 10% bias and 15% error.

A variational algorithm proposed by Hogan (2007) considers not only the beam-to-beam dependency of *α* and *β* for attenuation correction but also the gate-to-gate dependent variability of DSD inferred from *Z*_{DR}. The variational method for S-band radar observations (Furness 2005; Hogan 2007) demonstrated the usefulness of the method for detecting rain–hail regions and the estimation of attenuation and rain rate from S-band polarization radar. In this paper, a modified version of the variational technique for estimating attenuation in the X band that uses both Φ_{DP} and *Z*_{DR} is presented. The major objective of this paper is to investigate if relatively large values of attenuation at X band can be estimated by the variational method.

The variational algorithm uses a 1D retrieval method along the radar beam. The scheme is based on a linear estimation theory in which the background information and the observations are weighed proportional to the inverse of their respective error covariances. It is not uncommon to use predetermined error covariance matrices (Hogan 2007), but they fail to characterize the errors of a specific rain event. The performance of the variational scheme depends on the accuracy of the error covariance matrices of background terms and the observations.

Nevertheless, the quality of the polarimetric measurements is influenced by a range of factors under various measurement conditions. Ryzhkov (2007) shows that significant uncertainty or error in *Z*_{DR} up to ±0.7 dB, and in Φ_{DP} up to ±10°, can be introduced in the presence of nonuniform beam filling. Hubbert et al. (2010a,b) demonstrate in the case of simultaneous horizontally and vertically transmitted mode orientations that scatters cause cross coupling between horizontally and vertically polarized signals. Since all of the above-described errors vary spatiotemporally, it is necessary to use a case-dependent measurement error covariance matrix rather than a predetermined value as in the case of a standard variational algorithm.

The mathematical formulation of the variational method and estimation of the error covariance matrices are described in section 2. In section 3, an observing system simulation experiment (OSSE) evaluation of the variational algorithm is used to demonstrate the feasibility of the variational method at X band. The attenuation correction technique was applied to X-band radar measurements and the corrected X-band radar measurements were compared to collocated S-band radar measurements in section 4. An analysis of error characteristics of *Z*_{DR} and Φ_{DP} measurements is presented in section 5. Results are summarized in section 6.

## 2. Variational method and error covariance matrices

### a. Variational method

*J*and 2) a forward model

*Z*

_{DR}and

*K*

_{DP}) and the specific attenuation (

*A*

_{H}and

*A*

_{DP}) for a specified rain DSD (Vivekanandan et al. 1991; Bringi and Chandrasekar 2001). The forward model assumes a gamma DSD model (Ulbrich 1983) of the formThe gamma parameters, namely,

*N*

_{0}(intercept parameter; mm

^{−1}m

^{−3}),

*μ*(shape parameter; dimensionless), and

*D*

_{0}(median volume diameter; mm), are the inputs into the electromagnetic scattering and wave propagation model for computing

*Z*

_{HH},

*Z*

_{DR},

*K*

_{DP},

*A*

_{H}, and

*A*

_{DP}and for estimating the rainfall rate

*R*. The values of gamma DSD parameters are varied for simulating the natural variability of DSD based on observational statistics from the 2D-video disdrometer (2DVD; Schönhuber et al. 1997) of National Central University (NCU) in Taiwan. The

*D*

_{0}varied from 0.1 to 6 mm with an interval of 0.01 mm and the values of log

_{10}(

*N*

_{0}) varied from 1.0 to 16.0 with an interval of 1.0; meanwhile,

*μ*is fixed at 5.

In this paper, the forward model is simplified using the results from rigorous scattering model calculation. In a power-law relation between reflectivity and rainfall rate (*R*), *Z* = *aR*^{b}, where parameters *a* and *b* represent variations in the DSD (Steiner et al. 2004; Steiner and Smith 2000). Since the temperature dependency of *a* and *b* is relatively minor, it is not considered in this research. Large *μ* values (*μ* = 5) reduce the sensitivity of the exponent *b* to DSD in the *Z*–*R* relation (Steiner et al. 2004). Further, the multiplicative parameter *a* is much more sensitive to DSD variations than is the exponent parameter *b* (Bringi and Chandrasekar 2001; Hogan 2007). Hogan (2007) also indicated the variability of parameter *b* is small; therefore, the exponent *b* is fixed and it is equal to 1.5 (Steiner and Smith 2000; Doelling et al. 1998; Hogan 2007) based on observational statistics from NCU 2DVD. This particular type of *Z*–*R* relation represents rain DSDs controlled by variations in both number concentration and characteristic size (Steiner et al. 2004; Steiner and Houze 1997). Thus, the forward models use the ratio *Z*/*R* as a proxy for variation in DSD. Parameter *a* of *Z*–*R* is the target variable since the exponent *b* is fixed. Typically, the natural logarithm of *a* [i.e., ln(*a*)] is estimated to avoid retrieving an unrealistic negative value of *a*. Hereinafter, the symbol *ã* represents ln(*a*).

*Z*

_{DR},

*K*

_{DP},

*A*

_{H}, and

*A*

_{DP}as a function of

*Z*/

*R*, which are then smoothed by curve fitting and organized into a precalculated lookup table for values of

*Z*/

*R*between 10

^{0.6588}and 10

^{4.3045}, as shown in Fig. 1. The error in the gamma DSD-based forward model is examined by calculating the root-mean-square error (RMSE) defined asIn this case,

*y*represents the parameters of

*Z*

_{DR},

*K*

_{DP},

*A*

_{H}, and

*A*

_{DP}. The superscript true represents the parameters derived by rigorous electromagnetic scattering and propagation model simulation from 5 yr of NCU 2DVD-measured DSDs. The

*m*represents the number of DSDs. The superscript est. represents the parameters derived from the forward model with

*Z*/

*R*as input. The values of RMSE (

*σ*) are fairly low for all parameters, as shown in Table 1. Nevertheless, the actual uncertainty in the forward model should be larger, since the rain microphysics, namely, the DSD model, axis ratio of raindrops, temperature, and maximum size of raindrop, exhibit natural variability and they can only be approximated as mean values in the forward model. The uncertainty in the forward model due to natural variability in rain microphysics is considered to be part of the observation error in this research.

RMSE (σ) of the forward model. The RMSE of Φ_{DP} is derived using 400 gates with 0.125-km gate spacing.

*J*) is composed of three components, namely,

*Z*

_{DR}, Φ

_{DP}, and an optimal target variable

*x*

_{k}along the ray at gate

*k*. The contribution to the cost function from each component is weighted by the corresponding variance (

*σ*

^{2}). An array of radar measurements,

*n*gates, along a radar beam is processed one at a time. The background term

*ã*

^{background}is included in the cost function as an “a priori” so that an appropriate value of

*ã*is obtained in the regions of light rain where

*Z*

_{DR}and

*K*

_{DP}are too small or where no useful polarimetric radar measurements are available. These regions may have nonnegligible cumulative attenuation.

*Z*

_{HH}and

*Z*

_{DR}(dB) are related to the observed reflectivity

*A*

_{h}and

*A*

_{hv}are the two-way path-integrated

*A*

_{H}and

*A*

_{DP}, respectively. The two-way propagation phase Φ

_{DP}in degrees at range

*r*

_{2}can be derived from the Φ

_{DP}at range

*r*

_{1}and integration of the

*K*

_{DP}between the range

*r*

_{1}and

*r*

_{2}as

*ã*was obtained via minimizing the cost function. The Gauss–Newton iteration method is used (Rodgers 2000; Hogan 2007). The matrix form of the iterative solution iswhereandIn the above equations,

**x**

_{i}represents the target variable

*ã*at the

*i*th iteration,

**y**represents the observations

*Z*

_{DR}and Φ

_{DP}, the subscript “background” represents the background values of

*ã*, the

*Z*

_{DR}and Φ

_{DP}) and the background value, respectively. The

*ã*. The derivation of

The flowchart in Fig. 2 shows various steps in the variational attenuation retrieval scheme. After removing the system bias in *Z*_{HH} and *Z*_{DR}, as well as the nonmeteorological signal, the minimization process starts with a first guess of *ã* and the *A*_{H} is predicted via the forward model. As in Hogan (2007), the measured *Z*_{HH} is corrected cumulatively using the *A*_{H} (step 3). Using the corrected *Z*_{HH}, the forward model predicts intrinsic *Z*_{DR}, *A*_{DP}, and *K*_{DP} (step 4). The intrinsic *Z*_{DR} is reduced by the amount of *A*_{DP}. By integrating *K*_{DP} over the range, we are able to obtain Φ_{DP} (step 5). The corresponding *ã* for the next iteration are calculated in steps 7 and 8. The above-described steps are repeated until the solutions converge satisfactorily (step 6). A χ^{2} test is used to determine convergence. It usually requires 4–10 iterations for convergence. All of the computations are performed at range gate resolution and the measured data are not filtered or averaged over a range.

Predetermined diagonal matrices with fixed variance (σ^{2}) values of

### b. Estimating the background error covariance matrix

In practice, the background term is usually obtained from the climatological mean value of *ã* (Hogan 2007). Hence, the corresponding fixed value of the diagonal error covariance matrix of the background term (*ã*^{background} is 5.5 and the variance

For estimating an optimal *Z*_{DR} and Φ_{DP} compared to the radar observations, the “optimal”

There are three benefits of varying

### c. Estimating the observation error covariance matrices

*Z*

_{DR}and Φ

_{DP}(Desroziers et al. 2005). The diagnostic approach combines observation, analysis, and background information for tuning observation and background errors adaptively using innovation vectors. Innovation vectors of the background

**x**represents the

*ã*of

*Z*–

*R*. The superscripts ana., bg., and obs. represent the analyses (results from variational algorithm), background terms (i.e., climatological

*ã*of the

*Z*–

*R*relation), and the observation from radar, respectively. As described in section 2a,

**y**represents the

*Z*

_{DR}and Φ

_{DP}measurements. The observation, background, and analysis errors are

*ε*^{obs.},

*ε*^{bg.}, and

*ε*^{ana.}, respectively. As the variational method processes radar beams sequentially, the innovation vectors of the background

*Z*

_{DR}for each beam areandand vice versa for Φ

_{DP}, where

*k*is the available gates along a radar beam.

^{2}):Therefore, diagonal terms of

_{DR}and Φ

_{DP}. The

*k*× 2

*k*matrix:

There are two advantages to using the statistical diagnostic method: 1) predetermined observation error covariance matrices are no longer needed and 2) the true spatial distribution characteristics of radar measurement error are represented in

As suggested by Li et al. (2009), the proper values of

## 3. The observing system simulation experiments

The OSSE is designed to demonstrate the feasibility of the variational algorithm for estimating rain attenuation and rainfall rate at X band. The intrinsic polarimetric observations, *Z*_{HH}, *Z*_{DR}, Φ_{DP}, the attenuation, *A*_{H}, *A*_{DP}, and rainfall rate were simulated rigorously from 6-min-averaged DSDs of 5 yr of NCU 2DVD data. There are 6880 averaged DSDs available for OSSE studies. The procedures for modeling polarimetric observation beam are introduced in appendix B.

A set of DSDs with a maximum rainfall rate of 70.0 mm h^{−1} and that have a wide range in the coefficient *a* of *Z*–*R* were selected to demonstrate the simulated radar measurements of a representative precipitation system with natural variations in DSD, as shown in Fig. 3. The *A*_{h} and *A*_{hv} along the beam were 12.07 and 1.44 dB, respectively. The Φ_{DP} was obtained by integrating *K*_{DP} along the range (Fig. 3c). The OSSE was performed using two types of data: 1) assuming no error in measurements and 2) assuming observation error in *Z*_{HH}, *Z*_{DR}, and Φ_{DP}. The configurations of sensitivity studies are summarized in Table 2.

Configurations of the OSSEs. Each simulation contains 600 modeled polarimetric observation beams with the following values of standard errors (σ) of *Z*_{HH}, *Z*_{DR}, and Φ_{DP}.

### a. The OSSE without observation error

The variational algorithm was applied to the modeled X-band polarimetric data, as “no error” in Table 2. No background term was used in this case since the simulated data have no observational error. In theory, *Z*_{DR} and Φ_{DP} are diagonal with forward model error values of 0.05 dB and 0.20°, respectively (see section 2a; Table 1). Performance of the variational algorithm was examined as a function of range with gradually increasing *A*_{h}. The purpose of this sensitivity study is to investigate the performance of the variational algorithm for a range of *A*_{h}.

*A*

_{h},

*A*

_{hv}, and the logarithm of rainfall rate (dB

*R*), for both variational and Φ

_{DP}-based attenuation correction schemes, from 600 modeled polarimetric radar beams. The rainfall rate of the variational algorithm was derived from

*Z*–

*R*using the estimated

*a*and fixed

*b*. The rainfall rate from the Φ

_{DP}-based attenuation correction scheme was derived from

*R*–

*Z*

_{HH}–

*Z*

_{DR}asEven though regression coefficients

*α*and

*β*for attenuation estimation, and

*a*

_{1}and

*b*

_{1}for rainfall rate estimation were specifically tuned to the DSD data, RMSEs of dB

*R*,

*A*

_{h}, and

*A*

_{hv}obtained from the Φ

_{DP}-based schemes are higher than of variational algorithm-based results as shown in Fig. 4. The RMSEs in the Φ

_{DP}-based technique are larger as expected because DSD fluctuations were not characterized. The RMSE from the Φ

_{DP}-based technique deteriorates as the maximum range increases due to accumulated DSD variations. The values of RMSE increase from 0.7 to 2.8 dB for dB

*R*, from 0.08 to 0.21 dB for

*A*

_{h}, and from 0.02 to 0.05 dB for

*A*

_{hv}.

As no error was added to the observations, the RMSE of the variational algorithm is primarily caused by residual differences between the forward model based on simplified gamma DSD and measured DSD, as shown in Fig. 1. Despite the use of a simplified forward model, the lower RMSEs from variational-based algorithms indicate the simplified forward model has the capability to predict the polarimetric measurements and attenuation with sufficient accuracy.

### b. The OSSE with observation error

One of the advantages of the variational algorithm is that it is relatively less sensitive to observation error (Hogan 2007). The background term in the cost function could be used to constrain the result in the presence of observation error. Two sensitivity studies, “variable error in Z_{DR}” and “variable error in Φ_{DP}” in Table 2, investigate the impacts of *Z*_{DR} and Φ_{DP} observation error on the variational algorithm. Specified amounts of Gaussian standard errors (*σ*) were added to the simulated intrinsic polarization observables. The configurations of sensitivity studies with various values of standard errors (*σ*) of Z_{HH}, Z_{DR}, and Φ_{DP} are summarized in Table 2.

The procedure described in appendix A was used for estimating optimal *Z*_{DR} and Φ_{DP} for estimating the first-guess *Z*_{DR} and Φ_{DP} error are varied between 0.1 and 1.5 dB and 1.7° and 2.7°, respectively, the initial values of *Z*_{DR} and Φ_{DP} were assumed to be 0.1 dB and 1.0° respectively to verify whether the statistical diagnostic method is able to recover the proper values of _{DR} and Φ_{DP} with standard deviations (*σ*) of 1.1 dB and 2.5° is shown in Fig. 5. The diagnosed *Z*_{DR} and Φ_{DP} converge in four iterations to the specified error values.

In the case of the variable error in *Z*_{DR}, optimal *Z*_{DR} are greatly improved from the initial assumption of 0.1 dB, despite the *Z*_{DR} error being varied between 0.1 and 1.5 dB, as shown in Fig. 6a. The diagnosed *Z*_{DR} are close to the 1:1 line, as desired. Similarly, the diagnosed _{DP} error of variable error in Φ_{DP} case are improved from the initial assumption of 1.0° and they are comparable to true Φ_{DP} error in Fig. 6b. These results indicate that the statistical diagnostic method estimates the optimal values of *Z*_{DR} and Φ_{DP} fairly well. This methodology is used in the analysis of measurements, as variances of measurements are unknown.

In Fig. 7a, the values of the RMSEs of *A*_{h} and *A*_{hv} from the variational and Φ_{DP}-based algorithms are shown as a function of *Z*_{DR} observation error from the variable error in *Z*_{DR} case. Since the Φ_{DP}-based algorithm does not include *Z*_{DR}, the RMSEs of *A*_{h} and *A*_{hv} from the Φ_{DP}-based algorithm remain nearly constant as expected with values of 0.5 and 0.08 dB, respectively. While the corresponding values for the RMSEs of the variational algorithm are 0.1 and 0.02 dB. The RMSEs of dB*R* are also lower for the variational algorithm, as shown in Fig. 7b. Even in the presence of typical observation errors in *Z*_{HH}, *Z*_{DR}, and Φ_{DP}, the variational algorithm provides better estimations of dB*R*, *A*_{H}, and *A*_{DP}.

The OSSE studies have shown that the variational method can provide more accurate estimations of attenuation and rainfall rate, even in the presence of observational error in *Z*_{HH}, *Z*_{DR}, and Φ_{DP}. Moreover, errors in *Z*_{DR} and Φ_{DP} can be estimated accurately using the statistical diagnostic method. In the next section, the variational method will be applied to field measurements from an X-band radar.

## 4. Applications of the attenuation correction scheme to field measurements

The variational scheme was applied to actual X-band field measurements during the Southwest Monsoon Experiment/Terrain-Influenced Monsoon Rainfall Experiment (SoWMEX/TiMREX; Jou et al. 2011). The selected X-band polarimetric radar measurements collected by the Taiwan Experimental Atmospheric Mobile-Radar (TEAM-R) were examined. The specifications of TEAM-R are listed in Table 3. To verify the performance of the variational attenuation correction at X band, the collocated measurements from a synchronized RHI scan of the TEAM-R and the National Center for Atmospheric Research (NCAR) S-band dual-polarization Doppler radar (S-Pol) were analyzed.

Engineering specifications of TEAM-R.

Before applying the variational scheme, the raw TEAM-R radar data were quality controlled by filtering ground clutter and mixed-phase precipitation using *ρ*_{HV} (copolar correlation coefficient). Usually regions with *ρ*_{HV} < 0.7 correspond to either a mixed phase or ground clutter (Ryzhkov et al. 1994). System biases in *Z*_{HH} estimated by Vivekanandan et al. (2003) and in *Z*_{DR} estimated by vertical-pointing data were also removed before applying the variational correction. The system bias in *Z*_{HH} and *Z*_{DR} should be known within accuracies of ±0.5 and ±0.2 dB, respectively, to ensure the variational method outperforms the Φ_{DP}-based method according to OSSE studies (not shown). The melting level was about 4.6 km MSL based on sounding data. Therefore, radar data above 3.5 km MSL were not processed in order to avoid ice-phase and mixed-phase precipitation, as they are not accounted for in the forward model used in this study. The elevation angle dependence effect on polarimetric measurements *Z*_{DR}, *K*_{DP}, and *A*_{DP} was considered. The elevation angle dependency correction suggested by Bechini et al. (2008) was applied to the forward model.

Convergence of the target variable *ã* to an optimal value is achieved using the Gauss–Newton iteration method. Since attenuation at X band is much higher than at S band, a poor first guess of *ã* leads to physically unrealistic attenuation correction for *Z*_{HH} and causes numerical error. It is important to avoid a poor first guess of *ã*. The procedure for deriving a proper first guess of *a* in *Z*–*R* is described in appendix C.

An initial prescribed *σ*) of actual radar measurements. The *σ* of *Z*_{DR} was derived from the vertical-pointing data in a stratiform light rain. The *σ* of the Φ_{DP} observation error was computed using measurements from a light stratiform rain region where there was nearly no Φ_{DP} increment as a function of range. The values of *σ* for *Z*_{DR} and Φ_{DP} are 0.18 dB and 2.4°, as shown in Table 4.

The standard deviation (σ) of radar measurement errors estimated using statistical analysis and statistical diagnostic methods.

### a. Analysis of a single radar beam

Performance of the attenuation correction method for a single beam is examined by comparing corrected TEAM-R and S-Pol measurements. Figure 8 shows *Z*_{HH}, *Z*_{DR}, and Φ_{DP} for a radar beam at a 3.19° elevation angle from X- and S-band measurements. Higher values of *Z*_{HH} and *Z*_{DR}, as well as a large gradient in Φ_{DP} measurements (in about 30° increments), indicate convective precipitation between 5 and 13 km from radar. A stratiform precipitation system with flat Φ_{DP} exists between the ranges of 15 and 35 km. The raw X-band *Z*_{DR} values in the stratiform system varied between −2 and −1 dB due to cumulative differential attenuation.

For this radar beam, the variational scheme estimated *A*_{h} and *A*_{hv} as 11.1 and 1.5 dB, respectively. The intrinsic difference in *Z*_{DR} observations between X- and S-band wavelengths was removed by converting S-band *Z*_{DR} to X-band *Z*_{DR} (Chandrasekar et al. 2006). The attenuation-corrected *Z*_{HH} and *Z*_{DR} from TEAM-R are in good agreement with the *Z*_{HH} and *Z*_{DR} measurements from S-Pol. Good agreement among the *Z*_{DR} results in stratiform regions confirms that the differential attenuation was estimated correctly. In the convective region around 11-km range the significantly attenuated TEAM-R *Z*_{HH} and *Z*_{DR} are satisfactorily recovered as they compare favorably with the S-Pol measurements.

Figure 8c shows Φ_{DP} as measured by TEAM-R and predicted Φ_{DP} from the variational scheme. The agreement between the predicted and measured Φ_{DP} confirms the minimization procedure converged and the Φ_{DP} was predicted correctly, but there are differences at individual range gates. The measured Φ_{DP} exhibited significant fluctuations in both the convective and stratiform regions. In the convective region, the fluctuation was caused by the strong reflectivity gradient (Ryzhkov 2007) and backscattering Mie phase shift (*δ*). In the stratiform region, the fluctuation was caused by a low signal-to-noise ratio (SNR). Significant fluctuations in measured Φ_{DP} lead to poor estimation of attenuation by the Φ_{DP}-based algorithm. Due to the strong reflectivity gradient, the measured Φ_{DP} shows a localized minimum between 9 and 9.5 km. However, the predicted Φ_{DP} does not exhibit the local minimum. The variational method predicted a smooth Φ_{DP} with no fluctuation. This predicted Φ_{DP} from the variational scheme was derived from a physical approach rather than a mathematical approach, namely, filtering or smoothing of measured Φ_{DP} (Hubbert and Bringi 1995). The smoothing technique degrades the spatial resolution and gradient in Φ_{DP}. Since the predicted Φ_{DP} is free from measurement noise and backscattering Mie phase shift (*δ*), it is straightforward to estimate *K*_{DP} from predicted Φ_{DP} without losing any spatial resolution.

### b. Analysis of an RHI scan

The measurements of *Z*_{HH} and *Z*_{DR} from synchronized RHI scans of S-Pol and TEAM-R are shown in Fig. 9. Figure 9b shows the equivalent intrinsic X-band *Z*_{DR} derived from the measured S-band *Z*_{DR} (Chandrasekar et al. 2006). Figures 9c and 9d show that the measurements from TEAM-R were severely underestimated due to the attenuation effect. The attenuation-corrected TEAM-R measurements, *Z*_{HH} and *Z*_{DR} in Figs. 9e and 9f, are in good agreement with the S-Pol measurements in Figs. 9a and 9b. The strength of the rear part of the convection system (10–12 km) and stratiform precipitation (15–35 km) were enhanced by 10 dB. The negative *Z*_{DR} values at the stratiform region between 15 and 35 km were brought back to 0 dB, which is more appropriate for a drizzle region. In the convective precipitation region, minor discrepancies between attenuation-corrected X-band *Z*_{HH} and *Z*_{DR} and S-band radar measurements can be noticed. Considering the beamwidth of TEAM-R is 1.25°, this discrepancy is due to three possible reasons: 1) biases in *Z*_{HH} and *Z*_{DR} in the regions of large gradients due to partial beam filling (Ryzhkov 2007) limited the performance of the variational algorithm in estimating the attenuation accurately, 2) the differences in sample volumes between the TEAM-R and S-Pol radars amplified the *Z*_{HH} and *Z*_{DR} differences between the X- and S-band radar measurements, and 3) the rapidly evolving convective precipitation system might have also contributed to the majority of the difference in the *Z*_{DR} fields as the X- and S-band radar measurements were offset by 10 s.

The predicted Φ_{DP} and measured Φ_{DP} of the variational method are in good agreement, as shown in Fig. 10. The fluctuation in predicted Φ_{DP} is lower than in measured Φ_{DP}. The variational scheme underestimates Φ_{DP} at the lowest elevation angle when compared to the measured values. The elimination of the *Z*_{HH} and *Z*_{DR} measurements at low-elevation beams with poor data quality due to ground clutter contamination underestimated the predicted Φ_{DP}. In the case of noisy measurements, the variational method had to rely on the background term. However, the background term (i.e., climatology-based *ã*) is not good enough to represent local DSDs in this case. Underestimation of predicted Φ_{DP} can also be noticed between 5.5° and 8.1° in Fig. 10a, and also as stripes in the corrected Figs. 9e and 9f in the stratiform precipitation region.

To further quantify the differences between measured S-Pol *Z*_{DR} and attenuation-corrected *Z*_{DR} measurements, average values in convective and stratiform regions were computed. As shown in Table 5, the values from the variational method are closer to those of S-Pol than those of the Φ_{DP}-based method in the convective and stratiform regions. In this case the Φ_{DP}-based method used the value of *β* as 0.036, which was computed from SoWMEX/TiMREX disdrometer measurements. A value of 0 dB for *Z*_{DR} in stratiform is widely used as a physical constraint to reduce the uncertainty in the attenuation estimation by the Φ_{DP}-based method (Smyth and Illingworth 1998; Bringi et al. 2001; Park et al. 2005a,b). Nevertheless, the *Z*_{DR} value was not zero in the stratiform region based on S-Pol measurements in this case. The satisfactory performance of the variational method in this case reinforces its advantage, as no such constraint is required for the variational method.

Mean values of S-Pol and attenuation-corrected Z_{DR} from TEAM-R are listed. Mean values of Z_{DR} were computed for convective and stratiform regions.

### c. K_{DP}–A_{H} and K_{DP}–A_{DP} relations

A scatterplot showing attenuation (*A*_{H} and *A*_{DP}) and *K*_{DP} is presented in Fig. 11. Attenuation values as a function of *K*_{DP} were obtained using the variational scheme. The values of *A*_{H} and *A*_{DP} estimated by the variational scheme showed broader scatter compared to the Φ_{DP}-based method represented by solid and dash lines in Fig. 11. The data were categorized into two subgroups depending on the associated *Z*_{DR} values. For a specified *K*_{DP}, *A*_{H} and *A*_{DP} are higher when *Z*_{DR} is larger. This feature is consistent with the results from the electromagnetic scattering and wave propagation simulation (Carey et al. 2000). The results indicate that the variational algorithm can reproduce not only the broader distribution of the *A*_{H} and *A*_{DP}, but also the proper values of the *A*_{H} and *A*_{DP} that are consistent with the natural DSD. The improved accuracy in the variational method is mainly due to the usage of both Φ_{DP} and *Z*_{DR} measurements for the estimation of *A*_{H} and *A*_{DP} rather than using only Φ_{DP}.

## 5. Characteristics of *Z*_{DR} and Φ_{DP} measurement error

The results in previous sections show the variational algorithm produces better results than the Φ_{DP}-based algorithm. The major reason for this is that the variational method uses both *Z*_{DR}, Φ_{DP} and climatological values of *a* in *Z*–*R*. Moreover, the error covariance matrices of background term (*Z*_{DR} and Φ_{DP} (

_{DP}and

*Z*

_{DR}observations vary as a function of elevation angle, as shown in Fig. 12. These estimated values are consistently higher for elevation angles < 0.5° due to ground clutter contamination, as expected. The corresponding values of optimal

*Z*

_{DR}and Φ

_{DP}. This is consistent with higher values of

*Z*

_{DR}and Φ

_{DP}between 5.0 and 8.0 km and for elevation angles between 5.5° and 8.1°, as shown in Fig. 13. Unlike Eq. (16), which is used for deriving the standard error

*Z*

_{DR}and Φ

_{DP}further confirm the pronounced stripes in the predicted Φ

_{DP}, as well as the corrected

*Z*

_{HH}and

*Z*

_{DR}in Figs. 9 and 10 as the variational method failed to provide a reasonable estimation.

The mean values of *Z*_{DR} and Φ_{DP} in convective (distances between 5 and 13 km) and in stratiform (distances between 15 and 34 km) regions are summarized in Table 4. The mean values of diagnosed *Z*_{DR} and 2.2° for Φ_{DP} in the stratiform region are in good agreement with the prescribed values of the observation errors of the estimated 0.18-dB for *Z*_{DR} and 2.4° for Φ_{DP} in the stratiform region, as shown in Table 4. However, the mean values of diagnosed *Z*_{DR} and Φ_{DP} in the convective region are 0.38 dB and 2.88°, respectively, and are higher than the corresponding values in stratiform regions. The results indicate the statistical diagnostic method not only estimates accurate values of the error covariance of *Z*_{DR} and Φ_{DP}, but also characterizes the spatial distribution of the error covariance of the same.

## 6. Summary

A variational method originally designed for S-band radar has been modified for X-band radar data. Several modifications have been implemented in the original algorithm to overcome the difficulties caused by the high attenuation at X band. First, a discrete approximation method for estimating optimal initial values of *a* for each beam is used for smooth convergence of the Gauss–Newton minimization method. The results showed beam-dependent optimal initial values of *a* significantly improved estimation.

Second, instead of using fixed a priori values for error covariances, discrete approximation and statistical diagnostic methods were used for characterizing the temporal and spatial covariance matrices of the background term (

A modified variational method in this research has demonstrated several advantages compared to the Φ_{DP}-based algorithm: 1) more accurately corrected *Z*_{HH} and *Z*_{DR} compared to the Φ_{DP}-based algorithm, 2) physically derived Φ_{DP} for estimating *K*_{DP} without any loss of spatial resolution, 3) an estimated background error covariance (*Z*_{DR} and Φ_{DP}.

The variational method uses both the measurements of Φ_{DP} and *Z*_{DR} from X-band radar. The inclusion of *Z*_{DR} in the variational method improved the accuracy of the retrieved attenuation. The variational method yielded more accurate results than did the Φ_{DP}-based method even though the later method used retrieval coefficients specifically tuned to the particular disdrometer dataset. Since, DSD can change from event to event and even within the same storm, it is difficult to tune retrieval coefficients of the Φ_{DP}-based algorithm by not having a priori DSD information. The variational algorithm requires no knowledge about prior coefficients, yet it can estimate attenuation (*A*_{H} and *A*_{DP}) more accurately. Sensitivities of the variational correction algorithm to observation noise in *Z*_{HH}, *Z*_{DR}, and Φ_{DP} were also investigated using OSSEs. The results suggested that the variational method is relatively immune to random measurement errors when compared to the Φ_{DP}-based method.

Application of the method was demonstrated using radar measurements collected during the SoWMEX/TiMREX field program in Taiwan. Attenuation-corrected X-band measurements from TEAM-R using the variational method agreed with collocated S-band radar measurements. Moreover, the variational algorithm was able to reconstruct not only a broader, but also a more accurate distribution of *A*_{H} and *A*_{DP} for a specified *K*_{DP}. Inclusion of DSD information via *Z*_{DR} in the variational method significantly improved the accuracy of the retrieved attenuation.

In the process of estimating attenuation by the variational method, Φ_{DP} is predicted as a by-product. It is an alternative approach to a standard mathematical filtering technique for estimating smooth Φ_{DP} and *K*_{DP} that are free of observation noise and backscattering Mie phase shift (*δ*).

The error covariance matrices *Z*_{DR} and Φ_{DP} measurements obtained from the statistical diagnostic method are used for characterizing the dynamic nature of the spatial and temporal error characteristics instead of fixed a priori values. The value of optimal *σ*) of the actual radar measurements in a stratiform light rain. The results of diagnosed *Z*_{DR} and Φ_{DP} but can also characterize the spatial distribution of the error covariance of the same. The results show the observation errors were higher in a convective region than in a stratiform region. The spatial distributions of the measurement error further help to explain why the variational algorithm failed to retrieve reliable attenuation in the regions of noisy radar measurements.

It should be noted that the variational algorithm is designed for attenuation correction in rain only as the forward models are based on rain DSDs. The presence of hail can cause significant attenuation (Gu et al. 2011) in practice. The presence of hail (featured with low *Z*_{DR} and high *Z*_{HH}) can be identified by high values of *Z*_{DR}; thus, the Φ_{DP}-based algorithm can be used as an alternative to the variation method. The simplified background term of *ã* is used in this research (i.e., mean value of *ã*) and shows fairly promising results. It is recommended the background terms of *ã* be modified in correspondence with rain DSD climatology for favorable attenuation estimation.

## Acknowledgments

We thank Dr. Wen-Chau Lee of NCAR and Prof. Ben Jou of the National Taiwan University for sharing the use of S-Pol data and Robert Rilling and Scott Ellis for assistance with the S-Pol data quality and calibration. The National Center for Atmospheric Research is sponsored by the National Science Foundation. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation.

## APPENDIX A

### Implementation of the Discrete Approximation and Statistical Diagnostic Method

As shown in Fig. A1, the first guess of

## APPENDIX B

### Modeled Polarimetric Observation Beam Data

Six hundred polarimetric observation beams are modeled evenly from 5 yr of the NCU 2DVD dataset with different combinations of rainfall intensity to cover the natural variability of the DSD. Each beam contains 400 gates of intrinsic *Z*_{HH} and *Z*_{DR} with gate resolution of 0.125 km according to the configuration of most X-band radars. The total attenuation effects (*A*_{h} and *A*_{hv}) were included in intrinsic *Z*_{HH} and *Z*_{DR} cumulatively along the beam. The Φ_{DP} is obtained from integrating *K*_{DP} along the beam. The values of *A*_{h}, *A*_{hv}, and Φ_{DP} of these 600 modeled polarimetric observation beams are summarized in Fig. B1. Half of the modeled polarimetric observation beams have *A*_{h} > 8.1 dB, *A*_{hv} > 0.87 dB, and Φ_{DP} > 28°. For OSSE studies with observation error, Gaussian random errors of the standard distribution (*σ*) were added to the modeled polarization observables accordingly, as shown in Table 2.

## APPENDIX C

### Optimal First Guess of *a* of *Z*–*R*

A discrete approximation method is proposed to derive the optimal first guess of *a* of *Z*–*R*. Figure C1a shows that parameter *a* can vary between 20 and 650 as a function of rain intensity and natural variation in DSD. For a range of values of *a*, the predicted attenuated *Z*_{DR} and Φ_{DP} are derived. The total absolute difference between the forward model predicted parameters and the actual measurement is calculated. The value of *a* that corresponds to the minimum total absolute difference is chosen as the optimal initial value.

The initial value of *a* could change from one precipitation event to the next and even within an individual precipitation event. In Fig. C1b initial values of *a* for the range of elevation angles in a RHI scan shown in Fig. 9 are plotted. Initial values of *a* vary from one elevation to another due to the variability of the DSD. When there are differences in the optimal initial *a* based on *Z*_{DR} and Φ_{DP}, an average of these values is taken as the optimal initial *a.*

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