Estimation of X-Band Polarimetric Radar Attenuation and Measurement Uncertainty Using a Variational Method

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).

In the case of rain, the specific differential propagation phase (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 approximately

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The 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

The variational method consists of two major components: 1) the minimization of a cost function J and 2) a forward model . The essence of the variational algorithm is to derive an optimal set of state variables with the lowest sums of discrepancy between observations and forward model predictions. The forward model predicts radar observations as a function of reflectivity and rain intensity (Hogan 2007). The forward model for X-band radar is derived using a rigorous electromagnetic scattering and propagation model that is capable of predicting a suite of radar observables (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 form

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The gamma parameters, namely, N₀ (intercept parameter; mm⁻¹ m⁻³), μ (shape parameter; dimensionless), and D₀ (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 D₀ varied from 0.1 to 6 mm with an interval of 0.01 mm and the values of log₁₀(N₀) 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).

The forward models predict 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 as

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In 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.

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The X-band forward models based on a simplified electromagnetic scattering and propagation model: (a) Z_DR, (b) K_DP/Z_HH, (c) A_H/Z_HH, and (d) A_DP/Z_HH, all vs log(Z_HH/R). Gamma raindrop size distributions (RSDs) and measured video disdrometer DSDs were used as input into the electromagnetic scattering and propagation model. The black lines represent the forward models calculated using a gamma RSD with μ = 5 at 20°C. The gray dots represent the rigorous scattering model simulation corresponding to discretely measured DSDs based on the 5 yr of data from NCU 2DVD in northern Taiwan.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

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Table 1.

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

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The cost function (J) is composed of three components, namely, , as shown:

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They are functions of 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 (σ²). 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.

In the presence of attenuation, the intrinsic Z_HH and Z_DR (dB) are related to the observed reflectivity and differential reflectivity by the following equations:

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The 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 r₂ can be derived from the Φ_DP at range r₁ and integration of the K_DP between the range r₁ and r₂ as

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An optimal estimation of the target variable 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 is

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where

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and

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In the above equations, x_i represents the target variable ã at the ith iteration, y represents the observations Z_DR and Φ_DP, the subscript “background” represents the background values of ã, the and matrices represent the error covariance matrices of the observation (Z_DR and Φ_DP) and the background value, respectively. The and are diagonal matrices with fixed values. The matrix represents the forward model and the represents the Jacobian matrix, a matrix containing the partial derivative of each observation with respect to ã. The derivation of can be found in Hogan (2007).

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 and adjusted 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 χ² 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.

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Flowchart describing various numbered steps (at the top-right box corners) in the variational attenuation correction method. The dashed box represents a complete loop of the Gauss–Newton minimization process. See section 2a for a detailed description.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

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Predetermined diagonal matrices with fixed variance (σ²) values of and were used in Hogan (2007). However, the values of and are known only approximately as uncertainties in radar measurements and the natural variabilities of DSDs are unknown in practice. In X-band radar measurements with higher path-integrated attenuation, appropriate and have to be specified objectively to obtain satisfactory results by the variational method. In the following two subsections, procedures for estimating and are described.

c. Estimating the observation error covariance matrices

Instead of using a predetermined (Hogan 2007), a statistical diagnostic method is used for estimating the values of of 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 and analysis from observation are defined as

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and

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In the above equations, 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, is the forward operator. Variable 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 and analysis of Z_DR for each beam are

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and

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and vice versa for Φ_DP, where k is the available gates along a radar beam.

In general, background and analysis errors of observations are unbiased and mutually uncorrelated to each other statistically. Then, the mean value of the diagonal terms of the cross product of the innovation vector of background and analysis is the observation error variance (σ²):

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Therefore, diagonal terms of for each beam are obtained by calculating of Z_DR and Φ_DP. The is then obtained as a 2k × 2k matrix:

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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 . The observation error matrix takes into account uncertainty in radar measurements due to Mie scattering, the backscatter phase, and artifacts caused by gradients in reflectivity and nonuniform beam filling.

As suggested by Li et al. (2009), the proper values of and have to be derived simultaneously. The procedure shown in Fig. 2 for attenuation estimation is modified for estimating and adaptively. The modified procedure includes the discrete approximation and statistical diagnostic method is shown in appendix A.

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.

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The simulated X-band polarimetric radar observables for measured DSDs: (a) Z_HH and A_H, (b) Z_DR and A_hv, and (c) Φ_DP, R, and a, all vs distance. The DSDs were obtained from 5 yr of NCU 2DVD measurements. The modeled polarimetric observations without observation error were arranged along a radial with a gate resolution of 0.125 km. The total attenuation (A_h) and total differential attenuation (A_hv) for this 50-km beam are 12.07 and 1.44 dB, respectively.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

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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.

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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, contains radar measurement error and forward model error. In the absence of observation error, the for 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.

The mean RMSE as defined in Eq. (4) was calculated for A_h, A_hv, and the logarithm of rainfall rate (dBR), 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 as

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Even though regression coefficients α and β for attenuation estimation, and a₁ and b₁ for rainfall rate estimation were specifically tuned to the DSD data, RMSEs of dBR, 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 dBR, from 0.08 to 0.21 dB for A_h, and from 0.02 to 0.05 dB for A_hv.

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(a) RMSEs of attenuation (A_h) and differential attenuation (A_hv) derived from variational (var.) and Φ_DP-based methods vs distance. (b) As in (a), but for dBR; the values of dBR are defined as 10 log₁₀(R).

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

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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 and . The discrete approximation was first performed with prescribed observation error covariance matrices of Z_DR and Φ_DP for estimating the first-guess . Even though 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 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 . An example demonstrating the progression of convergence to known errors of Z_DR and Φ_DP with standard deviations (σ) of 1.1 dB and 2.5° is shown in Fig. 5. The diagnosed of Z_DR and Φ_DP converge in four iterations to the specified error values.

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The estimated values of the standard deviation (σ) of the measurement error of Z_DR (blue solid line) and Φ_DP (red solid line) as a function of the iteration number. The true values are shown as blue and red dashed lines, respectively.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

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In the case of the variable error in Z_DR, optimal values of 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 of Z_DR are close to the 1:1 line, as desired. Similarly, the diagnosed of Φ_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 of Z_DR and Φ_DP fairly well. This methodology is used in the analysis of measurements, as variances of measurements are unknown.

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(a) The relation between the estimated standard deviation of the measurement error of Z_DR (σ^est.–Z_DR) obtained from step 8 in Fig. A1 and the true value (σ^true–Z_DR) shown as a red line with the dashed line being a 1:1 line. (b) As in (a), but for Φ_DP.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

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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 dBR 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 dBR, A_H, and A_DP.

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As in Fig. 4, but vs the measurement error in Z_DR.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

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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.

Table 3.

Engineering specifications of TEAM-R.

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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 is required in the statistical diagnostic method for estimating optimal as discussed in section 2c. The values are determined by calculating the standard deviation (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.

Table 4.

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

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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.

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Radar data at a 3.19°-elevation angle from TEAM-R and S-Pol radar data: (a) Z_HH, (b) Z_DR, and (c) Φ_DP, all vs distance. The blue lines represent the measured raw values from TEAM-R and the red lines represent the attenuation-corrected Z_HH and Z_DR, and predicted Φ_DP based on the variational scheme. The black lines in (a) and (b) represent averaged measurements from the S-Pol radar. The distance between the TEAM-R and NCAR S-Pol radars was 50 m.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

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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.

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RHI scans of (left) Z_HH and (right) Z_DR measurements from NCAR S-Pol and TEAM-R. (a)–(d) Intrinsic data measured during TiMREX 2008 with attenuation correction for S-Pol and system bias correction methods for TEAM-R applied; a threshold of ρ_HV < 0.7 was applied to the X-band measurements, and no threshold was applied to the S-band measurements. Further, the intrinsic difference in the Z_DR observations between X-band and S-band wavelengths was removed by converting S-band Z_DR to X-band Z_DR (Chandrasekar et al. 2006). (e),(f) The variational attenuation-corrected Z_HH and Z_DR from TEAM-R, respectively. The solid lines represent the beams with elevation angles of 5.5° and 8.1°, respectively; the dashed line represents a beam of 3.19°.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

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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.

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(a) The predicted Φ_DP from the variational method and (b) the measured Φ_DP. The solid lines represent beam elevation angles of 5.5° and 8.1°, respectively, and the dashed lines represent beams of 3.19°.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

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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.

Table 5.

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.

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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.

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The Φ_DP attenuation correction with coefficients α = 0.374 and β = 0.052 for attenuation (A_H: black solid) and differential attenuation (A_DP: black dashed) vs K_DP. The gray dots represent the variational algorithm derived attenuation (A_H), differential attenuation (A_DP), and K_DP. The corresponding A_H (A_DP) and K_DP with Z_DR values between 0–1 dB and between 3–4 dB are shown by red and blue dots, respectively.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

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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 () and measurements of Z_DR and Φ_DP () were obtained objectively by the discrete approximation and statistical diagnostic method for each beam, rather than using the predetermined or a priori error covariance matrices.

It is interesting to note that the standard deviations of the Φ_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 are shown in Fig. 12c for evaluating how the analysis weighted the measurements and background term. The values of are relatively lower between elevation angles 5.5° and 8.1°, indicating the variational algorithm relied more on the background term than on observations of Z_DR and Φ_DP. This is consistent with higher values of for 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 of measurements for each beam, the for each range gate is estimated as

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Higher values of , 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.

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(a) The estimated optimal standard deviations of Z_DR (from steps 4–6 of Fig. A1) as a function of elevation angle. (b),(c) As in (a), but for Φ_DP and the optimal background error covariance matrices (), respectively.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

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(a) The diagnostic standard error of Z_DR (derived from step 8 of Fig. A1) vs distance obtained by using the statistical diagnostic method. (b) As in (a), but for Φ_DP. The solid lines represent elevation angles of 5.5° and 8.1°, respectively.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

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The mean values of for 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 of 0.27-dB for 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 for 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 () and observations (). The OSSE studies demonstrated that error covariance matrix observations () can be estimated with sufficient accuracy, even though rather underestimated prescribed values were used initially.

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 () for identifying the region with poor quality of measurement that failed to provide useful information to the variational algorithm, and 4) an estimated observation error covariance matrix () for characterizing the spatiotemporal error characteristics of polarimetric measurement 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 and of real radar measurements play a critical role in the variational attenuation correction method at X band. The discrete approximation ensured that is consistent with the data quality of the radar measurement. Higher weighting of the background term (i.e., lower value of ) was used in the variational method when noise in the radar measurements was high. The error covariance matrix () of the 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 obtained by the statistical diagnostic method is consistent with the standard deviation (σ) of the actual radar measurements in a stratiform light rain. The results of diagnosed indicate the statistical diagnostic method cannot only estimates accurate values of error covariance of 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 for 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.