## 1. Introduction

Monitoring of precipitation using high-frequency radar systems such as X band is becoming increasingly popular because of their lower cost compared to their counterpart at S band (Chandrasekar et al. 2004b; Park et al. 2005a). For urban or mountain hydrological applications, the X-band radar systems have been implemented in Europe and Japan (Delrieu et al. 1997; Testud et al. 2000; Park et al. 2005b). Meteorological radar systems operating at S-band frequencies are mostly not affected by attenuation resulting from precipitation, except in some regions of wet hail. The S-band radar systems with narrow beams (say 1° beamwidth) are typically expensive, with large antennas and high-power transmitters to cover large areas. Recently, networks of meteorological radar systems at higher frequencies such as X band have been pursued, especially for low-cost and targeted applications, such as coverage over a city or a small basin. However, at higher frequencies, the impact of attenuation resulting from precipitation needs to be resolved for successful implementation. Chandrasekar et al. (2002, 2004a) studied the relationship between the intrinsic radar variables of S and X bands in rain medium. Note that “intrinsic” refers to the radar variables that are nonattenuated and obtained by theoretical simulation. Extensive dual-polarization observations of precipitation at S band are available today from research radar facilities such as the Colorado State University’s (CSU’s) University of Chicago–Illinois State Water Survey (CHILL) radar and the National Center for Atmospheric Research (NCAR) S-band dual-polarized Doppler radar (S-POL).

This paper presents three methodologies to simulate X-band radar variables in rain from S-band data. These methodologies can simulate the realistic dual-polarization radar variables maintaining the natural spatial structure of the rainfall event. The simulated dataset with and without attenuation effect can be used effectively in the design of the X-band radar system as well as in the evaluation of algorithm development such as attenuation correction. The paper is organized as follows. In section 2 the theoretical background of the rain model is described, whereas the methods that simulate X-band variables from S-band data are discussed in section 3. The important results of this paper are summarized in section 4.

## 2. Rain model and polarimetric radar observables

*N*(

*D*) is the number of the raindrops per unit volume per unit size interval,

*D*(mm) is the volume equivalent spherical diameter, and

*N*

_{0}(intercept parameter, mm

^{−1−μ}m

^{−3}), Λ (slope parameter, mm

^{−1}), and

*μ*(shape parameter) are the parameters of the gamma distribution. One of disadvantage of this gamma model is that the unit of

*N*

_{0}depends on

*μ*. To study the shape of DSD with widely varying rainfall rates, gamma distribution can be expressed in a normalized form as (Sekhon and Srivastava 1971; Willis 1984; Testud et al. 2001; Bringi and Chandrasekar 2001) where

*D*

_{0}is the median volume diameter,

*μ*is a measure of the shape of the DSD, and

*N*(mm

_{w}^{−1}m

^{−3}) is the normalized intercept parameter of an equivalent exponential distribution with the same water content and

*D*

_{0}.

*Z*

_{h},

_{υ}at horizontal (h) and vertical (

*υ*) polarizations can be described as where

*λ*is the wavelength of the radar,

*σ*

_{h,υ}represents the radar cross sections at horizontal and vertical polarizations, and

*K*is the dielectric factor of water defined as

_{w}*K*= (

_{w}*ɛ*− 1)/(

_{r}*ɛ*+ 2), where

_{r}*ɛ*is the complex dielectric constant of water. Differential reflectivity can be expressed as the ratio of reflectivity factors at horizontal and vertical polarizations (Seliga and Bringi 1976), Specific differential phase is proportional to the real part of the difference in the complex forward-scatter amplitudes

_{r}*f*at horizontal and vertical polarizations. It can be defined as The two-way differential propagation phase

*ϕ*

_{dp}is defined as The measured differential propagation phase can be defined as where Δ is the backscattering propagation phase. Specific attenuation at two polarization states and differential attenuation are related to DSD as Two-way cumulative attenuation

*A*

_{h}and differential attenuation

*A*

_{dp}can be expressed as where

*s*is range for integration.

## 3. Relationship between X- and S-band radar variables

Three different methodologies for simulating Xband radar variables from S-band data will be discussed in the following, namely, the empirical conversion method, DSD sampling method, and DSD inversion method.

### a. Empirical conversion method

The intrinsic measurements of *Z*_{h}, *Z*_{dr} exhibit a nearly one-to-one relation between S and X bands. This principle was used by Chandrasekar et al. (2003) to characterize spaceborne radar observations at multiple frequencies. Figure 1 shows a scatterplot of *Z*_{h} and *Z*_{dr} at S and X bands for widely varying drop size distributions (0.5 ≤ *D*_{0} ≤ 3.5 mm, 3 ≤ log_{10}*N _{w}* ≤ 5, and −1 <

*μ*≤ 4 for

*R*< 300 mm h

^{−1}and

*Z*

_{h}< 55 dB

*Z*). The data were obtained by scattering simulation using the shape model proposed by Bringi et al. (2003), which combines the Andsager et al. (1999) fit and the Beard and Chuang (1987) model at a temperature of 10°C (henceforth referred as the ABC model). Under Rayleigh scattering assumptions, reflectivity will not change with frequency. However, at X band Rayleigh scattering assumptions are not strictly valid as shown in Fig. 1a. The comparison of

*Z*

_{dr}in Fig. 1b shows the non-Rayleigh scattering very well.

*X*and

*S*indicate simulated radar variables at X band and measured (assumed to be nonattenuated) radar measurements at S band, respectively. The relationship between X- and S-band radar data can be obtained by curve fitting using the data that come from the theoretical simulation using DSD parameters. This method is simple and reliable for reflectivity and differential reflectivity simulation. Figure 2a shows the scatterplot of intrinsic reflectivity obtained by theoretical simulation using widely varying DSD parameters versus simulated reflectivity by (11), whereas the scatterplot of intrinsic specific attenuation versus simulated specific attenuation by (12) is shown in Fig. 2b.

*Z*′

_{h,X}and attenuated differential reflectivity

*Z*′

_{dr,X}with attenuation effects resulting from precipitation can be generated with respect to radar variables simulated by (11)–(12) as where

*r*

_{0}is the first range that has precipitation echo, and

*r*is the range of echo (

*r*

_{0}<

*r*). Note that the attenuation impact by clouds and gases are not considered in (13), because attenuation from clouds and gases are negligible compared to attenuation resulting from rain. An example for simulation of attenuated X-band reflectivity is shown in Fig. 3. Figures 3a and 3b show S-band radar observations, whereas Figs. 3c and 3d show the simulated X-band radar variables obtained by (11) and (12) using the S-band dataset in Figs. 3a and 3b. Figure 3e shows the attenuated X-band reflectivity obtained by (13) using simulated X-band variables in Figs. 3c and 3d.

### b. DSD sampling method

It was shown by Scarchilli et al. (1996) that the triplet of measurements *Z*_{h}, *Z*_{dr}, and *K*_{dp} nearly lie on a three-dimensional surface. Therefore, once *Z*_{h} and *Z*_{dr} are specified, the choice of possible *K*_{dp} fall in a narrow range. As a result, if we choose a value of *K*_{dp} at S band corresponding to *Z*_{h} and *Z*_{dr}, then the *K*_{dp} value at X band can be obtained by direct frequency scaling, because *K*_{dp} is linearly proportional to frequency as (6). This procedure to choose *K*_{dp} at S band avoids the problem of direct *K*_{dp} estimation (Gorgucci et al. 2000), which can suppress peaks resulting from the slope estimation process, which is used to derive *K*_{dp} from *ϕ*_{dp}. The above principle is implemented in a detailed manner as explained in the following. A large dataset of *Z*_{h}, *Z*_{dr}, and *K*_{dp} values at S band are generated by the ABC model corresponding to a wide range of DSD parameters (0.5 ≤ *D*_{0} ≤ 3.5 mm, 3 ≤ log_{10}*N _{w}* ≤ 5, −1 <

*μ*≤ 4) under the constraints of

*Z*

_{h}< 55 dB and

*R*< 300 mm h

^{−1}. For a given set of

*Z*

_{h}and

*Z*

_{dr}, a search of this database provides possible choices of DSDs that satisfy the radar variables

*Z*

_{h}and

*Z*

_{dr}. One of those DSDs is randomly chosen to compute the X-band radar variables. Because the process is structured on DSD, the observed reflectivity and differential reflectivity can be computed according to (13). The block diagram in Fig. 4 provides a description of the simulation procedure. Figure 5a shows the range profile of reflectivity and differential reflectivity from the NCAR SPOL radar observed over central Florida. The simulated X-band profiles of

*Z*′

_{h,X}(

*r*),

*Z*

_{h,X}(

*r*) as well as

*Z*′

_{dr,X}(

*r*),

*Z*

_{dr,X}(

*r*) are shown in Figs. 5b and 5c, whereas the profiles of

*ϕ*

_{dp}at S and X bands are shown in Fig. 5d. A cursory glance of Fig. 5 shows that the simulation procedure produces reasonable range profiles of X-band radar variables. Once again, the purpose of this simulation is to simulate realistic range profiles of X-band dual-polarization variables in order to maintain the spatial correlation structure of the naturally occurring rainfall. It should be noted that the procedure is not to simulate the “exact” observation, but simulate profiles that fall within the range of observations.

### c. DSD inversion method

*Z*

_{h,}

*,*

_{S}*Z*

_{dr,}

*, and*

_{S}*K*

_{dp,}

*). The method is based on the concept of an effective mean axis ratio versus diameter model, which is a linear relationship (*

_{S}*r*= 1 −

*βD*). They developed an algorithm for estimating

*β*(magnitude of the slope of the shape–size relationship) using radar measurements at S band. For 10log

_{10}(

*Z*

_{h,}

*) ≥ 35 dB*

_{S}*Z*, 10log

_{10}(

*Z*

_{dr,}

*) ≥ 0.2 dB, and*

_{S}*K*

_{dp,}

*≥ 0.3 km*

_{S}^{−1},

*D*

_{0}and

*N*are retrieved as For other cases in which

_{w}*K*

_{dp}is noisy and 10log

_{10}

*Z*

_{h}is below 35 dB

*Z*, the retrieval method of

*D*

_{0}and

*N*proposed by Bringi et al. (2002) is applied. After retrieving DSD from convective event data observed by the CSU CHILL radar, X-band radar variables (

_{w}*Z*

_{h,}

*,*

_{X}*Z*

_{dr,}

*,*

_{X}*α*

_{h,}

*,*

_{X}*α*

_{dp,}

*,*

_{X}*K*

_{dp,}

*, and Δ*

_{X}_{co,}

*) with a realistic scenario of the precipitation event are simulated by theoretical simulation. Here*

_{X}*Z*′

_{h,X}and

*Z*′

_{dr,X}are generated by (13), whereas

*ψ*

_{dp,}

*is obtained from*

_{X}*K*

_{dp,}

*and Δ using (8). Figure 6 shows the S-band observations (*

_{X}*Z*

_{h,}

*,*

_{S}*Z*

_{dr,}

*) and parameters (*

_{S}*D*

_{0}and

*N*) of DSD retrieved from S-band observations. Simulated X-band radar variables and attenuated radar variables corresponding to S-band radar measurements in Fig. 6 are shown in Fig. 7. The results show that this method can produce reasonable X-band radar variables. Though one type of DSD retrieval is shown, this procedure can be applied with any DSD retrieval algorithm using dual-polarization radar data.

_{w}### d. Comparison of three methodologies

For comparison of the proposed methodologies, radar variables at X band are simulated using a ray profile observed by CSU CHILL radar. Observed reflectivity, differential reflectivity, and differential phase at S band are shown in Figs. 8a, 8c and 8e as a solid line, whereas simulated reflectivity, differential reflectivity, and differential phase at X band are shown in Figs. 8a, 8c and 8e according to the three methodologies. Note that the *ψ*_{dp} observations are filtered to remove measurement error. The corresponding attenuated reflectivity and differential reflectivity are shown in Figs. 8b and 8d. From the results of Fig. 8, we can see that all of the proposed methods can simulate reasonable radar variable profiles at X band falling within the range of observed S-band radar measurements. The empirical conversion method is based on a nearly one-to-one relation of the intrinsic radar measurements between S and X band. The method is simple and reliable, particularly for reflectivity and differential reflectivity. If the simulated *K*_{dp} or *ϕ*_{dp} is needed, the DSD sampling method or DSD inversion method will be useful. The DSD inversion method connects X-band variables and DSD retrieved from S-band radar measurements. This method is more sophisticated because of the DSD inversion procedure. The DSD sampling method lies between the empirical conversion method and the full DSD inversion method.

## 4. Summary and conclusions

Owing to the success of the dual-polarization methodology for attenuation correction in rain (Bringi et al. 1990; Testud et al. 2000; Bringi and Chandrasekar 2001; Delrieu et al. 2000; Anagnostou et al. 2004; Matrosov et al. 2005; Park et al. 2005b), X-band radars are becoming more viable for targeted short-range applications. To validate the performance of the radar retrieval algorithm during the test phase of algorithm development, it is necessary to use the empirical data based on actual precipitation event. One way to obtain that dataset for higher frequencies is to simulate it from radar observations of nonattenuated or low attenuated frequencies, such as at S band. This paper presents three such methodologies to simulate “realistic” dual-polarization radar variables at X-band. The conversion methods start from S-band dual-polarization radar observations and use the fundamental microphysical properties of rainfall, namely, size and shape distribution, to transform S-band into X-band variables. As a result, these methods maintain the connection between the realistic scenarios of rain events with the natural distribution of rainfall microphysical properties. The simulated X-band radar variables can give a possible scenario with wide varying drop size distribution. These simulations have been used in evaluating the performance of X-band radar designs and algorithms in our research.

This research was supported by the ERC program (0313747) and NSF ATM (0313881).

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