## Abstract

The CSU–CHILL radar is a dual-wavelength, dual-polarization weather radar system operating at S and X bands with coaxial beams. One of the capabilities of this radar system is the possibility of developing and/or validating algorithms across dual wavelengths and dual polarizations. This paper presents one such instance, showing how the rainfall field can be estimated either from the S- and X-band reflectivities or from the differential propagation phase at X band. To do so, the paper first presents a dual-wavelength attenuation correction method that uses the reflectivity measured at S band, as the constraint for the correction of the reflectivity measured at X band, and it describes how Mie scattering regions at X band may be detected from the retrieved path-integrated attenuation field. Then, the paper describes how the resulting specific attenuation field relates to rainfall and specific phase at X band, which can be obtained from dual-polarization data at a single wavelength as well, and shows examples. Finally, the paper looks at the relation between attenuation and the differential phase as a function of elevation angle for a few cases, which may be related to the drop size distribution and mean diameter, as well as temperature.

## 1. Introduction

The CSU–CHILL National Weather Radar Facility (Brunkow et al. 2000), located in Greeley, Colorado, is a research facility operated by Colorado State University (CSU) under the sponsorship of the National Science Foundation (NSF) and CSU. The CSU–CHILL radar went through a major transformation (Junyent et al. 2015) to add support for simultaneous dual-wavelength (S and X bands), dual-polarization (linear horizontal and vertical) radar operation, as well as high polarization purity single-wavelength operation at S or X band (see Table 1 for a list of the main radar characteristics).

Dual-wavelength radar systems have historically been used to estimate properties of the observed precipitation medium through the use of the dual-wavelength ratio (DWR), defined as the ratio between reflectivity at the longer wavelength and reflectivity at the shorter wavelength. The DWR signal captures the difference in propagation effects (i.e., attenuation) as the two wavelengths travel through the medium, as well as the difference in scattering magnitudes (particle size relative to wavelength). A lot of emphasis has been put on interpreting the scattering magnitude difference contribution to the DWR, as it could indicate the presence of hail (Atlas and Ludlam 1961; Eccles and Atlas 1973; Jameson and Srivastava 1978; Jameson and Heymsfield 1980; Bringi et al. 1986; Tuttle et al. 1989; Herzegh and Jameson 1992), while dealing with attenuation as an obstacle to get to the hail signal. This obstacle seemed to be tackled either through a least squares piecewise linear fit to the DWR to obtain a monotonically increasing attenuation signal (Jameson and Heymsfield 1980) or by using the DWR values at the farther edge of the storm to indicate the total attenuation at the shorter wavelength and apportioning it according to the unattenuated longer-wavelength reflectivity values measured along the signal path (Tuttle and Rinehart 1983).

Attenuation, however, is directly related to precipitation intensity and can be used to estimate rainfall (Eccles and Mueller 1971). This paper takes a complementary approach to that described in the previous paragraph by explicitly obtaining a dual-wavelength-derived attenuation field that makes use of all the reflectivity information along the propagation path at both S and X bands, instead of using an “end of ray”-type constraint or treating dual-wavelength-derived fields such as DWR with the embedded attenuation. The implemented attenuation correction procedure uses a cost function to minimize the difference between reflectivity at S and X bands, leveraging a well-known approach (Testud et al. 2000) to apportion attenuation along the propagation path and not making use of dual-polarization information. The obtained dual-wavelength attenuation field is used to estimate rainfall rate, investigate relations between attenuation and the propagation phase at different elevation angles and in different precipitation environments, and as means to detect areas of potential Mie scattering at X band.

## 2. Dual-wavelength attenuation correction

When using dual-polarization information to perform attenuation correction, the attenuation constraint is typically derived from a single value such as the differential propagation phase increment over the considered propagation region (Bringi et al. 1990; Testud et al. 2000) as expressed in

where the attenuation constraint, the one-way path-integrated attenuation has a value proportional to the differential propagation phase increment over the considered propagation region .

Relying on a single constraint value to represent the integrated effect over the propagation path could lead to erroneous results in certain situations, such as in the case of a differential propagation phase profile contaminated with the differential phase shift upon backscatter, or affected by noise. In the dual-wavelength case, this can also be true when using the DWR value in the far edge of the storm as an attenuation constraint. Additionally, there may be situations in which the dual-polarization variables may fail to perform as expected, such as in the case of hydrometeors presenting a large canting angle relative to the radar beam (i.e., when scanning at higher elevation angles) or in the case of vertically aligned particles due to the presence of electrical fields.

In the case of having simultaneous measurements of the reflectivity field at S and X bands, one can use the along-ray reflectivity observations at each wavelength, together with the path-integrated attenuation to be estimated, to build a cost function. Finding the minimum for that cost function will result in estimating the attenuation-corrected reflectivity at X band that best fits the measured reflectivity at S band under some norm.

Let be the reflectivity measured at S band and be the reflectivity measured at X band (both in mm^{6} m^{–3}), let be the one-way path-integrated attenuation (dB) experienced by the X-band reflectivity signal in the considered data ray region from to *r*, and we can write the cost function as follows:

where the brackets denote the L2 norm, is a set of weight coefficients along range, and the 2 factor accounts for the round-trip path-integrated attenuation. The weight coefficients give a mechanism to reduce the contribution to the cost function of those regions that may be suspect of having intrinsic differences in reflectivity—that is, differences not just due to attenuation but to other effects, such as Mie scattering in the presence of large hydrometeors.

Except for the possibility of adjusting the weights, the cost function assumes that the intrinsic reflectivities at S and X bands should be the same. In situations where this may not be the case, the affected gates will introduce a bias that will be spread across the data being processed. This could be due to the presence of Mie scattering at X band and/or attenuation at S band. However, we would assume these effects to be manageable, as areas of Mie scattering should be relatively small and localized compared to surrounding areas of rain, and significant areas of attenuation at S band typically imply signal extinction at X band. All of this leads to expecting the method’s best performance in rain.

Equation (2) can be solved by finding the PIA that minimizes its L2 norm using the propagation equation in Testud et al. (2000) and Bringi and Chandrasekar (2001). Following the formulation for the propagation equation as presented in Bringi and Chandrasekar (2001, chapter 7), one can see that the path-integrated attenuation along the data ray can be expressed as

where is specific attenuation and *b* is a constant parameter typically within 0.6–0.9 at microwave frequencies and close to 0.8 at X band (Delrieu et al. 1997; Testud et al. 2000; Park et al. 2005). Combining Eqs. (3) and (2), the resulting cost function can be solved for the total path-integrated attenuation (the control variable) that minimizes the difference between the S-band and X-band (attenuation corrected) reflectivities under the L2 norm, using a method for solving nonlinear least squares problems such as Levenberg–Marquardt (Levenberg 1944; Marquardt 1963). This dual-wavelength attenuation correction procedure is used throughout the paper to estimate and correct for attenuation in the X-band signal.

The method introduced should also be appropriate for attenuation correction of vertical-pointing (such as vertical profiling and satellite applications) and range–height indicator (RHI) scans since it does not depend on the relative canting angle between the hydrometeors and the antenna beam. The reflectivity measurements at each wavelength are affected equally by the beam-to-hydrometeor geometry, thus preserving the attenuation information. This is in contrast to the case of dual-polarization-based attenuation correction algorithms, where that geometry can have a strong effect on the algorithm performance. In the limit case of vertical pointing, dual-polarization variables such as the differential phase and differential reflectivity (typically used as constraints in dual-polarization attenuation correction schemes) may not be able to bring any additional information, as the hydrometeors will show essentially the same response at horizontal and vertical polarization independently of any attenuation in the path. This will improve as the antenna elevation gets lower, with the best case being when the beam is horizontal. However, the reflectivity measurements at each wavelength are affected equally by the relative orientation between antenna beam and hydrometeors, and therefore the presented dual-wavelength attenuation correction algorithm performance should be independent of antenna pointing angle, which is a benefit of this scheme.

## 3. Detection of Mie scattering at X band

One can use the computed path-integrated attenuation at X band to retrieve a Mie scattering signal from the dual-wavelength dataset. In general, multiple-wavelength radar systems rely on the fact that a given hydrometeor may have a different scattering behavior depending on the wavelength at which it is illuminated, as the relative size of the scatterer with respect to the illuminating wavelength dictates the scattering regime. A small scatterer (i.e., one that is small compared to the illuminating wavelength) will operate in the Rayleigh regime, and one that has a size comparable to the wavelength (a commonly used diameter threshold is ) will enter the Mie regime. The Mie scattering signal would indicate that the observed hydrometeor sizes are such that at X band they are in the Mie scattering regime, while at S band they would still be in the Rayleigh scattering regime, therefore creating a difference in their intrinsic reflectivity values. In general, one can use the following expression relating the measured reflectivities at S band and X band:

where is a difference signal that accounts for the difference between the measured reflectivity at S band and the attenuation corrected reflectivity at X band. That difference signal can be further decomposed into a component due to the possible difference in scattering regimes at the two wavelengths and the rest of the difference signal :

Assuming that the radar does not introduce any significant artifacts (including those that may be created by mismatched beam volumes) into the measurements [i.e., or ], the expression to estimate the Mie signal component can be rewritten as

The DWR, defined as the difference of the reflectivity values (dB*Z*) measured at S band and X band (with no attenuation correction), is introduced as it has been proposed to try to detect the areas of potential Mie scattering due to hail (Eccles and Atlas 1973). In the method presented here, subtracting the path-integrated attenuation signal portion from the dual-wavelength ratio signal should result in a signal that marks those areas where the reflectivity at S band and the reflectivity at X band (corrected for attenuation) differ due to the scattering magnitude (not the signal attenuation suffered during propagation). This difference in reflectivity due to scattering magnitude is typically associated with the X-band signal being in the Mie regime in areas of higher reflectivity values (i.e., indicating larger hydrometeor sizes).

## 4. Rainfall estimation from specific attenuation and specific differential phase

It is fairly well established that a good approximation for the relationship between and specific differential phase at X band is

as both and depend on the fourth moment of the drop size distribution (DSD; Bringi et al. 1990; Bringi and Chandrasekar 2001; Matrosov et al. 2002). Integrating over range on both sides, Eq. (7) can be rewritten as

In the CSU–CHILL radar, both the and the differential propagation phase data fields are available at X band, the first obtained through the dual-wavelength attenuation correction procedure previously described and the second directly measured by the radar, therefore allowing to retrieve the coefficient *α* from available radar data.

The availability of *α* allows for constructing and comparing rainfall estimators based on *A* and , which are related by that parameter as shown in Eq. (7). At X band, a rainfall estimator of the type

has been used with great success in the Center for Collaborative Adaptive Sensing of the Atmosphere (CASA) radar networks (Junyent et al. 2010), showing very good agreement with rain gauge comparisons (Wang and Chandrasekar 2009; Wang and Chandrasekar 2010). An equivalent rainfall estimator based on the *A* retrieved via the dual-wavelength attenuation correction algorithm can be built by substituting Eq. (7) into Eq. (9), resulting in

In some environments the estimator has been shown to be less sensitive to DSD variations than the equivalent estimator (Atlas and Ulbrich 1977; Matrosov 2005; Ryzhkov et al. 2014), and it should also have the advantage of operating at the resolution of the radar measurement. This is due to the fact that can be estimated at the radar range gate resolution, whereas is typically obtained using its definition by calculating the range derivative of the differential propagation phase, which introduces a trade-off between the measurement resolution and noise.

Implementation of the specific attenuation–based rainfall estimator in the presented dual-wavelength framework differs from other proposed implementations (Matrosov 2005; Ryzhkov et al. 2014) in the sense that is derived directly from the measured reflectivities at S band and X band (see section 2), and that it does not involve any preestablished relationship between attenuation and reflectivity or dual-polarization variable and/or constraint. Other methods have also been proposed to obtain at the radar measurement resolution, such as Lim et al. (2013).

## 5. Results

As the two wavelength components of the CSU–CHILL radar have different antenna beamwidths (see Table 1), the initial step in all data processing results shown throughout the paper is to synthesize X-band measurements that match the wider S-band beamwidth through the method described in the appendix. This is done with the intent of reducing wherever possible any potential errors due to beam mismatch.

On 21 July 2013, a thunderstorm was observed roughly 60 km southeast of the CSU–CHILL radar, over flat land that slowly rolls off to lower elevation away from the radar. This location is particularly well suited for physical algorithm evaluations as it produces very little ground clutter returns from both the 1° antenna beam at S band and the 0.3° antenna beam at X band.

Figure 1 presents dual-wavelength horizontal polarization reflectivity data, with Fig. 1a showing measured , Fig. 1b showing , and Fig. 1c showing . Comparing Figs. 1a and 1b makes readily apparent that the measured X-band reflectivity is reduced by attenuation, as the farther edge of the storm appears clearly weaker than that measured at S band. Figure 1c shows the result of applying the attenuation correction procedure previously described. Comparing Figs. 1a and 1c, there is a much-improved agreement between reflectivities at S and X bands, as the S-band contours match the attenuation-corrected X-band reflectivity values to a high degree.

Figure 2 presents the same dataset, and in the same layout as Fig. 1, for differential reflectivity. As expected, looking at Fig. 2b, one can see how the effect of differential attenuation makes the X-band differential reflectivity values at the farther edge of the storm go negative. Figure 2c shows the attenuation-corrected differential reflectivity at X band, where it is readily apparent that all negative values due to differential attenuation have been corrected. Comparing Figs. 2a and 2c reveals good visual agreement between the attenuation-corrected X-band values and S-band values, with the S-band contours generally following the attenuation-corrected differential reflectivity values at X band.

To get more insight into the attenuation correction performance, Fig. 3 shows a comparison of S- and X-band reflectivity density distributions and scatterplots before and after attenuation correction, corresponding to the dataset shown in Figs. 1 and 2. Figure 3a shows the measured reflectivity distributions at horizontal polarization, while Fig. 3b shows the corresponding scatterplot. The measured reflectivity distribution at X band is skewed toward lower reflectivity values when compared to the reflectivity distribution at S band, due to the effect of signal attenuation. This is also visible in the scatterplot, where at higher S-band reflectivity values the corresponding X-band values show a lot of dispersion and are generally lower. Figures 3c and 3d show the corresponding density distributions and scatterplot after attenuation correction. Looking at Fig. 3c, one can see the effect of the attenuation correction procedure in shifting points from the lower end of the distribution to fill in the higher-valued region, closely following the S-band distribution. The scatterplot clearly shows the improvement, with a fit slope *m* = 1.00 and correlation *r* = 0.99 after attenuation correction, indicating very good agreement between the two wavelength datasets. However, neither the distributions nor the scatterplots at the two wavelengths are expected to match exactly, since they come from independent measurements, taken through separate systems with different measurement and noise characteristics (see Table 1).

Similarly, Fig. 3e shows the measured differential reflectivity distributions at each wavelength, while Fig. 3f shows the corresponding scatterplot. One can see the effect of differential attenuation in the X-band measurements: the stronger attenuation at horizontal polarization versus vertical polarization creates larger negative values at X-band distribution than those in the S-band distribution, which can be seen in both the distribution and the scatterplot. The scatterplot also reveals a very low correlation between the two datasets. After attenuation correction, there is a significant improvement in the differential reflectivity matching across the two wavelengths. The plot in Fig. 3g shows how after attenuation correction the X-band distribution is no longer skewed toward negative values, whereas the scatterplot in Fig. 3h shows great improvement in terms of both correlation (from *r* = −0.09 to *r* = 0.81) and fit slope (from *m* = 0.12 to *m* = 0.75). The somewhat lower correlation when compared to the reflectivity results may be explained by both the higher noise component and the lower dynamic range of the differential reflectivity measurement, as well as intrinsic differences in the differential reflectivity measurement at each wavelength.

The results presented for both reflectivity and differential reflectivity show that the procedure described in section 2 is capable of retrieving horizontal polarization and differential path-integrated attenuation [and therefore the specific attenuation ] experienced by the X-band signal.

The same dual-wavelength attenuation correction procedure is now applied to an RHI scan at an azimuth angle of 157° through the same storm presented in Fig. 1, as a first step to Mie scattering detection as described in section 3 and also to explore the capability of the attenuation correction procedure when scanning in elevation.

Figure 4a shows , while Fig. 4b shows . The onset of signal attenuation at X band is clearly visible from a range of 65 km onward, as the signal goes through an intense core with reflectivities over 50 dB*Z* extending to heights close to 7 km. Figure 4c shows the resulting reflectivity at X band after attenuation correction , and Fig. 4d shows the computed two-way path-integrated attenuation. Comparing Figs. 4c to 4a, one can see how has very good general agreement with , as the S-band reflectivity contours overlaid on X-band attenuation corrected reflectivity follow the storm morphology from front to back with the notable exception of the area over 60 dB*Z* at around 70 km in range. Figure 4e shows the dual-wavelength ratio, where one can see how its signal trend closely follows that of the retrieved path-integrated attenuation in Fig. 4d, except for a local maximum over 50 dB located at around 70 km in range that appears collocated with the area of higher reflectivity at S band. The Mie signal in Fig. 4f (i.e., the difference between Figs. 4e and 4d) removes the upward monotonic trend of the attenuation component in the dual-wavelength ratio signal and exposes an area of large difference in scattering magnitude (peaking around 15 dB) collocated with the areas of higher reflectivity at around 70 km in range. This area of enhanced Mie signal corresponds with lowered copolar cross correlation (Fig. 4h) and near-zero differential reflectivity (Fig. 4g) at S band, which is consistent with the presence of hail.

Figure 5 shows a detailed comparison of data ray plots of the dual-wavelength ratio, path-integrated attenuation, and Mie scattering signal versus range, taken at 2.12° elevation, corresponding to the same dataset shown in Fig. 5. The arrows mark the point where DWR departs from PIA and the MIE signal shows a clear increase that could be associated with the presence of hail (as supported by the near-zero and lowered values in Fig. 4). It is worth noting that the presence of large areas of Mie scattering relative to the general area of Rayleigh scattering in the processed data can bias the final attenuation correction result and reduce the ability to the detect that Mie scattering region, as the attenuation correction cost function may treat the larger Mie scattering area as the “norm” and the remaining Rayleigh scattering area as the “perturbation” in a role reversal. This can be addressed by modifying the weight coefficients in the attenuation correction expression in Eq. (1). Lowering the value of the weight coefficients corresponding to the points with suspected Mie scattering (i.e., points where the reflectivity difference at S and X bands may exhibit large differences not due to attenuation) in the fit calculation will reach an attenuation correction solution that is less biased by those points.

The availability of both PIA (retrieved through the dual-wavelength attenuation correction procedure) and (from dual-polarization capability) at X band allows for estimating the *α* coefficient in Eq. (8). This coefficient relating accumulated attenuation to differential propagation phase is at the core of dual-polarization attenuation correction methods (Bringi et al. 1990; Testud et al. 2000).

Continuing with the PPI scan dataset shown in Figs. 1, 2, Fig. 6 shows the corresponding differential propagation phase at X band in Fig. 6a, whereas the two-way path-integrated attenuation is shown in Fig. 6b for the horizontal polarization signal, and Fig. 6c shows two-way path-integrated differential attenuation. One can readily see that all data fields have a qualitatively similar spatial distribution pattern as implied by Eq. (8). To estimate the slope of the linear relationship between *A* and , *α*, the data presented in Fig. 6 are used to construct scatterplots of two-way path-integrated attenuation (for both horizontal polarization reflectivity and differential reflectivity) versus differential propagation phase. Fitting a line to the resulting scatterplots allows for estimating *α* as the slope of the linear fit, as is shown in Fig. 7. The values obtained for this PPI scan dataset are in the case of horizontal polarization reflectivity and in the case of differential reflectivity, which are consistent with previously reported results in the literature (Chandrasekar et al. 1990; Testud et al. 2000; Park et al. 2005; Snyder et al. 2010; Matrosov et al. 2014). This is one of the few explicit experimental confirmations of the attenuation correction coefficient, and it can be taken as evidence that the dual-wavelength processing involved in obtaining attenuation and the dual-polarization processing involved in obtaining differential propagation phase are consistent and the radar behaves as expected.

A comparison between the rainfall estimators in Eqs. (9) and (10) is illustrated in Fig. 8, showing side-by-side panels of *A* and , together with the corresponding rainfall estimators [using *a* = 18.15 and *b* = 0.791 (as described in Wang and Chandrasekar 2010) for both estimators, and as retrieved in Fig. 7]. There is very good qualitative agreement on the location and relative intensity of the image features, but as expected the *A* field and associated rainfall estimator have a much sharper feature resolution than that of the and its related rainfall estimator. This is due to the numerical procedures involved in obtaining those fields. The attenuation field *A* is obtained from the method described in section 2, which assumes a parametric relationship between reflectivity and attenuation and operates at the resolution of the radar measurement (i.e., range gate size). However, the specific differential phase is obtained using its definition by calculating the range derivative of the differential propagation phase, which introduces a trade-off between the measurement resolution and noise. In this example, the specific differential phase is obtained through the adaptive spline fitting method described in Wang and Chandrasekar (2009), which offers a good trade-off between feature preservation in areas of higher values of and noise filtering in areas with lower values. When comparing the obtained rainfall fields, one can see that the peak values of the -based rainfall are below those of the *A* rainfall, probably due to the smoothing effect of the derivative calculation, and that there is some residual amount of value fluctuations in the areas of lower due to noise fluctuations and/or the differential phase shift upon backscatter in the differential propagation phase signal. This is illustrated in Fig. 9a, showing a ray comparison of and through the azimuth angle of 146.3° in the dataset shown in Fig. 8. Both estimators show the area of maximum rain intensity at approximately 65 km in range, but the dual-wavelength attenuation-based estimator shows a sharper peak that appears smoothed out in the -based estimator. Figure 9b shows a scatterplot of and for the dataset shown in Fig. 8, showing better agreement between the estimators at lower rainfall rates. As the rainfall rates increase, the attenuation-based estimator tends to report higher values, which is probably due to the better feature preservation previously discussed. It is worth noting that although the *a*,*b* coefficients used here are not obtained from a direct disdrometer comparison for that event, they should still be valid for comparing the relative behavior of the two rainfall estimators, as they are linked through the attenuation coefficient *α* (which is estimated independently).

As discussed throughout the paper, with the availability of dual-wavelength, dual-polarization observations, one can compute the scaling coefficient *α*, which relates specific attenuation to the specific differential phase (and path-integrated attenuation to differential propagation phase). It has been shown before (Bringi and Chandrasekar 2001; Carey et al. 2000) that in rain the value of the *α* coefficient is related to both the mean drop size of the observed drop size distribution and the physical temperature of the rain. Variability in *α* may therefore be related to changes in . This is investigated in the case of stratiform precipitation during the Colorado flooding event of September 2013 and in the convective precipitation case shown throughout the paper (starting in Fig. 1).

Figure 10 shows an RHI scan looking west obtained by the CSU–CHILL radar during the Colorado flooding rain event of September 2013 (Gochis et al. 2015). Figure 10a is reflectivity at X band, and Fig. 10b is the copolar correlation coefficient at X band. Both panels clearly show the presence of a melting layer at an approximate height of 3 km, seen as a “bright band” in the reflectivity picture collocated with a line of low correlation values. The low-elevation area with no signal starting approximately at 60-km range is due to signal blockage from the Front Range foothills.

Figure 11 shows the relevant radar data fields for dual-wavelength attenuation correction (S-band reflectivity in Fig. 11a, X-band attenuation-corrected reflectivity in Fig. 11b, X-band differential propagation phase in Fig. 11c, and X-band two-way path-integrated attenuation in Fig. 11d) and the coefficient (in Fig. 11e, obtained as the slope *m* of the line fit to the path-integrated attenuation vs differential propagation phase scatterplot) for the rain event introduced in Fig. 10, obtained at an antenna elevation of 0.8°. This same procedure is repeated to obtain *α* for two additional elevation angles, 1.7° and 2.3°, and in all cases the range is restricted to 70 km in order to make sure that all observations are below the melting layer.

Results are summarized in Table 2, where one can see the variation of with the reported scan elevation. At the highest measured elevation of 2.3°, the *α* coefficient has its lowest value of 0.23, which increases to 0.26 at 1.7° elevation and further increases to 0.28 at 0.8°. Although we do not have access to the actual drop size distribution for this event, Figure 11f attempts to reproduce this result using a *T*-matrix simulation (Leinonen 2014) of the *α* coefficient versus temperature assuming a 35 dB*Z* at 10°C exponential drop size distribution. We first use a mean diameter of 1.25 mm [as reported by Gochis et al. (2015) at the time the data were collected], which shows a trend opposite of that observed that could be explained by a change in drop size distribution with elevation plus uncertainty in the temperature profile (the atmospheric sounding temperature profile was retrieved from University of Wyoming 2013 for the closest location and time, and the temperature intervals corresponding to each elevation scan are overlaid in the plot). A second simulation with a mean diameter of 2.00 mm shows a trend of *α* increasing with temperature that is more consistent with the observed values.

Similarly, Fig. 12 presents measurements on the second of a set of six RHI scans spaced every 2° in azimuth in the convective precipitation case first introduced in Fig. 1. The dataset is further stratified into different elevation segments, from which a set of path-integrated attenuation versus differential propagation phase scatterplots are computed. The resulting coefficients, estimated as the slope *m* of the linear fit to the scatterplot, are summarized in Table 3. One can see that there is quite a bit of variability across the different elevation segments and less through the different azimuth angles. At the lowest elevation segment (0°–1.4°), the retrieved *α* is quite stable for all azimuth angles, with values around 0.30–0.31 consistent with rain. In the next elevation segment (1.4°–3.4°), the values increase quite a bit, coinciding with higher intensity reflectivity areas that probably include some measure of large raindrops and hail as discussed in Figs. 4 and 5. The following elevation segment (3.4°–5.4°) shows generally decreasing values between 0.14 and 0.29. Above 5.4° in elevation, the values are on the very low end near zero, as there seems to be very little attenuation despite the presence of some negative differential propagation phase, which could be consistent with vertically aligned ice particles in the higher portions of the storm. The variability seen in elevation could be explained by convective processes in the storm, which would sort the different drop sizes along the vertical axis, together with the associated temperature gradients. Variability in the different azimuth cuts seems to follow areas of higher reflectivity intensity, which could be related to larger drop sizes and counts.

## 6. Summary and conclusions

This paper has presented a new approach to applying dual-wavelength radar data to the study of precipitation. This approach relies on using the simultaneous measurements of reflectivity at S band and X band to obtain attenuation as the field that yields the best X-band-corrected reflectivity fit to the unattenuated reflectivity measurements at S band. This method should be well suited for applications at higher antenna elevation angles (such as RHI scans and vertical pointing), as it is not dependent on the relative angle and orientation of the hydrometeors with respect to the antenna, since it affects both wavelengths equally. The obtained *A* field can be used to generate high-spatial-resolution instantaneous rainfall estimates, analogous to that obtained through in dual-polarization systems but avoiding the need for computing a range derivative of a measured field. Also, it can be used to detect areas of Mie scattering where the dual-wavelength ratio departs significantly from the path-integrated attenuation. The paper also leverages the ability of the CSU–CHILL radar to measure dual-polarization variables at both wavelengths to study the relationship between attenuation and the differential phase for both horizontal polarization reflectivity and differential reflectivity, and in both a stratiform and a convective precipitation scenario. In both cases it is found that at low antenna elevation (close to horizontal), the *α* coefficient value is around 0.3, which is a value typically found in dual-polarization applications obtained from simulations and self-consistency data analysis (Chandrasekar et al. 1990; Testud et al. 2000; Park et al. 2005; Snyder et al. 2010; Matrosov et al. 2014). In the case of stratiform precipitation, and constraining the analysis below the melting layer, the *α* values found decrease slightly with elevation, which may be related to changes in the drop size distribution or larger mean diameters. In the case of convective rain, a much wider range of variations is observed between the more consistent values found at both low elevation (previously discussed around 0.3) and high elevations (with very little to no attenuation in the ice particle regions), which may be attributed to drop size sorting in convection and associated temperature gradients.

## Acknowledgments

This research is supported by the ASR Program and the NSF (Hazard Sees Program). The CSU–CHILL radar is operated under a cooperative agreement with NSF.

### APPENDIX

#### Procedure to Synthesize Matching Beams

By design, the 3-dB beamwidth of the CSU–CHILL radar at S band is 1°, which is roughly 3 times that of the X-band component at 0.3°. To minimize the occurrence of artifacts due to beam mismatch side effects, the data presented throughout this paper have been postprocessed to synthesize a beam resolution at X band that matches that of the measurement at S band in the antenna scanning plane dimension (azimuth for PPI-type scans, elevation for RHI-type scans) according to

The index *n* is used to indicate the data rays at X band that fall inside the integration limits of a data ray at S band. Term *P* is for signal power [from which reflectivity and the signal-to-noise ratio (SNR) can be obtained], and is the differential propagation phase (from which specific phase can be obtained).

## REFERENCES

*Polarimetric Doppler Weather Radar: Principles and Applications*. Cambridge University Press, 664 pp.

_{a}-band radars