1. Introduction
In dual-linear-polarization bases, the specific differential phase Kdp is defined as the slope of range profiles of the differential propagation phase shift Φdp between horizontal (H) and vertical (V) polarization states (Seliga and Bringi 1978; Jameson 1985; Bringi and Chandrasekar 2001). The specific differential phase is an important parameter for meteorological applications, because it is not affected by propagation attenuation and it is closely related to rain intensity, even in the presence of hail (Aydin et al. 1995; Chandrasekar et al. 2008).
However, the estimation of Kdp involves estimating the slope of Φdp profiles, which is known to be a noisy unstable computation. Evaluation of the derivative is essentially a high-pass filtering, and it expects a smooth and continuous function as the input. The range profile of total differential phase Ψdp contains both Φdp and differential backscatter phase shift δhv, as well as measurement fluctuation. The fluctuation in the estimates of Ψdp will be magnified during the differentiation, resulting in large variance in the estimates of Kdp. Furthermore, phase wrapping may exist in Φdp, and special care is generally required to unfold the wrapped phase profiles. The unambiguous range of Φdp usually is 180° in the alternate H/V transmission mode and 360° in the simultaneous H/V transmission mode. The system phase offset needs to be carefully adjusted to near the lower unambiguous limit for making use of the full unambiguous range. Nevertheless, for a long propagation path in rain or at higher-frequency bands such as X-band, Φdp values can easily exceed the unambiguous range. All these challenges have to be addressed to achieve a satisfactory estimation of Kdp, which can be applied for quantitative applications such as rainfall estimation. The artificial discontinuities introduced by phase wrapping have to be detected first from the noisy phase profiles and to be adjusted. Usually, a piecewise fitting is used to predict the local trend, and any phase sample deviated far from this trend can be attributed to phase wrapping. This solution is neither elegant nor stable because of the statistical fluctuations present in the Φdp field. Indeed, it is a common unfolding technique to estimate the slope first and then recover the intrinsic phase profiles based on the detected discontinuities. In this paper, a completely different approach to estimate Kdp is introduced to avoid the need for unfolding phases. The current techniques to reduce the variance of Kdp estimates include range filtering, linear fitting, or both (Golestani et al. 1989; Hubbert and Bringi 1995). All these techniques reduce the peak Kdp values in the estimation that could introduce biases. In addition, only limited adaptivity can be achieved with filtering or fitting to follow the steep slopes within intense rain cells and reduce the estimation variance in the rest of the segments simultaneously. This paper presents an adaptive scheme to estimate Kdp, which has better range resolution in intense rain cells to capture the small-scale variability. A regularization technique is introduced to control the balance between estimation bias and variance, and it incorporates the adaptivity to keep up with steep change of Φdp. The procedure is developed essentially to improve the robustness in real-time operations. This paper is organized as follows: the conventional estimation of Kdp is summarized first in section 2 with the state-of-the-art Colorado State University–University of Chicago–Illinois State Water Survey (CSU–CHILL) algorithm as an example. Then, the new estimator is described in section 3 to deal with wrapped phases, and an adaptive algorithm for its implementation is developed in section 4. Its application to radar observations from CSU–CHILL S-band radar and the Center for Collaborative Adaptive Sensing of the Atmosphere (CASA) Integrated Project 1 (IP1) X-band radars is presented to demonstrate the performance of the new algorithm.
2. Conventional estimation of specific differential phase
a. Phase unfolding
Phase wrapping can occur, depending on the system differential phase Φdp(0), even if the total Φdp accumulation is within the unambiguous range. For this reason, the system differential phase usually is intentionally set near the lower bound of the unambiguous range. Nevertheless, this system phase can drift, and Φdp can exceed the unambiguous range. Therefore, phase unfolding needs to be done first at the very beginning during the evaluation. In CSU–CHILL, a set of straightforward but complex logic is adopted to detect the wrapped phase and accordingly unfold the range profile of Φdp. The procedure is described in the following.
The first step is to determine the location and phase offset where the meaningful Φdp profile begins. Within the path free of weather echoes, low signal-to-noise ratio (SNR) and low copolar correlation result in large phase fluctuation, which should not be misjudged as phase wrapping. This decision is made based on the standard deviation of Φdp (σϕ) and the copolar correlation coefficient ρhv. A consensus of small σϕ and high ρhv over consecutive gates is used to ensure robustness in the detection of weather echoes.
Subsequently, σϕ and the slope of Φdp are continuously computed. A segment of profile with large σϕ is labeled as nonweather echoes. Otherwise, the estimated slope is accumulated into a smoothed Φdp profile for reference. Based on this knowledge, the current Φdp value is compared against the reference value of preceding Φdp to determine whether the current one needs to be unfolded.
The flowchart describing the current CSU–CHILL algorithm is shown in Fig. 1. The computation for σϕ and the slope is done to prevent false phase unfolding resulting from the contamination by noise. It should be noted that hard thresholds are used throughout the procedure that are obtained from the system metrics of the CSU–CHILL radar. It is also assumed that Φdp would not be wrapped at the beginning range.
b. Range filtering
c. Least square fitting
3. New estimator for specific differential phase
A practical phase unfolding approach has been described in the previous section. One feature of the procedure is somewhat complicated in the sense that a slope is first approximated for unfolding the phase profiles and then a more accurate slope is estimated for Kdp. Its performance also relies on the choice of several hard thresholds. Nevertheless, wrapped phase profiles have to be unfolded first before the subsequent filtering and fitting, because the discontinuities are not “physical” features. In this section, a new estimator will be proposed for estimating Kdp based on the complex-valued range profiles of the differential phase shift. To clarify the difference from the conventional description, hereafter the differential phase shift in the complex domain will be referred as the angular Φdp, whereas its counterpart in the real domain (e.g., from 0 to 2π) will be referred as the principal Φdp.
Numerical derivative errors can be reduced by using high-order estimation techniques. Linear finite difference is a genuine slope estimator for a linear function. However, the definition in (15) and (16) does require higher-order numerical differentiation because of the exponential term, unless the sampling interval is sufficiently small. As a simple example, Table 1 shows the numerical derivative errors in the angular Φdp profiles using different solutions. The principal Φdp profile is assumed linear from which all these estimators can accurately retrieve the true slope. However, as the slope increases, finite difference gives larger errors in the estimates of Kdp from the angular Φdp profiles. Numerical differentiation based on interpolation polynomials of a higher degree can substantially reduce the errors, simply because the high-degree polynomials can better approximate the underlying function. Note that cubic spline is also a piecewise interpolation polynomial with only extra constraint of continuities at the knots. It can be seen that cubic spline performs uniformly better than a four-degree Lagrange interpolation polynomial. Therefore, a feasible solution for Kdp estimates will be filtering angular Φdp profiles based on angular Ψdp statistics and estimating Kdp using cubic spline. Thus, phase unfolding is avoided, and the accuracy is also ensured. An adaptive estimation of Kdp is developed based on the principles described and is presented in the next section.
4. Adaptive estimation of specific differential phase
a. Adaptivity for statistical fluctuation
b. Adaptivity for Kdp
To follow the profiles in large Kdp, larger variation shall be allowed such that bias is minimized to avoid excessive smoothness. Therefore, let q−1 = 2Kdp, where the factor 2 is in place to compensate the two-way Φdp profile. With this adjustment, the smoothness requirement at large Kdp will be less emphasized. On the contrary, the smoothness at small Kdp will be heavily weighted. The scaling factor q−1 cuts off at a Kdp value of 0.1° km−1 for numerical stability.
c. Choice of the Lagrangian parameter
The adaptive approach is only left with the freedom in choosing the smoothing factor λ. Cross-validation can be used to find the optimal λ from a dataset (Craven and Wahba 1979). Such determination does not suit this application because of excessive ray-by-ray variation in radar observations. However, with the scaling scheme, a simple procedure is used to make the evaluation of smoothness and that of data fidelity equally important. The variation of angular Φdp has been normalized using w; therefore, a reasonable choice of λ should be close to the expected smoothness by evaluating (33). As an approximation, constant Kdp can be assumed in each range interval. At different Kdp levels, the smoothness is quantitatively assessed and shown in Table 2 by allowing a 10% variation in Kdp. The last column can be used as a baseline for choosing λ. The existence of extra Kdp in (33) requires a different smoothing factor for different peak Kdp values. For example, targeted at Kdp of 30° km−1, the smoothing factor should be around 1.1Δr.
Figure 3 shows a simulated Kdp profile, composed of a Gaussian function at 15 km, to demonstrate an infinitely differentiable profile, and a triangular function at 35 km, to demonstrate Kdp discontinuities. The profile exhibits sharp gradient on Kdp, which changes from 0° to 30° km−1 within a 3-km range. Statistical fluctuation is introduced to Φdp by setting the intrinsic ρhv at 0.99, the normalized spectral width at 0.12, and the integration length at 64 samples. The Lagrangian factor is chosen between 0.1Δr and 1.1Δr for comparison. The conventional Kdp estimation is also plotted from the filtered Φdp profiles with a six-order finite impulse response (FIR) filter (designed to keep the variation at 3-km scale). Near the Kdp peaks, it can be seen that at λ of 1.1Δr the estimation biases are almost completely removed and the standard deviation of the estimation increases but less than 1° km−1. In the nonstorm regions, both the biases and the standard deviation of the estimation are reduced. With smaller λ, oversmoothed Kdp estimates are obtained that exhibit large biases. It should be noted that the λ value is not guaranteed to be the optimal, but Table 2 can be used as a baseline to empirically pick an appropriate λ. The value at 1.1Δr is used for all the case studies that are presented in the next section.
The algorithm can be easily implemented in a real-time environment. Even though matrix inversion and matrix multiplication are involved in the solution, the computation complexity is well within the capability of modern-day computation power because all the matrices are band limited. Most importantly, the calculation for phase unfolding is completely avoided.
5. Evaluation
To solve these problems, a two-step iteration is adopted, in which a rough estimation of Kdp is made first using the nonadaptive cubic splines derived from (24) and then it is used in the adaptive cubic splines to perform the final Kdp estimation. This introduces a slight increase of computation time without any perceivable impact on real-time implementation. During the first step, a smaller Lagrangian parameter fixed at 0.1 is used to obtain the overall trend of the Kdp profile and the resulting scaling functions very well. This two-step iteration technique is adopted in this paper to estimate Kdp from radar observations.
a. Simulated storm
Chandrasekar et al. (2006) have developed a procedure to simulated X-band radar observation from S-band radar data with the natural variation of microphysical parameters preserved. X-band radar observations are simulated using the CSU–CHILL radar observations of a thunderstorm during the Severe Thunderstorm Electrification and Precipitation Study (STEPS). The simulated X-band observations are shown in Fig. 4, with Kdp of up to 7° km−1. Figures 4a–c present the radar reflectivity, the estimated Kdp using the conventional filtering and fitting approach, and the estimated Kdp using the new adaptive approach, respectively. Both the conventional approach and the adaptive approach result in a similar Kdp field, with the adaptive approach showing less “range streaks.” The adaptive approach presents better resolution, which captures the Kdp peaks as well as the storm texture that is exhibited in the radar reflectivity. In addition, the Kdp field is much cleaner with smaller variation in the regions of low radar reflectivities. Because the “true” radar parameters are available in the simulation, the scatterplot between the estimated Kdp and the intrinsic Kdp is shown in Fig. 4d. If the intrinsic Kdp is small, the new estimates are more stable; if the intrinsic Kdp is large, the new estimates have a larger standard deviation, but the mean bias is smaller. The results of Fig. 4 clearly illustrate the advantage of the new adaptive approach.
b. CSU–CHILL S-band radar observation
Figure 5 shows the S-band observations of a heavy rain event on 31 May 2003 UTC from the CSU–CHILL radar. A large area of thunderstorms developed southeast of the CSU–CHILL radar and some of the beams were continuously filled with heavy rain over range segments of nearly 50 km. The value of Φdp is accumulated over 200° around an azimuth of 128°, where phase is wrapped. Again, it can be found that both approaches give similar Kdp estimates in the overall pattern by comparing Figs. 5b,c. The Kdp peaks are well estimated by the adaptive estimator in the storm core, whereas the Kdp estimates outside of storm core are fairly smooth compared to these by the conventional estimator. Note that the storm is located at range of 50 km and beyond. The spatial resolution there is much lower than the previous simulated cases. The scatterplot between the estimated Kdp and radar reflectivity is presented in Fig. 5d. It can be seen once again that the large number of extraneous points from the conventional algorithm are eliminated.
c. CASA X-band radar observation
On 14 June 2007, an intense rainfall event developed over the CASA IP1 test bed in Oklahoma (Chandrasekar et al. 2007) for which a flash flood warning was issued by the National Weather Service (NWS). Reflectivity observations by the X-band IP1 radar in Rush Springs are shown in Fig. 6a. The estimated Kdp from the principal Φdp is shown in Fig. 6b, and the new Kdp estimates are shown in Fig. 6c. Visually, these two estimators give a comparable Kdp field in terms of the overall pattern and intensity. For this case, the radar has a range resolution of 48 m, and the conventional approach has better estimates for large Kdp observations than the case of lower range resolutions. However, the Kdp field in Fig. 6c matches the storm structure much better, and the negative Kdp in Fig. 6b is largely eliminated. The improved quality of the new adaptive Kdp is also shown in the Kdp–Zh scatterplot in Fig. 6d.
6. Discussion and summary
Challenges in the estimation of the specific differential propagation phase lie in the existence of phase wrapping and statistical fluctuations in the range profiles of the differential propagation phase, because differentiation is involved. Even though practical unfolding approach exists, the current implementation involves several ad hoc adjustments. Nevertheless, this procedure has been very useful so far and has been extensively used at CHILL. Phase wrapping is in fact an artifact coming from mapping an angular variable along unit circle into the real axis. In this paper, the estimation is shifted to the complex-valued range profile of the differential propagation phase exponentials. This yields a simpler estimation that is inherently immune to phase wrapping problem. Its numerical evaluation requires better accuracy as a result of the exponential form function. To this end, a regularization framework was developed, and a cubic smoothing spline was adopted to fit a better regression function.
The specific differential phase is an important parameter derived from dual-polarization radars. Using the specific differential phase can improve the accuracy of quantitative rainfall estimation. However, its estimates are highly susceptible to large variance in light and moderate rain, where Kdp is intrinsically small; on the other hand, its estimates are likely underestimated in heavy rain, where it is intrinsically large. These two factors could deter its wider use in quantitative rainfall mapping. A new algorithm is introduced to incorporate adaptivity through scaling the regression errors for estimation of Kdp.
Bias and variance are both typically a dilemma in inferring estimation from noisy data. During filtering, the bias and variance can be controlled through the cutoff frequency of the filter. As the sampling interval changes, new filter design needs to be in place. In the new adaptive algorithm, the bias and variance are controlled through a single Lagrangian parameter, or smoothing factor, which is also the only control parameter bounded. A simple method is proposed based on equilibrium principle to choose the smoothing factor. The proposed algorithm is well suitable for real-time implementation, primarily because of the banded matrix structure in the governing equations. It takes the same order of computation time as the conventional filtering or fitting procedure. The adaptive technique has been implemented in the CSU–CHILL radar, and it is expected to find importance in operational applications (Chandrasekar et al. 2008).
Acknowledgments
This research is supported by NASA via the Precipitation Measurement Mission (PMM) program and by Colorado State University. The authors acknowledge the opportunity to test the algorithm for CASA radar data. The CSU-CHILL radar facility is supported by the National Science Foundation through ATM-0118021.
REFERENCES
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Bringi, V. N., and Chandrasekar V. , 2001: Polarimetric Doppler Weather Radar: Principles and Applications. Cambridge University Press, 662 pp.
Chandrasekar, V., Lim S. , and Gorgucci E. , 2006: Simulation of X-band rainfall observations from S-band radar data. J. Atmos. Oceanic Technol., 23 , 1195–1205.
Chandrasekar, V., McLaughlin D. J. , Brotzge J. , Zink M. , Philips B. , and Wang Y. , 2007: Distributed collaborative adaptive radar network: The CASA IP-1 network and tornado observations. Preprints, 33rd Conf. Radar Meteor., Cairns, Australia, Amer. Meteor. Soc., 13A.3A.
Chandrasekar, V., Hou A. , Smith E. , Bringi V. N. , Rutledge S. A. , Gorgucci E. , Petersen W. A. , and Jackson G. S. , 2008: Potential role of dual- polarization radar in the validation of satellite precipitation measurements: Rationale and opportunities. Bull. Amer. Meteor. Soc., 89 , 1127–1145.
Craven, P., and Wahba G. , 1979: Smoothing noisy data with spline functions. Numer. Math., 31 , 377–403.
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APPENDIX A
Range Filter Design for Differential Propagation Phase
The Nyquist sampling rate in range is directly related to the along-range gate spacing. As shown in (11), the cutoff frequency depends on the gate spacing and the required cutoff scale of range. Because of these changes, the range filter needs to be redesigned for different gate spacings. As Δr decreases, the normalized passband width becomes narrower and more samples are available for the same pathlength. Using an FIR filter, its order can be correspondingly increased. The specifications of the FIR filters used in this paper are listed in Table A1, and their frequency responses are plotted in Fig. A1.
APPENDIX B
Solution of the Cubic Smoothing Spline
Flowchart of the phase unfolding algorithm used in the CSU–CHILL radar. The judgments and adjustments are valid in VH mode.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
Flowchart of the phase unfolding algorithm used in the CSU–CHILL radar. The judgments and adjustments are valid in VH mode.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
Flowchart of the phase unfolding algorithm used in the CSU–CHILL radar. The judgments and adjustments are valid in VH mode.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
Flowchart of decision for rain cells and good data mask, where MG and MB are constants set for consecutive “good” and “bad” samples, respectively. Usually, MB is smaller than MG.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
Flowchart of decision for rain cells and good data mask, where MG and MB are constants set for consecutive “good” and “bad” samples, respectively. Usually, MB is smaller than MG.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
Flowchart of decision for rain cells and good data mask, where MG and MB are constants set for consecutive “good” and “bad” samples, respectively. Usually, MB is smaller than MG.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
Kdp estimation from principal Φdp (black dashed line) using simple finite difference and from angular Φdp using adaptive cubic smoothing spline: (a) the mean and (b) standard deviation of estimated Kdp based on 100 realizations. The intrinsic Kdp profile (thick gray line) is simulated as (left) a narrow Gaussian function and (right) a triangular function, at gate spacing of 0.5 km. Fluctuation resulting from noise, Doppler spread, and copolar correlation is simulated.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
Kdp estimation from principal Φdp (black dashed line) using simple finite difference and from angular Φdp using adaptive cubic smoothing spline: (a) the mean and (b) standard deviation of estimated Kdp based on 100 realizations. The intrinsic Kdp profile (thick gray line) is simulated as (left) a narrow Gaussian function and (right) a triangular function, at gate spacing of 0.5 km. Fluctuation resulting from noise, Doppler spread, and copolar correlation is simulated.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
Kdp estimation from principal Φdp (black dashed line) using simple finite difference and from angular Φdp using adaptive cubic smoothing spline: (a) the mean and (b) standard deviation of estimated Kdp based on 100 realizations. The intrinsic Kdp profile (thick gray line) is simulated as (left) a narrow Gaussian function and (right) a triangular function, at gate spacing of 0.5 km. Fluctuation resulting from noise, Doppler spread, and copolar correlation is simulated.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
Simulated X-band radar observations at elevation angle of 0.5° and gate spacing of 0.1 km. (a) PPI plot of the radar reflectivity; (b) PPI plot of the estimated Kdp using conventional approach on the principal Φdp; (c) PPI plot of the estimated Kdp using the adaptive estimation on the angular Φdp; and (d) the scattering plot between the estimated Kdp and the intrinsic Kdp, where the black points illustrate the conventional estimates and the gray points illustrate the adaptive estimates.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
Simulated X-band radar observations at elevation angle of 0.5° and gate spacing of 0.1 km. (a) PPI plot of the radar reflectivity; (b) PPI plot of the estimated Kdp using conventional approach on the principal Φdp; (c) PPI plot of the estimated Kdp using the adaptive estimation on the angular Φdp; and (d) the scattering plot between the estimated Kdp and the intrinsic Kdp, where the black points illustrate the conventional estimates and the gray points illustrate the adaptive estimates.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
Simulated X-band radar observations at elevation angle of 0.5° and gate spacing of 0.1 km. (a) PPI plot of the radar reflectivity; (b) PPI plot of the estimated Kdp using conventional approach on the principal Φdp; (c) PPI plot of the estimated Kdp using the adaptive estimation on the angular Φdp; and (d) the scattering plot between the estimated Kdp and the intrinsic Kdp, where the black points illustrate the conventional estimates and the gray points illustrate the adaptive estimates.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
CSU–CHILL S-band radar observations at elevation angle of 0.5° and gate spacing of 0.15 km, at 0059 UTC 31 May 2003, Greeley, Colorado. (a) PPI plot of the attenuated reflectivity; (b) PPI plot of the estimated Kdp using conventional approach on the principal Φdp; (c) PPI plot of the estimated Kdp using the adaptive estimation on the angular Φdp; and (d) the scattering plot between the estimated Kdp and the attenuation-corrected Zh, where the black points illustrate the conventional estimates and the gray points illustrate the adaptive estimates.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
CSU–CHILL S-band radar observations at elevation angle of 0.5° and gate spacing of 0.15 km, at 0059 UTC 31 May 2003, Greeley, Colorado. (a) PPI plot of the attenuated reflectivity; (b) PPI plot of the estimated Kdp using conventional approach on the principal Φdp; (c) PPI plot of the estimated Kdp using the adaptive estimation on the angular Φdp; and (d) the scattering plot between the estimated Kdp and the attenuation-corrected Zh, where the black points illustrate the conventional estimates and the gray points illustrate the adaptive estimates.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
CSU–CHILL S-band radar observations at elevation angle of 0.5° and gate spacing of 0.15 km, at 0059 UTC 31 May 2003, Greeley, Colorado. (a) PPI plot of the attenuated reflectivity; (b) PPI plot of the estimated Kdp using conventional approach on the principal Φdp; (c) PPI plot of the estimated Kdp using the adaptive estimation on the angular Φdp; and (d) the scattering plot between the estimated Kdp and the attenuation-corrected Zh, where the black points illustrate the conventional estimates and the gray points illustrate the adaptive estimates.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
CASA X-band radar observations at elevation angle of 2.0° and gate spacing of 0.048 km, at 0708 UTC 14 Jun 2007, in Rush Springs, Oklahoma. (a) PPI plot of the attenuated reflectivity; (b) PPI plot of the estimated Kdp using conventional approach on the principal Φdp; (c) PPI plot of the estimated Kdp using the adaptive estimation on the angular Φdp; and (d) the scattering plot between the estimated Kdp and the attenuation-corrected Zh, where the black points illustrate the conventional estimates and the gray points illustrate the adaptive estimates.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
CASA X-band radar observations at elevation angle of 2.0° and gate spacing of 0.048 km, at 0708 UTC 14 Jun 2007, in Rush Springs, Oklahoma. (a) PPI plot of the attenuated reflectivity; (b) PPI plot of the estimated Kdp using conventional approach on the principal Φdp; (c) PPI plot of the estimated Kdp using the adaptive estimation on the angular Φdp; and (d) the scattering plot between the estimated Kdp and the attenuation-corrected Zh, where the black points illustrate the conventional estimates and the gray points illustrate the adaptive estimates.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
CASA X-band radar observations at elevation angle of 2.0° and gate spacing of 0.048 km, at 0708 UTC 14 Jun 2007, in Rush Springs, Oklahoma. (a) PPI plot of the attenuated reflectivity; (b) PPI plot of the estimated Kdp using conventional approach on the principal Φdp; (c) PPI plot of the estimated Kdp using the adaptive estimation on the angular Φdp; and (d) the scattering plot between the estimated Kdp and the attenuation-corrected Zh, where the black points illustrate the conventional estimates and the gray points illustrate the adaptive estimates.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
Fig. A1. Frequency responses of FIR range filters at various gate spacings.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
Fig. A1. Frequency responses of FIR range filters at various gate spacings.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
Fig. A1. Frequency responses of FIR range filters at various gate spacings.
Citation: Journal of Atmospheric and Oceanic Technology 26, 12; 10.1175/2009JTECHA1358.1
Numerical derivative errors in Kdp estimates from angular Φdp profile. The range spacing is assumed at 0.5 km, and the intrinsic Kdp is assumed constant. The derivative using higher-order polynomials is computed from piecewise Lagrangian interpolation polynomials.
Approximate quantitative evaluation of the smoothness of the cubic regression splines in single interval.
Table A1. Specification of FIR range filters to pass Φdp trend at 3-km scale.
