H. Remote Sensing of Precipitation

[Background] [Derivation of polarimetric rainfall estimators]
[Analytically Derived Relations for Rain Estimation]

---

1. Background

During the past year E. Brandes, J. Vivekanandan, and G. Zhang have continued their research seeking to improve the remote sensing of rainfall with polarimetric radar. Previously, data obtained during several field programs have been used to compare radar-reflectivity derived estimates of rainfall from collocated radars to evaluate the specific differential phase parameter for rainfall estimation in mountainous terrain and to evaluate “classical” polarimetric rainfall estimators. In the work reported here, a set of polarimetric rainfall estimators that are tuned for local drop-size in distributions and improved drop axis ratios are evaluated.

2. Derivation of polarimetric rainfall estimators

Polarimetric radar measurements and rainfall estimates derived from them are sensitive to the radar-apparent mean shape of rain drops illuminated by the radar beam. Oscillating drops in the free atmosphere tend to be more spherical in the mean than the equilibrium shapes used in many studies. An axis ratio relationship developed from a number of published observation studies is:

                        r = 0.9951+0.02510D-0.03644D²+0.005030D³-0.0002492D⁴ .

Using this relationship, available disdrometer measurements, and T-matrix calculations of scattering cross-sections, calculations were made for radar reflectivity at horizontal polarization (Z_H), specific differential phase (K_DP),  differential reflectivity (Z_DR), and the rain rate (R). The following set of fixed-form polarimetric rainfall estimators was then derived:

         Radar Reflectivity:                                   R = 2.62 x 10^-2 Z_H  ^0.687

         Specific Differential Phase:                      R = sign (K_DP) 54.3 | K_DP | ^0.806

         Spec. Diff.Phase/Diff.Reflectivity:            R = sign (K_DP) 136 | K_DP | ^0.968Z_DR ^-2.86

         Radar Reflectivity/Diff.Reflectivity:           R = 7.46 x 10^-3Z_H^0.945Z_DR^-4.76  .

The estimators were tested on a dataset consisting of 25 storm events involving 388 gauge-radar comparisons (Table 1). Comparative parameters are the mean bias factor (the total gauge-observed rainfall divided by the sum of the radar estimates at gauges reporting measurable rainfall), the bias factor range (the ratio of the largest individual storm event bias factor divided by the smallest value), the correlation coefficient between radar-estimated and gauge-observed rainfalls, and the root-mean-square error (RMSE) before and after removing the estimator residual bias. The utility of the polarimetric rainfall estimators for making climatological estimates of precipitation is indicated by the convergence of bias factors to a value near 0.95 (a small overestimate). If there were a significant radar hardware calibration error or if the radar beam were significantly blocked, the mean biases would not converge. Also, convergence would not occur if the axis ration relation were inappropriate. Consideration of more oblate drop shapes would result in a tendency to underestimate rainfall with the specific differential parameter and to overestimate rainfall with the reflectivity-differential reflectivity parameter pair. From the table it is clear that the most robust rainfall estimator is that for the reflectivity-differential reflectivity pair of measurements. This estimator has the smallest bias variation from storm to storm, the highest correlation between radar-estimated and gauge-observed rainfalls, and the smallest RMSEs.

Table 1: Comparison of polarimetric rainfall estimators for radar reflectivity [ℜ(Z_H)], specific differential phase [ℜ(K_DP)], specific differential phase-differential reflectivity [ℜ(K_DP,Z_DR)], and radar reflectivity-differential reflectivity [ℜ(Z_H, Z_DR)].

[top of page]

3. Analytically Derived Relations for Rain Estimation Using Polarimetric Radar Measurements

Accuracy of rain rate estimation by well-calibrated radar is limited by the lack of detail knowledge of drop size distribution (DSD). Rain rate is usually estimated from radar reflectivity using a Z-R relation based on convective or stratiform rain. The Z-R relation was obtained by fitting gauge measurements and radar reflectivity.  It is known that the Z-R relation changes from location to location and time to time depending on changes in DSD. Therefore, a fixed empirical Z-R relation cannot provide accurate rain estimation for various types of rain because it cannot handle variation in drop size information. The relation between radar reflectivity and rain rate is almost completely quantified only if the drop size distribution is specified because they are proportional to moments of DSD, namely, reflectivity is the 6^th moment and  rain rate is proportional to the 3.67^th moment of the drop spectrum. Accurate rain rate estimation requires detailed knowledge of rain DSD and hence various rain rate estimators are derived using polarimetric radar observation that includes reflectivity, differential reflectivity and propagation phase.

During the analysis of data measured by a video-disdrometer, Zhang, Vivek, and Brandes found that there is a high correlation between the shape (m) and slope (L) parameters, and derived a m-L relation from the video-disdrometer measurements collected during a special field experiment in east-central Florida to evaluate the potential for polarimetric radar to estimate rainfall in a subtropical environment.  The  m-L relation applied to three parameter Gamma DSD reduces to a  two parameter DSD and  is dubbed as a constrained Gamma DSD. Conceptually, DSD can be specified using a pair of independent radar observations, namely,  reflectivity and differential reflectivity (Z_DR) or  Specific propagation phase (K_DP) and Z_DR.  Thus radar-based rain estimators can be derived analytically rather than by a traditional numerical simulation of DSD. Numerical simulation might bias the rain rate estimator and the estimator may be valid only for a particular rain event that is represented by simulated DSD.

Figure H1 shows comparisons between radar estimated rain rate and the corresponding rain gauge measurements. The rain rate is plotted as a function of time for the rain gauge location. The radar estimated rain is calculated using both the classic relations and the newly derived relations for comparison. Radar-based rain estimators using numerically simulated relations (Figure H1a), both R(Z) and R(K_DP) underestimate rain by 40% while R(Z, Z_DR) overestimate rain by more than 20%. This result shows the inconsistency among the empirical relations.

Using the analytical relations derived from constrained Gamma DSD, the estimated rain rate was plotted in Figure H1b. The physically-based relations are shown to give consistent results and agree with the gauge measurement much better than that obtained using classic relations based on a fixed power law.  The constrained Gamma DSD based relations use the DSD information directly and hence the corresponding radar-based rain estimators are in  better agreement with gauge observations. The classic relations, however, were obtained using least-squares fitting, which depends on selection of  DSD data sets that were chosen for the fitting; for example,  DSDs with a certain range of DSD parameters and weight  might not  truly represent  natural rain drop size distribution.

 

 

Polarimetric measurements are sensitive to  DSD, shape and canting angle. Even though  the m - L relation simplify DSD representation, any difference between assumed and actual microphysical parameters such as shape and canting might introduce  significant uncertainties in polarization radar-based retrieval.  Equilibrium shape of raindrops is assumed in this study while some observations suggested that a more spherical shape should be adapted. The equilibrium shape of a raindrop is assumed for maintaining continuity with earlier studies. A more accurate model may be used in future studies when the axis ratio and canting angle are better understood.

[top of page]