Tree-based genetic programming approach to infer microphysical parameters of the DSDs from the polarization diversity measurements

Abstract

The use of polarization diversity measurements to infer the microphysical parametrization has remained an active goal in the radar remote sensing community. In view of this, the tree-based genetic programming (GP) as a novel approach has been presented for retrieving the governing microphysical parameters of a normalized gamma drop size distribution model D₀ (median drop diameter), N_w (concentration parameter), and μ (shape parameter) from the polarization diversity measurements. A large number of raindrop spectra acquired from a Joss–Waldvogel disdrometer has been utilized to develop the GP models, relating the microphysical parameters to the T-matrix scattering simulated polarization measurements. Several functional formulations retrieving the microphysical parameters-D₀ [f(Z_DR), f(Z_H, Z_DR)], log₁₀N_w [f(Z_H, D₀), f(Z_H, Z_DR, D₀), and μ[f(Z_DR, D₀), f(Z_H, Z_DR, D₀)], where Z_H represents reflectivity and Z_DR represents differential reflectivity, have been investigated, and applied to a S-band polarimetric radar (CAMRA) for evaluation. It has been shown that the GP model retrieved microphysical parameters from the polarization measurements are in a reasonable agreement with disdrometer observations. The calculated root mean squared errors (RMSE) are noted as 0.23–0.25 mm for D₀, 0.74–0.85 for log₁₀N_w (N_w in mm⁻¹ mm⁻³), and 3.30–3.36 for μ. The GP model based microphysical retrieval procedure is further compared with a physically based constrained gamma model for D₀ and log₁₀N_w estimates. The close agreement of the retrieval results between the GP and the constrained gamma models supports the suitability of the proposed genetic programming approach to infer microphysical parameterization.

Introduction

The use of orthogonal polarized signals from polarimetric radar measurements propagating through the precipitation media can provide significant microphysical information of raindrops, such as particle size, shape, orientation and thermodynamic state. This means, retrieval of microphysical parameters solely from the polarization diversity measurements such as reflectivity factors (Z_H), differential reflectivity (Z_DR), and specific differential phase shift (K_DP) can be possible.

The studies of Seliga and Bringi (1976) and Seliga and Bringi (1978) were among the first that showed a procedure to retrieve microphysical parameters from the orthogonal polarization measurements. At that time, in general, the microphysical drop size distributions (DSDs) were used to consider as exponential distributions. The follow on studies, for example, in early eighties by Ulbrich (1983), confirm that the DSDs are better represented by gamma distribution. This gamma distribution is described by three governing parameters, N₀ as a concentration parameter, μ as a shape parameter, and Δ as a slope parameter. Most recently, the gamma distribution has been advanced as a normalized gamma distribution with three governing parameters of N_w, D₀, and μ representing the concentration parameter, median drop diameter, and shape parameter respectively (Bringi et al., 2003).

Currently, it is a topic of interest in the remote sensing community to retrieve aforementioned governing gamma DSD parameters using polarization diversity measurements. Mainly, the reported works in the literature have dealt with ground based S-band (Brandes et al., 2004b, Bringi et al., 2002, Vivekanandan et al., 2004), C-band (Bringi et al., 2009, Gorgucci et al., 2001, Thurai et al., 2008), X-band (Anagnostou et al., 2008a, Gorgucci et al., 2008, Kim et al., 2010), and satellite based Ku–Ka band (Rose and Chandrasekar, 2006) radar observations. At higher frequency, for example C-band and above, the path integrated attenuation hinders the successful microphysical retrievals from the polarization measurements unless the reflectivity and differential reflectivity measurements are properly corrected for attenuation. In contrast, S-band has the advantage of having negligible path integrated attenuation and Mie scattering effects (Islam et al., 2012c, Testud et al., 2000).

Generally speaking, two types of microphysical retrieval procedures from polarization measurements are well established, the β method and the constrained gamma (CG) method (Brandes et al., 2004a). The β method was first developed in Gorgucci et al. (2001) for C-band, which was further adopted in Bringi et al. (2002) for S-band. Firstly, the method estimates β parameter from the combination of polarization measurements Z_H, Z_DR, and K_DP. This β parameter is then used to retrieve governing microphysical parameters D₀, N_w, and μ. However, estimation of the μ parameter with this method is subject to significant bias (Bringi et al., 2002). On the other hand, the CG method assumes that the microphysical parameters of the gamma model are not mutually independent. Two parameters D₀ and N_w are estimated first from the polarization measurements. The remaining parameter μ is then estimated assuming an empirical μ–Δ relation. However, in any of the two methods, there is no strict consensus at the moment about fitting relation between the governing gamma DSD parameters and the polarization measurements. From simple power laws (Bringi et al., 2002, Bringi et al., 2009) to different polynomial fits (Anagnostou et al., 2008b, Brandes et al., 2004b) have been included in the literature. Note that, in addition to these two methods (β and CG), there are also some data mining approaches of microphysical retrievals, for instance, by means of a neural network (Anagnostou et al., 2008b, Vulpiani et al., 2006) and a Bayesian approach (Cao et al., 2010).

This study focuses on the use of the tree-based genetic programming as a new approach to infer governing microphysical parameters through the formulation of relationships between the normalized gamma DSD parameters D₀, N_w, and μ and the polarization measurements Z_H and Z_DR. The genetic programming has been extensively used in artificial intelligence and engineering fields to solve a wide range of non-linear fitting problems (Ghorbani et al., 2010, Jafarian et al., 2010). Exploiting the capability of genetic programming to approximate non-linear fitting problems, this work is the first attempt to investigate the genetic programming based models for microphysical retrievals using polarization diversity measurements at S-band frequency. This paper is structured as follows. Section 2 describes the procedure for establishing genetic programming based models for microphysical retrievals through the use of disdrometer DSDs and T-matrix scattering simulations. The established GP models are then applied to the Chilbolton advanced meteorological radar (CAMRA) and evaluated in Section 3. Finally, Section 4 provides the conclusions of this work.