Atmospheric Radar Research Center (ARRC)

DSD models

A DSD model is required for the DSD retireval. The following DSD models are generally used.

• M-P model is a single parameter model with a slope parameter Λ. This well-known model was proposed by Marshall and Palmer (1948) and was widely used in the past 50 years. It was helpful in bulk-scheme rain parameterization and radar-rain estimation.
• Exponential model has two parameters: a concentration parameter N₀ and Λ. It is more flexible than M-P model since the latter is equivalent to the exponential model with a fixed N₀.
• In addition to N₀ and Λ, gamma distribution introduces a shape parameter μ. It has been widely accepted that the gamma model is flexible to represent the variability of natural DSDs.
• Gamma DSD is sometimes applied with its normalized form (e.g., Bringi et al. 2002). N_w is a normalized parameter, the value of which equals to the intercept parameter of an exponential DSD that has the same water content and the median volume diameter D₀.The advantage of a normalized gamma DSD is that parameters (N_w and D₀) have specific physical meaning.
• For log-normal distribution, N_T is the total number concentration. η and σ are the mean and standard deviation of Gaussian distribution, respectively. This model follows the assumption that parameters of the DSD can be modeled as random variables from a multivariate Gaussian distribution. It has a good explanation of DSD based on the probability theory, and the mathematical calculation is not that complicated. However, it might not be the best one matching observed DSDs.
• The constraint-gamma (C-G) DSD modole was recently proposed by Zhang et al. (2001). With the correlation between μ and Λ, it reduces the gamma DSD to a two-parameter model. It facilitates the DSD retrieval from two radar observations while still keeps a good flexibility to represent natural DSDs. The latest C-G was updated using rain date observed by disdrometer in central Oklahoma (Cao et al. 2008) as
  

  μ = -0.0201Λ² + 0.902Λ -1.718

Above figure shows an example of DSD models representing an observed DSD. It is evident that the M-P model has the worst performance. The DSD model, which has more freedom (i.e., more parameters), represent the observation better. The observed DSD in the figure is well modeled by the gamma function.

Retrieval Methods

Generally, the retrieval procedure is to find a DSD, based on which the calculated radar variables could match the radar observations. This fact implies that the calculation of radar variables should be a key factor to the DSD retrieval because it connects the DSD to the radar variables. The method of calculating radar variables is usually called the forward operator, which has designated the DSD model, the raindrop shape model, and other conditions for the scattering such as frequency and temperature. It is worth noting that the forward operator of radar variables could not reflect the real radar observations accurately. There would exist representing errors, which could affect the performance of DSD retrieval. These errors, including DSD model error and raindrop shape model error, can be viewed as the model error.

The straightforward way of DSD retrieval is to solve the unknown DSD parameters using the same number of radar observations. For example, if the exponential DSD model is used for the retrieval, two parameters N₀ and Λ need to be solved. If there are two observations Z_H and Z_DR available, we can use forward operators of Z_H and Z_DR, to solve these two DSD parameters. This approach assumes the forward operator results exactly match the observations and the DSD parameters are deterministically solved. The limitation is that the observation and model errors are not considered in this approach. Regardless of its defects, the direct DSD retrieval is convenient to be applied. It has been demonstrated with a good performance when an appropriate DSD model is used (Brandes et al. 2004, Zhang et al. 2006, Cao et al. 2008).

Other methods for the DSD retrieval work on the use of additional information for the optimization of retrieval. For example, the variational approach (Xue et al. 2009, Cao et al. 2009) provides a good way to combine multiple radar observations and spatial information to retrieve the DSD with minimizing the observation error effects. The long-term statistics of rain properties can also be used to improve the DSD retrieval (Cao et al. 2010). Although these methods are more complicated than the direct retrieval, their basses are same, i.e., they are primarily based on the forward operator.

Study Module

Objectives

• To enhance the understanding of DSD and its relation with different rain variables.
• To learn the method of retrieving DSD from radar measurements.
• To gain the knowledge of the difference between DSD models and their impacts on the rain retrieval.

Description

This module gives a practice of DSD retrieval using the direct approach. In brief, the retrieval uses two radar measurements Z_H and Z_DR to solve two unknown DSD parameters. Two DSD models are compared for the retrieval. One is the exponential model. The other is the C-G model. Both models are two-parameter based.

Procedures

• Download the example radar data Z_H, and Z_DR following the link.
• Assuming the DSD follows the exponential form, solve parameters N₀ and Λ from observed Z_H and Z_DR according to equations 1 and 2.
• Repeat this retrieval but use a C-G DSD model.
• Calculate rainfall rate and mean volume diameter using equations of R and D_m, respectively.
• Plot and analyze the results of N₀, Λ R, and D_m. Compare the retrieval results using different DSD models.