The specific differential phase

Apart from radar reflectivity (

A linear regression model has been developed to derive

X-band dual-polarization radars have drawn increasing attention in the radar meteorology community in recent years on account of their low cost,
fine resolution, and high sensitivity to light precipitation

The recent algorithms are focused on the improvement of estimating the mean

The paper is organized as follows. Section 2 provides background information about

The specific differential phase is the first
derivative of the differential phase shift

In this section, we introduce the physical interpretation of

For linear polarization,

By considering the Rayleigh–Gans scattering from identical and horizontally oriented oblate spheroids, such as raindrops,
the forward-scattering amplitudes are proportional to the inverse square of radar wavelength,
i.e.,

However,

Moreover, it is notable that the backscattering phase shift
is not negligible at the X band; thus the total propagation phase shift

The specific differential phase is a unique polarimetric variable in terms of statistical errors in the rain rate estimation, since it is the range
derivative of the phase measurement

The Gaussian mixture is a statistical model for data probability density estimation, assuming that the data points are
generated by a mixture of a finite number of Gaussian distributions associated with their weights

It is prevalent to use an Expectation–Maximization (EM) algorithm to estimate the parameters,

One of interpretations of the Gaussian mixture is to view each distribution as a cluster with a Gaussian probability density, while the individual data point
is attributed to a specific cluster or a weight toward the cluster, regarded as unsupervised learning

For the regression problem, the characteristics of the Gaussian mixture imply that the direct modeling of a regression function is very difficult. Nevertheless,
the joint probability of the measurements and the estimated parameters may be modeled as a Gaussian mixture,
leading to a regression function derived from the joint density model. Due to the asymptotic consistency of a Gaussian mixture model,
it is capable of estimating a general density function in

As part of the Missouri Experimental Project to Stimulate Competitive Research (EPSCoR), an X-band dual-polarization radar in the University
of Missouri (MZZU) was deployed at the South Farm Research Center (38.906

In this study, we analyze the data collected by the X-band MZZU dual-polarization radar. The maximum unambiguous range
of the MZZU radar is 94.64 km with a resolution of 260 m in range and 1

Characteristics of hourly rain gauge data at Bradford, Sanborn, Auxvasse, and Williamsburg between April 2016 and June 2018. Mean: mean values, SD: standard deviation, Max: maximum values, Total: sums of rain amounts, and Duration: sum of rainfall time.

Table

As discussed in Sect. 2, the joint probability density function (PDF) based on a Gaussian mixture can be used to derive the regression model
for

Flowcharts of

From the chart of LR in Fig.

The presence of clutter in the

In LR, the clutter is often eliminated by some criteria based on

Flowchart of data masking.

In contrast, GMM adopts sophisticated procedures, as depicted in Fig.

Examples of data masking:

Figure

In the previous section, it is shown that the

To estimate the relationship between

Equations (

Examples of

Figure

Figure

According to the continuity and consistency of the phase data, we can discern that some issues exist in the density estimation,
such as ambiguous

In LR,

Flowchart of the

On the other hand, the

As illustrated in Fig.

In addition to the ambiguous

The linear regression model often adopts an iterative filter technique, which generates a new

As shown in Fig.

It is clear from Fig.

As discussed in Sect. 2.1,

In Eq. (

Examples of

Figure

As discussed previously, the

Responses of finite impulse filter:

To obtain the reduced variance, we consider the filter as a number of weighting functions, denoted as

The red curves in Fig.

In this section, a case study is first presented to qualitatively analyze the storm structure and evolution based on

On 24 March 2016, a severe storm developed in central Missouri
and moved eastward across Columbia, MO, causing strong winds and heavy precipitation at the surface. When the storm became mature,
the S-band radars at Kansas City and St. Louis observed the storm structure at high levels, since each radar was about 150 km away
from the storm. Notably, the Kansas City radar showed positive and negative Doppler velocities in a small area (not shown),
indicating the occurrence of a downburst. On the other hand, the MZZU radar illustrated a bow echo of

A case study for GMM:

Figure

In this storm, the bow echo is shown as a number of convective cores embedded in a rain band, while the downbursts occurred at the leading
edge near the echo center. The bow echo can be considered as a mesoscale convection with a horizontal dimension of more than 60 km.
To gain further insight, Fig.

By taking a closer look at GMM

Comparison with the self-consistency relations:

To give a further evaluation of the GMM

Moreover, the computational time is crucial for the real-time application of the

In order to quantitatively evaluate the accuracy of GMM

Statistics for the comparison between the radar and gauge.
RMSE: root-mean-square error; NB: normalized bias;

Comparison between hourly radar and gauge data derived from GMM

Same as Fig.

Consistent with the data in Table

When compared to LR statistics as given in Table

To improve the accuracy of the radar rainfall estimation, we have optimized the

It can be found that the rain rates based on the GMM

In this study, we proposed a probabilistic method based on the Gaussian mixture model to estimate the specific differential phase

As an initial step of GMM, the data masking was performed to eliminate the residual clutter in the measurements of the total differential phase

Secondly, the joint probability density function (PDF) was obtained by fitting the data of

Thirdly, the ambiguous

Fourthly, the joint PDF of

In the final step, the expected values of

The experimental results with a severe storm observed by the X-band polarimetric radar in the University of Missouri (MZZU)
revealed the advantages of GMM. By studying the structure and evolution of a bow echo in the storm,
it was concluded that the GMM

The potential applications of GMM

The MZZU radar data can be made available upon request to the authors. The rain gauge data are available upon request to the University of Missouri Extension via Missouri Historical Agricultural Weather Database.

Let the total differential phase

In the linear regression, it is easy to find that

It is noted that the range gate

By taking the variance on both sides of Eq. (

We consider the range

First, we need to show that the derivative of the expected value of random variable

According to the conclusion in Eq. (

The first-order Taylor expansion is defined as

NIF designed the experiment and provided the radar data, GW developed the Gaussian mixture model and prepared the manuscript, GW and NIF performed the validation, and NF and PSM reviewed the paper.

The authors declare that they have no conflict of interest.

The authors would like to express our sincere thanks to the anonymous reviewers for their valuable comments and suggestions. This work was supported by Missouri Experimental Project to Stimulate Competitive Research (EPSCoR) of the National Science Foundation, under award number IIA-1355406. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

This research has been supported by the National Science Foundation (award number IIA-1355406).

This paper was edited by Gianfranco Vulpiani and reviewed by three anonymous referees.