# Real-Time Parameter Estimation of a Dual-Pol Radar Rain Rate Estimator Using the Extended Kalman Filter

^{*}

## Abstract

**:**

## 1. Introduction

^{b}) of the single-pol radar is to link the horizontal reflectivity (Z) and the ground rain rate (R). Z = 200R

^{1.6}by Marshall and Palmer (1948) is one of the most famous equations of this kind [8]. However, the two parameters A and b are found to vary locally, as well as to depend on the storm type [9]. Generally, these parameters are determined by analyzing the observed data from both radar and rain gauge, which is one of the key parts of quantitative precipitation estimation (QPE).

## 2. Methodology

#### 2.1. Rain Rate Estimator of the Dual-Pol Radar

_{h}is the horizontal reflectivity [mm

^{6}/m

^{3}], and Z

_{dr}is the differential reflectivity [no dimension]. R, Z

_{h}, and Z

_{dr}can also be represented using the dB (decibel) unit, which is based on the log-transformation, i.e., dBZ = 10log

_{10}(Z).

#### 2.2. Kalman Filter

_{k}at the current time step k by the state vector x

_{k−1}at the previous time step k−1 and the white noise w

_{k}. That is:

_{k−}

_{1}into x

_{k}. The white noise w

_{k}is assumed to follow the Gaussian distribution with its mean zero and covariance Q

_{k}.

_{k}cannot be directly observed in a real problem, a measurement equation is necessary. The measurement equation also linearly links the state vector x

_{k}and the measurement vector y

_{k}, along with another white noise v

_{k}[53]:

_{k}is a transition matrix to transform the state vector x

_{k}into the measurement vector y

_{k}and v

_{k}is also assumed to follow the Gaussian distribution with its mean zero and covariance R

_{k}.

_{k}and the measurement error covariance R

_{k}.

_{k}(−) and the measurement error covariance R

_{k}. The state estimate error covariance P

_{k}(−) is also dependent upon the system error covariance Q

_{k}. In the case that the system error covariance Q

_{k}is large, the state estimate error covariance P

_{k}(−) also becomes large, leading to the Kalman gain being close to 1.0. On the other hand, in the case that the measurement error covariance R

_{k}is large, as R

_{k}becomes larger than Q

_{k}, indicating the uncertainty of measurement is larger than that of the system estimate, the Kalman gain approaches 0.0. That is, a higher state uncertainty should lead to a larger Kalman gain, and hence, a bigger update. In this case, the new measurement dominates the state estimates in the next time step. More detailed information about the Kalman filter can be found in Kalman (1960), Gelb (1974), and Box et al. (2015) [53,56,57].

#### 2.3. Extended Kalman Filter

_{k}is a controlling input vector, which is considered if needed. Both the system equation f(x

_{k}, u

_{k}) and the measurement equation h(x

_{k}) are nonlinear, and need to be transformed to be linear. In most studies, the Jacobian matrix is used as a linearization method. The Jacobian matrix is especially useful when the system is defined to be partial differentiable. The Jacobian matrices of the nonlinear functions f(x

_{k}, u

_{k}) and h(x

_{k}) are as follows:

_{k}and H

_{k}are assumed to be transition matrices of the linear system and measurement equations. Even though the linearization procedure is introduced in the extended Kalman filter, its theoretical background is exactly the same as that of the Kalman filter. The linearization procedure is simply to transform the relation between the independent variable and the dependent variable to be linear, to make the Kalman filter applicable. Table 2 also compares the Kalman filter and the extended Kalman filter.

## 3. Derivation of the Model

#### 3.1. State-Space Model for Dual-Pol Radar Rain Rate Estimator

_{10}of both sides, a new linear equation can be obtained in decibel (dB):

_{h}, and dBZ

_{dr}are the measurement variables, while A, b, and c are the state variables.

_{k}is a state vector at time step k and w

_{k}is a system error following the Gaussian distribution with its mean zero and covariance Q

_{k}. Equation (9) can also be represented by a matrix form, which is directly applied to the Kalman filter:

_{A}, w

_{b}, and w

_{c}represent the error of the state variables A, b, and c. The error covariance matrix is composed of the variances ${\sigma}_{A}^{2}$, ${\sigma}_{b}^{2}$, and ${\sigma}_{c}^{2}$ and co-variances ${\sigma}_{A,b}^{2}$, ${\sigma}_{A,c}^{2}$, and ${\sigma}_{b,c}^{2}$ of the state variables A, b, and c. That is:

_{k}represents the error term following the Gaussian distribution with its mean zero and covariance R

_{k}. The function h(A, b, c)

_{k}, a function of state variables A, b, and c, is given as follows:

_{k}transforms the state variable in the measurement equation into the measurement variable dBR. The Jacobian matrix is used as a transition H

_{k}, which is nothing but the partial derivative of the function h(A, b, c)

_{k}by the state variables A, b, and c. The Jacobian matrix of Equation (13) is derived as follows:

#### 3.2. Error Covariance and Initial Condition of the Extended Kalman Filter

_{0}, measurement error covariance R

_{0}, and system error covariance Q

_{0}. First, the initial condition of the state variable was assumed to be the parameters proposed by Chandrasekar and Bringi (1988) [41]. This initial condition is one of the most popular options to be considered. Second, the uncertainty of the state variable was assumed to be the variance of the parameter determined based on the least squares method.

## 4. Application, Results, and Discussion

#### 4.1. Study Area and Data

#### 4.1.1. Radar and Rain Gauges Located at Study Area

#### 4.1.2. Rain Gauge and the Corresponding Radar Data for the Storm Events Used in This Study

_{h}) ranges about (−10–50), while the differential reflectivity (i.e., dBZ

_{dr}) ranges about (−8–8). The range of the rain rate in the dB unit (i.e., dBR) is about (−4–14). The data concentration around the −3 dBR value is mainly due to the minimum detectable rain rate of the rain gauge being 0.5 mm/h. The trend of dBR was found to be similar to that of dBZ

_{h}, but when the rain rate was high, dBZ

_{dr}seemed to approach zero.

_{h}and dBR. The theoretically obvious positive relation was not so clear in this scatter plot, as the variation was too high. On the other hand, the relation between dBZ

_{dr}and dBR was more obvious. That is, as dBR increases, dBZ

_{dr}seems to converge to zero; but when dBR is rather low, the variation was still high. In fact, this relation was also found in other studies, such as Lee (2006), Chen et al. (2017), and Gou et al. (2019) [48,71,72]. The relation between dBZ

_{dr}and dBZ

_{h}was more interesting. A very strong and positive correlation could be found between dBZ

_{dr}and dBZ

_{h}. When removing the variation where dBZ

_{h}is around zero, their relation seems very solid. The reason for this strong correlation could be explained by the positive relation between rainfall intensity and the characteristics of the raindrops (i.e., the drop size and drop shape). As dBZ

_{h}increases from 0 to 50, dBZ

_{dr}moves from 0 to 1.5.

#### 4.2. Case of Shinpo Rain Gauge (No. 936) and the Corresponding Radar Data for the Event Occurred in 2016

#### 4.2.1. Initial Condition and Result

_{0}were set up as explained earlier. The initial conditions of the state variable were assumed to follow Chandrasekar and Bringi (1988) (i.e., A = −26.20, b = 0.94, and c = −1.08) [41], and that of the state estimate error covariance was assumed to be the error covariance of these three state variables estimated by applying the least squares method. The system error covariance Q

_{k}and the measurement error covariance R

_{k}vary at every time step, whose values were determined by considering both the current and previous information. In this study, a total of six different datapoints (i.e., data collected at the previous five time steps and the current data) were considered in the estimation of the system and measurement error covariance.

#### 4.2.2. Comparison with the Least Squares Method

#### 4.2.3. Radar Rain Rate vs. Ground Rain Rate

#### 4.3. Case of Considering All Rain Gauges and the Corresponding Radar Data

## 5. Summary and Conclusions

_{h}, and dBZ

_{dr}were set to be measurement variables, while A, b, and c were set to be state variables. Additionally, the initial condition of the state variable was assumed to be the parameters proposed by Chandrasekar and Bringi (1988) [41]. The update of the error covariance in this study was made by considering the variation of the state variable and the measurement during the last five steps, which was available without any computation burden.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Location of Beaslesan radar in Korea and 10 rain gauges selected in this study within the radar umbrella.

**Figure 2.**Time series plots of mean rain rate observed at 10 rain gauge stations located within the Beaslesan radar umbrella during (

**a**) 1–6 July 2016, (

**b**) 25–28 August 2018, (

**c**) 1–3 October 2019, and (

**d**) 22–25 July 2020.

**Figure 3.**Time series plots and scatter plots of rain rate observed at the Shinpo rain gauge station (No. 936) for the storm event that occurred in 2016 and the corresponding radar data: (

**a**) time series plots, (

**b**) three-dimensional scatter plot.

**Figure 4.**Rain rate, variance and covariance of state variables, measurement error covariance, and behavior of Kalman gain at the Shinpo rain gauge station (No. 936).

**Figure 5.**Resulting time series plots of parameters at the Shinpo rain gauge station (No. 936) estimated by applying the extended Kalman filter: (

**a**) parameter A, (

**b**) parameter b, (

**c**) parameter c.

**Figure 6.**Box-whisker plots of the parameters and Kalman gains at the Shinpo rain gauge station (No. 936) estimated by applying the extended Kalman filter: (

**a**) parameters, (

**b**) Kalman gains.

**Figure 7.**Comparison of time series plots of estimated parameters at the Shinpo rain gauge station (No. 936) by applying the extended Kalman filter (EKF) and the least squares method (LSM): (

**a**) parameter A, (

**b**) parameter b, (

**c**) parameter c.

**Figure 8.**Comparison of box-whisker plots of the estimated parameters at the Shinpo rain gauge station (No. 936) by applying the extended Kalman filter (EKF) and the least squares method (LSM): (

**a**) parameter A, (

**b**) parameter b, (

**c**) parameter c.

**Figure 9.**Comparison of the ground rain rate and radar rain rate at the Shinpo rain gauge station (No. 936) by applying the parameters of radar rain rate estimator in this study (EKF and LSM) and several other studies: (

**a**) EKF, (

**b**) Chandrasekar and Bringi (1988) [41], (

**c**) Chandrasekar et al. (1990) [43], (

**d**) Lee (2006) [48], (

**e**) Kwon et al. (2015) [51], (

**f**) LSM.

**Figure 10.**Comparison of box plots for three estimated parameters at 10 rain gauge stations for four storm events by applying the extended Kalman filter: (

**a**) parameter A, (

**b**) parameter b, (

**c**) parameter c.

References | Parameters | ||
---|---|---|---|

A | b | c | |

Chandrasekar and Bringi (1988) [41] | −26.20 | 0.94 | −1.08 |

Aydin et al. (1989) [42] | −26.78 | 0.96 | −1.17 |

Chandrasekar et al. (1990) [43] | −27.03 | 0.97 | −1.08 |

Aydin and Giridhar (1992) [44] | −26.25 | 0.95 | −1.17 |

Gorgucci et al. (1995) [45] | −20.00 | 0.92 | −0.37 |

Bringi and Chandrasekar (2001) [46] | −21.74 | 0.71 | −3.43 |

Ryzhkov et al. (2005) [47] | −17.99 | 0.74 | −1.03 |

Lee (2006) [48] | −24.79 | 0.94 | −1.13 |

Cifelli et al. (2011) [49] | −21.74 | 0.93 | −3.43 |

WRC (2014) [50] | −20.91 | 0.91 | −4.25 |

Kwon et al. (2015) [51] | −22.10 | 0.95 | −5.55 |

Zhang et al. (2018) [52] | −20.76 | 0.93 | −0.41 |

Steps | Kalman Filter | Extended Kalman Filter |
---|---|---|

Initial values | ${\widehat{x}}_{0}$, ${P}_{0}$, ${R}_{0}$, ${Q}_{0}$ | ${\widehat{x}}_{0}$, ${P}_{0}$, ${R}_{0}$, ${Q}_{0}$ |

State estimate prediction ${\widehat{x}}_{k}(-)$ | ${\widehat{x}}_{k}(-)={\varphi}_{k-1}{\widehat{x}}_{k-1}(+)$ | ${\widehat{x}}_{k}(-)=f\left({\widehat{x}}_{k-1}(+),{u}_{k-1}\right)$ |

State estimate error covariance prediction ${P}_{k}(-)$ | ${P}_{k}(-)={\varphi}_{k-1}{P}_{k-1}(+){\varphi}_{k-1}^{T}+{Q}_{k-1}$ | ${P}_{k}(-)={F}_{k-1}{P}_{k-1}(+){F}_{k-1}^{T}+{Q}_{k-1}$ |

Kalman gain ${K}_{k}$ | ${K}_{k}={P}_{k}(-){H}_{k}^{T}{\left[{H}_{k}{P}_{k}(-){H}_{k}^{T}+{R}_{k}\right]}^{-1}$ | |

State estimate update ${\widehat{x}}_{k}(+)$ | ${\widehat{x}}_{k}(+)={\widehat{x}}_{k}(-)+{K}_{k}\left[{y}_{k}-{H}_{k}{\widehat{x}}_{k}(-)\right]={\widehat{x}}_{k}(-)+{K}_{k}{\upsilon}_{k}$ | |

State estimate error covariance update ${P}_{k}(+)$ | ${P}_{k}(+)=(I-{K}_{k}{H}_{k}){P}_{k}(-)$ |

No. | Duration | Type | Total Rainfall (mm) | Maximum Rainfall Intensity (mm/h) |
---|---|---|---|---|

1 | 1–6 July 2016 | Frontal | 177.8 | 10.4 |

2 | 25–28 August 2018 | Typhoon | 190.1 | 13.1 |

3 | 1–3 October 2019 | Typhoon | 203.9 | 22.8 |

4 | 22–25 July 2020 | Frontal | 167.8 | 12.8 |

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## Share and Cite

**MDPI and ACS Style**

Na, W.; Yoo, C.
Real-Time Parameter Estimation of a Dual-Pol Radar Rain Rate Estimator Using the Extended Kalman Filter. *Remote Sens.* **2021**, *13*, 2365.
https://doi.org/10.3390/rs13122365

**AMA Style**

Na W, Yoo C.
Real-Time Parameter Estimation of a Dual-Pol Radar Rain Rate Estimator Using the Extended Kalman Filter. *Remote Sensing*. 2021; 13(12):2365.
https://doi.org/10.3390/rs13122365

**Chicago/Turabian Style**

Na, Wooyoung, and Chulsang Yoo.
2021. "Real-Time Parameter Estimation of a Dual-Pol Radar Rain Rate Estimator Using the Extended Kalman Filter" *Remote Sensing* 13, no. 12: 2365.
https://doi.org/10.3390/rs13122365