Mathematical Problems in Engineering

Volume 2018, Article ID 3083258, 9 pages

https://doi.org/10.1155/2018/3083258

## Improving Bias Estimation Precision via a More Accuracy Radar Bias Model

^{1}Aviation Maintenance NCO Academy, Air Force Engineering University, Xinyang, Henan, China^{2}Aeronautics Engineering College, Air Force Engineering University, Xi’an, Shaanxi, China

Correspondence should be addressed to Xiaoju Yong; moc.621@7891ujoaixgnoy

Received 28 February 2018; Revised 3 July 2018; Accepted 23 August 2018; Published 10 September 2018

Academic Editor: R. Aguilar-López

Copyright © 2018 Xiaoju Yong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Radar bias estimation is an important part of a multisensor tracking system. Because the radar bias models in current efforts are not sufficiently accurate, the bias estimation accuracy is not satisfied. First, a general radar bias model, in which the importance of the sensor-target pair-related bias is emphasized, is proposed in this paper. Moreover, the target-sensor pair-related bias is considered on the grounds of the impact of atmospheric refraction on the electromagnetic wave propagation path. Second, by using the augmented state vector with the target state and radar bias, a dynamic equation and measurement equation under the IMM framework are obtained. Last, the IMM-UKF estimator is utilized to obtain the target state and radar bias simultaneously. The numerical simulations show that the accuracy of the bias estimation together with the target states estimation can be improved via the proposed radar bias model.

#### 1. Introduction

The measurements from multisensors must be corrected through a sensor registration algorithm. Multisensor fusion can greatly improve the target tracking precision. It is worth noting that the measurements that do not have bias correction could potentially lead to bad fusion performance. Thus, the problem of sensor registration is an important, yet often neglected [1], part of a multisensor tracking system.

Owing to this problem, much work has been devoted to estimating the sensor errors [2–6]. As is generally known, system errors are composed of four types. However, it is not a successful approach to consider all four types of errors simultaneously. Additionally, it has been proven that a satisfactory estimation performance is obtained when only one type of error is estimated [3]. From this point forward, the errors from the bias of the remote platform and the bias of the sensor measurements are accounted for in this paper.

Sensor registration algorithms can be divided into offline methods and online methods. Among the offline methods, the least square algorithm is usually utilized to solve this problem [4]. By transforming the measurements into the Earth Center Earth Fixed (ECEF) coordinates, the measurements can be expressed in a linear function with respect to the radar bias. Then, by adopting the LS algorithm, the radar error can be obtained. In the offline method, there are three main approaches to solving this problem. One approach is based on the ECEF coordinates [7], after obtaining a pseudo measurements equation that is independent of the target state, the Kalman Filter (KF) is utilized to estimate the radar bias. Another approach is to use a KF or Extended Kalman Filter (EKF) to provide online estimation of the sensor biases of a secondary sensor or the orientation errors of that secondary sensor with respect to a primary sensor [8]. The third method is to augment the state vector, combining the target and sensor uncertainty states, and the KF or EKF is then used to estimate the radar errors and target states simultaneously. The representative work of this method was proposed by Nabba and Bishop [9].

Regardless of whether the sensor registration is processed in an online method or offline fashion, in most of these studies, the radar bias is assumed to be constant or to change slowly over time. In fact, the radar bias is related to the range and elevation between the target and radar as well as the characteristics of different transponders. Great efforts have been made by Besada to establish a more precise bias model for sensor registration. In [10], the errors are divided into three parts, which are sensor-related bias that has the same value independently of the target, target-related bias, which is equal for every sensor of the same type, independent of the sensor, and sensor-target pair-related bias, which is different for each sensor-target pair. Most of the studies above focus on the sensor-related biases and neglect the importance of the target-related biases. In [11], Besada describes these errors under different situations in detail.

Indeed, most of the efforts do not provide an accurate model for the sensor-target pair-related bias. In [12], a complete systematic error model of SSR is established to for sensor registration in ATC surveillance networks. But the model in that paper is too complex with too many parameters, which is not suit for online estimation and will leads to computational complex. To our knowledge, the sensor-target pair-related bias is caused by the atmospheric refraction and is related to the range and elevation between the target and sensor [13–18]. In this respect, the accurate model of the sensor-target pair-related bias is considered by the analysis of the propagation of the electromagnetic wave in the atmosphere. Then, the sensor registration can be performed on the grounds of this newly obtained model.

#### 2. Accurate Model of the Radar Bias

In this section, an accurate model of the radar bias is proposed. Under the framework of the bias model, the target-sensor pair-related bias caused by atmospheric refraction is emphasized.

##### 2.1. Radar Bias Model

Consider a 3D radar, the measurements of which include the range, elevation, and azimuth. The model of the measurements is described as follows:where are the measurements of the range, elevation, and azimuth, respectively. Here, are the measurements of the ideal target position. Additionally, and stand for the target-sensor pair-related radar biases on the range and elevation that were introduced by the atmospheric refraction. The variable means the transponder-induced bias, and are the constant elevation bias and azimuth bias, respectively, and they are different for each radar. The stochastic noise is represented by , , and . All of these variables are expressed in local spherical coordinates.

##### 2.2. Target-Sensor Pair-Related Bias Model

In this paper, we are concerned with the target-sensor pair-related bias that is caused by atmospheric refraction. In the traditional method, this bias is treated as a linear changed variable of range or elevation, but this treatment is not so accurately. In this respect, to accurately model this type of bias, we should review the principle of how the atmospheric refraction influences the radar error.

To see the radar bias that is introduced by the atmosphere, the spherical stratification method is mainly used to calculate the bias. Moreover, by using this method, the model of the atmospheric refraction index is indispensable. To our knowledge, the segmented model and Hopfield model are the common algorithms. Different models suit different area characteristics. Considering the atmospheric character of China, the segmented model proposed by Bean and Thayer [19] is chosen in this paper. The segmented model is described as follows:where is the altitude of the surface in km. is the grads of the atmospheric refraction rate 1 km above the surface, the measurement of which is made in . is a measure of the refractivity in N-units at the surface. and are measures of the refractivity in N-units at 1 km and 9 km above the surface, respectively. The variables and are the attenuation index of the atmospheric index with the altitude between 1 km and 9 km and higher than 9 km, respectively, the measurements of which are made in .

In what follows, the spherical stratification method [20, 21] is adopted to analyze the radar system error that is caused by atmospheric refraction.

As shown in Figure 1, and are the positions of the radar and target, respectively. Here, is the propagation path of the radar electromagnetics, is the apparent position of the target, and means the horizontal distance between the radar and target. The propagation of radar electromagnetic waves is not still a line, and thus, the apparent range between the target and radar iswhere is the propagation path. denotes the atmospheric refraction rate on the propagation path. The variable stands for the refraction index, and . Here, is the elevation of the propagation path. And is the radius of the earth.