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

Authors: Xiaoju Yong; Youli Wu; Mingwu Tu; Xiaowei Du; Shuang Zhang
Journal: Mathematical Problems in Engineering
 (2018)

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## 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.


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## Body

## 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:(1)R=rt+Δrc+Δre+nr(2)φ=φt+Δφc+Δφ+nφ(3)θ=θt+Δθ+nθwhere R,φ,θ are the measurements of the range, elevation, and azimuth, respectively. Here, rt,φt,θt are the measurements of the ideal target position. Additionally, Δrc and Δφc stand for the target-sensor pair-related radar biases on the range and elevation that were introduced by the atmospheric refraction. The variable Δre means the transponder-induced bias, Δφ and Δθ are the constant elevation bias and azimuth bias, respectively, and they are different for each radar. The stochastic noise is represented by nr, nφ, and nθ. 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:(4)Nh=N0-Gh-h0,h0≤h<h0+1N1exp⁡-c1h-h0-1,h0+1≤h<9N9exp⁡-c9h-9,9≤h≤60where h0 is the altitude of the surface in km. G is the grads of the atmospheric refraction rate 1 km above the surface, the measurement of which is made in N/km. N0 is a measure of the refractivity in N-units at the surface. N1 and N9 are measures of the refractivity in N-units at 1 km and 9 km above the surface, respectively. The variables c1 and c9 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 km-1.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 Figure1, O and T are the positions of the radar and target, respectively. Here, s is the propagation path of the radar electromagnetics, T′ is the apparent position of the target, and L 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 is(5)Re=10-6∫0sNa+hds=∫hrhTna+hcsc⁡φa+hdhwhere s is the propagation path. N denotes the atmospheric refraction rate on the propagation path. The variable n stands for the refraction index, and n=1+N×10-6. Here, φ is the elevation of the propagation path. And a is the radius of the earth.Figure 1
The geometry sketch map on the impact of atmosphere refraction on the propagation of a radar electromagnetic wave.Assume that the atmosphere is horizontally stratified, and using Snell’s law, the following can be obtained:(6)csc⁡φa+h=na+hn2a+h2-n2a+hr2cos2⁡φrwhere hr and φr are said to be the altitude of the radar and the elevation between target and radar, respectively.Based on the geometric relation of the target and radar,(7)tan⁡φr=hT-hrLwhere L means the horizontal distance between the radar and target and hT is the altitude of target.Based on (5), the apparent distance is(8)Re=∫hrhTn2a+hn2a+h2-n2a+hr2cos2⁡φrdhAs shown in Figure1, the target geocentric field angle φg is(9)φg=∫hrhTcot⁡φa+hdh=na+hrcos⁡φr×∫hrhT1a+hn2a+h2-n2a+hr2cos2φrdhNext, the true elevation can be calculated according to the law of sines:(10)φ0=arctana+hTcos⁡φ-a+hra+hTsin⁡φgAdditionally, the true range can also be calculated according to the law of sines:(11)R0=a+hTsin⁡φgcos⁡φ0Now, the target-sensor pair-related error on the range and elevation caused by atmospheric refraction is(12)Δφc=φe-φ0Δrc=Re-R0From (5) to (12), we can find the fact that the errors are related to the radar altitude, target altitude, distance, and elevation between the target and radar. However, this method only supplies us a method for calculating the bias, but we cannot utilize (5) to (12) for online sensor registration at every step. What we need is a simple function to calculate the bias directly. For this purpose, in what follows, a convenient function for calculating the target-sensor pair-related bias is proposed.Polynomial fitting is used to obtain the bias function, but before the polynomial fitting, to simplify the function expression, we should find out the less sensitive variable and ignore it. Here, we will use numeric simulations to compare the impacts of these variables on the errors. Then, we will attempt to obtain the bias function of these main items to calculate the system errors.Under the assumption that the range between the target and radar changes from 50 km to 400 km, the altitude of the radar changes from 0.5 km to 6 km and the elevation between the target and radar changes from 5° to 50°; the errors on the range and elevation are shown in Figures2 and 3.Figure 2
The radar system error on range.Figure 3
The radar system error on elevation.Fortunately, we can find a factor that is not sensitive to the errors. Considering the color that represents the radar system errors, as the altitude of the radar changes from 1 km to 6 km, if the range and elevation do not change, then the color stays almost the same, which means that the radar system errors do not change. Thus, we can draw the conclusion that the radar system errors are not sensitive to the altitude of the radar. In contrast, the errors are sensitive to the elevation and range between the radar and target.The radar system errors are impacted by the range and elevation between the radar and target and the altitude of the radar, but from the analysis above, the main factors are the range and elevation. As a consequence, in what follows, we consider only the range and elevation between the target and radar.Adopting a quadric multinomial function, the form of the expression is as follows:(13)Δrc=fR,φΔφc=gR,φ(14)ΔR=a11+a12φ+a13φ2+a14R+a15Rφ+a16Rφ2+a17R2+a18R2φ+a19R2φ2Δφ=a21+a22φ+a23φ2+a24R+a25Rφ+a26Rφ2+a27R2+a28R2φ+a29R2φ2A grid was formed for the variablesR and φ, and m,n are the size of R and φ, respectively.The least square rule is used to fit the bias function, and the objective function is(15)E=∑i=1m∑j=1nzRi,j-ΔRi,j2and(16)E=∑i=1m∑j=1nzφi,j-Δφi,j2where zR and zφ are the radar system errors that are acquired from (5) to (12), respectively. ΔR(i,j) and Δφ(i,j) are the range bias and elevation bias calculated by (14). And the altitude of radar is set to be a constant. The range, elevation, and altitude are measured by km, mrad, and km, respectively. The radar system errors on the range and elevation are measured by m and mrad, respectively.Here, we take the functionΔrc as an example to solve the problem. The coefficient vector a=a11a12a13a14a15a16a17a18a19T should satisfy(17)∂E∂a1i=0Substitute (14) and (15) into (17); then(18)Ha=Zwhere(19)H=λ0,0λ0,1λ0,2λ1,0λ1,1λ1,2λ2,0λ2,1λ2,2λ0,1λ0,2λ0,3λ1,1λ1,2λ1,3λ2,1λ2,2λ2,3λ0,2λ0,3λ0,4λ1,2λ1,3λ1,4λ2,2λ2,3λ2,4λ1,0λ1,1λ1,2λ2,0λ2,1λ2,2λ3,0λ3,1λ3,2λ1,1λ1,2λ1,3λ2,1λ2,2λ2,3λ3,1λ3,2λ3,3λ1,2λ1,3λ1,4λ2,2λ2,3λ2,4λ3,2λ3,3λ3,4λ2,0λ2,1λ2,2λ3,0λ3,1λ3,2λ4,0λ4,1λ4,2λ2,1λ2,2λ2,3λ3,1λ3,2λ3,3λ4,1λ4,2λ4,3λ2,2λ2,3λ2,4λ3,2λ3,3λ3,4λ4,2λ4,3λ4,4Z=η0,0η0,1η0,2η1,0η1,1η1,2η2,0η2,1η2,2Tλp,q=∑i=1m∑j=1nRpi,jφqi,j,ηp,q=∑i=1m∑j=1nzRi,jλp,q.Thus, the coefficient vectora is(20)a=H-1ZUsing the data in the simulation above, the coefficients of the bias function were obtained, and they are shown as follows:(21)a11=138.3313,a12=-0.1370,a13=3.5907×10-5,a14=-1.5251,a15=0.0017,a16=-5.2007×10-7,a17=0.0049,a18=-5.8185×10-6,a19=2.3228×10-9.Adopt the method above similarly, and the coefficients of the functionΔθ can be obtained; they are shown as follows:(22)a21=-31.5830,a22=0.0344,a23=-5.7131×10-6,a24=0.4947,a25=-4.2515×10-4,a16=6.9947×10-8,a27=-0.0012,a28=9.4424×10-7,a19=-2.0144×10-10.Now, we can obtain a bias function to calculate the target-sensor pair-related radar system errors caused by atmospheric refraction.


## 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:(1)R=rt+Δrc+Δre+nr(2)φ=φt+Δφc+Δφ+nφ(3)θ=θt+Δθ+nθwhere R,φ,θ are the measurements of the range, elevation, and azimuth, respectively. Here, rt,φt,θt are the measurements of the ideal target position. Additionally, Δrc and Δφc stand for the target-sensor pair-related radar biases on the range and elevation that were introduced by the atmospheric refraction. The variable Δre means the transponder-induced bias, Δφ and Δθ are the constant elevation bias and azimuth bias, respectively, and they are different for each radar. The stochastic noise is represented by nr, nφ, and nθ. 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:(4)Nh=N0-Gh-h0,h0≤h<h0+1N1exp⁡-c1h-h0-1,h0+1≤h<9N9exp⁡-c9h-9,9≤h≤60where h0 is the altitude of the surface in km. G is the grads of the atmospheric refraction rate 1 km above the surface, the measurement of which is made in N/km. N0 is a measure of the refractivity in N-units at the surface. N1 and N9 are measures of the refractivity in N-units at 1 km and 9 km above the surface, respectively. The variables c1 and c9 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 km-1.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 Figure1, O and T are the positions of the radar and target, respectively. Here, s is the propagation path of the radar electromagnetics, T′ is the apparent position of the target, and L 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 is(5)Re=10-6∫0sNa+hds=∫hrhTna+hcsc⁡φa+hdhwhere s is the propagation path. N denotes the atmospheric refraction rate on the propagation path. The variable n stands for the refraction index, and n=1+N×10-6. Here, φ is the elevation of the propagation path. And a is the radius of the earth.Figure 1
The geometry sketch map on the impact of atmosphere refraction on the propagation of a radar electromagnetic wave.Assume that the atmosphere is horizontally stratified, and using Snell’s law, the following can be obtained:(6)csc⁡φa+h=na+hn2a+h2-n2a+hr2cos2⁡φrwhere hr and φr are said to be the altitude of the radar and the elevation between target and radar, respectively.Based on the geometric relation of the target and radar,(7)tan⁡φr=hT-hrLwhere L means the horizontal distance between the radar and target and hT is the altitude of target.Based on (5), the apparent distance is(8)Re=∫hrhTn2a+hn2a+h2-n2a+hr2cos2⁡φrdhAs shown in Figure1, the target geocentric field angle φg is(9)φg=∫hrhTcot⁡φa+hdh=na+hrcos⁡φr×∫hrhT1a+hn2a+h2-n2a+hr2cos2φrdhNext, the true elevation can be calculated according to the law of sines:(10)φ0=arctana+hTcos⁡φ-a+hra+hTsin⁡φgAdditionally, the true range can also be calculated according to the law of sines:(11)R0=a+hTsin⁡φgcos⁡φ0Now, the target-sensor pair-related error on the range and elevation caused by atmospheric refraction is(12)Δφc=φe-φ0Δrc=Re-R0From (5) to (12), we can find the fact that the errors are related to the radar altitude, target altitude, distance, and elevation between the target and radar. However, this method only supplies us a method for calculating the bias, but we cannot utilize (5) to (12) for online sensor registration at every step. What we need is a simple function to calculate the bias directly. For this purpose, in what follows, a convenient function for calculating the target-sensor pair-related bias is proposed.Polynomial fitting is used to obtain the bias function, but before the polynomial fitting, to simplify the function expression, we should find out the less sensitive variable and ignore it. Here, we will use numeric simulations to compare the impacts of these variables on the errors. Then, we will attempt to obtain the bias function of these main items to calculate the system errors.Under the assumption that the range between the target and radar changes from 50 km to 400 km, the altitude of the radar changes from 0.5 km to 6 km and the elevation between the target and radar changes from 5° to 50°; the errors on the range and elevation are shown in Figures2 and 3.Figure 2
The radar system error on range.Figure 3
The radar system error on elevation.Fortunately, we can find a factor that is not sensitive to the errors. Considering the color that represents the radar system errors, as the altitude of the radar changes from 1 km to 6 km, if the range and elevation do not change, then the color stays almost the same, which means that the radar system errors do not change. Thus, we can draw the conclusion that the radar system errors are not sensitive to the altitude of the radar. In contrast, the errors are sensitive to the elevation and range between the radar and target.The radar system errors are impacted by the range and elevation between the radar and target and the altitude of the radar, but from the analysis above, the main factors are the range and elevation. As a consequence, in what follows, we consider only the range and elevation between the target and radar.Adopting a quadric multinomial function, the form of the expression is as follows:(13)Δrc=fR,φΔφc=gR,φ(14)ΔR=a11+a12φ+a13φ2+a14R+a15Rφ+a16Rφ2+a17R2+a18R2φ+a19R2φ2Δφ=a21+a22φ+a23φ2+a24R+a25Rφ+a26Rφ2+a27R2+a28R2φ+a29R2φ2A grid was formed for the variablesR and φ, and m,n are the size of R and φ, respectively.The least square rule is used to fit the bias function, and the objective function is(15)E=∑i=1m∑j=1nzRi,j-ΔRi,j2and(16)E=∑i=1m∑j=1nzφi,j-Δφi,j2where zR and zφ are the radar system errors that are acquired from (5) to (12), respectively. ΔR(i,j) and Δφ(i,j) are the range bias and elevation bias calculated by (14). And the altitude of radar is set to be a constant. The range, elevation, and altitude are measured by km, mrad, and km, respectively. The radar system errors on the range and elevation are measured by m and mrad, respectively.Here, we take the functionΔrc as an example to solve the problem. The coefficient vector a=a11a12a13a14a15a16a17a18a19T should satisfy(17)∂E∂a1i=0Substitute (14) and (15) into (17); then(18)Ha=Zwhere(19)H=λ0,0λ0,1λ0,2λ1,0λ1,1λ1,2λ2,0λ2,1λ2,2λ0,1λ0,2λ0,3λ1,1λ1,2λ1,3λ2,1λ2,2λ2,3λ0,2λ0,3λ0,4λ1,2λ1,3λ1,4λ2,2λ2,3λ2,4λ1,0λ1,1λ1,2λ2,0λ2,1λ2,2λ3,0λ3,1λ3,2λ1,1λ1,2λ1,3λ2,1λ2,2λ2,3λ3,1λ3,2λ3,3λ1,2λ1,3λ1,4λ2,2λ2,3λ2,4λ3,2λ3,3λ3,4λ2,0λ2,1λ2,2λ3,0λ3,1λ3,2λ4,0λ4,1λ4,2λ2,1λ2,2λ2,3λ3,1λ3,2λ3,3λ4,1λ4,2λ4,3λ2,2λ2,3λ2,4λ3,2λ3,3λ3,4λ4,2λ4,3λ4,4Z=η0,0η0,1η0,2η1,0η1,1η1,2η2,0η2,1η2,2Tλp,q=∑i=1m∑j=1nRpi,jφqi,j,ηp,q=∑i=1m∑j=1nzRi,jλp,q.Thus, the coefficient vectora is(20)a=H-1ZUsing the data in the simulation above, the coefficients of the bias function were obtained, and they are shown as follows:(21)a11=138.3313,a12=-0.1370,a13=3.5907×10-5,a14=-1.5251,a15=0.0017,a16=-5.2007×10-7,a17=0.0049,a18=-5.8185×10-6,a19=2.3228×10-9.Adopt the method above similarly, and the coefficients of the functionΔθ can be obtained; they are shown as follows:(22)a21=-31.5830,a22=0.0344,a23=-5.7131×10-6,a24=0.4947,a25=-4.2515×10-4,a16=6.9947×10-8,a27=-0.0012,a28=9.4424×10-7,a19=-2.0144×10-10.Now, we can obtain a bias function to calculate the target-sensor pair-related radar system errors caused by atmospheric refraction.


## 3. Bias Estimation Algorithm Based on UKF

If we only use the established model to estimate the biases, other kinds of biases, asΔre, Δφ, and Δθ, will still stay and need to be estimated. So a method to estimate all the biases is requisite. This section presents the framework of the registration algorithm to estimate both the target state and the radar biases. The augmented state vector, with a target state and radar bias, will be estimated by the IMM-UKF algorithm.


### 3.1. Dynamic Model and Measurement Model

As in the previous description, the state vector consists ofXt and ΔX, where Xt is the vector of the target states and ΔX is the vector that contains the radar biases to be estimated, respectively. Furthermore, the content of the target state Xt depends on the dynamic model for the target motion.The dynamic equation is defined as(23)Xik+1=FiXik+ndwhere F(i) is the state transition function, which is different for different target kinematic models. When Δrc=f(R,φ) and Δφc=g(R,φ) are nonlinear functions, the dynamic equation should be a set of nonlinear equations. However, because the state vector is augmented with the target state and radar bias, the state transition function can be separated into two corresponding parts. One part is related to the target motion, and the other part involves the radar bias. Fortunately, the first part is usually a linear function. In turn, the first part can be expressed by a matrix, which is the same as that in the target state estimation problems.Consider IMM-UKF estimator with CA and CT models. Undertaking the CA model, the parameters are defined as follows:X ( 1 ) = [ X t ( 1 ) , Δ X ] T , X t ( 1 ) = x , y , z , v x , v y , v z , a x , a y , a z T is the target state, and the radar bias is denoted by ΔX=Δrc1,Δre1,Δφc1,Δφ1,Δθ1,Δrc2,Δre2,Δφc2,Δφ2,Δθ2T. The state transition function is F(1)=Φ(1)f(1)(·), where Φ1=I3×3T·I3×3T2/2·I3×30I3×3T·I3×300I3×3 stands for the transition matrix for the target state with CV model. The notation f(1)(·)=f1,f2,f3,f4,f5,f6,f7,f8,f9,f10T is said to be the transition function with respect to the radar bias and is defined as follows:(24)f1=frt1+Δrc1+Δre1,φt1+Δφc1+Δφ1=frt1+X10+X11,φt1+X12+X13,f2=Δre1=X11,f3=grt1+Δrc1+Δre1,φt1+Δφc1=grt1+X10+X11,φt1+X12+X13,f4=Δφ1=X13f5=Δθ1=X14,f6=frt2+Δrc2+Δre2,φt2+Δφc2+Δφ2=frt2+X15+X16,φt2+X17+X18,f7=Δre2=X16,f8=grt2+Δrc2+Δre2,φt2+Δφc2+Δφ2=grt2+X15+X16,φt2+X17+X18,f9=Δφ2=X18.f10=Δθ2=X19.rt1=(x-xs1)2+(y-ys1)2+(z-zs1)2, rt2=(x-xs2)2+(y-ys2)2+(z-zs2)2, xs1,ys1,zs1, and xs2,ys2,zs2 are the positions of radar #1 and radar #2 in the ECEF coordinates, respectively.With the CT model, the parameters are defined as follows:(25)X2=Xt2,ΔXT,Xt2=x,y,z,vx,vy,vz,ax,ay,azT,F2=Φ2f2,Φ2=I3×3sin⁡ωTω·I3×31-cos⁡ωTω2·I3×30cos⁡ωT·I3×3sin⁡ωTω·I3×30-ωsin⁡ωT·I3×3cos⁡ωT·I3×3.whereωistheturningrateoftarget.And f(2)=f1,f2,f3,f4,f5,f6,f7,f8,f9,f10T, because the bias model in CT model is the same as that in CA motion, so the definition of f(2) is the same as f(1).The measurement equation is given as follows:(26)Yk=HiXik+nzwhere Y(k)=R1,θ1,φ1,R2,θ2,φ2T is the measurement vector, and R1,θ1,φ1 and R2,θ2,φ2are the measurements of radar #1 and radar #2 in the local polar coordinates.H ( 1 ) · = H ( 2 ) · = h 1 , h 2 , h 3 , h 4 , h 5 , h 6 T is the measurement function with respect to the state vector, where(27)h1=X1-xs12+X2-ys12+X3-zs12+X10+X11h2=arctanX2-ys1X1-xs1+X14h3=arctanX3-zs1X1-xs12+X2-ys12+X12+X13h4=X1-xs22+X2-ys22+X3-zs22+X15+X16h5=arctanX2-ys2X1-xs2+X19h6=arctanX3-zs2X1-xs22+X2-ys22+X17+X18


### 3.2. IMM-UKF Estimator

To apply the IMM-UKF [22] filter to estimate the target state and radar bias, the same important conditions should be specified, as follows:( 1 ) The chosen coordinates: the measurements are under the local spherical coordinates, the target states are under the ECEF coordinates, and the positions of the radars are also under the ECEF coordinates.( 2 ) The measurements are from two radars and are attributed to a common target.


## 3.1. Dynamic Model and Measurement Model

As in the previous description, the state vector consists ofXt and ΔX, where Xt is the vector of the target states and ΔX is the vector that contains the radar biases to be estimated, respectively. Furthermore, the content of the target state Xt depends on the dynamic model for the target motion.The dynamic equation is defined as(23)Xik+1=FiXik+ndwhere F(i) is the state transition function, which is different for different target kinematic models. When Δrc=f(R,φ) and Δφc=g(R,φ) are nonlinear functions, the dynamic equation should be a set of nonlinear equations. However, because the state vector is augmented with the target state and radar bias, the state transition function can be separated into two corresponding parts. One part is related to the target motion, and the other part involves the radar bias. Fortunately, the first part is usually a linear function. In turn, the first part can be expressed by a matrix, which is the same as that in the target state estimation problems.Consider IMM-UKF estimator with CA and CT models. Undertaking the CA model, the parameters are defined as follows:X ( 1 ) = [ X t ( 1 ) , Δ X ] T , X t ( 1 ) = x , y , z , v x , v y , v z , a x , a y , a z T is the target state, and the radar bias is denoted by ΔX=Δrc1,Δre1,Δφc1,Δφ1,Δθ1,Δrc2,Δre2,Δφc2,Δφ2,Δθ2T. The state transition function is F(1)=Φ(1)f(1)(·), where Φ1=I3×3T·I3×3T2/2·I3×30I3×3T·I3×300I3×3 stands for the transition matrix for the target state with CV model. The notation f(1)(·)=f1,f2,f3,f4,f5,f6,f7,f8,f9,f10T is said to be the transition function with respect to the radar bias and is defined as follows:(24)f1=frt1+Δrc1+Δre1,φt1+Δφc1+Δφ1=frt1+X10+X11,φt1+X12+X13,f2=Δre1=X11,f3=grt1+Δrc1+Δre1,φt1+Δφc1=grt1+X10+X11,φt1+X12+X13,f4=Δφ1=X13f5=Δθ1=X14,f6=frt2+Δrc2+Δre2,φt2+Δφc2+Δφ2=frt2+X15+X16,φt2+X17+X18,f7=Δre2=X16,f8=grt2+Δrc2+Δre2,φt2+Δφc2+Δφ2=grt2+X15+X16,φt2+X17+X18,f9=Δφ2=X18.f10=Δθ2=X19.rt1=(x-xs1)2+(y-ys1)2+(z-zs1)2, rt2=(x-xs2)2+(y-ys2)2+(z-zs2)2, xs1,ys1,zs1, and xs2,ys2,zs2 are the positions of radar #1 and radar #2 in the ECEF coordinates, respectively.With the CT model, the parameters are defined as follows:(25)X2=Xt2,ΔXT,Xt2=x,y,z,vx,vy,vz,ax,ay,azT,F2=Φ2f2,Φ2=I3×3sin⁡ωTω·I3×31-cos⁡ωTω2·I3×30cos⁡ωT·I3×3sin⁡ωTω·I3×30-ωsin⁡ωT·I3×3cos⁡ωT·I3×3.whereωistheturningrateoftarget.And f(2)=f1,f2,f3,f4,f5,f6,f7,f8,f9,f10T, because the bias model in CT model is the same as that in CA motion, so the definition of f(2) is the same as f(1).The measurement equation is given as follows:(26)Yk=HiXik+nzwhere Y(k)=R1,θ1,φ1,R2,θ2,φ2T is the measurement vector, and R1,θ1,φ1 and R2,θ2,φ2are the measurements of radar #1 and radar #2 in the local polar coordinates.H ( 1 ) · = H ( 2 ) · = h 1 , h 2 , h 3 , h 4 , h 5 , h 6 T is the measurement function with respect to the state vector, where(27)h1=X1-xs12+X2-ys12+X3-zs12+X10+X11h2=arctanX2-ys1X1-xs1+X14h3=arctanX3-zs1X1-xs12+X2-ys12+X12+X13h4=X1-xs22+X2-ys22+X3-zs22+X15+X16h5=arctanX2-ys2X1-xs2+X19h6=arctanX3-zs2X1-xs22+X2-ys22+X17+X18


## 3.2. IMM-UKF Estimator

To apply the IMM-UKF [22] filter to estimate the target state and radar bias, the same important conditions should be specified, as follows:( 1 ) The chosen coordinates: the measurements are under the local spherical coordinates, the target states are under the ECEF coordinates, and the positions of the radars are also under the ECEF coordinates.( 2 ) The measurements are from two radars and are attributed to a common target.


## 4. Numerical Simulation

In this section, the validity of the model of the target-sensor pair-related bias will be checked out first. Then, the impact of the better radar bias model on the bias estimation will be verified by numerical simulations.


### 4.1. The Radar Bias Model

In this section, the validity of the function in Section3.2 will be checked out. Assume that the altitude is 6 km, the distance between the radar and target is 50 km to 400 km, and the elevation between the target and radar is 5° to 50°; then, the radar system errors calculated based on (5) to (12) are shown in Figures 4 and 6. The radar system errors were calculated in relation to the proposed algorithm and are shown in Figures 5 and 7.Figure 4
The radar system error on range calculated from (5) to (12).Figure 5
The radar system error on range calculated based on the proposed algorithm in this paper.Figure 6
The radar system error on the elevation calculation from (5) to (12).Figure 7
The radar system error on the elevation calculated based on the proposed algorithm in this paper.As shown in Figures4 and 6 and in Figures 5 and 7, we can draw the conclusion that the target-sensor bias model proposed in this paper can accurately describe the errors that are caused by the atmospheric refraction.


### 4.2. The Radar Bias Estimation

Assume that the target takes a CV motion during 0~15 s, takes a CT motion during 15~25 s, and takes a CV motion during 25~40 s. The initial position of target is5000,5000,2000 in the ECEF coordinates. The positions of sensor A and sensor B in the ECEF coordinates are 0,0,0 and 10000,10000,5000, respectively. The offset bias of sensor A is 500m,0.01rad,0.01radT. The stochastic noise is zero-mean, white with variance RA=diag⁡1020.00220.0022. The offset bias of sensor B is ξB(k)=-750m,0.05rad,-0.02radT. The time varying errors are calculated based on (5) to (12).Adopting the IMM-UKF algorithm to solve the problem of sensor registration, the IMM-UKF estimator consists of two models, namely, a constant acceleration model and a constant turn model. The Markov chain transition matrix for this IMM estimator is(28)pij=0.950.050.050.95,The RMSEs of the estimation results are shown in Figures 8–10. The results are based on 100 Monte-Carlo runs.Figure 8
RMSE of the bias estimation on range.Figure 9
RMSE of the bias estimation on elevation.Figure 10
RMSE of the bias estimation on azimuth.As shown in Figures8–10, the proposed algorithm can accurately estimate the radar bias and track the maneuver target with high precision, as expected. The RMSE of range bias estimation of the proposed method is about 42m for radar #1 and 80m for radar #2, respectively. As for the traditional method, the RMSEs are about 650m. The RMSEs of azimuth bias estimation of the proposed method and traditional method are 0.0011rad(radar #1), 0.0006rad(radar #2), 0.0025rad(radar #1), and 0.008rad(radar#2), respectively. As to elevation, the RMSEs of proposed method are 0.005rad(radar #1) and 0.02rad(radar#2), respectively; the RMSEs of the traditional method are 0.08rad and 0.04rad, respectively.The estimation of the proposed method is better than the traditional method, the reason is that the proposed radar bias model is more accurate than the traditional one. We can also find that accuracy of the azimuth bias estimation is higher than range and elevation bias estimation, the reason is that in the bias model the atmospheric is assumed to be horizontally stratified, so the azimuth only contain a constant bias.


## 4.1. The Radar Bias Model

In this section, the validity of the function in Section3.2 will be checked out. Assume that the altitude is 6 km, the distance between the radar and target is 50 km to 400 km, and the elevation between the target and radar is 5° to 50°; then, the radar system errors calculated based on (5) to (12) are shown in Figures 4 and 6. The radar system errors were calculated in relation to the proposed algorithm and are shown in Figures 5 and 7.Figure 4
The radar system error on range calculated from (5) to (12).Figure 5
The radar system error on range calculated based on the proposed algorithm in this paper.Figure 6
The radar system error on the elevation calculation from (5) to (12).Figure 7
The radar system error on the elevation calculated based on the proposed algorithm in this paper.As shown in Figures4 and 6 and in Figures 5 and 7, we can draw the conclusion that the target-sensor bias model proposed in this paper can accurately describe the errors that are caused by the atmospheric refraction.


## 4.2. The Radar Bias Estimation

Assume that the target takes a CV motion during 0~15 s, takes a CT motion during 15~25 s, and takes a CV motion during 25~40 s. The initial position of target is5000,5000,2000 in the ECEF coordinates. The positions of sensor A and sensor B in the ECEF coordinates are 0,0,0 and 10000,10000,5000, respectively. The offset bias of sensor A is 500m,0.01rad,0.01radT. The stochastic noise is zero-mean, white with variance RA=diag⁡1020.00220.0022. The offset bias of sensor B is ξB(k)=-750m,0.05rad,-0.02radT. The time varying errors are calculated based on (5) to (12).Adopting the IMM-UKF algorithm to solve the problem of sensor registration, the IMM-UKF estimator consists of two models, namely, a constant acceleration model and a constant turn model. The Markov chain transition matrix for this IMM estimator is(28)pij=0.950.050.050.95,The RMSEs of the estimation results are shown in Figures 8–10. The results are based on 100 Monte-Carlo runs.Figure 8
RMSE of the bias estimation on range.Figure 9
RMSE of the bias estimation on elevation.Figure 10
RMSE of the bias estimation on azimuth.As shown in Figures8–10, the proposed algorithm can accurately estimate the radar bias and track the maneuver target with high precision, as expected. The RMSE of range bias estimation of the proposed method is about 42m for radar #1 and 80m for radar #2, respectively. As for the traditional method, the RMSEs are about 650m. The RMSEs of azimuth bias estimation of the proposed method and traditional method are 0.0011rad(radar #1), 0.0006rad(radar #2), 0.0025rad(radar #1), and 0.008rad(radar#2), respectively. As to elevation, the RMSEs of proposed method are 0.005rad(radar #1) and 0.02rad(radar#2), respectively; the RMSEs of the traditional method are 0.08rad and 0.04rad, respectively.The estimation of the proposed method is better than the traditional method, the reason is that the proposed radar bias model is more accurate than the traditional one. We can also find that accuracy of the azimuth bias estimation is higher than range and elevation bias estimation, the reason is that in the bias model the atmospheric is assumed to be horizontally stratified, so the azimuth only contain a constant bias.


## 5. Conclusions

In this paper, the more accurate radar bias model is established. In the proposed model, the target-sensor pair-related bias caused by atmospheric refraction is emphasized. A convenient function for calculating this type of bias is obtained by using the least square algorithm. By adopting the proposed model, the dynamic model with respect to an augmented state vector with a target state and radar bias is obtained. Then, an IMM-UKF is utilized to estimate the target state and radar bias simultaneously. The limitation of this paper is that only a two-sensor one-target scenario is under consideration. The multisensor multitarget situation will be considered in the future. Moreover, in practical situations this model will not always hold; for example, in locations with interfaces between land and substantial bodies of water (lakes, swamps, etc.), or in the presence of assorted weather conditions, there may be vertical stratification, and this stratification may even vary—and not necessarily slowly—over time. Furthermore, from a practical perspective, even when the atmosphere is indeed horizontally stratified, one may have that, due to installation issues, sagging or settling ground, etc., the radar antenna may be somewhat rotated about boresight, meaning that the radar’s purported azimuthal measurements will also contain elevation components and will thus be subject to path-distortion effects. Additionally, the method proposed in our paper can only suit for the situation of single target. In multitargets situation, as to estimate the radar biases, selecting two radars which detect a common target is the first step. And our method can be extended to multisensor multitarget situation, which is the next work for our team.



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*Source: 3083258-2018-09-10.xml*