- Email: [email protected]
Coordinate registration algorithms for over-the-horizon radar*
Journal of Systems Engineering and Electronics, Vol. 17, No.4, 2006, pp.725^-730
Coordinate registration algorithms for over-the-horizon radar* Kong Min , Wang Guohong & Wang Yongcheng 1. Research Inst. of Information Fusion, Naval Aeronautical Engineering Inst, Yantai 264001, P. R. China; 2. The 14th Inst. of China Electronics & Technology Cooperation, Nanjing 210013, P. R. China (Received May 15,2005) Abstract: Different from the other conventional radars, the over the horizon radar (OTHR) faces complicated nonlinear coordinate transform due to electromagnetic wave propagation and reflection in ionospheres. A significant problem is the phenomenon of multi-path propagation. Considering it, the coordinate registration algorithms of planar measurement model and spherical measurement model are respectively derived in detail. Noticeably, a new transforming expression of apparent azimuth and an integrated form of transforming expressions from measurement vector to ground state vector in coordinate registration algorithm of spherical measurement model are proposed. And then simulations are made to verify the correctness of the proposed algorithms and expression. Besides this, the transforming error rate of slant range, Doppler and apparent azimuth of the two kinds of models are given respectively. Then the quantitative analysis of error rate is also given. It can be drawn a conclusion that the coordinate registration algorithms of planar measurement model and spherical measurement model are both correct. Keywords: over-the-horizon radar, coordinate registration, error rate. 1.
INTRODUCTION
Over-the-horizon radar, which works in shortwave band, performs over-the-horizon wide area surveillance by reflecting via ionosphere. A significant problem is the phenomenon of multi-path propagation[1], which arises from the presence of layers or regions of relatively high electron density in the ionosphere. Radar signals scattered from the same target arrive at the receiver via different propagation paths. Presently, the common method in OTHR tracking system is multi-path probability data association (MPDA) algorithm, which includes coordinate registration according to estimation of propagation mode. By transforming measuring coordinate to ground coordinate, the method performs data association and state estimation in ground coordinate^. So, coordinate registration is one important part of multipath tracking. Currently, there are two measurement models of coordinate registration. One is planar measurement model, which considers that transmitter and receiver of the radar and the movement of target are lying in the same plane; and the other is spherical model which takes the curvature of earth into account. The coordinate registration algorithms of planar measurement model and spherical measurement model will be discussed respectively in this paper. For planar *
measurement model, the expression of apparent azimuth in Ref. [2] is dependent on the elevation of receiving beam; however the elevation is not measured or calculated in Ref. [2]. So, the expression of apparent azimuth is amended in this paper. In spherical measurement model, a new expression of apparent azimuth is given based on different geometrical relation compared with Ref. [3] in this paper. Simulations show that the expression given in this paper is also correct and feasible. Further more, in Refs. [2] and [3], measurement vectors include the change rate of slant range, in this paper, measurement vector and ground state vector include Doppler frequency directly, substitute for the change rate of slant range. Moreover, the transforming error rates of two measurement models are compared . 2.
MULTI-PATH PROPAGATION PRINCIPLE
The principle of multi-path propagation is shown in Fig.l. The transmitting beam reflected from transmitter to target by ionosphere, then the receiving beam reflected from target to receiver by ionosphere. For simplicity, it is assumed that there are only two ionospheres, say, E and F, with constant heights hE and hF . In this case, the propagation modes are: EE- transmit on E and receive on E\ EF- transmit on E and receive on F\ FE- transmit on F and receive on E\ FF- transmit on F and receive on F.
This project was supported by the Foundation for the Author of National Excellent Doctoral Dissertation of China(200443).
726
Kong Min, Wang Guohong & Wang Yongcheng
Fig. 1
3.
Principle of multi-path propagation
GEOMETRICAL M E A S U R E M E N T M O D E L OF COORDINATE REGISTRATION
3.1 Planar Measurement Model of Coordinate Registration The planar OTHR measurement geometry of Fig.2 assumes that the transmitter is located on the axis of the receiver array at a distance d from it and that the motion of target is in the same plane. Of course, a more accurate model would take the curvature of the Earth into account and possibly the altitude of target above the Earth's surface. Receiver and transmitter both locate on X axis. Receiver is at origin (0\), with transmitter (0 2 ) situated distance d from it. Z axis is vertical direction. The location of target is T. Target motion confined to (X, Y) plane. The ray paths from the transmitter to the target and from the target to the receiver are assumed to be reflected from idealized ionosphere at virtual heights ht and hr, respectively. The heights are assumed to be known. The elevation of transmitter and receiver are ψι and ψ\ respectively. The distance between target and receiver is pu and the distance between target and transmitter is pi. The angle between receiving beam and plane 0\ YZ is Az. The radar measurements consist of (a) Slant range P=P\+P2 (half the path length); (b) Doppler^,=2 P Ιλ (difference between transmitting frequency and receiving frequency, where λ is wavelength of radar and P is change rate of slant range); (c) Apparent azimuth Α^π/2-θ (angle between ΒΟχ and plane OxY7)\ (d) SNR. The measured azimuth (Az) is not the same as the target ground bearing (A), since the former includes an unknown elevation component. Because of ionosphere
propagation effects, antenna and patterns and variation of radar cross section, SNR tends to be an unreliable measurement. So SNR is usually used as peak detection. Note that the inward elevation ψ\ cannot be measured by a receiver with one-dimensional array. Ignoring SNR,measurement vector has a form as follows Y=[p,fP,Azy The state vector of target in ground coordinates consists of (a) ground distance between target and receiver : D (shown as p\ in Fig.2); (b) Doppler//=2Z)//l (difference between transmitting frequency and receiving frequency, where λ is wavelength of radar and D is change rate of ground range); (c) bearing^; (d) A (the change rate of A). So, the ground state vector has a form as X=[DffAA,ÄY Considering the OTHR system shown in Fig.2, the expression is given in Ref. [2]. However the definitions of vectors X and Y are different from that in this paper. In this paper, measurement vector and ground state vector include Doppler frequency directly, substitute for the rate of change of distance. For simplicity, here it will not be derived in detail. From Fig.2, it shows that the mapping can be expressed as {P=PX+P2
(1)
\fP=/AD/P^l/P2)/4 [Az = arcsin(D sin A /(2Pl))
Target Fig.2
Planar OTHR measurement model
where : η^Ό-d
ύηΑ , Px = ^{DI2f
^{dl2)2-DdunAI2
+ (DI2)2+h? .
+ hr2 , P2 =
From Eq.(l), one has Eq.(2). Since the distance between target and receiver is more than thousands kilometers, the change of azimuth is very little, and
727
Coordinate registration algorithms for over-the-horizon radar A is approximately regarded as zero. D = 2^P2-hr2 \/ά=Λ/ρΙφΙΡ{+ηΙΡ2)
(2)
A = arc sin(2/J sin Az ID) where ρχ =(p2 +hr2
and plane 0\ YZ is Az. The definitions of measurement vector Y=[P, fp, Az]' and ground state vector X=[D, fd, A,A]' are same as that of planar measurement model. It' s considered that the radial distance in ground is a segment of a circle. So BO' = hr + μ, and the radius of earth is μ at 0\. According to Fig.3, we have
-h2-(d/2)2)/(2P-dsmAz)
P2=P-P, The expression of azimuth in Ref. [2] is A=arc sin (sin/iz/cosy/i). However there is no explaining about the value of elevation ψ\. In the actual OTHR system, the measurement vector does not include elevation. So, the expression of A in Eq.(2) is adopted.
Ρλ=^+(μ
P2 = Ιμ2 + (μ + h,)2 -2μ(μ + h,)cos(^-) 2μ
R
St
(4)
Ρ = Ρλ+Ρ2 =
In the case of considering curvature of earth, earth is considered as an ellipsoid according to WGS—84 coordinates. The spherical OTHR measurement geometry of Fig. 3 assumes that the transmitter is located on the axis of the receiver array at a distance d from it. Ref. [3] discussed this case coarsely.
Receiver Ολ
(3)
For the distance between target and receiver is more than one thousand kilometers, it is considered that the radius of 0\ is equal to the radius of 02· Thus, Eqs.(4)~(6) can be got
3.2 Spherical Measurement Model of Coordinate Registration
Z\
+ Κ)2-2μ{μ+Ην)οο*{^)
Broesight Y
Transmitter 0:
Fig. 3 Spherical OTHR measurement model
Receiver and transmitter both locate on X axis. Receiver is at origin (0\), with transmitter (O2) situated distance d from it. Z axis is vertical direction. The target is located at T. The earth's core locates at 0\ and the radius of earth is μ. The central angle is α=Ό/(2μ). The ray paths from the transmitter to the target and from the target to the receiver are assumed to be reflected from idealized ionosphere at virtual heights ht and hn respectively. The heights are assumed to be known. The distance between target and receiver is p\, and the distance between target and transmitter is pi. The angle between receiving beam
μ2+(μ + hrf -2μ{μ + hr)cos(—) + L 2 +Gu + Ä , ) 2 - 2 ^ + Ä,)cos(i?-)
(5)
/=2P/A =^ s i n ( ^ ) ( ^ + ^ ) = λ 2μ Ρλ (6) 1 , . ,D.M + hr μ + h, r -fd sin(—)(-+ '- ) Jd ν 2 2μ P, P2 Equation (5) is same as Ref.[3]. Eq. (6) gives the expression of fp, and Ref. [3] gives the expression of the rate of distance change P . The expression of Az is different from Ref.[3]. The expression of Az in Ref.[3] is Az = arcsin(^ + hr) sin(—) sin A/P.) (7) 2μ In this paper, the expression of Az is given as Eq. (8) Az - arcsin(^ arcsin(sin(—) sin A) I P}) (8) 2μ The extending can be given as follows. In Fig.3, draw a plumb line from point E to axis X, the point of intersection is Q. It can be easily proved that BQ LX axis. So sin ^z=cos&=OxQIP\ . Let OxQ =Ad, so sin Az=Ad/P\. In spherical orthogonal triangle 0\QE, use the principle of spherical orthogonal triangle, thus sin(—) = sin(—) sin A μ 2μ
(9)
728
Kong Min, Wang Guohong & Wang Yongcheng Ad = /iarcsin (sin(—)sin;4) 2μ
(10)
sin Az - μarc sin (sin(—) sin A) I P] (11) 2μ Considering Eqs.(5), (6) and (11), the transforming expressions from ground state vector to measurement vector can be gotten.
P= L· +^ + hrf -2μ{μ + Κ)^{ψ) + μ1 + (μ + h, f -2μ(μ D ^
^
^
8
μ + h,
^
2μ
(12)
μ + h,L ) +J
Since the distance between target and receiver is more than one thousand kilometers, the change of azimuth is very small, so A is considered as zero. According to Eq.(12), we have
fä =
2μ(μ + Κ)
)
2Λ sin(—)(-
r
-+—
L
)
(13)
. . . P1sinA2 . ,D A = arcsin(sin(-^ -) / sm(—)) μ 2μ where
„ = Ρ(μ±Μ±
h-h 4Ρ {μ+Κ){μ+Ηι)-Α{Κ-Η,)2{μ(Κ+Κ) 1
κ-κ
+
2μ(μ+Α)
)
ff 2 δ ί η Α ( μ + Λ) 2μ
A = arcsin(sin(
4.
+ Ä, )cos(—)
μ2+(μ+Κ)2_ρ2
fä
(P/2)2
2μ
(14) ^)/8ΐη(—)) 2μ
Ä=0
D sin A) I P ) Az = arcsin(/i arcsin(sin(—) x 2μ
Ό-2μ arccos(
Ό = 2μ arccos(l +
ΚΗί)
SIMULATION AND ANALYSIS
4.1 Simulation and Conclusion of Planar Measurement Model Assume there exists a single target with a constant velocity in 2-dimentional plane, and its initial state vector in polar coordinates is X=[l 100km, 0.15 km/s, 0.104 72 rad, 8.726 65e-05 rad/s] [2] . Assume that there are only two ionospheres, E and F. In this case, there are only four propagation modes: EE, EF, FE, and FF. With each mode, transform target state vector in ground coordinates to measurement vector in measurements coordinates according to Eq.(l), then transform measurement vector in measurements coordinates back to target state vector in ground coordinates according to Eq.(2). Subsequently consider the error rate between outcome and original target state and see whether the error rate is within an acceptable error range. Figures 4-6 show transforming error rates of slant range, Doppler and apparent azimuth respectively. According to these figures, it can be seen that the maximal transforming error rates of slant range, Doppler and apparent azimuth are all less than 10"14. By calculating, we know that the three kinds of magnitudes of average transforming error rates are all 10"16. The three kinds of transforming error rates are
Ρ2=Ρ-Ρ, Assume that there are only two ionospheres, say, E and F. In this case, there are only four propagation modes: EE, EF, FE, and FF. In above case, it is supposed hr^hh namely under mode EF and FE. Under the mode EE and FF, there is hr=hf=K so P{=P2=P/2. Then Eq. (13) can be simplified to Eq.(14). Expression (13) is not given in Ref. [3], however expression (14) is given in Ref. [3], say, the simplified expression under EE and FF modes. Fig.4
Error rate of slant range
729
Coordinate registration algorithms for over-the-horizon radar
Fig. 6 Error rate of apparent azimuth
all very little. All these prove that the coordinate registration expressions of planar measurement model, i.e. Eqs. (1) and (2) are right. 4.2
Simulation and Conclusion of Spherical Measurement Model
The initial state of target and system parameters are set same as the scenario as Section 4.1. With each mode of EE, EF, FE and FF, transform target state vector in ground coordinates to measurement vector in measurements coordinates according to Eq.(12), then transform measurement vector in measurements coordinates back to target state vector in ground coordinates according to Eq. (13). Subsequently consider the error rate between outcome and original target state and see whether the error rates is within an acceptable error range. Figures 7-9 show error rates of slant range, Doppler and apparent azimuth respectiv ely. Adopted the expression of azimuth in Ref. [3], we get the error rate of apparent azimuth as Fig. 10. There is almost no significant difference between Fig.9 and Fig. 10. Further, the quantitative analysis is shown in Table 1.
Fig. 10 Error rate of apparent azimuth according to Ref. [3]
730
Kong Min, Wang Guohong & Wang Yongcheng Table 1 Quantitative analysis of transmitting error rate /% Variable
P
f, A Λ
ζ
(By this paper)
Ά
Δζ
(By Ref.[3J)
Error kind
EE
EF
FE
FF
Maximal error
3.475e-012
3.227 2e-012
3.893 5e-012
3.893 5e-012
Mean error
-4.890 7e-014
-7.243 5e-014
-1.664 8e-014
1.438e-014
MSE error
1.108 8e-012
1.014 5e-012
9.865 8e-013
1.113e-012
Maximal error
3.562 5e-012
3.458 2e-012
3.052 4e-012
3.429 7e-012
Mean error
4.991 2e-014
7.403 2e-014
1.901e-014
-1.399 9e-014
MSE error
1.105 8e-012
1.026 4e-012
9.685 6e-013
1.110 3e-012
Maximal error
4.490 4e-012
3.523 le-012
3.425 9e-012
4.604 5e-012
Mean error
5.786 9e-014
3.316 2e-014
7.584 7e-014
-1.781 3e-014
MSE error
1.275e-012
1.086 8e-012
1.008 6e-012
1.287 8e-012
Maximal error
4.490 4e-012
3.523 le-012
3.470 5e-012
4.617 5e-012
Mean error
5.936 4e-014
3.475e-014
7.598 3e-014
-1.609 7e-014
MSE error
1.274 9e-012
1.089 le-012
1.006 9e-012
1.286 4e-012
According to these figures and Table 1, we know that the maximal transforming error rates of Slant Range, Doppler and apparent azimuth are all less than 10"12. By calculating, it can be seen that the three kinds of magnitudes of average transforming error rates are all 10"14. All these prove that the coordinate registration expressions of spherical measurement model, i.e. Eqs. (12) and (13) are right. The transforming expression of apparent azimuth in Ref. [3] and proposed in this paper and are both satisfying with the requirement. Under EE, EF and FE modes, the error rate of apparent azimuth according to this paper is a little lower than the error rate according to Ref. [3]. A new transforming expression of apparent azimuth is given. As a result of considering the curvature of earth, the transforming error rate of spherical measurement model is lager than that of planar measurement model; however it is also within an acceptable error range. 5. CONCLUSIONS
In this paper, the coordinate registration algorithms of planar measurement model and of spherical measurement model are respectively derived and demonstrated in detail, and simulations are designed for the multi path propagation phenomenon of OTHR. The simulations prove the correctness of coordinate regist-
ration algorithms. At the same time, the transforming error rates of the two kinds of measurement models are given respectively. As a result of simplified hypothesis, the transforming error rate is lower in the case of planar measurement model than in the case of spherical measurement model. However they are all within an acceptable error range. So, it can be drawn a conclusion that the coordinate registrations of two models are both correct and feasible. Further, a new transforming expression of apparent azimuth is given. Via simulation, it is testified to be correct. REFERENCES [1] Bogner R E , Bouzerdoum A, Pope K J,et al. Association of tracks from over the horizon radar. IEEE AES Systems Magazine, 1998:31-35. [2] Pulford G W, Evans R J. A multipath data association tracker for over-the-horizon radar. IEEE Trans, on Aerospace and Electronic Syste/ws, 1998,34(4): 1165-1182. [3] Percival D J, White K A B. Multihypothesis fusion of multipath over-the-orizon radar tracks. Proc. of
SPIE
3373, Orlando, FL, 1998;440~451.
Kong Min was born in 1975. She is a Ph. D. candidat in Naval Aeronautical Engineering Instiute. Her research interest is radar data processing. E-mail:[email protected]
Coordinate registration algorithms for over-the-horizon radar* Kong Min , Wang Guohong & Wang Yongcheng 1. Research Inst. of Information Fusion, Naval Aeronautical Engineering Inst, Yantai 264001, P. R. China; 2. The 14th Inst. of China Electronics & Technology Cooperation, Nanjing 210013, P. R. China (Received May 15,2005) Abstract: Different from the other conventional radars, the over the horizon radar (OTHR) faces complicated nonlinear coordinate transform due to electromagnetic wave propagation and reflection in ionospheres. A significant problem is the phenomenon of multi-path propagation. Considering it, the coordinate registration algorithms of planar measurement model and spherical measurement model are respectively derived in detail. Noticeably, a new transforming expression of apparent azimuth and an integrated form of transforming expressions from measurement vector to ground state vector in coordinate registration algorithm of spherical measurement model are proposed. And then simulations are made to verify the correctness of the proposed algorithms and expression. Besides this, the transforming error rate of slant range, Doppler and apparent azimuth of the two kinds of models are given respectively. Then the quantitative analysis of error rate is also given. It can be drawn a conclusion that the coordinate registration algorithms of planar measurement model and spherical measurement model are both correct. Keywords: over-the-horizon radar, coordinate registration, error rate. 1.
INTRODUCTION
Over-the-horizon radar, which works in shortwave band, performs over-the-horizon wide area surveillance by reflecting via ionosphere. A significant problem is the phenomenon of multi-path propagation[1], which arises from the presence of layers or regions of relatively high electron density in the ionosphere. Radar signals scattered from the same target arrive at the receiver via different propagation paths. Presently, the common method in OTHR tracking system is multi-path probability data association (MPDA) algorithm, which includes coordinate registration according to estimation of propagation mode. By transforming measuring coordinate to ground coordinate, the method performs data association and state estimation in ground coordinate^. So, coordinate registration is one important part of multipath tracking. Currently, there are two measurement models of coordinate registration. One is planar measurement model, which considers that transmitter and receiver of the radar and the movement of target are lying in the same plane; and the other is spherical model which takes the curvature of earth into account. The coordinate registration algorithms of planar measurement model and spherical measurement model will be discussed respectively in this paper. For planar *
measurement model, the expression of apparent azimuth in Ref. [2] is dependent on the elevation of receiving beam; however the elevation is not measured or calculated in Ref. [2]. So, the expression of apparent azimuth is amended in this paper. In spherical measurement model, a new expression of apparent azimuth is given based on different geometrical relation compared with Ref. [3] in this paper. Simulations show that the expression given in this paper is also correct and feasible. Further more, in Refs. [2] and [3], measurement vectors include the change rate of slant range, in this paper, measurement vector and ground state vector include Doppler frequency directly, substitute for the change rate of slant range. Moreover, the transforming error rates of two measurement models are compared . 2.
MULTI-PATH PROPAGATION PRINCIPLE
The principle of multi-path propagation is shown in Fig.l. The transmitting beam reflected from transmitter to target by ionosphere, then the receiving beam reflected from target to receiver by ionosphere. For simplicity, it is assumed that there are only two ionospheres, say, E and F, with constant heights hE and hF . In this case, the propagation modes are: EE- transmit on E and receive on E\ EF- transmit on E and receive on F\ FE- transmit on F and receive on E\ FF- transmit on F and receive on F.
This project was supported by the Foundation for the Author of National Excellent Doctoral Dissertation of China(200443).
726
Kong Min, Wang Guohong & Wang Yongcheng
Fig. 1
3.
Principle of multi-path propagation
GEOMETRICAL M E A S U R E M E N T M O D E L OF COORDINATE REGISTRATION
3.1 Planar Measurement Model of Coordinate Registration The planar OTHR measurement geometry of Fig.2 assumes that the transmitter is located on the axis of the receiver array at a distance d from it and that the motion of target is in the same plane. Of course, a more accurate model would take the curvature of the Earth into account and possibly the altitude of target above the Earth's surface. Receiver and transmitter both locate on X axis. Receiver is at origin (0\), with transmitter (0 2 ) situated distance d from it. Z axis is vertical direction. The location of target is T. Target motion confined to (X, Y) plane. The ray paths from the transmitter to the target and from the target to the receiver are assumed to be reflected from idealized ionosphere at virtual heights ht and hr, respectively. The heights are assumed to be known. The elevation of transmitter and receiver are ψι and ψ\ respectively. The distance between target and receiver is pu and the distance between target and transmitter is pi. The angle between receiving beam and plane 0\ YZ is Az. The radar measurements consist of (a) Slant range P=P\+P2 (half the path length); (b) Doppler^,=2 P Ιλ (difference between transmitting frequency and receiving frequency, where λ is wavelength of radar and P is change rate of slant range); (c) Apparent azimuth Α^π/2-θ (angle between ΒΟχ and plane OxY7)\ (d) SNR. The measured azimuth (Az) is not the same as the target ground bearing (A), since the former includes an unknown elevation component. Because of ionosphere
propagation effects, antenna and patterns and variation of radar cross section, SNR tends to be an unreliable measurement. So SNR is usually used as peak detection. Note that the inward elevation ψ\ cannot be measured by a receiver with one-dimensional array. Ignoring SNR,measurement vector has a form as follows Y=[p,fP,Azy The state vector of target in ground coordinates consists of (a) ground distance between target and receiver : D (shown as p\ in Fig.2); (b) Doppler//=2Z)//l (difference between transmitting frequency and receiving frequency, where λ is wavelength of radar and D is change rate of ground range); (c) bearing^; (d) A (the change rate of A). So, the ground state vector has a form as X=[DffAA,ÄY Considering the OTHR system shown in Fig.2, the expression is given in Ref. [2]. However the definitions of vectors X and Y are different from that in this paper. In this paper, measurement vector and ground state vector include Doppler frequency directly, substitute for the rate of change of distance. For simplicity, here it will not be derived in detail. From Fig.2, it shows that the mapping can be expressed as {P=PX+P2
(1)
\fP=/AD/P^l/P2)/4 [Az = arcsin(D sin A /(2Pl))
Target Fig.2
Planar OTHR measurement model
where : η^Ό-d
ύηΑ , Px = ^{DI2f
^{dl2)2-DdunAI2
+ (DI2)2+h? .
+ hr2 , P2 =
From Eq.(l), one has Eq.(2). Since the distance between target and receiver is more than thousands kilometers, the change of azimuth is very little, and
727
Coordinate registration algorithms for over-the-horizon radar A is approximately regarded as zero. D = 2^P2-hr2 \/ά=Λ/ρΙφΙΡ{+ηΙΡ2)
(2)
A = arc sin(2/J sin Az ID) where ρχ =(p2 +hr2
and plane 0\ YZ is Az. The definitions of measurement vector Y=[P, fp, Az]' and ground state vector X=[D, fd, A,A]' are same as that of planar measurement model. It' s considered that the radial distance in ground is a segment of a circle. So BO' = hr + μ, and the radius of earth is μ at 0\. According to Fig.3, we have
-h2-(d/2)2)/(2P-dsmAz)
P2=P-P, The expression of azimuth in Ref. [2] is A=arc sin (sin/iz/cosy/i). However there is no explaining about the value of elevation ψ\. In the actual OTHR system, the measurement vector does not include elevation. So, the expression of A in Eq.(2) is adopted.
Ρλ=^+(μ
P2 = Ιμ2 + (μ + h,)2 -2μ(μ + h,)cos(^-) 2μ
R
St
(4)
Ρ = Ρλ+Ρ2 =
In the case of considering curvature of earth, earth is considered as an ellipsoid according to WGS—84 coordinates. The spherical OTHR measurement geometry of Fig. 3 assumes that the transmitter is located on the axis of the receiver array at a distance d from it. Ref. [3] discussed this case coarsely.
Receiver Ολ
(3)
For the distance between target and receiver is more than one thousand kilometers, it is considered that the radius of 0\ is equal to the radius of 02· Thus, Eqs.(4)~(6) can be got
3.2 Spherical Measurement Model of Coordinate Registration
Z\
+ Κ)2-2μ{μ+Ην)οο*{^)
Broesight Y
Transmitter 0:
Fig. 3 Spherical OTHR measurement model
Receiver and transmitter both locate on X axis. Receiver is at origin (0\), with transmitter (O2) situated distance d from it. Z axis is vertical direction. The target is located at T. The earth's core locates at 0\ and the radius of earth is μ. The central angle is α=Ό/(2μ). The ray paths from the transmitter to the target and from the target to the receiver are assumed to be reflected from idealized ionosphere at virtual heights ht and hn respectively. The heights are assumed to be known. The distance between target and receiver is p\, and the distance between target and transmitter is pi. The angle between receiving beam
μ2+(μ + hrf -2μ{μ + hr)cos(—) + L 2 +Gu + Ä , ) 2 - 2 ^ + Ä,)cos(i?-)
(5)
/=2P/A =^ s i n ( ^ ) ( ^ + ^ ) = λ 2μ Ρλ (6) 1 , . ,D.M + hr μ + h, r -fd sin(—)(-+ '- ) Jd ν 2 2μ P, P2 Equation (5) is same as Ref.[3]. Eq. (6) gives the expression of fp, and Ref. [3] gives the expression of the rate of distance change P . The expression of Az is different from Ref.[3]. The expression of Az in Ref.[3] is Az = arcsin(^ + hr) sin(—) sin A/P.) (7) 2μ In this paper, the expression of Az is given as Eq. (8) Az - arcsin(^ arcsin(sin(—) sin A) I P}) (8) 2μ The extending can be given as follows. In Fig.3, draw a plumb line from point E to axis X, the point of intersection is Q. It can be easily proved that BQ LX axis. So sin ^z=cos&=OxQIP\ . Let OxQ =Ad, so sin Az=Ad/P\. In spherical orthogonal triangle 0\QE, use the principle of spherical orthogonal triangle, thus sin(—) = sin(—) sin A μ 2μ
(9)
728
Kong Min, Wang Guohong & Wang Yongcheng Ad = /iarcsin (sin(—)sin;4) 2μ
(10)
sin Az - μarc sin (sin(—) sin A) I P] (11) 2μ Considering Eqs.(5), (6) and (11), the transforming expressions from ground state vector to measurement vector can be gotten.
P= L· +^ + hrf -2μ{μ + Κ)^{ψ) + μ1 + (μ + h, f -2μ(μ D ^
^
^
8
μ + h,
^
2μ
(12)
μ + h,L ) +J
Since the distance between target and receiver is more than one thousand kilometers, the change of azimuth is very small, so A is considered as zero. According to Eq.(12), we have
fä =
2μ(μ + Κ)
)
2Λ sin(—)(-
r
-+—
L
)
(13)
. . . P1sinA2 . ,D A = arcsin(sin(-^ -) / sm(—)) μ 2μ where
„ = Ρ(μ±Μ±
h-h 4Ρ {μ+Κ){μ+Ηι)-Α{Κ-Η,)2{μ(Κ+Κ) 1
κ-κ
+
2μ(μ+Α)
)
ff 2 δ ί η Α ( μ + Λ) 2μ
A = arcsin(sin(
4.
+ Ä, )cos(—)
μ2+(μ+Κ)2_ρ2
fä
(P/2)2
2μ
(14) ^)/8ΐη(—)) 2μ
Ä=0
D sin A) I P ) Az = arcsin(/i arcsin(sin(—) x 2μ
Ό-2μ arccos(
Ό = 2μ arccos(l +
ΚΗί)
SIMULATION AND ANALYSIS
4.1 Simulation and Conclusion of Planar Measurement Model Assume there exists a single target with a constant velocity in 2-dimentional plane, and its initial state vector in polar coordinates is X=[l 100km, 0.15 km/s, 0.104 72 rad, 8.726 65e-05 rad/s] [2] . Assume that there are only two ionospheres, E and F. In this case, there are only four propagation modes: EE, EF, FE, and FF. With each mode, transform target state vector in ground coordinates to measurement vector in measurements coordinates according to Eq.(l), then transform measurement vector in measurements coordinates back to target state vector in ground coordinates according to Eq.(2). Subsequently consider the error rate between outcome and original target state and see whether the error rate is within an acceptable error range. Figures 4-6 show transforming error rates of slant range, Doppler and apparent azimuth respectively. According to these figures, it can be seen that the maximal transforming error rates of slant range, Doppler and apparent azimuth are all less than 10"14. By calculating, we know that the three kinds of magnitudes of average transforming error rates are all 10"16. The three kinds of transforming error rates are
Ρ2=Ρ-Ρ, Assume that there are only two ionospheres, say, E and F. In this case, there are only four propagation modes: EE, EF, FE, and FF. In above case, it is supposed hr^hh namely under mode EF and FE. Under the mode EE and FF, there is hr=hf=K so P{=P2=P/2. Then Eq. (13) can be simplified to Eq.(14). Expression (13) is not given in Ref. [3], however expression (14) is given in Ref. [3], say, the simplified expression under EE and FF modes. Fig.4
Error rate of slant range
729
Coordinate registration algorithms for over-the-horizon radar
Fig. 6 Error rate of apparent azimuth
all very little. All these prove that the coordinate registration expressions of planar measurement model, i.e. Eqs. (1) and (2) are right. 4.2
Simulation and Conclusion of Spherical Measurement Model
The initial state of target and system parameters are set same as the scenario as Section 4.1. With each mode of EE, EF, FE and FF, transform target state vector in ground coordinates to measurement vector in measurements coordinates according to Eq.(12), then transform measurement vector in measurements coordinates back to target state vector in ground coordinates according to Eq. (13). Subsequently consider the error rate between outcome and original target state and see whether the error rates is within an acceptable error range. Figures 7-9 show error rates of slant range, Doppler and apparent azimuth respectiv ely. Adopted the expression of azimuth in Ref. [3], we get the error rate of apparent azimuth as Fig. 10. There is almost no significant difference between Fig.9 and Fig. 10. Further, the quantitative analysis is shown in Table 1.
Fig. 10 Error rate of apparent azimuth according to Ref. [3]
730
Kong Min, Wang Guohong & Wang Yongcheng Table 1 Quantitative analysis of transmitting error rate /% Variable
P
f, A Λ
ζ
(By this paper)
Ά
Δζ
(By Ref.[3J)
Error kind
EE
EF
FE
FF
Maximal error
3.475e-012
3.227 2e-012
3.893 5e-012
3.893 5e-012
Mean error
-4.890 7e-014
-7.243 5e-014
-1.664 8e-014
1.438e-014
MSE error
1.108 8e-012
1.014 5e-012
9.865 8e-013
1.113e-012
Maximal error
3.562 5e-012
3.458 2e-012
3.052 4e-012
3.429 7e-012
Mean error
4.991 2e-014
7.403 2e-014
1.901e-014
-1.399 9e-014
MSE error
1.105 8e-012
1.026 4e-012
9.685 6e-013
1.110 3e-012
Maximal error
4.490 4e-012
3.523 le-012
3.425 9e-012
4.604 5e-012
Mean error
5.786 9e-014
3.316 2e-014
7.584 7e-014
-1.781 3e-014
MSE error
1.275e-012
1.086 8e-012
1.008 6e-012
1.287 8e-012
Maximal error
4.490 4e-012
3.523 le-012
3.470 5e-012
4.617 5e-012
Mean error
5.936 4e-014
3.475e-014
7.598 3e-014
-1.609 7e-014
MSE error
1.274 9e-012
1.089 le-012
1.006 9e-012
1.286 4e-012
According to these figures and Table 1, we know that the maximal transforming error rates of Slant Range, Doppler and apparent azimuth are all less than 10"12. By calculating, it can be seen that the three kinds of magnitudes of average transforming error rates are all 10"14. All these prove that the coordinate registration expressions of spherical measurement model, i.e. Eqs. (12) and (13) are right. The transforming expression of apparent azimuth in Ref. [3] and proposed in this paper and are both satisfying with the requirement. Under EE, EF and FE modes, the error rate of apparent azimuth according to this paper is a little lower than the error rate according to Ref. [3]. A new transforming expression of apparent azimuth is given. As a result of considering the curvature of earth, the transforming error rate of spherical measurement model is lager than that of planar measurement model; however it is also within an acceptable error range. 5. CONCLUSIONS
In this paper, the coordinate registration algorithms of planar measurement model and of spherical measurement model are respectively derived and demonstrated in detail, and simulations are designed for the multi path propagation phenomenon of OTHR. The simulations prove the correctness of coordinate regist-
ration algorithms. At the same time, the transforming error rates of the two kinds of measurement models are given respectively. As a result of simplified hypothesis, the transforming error rate is lower in the case of planar measurement model than in the case of spherical measurement model. However they are all within an acceptable error range. So, it can be drawn a conclusion that the coordinate registrations of two models are both correct and feasible. Further, a new transforming expression of apparent azimuth is given. Via simulation, it is testified to be correct. REFERENCES [1] Bogner R E , Bouzerdoum A, Pope K J,et al. Association of tracks from over the horizon radar. IEEE AES Systems Magazine, 1998:31-35. [2] Pulford G W, Evans R J. A multipath data association tracker for over-the-horizon radar. IEEE Trans, on Aerospace and Electronic Syste/ws, 1998,34(4): 1165-1182. [3] Percival D J, White K A B. Multihypothesis fusion of multipath over-the-orizon radar tracks. Proc. of
SPIE
3373, Orlando, FL, 1998;440~451.
Kong Min was born in 1975. She is a Ph. D. candidat in Naval Aeronautical Engineering Instiute. Her research interest is radar data processing. E-mail:[email protected]