# Estimating Target Heights Based on the Earth Curvature Model and Micromultipath Effect in Skywave OTH Radar

Authors: Hou Chengyu; Wang Yuxin; Chen Jiawei
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

Skywave over-the-horizon (OTH) radar systems have important long-range strategic warning values. They exploit skywave propagation reflection of high frequency signals from the ionosphere, which provides the ultra-long-range surveillance capabilities to detect and track maneuvering targets. Current OTH radar systems are capable of localizing targets in range and azimuth but are unable to achieve reliable instantaneous altitude estimation. Most existing height measurement methods of skywave OTH radar systems have taken advantage of the micromultipath effect and been considered in the flat earth model. However, the flat earth model is not proper since large error is inevitable, when the detection range is over one thousand kilometers. In order to avoid the error caused by the flat earth model, in this paper, an earth curvature model is introduced into OTH radar altimetry methods. The simulation results show that application of the earth curvature model can effectively reduce the estimation error.


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## Body

## 1. Introduction

Skywave OTH radar works in the high frequency band (3–30 MHz), which uses the ionosphere to scatter the electromagnetic wave, observing the air and ground targets from top to bottom. Thus, skywave OTH radar has important long-range strategic warning value and a wide range of applications [1].Skywave OTH radar systems are capable of localizing targets in range and azimuth but are unable to achieve reliable altitude estimation [2]. The altitude parameter is an important parameter for target recognition and tracking. Most existing altitude-finding theories for skywave OTH radar systems are based on the micromultipath effect. This effect refers to the fact that the radar signals have at least four kinds of paths through the relatively stable ionosphere and ground or sea surface reflection, when transmitted from the transmitting antenna and scattered by targets back to receiving antenna [3].Currently, the methods that skywave OTH radar systems used to estimate the instantaneous altitude of the aircrafts can be divided into four kinds. (1) The superresolution method: this method uses modern spectral analysis method to distinguish echo delay of each propagation path and then estimates the target altitude. By this method, the height estimation is divided into three types: low altitude, intermediate altitude, and high altitude. But this technique can only be applied to analog data [3]. (2) The state-space multipath fading model: after a series of delays, the micromultipath echoes interfered by each other will cause distance-Doppler peak attenuation. Then, using of the Fourier analysis to the amplitude attenuation echoes, we can obtain altitude information [4]. (3) The high-resolution ionospheric sounder: through the experiment data by the high-resolution ionosphere sounder, the relation between the micromultipath effect and the target altitude is studied [5]. (4) The matched-field processing [6, 7]: the echo signal is used to rematch delay-Doppler space. And then the maximum likelihood estimation (MLE) is adopted to estimate the target altitude [8].The methods mentioned above mostly consider the radar altimetry problem in the flat earth model. However, for skywave OTH radar systems, the detection range is thousands of kilometers, it will cause large errors to estimate target altitude with the flat earth model. Therefore, this paper will focus on how to introduce the earth curvature model into the altimetry method.


## 2. Signal Model and Maneuvering Target Model

### 2.1. Establishing Echo Signal Model

In this paper, a monostatic multiple-input multiple-output (MIMO) radar system is considered, which consists ofNt closely spaced transmit antennas and Nr closely spaced receive antennas. Denote S∈CNt×T as the orthogonal narrowband waveform matrix which is transmitted from the Nt transmit antennas over the pulse repetition period T. Assume that the orthogonal waveform is completed in the coherent integral time. si is the ith row of matrix S. When i≠m, si is orthogonal to sm; then SSH=INt. So, for the kth pulse, the signal received at the mth receive antenna is expressed as [8, 9]
(1)um,k=∑i=1I∑n=1Ntρi,ksnejϕi,m,k[A]ejϕi,n,k[D]+wm,k,
where I=4 is the number of the multipath components and ρi,k is the complex response of the ith multipath that considers the target reflection and propagation delays. The superscripts [D] and [A] denote the departure and the arrival modes, respectively. dm[A] is the location of receive antennas. ϕi,m,k[A] is determined by the mth receive antenna and is expressed as
(2)ϕi,m,k[A]=(2πλ)dm[A]sin(θi,k[A]),
where θi,k[A] is the elevation angle of arrival of the ith path. Similarly, ϕi,n,k[D] is determined by the location of the nth transmit antenna dn[D], and ϕi,n,k[D] is expressed as
(3)ϕi,n,k[D]=(2πλ)dn[D]sin(θi,k[D]),
where θi,k[D] is the elevation angle of departure of the ith path. For simplicity, we assume that the clutter has been filtered. Thus wm,k is the additive noise, which is independent and identically distributed complex Gaussian random variable and independent of the target echo signal.So thekth pulse echo, which is transmitted by the mth transmit antenna and received by the nth receive antenna, can be expressed as
(4)rm,n,k=∑i=1Iρi,kejϕi,m,k[A]ejϕi,n,k[D]+wm,n,k.Denoteφi,k=[e(jϕi,1,k[A]+jϕi,1,k[D]),…,e(jϕi,m,k[A]+jϕi,n,k[D])]T∈CN×1 as the signal steering vector of the virtual array corresponding to the ith path at the kth pulse, where N=NtNr. Denote Ak=[φ1,k,…,φI,k], ρk=[ρ1,k,…,ρI,k]T, and Wk=[w1,1,k,…,wNt,Nr,k]T, so the measurement radar data vector can be expressed as
(5)rk=[r1,1,k,…,rNt,Nt,k]T=Akρk+Wk.In order to be consistent with the motion model, thekth pulse of the measurement data is expressed with the time t corresponding to the kth pulse.


### 2.2. Establish Maneuvering Target Model

In this paper, for generality, we consider a maneuvering aircraft which makes a 180° circular turn inT=30 s [10]. The motion parameters are listed in Table 1. All the multipath signals are considered to fall within the same range cell. The ground distance x(t) and flight altitude of the target z(t), respectively, are expressed as
(6)x(t)=x(0)-vx(0)Tπsin(tπT),z(t)=z(0)+vz(0)Tπ(1-cos⁡(tπT)).Table 1
The parameters of the motion target.
Parameter
Notation
Value
Initial ground range
x
0
2000 km
Ionosphere height
H
350 km
Aircraft initial height
z
0
10 km
Maximum horizontal velocity
v
x
(
0
)
500 km/h
Maximum descending velocity
v
z
max
⁡
90 km/h
Carrier frequency
f
C
20 MHz
Impulse repetition frequency
f
S
50 HzThe target tracks in two dimensions are shown in Figures1 and 2.Figure 1
The target’s track in the range direction.Figure 2
The target’s track in the altitude direction.


## 2.1. Establishing Echo Signal Model

In this paper, a monostatic multiple-input multiple-output (MIMO) radar system is considered, which consists ofNt closely spaced transmit antennas and Nr closely spaced receive antennas. Denote S∈CNt×T as the orthogonal narrowband waveform matrix which is transmitted from the Nt transmit antennas over the pulse repetition period T. Assume that the orthogonal waveform is completed in the coherent integral time. si is the ith row of matrix S. When i≠m, si is orthogonal to sm; then SSH=INt. So, for the kth pulse, the signal received at the mth receive antenna is expressed as [8, 9]
(1)um,k=∑i=1I∑n=1Ntρi,ksnejϕi,m,k[A]ejϕi,n,k[D]+wm,k,
where I=4 is the number of the multipath components and ρi,k is the complex response of the ith multipath that considers the target reflection and propagation delays. The superscripts [D] and [A] denote the departure and the arrival modes, respectively. dm[A] is the location of receive antennas. ϕi,m,k[A] is determined by the mth receive antenna and is expressed as
(2)ϕi,m,k[A]=(2πλ)dm[A]sin(θi,k[A]),
where θi,k[A] is the elevation angle of arrival of the ith path. Similarly, ϕi,n,k[D] is determined by the location of the nth transmit antenna dn[D], and ϕi,n,k[D] is expressed as
(3)ϕi,n,k[D]=(2πλ)dn[D]sin(θi,k[D]),
where θi,k[D] is the elevation angle of departure of the ith path. For simplicity, we assume that the clutter has been filtered. Thus wm,k is the additive noise, which is independent and identically distributed complex Gaussian random variable and independent of the target echo signal.So thekth pulse echo, which is transmitted by the mth transmit antenna and received by the nth receive antenna, can be expressed as
(4)rm,n,k=∑i=1Iρi,kejϕi,m,k[A]ejϕi,n,k[D]+wm,n,k.Denoteφi,k=[e(jϕi,1,k[A]+jϕi,1,k[D]),…,e(jϕi,m,k[A]+jϕi,n,k[D])]T∈CN×1 as the signal steering vector of the virtual array corresponding to the ith path at the kth pulse, where N=NtNr. Denote Ak=[φ1,k,…,φI,k], ρk=[ρ1,k,…,ρI,k]T, and Wk=[w1,1,k,…,wNt,Nr,k]T, so the measurement radar data vector can be expressed as
(5)rk=[r1,1,k,…,rNt,Nt,k]T=Akρk+Wk.In order to be consistent with the motion model, thekth pulse of the measurement data is expressed with the time t corresponding to the kth pulse.


## 2.2. Establish Maneuvering Target Model

In this paper, for generality, we consider a maneuvering aircraft which makes a 180° circular turn inT=30 s [10]. The motion parameters are listed in Table 1. All the multipath signals are considered to fall within the same range cell. The ground distance x(t) and flight altitude of the target z(t), respectively, are expressed as
(6)x(t)=x(0)-vx(0)Tπsin(tπT),z(t)=z(0)+vz(0)Tπ(1-cos⁡(tπT)).Table 1
The parameters of the motion target.
Parameter
Notation
Value
Initial ground range
x
0
2000 km
Ionosphere height
H
350 km
Aircraft initial height
z
0
10 km
Maximum horizontal velocity
v
x
(
0
)
500 km/h
Maximum descending velocity
v
z
max
⁡
90 km/h
Carrier frequency
f
C
20 MHz
Impulse repetition frequency
f
S
50 HzThe target tracks in two dimensions are shown in Figures1 and 2.Figure 1
The target’s track in the range direction.Figure 2
The target’s track in the altitude direction.


## 3. Analysis of Micromultipath Effect Based on the Flat Earth Model

### 3.1. Analysis of Micromultipath Effect

An OTH radar system considering the micromultipath effect based on the flat earth model is illustrated in Figure3.Figure 3
The micromultipath effect in the flat earth model.Since our purpose in this paper is to estimate the instantaneous height of the target, for simplicity and without loss of generality, we only consider the 2D position and velocity (range directionx and altitude direction z). Then the target state at the time t is expressed as
(7)x~(t)=[x(t)z(t)vx(t)vz(t)]T,
where vx(t) and vz(t) are the velocity of the range direction and the altitude direction, respectively.As the effect of the cross-range array apertures on the height estimation is small, only the effect of array apertures which lies in the range direction will be considered. Correspondingly both the transmit arrays and receive arrays are considered to be linear and located in thex-axis.As the micromultipath effect exists, the combination of the transmit path and receive path has four components: Path I (l1:l1), Path II (l2:l2), Path III (l1:l2), and Path IV (l2:l1). Path III and Path IV are with the same length [6]. The slant range l1(t) and l2(t) is expressed with the distance x(t), the height of the ionosphere H, and the target height z(t) as
(8)l1(t)=x(t)2+(2H-z(t))2,l2(t)=x(t)2+(2H+z(t))2.
As x≫H≫z, l1(t) and l2(t) are simplified as
(9)l1(t)≈x(t)+2H2-2Hz(t)x(t),l2(t)≈x(t)+2H2+2Hz(t)x(t).
Then L1(t) and L2(t), the length of Path I and Path II, are expressed as
(10)L1(t)=2l1(t)≈2x(t)+4H2-4Hz(t)x(t),L2(t)=2l2(t)≈2x(t)+4H2+4Hz(t)x(t).L
3
(
t
) and L4(t), the length of Path III and Path IV, are written as
(11)L3(t)=L4(t)=l1(t)+l2(t)≈2x(t)+4H2x(t).


### 3.2. Instantaneous Target Altitude Estimation

In (4), ρi(t) comprises the targets’ Doppler information. Here we express ρi(t) as
(12)ρi(t)=σi(t)e-j2πLi(t)/λ,
where i=1,…,4. i is the index of the path. λ=c/fc is the wavelength corresponding to carrier frequency fc, and σi(t) represents the combined effect of transmit power, target reflection, which is a function of radar cross-section (RCS), and ionospheric effect. Li(t) is the two-way slant range, and the expressions of Li(t) are shown in the previous section.In general, the target’s track consists of the movement in the range and altitude direction. This section will discuss the Doppler frequency of the target flight in two directions and then deduce the instantaneous expression of the target height.Suppose that both the range between the target and radar and the height of the target are the functions of time. As the height of the ionosphereH is changing slowly, we assume that H is a constant value in the coherent time. So we can get
(13)dl1(t)dt≈J(t)vx(t)-2Hx(t)vz(t),dl2(t)dt≈J(t)vx(t)+2Hx(t)vz(t),
where J(t)=(1-2H2/x2(t)), vx(t)=dx(t)/dt is the target’s velocity of the range direction, and vz(t)=dz(t)/dt is the target’s velocity of the altitude direction. So the Doppler frequencies of the four paths are
(14)f1(t)=2fccdl1(t)dt≈2fccJ(t)vx(t)-4fcHx(t)cvz(t),(15)f2(t)=2fccdl2(t)dt≈2fccJ(t)vx(t)+4fcHx(t)cvz(t),(16)f3(t)=f4(t)=fccdl1(t)+dl2(t)dt≈2fccJ(t)vx(t).From (14)–(16), we find that the Doppler frequencies of the four paths have the same part 2fcJ(t)vx(t)/c, which is determined by vx(t). We call this the dominant Doppler component. c is the speed of light. f1 and f2 also have the part caused by vz(t). We call this the minor Doppler component, which contains the important information of the target movement in the altitude direction. Let f2(t)-f1(t); then vz(t) is obtained by the expression
(17)vz(t)=[f2(t)-f1(t)]x(t)c8fcH.To get the target height, it is necessary to know the target initial height. In practice, we can getL1(t), L2(t) and L3(t) by the time delay, and the transmit angle α and the ionospheric height H can also be known by the other measurement equipment. Then these parameters can be used to calculate the initial height z0. According to (8), we can get two estimated values z0 by
(18)z0=2H-(L1(0)2)2-x02,z0=(L2(0)2)2+x02-2H.However,z0 gotten by (18) is quite different from the actual value. It is because z0 is very small compared to x(t). If x(t) is with a small error, there will be a great impact on z0.For elimination of the impact by the error ofx(t), we can use the following to calculate z(t):
(19)x(t)2+[2H-z(t)]2=L12,x(t)2+[2H+z(t)]2=L22.z(t) is obtained by (19) as
(20)z(t)=L2(t)2-L1(t)28H.
Let t=0, z0 be gotten. Then z(t) can be written as
(21)z(t)=L2(0)2-L1(0)28H+[f2(t)-f1(t)]x(t)c8fcH×t.


## 3.1. Analysis of Micromultipath Effect

An OTH radar system considering the micromultipath effect based on the flat earth model is illustrated in Figure3.Figure 3
The micromultipath effect in the flat earth model.Since our purpose in this paper is to estimate the instantaneous height of the target, for simplicity and without loss of generality, we only consider the 2D position and velocity (range directionx and altitude direction z). Then the target state at the time t is expressed as
(7)x~(t)=[x(t)z(t)vx(t)vz(t)]T,
where vx(t) and vz(t) are the velocity of the range direction and the altitude direction, respectively.As the effect of the cross-range array apertures on the height estimation is small, only the effect of array apertures which lies in the range direction will be considered. Correspondingly both the transmit arrays and receive arrays are considered to be linear and located in thex-axis.As the micromultipath effect exists, the combination of the transmit path and receive path has four components: Path I (l1:l1), Path II (l2:l2), Path III (l1:l2), and Path IV (l2:l1). Path III and Path IV are with the same length [6]. The slant range l1(t) and l2(t) is expressed with the distance x(t), the height of the ionosphere H, and the target height z(t) as
(8)l1(t)=x(t)2+(2H-z(t))2,l2(t)=x(t)2+(2H+z(t))2.
As x≫H≫z, l1(t) and l2(t) are simplified as
(9)l1(t)≈x(t)+2H2-2Hz(t)x(t),l2(t)≈x(t)+2H2+2Hz(t)x(t).
Then L1(t) and L2(t), the length of Path I and Path II, are expressed as
(10)L1(t)=2l1(t)≈2x(t)+4H2-4Hz(t)x(t),L2(t)=2l2(t)≈2x(t)+4H2+4Hz(t)x(t).L
3
(
t
) and L4(t), the length of Path III and Path IV, are written as
(11)L3(t)=L4(t)=l1(t)+l2(t)≈2x(t)+4H2x(t).


## 3.2. Instantaneous Target Altitude Estimation

In (4), ρi(t) comprises the targets’ Doppler information. Here we express ρi(t) as
(12)ρi(t)=σi(t)e-j2πLi(t)/λ,
where i=1,…,4. i is the index of the path. λ=c/fc is the wavelength corresponding to carrier frequency fc, and σi(t) represents the combined effect of transmit power, target reflection, which is a function of radar cross-section (RCS), and ionospheric effect. Li(t) is the two-way slant range, and the expressions of Li(t) are shown in the previous section.In general, the target’s track consists of the movement in the range and altitude direction. This section will discuss the Doppler frequency of the target flight in two directions and then deduce the instantaneous expression of the target height.Suppose that both the range between the target and radar and the height of the target are the functions of time. As the height of the ionosphereH is changing slowly, we assume that H is a constant value in the coherent time. So we can get
(13)dl1(t)dt≈J(t)vx(t)-2Hx(t)vz(t),dl2(t)dt≈J(t)vx(t)+2Hx(t)vz(t),
where J(t)=(1-2H2/x2(t)), vx(t)=dx(t)/dt is the target’s velocity of the range direction, and vz(t)=dz(t)/dt is the target’s velocity of the altitude direction. So the Doppler frequencies of the four paths are
(14)f1(t)=2fccdl1(t)dt≈2fccJ(t)vx(t)-4fcHx(t)cvz(t),(15)f2(t)=2fccdl2(t)dt≈2fccJ(t)vx(t)+4fcHx(t)cvz(t),(16)f3(t)=f4(t)=fccdl1(t)+dl2(t)dt≈2fccJ(t)vx(t).From (14)–(16), we find that the Doppler frequencies of the four paths have the same part 2fcJ(t)vx(t)/c, which is determined by vx(t). We call this the dominant Doppler component. c is the speed of light. f1 and f2 also have the part caused by vz(t). We call this the minor Doppler component, which contains the important information of the target movement in the altitude direction. Let f2(t)-f1(t); then vz(t) is obtained by the expression
(17)vz(t)=[f2(t)-f1(t)]x(t)c8fcH.To get the target height, it is necessary to know the target initial height. In practice, we can getL1(t), L2(t) and L3(t) by the time delay, and the transmit angle α and the ionospheric height H can also be known by the other measurement equipment. Then these parameters can be used to calculate the initial height z0. According to (8), we can get two estimated values z0 by
(18)z0=2H-(L1(0)2)2-x02,z0=(L2(0)2)2+x02-2H.However,z0 gotten by (18) is quite different from the actual value. It is because z0 is very small compared to x(t). If x(t) is with a small error, there will be a great impact on z0.For elimination of the impact by the error ofx(t), we can use the following to calculate z(t):
(19)x(t)2+[2H-z(t)]2=L12,x(t)2+[2H+z(t)]2=L22.z(t) is obtained by (19) as
(20)z(t)=L2(t)2-L1(t)28H.
Let t=0, z0 be gotten. Then z(t) can be written as
(21)z(t)=L2(0)2-L1(0)28H+[f2(t)-f1(t)]x(t)c8fcH×t.


## 4. Analysis of Micromultipath Effect Based on Earth Curvature Model

### 4.1. Analysis of Micromultipath Effect

In this section, the micromultipath effect model, which is under the influence of the earth curvature, will be discussed. The micromultipath effect model in an OTH radar system of the earth curvature model is illustrated in Figure4. x(t) is expressed as the ground range between the target and the radar. In order to be consistent with the definition in the flat earth model, z(t) indicates the vertical height, H is the height of the ionosphere, and R is the earth radius (it is set as a constant value in the paper; R=6372km). Same as the flat earth model, the radar signal propagation paths in the earth curvature model have four combinations: Path I (l1:l1), Path II (l2:l2), Path III (l1:l2), and Path IV (l2:l1).Figure 4
The micromultipath effect in the earth curvature model.Path I is shown in Figure5, in which α1 indicates the departure angle of radar transmitting antenna, P1 is the distance of electromagnetic signal from radar transmitting antenna to ionosphere, m1 is the extended line, and β1, γ1 indicate the auxiliary angles.Figure 5
Path I in the earth curvature model.According to the geometrical relationship,α1 meets the following condition:
(22)x(t)R+α1+2arcsin[Rcos⁡(α1)R+H]-arcsin[RR+zcos⁡⁡(α1)]=π2.The accurate value ofα1 can be calculated by (22). The auxiliary angles β1 and γ1 can be expressed as the function of α1:
(23)β1=arcsin[Rcos⁡(α1)R+H],γ1=arcsin[Rcos⁡(α1)R+z(t)].P1 and m1 can be, respectively, expressed as
(24)P1=Rcos⁡(α1+β1)sinβ1,m1=P1+H+Rsinγcos⁡[x(t)R+α1+β]
Then l1(t) is written as
(25)l1(t)=2P1-m1=Rcos⁡(α1+β1)sinβ1IIIIIIIIII-H+Rsinγ1cos⁡[x(t)R+α1+β1].Path II is illustrated in Figure6. Similarly, α2 indicates the departure angle of radar transmitting antenna, P2 is the distance of electromagnetic signal from radar transmitting antenna to ionosphere, m2 is the extended line, and β2 and γ2 indicate the auxiliary angles.Figure 6
Path II in the earth curvature model.From Figure6, α2 should meet the following conditions:
(26)x(t)R+arcsin(RR+zcos⁡α2)+2arcsin(RR+Hcos⁡α2)+3α2=3π2.Thenα2 can be calculated by (26). The auxiliary angles β2 and γ2 can be expressed as the function of α2:
(27)β2=arcsin(Rcos⁡α2R+H),γ2=arcsin[Rcos⁡α2R+z(t)].P2 and m2 can be, respectively, expressed as
(28)P2=Rcos⁡(α2+β2)sinβ2,m2=cos⁡(α2+γ2)cos⁡α2[R+z(t)].
Then the expression of transmission oblique diameter l2(t) is as follows:
(29)l2(t)=2P2+m2=2Rcos⁡(α2+β2)sinβ2IIIIIIIIII+cos⁡(α2+γ2)cos⁡α2[R+z(t)].
So L1(t) and L2(t), the length of Path I and Path II, are
(30)L1(t)=2l1(t)=2Rcos⁡(α1+β1)sinβ1IIIIIIIIIII-2H+Rsinγ1cos⁡[x(t)R+α1+β1],L2(t)=2l2(t)=4Rcos⁡(α2+β2)sinβ2IIIIIIIIIII+2cos⁡(α2+β2)cos⁡α2[R+z(t)].
And L3(t) and L4(t), the length of Path III and Path IV, are shown as
(31)L3(t)=L4(t)=l1(t)+l2(t)IIIIIII=Rcos⁡(α1+β1)sinβ1-H+Rsinγ1cos⁡[x(t)R+α1+β1]IIIIIIIIIII+2Rcos⁡(α2+β2)sinβ2+cos⁡(α2+γ2)cos⁡α2[R+z(t)].


### 4.2. Instantaneous Target Altitude Estimation

Consistent with the processing methods in the flat earth model, we suppose thatH is constant in the coherent time. Then the time derivatives of the l1(t), l2(t), l3(t), and l4(t) are obtained by
(32)dl1(t)dt=sin[α1+β1+x(t)/R][R+z(t)](H+R)R2cos⁡α1vx(t)iiiiiiiiiiiiii-cos⁡[α2/2+β1+x(t)/R](H+R)Rcos⁡α1vz(t)iiiiiiiiii=l1xvx(t)+l1zvz(t),dl2(t)dt=((cos⁡[α22-π4+γ2+x(t)2R](H+R))iiiiiiiiiiiii×(Rcos⁡α2)-1[α22-π4+γ2+x(t)2R])vx(t)iiiiiiiiiii+{×1-[Rcos⁡α2R+z(t)]2)-1cos⁡(α2+γ2)cos⁡α2+Rsin(α2+γ2)[R+z(t)]cos⁡γ2iiiiiiiiiiiiiiiii-(Rcos⁡[α22-π4+γ2+x(t)2R](H+R))iiiiiiiiiiiiiiiii×(1-[Rcos⁡α2R+z(t)]2[R+z(t)]2)-1}vz(t)dl2(t)dt=l2xvx(t)+l2zvz(t),dl3(t)dt=dl4(t)dt=12[dl1(t)dt+dl2(t)dt].l1x is the coefficient of the part of dl1(t)/dt caused by vx(t). l1z is the coefficient of the part of dl1(t)/dt caused by vz(t). The expressions of l2x and l2z are similar to l1x and l1z, but replace l1 with l2. So the Doppler frequencies produced by the moving target of the four paths can be, respectively, written as
(33)f1(t)=2fccdl1(t)dt,f2(t)=2fccdl2(t)dt,f3(t)=f4(t)=12[f1(t)+f2(t)].The four Doppler frequencies both consist of the components caused byvx(t) and vz(t). The ratio of l1x to l2x under the condition that x(t) is made to change from 1000 km to 3000 km is shown in Figure 7. From the figure, it can be found that the values of Doppler frequency components caused by vx(t) in f1(t) and f2(t) are very close. And it indicates that the Doppler frequency component in each path caused by vx(t) is approximately equal. This conclusion is same as the one in the flat earth model. Therefore, vz(t) can be obtained by f2(t)-f1(t) as
(34)vz(t)=c2fcf2(t)-f1(t)l2z-l1z.Figure 7
The ration of 11x to 12x.To get the instantaneous height of the moving target, the initial height of the targetz0 is needed. Different from the situation in the flat earth model, z0 can be calculated accurately by using L1(t) or L2(t). In order to reduce random errors, L1(t) and L2(t) are used to calculate the initial heights z01, z02 and then make the average of z01, z02 as z0.For Path I,z01 can be shown as follows:
(35)z01=[L1(0)2-sinθ1sinβ1]sinβ1sinθ2-R,
where
(36)β1=arcsin(Rcos⁡α1H+R),θ2=x(t)R-π2+α1+β1.For Path II,z02 can be written as
(37)z02=Rcos⁡α2sinγ0-R,
where
(38)γ2=3π2-x(t)R-2α2-2β2,β2=arcsin(RR+H)×cos⁡α2.
Then z0 is the average value of z01 and z02:
(39)z0=([L1(0)2-sinθ1sinβ1]sinβ1sinθ2+Rcos⁡α2sinγ0-2R)iiiiii×(2)-1.
So the instantaneous height z(t) can be expressed as
(40)z(t)=z0+vz(t)×t.


## 4.1. Analysis of Micromultipath Effect

In this section, the micromultipath effect model, which is under the influence of the earth curvature, will be discussed. The micromultipath effect model in an OTH radar system of the earth curvature model is illustrated in Figure4. x(t) is expressed as the ground range between the target and the radar. In order to be consistent with the definition in the flat earth model, z(t) indicates the vertical height, H is the height of the ionosphere, and R is the earth radius (it is set as a constant value in the paper; R=6372km). Same as the flat earth model, the radar signal propagation paths in the earth curvature model have four combinations: Path I (l1:l1), Path II (l2:l2), Path III (l1:l2), and Path IV (l2:l1).Figure 4
The micromultipath effect in the earth curvature model.Path I is shown in Figure5, in which α1 indicates the departure angle of radar transmitting antenna, P1 is the distance of electromagnetic signal from radar transmitting antenna to ionosphere, m1 is the extended line, and β1, γ1 indicate the auxiliary angles.Figure 5
Path I in the earth curvature model.According to the geometrical relationship,α1 meets the following condition:
(22)x(t)R+α1+2arcsin[Rcos⁡(α1)R+H]-arcsin[RR+zcos⁡⁡(α1)]=π2.The accurate value ofα1 can be calculated by (22). The auxiliary angles β1 and γ1 can be expressed as the function of α1:
(23)β1=arcsin[Rcos⁡(α1)R+H],γ1=arcsin[Rcos⁡(α1)R+z(t)].P1 and m1 can be, respectively, expressed as
(24)P1=Rcos⁡(α1+β1)sinβ1,m1=P1+H+Rsinγcos⁡[x(t)R+α1+β]
Then l1(t) is written as
(25)l1(t)=2P1-m1=Rcos⁡(α1+β1)sinβ1IIIIIIIIII-H+Rsinγ1cos⁡[x(t)R+α1+β1].Path II is illustrated in Figure6. Similarly, α2 indicates the departure angle of radar transmitting antenna, P2 is the distance of electromagnetic signal from radar transmitting antenna to ionosphere, m2 is the extended line, and β2 and γ2 indicate the auxiliary angles.Figure 6
Path II in the earth curvature model.From Figure6, α2 should meet the following conditions:
(26)x(t)R+arcsin(RR+zcos⁡α2)+2arcsin(RR+Hcos⁡α2)+3α2=3π2.Thenα2 can be calculated by (26). The auxiliary angles β2 and γ2 can be expressed as the function of α2:
(27)β2=arcsin(Rcos⁡α2R+H),γ2=arcsin[Rcos⁡α2R+z(t)].P2 and m2 can be, respectively, expressed as
(28)P2=Rcos⁡(α2+β2)sinβ2,m2=cos⁡(α2+γ2)cos⁡α2[R+z(t)].
Then the expression of transmission oblique diameter l2(t) is as follows:
(29)l2(t)=2P2+m2=2Rcos⁡(α2+β2)sinβ2IIIIIIIIII+cos⁡(α2+γ2)cos⁡α2[R+z(t)].
So L1(t) and L2(t), the length of Path I and Path II, are
(30)L1(t)=2l1(t)=2Rcos⁡(α1+β1)sinβ1IIIIIIIIIII-2H+Rsinγ1cos⁡[x(t)R+α1+β1],L2(t)=2l2(t)=4Rcos⁡(α2+β2)sinβ2IIIIIIIIIII+2cos⁡(α2+β2)cos⁡α2[R+z(t)].
And L3(t) and L4(t), the length of Path III and Path IV, are shown as
(31)L3(t)=L4(t)=l1(t)+l2(t)IIIIIII=Rcos⁡(α1+β1)sinβ1-H+Rsinγ1cos⁡[x(t)R+α1+β1]IIIIIIIIIII+2Rcos⁡(α2+β2)sinβ2+cos⁡(α2+γ2)cos⁡α2[R+z(t)].


## 4.2. Instantaneous Target Altitude Estimation

Consistent with the processing methods in the flat earth model, we suppose thatH is constant in the coherent time. Then the time derivatives of the l1(t), l2(t), l3(t), and l4(t) are obtained by
(32)dl1(t)dt=sin[α1+β1+x(t)/R][R+z(t)](H+R)R2cos⁡α1vx(t)iiiiiiiiiiiiii-cos⁡[α2/2+β1+x(t)/R](H+R)Rcos⁡α1vz(t)iiiiiiiiii=l1xvx(t)+l1zvz(t),dl2(t)dt=((cos⁡[α22-π4+γ2+x(t)2R](H+R))iiiiiiiiiiiii×(Rcos⁡α2)-1[α22-π4+γ2+x(t)2R])vx(t)iiiiiiiiiii+{×1-[Rcos⁡α2R+z(t)]2)-1cos⁡(α2+γ2)cos⁡α2+Rsin(α2+γ2)[R+z(t)]cos⁡γ2iiiiiiiiiiiiiiiii-(Rcos⁡[α22-π4+γ2+x(t)2R](H+R))iiiiiiiiiiiiiiiii×(1-[Rcos⁡α2R+z(t)]2[R+z(t)]2)-1}vz(t)dl2(t)dt=l2xvx(t)+l2zvz(t),dl3(t)dt=dl4(t)dt=12[dl1(t)dt+dl2(t)dt].l1x is the coefficient of the part of dl1(t)/dt caused by vx(t). l1z is the coefficient of the part of dl1(t)/dt caused by vz(t). The expressions of l2x and l2z are similar to l1x and l1z, but replace l1 with l2. So the Doppler frequencies produced by the moving target of the four paths can be, respectively, written as
(33)f1(t)=2fccdl1(t)dt,f2(t)=2fccdl2(t)dt,f3(t)=f4(t)=12[f1(t)+f2(t)].The four Doppler frequencies both consist of the components caused byvx(t) and vz(t). The ratio of l1x to l2x under the condition that x(t) is made to change from 1000 km to 3000 km is shown in Figure 7. From the figure, it can be found that the values of Doppler frequency components caused by vx(t) in f1(t) and f2(t) are very close. And it indicates that the Doppler frequency component in each path caused by vx(t) is approximately equal. This conclusion is same as the one in the flat earth model. Therefore, vz(t) can be obtained by f2(t)-f1(t) as
(34)vz(t)=c2fcf2(t)-f1(t)l2z-l1z.Figure 7
The ration of 11x to 12x.To get the instantaneous height of the moving target, the initial height of the targetz0 is needed. Different from the situation in the flat earth model, z0 can be calculated accurately by using L1(t) or L2(t). In order to reduce random errors, L1(t) and L2(t) are used to calculate the initial heights z01, z02 and then make the average of z01, z02 as z0.For Path I,z01 can be shown as follows:
(35)z01=[L1(0)2-sinθ1sinβ1]sinβ1sinθ2-R,
where
(36)β1=arcsin(Rcos⁡α1H+R),θ2=x(t)R-π2+α1+β1.For Path II,z02 can be written as
(37)z02=Rcos⁡α2sinγ0-R,
where
(38)γ2=3π2-x(t)R-2α2-2β2,β2=arcsin(RR+H)×cos⁡α2.
Then z0 is the average value of z01 and z02:
(39)z0=([L1(0)2-sinθ1sinβ1]sinβ1sinθ2+Rcos⁡α2sinγ0-2R)iiiiii×(2)-1.
So the instantaneous height z(t) can be expressed as
(40)z(t)=z0+vz(t)×t.


## 5. Comparison of the Difference of Micromultipath Effects between Two Kinds of Models

The aim of this section is to analyze and compare the difference of micromultipath effects between the flat earth model and the earth curvature model. The moving target model established in Section2.2 is used in this section.


### 5.1. The Difference in the Doppler Frequencies Produced by the Target’s Movement

The curves of Doppler frequencies of the four paths produced by the target’s movement in the flat earth model and the earth curvature model are shown in Figure8 and Figure 9, respectively. As can be seen from the figures, the target turns an angle of 180 degrees on the ground, so the value of Doppler frequency reduces from the maximum to the minimum. At the start and the end points of the movement period, for the altitude velocities are both zero, the Doppler frequency of each path is the same. For the absolute value of vx(t) is maximal, the absolute value of Doppler frequency comes to the maximum.Figure 8
The Doppler frequency curves of the moving target in the flat earth model.Figure 9
The Doppler frequency curves of the moving target in the earth curvature model.At the momentT/2, vx(t) comes to zero, and the Doppler frequencies of every path are close to zero, while vz(t) reaches its maximum. Therefore, the Doppler frequencies of every path come to its maximum difference value at this moment, which is the best moment to obtain the Doppler frequency differences produced by vz(t).By comparison of Figures8 and 9, the Doppler frequency differences of the every path in the earth curvature model are smaller than the corresponding results in the flat earth model. Figures 10 and 11 illustrate, respectively, the values of the Doppler frequency differences between Path I and Path II in the flat earth model and the earth curvature model within one movement period. As seen, the maximum value of Doppler frequency difference in the flat earth model is 2.4 Hz and the one in the earth curvature model only is 1.3 Hz. It indicates that the error of the Doppler frequency of the flat earth model is close to 1.1 Hz, which will bring larger error in the altitude estimation.Figure 10
The Doppler frequency difference between Path I and Path II in the flat earth model.Figure 11
The Doppler frequency difference between Path I and Path II in the earth curvature model.


### 5.2. The Difference in the Target Track

Figures12(a) and 12(b) illustrate Path I and Path II in two kinds of models, in which the solid line represents the propagation path in the earth curvature model and the dotted line represents the propagation path in the flat earth model. The superscript * denotes the reference planes in the flat earth model. From these figures, we cannot determine directly in which model the propagation path is longer. So the tracks within one movement period are shown in Figure 13. In this figure, the varied tendencies of the signal propagation path in two kinds of models are basically identical, the length of Path I in the earth curvature model is about 94 km longer than it is in the flat earth model, and the length of Path II in the earth curvature model is about 89 km longer than it is in the flat earth model.The Path I and Path II in two kinds of models.
(a)
Path I
(b)
Path IIThe simulation results of Path I and Path II in two kinds of models.
(a)
Path I in the flat earth model
(b)
Path II in the flat earth model
(c)
Path I in the earth curvature model
(d)
Path II in the earth curvature modelThe Doppler frequency differencefdiff, between Path I and Path II in the flat earth model, can be obtained by (14) and (15) as
(41)fdiff=8fcHx(t)cvz(t).From (41), fdiff is inversely proportional to x(t). For the signal propagation path in the earth curvature model is longer than the one in the flat earth model; x(t) in the earth curvature model is larger than the one in the flat earth model. This is why the Doppler frequency differences of each path in the earth curvature model are smaller than the one in the flat earth model.Figure14 illustrates the comparison of the measured values v^z(t) of the height velocity in two kinds of earth models with its actual value vz(t). It can be seen that v^z(t) calculated in the earth curvature model is identical to the actual value, and v^z(t) obtained in the flat earth model has great differences from the actual one. The error of v^z(t) comes to 12 m/s when vz(t) is with the maximum value, so the altitude estimation would have great bias if the flat earth model is adopted.Figure 14
The comparison ofvz calculated by two kinds of earth models.Figure15 shows the track in the altitude direction with two kinds of models. In the earth curvature model, v^z(t) has no error, so the varied tendency of the calculated track (the dotted line) coincides with the actual one (the real line). The initial height z0 calculated by (39) has a slight bias about 120 m. So the whole track has a fixed error with the true track. While v^z(t) and z0 calculated in the flat earth model both have a larger error, the varied tendency of the whole movement track and the altitude value at every moment have great difference from the real value.Figure 15
The comparison of the movement tracks calculated in two kinds of earth models.


### 5.3. The Application Condition of the Earth Curvature Model

#### 5.3.1. The Requirement of the Skywave Radar Altimetry to the Transmission Antenna Beam WidthΩ

In order to use the effect of micromultipath, the transmission beam is requested to have a certain width to receive the echoes of four paths. Therefore, the transmission antenna beam width Ω must be larger than two times of the differenceαdiff between the emission angle α1 and α2 corresponding to Path I and Path II. In Figure 16, the changing curves of the emission angle α1 (the real line) and α2 (the dotted line) corresponding to Path I and Path II are shown when the ground distance x(t) increases from 1000 km to 4000 km. Figure 17 illustrates the curve of the difference αdiff between the emission angles of two paths changing with x(t). We can find that the difference αdiff between the emission angles corresponding to Path I and Path II changes little (its biggest value is about 1.2°) within the detection range from 1000 km to 3000 km. When the detection range exceeds 3000 km, αdiff rises sharply, and it reaches about 3.7° when the detection range is 4000 km. Accordingly, when the skywave radar use micromultipath effect to measure the height, its beam width needs to be larger than 2.4° if the detection range is smaller 3000 km and the beam width of the transmission antenna has to be larger than 7.4° when the detection range exceeds 3000 km.Figure 16
The curve of the emission anglesα1 and α2 changing with the ground distance.Figure 17
The curve of the differenceαdiff changing with x(t).


#### 5.3.2. The Detection Range of the Earth Curvature Model

The detection range of the skywave radar is limited in practice, for the emission angleα can only change in a certain range. The maximum value and the minimum value of α are written, respectively, as αmax⁡ and αmin⁡.When the target’s altitude is close to the ground and the emission angle isαmax⁡, the sky wave radar can realize the detection of the minimum range, as shown in Figure 18. When the emission angle is αmin⁡, the maximum detection is obtained, as shown in Figure 19.Figure 18
The illustration of the minimum detection range of the skywave radar.Figure 19
The illustration of the maximum detection range of the skywave radar.The emission angleαmax⁡ is generally taken as 30°. Then the minimum detection range xmin⁡ can be derived as follows:
(42)xmin⁡=2πR3-2Rarcsin(3R2(R+H)).The emission angleαmin⁡ is always 0°; then the maximum detection range xmax⁡ can be derived as
(43)xmax⁡=πR-2Rarcsin(RR+H).Taking the earth radius asR=6372km and the ionosphere height as H=350km, then xmin⁡=1072km and xmax⁡=4131km. The results conform to the practical detection range of the skywave radar.


## 5.1. The Difference in the Doppler Frequencies Produced by the Target’s Movement

The curves of Doppler frequencies of the four paths produced by the target’s movement in the flat earth model and the earth curvature model are shown in Figure8 and Figure 9, respectively. As can be seen from the figures, the target turns an angle of 180 degrees on the ground, so the value of Doppler frequency reduces from the maximum to the minimum. At the start and the end points of the movement period, for the altitude velocities are both zero, the Doppler frequency of each path is the same. For the absolute value of vx(t) is maximal, the absolute value of Doppler frequency comes to the maximum.Figure 8
The Doppler frequency curves of the moving target in the flat earth model.Figure 9
The Doppler frequency curves of the moving target in the earth curvature model.At the momentT/2, vx(t) comes to zero, and the Doppler frequencies of every path are close to zero, while vz(t) reaches its maximum. Therefore, the Doppler frequencies of every path come to its maximum difference value at this moment, which is the best moment to obtain the Doppler frequency differences produced by vz(t).By comparison of Figures8 and 9, the Doppler frequency differences of the every path in the earth curvature model are smaller than the corresponding results in the flat earth model. Figures 10 and 11 illustrate, respectively, the values of the Doppler frequency differences between Path I and Path II in the flat earth model and the earth curvature model within one movement period. As seen, the maximum value of Doppler frequency difference in the flat earth model is 2.4 Hz and the one in the earth curvature model only is 1.3 Hz. It indicates that the error of the Doppler frequency of the flat earth model is close to 1.1 Hz, which will bring larger error in the altitude estimation.Figure 10
The Doppler frequency difference between Path I and Path II in the flat earth model.Figure 11
The Doppler frequency difference between Path I and Path II in the earth curvature model.


## 5.2. The Difference in the Target Track

Figures12(a) and 12(b) illustrate Path I and Path II in two kinds of models, in which the solid line represents the propagation path in the earth curvature model and the dotted line represents the propagation path in the flat earth model. The superscript * denotes the reference planes in the flat earth model. From these figures, we cannot determine directly in which model the propagation path is longer. So the tracks within one movement period are shown in Figure 13. In this figure, the varied tendencies of the signal propagation path in two kinds of models are basically identical, the length of Path I in the earth curvature model is about 94 km longer than it is in the flat earth model, and the length of Path II in the earth curvature model is about 89 km longer than it is in the flat earth model.The Path I and Path II in two kinds of models.
(a)
Path I
(b)
Path IIThe simulation results of Path I and Path II in two kinds of models.
(a)
Path I in the flat earth model
(b)
Path II in the flat earth model
(c)
Path I in the earth curvature model
(d)
Path II in the earth curvature modelThe Doppler frequency differencefdiff, between Path I and Path II in the flat earth model, can be obtained by (14) and (15) as
(41)fdiff=8fcHx(t)cvz(t).From (41), fdiff is inversely proportional to x(t). For the signal propagation path in the earth curvature model is longer than the one in the flat earth model; x(t) in the earth curvature model is larger than the one in the flat earth model. This is why the Doppler frequency differences of each path in the earth curvature model are smaller than the one in the flat earth model.Figure14 illustrates the comparison of the measured values v^z(t) of the height velocity in two kinds of earth models with its actual value vz(t). It can be seen that v^z(t) calculated in the earth curvature model is identical to the actual value, and v^z(t) obtained in the flat earth model has great differences from the actual one. The error of v^z(t) comes to 12 m/s when vz(t) is with the maximum value, so the altitude estimation would have great bias if the flat earth model is adopted.Figure 14
The comparison ofvz calculated by two kinds of earth models.Figure15 shows the track in the altitude direction with two kinds of models. In the earth curvature model, v^z(t) has no error, so the varied tendency of the calculated track (the dotted line) coincides with the actual one (the real line). The initial height z0 calculated by (39) has a slight bias about 120 m. So the whole track has a fixed error with the true track. While v^z(t) and z0 calculated in the flat earth model both have a larger error, the varied tendency of the whole movement track and the altitude value at every moment have great difference from the real value.Figure 15
The comparison of the movement tracks calculated in two kinds of earth models.


## 5.3. The Application Condition of the Earth Curvature Model

### 5.3.1. The Requirement of the Skywave Radar Altimetry to the Transmission Antenna Beam WidthΩ

In order to use the effect of micromultipath, the transmission beam is requested to have a certain width to receive the echoes of four paths. Therefore, the transmission antenna beam width Ω must be larger than two times of the differenceαdiff between the emission angle α1 and α2 corresponding to Path I and Path II. In Figure 16, the changing curves of the emission angle α1 (the real line) and α2 (the dotted line) corresponding to Path I and Path II are shown when the ground distance x(t) increases from 1000 km to 4000 km. Figure 17 illustrates the curve of the difference αdiff between the emission angles of two paths changing with x(t). We can find that the difference αdiff between the emission angles corresponding to Path I and Path II changes little (its biggest value is about 1.2°) within the detection range from 1000 km to 3000 km. When the detection range exceeds 3000 km, αdiff rises sharply, and it reaches about 3.7° when the detection range is 4000 km. Accordingly, when the skywave radar use micromultipath effect to measure the height, its beam width needs to be larger than 2.4° if the detection range is smaller 3000 km and the beam width of the transmission antenna has to be larger than 7.4° when the detection range exceeds 3000 km.Figure 16
The curve of the emission anglesα1 and α2 changing with the ground distance.Figure 17
The curve of the differenceαdiff changing with x(t).


### 5.3.2. The Detection Range of the Earth Curvature Model

The detection range of the skywave radar is limited in practice, for the emission angleα can only change in a certain range. The maximum value and the minimum value of α are written, respectively, as αmax⁡ and αmin⁡.When the target’s altitude is close to the ground and the emission angle isαmax⁡, the sky wave radar can realize the detection of the minimum range, as shown in Figure 18. When the emission angle is αmin⁡, the maximum detection is obtained, as shown in Figure 19.Figure 18
The illustration of the minimum detection range of the skywave radar.Figure 19
The illustration of the maximum detection range of the skywave radar.The emission angleαmax⁡ is generally taken as 30°. Then the minimum detection range xmin⁡ can be derived as follows:
(42)xmin⁡=2πR3-2Rarcsin(3R2(R+H)).The emission angleαmin⁡ is always 0°; then the maximum detection range xmax⁡ can be derived as
(43)xmax⁡=πR-2Rarcsin(RR+H).Taking the earth radius asR=6372km and the ionosphere height as H=350km, then xmin⁡=1072km and xmax⁡=4131km. The results conform to the practical detection range of the skywave radar.


## 5.3.1. The Requirement of the Skywave Radar Altimetry to the Transmission Antenna Beam WidthΩ

In order to use the effect of micromultipath, the transmission beam is requested to have a certain width to receive the echoes of four paths. Therefore, the transmission antenna beam width Ω must be larger than two times of the differenceαdiff between the emission angle α1 and α2 corresponding to Path I and Path II. In Figure 16, the changing curves of the emission angle α1 (the real line) and α2 (the dotted line) corresponding to Path I and Path II are shown when the ground distance x(t) increases from 1000 km to 4000 km. Figure 17 illustrates the curve of the difference αdiff between the emission angles of two paths changing with x(t). We can find that the difference αdiff between the emission angles corresponding to Path I and Path II changes little (its biggest value is about 1.2°) within the detection range from 1000 km to 3000 km. When the detection range exceeds 3000 km, αdiff rises sharply, and it reaches about 3.7° when the detection range is 4000 km. Accordingly, when the skywave radar use micromultipath effect to measure the height, its beam width needs to be larger than 2.4° if the detection range is smaller 3000 km and the beam width of the transmission antenna has to be larger than 7.4° when the detection range exceeds 3000 km.Figure 16
The curve of the emission anglesα1 and α2 changing with the ground distance.Figure 17
The curve of the differenceαdiff changing with x(t).


## 5.3.2. The Detection Range of the Earth Curvature Model

The detection range of the skywave radar is limited in practice, for the emission angleα can only change in a certain range. The maximum value and the minimum value of α are written, respectively, as αmax⁡ and αmin⁡.When the target’s altitude is close to the ground and the emission angle isαmax⁡, the sky wave radar can realize the detection of the minimum range, as shown in Figure 18. When the emission angle is αmin⁡, the maximum detection is obtained, as shown in Figure 19.Figure 18
The illustration of the minimum detection range of the skywave radar.Figure 19
The illustration of the maximum detection range of the skywave radar.The emission angleαmax⁡ is generally taken as 30°. Then the minimum detection range xmin⁡ can be derived as follows:
(42)xmin⁡=2πR3-2Rarcsin(3R2(R+H)).The emission angleαmin⁡ is always 0°; then the maximum detection range xmax⁡ can be derived as
(43)xmax⁡=πR-2Rarcsin(RR+H).Taking the earth radius asR=6372km and the ionosphere height as H=350km, then xmin⁡=1072km and xmax⁡=4131km. The results conform to the practical detection range of the skywave radar.


## 6. Conclusion

Skywave OTH radar systems can make use of the micromultipath effect to estimate the instantaneous altitude of maneuvering targets. The flat earth model is often used. However, ignoring the impact of the curvature of the earth, the analytical expressions of instantaneous target altitude are relatively simple, making it suitable for descriptive analysis. And the initial altitude has a great difference from actual altitude (600 meters) in the flat earth model. Moreover, the curvature of the earth should not be ignored for the thousands of kilometers’ detection in the skywave radar.This paper focuses on the theoretical derivations about how to introduce the earth curvature model into the estimation of the target’s altitude. The expressions of the Doppler frequency in four paths are deduced. And the difference of two kinds of models is compared. Although the analytical expressions are quiet complicated, the estimation of the initial altitude is close to the actual one. Therefore, any altimetry methods that are based on real data can improve the estimation results by exploiting the altimetry correction factors derived from earth curvature model.



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*Source: 424191-2014-07-14.xml*