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

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.

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 of 𝑁_𝑡 closely spaced transmit antennas and 𝑁_𝑟 closely spaced receive antennas. Denote 𝑆∈𝐶^{𝑁𝑡×𝑇} as the orthogonal narrowband waveform matrix which is transmitted from the 𝑁_𝑡 transmit antennas over the pulse repetition period 𝑇. Assume that the orthogonal waveform is completed in the coherent integral time. 𝑠_𝑖 is the 𝑖th row of matrix 𝑆. When 𝑖≠𝑚, 𝑠_𝑖 is orthogonal to 𝑠_𝑚; then 𝑆𝑆^𝐻=𝐼_𝑁𝑡. So, for the 𝑘th pulse, the signal received at the 𝑚th receive antenna is expressed as [8, 9] 𝑢_𝑚,𝑘=^𝐼_𝑖=1^𝑁𝑡_𝑛=1𝜌_𝑖,𝑘𝑠_𝑛𝑒^{𝑗𝜙[𝐴]𝑖,𝑚,𝑘}𝑒^{𝑗𝜙[𝐷]𝑖,𝑛,𝑘}+𝑤_𝑚,𝑘, (1) where 𝐼=4 is the number of the multipath components and 𝜌_𝑖,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. 𝑑_𝑚^[𝐴] is the location of receive antennas. 𝜙^[𝐴]_{𝑖,𝑚,𝑘} is determined by the 𝑚th receive antenna and is expressed as 𝜙^[𝐴]_{𝑖,𝑚,𝑘}=2𝜋𝜆𝑑_𝑚^[𝐴]sin𝜃^[𝐴]_𝑖,𝑘, (2) where 𝜃^[𝐴]_𝑖,𝑘 is the elevation angle of arrival of the 𝑖th path. Similarly, 𝜙^[𝐷]_{𝑖,𝑛,𝑘} is determined by the location of the 𝑛th transmit antenna 𝑑_𝑛^[𝐷], and 𝜙^[𝐷]_{𝑖,𝑛,𝑘} is expressed as 𝜙^[𝐷]_{𝑖,𝑛,𝑘}=2𝜋𝜆𝑑_𝑛^[𝐷]sin𝜃^[𝐷]_𝑖,𝑘, (3) where 𝜃^[𝐷]_𝑖,𝑘 is the elevation angle of departure of the 𝑖th path. For simplicity, we assume that the clutter has been filtered. Thus 𝑤_𝑚,𝑘 is the additive noise, which is independent and identically distributed complex Gaussian random variable and independent of the target echo signal.

So the 𝑘th pulse echo, which is transmitted by the 𝑚th transmit antenna and received by the 𝑛th receive antenna, can be expressed as 𝑟_{𝑚,𝑛,𝑘}=^𝐼_𝑖=1𝜌_𝑖,𝑘𝑒^{𝑗𝜙[𝐴]𝑖,𝑚,𝑘}𝑒^{𝑗𝜙[𝐷]𝑖,𝑛,𝑘}+𝑤_{𝑚,𝑛,𝑘}. (4)

Denote 𝜑_𝑖,𝑘=[𝑒^{(𝑗𝜙[𝐴]𝑖,1,𝑘+𝑗𝜙[𝐷]𝑖,1,𝑘)},…,𝑒^{(𝑗𝜙[𝐴]𝑖,𝑚,𝑘+𝑗𝜙[𝐷]𝑖,𝑛,𝑘)}]^𝑇∈𝐶^𝑁×1 as the signal steering vector of the virtual array corresponding to the 𝑖th path at the 𝑘th pulse, where 𝑁=𝑁_𝑡𝑁_𝑟. Denote 𝐴_𝑘=[𝜑_1,𝑘,…,𝜑_𝐼,𝑘], 𝜌_𝑘=[𝜌_1,𝑘,…,𝜌_𝐼,𝑘]^𝑇, and 𝑊_𝑘=[𝑤_1,1,𝑘,…,𝑤_{𝑁𝑡,𝑁𝑟,𝑘}]^𝑇, so the measurement radar data vector can be expressed as 𝑟_𝑘=𝑟_1,1,𝑘,…,𝑟_{𝑁𝑡,𝑁𝑡,𝑘}^𝑇=𝐴_𝑘𝜌_𝑘+𝑊_𝑘. (5)

In order to be consistent with the motion model, the 𝑘th 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 in 𝑇=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 𝑥(𝑡)(0)−𝑣=𝑥^𝑥(0)𝑇𝜋sin𝑡𝜋𝑇,𝑧(𝑡)(0)+𝑣=𝑧^𝑧(0)𝑇𝜋1−cos𝑡𝜋𝑇. (6)

Table 1 

The parameters of the motion target.

The target tracks in two dimensions are shown in Figures 1 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 Figure 3.

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 direction 𝑥 and altitude direction 𝑧). Then the target state at the time 𝑡 is expressed as 𝑥(𝑡)=𝑥(𝑡)𝑧(𝑡)𝑣_𝑥(𝑡)𝑣_𝑧(𝑡)^𝑇, (7) where 𝑣_𝑥(𝑡) and 𝑣_𝑧(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 the 𝑥-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 𝑙₁(𝑡)=𝑥(𝑡)²+(2𝐻−𝑧(𝑡))²,𝑙₂(𝑡)=𝑥(𝑡)²+(2𝐻+𝑧(𝑡))². (8) As 𝑥≫𝐻≫𝑧, 𝑙₁(𝑡) and 𝑙₂(𝑡) are simplified as 𝑙₁(𝑡)≈𝑥(𝑡)+2𝐻²−2𝐻𝑧(𝑡)𝑥(𝑡),𝑙₂(𝑡)≈𝑥(𝑡)+2𝐻²+2𝐻𝑧(𝑡)𝑥(𝑡). (9) Then 𝐿₁(𝑡) and 𝐿₂(𝑡), the length of Path I and Path II, are expressed as 𝐿₁(𝑡)=2𝑙₁(𝑡)≈2𝑥(𝑡)+4𝐻²−4𝐻𝑧(𝑡)𝑥(𝑡),𝐿₂(𝑡)=2𝑙₂(𝑡)≈2𝑥(𝑡)+4𝐻²+4𝐻𝑧(𝑡)𝑥(𝑡). (10)

𝐿₃(𝑡) and 𝐿₄(𝑡), the length of Path III and Path IV, are written as 𝐿₃(𝑡)=𝐿₄(𝑡)=𝑙₁(𝑡)+𝑙₂(𝑡)≈2𝑥(𝑡)+4𝐻²𝑥(𝑡). (11)

3.2. Instantaneous Target Altitude Estimation

In (4), 𝜌_𝑖(𝑡) comprises the targets’ Doppler information. Here we express 𝜌_𝑖(𝑡) as 𝜌_𝑖(𝑡)=𝜎_𝑖(𝑡)𝑒^{−𝑗2𝜋𝐿𝑖(𝑡)/𝜆}, (12) where 𝑖=1,…,4. 𝑖 is the index of the path. 𝜆=𝑐/𝑓_𝑐 is the wavelength corresponding to carrier frequency 𝑓_𝑐, and 𝜎_𝑖(𝑡) represents the combined effect of transmit power, target reflection, which is a function of radar cross-section (RCS), and ionospheric effect. 𝐿_𝑖(𝑡) is the two-way slant range, and the expressions of 𝐿_𝑖(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 ionosphere 𝐻 is changing slowly, we assume that 𝐻 is a constant value in the coherent time. So we can get 𝑑𝑙₁(𝑡)(𝑡)𝑣𝑑𝑡≈𝐽_𝑥(𝑡)−2𝐻𝑥(𝑡)𝑣_𝑧(𝑡),𝑑𝑙₂(𝑡)(𝑡)𝑣𝑑𝑡≈𝐽_𝑥(𝑡)+2𝐻𝑥(𝑡)𝑣_𝑧(𝑡), (13) where 𝐽(𝑡)=(1−2𝐻²/𝑥²(𝑡)), 𝑣_𝑥(𝑡)=𝑑𝑥(𝑡)/𝑑𝑡 is the target’s velocity of the range direction, and 𝑣_𝑧(𝑡)=𝑑𝑧(𝑡)/𝑑𝑡 is the target’s velocity of the altitude direction. So the Doppler frequencies of the four paths are 𝑓₁(𝑡)=2𝑓_𝑐𝑐𝑑𝑙₁(𝑡)≈𝑑𝑡2𝑓_𝑐𝑐𝐽(𝑡)𝑣_𝑥(𝑡)−4𝑓_𝑐𝐻𝑥(𝑡)𝑐𝑣_𝑧(𝑡),𝑓₂(𝑡)=2𝑓_𝑐𝑐𝑑𝑙₂(𝑡)≈𝑑𝑡2𝑓_𝑐𝑐𝐽(𝑡)𝑣_𝑥(𝑡)+4𝑓_𝑐𝐻𝑥(𝑡)𝑐𝑣_𝑧(𝑡),𝑓₃(𝑡)=𝑓₄(𝑡)=𝑓_𝑐𝑐𝑑𝑙₁(𝑡)+𝑑𝑙₂(𝑡)≈𝑑𝑡2𝑓_𝑐𝑐𝐽(𝑡)𝑣_𝑥(𝑡). (14)(15)(16)

From (14)–(16), we find that the Doppler frequencies of the four paths have the same part 2𝑓_𝑐𝐽(𝑡)𝑣_𝑥(𝑡)/𝑐, which is determined by 𝑣_𝑥(𝑡). We call this the dominant Doppler component. 𝑐 is the speed of light. 𝑓₁ and 𝑓₂ also have the part caused by 𝑣_𝑧(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 𝑣_𝑧(𝑡) is obtained by the expression 𝑣_𝑧(𝑡)=𝑓₂(𝑡)−𝑓₁(𝑡)𝑥(𝑡)𝑐8𝑓_𝑐𝐻. (17)

To get the target height, it is necessary to know the target initial height. In practice, we can get 𝐿₁(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 𝑧₀=2𝐻−𝐿₁(0)2²−𝑥₀²,𝑧₀=𝐿₂(0)2²+𝑥₀²−2𝐻. (18)

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 of 𝑥(𝑡), we can use the following to calculate 𝑧(𝑡): 𝑥(𝑡)²+[2𝐻−𝑧(𝑡)]²=𝐿₁²,𝑥(𝑡)²+[2𝐻+𝑧(𝑡)]²=𝐿₂². (19)𝑧(𝑡) is obtained by (19) as 𝑧(𝑡)=𝐿₂(𝑡)²−𝐿₁(𝑡)².8𝐻 (20) Let 𝑡=0, 𝑧₀ be gotten. Then 𝑧(𝑡) can be written as 𝑧(𝑡)=𝐿₂(0)²−𝐿₁(0)²+𝑓8𝐻²(𝑡)−𝑓₁(𝑡)𝑥(𝑡)𝑐8𝑓_𝑐𝐻×𝑡. (21)

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 Figure 4. 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 Figure 5, 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: 𝑥(𝑡)𝑅+𝛼₁𝛼+2arcsin𝑅cos₁𝑅𝑅+𝐻−arcsin𝛼𝑅+𝑧cos₁=𝜋2. (22)

The accurate value of 𝛼₁ can be calculated by (22). The auxiliary angles 𝛽₁ and 𝛾₁ can be expressed as the function of 𝛼₁: 𝛽₁𝛼=arcsin𝑅cos₁𝑅+𝐻,𝛾₁𝛼=arcsin𝑅cos₁(𝑡).𝑅+𝑧 (23)𝑃₁ and 𝑚₁ can be, respectively, expressed as 𝑃₁=𝑅cos𝛼₁+𝛽₁sin𝛽₁,𝑚₁=𝑃₁+𝐻+𝑅𝑥(𝑡)sin𝛾cos𝑅+𝛼₁+𝛽 (24) Then 𝑙₁(𝑡) is written as 𝑙₁(𝑡)=2𝑃₁−𝑚₁𝛼=𝑅cos₁+𝛽₁sin𝛽₁−𝐻+𝑅sin𝛾₁𝑥(𝑡)cos𝑅+𝛼₁+𝛽₁. (25)

Path II is illustrated in Figure 6. 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 Figure 6, 𝛼₂ should meet the following conditions: 𝑥(𝑡)𝑅𝑅+arcsin𝑅+𝑧cos𝛼₂𝑅+2arcsin𝑅+𝐻cos𝛼₂+3𝛼₂=3𝜋2. (26)

Then 𝛼₂ can be calculated by (26). The auxiliary angles 𝛽₂ and 𝛾₂ can be expressed as the function of 𝛼₂: 𝛽₂=arcsin𝑅cos𝛼₂𝑅+𝐻,𝛾₂=arcsin𝑅cos𝛼₂(𝑡).𝑅+𝑧 (27)𝑃₂ and 𝑚₂ can be, respectively, expressed as 𝑃₂=𝑅cos𝛼₂+𝛽₂sin𝛽₂,𝑚₂=cos𝛼₂+𝛾₂cos𝛼₂[𝑅+𝑧(𝑡)]. (28) Then the expression of transmission oblique diameter 𝑙₂(𝑡) is as follows: 𝑙₂(𝑡)=2𝑃₂+𝑚₂𝛼=2𝑅cos₂+𝛽₂sin𝛽₂+cos𝛼₂+𝛾₂cos𝛼₂[𝑅+𝑧(𝑡)]. (29) So 𝐿₁(𝑡) and 𝐿₂(𝑡), the length of Path I and Path II, are 𝐿₁(𝑡)=2𝑙₁(𝑡)𝛼=2𝑅cos₁+𝛽₁sin𝛽₁−2𝐻+𝑅sin𝛾₁𝑥(𝑡)cos𝑅+𝛼₁+𝛽₁,𝐿₂(𝑡)=2𝑙₂(𝑡)𝛼=4𝑅cos₂+𝛽₂sin𝛽₂𝛼+2cos₂+𝛽₂cos𝛼₂[𝑅+𝑧(𝑡)]. (30) And 𝐿₃(𝑡) and 𝐿₄(𝑡), the length of Path III and Path IV, are shown as 𝐿₃(𝑡)=𝐿₄(𝑡)=𝑙₁(𝑡)+𝑙₂(𝑡)𝛼=𝑅cos₁+𝛽₁sin𝛽₁−𝐻+𝑅sin𝛾₁𝑥(𝑡)cos𝑅+𝛼₁+𝛽₁𝛼+2𝑅cos₂+𝛽₂sin𝛽₂+cos𝛼₂+𝛾₂cos𝛼₂[𝑅+𝑧(𝑡)]. (31)

4.2. Instantaneous Target Altitude Estimation

Consistent with the processing methods in the flat earth model, we suppose that 𝐻 is constant in the coherent time. Then the time derivatives of the 𝑙₁(𝑡), 𝑙₂(𝑡), 𝑙₃(𝑡), and 𝑙₄(𝑡) are obtained by 𝑑𝑙₁(𝑡)=𝛼𝑑𝑡sin₁+𝛽₁+𝑥(𝑡)/𝑅[𝑅+𝑧(𝑡)](𝐻+𝑅)𝑅²cos𝛼₁𝑣_𝑥(𝑡)−cos𝛼₂/2+𝛽₁+𝑥(𝑡)/𝑅(𝐻+𝑅)𝑅cos𝛼₁𝑣_𝑧(𝑡)=𝑙_1𝑥𝑣_𝑥(𝑡)+𝑙_1𝑧𝑣_𝑧(𝑡),𝑑𝑙₂(𝑡)=𝛼𝑑𝑡cos²2−𝜋4+𝛾2+𝑥(𝑡))×2𝑅(𝐻+𝑅𝑅cos𝛼₂⁻¹𝑣_𝑥(𝑡)+cos𝛼₂+𝛾₂cos𝛼₂+𝑅sin𝛼₂+𝛾₂[𝑅+𝑧(𝑡)]cos𝛾₂−𝛼𝑅cos²2−𝜋4+𝛾2+𝑥(𝑡))×2𝑅(𝐻+𝑅1−𝑅cos𝛼₂𝑅+𝑧(𝑡)²[𝑅+𝑧(𝑡)]²⁻¹𝑣_𝑧(𝑡)=𝑙_2𝑥𝑣_𝑥(𝑡)+𝑙_2𝑧𝑣_𝑧(𝑡),𝑑𝑙₃(𝑡)=𝑑𝑡𝑑𝑙₄(𝑡)=1𝑑𝑡2𝑑𝑙₁(𝑡)+𝑑𝑡𝑑𝑙₂(𝑡).𝑑𝑡 (32)𝑙_1𝑥 is the coefficient of the part of 𝑑𝑙₁(𝑡)/𝑑𝑡 caused by 𝑣_𝑥(𝑡). 𝑙_1𝑧 is the coefficient of the part of 𝑑𝑙₁(𝑡)/𝑑𝑡 caused by 𝑣_𝑧(𝑡). The expressions of 𝑙_2𝑥 and 𝑙_2𝑧 are similar to 𝑙_1𝑥 and 𝑙_1𝑧, but replace 𝑙₁ with 𝑙₂. So the Doppler frequencies produced by the moving target of the four paths can be, respectively, written as 𝑓₁(𝑡)=2𝑓_𝑐𝑐𝑑𝑙₁(𝑡),𝑓𝑑𝑡₂(𝑡)=2𝑓_𝑐𝑐𝑑𝑙₂(𝑡),𝑓𝑑𝑡₃(𝑡)=𝑓₄(𝑡)=12𝑓₁(𝑡)+𝑓₂(𝑡). (33)

The four Doppler frequencies both consist of the components caused by 𝑣_𝑥(𝑡) and 𝑣_𝑧(𝑡). The ratio of 𝑙_1𝑥 to 𝑙_2𝑥 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 𝑣_𝑥(𝑡) in 𝑓₁(𝑡) and 𝑓₂(𝑡) are very close. And it indicates that the Doppler frequency component in each path caused by 𝑣_𝑥(𝑡) is approximately equal. This conclusion is same as the one in the flat earth model. Therefore, 𝑣_𝑧(𝑡) can be obtained by 𝑓₂(𝑡)−𝑓₁(𝑡) as 𝑣_𝑧(𝑡)=𝑐2𝑓_𝑐𝑓₂(𝑡)−𝑓₁(𝑡)𝑙_2𝑧−𝑙_1𝑧. (34)

Figure 7 

The ration of 11x to 12x.

To get the instantaneous height of the moving target, the initial height of the target 𝑧₀ 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: 𝑧₀₁=𝐿₁(0)2−sin𝜃₁sin𝛽₁sin𝛽₁sin𝜃₂−𝑅, (35) where 𝛽₁=arcsin𝑅cos𝛼₁,𝜃𝐻+𝑅₂=𝑥(𝑡)𝑅−𝜋2+𝛼₁+𝛽₁. (36)

For Path II, 𝑧₀₂ can be written as 𝑧₀₂=𝑅cos𝛼₂sin𝛾₀−𝑅, (37) where 𝛾₂=3𝜋2−𝑥(𝑡)𝑅−2𝛼₂−2𝛽₂,𝛽₂𝑅=arcsin𝑅+𝐻×cos𝛼₂. (38) Then 𝑧₀ is the average value of 𝑧₀₁ and 𝑧₀₂: 𝑧₀=𝐿₁(0)2−sin𝜃₁sin𝛽₁sin𝛽₁sin𝜃₂+𝑅cos𝛼₂sin𝛾₀×(2)−2𝑅₋₁. (39) So the instantaneous height 𝑧(𝑡) can be expressed as 𝑧(𝑡)=𝑧₀+𝑣_𝑧(𝑡)×𝑡. (40)

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 Section 2.2 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 Figure 8 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 𝑣_𝑥(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 moment 𝑇/2, 𝑣_𝑥(𝑡) comes to zero, and the Doppler frequencies of every path are close to zero, while 𝑣_𝑧(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 𝑣_𝑧(𝑡).

By comparison of Figures 8 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

Figures 12(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.

(a) Path I

(b) Path II

Figure 12 

The 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 model

Figure 13 

The simulation results of Path I and Path II in two kinds of models.

The Doppler frequency difference 𝑓_diff, between Path I and Path II in the flat earth model, can be obtained by (14) and (15) as 𝑓_diff=8𝑓_𝑐𝐻𝑥(𝑡)𝑐𝑣_𝑧(𝑡). (41)

From (41), 𝑓_diff 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.

Figure 14 illustrates the comparison of the measured values 𝑣̂_𝑧(𝑡) of the height velocity in two kinds of earth models with its actual value 𝑣_𝑧(𝑡). It can be seen that 𝑣̂_𝑧(𝑡) calculated in the earth curvature model is identical to the actual value, and 𝑣̂_𝑧(𝑡) obtained in the flat earth model has great differences from the actual one. The error of 𝑣̂_𝑧(𝑡) comes to 12 m/s when 𝑣_𝑧(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 of 𝑣𝑧 calculated by two kinds of earth models.

Figure 15 shows the track in the altitude direction with two kinds of models. In the earth curvature model, 𝑣̂_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 𝑣̂_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 Width Ω

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 difference 𝛼_diff 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 𝛼_diff between the emission angles of two paths changing with 𝑥(𝑡). We can find that the difference 𝛼_diff 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, 𝛼_diff 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 angles 𝛼1 and 𝛼2 changing with the ground distance.

Figure 17 

The curve of the difference 𝛼diff 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 angle 𝛼 can only change in a certain range. The maximum value and the minimum value of 𝛼 are written, respectively, as 𝛼_max and 𝛼_min.

When the target’s altitude is close to the ground and the emission angle is 𝛼_max, the sky wave radar can realize the detection of the minimum range, as shown in Figure 18. When the emission angle is 𝛼_min, 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 angle 𝛼_max is generally taken as 30°. Then the minimum detection range 𝑥_min can be derived as follows: 𝑥_min=2𝜋𝑅3√−2𝑅arcsin3𝑅2().𝑅+𝐻 (42)

The emission angle 𝛼_min is always 0°; then the maximum detection range 𝑥_max can be derived as 𝑥_max𝑅=𝜋𝑅−2𝑅arcsin.𝑅+𝐻 (43)

Taking the earth radius as 𝑅=6372km and the ionosphere height as 𝐻=350km, then 𝑥_min=1072km and 𝑥_max=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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

At the point of finishing this paper, the authors would like to express their sincere thanks to the National Natural Science Foundation (no. 61201303) and the Fundamental Research Funds for the Central Universities (HIT.NSRIF.2013027).