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Hou Chengyu, Wang Yuxin, Chen Jiawei, "Estimating Target Heights Based on the Earth Curvature Model and Micromultipath Effect in Skywave OTH Radar", Journal of Applied Mathematics, vol. 2014, Article ID 424191, 14 pages, 2014. https://doi.org/10.1155/2014/424191
Estimating Target Heights Based on the Earth Curvature Model and Micromultipath Effect in Skywave OTH Radar
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.
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 .
Skywave OTH radar systems are capable of localizing targets in range and azimuth but are unable to achieve reliable altitude estimation . 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 .
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 . (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 . (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 . (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 .
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 closely spaced transmit antennas and closely spaced receive antennas. Denote as the orthogonal narrowband waveform matrix which is transmitted from the transmit antennas over the pulse repetition period . Assume that the orthogonal waveform is completed in the coherent integral time. is the th row of matrix . When , is orthogonal to ; then . So, for the th pulse, the signal received at the th receive antenna is expressed as [8, 9] where is the number of the multipath components and is the complex response of the th multipath that considers the target reflection and propagation delays. The superscripts and denote the departure and the arrival modes, respectively. is the location of receive antennas. is determined by the th receive antenna and is expressed as where is the elevation angle of arrival of the th path. Similarly, is determined by the location of the th transmit antenna , and is expressed as where is the elevation angle of departure of the th path. For simplicity, we assume that the clutter has been filtered. Thus is the additive noise, which is independent and identically distributed complex Gaussian random variable and independent of the target echo signal.
So the th pulse echo, which is transmitted by the th transmit antenna and received by the th receive antenna, can be expressed as
Denote as the signal steering vector of the virtual array corresponding to the th path at the th pulse, where . Denote , , and , so the measurement radar data vector can be expressed as
In order to be consistent with the motion model, the th pulse of the measurement data is expressed with the time corresponding to the th pulse.
2.2. Establish Maneuvering Target Model
In this paper, for generality, we consider a maneuvering aircraft which makes a 180° circular turn in s . The motion parameters are listed in Table 1. All the multipath signals are considered to fall within the same range cell. The ground distance and flight altitude of the target , respectively, are expressed as
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.
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 and altitude direction ). Then the target state at the time is expressed as where and 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 -axis.
As the micromultipath effect exists, the combination of the transmit path and receive path has four components: Path I (), Path II , Path III , and Path IV . Path III and Path IV are with the same length . The slant range and is expressed with the distance , the height of the ionosphere , and the target height as As , and are simplified as Then and , the length of Path I and Path II, are expressed as
and , the length of Path III and Path IV, are written as
3.2. Instantaneous Target Altitude Estimation
In (4), comprises the targets’ Doppler information. Here we express as where . is the index of the path. is the wavelength corresponding to carrier frequency , and represents the combined effect of transmit power, target reflection, which is a function of radar cross-section (RCS), and ionospheric effect. is the two-way slant range, and the expressions of 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 is changing slowly, we assume that is a constant value in the coherent time. So we can get where , is the target’s velocity of the range direction, and is the target’s velocity of the altitude direction. So the Doppler frequencies of the four paths are
From (14)–(16), we find that the Doppler frequencies of the four paths have the same part , which is determined by . We call this the dominant Doppler component. is the speed of light. and also have the part caused by . We call this the minor Doppler component, which contains the important information of the target movement in the altitude direction. Let ; then is obtained by the expression
To get the target height, it is necessary to know the target initial height. In practice, we can get , and by the time delay, and the transmit angle and the ionospheric height can also be known by the other measurement equipment. Then these parameters can be used to calculate the initial height . According to (8), we can get two estimated values by
However, gotten by (18) is quite different from the actual value. It is because is very small compared to . If is with a small error, there will be a great impact on .
For elimination of the impact by the error of , we can use the following to calculate : is obtained by (19) as Let , be gotten. Then can be written as
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. 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, indicates the vertical height, is the height of the ionosphere, and is the earth radius (it is set as a constant value in the paper; ). Same as the flat earth model, the radar signal propagation paths in the earth curvature model have four combinations: Path I (), Path II , Path III , and Path IV .
Path I is shown in Figure 5, in which indicates the departure angle of radar transmitting antenna, is the distance of electromagnetic signal from radar transmitting antenna to ionosphere, is the extended line, and , indicate the auxiliary angles.
According to the geometrical relationship, meets the following condition:
The accurate value of can be calculated by (22). The auxiliary angles and can be expressed as the function of : and can be, respectively, expressed as Then is written as
Path II is illustrated in Figure 6. Similarly, indicates the departure angle of radar transmitting antenna, is the distance of electromagnetic signal from radar transmitting antenna to ionosphere, is the extended line, and and indicate the auxiliary angles.
From Figure 6, should meet the following conditions:
Then can be calculated by (26). The auxiliary angles and can be expressed as the function of : and can be, respectively, expressed as Then the expression of transmission oblique diameter is as follows: So and , the length of Path I and Path II, are And and , the length of Path III and Path IV, are shown as
4.2. Instantaneous Target Altitude Estimation
Consistent with the processing methods in the flat earth model, we suppose that is constant in the coherent time. Then the time derivatives of the , , , and are obtained by is the coefficient of the part of caused by . is the coefficient of the part of caused by . The expressions of and are similar to and , but replace with . So the Doppler frequencies produced by the moving target of the four paths can be, respectively, written as
The four Doppler frequencies both consist of the components caused by and . The ratio of to under the condition that 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 in and are very close. And it indicates that the Doppler frequency component in each path caused by is approximately equal. This conclusion is same as the one in the flat earth model. Therefore, can be obtained by as
To get the instantaneous height of the moving target, the initial height of the target is needed. Different from the situation in the flat earth model, can be calculated accurately by using or . In order to reduce random errors, and are used to calculate the initial heights , and then make the average of , as .
For Path I, can be shown as follows: where
For Path II, can be written as where Then is the average value of and : So the instantaneous height can be expressed as
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 is maximal, the absolute value of Doppler frequency comes to the maximum.
At the moment , comes to zero, and the Doppler frequencies of every path are close to zero, while 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 .
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.
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
(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
From (41), is inversely proportional to . For the signal propagation path in the earth curvature model is longer than the one in the flat earth model; 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 of the height velocity in two kinds of earth models with its actual value . It can be seen that calculated in the earth curvature model is identical to the actual value, and obtained in the flat earth model has great differences from the actual one. The error of comes to 12 m/s when is with the maximum value, so the altitude estimation would have great bias if the flat earth model is adopted.
Figure 15 shows the track in the altitude direction with two kinds of models. In the earth curvature model, 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 calculated by (39) has a slight bias about 120 m. So the whole track has a fixed error with the true track. While and 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.
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
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 between the emission angle and corresponding to Path I and Path II. In Figure 16, the changing curves of the emission angle (the real line) and (the dotted line) corresponding to Path I and Path II are shown when the ground distance increases from 1000 km to 4000 km. Figure 17 illustrates the curve of the difference between the emission angles of two paths changing with . We can find that the difference 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, 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.
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 can only change in a certain range. The maximum value and the minimum value of are written, respectively, as and .
When the target’s altitude is close to the ground and the emission angle is , the sky wave radar can realize the detection of the minimum range, as shown in Figure 18. When the emission angle is , the maximum detection is obtained, as shown in Figure 19.
The emission angle is generally taken as 30°. Then the minimum detection range can be derived as follows:
The emission angle is always 0°; then the maximum detection range can be derived as
Taking the earth radius as and the ionosphere height as , then and . The results conform to the practical detection range of the skywave radar.
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.
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).
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