Method for estimating tropospheric bias in passive detection system

Electromagnetic wave propagated via troposcatter is a valuable candidate for beyond line-of-sight detection. To improve the location accuracy of passive detection system based on troposcatter, path delay and angle bias caused by tropospheric refraction effect are estimated through the ray tracing method. As well, the Hopfield model is introduced to overcome ray tracing method's dependence on radiosonde data. Finally, consequences indicate that the Hopfield model can describe the tropospheric refraction better than other models. As well, the ray tracing method improved by the Hopfield model can effectively estimate path delay and angle bias.


Introduction
Detecting and suppressing enemy early-warning radar plays an important role in modern war [1][2][3]. As the horizons of conventional ground-based detection systems are limited by the curvature of earth, satellite detection systems have been widely used. However, satellite links have security problems and may be interrupted by hostile jamming. As well, image recognition technology generally used in satellite system is relatively powerless to detect radars in disguise or in blind areas [2]. Therefore, progressive detection mechanism to detect and locate the early-warning radar is in urgent need.
Conventional radar must face the threat coming from antiradiation missile. Passive mechanism is an effective approach to improve the survivability of radar system. Nowadays, passive radar system can use non-cooperative illuminators to detect and track a target, such as television signal and radio signal [3]. However, the coverage areas of such illuminators are limited. Most early-warning radars utilise electromagnetic (EM) wave in the microwave bandwidth to detect their targets. Microwave can be randomly scattered and realise beyond line-of-sight (b-LoS) propagation as it passes through troposphere. This propagation mechanism is widely known as troposcatter. With advantages of long single-hop span, anti-nuclear and anti-interception, troposcatter is a promising candidate for b-LoS links [4][5][6]. Therefore, signals of hostile radar propagated by troposcatter can be utilised for b-LoS detection [4].
Recently, characteristics of troposcatter have been well studied. The authors in [4,5] present a method for predicting total tropospheric attenuation. Wang et al. [ 6] propose a troposcatter passive ranging mechanism based on signal group delay. As well, troposphere is inhomogeneous and has obvious refraction effect. Dinc and Akan [7] analyse the time delay in satellite system resulting from the inhomogeneous troposphere. To better design passive detection system, negative effects caused by refraction effect must be obtained. We utilise the ray tracing method based on the Snell theorem to describe the path of EM wave propagated by troposcatter. Meanwhile, path delay and angle bias of EM wave are estimated. The Hopfield model is also introduced to improve ray tracing method through calculating the key refractivity.
The rest of this paper is organised as follows. Passive detection system based on troposcatter is described in the next section. In Section 3, ray tracing method improved by the Hopfield model is introduced. Finally, some conclusions are drawn in Section 4. When enemy radar searches its targets, troposcatter channel can allow EM wave to arrive at the passive detection system. Location of enemy radar can be obtained with an appropriate algorithm. Conventional location algorithm depends on path length and elevation angle of EM wave. However, refraction effect of troposcatter can contribute obvious bias to path length and elevation angle. To improve the location accuracy, we introduce an effective model to estimate the path delay and angle bias.

Ray tracing method
Heterogeneous troposphere consists of dry gases and water vapours, which cannot only cause a curving path, but also slow down the speed of EM wave [7,8]. EM wave transmitted by troposcatter channel is shown in Fig. 2. In Fig. 2, B is the scatter point, α and β are the measured and real incidence angle, θ is the geocentric angle, r the distance between any spot in true trajectory and earth's core, r 1 the radius of earth, r 2 the distance between point B and earth's core, h θ the altitude of point B to describe the curve path and lower speed of wave, path delay and bias of incidence angle can be estimated according to the ray tracing method [9], which is given as where ΔS denotes path delay, Δε the bias of incidence, and n(r) the refractive index of passing medium. β can be obtained on the basis of sine theorem. Geocentric angle can be expressed as In (1) and (2), n(r)=1+10 −6 N(h), where N(h) denotes the refractivity, which can be expressed as the sum of N d and N w in N-units. Also, h denotes the altitude. Empirical relationship between N d(w) and meteorological parameters can be given as where N d and N w refer to the dry and wet term of the refractivity, P the atmospheric pressure in hPa, ew the water vapour pressure in hPa, and T the absolute temperature in K. Above meteorological parameters can be measured through radiosonde balloon. However, these parameters are time-variant and cannot be acquired at any places in real time only through the radiosonde balloon.

Ray tracing method improved by reflectivity model
To precisely and conveniently calculate the tropospheric refractivity, several meteorological models are proposed. In ITU-R P. 835-5 [10], the troposphere model, including atmospheric pressure, temperature, and water vapour reference model, is given based on latitude areas (low-latitude, mid-latitude, and high-latitude) and seasons (summer and winter). For example, the mid-latitude troposphere model in summer can be given as Relatively poor temporal and spatial resolution of above troposphere model limits its application in high precision field. To demonstrate the defects of this troposphere model in more detail, three observations in different regions are selected; information of them is shown in Table 1.
Meteorological data of them during a year are provided to calculate refractivity according to (3) . Fig. 3 shows the consequences.
Values of refractivity are different and change with time. The maximum appears in summer, which refers to the unique meteorological parameters. However, values of refractivity are constant during whole summer or winter according to ITU-R P. 835-5. Therefore, changing refractivity effectively expose the shortcomings of the troposphere model proposed in ITU-R P. 835-5.
According to double exponential model, we can estimate N(h)as where N d0 and N w0 represent the dry and wet term of surface refractivity, and H d and H w characteristic height of dry and wet atmosphere. In (5), the way to estimate H d and H w depends on a large number of measure data, which limits its application. In ITU-R P.834-6 [11], exponential profile model is given as where H can be expressed as where DS V = 2.27P 0 + 1000f · RH 0 , where f is an empirical parameter, which can be derived from [9]. RH 0 denotes the relative humidity in percentage and can be defined as The Hopfield model also can calculate the value of N(h). According to the Hopfield model [12], N(h) changing with h can be expressed as In the Hopfield model, H dry and H wet denote the equivalent height for hydrostatic and wet refractivity, which can be given as  As analysed above, the Hopfield model does not need to define any initial parameters. Zenith tropospheric delay (ZTD) is introduced as a reference to evaluate different troposphere models. ZTD is the important factor in global navigation satellite system, which can be estimated as ZTD =10 −6 × N (h)dh (11) The Global Geodetic Observing System (GGOS) can provide the actual value of ZTD for doing research. The observations of GGOS are located all the earth. Considering the practicability of different models, we only list annual mean bias of the Hopfield model and exponential profile model in Fig. 4. The information of remaining observations in Fig. 4 is demonstrated in Table 2.
As shown in Fig. 4, the Hopfield model has obvious superiority to other models, not only in initial parameters settings, but also in accuracy. Therefore, the Hopfield model is introduced to improve ray tracing method.

Example analysis
Meteorological parameters of BJFS on 1 January 2017 are selected. Path delay and the angle bias estimated by the improved ray tracing method are shown in Fig. 5. The scatterer height is 5 km, and path delay and angle bias changing with time are also shown in Fig. 5.
Figures 5a and b demonstrate that higher scatter point can bring more obvious bias to path length and elevation angle. As well, with the elevation angle getting bigger, negative effects caused by troposcatter become weaker. As shown in Figs. 5c and d, values of all bias change during the year. Also, unique meteorological parameters in summer lead to more severe refraction effect than other time.

Conclusion
To improve the location accuracy of passive detection system based on troposcatter, path delay and angle bias are estimated on the basis of ray tracing method. Furthermore, the Hopfield model is introduced to estimate the key parameter in ray tracing method. Actual meteorological data are utilised to demonstrate the improved ray tracing method. Consequences indicate that ray tracing method improved by the Hopfield model can effectively estimate path delay and angle bias.

Acknowledgments
We are grateful to Global Geodetic Observing System (GGOS) for providing the related meteorological data and ZTD. We also gratefully acknowledge anonymous reviewers who read and made many insightful and constructive suggestions. This paper is supported by the National Natural Science Foundation of China under grant no. 61671468 and 61701525.