A Propagation Environment Modeling in Foliage
© Jing Liang et al. 2010
Received: 16 September 2009
Accepted: 22 January 2010
Published: 21 March 2010
Foliage clutter, which can be very large and mask targets in backscattered signals, is a crucial factor that degrades the performance of target detection, tracking, and recognition. Previous literature has intensively investigated land clutter and sea clutter, whereas foliage clutter is still an open-research area. In this paper, we propose that foliage clutter should be more accurately described by a log-logistic model. On a basis of pragmatic data collected by ultra-wideband (UWB) radars, we analyze two different datasets by means of maximum likelihood (ML) parameter estimation as well as the root mean square error (RMSE) performance. We not only investigate log-logistic model, but also compare it with other popular clutter models, namely, log-normal, Weibull, and Nakagami. It shows that the log-logistic model achieves the smallest standard deviation (STD) error in parameter estimation, as well as the best goodness-of-fit and smallest RMSE for both poor and good foliage clutter signals.
1. Introduction and Motivation
Detection and identification of military equipment in a strong clutter background, such as foliage, soil cover, or building has been a long-standing subject of intensive study. It is believed that solving the target detection through foliage environment will significantly benefit sense-through-wall and many other subsurface sensing problems. However, to this date, the detection of foliage-covered targets with satisfied performances is still a challenging issue. Recent investigations in environment behavior of tree canopies have shown that both signal backscattering and attenuation are significantly influenced by tree architecture . Therefore using the return signal from foliage to establish the clutter model that accounts for environment effects is of great importance for the sensing-through-foliage radar detection.
Clutter is a term used to define all unwanted echoes from natural environment . The nature of clutter may necessarily vary on a basis of different applications and radar parameters. Most previous studies have investigated land clutter or sea clutter, and some conclusions have been reached. For example, log-normal, Weibull, and K-distributions have been proven to be better suited for the clutter description other than Rayleigh and Rician models in high-resolution radar systems. Fred did statistical comparisons and found that sea clutter at low-grazing angles and high range resolution is spiky based on the data measured from various sites in Kauai and Hawaii . David generalized radar clutter models using noncentral chi-square density by allowing the noncentrality parameter to fluctuate according to the gamma distribution . Furthermore, Leunget al. used a Neural-Network-based approach to predict the sea clutter model [5, 6].
However, as far as clutter modeling in forest is concerned, it is still of great interest and will be likely to take some time to reach any agreement. A team of researchers from MIT  and U. S. Army Research Laboratory (ARL) [8, 9] have measured ultra-wideband (UWB) backscatter signals in foliage for different polarizations and frequency ranges. The measurements show that the foliage clutter is impulsively corrupted with multipath fading, which leads to inaccuracy of the K-distributions description . The US Air Force Office of Scientific Research (AFOSR) has conducted field measurement experiment concerning foliage penetration radar since 2004 and noted that metallic targets may be more easily identified with wideband than with narrowband signals.
In this investigation, we will apply ultra-wideband (UWB) radar to model the foliage clutter. UWB radar emissions are at a relatively low frequency typically between 100 MHz and 3 GHz. Additionally, the fractional bandwidth of the signal is very large (greater than 0.2). Such a radar sensor has exceptional range resolution, as well as the ability to penetrate many common materials (e.g., walls). Law enforcement personnel have used UWB ground-penetrating radars (GPRs) for at least a decade. Like the GPR, sensing-through-foliage radar takes advantage of UWB's very fine resolution (time gating) and the low frequency of operation.
In our work, we investigate the log-logistic distribution (LLD) to model foliage clutter and illustrate the goodness-of-fit to real UWB clutter data. Additionally, we compare the goodness-of-fit of LLD with existing popular models, namely, log-normal, Weibull, and Nakagami by means of maximum likelihood estimation (MLE) and the root mean square error (RMSE). The result shows that log-logistic model provides the best fit to the foliage clutter. Our contribution is not only the new proposal on the foliage clutter model with detailed parameter estimation, but also providing the criteria and approaches based on which the statistical analysis is obtained. Further, based on LLD the theoretical study about the probability of detection as well as the probability of false alarm are discussed.
The rest of this paper is organized as follows. Section 2 provides a review on statistical models of log-logistic, log-normal, Weibull, and Nakagami, and discusses their applicability for foliage clutter modeling. Section 3 summarizes the measurement and the two sets of clutter data that are used in this paper. Section 4 discusses estimation on parameters and the goodness-of-fit for log-logistic, log-normal, Weibull and Nakagami models, respectively. Section 5 analyzes the performance of radar detection in the presence of foliage clutter. Finally, Section 6 concludes this paper and describes some future research topics.
2. Clutter Models
Many radar clutter models have been proposed in terms of distinct statistical distributions; most of which describe the characteristics of clutter amplitude or power. Before detailed analysis on our measurement, we would like to discuss the properties and applicability of log-logistic, log-normal, Weibull, and Nakagami statistic distributions, which are designated as "curve-fit" models in Section 4, since they are more likely to provide good fit to our collections of pragmatic clutter data. Detailed explanations would be given in the following subsections.
2.1. Log-Logistic Model
Recently Log-logistic model has been applied in hydrological analysis. This distribution is a special case of Burr's type-XII distribution  as well as a special case of the kappa distribution proposed by Mielke andJohnson . Lee et al. employed the log-logistic distribution (LLD) for frequency analysis of multiyear drought durations , whereas Shoukri et al. employed LLD to analyze extensive Canadian precipitation data , and Narda and Malik used LLD to develop a model of root growth and water uptake in wheat . In spite of its intensive application in precipitation and stream-flow data, the log-logistic distribution (LLD)  statistical model, to the best of our knowledge, has never been applied to radar foliage clutter. The motivation for considering log-logistic model is based on its higher kurtosis and longer tails, as well as its probability density function (PDF) similarity to log-normal and Weibull distributions. It is intended to be employed to estimate how well the model matches our collected foliage clutter statistics.
2.2. Log-Normal Model
The log-normal distribution is most frequently used when the radar sees land clutter  or sea clutter  at low-grazing angles ( 5 degrees) since log-normal has a long tail. However, it has been reported that the log-normal model tends to overestimate the dynamic range of the real clutter distribution . Furthermore, whether it is applicable to model foliage clutter still requires detailed analysis.
2.3. Weibull Model
The Weibull distribution, which is named after Waloddi Weibull, can be made to fit clutter measurements that lie between the Rayleigh and log-normal distribution . It has been applied to land clutter [22, 23], sea clutter [24, 25] and weather clutter . However, in very spiky sea and foliage clutter, the description of the clutter statistics provided by Weibull distributions may not always be sufficiently accurate .
2.4. Nakagami Model
In the foliage penetration setting, the target returns suffer from multipath effects corrupted with fading. As Nakagami distribution is used to model scattered fading signals that reach a receiver by multiple paths, it is natural to investigate how well it fits the foliage clutter statistics.
3. Experiment Setup and Data Collection
The principle pieces of equipment are
Dual antenna mounting stand,
A trihedral reflector target,
Barth pulse source (Barth Electronics, Inc. model 732 GL) for UWB,
Tektronix model 7704 B oscilloscope,
HP signal Generator,
Custom RF switch and power supply,
Weather shield (small hut).
4. Statistical Analysis of the Foliage Clutter
4.1. Maximum Likelihood Estimation
Using the collected clutter data mentioned above, we apply Maximum Likelihood Estimation (MLE) approach to estimate the parameters of the log-logistic, log-normal, Weibull, and Nakagami models. MLE is often used when the sample data are known and parameters of the underlying probability distribution are to be estimated [29, 30]. It is generalized as follows.
The maximum likelihood estimate of , , , is the set of values , , , that maximize the likelihood function .
Estimated parameters for dataset I.
Averaged estimated parameters for dataset I.
From Tables 1 and 2, it can be easily seen that STD errors for log-logistic and log-normal parameters are less than 0.02 and the estimated parameters for these two models vary little from data to data compared to parameters of Weibull and Nakagami. It is obvious that log-logistic model provides the smallest STD error for all the 10 collections compared to log-normal. Although accurate shape parameter estimation can be achieved by both Weibull and Nakagami models, their scale parameters are not acceptable.
Estimated and averaged parameters for dataset ii.
In view of smaller error in parameter estimation, log-logistic model fits the collected data best compared to log-normal, Weibull, and Nakagami. Log-normal model is also acceptable.
4.2. Goodness-of-Fit in Curve and RMSE
Here we apply n 100 for each model and k is the number of data collections for each set.
Similarly, in Figure 11 histogram bars denote the PDF of the absolute amplitude of one collection of clutter data from set II. Compared to Figure 10, the log-logistic and the log-normal provide similar extent of goodness-of-fit. Weibull is worse since it cannot fit well in either kurtosis or tail, while Nakagami is the worst and unacceptable. Also, we obtain , , , . This illustrates that for clutter backscattering dataset II, the log-logistic model still fits the best.
5. Target Detection Performance
One of the primary goals for a radar is target detection; therefore based on clutter models that have been investigated in the previous sections, we apply a special case of the Bayesian criterion named Neyman-Parson criterion to analyze the target detection performance in the foliage environment.
where and represent the random variable of clutter and noise, respectively. follows log-logistic model with both parameters and , and is Gaussian noise with zero mean and variance . is the target signal, which assumes to be a constant for simplicity.
Therefore and mean that
(i) PDF of given that a target was not present,
(ii) PDF of given that a target was present.
On a basis of two sets of foliage clutter data collected by a UWB radar, we show that it is more accurate to describe the amplitude of foliage clutter using log-logistic statistic model ratherthanlog-normal, Weibull, or Nakagami. Log-normal is also acceptable. The goodness-of-fit for Weibull is worse whereas that of Nakagami is the worst. Our contribution is not only the new proposal on the foliage clutter model with detailed parameters, but also providing the criteria and approaches based on which the statistical analysis is obtained. Further, the theoretical study on the probability of detection and the probability of false alarm in the presence of log-logistic foliage clutter are discussed. Future research will investigate the characteristics of targets and the design of radar receivers for the log-logistic clutter so as to improve the performance of target detection, tracking and identification in foliage.
This work was supported in part by the U.S. Office of Naval Research under Grant N00014-07-1-0395 and Grant 00014-07-1-1024 and in part by the National Science Foundation under Grant CNS-0721515, Grant CNS-0831902, Grant CCF-0956438, and Grant CNS-0964713.
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