## 1. Introduction

[2] In a high frequency surface wave radar (HFSWR) system, the transmitted high frequency (HF) radar waveform propagates primarily along the surface of the Earth, typically over the ocean. The signal diffracts around the curvature of the Earth to illuminate ship and low-flying aircraft targets beyond the line-of-sight horizon [*Sevgi et al.*, 2001a, 2001b]. However, during transmission, a small amount of radiation is unavoidably radiated vertically and the direct reflection from the ionosphere is observed as intense radar clutter [*Chan*, 2003]. This clutter imposes detection range limitations in long-range HFSWR systems. In this paper we refer to this type of clutter as “ionospheric” clutter, following HFSWR literature conventions.

[3] In an over-the-horizon radar (OTHR) system, the ionosphere is used as a reflecting surface to permit illumination of targets at distances beyond the horizon [*Headrick*, 1990]. The transmitter beams a radar signal at an oblique angle to the ionosphere, the signal reflects from the ionosphere, and the target, such as an airplane or a ship, is illuminated by the signal. The target scatters the signal back to the transmitting location along a similar propagation path. At high latitudes, the target signals are often masked by radar clutter comprising echoes from the Earth's aurora zone [*Elkins*, 1980], which we denote as “auroral clutter”.

[4] For successful target detection, HFSWR and OTHR systems require target signals to be well separated from clutter in range, Doppler, or angle-of-arrival. In quiet ionospheric conditions, this separation can often be achieved. However, in disturbed ionospheric conditions, plasma density irregularities in the ionosphere can lead to the spreading of the clutter signal in the Doppler and angle-of-arrival domains [*Vallieres et al.*, 2004]. The ionospheric plasma density irregularities represent refractive index fluctuations, which serve to scatter the radar signal wave packet. The spread clutter signal can occupy a large number of radar range-Doppler-angle resolution cells, leading to the masking of targets.

[5] Given the size and cost of these radar systems, target detection analysis based purely on measured data is difficult. For example, one approach to evaluating clutter mitigation schemes is to compare probability of detection versus signal-to-noise ratio (SNR) curves based on a common false alarm rate and a corresponding detection threshold. However, it is unusual to have at hand adequate measured data to accurately estimate these parameters. Quantifying the impact of clutter on target detection schemes therefore requires simulations. In turn, these simulations must be based on a quantitative model of the space-time radar clutter properties, such as the Doppler and angle-of-arrival spectra of the clutter.

[6] It is important to distinguish the source mechanism of the clutter versus the mechanism by which the clutter signal is spread in Doppler and angle-of-arrival. In the case of HFSWR, the clutter source mechanism is the reflection from the bottomside of the Earth's ionosphere. In the case of OTHR, the source mechanism is the Bragg backscatter from the aurora. However, the mechanism of spreading in Doppler or angle-of-arrival is of much greater consequence for target detection. It is the existence of forward scattering during the propagation of radar signals through volumes of plasma irregularities that leads to the spreading phenomenon. More precisely, in phased-array radar systems, the Doppler spread is due to the randomization of the signal pulse-to-pulse temporal phase relationships and the angle-of-arrival spread arises from a similar randomization of the signal sensor-to-sensor spatial phase relationships. Thus the central feature of the propagation models is the phase scintillation experienced by the radar signal upon transiting the ionospheric medium. It should also be noted that some radars suffer from range ambiguity such that range-folded ground clutter can mask non-range-folded targets. While this is a waveform issue and is beyond the scope of this paper, it should be noted that ground clutter spreading could be modeled with the same tools that are provided in this paper.

[7] The modeling of wave propagation through random media is a well-studied problem. As already mentioned, we use the term “propagation” in a broad context to include both the movement of a radar wave packet in a smooth medium and the forward-scattering effects that occur when the wave packet moves through a medium containing irregularities. A recent monograph on the subject [*Wheelon*, 2001] promulgates a three-level hierarchy for describing the methods that have been applied to the problem, which we briefly discuss here. The first and most basic method in the hierarchy is geometric optics, whereby the scattered wavefield can be expressed in terms of a line integral of the refractive index fluctuations [*Tatarskii*, 1971; *Rufenach*, 1975; *Coleman*, 1996]. The line integral is taken along the zero-order signal ray trajectory, namely the ray that would occur in the absence of first-order refractive index fluctuations. Geometric optics provides good modeling of the resulting fluctuations of the signal phase and angle-of-arrival, but does not inherently contain diffractive effects. It is thus unable to accurately model fluctuations in signal amplitude or intensity. Geometric optics can be extended in an ad hoc manner to include diffractive effects by modeling the ionosphere as a series of phase screens interpolated by free space [*Dana and Wittwer*, 1991; *Nickisch*, 1992; *Nickisch et al.*, 2011].

[8] As a result of this paper's focus on pulse-to-pulse and sensor-to-sensor signal correlation, our main interest is in the signal phase. A geometric optics approach is thus suitable for our application. As a supporting example from the literature, if we take the plane wave limit of*Knepp* [1983, equation 30] (by setting the transmitter-phase screen distance*z*_{t} to infinity), it is clear that the spatial and temporal correlation lengths are controlled dominantly by the phase variance *σ*_{ϕ}^{2}. However, if there was interest in amplitude or intensity fluctuations, for the purpose of modeling signal fading for instance, one could consider the second method in the hierarchy of [*Wheelon*, 2001], referred to as the method of smooth perturbations [*Tatarskii*, 1971] or the Rytov method [*Gherm et al.*, 2005]. In this method, the scattered wavefield is expressed as an integral over a volume of refractive index fluctuations, which allows for the accounting of diffractive effects. A key feature of this method is the expression of the scattered wavefield as an exponential of a function to be determined. By allowing the function to be complex-valued, both amplitude and phase effects are captured in a straightforward manner.

[9] For completeness, we mention the third method in the hierarchy, which is referred to as the method of strong fluctuations, where the scattered field is expressed as a functional integral with respect to the refractive index fluctuations [*Rytov et al.*, 1989]. As the name suggests, this method handles large-amplitude refractive index fluctuations where the other methods do not provide convergent solutions.

[10] In this paper, we report on the use of geometric optics models developed to simulate radar data. These data are tested using measurements of ionospheric and auroral clutter. In addition, as an application of the models, we use simulated data to test the efficacy of a newly developed clutter mitigation space-time adaptive processing (STAP) algorithm named the fast fully adaptive (FFA) algorithm [*Saleh et al.*, 2009]. The aim of this algorithm is to reduce the radar clutter sample support requirements of the adaptive processor. Sample support is required to estimate radar clutter sensor covariances matrices, which are the essential components for most STAP schemes. In particular, HF radar STAP schemes are vulnerable to sample support shortages, since the relatively coarse resolution severely restricts the number of independent samples available for clutter covariance estimation. The FFA alleviates the sample support problem by casting the STAP algorithm as a multistage process, where each stage operates with a greatly reduced number of degrees of freedom with similarly reduced requirement for sample support.

[11] The rest of the paper is organized as follows. The clutter models for the HFSWR and OTHR scenarios are presented in sections 2 and 3, respectively, and a brief comparison with measured data is provided. In section 4, the suite of available FFA algorithms is summarized. In section 5, results of simulations based on the data models are presented and the performance of FFA approaches in terms of the probability of target detection versus SNR is evaluated. In section 6, the paper concludes.