**1. Introduction**

High frequency (HF) radars have been efficient tools for ocean current, wave, and wind measurement, as well as target detection in the past four decades [1–4]. The interpretation of HF radio scattering from the ocean surface in monostatic and bistatic mode has been developed for several decades. A monostatic HF radar system consisting of a colocated transmitter and receiver operates in backscattering case. The first-order and second-order scattering coefficients for monostatic HF radar, derived by Barrick [5,6] based on Rice's work [7], have been widely accepted [8,9]. Subsequently, new monostatic HF electromagnetic scattering coefficients were proposed by Walsh using generalized functions from rough surfaces [10]. Hisaki and Tokuda also presented the monostatic results using the perturbation method when the illuminated area is finite [11,12]. In contrast to monostatic mode, bistatic HF radar system operates in the non-backscattering case in general. Johnstone presented the scattering coefficients of bistatic HF radar [13]. Anderson et al. [14] obtained a general solution to the bistatic scattering problem and have published a number of papers applying the formulae to various configurations and presenting computed spectra (e.g., [15]); however, they did not publish the details of their derivation. Other theoretical results were proposed by Anderson et al. and validated by field experiments [16–18]. Gill and Walsh developed the first-order and second-order scattering coefficients for land-based bistatic HF radar based on a generalized function [19–21]. Some theoretical results were

validated by Huang et al. using the wind direction measurements from the land-based bistatic HF radar [22]. Recently, Bernhardt gave an incoherent scattering coefficient related to the wave-height spectra for HF Ground-Ionosphere-Ocean-Space (GIOS) system [23].

Air-borne radars have the ability to detect large areas of the sea, which is meaningful to extend coverage for ocean radars [23–25]. To extend the scope of HF radio oceanography and meet the demands for large-area ocean observation, a new bistatic radar model so-called shore-to-air bistatic HF radar is designed for ocean observation [26]. The configuration is shown in Figure 1: The transmitter installed on the coast emits vertically polarized and narrow-beam electromagnetic waves to illuminate the ocean patch in a grazing incidence; the electromagnetic waves are scattered to a radar receiver deployed at an air platform (airplane or airship) owing to the rough sea surface; then the power spectra are estimated from radar echoes to extract the sea state information.

**Figure 1.** Model of shore-to-air bistatic high frequency (HF) radar.

In this paper, beginning with the establishment of the geometry of shore-to-air bistatic HF radar, the electric field intensity is obtained for the perfectly conducting rough ocean surface. Then the first-order and second-order electric field intensities are derived using Rice's method. The electric field near the observation point is obtained based on Kirchhoff theory [27]. The first-order and second-order electromagnetic scattering coefficients are given. Finally, the second-order scattering coefficient is obtained in conjunction with the contribution of hydrodynamic coupling. In order to validate the proposed scattering coefficients, the Doppler spectra are simulated in various scattering and azimuthal angles, operating frequencies, wind speeds, and wind directions.

This paper is organized as follows. The derivation of the first-order scattering coefficient and second-order scattering coefficient is given in Section 2. In Section 3, the simulated Doppler spectra in various operating modes and sea states are presented and analyzed. In Section 4, the singularities that occur in the simulated Doppler spectra are discussed. Conclusions are drawn in Section 5.
