This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE JOURNAL OF OCEANIC ENGINEERING 1 Nonlinear Extraction of Directional Ocean Wave Spectrum From Synthetic Bistatic High-Frequency Surface Wave Radar Data Murilo Teixeira Silva , Student Member, IEEE, Reza Shahidi, Member, IEEE, Eric W. Gill, Senior Member, IEEE, and Weimin Huang , Senior Member, IEEE Abstract—This paper proposes the use of a nonlinear inversion technique for the extraction of the directional ocean wave spec- trum from bistatic high-frequency surface wave radar (HFSWR) Doppler data. The extraction method is combined with nonlinear regression expressions, solely based on the Doppler data, to retrieve wind speed and direction. Once the initialization parameters have been defined, a blind iterative algorithm based on Tikhonov reg- ularization in Hilbert Scales is used to extract the nondirectional spectrum. The extracted spectrum is then used to determine the directional factor, which is assumed to be described by a cosine- power model. The proposed method yields fairly good results with synthetic noise-contaminated HFSWR data with a priori regular- ization parameters. Index Terms—Bistatic radar, directional spectrum, HF radar, nonlinear extraction. I. INTRODUCTION S INCE the discovery of the Bragg scattering mechanism over the ocean surface by Crombie [1], high-frequency surface wave radars (HFSWRs) have been used in various ocean engi- neering applications, from target detection to the measurement of the directional ocean wave spectrum [2], [3]. High-frequency radar operation can be generally divided into two main types depending on the geometry: monostatic, where transmitter and receiver are collocated, i.e., within a small fraction of the dis- tance to the scattering patch, and bistatic, where the distances from the transmitter to the receiver, and to the scattering patch, are comparable [4]. In the past decades, a number of methods have been devel- oped to extract the directional ocean-wave spectrum from the returns of a monostatic HFSWR. Lipa [5], [6] was the first to Manuscript received August 20, 2018; revised December 16, 2018 and March 25, 2019; accepted April 3, 2019. This work was supported by the Natural Sci- ences and Engineering Research Council of Canada (NSERC) under Discovery Grants to W. Huang (NSERC RGPIN-2017-04508 and RGPAS-2017-507962) and E. W. Gill (NSERC RGPIN-2015-05289). This work was presented in part at the MTS/IEEE OCEANS Conference, Aberdeen, Scotland, June 19–22, 2017 and at the MTS/IEEE OCEANS Conference and Exposition, Charleston, SC, USA, October 22-25, 2018. (Corresponding author: Murilo Teixeira Silva.) Associate Editor: R. Romeiser The authors are with the Department of Electrical and Computer Engi- neering, Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, NL A1B 3X5, Canada (e-mail: murilots@mun.ca; d97rs@mun.ca; ewgill@mun.ca; weimin@mun.ca). Digital Object Identifier 10.1109/JOE.2019.2909961 extract the ocean spectrum by treating the monostatic second- order radar cross section proposed by Barrick [7]. Later, Wy- att [8] extended the Chahine–Twomey relaxation method to in- vert Barrick’s cross-section equations from measured radar data. Meanwhile, Howell and Walsh [9] developed an algorithm to ex- tract ocean wave information from data obtained from single, or multiple narrow beam HF-radar systems, using both the Barrick and Walsh et al. [10] cross-section formulations. Hisaki [11] was the first to perform the nonlinear extraction of ocean spectrum from monostatic HFSWR data. He developed a nonlinear optimization algorithm to invert Barrick’s cross sec- tion, applying several constraints due to the ill-posedness of the problem. A more recent approach to the nonlinear extraction problem was proposed by Shahidi and Gill [12], where Bar- rick’s second-order cross section was further reduced to a sum of single-integral nonlinear Fredholm operators of the first kind, and then nonlinear least-squares optimization using Levenberg– Marquard was applied to simulated HF-Radar data to extract the ocean-wave spectrum. While significant effort has been expended on inversion meth- ods for monostatic HFSWR, the same variety of extraction meth- ods does not exist for the bistatic case. According to Teague et al. [13], the first efforts to implement a bistatic HF surface- radar system came in the late 1960s, in a collaboration between Peterson, Munk, and Nieremberg. Peterson would later work with Teague and Tyler to conduct experiments that would lead to the first measurements of the directional ocean-wave spec- tra for swell using bistatic HFSWR [14]. Further observations by Teague [15] concluded that a more refined, multifrequency experiment would be necessary to obtain the directional ocean- wave spectrum for the local sea. However, due to the ubiquity of monostatic radar for HF ground wave propagation, research on the bistatic case has not evolved at the same pace. In 2001, Gill and Walsh [16] presented the first- and second-order radar cross sections using the generalized-function approach of Walsh et al. [10], which was later used by Zhang and Gill [17] to extract the ocean wave spectrum by adapting the technique of Howell and Walsh [9]. This paper presents a nonlinear method for the extraction of the directional ocean-wave spectrum from bistatic HFSWR data, based on the bistatic cross-section formulation proposed by Gill and Walsh [16], and the nonlinear extraction technique pre- sented by Shahidi and Gill [12]. In a previous work, an adaptive 0364-9059 © 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.