532 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 38, NO. 1, JANUARY 2000 Detection of Ship Wakes Using an Airborne Magnetic Transducer Nan Zou and Arye Nehorai, Fellow, IEEE Abstract—Wakes induced by the motion of vessels may extend for tens of kilometers and exist for hours under certain conditions in the open sea. This provides a useful feature for long-range ship detection. We present a method for passively detecting a ship wake using measurements obtained by an airborne SQUID magnetic transducer that measures the first order gradients of the magnetic signature induced by the wake. Analytical formulas for wake magnetic gradients are derived to provide guidelines for using the airborne detector. We also derive bounds on the probability of wake detection for crosscorrelation and square-law detectors, which can be used to predict the expected performance. Numerical examples are used to demonstrate the applicability of our results. Index Terms—Detection, magnetic transducer, wakes. I. INTRODUCTION W AKES generated by submerged or surface vessels in open sea may extend for tens of kilometers and exist for hours under certain conditions [1]. Sea water is a conductor sit- uated in the Earth’s magnetic field. Electrical current is induced in the sea by the motion of water waves in the presence of this magnetic field. The current induces its own magnetic field that can be sensed by a high-sensitivity magnetic transducer [2]–[5]. Previously, measurements of magnetic fields over oceans were limited to mapping large-scale irregular features of the Earth’s field because of limitations on sensor sensitivity and the diffi- culty of separating spatial and temporal fluctuations produced by ocean waves, ionospheric currents, and geological sources [6], [7]. However, the advent of SQUID’s (superconducting quantum interference devices) has offered unprecedented sensitivity [8], [9]. The invention of high-temperature SQUID (HTS) has further reduced the dimensions and logistic support of the SQUID sensor and has made mobile detectors more practical than ever. An HTS gradiometer with a sensitivity of 300 at 0.1 Hz and recharge time of more than one month has been reported [10]. In this paper, we introduce an approach for passive sensing of a wake using a high-sensitivity SQUID airborne detector. Some guidelines as to platform speed and encounter angle of the detector will be presented. We consider two types of detec- tors, namely cross-correlation and square-law detectors to eval- Manuscript received September 10, 1998; revised May 24, 1999. The work of A. Nehorai was supported by the Office of Air Force of Scientific Research under Grant F49620-97-1-0481, the National Science Foundation under Grant MIP-9615590, and the Office of Naval Research under Grant N00014-98-1- 0542. N. Zou is with the DSO National Laboratories, Republic of Singapore (e-mail: znan@dso.org.sg). A. Nehorai is with the EECS Department (M/C 154), The University of Illi- nois at Chicago, Chicago, IL 60607-7053 USA (e-mail: nehorai@eecs.uic.edu). Publisher Item Identifier S 0196-2892(00)00421-6. uate the system performance. Detection probabilities based on a binary hypothesis test will be derived for these detectors. Sec- tion II discusses the theory of electromagnetic fields generated by sea water motion. We introduce an analytical second-order ordinary differential equation (ODE) to describe the magnetic signal. Section III describes the hydrodynamic theory of ship wakes and their wave number spectra. Section IV details the detection scenario. We derive a spatio-temporal spectra-conver- sion using the motion parametric equations of the airborne de- tector. Analytical formulas are derived for the magnetic gradient tensor. Section V contains two detection statistics that can be used to estimate upper and lower values of detection probability, and Section VI contains numerical results based on the models discussed in Sections VI and V. II. ELECTROMAGNETIC FIELD GENERATED BY MOVING SEA WATER We assume that the ocean surface is perfectly planar with the Cartesian coordinate system oriented such that is the local ver- tical, where defines the region above the ocean surface. The horizontal axis is directed in the opposite direction to the ship’s course; the axis, perpendicular to the ship’s course. The ship sails with velocity and magnitude of . The Earth’s ge- omagnetic field vector is inclined at an angle between the axis and magnetic north with dip angle . The airborne de- tector, flying parallel to the sea surface at height and uniform speed making an acute angle with the axis, measures the ship-induced magnetic field immediately behind the vessel. Fig. 1 illustrates this geometry. Let be the magnitude of the geomagnetic field . Then (2.1) where denotes the unit vector pointing in the direction of (2.2) where , , and are unit vectors pointing in the directions of the , , and axes. Let be the velocity of the water wave and and be the water’s electromagnetic conductivity and dielectricity, respec- tively (normally, sea water has a conductivity of about 4 Mho/m and a relative dielectricity of about 80). The wake-induced elec- tric and magnetic fields are denoted by and , respectively. Our analytical model neglects attenuation due to the prop- agation of electromagnetic waves through sea water, which is justified because of the low frequencies involved and the cor- responding large wave lengths as follows. The electromagnetic waves are characterized by their wavelength, . For a good con- 0196–2892/00$10.00 © 2000 IEEE