David V. Arnold

Teaching electromagnetic field theory using differential forms

The calculus of differential forms has significant advantages over traditional methods as a tool for teaching electromagnetic (EM) field theory. First, films clarify the relationship between field intensity and flux density, by providing distinct mathematical and graphical representations for the two types of fields. Second, Amperes and Faradays laws obtain graphical representations that are as intuitive as the representation of Gausss law. Third, the vector Stokes theorem and the divergence theorem become special cases of a single relationship that is easier for the student to remember, apply, and visualize than their vector formulations. Fourth, computational simplifications result from the use of forms: derivatives are easier to employ in curvilinear coordinates, integration becomes more straightforward, and families of vector identities are replaced by algebraic rules. In this paper, EM theory and the calculus of differential forms are developed in parallel, from an elementary, conceptually oriented point of view using simple examples and intuitive motivations. We conclude that because of the power of the calculus of differential forms in conveying the fundamental concepts of EM theory, it provides an attractive and viable alternative to the use of vector analysis in teaching electromagnetic field theory.

Extending the phase gradient autofocus algorithm for low-altitude stripmap mode SAR

The phase gradient autofocus (PGA) algorithm has been widely used in spotlight synthetic aperture radar (SAR) to remove motion-induced blurs in the images. The PGA algorithm has been proven to be a superior autofocus method. This algorithm is extended for application to low-altitude stripmap mode SAR. Standard PGA algorithms are formulated only for spotlight-mode SAR. We describe a new algorithm which allows PGA to be applied to stripmap-mode SAR. Three SAR images with different characteristics are used in demonstrating this algorithm.

Electromagnetic Green functions using differential forms

In this paper we redevelop the scalar and dyadic Green functions of electromagnetic theory using differential forms. The Green dyadic becomes a double form, which is a differential form in one space with coefficients that are forms in another space, or a differential form-valued form. The results presented here correspond closely with the usual dyadic treatment, but are clearer and more intuitive. Many of the usual expressions using green functions in vector notation require a surface normal; with the Green forms the surface normal is unnecessary. We illustrate the formalism by computing scattering from a randomly rough conducting surface and deriving the Green form for a dielectric half-space. We also define the interior derivative, which is equivalent to the coderivative but for a constant metric has a computational rule dual to that of the exterior derivative and simplifies calculation in coordinates. This work makes available some of the tools that have not yet been presented in the language of differ...

Green Forms for Anisotropic, Inhomogeneous Media

The dyadic Green function for an anisotropic, inhomogenous medium is reformulated as a double differential form by embedding material properties in the Hodge star operator. This formalism simplifies the manipulation of differential operators in the anisotropic case. An integral equation is obtained which relates the Green form for the electric field to a generalization of the usual free space scalar Green function.

Dependence of the normalized radar cross section of water waves on Bragg wavelength-wind speed sensitivity

Measurements of the normalized radar cross section (/spl sigma//spl deg/) made by the YSCAT ultrawideband scatterometer during an extended deployment on the Canada Centre for Inland Waters (CCIW) Research Tower located at Lake Ontario are analyzed and compared with anemometer wind measurements to study the sensitivity of /spl sigma//spl deg/ to the wind speed as a function of the Bragg wavelength. This paper concentrates on upwind and downwind azimuth angles in the wind speed range of 4.5-12 m/s. While YSCAT collected measurements of /spl sigma//spl deg/ at a variety of frequencies and incidence angles, this paper focuses on frequencies of 2.0, 3.05, 5.30, 10.02, and 14.0 GHz and incidence angles within the Bragg regime, 30-50/spl deg/. Adopting a power law model to describe the relationship between /spl sigma//spl deg/ and wind speed, both wind speed exponents and upwind/downwind (u/d) ratios of /spl sigma//spl deg/ are found using least squares linear regression. The analysis of the wind speed exponents and u/d ratios show that shorter Bragg wavelengths (/spl Lambda/<4 cm) are the most sensitive to wind speed and direction. Additionally, vertical polarization (V-pol) /spl sigma//spl deg/ is shown to be more sensitive to wind speed than horizontal polarization (H-pol) /spl sigma/, while the H-pol u/d ratio is larger than the V-pol u/d ratio.

Development of a low cost, FM/CW transmitter for remote sensing

A generic, low cost, compact Frequency Modulated Continuous Wave (FM/CW) transmitter has been developed for use from 2 to 18 GHz with a transmit bandwidth of up to 1 GHz. This generic design is currently being used in a railway hazard detection system, a synthetic aperture radar, and an altimeter. Capitalizing on the functionality and low cost of wireless communication parts, an inexpensive broad band linear frequency generator was developed which is extremely small in size and lightweight. This design incorporates the functionality of a microcontrollers counter to measure the frequency at a given voltage and thus calibrate a digital waveform to drive a VCO with linear frequency modulation. The microcontroller communicates with a host using a RS232 serial port, enabling the user to set parameters such as center frequency, bandwidth and repetition count. A trigger is used to synchronize receiver analog to digital conversions with the transmitted signal. Tests show the transmitter waveform has low deviation from linearity and level output power over the entire band.

Generalization of the geometrical-optics scattering limit for a rough conducting surface

We consider the backscattering coefficient of a perfectly conducting, one-dimensional random rough surface in the physical-optics approximation. The high-frequency limit of physical optics yields the geometrical-optics scattering coefficient with Gaussian dependence on incidence angle. We demonstrate that for finite frequencies and surfaces with infinite slope variance the Gaussian form of the geometrical-optics limit generalizes to an α-stable distribution function. The proof of this result employs an asymptotic method that can be interpreted as a refinement of the central-limit theorem of probability theory for infinite-variance random variables. The theory leads to an effective cutoff of the surface-height power spectral density. The backscatter is not sensitive to surface components with wave number above this spectral cutoff, thus eliminating the nonphysical dependence of geometrical optics on surface features much smaller than the wavelength of the incident field. The composite or two-scale surface model is also derived as a term in a series expansion of the stable distribution. Comparison with numerical results shows that the approximation, although asymptotic, remains accurate for relatively low values of the surface roughness parameter.

Interferometric radar principles in track hazard detection to improve safety

The IN-RAIL track hazard detection system uses radar interferometry to improve railroad safety by detecting dangerous situations that occur on railways. Common detectable hazards on railroads include rockslides and washouts. The IN-RAIL radar system provides an efficient and cost effective way to detect these problems, thus, improving railway safety. The IN-RAIL radar consists of one transmit antenna and two receive antennas. The transmit antenna sends out a linearized frequency modulated chirp. Two horn antennas receive the reflected signal from the scene. The received signal is amplified, passed through a matched filter, and sampled. The authors process the sampled data using a Fast-Fourier-Transform algorithm which gives frequency and phase information. Frequency in the signal is proportional to range in the observed scene. The phase of each range bin contains the topographical information of the scene. Other algorithms are then used to determine the topography of the scene. By comparing the topography given by the current set of radar data to a known safe topography the authors can then determine if there is a hazard on the railroad tracks.

YSAR: a compact, low-cost synthetic aperture radar

The Brigham Young University Synthetic Aperture Radar (YSAR) is a compact, inexpensive SAR system which can be flown on a small aircraft. The system has exhibited a resolution of approximately 0.8 m by 0.8 m in test flights in calm conditions. YSAR was used to collect data over archeological sites in Israel. Using a relatively low frequency (2.1 GHz), the authors hope to be able to identify walls or other archeological features to assist in excavation. A large data set of radar and photographic data were collected over sites at Tel Safi, Qumran, Tel Micnah, and the Zippori National Forest in Israel. The authors show sample images from the archeological data. They are currently working on improved autofocus algorithms for this data and are developing a small, low-cost interferometric SAR system (YINSAR) for operation from a small aircraft.

A low cost, radio controlled blimp as a platform for remote sensing

The authors propose using a radio-controlled blimp as a new platform for microwave remote sensing. Although blimps have been used for aerial photography, they have not been applied in precision remote sensing. Among the advantages suggested are low cost, low maintenance, ease of deployment, and minimal personnel requirements. Applications are given of typical radar systems for which a remotely controlled blimp is a suitable platform. Among these systems, the authors suggest microwave radar systems such as synthetic aperture radars (SAR), altimeters, and scatterometers. Such radar systems are currently used for archaeological and geological studies typically from aircraft or satellites, but they propose using a blimp as a low cost alternative for these systems. As an example, the authors present an altimeter system designed to support NASAs Jason-1 launch. In support of this launch, testing is scheduled for work on landslide detection and mean sea level measurement. For the altimeter the remote controlled blimp will be used as the platform for these measurements. They acknowledge the funding support of NASA for this study.