Richard H. Anderson

Maximum likelihood coordinate registration for over-the-horizon radar

Over-the-horizon radar exploits the refractive and multipath nature of high-frequency propagation through the ionosphere to achieve wide-area surveillance. The coordinate registration process converts the group delays and azimuths (i.e., slant coordinates) from a set of multipath target returns to an estimate of its location (i.e., ground coordinates). This is performed by associating the target returns with ray modes determined using a computational electromagnetic propagation model. Not surprisingly, errors in the estimates of down-range ionosphere parameters can seriously degrade the accuracy of the target location estimate. The coordinate registration method presented is designed to achieve improved accuracy by employing a statistical model for uncertainties in the ionosphere. Modeling down-range ionospheric parameters as random variables with known statistics facilitates maximum likelihood (ML) target location estimation, which is more robust to errors in the measured ionospheric conditions. The statistics of down-range ionospheric parameters can be determined using current and historical soundings of the ionosphere. ML target localization consists of determining the most likely target ground coordinates over an ensemble of ionospheric conditions consistent with the data. For greater computational efficiency, the likelihood function is approximated by a hidden Markov model (HMM) for the probability of a sequence of observed slant coordinates given a hypothesized target location. The parameters of the HMM are determined via Monte Carlo execution of a ray tracing propagation model for random realizations of the ionosphere. A simulation study performed using a random ionospheric model derived from ionogram measurements made at Wallops Island suggests that the ML method can potentially achieve average absolute miss distances as much as five times better than a conventional coordinate registration technique.

A generalized Karhunen-Loeve basis for efficient estimation of tropospheric refractivity using radar clutter

In this paper, we consider the problem of obtaining a reduced-dimension parameterization of a propagation medium for the purpose of estimating the medium from transmission data. The application addressed is microwave remote sensing of tropospheric index-of-refraction profiles over the sea surface, using radar clutter returns. The proposed parameterization balances the desire to represent features prominent in the a priori statistics of the profiles versus the need to capture elements of the profile that significantly affect the observed clutter data. In linear estimation problems, basis vectors for the unknown parameter vector that optimizes this tradeoff have been derived as the reduced-rank Wiener filter or, equivalently, the generalized Karhunen-Loeve transform (GKLT). In this paper, we reinterpret the linear result, producing an extension to the nonlinear refractivity estimation problem. The resulting procedure produces basis vectors for tropospheric refractivity that are less dependent on features that have little effect on the clutter measurements. This results in a more efficient parameterization and reduces mean-square estimation error relative to an approach driven purely by the statistical prior. Application of the generalized KL technique to finding efficient basis vectors for refractivity profiles taken off the southern California coast is presented.

Over‐the‐horizon radar target localization using a hidden Markov model estimated from ionosonde data

Uncertainty about the downrange ionospheric conditions is a well-known source of localization errors in over-the-horizon radar. Statistical modeling of ionospheric parameters has recently been proposed in order to derive a maximum likelihood (ML) localization method which is more robust to ionospheric variability. Maximum likelihood coordinate registration consists of determining the most likely target ground coordinates over an ensemble of ionospheric conditions consistent with the data. For greater computational efficiency the likelihood function is approximated by a hidden Markov model (HMM) for the probability of a sequence of observed slant coordinates given a hypothesized target location. In previous work, estimation of the HMM parameters was achieved assuming that the statistics of the underlying ionosphere were known precisely. This paper addresses the problem of estimating the parameters of the HMM from contemporaneous ionospheric sounder measurements. The approach taken here is to treat the plasma frequency profile as a homogeneous random process over the region around the midpoint between the radar and the dwell illumination region. In particular, spatial sampling of a three-dimensional (3-D) ionospheric model, fitted to ionosonde measurements, is used to generate quasi 2-D plasma frequency profile realizations. Estimates of the hidden Markov model parameters are then obtained by using smoothed bootstrap Monte Carlo resampling. A comparison of ML localization and conventional methods, using full 3-D ionospheric modeling and 2-D ray tracing, are given using real data from a known target at a ground range of 2192 km. Results for over 250 radar dwells indicate that the ML localization technique achieves better than a factor of 2 improvement over conventional methods.

Robust altitude estimation for over-the-horizon radar using a state-space multipath fading model

In previous work, a matched-field estimate of aircraft altitude from multiple over-the-horizon (OTH) radar dwells was presented. This approach exploits the altitude dependence of direct and surface reflected returns off the aircraft and the relative phase changes of these micro-multipath arrivals across radar dwells. Since this previous approach assumed high dwell-to-dwell predictability, it has been found to be sensitive to mismatch between modeled versus observed micro-multipath phase and amplitude changes from dwell-to-dwell. A generalized matched-field altitude estimate is presented here based on a state-space model that accounts for random ionospheric and target-motion effects that degrade the dwell-to-dwell predictability of target returns. The new formulation results in an efficient, robust recursive maximum likelihood (ML) estimation of aircraft altitude. Simulations suggest that the proposed technique can achieve accuracy within 5,000 ft of the true aircraft altitude, even with relatively high levels of uncertainty in modeling of dwell-to-dwell changes in the target return. A real data result is also presented to illustrate the technique.

Maximum a posteriori refractivity estimation from radar clutter using a Markov model for microwave propagation

This paper addresses the problem of estimating range-varying parameters of the height-dependent index of refraction over the sea surface in order to predict ducted microwave propagation loss. Refractivity estimation is performed using a Markov model for microwave radar clutter returns from the sea surface. Specifically, the parabolic approximation for numerical solution of the wave equation is used to formulate the problem within a non-linear recursive Bayesian state estimation framework. Solution for the maximum a posteriori (MAP) sequence of range-varying refractivity parameters, given log-amplitude clutter versus range data, is achieved using a technique based on the Viterbi algorithm. Simulation and real data results based on experiments performed off Wallops Island, Virginia are presented which quantify the techniques ability to predict propagation loss at 3 GHz.

Multipath track association for over-the-horizon radar using a bootstrapped statistical ionospheric model

Over-the-horizon (OTH) radar uses the refractive properties of high-frequency radiowave propagation through the ionosphere for wide-area surveillance at long ranges. Ionospheric propagation often gives rise to multiple raymodes between the OTH radar and a target which results in multiple slant tracks from a single target. Multipath and multiple target ambiguities are typically resolved by assuming that the down-range ionosphere is known precisely and then using ray tracing to determine the coordinate registration (CR) transformations from slant coordinates to target locations in ground coordinates. To achieve greater robustness to the uncertainty in down-range ionospheric conditions, this paper presents a maximum a posteriori (MAP) mode linking method for track association that employs statistical modeling of the down-range plasma frequency profile and corresponding multipath slant track measurements. To determine the statistical model parameters from quasi-vertical incidence (QVI) ionogram and wide-sweep backscatter ionogram (WSBI) measurements, the plasma frequency profile is approximated as a homogeneous random process over the region near the mid-point between the radar and the dwell illumination region. Using samples of a 3-D ionospheric model fitted to the and QVIs and WSBIs, statistical propagation model parameters are obtained by smoothed bootstrap resampling combined with Monte Carlo evaluation of a ray tracing propagation model. Simulation results indicate that the MAP mode linking method can achieve nearly a 3:1 improvement in ground coordinate accuracy over conventional mode linking methods with much higher probabilities of correct raymode identification and slant-track-to-target assignment. Real data results from roughly 90 minutes of OTH radar slant track data demonstrate that the MAP mode linking method can provide as much as a 4:1 improvement in ground coordinate accuracy over conventional methods.

The performance of maximum likelihood over-the-horizon radar coordinate registration

A well-known source of target localization errors in over-the-horizon radar is the uncertainty about downrange ionospheric conditions. Maximum likelihood (ML) coordinate registration, using statistical modeling of ionospheric parameters, has recently been proposed as a method which is robust to ionospheric variability. This paper reports ML performance results for real data from a known target using estimates of ionospheric statistics derived from ionosonde measurements. Bootstrap samples derived from these statistics are then used in a hidden Markov model approximation to the ground range likelihood function. Comparison of the ML and conventional methods for over 250 radar dwells indicates the new technique achieves better than a factor of two improvement in ground range accuracy.

Map sequence estimation of microwave refractivity from radar clutter via a particle filtering implementation of the Viterbi algorithm

This paper addresses the problem of predicting microwave propagation loss under ducting conditions by means of estimating the range and height varying index of refraction from observations of radar sea clutter returns. Specifically, the Fourier split-step solution to the parabolic equation for wave propagation is used to formulate the problem into a non-linear dynamic state-estimation framework. Solution for the maximum a posteriori (MAP) sequence estimate of the range-varying refractivity is achieved by extending notions of particle filtering framework to the Viterbi algorithm for state estimation. Real data results based on experiments performed off Wallops Island, Virginia are presented which quantify the proposed methods ability to predict propagation loss at 3 GHz.

Robust altitude estimation for over-the-horizon radar using a state-space model for multipath fading

In previous work, a matched-field estimate of aircraft altitude from multiple over-the-horizon radar dwells was presented. This approach exploits the altitude dependence of direct and surface reflected returns off the aircraft and the relative phase changes of these micro-multipath arrivals across radar dwells. Since this previous approach assumed high dwell-to-dwell predictability, it is sensitive to mismatch between modeled versus observed micro-multipath phase and amplitude changes from dwell-to-dwell. A generalized matched-field altitude estimate is presented based on a state-space model that accounts for random ionospheric and target-motion effects which degrade the dwell-to-dwell predictability of target returns. The new formulation results in an efficient, robust recursive maximum likelihood altitude estimate. Simulation and real data results suggest that the proposed technique can achieve an accuracy within 5000 ft. using 10-20 dwells, even with relatively high levels of uncertainty in modeling of dwell-to-dwell changes in the target return.