Signal inversion for target extraction and registration (SIFTER) is a technique for enhanced radar target detection and coordinate registration (CR). The technique is currently being developed for over-the-horizon radar (OTHR). The power received by a radar as a function of radar coordinates and Doppler shift may be expressed as an integral operator applied to the field of backscatter cross section (FBC) per unit area of the illuminated region. The FBC is a function of the geographic position and the velocity vector of the scattering point. The kernel of the integral operator is treated as a known function that is determined by the model of the propagation channel, antenna patterns, and the radar signal processing parameters. The relationship between the FBC and the received power is treated as an integral equation to be resolved with respect to the FBC. This inversion is accomplished using techniques for ill-posed problems. The obtained FBC is subsequently analyzed for peaks. The location of a peak provides an estimate for actual ground coordinates and velocity vector of the target. Gain in sensitivity and CR is achieved by introducing a continuity equation, which evolves the FBC between radar revisit cycles. The evolved FBC is corrected using data from each subsequent revisit. Two versions of the SIFTER algorithm are presented. One is based on Tikhonov’s method, and the other is based on the Kalman filter method. SIFTER concepts and performance are demonstrated using modeled data.
 In the traditional radar “peak picking” approach to target detection, the target must be large enough in cross section to provide a peak (or multimode peaks) significantly above the noise level at the Doppler frequency associated with the target’s radial velocity. Conventional peak picking cannot discern weak target returns from noise. No attempt is made to interpret the signal below threshold as potential multimode partners of detected peaks. Detection of small targets is further impeded by the presence of clutter. Slow targets may be obscured by clutter from the ground or ocean surface. Once peaks associated with a target are detected, the geographic location of the target must be determined. Typically each detected peak is assigned to an available propagation mode using a set of predetermined rules (mode linking). Given the chosen mode assignments for each target, a weighted centroid of predicted ground locations for the mode set is taken as the target position.
 Instead of simple threshold-based peak picking of individual multimode returns in amplitude-range-Doppler (ARD) space, the signal inversion for target extraction and registration (SIFTER) method exploits the knowledge of propagation channels, radar antenna pattern, and signal processing to interpret the entire ARD map for all azimuth beams simultaneously, solving for the distribution of scatterers that yields the measured ARDs. This distribution of scatterers is also called the “scattering surface.” Because of the highly correlated nature of returns produced in overlapping beams by localized targets compared to the weakly correlated returns from clutter sources (which are continuously distributed over the ground surface), targets are enhanced over clutter in the inverted distribution of scatterers. Further target detection gain is realized because distributed power in the ARDs from multiple propagation modes is, in effect, combined by the method before peak detection occurs. All propagation modes predicted to exist by the propagation model are used. This enhances coordinate registration (CR) as well because only a single peak is returned per target that is positioned geographically to reproduce the measured ARD. The error-prone step of mode linking is eliminated, and with it go the large CR errors associated with misidentified modes.
 The SIFTER solutions can be evolved in time for yet further gain. Spurious noise peaks that do not evolve in time in a way consistent with their Doppler velocity are quickly smoothed away by the time-evolution form of SIFTER, exposing even quite weak targets.
 In section 2 we describe in detail the concepts of SIFTER and its algorithms. Section 3 contains a demonstration of the sensitivity and accuracy of the SIFTER technique using simulated data.
2. SIFTER Concepts
2.1. Equivalent Scattering Surface
 Electromagnetic returns received by a surface radar such as over-the-horizon radar (OTHR) are composed of waves backscattered by objects and structures located on or near the ground. These scatterers include boats, aircraft, ocean waves, mountain ridges, buildings, etc. The received power may be expressed in terms of the backscatter cross section per unit area of the Earth’s surface, σ(ρ, ϕ), where ρ and ϕ are the range and azimuth (geographic position) of the scattering point. Consequently, we shall use σ(ρ, ϕ) to characterize the distribution of scatterers. Note that targets produce peaks in σ(ρ, ϕ). The measured received power Y is a function of slant range ρs and slant azimuth ϕs, and its relation to σ(ρ, ϕ) has the general form [Croft, 1967]
where the kernel K(ρs, ϕs, ρ, ϕ) is determined entirely by ray-tracing calculations in the model of the environment (an ionospheric model in case of OTHR), by the radar receive and transmit antenna patterns, and by the shape of range window. For simplicity we have ignored here the Doppler dimension of the data in equation (1).
 A comprehensive model of the environment combined with complete ray-tracing algorithms and radar response functions contains everything necessary to determine K(ρs, ϕs, ρ, ϕ). In equation (1), Y and K are known functions and σ is unknown. In this sense it is an integral equation for σ. An alternative to the conventional method for target detection and CR is to solve (or invert) this equation. That is, if we find a solution for σ(ρ, ϕ) that corresponds to Y(ρs, ϕs), then targets will appear as peaks in σ(ρ, ϕ). The location of a peak provides an estimate for the actual range and azimuth of the target. We call such a solution the “equivalent scattering surface.” The word “equivalent” stresses that σ(ρ, ϕ) does not necessarily coincide with the actual scattering surface because the solution of equation (1) may not be unique. This is the essence of the SIFTER technique for a single dwell. Gain in sensitivity is achieved by effectively combining simultaneous data from different bins of the slant space.
Equation (1) is a Fredholm equation of the first kind. Solutions of such equations are known to be unstable, and sophisticated regularization techniques must be applied to obtain sensible results. A very powerful method of regularization (Tikhonov’s method) for multidimensional problems has been previously developed by the authors [Fridman, 1998; Nickisch et al., 1998; Fridman and Nickisch, 1999]. These highly developed algorithms can be directly used to solve equation (1).
2.2. Continuity Equation
 Further gain in sensitivity may be achieved by accumulating the SIFTER solution in time using the Kalman filter method. In our approach the whole scattering surface is treated as the state vector of the dynamic system being observed.
 The assumption used for evolving the SIFTER solution in time is that the cross sections of the targets of interest are a slowly varying function of time, so that to a good degree of accuracy we have
The mechanical state and the short-term motion of a target are characterized by its position and velocity vectors. Thus, for tracking the evolution of the scattering surface, we assume that it is a function of the range, azimuth, range rate (vρ = dρ/dt), and azimuth rate (vϕ = dϕ/dt) variables:
The total time derivative of the scattering surface is given by
where aρ = dvρ/dt and aϕ = dvϕ/dt are components of target acceleration.
 For the majority of surface radar applications, there is no explicit relationship between the target state vector (ρ, ϕ, vρ, vϕ) and target acceleration. Surface targets may occasionally change course or speed in a manner that is hard to predict. We choose to treat the terms with acceleration in equation (4) as random noise (if accelerations of a particular kind are of interest, the specified acceleration information could be built into equation (4) to effectively filter only targets with that kind of acceleration). Within this line of reasoning, we combine equations (2) and (4) to produce the continuity equation,
Here τ is the relaxation time and η(t, ρ, ϕ, vρ, vϕ) is the random source of scatterers that effectively accounts for targets that unpredictably change course or speed. The same random noise term may also account for the unpredictable fluctuations of the scattering cross section. The changes in scattering cross section may occur, for example, due to Faraday rotation or aspect angle variations. The empirical relaxation parameter τ sets an effective limit on the duration during which a predicted target trajectory can be trusted without updating. Equation (5) postulates a model of target dynamics. In this model, targets are allowed to be in any point of the ground space, to move in all possible directions with all possible velocities, and to possess arbitrary scattering cross section. Targets maintain constant velocities, but their scattering properties evolve in accordance with equation (5), which can be rewritten as
This form of the continuity equation explicitly shows the meaning of the parameter τ as the relaxation time (one can see that in the absence of random sources any initial distribution of scatterers will decay proportional to exp(−t/τ)).
 In practical applications the attenuation time is set to comparatively large values, while the random source η is small.
Equation (5) governs the evolution of the scattering surface. It can be solved using the method of characteristics. Given the state of the scattering surface at the instant t0, σ(t0, ρ, ϕ, vρ, vϕ), the surface at t > t0 may be expressed as
where is the random component of σ,
2.3. Tracking the Scattering Surface
 The relationship between the scattering surface and the radar measurements discussed in section 2.1 along with the continuity equation discussed in the previous section provide the framework for formulating the problem of tracking the scattering surface.
Equations (1) and (6) may be used to formulate a discrete time Kalman-filter-like tracking problem. Indeed, for the nth radar revisit we can write the following equations:
Here Yn (ρs, ϕs, vD) is the observed power for the range-azimuth-Doppler bin specified by the parameters ρs, ϕs, and vD, correspondingly; χn(ρs, ϕs, vD) is the measurement noise for the same bin; Δt = tn − tn−1; and ηn(ρ, ϕ, vρ, vϕ) is associated with the random noise in the continuity equation. Equation (9) is an obvious generalization of equation (1), where we incorporated the dependence of the scattering surface on the velocity components and the dependence of the received power on Doppler velocity vD. The kernel M(ρs, ϕs, vD, ρ, ϕ, vρ, vϕ) may be found with the help of ray-tracing calculations and is considered here a known function. Also known are the parameter τ and certain statistical characteristics of the noise terms χn(ρs, ϕs, vD) and ηn(ρ, ϕ, vρ, vϕ). The measurement noise term may also incorporate the errors associated with the inaccuracy of the kernel caused by the uncertainty of the propagation model.
 The crucial element of the SIFTER analysis is estimation of the scattering surface. Discrete targets are subsequently detected as peaks that eventually form on the surface. An advanced procedure for estimating (or tracking) the scattering surface σn must utilize the time series of available observations (Yn, Yn−1 …), the relationship between σn and the observations equation (9), as well as the knowledge of the evolution equation (10). We have developed and tested two approaches to tracking the scattering surface. The first is a straightforward application of the Kalman filter method. The second approach is based on the Tikhonov’s inversion technique.
2.3.1. Kalman Filter Tracking of the Scattering Surface
 The Kalman filter technique [Kalman, 1960] allows one to obtain maximum likelihood estimates for unknown parameters of a linear system. In order to apply this method we introduce a four-dimensional grid in the target phase space (ρ, ϕ, vρ, vϕ). Then σn (ρ, ϕ, vρ, vϕ) is approximated by a vector σn that contains the values of the scattering surface at the nodes of this grid, the integral in equation (9) is approximated by a finite sum, and σn−1 (ρ − vρ Δt, ϕ − vϕ Δt, vρ, vϕ) in the right-hand-side of equation (10) is expressed in terms of grid values stored in σn−1 with the help of appropriate interpolation. After that, equations (9) and (10) take the following form:
Here Yn is the observed vector that contains measured power for all range-azimuth-Doppler bins, and Mn and Φn are known matrices. The vector σn is treated as the state vector of a dynamic system governed by equation (12). It is further postulated that χn and ηn are Gaussian random variables such that
where covariance matrices Sn and Qn are known.
 Relationships (11)–(13) constitute the formulation of the Kalman filtering problem. The optimum estimate of the state vector and its covariance matrix Pn may be found recursively using the following relationships:
This procedure for Kalman filter tracking of the scattering surface will be tested on simulated data in section 4. An important practical problem associated with initializing the covariance matrix of the solution will be addressed there as well.
2.3.2. Tikhonov Filter Tracking of the Scattering Surface
 We will introduce here a somewhat generalized interpretation of the original regularization technique developed by Tikhonov and Arsenin . This interpretation will explicitly show similarities between the Tikhonov method and the Kalman filter approach.
 The Tikhonov method offers a robust procedure for building a stable solution of a system of linear equations
where Y is a known (measured) vector and χ is a vector representing random noise with known covariance matrix S:
The size of the unknown vector σ is usually larger than the size of the measured vector (dim σ > dim Y).
 In order to build the regularized solution, a symmetric positive-definite matrix P with rank P = dim σ is postulated. The quadratic form
is called the stabilizing functional or stabilizer. The solution that satisfies the condition
and minimizes the stabilizer
provides the regularized solution. Note that the meaning of equation (21) is that the error of the fit provided by is consistent with the errors of measurements that are characterized by S.
 The positive-definite matrix P is usually selected in such a way that the numerical value of the stabilizing functional may be interpreted as a “measure of smoothness” of the solution . So that the smoother is the solution, the smaller is the stabilizing functional, and is the smoothest possible solution. Thus solving the problem stated by equations (21) and (22) may be interpreted as selecting the smoothest solution (requirement expression (22)) among all that agree with experimental data within the errors of measurements (condition (21)). The requirement of smoothness is imposed in order to eliminate solutions that contain numerous peaks, which do not correspond to real targets. A concrete example of designing a suitable stabilizing functional is presented in section 4 below (expression (38) and related discussion).
 The problem stated by equations (21) and (22) is a conditional extremum problem, and it may be solved using the method of Lagrange multipliers as follows:
 Find σ as a function of a scalar parameter α (denoted below as σα) by minimizing the expression
 Resolve the following equation with respect to
Comparison of equations (14) and (17) (put σn-1 = 0 for simplicity) with equations (26) and (27) suggests that αP in the Tikhonov method plays a role similar to the solution covariance matrix in the Kalman filter approach. This observation suggests a modification of the Kalman filter algorithm as described below (we will name this algorithm the Tikhonov filter).
 Given a sequence of measurements Yn on a dynamic system
the evolution equation of the system
the covariance matrix of the errors of measurements
and a positive definite matrix P (the stabilizing matrix), we build a recurrent procedure for estimating the system’s state vector based on the following principles. Given the state vector estimate , the next estimate is found from conditions similar to equations (21) and (22):
Solution of this problem yields
where Kα is given by equation (27) and is determined as a solution of the scalar equation
Thus each subsequent update of the state vector is obtained by introducing the minimal (in the sense of the norm defined by equation (31)) change to the predicted state vector so that the measurements agree with the model within error tolerances specified by Sn.
 The Tikhonov filter process may be viewed as a modified Kalman filter. The modification amounts to fixing the shape of the process covariance matrix (the postulated matrix P defines the shape) and adjusting the strength of the process covariance matrix (the strength is proportional to α) in order to achieve reasonable agreement with the measurements.
 Limited practical experience shows that the computational burden is much less heavy for the Tikhonov filter than for the Kalman filter (the Kalman filter requires repeated calculations of matrix products in equation (17), while in the Tikhonov filter similar matrix products need to be computed only once because matrix P is fixed). The Kalman filter is able to provide a comprehensive characterization of the system. It provides an estimate of the state vector along with its covariance matrix. The Tikhonov filter gives only the state vector. On the other hand, the Tikhonov filter is less restrictive on data statistics (Gaussian statistics are not required), and it is prone to substantially fewer uncertainties in the initialization stage (it does not need covariance matrices Qn and P0 for its initialization and operation).
3. SIFTER Compared to Track-Before-Detect
 One of the distinguishing features of the SIFTER technique is that it processes the whole volume of radar data without applying any threshold to separate signal from noise. The goal is to extract weak targets that would normally be confused with noise. The same goal was pursued by the developers of the so-called track-before-detect (TBD) techniques. We shall briefly explain here what distinguishes SIFTER from TBD algorithms without going into a full discussion of various modifications of TBD techniques.
 A comprehensive review of TBD algorithms is given by by Blackman and Popoli [1999, chapter 7]. The unifying idea behind TBD methods is to hypothesize all possible target trajectories and to use observations (usually without any thresholding) with the goal to accumulate the likelihood (or similar quantitative characteristic of goodness) for each hypothetical target track. Tracks with sufficiently high likelihood are declared as real. The likelihood is generally accumulated on the basis of signal strength along the projection of a hypothetical track in slant space.
 In SIFTER we reconstruct and track the scattering surface. This evolving scattering surface creates a radar response that matches the radar data in the whole slant space. The SIFTER ideology for target detection is that in the course of tracking the scattering surface, localized targets will form peaks on the surface and the peaks will grow more and more distinct as tracking of the scattering surface proceeds.
 We think that the most important difference between SIFTER and TBD is that SIFTER works with the whole radar response and attempts to replicate the whole radar response, while TBD still attempts to (partially) associate slant cells with hypothetical tracks.
 Another distinguishing feature is that there is no limitation on the number of targets in SIFTER and it works with several targets in the same way as with a single one; the computational load is not affected by the number of targets. In TBD algorithms one has to hypothesize the number of possible targets before starting the processing. TBD analysis becomes more cumbersome as the number of potential targets increases.
4. Testing the SIFTER Time-Evolution Algorithm
 Evolving the SIFTER solution in time allows information from previous dwells to be carried into subsequent revisits. This provides a sensitivity enhancement over the single-dwell inversion by providing a sort of momentum that helps the SIFTER inversion to “hang on” to targets that may be Faraday-fading away. Furthermore, it removes spurious noise-related peaks that do not evolve in a way that is consistent with the Doppler velocity of a given scattering surface cell. In this section we will use simulated data to demonstrate how the SIFTER concept works. We will concentrate on the approach for Kalman filter tracking of the scattering surface.
 We have prepared a demonstration of the time evolution algorithm using simulated data. For this demonstration we have only implemented a single beam, but include all Doppler frequencies. Thus this version does not have the benefit of coherent target returns in adjacent beams; the detection gain shown here is due entirely to the time evolution component of SIFTER. This limits target motion to range and range rate. The following analog of the evolution equation (10) was used here to model the dynamics of the scattering surface:
We created a FORTRAN code that realizes the Kalman filter and takes advantage of the sparse character of the matrices in our problem.
 A sequence of signal plus noise data was simulated on a grid of 21 slant range bins by 21 Doppler bins. Range and Doppler processing windows were assumed Gaussian,
with Dρ = 2 and Dv = 1 when expressed in appropriate slant-bin-size units. The simulated received field (serving as input to the Kalman filter) was calculated as
Here ηn(ρs, vD) is Gaussian white noise, si are scattering coefficients of the targets, and ρi(tn) and vi(tn) specify position and velocity of simulated targets as functions of time.
 The Kalman filter was initialized with σ(ρ, v) = 0. In order to initialize the solution covariance matrix P00 we perform a Tikhonov solution of the equation Y0 = Mσ with the stabilizer defined by equation (22), determine the regularization parameter α, and set P00 = αP. In view of the similarities between the methods of Tikhonov and Kalman discussed in section 2.3, this procedure ensures that from the first step the output of the Kalman filter is in reasonable agreement with measurements. The stabilizing matrix P that we used for this example was defined by the condition
In other words, matrix P−1 provides a finite difference approximation of the integral expression shown above. So defined stabilizing functional may be used as a measure of smoothness of the solution. Indeed, the terms with derivatives in equation (38) will ensure that smoother solutions will generally produce smaller values of Ω and vice versa.
 The simulated example is for two targets that are initially close together in slant range and Doppler and eventually cross each other. One target has a signal-to-noise ratio (SNR) of about 6 dB, and the other is at 0 dB (that is, its signal strength is at the noise level). The 6-dB target is accelerating. Thus this example shows (1) SIFTER's capability to resolve two targets that are initially overlapped, (2) the great sensitivity of the time evolution version of SIFTER to extract very weak targets, and (3) SIFTER's ability to handle accelerating targets.
Figure 1 shows the simulated input amplitude data in slant-range/Doppler space. In Figure 1 we have Doppler on the vertical axis and slant range on the horizontal axis, both in arbitrary units. Each panel is one dwell, advancing in time from left to right and top to bottom at an arbitrary revisit rate. The weaker of the two targets cannot be distinguished from the noise. A slight hint of the stronger target appears occasionally, but this target always remains far below typical radar threshold for detection.
Figure 2 displays the scattering cross section surfaces output by the SIFTER time-evolution algorithm when applied to the input data of Figure 1. Here the vertical axis is radial velocity and the horizontal axis is geographic ground range, both in arbitrary units. The true target positions (in radial velocity and ground range) are overlain in symbols. After only a couple of dwells, the overlapping targets emerge clearly. As time goes on, the targets cross and then separate from each other, and distinct SIFTER peaks emerge for each target. SIFTER successfully holds on to the accelerating target even as it leaves the illuminated area.
 A new signal processing algorithm has been developed that provides enhanced detection of weak targets in noise and clutter while simultaneously yielding the geographic location of the targets (coordinate registration). This new method, called signal inversion for target extraction and registration (SIFTER), accomplishes this by solving for the distribution of scatterers in ground coordinates (the scattering surface) that reproduces the radar’s “slant space” measurements. The enhanced detection sensitivity provided by SIFTER over conventional methods arises because extra information is used. In particular, the knowledge of the propagation channels is used in deriving the SIFTER solution. Also, effective use is made of the coherence of targets across overlapping beams and a requirement is made for the consistency of target-like motion from dwell to dwell.
 While the SIFTER approach may have some conceptual similarity to track-before-detect methods, it is significantly different; it operates in geographic space (incorporating radar sounding information) and its computational burden does not grow with the number of targets.
 We have described in detail the mathematical algorithms of SIFTER and have shown through simulation the degree of detection sensitivity possible; in simulation we have successfully detected targets with SNR = 0 dB.
 A portion of the technology developed under the SIFTER program was funded by the DOD Counterdrug Technology Development Program Office at Dahlgren, Virginia (540-653-2374).