Liliana Borcea

We introduce a method for Multiple Input Multiple Output (MIMO) radar imaging of moving targets in a strongly reflecting, complex stationary scenery (clutter). The radar system has fixed nearby antennas that play the dual role of sources and receivers. It gathers data either by emitting probing pulses from one antenna at a time, or by sending from all the antennas non-coherent, possibly orthogonal, waveforms. We show how to obtain from the measurements an imaging function that depends on search position and velocity and is approximately separable in these variables, for a single moving target. For multiple moving targets in clutter, the imaging function is a sum of separable functions. By sampling this imaging function on a position velocity grid we obtain an imaging matrix whose Singular Value Decomposition (SVD) allows the separation of the clutter and the targets moving at different velocities. The decomposition also leads directly to estimates of the locations and motion of the targets. The imaging method is designed to work in strong clutter, with unknown and possibly heterogeneous statistics. It does not require prior estimation of the covariance matrix of the clutter response or of its rank. We give an analysis of the imaging method and illustrate how it works with numerical simulations.

We present an analysis of enhanced wave transmission through random media with mirror symmetry about a reflecting barrier. The mathematical model is the acoustic wave equation, and we consider two setups, where the wave propagation is along a preferred direction: in a randomly layered medium and in a randomly perturbed waveguide. We use the asymptotic stochastic theory of wave propagation in random media to characterize the statistical moments of the frequency-dependent random transmission and reflection coefficients, which are scalar-valued in layered media and matrix-valued in waveguides. With these moments, we can quantify explicitly the enhancement of the net mean transmitted intensity, induced by wave interference near the barrier.

The inverse wave scattering problem seeks to estimate a heterogeneous, inaccessible medium, modeled by unknown variable coefficients in wave equations, from transient recordings of waves generated by probing signals. It is a widely studied inverse problem with important applications, that is typically formulated as a nonlinear least squares data fit optimization. For typical measurement setups and band-limited probing signals, the least squares objective function has spurious local minima far and near the true solution, so Newton-type optimization methods fail. We introduce a different approach, for electromagnetic inverse wave scattering in lossless, anisotropic media. Our reduced order model (ROM) is an algebraic, discrete time dynamical system derived from Maxwell's equations with four important properties: (1) It is data driven, without knowledge of the medium. (2) The data to ROM mapping is nonlinear and yet the ROM can be obtained in a non-iterative fashion. (3) It has a special algebraic structure that captures the causal Wave propagation. (4) The ROM interpolates the data on a uniform time grid. We show how to obtain from the ROM an estimate of the wave field at inaccessible points inside the unknown medium. The use of this wave is twofold: First, it defines a computationally inexpensive imaging function designed to estimate the support of reflective structures in the medium, modeled by jump discontinuities of the matrix valued dielectric permittivity. Second, it gives an objective function for quantitative estimation of the dielectric permittivity, that has better behavior than the least squares data fitting objective function. The methodology introduced in this paper applies to Maxwell's equations in three dimensions. To avoid high computational costs, we limit the study to a cylindrical domain filled with an orthotropic medium, so the problem becomes two dimensional.

We present an analysis of enhanced wave transmission through random media with mirror symmetry about a reflecting barrier. The mathematical model is the acoustic wave equation and we consider two setups, where the wave propagation is along a preferred direction: in a randomly layered medium and in a randomly perturbed waveguide. We use the asymptotic stochastic theory of wave propagation in random media to characterize the statistical moments of the frequency dependent random transmission and reflection coefficients, which are scalar valued in layered media and matrix valued in waveguides. With these moments, we can quantify explicitly the enhancement of the net mean transmitted intensity, induced by wave interference near the barrier.

We introduce a novel approach to waveform inversion, based on a data driven reduced order model (ROM) of the wave operator. The presentation is for the acoustic wave equation, but the approach can be extended to elastic or electromagnetic waves. The data are time resolved measurements of the pressure wave gathered by an acquisition system which probes the unknown medium with pulses and measures the generated waves. We propose to solve the inverse problem of velocity estimation by minimizing the square misfit between the ROM computed from the recorded data and the ROM computed from the modeled data, at the current guess of the velocity. We give the step by step computation of the ROM, which depends nonlinearly on the data and yet can be obtained from them in a non-iterative fashion, using efficient methods from linear algebra. We also explain how to make the ROM robust to data inaccuracy. The ROM computation requires the full array response matrix gathered with collocated sources and receivers. However, we show that the computation can deal with an approximation of this matrix, obtained from towed-streamer data using interpolation and reciprocity on-the-fly. While the full-waveform inversion approach of nonlinear least-squares data fitting is challenging without low frequency information, due to multiple minima of the data fit objective function, we show that the ROM misfit objective function has a better behavior, even for a poor initial guess. We also show by an explicit computation of the objective functions in a simple setting that the ROM misfit objective function has convexity properties, whereas the least squares data fit objective function displays multiple local minima.

We introduce a novel approach to waveform inversion based on a data-driven reduced order model (ROM) of the wave operator. The presentation is for the acoustic wave equation, but the approach can be extended to elastic or electromagnetic waves. The data are time resolved measurements of the pressure wave gathered by an acquisition system that probes the unknown medium with pulses and measures the generated waves. We propose to solve the inverse problem of velocity estimation by minimizing the square misfit between the ROM computed from the recorded data and the ROM computed from the modeled data, at the current guess of the velocity. We give a step by step computation of the ROM, which depends nonlinearly on the data and yet can be obtained from them in a noniterative fashion, using efficient methods from linear algebra. We also explain how to make the ROM robust to data inaccuracy. The ROM computation requires the full array response matrix gathered with colocated sources and receivers. However, we find that the computation can deal with an approximation of this matrix, obtained from towed-streamer data using interpolation and reciprocity on-the-fly. Although the full-waveform inversion approach of nonlinear least-squares data fitting is challenging without low-frequency information, due to multiple minima of the data fit objective function, we find that the ROM misfit objective function has better behavior, even for a poor initial guess. We also find by explicit computation of the objective functions in a simple setting that the ROM misfit objective function has convexity properties, whereas the least-squares data fit objective function displays multiple local minima.

We introduce a novel, computationally inexpensive approach for imaging with an active array of sensors, which probe an unknown medium with a pulse and measure the resulting waves. The imaging function is based on the principle of time reversal in non-attenuating media and uses a data driven estimate of the ‘internal wave’ originating from the vicinity of the imaging point and propagating to the sensors through the unknown medium. We explain how this estimate can be obtained using a reduced order model (ROM) for the wave propagation. We analyze the imaging function, connect it to the time reversal process and describe how its resolution depends on the aperture of the array, the bandwidth of the probing pulse and the medium through which the waves propagate. We also show how the internal wave can be used for selective focusing of waves at points in the imaging region. This can be implemented experimentally and can be used for pixel scanning imaging. We assess the performance of the imaging methods with numerical simulations and compare them to the conventional reverse-time migration method and the ‘backprojection’ method introduced recently as an application of the same ROM.