  We introduce a novel algorithm for nonlinear processing of data gathered by
an active array of sensors which probes a medium with pulses and measures the
resulting waves. The algorithm is motivated by the application of array
imaging. We describe it for a generic hyperbolic system that applies to
acoustic, electromagnetic or elastic waves in a scattering medium modeled by an
unknown coefficient called the reflectivity. The goal of imaging is to invert
the nonlinear mapping from the reflectivity to the array data. Many existing
imaging methodologies ignore the nonlinearity i.e., operate under the
assumption that the Born (single scattering) approximation is accurate. This
leads to image artifacts when multiple scattering is significant. Our algorithm
seeks to transform the array data to those corresponding to the Born
approximation, so it can be used as a pre-processing step for any linear
inversion method. The nonlinear data transformation algorithm is based on a
reduced order model defined by a proxy wave propagator operator that has four
important properties. First, it is data driven, meaning that it is constructed
from the data alone, with no knowledge of the medium. Second, it can be
factorized in two operators that have an approximately affine dependence on the
unknown reflectivity. This allows the computation of the Fr\'{e}chet derivative
of the reflectivity to the data mapping which gives the Born approximation.
Third, the algorithm involves regularization which balances numerical stability
and data fitting with accuracy of the order of the standard deviation of
additive data noise. Fourth, the algebraic nature of the algorithm makes it
applicable to scalar (acoustic) and vectorial (elastic, electromagnetic) wave
data without any specific modifications.
