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We study the paraxial wave equation with a randomly perturbed index of refraction, which can model the propagation of a wave beam in a turbulent medium. The random perturbation is a stationary and isotropic process with a general form of the covariance that may or may not be integrable. We focus attention mostly on the nonintegrable case, which corresponds to a random perturbation with long-range correlations, that is, relevant for propagation through a cloudy turbulent atmosphere. The analysis is carried out in a high-frequency regime where the forward scattering approximation holds. It reveals that the randomization of the wave field is multiscale: The travel time of the wave front is randomized at short distances of propagation, and it can be described by a fractional Brownian motion. The wave field observed in the random travel time frame is affected by the random perturbations at long distances, and it is described by a Schr\"odinger-type equation driven by a standard Brownian field. We use these results to quantify how scattering leads to decorrelation of the spatial and spectral components of the wave field and to a deformation of the pulse emitted by the source. These are important questions for applications, such as imaging and free space communications with pulsed laser beams through a turbulent atmosphere. We also compare the results with those used in the optics literature, which are based on the Kolmogorov model of turbulence. We show explicitly that the commonly used approximations for the decorrelation of spatial and spectral components are appropriate for the Kolmogorov model but fail for models with long-range correlations. 

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.