## Dynamically Orthogonal Narrow-Angle Parabolic Equations for Stochastic Underwater Sound Propagation. Part I: Theory and Schemes

Ali, W.H., and P.F.J. Lermusiaux, 2024a. *Dynamically Orthogonal Narrow-Angle Parabolic Equations for Stochastic Underwater Sound Propagation. Part I: Theory and Schemes.* Journal of the Acoustical Society of America 155(1), 640-655. doi:10.1121/10.0024466

Robust informative acoustic predictions require precise knowledge of ocean physics, bathymetry, seabed, and acoustic parameters. However, in realistic applications, this information is uncertain due to sparse and heterogeneous measurements and complex ocean physics. Efficient techniques are thus needed to quantify these uncertainties and predict the stochastic acoustic wave fields. In this work, we derive and implement new stochastic differential equations that predict the acoustic pressure fields and their probability distributions. We start from the stochastic acoustic parabolic equation (PE) and employ the instantaneously-optimal Dynamically Orthogonal (DO) equations theory. We derive stochastic DO-PEs that dynamically reduce and march the dominant multi-dimensional uncertainties respecting the nonlinear governing equations and non-Gaussian statistics. We develop the dynamical reduced-order DO-PEs theory for the Narrow-Angle PE (NAPE) and implement numerical schemes for discretizing and integrating the stochastic acoustic fields.