Towards remote sensing of sub-surface turbulence from surface-only measurements with the SHRED machine learning framework

Near-surface turbulent fluid flows beneath a free surface are reconstructed from sparse measurements of the surface only. We study data from direct numerical simulations (DNS) as well as a laboratory experiment.  Fast and economical measurements of the turbulent flow near the free surface of natural flows is of high importance, for estimation and monitoring of a range of environmental factors. Gas evasion from rivers make a large and poorly constrained contribution to the total CO2 emissions, the transfer rates of gas and heat between water and atmosphere transfer are controlled by near-surface turbulent mixing. Transport of microplastics and nutrients and the living conditions of phytoplankton depend on turbulent mixing. The ability to estimate, e.g., the rate of gas transfer from rivers based on video footage taken from drones would enable coverage of large areas, much faster and at much lower cost than state-of-the-art in situ measurements.We employ a machine learning approach to build on recent progress in quantifying sub-surface turbulent flow from surface-only observations, such as utilising surface imprints to identify strong sub-surface turbulent flow structures [1]. A previous machine learning approach showed promise, using the same DNS data that we also employ [2].We apply a recently developed method, the Shallow Recurrent Decoder (SHRED) neural network [3], to free-surface turbulent flows. It combines a recurrent network, which learns a latent representation of the temporal dynamics of the system, with a shallow decoder network, that transforms this latent space back to real-state space. The algorithm is applied to DNS cases and experimental cases of different turbulence levels, with several horizontal subsurface velocity planes measured simultaneously as the surface. The temporal dynamics of subsurface planes are successfully reconstructed from as little as three time-resolved sensors at the surface, with low-rank features matching well with ground truth data, as well as matching turbulence spectra in the low-wavenumber regime. Depth profiles of selected error metrics suggest reasonable velocity field reconstructions, although the performance generally decreases with depth. Our results amount to a proof of concept of a method with potential to facilitate remote sensing of sub-surface flow from e.g. video images.[1] J. R. Aarnes, O.M. Babiker, A. Xuan, L. Shen, and S.Å. Ellingsen (2025). “Vortex structures under dimples and scars in turbulent free-surface flows”. J. Fluid Mech., accepted, Preprint: https://doi.org/10.48550/arXiv.2409.05409[2] A.Xuan and L.Shen (2023) “Reconstruction of three-dimensional turbulent flow structures using surface measurements for free-surface flows based on a convolutional neural network” J. Fluid Mech. 959 A34.[3] J. P. Williams, O. Zahn, and J. N. Kutz (2024), “Sensing with shallow recurrent decoder networks,” Proc. R. Soc. A, 480, no. 2298.