Jo Eidsvik, Department of Mathematical Sciences, NTNU
The vast oceans will always be under-sampled, but with the availability of satellite data and physics-based numerical ocean models one can gain nuanced insight about a variety of ocean variables. This is particularly so when these data sources are calibrated and combined with in-situ measurements using statistical methods. Funded by the Norwegian Research Council, Maritime Autonomous Sampling and Control (MASCOT, https://wiki.math.ntnu.no/mascot ) is an inter-disciplinary project led by the Department of Mathematical Science at NTNU in a collaboration with Sintef in Norway and the Underwater Systems and Technology Laboratory (LSTS) in Portugal.
Background
The MASCOT project aims to develop statistical methods for space-time sampling with autonomous robots operating in the upper water column. Sampling here refers to the design of observational strategies, enabling autonomous platforms to decide on where, when and what to measure (Eidsvik et al., 2015) to increase our knowledge of dynamic environments.
While the project’s focus is on methodological development of algorithms, we leverage on complex oceanographic models on shore for training statistical surrogate models and generate a strategy for sampling, and then embed these models and algorithms for updating them onboard autonomous underwater vehicles (AUVs). The numerical ocean models used in the project are provided by Sintef via their SINMOD project (https://www.sintef.no/en/ocean/initiatives/sinmod/ ). The trained statistical model forms a prior model, which is easily updated so that it can run onboard the AUV. Moreover, this model enables fast calculations of expected value functions for evaluating multiple sampling designs, among which the most optimal design is chosen for exploration by the AUV.
Figure 1: Three-dimensional view of an AUV sampling the water column. Its onboard computing unit enables AI-based adaptive sampling where the statistical model is updated over time and informs the AUV about valuable path for continued sampling.
Figure 1 illustrates the core idea with an AUV in the water, having an onboard statistical model representing its knowledge of the ocean variables, and informing it about where to sample. The LSTS lab (https://lsts.fe.up.pt ) and the AURLab at NTNU (https://www.ntnu.edu/aur-lab ) have made AUVs available for us. MASCOT is motivated by earlier work of e.g. Fossum et al. (2019) who demonstrated AUV sampling operations for adaptive mapping plankton abundance.
Non-stationary spatial modeling
Berild and Fuglstad (2023) used multiple realizations from the numerical ocean model (SINMOD) to train a statistical surrogate model with realistic mean, variance and correlation structures. Their model is a Gaussian Markov random field (GMRF) which can incorporate complex non-stationarity as seen in the ocean model outputs for the river plume that they studied. The GMRF model also gives sparse precision matrices which facilitate efficient computations, even in three-dimensional space. Figure 2 shows the salinity profile at six different depths near the Nidelva river plume, Norway. Salinity in the fjord is about 32 g/kg, while it is only about 5 g/kg in the river.
Figure 2: Building on numerical ocean models from SINTEF (left), statistical methods are used to train a surrogate model for the fjord salinity. The prior variance (middle) is larger near the river plume in the south-east and the non-stationary model has flexibility to incorporate complicated realistic correlations (right). (Fig. from Berild and Fuglstad (2023).)
Adaptive sampling
Ge et al. (2023) developed adaptive sampling methods for AUV operations sampling the river plume front in three spatial dimensions. Within the realm of “sense-plan-act”, the AUV is first measuring in-situ salinity, then it updates the onboard model and plans where to go next (based on the design criterion), and finally acts to continue its trajectory in the optimal direction. Extending the work of Fossum et al. (2021), they incorporated a three-dimensional Gaussian process model onboard the AUV and developed algorithms for efficient data assimilation and design calculations in this setting. Figure 3 shows excursion probabilities calculated in a field test in the Nidelva river plume, Norway. In the shallow layers, the excursion probabilities are rather large indicating that this is most likely freshwater from the river, while the deeper water masses are likely to be salt fjord water. The AUV path is shown as the black line. Here, the AUV nicely adapts to map the front of the river plume.
Figure 3: Excursion probabilities at the end of an adaptive AUV mission conducting three-dimensional sampling in the Nidelva river plume front, Trondheim Fjord, Norway. (Fig. from Ge et al. (2023).)
The combination of a nonstationary GMRF model and adaptive three-dimensional sampling was conducted in Berild et al. (2024). Using the more nuanced GMRF model, one gets more realistic covariance structures in the onboard model, and this helps guide the AUV in more promising directions. Also, the sparse precision matrices of the GMRF means that larger three-dimensional grids can be considered.
New directions
Future plans of the MASCOT project include adaptive sampling to map plankton in space and time. The onboard sensor is here a camera mounted on the AUV, and the data assimilation is preceded by automatic image processing to count plankton in the sample. For related camera data, we are also implementing algorithms for effective mapping of corals.
References
Berild, M. O., & Fuglstad, G. A. (2023). Spatially varying anisotropy for Gaussian random fields in three-dimensional space. Spatial Statistics, 55, 100750. https://doi.org/10.1016/j.spasta.2023.100750
Berild, M. O., Ge, Y., Eidsvik, J., Fuglstad, G. A., & Ellingsen, I. (2024). Efficient 3D real-time adaptive AUV sampling of a river plume front. Frontiers in Marine Science. https://doi.org/10.3389/fmars.2023.1319719
Eidsvik, J., Mukerji, T., & Bhattacharjya, D. (2015). Value of information in the earth sciences: Integrating spatial modeling and decision analysis. Cambridge University Press. https://doi.org/10.1017/CBO9781139628785
Fossum, T. O., et al. (2019). Toward adaptive robotic sampling of phytoplankton in the coastal ocean. Science Robotics, 4(27), eaav3041. https://doi.org/10.1126/scirobotics.aav3041
Fossum, T. O., Travelletti, C., Eidsvik, J., Ginsbourger, D., & Rajan, K. (2021). Learning excursion sets of vector-valued Gaussian random fields for autonomous ocean sampling. The annals of applied statistics, 15(2), 597-618.https://doi.org/10.1214/21-AOAS1451
Ge, Y., Eidsvik, J., & Mo-Bjørkelund, T. (2023). 3-D Adaptive AUV Sampling for Classification of Water Masses. IEEE Journal of Oceanic Engineering. 626-639. https://doi.org/10.1109/JOE.2023.3252641
