The PIs will develop a methodology for improving estimate and prediction of the state of a dynamical system, with particular focus on analyzing ocean dynamics. The primary goals of this project are thus to develop innovative approaches for representation and manipulation of data uncertainty and model error using a fuzzy set formulation and to then apply these approaches for the data and model fusion formulated as the global optimization problem. Convenient and fast numerical algorithms will be developed to solve the problem using high-performance parallel computing. Such an approach differs from usual statistical estimates but with advantages and drawbacks of its own. The general mathematical theory will be applied to a long-standing but important problem of improving estimates and prediction of the state of the ocean. In particular, the proposed study targets a synthesis of submesoscale/mesoscale fronts, jets and eddies by fusing satellite observations, float and shipboard data of lower resolution, as well as ROMS simulation results for Central California. The theory should provide new tools to be applied in oceanography, meteorology, climatology, artificial intelligence, computer science, control engineering, decision theory, expert systems, operational research and pattern recognition. As the first step in using these tools for broader oceanography community goals, the fusion approach will be applied to different data bases to understand and quantify heat storage and carbon content of the North Atlantic in collaboration with scientists from Great Britain and Germany and to allow junior scientists to obtain excellent training and learning in cross disciplinary/multi-disciplinary areas of great scientific and practical importance.
The PIs will address a long-standing but important problem involved with improving the estimation and prediction of the state of the ocean. The primary goals of this project are to develop an innovative approach for representation and manipulation of uncertainty coming from a wide variety of sources such as sensor outputs, model outputs, aggregating expert opinions as well as merging different databases and data even when distinct pieces of information are contradictory, and to suggest methods to fuse this information in decision making goals. The study will provide new mathematical theory and tools relevant for this problem, but also for more general applications in oceanography, meteorology and climatology. Mathematically the approach uses a fuzzy set formulation which originated in pure mathematics and which will be adapted for representing and manipulating data uncertainty and ocean model error. Results of the work will advance development of new forecast metrics in terms of fuzzy sets as well as new methods for quantification of model predictability through data-model and model-model comparisons at weather and climatic scales. As the first step in using these tools for broader oceanography community goals, the approach will be applied to different data bases which relate to quantifying heat storage and carbon content of the North Atlantic. The PIs will collaborate with scientists from Great Britain and Germany. Junior scientists involved in the project will obtain excellent training and learning in cross disciplinary/multi-disciplinary areas of great scientific and practical importance.
Outcomes. The main outcomes of our project are divided in four groups and most essential achievements in each group are highlighted. Almost all the results are published in peer review journals and relevant references are provided. 1. Developing a theoretical background for data fusion based on fuzzy logic ideas, [1,2] It was rigorously proven and illustrated by simulations that a fuzzy estimate of location parameter from two short biased samples is more efficient than traditional statistics based on least squares or ranking approach 2. Estimating surface velocities by combining data from different sources. Efficient, stable, and fast algorithms have been developed, tested on synthetic and in some cases on real data for estimating surface velocities by using the following combinations of data. (a) Model output and tracer observations such as sea surface temperature or color, [3,4] (b) Tracer observations and direct velocity measurements (ADCP), [5] (c) Two tracers (SST and color) and HF radar measurements (d) Drifter trajectories and HF radar 3. Estimating Lagrangian characteristics by combining data from different sources. (a) Relative diffusivity from an Eulerian model output and tracer observations, [6] An efficient portable procedure has been developed and tested on synthetic data for estimating the absolute and relative diffusivity from a rough estimate of the underlying Eulerian velocity field and noisy tracer observations. (b) Finite size Lyapunov exponent (FSLE) from a circulation model and drifter observations, [7.8] FSLE has been estimated in the Gulf Stream zone based on a circulation model (HYCOM) and a Lagrangian submesoscale stochastic model with parameters estimated from drifter data. The result turned out to be in good agreement with similar estimates from real observations. 4. Identifying patterns in thermohaline fields by combining glider data and ship CTD. (a) Reconstructing a front evolution from glider transects and ship CTD, [9] We provided a general framework to illustrate the Doppler smearing effect in spatial and temporal variability and introduce a methodology to unfold the uncertainty. Application to the Ligurian Current showed that the retrieved frontal zone evolution was consistent with independent observations. (b) Identifying 3D structures from glider and mooring CTD A theoretical approach have been developed for estimating parameters of thermohaline patterns such as intrusion and lenses by combining glider transects and vertical T-S profiles obtained from ships or platforms. Intellectual merit and broader impact The completed project contributed to understanding and developing novel mathematical methods for ocean state estimation in multiple metrics with representation of data and model uncertainty in terms of the fuzzy logic. This is a new approach to uncertainty of oceanographic data and errors of ocean modeling, and a new way to improve the ocean state nowcast and forecast through fusion of historical data, model results, sensor outputs, aggregation of expert opinions as well as merging databases. A set of mathematical tools for representation and manipulation of oceanographic information and merging different types of data have been developed accounting for specific features of the ocean variability and observations. The completed research could help to better understand submeso/mesoscale variability induced by eddies , jets and tides in complex areas of the world ocean (on examples of the Ligurian Sea and Gulf Stream zone). In broader aspects the developed mathematical tools also contributed to some areas of statistics such as a classical problem of estimating location parameter from short biased samples. Our findings in the study of finite size Lyapunov exponent could be viewed as an advance in the dynamical system theory. Published papers [1] L.I.Piterbarg, (2011), Parameter estimation from small biased samples: statistics vs fuzzy logic, Fuzzy Sets and Systems, 170, 121 [2] L.I. Piterbarg, (2013), Estimation of location parameter from two biased samples, Applied Mathematics, Vol.4 No.9, pp. 1269-1277. doi: 10.4236/am.2013.49171 [3] A. Mercatini , A. Griffa , L. Piterbarg, E. Zambianchi , M.G. Magaldi, (2010), Estimating surface velocities from satellite data and numerical models: Implementation and testing of a new simple method, Ocean Modeling, 33, 190-203 [4] L.I. Piterbarg and L. Ivanov, (2011), Fuzzy-logic based algorithm for estimating circulation patterns, Current Applied Mathematics, v.1, n.1, 17-39 [5] L.I. Piterbarg and L.M. Ivanov, (2013), Estimating Circulation Patterns by Combining Velocity and Tracer Observations, Open Journal of Applied Sciences, Vol.3, No.1, pp. 8-14. doi: 10.4236/ojapps.2013.31002. [6] L.I. Piterbarg, (2010), Computing Lagrangian statistics from tracer observations and a model output, Applied Mathematical Modeling, 10.1016/j.apm.2010.03.017 [7] L.I. Piterbarg, (2012), Finite size Lyapunov exponent for some simple models of turbulence, Applied Mathematical Modelling, v.36, n.8, 34643476 [8] A. Haza, T. Ozgokmen, A. Griffa, Z. Garaffo, L. Piterbarg, (2012), Parameterization of Submesoscale Transport in the Gulf Stream Region Using Lagrangian Subgridscale Models, Ocean Modelling, v.42, 3; 31-49 [9] L.I. Piterbarg, V. Taillandier, and A. Griffa (2013), Investigating frontal variability from repeated glider transects in the Ligurian Current (North West Mediterranean Sea), Journal of Marine Systems, doi: 10.1016/j.jmarsys.2013.08.003
