Assimilating Eulerian and Lagrangian data to quantify flow uncertainty in testbed oceanography models
Doctoral thesis
English
OPEN
McDougall, Damon
Data assimilation is the act of merging observed data into a mathematical model.\ud This act enables scientists from a wide range of disciplines to make predictions. For\ud example, predictions of ocean circulations are needed to provide hurricane disaster\ud maps. Alternatively, using ocean current predictions to adequately manage oil spills\ud has significant practical applications. Predictions are uncertain and this uncertainty\ud is encoded into a posterior probability distribution. This thesis aims to explore two\ud overarching aspects of data assimilation, both of which address the influence of the\ud mathematical model on the posterior distribution.\ud The first aspect we study is model error. Error is always present in mathematical\ud models. Therefore, characterising posterior flow information as function of model\ud error is paramount in understanding the practical implications of predictions. In\ud a model describing advective transport, we make observations of the underlying\ud flow at fixed locations. We characterise the mean of the posterior distribution as a\ud function of the error in the advection velocity parameter. When the error is zero,\ud the model is perfect and we reconstruct the true underlying flow. Partial recovery of\ud the true underlying flow occurs when the error is rational, the denominator of which\ud dictates the number of Fourier modes present in the reconstruction. An irrational\ud error leads to retrieval only of the spatial mean of the flow.\ud The second aspect we study is the control of ocean drifters. Commonplace in\ud oceanography is the collection of ocean drifter positions. Ocean drifters are devices\ud that sit on the surface of the ocean and move with the flow, transmitting their\ud position via GPS to stations on land. Using drifter data, it is possible to obtain a\ud posterior on the underlying flow. This problem, however, is highly underdetermined.\ud Through controlling an ocean drifter, we attempt to improve our knowledge of the\ud underlying flow. We do this by instructing the drifter to explore parts of the flow\ud currently uncharted, thereby obtaining fresh observations. The efficacy of a control\ud is determined by its e↵ect on the variance of the posterior distribution. A smaller\ud variance is interpreted as a better understanding of the flow. We show a systematic\ud reduction in variance can be achieved by utilising controls that allow the drifter to\ud navigate new or ‘interesting’ flow structures, a good example of which are eddies.

References
(3)
A Apte, M Hairer, A M Stuart and J Voss (2007). Sampling the posterior: An approach to nonGaussian data assimilation. Physica D: Nonlinear Phenomena 230 (12), 5064.
A Apte, C K R T Jones and A M Stuart (2008)a. A Bayesian approach to Lagrangian data assimilation. Tellus A 60 (2), 336347a.
A Apte, C K R T Jones, A M Stuart and J Voss (2008)b. Data assimilation: Mathematical and statistical perspectives. International Journal for Numerical Methods in Fluids 56, 10331046b.

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