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Data assimilation Preliminary remarks Tony O’Hagan

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Framework We have a dynamic simulator f(x t, c, z t+1 ) Inputs: the current state vector x t, parameters c and current forcing inputs z t+1 Outputs: the new state vector x t+1 At various time points t We make an observation y t That we model as a noisy function of x t I’ll assume y t = g(x t ) + ε t What can we learn from such observations? In order to address that question, we have to acknowledge one important truth …

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All models are wrong The output of the simulator is not x t+1 Even if all the inputs have their true values In particular, even if we know the true x t We need to acknowledge model discrepancy The correct formulation is x t+1 = f(x t, c, z t+1 ) + δ (x t,z t+1 ) Where δ (x t,z t+1 ) is model discrepancy Note that it doesn’t formally depend on c Model discrepancy is hugely important Without it,, there would be little need for data assimilation And yet almost all theory of DA ignores it!

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What can we learn? We have the following system y t = g(x t ) + ε t x t = f(x t-1, c, z t ) + δ (x t-1,z t ) In which there a number of uncertain quantities Conventional DA Ignores uncertainty in c and z t Treats model discrepancy as a noise term Only learns about x t But the observations are also informative about c, z t and δ(.,.) DA should be holistic

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Why does this matter? Because when at every step we need to correct the state vector we should be learning about model discrepancy And about c (calibration) So that DA becomes more efficient, but more importantly So that we can forecast into the future Or in different contexts DA needs to stop firefighting and learn fire prevention

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