### Data Assimilation Part 1

parent 92ce9f0f
 ... @@ -68,4 +68,104 @@ Olivier Asselin: ... @@ -68,4 +68,104 @@ Olivier Asselin: Nathan Grivault: UoAlberta w/ Paul Myers Nathan Grivault: UoAlberta w/ Paul Myers - Freshwater in the Arctic: Arctic -> Lower latitudes - Freshwater in the Arctic: Arctic -> Lower latitudes - Impact on circulation: Export vs Forcing - Impact on circulation: Export vs Forcing \ No newline at end of file ## Data Assimilation ### Development of data assimilation - developped for weather forecasting (fast time scale, highly nonlinear) - applied for operational oceanography (slower time scales, poorly observed) - starting to be used in sea ice ### Why? - Obtain an improved state that can be used to initialize a forecast - Obtain consistent states that can be used in process studies or hindcasting - To challenge models with data and vice versa, leading to improvements in models and/or observational methods (OSSEs) ### Assimilation cycle 1 Observation 2 Assimilation system 3 Analysis 4 Forecast model 5 Background 6 Repeat Note: Need really good observations otherwise they can contaminate runs ### Components of a DA system - DA is a state estimation problem - Need a prognostic model to solve time evolution of the state - the state is all you need to characterize the system - prevent model drift ### DA problem given a mapping from a state to observations, try to get from observation to state for y = H(x), try to get x = H^-1(y) but size(x) and size(y) don't match so it's not invertible ### Best Linear Unbiased Estimator (BLUE) x_a = L x_b + K y a: state estimate at a given time b: background (state of the model for a given time) y: obs for a given time Best estimate minimizes analysis error. E_a = x_a - x_t E_b = x_b - x_t E_o = y - x_t x_t + E_a = L(x_t + E_b) + K(E_o + H(x_t)) assume errors are unbiased: = 0 = L + KH = (L+KH) --> L+KH = I --> L = I - KH Plug back in: x_a = (I-KH)x_b + Ky = x_b + K (y - H x_b) <== Kalman filter equation ** -Need to get K -'best' K is the one that minimizes trace of A where A = 1 subtract x^t from both sides of ** to get an eqn for E^a 2 A = 3 A(K+dK) - A(K) = A(dK) 4 Trace A(dK) = 0 -2[(I-KH)BH^t + KR]dK = 0 for any dK => (I-KH)BH^t + KR = 0 ==> BH^t - KHBH^t - KR =0 ===> K(HBH^t + R) = BH^t ====> K = BH^t (HBH^t+R)^-1 Try H = I, ie you have 1 observation for every state variable Try B = \sigma^2_b I and R = \sigma^2_o I xa = xb + BH^t (HBH^t+R)^-1 (y - Hx^b) xa = xb + (sigma^2_b)(y-xb)/(sigma^2_o + sigma^2_b)*** IF \sigma^2_o >> \sigma^2_b => xa -> xb because sig/(sig+sig) -> 0 IF \sigma^2_b >> \sigma^2_o => xa -> y because sig/(sig+sig) -> 1 If model is biased, need to include bias in *** and have a coupled error/bias data assimilation Generally treat things as a minimization problem. ## Getting B and R: Error covariance matrices: B: Model error covariance matrix for the model: how state variables vary with each others R: Obs error covariance matrix Innovations: y - Hxb Correlated B&R act as low&high pass filters of innovations. Using ensembles to create error correlation function: Not bad but spurious noise at longer range. => Use localization. ## Case studies T&S from buoy in Labrador sea using EnOI -> Improved profiles Sea ice concentration from passive microwave Look at innovations (PDF, Space-time distributions) to diagnose DA Can take a while for assimilation to kick in \ No newline at end of file
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