diff --git a/README.md b/README.md index cdb90ef205849597c9bc6f02254986cdde1ea863..5c86dde2155e911972c6d17eb334ee8c45743e2e 100644 --- a/README.md +++ b/README.md @@ -68,4 +68,104 @@ Olivier Asselin: Nathan Grivault: UoAlberta w/ Paul Myers - Freshwater in the Arctic: Arctic -> Lower latitudes -- Impact on circulation: Export vs Forcing \ No newline at end of file +- Impact on circulation: Export vs Forcing + +## 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