Data Assimilation Part 1

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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
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+- 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: <E_?> = 0
+
+<x_t> = L<x_t> + KH<x_t>
+<x_t> = (L+KH)<x_t>  --> 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 = <E^a E^a*>
+
+1 subtract x^t from both sides of ** to get an eqn for E^a
+2 A = <E^a E^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
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