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Data Assimilation Part 1

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README.md
... ... @@ -68,4 +68,104 @@ Olivier Asselin:
68 68  
69 69 Nathan Grivault: UoAlberta w/ Paul Myers
70 70 - Freshwater in the Arctic: Arctic -> Lower latitudes
71   -- Impact on circulation: Export vs Forcing
72 71 \ No newline at end of file
  72 +- Impact on circulation: Export vs Forcing
  73 +
  74 +## Data Assimilation
  75 +
  76 +### Development of data assimilation
  77 +- developped for weather forecasting (fast time scale, highly nonlinear)
  78 +- applied for operational oceanography (slower time scales, poorly observed)
  79 +- starting to be used in sea ice
  80 +
  81 +### Why?
  82 +- Obtain an improved state that can be used to initialize a forecast
  83 +- Obtain consistent states that can be used in process studies or hindcasting
  84 +- To challenge models with data and vice versa, leading to improvements in models and/or observational methods (OSSEs)
  85 +
  86 +### Assimilation cycle
  87 +1 Observation
  88 +2 Assimilation system
  89 +3 Analysis
  90 +4 Forecast model
  91 +5 Background
  92 +6 Repeat
  93 +
  94 +Note: Need really good observations otherwise they can contaminate runs
  95 +
  96 +### Components of a DA system
  97 +- DA is a state estimation problem
  98 +- Need a prognostic model to solve time evolution of the state
  99 +- the state is all you need to characterize the system
  100 +- prevent model drift
  101 +
  102 +### DA problem
  103 +given a mapping from a state to observations, try to get from observation to state
  104 +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
  105 +
  106 +### Best Linear Unbiased Estimator (BLUE)
  107 +x_a = L x_b + K y
  108 +a: state estimate at a given time
  109 +b: background (state of the model for a given time)
  110 +y: obs for a given time
  111 +
  112 +Best estimate minimizes analysis error.
  113 +E_a = x_a - x_t
  114 +E_b = x_b - x_t
  115 +E_o = y - x_t
  116 +
  117 +x_t + E_a = L(x_t + E_b) + K(E_o + H(x_t))
  118 +
  119 +assume errors are unbiased: <E_?> = 0
  120 +
  121 +<x_t> = L<x_t> + KH<x_t>
  122 +<x_t> = (L+KH)<x_t> --> L+KH = I --> L = I - KH
  123 +
  124 +Plug back in:
  125 +x_a = (I-KH)x_b + Ky
  126 + = x_b + K (y - H x_b) <== Kalman filter equation **
  127 +
  128 +-Need to get K
  129 +-'best' K is the one that minimizes trace of A where A = <E^a E^a*>
  130 +
  131 +1 subtract x^t from both sides of ** to get an eqn for E^a
  132 +2 A = <E^a E^a*>
  133 +3 A(K+dK) - A(K) = A(dK)
  134 +4 Trace A(dK) = 0
  135 +
  136 +-2[(I-KH)BH^t + KR]dK = 0 for any dK
  137 +=> (I-KH)BH^t + KR = 0
  138 +==> BH^t - KHBH^t - KR =0
  139 +===> K(HBH^t + R) = BH^t
  140 +====> K = BH^t (HBH^t+R)^-1
  141 +
  142 +Try H = I, ie you have 1 observation for every state variable
  143 +Try B = \sigma^2_b I and R = \sigma^2_o I
  144 +
  145 +xa = xb + BH^t (HBH^t+R)^-1 (y - Hx^b)
  146 +
  147 +xa = xb + (sigma^2_b)(y-xb)/(sigma^2_o + sigma^2_b)***
  148 +
  149 +IF \sigma^2_o >> \sigma^2_b => xa -> xb because sig/(sig+sig) -> 0
  150 +IF \sigma^2_b >> \sigma^2_o => xa -> y because sig/(sig+sig) -> 1
  151 +
  152 +If model is biased, need to include bias in *** and have a coupled error/bias data assimilation
  153 +
  154 +Generally treat things as a minimization problem.
  155 +
  156 +## Getting B and R:
  157 +Error covariance matrices:
  158 +B: Model error covariance matrix for the model: how state variables vary with each others
  159 +R: Obs error covariance matrix
  160 +
  161 +Innovations: y - Hxb
  162 +
  163 +Correlated B&R act as low&high pass filters of innovations.
  164 +Using ensembles to create error correlation function: Not bad but spurious noise at longer range.
  165 +=> Use localization.
  166 +
  167 +## Case studies
  168 +T&S from buoy in Labrador sea using EnOI -> Improved profiles
  169 +Sea ice concentration from passive microwave
  170 +
  171 +Look at innovations (PDF, Space-time distributions) to diagnose DA
  172 +Can take a while for assimilation to kick in
73 173 \ No newline at end of file
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