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DataAssimilation

### Data Assimilation Part 1

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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