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lec21 (1) - R G Prinn 12.806/10.571 Atmospheric Physics...

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R. G. Prinn, 12.806/10.571: Atmospheric Physics & Chemistry, April 27, 2006 Estimating Surface/Internal Sources and Sinks Figure by MIT OCW.
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Measurement Equation In Lagrangian framework: Change in mole fraction from its initial condition ( ) y s,t ( ) y* 0,0 is given by ( ) ( ) ( ) t 0 s 0 y s,t y* s,t y* 0,0 xdt ' x d ds' v v d = = = = s' t ' where x is the net chemical production (ignoring molecular diffusion and assuming perfect definition of back trajectory) Real measurements of y have errors ε ( ) ( ) s t 0 0 x y s,t ds' s,t v = + ε Real measure- ment True value if perfect model Measure- ment error In discrete form: 0 t i ij j j y h x = + i ε ε In vector-matrix form for multiple observing sites i 0 t y Hx = + measurement equation where is the ij H h = partial derivative (or observation ) matrix , and Figure by MIT OCW. Observing Station Back Trajectory s, t, l, k v(s',t') s',t', j (0,0) Position Velocity Time
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0 i i ij t j j y y h x x = = compute in model where ( ) i j y ,x denotes an estimate of ( ) 0 t i j y ,x . Eulerian models : Here x refers to grid points in the Eulerian model rather than to points along a back trajectory. To utilize the above equations we must define: ( ) ( ( ) ( x x grid point x ref,grid point y y grid point y ref,grid point = = ) ) where y(ref) is the mole fraction computed in a reference run using best available estimates of the state vector prior to estimating . ( x ref ) t x The measurement equation expresses an apparent linear relation between the observation vector and the unknowns contained in the 0 y state vector . It is an expression of the t x forward problem . The theory of the Linear Kalman Filter allows us to perform the inverse problem in which we optimally estimate given a model and a time series of observations at one or more sites. t x 0 y Optimal estimation – the “cost” function (J) Seek the estimate x of that minimizes J (i.e. t x J 0 x = ) (Notation : probability distribution function of ( ) p = ( ) ) (a) ( ) ( ) t 0 No knowledge of p x , p y J = sum of squares of differences between predictions ( ) y Hx = and observations ( ) 0 y ( ) ( ) T 0 0 y Hx y Hx = (called “least squares” minimization) (b) ( ) 0 T Know p y and hence R expectation
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