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 () ys , t ( ) y* 0,0 is given by () () ( ) t 0 s 0 , t y*s , t y*0 ,0 xdt ' xd ds' v vd =− = ⎛⎞ == ⎜⎟ ⎝⎠ 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: 0t ii j j j yh x i ε ε In vector-matrix form for multiple observing sites i yH x measurement equation where is the ij Hh ⎡⎤ = ⎣⎦ 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 ii ij t jj yy h xx ∂∂ == compute in model where ( ) ij y,x denotes an estimate of ( ) 0t . 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 . ( xr e f ) 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) () () t0 No knowledge of p x , p y J = sum of squares of differences between predictions ( ) yH x = and observations ( ) 0 y T 00 x x (called “least squares” minimization) (b) 0T Know p y and hence R expectation
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This note was uploaded on 11/27/2011 for the course CHEMICAL E 20.410j taught by Professor Rogerd.kamm during the Spring '03 term at MIT.

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

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