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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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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 vectormatrix 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
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.
 Spring '03
 RogerD.Kamm
 pH

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