aosc 652: week 6, day 1 - university of maryland · this material may not be reproduced or...
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Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
Numerical Integration
Week 6, Day 1
6 Oct 2014
Analysis Methods in Atmospheric and Oceanic Science
AOSC 652
1
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Also known as “quadrature”
2
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Also known as “quadrature”
Why might you need to compute an integral ?
3
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSC
4
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AOSC 652: Analysis Methods in AOSC
OH Column: Units of 1012 molecules cm−2 OR1012 cm−2
Trop: Troposphere
5
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSC
OH Column: Units of 1012 molecules cm−2 OR1012 cm−2
How is Trop OH column from a model such as theGoddard “Global Modeling Initiative” (GMI)
Chemical Transport Model (CTM) found ???
Trop: Troposphere
6
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSC
OH Column: Units of 1012 molecules cm−2 OR1012 cm−2
[ ]Trop OH column OH( ) Tropopause
Surface
z dz= ∫
7
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSC
[ ]Trop OH column OH( ) Tropopause
Surface
z dz= ∫
8
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[ ]
Mixing Ratio
Trop OH Column
OH( )
OH ( )
Tropopause
Surface
Tropopause
Surface
z dz
c p dp
=
=
∫
∫
9
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AOSC 652: Analysis Methods in AOSC
τ CH4 : CH4 Lifetime with respect to loss by reaction w/ tropospheric OH
10
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSC
τ CH4 : CH4 Lifetime with respect to loss by reaction w/ tropospheric OH
How found ?!?
11
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AOSC 652: Analysis Methods in AOSC
)( )( )][][(
)( )( )]([
0 0 44
0 0 4
4latdlonddpmassCHOHk
latdlonddpmassCHnlats nlons Tropopause
Surface airCHOH
nlats nlons Tropopause
Surface air
CH
∫ ∫ ∫∫ ∫ ∫
×××
×=
+
τ
12
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GMI
CAM-Chem JAN
GEOS-Chem
13
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GMI
CAM-Chem FEB
GEOS-Chem
14
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GMI
MARCAM-Chem
GEOS-Chem
15
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GMI
APRCAM-Chem
GEOS-Chem
16
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GMI
MAYCAM-Chem
GEOS-Chem
17
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GMI
JUNCAM-Chem
GEOS-Chem
18
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GMI
JULCAM-Chem
GEOS-Chem
19
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GMI
AUGCAM-Chem
GEOS-Chem
20
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GMI
SEPCAM-Chem21
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GMI
OCTCAM-Chem
GEOS-Chem
22
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GMI
NOVCAM-Chem
GEOS-Chem
23
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GMI
DECCAM-Chem
GEOS-Chem
24
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Air Quality Model Satellite Data
AOSC 652: Analysis Methods in AOSC
25
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
Air Quality Model Satellite Data
AOSC 652: Analysis Methods in AOSC
26
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
22 March
NASA Aura MLS ClO490 K pot’l temp
~18 km22 March
2010 2011
ppb ppb
27
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2 October 2011
28
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
2 October 2011
{ }HIGH
LOW
*3
θ*
3θ
3
3
O (θ) is measure
Ozone Deficit = Factor O (θ) O (θ) θ
where
O (θ) is profile withou and t any
d prof
chemi
ile (
cal l
RED
oss (BLUE)
)
d× −∫
29
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Why else might you need to compute an integral ?
Calculate carbon emissions to compare with change in atmospheric CO2
30
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Why else might you need to compute an integral ?
Calculate carbon emissions to compare with change in atmospheric CO2
CO2 emission, 1959 to 2012 = 305.4 Gt C
31
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSC
Legacy of Charles Keeling, Scripps Institution of Oceanography, La Jolla, CAhttp://www.esrl.noaa.gov/gmd/ccgg/trends/co2_data_mlo.html
Numerical IntegrationWhy else might you need to compute an integral ?
Calculate carbon emissions to compare with change in atmospheric CO2
Change in atmospheric CO2, 1959 to 2012 = 77.9 ppm
CO2 emission, 1959 to 2012 = 305.4 Gt C
32
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSC
Legacy of Charles Keeling, Scripps Institution of Oceanography, La Jolla, CAhttp://www.esrl.noaa.gov/gmd/ccgg/trends/co2_data_mlo.html
Numerical IntegrationWhy else might you need to compute an integral ?
Calculate carbon emissions to compare with change in atmospheric CO2
Change in atmospheric CO2, 1959 to 2012 = 77.9 ppm= 171.3 Gt C
CO2 emission, 1959 to 2012 = 305.4 Gt C
33
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSC
About half of the CO2 released by fossil fuelcombustion stays in the atmosphere:
rest taken up by land biosphere and oceans
Legacy of Charles Keeling, Scripps Institution of Oceanography, La Jolla, CAhttp://www.esrl.noaa.gov/gmd/ccgg/trends/co2_data_mlo.html
Numerical IntegrationWhy else might you need to compute an integral ?
Calculate carbon emissions to compare with change in atmospheric CO2
Change in atmospheric CO2, 1959 to 2012 = 77.9 ppm= 171.3 Gt C
CO2 emission, 1959 to 2012 = 305.4 Gt C
34
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSC
About half of the CO2 released by fossil fuelcombustion stays in the atmosphere:
rest taken up by land biosphere and oceans
Legacy of Charles Keeling, Scripps Institution of Oceanography, La Jolla, CAhttp://www.esrl.noaa.gov/gmd/ccgg/trends/co2_data_mlo.html
Numerical IntegrationWhy else might you need to compute an integral ?
Calculate carbon emissions to compare with change in atmospheric CO2
Change in atmospheric CO2, 1959 to 2012 = 77.9 ppm= 171.3 Gt C
CO2 emission, 1959 to 2012 = 305.4 Gt C
35
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Why might you need to compute an integral ?
Heat Balance in Ocean:
( ) /
0
0 −
− − −
=
∫∫ OCEAN WARMING
t
SOLAR IR RADIATION EVAPORATION SENSIBLE HEAT LOSS GAIN
t
Q Q Q Q dt
Q dt
36
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Why might you need to compute an integral ?
Heat Balance in Ocean:
( ) /
0
0 −
− − −
=
∫∫ OCEAN WARMING
t
SOLAR IR RADIATION EVAPORATION SENSIBLE HEAT LOSS GAIN
t
Q Q Q Q dt
Q dt
This heat then affects temperature in a particular ocean layer via:
T Z
OCEAN WARMING p
tc dzQ dt ρ− ∆= ∫∫ 00
37
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Also known as “quadrature”
Why might you need to compute an integral ?
Find pressure based on the mass of the overlying atmosphere:
Find dynamic height D of ocean based on specific volume α:
( ) dz
p z g zρ∞
= ∫
2
1
dp
p
D pα= ∫
38
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Trapezoidal Rule:
1
11
11
( ) + ( )( ) ( ) 2
NNx i i
i ix i
f x f xf x dx x x−
++
=
≈ −Σ∫
39
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Trapezoidal Rule:
1
11
11
( ) + ( )( ) ( ) 2
NNx i i
i ix i
f x f xf x dx x x−
++
=
≈ −Σ∫
Analysis Methods for Engineers, Ayyub and McCuen
40
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Trapezoidal Rule:
1
11
11
( ) + ( )( ) ( ) 2
NNx i i
i ix i
f x f xf x dx x x−
++
=
≈ −Σ∫
Base of trapezoid
Analysis Methods for Engineers, Ayyub and McCuen
41
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Trapezoidal Rule:
1
11
11
( ) + ( )( ) ( ) 2
NNx i i
i ix i
f x f xf x dx x x−
++
=
≈ −Σ∫Avg height of end points
Analysis Methods for Engineers, Ayyub and McCuen
42
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Trapezoidal Rule:
1
11
11
( ) + ( )( ) ( ) 2
NNx i i
i ix i
f x f xf x dx x x−
++
=
≈ −Σ∫
Analysis Methods for Engineers, Ayyub and McCuen
Case where integration using the trapezoidal rule should work well:
43
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Trapezoidal Rule:
1
11
11
( ) + ( )( ) ( ) 2
NNx i i
i ix i
f x f xf x dx x x−
++
=
≈ −Σ∫
Analysis Methods for Engineers, Ayyub and McCuen
Case where integration using the trapezoidal rule may not work well:
44
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Trapezoidal Rule:
1
11
11
( ) + ( )( ) ( ) 2
NNx i i
i ix i
f x f xf x dx x x−
++
=
≈ −Σ∫
Error order d2f/dx2: i.e., second derivative of function evaluated at some placein the interval
Hence, trapezoidal rule is exact for any function whose second derivativeis identically zero.
45
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Simpson’s Rule:
1
22 2 1
1, 3, 5
( ) + 4 ( ) ( )( ) 2 3
NNx i i i i i
x i
x x f x f x f xf x dx−
+ + +
=
− +≈ Σ∫
What is the basis of this formula?
46
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Simpson’s Rule:
1
22 2 1
1, 3, 5
( ) + 4 ( ) ( )( ) 2 3
NNx i i i i i
x i
x x f x f x f xf x dx−
+ + +
=
− +≈ Σ∫
What is the basis of this formula? each point fit by “quadratic” Lagrange polynomials
Analysis Methods for Engineers, Ayyub and McCuen
47
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Simpson’s Rule:
1
22 2 1
1, 3, 5
( ) + 4 ( ) ( )( ) 2 3
NNx i i i i i
x i
x x f x f x f xf x dx−
+ + +
=
− +≈ Σ∫
What is the basis of this formula? each point fit by “quadratic” Lagrange polynomials
Analysis Methods for Engineers, Ayyub and McCuen
See pages 187 to 190 of Numerical Analysis, Burden and Faires,for a derivation of Simpson’s Rule
in terms of Quadratic Lagrange Polynomials
48
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Simpson’s Rule:
1
22 2 1
1, 3, 5
( ) + 4 ( ) ( )( ) 2 3
NNx i i i i i
x i
x x f x f x f xf x dx−
+ + +
=
− +≈ Σ∫
Error order d4f/dx4: i.e., fourth derivative of function evaluated at some placein the interval
Hence, Simpson’s rule is exact for what order polynomial?
49
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
AOSC 652: Analysis Methods in AOSCNumerical Integration
Bode’s Rule:
1
44 4 3 2 1
1, 5, 11, ...
( )
14 ( ) 64 ( ) 24 ( ) + 64 ( ) 14 ( ) 2 45
−+ + + + +
=
≈
− + + +
∫
Σ
Nx
x
Ni i i i i i i
i
f x dx
x x f x f x f x f x f x
Error order d6f(x)/dx6: i.e., sixth derivative of function evaluated at some placein the interval
Hence, Bode’s rule is exact for what order polynomial?
50
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AOSC 652: Analysis Methods in AOSCNumerical Integration
Gaussian Quadrature:
y1, y2, … yn nodes are not uniformly spacedc1, c2, … cn Gauss coefficients are determined once n is specified
Nice descriptions of theory at http://en.wikipedia.org/wiki/Gaussian_quadratureand http://www.efunda.com/math/num_integration/num_int_gauss.cfm
( )n1
1i=1
1 ( ) = ( ( )) ( ( )) 2
b
i ia
dxf x dx f g y dy b a c f g ydy−
≈ − ∑∫ ∫
Note:
produces exact result for polynomials of degree 3 or less
1
1
3 3( ) 3 3−
−≈ +
∫ f y dy f f
51
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Numerical Integration: Derivation of Simpson’s rule, page 1
R. L. Burden and J. D. Faires, Numerical Analysis, 8th edition 52
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
Numerical Integration: Derivation of Simpson’s rule, page 2
R. L. Burden and J. D. Faires, Numerical Analysis, 8th edition 53
Copyright © 2014 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 6 Oct 2014
Numerical Integration: Derivation of Simpson’s rule, page 3
R. L. Burden and J. D. Faires, Numerical Analysis, 8th edition 54