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Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
1
Numerical Integration
Week 6, Day 1
3 Oct 2016
Analysis Methods in Atmospheric and Oceanic Science
AOSC 652
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
2
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
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
3
AOSC 652: Analysis Methods in AOSC
τ CH4 : CH4 Lifetime with respect to loss by reaction w/ tropospheric OH
How found ?!?
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
4
GMIGEOS-Chem
DECCAM-Chem
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
5
Air Quality Model Satellite Data
AOSC 652: Analysis Methods in AOSC
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
6
22 March
NASA Aura MLS ClO490 K pot’l temp
~18 km22 March
2010 2011
ppb ppb
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
7
2 October 2011
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
8
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× −∫
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
9
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 2014 = 323.6 Gt C
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
10
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
Change in atmospheric CO2, 1959 to 2014 = 82.64 ppm82.64 ppm × 2.18 Gt C / ppm CO2 = 180.16 Gt C
CO2 emission, 1959 to 2014 = 323.6 Gt C
Legacy of Charles Keeling, Scripps Institution of Oceanography, La Jolla, CAhttp://www.esrl.noaa.gov/gmd/ccgg/trends/co2_data_mlo.html
About half of the CO2 released by combustion offossil fuels stays in the atmosphere:
rest taken up by land biosphere and oceans
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
11
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
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
12
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
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
13
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= ∫
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
14
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∫
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
15
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
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
16
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
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
17
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
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
18
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:
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
19
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:
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
20
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.
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
21
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?
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
22
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
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
23
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
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
24
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?
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
25
AOSC 652: Analysis Methods in AOSCNumerical Integration
Boole’s Rule (aka as 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, Boole’s rule is exact for what order polynomial?
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
26
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
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
27
Numerical Integration: Derivation of Simpson’s rule, page 1
R. L. Burden and J. D. Faires, Numerical Analysis, 8th edition
Copyright © 2016 University of Maryland.This material may not be reproduced or redistributed, in whole or in part, without written permission from Ross Salawitch. 3 Oct 2016
28
Numerical Integration: Derivation of Simpson’s rule, page 2
R. L. Burden and J. D. Faires, Numerical Analysis, 8th edition