Atmospheric Inventories for Atmospheric Inventories for Regional BiogeochemistryRegional Biogeochemistry

Scott Denning

Colorado State University

The Global Carbon CycleThe Global Carbon Cycle

Atmosphere

750 + 3/yr

Ocean

38,000

Plants and Soils

2000

Fossil Fuel

~90

~120

~120

6

~90

• Global trend known very accurately

• Provides an integral constraint on the total carbon budget

• Interannual variability is of the same order as anthropogenic emissions!

Meridional Gradients Meridional Gradients (Tans et al, 1990)(Tans et al, 1990)

Simulated gradient way too steep for most emission scenarios

Diurnal VariationsDiurnal Variations

• Diurnal cycle of 70 ppm over an active forest is 5x as strong as seasonal cycle

• Vertical gradient at sunrise is 15x as strong as pole-to-pole gradient

• Vertical gradient reverses in daytime, much weaker0 5 10 15 20 25

local time [hours]

340

360

380

400

420

CO2 [ppm]

tower levels:11 m30 m76 m122 m244 m396 m (1)396 m (2)

WLEF July 1997 diurnal composite

Diurnal Variations

• Advection: – Stuff that was “upstream” moves here with the wind:

– To predict the future amount of q, we simply need to keep track of gradients and wind speed in each direction

• Turbulence– Tracer transport by “unresolved” eddies– Behaves like down-gradient diffusion or “mixing”

• Moist (cumulus) convection– Vertical transport by updrafts and downdrafts in clouds

Atmospheric TransportAtmospheric Transport

( )q q q q

u v w Source qt x y z

∂ ∂ ∂ ∂=− − − +

∂ ∂ ∂ ∂

∂q∂t

=−∂∂x

k∂q∂x

⎛

⎝⎜⎞

⎠⎟−

∂∂y

k∂q∂y

⎛

⎝⎜⎞

⎠⎟−

∂∂z

k∂q∂z

⎛

⎝⎜⎞

⎠⎟

Cumulus Transport (example)Cumulus Transport (example)

• Cloud “types” defined by lateral entrainment

• Separate in-cloud CO2 soundings for each type

• Detrainment back into environment at cloud top

Inverse Modeling of Atmospheric Inverse Modeling of Atmospheric COCO22

Air Parcel Air Parcel

Air Parcel

Sources Sinks

transport transport

(model) (solve for)

concentration transport sources and sinks(observe)

Sample Sample

Synthesis Inversion ProcedureSynthesis Inversion Procedure(“Divide and Conquer”)(“Divide and Conquer”)

1. Divide carbon fluxes into subsets based on processes, geographic regions, or some combination1. Spatial patterns of fluxes within regions?2. Temporal phasing (e.g., seasonal, diurnal,

interannual?)

2. Prescribe emissions of unit strength from each “basis function” as lower boundary forcing to a global tracer transport model

3. Integrate the model for three years (“spin-up”) from initially uniform conditions to obtain equilibrium with sources and sinks

4. Each resulting simulated concentration field shows the “influence” of the particular emissions pattern

5. Combine these fields to “synthesize” a concentration field that agrees with observations

Consider Linear RegressionConsider Linear Regression

• Given N measurements yi for different values of the independent variable xi, find a slope (m) and intercept (b) that describe the “best” line through the observations– Why a line?– What do we mean by “best”– How do we find m and b?

• Compare predicted values to observations, and find m and b that fit best

• Define a total error (difference between model and observations) and minimize it!

• Our model is

• Error at any point is just

• Could just add the error up:

• Problem: positive and negative errors cancel … we need to penalize signs equally

• Define the total error as the sum of the Euclidean distance between the model and observations (sum of square of the errors):

Defining the Total ErrorDefining the Total Error

prey mx b= +

i i ie y mx b= − −

1 1

( )N N

i i ii i

e y mx b= =

= − −∑ ∑

2 2 2

1 1 1

( ) ( )N N N

prei i i i i

i i i

SSE e y y y mx b= = =

≡ = − = − −∑ ∑ ∑

Minimizing Total Error (Least Minimizing Total Error (Least Squares)Squares)

0

10

20

30

40

50

60

0 5 10 15 20 25 30 35

y

ei

yi

ypre

2 2 2

1 1 1

( ) ( )N N N

prei i i i i

i i i

SSE e y y y mx b= = =

≡ = − = − −∑ ∑ ∑

Take partial derivs

Set to zero

Solve for m and b

( ) ( ),

SSE SSE

m b

∂ ∂∂ ∂

Geometric View of Linear Geometric View of Linear RegressionRegression

• Any vector can be written as a linear combination of the orthonormal basis set

• This is accomplished by taking the dot product (or inner product) of the vector with each basis vector to determine the components in each basis direction

• Linear regression involves a 2D mapping of an observation vector into a different vector space

• More generally, this can involve an arbitrary number of basis vectors (dimensions)

( , , )a x y z=r

ˆˆ ˆ( , , )i j k

Generalized Least SquaresGeneralized Least Squares

mur

=

1m

2m..

Mm

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

1d2d

.

.

Nd

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

=

11G 12G . . 1MG21G 22G . . .

. . . . .

. . . . .

N1G N2G . . NMG

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

1m2m

.

.

Mm

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

The G's can be thought of as partial derivatives and the whole matrix as a Jacobian.

1 11 1 12 2 1d G m G m G mM M= + + +.....

m G d=- 1$

““Determinedness”Determinedness”

• Even-determined– Exact solution, zero prediction error

• Over-determined– Single "best" solution, finite prediction error

• Underdetermined– Infinite solutions, zero prediction error

• Mixed-determined– Many solutions, finite prediction error

TransCom Inversion TransCom Inversion IntercomparisonIntercomparison

• Discretize world into 11 land regions (by vegetation type) and 11 ocean regions (by circulation features)

• Release tracer with unit flux from each region during each month into a set of 16 different transport models

• Produce timeseries of tracer concentrations at each observing station for 3 simulated years

Synthesis InversionSynthesis Inversion

• Decompose total emissions into M “basis functions”

• Use atmospheric transport model to generate G– Fill by columns using

forward model, or– Fill by rows using

backward (adjoint) model

• Observe d at N locations• Invert G to find m

1d2d

.

.

Nd

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

=

11G 12G . . 1MG21G 22G . . .

. . . . .

. . . . .

N1G N2G . . NMG

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

1m2m

.

.

Mm

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

data transport fluxes

d1 = G11 m1 + G12 m2 + … + G1N mN

CO2 sampled at location 1 Strength of emissionsof type 2

partial derivative of CO2 at location 1 with respect to emissions of type 2

Rubber BandsRubber Bands

• Inversion seeks a compromise between detailed reproduction of the data and fidelity to what we think we know about fluxes

• The elasticity of these two “rubber bands” is adjustable

Prior estimates of regional fluxes

Observational Data

Solution:Fluxes, Concentrations

MODELMODEL

a prioriemissiona prioriemission

uncertaintyuncertainty

concentrationdata

concentrationdata

uncertaintyuncertainty

emissionestimate emissionestimate

uncertaintyuncertainty

Bayesian Inversion Bayesian Inversion TechniqueTechnique

MODELMODEL

a prioriemissiona prioriemission

uncertaintyuncertainty

uncertaintyreduction

concentrationdata

concentrationdata

uncertaintyuncertainty

emissionestimate emissionestimate

uncertaintyuncertainty

Bayesian Inversion Bayesian Inversion TechniqueTechnique

( ) ( )1 11

1ˆ ˆˆ ˆ ˆ ˆ ˆT Test p pobsd m dG Gm m mdG C C G C

− −−

−= + + −rr r r

Our problem is ill-conditioned. Apply prior constraints and minimize a more generalized cost function:

Bayesian Inversion FormalismBayesian Inversion Formalism

Solution is given by

Inferred flux

PriorGuess

DataUncertainty

FluxUncertainty

observations

data constraint prior constraint

Uncertainty in Flux EstimatesUncertainty in Flux Estimates

• A posteriori estimate of uncertainty in the estimated fluxes

• Depends on transport (G) and a priori uncertainties in fluxes (Cm) and data (Cd)

• Does not depend on the observations per se!

( ) 11 1* ˆ ˆT

mm dC C CG G−

− −= +

( ) ( )1 11

1ˆ ˆˆ ˆ ˆ ˆ ˆT Test p pobsd m dG Gm m mdG C C G C

− −−

−= + + −rr r r

Covariance MatricesCovariance Matrices

• Inverse of a diagonal matrix is obtained by taking reciprocal of diagonal elements

• How do we set these?

• (It’s an art!)

1

2

2

2

2

0 . . 0

0 . . 0

. . . . .

. . . . .

0 0 . 0N

d

d

d

σσ

σ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

dC

1

2

2

2

2

0 . . 0

0 . . 0

. . . . .

. . . . .

0 0 . 0M

m

mm

m

σσ

σ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

C

Accounting for “Error”Accounting for “Error”

• Sampling, contamination, analytical accuracy (small)

• Representativeness error (large in some areas, small in others)

• Model-data mismatch (large and variable)

• Transport simulation error (large for specific cases, smaller for “climatological” transport)

• All of these require a “looser” fit to the data

Regional Problem is Harder!Regional Problem is Harder!

• Higher density of observations over central US or Europe allows more detailed analysis

• High-resolution transport modeling typically requires specifying inflow!

• How to treat tracer variations on lateral boundaries?

• Can “nest” models of various scales, but how to treat errors?

• The smaller the region, the less trace gas variations are determined inside it!

QuickTime™ and aPNG decompressor

are needed to see this picture.

Ingredients for InversionsIngredients for Inversions

• Observations!

• Choices of domain, spatial & temporal structure of solution to seek

• Transport model to specify winds, turbulence, cloud motions

• Source-receptor matrix linking fluxes to observable concentrations

• An optimization scheme

• Error covariance statistics for expected model-data mismatch

• Numerical matrix algebra