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Improving Understanding of
Global and Regional Carbon Dioxide Flux
Variability through Assimilation of in Situ
and Remote Sensing Data in a
Geostatistical Framework
Anna M. Michalak
Department of Civil and Environmental Engineering
Department of Atmospheric, Oceanic and Space Sciences
The University of Michigan
Outline
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
Introduction to geostatistics
Inverse modeling approaches to estimating flux
distributions
Geostatistical approach to quantifying fluxes:
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Global flux estimation
Use of auxiliary data
Regional scale synthesis
Spatial Correlation

Measurements in close proximity to each other generally
exhibit less variability than measurements taken farther
apart.

Assuming independence, spatially-correlated data may
lead to:
1.
2.
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Biased estimates of model parameters
Biased statistical testing of model parameters
Spatial correlation can be accounted for by using
geostatistical techniques
Parameter Bias Example
map of an alpine basin
Q: What is the mean snow depth in
the watershed?
snow depth measurements
600
kriging estimate of mean snow depth
(assumes spatial correlation)
mean of snow depth measurements
(assumes spatial independence)
snow depth [cm]
500
400
300
200
100
0
0
200
400
600
x [m]
800
1000
5% H0
H0
Rejected
Rejected!
H0 is TRUE
5% H0 rejected
H0
H5%
0
Not Rejected
Rejected
Variogram Model
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Used to describe spatial correlation
1
2
3
4
Geostatistics in Practice

Main uses:
Data integration
Numerical models for prediction
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6
6
5
5
4
4
3
3
2
2
1
2
4
1
6
2
4
6
Numerical assessment (model) of uncertainty
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6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
2
4
6
1
2
4
6
1
2
4
6
1
2
4
6
Geostatistical Inverse Modeling
Actual flux history
Available data
15
5
Actual release history
Prior guess
Actual plume
Measurement locations and values
4
Concentration
Concentration
10
5
0
-5
0
3
2
1
50
100
Time
150
200
0
50
100
150
200
250
Location downstream
300
350
Geostatistical Inverse Modeling
Bayesian / Independent Errors
10
10
8
8
6
6
Concentration
Concentration
Geostatistical
4
2
4
2
0
0
-2
-2
-4
0
50
100
Time
150
200
-4
0
31 data
201
101
41
21
11 fluxes
fluxes
50
100
Time
150
200
Key Points
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If the parameter(s) that you are modeling exhibits
spatial (and/or temporal) autocorrelation, this feature
must be taken into account to avoid biased solutions
Spatial (and/or temporal) autocorrelation can be used
as a source of information in helping to constrain
parameter distributions
The field of geostatistics provides a framework for
addressing the above two issues
ASIDE: CO2 Measurements from Space
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
Factors such as clouds,
aerosols and computational
limitations limit sampling for
existing and upcoming
satellite missions such as the
Orbiting Carbon Observatory
A sampling strategy based
on XCO2 spatial structure
assures that the satellite
gathers enough information
to fill data gaps within
required precision
Alkhaled et al. (in prep.)
XCO2 Variability
Regional spatial covariance
structure is used to evaluate:
 Regional sampling
densities required for a set
interpolation precision
 Minimum sampling
requirements and optimal
sampling locations

x104 km
2
2.5 ppm
Variance
2
Correlation Length
2
1.5
o
o
60 N
60 N
o
1.5
30 N
o
o
30 N
1
o
0
0
1
o
o
30 S
30 S
o
60 S
0.5
o
180 W
o
120 W
o
60 W
o
0
o
60 E
o
120 E
o 0
180 W
0.5
o
60 S
o
180 W
o
120 W
o
60 W
o
0
o
60 E
o
120 E
0
o
180 W
Source: NOAA-ESRL
What Surface Fluxes do Atmospheric
Measurements See?
Latitude
Height Above Ground Level (km)
24 June 2000: Particle Trajectories
Longitude
Source: Arlyn Andrews, NOAA-GMD
-24 hours
-48 hours
-72 hours
-96 hours
-120 hours
Longitude
Need for Additional Information
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Current network of atmospheric sampling sites
requires additional information to constrain fluxes:
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Problem is ill-conditioned
Problem is under-determined (at least in some areas)
There are various sources of uncertainty:
 Measurement error
 Transport model error
 Aggregation error
 Representation error
One solution is to assimilate additional information
through a Bayesian approach
Bayesian Inference Applied to Inverse
Modeling for Surface Flux Estimation
Posterior probability
of surface flux
distribution
Likelihood of fluxes given
atmospheric distribution
ps|y  
Prior information
about fluxes
py|s  ps 
 py|s ps ds
y : available observations (n×1)
s : surface flux distribution (m×1)
p(y) probability of
measurements
Synthesis Bayesian Inversion
Biospheric
Model
Prior flux
estimates (sp)
Auxiliary
Variables
CO2
Observations
(y)
?
Inversion
Transport
Model
Sensitivity of
observations to
fluxes (H)
Meteorological
Fields
Residual
covariance
structure (Q, R)
?
Flux estimates
and covariance
ŝ, Vŝ
Large Regions Inversion
TransCom, Gurney et al. 2003
Transport Model Gridscale Inversions
Rödenbeck et al. 2003
Variogram Model
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Used to describe spatial correlation
1
2
3
4
Geostatistical Approach to Inverse Modeling
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Geostatistical inverse modeling objective function:
Ls , β 
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1
1
(y  Hs)T R 1 (y  Hs)  (s  Xβ)T Q 1 (s  Xβ)
2
2
H = transport information, s = unknown fluxes,
y = CO2 measurements
X and  define the model of the trend
R = model data mismatch covariance
Q = spatio-temporal covariance matrix for the flux deviations
from the trend
Deterministic
component
Stochastic
component
Synthesis Bayesian Inversion
Biospheric
Model
Prior flux
estimates (sp)
Auxiliary
Variables
CO2
Observations
(y)
Inversion
Transport
Model
Sensitivity of
observations to
fluxes (H)
Meteorological
Fields
Residual
covariance
structure (Q, R)
Flux estimates
and covariance
ŝ, Vŝ
Geostatistical Inversion
select significant variables
Auxiliary
Variables
Variance
Ratio
Test
CO2
Observations
(y)
Flux estimates
and covariance
ŝ, Vŝ
Inversion
Transport
Model
Sensitivity of
observations to
fluxes (H)
Meteorological
Fields
Residual
covariance
structure (Q, R)
Trend estimate
and covariance
β, Vβ
RML
Optimization
optimize covariance parameters
Key Questions
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Can the geostatistical approach estimate:





If so, what do we learn about:

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

Sources and sinks of CO2 without relying on prior
estimates?
Spatial and temporal autocorrelation structure of
residuals?
Significance of available auxiliary data?
Relationship between auxiliary data and flux
distribution?
Flux variability (spatial and temporal)
Influence of prior flux estimates in previous studies
Impact of aggregation error
What are the opportunities for further expanding this
approach to move from attribution to diagnosis and
prediction?
Fluxes Used in Pseudodata Study
Michalak, Bruhwiler & Tans (JGR, 2004)
Recovery of Annually Averaged Fluxes
Best estimate
Michalak, Bruhwiler & Tans (JGR, 2004)
“Actual” fluxes
Recovery of Annually Averaged Fluxes
Best estimate
Michalak, Bruhwiler & Tans (JGR, 2004)
Standard Deviation
Key Questions

Can the geostatistical approach estimate





If so, what do we learn about:




Sources and sinks of CO2 without relying on prior
estimates?
Spatial and temporal autocorrelation structure of
residuals?
Significance of available auxiliary data?
Relationship between auxiliary data and flux
distribution?
Flux variability (spatial and temporal)
Influence of prior flux estimates in previous studies
Impact of aggregation error
What are the opportunities for further expanding this
approach to move from attribution to diagnosis and
prediction?
Auxiliary Data and Carbon Flux Processes
Other:
Spatial trends
Anthropogenic
Flux:
Fossil fuel
combustion
(sine latitude,
absolute value
latitude)
(GDP density,
population)
Environmental
parameters:
(precipitation,
%landuse, Palmer
drought index)
Oceanic Flux:
Gas transfer
Terrestrial Flux:
Photosynthesis
(sea surface
(FPAR, LAI, NDVI)
temperature, air
temperature)
Respiration
(temperature)
Image Source: NCAR
Sample Auxiliary Data
Gourdji et al. (in prep.)
Which Model is Best?
5
10
0
5
-5
0
-5
0
2
4
6
8
Available data
Real (unknown) determininistic component
Constant mean
Linear trend
Linear + Quadratic
Linear+Quadratic+Cubic
0
2
4
6
8
Available data
Real (unknown) determininistic component
Constant mean
Linear trend
Linear + Quadratic
Linear+Quadratic+Cubic
Geostatistical Approach to Inverse Modeling

Geostatistical inverse modeling objective function:
Ls , β 




1
1
(y  Hs)T R 1 (y  Hs)  (s  Xβ)T Q 1 (s  Xβ)
2
2
H = transport information, s = unknown fluxes,
y = CO2 measurements
X and  define the model of the trend
R = model data mismatch covariance
Q = spatio-temporal covariance matrix for the flux deviations
from the trend
Deterministic
component
Stochastic
component
Global Gridscale CO2 Flux Estimation

Estimate monthly CO2 fluxes (ŝ) and their uncertainty on
3.75° x 5° global grid from 1997 to 2001 in a geostatistical
inverse modeling framework using:
 CO2 flask data from NOAA-ESRL network (y)
 TM3 (atmospheric transport model) (H)
 Auxiliary environmental variables correlated with CO2
flux
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Three models of trend flux (Xβ) considered:
 Simple monthly land and ocean constants
 Terrestrial latitudinal flux gradient and ocean constants
 Terrestrial gradient, ocean constants and auxiliary
variables
Measurement Locations
Gourdji et al. (in prep.)
Mueller et al. (in prep.)
Selected Auxiliary Variables
Combine physical understanding with results of VRT to choose final
set of auxiliary variables:
LAI
% Ag
LAI
SST
fPAR
% Forest
fPAR
dSSt/dt
% Shrub
%
Shrub
NDVI
Palmer Drought Index
GDP Density
% Grass
Precipitation
GDP
Density
Land Air
Land
AirTemp.
Temp.
Population Density
Inversion estimates drift coefficients (β):

CV
X
(GtC/yr)

GDP

LAI

fPAR
GDP
0.09
0.247
2.4
1
0.01
-0.19
0.24
0.10
LAI
-0.67
0.094
-44.6
---
1
-0.93
0.03
-0.05
fPAR
0.60
0.094
49.3
---
---
1
-0.15
-0.15
% Shrub
-0.11
0.175
-4.4
---
---
---
1
0.02
LandTemp
0.06
0.485
1.7
---
---
---
---
1
Aux.
Variable
Gourdji et al. (in prep.)


% Shrub L. Temp
Building up the best estimate in January 2000
Deterministic
Gourdji et al. (in prep.) component
sˆ  X  Q HT 
Stochastic
component
A posteriori uncertainty for January 2000
Gourdji et al. (in prep.)
Transcom Regions
TransCom, Gurney et al. 2003
Regional comparison of seasonal cycle
Gourdji et al. (in prep.)
Regional comparison of seasonal cycle #2
Gourdji et al. (in prep.)
Comparison of annual average non-fossil fuel flux
Transcom Land Regions
3
Transcom Land Regions
Variable Trend Best Estimates +/- 2 
Simple Trend Best Estimates
Modified Trend Best Estimates
Transcom (Baker et al., 2006) +/- 2 
Rodenbeck et al. (2003) +/- 2 
2.5
-1
-1.5
-2
Gourdji et al. (in prep.)
Euro
BNAm
-2
-0.5
Aus t
-1.5
TrAs
-1
0
Te As
-0.5
0.5
BoAs
0
1
SoAf
0.5
1.5
NoAf
1
2
SoAm
1.5
2.5
TrAm
2
TNAm Average Non-Fossil Fuel Flux (GtC/yr)
Annual
Annual Average Non-Fossil Fuel Flux (GtC/yr)
3
Key Questions

Can the geostatistical approach estimate





If so, what do we learn about:




Sources and sinks of CO2 without relying on prior
estimates?
Spatial and temporal autocorrelation structure of
residuals?
Significance of available auxiliary data?
Relationship between auxiliary data and flux
distribution?
Flux variability (spatial and temporal)
Influence of prior flux estimates in previous studies
Impact of aggregation error
What are the opportunities for further expanding this
approach to move from attribution to diagnosis and
prediction?
Opportunities for Regional Synthesis
Continuous tall-tower data
available
More consistent relationship to
auxiliary variables
Flux tower and aircraft
campaign data available for
validation
NACP offers opportunities for
intercomparison / collaborations
B. Stephens,
UND
crew, COBRA
WLEF Photo
tall credit:
tower
(447m)
inCitation
Wisconsin
with CO2
mixing ratio measurements at 11, 30,
76, 122, 244 and 396 m
North American CO2 Flux Estimation

Estimate North American
CO2 fluxes at 1°x1°
resolution &
daily/weekly/monthly
timescales using:
 CO2 concentrations
from 3 tall towers in
Wisconsin (Park Falls),
Maine (Argyle) and
Texas (Moody)
 STILT – Lagrangian
atmospheric transport
model
 Auxiliary remotesensing and in situ
environmental data
Pseudodata and recovered fluxes
(Source: Adam Hirsch, NOAA-ESRL)
Assimilation of Remote Sensing and
Atmospheric Data
Analysis steps:
Compile auxiliary variables
Select significant variables to
include in model of the trend
Estimate covariance
parameters:
Model-data mismatch
Flux deviations from
overall trend.
Perform inversion, estimating
both (i) the relationship
between auxiliary variables
and flux , and (ii) the flux
distribution s.
A posteriori covariance
includes the uncertainties of
fluxes, trend parameters,
and all cross-covariances
Key Questions

Can the geostatistical approach estimate





If so, what do we learn about:




Sources and sinks of CO2 without relying on prior
estimates?
Spatial and temporal autocorrelation structure of
residuals?
Significance of available auxiliary data?
Relationship between auxiliary data and flux
distribution?
Flux variability (spatial and temporal)
Influence of prior flux estimates in previous studies
Impact of aggregation error
What are the opportunities for further expanding this
approach to move from attribution to diagnosis and
prediction?
Conclusions




Atmospheric data information content is sufficient to:
 Quantify model-data mismatch and flux covariance
structure
 Identify significant auxiliary environmental variables and
estimate their relationship with flux
 Constrain continental fluxes independently of biospheric
model and oceanic exchange estimates
Uncertainties at grid scale are high, but uncertainties of
continental and global estimates are comparable to
synthesis Bayesian studies
Auxiliary data inform regional (grid) scale flux variability;
seasonal cycle at larger scales is consistent across models
Use of auxiliary variables within a geostatistical framework
can be used to derive process-based understanding directly
from an inverse model
Acknowledgements



Collaborators:
 Research group: Alanood Alkhaled, Abhishek Chatterjee, Sharon
Gourdji, Charles Humphriss, Meng Ying Li, Miranda Malkin, Kim
Mueller, Shahar Shlomi, and Yuntao Zhou
 NOAA-ESRL: Pieter Tans, Adam Hirsch, Lori Bruhwiler and Wouter
Peters
 JPL: Bhaswar Sen, Charles Miller
 Kevin Gurney (Purdue U.), John C. Lin (U. Waterloo), Ian Enting (U.
Melbourne), Peter Curtis (Ohio State U.)
Data providers:
 NOAA-ESRL cooperative air sampling network
 Seth Olsen (LANL) and Jim Randerson (UCI)
 Christian Rödenbeck, MPIB
 Kevin Schaefer, NSIDC
Funding sources:
QUESTIONS?
[email protected]
http://www.umich.edu/~amichala/