Geostatistical structural analysis of
TransCom data for development of
time-dependent inversion
Erwan Gloaguen, Emanuel Gloor, Jorge
Sarmiento and TransCom modelers
Plan
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Motivation
Mathematical statement
Methodology
Regularization and warning
TransCom synthetic data
Structural analysis of TransCom data
Motivations and goal
 Long time series inversions becomes
computer intensive.
 Sliding window inversions are commonly
used in many sciences (Kalman filter,
ARMA filter, etc…) and have been recently
used in CO2 inversion (Bruhwiler et al.,
2004).
 Geostatistical structural analysis for
stochastic inversion.
Mathematical statement of the
problem
We can write the system we are working on, A(f) = c , as a matrix equation:
As = c
Where,
A (describing the forward modelling function) is an N x M matrix, obtained
by Transport Simulations.
s is the vector of model parameters with M elements, the CO2 fluxes
c is the vector of data with N elements, the CO2 measured concentrations
Least-squares and regularization
Encountered problems in CO2 flux inversion:
- ill-posed
- generalized inverse numerically unstable.
Regularization allows to compute an inverse of A:
||As – c||2 + g ||K s||2
where K is any definite positive matrix
g is a scalar.
Regularized inversion tells the truth
you want to hear
Gurney
Jacobson
A simple example
We present here an example of the nonuniqueness of underdetermined problems using a
simple pair of linear equations. Consider the system described by
m1 + 2m2 - m3 + m4 = 6
-m1 + m2 + 2m3 - m4 = 2
or, equivalently:
Suite…
This system of equations has
•
Four unknowns (m1, m2, m3, m4 ).
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Two data (6, 2).
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It is an underdetermined system.
•
There is no unique solution.
Here are four solutions that all will satisfy this system of equations:
mA = ( 2.000,
mB = ( 0.444,
mC = (-2.408,
mD = ( 2.002,
2.000,
2.622,
2.630,
2.846,
2.000,
0.134,
0.109,
-0.537,
2.000 )
0.446 ) => How to choose which model is "best"?
3.256 )
-2.230 )
Dealing With Nonuniqueness:
Norms & Model Objective Functions
Given multiple solutions, how do we choose one that is useful? We need a quantitative way
to distinguish between acceptible models.
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The solution is to find a solution that is "largest" or "smallest".
•
Norms are mathematical rulers to measure "length".
We will define m to be the norm of the model. m will be called the model objective function.
The procedure for selecting one model will be:
1
Define the model objective function.
2
Choose the shortest; i.e. minimize this function.
As examples, one could:
•
Find the solution with smallest magnitude by minimizing (eqn. 1) ,
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Or find the solution that is flattest by minimizing , (eqn. 2).
Impact of the choice of then norm
The minimum model as specified by the objective function is highlighted in colour.
Using a smallest model objective function (eqn. 1)
mA = ( 2.000,
mB = ( 0.444,
mC = (-2.408,
mD = ( 2.002,
2.000,
2.622,
2.630,
2.846,
2.000,
0.134,
0.109,
-0.537,
2.000 ) small = 16.00
0.446 ) small = 7.23
3.256 ) small = 23.33
-2.230 ) small = 17.36
Using a flattest (most featureless) model objective function (eqn. 2)
mA = ( 2.000,
mB = ( 0.444,
mC = (-2.408,
mD = ( 2.002,
2.000,
2.622,
2.630,
2.846,
2.000,
0.134,
0.109,
-0.537,
2.000 ) flat = 0.0
0.446 ) flat = 11.02
3.256 ) flat = 41.61
-2.230 ) flat = 15.00
Conclusions on regularization
The choice of Om determines the outcome, and if the "right" model objective function is
chosen, a solution close to the "true" fluxes is obtained.
Just what exactly is the "right" model objective function is the next obvious question. It will
be tackled in the section entitled A Generic Model Objective Function. First, however, we
must discuss the important general issue of how close predicted data must match
observations. This is referred to as the "data misfit".
This implies the importance of exploring the data and model spaces.
Structural analysis allows to regularize the solution without any a
priori.
Sliding window cokriging as a
regularization tool
 Cokriging is a mathematical tool that allows
to interpolate an unsample variable (here,
the fluxes) using a secondary measured
variable (here, the concentration).
 Fluxes cokriging needs the spatial and
temporal covariances to be known.
Slowness covariance modelisation
based on measured times
Covariances of linearly related data are related with:
cov(c,c) = H cov(f,f) HT + Co
• If E[c] =0, then cov(c,c) = E[c,cT]
• Their exists several covariance functions that allow
the modelization of cov(f,f).
Cokriging
As cov(s,s) has been modelized, the slowness
field can be cokriged.
The cokriging estimator is
L = (Hcov(f,f)HT +Co)-1 * c
Sck = LT H cov(s,s)
TransCom Data
 Synthetic CO2 fluxes using fossil fuel, Net
Ecosystem Productivity and Takahashi
ocean’s fluxes.
 Synthetic CO2 concentrations from 253
TransCom stations.
 Integration of sampled fluxes on the 22
TransCom regions.
Latitudes
TransCom Regions and measurement
sites
Longitudes
Synthetic CO2 fluxes of the 22
TransCom regions (Michalak, 2004)
The synthetic monthly fluxes
What can we say?
 Fluxes vary strongly in time.
 The « shape » of the fluxes varies in time.
 Consecutive fluxes seems to be more
correlated than fluxes farther apart in time.
=> Structural analysis of the fluxes
Structural analysis
 Cross-covariance of variables Y and Z:
Czy(h) = cov(Z(x),Y(x+h))
where h is a distance separating 2 samples
 Cross-variogram of variables Y and Z:
zy(h) = 0.5*cov((Z(x)-Z(x+h)), (Z(x)-Z(x+h)))
Exemple of CO2 flux covariance
Nugget
+ sill
Features of the experimental
variogram
Features of the covariogram:
Sill: maximum semi-variance; represents variability
in the absence of spatial dependence.
Range: separation between point-pairs at which the
sill is reached; distance at which there is no
evidence of spatial dependence
Nugget: semi-variance as the separation
approaches zero; represents variability at a point
that can’t be explained by spatial structure.
Structural
analysis of the
monthly CO2
fluxes.
It appears that the
covariances of the
fluxes vary in time.
Cross-covariances
between january fluxes
and the other eleven
months.
After 4 months, the
fluxes are uncorrelated.
Covariance of the
monthly CO2
concentrations
computed using
TransCom fluxes.
After 4 months, the
spatial structure
changes dramatically!
Cross-covariance of
the January CO2
concentrations and
other months
computed using
TransCom fluxes.
After 4 months, the
concentrations are
uncorrelated.
CO2 flux anisotropy…
CO2 concentration anisotropy…
What about real data?
Cross-covariances
between june 199
fluxes and the other
eleven months.
After 2 months, the
fluxes are uncorrelated.
CO2 concentration anisotropy for
real data
Conclusions
• After 4 months synthetic CO2 fluxes and
concentrations are uncorrelated
• Synthetic fluxes show an spatial anisotropy
• These results can be used to performed timedependent sliding windows stochastic
inversion and/or cosimulation.