Uncertainty in
eddy covariance flux measurements
Andrew Richardson, University of New Hampshire
David Hollinger, USDA Forest Service
NSF Workshop on Data-Model Assimilation in Ecology:
Techniques and Applications
Norman, Oklahoma, October 22-24, 2007
Research at the Howland AmeriFlux site is supported
by the Department of Energy’s TCP, CS and NICCR programs.
"To put the point provocatively,
providing data and allowing
another researcher to provide the
uncertainty is indistinguishable
from allowing the second
researcher to make up the data
in the first place."
Raupach et al. (2005). Model data synthesis in
terrestrial carbon observation: methods, data
requirements and data uncertainty specifications. Global
Change Biology 11:378-97.
Measurement uncertainty
and parameter uncertainty
• The observed data are just
one realization, drawn from
a “statistical universe of data
sets” (Press et al. 1992)
• Different realizations of the
random draw lead to
different estimates of “true”
model parameters
• Parameters estimates are
therefore themselves
uncertain (but we want to
describe their distributions!)
A statistical universe
What uncertainty?
• A measurement is never perfect - data are not “truth”
(corrupted truth?)
• “Uncertainty” describes the inevitable error of a
measurement:
x´ = x +  + 
• x´ is what is actually measured;
it includes both systematic ()
and random () components
• typically,  is assumed Gaussian,
and is characterized by its
standard deviation, ()
Systematic errors
•
examples:
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Random errors
•
nocturnal biases
imperfect spectral response
advection
energy balance closure
operate at varying time scales: fully
systematic vs. selectively systematic
variety of influences: fixed offset vs.
relative offset
cannot be identified through
statistical analyses
can correct for systematic errors
(but corrections themselves are
uncertain)
uncorrected systematic errors will
bias DMF analyses
examples:
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surface heterogeneity and time
varying footprint
turbulence sampling errors
measurement equipment (IRGA and
sonic anemometer)
random errors are stochastic;
characteristics of pdf can be
estimated via statistical analyses
(but may be time-varying)
affect all measurements
cannot correct for random errors
random errors limit agreement
between measurements and models,
but should not bias results
Uncertainty at various time scales
• Systematic errors accumulate
linearly over time (constant
relative error)
• Random errors accumulate in
quadrature (so relative uncertainty
decreases as flux measurements
are aggregated over longer time
periods)
• Monte Carlo simulations suggest that uncertainty in annual NEE
integrals uncertainty is ± 30 g C m-2 y-1 at 95% confidence
(combination of random measurement error and associated
uncertainty in gap filling)
Random flux uncertainties are
non-Gaussian with non-constant variance
1.
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3.
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Simultaneous measurements at 2 towers (Howland)
Hollinger & Richardson (2005) Tree Physiology.
Single tower, next day comparisons (Howland,
Harvard, Duke, Lethbridge, WLEF, Nebraska)
Richardson et al. (2006) Agric. & For. Meteorol.
Data-model residuals
Hagen et al. (2006) JGR
Richardson & Hollinger (2005) Agric. & For. Meteorol.
Richardson et al. (2007) Agric. & For. Meteorol.
These characteristics violate two key
assumptions of least-squares fitting!
A method to estimate the distribution of 
Repeated measurements:
Use simultaneous, but
independent, measurements
from two towers, x´1 and x´2
Howland: Main and West towers
– same environmental conditions
– similar patches of forest
– non-overlapping footprints
(independent turbulence)
Then:
If x´ = x +  and cov(x´1, x´2)=0
(x´1 – x´2)
= x´1) + x´2) – 2cov(x´1,
x´2)
() = (x´1 – x´2)/2
x´1 ~800m x´2
A double exponential (Laplace) pdf
better characterizes the uncertainty
Strong central peak &
heavy tails (leptokurtic)
 non-Gaussian pdf
Better: double-exponential pdf,
f(x) = exp(|x/b|)/2b
The double-exponential is characterized by the parameter b
 (x)  2b where b 
1
xi – x

N
The standard deviation of the uncertainty
scales with the magnitude of the flux
• Larger fluxes are more uncertain than
small fluxes
• Relative error decreases with flux
magnitude
• Large errors are not uncommon
• 95% CI = ± 60%
• 75% CI = ± 30%
To obtain maximum likelihood parameter estimates, cannot use
OLS: must account for the fact that the flux measurement errors
are non-Gaussian and have non-constant variance.
Generality of results
• Scaling of uncertainty with flux magnitude has been validated
using data from a range of forested CarboEurope sites: y-axis
intercept (base uncertainty) varies among sites (factors: tower
height, canopy roughness, average wind speed), but slope
constant across sites (Richardson et al., 2007)
• Similar results (non-Gaussian, heteroscedastic) have been
demonstrated for measurements of water and energy fluxes (H
and LE) (Richardson et al., 2006)
• Results are in agreement with predictions of Mann and
Lenschow (1994) error model based on turbulence statistics
(Hollinger & Richardson, 2005; Richardson et al., 2006)
Generality of results
Maximum likelihood paradigm:
“what model parameters values are most likely to have generated the observed
data, given the model and what is known about measurement errors?”
Assumptions about errors affect specification of
the ML cost function:
N
cost  
i
N
cost  
i
(yi  y pred )
2
 2 (i )
yi  y pred
 (i )
For Gaussian data
(“weighted least squares”)
For double exponential data
(“weighted absolute deviations”)
Other cost functions are possible!
Specifying a different cost function affects
optimal parameter estimates
• Lloyd & Taylor (1994) respiration model: Reco  AeE
• model parameters differ depending
on how the uncertainty is treated
(explanation: nocturnal errors have
slightly skewed distribution)
• Why? error assumptions influence
form of likelihood function
LS
AD
A
24.9
43.9
T0
263.9
259.5
E0
33.6
58.5
0
TT0 
Reco –respiration
T –soil temperature
A, E0, T0 – parameters

Influence of cost function specification
on model predictions
• Half-hourly model predictions depend on parameter-ization;
integrated annual sum decreases by ~10% decrease (≈40% of
NEE) when absolute deviations is used
• Influences NEE partitioning, annual sum of GPP
• Trivial model but relevant example
Another example: Q10 model
Summary
• Knowledge of measurement uncertainty is
critical for inverse modeling and advanced
analysis techniques
• Random flux measurement error is
reasonably well described by a double
exponential pdf (it is not Gaussian!)
• Flux measurement error is heteroscedastic –
uncertainty scales with magnitude of flux (it
is not constant!)
Conclusion
• “Data uncertainties are as important as the
data themselves” (Raupach et al. 2005)
because specification of uncertainties affects
the uncertainty of the model as well as the
model predictions themselves.
• Should use weighted absolute deviations
rather than ordinary least squares when
specifying the cost function for most DMF
exercises involving eddy flux data (aside:
implications of CLT for aggregated data).