Uncertainty in eddy covariance flux measurements

Andrew Richardson, University of New HampshireDavid 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

• A measurement is never perfect - data are not “truth” (corrupted truth?)

• “Uncertainty” describes the inevitable error of a measurement:x´ = x + +

What uncertainty?

• 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:– 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:– 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

Random errors

Uncertainty at various time scales

• 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)

• 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)

These characteristics violate two keyassumptions of least-squares fitting!

Random flux uncertainties are non-Gaussian with non-constant

variance 1. Simultaneous measurements at 2 towers

(Howland) – Hollinger & Richardson (2005) Tree Physiology.

2. Single tower, next day comparisons (Howland, Harvard, Duke, Lethbridge, WLEF, Nebraska)

– Richardson et al. (2006) Agric. & For. Meteorol.3. Data-model residuals

– Hagen et al. (2006) JGR– Richardson & Hollinger (2005) Agric. & For. Meteorol.– Richardson et al. (2007) Agric. & For. Meteorol.

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

The double-exponential is characterized by the parameter

A double exponential (Laplace) pdf better characterizes the uncertainty

(x) 2 where 1N

xi – x

Strong central peak &heavy tails (leptokurtic)

non-Gaussian pdfBetter: double-exponential

pdf, f(x) = exp(|x/|)/2

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.

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%

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

Assumptions about errors affect specification of the ML cost function:

Other cost functions are possible!

cost (yi ypred )2

2(i )i

N

cost yi ypred

(i )i

N

For Gaussian data(“weighted least squares”)

For double exponential data(“weighted absolute deviations”)

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?”

• Lloyd & Taylor (1994) respiration model:Reco –respirationT –soil temperatureA, E0, T0 – parameters

LS AD

A 24.9 43.9

T0 263.9 259.5

E0 33.6 58.5

Specifying a different cost function affects optimal parameter estimates

Reco Ae E0 T T0

• 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

• 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

Influence of cost function specification

on model predictions

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).