Uncertainty in eddy covariance data and its relevance to gap filling

David Hollinger Andrew Richardson

USDA Forest Service University of New Hampshire

Durham, NH 03824 USA

The Howland flux research was supported by the USDA Forest Service Northern Global Change Program, the Office of Science (BER), U.S. Department of Energy, through the Northeast Regional Center of the National Institute for Global Environmental Change under Cooperative Agreement No. DE-FC03-90ER61010, and by the Office of Science (BER), U.S. Department of Energy, Interagency Agreement No. DE-AI02-00ER63028

• All measurements are corrupted by random error:

We want the flux, F, but we measure F + +

where is the random error and is a bias

• The random error is characterized by its PDF; the first moment () referred

to as

“uncertainty”

What is uncertainty and why do we care?

• Uncertainty information is required to properly fit models to data; e.g. for estimating the parameters of models used to fill gaps. Maximum Likelihood Data assimilation

(Kalman filter) Bayesian methods

• Uncertainty useful for assessing model fits, annual uncertainties, risk analysis, etc. -10 -5 0 5 10

"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.

• Compare independent measurements of the same thing (x and y) - The surface must be homogenous

• If x and y are independent and have the same random measurement uncertainty:

Determining measurement uncertainty:Two-tower method:

2

then assume y

22

qx

x

yxq

yxq

Hollinger & Richardson (2005), Tree Physiology 25:873-85.

• A “one-tower” method trades time for space (Compare with next day values if environmental conditions similar) Howland

=28.7

=31.6

=2.7

n

yy i

ˆ where,2

• a double exponential PDF better represents the random error distribution of eddy fluxes

Flux Uncertainties are non-Gaussian with non-constant variance

• Simultaneous measurements at 2 towers (Howland)1

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

• Data-model residuals3

1. Hollinger & Richardson (2005) Tree Physiology 25: 873-885. 2. Richardson et al. (2006) Ag. Forest Met. 136: 1-18.3. Hagen et al. (2006), Journal of Geophysical Research 111,

D08S03.

Richardson et al. (2006) Ag. Forest Met. 136: 1-18.

Flux uncertainty increases linearly with flux magnitude

~proportional to flux(for least squares uncertainty is constant)

Forests

FC > 0

= 0.62 + 0.63*FC

FC < 0

= 1.42 - 0.19*FC

(reduce one-tower estimates by ~25%)

Maximum likelihood – “given the data, what are the most likely model

coefficients?”

• Determined by minimizing the difference between data and model:

N xyy

i2

2ii2

i

))((

N xyy

i i

ii2

2

)(

For Gaussian data

For double exponential data

Relevance of uncertainty to gap filling:

1. Information needed to correctly determine model parameters (likelihood function).

2. The data we are trying to “fill” are contaminated by random noise so there is a minimal MAE or MSE error we can achieve, even with a perfect model!

Summary

• Uncertainty is a characterization of measurement error (and PDF)

• Flux measurement error is not well described by a Gaussian model (least squares inappropriate)– distribution is peaked with long tails– heteroscedastic (uncertainty ↑ with magnitude of

flux)• Double exponential model better (median)• Uncertainty can be estimated by the difference

between measurements made at 1 tower under similar conditions (reduce by ~20-25%)