63 statistical properties of random co2 ﬂux

8/13/2019 63 Statistical properties of random CO2 flux

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Statistical properties of random CO2flux measurement

uncertainty inferred from model residuals

Andrew D. Richardson a,*, Miguel D. Mahecha b, Eva Falge c, Jens Kattge b,Antje M. Moffatb, Dario Papale d, Markus Reichstein b, Vanessa J. Stauch e,Bobby H. Braswell a, Galina Churkina b, Bart Kruijtf, David Y. Hollingerg

a University of New Hampshire, Complex Systems Research Center, Morse Hall, 39 College Road, Durham, NH 03824, USAb Max Planck Institute for Biogeochemistry, Hans-Knoll-Strasse 10, 07745 Jena, Germany

c Max Planck Institute for Chemistry, Biogeochemistry Department, J.J.v. Becherweg 27, 55128 Mainz, Germanyd DISAFRI, University of Tuscia, via C. de Lellis, 01100 Viterbo, Italye Computational Landscape Ecology, Centre for Environmental Research (UFZ), Leipzig, GermanyfAlterra, Wageningen University and Research Centre, 6700 AA Wageningen, The NetherlandsgUSDA Forest Service, Northern Research Station, Durham, NH 03824, USA

a g r i c u l t u r a l a n d f o r e s t m e t e o r o l o g y 1 4 8 ( 2 0 0 8 ) 3 8 5 0

a r t i c l e i n f o

Article history:

Received 26 March 2007

Received in revised form

25 August 2007

Accepted 3 September 2007

Keywords:

Data-model fusion

Eddy covariance

FLUXNET

Measurement error

Pink noise

Spectral analysis

Uncertainty

a b s t r a c t

Information about the uncertainties associated with eddy covariance measurements of sur-

faceatmosphere CO2 exchange is needed for data assimilation and inverse analyses to

estimate model parameters, validation of ecosystem models against flux data, as well as

multi-site synthesis activities (e.g., regional to continental integration) and policy decision-

making. While model residuals (mismatch between fitted model predictions and measuredfluxes) can potentially be analyzed to infer data uncertainties, the resulting uncertainty

estimates may be sensitive to the particular model chosen. Here we use 10 site-years of data

fromthe CarboEurope program, and compare the statistical propertiesof the inferred random

flux measurement error calculated first using residuals from five different models, and

secondly using paired observations made under similar environmental conditions. Spectral

analysis of the model predictions indicated greater persistence (i.e., autocorrelation or

memory) compared to the measured values. Model residuals exhibited weaker temporal

correlation, but were not uncorrelated white noise. Random flux measurement uncertainty,

expressed as a standard deviation, was found to vary predictably in relation to the expected

magnitude of the flux, in a manner that was nearly identical (for negative, but not positive,

fluxes) to that reported previously for forested sites. Uncertainty estimates were generally

comparable whether the uncertainty was inferred from model residuals or paired observa-

tions, although the latter approach resulted in somewhat smaller estimates. Higher ordermoments (e.g., skewness and kurtosis) suggested that for fluxes close to zero, the measure-

ment error is commonly skewed and leptokurtic. Skewness could not be evaluated using the

paired observation approach, because differencing of paired measurements resulted in a

symmetricdistributionof the inferred error. Patterns were robust and not especially sensitive

to the model used, although more flexible models, which did not impose a particular func-

tional form on relationships between environmental drivers and modeled fluxes, appeared to

give the best results. We conclude that evaluation of flux measurement errors from model

residuals is a viable alternative to the standard paired observation approach.

# 2007 Elsevier B.V. All rights reserved.

* Corresponding author. Tel.: +1 603 868 7654; fax: +1 603 868 7604.E-mail address:[email protected](A.D. Richardson).

a v a i l a b l e a t w w w . s c i e n c e d i r e c t . c o m

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a g r f o r m e t

0168-1923/$ see front matter # 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.agrformet.2007.09.001
mailto:[email protected]://dx.doi.org/10.1016/j.agrformet.2007.09.001http://dx.doi.org/10.1016/j.agrformet.2007.09.001mailto:[email protected]


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1. Introduction

The eddy covariance method is used at research sites around

the world to measure surfaceatmosphere exchanges of

carbon, water and energy (Baldocchi et al., 2001b). Increas-

ingly, these flux data are being used to constrain large-scale

carbon cycle models (e.g.,Papale and Valentini, 2003; Friend

et al., 2006). In the context of this so-called model-datasynthesis,Raupach et al. (2005)have argued the importance

of accurately characterizing data uncertainties, suggesting

that uncertainties are as important as the data values

themselves. This is due to the fundamental role of uncertain-

ties in specifying an objective criterion (commonly referred to

as the cost function; see Raupach et al., 2005), quantifying

the mismatch or distance between model and data, used as

the basis for optimization or synthesis (Trudinger et al., 2007).

If the true flux is F0i, and the measurement error is a random

variable di (the measured flux is thus Fi F0i di), then a

convenient way to characterize the uncertainty is s(di), i.e., as

thestandard deviationof the randomerror.Accordingly, 1/s(di)

is a quantitative measure of our confidence in the measureddata (see Raupach et al., 2005). In conducting model-data

syntheses, data values are typically weighted in the cost

function by the reciprocal of their uncertainties, i.e., as 1/s(di)

(possiblyraisedto a higher power). An important consequence

of this is that estimated model parameters or states depend on

both measurements and uncertainties.

There is a range of other applications in which knowledge

of flux data uncertainties is also critical. These include, for

example, ecosystem model validation against flux data (e.g.,

Thornton et al., 2002; Churkina et al., 2003), because

uncertainty estimates are required to make statistically

rigorous comparisons between measurements and model

predictions. Additionally, flux data uncertainties are neededfor multi-site syntheses (e.g.,Law et al., 2002; Churkina et al.,

2005) and regional-to-contintental integration efforts, espe-

cially with regard to carbon accounting and policy decision-

making (Wofsy and Harriss, 2002). In both of these latter

examples, sites with smaller data uncertainties can be

accorded more influence than sites with larger uncertainties.

Flux measurement errors can be broadly classified into

random and systematic sources of uncertainty; whereas

statistical analyses can be used to estimate the size of random

errors, systematic errors can be difficult to detect or quantify

in data-based analyses (Moncrieff et al., 1996; but see Lee,

1998). Random flux measurement uncertainties (e.g.,Hollinger

and Richardson, 2005), which are stochastic, represent theaggregate uncertainty due to surface heterogeneity and a

time-varying flux footprint, turbulence sampling errors

(especially problematic over tall vegetation), and errors

associated with the measurement equipment (e.g., gas

analyzer and sonic anemometer). Systematic errors, on the

other hand, operate at varying time scales (e.g., fully

systematic vs. selectively systematic, see Moncrieff et al.,

1996), and can influence measurements in a variety of ways

(e.g., fixed offsets vs. relative offsets and positive vs. negative

biases). Systematic errors due to the imperfect spectral

response of the measurement system, nocturnal biases

resulting from inadequate mixing, issues related to advection

and non-flat terrain, and problems of energy balance closure,

are discussed elsewhere (Goulden et al., 1996; Moncrieff et al.,

1996; Lee, 1998; Massman and Lee, 2002; Wilson et al., 2002;

Aubinet et al., 2005; Loescher et al., 2006).

While any attempt to quantifying the total flux measure-

ment uncertainty must account for both random and

systematic errors, our goal here is to build on previous

statistically-based analytical efforts to quantify the random

component. For example, Hollinger et al. (2004) differencedpaired measurements from two flux towers in an effort to

estimate the random flux measurement uncertainty, which

was then characterized, as s(d), by the standard deviation of

the differenced observations (divided by the square root of

two). The two towers were located in similar old-growth Picea

stands and instrumented identically, but were separated by

775 m and had non-overlapping footprints and largely

independent turbulence (cf. the more recent study byRannik

et al., 2006, where the30 m separation of two towers resulted

in paired measurements that were not fully independent).

Results of the above studies were in close agreement with (but

somewhat higher than) predictions of the Lenschow et al.

(1994)and Mann and Lenschow (1994)error model based onturbulence statistics.

In a subsequent paper, Hollinger and Richardson (2005)

proposed that paired measurements at one tower, separated

by 24 h and made under similar environmental conditions

could also be used to estimate the random flux measurement

uncertainty; this method was applied by Richardson et al.

(2006a) to data from seven sites (one grassland, an agricultural

site, and five forested sites) in the first multi-site synthesis of

random measurement uncertainty. The one-tower approach

may result in lower uncertainty estimates compared to the

two-tower approach because spatial heterogeneity across an

ecosystem is likely to be larger than that across the time-

varying footprint of a single tower (seeKatul et al., 1999; Orenet al., 2006). Conversely, however, imperfect matching

(between days) in environmental and turbulent conditions

may result in slight over-estimation of uncertainty using the

one-tower approach.

We consider here a third method to quantify eddy flux

random measurement uncertainty. In the hypothetical

instance of a perfect model and no systematic measurement

errors, then the concordance between model predictions and

measured data is ultimately limited by the random flux

measurement uncertainty. Thus an alternative means by

which the random flux measurement uncertainty might be

evaluated is through posterior analysis of residuals from a

fitted model (Stauch et al., in press). However, models are animperfect representation of reality, andthus a disadvantage of

this approach is that the resulting uncertainty estimates will

be a combination of both measurement error and model error.

With a particularly poor model (inappropriate functional form,

mis-representation of processes and driving variables, or

suboptimal parameter values), model error could potentially

overwhelm the random measurement error. While estimates

of measurement error inferred from model residuals will

depend somewhat on the model itself, there is some evidence

that this may be a reasonable approach to quantifying

uncertainties. For example, Richardson and Hollinger (2005)

reported that the standard deviationof model residuals forthe

Lloyd and Taylor (1994) respiration model, fit to nocturnal

a g r i c u l t u r a l a n d f o r e s t m e t e o r o l o g y 1 4 8 ( 2 0 0 8 ) 3 8 5 0 39


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data, was only 14% larger than the random flux measurement

uncertainty estimated using paired measurements, suggest-

ing that model error is potentially relatively small. In that

study, the statistical properties of the residuals were other-

wise similar to the uncertainty estimated using paired

measurements, i.e., non-Gaussian and leptokurtic (see also

Hagen et al., 2006), with comparable patterns in relation to

wind speed and flux magnitude.A number of other studies have also conducted analyses of

model residuals for the purpose of evaluating flux measure-

ment uncertainty. Using data from a range of CO2 flux

measurement sites,Chevallier et al. (2006) noted that at the

daily time step, ORCHIDEE model residuals were non-

Gaussian. At least two related studies have purposely avoided

making assumptions about the specific probability distribu-

tion function for the random flux measurement uncertainty.

For example,Hagen et al. (2006)used bootstrap resampling of

model residuals to estimate uncertainties in gross productiv-

ity at different time scales (from 30 min. to annual), whereas

Stauch et al. (in press) used a stochastic model to partition

signal and noise in measured net fluxes, and character-ized the noise (i.e., model residuals) using a non-parametric

statistical model based on time-varying cubic splines (As

noted by both of these authors, annual flux sums are

approximately Gaussian, despite the non-Gaussian uncer-

tainties in the 30 min. data; this result follows directly from

the Central Limit Theorem).

In the present study, we use a wide array of data and

models to investigate whether an approach based on model

residuals results in uncertaintyestimates that are comparable

to those derived from paired observations. We optimize model

parameters (on a site-by-site basis) to maximize the agree-

ment between model and measurements, and thus obtain

model predictions that are informed by the data, rather thanindependent of the data. In this manner the models can be

used as tools to predict the true flux, F 0i, conditional on the

measured fluxes, environmental drivers, and implicit (in the

model) relationships between these two.

To evaluate the degree to which model-based estimates of

uncertainty are influenced by the particular model chosen

(i.e., the degree to which model error confounds random

measurement error), we compare results from five different

models, ranging from simple nonlinear regressions to more

highly parameterized artificialneuralnetworks and a process-

based ecosystem model. Some of the models (e.g., neural

network and look-up table) contain few, if any, assumptions

about the underlying functional form of relationships between

CO2 flux and environmental drivers. Parameterization of some

of the models (look-up table and nonlinear regression) was

time-varying to account for seasonal changes in the flux-

driver relationships.

We begin by evaluating, first in the temporal domain and

then in the frequency domain, model predictions and modelresiduals in relation to the measured CO2flux time series (i.e.,

the net ecosystem exchange, NEE). By using new tools

(improvements over traditional Fourier power analysis) to

probe the spectral properties of both the measured fluxes and

the inferred random error, we are able to investigate and

characterize the correlation structure inherent in each of

these time series. This analysis complements the more

traditional analysis which follows, where we focus on the

moments (standard deviation, skewness and kurtosis) of the

probability distribution of the inferred random error. We

compare inferred uncertainties calculated from paired obser-

vations and model residuals, and look for similarities across

sites. Finally, relationships between the standard deviation ofthe random flux measurement uncertainty and flux magni-

tude developed here for six forested European sites are

compared to those presented previously byRichardson et al.

(2006a)for four forested AmeriFlux sites.

2. Data and method

The present analysis is based on 10 site-years of data from the

CarboEurope database. The sites (seeTable 1) span a range of

forest types, including deciduous (FR1, DE3), coniferous (FI1),

mixed deciduous/coniferous (BE1), and Mediterranean (FR4,

IT3). The data were assembled as part of a data set used forevaluation of a standardized data processing algorithm for

half-hourly data (Papale et al., 2006), a comprehensive gap

filling comparison (Moffat et al., 2007), and an evaluation of

different algorithms for partitioning NEE to its component

fluxes, gross productivity and ecosystem respiration (Desai

et al., unpublished). Data were corrected for canopy storage

but not energy balance closure; quality control procedures, u*

filtering, and gap filling of meteorological data are discussed in

the companion paper byMoffat et al. (2007). Measured CO2fluxes are here denoted Fc, with units of mmol m

2 s1. The

sign convention is that a negative flux indicates carbon uptake

Table 1 Sites and years of data used for the uncertainty analysis. MAT = mean annual temperature

Site Species Climate (MAT) Years Latitude Longitude Reference

BE1 (Vielsalm, Belgium) Fagus sylvatica,

Pseudotsuga menziesii

Temperate-continental

(7.5 8C)

2000, 2001 50.308N 5.988E Aubinet et al. (2001)

DE3 (Hainich, Germany) Fagus sylvatica Temperate-continental

(6.8 8C)

2000, 2001 51.078N 10.458E Knohl et al. (2003)

FI1 (Hyytiala, Finland) Pinus sylvestris Boreal (3.8 8C) 2001, 2002 61.838N 24.288E Suni et al. (2003)

FR1 (Hesse, France) Fagus sylvatica Temperate-suboceanic

(9.9 8C)

2001, 2002 48.678N 7.058E Granier et al. (2000)

FR4 (Puechabon, France) Quercus ilex Mediterranean (13.5 8C) 2002 43.738N 3.588E Rambal et al. (2004)

IT3 (Roccarespampani, Italy) Quercus cerris Mediterranean (15.28

C) 2002 42.408

N 11.928

E Tedeschi et al. (2006)



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by the ecosystem, whereas a positive flux indicates carbon

release.

For each site-year, fitted model predictions from five

different models selected from among those submitted to

the gap filling comparison were used. The models (described

fully byMoffat et al., 2007) were as follows: an artificial neural

network model (ANN), a non-linear regression model (NLR), a

process-based ecosystem model (BETHY), a semi-parametricmodel (SPM), and a marginal distribution sampling method

(MDS). The ANN (Papale and Valentini, 2003) is essentially a

multi-stage, empirical, non-linear regression model. An

advantage of neural networks is that no a priori decisions

about the functional form of relationships between data

inputs (e.g., meteorological drivers and time of day/year as a

fuzzy variable) and outputs (measured fluxes) are required.

The particularNLR used here (Falge et al., 2001)wasbasedona

MichaelisMenten light response function for gross photo-

synthesis and aLloyd and Taylor (1994)temperature function

for ecosystem respiration; model parameters were fit twice-

monthly. BETHY (Knorr and Kattge, 2005) is a terrestrial

biosphere model that predicts CO2, waterand energy fluxes. Inaddition to meteorological drivers, required input data include

site-specific information about soil and canopy properties

(e.g., leaf area index, estimated from remote sensing pro-

ducts). SPM (Stauch and Jarvis, 2006) is analogous to a multi-

dimensional look-up table, with an underlying semi-para-

metric interpolation defined by three-dimensional cubic

splines (time-varying functions of PPFD and temperature).

For MDS (Reichstein et al., 2005), which is essentially a moving

look-up table, fluxes are averaged from similar meteorological

conditions (global radiation, air temperature and vapor

pressure deficit within pre-specified criteria) across time-

windows as small as possible. Thus the models ranged from

data based (e.g., ANN) to process based (e.g., BETHY), and theadaptation to measurements ranged from time constant (e.g.,

BETHY) to time varying parameterization (e.g., MDS). While

our goal here is not to identify any one of these as the best

model, the application of objective model selection criteria

(e.g., Akaikes Information Criterion) to similar models of CO2exchange data has been described previously (Richardson

et al., 2006b).

2.1. Evaluation of spectral properties

The spectral properties (i.e., correlation structure) of a time

series can be summarizedby calculating the scaling exponent,

a, of the power spectrum P(f) =fa (Mandelbrot, 1999). Here frefersto the frequencies contained in a given timeseries, anda

is conventionally estimated as the slope of the log-log

transformed relationship between frequency and power

(i.e., the Fourier power spectrum). Uncorrelated white noise

has a= 0 (equal power across all frequencies), whereas a

strong autocorrelation structure, such as characterizes Brow-

nian motion, has a 2. The power spectrum of this red

noise is dominated by the lower frequencies, which imparts

the signal with substantial memory. Intermediate between

white and red noise is pink noise, for which the correlation

structure is moderate and a 1.

With the goal of evaluating whether the different models

were able to reproduce the temporal correlation structure of

the measured NEE time series, and to characterize the spectral

properties of the model residuals, we used the Lomb-Scargle

periodogram (L-S, Lomb, 1976; Scargle, 1982) and the

multiple segmentation method (MSM, Miramontes and

Rohani, 2002; Rohani et al., 2004) to estimate a for the

measured fluxes, model predictions, and model residuals. The

L-S method (following Press et al., 2002) estimates the

equivalent of a Fourier power spectrum for unevenly-spacedtime series data, and is thus suited to analysis of fragmented

time series with up to 20% missing observations; a is

calculated from the resulting L-S power spectrum using the

conventional method.

Since a reasonable estimate of the scaling exponent a is

only possible forvery long time series when standard methods

are applied, new methods (e.g., MSM) have been developed to

yield statistically robust estimates ofa for short time series.

The MSMmethod subdivides a timeseries into segments, with

segment lengths defined by powers of two. The scaling

exponent for each segment is calculated as the slope of the

power spectrum of the sub-series. An overall estimate ofa is

obtained by fitting a linear model to the relationship betweensegment length and the mean of the scaling exponents found

at each segment size. In the presentstudythe powerspectra of

the sub-series in MSM were determined using theL-S method.

With this hybrid approach that blends MSM and L-S, we could

estimate a notonly forshort,but also forshort andfragmented

time series. Thus we were able to estimate the scaling

exponent for measured fluxes, without having to resort to

gap-filled time series.

2.2. Inferred random flux measurement error

For each site-year of data, the paired observation approach to

estimating uncertainty was implemented as described inRichardson et al. (2006a). Briefly, this involved considering as a

pair anytwo fluxmeasurements made exactly 24 h apart, with

mean half-hourly photosynthetically active photon flux

densities within 75 mmol m2 s1, air temperatures within

3 8C, and vapor pressure deficits within 0.2 kPa. Across sites,

between 19% and 27% (overall mean, 23%) of half-hourly

measurements met these criteria. For this method, the

inferred error, di, equals the difference between the two

measurements divided by the square root of two (see also

Hollinger and Richardson, 2005).

The model-based approach to estimating uncertainty was

implemented using the five models named above. The

inferred random error was taken to be the residual, ei,between the predicted value of the fitted model and the

measured flux. With data coverage averaging 69% across all

sites, this method provided roughly three-fold as many

estimates of the inferred random error, compared to the

paired observation approach.

The statistical properties (i.e., moments of the distribution)

of the inferred random error were analyzed with respect to

flux magnitude. Based on predictions of the ANN model (the

standard deviation of the residuals was smallest for this

model), observations were grouped into bins of comparable

flux magnitude (bin width = 5 mmol m2 s1) and the mean,

standard deviation, skewness, and kurtosis of the inferred

random error calculated for each bin; as in the past, we



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characterize therandom fluxmeasurement uncertainty by the

second moment of the inferred random error, i.e., as the

standard deviation, s(d). Moments were only calculated for

bins containing at least 10 observations.

3. Results

3.1. Time series of model residuals

Model residuals were well-correlated across models, with

pairwise linear correlations generally 0.80, suggesting that

the dominant source of variability in residualsis relatedto flux

measurement errors (which would affect residual time series

from all models similarly), rather than model error. To use the

FR1 2001 data as an example, MDS and BETHY residuals were

more weakly correlated (r= 0.80) than, for example, ANN and

SPM residuals (r= 0.94). Visual inspection of the raw residual

data (Fig. 1) suggested only modest differences among models,

but when the mean bias was calculated over weekly periods, it

became apparent that predictions of some models (e.g.,BETHY) were consistently biased at certain times of the year,

whereas other models (e.g., MDS) exhibited little or no

persistent bias.

3.2. Spectral properties of measured fluxes, model

predictions, and model residuals

The Lomb-Scargle periodogram suggested that all models

did a reasonable job at describing the higher-frequency

components of the data, but due to time constant para-

meterization, some models (e.g. BETHY) were less successful

at describing the lower-frequency components than other

models (e.g., MDS) that used time-varying parameterization.

All residual time series exhibited a spectral peak at

frequency of 1/day. This does not necessarily mean that

the models fail to adequately capture the diurnal cycle;

rather, it just means that there is an underlying structure to

the residuals, which could arise from diurnal flux patterns

(e.g., day vs. night) and corresponding patterns to the

uncertainty.

Similar estimates of the scaling exponent, a, wereobtained with both the L-S and hybrid MSM/L-S methods,

except that the latter methodtendedto result in more clearly

defined differences ina amongthedifferentmodels.Wefocus

here on the hybrid MSM/L-S results. For the measured NEE

fluxes, a varied among sites between 0.6 and 1.1 (Fig. 2),

indicating pink noise (an explanation for this variation

across sites is given below). On the other hand, the scaling

exponent for each model was more typically 1.25,

indicatingthatthemodeledtimeserieshavealongermemory

effect (i.e., tending towards red noise) than the measured

NEEtime series. The site-to-site variationin aofthemeasured

fluxes was not captured by any model. The MDS model

consistently came closest to reproducing the spectral proper-ties of the measured fluxes, probably because the narrow

sliding window at which the algorithm operates minimizes

persistent biases that could result from more slowly varying

parameters.

The a of model residuals was generally between 0.2 and

0.8 (although for one site, BE1 2000, the values ranged from

0.1 to 0.2), rather than zero as would be expected if

residuals were uncorrelated white noise (Fig. 2). The fact that

the a of theresiduals varied among sites, and in a manner that

was similar to that for the NEE series (but not captured by any

model), suggests that the residuals maintain a certain

temporal correlation structure that is characteristic to each

site.

Fig. 1 Comparison of model residuals for MDS and BETHY

models. (A) Half hourly residuals; (B) weekly mean of

model residuals. From (B) it is clear that BETHY results in

stronger and more persistent biases (e.g., negative bias at

days 120 and 240, positive biases at days 150, 180, and

210) than MDS.

Fig. 2 fa scaling exponents estimated from time series of

measured (+) and modeled (filled circles) net ecosystem

exchange of CO2 (NEE), and the residual (open circles)

between these two. Models are as follows: artificial neural

network model (ANN), non-linear regression model (NLR),

process-based ecosystem model (BETHY), semi-

parametric model (SPM), and a look-up table based on

marginal distribution sampling (MDS); sites (y-axis) are as

identified inTable 1.



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3.3. General properties of the uncertainty

The statistical properties of the aggregated (i.e., calculated

across all site-years) data set of inferred random errors for

the different methods are shown in Table 2A. In all cases,both the mean model residual, and the mean difference

between paired observations, was relatively close to zero.

The standard deviation of the inferred random error, s(d),

was 20% smaller for the paired observation approach

(2.19 mmol m2 s1) than for the best of the model-based

approaches (2.733.10 mmol m2 s1), presumably because

uncertainties estimated via the paired observation approach

do not incorporate additional model error (conversely, these

results also suggest that ANN and MDS have the lowest

model error, while BETHY and NLR have the highest model

error). The skewness was consistently close to zero,

suggesting a relatively symmetric error distribution. How-

ever, the strong positive kurtosis indicated by all methods,approximately10, is characteristic of a leptokurtic error

distribution, i.e., one characterized by a more strongly

defined central peak and heavier tails than a normal

distribution.

When a single model was used, and moments compared

across sites (results shown for ANN model inTable 2B), the

most obvious differences tended to be in terms of the standard

deviation of the inferred random error, s(d), which was

smallest for site-years FI1 2001 and 2002, and largest for

site-years BE1 2001and 2000, and FR1 2001and 2002. Skewness

was not especially prominent at any particular site. All sites

had kurtosis of3 or more. For individual sites, skewness and

kurtosis estimates were similar for all models.

3.4. Moments in relation to flux magnitude

Results were somewhat different when the data were broken

down into bins of different flux magnitude (Fig. 3). Within a

site, patterns were quite similar regardless ofthe model used,and the model-based approach was largely consistent with

the paired observation approach, as illustrated by results

from FR1 2001 (seeFig. 3AC; this site-year was chosen for

illustration because it had the fewest missing observations,

21%). Thus, the standard deviation of the inferred random

error consistently increased with the absolute magnitude of

theflux,and there were only small differences among models

in the nature of this relationship (Fig. 3A). For this particular

site-year, the skewness was generally negligible, i.e., 1, and

didnot show anyclearpatternsof variation in relation to flux

magnitude(Fig.3B).Forthe0mmol m2 s1 Fc bin,skewnessof

random error inferred from NLR model residuals was much

higher (1.6) than any of the other models, which wereotherwise in the range 0.00.6 (note that while a similar

pattern was seen for the other year of data for this site, FR1

2002, the pattern was not repeated across sites). The kurtosis

was most pronounced, i.e., consistently 5, for the

0 mmol m2 s1 Fcbin, and appeared to become smaller, i.e.,

closer to zero, with increasing absolute flux magnitude

(Fig. 3C).

Across sites, patterns were consistent when a single

model was used, as illustrated for the ANN model (Fig. 3DF).

The rate at which the standard deviation of the flux

uncertainty increased with the magnitude of the flux (slope)

was quite similar across sites, but the base level of

uncertainty (y-axis intercept), i.e., at Fc= 0 mmol m2 s1,

Table 2 Statistical properties of inferred random flux measurement error (CO2fluxes measured in mmol mS2 sS1)

Method N Mean Standard deviation Skewness Kurtosis

(A) All sites, by method

ANN 121217 0.004 2.73 0.22 11.61

NLR 121217 0.001 3.10 0.05 9.22

BETHY 121217 0.048 3.10 0.16 8.37

MDS 121217 0.067 2.80 0.04 10.23

SPM 121217 0.000 2.93 0.09 9.66

Paired 40563 0.029 2.19 0.18 10.52

Site N Mean Standard deviation Skewness Kurtosis

(B) ANN method, by site-year

BE1 2000 12519 0.043 3.21 0.03 3.55

BE1 2001 12287 0.000 2.89 0.32 7.49

DE3 2000 11452 0.014 2.46 0.16 7.01

DE3 2001 11731 0.005 2.17 0.20 6.29

FI1 2001 11236 0.002 1.28 0.21 7.31

FI1 2002 11427 0.015 1.32 0.32 8.46

FR1 2001 13744 0.010 3.60 0.20 8.84

FR1 2002 13651 0.029 4.12 0.26 7.73

FR4 2002 11190 0.010 1.81 0.31 2.99

IT3 2002 11980 0.005 2.29 0.50 6.76

(A) Two different approaches were used, one based on model residuals and the other on paired observations. For the former, a total of five

different models were used (ANN, artificial neural network; NLR, nonlinear regression; BETHY, a process model; MDS, marginal distribution

sampling; SPM, semi-parametric model). For the latter, the random error was inferred from the difference between paired observations made

24 h apart under similar environmental conditions (Paired). Results based on 10 site-years of data, as described in Methods section. (B) ANN

model only, with results broken down by site-year.



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varied among sites (Fig. 3D; note the consistency with

patterns reported byRichardson et al., 2006a, as indicated

by the heavy blue shaded line; see alsoTable 3). Many siteshad skewness 0 regardless of the magnitude of the flux, but

for some site-years (in particular, FI1 2002 and IT3 2002,

although these are not specifically identified in the figure),

there was pronounced skewness (resulting from long,

asymmetric, tails of the inferred random error), especially

for the 0 mmol m2 s1 Fc bin (Fig. 3E). The kurtosis pattern

reported above for FR1 2001 appeared to be a more general

result, as distributions became more mesokurtic (i.e.,

kurtosis approached zero) as the absolute magnitude ofFcincreased, and were in fact only strongly leptokurtic for the

0 mmol m2 s1 Fc bin. The strong kurtosis (as well as the

skewness) associated with the 0 mmol m2 s1 Fcbin was due

almost entirely to the extreme tails of the inferred random

error. Trimming the top and bottom 1% of residuals resulted

in a nearly symmetric distribution with less pronounced

kurtosis.These results were generally comparable (compareFig. 3G

and D, or I and F) to those obtained using the paired

observation approach. However, although estimates of both

the standard deviation and the kurtosis of the random

measurement error (calculated for each flux magnitude bin

within each site-year) were strongly correlated for model-

based (ANN) and paired observation approaches (Fig. 4A and

C), the skewness estimates for these two approaches were not

at all correlated (Fig. 4B). This discrepancy is attributed to the

inability of the paired observation approach to characterize

odd-order moments of the distribution, because it is impos-

sible to differentiate between a positive error in one

measurement versus a negative error in the other measure-

Fig. 3 Comparison of statistical properties (standard deviation, skewness and kurtosis) of inferred flux measurement error

in relation to CO2flux magnitude. Two different methods, described more fully in the text, were used to estimate the error:

(1) residuals between measured values and fitted models used for gap-filling (Residual), and (2) differences between

paired observations made exactly 24 h apart under similar environmental conditions (Paired). Five different models were

used for (1). In the left column (AC), results are shown for all five models (thin lines), and the paired method (thick red line),

for a single site-year (FR1_2001; Hesse, France). In the middle column (DF), results are shown across all 10 site-years for a

single model (ANN, artificial neural network). In the right column (GI), results are shown across all 10 site-years for the

paired observations method. In panels (D) and (G), the thick blue line shows the uncertainty estimates reported by

Richardson et al. (2006a)for four forested AmeriFlux sites, estimated using the paired observation method. (For

interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article.)



8/13

ment. Thus, the paired observation approach should not be

used to evaluate bias or skewness.

Two strong conclusions emerge from the above results.

First, flux measurement errors become more Gaussian asfluxes increase in magnitude. For fluxes close to

0 mmol m2 s1, skewness, and especially pronounced kur-

tosis, are observed. Second, flux measurement errors are

clearly heteroscedastic, meaning that the error variance is

not constant but rather varies, particularly in relation to flux

magnitude. We calculated the relationship between flux

magnitude and flux measurement uncertainty for each site

using the model-based approach (see Table 3A for ANN

results). Across the 10 site-years, they-axis intercept of each

regression was strongly correlated (r= 0.95, P < 0.001) with

the overall standard deviation of the inferred random

measurement errors given inTable 2B, but the differences

in slope among sites was not significant (P= 0.10). When datawere pooled for all site-years, and relationships calculated

separately for model-based and paired observation

approaches (Table 3B), the regression lines could not be

statistically distinguished (P > 0.10). Furthermore, neither of

theserelationships couldbe distinguished(P > 0.10)fromthat

reported by Richardson et al. (2006a) for Fc 0. However,

whereasRichardson et al. (2006a)found that the uncertainty

increased more rapidly with flux magnitude for positive

fluxes (nocturnal release) than negative fluxes (daytime

uptake), the present results did not support this, as the

difference in slopes for Fc 0 and Fc 0 was not significant

(P > 0.30) for either model-based or paired observation

approaches (see alsoStauch et al., in press). This difference

may be a result of more stringent nocturnal rejection criteria

(u*filtering, quality flags and other data cleaning, see Papale

etal.,2006)attheEuropeansitescomparedtothesitesusedby

Richardson et al. (2006a).

Table 3 Relationship between CO2flux magnitude (Fc, inmmol mS2 sS1) and the standard deviation of the inferredrandom flux measurement error, s(d)

Method Relationship

(A) By site

BE1 2000 2.91(0.30) + 0.09(0.03) jFcj

BE1 2001 2.27(0.17) + 0.14(0.02) jFcj

DE3 2000 1.31(0.33) + 0.18(0.02) jFcjDE3 2001 1.13(0.21) + 0.16(0.01) jFcj

FI1 2001 0.87(0.24) + 0.12(0.03) jFcj

FI1 2002 0.91(0.17) + 0.12(0.02) jFcj

FR1 2001 2.62(0.34) + 0.18(0.02) jFcj

FR1 2002 3.49(0.38) + 0.13(0.02) jFcj

FR4 2002 1.28(0.41) + 0.09(0.05) jFcj

IT3 2002 1.79(0.07) + 0.11(0.01) jFcj

(B) All sites

Model residuals (ANN) 1.69(0.20) + 0.16(0.02) jFcj

Paired observations 1.47(0.22) + 0.17(0.02) jFcj

(C)Richardson et al. (2006a)

Fc 0 1.42(0.31) + 0.19(0.02) jFcj

Fc 0 0.62(0.73) + 0.63(0.09) jFcj

Results are based on 10 site-years of data from a range of forested

sites, as described in Methods. In (A), results are calculated

separately for each site-year, based on ANN model residuals. In

(B), results are calculated for residual and paired observation

approaches, using data for all sites together. In (C), previously

published results for forested sites are shown for comparison.

Standard errors on estimated coefficients are given in parentheses.

Fig. 4 Comparison of statistical properties (standard

deviation, skewness and kurtosis) of inferred flux

measurement error calculated using two different

approaches: the residual between measured values and a

fitted artificial neural network model (ANN), and the

difference between paired observations made exactly 24 h

apart under similar environmental conditions (Paired).

The 1:1 line is shown in all three panels. Reported

correlation coefficients are for all site-years (each site-year

shown by a different color) of data.



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3.5. Linking flux measurement uncertainty and spectral

properties

Additional analysis of the above results suggested some

interesting relationships between the scaling exponent a, and

the estimated random flux measurement uncertainty as

characterized by the overall standard deviation of the inferred

random measurement error, s(d). First, the site-specificvariation in a of the measured NEE time series (+ symbols

inFig. 2) was positively correlated with the standard deviation

of model residuals for each site. Results are shown inFig. 5A

for the ANN model (r= 0.87), but similar results were obtained

for all other models (rbetween 0.80 and 0.86). Thus, it should

be feasible to estimate site-specific average uncertainties

purely on the basis of the calculated scaling exponent, i.e., as:

sd 6:

590:

72 4:

750:

82 a(Coefficients were calculated using reduced major axis regres-

sion, with standard errors on coefficients given in parenth-

eses). In thesame vein, a was also strongly correlated(r= 0.84)

with the y-intercept values reported in Table 3A, and given

that results above indicated that the slope of the relationship

between flux magnitude and flux uncertainty did not vary

among sites, then this also provides a mechanism for devel-

oping site-specific estimates of thisrelationship, purely on the

basis ofa, i.e., as:

sd 5:820:79 4:610:90 a 0:17 jFcj

Second, the variation in a of model residuals (open circles

in Fig. 2) was negativelycorrelated with the standard deviationof residuals for each model. Results shown in Fig. 5B are based

on averages (across all sites) calculated for each model

(r= 0.89), but similar results were obtained for individual

sites as well (rbetween 0.72 and 0.98).

4. Discussion

4.1. General characteristics of the measurement error

Random flux measurement uncertainties are characterized by

non-stationary probability distributions, in that the statistical

properties of the error vary over time and in relation to themagnitude ofthe flux.Fromthisstudywe find thatthe manner

in which theuncertainty scales with the magnitude of theflux

(Table 3) is similar to that presented previously (e.g.,Hollinger

and Richardson, 2005; Richardson et al., 2006a; Stauch et al., in

press). This provides a solid foundation for implementing a

weighted optimization scheme in conducting model-data

syntheses, whereby observations measured with greater

confidence (lower s(di)) receive more weight during the

optimization (i.e., in the cost function) than observations

measured with less confidence (higher s(di)). Note that the

non-zero intercept in the relationship between s(di) and jFcj

means that large-magnitude fluxes have a better signal-to-

noise ratio, and will still exert a greater influence during theoptimization, than small fluxes.

The results presented in Fig. 4 indicate more complex

patterns of variation in the statistical properties of the flux

uncertainty compared to what has been previously reported

(cf. Hollinger and Richardson, 2005). In particular, it would

appear that while random measurement errors tend to be

approximately Gaussian (symmetric and mesokurtic) for

jFcj 0 mmol m2 s1, they are frequently non-Gaussian for

jFcj 0 mmol m2 s1. In this regard, it may be considerably

more challenging than has been previously acknowledged to

properly implement a maximum likelihood optimizations

scheme when inverting ecosystem models against flux data,

because the error model needs to account forthese differences

Fig. 5 Relationship between fa scaling exponents and

estimated random flux measurement uncertainty,

characterized by the standard deviation of model residuals

(models are described fully in text). In (A), the scaling

exponent of raw NEE time series is plotted against the

standard deviation of ANN model residuals, with each

data point representing one site-year of data. In (B), the

scaling exponent of model residual time series is plotted

against the standard deviation of the model residuals,

with each data point representing the average across all

site-years for an individual model. In both panels, the

fitted line is based on reduced major axis regression.



10/13

in the probability distribution function of the measurement

errors.

If the random fluxmeasurement uncertainty doesnot follow

a Gaussian distribution, then what distribution does it follow?

Hollinger and Richardson (2005) suggested that the distribution

of inferred flux measurement errors was better approximated

by a double exponential (Laplace) distribution, which has botha

prominent central peak and heavy tails, and kurtosis of 3. Bycomparison, Chevallier et al. (2006) suggested that residuals

between the ORCHIDEE model and flux data from a range of

sites followed a Cauchy distribution. (From the perspective of

characterizing uncertainty for the purposes of data-model

synthesis,the Cauchy distribution is a poorchoice,since its first

fourmoments are undefined). In a recent data-model synthesis

Owen et al. (2007)implicitly assumed an error model where a

large part of the data is Gaussian and high kurtosis is only

introduced by a few high-leverage points (outliers that strongly

influence the fourth moment). In the present study, our results

indicated pronounced error kurtosis for Fc 0 mmol m2 s1,

although trimming outliers that accounted for the top and

bottom 1% of theerror distribution resulted in kurtosis that wasintermediate between a Gaussian (kurtosis = 0) and double-

exponential (kurtosis = 3) distribution.

Hollinger and Richardson (2005)andStauch et al. (in press)

suggested that heteroscedasticity, combined with the varying

frequency of different fluxmagnitudes, could result in an error

distribution that appeared non-Gaussian, even if each error

was drawn from a Gaussian distribution. As an example, take

8000 draws fromX1 N(0,1), and 2000 draws from X2 N(0,5):

the distribution of these 10,000 draws is characterized by a

strong peak (predominantly because of X1) and heavy tails

(predominantly because of X2), and thus appears non-

Gaussian despite the normality of bothX1and X2. While this

addition of distributions no doubt contributes to fact that ateach site the overall kurtosis is always3 or larger, it does not

explain why when the analysis is restricted to relatively

narrow Fc bins, the kurtosis is consistently >5 for

jFcj 0 mmol m2 s1. The present analysis gives additional

support for the idea that, at least for near-zero fluxes, the

uncertainty is better characterized by a double exponential

than a Gaussian distribution.

4.2. Patterns across sites

Random flux measurement uncertainties have been shown to

vary among sites, especially in relation to vegetation type. For

example,Richardson et al. (2006a)reported that uncertaintiesin both carbon and energy fluxes were smaller at a grassland

site compared to forested sites. However, even among forested

sites there may be substantial variation in uncertainty; we

found, forexample, that uncertainties were roughlythree-fold

larger at FR1 than FI1 (Table 3A). The results presented in

Table 3A areconsistent with those shown previously by Moffat

et al. (2007) and Stauch et al. (in press), i.e., for a given Fc, there

is greater uncertainty at FR1 than either DE3 or FR4.

Differences across sites in s(d) can be attributed to a range

of factors, such as measurement height, surface roughness,

wind speed, surface heterogeneity, and measurement system

errors. These appear to affect the base level of uncertainty, but

not the manner in which s(d) scales with Fc, as only the

intercept, but not the slope, of regressions presented in

Table 3A were found to vary significantly among these

forested sites.

4.3. Uncertainty and the color of flux data time series

Spectral analysis methods enable researchers to view time

series in the frequency domain rather than the standardtemporal domain; in this manner, the time scales (from

minutes to years) at which relevant variation occurs can be

identified. Previous frequency domain analyses have made

use of orthonormal wavelet transformations (e.g.,Katul et al.,

2001) and Fourier analysis (e.g.,Baldocchi et al., 2001a). Here

we used an alternative approach, based on the Lomb-Scargle

periodogram (Lomb, 1976; Scargle, 1982) and the multiple

segmentation method of Miramontes and Rohani (2002),

which is naturally suited to investigating the correlation

structure of short time series with missing data (i.e., gaps).

This analysis showed that while the fa scaling exponent for

most sites indicated a moderate correlation structure (pink

noise), there were some sites that tended more towards rednoise (e.g., FR4) and other sites that tended towards white

noise (FR1, BE1). An interesting finding was that there was a

robust connection between the estimated random flux

measurement uncertainty and the scaling exponent, i.e., a

strong correlation between a and s(d) (Fig. 5A). A likely

explanation for this result is that at sites with larger

measurement uncertainties, the respective white noise

components of the NEE measurements tends to sufficiently

obscure the underlying signal such that persistence is much

less pronouncedcomparedto sites with smaller measurement

uncertainties. Thus, larger random flux measurement errors

lead to whiter NEE time series.

Similarly, variation among models in the aof residual timeseries was found to be correlated with the standard deviation

of residuals for each model (Fig. 5B). A possible interpretation

for this pattern is that models that fit the NEE time series less

well tend to have not only larger residuals (and hence higher

s(d)), but also largerand more long-livedsystematic biases (i.e.,

greater persistence and hence more negative values of a).

Thus, increasing model error (compare BETHY and MDS)

results in reddened model residuals.

The most flexible models used here, ANN and MDS, yielded

inferred random errors that were closest to white noise and

also had statistical properties that were most similar to those

of differenced paired measurements. An explanation for this

is that predictions of ANN and MDS models for time periodieffectively represent the mean of all fluxes measured under

similar environmental conditions and at a similar time of day

and time of the year as those that prevail at time i. Residuals

from these models therefore can be seen as analogues of the

paired measurement approach; the difference being that the

model residual is the difference between a measurement (Fi)

and the mean of a population of measurements (mi), rather

than the difference between two pure measurements.

4.4. Systematic errors and biases

Until now, ouranalysisand discussion has focused on random

flux measurement errors. Systematic flux measurement



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errors, or biases, are often dealt with by flagging according

to data quality, after applying pre-defined filters (Foken and

Wichura, 1996). Poor quality data often occur when low-

frequency turbulent motions or non-stationary conditions

occur. This is because there is substantial uncertainty about

the extent to which low-frequency covariances contribute to

the actual exchange between surface and atmosphere (von

Randow et al., 2002). Apart from these, a wide range of errorsources contributes to bias, ranging from poor instrument

maintenance and uncertain delays in sampling tubes to

uncertainty on the height of the zero-plane displacement

(which determines the magnitude of several frequency

corrections to the data) (Kruijt et al., 2004). A number of

systematic errors are uncertain even in sign, while for others

the corrections to be made are themselves highly uncertain.

This suggests that systematic errors (and attempts to correct

for them) have a random component to them, although

usually the error source varies on a (much) larger time scale

(compared to the purely random measurement errors we have

quantified here). Kruijt et al. (2004) assessed the importance of

such errors by assuming ignorance on their magnitude and re-calculating fluxes from rawdata using a range of configuration

settings (note also that Kruijt et al. expressed uncertainties in

terms of percentages, rather than absolute amounts, which

implies that systematic errors are heteroscedastic as well,

similar to results reported here for random errors). The

resulting standard errors can be used to assess random

uncertainty, but, as they actat longertime scales, error will not

reducewith the numberof data taken as would be expected for

independent data points. Rather, for each error source the

appropriate number of degrees of freedom in the data should

be assessed, by assessing the periodicity of, for example,

maintenance, wind directions, and other factors that affect

slow errors. The methodology to assess such ranges oferrors at different time scales still needs to be developed, but

in general this suggests that, instead of distinguishing

random and systematic error, there is a range of time

scales at which errors should be assessed.

5. Summary and conclusions

We evaluated the statistical properties of random flux

measurement errors inferred first from model residuals, and

second, using a standard paired measurement approach. The

five models used in the present analysis were quite different

with respect to structure and parameterization, but alladequately captured the patterns of variation in the flux data

time series across a range of temporal scales. However, the

power spectra of the model predictions indicated too little

power at the highest frequencies (stochastic variation that is

due to random measurement errors) and too much power at

lower frequencies. There was a tendency for the residuals of

some models to exhibit greater persistence than other models,

and this memory effect (reddened model residuals) was

attributed to model error which resulted in larger and more

long-lived systematic biases.

Two models in particular, ANN and MDS, made no prior

assumptions about the functional form of relationships

between environmental drivers and CO2 fluxes, and the

resulting uncertainty estimates were in closest agreement

with the paired observation approach. Compared to the paired

observation approach, an advantage of inferring uncertainties

from model residuals is that many more data points are

available(an average of three times as many for ourdata) from

which the statistics of the distributioncan be estimated;this is

especially relevant with short flux time series, where there

may be an insufficient number of paired observations meetingthe criteria for environmental similarity. While the model

residual method does not require two towers, or arbitrary

limits for what constitutes similar environmental condi-

tions on successive days, care must be taken to ensure that a

good model is used, so that model error does not confound

uncertainty estimates.

In all cases, the flux measurement uncertainty increased

with flux magnitude. The base level of uncertainty differed

markedly among sites, and patterns of variation were

consistent regardless of whether residuals or paired measure-

ments were used to infer the error, suggesting that both

methods yield robust uncertainty estimates that are sensitive

to real underlying differences in uncertainty (caused by,among other things, site heterogeneity and measurement

system characteristics). We were able to link the site-specific

uncertainties to the spectral properties (fa scaling) of the

original NEE time series: these results indicated the potential

for estimatings(d) on the basis of the scaling coefficient a .

The results presented here offer a basis by which the

random flux measurement errors can be specified for a variety

of applications, including data-model fusion efforts, statisti-

cally rigorous model evaluation, and regional-to-continental

integration and synthesis projects. A logical next step would

be to reconcile randomand systematicerrorsundera common

framework, so that the total uncertainty in measured fluxes

(and annual integrals) might be more accurately specified.

Acknowledgements

A.D.R. was supported by the Office of Science (BER), U.S.

Department of Energy, through Interagency Agreement No.

DE-AI02-07ER64355. D.P. was supported by the CarboEurope-IP

research program, funded by the European Commission. The

Max Planck Institute for Biogeochemistry provided funding

and support for the gap filling comparison workshop held in

Jena in September 2006. Site PIs Marc Aubinet (Vielsalm),

WernerKutsch (Hainich), Andre Granier (Hesse), Serge Rambal

(Puechabon), Riccardo Valentini (Roccarespampani) and TimoVesala (Hyytiala) are thanked for making their data available.

Ankur Desai and three anonymous reviewers are all thanked

for their constructive criticism.

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