agricultural and forest meteorology 148 (2008) 38–50
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/agrformet
Statistical properties of random CO2 flux measurement
uncertainty inferred from model residuals
Andrew D. Richardson a,*, Miguel D. Mahecha b, Eva Falge c, Jens Kattge b,
Antje M. Moffat b, Dario Papale d, Markus Reichstein b, Vanessa J. Stauch e,
Bobby H. Braswell a, Galina Churkina b, Bart Kruijt f, David Y. Hollinger g
a
University of New Hampshire, Complex Systems Research Center, Morse Hall, 39 College Road, Durham, NH 03824, USA
Max Planck Institute for Biogeochemistry, Hans-Knöll-Strasse 10, 07745 Jena, Germany
c
Max Planck Institute for Chemistry, Biogeochemistry Department, J.J.v. Becherweg 27, 55128 Mainz, Germany
d
DISAFRI, University of Tuscia, via C. de Lellis, 01100 Viterbo, Italy
e
Computational Landscape Ecology, Centre for Environmental Research (UFZ), Leipzig, Germany
f
Alterra, Wageningen University and Research Centre, 6700 AA Wageningen, The Netherlands
g
USDA Forest Service, Northern Research Station, Durham, NH 03824, USA
b
article info
abstract
Article history:
Information about the uncertainties associated with eddy covariance measurements of sur-
Received 26 March 2007
face–atmosphere CO2 exchange is needed for data assimilation and inverse analyses to
Received in revised form
estimate model parameters, validation of ecosystem models against flux data, as well as
25 August 2007
multi-site synthesis activities (e.g., regional to continental integration) and policy decision-
Accepted 3 September 2007
making. While model residuals (mismatch between fitted model predictions and measured
fluxes) 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
Keywords:
from the CarboEurope program, and compare the statistical properties of the inferred random
Data-model fusion
flux measurement error calculated first using residuals from five different models, and
Eddy covariance
secondly using paired observations made under similar environmental conditions. Spectral
FLUXNET
analysis of the model predictions indicated greater persistence (i.e., autocorrelation or
Measurement error
‘‘memory’’) compared to the measured values. Model residuals exhibited weaker temporal
Pink noise
correlation, but were not uncorrelated white noise. Random flux measurement uncertainty,
Spectral analysis
expressed as a standard deviation, was found to vary predictably in relation to the expected
Uncertainty
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 observations, although the latter approach resulted in somewhat smaller estimates. Higher order
moments (e.g., skewness and kurtosis) suggested that for fluxes close to zero, the measurement error is commonly skewed and leptokurtic. Skewness could not be evaluated using the
paired observation approach, because differencing of paired measurements resulted in a
symmetric distribution of the inferred error. Patterns were robust and not especially sensitive
to the model used, although more flexible models, which did not impose a particular functional 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).
0168-1923/$ – see front matter # 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.agrformet.2007.09.001
agricultural and forest meteorology 148 (2008) 38–50
1.
Introduction
The eddy covariance method is used at research sites around
the world to measure surface–atmosphere exchanges of
carbon, water and energy (Baldocchi et al., 2001b). Increasingly, 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-data
synthesis’’, 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 uncertainties 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 ¼ F0i þ di ), then a
convenient way to characterize the uncertainty is s(di), i.e., as
the standard deviation of the random error. Accordingly, 1/s(di)
is a quantitative measure of our confidence in the measured
data (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)
(possibly raised to 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 needed
for multi-site syntheses (e.g., Law et al., 2002; Churkina et al.,
2005) and regional-to-contintental integration efforts, especially with regard to carbon accounting and policy decisionmaking (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 the
aggregate 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,
39
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 measurement 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) differenced
paired 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 by Rannik
et al., 2006, where the 30 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 on
turbulence 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 timevarying footprint of a single tower (see Katul et al., 1999; Oren
et 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 an
imperfect representation of reality, and thus 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 deviation of model residuals for the
Lloyd and Taylor (1994) respiration model, fit to nocturnal
40
agricultural and forest meteorology 148 (2008) 38–50
data, was only 14% larger than the random flux measurement
uncertainty estimated using paired measurements, suggesting that model error is potentially relatively small. In that
study, the statistical properties of the residuals were otherwise 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 measurement 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 nonGaussian. At least two related studies have purposely avoided
making assumptions about the specific probability distribution function for the random flux measurement uncertainty.
For example, Hagen et al. (2006) used bootstrap resampling of
model residuals to estimate uncertainties in gross productivity 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 characterized 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 uncertainties 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 uncertainty estimates that are comparable
to those derived from paired observations. We optimize model
parameters (on a site-by-site basis) to maximize the agreement between model and measurements, and thus obtain
model predictions that are informed by the data, rather than
independent of the data. In this manner the models can be
used as tools to predict the ‘‘true’’ flux, F0i , 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 artificial neural networks and a processbased 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 fluxdriver relationships.
We begin by evaluating, first in the temporal domain and
then in the frequency domain, model predictions and model
residuals in relation to the measured CO2 flux 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 observations and model residuals, and look for similarities across
sites. Finally, relationships between the standard deviation of
the random flux measurement uncertainty and flux magnitude developed here for six forested European sites are
compared to those presented previously by Richardson 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 (see Table 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 for
evaluation 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 by Moffat et al. (2007). Measured CO2
fluxes are here denoted Fc, with units of mmol m2 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
BE1 (Vielsalm, Belgium)
DE3 (Hainich, Germany)
Species
Climate (MAT)
Years
Fagus sylvatica,
Pseudotsuga menziesii
Fagus sylvatica
Temperate-continental
(7.5 8C)
Temperate-continental
(6.8 8C)
Boreal (3.8 8C)
Temperate-suboceanic
(9.9 8C)
Mediterranean (13.5 8C)
Mediterranean (15.2 8C)
2000, 2001
50.308N
5.988E
Aubinet et al. (2001)
2000, 2001
51.078N
10.458E
Knohl et al. (2003)
2001, 2002
2001, 2002
61.838N
48.678N
24.288E
7.058E
Suni et al. (2003)
Granier et al. (2000)
2002
2002
43.738N
42.408N
3.588E
11.928E
Rambal et al. (2004)
Tedeschi et al. (2006)
FI1 (Hyytiala, Finland)
FR1 (Hesse, France)
Pinus sylvestris
Fagus sylvatica
FR4 (Puechabon, France)
IT3 (Roccarespampani, Italy)
Quercus ilex
Quercus cerris
Latitude Longitude
Reference
agricultural and forest meteorology 148 (2008) 38–50
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 by Moffat 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-parametric
model (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 particular NLR used here (Falge et al., 2001) was based on a
Michaelis–Menten light response function for gross photosynthesis and a Lloyd and Taylor (1994) temperature function
for ecosystem respiration; model parameters were fit twicemonthly. BETHY (Knorr and Kattge, 2005) is a terrestrial
biosphere model that predicts CO2, water and energy fluxes. In
addition to meteorological drivers, required input data include
site-specific information about soil and canopy properties
(e.g., leaf area index, estimated from remote sensing products). SPM (Stauch and Jarvis, 2006) is analogous to a multidimensional look-up table, with an underlying semi-parametric 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 timewindows as small as possible. Thus the models ranged from
data based (e.g., ANN) to process based (e.g., BETHY), and the
adaptation 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., Akaike’s Information Criterion) to similar models of CO2
exchange 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 summarized by calculating the scaling exponent,
a, of the power spectrum P( f) = fa (Mandelbrot, 1999). Here f
refers to the frequencies contained in a given time series, and a
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 Brownian 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
41
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-spaced
time 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 for very long time series when standard methods
are applied, new methods (e.g., MSM) have been developed to
yield statistically robust estimates of a for short time series.
The MSM method subdivides a time series 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 of a is
obtained by fitting a linear model to the relationship between
segment length and the mean of the scaling exponents found
at each segment size. In the present study the power spectra of
the sub-series in MSM were determined using the L-S method.
With this hybrid approach that blends MSM and L-S, we could
estimate a not only for short, but also for short and fragmented
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 in
Richardson et al. (2006a). Briefly, this involved considering as a
pair any two flux measurements 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
42
agricultural and forest meteorology 148 (2008) 38–50
characterize the random flux measurement 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 residuals is related to 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-
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), semiparametric model (SPM), and a look-up table based on
marginal distribution sampling (MDS); sites (y-axis) are as
identified in Table 1.
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, were
obtained with both the L-S and hybrid MSM/L-S methods,
except that the latter method tended to result in more clearly
defined differences in a among the different models. We focus
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,
indicating that the modeled time series have a longer memory
effect (i.e., tending towards ‘‘red’’ noise) than the measured
NEE time series. The site-to-site variation in a of the measured
fluxes was not captured by any model. The MDS model
consistently came closest to reproducing the spectral properties 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 the residuals 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.
43
agricultural and forest meteorology 148 (2008) 38–50
Table 2 – Statistical properties of inferred random flux measurement error (CO2 fluxes measured in mmol mS2 sS1)
Method
(A) All sites, by method
ANN
NLR
BETHY
MDS
SPM
Paired
Site
N
Mean
Standard deviation
Skewness
121217
121217
121217
121217
121217
40563
0.004
0.001
0.048
0.067
0.000
0.029
2.73
3.10
3.10
2.80
2.93
2.19
0.22
0.05
0.16
0.04
0.09
0.18
Mean
Standard deviation
Skewness
0.043
0.000
0.014
0.005
0.002
0.015
0.010
0.029
0.010
0.005
3.21
2.89
2.46
2.17
1.28
1.32
3.60
4.12
1.81
2.29
0.03
0.32
0.16
0.20
0.21
0.32
0.20
0.26
0.31
0.50
N
(B) ANN method, by site-year
BE1 2000
12519
BE1 2001
12287
DE3 2000
11452
DE3 2001
11731
FI1 2001
11236
FI1 2002
11427
FR1 2001
13744
FR1 2002
13651
FR4 2002
11190
IT3 2002
11980
Kurtosis
11.61
9.22
8.37
10.23
9.66
10.52
Kurtosis
3.55
7.49
7.01
6.29
7.31
8.46
8.84
7.73
2.99
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.
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 (2.73–3.10 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. However, the strong positive kurtosis indicated by all methods,
approximately 10, 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 in Table 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 2001 and 2000, and FR1 2001 and 2002. Skewness
was not especially prominent at any particular site. All sites
had kurtosis of 3 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 of the model used,
and the model-based approach was largely consistent with
the paired observation approach, as illustrated by results
from FR1 2001 (see Fig. 3A–C; 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
the flux, 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
did not show any clear patterns of variation in relation to flux
magnitude (Fig. 3B). For the 0 mmol m2 s1 Fc bin, skewness of
random error inferred from NLR model residuals was much
higher (1.6) than any of the other models, which were
otherwise in the range 0.0–0.6 (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 Fc bin, 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. 3D–F).
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,
44
agricultural and forest meteorology 148 (2008) 38–50
Fig. 3 – Comparison of statistical properties (standard deviation, skewness and kurtosis) of inferred flux measurement error
in relation to CO2 flux 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 (A–C), 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 (D–F), results are shown across all 10 site-years for a
single model (ANN, artificial neural network). In the right column (G–I), 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.)
varied among sites (Fig. 3D; note the consistency with
patterns reported by Richardson et al., 2006a, as indicated
by the heavy blue shaded line; see also Table 3). Many sites
had 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 of Fc
increased, 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 Fc bin 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 (compare Fig. 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 modelbased (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 impossible to differentiate between a positive error in one
measurement versus a negative error in the other measure-
agricultural and forest meteorology 148 (2008) 38–50
45
Table 3 – Relationship between CO2 flux magnitude (Fc, in
mmol mS2 sS1) and the standard deviation of the inferred
random flux measurement error, s(d)
Method
Relationship
(A) By site
BE1 2000
BE1 2001
DE3 2000
DE3 2001
FI1 2001
FI1 2002
FR1 2001
FR1 2002
FR4 2002
IT3 2002
2.91(0.30) + 0.09(0.03) jFcj
2.27(0.17) + 0.14(0.02) jFcj
1.31(0.33) + 0.18(0.02) jFcj
1.13(0.21) + 0.16(0.01) jFcj
0.87(0.24) + 0.12(0.03) jFcj
0.91(0.17) + 0.12(0.02) jFcj
2.62(0.34) + 0.18(0.02) jFcj
3.49(0.38) + 0.13(0.02) jFcj
1.28(0.41) + 0.09(0.05) jFcj
1.79(0.07) + 0.11(0.01) jFcj
(B) All sites
Model residuals (ANN)
Paired observations
1.69(0.20) + 0.16(0.02) jFcj
1.47(0.22) + 0.17(0.02) jFcj
(C) Richardson et al. (2006a)
Fc 0
Fc 0
1.42(0.31) + 0.19(0.02) jFcj
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.
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 as
fluxes increase in magnitude. For fluxes close to
0 mmol m2 s1, skewness, and especially pronounced kurtosis, 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, the y-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 in Table 2B, but the differences
in slope among sites was not significant (P = 0.10). When data
were 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
these relationships could be distinguished (P > 0.10) from that
reported by Richardson et al. (2006a) for Fc 0. However,
whereas Richardson 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 also Stauch et al., in press). This difference
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.
may be a result of more stringent nocturnal rejection criteria
(u* filtering, quality flags and other data cleaning, see Papale
et al., 2006) at the European sites compared to the sites used by
Richardson et al. (2006a).
46
agricultural and forest meteorology 148 (2008) 38–50
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-specific
variation in a of the measured NEE time series (‘+’ symbols
in Fig. 2) was positively correlated with the standard deviation
of model residuals for each site. Results are shown in Fig. 5A
for the ANN model (r = 0.87), but similar results were obtained
for all other models (r between 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:
sðdÞ ¼ 6:59ð0:72Þ þ 4:75ð0:82Þ a
(Coefficients were calculated using reduced major axis regression, with standard errors on coefficients given in parentheses). In the same 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 developing site-specific estimates of this relationship, purely on the
basis of a, i.e., as:
sðdÞ ¼ 5:82ð0:79Þ þ 4:61ð0:90Þ a þ 0:17 jFc j
Second, the variation in a of model residuals (open circles
in Fig. 2) was negatively correlated with the standard deviation
of 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 (r between 0.72 and 0.98).
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.
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 the
magnitude of the flux. From this study we find that the manner
in which the uncertainty scales with the magnitude of the flux
(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-tonoise ratio, and will still exert a greater influence during the
optimization, 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 for these differences
agricultural and forest meteorology 148 (2008) 38–50
in the probability distribution function of the measurement
errors.
If the random flux measurement uncertainty does not 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 both a
prominent central peak and heavy tails, and kurtosis of 3. By
comparison, 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 poor choice, since its first
four moments 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 the error distribution resulted in kurtosis that was
intermediate between a Gaussian (kurtosis = 0) and doubleexponential (kurtosis = 3) distribution.
Hollinger and Richardson (2005) and Stauch et al. (in press)
suggested that heteroscedasticity, combined with the varying
frequency of different flux magnitudes, 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 from X1 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 nonGaussian despite the normality of both X1 and X2. While this
‘‘addition of distributions’’ no doubt contributes to fact that at
each site the overall kurtosis is always 3 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 uncertainties
in 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, for example, that uncertainties were roughly three-fold
larger at FR1 than FI1 (Table 3A). The results presented in
Table 3A are consistent 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
47
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 standard
temporal 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 ‘‘red’’
noise (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 pronounced compared to sites with smaller measurement
uncertainties. Thus, larger random flux measurement errors
lead to ‘‘whiter’’ NEE time series.
Similarly, variation among models in the a of residual time
series 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 larger and more long-lived systematic 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 period i
effectively 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, our analysis and discussion has focused on random
flux measurement errors. Systematic flux measurement
48
agricultural and forest meteorology 148 (2008) 38–50
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 lowfrequency 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 error
sources 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 recalculating fluxes from raw data 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 act at longer time scales, error will not
reduce with the number of 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 of
errors 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 all
adequately 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 our data) from
which the statistics of the distribution can be estimated; this is
especially relevant with short flux time series, where there
may be an insufficient number of paired observations meeting
the criteria for environmental similarity. While the model
residual method does not require two towers, or arbitrary
limits for what constitutes ‘‘similar’’ environmental conditions 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 measurements 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 estimating s(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, statistically rigorous model evaluation, and regional-to-continental
integration and synthesis projects. A logical next step would
be to reconcile random and systematic errors under a 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),
Werner Kutsch (Hainich), André Granier (Hesse), Serge Rambal
(Puechabon), Riccardo Valentini (Roccarespampani) and Timo
Vesala (Hyytiälä) are thanked for making their data available.
Ankur Desai and three anonymous reviewers are all thanked
for their constructive criticism.
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