```Errors and uncertainties
in measuring and modelling
surface-atmosphere exchanges
Andrew D. Richardson
University of New Hampshire
NSF/NCAR Summer Course on Flux Measurements
Niwot Ridge, July 17 2008
Outline
• Introduction to errors in data
• Errors in flux measurements
• Different methods to quantify flux errors
• Implications for modeling
32,000 ± how much?
(note the super calculation error!)
Introduction to errors in data
Errors are unavoidable,
but errors don’t have to cause disaster
(Gare Montparnasse, Paris, 22 October 1895)
Errors in data
• Why do we have measurement errors?
Errors in data
• Why do we have measurement errors?
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Instrument errors, glitches, bugs
Instrument calibration errors
Imperfect instrument design, less-than-ideal application
Instrument resolution
“Problem of definition”: what are we trying to measure, anyway?
• Errors are unavoidable and inevitable, but they can always be
reduced
• Errors are not necessarily bad, but not knowing what they are,
or having an unrealistic view of what they are, is bad
Contemporary political perspective
“There are known knowns. These are things
we know that we know. There are known
unknowns. That is to say, there are things
that we know we don't know. But there are
also unknown unknowns. There are things
we don't know we don't know.”
Donald Rumsfeld
(February 12, 2002)
What measurement uncertainty?
• A measurement is never perfect - data are not “truth”
(corrupted truth?)
• “Uncertainty” describes the inevitable error of a measurement:
x´ = x +  + 
• x´ is what is actually measured;
it includes both systematic ()
and random () components
• typically,  is assumed Gaussian,
and is characterized by its
standard deviation, ()
Types of errors
• Random error
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–
–
–
Unpredictable, stochastic
Scatter, noise, precision
Cannot be corrected (because they are stochastic)
Example: noisy analyzer (electrical interference)
• Systematic error
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Deterministic, predictable
Bias, accuracy
Can be corrected (if you know what the correction is)
Propagation of errors
• Random errors:
– true value, x, measure xi’ = x + i, where i is a random variable
~N(0,i)
– ‘average out’ over time, thus errors accumulate ‘in quadrature’
– expected error on (x1’+ x2’) is    , which is  (   )
2
1
• Systematic errors:

2
2
1
2

– fixed biases don’t average out, but rather accumulate linearly
– measure xi’ = x + i,where i is not a random variable
– expected error on (x1’+ x2’) is just (1 + 2)
•
So random and systematic errors are fundamentally different
in how they affect data and interpretation
Precision and Accuracy:
Target analogy
• Accuracy: how close to center
• Precision: how close together
High accuracy, low precision
High precision, low accuracy
Evaluating Errors
• Random errors and precision
– Make repeated measurements of the same thing
– What is the scatter in those measurements?
• Systematic errors and accuracy
– Measure a reference standard
– What is the bias?
– Not always possible to quantify some systematic
errors (know they’re there, but don’t have a
standard we can measure)
What can we do about errors?
• (Honestly) quantify sources of error
– Random
– Systematic
• Minimize/eliminate them
– Identify ways to reduce specific sources of
error
• Examples: more frequent calibration, better QC
procedures, reduce instrument noise
– Correct for biases where possible
• Evaluate reductions in error
Why do we care
Why do we care
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How much confidence do we have in the data?
How certain are we about a particular measurement?
How close to the “true” value are we?
Are our data biased? In which direction?
Errors influence our interpretation of data
Errors reduce the usefulness of the information in our
data
• Errors in data propagate to subsequent analyses
Random errors
• The observed data are just one
realization, drawn from a
“statistical universe of data sets”
(Press et al. 1992)
• Different realizations of the
estimates of “true” model
parameters (or other statistics
calculated from data)
• Parameters estimates are therefore
themselves uncertain (but we
want to describe their
distributions!)
A statistical universe
Monte Carlo Example
• Two options for propagating errors
– Complicated mathematics, based on theory and
first principles
– “Monte Carlo” simulations
1. Characterize uncertainty
2. Generate synthetic data (model + uncertainty) i.e. new
realization from “statistical universe”
3. Estimate statistics or parameters, P, of interest
4. Repeat (2 & 3) many times
5. Posterior evaluation of distribution of P
Viva Las Vegas!
“Offered the choice between mastery of
a five-foot shelf of analytical statistics
books and middling ability at
performing statistical Monte Carlo
simulations, we would surely choose to
have the latter skill.”
William H. Press,
Numerical Recipes in Fortran 77
Why you should learn a programming language (even fossil languages like BASIC):
It’s really easy and fast to do MC simulations. In spreadsheet programs, it is extremely
tedious (and slow), especially with large data sets (like you have with eddy flux data!).
What we want to know
• What are the characteristics of the error
– What are the sources of error, and do the sources
of error change over time?
– How big are the errors (10±5 or 10.0001±0.0001),
and which are the biggest sources?
– Are the errors systematic, random, or some
combination thereof?
– Can we correct or adjust for errors?
– Can we reduce the errors (better instruments, more
careful technician, etc.?)
Characteristics of interest
• How is the error distributed?
– What is the (approximate) pdf
• What are its moments?
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–
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First moment:
Second moment:
Third moment:
Fourth moment:
mean (average value)
standard deviation (how variable)
skewness (how symmetric)
kurtosis (how peaky)
Random error distributions
• Assumptions:
– Normal distribution
– Constant variance
– Independent errors
• Reality:
– Other distributions
are possible!
– Variance may not be
constant
– Independence?
Normal
Laplace
Lognormal
Uniform
Constant variance?
• Homoscedastic (constant
variance) vs.
Heteroscedastic
(nonconstant variance)
• Assume homoscedastic, but
commonly: error variance
scales with measurement
• Large outliers more likely
when error variance is large
Independence
• Measurement error in period (t) are uncorrelated with errors in
period (t-1)
• Independent
– coin toss (white noise)
• Negative autocorrelation
– growth rate estimated from size measurements
• Positive autocorrelation
– weekly biomass estimates confounded by seasonal variation in other
factors (e.g. summer drought and shrinking xylem)
• Difficult to detect or test for independence without a very good
underlying model (with poor model, apparent autocorrelation
may be due to model structure and not error structure!)
Take-home messages
• For modelling,
– More noise in data = more uncertainty in estimated
model parameters
– Systematic error in data = biased estimates of
model parameters
• Monte Carlo simulations as an easy way to
evaluate impact of errors
Errors in flux measurements
Why characterize flux
measurement uncertainty?
• Uncertainty information needed to compare measurements,
measurements and models, and to propagate errors (scaling up
in space and time)
• Uncertainty information needed to set policy & for risk
analysis (what are confidence intervals on estimated C sink
strength?)
• Uncertainty information needed for all aspects of data-model
fusion (correct specification of cost function, forward
prediction of states, etc.)
• Small uncertainties are not necessarily good; large
uncertainties are not necessarily bad: biased prediction is ok if
‘truth’ is within confidence limits; if truth is outside of
confidence limits, uncertainties are under-estimated
Challenge of flux data
• Complex
– Multiple processes (but only measure NEE)
– Diurnal, synoptic, seasonal, annual scales of variation
– Gaps in data (QC criteria, instrument malfunction,
unsuitable weather, etc.)
• Multiple sources of error and uncertainty (known
unknowns as well as unknown unknowns!)
• Random errors are large but tolerable
• Systematic errors are evil, and the corrections for them are largely
uncertain (sometimes even in sign)
• Many sources of systematic error (sometimes in different
directions)
Systematic errors
Random errors
Systematic errors
•
examples:
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•
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Random errors
•
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nocturnal biases
imperfect spectral response
energy balance closure
operate at varying time scales: fully
systematic vs. selectively systematic
variety of influences: fixed offset vs.
relative offset
cannot be identified through statistical
analyses
can correct for systematic errors (but
corrections themselves are uncertain)
uncorrected systematic errors will bias
DMF analyses
examples:
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•
•
•
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surface heterogeneity and time varying
footprint
turbulence sampling errors
measurement equipment (IRGA and
sonic anemometer)
random errors are stochastic;
characteristics of pdf can be estimated
via statistical analyses (but may be
time-varying)
affect all measurements
cannot correct for random errors
random errors limit agreement between
measurements and models, but should
not bias results
How to estimate distributions of
random flux errors
Why focus on random errors?
• Systematic errors can’t be identified through
analysis of data
• Systematic errors are harder to quantify (leave
that to the geniuses)
• For modeling, must correct for systematic
errors first (or assume they are zero)
• Knowing something about random errors is
much more important from modeling
perspective
Two methods
Two methods
• Repeated measurements of the “same thing”
– Paired towers (rarely applicable)
• Hollinger et al., 2004 GCB ; Hollinger and Richardson, 2005 Tree Phys
– Paired observations (applicable everywhere)
• Hollinger and Richardson, 2005 Tree Phys; Richardson et al. 2006 AFM
• Comparison with “truth”
– Model residuals (assume model = “truth”)
• Richardson et al., 2005 AFM; Hagen et al. JGR 2006; Richardson
et al. 2008 AFM
Paired tower approach
Repeated measurements: Use
simultaneous but independent
measurements from two
towers, x1 and x2
Howland: Main and West towers
- same environmental conditions
- located in similar patches of forest
- non-overlapping footprints
(independent turbulence)
Main
West
~800m
Paired measurements
Assume we have measurements x1, x2 from two towers
var(x1 – x2) = var(x1) + var(x2) – 2 covar (x1, x2)
Since x1 and x2 are assumed independent, covar(x1, x2) = 0
Also, var(x1) = var(x2) = var(), where  is the random error
So var(x1 – x2) = 2 var()
And thus
 ( ) 
 ( x1  x 2 )
2
Use multiple pairs x1, x2 to infer distribution of !
Alternatively…
• Earlier, suggested quantifying random error by
the standard deviation () of multiple
independent measurements of the same thing
(xi):
• For i = 1,2, reduces to
 
1
2
( x1  x 2 )
2
or
x1  x 2
2
• If following this approach, mean  across multiple x1,
2
2
x2 pairs is calculated
as



(

,
...,

)
1

 i
(i.e. as a geometric mean, or square root of the mean variance).
Another paired approach…
• Two tower approach can only rarely be used
• Alternative: substitute time for space
– Use x1, x2 measured 24 h apart under similar
environmental conditions
• PPFD, VPD, Air/Soil temperature, Wind speed
– Tradeoff: tight filtering criteria = not many paired
measurements, poor estimates of statistics; loose
filtering = other factors confound uncertainty
estimate
Model residuals
• Common in many fields (less so in “flux world”) to
conduct posterior analyses of residuals to investigate
pdf of errors, homoscedasticity, etc.
– Model must be ‘good’ or uncertainty estimates confounded
by model error
– can evaluate asymmetry in error distribution (not possible
with paired approach)
– many data points with which to estimate statistics
A double exponential (Laplace) pdf
better characterizes the uncertainty
Strong central peak &
heavy tails (leptokurtic)
 non-Gaussian pdf
Better: double-exponential pdf,
f(x) = exp(|x/|)/2
The double-exponential is characterized by the scale parameter 
 (x) 
2  where  
1
N

xi – x
The standard deviation of the uncertainty
scales with the magnitude of the flux
• Larger fluxes are more uncertain
than small fluxes
• Relative error decreases with flux
magnitude (even when flux = 0
there is still some uncertainty)
• Large errors are not uncommon
• 95% CI = ± 60%
• 75% CI = ± 30%
To obtain maximum likelihood parameter estimates, cannot use OLS:
must account for the fact that the flux measurement errors are nonGaussian and have non-constant variance.
• Scaling of uncertainty with flux magnitude has been validated
using data from a range of forested CarboEurope sites: y-axis
intercept (base uncertainty) varies among sites (factors: tower
height, canopy roughness, average wind speed), but slope
constant across sites (Richardson et al., 2007)
• Similar results (non-Gaussian, heteroscedastic) have been
demonstrated for measurements of water and energy fluxes (H
and LE) (Richardson et al., 2006)
• Results are in agreement with predictions of Mann and
Lenschow (1994) error model based on turbulence statistics
(Hollinger & Richardson, 2005; Richardson et al., 2006)
Generality of
s(H) = 19.5 W m-2
s(LE) = 16.5 W m-2
s(FCO2) = 2.0 mol m-2 s-1
Uncertainties of all fluxes increase
with flux magnitude.
Comparison of approaches
• Error estimates vary by ≈ 10% across models, are ≈20% lower
for paired approach than for ‘best’ model
• Errors are “more Gaussian” for large uptake fluxes and “less
Gaussian” for fluxes ≈0 mol m-2 s-1
Uncertainty at various time scales
• Systematic errors accumulate linearly
over time (constant relative error)
• Random errors accumulate in
decreases as flux measurements are
aggregated over longer time periods)
• role of Central Limit Theorem as fluxes are aggregated
Implications for modeling
"To put the point provocatively, providing
data and allowing another researcher to
provide the uncertainty is indistinguishable
from allowing the second researcher to
make up the data in the first place."
– Raupach et al. (2005). Model data synthesis in terrestrial
carbon observation: methods, data requirements and data
uncertainty specifications. Global Change Biology 11:37897.
Why does it matter for modeling?
• Cost function (Bayesian or not) depends on error
structure
– likelihood function: the probability of actually observing the data,
given a particular parameterization of model
– appropriate form of likelihood function depends on pdf of errors
– maximum likelihood optimization: determine model parameters that
would be most likely to generate the observed data, given what is
known or assumed about the measurement error
• Ordinary least squares generates ML estimates only
when assumptions of normality and constant variance
are met
“what model parameters values are most likely to have generated the observed data, given
the model and what is known about measurement errors?”
Assumptions about errors affect specification of the ML cost
function:
N
cost 

i
N
cost 

i
( y i  y pred )
 ( i )
2
y i  y pred
 ( i )
2
For Gaussian data
(“weighted least squares”)
For double exponential data
(“weighted absolute deviations”)
Other cost functions are possible—depends on error structure!
Specifying a different cost function affects
optimal parameter estimates
• Lloyd & Taylor (1994) respiration model:
• model parameters differ depending
on how the uncertainty is treated
(explanation: nocturnal errors have
slightly skewed distribution)
• Why? error assumptions influence
form of likelihood function
LS
A
24.9
43.9
T0
263.9
259.5
E0
33.6
58.5
R eco  Ae
 E 0 T T 0 
Reco –respiration
T –soil temperature
A, E0, T0 – parameters

Influence of cost function specification
on model predictions
• Half-hourly model predictions depend on parameter-ization;
integrated annual sum decreases by ~10% decrease (≈40% of
NEE) when absolute deviations is used
• Influences NEE partitioning, annual sum of GPP
• Trivial model but relevant example
and also…
• Random errors are stochastic noise
– do not reflect “real” ecosystem activity
– cannot be modeled because they are stochastic
– ultimately limit agreement between models and
data
– make it difficult
• to obtain precise parameter estimates (as shown by
previous Monte Carlo example)
• to select or distinguish among candidate models (more
than one model gives acceptably good fit)
Summary
• Two types of error, random and systematic
• Random errors can be inferred from data
• Flux measurement errors are non-Gaussian and
have non-constant variance
• These characteristics need to be taken into
account when fitting models, when comparing
models and data, and when estimating
statistics from data (annual sums,
physiological parameters, etc.)
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