The Propagation of Errors in Long-term Measurements

Global Change Biology (1996) 2, 231-240

The propagation of errors in long-term measurements ofland-atmosphere fluxes of carbon and water

J.B.MONCRrEFF*, Y. MALHI* andR. LEUNING +*Jnstitute of Ecology and Resource Management, University of Edinburgh, Edinburgh EH9 3JU, UK. fComTnonwealth Scientificand Industrial Research Organization, Centre for Environmental Mechanics, PO Box 821, Canberra ACT 2601, Australia

Abstract

For surface Ouxes of carbon dioxide, the net daily flux is the sum of daytime and night-time fluxes of approximately the same magnitude and opposite direction. The net fluxis therefore significantly smaller than the individual flux measurements and errorassessment is critical in determining whether a surface is a net source or sink of carbondioxide. For carbon dioxide flux measurements, it is an occasional misconception thatthe net flux is measured as the difference between the net upward and downward fluxes<i.e. a small difference between large terms). This is not the case. The net flux is thesum of individual (half-hourly or hourly) flux measurements, each with an associatederror term. The question of errors and uncertainties in long-term flux measurements ofcarbon and water is addressed by first considering the potential for errors in fluxmeasuring systems in general and thus errors which are relevant to a wide range oftimescales of measurement. We also focus exclusively on flux measurements made bythe micrometeorological method of eddy covariance. Errors can loosely be divided intorandom errors and systematic errors, although in reality any particular error may be acombination of both types. Systematic errors can be fully systematic errors (errors thatapply on all of the daily cycle) or selectively systematic errors (errors that apply to onlypart of the daily cycle), which have very different effects. Random errors may also befull or selective, but these do not differ substantially in their properties. We describean error analysis in which these three different types of error are applied to a long-termdataset to discover how errors may propagate through long-term data and which can beused to estimate the range of uncertainty in the reported sink strength of the particularecosystem studied.

Keywords: carbon balance, eddy covariance, enrors, long-term measurements, micrometeorology.

Received 18 August 1995; revision accepted 6 January 1996

1 Introduction

All long-term fluxes are estimated from a series ofmeasurements made at shorter intervals. Even withina campaign of long-term flux measurements for carbondioxide and water vapour there will be a range ofscientific objectives relevant to differing periods overwhich answers are sought. At short timescales (hoursto days) interest may focus on the control of surfacefluxes by the biology, soil and atmosphere. At longertimescales (weeks to years) we may seek to determinewhether the ecosystem is a net source or sink forcarbon, and to describe how evaporation from the

Correspondence: Dr J. B. Moncrieff, tel + 44-(0)l31-650-5402,fax + 44-(0)l31-662-0478, e-mail [email protected]

system responds to seasonal fluctuations in soil watersupply and atmospheric demand. Errors occur at eachstage of measurement and the question of errorpropagation from short- to long-term needs to beaddressed.

In this paper we briefly examine errors assodated withthe measurement of fluxes using the eddy covariancetechnique at short timescales, and discuss the con-sequences of these errors in long-term measurements ofCO2 and HjO exchange. The error analysis is then usedto assess the confidence with which a mean source/sinkstrength can be stated for a particular ecosystem giventhe nature of the instrumentation, and underlyingassumptions in the eddy covaritince technique.

© 1996 Blackwell Sdence Ltd.

232 J.B.MONCRIEEE etai

2 The eddy covariance technique

2.1 Basic assumptions

The eddy covariance technique measures the flux of ascalar (heat, mass) or momentum at a point centred oninstruments placed at some height above the surface. Forthese measurements to be identical to the flux at theunderlying surface, the instruments must be located inthe internal boundary layer where the flux is constantwith height. A clear understanding of the conditionsrequired for a constant flux layer is obtained by examiningthe conservation equation for a scalar quantity in a controlvolume placed over the surface:

as

dt-D si.

dx]— S

In this equation s is the density of the scalar, t is time, u,is the wind velocity component in the orthogonal direc-tions Xi (i = 1,2,3), D is the molecular diffusivity forquantity s in air, S is the volumetric source/sink strength,and the overbar represents a time average. The first termon the left is the mean rate of change of mass per volume,while the second and third terms are the flux divergencearising from net advection and molecular diffusionthrough the sides of the control volume. A constant fluxlayer requires stationarity (ds/dt = 0), the absence ofsources or sinks between the surface and the instruments(S = 0), and a horizontally homogeneous surface for asufficiently long upwind area, so that d(u^)/dx, = 0 andDd^/dx^ = 0(i = 1, 2). When these assumptions are met,we obtain

dz 322= 0, (2)

where w = ujis the vertical windspeed, and z = X3 is thevertical coordinate. Turbulence is suppressed near thesurface while turbulent transport is many orders ofmagnitude greater than molecular diffusion at height zabove the surface (Businger 1986). Thus integrating thisequation and using the Reynolds convention that w =w' +w and s = s' + s, yields

fn=-Ddz 0

(3)

where FQ is the flux resulting from molecular diffusionat the underlying soil and leaf surfaces, and F is theturbulent eddy flux at height 2. Note that the eddy fluxcan also be written as

* ? (4)

where r^s is the correlation coefficient and c^^ andthe standard deviations of w and s, respectively.

are

2.2 Errors associated with the eddy covariance method

Figure 1 is a schematic diagram which summarizes thefactors micrometeorologists consider when assessing theaccuracy of their field measurements (Businger 1986).This shows that errors in the eddy covariance techniqueare assodated with instrumentation as well as a failureto meet the assumptions required for Eq. (3). We firstdiscuss errors inttoduced by non-steady state conditions,advection and complex terrain, and then sources ofinstrumentation error. This is followed by an examinationof the impact of random and systematic errors on long-term flux measurements.

Under non-steady state but horizontally homogeneousconditions, Eq. (1) may be integrated with respect toheight to give

f.-F.= -ds

dtdz. (5)

The term on the right represents the change in storageof s in the air mass between the surface and the heightz. Baldocchi et ai (1988) showed that these errors aregenerally small during the day but can be significant atdusk, overnight and at dawn when turbulent mixing islow. This is particularly relevant to fiux measurementsover forests, as discussed later.

Ideally the measurement site should be extensive,flat and horizontally uniform to ensure that fluxes aremeasured in the constant flux layer. Under stationary butnon-homogeneous conditions, error in the flux measure-ment at height z due to advection is given by

j- dus 'r ds

i dx i dxdz. (6)

To a first-order approximation, the error Fj - FQ, follow-ing a change in surface with a flux of f, to one with fluxFQ may be estimated using

= 10z ln(z/2o)

X ln(X/10zo)(7)

where X is the upwind distance from the change insurface and ZQ is the roughness length of the downwindsurface. To derive this equation from Eq. (6), we haveassumed that the internal boundary layer height, h « O.IX(see Kaimal and Finnigan 1994; p. 115), and that windprofiles are logarithmic, i.e. neutral stability. It is clearfrom this expression that the error in the flux measure-ment fj - FQ, is proportional to the difference in fluxes

© 19% Blackwell Sdence Ltd., Global Change Biology. 2. 231-240

E R R O R S IN L O N G - T E R M FLUX M E A S U R E M E N T S 233

Fig. 1. Schematic of the main processesand terms which have to be consideredwhen assessing the validity of fluxmeasurements made by an eddycovariance system.

ATMOSPHERE- stationarity

flux footprint Mean horizontaJ wind

INSTRUMENTATION- one point sampling- time response- sensor separation- tube losses- digital Filtept- suppqptirig structi

SURFACE- heterogeneity- complex tertain

katabatic flow

of the upwind and downwind surfaces, F| - FQ. The erroralso diminishes as z / X decreases, and is < 5% of F-[ - FQfor z/X < 0.01, which conforms with the micrometeoro-logical rule-of-thumb that the measurement height shouldbe < X/100 to ensure accurate measurements. This rationeeds to be increased for stable conditions and maybe relaxed under unstable stratification (Kaimal andFinnigan 1994). The flux footprint schemes of Schmidand Oke (1990) and Schuepp et al. (1990) and the workof Mulheam (1977) provide a more detailed analysisof the sensitivity of flux measurements to advectiveconditions.

Instrumentation must have a suffidently high fre-quency response to measure all the turbulent fluctuationsof w and s that contribute to the flux (Eq. 3), and theproduct of w and s must be formed over a suitable lengthof time to capture the low frequency parts of the spectrum.Errors assodated with the instrumentation have beenwidely studied as they are more easily quantified andamenable to correction. Errors associated with the threemain compranents of an eddy covariance system, namelysonic anemometer, gas analyser, and software have beenpublished frequently, usually as a description of a particu-lar system. Examples for sonic anemometers include thepapers by Kaimal and Gaynor (1991) and de Bruin et a!.(1993); Businger and Delany (1990) present nomogramsfor the chemical resolution required of a sensor to achievea certain level of accuracy when used in an eddy covari-ance system; for software see Shuttleworth (1988). Adiscussion of errors associated with open- and closed-path sensors for measuring trace gas concentrations ispresented by Leuning and Judd (1996, this volume, p.241).

It is implidt in Eq. (3) that fluctuations in the verticalwind and scalar are measured at the same point in space.This is generally impossible physically and often leadsto underestimates in the measured flux because thecorrelation coeffident r^s decreases with increasinginstrument separation (Eq. 4). Inadequate instrumentfrequency response, caused by averaging along finite

path-lengths of sonic anemometers or by air flow throughtrace gas analysers, also reduces the measured fluxthrough underestimates of the standard deviations ineqn (4). Corrections to the measured flux may be estim-ated provided spectral transfer functions can be definedfor any particular set of instrumentation, including hard-ware and processing software (Moore 1986; Moncrieffet al. 1996). Such correction terms of course, have uncer-tainties assodated with them as they are usually basedeither on engineering formulae which are partly empiricalor rely on similarity theory which is itself dependent onatmospheric and surface properties which may be difficultto spedfy exactly. Nonetheless, such transfer functionsdo permit the approximate correction of flux estimatesfor a wide variety of error terms assodated with usingthe inevitable compromises assodated with field instru-mentation. However, even when seemingly identical eddycovariance sensors are intercompared at an ideal site,random errors tire present and this generally sets a lowerlimit of several percent on the overall accuracy of themeasurements (Moncrieff et al. 1992).

Most micrometeorologists accept that variability inthe natural environment (atmosphere and surface) andinstrumentation errors restricts the accuracy of an indi-vidual turbulent flux measurement to between 10 and20% (Wesely and Hart 1985). Baldocchi et al. (1988) pointout that, rather than having fixed guidelines about whereand for how long measurements should be made, therehas grown up a number of rules-of-thumb. One of themost widely quoted of these guidelines is concerned withthe degree of departure possible from level terrain. Insloping ground, an apparent mean vertical velodty maybe recorded by the sonic anemometer unless the co-ordinate system is rotated to ensure the turbulent flux iscalculated with respect to the perpendicular to the localstreamlines (Dyer et al. 1982; McMillen 1988). It is gener-ally accepted that once coordinate rotation of the axes isperformed, it is possible to make reliable measurementson slopes of up to about 15%. Dyer et al. (1982) pointed

© 1996 Blackwell Sdence Ltd., Global Change Biology, 2, 231-240

234 j . B . M O N C R I E F F etai

ao

U

Chamber

Micro mel

Fig. 2. Cumulative carbon uptake by scaled-up chamber measurements and by eddy covariance (labelled micromet in the figure) atan old black spruce stand during BOREAS (rc-drawn from Jarvis et al. 1995).

out that bit errors for momentum and scalar Suxes aredifferent, i.e. momentum flux would be in error by about14% per degree of tilt but only about 3% per degree oftilt for scalar fluxes.

When tissessing the representativeness of point meas-urements of fluxes in the surface boundary layer it canbe difficult to differentiate between an uncertainty causedby the variability of natural surfaces and uncertaintiescaused when there is no independent check of the meas-urements available, as is often the case. For the sum offluxes of sensible and latent heat it is possible to attemptto close the energy balance over a daily cycle or overseveral days, by comparing eddy covariance measure-ments against independent measurements of net radi-ation. Satisfactory closure can give confidence in theoverall measurement accuracy of the whole system (Lloydet al. 1984). This may not always work, however, interrain which is extremely variable in surface cover asthe sources and sinks for heat, water and energy mayinteract in a non-linear fashion (Lloyd 1995). For fluxessuch as carbon dioxide it is difficult to conceive of anysort of closure statistic which could be used in a similarfashion to energy-balance closure. One possible way toclose the carbon balance of a stand would be to measureall the separate components contributing to the net flux,i.e. leaf photosynthesis, soil and stem respiration andthen scale them up. Figure 2 shows a comparison betweenthe net CO2 flux as measured by an eddy covariancesystem over an old black spruce stand during theBOREAS exfjeriment of 1994, and the sum of the simultan-eous scaled-up chamber measurements made on the soil,leaves and stems (Jarvis et al. 1995). The agreement isgood but, presumably, the errors which occur in thescaling up of chamber measurements will account for alarge part of the discrepancy.

si

Ou

true flux^ ^ " measured flux

^ A T'

^ V ^

V

Randome-g- stochastic natureof one-point samplingvarying flux footprint

Time

FullySystematic

e.g. consistently missinghigh and/or low frequencycomponents of cospeclnim

Time

ou

Time

Selective-Systematic

e.g. under-reading ofnighl-Eime fluxes becauseof drainage flows, differentturbulence spectra

Fig. 3. An illustration of the main types of error: random, fullyand selective systematic errors with examples of the processeswhich might bring them about.

1996 Blackwell Sdence Ltd., Global Change Biology, 2, 231-240

ERRORS IN LONG-TERM FLUX MEASUREMENTS 235

Table 1 Examples of random and systematic errors in an eddycovariance system

Random Systematic

one-point sampling ofturbulence;

varying size of flux footprintand surface heterogeneity;

inadequate length ofsampling interval;

random noise in the signal;

inadequate sensor response;

limited fetch andnonstationarity;

consistent over- or under-reading of fluxes bycalibration error;use of incorrect spectralforms in calculation oftransfer functions;under reading of flux at nightbecause of katabatic flowbelow sensor;incorrect application of thecorrections due tosimultaneous fluxes of heatand water vapour {the Webb,Pearman and Leuning (1980)correction);

inadequate sensor responseor fiow distortion;inadequate height above thesurface;

3 Random and systematic error

For our purposes, errors can loosely be divided intorandom errors and systematic errors, and the essentialdifferences as they impact on flux studies is shownschematically in Fig. 3 with examples of the type ofmechanism which brings about each type of error. It isaccepted that in reality, however, any particular errormay be a combination of both types. Systematic errorscan be fully systematic errors (errors that apply on all thediumal cycle) or selectively systematic errors (errors thatapply to only part of the daily cycle), which, as we shalldemonstrate, can have very different effects. Randomerrors may also be full or selective, but these do notdiffer substantially in their properties. Table 1 lists theerror terms which were illustrated in Fig. 1 as eitherrandom or systematic.

When errors are random, errors in estimated meansand variances diminish with increasing size of data setaccording to 1 / VW, where N is the number of data points(Barlow 1989). Thus random errors can be detected,estimated, and minimized by examining the convergenceof calculations of the net flux with increasing size of dataset (as long as the data set is not so enlarged that, forexample, seasonal trends become important). In contrast,systematic errors are not affected by increasing data setsize ('multiplying your mistakes produces no reward').This is because, being persistent offsets or multipliers tothe data, they simply add in a linear fashion. These errorscan be very difficult to detect.

For the daily cycles illustrated in Fig. 4, we can apply


E 5 - -

22 24

Fig, 4. The mean daily flux density of carbon dioxide (a) andwater vapour (b) obtained from 44 days of data in Rondonia asdescribed by Grace ek ai (1995). The dashed lines are upper andlower quartiles. The x-ajris is local time.

a random error, p^, to each half-hour datum. The randomerror in the mean flux for one mean daily cycle is

= PrN

where N is the number of flux measurements In one day,and F, is the ith flux measurement of the daily cycle. Fora data set consisting of several days of data, the randomerror is further reduced, such that E,(Nd) = Er{l)/VN2where Nij is the number of days.

If instead we apply a systematic error, p,, to each half-hour datum, the systematic error on one mean diumalcycle is simply the sum of the individual systematicerrors, such that

This systematic error is identical to the overallsystematic error on the data, Es(N^), irrespective of the

236 J . B . M O N C R I E F F etai.

1.0

•r 0.5en

C-0.5

O

-2.0

\1\ ,

Mean Daily Rux

1

. . . - - " " • I - . -

I...

0.1 1 10 100

Number of Days' Data, W

size of flie data set. In the case of a selective systematicerror, eqn (9) should only be applied to the relevantportion of the daily cycle.

We have only considered fractional systematic errorshere, as these are the type most likely to be encounteredin field measurements (e.g. persistent non-measurementof a certain fraction of the flux). However, it is conceivablethat some systematic errors may be fixed rather thanpercentage offsets, and a similar analysis ctin be appliedin that situation.

As part of a review of measurements of dry deposition,Businger (1986) presented a table of the magnitude oftypical random and systematic errors assodated with sixeddy covariance systems reported in the literature. Heidentified ID sources of error, most of which have beendiscussed in relation to Fig. 1. Although there was quiteconsiderable variability in whether or not some errorterms had been accounted for in different systems, ingeneral systematic errors on the order of 30% or moreand random errors of the same magnitude were found.

4 Case study

How do errors in flux estimates for individual half-hourperiods translate into those for a week, a month or ayear? To examine this question we shall take some datafrom a field campaign in the Brazilian Amazon reportedby Grace et al. (1995, 1996), where details of the measure-ments can be found. Our task is to establish the level ofconfidence in the net flux and to decide whether thereported long-term sink strength was real or whether itcould have come about through some combination ofuncertainties assodated with the methodology. This casestudy will allow similar assessment of results from otherfield studies. The data used here were collected during

5%

10%

20%

50%

1000

Fig. 5. The effect of applying randomerrors of varying magnitudes on themean daily CO^ flux for the Rondoniadata set. The solid, horizontal line isthe mean, net daily CC)2 flux of-0.92 ^ ^

1.0

0.5 -

„ 0.0

E -0.5 - - -

-t.O -

-1.5 - -

-2.0 - -

-2.5 -

-3.0 -

-3.5 - -

-4.0

. , . - . 1 . - ^ J

I I

Mean Daily Flui

..-A-.-i 4 ) f.

1 .^ 1,_ _ ..^ j _i- t

. , J ,

T

Fig.

-100 -80 -60 -40 -20 0 20 40 60 80 100

Percentage Systematic Error, p,

6. As for Fig. 5 but for fully systematic errors.

April and May 1995 over tropical rain forest in Rondonia,Brazil. Half-hour averaged values of turbulent fluxes ofheat, water vapour and carbon dioxide were measuredcontinuously over a period of 44 days using the Universityof Edinburgh's EdiSol eddy covariance system (Moncrieffet al. 1996). The measurements indicated that over thisperiod, there was a net flux of carbon dioxide into theforest of about 0.92 ^unol COj m ' V (= 0.95 g C m'^d"'), suggesting that the forest was a substantial sinkof carbon.

We will draw attention in particular to the effects oferrors that are important during one phase of the dailycycle (e.g. at night). To examine these and other types of



error, a useful technique is to examine the properties ofthe mean daily cycle, constructed by taking the meanaccording to time of day of the entire data set. The netflux over the mean daily cycle is directly proportional tothe overall net flux over the measurement period, andby introducing 'errors' at specific times of day one caninvestigate the effects of these errors on the overall dataset. This approach is valid as long as the basic characterof the daily cycle does not vary significantly from day today. An alternative approach which has been used on amuch longer dataset, is described by Goulden et ai (1996).

4.1 Diumal cycles of fluxes of carbon dioxide andwater vapour far the Ronddnia 1994 data set

The mean daily cycle for carbon dioxide fluxes is shownin Fig. 4a. The upper and lower quartiles of the data arealso plotted, giving an indication of the day-to-dayvariability in each phase of the daily cycle. The flux isnegative (i.e. into the canopy) during daytime, whenphotosynthetic uptake dominates over respiration, andpositive at night, when respiration, particularly from thesoil, is the only significant process. At night the turbulentflux is partially inhibited by the stable stratification ofthe atmospheric surface layer above the canopy; thusthere is an overnight accumulation of carbon dioxidewithin the canopy, a large proportion of which is 'flushedout' after sunrise with the onset of convection, resultingin the prominent 'spike' in the CO2 flux between 07.00and 09.00 hours. The magnitude of the spike variessignificantly from day to day, being dependent on theamount of within-canopy CO2 accumulation, and thuson the overnight wind speed and stratification.

The mean daily turbulent flux is the average of daytimeand night-time fluxes of approximately the same magni-tude and opposite direction. In this case the mean fluxis -0.92 fimol COT m " V , significantly smaller than mostof the individual flux measurements. It is an occasionalmisconception that each value of the CO; flux is measuredas the difference between the net upward and downwardfluxes (i.e. a small difference between large terms), whichwould have drastic consequences for sensitivity to errors.While this is true in terms of the competing biologicalprocesses (photosynthesis and respiration) generating thenet flux, this is not true for any actual measurement ofnet flux. The mean daily flux is the sum of individual(e.g. half-hourly or hourly) flux measurements, each withan associated error term.

The mean daily cycle for fluxes of water vapour isillustrated in Fig. 4b. As before, the upper and lowerquartiles of the data are also plotted. The flux is largeand positive during daytime, when evapotranspirationis driven by solar heating of the canopy, and muchsmaller but still positive at night, when there is no solar

-4.0 -

Daytime Error OnlyNight-time Error Only

-100 -80 -60 -40 -20 0 20 40 60 80 100

Percentage Systematic Error, p.

Fig. 7. The effect of applying selective systematic errors (by dayand by night) of varying magnitudes on the mean daily CO;flux for the Rondonia data set. The solid, horizontal line is themean net daily CO2 flux of -0.92 |imol m" s^'.

heating and, moreover, turbulent fluxes are inhibited bystable stratification. The mean daily water vapour fluxout of the canopy is 1.64 mmol H2O m" s"', correspondingto 2.55 kg m" d"^ Thus, for water vapour the net dailyflux is the sum of large daytime and small night-timefluxes and is effectively unidirectional. The mean fluxis therefore approximately half the magnitude of thedaytime fluxes.

4.2 Consequences of each type of error

4.2.1 Carbon dioxide fluxes. Figure 5 shows the results ofapplying various values of percentage random error, p,,to each half-hour CO2 flux of the Rondonia data set. Thesolid horizontal line represents the net daily carbondioxide flux measured at the site, -0.92 ^mol CO2 m~s'^ The symmetrical sets of lines above and below thisline represent the overall random error on this net flux,dependent on the magnitude of p^ and the number ofdays data collected. The magnitude of the random errorsdecrease with increasing data set size, but at an ever-decreasing rate (Eq. 8). This graph allows an estimationof how many days' data are required to ensure that ameasured net CO2 balance is not an artifact of randomerrors. For example, assuming 20% random errors oneach half-hour data point, we can see that 1 day's datawould not be sufficient to resolve a net sink of 0.92 |imolCO2 m" s"V On the other hand, 10 days' data wouldresolve a net sink of about 0.4 jimol CO2 m"^"V and 100


238 J.B.MONCRIEPF etai

3.0

2 .5 - . ,

2.0 - =

u. 1.5 --

0.0

\\

\

U.

I

•r'

-y-/

0.5 H / -

/0.1 1 10 100

Number of Days' Data. N^

days' data would resolve a net sink of 0.2 ^mol CO2m"^"'. In the particular case of the Ronddnia data set,with 44 days' data and an assumption (possibly over-pessimistic) of 20% random errors per half-hour value,the overall random error on the net carbon dioxide fluxis 0.40 ^mol CO2 m"V^ (i.e. 44% random error).

Figure 6 shows the consequences of applying variousmagnitudes of percentage systematic error, p^, to the samedata set. As before, the solid horizontal line representsthe measured mean CO2 flux. The dashed line representsthe actual net CO2 flux (vertical axis), for a given percent-age error, p^ (horizontal axis), on each measured half-hourflux. For example, if there is a tendency for each half-hour flux measurement to underestimate the actual fluxby 20% (i.e. p^ = -20%), and the measured mean flux is-0.92 imol CO2 m-2 s"', then the actual net flux is-1.23 ^mol CO2 m~^s"^ (i.e. there is a net systematic errorof -26%).

Note that, in the case of CO2 fluxes, full systematicerrors are not as severe in their consequences as mightbe expected. This is because the CO2 flux has oppositesigns by day and night; an over- or underestimation ofdaytime flux is to some extent compensated for by anover- or underestimation of night-time flux. The shapeof the plot in Fig. 6 is very sensitive to the asymmetrybetween day and night of the diumal cycle shown inFig. 4; if the cycle were totally symmetrical, the netsystematic error would always be zero. Another featureto note is that a percentage full systematic error on itsown cannot force the net CO2 flux to change sign (e.g. itcannot make an actual CO2 source appear as an apparentCO2 sink).

While partial synnmetry reduces the impact of fullsystematic errors, this Is not the case for selective system-atic errors that exist for only part of the daily cycle.

Pr

5%

10%

20%

50%

1000

Fig. 8. The effect of applynng randomerrors of varying magnitudes on themean daily HjO flux for the Ronddniadata set. The solid, horizontal line isthe mean, net daily H2O flux of1.64 mmol m~ s"

Figure 7 shows the consequences of applying variousmagnitudes of selective percentage systematic error, p^to the data shown in Fig. 4. The thin, solid line indicatesthe actual net flux if a given percentage systematic errorapplies only to night-time fluxes, and the dotted lineindicates the same if the error applies only to daytimefluxes.

The slopes of the lines are greater than in Fig. 6,indicating the greater importance of selective systematicerrors. Another notable feature is that the lines crossthe zero-axis; thus a selective systematic error can, forexample, convert an actual net CO2 source into an appar-ent net CO2 sink. As another example, if the actual CO2flux balance of a surface is zero, the apparent measuredCO2 sink of 0.92 ^mol CO2 m" s"' could be generatedby underestimation of night-time flux by about 40% (by,for example, the presence of lateral within-canopy flow),or by overestimation of daytime fluxes by 60% (by, forexample, the presence of an anomalous CO2 sourceupwind). An analysis such as this gives an estimate ofthe magnitude of the possible error factors that need tobe searched for.

4.2.2 Water vapour fluxes. Figure 8 illustrates the influenceof random errors on the measurement and calculation ofnet water vapour flux. Because the random errors oneach half-hour flux measurement are generally smallerthan the mean water vapour flux, the overall randomerrors are much smaller than for CO2. In the particularcase of the Rondonia data set, with 44 days' data andthe assumption of 20% random errors per half-hourvalue, the overall random error on the net water vapourflux is 0.21 mmol H2O m" s~ (i.e. 13% random error, asignificantly smaller percentage than for the CO2 flux).

The daily cycle of the water vapour flux is unidirec-

1996 Blackwell Sdence Ltd., Global Change Biology, 2. 231-240


tional and highly asymmetrical between day and night.Systematic errors can be important, and are simply thesame percentage of the net flux as p , the percentage erroron each individual half-hour measurement. Thus, forexample, a systematic underestimate of the flux by 10%in each half-hour flux measurement (due to, say, non-measurement of the high-frequency component of theflux) results in a systematic error of -10% on the valueof the flux.

As the night-time flux of water vapour is negligiblecompared to the daytime flux, any selective systematicerror occurring only at night (e.g. the presence of drainageflows) would have little effect on the net water vapourflux. For the same reason, any selective systematic errorbiased towards daytime will have exactly the sameinfluence as a full systematic error.

5 Discussion

In any long-term flux measurement, a sensitivity analysisof the effects of errors on an averaged day can be a keytool in assessing the likelihood of certain errors producingspurious results. Thus, for the Rondonia data set, if weallow for a 20% random error on each half-hour datavalue, the total random error on the net flux is 0.40 pjnolCO2 m" s" (i.e. 44% of the net flux). If we also allow forsystematic errors of 10% and that they may be selectiveby day or night, we see from Fig. 7 that the totalsystematic error on the net flux is 0.25 jimoi CO2 m'^ s"'(i.e. 27%). Adding these two errors in quadrature, thetotal uncertainty of the net flux is 0.47 ^mol CO2 m^^s"'(i.e. 53% of the total net flux). Thus we can conclude thatthe observation that the forest appeared to be a net sinkof CO2 over the experimental period was valid.

Using the error assessments shown in Figs 4-8, it ispossible to establish (i) how many days' data are neededbefore random errors are likely to be significantly smallerthan the mean measured net flux; (ii) the magnitude ofnocturnal under-measurement of flux that could lead toa spurious apparent sink term and whether this magni-tude of underestimation is likely; and (iii) whether errorsin high-frequency flux corrections have a significantimpact on the overall result.

The relationship between the errors on individual half-hour flux measurements and the net flux to or from asurface depend strongly on the shape of the daily cycleof the flux, in particular on the uru- or bidirectionalty ofthe flux and on the degree of symmetry between thedaytime and night-time fluxes. Therefore, this relation-ship is very different for CO2 and water vapour fluxes.For daily variation in fluxes that are approximatelysymmetrical about zero (as for CO2), full systematic errorspartially cancel. Therefore, they are not as important asmight be expected. This is not the case for heat or water

vapour fluxes where fluxes are generally much largerduring the day than at night. Partial systematic errors (e.g.under-reading of night-time fluxes because of drainageflows, or different turbulence spectra) do not cancel foreither quantity. Therefore, they are potentially the mostimportant type of error and should be considered withthe greatest attention when planning any field experimentor analysing any field data.

For carbon dioxide fluxes, tihe net daily flux is thedifference between daytime and night-time fiuxes inopposite directions. Therefore, it is essential to calculateerror estimates when presenting any values for the netcarbon balance of a surface. It is a common conventionin physical statistics to quote values as X ± a ±b,where X is the mean value of the result, a is the estimateof stochastic (random) uncertainty, and fe is an assessmentof systematic uncertainty because of possible systematicerrors in the system. This might be a good convention toadopt in quoting results from studies employing eddycovariance systems.

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The Propagation of Errors in Long-term Measurements