Consequences of Incomplete Surface Energy Balance Closure for CO2 Fluxes from Open-Path CO2/H2O Infrared Gas Analysers

We present an approach for assessing the impact of systematic biases in measured energy fluxes on CO2 flux estimates obtained from open-path eddy-covariance systems. In our analysis, we present equations to analyse the propagation of errors through the Webb, Pearman, and Leuning (WPL) algorithm [Quart. J. Roy. Meteorol. Soc. 106, 85–100, 1980] that is widely used to account for density fluctuations on CO2 flux measurements. Our results suggest that incomplete energy balance closure does not necessarily lead to an underestimation of CO2 fluxes despite the existence of surface energy imbalance; either an overestimation or underestimation of CO2 fluxes is possible depending on local atmospheric conditions and measurement errors in the sensible heat, latent heat, and CO2 fluxes. We use open-path eddy-covariance fluxes measured over a black spruce forest in interior Alaska to explore several energy imbalance scenarios and their consequences for CO2 fluxes.


Introduction
The eddy-covariance technique has been widely used to directly measure energy, water vapour, and carbon exchange between terrestrial ecosystems and the atmosphere (Aubinet et al., 2000;Baldocchi et al., 2001). A critical issue with this approach is to quantify the uncertainties related to longterm eddy-covariance measurements of carbon exchange so that reliable assessments of carbon sources or sinks can be made over time scales ranging from hours to years (Goulden et al., 1996;Massman and Lee, 2002;Baldocchi, 2003). In the micrometeorological community, the evaluation of surface energy balance closure is accepted as an important procedure in assessing data quality (Aubinet et al., 2000;Wilson et al., 2002). However, direct measurements of sensible (H ) and latent (LE) heat fluxes, net radiation (Rn), soil heat flux, and storage almost always indicate an incomplete energy balance closure (Aubinet et al., 2000;Oncley et al., 2002;Wilson et al., 2002;Culf et al., 2004). Experimental evidence has also shown that the energy imbalance (typically ranging from about 5 to 30%) and its causes are site dependent (Aubinet et al., 2000;Twine et al., 2000;Oncley et al., 2002;Turnipseed et al., 2002;Wilson et al., 2002). It is generally accepted that methodological and instrumental problems exist that cause energy flux loss, especially over non-ideal surfaces (Foken and Oncley, 1995;Lee, 1998;Panin et al., 1998;Finnigan, 1999Finnigan, , 2004Massman and Lee, 2002;Baldocchi, 2003;Finnigan et al., 2003). Considerable work has gone into seeking corrections for these flux losses (Webb et al., 1980 -WPL;Moore, 1986;Kramm et al., 1995;Massman, 2000;Paw U et al., 2000;Fuehrer and Friehe, 2002;Massman and Lee, 2002;Foken, 2003, 2004;Liu, 2005;Massman, 2004). Since CO 2 fluxes are also measured by the same system that often underestimates sensible and latent heat fluxes, several studies have addressed possible links between surface energy imbalance and uncertainties in estimates of CO 2 fluxes (Twine et al., 2000;Wilson et al., 2002). Twine et al. (2000) provide experimental evidence suggesting that the magnitude of CO 2 fluxes measured using closed-path H 2 O/CO 2 infrared gas analysers (IRGA) may be underestimated by the same factor as latent heat fluxes. In contrast, other investigators argue that the magnitude of CO 2 fluxes using open-path IRGAs are largely overestimated because sensible heat flux, and thus the WPL algorithm associated with this flux, is underestimated (Anthoni et al., 2002b). Alternatively, using the data measured by both open-path and closed-path IRGAs at 22 sites in FLUX-NET, Wilson et al. (2002) proposed that CO 2 fluxes are underestimated when turbulent fluxes are underestimated relative to available energy. As illustrated by these contrasting findings, uncertainties in our understanding of the links between surface energy imbalance and CO 2 fluxes still remain.
Despite the large differences in the types of measurement errors associated with open-path and closed-path IRGAs (Baldocchi, 2003), the relationship between incomplete energy balance closure and the uncertainties of CO 2 fluxes has not been separately and quantitatively addressed in previous studies using open-path and closed-path systems.
There are two primary types of errors embedded in the measurements of CO 2 fluxes with an open-path IRGA. The first is the measurement error in raw CO 2 fluxes caused by multiple factors (Goulden et al., 1996;Wilson et al., 2002), which may be caused by the uncertainties in instrument calibration or errors resulting from methodological aspects such as loss of high and/or low frequency contributions to fluxes, or the neglect of horizontal advection, vertical divergence, or storage terms (Wilson et al., 2002). The second type of error arises from the propagation of systematic or random errors in raw sensible heat flux (w T ) (w T is actually the kinematic heat flux), latent heat flux (w ρ v ) (w ρ v is actually the kinematic moisture flux), and CO 2 flux (w ρ c ) through the WPL algorithm. When open-path IRGAs are used in eddy-covariance systems to measure CO 2 flux (hereafter, raw CO 2 flux), the WPL algorithm (Webb et al., 1980) is usually applied to account for the density effects caused by heat and water vapour transfer to obtain corrected CO 2 fluxes (hereafter, final CO 2 flux). Since the WPL algorithm relates the final CO 2 fluxes to the raw eddy-covariance fluxes, any errors in the raw sensible and latent heat fluxes as well as raw CO 2 fluxes will be propagated through the WPL algorithm to generate final CO 2 fluxes. Hereafter, this type of error is referred to as a propagation error. Although this additional propagation error has potential impacts on CO 2 fluxes, it has received little attention (Anthoni et al., 2002b;Baldocchi, 2003). Instead, it is common that the raw measurement errors are used in discussing errors of final CO 2 fluxes or assessing uncertainties in long-term net ecosystem exchange (NEE) regardless of the difference in IRGA types (i.e. open-path and closed-path IRGAs).
The interaction of measurement errors of raw CO 2 fluxes directly related to incomplete energy balance closure and propagation errors indirectly related to incomplete energy balance closure through the WPL algorithm presents a challenge to assessing the precision and accuracy of final CO 2 fluxes and the error sources. Understanding the role of propagation error is important because such errors could lead to large biases in the estimation of long-term NEE. Here we present an approach for assessing the influence of measurement errors in energy balance on CO 2 fluxes through the WPL algorithm when open-path IRGAs are used. We then illustrate the role of these propagation errors using eddy-covariance data measured from a flux tower in interior Alaska. We show, in a quantitative way, that errors (or biases) in the measurement of eddy-covariance fluxes do not necessarily lead to proportional errors in the CO 2 fluxes computed through the WPL algorithm; instead, the impact of these errors depends strongly on the magnitude of the sensible heat flux.

Theoretical Considerations
When an eddy-covariance system with an open-path IRGA is used for measuring water vapour and CO 2 fluxes, the WPL algorithm is usually applied to account for the density effects. We rewrite the WPL algorithm as follows (Webb et al., 1980), where w T , w ρ v , and w ρ c are the sensible heat flux, latent heat flux, and CO 2 flux measured by eddy-covariance systems, respectively. Although other corrections (e.g., a frequency response correction) may be made to these fluxes, we still refer them as the raw fluxes. E and F c are the latent heat flux and the CO 2 flux after application of the WPL algorithm (hereafter, referred as to the final fluxes), respectively;ρ c ,ρ a , andρ v are the densities of CO 2 , dry air, and water vapour, respectively; µ = m a /m v and σ =ρ v /ρ a ; m a and m v are the molecular mass of dry air and water vapour, respectively. T is the air temperature. Differentiating Equations (1) and (2), we obtain Note from Equation (3) that a change in final latent heat flux (δE/E) is sensitive to changes in raw latent heat fluxes ω v = δw ρ v /w ρ v and to changes in raw sensible heat fluxes ω T = δw T /w T . Equation (4) indicates that a change in the final CO 2 fluxes (δF c /F c ) is related not only to a change in raw CO 2 fluxes (ω c = δw ρ c /w ρ c ) but also to a change in raw sensible and latent heat fluxes (i.e. ω T and ω v ).
These equations allow for an assessment of the impact of systematic and random errors incurred in measurements of raw sensible and latent heat fluxes and CO 2 fluxes on the final CO 2 fluxes (as propagated through the WPL algorithm).
We identify the potential magnitudes of raw systematic errors for our open-path eddy-covariance system (i.e. ω T , ω v and ω c ) through evaluation of energy balance closure and other techniques in Section 4.1. We then make three case studies to further explore consequences of error propagation through the WPL algorithm for the CO 2 , NEE, and latent heat flux estimates obtained from our site (Sections 4.2,4.3,and 4.4). Finally, in Section 4.5, we explore how random errors may be amplified or damped through the WPL algorithm, specifically examining the effect of coordinate rotation.

Site Description and Instruments
Our experimental study was conducted at a black spruce forest near Delta Junction (63 • 54 N, 145 • 40 W) located about 100 km south-east of Fairbanks, Alaska. This site was dominated by relatively homogeneous old black spruce stands (Picea mariana) about 80 years old; the mean canopy height was approximately 4 m above the surface , and the canopy coverage was about 60%. The terrain was generally flat on glacial outwash in the Tanana River drainage of interior Alaska (Manies et al., 2004), and the fetch exceeded 1 km to the south, west, and north, and was approximately 200 m to the east. The dominant ground cover species were feathermoss (Pleurozium schreberi and Rhytidium rugosum) and lichens (Cladonia spp. and Stereocaulon spp.) that reached a depth of 0.1-0.15 m .
Eddy-covariance and microclimate measurements were made at this site during , and the 2002 summer data were used here. The eddy-covariance system consisted of a three-dimensional sonic anemometer (model CSAT3;Campbell Scientific,Inc.) and an open-path CO 2 /H 2 O IRGA (model LI-7500; LI-COR, Inc.) that were mounted at a height of 9.5 m (more than twice the mean canopy height). The sonic anemometer was used to determine wind velocity and sonic temperature fluctuations, and the IRGA was used to measure the fluctuations of water vapour and CO 2 density. Water vapour calibration was made regularly using compressed air with a magnesium perchlorate desiccant as a zero gas and output from a dew point hygrometer (model LI-610, LI-COR, Inc.) as the water vapour span. CO 2 calibration was made by passing a standard CO 2 gas as a span and passing compressed air through desiccants and a scrubber (soda lime) as a zero reference. The distance between the IRGA and the sonic anemometer was about 0.25 m .
Signals from the sonic anemometer and the IRGA were sampled at 10 Hz, and fluxes of sensible heat (H ), latent heat (LE), and CO 2 (F c ) were calculated using the 30-min covariances of vertical wind velocity with virtual temperature, water vapour density, and CO 2 concentration. The planar fit method was adopted for rotating the coordinate system to streamline (Wilczak et al., 2001), and the fluxes were computed using a block average rather than a linear detrending method. Because the sonic temperature is used, and because the crosswind effect should be taken into account (Schotanus et al., 1983;Kaimal and Finnigan, 1994), we converted the buoyancy flux derived directly from the sonic temperature to sensible heat flux according to Liu et al. (2001). Latent heat and CO 2 fluxes obtained by the open-path IRGA were calculated following Webb et al. (1980). Data were rejected when winds were blowing through the tower or from the east, or when the data did not pass a quality check (Foken and Wichura, 1996;Foken et al., 2004).
We also measured microclimate variables as 30-min averages of 1-s readings. These variables included net radiation (Rn; model Q-7.1, Radiation and Energy Balance Systems (REBS), Inc.), incoming and reflected shortwave radiation (model Precision Spectral Pyranometer, Eppley Laboratories), and photosynthetic photon flux density (model LI-190SB, LI-COR, Inc.). Air temperature and relative humidity were measured at 10, 6 and 2 m with temperature/humidity probes (model HMP45C, Vaisala, Inc.). We estimate the heat storage in the canopy air space (S a ) using the air temperature measurements (HMP45C) at 2 m and a mean canopy height of 4 m. A 3-cup anemometer and wind vane (model 03001, RM Young, Inc.) was mounted at 12 m to measure wind speed and wind direction, and an additional wind speed sensor (Model 03101, RM Young, Inc.) was mounted at 6 m to measure wind speed. Additionally, thermocouples and soil moisture probes (model CS615, Campbell Scientific, Inc.) were buried at 0-0.34 m below the surface and soil heat flux (S) plates (model HFT3, REBS, Inc.) were buried at 0.1 m below the surface in three microenvironments near the tower, including one in an open patch and another under a shrub. The 30-min heat storage (S) in the soil above the soil heat flux plates was estimated using the combination method (Oke, 1987), using volume fractions of moss/lichen community mineral, organic and water content of the surface soil for calculating surface heat capacity (Beringer et al., 2001;Chambers and Chapin, 2002).

Eddy-covariance fluxes and closure of the surface energy budget
Closure of the surface energy budget is commonly accepted as an important factor in assessing the quality of eddy-covariance data. We used a straight-line regression with errors in both coordinates (i.e. H + LE vs. . This site has a slope of 0.85 and a zero offset of 2.8 W m −2 (r 2 = 0.87, n = 3302) before application of the frequency response corrections (denoted with circles). The slope increased to 0.89 and the offset changed to 3.6 W m −2 (r 2 = 0.88, n = 3302) after application of the frequency response corrections (denoted with crosses).
above the soil heat flux plates, S a : heat storage at the canopy air space) to estimate the slope and intercept of the regression line (Press et al., 1992). Using the data from the 2002 summer season from June 15 to August 15, the slope and intercept between H + LE and Rn − G − S − S a were 0.85 and 2.8 W m −2 respectively for our boreal forest site (r 2 = 0.87) (Figure 1). Generally, the closure of the energy budget was higher during the day (0.86) and lower during the night (0.70). The ratio of the sum of H + LE and the sum of Rn − G − S − S a over the whole period is considered as an alternative method to evaluate closure (Wilson et al., 2002). This ratio for our site was 0.83. All eddy-covariance systems encounter attenuation of turbulent fluxes at high and/or low frequencies (Moore, 1986;Massman and Lee, 2002). Based on the theoretical approach described by Moore (1986), Eugster and Senn (1995), and Horst (1997Horst ( , 2000, we calculated the diurnal variation in the correction factors for sensible heat flux, latent heat flux, and CO 2 flux to account for flux losses due to sensor separation, line average, and time  response ( Figure 2). In general, the spectral correction factor for latent heat and CO 2 fluxes was slightly greater than that for sensible heat fluxes. During the day, spectral response loss decreased latent heat and CO 2 fluxes by about 5% while it decreased sensible heat fluxes by about 4%. The magnitudes of these correction factors are comparable to the estimates by Leuning and Judd (1996) and Chambers and Chapin (2002), who used closed-path IRGAs that encountered high-frequency losses due to tube attenuation of turbulent fluctuations. The dominant part of the turbulent spectrum shifted from a low frequency range (about 0.005-0.05 Hz) during the day to a higher frequency range (about 0.01-0.5 Hz) during the night when the atmosphere was stable. This led to larger spectral correction factors at night (about 12% for latent heat and CO 2 fluxes and about 7% for sensible heat fluxes) due to high frequency loss (Figure 2). After frequency response corrections of the raw flux data, we obtained a slope and intercept of the regression line between corrected H + LE and Rn − G − S − S a of 0.89 and 3.6 W m −2 respectively (r 2 = 0.88) (Figure 1). We did not measure the canopy heat storage (in boles and other components of the above ground vegetation) although we believe this term to be small because of the relatively low height and sparse nature of the canopy (with a mean height of 4 m and mean surface coverage of 60%). Taking account of heat storage in the canopy air space (S a ) accounted for a small increase in the slope. Combining the frequency response losses and heat storage in the canopy air appeared to account for some, but not all, of the energy loss identified in our analysis of the energy budget closure. Other unidentified error sources must contribute to the remainder of the imbalance.
A lack of closure of the surface energy budget by 10% or more is not uncommon at eddy flux sites that are used to study ecosystem processes in remote areas under conditions that are not perfectly ideal. The causes for this remaining imbalance in the surface energy budget (after application of all appropriate corrections) still remain unclear (Wilson et al., 2002).
One possibility is that these heterogeneities introduce both horizontal and vertical advective flow terms that are not resolvable with our single point vertical flux tower. If these advective terms contribute to vertical fluxes at our site, non-closure of the surface energy balance would be inevitable even though we made the appropriate adjustments for high/low frequency losses and storage. Another possibility could be the effect of the roughness sublayer on the flux measurements. For our site with the mean canopy height of 4 m , the roughness sublayer with its height of about one to two times the mean canopy height (Garratt, 1994) might have a small impact on our measurements at 9.8 m.
The intercomparison experiment with different eddy-covariance systems from six groups in the Energy Balance Experiment (EBEX) indicates that the uncertainties for the sensible and latent fluxes could be reduced to around 5% when the eddy-covariance systems are well calibrated and maintained, as well as the measurement field being well selected to meet the requirements of the similarity theory (Mauder et al., submitted to B.L.M.). Using the same dataset, but with different post-field data processing methods from the EBEX participants, gives the sensible and latent heat fluxes with their uncertainties of 10 -15% as a result of the difference in correction methods and physical parameters used (Mauder et al., ibid). Although all terms of the energy budget are taken into consideration in the EBEX, including net radiation, sensible and latent heat fluxes, heat storage by plant and air mass within the canopy, heat flux at the top of the soil canopy heat storage, and vertical flux divergence/horizontal advection from multi-layer measurements, the EBEX dataset still contained an imbalance in the surface energy budget of around 10% that the EBEX community is unable to explain (Oncley et al., submitted to B.L.M.). Both instrumental and methodological problems are believed to be responsible for these uncertainties since both the performance of instruments and flux correction methods are not perfect.
Although the question of the surface energy imbalance is well known, the implications for CO 2 flux measurements are not yet fully understood (Wilson et al., 2002;Baldocchi, 2003). In this context, it is useful to understand how (even small) losses of turbulent energy fluxes influence CO 2 fluxes derived from open-path analysers via application of the WPL algorithm. In the following sections, we generate three cases to quantitatively illustrate how it is important to apply simultaneously all corrections for high and low frequency losses to sensible and latent heat fluxes and CO 2 fluxes prior to application of the WPL algorithm. In particular, we stress that applying corrections solely to CO 2 fluxes, but not to sensible and latent heat fluxes, is not a good strategy for obtaining high-quality CO 2 flux data. Instead, large biases in the final CO 2 fluxes may be introduced in this scenario.

Case 1: Raw Sensible Heat and Moisture Fluxes and CO 2 Fluxes
Without Correction We use the information from the energy balance closure (Section 4.1) construct the first case for analysis to illustrate the effects of the imbalance on CO 2 fluxes in the case where no corrections are made. The objective of this paper is not to show how well we can achieve energy balance closure for our site, but rather, to show that non-closure, even if quite small, may have negatively large impacts on the accuracy of CO 2 fluxes through the WPL algorithm.
Starting with our assumption that sensible and latent heat fluxes were underestimated by about 15% for daytime periods and by about 30% at nighttime according to the above energy balance closure, we assumed that raw CO 2 fluxes were also underestimated by this same amount. Specifically, to correct the fluxes for these losses, we obtain ω c = −15%, ω v = −15%, and ω T = −15% for the daytime measurements (Table I, Case 1), and ω c = −30%, ω v = −30%, and ω T = −30% for the nighttime measurements (Table II, Case 1).
It was expected that the errors of final CO 2 fluxes would vary with time, depending on the relative contributions from the individual error sources (i.e. ω T , ω v and ω c ) and their corresponding coefficients (i.e. C 1 = w ρ c /F c , C 2 = (µw ρ v /F c )(ρ c /ρ a ) and C 3 = ρ c w T (1 + µσ )/T F c ) in Equation (4) under different atmospheric conditions. Using the data obtained over the black spruce stands during the 2002 summer season, we obtained the variation of the coefficients C 1 , C 2 , and C 3 with sensible heat fluxes   Figure 3). Except for the large scattering when the sensible heat fluxes were close to zero, which was normal when F c and other fluxes approached zero during the morning and evening transition periods, there were strong relationships between the coefficients (i.e. C 1 , C 2 and C 3 ) and sensible heat fluxes. We made second-order polynomial regressions for the data in Figure 3 and then obtained the magnitudes of the coefficients from the relationship between the fit curves and the sensible heat fluxes. On average, during the day coefficient C 1 ranged from 1.7 to 5.6, C 2 from −0.15 to −0.35, and C 3 from −0.77 to −7.1, while during the night, coefficient C 1 ranged from 1.45 to 2.6, C 2 from 0.04 to 0.09, and C 3 from −0.9 to −1.8. The variation in the coefficients listed in Table I suggests that the daytime errors of final CO 2 fluxes were mainly controlled by the measurement errors in raw CO 2 fluxes and raw sensible heat fluxes. For Case 1 during the day, even though we assumed the raw flux error was −15%, the final CO 2 flux errors (δF c /F c ) varied from −13% to +28%.
For Case 1 during the night, the final nighttime CO 2 flux errors (δF c /F c ) varied from −16% to −26% (Table II). It was likely that the combination of the different error sources on the right-hand terms of Equation (4) reduced the errors for the final nighttime CO 2 fluxes through the WPL algorithm.
For Case 2, the biases of the daytime final CO 2 fluxes (δF c /F c ) varied from −8.5% to +18.5% (Table I), again with the application of Equation (4) always leading to a smaller inferred carbon uptake rate by the ecosystem as compared with the initial guess at the raw CO 2 error of −10%. Similarly, the nighttime final CO 2 fluxes were always underestimated (about −5.5%; Table II). Clearly, the errors of the final CO 2 fluxes were substantially reduced due to the simultaneous corrections on the sensible and latent heat fluxes and CO 2 fluxes before the WPL algorithm was applied. Obviously, increasing the CO 2 fluxes by the same fractional percentage as required to bring the H and LE into agreement with the observed available energy fluxes is also flawed. Rather, uncertainties in the final CO 2 fluxes are the combination of uncertainties in raw CO 2 , H , and LE, invariant with changes in atmospheric conditions.

Case 3: Frequency Response Corrections for Raw CO 2 Flux
(but not for H or LE) We also analysed a third case that describes the situation where accurate measurements or careful corrections are made for one individual quantity (e.g., CO 2 fluxes), but not for other fluxes (e.g., sensible and latent heat fluxes). This situation may occasionally arise in studies that focus solely on carbon fluxes with the accuracy of the energy fluxes being inadvertently ignored. In this scenario, we improved our estimate of CO 2 fluxes from Case 1 by applying frequency response correction for both day and night periods, but left the errors in sensible and latent heat fluxes at the same values. Specifically, in this case, ω c = −10% (−18%), ω v = −15% (−30%), and ω T = −15% (−30%) for the day (and night) measurements.
Corrections applied solely to CO 2 fluxes (Case 3) had the highest daytime sensitivity to sensible heat fluxes (Table I). Specifically, the daytime CO 2 fluxes were either underestimated or largely overestimated. At night, the propagation of errors through the WPL algorithm always led to positive errors in the final CO 2 fluxes (Table II). This case strongly suggests that special attention should be paid not only to the accuracy of raw CO 2 flux measurements but also to the accuracy of raw energy flux measurements (especially the sensible heat fluxes). Additionally, the accuracy of the raw sensible heat fluxes is even more important than the accuracy of the raw CO 2 fluxes under some circumstances when the correction magnitudes through the WPL algorithm are small as indicated in Equation (4).

Influence of incomplete energy closure on nee estimates
The adjustments to the final CO 2 fluxes as measured by the open-path systems varied widely after variations in C 1 , C 2 and C 3 were taken into consideration. In terms of the influence of energy imbalance on CO 2 fluxes, comprehensive evaluation of the uncertainties for long-term NEE should be made using the estimates from Equation (4) on a half-hour basis. Although we do not advocate adjusting CO 2 fluxes based on remaining surface energy imbalances (after all appropriate corrections have been applied), it is interesting to consider how these adjustments would propagate through the WPL algorithms. For example, during the summer period of our measurements we obtained a daily NEE of −1.27 gC m −2 day −1 after application of the WPL algorithm. If we correct this value for the lack of energy balance closure during day and night (losses of 15% and 30% as described by case 1) we obtain a value of −1.38 gC m −2 day −1 . A more complete treatment for the lack of energy balance closure that accounted for propagation error through the WPL algorithm (Equation (4)) yields a daily integral of −1.33 gC m −2 day −1 . Thus, consideration of Equation (4) decreases the size of the inferred summer carbon sink in calculations that attempt to correct for energy losses. In other ecosystems with larger sensible heat fluxes we would expect differences in NEE inferred for the two approaches to be even greater.
NEE bias errors depend on the accuracy of daytime and nighttime eddy-covariance measurements that diurnally vary due to different daytime and nighttime error sources. Recently there has been considerable progress in the analysis and correction of nighttime CO 2 flux measurements (Goulden et al., 1996;Aubinet et al., 2000;Anthoni et al., 2002b). In con-trast, the correction of daytime CO 2 flux measurements has received little attention although the influences of vertical advection, low and high frequency filtering and other factors on flux losses are known to be important (Lee, 1998;Massman and Lee, 2002;Wilson et al., 2002). As a result, applying an empirical correction for underestimated nighttime CO 2 fluxes, while ignoring the correction for the daytime bias of raw CO 2 flux measurements, could potentially enhance uncertainties of long-term NEE integrals obtained using open-path systems. Our analysis suggests that flux corrections should be made simultaneously and carefully for all raw flux components (i.e. sensible heat, latent heat and CO 2 fluxes) before applying the WPL algorithm. Any selective corrections for individual components of these raw fluxes may induce additional systemic errors and thus may enhance the uncertainties in the final fluxes.
Current extensive comparisons between open-path and closed-path IRGAs in the flux community have illustrated consistently higher daytime NEE and lower nighttime NEE by open-path IRGAs compared to closepath IRGAs (e.g., Anthoni et al., 2002a;Liang et al., 2004) except for some special cases with small differences between two systems (e.g., Ham and Heilman, 2003). Anthoni et al. (2002a) compared the response of a closed-path IRGA (i.e.  and an open-path IRGA (i.e. LI-7500) over Pondersoa pine sites in Oregon, U.S.A. Concentrations measured by the open-path IRGA and the closed-path IRGA generally fell on a 1:1 line, but NEE estimated using the closed-path IRGA was about 20% higher than that using the open-path IRGA at night. In contrast, NEE estimated using the closed-path IRGA was about 20% less negative than that using the open path IRGA during the daytime. But NEE estimates from the closed-path IRGA remained about 35% more positive when integrating daily totals of NEE during summer periods with long day lengths (Anthoni et al., 2002a). Liang et al. (2004) found that the total integral of NEE over a larix forest was −2.1 to −2.4 t C ha −1 year −1 as measured using a closed-path IRGA (LI-6262) and −4.7 to −5.4 t C ha −1 year −1 as measured using an open-path IRGA (LI-7500) due to 20% more negative daytime NEE and 15% lower nighttime NEE as measured using the open-path IRGA. Our analysis suggests that more negative daytime NEE and lower nighttime NEE measured using open-path IRGAs relative to closed-path IRGAs may be partly caused by inappropriately accounting for the energy losses (especially underestimated sensible heat fluxes) through the WPL algorithm.
It is obvious that if these additional errors were corrected for in the final CO 2 fluxes obtained by the open-path system, the agreement between the final daytime CO 2 fluxes of the two systems could be greatly improved.

Influence of incomplete energy balance closure on latent heat fluxes
Since the latent heat fluxes were positive both during the day and at night for our site, the effect of the WPL algorithm (Equation (1)) was to increase the magnitudes of the raw latent heat fluxes during daytime and to reduce the magnitudes of the raw latent heat fluxes at night. Using the biases from Case 1 in Equation (3) (i.e. ω v = −15%, and ω T = −15% for daytime measurements), we obtained an average δE/E of −23% (ranging from −13 to −28%), indicating an additional reduction in latent heat flux obtained through use of the WPL algorithm.

Impact of random errors in raw fluxes on final co 2 fluxes
Random errors in sensible heat fluxes and CO 2 fluxes may also affect the final CO 2 fluxes through the WPL algorithm. As an example, we investigated the influence of the coordinate rotation on the fluxes. Instead of using double rotation (DR) that is thought to induce large errors due to the sampling uncertainties of the mean vertical velocity (Wilczak et al., 2001) or possible low frequency loss, the planar fit (PF) technique was used in our study to rotate the sonic anemometer's coordinate system to make its vertical axis perpendicular to the mean air flow. We found no systematic errors in sensible heat, latent heat and CO 2 fluxes when comparing the PF coordinate system with the DR coordinate system (Figure 4). Instead, large scattering of fluxes was observed (about ±5% for sensible heat flux and about ±10% for latent heat flux) especially when wind speeds were small, which was demonstrated by the large scatter in friction velocity (u * ; m s −1 ) (about ±40%) when wind speeds were low (u * < 0.4 m s −1 ) (Wilczak et al., 2001;Foken et al., 2004). In particular, it was likely that the DR method induced much large random errors in CO 2 fluxes (up to about ±30%) (Finnigan et al., 2003) when we compared the standard deviation for CO 2 fluxes of 2.93 for the DR coordinate and 2.84 for the PF technique. Additionally, large random errors in sensible and latent heat fluxes, as well as raw CO 2 fluxes, resulting from the double rotation of the coordinate system were amplified as a result of propagation through the WPL algorithm. In this case, uncertainties increased in the estimate of long-term NEE. Our study supports the use of the planar fit method (Wilczak et al., 2001) in flux calculations. As demonstrated in our results, the uncertainties in the final CO 2 fluxes are related to both systematic and random errors in sensible heat fluxes. From the practical point of view, any effort that reduces the measurement errors in sensible heat fluxes can largely decrease the errors in final CO 2 fluxes. Even the conversion from buoyancy fluxes obtained from sonic temperature could yield a 5-10% difference when compared with the sensible heat fluxes obtained directly from fine-wire thermocouples (Liu et al., 2001), one of the best options is still to use fast-response thermometers to directly measure sensible heat fluxes from this point of view.

Conclusions
The mechanisms responsible for incomplete energy balance closure, including the partitioning of loss terms across sensible and latent heat components, have important consequences for the calculation of CO 2 fluxes when open-path IRGAs are used. Although energy imbalance may lead to underestimation of raw CO 2 fluxes, final CO 2 fluxes after application of the WPL algorithm may be either underestimated or overestimated. Some corrections (e.g., frequency response corrections) of all raw fluxes (i.e. CO 2 flux, sensible and latent heat fluxes) are quite effective in reducing the errors in the final CO 2 fluxes, and should be made simultaneously and diurnally for all raw fluxes before applying the WPL algorithm. Any selective corrections for individual raw fluxes (e.g., CO 2 fluxes only), while neglecting the accuracy of turbulent fluxes, could increase the uncertainties in the final CO 2 fluxes. However, since no correction method is perfect (Massman, 2004), therefore, uncertainties and biases in the raw sensible heat, latent heat, and CO 2 fluxes are inevitable. Consequently, the propagation error through the WPL algorithm is general, and may be amplified or dampened. Since the errors in the final CO 2 fluxes are quite sensitive to the errors in sensible heat fluxes as well as raw CO 2 fluxes, it is necessary to consider the use of more robust fast-response thermometers, or the use of a careful conversion approach of the buoyant heat flux when sonic temperature is employed (e.g., Schotanus et al., 1986;Liu et al., 2001). Additionally, the methodology in calculating fluxes highlighted in this study supports the use of the planar fit algorithm for coordinate rotation to reduce random errors of CO 2 fluxes. Large run-to-run variations in CO 2 fluxes do not necessarily reflect variations in ecosystem functioning but may result from the combined effects of various random errors.
Although the adjustment of CO 2 fluxes based on a lack of energy closure remains controversial, it may be required for some applications. Here we show that the adjustment approach for open-path systems is fundamentally different from what is appropriate for closed-path systems, because of the consideration of the WPL algorithm.