a. Flux IRGA calibration

The flux IRGA output voltages corresponding to CO₂ and H₂O vapor are calibrated using data from the profiler system and the temperature/humidity probes. The calibration method results in flux signals of CO₂ and H₂O in terms of mixing ratios. Fluxes calculated with mixing ratios do not require correction by the heat flux (and H₂O vapor flux for the CO₂ case) (Webb et al. 1980).

The first step in the flux IRGA calibration is to perform a linear fit of the function that represents the CO₂ or H₂O vapor mixing ratio in terms of the IRGA parameters (voltage, pressure, temperature, and in the case of CO₂, the H₂O vapor mixing ratio—all known at 5 Hz) versus the CO₂ or H₂O mixing ratios from the profiler system or the temperature/humidity probes, respectively, for every 2 days of data.

A second step is to perform a weighted running average of the slopes and intercepts over a period longer than 2 days (we are currently using 30 days). The weights are based on the statistical properties of the 2-day fits, with better fits receiving higher weight. Daily slope and intercept values determined by the weighted running average are used to convert the 5-Hz IRGA voltages for that day into mixing ratios for CO₂ and H₂O vapor. These high-rate time series are used to calculate the fluxes.

Since the partial density of CO₂ is negligible compared to that of dry air and H₂O vapor, we can use the ideal gas law to obtain

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where p is the total pressure; M_d and M_w are the molecular weights of dry air and water, respectively; T_υ is the virtual temperature, R is the universal gas constant;e is the H₂O vapor pressure; and T is the air temperature.

By connecting the reference and sample cell output tubes immediately after the cells, and because the input tubes are in the same temperature environment, we assume that p_r = p_s = p and T_r = T_s = T, respectively. These conditions are estimated to hold within less than 1 hPa for the pressure and 0.5°C for temperature, resulting in less than 0.6% combined error in the flux of CO₂ when the system is running in “differential” mode (i.e., r_cr ≠ 0). The system was converted to “absolute” mode (r_cr = 0) in mid-1998, which eliminates the issue of unequal pressure and temperature between the reference and sample cell. Also, e_r = 0 because the reference gas is dry. Thus, if we use standard relationships for H₂O vapor pressure and virtual temperature, we can rewrite (2) as

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The quantity r_qs is the H₂O vapor mixing ratio in the sample cell (obtained below) and ɛ = M_w/M_d = 0.622. The quantities T₀ and p₀ are arbitrary reference values for temperature and pressure, respectively, that have been introduced to normalize the temperature and pressure. We have chosen T₀ = 273.15 K and p₀ = 1000 hPa.

We assume that (4) has a linear functional form over a narrow range of CO₂ so that

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where R and M_d are absorbed into the slope m̂_c. We determine the values of the coefficients m̂_c and b̂_c by performing a linear regression with the unknown sample cell CO₂ mixing ratio, r_cs, replaced by the value measured simultaneously by the profiler system.

Finally, we rearrange (5) to yield the mixing ratio for CO₂ as

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with the slope m_c = 1/m̂_c and intercept b_c = −b̂_c/m̂_c. The statistics from the regressions used to find the hatted coefficients provide a straightforward way to determine weights for an averaging procedure that smooths variation in the coefficients over periods longer than that over which the fit is performed.

Similarly, the H₂O vapor mixing ratio calculated from the relative humidity and temperature obtained from the temperature/humidity probes and atmospheric pressure measured at the base with an analog barometer (Atmospheric Instrumentation Research, Inc., model AB-2A) and extrapolated to the tower level using the altimeter equation (2.35) of Wallace and Hobbs (1977) should match the H₂O vapor mixing ratio from the flux IRGA. Since the reference cell gas contains no H₂O vapor, the calibration equation for H₂O vapor mixing ratio can be written as

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It may be noted that, excluding the water vapor correction, (4) has the same form as presented in the IRGA operation manual [Li-Cor 1996, Eq. (3–4)]. Because the IRGA response is nearly linear in the range encountered over the 2-day fit period, we find it only necessary to calibrate using a linear function instead of the higher-order calibration polynomial function suggested by the manufacturer. A 2-day interval has been found to be long enough to capture sufficient variation in the mixing ratios to determine a fit, and short enough to allow the instrument to be locally calibrated within a linear range and to track instrument drift. Seasonal variability of the range (and thus the fit slope) is most prominent for H₂O vapor.

The slopes m_c and m_q in (6) and (7) are nearly proportional to the variances of r_cs and r_qs, respectively. This is because most of the variance is contained in the voltage while the IRGA pressure and temperature are fairly stable. Thus, to accurately calculate fluxes of CO₂ and H₂O vapor, it is more important to accurately determine the slope than the intercept of the fits.

The precision of the CO₂ measurement from the profiler that is used to calibrate the fast IRGA should be better than 0.1 ppm (Bakwin et al. 1998). The precision of the humidity probe has been estimated to be approximately ±0.15% RH (Ivan Bogoev 2000, personal communication), or in terms of the tower environment ±0.08 g kg⁻¹ in the worst case. Since these errors should be random and the calibration is performed with 2-min averages over 2 days, the slope error due to the precision has been estimated to be less than 0.1% for H₂O, and even smaller for CO₂. The 2-min averaging time for each data point should also minimize error due to hysteresis of the humidity probe.

The motivation for the calibration method described derives from the availability of the high precision CO₂ and H₂O vapor values used for determining the slopes. The cost and complexity of running calibration gases through the fast IRGAs were felt to be unneccesary since the CO₂ profile system was already in place and the temperature/humidity probes are relatively inexpensive.

Figures 2a,b show typical 2-day fits (15–16 May 1998) of data used for finding the slopes and intercepts. During the growing season, when fluctuations in CO₂ and H₂O vapor are large, the correlation coefficients (r²) for the fits are usually near unity. The r² values drop somewhat during the winter months when the fluctuations are smaller. Figures 2c,d show the slopes from the 2-day fits superimposed on the 30-day weighted running average for CO₂ and H₂O, respectively. The seasonal change in the H₂O calibration coefficients due to the IRGA response can be clearly seen.

The uncertainty in calculated CO₂ and H₂O fluxes due to random error in the calibration is estimated to be approximately 2%–3%. This is a standard error based on the distribution of the 2-day fits about the weighted running average and assuming that this distribution is steady over the year. Slightly less scatter is observed in the fits during the growing season, therefore the uncertainty should be less during the portion of the year when fluxes are largest. The uncertainty may be larger during the winter months, but fluxes are small at this time. The uncertainty will increase if the number of usable 2-day fits in the running average window decreases.

b. Sonic anemometer rotation correction

Wind velocities from the sonic anemometers are rotated into a coordinate system aligned with the local mean streamlines. McMillen (1988) and Baldocchi et al. (1988) discuss these rotations in detail.

The rotation is defined by two angles. Using 1-h of data, the mean wind direction ϕ is found with the horizontal wind components u and υ, then a tilt angle θ is found by rotating about the mean cross-wind direction to where the average vertical velocity w becomes zero. A misaligned sonic anemometer in a uniform flow would exhibit a perfect sinusoid to describe tilt angle versus wind direction. Therefore, a sine function is first fitted to the hourly wind direction and tilt angle data. To account for curvature in the streamlines due to local topography and the tower, the sine function is then subtracted from the data and a third-order polynomial function is fitted to the residual. The sine function and polynomial are added to obtain the complete rotation fit, which is then used to align each hour of sonic anemometer data with the local mean streamlines. This procedure results in a finite average vertical wind velocity, w. Preliminary analysis shows that although hourly values may not have enough accuracy (σ_w = 0.1 m s⁻¹) for determining synoptic-scale lifting or subsidence at that hour, averages over 1–2 days should be suitably accurate (σ_w = 0.01 m s⁻¹) to detect synoptic motions and may provide insight into vertical advection issues in the computation of NEE of CO₂ (Yi et al. 2000).

Figure 3 shows an example of the resulting fits for several months of data. When the wind blows from the north, the tower impedes the flow. This is the most likely reason for the larger variance of hourly θ values for wind in this direction; however, the influence of the tower is not observed at the 30- and 122-m levels. The distortion at 396 m may be due to a broadcast antenna mounted to the tower near the instrumentation only at that level.

c. Lag times

Calculating eddy covariance fluxes requires simultaneous measurements of wind and scalar quantities. The long tubes from the air sample inlets to the IRGAs at the base of the tower induce substantial delay in the signals used for calculating the CO₂ and H₂O fluxes. This delay can be determined by lagging the rotated wind signal until maximum covariance is obtained (Fan et al. 1990). However, this delay may change slightly due to changing flow rates in the tubes from factors such as pumping variation or density and viscosity changes. Lag times may also change in our case, due to poor synchronization of the remote computer that acquires the sonic anemometer data and the main data acquisition computer that acquires the trailer IRGA data and stores all of the data. However, this synchronization error only results in a fixed lag time between the start time of a particular hour-long data file for the sonic and the start time of an hour file of IRGA data. The dis-synchronization can vary from hour to hour up to approximately 10-s maximum. Although the synchronization-induced lag variation is not ideal, it is not considered to be a serious problem since the lag time can be determined. The computer synchronization has recently been improved.

The separation distance from the air sample inlet to the sonic anemometer is approximately 1 m. Lag times resulting from this displacement should be <0–2 s depending on wind speed and direction, but this lag will be included in the lag time determined with the maximum covariance method. Because the turbulence scales contributing to the covariance are much larger than 1 m (see section 3e), the separation will have a minimal effect on the estimation of fluxes.

Figure 4 shows a typical example of the variation in the lag time between the vertical velocity signal and the CO₂ signal from the trailer IRGA for the 30-m level. The lag times include the computer dis-synchronization time. All points in Fig. 4 are from periods where the lagged covariances were considered to be unambiguous, that is, a discernable peak was found within a window of lag times expected for that level. It is important to keep track of these lags on an hourly basis to ensure correctly calculated fluxes; one fixed lag time is insufficient. If a flux is not large enough to allow detection of the lag time, as sometimes occurs at night due to low turbulence levels, adjacent hours are used to estimate the lag.

When the lagged covariance shows a clearly detectable peak, a lag time can be determined within approximately ±1 s for CO₂ and ±3 s for H₂O. These times correspond to variation in the calculated flux of 1.5%–2% and 1% for CO₂ and H₂O, respectively. The variation in CO₂ flux is proportionally larger because the CO₂ lagged covariance typically has a sharper peak than H₂O.

d. CO₂ and H₂O signal spectral corrections

Spectral degradation of the CO₂ and H₂O flux signals is expected due to the long tubes that carry sample air to the trailer IRGAs. This problem can be attributed primarily to nonuniform velocity across the cross section of the tube as well as interaction with the tubing walls. These diffusive mechanisms will attenuate high frequencies (smaller scales) more than low frequencies (larger scales) and result in underestimation of CO₂ and H₂O fluxes.

Figures 5a,b show typical spectra for CO₂ and H₂O from the trailer IRGAs for all levels. In order to match the levels of the spectra at low frequency, each spectrum is normalized by dividing every value in the spectrum by the variance contributed by the lowest frequencies. We chose f ≈ 6 × 10⁻³ Hz as the upper limit for computing this “partial” variance. The CO₂ spectra in Fig. 5a show no relative degradation, at least up to frequencies greater than 0.2 Hz where the signal becomes dominated by instrumental noise. The H₂O spectra in Fig. 5b, however, clearly show increasing attenuation with increasing tower height (i.e., tubing length). Comparison of the low-frequency portion of CO₂ and H₂O nonnormalized spectra from the trailer IRGAs with that from the tower IRGAs found no difference in spectral power (not shown) confirming that there is little low-frequency degradation.

Spectra of temperature, H₂O vapor, and CO₂ from a given location should be similar in shape (Ohtaki 1985) if they are perfectly measured. Virtual temperature obtained from the sonic anemometers should give the best representation of the true atmospheric spectra for all scalars since it is from a fast-response, open-path instrument. Humidity and cross-wind corrections to the temperature signal are described by Hignett (1992) and we currently employ the humidity correction. The crosswind correction has been estimated to affect the measured sensible heat flux by no more than 1% when the flux is larger than 5–10 W m² and spectral degradation of the temperature due to crosswinds is minimal. The crosswind correction will be implemented in the near future. Spectral degradation due to line averaging error [the effect of measuring the temperature over the distance between the sonic transducers and discussed by Moore (1986)] has been estimted to be less than 2% at 0.3 Hz (the typical highest frequency of scales contributing to the fluxes at the lowest level, see section 3e), and far less at lower frequencies.

Figure 6 compares the CO₂ and H₂O spectra to the temperature spectrum at the 396-m level and is representative of the other levels. Similarity of the CO₂ and temperature spectra confirms that CO₂ is not seriously degraded, at least up to the frequency of the instrumental noise.

Early work studying diffusion in tubes with laminar and turbulent flows was done by Taylor (1953, 1954). His work was used by Lenschow and Raupach (1991) to analytically estimate spectral degradation due to sampling tubes in order to determine the transfer function ϕ(f) that gives the attenuation at each frequency f such that

Gϕ²fG_d(8)

where G is the perfectly measured spectrum of some scalar quantity and G_d is the degraded spectrum. Using Taylor’s representation of the diffusion in the tube for turbulent flow, ϕ is found to be

ϕfKf²(9)

with

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for laminar flow in the tube, and

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for turbulent flow. Here K is a function of the tubing radius a; tubing length L; mean flow velocity U; the Reynolds number Re = 2aU/ν, where ν is the molecular viscosity; and in the case of turbulent flow, the Schmidt number Sc = ν/ν_D, where ν_D is the coefficient of molecular diffusion.

Massman (1991) presents a numerical analysis of spectral attenuation including tabulated values that may be used to calculate the value of K in (9). He describes an attenuation coefficient based on Reynolds number that differs for particular scalars to take into account differing Schmidt number.

Goulden et al. (1996b) discuss an alternative method for correction of the CO₂ fluxes. Assuming that the temperature signal is nearly perfect, they empirically determine the time constant for a low-pass filter that allows them to mathematically degrade the temperature signal to mimic the degradation of the CO₂ signal by the IRGA and tubes as a system. Then the ratio of the nondegraded to degraded heat flux should match the ratio of the corrected to uncorrected CO₂ flux such that

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where w′, c′, T′, and T^′_d are the fluctuating components of the vertical velocity, the CO₂ mixing ratio, the temperature and degraded temperature, respectively, and the overbar denotes time averaging. This presents a simple and effective way to repair CO₂ and H₂O fluxes if the appropriate filter can be found.

Based on typical values of flow through the sample tubes, the range of K for all tower levels is estimated to be between 0.7 and 2.7 s² based on the Lenschow and Raupach formulation. Using the tabulation of Massman, K is estimated to be between 4.6 and 20.9 s² for H₂O and 6.1 and 28.8 s² for CO₂. Figure 7 shows T and H₂O spectra for a typical midday hour at the 122-m level. Also shown is the theoretical curve of the degraded temperature generated using K = 14.2 s², the maximum attenuation for that level based on the tabulation for H₂0 of Massman for Re = 2300. For the case shown, the actual Re = 2850 so the curve should overestimate the degradation, but it is clear that even when ϕ is overestimated the theory does not capture the attenuation actually seen in the H₂O spectrum for our instrumentation. However, it is possible to determine a ϕ based directly on the spectra. Fits of the normalized spectra give ϕ² as

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where G represents the one-sided spectrum of H₂O vapor (denoted by q) or temperature (T), σ² is the partial variance for H₂O or temperature, and the FIT subscript denotes that the quantity is a curve fit to the data by least squares. The coefficients for a second-order polynomial fit of the H₂O spectrum are found by fitting the spectrum from the minimum frequency (f_min in Fig. 7) up to the frequency where the noise becomes apparent (f_knee). Finding f_knee is done using a wavelet edge detector similar to that described by Davis et al. (2000). The fitted H₂O spectrum is then generated across the full frequency band (f_min to f_max where f_max is the Nyquist frequency) and used in (15). The full frequency band is used to find the coefficients for a second-order fit of the temperature. Below the highest frequency where the fits of the spectra are equal (f_lo), ϕ is assumed to be unity. Once ϕ is determined, the degraded temperature can be found with (14) and the corrected latent heat flux found with an equation analogous to (12).

Although suitable for correcting the H₂O fluxes, the spectral fit method of finding ϕ outlined above cannot be used for CO₂ because any degradation to the CO₂ signal is hidden in high-frequency noise. Fluxes for CO₂ are currently being corrected by using ϕ calculated with the theory of Lenschow and Raupach (1991) using (9)–(11).

In addition to spectral attenuation, phase shift (dispersion) may occur due to the tubes. The lag time mentioned in section 3c (a fixed time delay) imparts a linear phase shift with frequency, but if the lag is removed by shifting the time series, phase shift from other mechanisms may remain. Shaw et al. (1998) discuss this issue for a system modeled by a time constant τ. For the recursive low-pass digital filter given by (13) the phase relationsip is

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Although not mentioned explicitly in Goulden et al. (1996b), according to S. C. Wofsy and J. W. Munger (2000, personal communication), they use a subtle variation to (12) where the vertical velocity is degraded similarly to T using (13) (w_d) and it is substituted into the degraded heat flux w^′_dT^′_d and the uncorrected CO₂ flux w^′_dc^′_uncorrected. This keeps the signals in phase and maximizes the degraded flux correlations. Although this changes the nondegraded to degraded flux ratios, the corrected CO₂ (or H₂O) flux should be the same as that given by (12) on average.

We have found that the phase between the velocity and scalar quantities is minimally altered by our system. This implies that degrading the temperature signal using the spectral fit method (which does not alter the phase) and then using (12) to determine the correction is appropriate. Figure 8 shows phase spectra from a typical midday hour between temperature and H₂O, −CO₂ (which provides a more clear plot since +CO₂ is 180° out of phase with temperature at this time period), the temperature degraded using (13) with τ = 6 s (the τ value necessary to make the degraded temperature spectrum match the H₂O spectrum), and the corresponding ω_τ from (16). It is clear that H₂O and CO₂ are not appreciably degraded in phase up to at least 0.1 Hz where the calculated phase passes from ±180° to ∓180° (“wrapping”), and that our system is not modeled well by the filtered signal. For the case shown, the flux contributed by scales corresponding to f > 0.1 Hz is approximately 4% of the total. Additionally, the heat flux calculated from the degraded temperature obtained by the spectral fit method (no phase shift) differs from the flux using the filter method [phase shifted by (13)] by only 2%. This indicates that the error due to phase degradation is minimal. Additional justification for neglecting the phase is given by Leuning and Moncrieff (1990) and Leuning and King (1992), who discuss dispersion effects studied by Philip (1963), concluding that the speeds of different scales through a tube are nearly equal if Ω < 10 with Ω = 2πfa²/ν_D. For our system we have estimated Ω ⩽ 4.0 for f < 1 Hz.

Figures 9a,b show a comparison of spectrally corrected and uncorrected diurnally averaged fluxes for CO₂ and H₂O vapor from both the trailer and tower IRGA signals at the 122-m level. Note that H₂O presents good agreement between the corrected fluxes, with the uncorrected flux from the trailer IRGA signal being the most degraded. For the CO₂ flux, the corrected and uncorrected curves are virtually identical. However, the trailer and tower show some disagreement. Calibration error may account for some of this discrepancy. Table 2 lists the approximate underestimation of CO₂ and H₂O fluxes for the trailer and tower IRGAs from examination of long-term spectral corrections. The ϕ of Lenschow and Raupach strongly attenuates small turbulence scales, thus the night values for CO₂ flux correction are larger due to smaller turbulence scales typically dominating at night. The correction to H₂O, however, is influenced by much larger scales and has been found to have little diurnal sensitivity. The reduction in flux underestimation with increasing height is also due to scales, where scale size increases with height. The trailer IRGAs consistently require more correction than those on the tower, although the underestimation of H₂O flux from the tower IRGAs is not negligible (≈12%).

e. Scale and random error analysis of fluxes

The eddy covariance method requires that the flow be stationary and that all flux-carrying eddies are sampled for an accurate estimation of the flux. Provided that the flow is stationary, an ogive provides a check on the necessary condition of sampling all flux-carrying scales (Desjardins et al. 1989; Friehe et al. 1991). An ogive is defined as the cumulative sum of a cospectrum from high to low frequency and thus has a value equal to the covariance or flux at the lowest frequency. Ogives that have an asymptotic shape toward the highest and lowest frequencies suggest that all flux-carrying scales are contained in the sample period. Figure 10 presents ogives from all tower levels for sensible heat flux based on 2-h sample periods centered at every hour of a day except 0000 and 2400 UTC. Daytime heat fluxes are large and positive, and are the result of the contribution of a wide range of scale sizes. Nighttime heat fluxes are small and negative, and the range of flux-contributing scale sizes is decreased. Flux-carrying scale size also increases with altitude. These observations are consistent with buoyancy dominating the daytime convective regime, shear dominating at night, and scales decreasing in size closer to the surface. Dashed lines denoting 2-, 1-, and 0.5-h sample times show that although 0.5 h may be an acceptable period for flux calculations at the lowest level, at least 1 h is necessary to allow the ogives to converge at the highest level. This supports the choice of 1 h as the period for flux calculations at all tower levels at WLEF. It also adds confidence to fluxes calculated over 0.5 h at the many canopy-level flux towers being used throughout the world. Additionally, Fig. 10 makes it clear that the sample frequency of 5 Hz is adequate for resolving the high-frequency scales contributing to the flux.

Based on theory developed by Lenschow and Stankov (1986), the relative error, a, of a flux value can be estimated using the integral length scale of the flux, the cross-correlation coefficient for the flux variables, and the length over which the flux measurement is made. The relative error of flux F is defined as a ≡ σ_F/〈F〉, where 〈F〉 is the ensemble average. Figures 11a,b show how a varies with height normalized by the boundary-layer height, z/z_i, for sensible heat flux and CO₂ flux determined over hourly periods during the convective times of 10–11 May 1998. Especially at higher levels in the boundary layer, the relative error can be quite large, on the order of the flux value itself. This result is not unexpected, Lenschow and Stankov show that large averaging times are necessary to reach low relative error levels, with momentum flux requiring the longest averaging length. At the lower levels, relative error for heat flux is less than 20%; however, for CO₂ flux, the relative error can be up to 40%–50%. These relative error values should be similar for measurements over any forest canopy. The only way to reduce the relative error is to increase the length over which the turbulence is averaged; however, care must be taken to avoid averaging times that exceed the time over which the flow is stationary, such as the transition from evening to daytime regimes. It should be noted, however, that even though the relative error may be fairly large for a single flux measurement, the long-term average of the flux will have significantly less error because the relative errors are random (Moncrieff et al. 1996). Another important aspect of the relative error with respect to our site, where multiple levels are used, is that the measurements at different levels are not completely independent; the relative error in the difference between the fluxes from two different levels should be less than that implied by the independent relative error of each measurement. The relative error of the difference will depend on the coherence of the turbulence between the levels. This discussion is beyond the scope of this paper.