Post-processing: spectral corrections

It is not possible to construct a measuring system which detects all fluctuations with zero response time, and wich does not affect the measured signal itself. Moreover, introducing a closed-path gas analyzer to the system causes extra demands with regards to corrections. A part of these corrections -- e.g. the utilization of the lag time -- was already described.

Spectral corrections are supposed to be applied in order to account for the damping of fluctuations caused by the long air sample tubes, limited sensor response time, sensor line averaging and sensor separation, and sampling. One possible correction method is the application of a series of transfer functions defined for each correction term (Moore, 1986; Leuning and Moncrieff, 1990; Lenschow and Raupach, 1991; Massman, 1991; Leuning and King, 1992; Suyker and Verma, 1993; Moncrieff et al., 1997; Aubinet et al., 2000).

One other scheme which is quite frequently used in micrometeorology is the application of the spectral corrections based on the comparison of the (hopefully) ideally measured sensible heat cospectra (Co, see e.g. for the definition of cospectra), and the degraded water vapor and carbon dioxide cospectra (Högström et al., 1989; Grelle, 1997; Berger et al., 2001). The method is quite attractive, however it is not to apply to our system since the temperature signal is heavily degraded. Both method is based on the assumption of spectral similarity between temperature, HO and CO (Anderson and Verma, 1984; Ohtaki, 1985).

The scheme of Moore (1986) is applied through the use of transfer functions for each correction term. The fractional error of the measured flux can be written as:

$$\frac{\Delta F_{s}}{F_{s}}=1-\frac{\int _{_{0}}^{^{\infty }}... ...t) dn}{\int ^{^{\infty }}_{_{0}}Co_{ws}\left( n\right) dn}\: ,$$ (2.49)

where is the convolution of all transfer functions associated with sensors of vertical wind speed and the scalar quantity in question, is the cospectrum of the scalar flux , and is natural frequency.

To calculate , we need to determine each individual transfer functions. In our system, corrections are applied in order to take into account the effect of the signal damping inside the air inlet tube, the limited time response of the IRGA, the sensor line averaging inside the IRGA's optical path, the sensor line averaging caused by the sonic anemometer, the sensor separation caused by the separation of the sensors (i.e. the separation of the air inlet and the sonic anemometer), and the effect of the discretized sampling of the continuously fluctuating flow field (Moore, 1986). These transfer functions are described in the followings.

a) Spectral attenuation

The signal damping (i.e. spectral attenuation) caused by the long air inlet tubes can be approximated if the type of the flow (laminar or turbulent) is known and the parameters of the tubing are measured (Leuning and Moncrieff, 1990; Massman, 1991; Leuning and King, 1992; Moncrieff et al., 1997; Suyker and Verma, 1993). Caused by the change in the lag time of our eddy covariance system, the type of the flow is not constans in time, but dependent on the date. Massman (1991) provides transfer functions for both laminar and turbulent flow rates.

The general transfer function has the form of

$$T_{t}\left( n\right) =\exp \left( -4\pi ^{2}n^{2}\Lambda Lau^{-2}_{t}\right) ,$$ (2.50)

where is natural frequency, is the attenuation coefficient (different for laminar and turbulent flow), is the tube length (115 m in our case), is the (internal) radius of the tube (=0.00295 m) and is the velocity of air inside the tube (calculated as divided by the average lag time determined during the calibration procedure).

In case of laminar flow rate ( 2300, where the kinematic viscosity of air is calculated as  m s, with the air temperature in Celsius), the tube attenuation coefficient can be written as (Philip, 1963):

$$\Lambda =0.0104\nu _{air}ReD^{-1}$$ (2.51)

where is the molecular diffusivity of water vapor and carbon dioxide. These values are taken directly form Massman's web site:  = 0.217810 m s,  = 0.138110 m s. It should be noted that the transfer function for laminar flow calculated with inserting eq. into eq. is valid for the case when and . In our case, when the highest frequency to be considered is the Nyquist frequency (2.08 Hz), both criteria is fulfilled. Considering the first criteria, the expression equals to 8.235 at =2.08 Hz. The second criteria is easily fulfilled since equals to 38.983 while the right hand side equals to 120 with =2200. Consequently, the transfer function is valid for our system with laminar flow configuration.

In case of turbulent flow inside the sampling tube Massman (1991) provides values of and minimum values for different Reynolds numbers (Table 1. in Massman's paper). Practically, the attenuation coefficient is determined with interpolation using the data provided by Massman (1991) for any given Reynolds number.

Comparison of the transfer functions associated with laminar and turbulent flow shows that turbulent flow field inside the air inlet tube causes less severe signal damping compared to the laminar case (Lenschow and Raupach, 1991; Massman, 1991; Suyker and Verma, 1992), hence this configuration is the desirable in eddy covariance systems using long air inlet tubes.

b) Limited time response of the IRGA

The transfer function associated with the limited time response of the LI-COR 6262 IRGA (=0.1 sec, Moncrieff et al., 1997) has the following form (Moore, 1986; Moncrieff et al., 1997):

$$G\left( n\right) =\frac{1}{\sqrt{1+\left( 2\pi n\tau \right) ^{2}}}\: ,$$ (2.52)

where is natural frequency, as usually.

c) Line averaging

If a sensor measures the turbulent flow field over a finite sampling path, the signal of turbulent eddies with size comparable with the sensor path is averaged. In order to take it into account transfer functions should be applied. This function has a different form for scalar and vector quantities (Moore, 1986).

In our system, the LI-COR 6262 IRGA has an optical path with a length of 15.2 cm. To take into account its line averaging effect, the following transfer function should be considered:

$$T_{p}\left( f_{p1}\right) =\frac{\sin ^{2}\left( \pi f_{p1}\right) }{\left( \pi f_{p1}\right) ^{2}}\: ,$$ (2.53)

where is the normalized frequency, where the averaging path is given by and the average horizontal wind speed is given by . Note that eq. is different from the general case (eq. 7. in Moore, 1986), because the flow inside the optical bench is parallel to the averaging path. This transfer function has a very slight contribution to the overall transfer function.

A different transfer function is applied to the sonic anemometer data which has a transducer head-to-head averaging path of 15 cm. According to Moore (1986), this function is as follows:

$$T_{w}\left( f_{p2}\right) =\frac{2}{\pi f_{p2}}\left( 1+\fra... ...-\exp \left( -2\pi f_{p2}\right) \right) }{4\pi f}\right) \: ,$$ (2.54)

where is again the normalized frequency, where the averaging path length of the GILL sonic anemometer given by .

d) Sensor separation

The sensor separation can be hadled using the general transfer function for lateral and longitudinal sensor separation (Moore, 1986). The transfer function has the form of

$$T_{ss}\left( f_{ss}\right) =\exp \left( -9.9f^{1.5}_{ss}\right) \: ,$$ (2.55)

where , where the separation distance is given by (=0.8 m is used here).

e) Discretized sampling

The effect of the discretized sampling of the continuously fluctuating flow field using the analog to digital conversion can be approximated using the following equation (Moore, 1986):

$$T_{a}\left( n\right) =1+\left( \frac{n}{n_{s}-n}\right) ^{3}\: ,$$ (2.56)

where is the sampling frequency.

The whole data acquisition system's transfer function can be calculated utilizing all transfer functions described above (subscript refers to CO or HO):

$$T_{ws}\left( n\right) =T_{a}\left( n\right) T_{t}\left( n\ri... ...sqrt{T_{p}\left( f_{p1}\right) T_{w}\left( f_{p2}\right) }\: .$$ (2.57)

Note that the only difference between the whole system's transfer function for CO and HO resides in the different signal damping inside the air inlet tubes described by the different transfer functions, which is caused by the different molecular diffusivity () values for water vapor and carbon dioxide in eq. and .

In an earlier stage of the data processing moving averages trend removal technique was applied, which needed further spectral correction described by Kaimal et al. (1968). As it was described in section , linear trend removal is applied currently, which does not cause spectral modification of the turbulent characteristics.

Once the system's transfer function is determined, we have to obtain adequate cospectral models which can be applied in eq. to estimate the loss in the system. The modified forms of model spectra of Kaimal et al. (1972) is used for this purpose (Moore, 1986, Moncrieff et al., 1997). These normalized model spectras describe the stability dependent behaviour of the atmospheric spectra and cospectra of wind speed, temperature or any other scalar. The integrals of the model cospectra are equal to unity, thus they can easily be modified to simulate the real atmospheric spectra of the parameter in question.

The model spectra definitions are based on the normalised frequency , where is the measuring height above the zero plane displacement , and is the average horizontal wind speed. Normalization is neccessary since high wind speed causes the turbulent signal biased towards the higher frequency range accorning to Taylor's hypothesis (Stull, 1988).

The functional forms of the model spectra are as follows. During stable conditions it is written as

$$Co_{ws}\left( n\right) =\frac{1}{n}\frac{f}{A_{ws}+B_{ws}f^{2.1}}\: ,$$ (2.58)

where

$$A_{ws}=0.284\left[ 1+6.4\left( \frac{z-d}{L}\right) \right] ^{0.75}\: ,$$ (2.59)

and

$$B_{ws}=2.34A^{-1.1}_{ws}$$ (2.60)

(Moncrieff et al., 1997). Here denotes the Obukhov length, and is the scalar quantity (CO or HO).

During unstable conditions the model spectras are as follows:

$$Co_{ws}\left( n\right) =\frac{1}{n}\frac{12.92f}{\left( 1+26.7f\right) ^{1.375}}\: ,\quad f<0.54\: ,$$ (2.61)

$$Co_{ws}\left( n\right) =\frac{1}{n}\frac{4.378f}{\left( 1+3.8f\right) ^{2.4}}\: ,\quad f\geq 0.54\: .$$ (2.62)

As it was mentioned at the end of subsection , Obukhov length is calculated in each hourly period to provide stability information for the model spectra calculations. Flux loss caused by the eddy covariance system is estimated substituting the model cospectra and the overall system transfer function into .

Since the manipulation described above are based on theoretical transfer functions published in the literature, it is useful to prove that the cospectra behaves the same way as it is expected.

Figure: Theoretical (dashed line), measured (dotted line) and simulated cospectra (solid line) of carbon dioxide. The measured cospectra is an average of two hourly cospectra measured in 5 August, 1998 (10h-12h UTC+1). The mean wind speed was 8.2 m s. The theoretical cospectra is calculated from eq. and .

Figure shows the cospectra of the vertical wind speed fluctuations with the carbon dioxide mixing ratio fluctuations measured in 5 August, 1998 between 10h and 12h UTC+1 together with the theoretical Kaimal cospectra for unstable situations, and the same cospectra multiplied by the total measuring system transfer function. The average horizontal wind speed was 8.2 m s. The measured cospectra follows the simulated one well indicating that the theoretical transfer functions describe the behaviour of the system adequately. There is a slight deviation from the theoretical curve at the highest frequencies, but the contribution of the shortest wavelengths to the total flux is very little (e.g. the area under the curve at that specific interval is little).

Figure shows the relative frequency distribution of the CO flux loss calculated from the theoretical considerations. The histogram was created using all available data.

Figure: Relative frequency distribution of the CO flux loss for the 82 m eddy covariance system.

Average losses of the eddy covariance system are about 6.6% for CO and 5.9% for HO. In 59% of the cases for HO and in 55% for CO the loss is less than 3%. In 84% of the cases for HO and in 82% for CO the loss is less than 10%. Flux loss is higher during stable conditions. Low values of the Obukhov length (, defined by eq. ) or high wind speed conditions (), cause extra loss. For example, =10 m and =5 m s causes 21.7% and 23.4% loss for HO and CO, respectively. In case of =20 m and =15 m s, the loss is 26.4% for HO and 29.5% for CO.

As it was mentioned before, it appears that the water vapor signal suffers from excess spectral degradation compared to the theoretical spectral damping because of the long tubing of the system which causes extra loss of water vapor flux, hence needs more investigation, but this is out of scope of this work.