ESPM 228, Advanced Topics in Biometeorology and ......ESPM 228, Advanced Topics in Biometeorology...

ESPM 228, Advanced Topics in Biometeorology and Micrometeorology

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Lecture 4 Micrometeorological Flux Measurements, Eddy Covariance, Application, Part 2 Instructor: Dennis Baldocchi, Professor of Biometeorology Ecosystems Science Division Department of Environmental Science, Policy and Management 345 Hilgard [email protected] 642-2874 February 11, 2016 Evaluating the Flux Covariance The power spectrum (also called the variance or energy spectrum) quantifies the amount of variance (or energy) associated with particular frequency or wavelength scales. The power spectrum of a scalar or a wind velocity component is derived through a Fourier transformation of a temporal (or spatial) series of a given variable. The Fourier transform converts the time (or space) series into a frequency (or wavelength) domain by representing it as an infinite sum of sine and cosine terms. Integration of spectral energy densities (Sww) across the whole range of significant frequencies (or wavelengths) yields the total variance of the velocity component (w’w’) or scalar quantity under scrutiny.

0

' ' ( )www w S d

In a similar manner the real component of the cross-spectrum, the co-spectrum (Swc), quantifies the amount of ‘flux’ that is associated with a particular scale.

0

' ' ( )wcF w c S d

A distinct property of turbulence in the natural environment is a wide spectrum of motion scales (eddies) associated with the fluid flow. The largest scales of turbulence are produced by forces driving the mean fluid flow. Dynamically instable, these large eddies break down into progressively smaller and smaller scales, via an inertial cascade. This cascading breakdown of eddy size continues until the eddies are so small that energy is consumed by working against viscous forces that convert kinetic energy into heat. Hence, in application, the method must acquire both short and long scale contributions to the turbulent flux.


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The dependence of the variance or flux covariance on a spectrum of eddies imposes several constraints on instrument design, measurement principles and sensor sampling. Reviews and critiques of the eddy covariance method have been produced by numerous investigators. The reader is referred to them for further details [Aubinet et al., 2000; D.D. Baldocchi, 2003; D. D. Baldocchi et al., 1988; Businger, 1986; Foken and Wichura, 1996; Goulden et al., 1996; Loescher et al., 2006; Massman and Lee, 2002; McMillen, 1988]. With regard to sensor sampling attention and care must be paid towards: a. sampling duration; b. sampling frequency; c. averaging method. With regard to instrumental placement, design and implementation, the accuracy of any flux measurement will be influenced by: 1) sensor size 2) separation of instruments 3) placement, in terms of height and position in the constant flux layer 4) mechanical filtering or distortion of turbulence, as when sampling through a tube or when a tower or anemometer head interferes with the wind. 5) Rotation of coordinates to compute fluxes orthogonal to mean streamlines. 6) Flow interference by towers and booms. A filtering of covariance signals occurs for several reasons. High frequency contributions are attenuated because sensors have a finite response time, their transducer may have a significant sampling volume or integral scale length (relative to scales of turbulence) or the pumping of air through a tube will distort the structure of eddies. Filtering of the turbulent fluctuations can be imposed through data acquisition because data acquisition systems use a discrete sampling interval. As a result they possess a finite cut-off, denoted at the high frequency end of the spectrum as the Nyquist frequency. In practice, a measured covariance is a function of the true cospectrum (Cowc) and a spectral transfer function (H), that arises from the issues discussed above:

0

' ' ( ) ( )measured wcw c H Co d


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Figure 1co-spectrum from study over peatland on Sherman Island. Co-

spectra are shown for temperature, water vapor, CO2 and methane. Data of Detto and Baldocchi. The transfer function H() is determined by the product of numerous filtering effects [Massman, 2000; Moore, 1986]:

1 21

( ) ( ) ( )... ( ) ( )N

n nn

H H H H H

The most important transfer functions that are applied to eddy covariance measurements include:

1. high pass filtering 2. low pass filtering 3. digital sampling at a limited frequency 4. sensor response time 5. fluctuation attenuation by sampling through a tube 6. Line or volume sampling 7. sensor separation

When computing filter and transfer functions one must be careful and not to confuse terminology. Moore [Moore, 1986] reports filtering gain functions (G) and transfer functions (T). When applied to correct a power spectrum the gain filters are squared. Time Averaging, Detrending and High Pass Filtering Since fluctuations from the mean are computed as the difference between instantaneous and mean values, we must assess the time series mean. This is not a trivial exercise due


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to the multiple time scales associated with a time series. The basic rules of Reynolds averaging use arithmetic means. The application of Reynold’s decomposition upon a time series, using a finite mean, imposes a band pass filter on the data [Kaimal and Finnigan, 1994]. Arithmetic means or digital recursive filters pass high frequency fluctuations but attenuate low frequency components in their attempt to assess means for mean removal calculations. Moore [Moore, 1986], Horst [Horst, 1997] and Massman [Massman, 2000] have developed schemes for estimating the amount of flux density lost by these ‘filtering’ effects. Since concentrations and velocities experience a diurnal pattern, they are never at steady state. Some investigators, therefore, prefer to detrend a time series and using a trend line to remove the mean. The rules of Reynolds averaging, however, say nothing about detrending (K.T. Paw U, personal communication). They are based on arithmetic means. In my opinion, if there is a trend we should treat this with a modification of the Conservation Budget and account for it as a storage term.

Figure 2 Representations of mean removal of turbulence time series

One can visualize the effects of removing means from time series by using transforms, such as the Fourier Transform, to convert time series from a temporal to frequency basis. In this transformed basis, we can examine the frequency at which fluctuations are passed or filtered.

M eanR em ova l

T rendR em ova l


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Filters of interest include low-pass, high-pass and band-pass filters. There is a simple but distinct difference between the high and low pass filters. A low pass filter allows low frequencies to pass, but stops high frequencies. The simplest low pass filter is defined as: H(f)low =1, f< fc H(f)low=0, f > fc A high pass filter is the opposite of the low pass filter, H(f)high=1-H(f)low. An example of a high and low pass filter function is given in Figure 2. Note the low pass filter passes all signals with a frequency lower than a critical frequency, fcrit; in contrast the high pass filter passes frequencies higher than the critical value. An additional point of information is that most band pass filters do not have such a perfect cut off (see Hamming, Digital Filters). This reality itself leads to numerical errors in the application of such filters. This point is discussed next.

Frequency

0.00001 0.0001 0.001 0.01 0.1 1 10 100

T(f

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Low Pass FilterHigh Pass Filter

Figure 3 Example of low and high pass filter functions

The act of block-averaging a time series and using it to construct a series of fluctuation is equivalent to applying a square wave transfer function in temporal space.


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( ) 1,

2 2

h t

T Tt

( ) 0,

2 2

h t

T Tt

where T is the averaging period, which typically ranges between 30 and 60 minutes. The Fourier transform of a square wave signal produces a transfer function that is a function of the sine function of the angular frequency. So a presumably ‘clean’ square wave averaging introduces sinusoidal noise at high frequencies. The Fourier transform of the Reynolds decomposition relations with an arithmetic mean yields a low pass transfer function:

sin( )( )

f TF f

f T

Figure 4 Transfer function for one hour average, T=3600

10-5 10-4 10-3 10-2 10-1 100 101

frequency

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

sin(

f T

)/(

f T

)


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When the filter function is applied to correct a power spectrum or co-spectrum the filter function is squared: H(f)=F(f)2. If we want to compute fluctuations from the mean then we are subtract a mean value from the instantaneous values. This action imposes a high pass filter upon the data. The resulting high pass transfer function is [Kaimal and Finnigan, 1994]:

2_ ( ) (1 ( ))high passH f H f

2

20

sin ( )' ' [1 ] ( ) ( )

( )measured wc

f Tw c H Co d

f T

Applying moving averages of shorter duration produces the following transfer functions.

Figure 5 Transfer function for block averaging. In this case the transfer function

for a low pass filter is shown. Plotting 1-H(w) would provide information for the high pass filter. Alternative Approaches to Computing Means

sin(

f T

)/(

f T

)


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At the advent of eddy flux measurements, storage capacity of computers were quite small, or scientists had to use analog computers. One clever means of avoiding the storage of raw data, and later processing of means and covariances, was to compute means real-time with a digital recursive filter [Dyer and Hicks, 1972; Massman, 2000; McMillen, 1988]. Turbulent fluctuations are decomposed from the mean using as the difference between

instantaneous (xi) and mean ( ix ) quantities. An arithmetic mean or a running mean can

be used to compute fluctuations. Computing the arithmetic mean, however, requires post processing of the data. Mean values were determined in real-time, using a digital recursive filter:

1 (1 )i i ix x x (1

where exp( )t

, t is the sampling time increment andis the filter time constant.

Values of alpha for a range of time constants, assuming a 1/10 th second sampling interval (10 Hz) is listed below.

50 0.998002 100 0.999 200 0.9995 400 0.99975 800 0.999875

1600 0.999938

Theoretically, an optimal time constant can be chosen with the aid of a Fourier transform of the digital recursive filter. Massman [Massman, 2000] derived a spectrally dependent version of the transfer function as

2

2

[ ][1 cos( / )]( )

(1 2 cos( / ) )s

s

fH

f


9

Figure 6 High pass transfer function of a digital recursive filter with various time

constant The algorithm is from Massman [Massman, 2000]. Spectral cut off is not perfectly clean at low frequencies, but it lets some energy pass. There remains uncertainty as to what the preferred value of the digital filter time constant should be inside a plant canopy. We can examine this question by comparing eddy covariances computed with the various digital times constant against those computed with conventional Reynolds’ decomposition and averaged over one-hour. We observe little differences (within 5%) between the two methods of compute flux covariances. Extending this analysis one step further, we observed that mass and energy flux covariance computations exhibit some sensitivity to the choice of filter time constant. Flux covariance computed with a 600s digital time constant is most ideal, while time constants between 400 and 800 s yielded covariance values that agreed within 5% of the Reynolds’ flux covariance [McMillen, 1988]. Greatest numerical errors are associated with filter time constants less than 100s and more than 1000 s. This analysis was based on regressing the independent and dependent variables against one another, rather than plotting the ratios, which are numerically unstable when fluxes are small.

High Pass Filters

frequency

0.00001 0.0001 0.001 0.01 0.1 1 10

|H(

)|

0.01

0.1

1

200 s 300 s 400 s 800 s

Hf

fs

s

( )[ ][ cos( / ]

( cos( / ) )

2

2

1

1 2


10

Figure 7 One to one plot of sensible heat flux covariance computed with

conventional Reynolds averaging and with a digital recursive filter with a 400 s time constant.

Jack Pine Forest

w'T' (Reynolds Ave, 3600s)

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6

w'T

' (=

400s

)

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

b[0] 1.744e-3b[1] 0.986r ² 0.979


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Figure 8 Slope of the relation of fluxes computed with a digital recursive filter

and varying time constant and a 3600 s long time series, to which conventional Reynolds averaging was applied In summary, a 400 to 600 s digital time constant is adequate to mimic the behavior of averaging over one hour. Interestingly, the critical time constant is close to T/2, or 572 s. One is not to confuse the concept of digital time constant with the averaging interval. Yet, there are many cases cited in the literature where one has used a 1000 s plus time constant. We have observed that exceeding long digital time constants can be problematic and error prone, too. This is a major reason why we have examined the Fourier transform of the filter to dispel this delusion. Evaluating Short Term Fluctuations Digital Sampling at Limited Frequency Computerized data acquisition systems digitize analog signals at discrete intervals. Discrete sampling of a time series, one which may consist of higher frequency contributions, limits the spectrum that can be resolved. The highest frequency is noted as the Nyquist frequency. It is defined as: fNyquist = fsampling/2. Ideally, we want to know the bandwidth of the turbulence spectra a priori and design the sampling strategy so we are able to sample at a frequency that is twice that of the highest frequency that significantly contributes to flux or variance. The following figure shows a typical spectrum for variance and flux. Over a tall forest the cut-off frequency may be as low as a few cycles per second.

time constant

0 500 1000 1500 2000 2500

slop

e w

T(t

) vs

wt(

3600

)

0.7

0.8

0.9

1.0

1.1

1.2

1.3


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Figure 9 Vertical velocity power spectrum and w-T cospectrum rice in

California. The non-dimensional cutoff frequency is a function of measurement height, wind speed, stability and the natural spectral cutoff frequency: fc=nc (z-d)/u). One can note from the figures above that the critical frequency of the co-spectrum occurs at a lower frequency than for the variance. This is because turbulence is isotropic in the inertial subrange of the power spectrum. If eddies have equal length and velocity scales in all direction, no material can be transferred at that scale. It is like a leaf in a whirl pool. It goes no where except round and round. The minimum sampling rate can be adjusted by placing the sensors at a higher level if there are sensor time response constraints. Under neutral conditions the non-dimensional spectral cut-off for a power spectrum is on the order of about 5 to 10 Hz. What are typical values of the natural cutoff frequency, the one we must sample at in the field? U z n

1 1 5 2 1 10 3 1 15 5 1 25

nCo/

(w'x

')


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10 1 50 1 2 2.5 2 2 5 3 2 7.5 5 2 12.5

10 2 25 1 5 1 2 5 2 3 5 3 5 5 5

10 5 10 1 10 0.5 2 10 1 3 10 1.5 5 10 2.5

10 10 5

With regard to the cospectrum and flux measurements, isotropy associated with the inertial subrange (no material is transferred because motions in the x, y and z planes are equivalent), pushes significant eddies towards lower frequencies. Sampling restrictions are not as severe. Sampling too slow leads to aliasing problems. Aliasing occurs when a high frequency signal appears as a lower frequency signal, since the harmonics of the high frequency signal are folded back on the lower frequency signals in the band between fs/2 and 0. This effect will distort the shape of the measured spectrum. A common example of aliasing is the appearance of wagon wheels rotating backward on Western movies. Aliasing results in spurious energy or power being attributed to lower frequency eddies. In order to minimize aliasing the sampling rate should be at least 2 times the highest frequency of interest. This rule of thumb is derived from Shannon’s sampling theorem says that at least 2 samples per cycle are needed to define the frequency component of that cycle. We attempt to measure the whole spectrum of turbulence when designing and conducting an experiment. However, field measurements can suffer from the aliasing effects of 60 Hz AC noise if this information is significant and if it is not pre-filtered by analog means before digitization.


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Figure 10 Visualization of aliasing, where high frequency components match and

add to low frequency components. Shows the importance of high pass filtering before digitization of eddy flux data. Analog pre-filtering using Butterworth, RC or Chebychev filters is helpful for removing environmental and electrical (AC 50 or 60 Hz) noise. This filtering was commonly done in early systems. Modern systems tend to already be filtered, so banks of analog filter systems tend not to be components of many systems today. But this is a feature to consider when designing and fabricating a system.

A filter function can be applied to compensate for the impact of aliasing [Moore, 1986].

3( ) 1 ( )digitizations

fH f

f f

This relation is valid for frequencies less than one-half the sampling frequency, f<= fs/2 Electronic low pass filtering is performed to minimize aliasing. In action, it minimizes the folding of ‘energy’ from frequencies higher than the Nyquist frequency onto lower frequencies. This filter is not to be confused with the digital recursive filter we will use to remove running means. That one is applied at another end of the spectrum:

4 1

0

( ) [1 ( ) ]lowpass

fH f

f

Massman (2000), among others, argues that it is wrong to filter and correct for aliasing, as it is an artifact of digitization. Aliasing distorts the spectrum, but does not contribute

0.0000 3.1415 6.2830 9.4245

-2

-1

0

1

2


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to power or covariance; my mentor, Shashi Verma, also firmly stated that aliasing cannot be removed once an analog signal has been digitized.

Figure 11 Low pass transfer function and spectrum

Sensor response Mechanical instruments, like a cup anemometer, can impose a filtering effect on the process it measures. A cup anemometer will have a stall speed, below which it will not respond. If there is a dynamic pulse, there is a distance or time constant to which the instrument will react. A typical filtering function for a first order response is:

2 2 1/2_ ( ) (1 (2 ) )sensor response cH f f

Key factors are the frequency and sensor time constant. Tube dampening Often, biometeorologists use a closed path sensor with a small tube that extends to near the volume of the sonic anemometer. Classic examples pertain to the measurement of CO2, methane, ozone, SO2 and water vapor. Ideas on sampling through a tube originated with theoretical studies by Taylor (1920s), Philip (1963). More modern treatments have been produced by Massman [Massman, 1991], Raupach and Lenschow [Lenschow and Raupach, 1991]and Leuning and Moncrieff[R. Leuning and Moncrieff, 1990].

f

0.1 1 10

H(f

)

0.01

0.1

1

10 digitization, 10 Hz sampling low pass


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The transfer function associated with samping through a tube is dependent on the tube radius (r), frequency (f) and diffusivity (D). If the dimensionless quantity, derived from these factors, is less than a critical value:

2210

x

fr

D

_ ( ) exp( / 6tube dampening xH f x D u

else the transfer function is one H(f)=1 (see Leuning and King 1992). Massman reports a slightly different transfer function for turbulence flow:

2 2 2_ ( ) exp( 4 )tube dampeningH f f Lru

L is tube length, For Re > 2300

1 10.5 Re | |i D

This case is valid for laminar flow, there is no dispersion of density fluctuations and they flow together with a velocity, denoted u. The problem with using laminar flow is that tube lengths and residence time in the tube needs to be short. Otherwise ‘packets’ of fluid start to diffuse and lose their coherence. For laminar flow Re < 2100, Massman uses:

10.0104 Re D The tube attenuation cutoff frequency

2 1 1/2(0.008779 ( ) )cf u Lr

Another attribute of sampling with turbulent flow is that a significant pressure drop is needed. In this situation, air is less apt to condense on the walls of the tube (Mike Goulden, personal communications). The tube dampening correction does not correct for potential absorption/desorption of moisture by hygroscopic dirt particles in the tube or the diffusional loss of CO2 or water vapor through a tube.


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The impact of sampling air with a closed or open path sensor has been quantified by several investigators [R. Leuning and King, 1992; R. Leuning and Judd, 1996; Suyker and Verma, 1993]. Sukyer and Verma have conducted extensive tests of open vs closed path sensors. There seems to be extensive line loss of water vapor. The problem is a function of tubing type and how dirty a tube is. Hygroscopic particles on a tube may be a reason. Today, no one has performed theoretical calculations about loss of water vapor as it flows down a tube due to permeability of the tube or absorption/desorption of hygroscopic particles. Leave this as a challenge for a precocious student.

Figure 12 Suyker and Verma. Spectral characteristics of eddy covariance

computed with an open and closed path infrared gas analyzer. In 1992, we conducted a study and compared open and closed IRGAs. In this case the agreement was reasonable, but the tube was short and the study was conducted only over 3 weeks, so the tube was relatively clean. Leuning et al reports that corrections for the effect of T fluctuations are not needed when one samples through a tube if temperature fluctuations are attenuated.


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Figure 13 Comparision of an open and closed path IRGA to measure CO2 flux

densities. (Baldocchi and Guenther, unpublished). More recent and extensive measurements comparing open and closed path IRGAs by our group shows clear attenuation of <w’q’> and <w’c’> at a windy site and over an actively photosynthesizing and evaporating rice paddy. Here in this wet environment we see 20% reduction in water fluxes measured with closed path sensors, due to line attenuation. This hygroscopic filtering is hard to fix. My colleagues in France, so gradual degradation after a few days of using a new tube. The new LI7200 places the sensor on the tower and so it uses a very short tube. This may be a good solution to this otherwise nagging problem.

Twitchell Island

<w'q'> open-path

0 2 4 6 8 10 12 14 16 18

<w

'q'>

clo

sed

pa

th

0

2

4

6

8

10

12

-1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

wit

ho

ut

We

bb

T c

orr

ec

tio

n

slope=1.09r2=0.99

Deciduous forest

Fc

clo

sed

pat

h (

mg

m-2

s-1)

Fc open path (mg m-2 s-1)


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Twitchell Island

<w'c'> open-path

0 2 4 6 8 10

<w

'c'>

clo

sed

pat

h

0

2

4

6

8

10

Comparison of the attenuation of sampling methane co-spectrum through a closed and open sensors is seen below.

On annual time scales, Haslwanter et al report [Haslwanter et al., 2009] that the closed path system yielded more positive net ecosystem exchange (25 gC m-2 y-1) than an open


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path system (0 gC m-2 y-1), and lower evaporation totals (465 mm yr-1) compared with the open path system (549 mm y-1). In practice more data from open path systems will be excluded over a year due to rain, dew, fog etc. and the closed path system can be calibrated regularly. Sensor Line Averaging Sonic anemometers and infrared gas analyzers have finite sensor paths. These path lengths smear smaller eddies. One cannot detect frequencies smaller than the pathlength, which is why small hot wire anemometers are used to measure finer scale turbulence. One proposed transfer function for sensor line averaging was proposed by Van den Hurk (1995):

1 1 exp( 2 )( ) (3 exp( 2 ) 4 )

2 2pathave

fH f f

f f

here f is a normalized frequency, nd/u, where d is the path length. Sensor Separation The velocity and scalar sensors should be co-located to minimize a decorrelation as an eddy of a given size passes through the sensor array. Yet care must be made to minimize the scalar sensor from distorting the flow sensed by the anemometer. Kristensen [Kristensen et al., 1997] recommends that the separation distance d < (z-d)/5 Kaimal recommends a more conservative metric, d=(z-d)/6. More recently, Lee and Black report that error is less than 3% if the ratio between separation distance and z-d is less than 5%. If one is interested in computing the transfer function for sensor separation, one can simulate it with the following algorithm:

1.5_( ) exp( 9.9 )sensor separationH x x

x is a normalized frequency (fs/u) There are new questions about where to place an open path irga relative to a sonic anemometer. Kristensen et al [Kristensen et al., 1997] recommends placing it below the anemometer, but in the same vertical access. If it is displaced horizontally, there will be a time lag (d/u) based on the wind speed and the spatial separation. They state that the loss of flux will be less than 1% if the displacement at 10 m is 0.2 m, but it can be 13% if the instrument is at 1 m. In the vertical, the loss is 18% if the scalar sensor is displaced above the sonic, but only 2% if the scalar sensor is 0.2 m below the sonic anemometer. They conclude that when sampling near the ground vertical separation is preferred with the scalar sensor below the anemometer. This keeps a symmetric configuration.


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The Total Transfer Function Moore reports net transfer functions as

_ sin _ _data acquisition alia g high pass low passT H H H

_ _ _ _ _ _ _ _measurement sensor separation instrument response path ave chem sensor path ave w tube dampeningT H H H H H

So the total transfer function for CO2 flux would be

A basic version of the Moore code is provided on the course web page for use and implementation,

Figure 14 Examples of integrated transfer function for heat, water vapor and

Improper spectral response can be compensated two ways. One is to correct the flux covariance by the ratios of the observed spectrum and some ‘perfect’ measure, such as the acoustic temperature co-spectrum. This method is used by the group at Harvard Forest, for example. Two down sides with this approach. It does not account for line

n(z-d)/U

0.001 0.01 0.1 1 10

Tra

nsfe

r F

unct

ion

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

w'T'w'q'w'c'

_( ) ( ) ( )total data acquisition measurementT T T


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averaging, which occurs because of the fit distance of the anemometer path, and fails with sensible heat flux density is near zero. The other approach involves quantifying the appropriate transfer functions and using them to correct the eddy covariance method. This approach was developed by Moore in 1986 and has recently been modified by Massman [Massman, 2000]. It is based on certain assumptions about the co-spectral shape of turbulent transfer. Most recently, Massman (2000) derived an analytical version of the numerical transfer functions originally posited by Moore (1986). The general equation is:

' ' 1

11 1 1 1' '

mw c ab ab p

a b a p b p p a a pw c

where 2 x ha f and is a function of the time constant for the trend removal

2 x bb f and is a function of the time constant for the block average removal

2 x ep f and is a function of the first order time constant

xx

n uf

z

For z/L < 0 nx = 0.085 Massman’s relation does not account for aliasing or vertical displacement of sensors.

The transfer functions are then used estimate and correct the relative error in an eddy covariance measurement. With this approach one computes:

0

0

( ) ( )

1

( )

wc wc

c

cwc

H S dF

FS d

For engineering purposes, Kaimal et al. derived equations for predicting spectral shapes under near neutral conditions

2 5/3*

( ) 2.1

1 5.3wnS n n

u n


23

2 5/3*

( ) 102

(1 33 )unS n n

u n

2 5/3*

( ) 17

(1 9.5 )vnS n n

u n

The spectrum of turbulence also scales with stability. The spectral peak shifts toward larger wavelengths with convective conditions. Thermals scale with the depth of the pbl, hence the shift towards longer wavelengths.

Figure 15 Computations of Spectra as a function of stability, using the Kaimal

functions

Numerous functions exist in the literature for co-spectra, too, starting with the famous Kansas experiment.

2.1( )wx

fnS n

A Bf

nz/u

0.0001 0.001 0.01 0.1 1 10 100

nSw

w(n

)/

w2

0.0001

0.001

0.01

0.1

1

z/L=0 z/L=1 z/L=-1


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The coefficients A and B very with stability. We also assume co-spectral similarity, in that the spectra for heat, water and CO2 are identical. Accumulating data taken over tall forests also show evidence of a spectral shift, as compared to idea conditions simulated by the Kaimal spectra. A few new comments should be raised about the classical Kaimal spectra. First they were developed over very flat, ideal landscape. Secondly they were developed on the basis of about 45 hours of data. More recently, Kai Morgenstern has used data from the Fluxnet project to examine turbulence spectra over 100,000s+ hours of data and from about 20 field sites. Detto et al. [Detto et al., 2010] provide the newest set of algorithms for spectra under stable, near neutral and unstable thermal stratification from thousands of hours of measurements and for u, T, q, CO2 and CH4. In terms of wavenumber the spectral equation is

0( ) ( )(1 )mS S

For shorthand we provide the matices of coefficients for u, T, H2O, CO2, CH4, indexed j=1,2,3,4,5, and for stable, near neutral, unstable thermal stratification, indexed i = 1,2,3.... % S0 power spectra S0(1, :) = [4,2.5,3.1,1.6,0.4]; % stable S0(2,:) = [8,5.3,3.8,2.9,0.2]; % near neutral S0(3,:) =[8.9,6.8,5.2,3.9,2.3]; % unstable % km power spectra (m^-1) km(1,:)= [4,3.1,4.3,3.1,2.1]; % stable km(2,:)=[11.1,12.3,9.3,10.5,1.4]; % near neutral km(3,:)=[14.2,16.8,12.2,14.8,15.2]; % unstable % gamma, v, power spectra v(1,:)=[1.7,1.5,1.6,1.3,1]; % stable v(2,:)=[1.6,1.2,1.4,1.2,1]; % near neutral v(3,:)=[1.6,1.3,1.4,1.2,1.3]; % unstable % S0 c spectra Co.S0(1,:) = [4,2.5,3.1,1.6,0.4]; % stable Co.S0(2,:) = [8,5.3,3.8,2.9,0.2]; % near neutral Co.S0(3,:) =[8.9,6.8,5.2,3.9,2.3]; % unstable % km power co spectra Co.km(1,:)= [4,3.1,4.3,3.1,2.1]; % stable Co.km(2,:)=[11.1,12.3,9.3,10.5,1.4]; % near neutral Co.km(3,:)=[14.2,16.8,12.2,14.8,15.2]; % unstable % gamma, v, co spectra


25

Co.v(1,:)=[1.7,1.5,1.6,1.3,1]; Co.v(2,:)=[1.6,1.2,1.4,1.2,1]; Co.v(3,:)=[1.6,1.3,1.4,1.2,1.3];


26


27

Figure 16 Su et al 2004 BLM

Averaging Interval A proper time averaging duration is needed to assess the mean variables and the fluctuations from the mean. No one averaging time is adequate for all statistical moments! Ideally we want an averaging time that is long enough to sample all the energy under the power or co-spectra, but not so long that red noise is encountered as we experience the diurnal change in wind, temperature, light and humidity, as the sun moves across the sky. Co-spectral analysis is a key means for identifying the length of the averaging period. Lumley and Panofsky (1964) were one of the first researchers to discuss this topic. The ensemble average of the variance of a meteorological variable is a function of the averaging time (T) and integral time scale (:

22 2 xx T

With this knowledge, and information on the sampling error, , we can compute the desired averaging time.

2

2 2

2 xTx

They reported that the time duration for a flux covariance during neutral conditions is a function of the height above the ground, the mean wind speed and the desirable tolerance for error:

T z d u 200 2( ) / ( ) Typically, the averaging duration to capture all the components that contribute to the turbulent flux is between 30 minutes and one hour. Longer averaging periods are ill advised because the assumption of temporal stationarity breaks down. Others, Wyngaard (1976), [Sreenivasan et al., 1978] and Lenschow et al. (1994), have treated this subject in greater detail. Sreenivasan et al. (1978), in particular, provide theory for the averaging time criteria for 1st, 2nd and third order moments.


28

2 2

2 2

2x

x xT

x

is the relative error and is the turbulence time scale. Typical averaging times for 2nd and 4th moments is listed in the Table.

' 'u u After Sreenivasan et al. BLM 1978 variable Sampling time,

10% error (min) Sampling time, 20% error (min)

' 'u u 12.1 3

' 'w w 3.4 .9

' 'T T 18.1 4.5

' 'q q 10.8 2.7

' ' ' 'u u u u 53.1 13.3

w'4 17.6 4.4

T '4 56.2 14.1

q'4 36.8 9.2

( ' ' )w u 2 20.4 5.1

( ' ' )w T 2 15.8 3.9

( ' ' )w q 2 10.3 2.57

With regards to the integration times for flux covariances, Sreenivasan et al. recommends:

T z d uwu 30 2( ) / ( )

T z d uwT 64 2( ) / ( )

T z d uwq 44 2( ) / ( )

Assuming 4 m height, 3 m/s wind speed and 10% error yields 8533s for <w’T’> and 5866 s for <w’q’>. If we reduce the error tolerance to 15% we obtain recommend sampling times of 3792 and 2607 s for heat and water vapor fluxes, respectively. Flux Variance


29

Since turbulence signals are inherently fluctuating, one will acquire a given amount of sampling error and statistical noise in sampling a flux covariance. Lenschow, Wesely, among others have treated this topic and have produced formula for guidance. Lenschow evaluates the coefficient of variance of a flux covariance as a function of the correlation coefficient, the turbulence time scale and sampling duration F wc

wcF T

r

r

( ) ( )/ /2 11 2

2

21 2

Wesely and Hart (1985) evaluated flux variance as a function of instrument noise and sampling error w c

c

w c

cF

R b

FTuz

' '/

/

( )

( )

1 2 2 1 2

1 2

Analysis of flux sampling errors by Finkelstein and Sims (2001) show they are about

10% for H and 25 to 30% for trace gases. Sensor Alignment and Coordinate rotation Ideally, we intend to measure flux densities across the mean streamlines of the wind. From this statement we must realize the the vertical velocity measured by a three dimensional sonic anemometer is not the true vertical velocity of the atmosphere, due to errors imposed by improper instrument alignment and distortion of the air streamlines by the underlying surface. As a stopgap people tend to align the instrument with the geopotential. The simple Reynolds average equation suggests that the mean vertical velocity, w, is zero, which is true near the surface as vertical motions must go to zero at the ground surface. Over flat land the mean vertical velocity is close to zero and over long times indeed averages to zero Misalignment of the vertical velocity sensor can cause an apparent mean vertical velocity. Tilt errors are about 3-4% per degree for scalar fluxes and about 14% per degree for momentum fluxes [Ray Leuning et al., 2012](Dyer et al). Tilt errors can be minimized by rotating the coordinates of the wind velocity vectors so the vertical axis is orthogonal to the mean wind streamline, not the Earth’s geopotential. Errors can also


30

result from transducer shadowing of the sonic anemometers. Modern sensors tend to correct such effects, but they are empirical and vary from sensor to sensor.

Figure 17 Schematic on the need to rotate coordinates

To remove mean w from the covariance calculation, a coordinate rotation is applied to the flux covariance measurements. To do so we define several trigonometric values that are deduced from the orthogonal wind vectors:

2 2 0.5 2 2 2 0.5

2 2 2 0.5

2 2 0.5

2 2 0.5

cos ( ) / ( )

sin / ( )

cos / ( )

sin / ( )

U V U V W

W U V W

U U V

V U V

cos cos cos sin sinu U V W

cos sinv V U

cos sin cos sin sinw W U V

Here, capital letters denote covariances measured in the original coordinate system of a three dimensional anemometer and the lower case vectors are the rotated coordinates.

WU

U

U


31

The first rotation is about the z axis and aligns u into the x direction of the x-y plane. The second rotation is about the y axis and aligns w into the z direction yielding a mean value of w and v equal to 0. The u vector is equal to the mean scalar wind speed

2 2 2 0.52 ( )u U V W .

If v’w’ is non zero a third rotation is needed along the z-y plane[Detto et al., 2008].

Applying simple Reynold’s averaging rules to the algebraic sums we obtain a new equation for the flux covariance:

' ' ' ' cos ' ' sin cos ' ' sin sinw c W C U C V C

First

u

v U (u2 v2 )1/2

U

U

Second

w S (u2 v2 w2 )1/2

S

w 'c ' W 'C ' cos U 'C ' sin cos V 'C ' sin sin

cos u

(u2 v2 )1/2

sin v

(u2 v2 )1/2

cos U

(u2 v2 w2 )1/2

sin w

(u2 v2 w2 )1/2


32

With MATLAB and other matrix based programming languages it is instructive to recast the coordinate rotation concept in Vector notation

Basic Law

1

1 , ,

1

m

x y z m

m

x x

y R y

z z

11 12 13 11 12 13

21 22 23 21 22 23

31 32 33 31 32 33

a a a x xa ya za

a a a y xa ya za

a a a z xa ya za

1. Rotate around the vertical axis, z

1 1 mU R U

1 cos sinm mx x y

1 sin cosm my x y

sin( )tan( )

cos( )

V

U

1tan ( )

1

cos sin 0

sin cos 0

0 0 1

R

1

1

1

cos sin 0

sin cos 0

0 0 1

m

m

m

u U

v V

w W


33

Rotate about y

2 2 1U R U

x x zm m1 cos sin

z x zm m1 sin cos

2 2 1/2

sin( )tan( )

cos( ) ( )

W

U V

1tan ( )

2

cos 0 sin

0 1 0

sin 0 cos

R

2 1

2 1

2 1

cos 0 sin

0 1 0

sin 0 cos

u u

v v

w w

A third rotation is small, but can be done by rotating about x to make ,w’v’ equal to zero.. Use of coordinate rotation 1. Accounts for slight slope in terrain; flux is orthogonal to mean streamline, not geopotential. 2. It is difficult to orient sonic anemometers normal to the mean streamline from towers 3. First order correction for streamline and structure flow distortion.

4. Does not apply if flow separation occurs, which is when slope is about 15 degrees Impact of non-zero w.


34

Ideally, we want to rotate our coordinate system to compute flux covariances that are orthogonal to the mean streamlines flowing over the landscape. Over level terrain and in the absence of mesoscale circulations, this condition is attained naturally, as mean vertical velocities equal zero and are independent of wind direction. Over rolling terrain or under conditions where mesoscale circulations persist, this ideal behavior is not expected or observed. There is also no guarantee that there is not an electronic bias on the mean vertical velocity. Any bias error on w will force a rotation to the wrong streamline [D. D. Baldocchi et al., 2000; Lee, 1998; Wilczak et al., 2001; Yuan et al., 2011].

The vertical velocity, denoted mw , is the temporally averaged, vertical velocity, which is measured over a 30 minute interval with a sonic anemometer. Another vertical velocity (

Tw ) is a function of wind direction (hence, topography) and instrument orientation and

bias offsets attributed to the tower and anemometer.

m r Tw w w

Normalized vertical bias velocity ( Tw ) is a quasi-sinusoidal function of wind direction.

In principle this behavior results because vertical velocity is positive when wind blows up the hill, it is negative when if blows down the slope and is zero when wind is aligned across the slope[Rannik, 1998; Wilczak et al., 2001]. At this complex site, the majority of data come from the southwest and northeast quadrants. From these quadrants the rotation angle is typically less than 10 degrees, which is not too severe to cause flow separation and to shed wake vortices. The other feature to be noted in the Figure is how the data scatter along the regression line. This scatter denotes the functional behavior of wr. These deviations are presumed to be due to drainage flow or localized convection (Avissar and Pielke, 1989; Lee, 1998).


35

Figure 18 Impact of terrain on vertical velocity. The bias value of w is a function

of wind direction as wind blows up, across or down a gentle slope. Rotating these data to zero, may not be valid. (after Baldocchi et al., 2000) In the figure we see that the mean vertical velocity is the sum of a wind direction dependent mean value (the regression through the data set) plus fluctuations from the mean regression. Over flat terrain the mean value is zero and we simply rotate w fluctuations to zero. Over complex terrain we rotate the vertical velocity fluctuations to the mean value, then apply the residual vertical velocity in the advection relation [D. D. Baldocchi et al., 2000; Lee, 1998].

wind direction

0 40 80 120 160 200 240 280 320 360

w/u

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5


36

Figure 19 Mean mesoscale vertical velocity over a deciduous forest in rolling

terrain. [D. D. Baldocchi et al., 2000]. Over a year the mean velocity is less than 0.005 m s-1.

Other sites will produce a different patter of w with wind direction. Next is an example from the Vaira grassland in California. In this case a sinusoidal model will not fit the data well enough. So like the analysis of Yuan et al. one may need to treat each sector separately.

Day

0 50 100 150 200 250 300 350

wm

eso

(m

s-1

)

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6


37

Figure 20 vertical velocities from a relatively flat grassland site (Ione, CA). There

is some slope to the east. Notice that the range of values are constrained less than 5 cm s-1, for the most part, while over the forest values range +/- 30 cm/s (unpublished data of Baldocchi and Xu) In recent papers by Finnigan et al [2003], Massman [2002], Detto et al [2008], Yuan et al [Yuan et al., 2011] and formal discussions by the FLUXNET community a consensus is growing that investigators should use the planar fit method of rotating coordinates. This approach will account for offsets of the instrumentations, flow distortion by towers and by the terrain. Our experience shows there are some trades offs by adopting this method. For simple cases such as our grassland and savanna we see negligible differences using either method. There is also the additional issue of introducing new sources of error by adding an additional correction term, that is itself uncertain and error prone.

Wind Direction

0 50 100 150 200 250 300 350

w/u

-0.10

-0.05

0.00

0.05

0.10

Grassland


38

Fc, 1d rotation

-30 -20 -10 0 10

Fc

3D r

otat

ion

-30

-20

-10

0

10

Figure 21 Comparison between fluxes computed with a 1d vs 3D rotation for a

complex landscape with sloping terrain. Regression slope in this case is 0.975. But over a year this 2% difference adds to 15 gC m-2. Sonic Temperature There are two ways to measure temperature fluctuations [Kaimal and Finnigan, 1994]. One involves a microbead thermistor, with its separate electronics and sensor. Temperature fluctuations can also be deduced from sonic anemometer measurements . To a first approximation the temperature measured by a sonic anemometer is related to the virtual temperature [Schotanus et al., 1983]. It is derived by knowing information on the time it takes for a sound pulse to travel to and from transducers one and two and knowing the distance between transducers. Hence a sonic anemometer deduces wind velocity by the difference in the transmission of sound between two sources and receivers:

1 2

1 1( )

2

dV

t t

By taking the difference between two transit times, we eliminate need to know the speed of sound directly. But the answer is sensitive on knowing the distance between the transmitters and receivers. And if we know the wind speed and the transit time of sound pulses we can compute the speed of sound and deduce the virtual temperature:


39

22 2 2

1 2

1 1( )

4 n

dc V

t t

2 2 2n x yV V V

We also know the relation between virtual temperature and the speed of sound

2 (1 0.51 ) vc RT q RT

Algebraic substitution of an equal number of equations and unknowns produces an estimate of virtual temperature:

R= 403 m2 s-2 By manipulating equations we arrive at an equation for mean sonic temperature, as a function of air temperature and specific humidity.

(1 0.5 )sT T q

The flux covariance measured by the sonic anemometer is a function of the actual temperature-w covariance and terms related to vapor and momentum flux density [Schotanus et al., 1983].

'

2' ' ' 0.51 ' ' 2 ' 's

T qw T w T T w q w u

c

With the development of new sonic anemometers of various geometries, Liu et al. [Liu et al., 2001] revisited the Schotanus equations and have produced a new version of this correction;

2 2 21 2 31

(1 0.51 ) ( )3 403 403 403

n n ns

V V VT T q

2 22

1 2

1 1

4n

s

VdT

R t t R


40

2 2 21 1 1 1nV Au B v C uv

2 2 22 2 2 2nV A u B v C uv 2 2 23 3 3 3nV A u B v C uv

The A, B and C coefficients depend upon the geometry of the sonic anemometer.

For the Windmaster Pro sonic anemometer, we use, (Gill instruments) 45 with 3

axes angle ( 30 , 150 , 270 respectively for axis 1, 2 and 3 (note that the sign of the

angles is irrelevant) leading to:

2 2 23 4 3 4 1 2corr uncorrs s

d d

U V WT T

R

(5)

And the corrected kinematic heat flux density is:

0.51

1 0.51s r

r

w T T w qw T

q

In a closing comment, one has to be careful when applying these equations, as some sonic software already considers the momentum effect. We are finding it is extremely critical to maintain a constant path length, d. With long term operation of sonic anemometers in the field the head can get distorted as birds land on the array. By inspection we can show that a 1 mm change in path length can cause a 4 K change in sonic temperature, because it is a function of d squared!

TOP VIEW

X

Y


41

It is critical to calibrate and test all instruments, even if they seem absolute. We find thatsonic temperature does not match virtual temperature. Sensor placement Instrument array should be placed within the constant flux layer, but out of the roughness sublayer. Placing the flux array between 1.3 and 1.5 times canopy height usually suffices. Sensor sensor placement is also affected by field size, as we do not want to place a sensor so high that it is out of the internal boundary layer. Yet, higher instruments measure larger and low frequency eddies. Higher instrument placement is necessary Sensor height and field size We desire to make flux measurements in the constant flux layer. The depth of the constant flux layer can be calculated from boundary layer theory as:

0.80.1 0.75 ( / )o oz x z

This analytical relation assumes near neutral thermal stratification and ignores the zero plane displacement, which is non-zero over tall vegetation. There are limits to how high one can place a sensor, even over extremely large fields. One still needs to place the sensors in the lower portion of the boundary layer. As one goes higher and exceeds a few percent of the height of the planetary boundary layer, flux divergence and advection occurs. Plus storage is non negligible. Horizontal homogeneity and non-establishment of an internal boundary layer can cause problems (show advection data from potatoes). So either we need to limit how high we place our instruments or restrict how small the underlying field is. Rule of thumb is to have a 100 to 1 fetch to height ratio. This rule is very conservative and in actuality is a function of surface roughness. A shorter fetch to height ratio is needed for the transition from a smooth to rough surface than vice versa. Gash provides the following equation for 90% development of the internal boundary layer under neutral conditions.

2

(ln( / ) 1 / )

ln(0.90)o o

f

z z z z zx

k


42

Figure 22 Calculations of fetch extent for 2 different measurement heights, using

the analytical algorithm of Gash Since the cutoff frequency is fc=2=nc (z-d)/u, this information can be used define the height to place ones sensors based on the physical limitations of the instruments and the wind climate where the experiment is taking place.

(z-d)=2 u/nc If maximum winds are rarely about 5 m/s and the instrument responds to 5 Hz, then a placement of 2 m is adequate. We typically place our sensors about 4 m above short statured crops, at 20 m over a 13 m pine forest and at 35 m over a 25 m deciduous forest Flow Distortion If one is mounting ones instruments on a walkup scaffold tower, as used in many forest meteorology applications how far should the instruments be away from the tower? The answer to the question depends on the type of anemometer. Some are horizontally displaced and others are vertically displaced. There are advantages and disadvantages to each design configuration.

distance (m)

0.1 1 10 100 1000 10000

f(x,

z m)

1e-5

1e-4

1e-3

1e-2 zm equal 1 m

zm equal 5 m

Gash footprint model


43

The horizontal anemometer is placed into the wind. Its electronics and case are behind, so the head is aerodynamically clean. But this sensor must be mounted horizontal to a tower. One rule of thumb is to place the sensor about 3 times the crosswind length scale of mast. Of course this rule of thumb will vary with tower porosity. In general, one should ignore data when wind is blowing through the tower. One can visualize flow distortion by comparing the drag coefficient with wind direction.

Vaira 2004

Wind Direction

0 100 200 300 400

u*/

u

0.01

0.1

1

10

Figure 23 Drag Coefficient over grassland. This is for an omni-directional sonic

anemometer and note little effect is seen with wind direction as opposed with a horizontal sonic anemometer which will suffer when wind flows through the tower infrastructure behind the sensor. Other scientists like to place sonic anemometers on top of mast. The advantage is equal wind direction acceptance. But the sensor head is placed on top of a volume of electronics. Wyngaard [Wyngaard, 1981]warns that there can be asymmetric distortion of the eddies that transfer material (causing cross-talk errors). Sensor Noise and Minimum Detectable Flux Density Sensor noise and statistical variance from natural variability of turbulence affects the accuracy of the flux measured. From Lenschow and Kristensen [Lenschow and


44

Kristensen, 1986], the error variance of a scalar flux is finite due to length of the sample period (T), the variances of c and w and and the integral time scales for the scalar and w ():

2 2 2 min( , )( , ) 4 c w

c wF TT

Additional flux variance can be introduced by sensor noise. Ritter et al. (1990) assessed the total error variance due to turbulence and sensor noise.

2 2 2 min( , )( , ) 4 c w

c wF TT

it is important to make this assessment to determine whether or not a measured flux is significantly different from zero. Averaging numerous runs also helps reduce run-to-run variability introduced by noisy sensors From elementary statistics the necessary sample size to determine a flux within 5% of the actual mean (0.05 d) (I assume a 20% standard deviation (0.2 s) due to geophysical variability and the intermittency of turbulence and prescribe a t value of 1.645) is:

n=t2 s2/d2

If we desire a flux within 5% of the actual mean (0.05 d) when the standard deviation due to geophysical variability and the intermittency of turbulence is 20% and the prescribed t value of 1.645), 43 runs are needed. Of course this statistical analysis is over simplistic and presumes stationarity and repeatablitity of turbulence conditions. In some respects these conditions can be met by looking at the behavior of diurnal trends that have been bin averaged. When binned and averaged over many days the standard deviation and error become progressively smaller.

Appendix


45

The Fourier transform (Sxx()) at a particular angular frequency (=2fradians

per second; f is natural frequency; cycles per second) of a stochastic time series (x(t)) is

defined as:

( ) ( ) exp( )xxS x t i t dt

(1

In Equation 1, i is the imaginary number, i . One attribute of examining Fourier

transforms is that, according to Parseval’s theorem, the variance (x2) is related to the

integral of the power spectrum with respect to angular frequency:

2 2| ( ) |x xxS d

(2

It thereby allows us to examine the amount of variance associated with specific

frequencies.

Remember

exp( ) cos sinix x i x

exp( ) cos sinix x i x

| exp | 1ix

1cos [exp( ) exp( )]

2x ix ix

1sin [exp( ) exp( )]

2x ix ix

So the Fourier expansion series:

0

1 1

( ) cos sin2

N N

k kk k

Af A k B k

Fourier Transform Pairs


46

1( ) ( ) exp( )

2xx xxR S i d

1( ) ( ) exp( )

2xx xxS R i d

The spectral relation between two independent, but simultaneous, time series was

quantified with a co-spectral analysis. The co-spectra derived from the cross spectrum

(Sxy()) between two time series, x(t) and y(t). The cross spectrum is a function of the

cross-correlation function, Rxy:

1( ) ( )exp( )xy xyS R i d

(3

The cross-correlation between x(t) and y(t+) is computed as:

lim 1( ) ( ) ( ) ( )

2

T

xy

T

R T x t y t dtT

(4

The cross-spectrum has an even and odd component:

( ) ( ) ( )xy xy xyS Co iQ (5

The even component of the cross spectrum yields the co-spectrum, Coxy()):

1( ) ( ) cos( )xy xyCo R d

(6

and the odd component yields the quadrature, Qxy()), spectrum:

1( ) ( )sin( )xy xyQ R d

Fundamentally, these calculations are performed on discrete and evenly-spaced, time

series. The specific frequencies that can be decomposed from such a time series are

defined from / ( )nf n N t , where the time step between samples is t , the total number


47

of samples is denoted as N and the index n varies from –N/2 to +N/2. The discrete

Fourier transform (Fx) for for a time series (f(n)) at a time index number k is:

1

0

( ) ( ) exp( 2 )N

xn

F k f n i nk N

(9

The power spectrum is a function of the Fourier transform and its complex conjugate *( ) ( ) ( )x x x

tS k F k F k

N

. The co-spectrum and quadrature spectrum between two

variables, x and y, are computed in a related manner, with respect to the real (*( ) Re( ( ) ( ))xy x y

tCo k F k F k

N

) and imaginary ( *( ) Im( ( ) ( ))xy x y

tQ k F k F k

N

) components.

Bibliography Denmead, O.T. 1983. Micrometeorological methods for measuring gaseous losses of

nitrogen in the field. In: Gaseous Loss of Nitrogen from plant-soil systems. eds. J.R. Freney and J.R. Simpson. pp 137-155.

Lenschow, DH. 1995. Micrometeorological techniques for measuring biosphere-

atmosphere trace gas exchange. In: Biogenic Trace Gases: Measuring Emissions from Soil and Water. Eds. P.A. Matson and R.C. Harriss. Blackwell Sci. Pub. Pp 126-163.

Wesely, M.L. 1970. Eddy correlation measurements in the atmospheric surface layer

over agricultural crops. Dissertation. University of Wisconsin. Madison, WI. Wesely, M.L., D.H. Lenschow and O.T. 1989. Flux measurement techniques. In: Global

Tropospheric Chemistry, Chemical Fluxes in the Global Atmosphere. NCAR Report. Eds. DH Lenschow and BB Hicks. Pp 31-46.

Endnote References Aubinet, M., et al. (2000), Estimates of the annual net carbon and water exchange of Europeran forests: the EUROFLUX methodology, Advances in Ecological Research, 30, 113-175. Baldocchi, D. D. (2003), Assessing the eddy covariance technique for evaluating carbon dioxide exchange rates of ecosystems:past, present and future., Global Change Biology, 9, 479-492. Baldocchi, D. D., B. B. Hicks, and T. P. Meyers (1988), Measuring biosphere-atmosphere exchanges of biologically related gases with micrometeorological methods, Ecology., 69, 1331-1340.


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