The Eddy Covariance (EC) method is a statistical tool. Used to analyse high frequency wind and scalar atmospheric data series, it yields values of fluxes of these properties representing quite large areas.

 Transport in the boundary layer of heat, moisture, momentum and pollutants is governed almost entirely by turbulence (See section 4.1). Using the Reynolds decomposition it is possible to quantify turbulent transport given a high enough sampling rate and fast response instruments.

Recall that a signal (any signal) can be split up into mean and turbulent parts:

 Where is the instantaneous signal, is the mean of the signal evaluated over a suitable time period (section 4.1), and is the instantaneous perturbation from the mean. It is easy to show (dimensionally) that, for example, an advective aerosol number flux is given by:

 Where = Aerosol number concentration (in this case in units of per cubic metre). This gives a flux in units of (N = Number of aerosol) as long as the mean wind direction equals the direction. This is quite straightforward to ensure (see later section on rotation).

 In the boundary layer, however, we are often primarily interested in vertical fluxes. Since mean vertical wind speeds in the boundary layer are very close to zero under most circumstances, the vertical advective flux is usually found to be very small. In fact, since co-ordinate systems are often rotated to make the average vertical wind component zero (making vertical advection zero by definition), the advective flux is of little use.

We have stated that transport in the boundary layer is dominated by turbulence. Because of the way in which signals are decomposed (see section 4.1) this means we are interested in the perturbation parts of any signal. To see how the perturbation terms are used to derive a turbulent flux, look at an example.

 Imagine an eddy moving over a city centre with the boundary layer convecting. The city is emitting aerosol, giving rise to a negative number concentration gradient. The eddy mixes air from above downwards, and from below upwards. The air from above has a lower number concentration, and as it is mixed downwards causes a negative perturbation in a concentration time series. The air from below has a higher aerosol number concentration, so as it moves up it causes a positive perturbation in a time series being taken at the higher level.

Now, for the upward moving air is positive and is positive. For the downward moving air, is negative and is negative. This means that the perturbation product is always positive. It appears that this flux can be represented by:

This is still dimensionally correct, and represents the upward flux of aerosol in our example as a positive flux. Thinking carefully about the nature of turbulent transport, it becomes clear that on average, eddies can only mix quantities in the direction of the mean gradient of that quantity. That is, there is no turbulent process that could increase the number concentration in our example at the expense of .

We now have, in principle, a means of quantifying vertical turbulent fluxes of any measurable quantity. To complete this definition, we note that the right hand side of the equation above has an analogue in statistics. The definition of a statistical covariance is (with A and B representing any variable):

 Which from the first equation of this section (Reynolds’ decomposition) is:

From the definition of the mean.

 This means that we can find fluxes from covariances. Specifically, we find vertical turbulent fluxes from the covariances of with the variable of interest.

The eddy correlation technique is very simple in theory, but quite difficult to put into practice. A major problem has been availability of sufficiently fast instruments to measure the fluctuating signals. Unless an instrument has a sufficiently fast response and logging time to resolve the fastest eddies contributing to the turbulent flux, this portion of the flux will not be recorded. Another effect of slow logging is that the faster eddies may fold around the Nyquist frequency of the logger or instrument and show up as lower frequency turbulence which did not exist, again skewing the results.

This effect, known as aliasing cannot be detected or corrected for during data analysis. This means it is essential both to know the frequency response of the instrument used, and the likely scales of turbulence being encountered (hence the frequency) before taking data. These must be compared, and a suitable margin of error allowed for. The "suitable margin of error" will depend upon the likely development of the weather over the course of the measurements, as well as existing knowledge of turbulence scales over the surface in question.

As an aside, if a fast response instrument cannot be logged at a high enough speed, the ill effects on the data can at least be reduced. High frequency components of the flux will always be lost at unsuitably slow logging speeds, but the folding back of higher frequency perturbations can be avoided. This is done by either block averaging many data points within each logging period (unlikely to be practical if logging speed is a problem), or by damping the response of the instrument to match the logging frequency (usually with analogue electronics).

The frequency required for the Eddy Covariance method depends upon the mean eddy size at the sensors. This is a function of surface roughness length z₀ (see section 4.2.4) and measurement height z. Specifically, larger values of z₀ give larger mean eddy sizes and ease the frequency requirements for the system. Equally, larger values of z (larger measurement heights) give larger mean eddy sizes, having the same effect.

For example, during the SASUA I experiment, equipment was mounted on masts, on top of a large tower, on top of a hill in the centre of Edinburgh. This made for an effective measurement height of more than 65 m above street level. Although the measured values of z₀ are still under debate, the formula of Lettau (1969) presented in section 4.2.4 would suggest a large value. The result of this was that the turbulence contributing to particle fluxes was resolved at around 1 Hz.

Nemitz (1998) reports average eddy sizes equating to >30 Hz within small scale canopies. This is expected since in this situation we find both small measurement heights (z), and very small roughness lengths (z₀).

Ultrasonic Doppler anemometers are normally used to measure the three components of wind. These instruments are easily fast enough to resolve turbulence under most practical conditions, - they have maximum response and reporting frequencies up to around 100 Hz. The problem with such anemometers tends to be designing logging systems fast enough to log the data.

The potential difficulty in designing working Eddy Covariance systems is often in finding the right instrument to detect the variable of interest to be correlated with the vertical wind fluctuations (the scalar variable). Some instruments simply are not fast enough, for example the DMPS (see section 7), and many chemiluminescent analysers. Also, measuring certain gases is fundamentally problematic, as some ("sticky") gases can adhere to the inside of inlet lines before reaching the analyser. This damps the signal, giving the same problem as a poor frequency response instrument (although without the frequency folding).

The final factor affecting response time of eddy covariance systems is the size of the sensor. Eddies smaller than the size of the transducer array of the anemometer cannot be detected effectively. This places an upper frequency limit on response according to Taylor’s hypothesis (see section 4.1). The effect of this limit has not been observed in measurements referred this work.

Another problem particular to aerosol Eddy Covariance measurements is the instrument sample flow rate. Unlike a gas, aerosol are non-continuous, and there is only a limited, finite number of aerosol in a given volume of air. This number is often of the order of thousands per cc, as opposed to perhaps tens of millions of gas molecules.

A class of instrument particularly well suited to the response time requirements of the Eddy Covariance technique is Optical Particle Counters (OPCs). These generally work by passing a sample air flow through a cavity with a clean sheath flow around it. The aerosol breaks a laser beam crossing the cavity, and pulses of scattered light are counted to determine the number concentration of aerosol. Many models can even size the particles using fast electronics to analyse the intensity of scattering.

Most OPCs have a problem which normally only becomes apparent in exceptionally clean air, or when used in an application as demanding as Eddy Covariance. One reason for this is that each is designed to measure only a comparatively narrow band of aerosol sizes. Combined with the fact that most cannot detect aerosol in the nucleation mode (<100 nm) which is where a significant proportion of the number of aerosol are to be found, there are often not many aerosol available to be detected by such counters.

OPCs generally also have low sample flow rates (of the order of a few cc per second). When used to measure ambient concentration the low flow rate and size range is not a problem, as data can be block averaged to obtain a statistically significant sample. However, in Eddy Correlation applications the scalar measuring instrument is normally required to report at around 10 – 30 Hz, and no averaging up is possible. From basic statistics it is clear that the error (N = Number of samples) is likely to become a complicating factor.

It has been suggested (Fairall, 1984) that the minimum number of counts required to derive a statistically valid covariance is:

Where: Time period of the measurement

Zero plane displacement (see section 4.2.5)

Friction velocity (see section 4.2.1)

Deposition velocity (see section 4.2.6)

Nemitz (1998) notes that the associated error in deposition velocity is given by:

Where: Standard deviation in W

Number of particles detected over the time of the V_d calculation

And then goes on to solve the N_Min equation for the minimum sample flow rate required under certain conditions. Note that for OPCs the flow rate problem is even more acute than it appears, since N, N_Tot, And N_Min refer to counts within a relatively narrow band of aerosol sizes.

The fetch describes the terrain upwind of the measurement location. It is an important concept as the rate of deposition to a surface depends upon the nature of the surface. The footprint is defined as the area upwind of the measurement site contributing to the measured flux. It is possible to estimate the footprint of an Eddy Covariance measurement using the roughness length and an assumed logarithmic wind profile. However it is simpler and more practical to ensure that the part of the fetch of interest is large enough that the whole of the footprint must fall within it.

An easy way to check that the footprint falls within the desired part of the fetch is suggested by Monteith and Unsworth, (1990). Imagine a surface discontinuity such as a change in the type of surface vegetation, a group of houses, a wall etc. (Figure 2). A new "inner boundary layer" (Nemitz, 1998) grows, inside which the exchange properties are determined by the new surface.

According to Monteith and Unsworth, if the length of the fetch to the next upwind discontinuity is x_f, then the height of the new inner boundary layer is found in the range:

This is clearly only a rule of thumb, as the height of the new layer must depend upon the wind speed and other factors such as stability, u_*, and z₀ for both surfaces etc. It’s use must therefore be restricted to selection of an appropriate fetch to begin with, rather than detailed calculations for analysis purposes.

One of the requirements for a valid Eddy Covariance flux measurement is that the mean particle concentration () should not change significantly over the averaging time used to determine the mean. This is called the Stationarity requirement. It is clear that if there is a marked trend of increasing concentration over this period, the earlier perturbation parts are underestimated and the later ones overestimated. The converse is true for trends of decreasing concentration. Figure 3 shows a hypothetical increasing concentration situation.

  Figure 3: Schematic of the effect of an increasing concentration trend

It can be seen that the perturbation parts used in the covariance calculation can only have the correct values exactly in the middle of the averaging period. This produces incorrect assessments of the vertical turbulent flux. An example of this problem is shown below. Figure 4 shows data taken from the city centre tower site during SASUA I in May 1999.

The concentration shown in Figure 4 increases dramatically (by around 80%) over the course of ten minutes. Analysis of this data would certainly be complicated by the factors outlined above. (Lamaud et. al., 1994)

A way of dealing with such linear instationarities is to us some kind of low pass filter during the analysis stage. Known as de-trending, such an approach would involve shifting the values of concentration so that the mean remained valid throughout the time period. The simplest approach would be linear detrending (Buzorius et. al., 1998). They report that linear detrending causes a slight underestimation of the flux, but that this can in turn be estimated from spectral analysis. Underestimates of less than 10% of magnitude were reported.

Horizontal concentration gradients may also lead to perturbation calculation errors, which would propagate into the flux calculation, again causing errors. An example of such an effect can be seen at approximately 13:09 GMT in figure 4. Unless horizontal concentration gradients are explicitly measured at the same time as the flux, there is no effective way of correcting for this (Nemitz, 1998). Figure 5 shows an advection effect more clearly than figure 4. In figure 5, there is no concurrent trend as such in the data.

Buzorius et. al. (1998) describe the effect in figure 5 as a strong instationarity. They found detrending less effective in these circumstances, in some cases finding different directions of flux using different detrending methods. They noted, however, that the magnitude of the flux was always small in these cases.

One useful approach may be to Fourier transform the data, and despike the transform. This should remove the unwanted frequencies (associated with the instationarity) while leaving the transporting turbulence unaltered.

As stated in section 4.2, an assumption is made that the flux in the surface layer is constant with height (in fact this is how the surface layer is defined). While the Eddy Covariance method always yields the correct flux at the measurement point (allowing for the factors described above), the interpretation of results can be complicated by processes creating or destroying particles in the vicinity of the sensors.

For example, an upward flux may mean either that particles are being produced below the measurement height, that particles are coagulating above the measurement height, or both.

As stated previously, this is a matter of interpretation, and has no real effect upon the theory of Eddy Covariance.

As stated previously, mean vertical wind speeds in the surface layer are close to zero. It is difficult to check the orientation of an anemometer with any great accuracy, and the "airflow horizontal" and local actual horizontal may not coincide, depending upon the terrain being measured over. For these reasons, there is a convention when using the Eddy Covariance technique that the mean vertical wind speed is zero. To simplify calculations (such as friction velocity, ) the mean crosswind component is also zero. i.e.

No matter how carefully the anemometer is installed and subsequently rotated, this is unlikely ever to be the case. As part of the analysis procedure, the co-ordinate system must be rotated to make these conditions true. The rotation is simply implemented as part of the analysis software. Two methods are presented. All the quantities below are unrotated, the rotated flux of generic scalar "C" being:

Where: and

And since:

Note that this method applies a correction after the unrotated flux has been calculated, (the averages must be calculated anyway to do this). Equally valid would be to work out the averages, and use these to calculate and (above) and apply rotational corrections to the entire dataset before calculating the covariance:

These equations are proved geometrically, with no reference to covariances or fluxes (K. Beswick, pers. comm.).