ON FREQUENCY RESPONSE CORRECTIONS FOR EDDY
COVARIANCE FLUX MEASUREMENTS
Research Note
T. W. HORST
National Center for Atmospheric Research, Boulder, CO, U.S.A.
(Received in final form 28 October 1999)
Abstract. Several potentially significant errors are noted in published frequency-response corrections for eddy-covariance flux measurements.
Keywords: Frequency-response corrections, Eddy covariance, Flux measurement.
The importance of the role of surface-exchange fluxes in the dynamics of the
environment is becoming widely appreciated. At the same time, fast-response
anemometers, scalar sensors, and data acquisition systems are becoming more reliable, more affordable, and easier to use. As a consequence, environmental scientists
are increasingly using eddy covariance as a routine tool for the measurement of
surface fluxes of momentum, sensible heat, water vapor, and trace atmospheric constituents. These new practitioners, particularly non-micrometeorologists, rely on
the published literature to impart the procedures necessary for correct application
of eddy covariance.
One important aspect of micrometeorological flux measurements is assessment of and correction for attenuation of the measured covariance caused by
inadequate frequency response of the sensors and data acquisition system. Unfortunately, several errors have crept into published frequency-response corrections
and these errors appear to be proliferating. Moore (1986; hereafter M86) presents
a comprehensive discussion of frequency-response losses associated with fastresponse sensors and data systems, as well as recommending methods to correct
for these losses during data post-processing. There are two apparent errors in
this commonly-cited reference, one of which was discussed in Blanford and Gay
(1992); the other was discussed in Eugster and Senn (1995) and noted again in
Horst (1997). However, both errors are repeated in Rißman and Tetzlaff (1994;
hereafter RT94), along with the introduction of additional questionable approximations. These errors were recently brought to the attention of the present author
when they all appeared as established facts in a manuscript reviewed for another
journal. The quantitative consequences of these covariance correction errors can
range from negligible to significant, depending on the particular eddy-covariance
instrumentation and how it is employed, as well as on meteorological conditions.
Boundary-Layer Meteorology 94: 517–520, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
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T. W. HORST
The purpose of this note is to call attention to these errors in an attempt to curtail
their propagation.
M86 and RT94 estimate covariance attenuation coefficients Axy by integrating
known empirical formulas for the cospectral distribution of turbulence covariance
Coxy (f ), multiplied by a frequency-dependent system transfer function Txy (f ),
R∞
hx 0 y 0 im
0 Txy (f )Coxy (f )df
R∞
Axy ≡
=
,
(1)
hx 0 y 0 i
0 Coxy (f )df
where f denotes frequency, x 0 and y 0 are turbulent fluctuations of atmospheric
variables, brackets h i denote time-averaging of the data, and the subscript m
refers to a measured quantity. The system transfer function Txy is calculated as
the product of several independent transfer functions which account individually
for sensor frequency response, sensor spatial separation, frequency response of the
data acquisition system, etc. Thus, the corrected covariance is calculated as
hx 0 y 0 i = hx 0 y 0 im /Axy ,
(2)
where Axy depends on the response characteristics of the sensors and data system
as well as on measurement height, wind speed, and atmospheric stability.
One conceptual error in M86 and RT94 is neglect of the phase shift inherent in
applying a frequency-dependent filter to time-series data. This leads to the oversimplified assumption that the transfer function for the covariance of two variables
x and y is equal to the square root of the product of the transfer functions for the
variances of x and y,
p
Txy = Txx Tyy .
(3)
A simple example is two sensors that both have first-order frequency response, each
characterized by a different time constant τ . The transfer function for the variance
of x, and similarly for y, is
Txx =
1
,
1 + ω2 τx2
(4)
while the transfer function for the covariance is equal to
Txy =
(1 + ω2 τx τy ) + ω(τx − τy )Q/Co
,
(1 + ω2 τx2 )(1 + ω2 τy2 )
(5)
(Hicks, 1972; Kristensen and Lenschow, 1988; Horst, 1997), where ω ≡ 2πf and
Q is the quadrature spectrum. The quadrature spectrum of two collocated variables
is commonly assumed to be negligible, but even with this simplification Equation
(3) is valid only for the special case τx = τy . For the case τx = 0, Equation (3)
EDDY COVARIANCE FLUX MEASUREMENTS
519
is the square root of Equation (5) and the half-power
point for Equation (3) occurs
√
at a frequency that is higher by a factor of 3. A second example is found in
Horst (1997), for the case of a sonic anemometer measurement of vertical velocity
and a first-order-response scalar sensor. See Eugster and Senn (1995) for a short
pedagogical discussion of this issue.
A second error found in M86 and RT94 is the suggestion that the measured
covariance must be corrected for aliasing associated with digital sampling of analog
sensors at a frequency fs . A frequency-dependent correction needs to be made
for aliasing only for an explicit spectral analysis of the data, where covariance
at frequencies above the Nyquist frequency, fs /2, is aliased or folded at that frequency so that it appears in the cospectrum at a lower frequency, e.g., fs − f
for fs /2 ≤ f ≤ fs , thus distorting the calculated cospectrum from its true value
(Kaimal and Finnigan, 1994). However, when the covariance is determined directly
in the time domain, as hx 0 y 0 im , its frequency distribution is not calculated and thus
no correction should be made for aliasing (Blanford and Gay, 1992).
Additional questionable approximations were introduced in RT94. The more
consequential is the function suggested to account for high-pass filtering associated
with block averaging of the covariance over a data record of finite duration 1T ,
0, f ≤ 1/1T
Td =
(6)
1, f > 1/1T .
The correct form of the filter function is well known,
Td = 1 −
sin2 (πf 1T )
,
(πf 1T )2
(7)
e.g., Pasquill (1974). The frequency of the half-power point of Equation (7) equals
0.44/1T , and thus the oversimplified approximation of Equation (6) predicts an
attenuation that is too large.
Second, and less consequential, RT94 account for spatial averaging over the
horizontal area sampled by a propeller rotating about a vertical axis by using a
transfer function for vertical line averaging of the vertical velocity (as for a sonic
anemometer). Although RT94 acknowledge that this is an approximation, it is
strictly bare speculation. This approximation is most likely adequate for making
a judgement that the attenuation associated with spatial averaging by the propeller
is negligible compared to the attenuation caused by the dynamic response of the
propeller, as noted by RT94. However, use of this formula to explicitly correct the
measured covariance, as also suggested by RT94, is unjustified.
It is appropriate that non-micrometeorologists who wish to make eddycovariance flux measurements rely on the literature to determine how to apply
the technique correctly. It is also understandable that in many cases new practitioners accept recommendations in the literature verbatim without reviewing the
basis (or lack of basis) for their validity. Consequently the micrometeorological
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T. W. HORST
community needs to diligently assure that published descriptions of measurement
techniques are clear and correct, that relevant assumptions are clearly stated, and
that approximations are clearly identified and justified.
Acknowledgements
The author is grateful to Leif Kristensen and Don Lenschow for helpful comments
on an initial draft of this note. The National Center for Atmospheric Research is
sponsored by the National Science Foundation.
References
Blanford, J. H. and Gay, L. W.: 1992, ‘Tests of a Robust Eddy-Correlation System for Sensible
Heat-Flux’, Theor. Appl. Climatol. 46, 53–60.
Eugster, W. and Senn, W.: 1995, ‘A Cospectral Correction Model for Measurement of Turbulent NO2
Flux’, Boundary-Layer Meteorol. 74, 321–340.
Hicks, B. B.: 1972, ‘Propeller Anemometers as Sensors of Atmospheric Turbulence’, BoundaryLayer Meteorol. 3, 214–228.
Horst, T. W.: 1997, ‘A Simple Formula for Attenuation of Eddy Fluxes Measured with First-Order
Response Scalar Sensors’, Boundary-Layer Meteorol. 82, 219–233.
Kaimal, J. C. and Finnigan, J. J.: 1994, Atmospheric Boundary Layer Flows – Their Structure and
Measurement, Oxford University Press, New York, 289 pp.
Kristensen, L. and Lenschow, D. H.: 1988, ‘The Effect of Nonlinear Dynamic Sensor Response on
Measured Means’, J. Atmos. Oceanic Tech. 5, 34–43.
Moore, C. J.: 1986, ‘Frequency Response Corrections for Eddy Correlation Systems’, BoundaryLayer Meteorol. 37, 17–36.
Pasquill, F.: 1974, Atmospheric Diffusion, John Wiley, New York, 429 pp.
Rißman, J. and Tetzlaff, G.: 1994, ‘Application of a Spectral Correction Method for Measurements
of Covariances with Fast-Response Sensors in the Atmospheric Boundary Layer up to a Height
of 130 m and Testing of the Corrections’, Boundary-Layer Meteorol. 70, 293–305.