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. 518 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 520 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.