A simple method for estimating frequency response corrections for eddy covariance systems

doi:10.1016/S0168-1923(00)00164-7

Agricultural and Forest Meteorology

Volume 104, Issue 3, 28 August 2000, Pages 185-198

https://doi.org/10.1016/S0168-1923(00)00164-7 Get rights and content

Abstract

A simple analytical formula is developed for estimating the frequency attenuation of eddy covariance fluxes due to sensor response, path-length averaging, sensor separation, signal processing, and flux averaging periods. Although it is an approximation based on flat terrain cospectra, this analytical formula should have broader applicability than just flat-terrain providing the peak frequencies of the logarithmic cospectra are known. Comparing the integral and analytical formulations for momentum flux, heat flux, vapor flux, and closed-path and open-path CO₂ eddy covariance systems demonstrates that, except for a relatively uncommon atmospheric condition, the absolute difference between the integral and approximate correction factors is less than ±0.06 for both stable and unstable atmospheric conditions (0≤z/L≤2). Because closed-path systems can have the tube entrance separated longitudinally from the sonic anemometer, a cospectral transfer function is developed for the phase shift caused by the intrinsic time constant of a first-order scalar instrument and the longitudinal separation of the mouth of the tube and the sonic anemometer. The related issues of tube lag time and other spectral transfer functions are also discussed. In general, it is suggested that the simple formula should be quite useful for experimental design and numerical correction of eddy covariance systems for frequency attenuation.

Introduction

The eddy covariance technique is now used routinely for direct measurements of surface layer fluxes of momentum, heat, and trace gases (CO₂, H₂O and O₃) between the surface and the turbulent atmosphere. This technique employs a sonic anemometer for vertical velocity fluctuations, sonic thermometry for virtual temperature fluctuations, and a scalar sensor for density fluctuations. However, all sensors display some high frequency attenuation caused by the relatively slow response of the scalar sensors (i.e. first-order instruments often characterized by time constants of 0.1 s or greater), the spatial separation of the instruments, and line or volume averaging effects associated with sensor design. Furthermore, low frequencies are also attenuated when the flux is estimated by block averaging over a finite length of time (usually between 5 and 40 min or so, e.g. Panofsky, 1988, Kaimal et al., 1989, by high-pass recursive digital filtering (often incorporated as part of the data acquisition system, e.g. McMillen, 1988), or by linear detrending of the raw data time series (e.g. Gash and Culf, 1996, Rannik and Vesala, 1999.

Although some flux loss is inevitable with any eddy covariance system, there are a variety of methods, each having its own strength and weakness, which can be used either to correct the measured fluxes or to minimize flux losses through experimental design. For example, it is possible to correct flux measurements in situ (e.g. Laubach and McNaughton, 1999). This method has the advantage of being relatively free of cospectral shape, even though it assumes cospectral similarity between heat and water vapor fluxes. However, it requires more than one measurement of the virtual temperature flux ( $w′T_{v} ′$ ), and it does not correct for finite acoustic path length (sonic line averaging). In addition, because $w′T_{v} ′$ is the standard by which all other scalar fluxes are corrected, this method becomes less reliable as $w′T_{v} ′$ approaches zero. Other methods employ spectral transfer functions, which have the advantage of being relatively comprehensive (e.g., Moore, 1986), but require a priori assumptions about the cospectral shape. If the true cospectrum resembles the assumed shape, Moore’s approach (Moore, 1986) does give reasonable estimates of the correction factors (Leuning and King, 1992). However, if the true cospectrum departs significantly from the assumed shape, then the correction factor can be in error (Laubach and McNaughton, 1999). Another possibility is to estimate a cospectrum for each block averaging period by Fourier transform, correct the cospectrum, and then integrate the corrected cospectrum to obtain the desired flux. This Fourier transform method may be the best method of all because it requires the fewest assumptions. However, it is numerically intensive and, therefore, impractical for long duration experiments comprised of many block averaged periods. Finally, Horst (1997) suggested a simple analytical alternative to Moore’s comprehensive numerical approach (Moore, 1986), but, because Horst’s development focuses on the (usually) slower responding scalar sensor, it does not include the effects of line averaging, sensor separation, or the data acquisition system.

The present study, which incorporates and extends Horst’s (1997) approach develops and tests a general analytical formula for estimating the flux loss caused by attenuation effects associated with the sonic anemometer, the scalar sensor, sensor separation and design, and the data acquisition system. The initial formulation of this analytical method is in terms of the flat terrain cospectra of Kaimal et al. (1972). But, because the approximations developed for this study result in flux loss parameterizations that are functions of the maximum frequency f_x of the logarithmic cospectrum, fCo(f), they can be used with cospectra that differ from the flat terrain cospectra. Consequently, because the present methods assume a relatively smooth cospectra they require either an in situ determination of f_x or a reasonable parameterization for it. The primary focus of the present study is on the most challenging scenarios: the extremum cases for the analytical approximation, i.e. the heat flux as measured by sonic thermometry (smallest corrections) and the closed-path flux system (largest corrections). Nevertheless, the approximation is also tested for momentum and water vapor flux measurements. An additional correction term to the formal analytical approximation is developed to improve the analytical correction factors for the relatively infrequent situation of fluxes measured during windy, stable atmospheric conditions using first-order scalar sensors with time constants ≥0.1 s.

Section 2 discusses the mathematical issues related to this study and summarizes many of the transfer functions used with eddy covariance. Section 3 compares the eddy covariance correction factors estimated by the simple analytical model with the complete integral formulation. The final section summarizes the results of this study and provides suggestions and recommendations that can be drawn from it.

Section snippets

Integral expression

The true eddy flux, $w′β′$ , can be represented as the integral over frequency f of the one-sided cospectrum Co_wβ(f): $w′β′ = ∫ 0 ∞ Co_{wβ} (f) d f$ where w′ and b′ are the fluctuations of vertical velocity and either horizontal wind speed or scalar concentration. However, the measured flux, $(w′β′)_{m}$ , is usually limited by the effects of sonic line averaging, sensor separation, block averaging when computing the fluxes, discrete time sampling, anti-noise filters, etc. The influence of these limitations is usually

L_t/U_t phase effects have been removed by digital time shifting

Five different eddy covariance scenarios were tested for this study: momentum flux, virtual temperature flux (sonic thermometry), water vapor flux with an open path Krypton hygrometer, and both open- and closed-path CO₂ systems. This study focuses on the latter two scenarios and the sonic thermometry virtual temperature flux because they represent the extremum cases for the analytical approach and they are probably of somewhat greater interest in general. For these simulations the following

Summary and recommendations

The primary purpose of this study is to develop and test an analytical approximation for estimating eddy flux corrections. In the process of attempting this goal it proved necessary to clarify some scientific aspects of making these corrections and to summarize or derive transfer functions that either had not been previously considered in the literature or have not received sufficient attention. A secondary goal is to clarify some aspects of the general methodology used for deriving transfer

Acknowledgements

The author would like to thank Dr. T. Horst for his comments on earlier drafts of this manuscript and for many helpful discussions on eddy covariance transfer functions.

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Published by Elsevier Ltd.

A simple method for estimating frequency response corrections for eddy covariance systems

Abstract

Introduction

Section snippets

Integral expression

Lt/Ut phase effects have been removed by digital time shifting

Summary and recommendations

Acknowledgements

Boundary-Layer Meteorol.

Atmos. Environ.

The effects of volume averaging on spectra measured with a Lyman-Alpha hygrometer

J. Appl. Meteorol.

The dispersion of solute from time-dependent releases in parallel flow

J. Fluid Mech.

On the longitudinal dispersion of dye whose concentration varies harmonically with time

J. Fluid Mech.

Applying a linear detrend to eddy correlation data in real time

Boundary-Layer Meteorol.

Physiological responses of a black spruce forest to weather

J. Geophys. Res.

The pulsation spectra of the vertical component of the wind velocity and their relations to micrometeorological conditions

Izvestiya Atmos. Oceanic Phys.

Propellor anemometers as sensors of atmospheric turbulence

Boundary-Layer Meteorol.

A simple formula for attenuation of eddy fluxes measured with first-order response scalar sensors

Boundary-Layer Meteorol.

On frequency response corrections for eddy covariance flux measurements

Boundary-Layer Meteorol.

Spectral characteristics of surface-layer turbulence

Q. J. R. Meteorol. Soc.

Deriving power spectra from a three-component sonic anemometer

J. Appl. Meteorol.

Effect of finite sampling on atmospheric spectra

Boundary-Layer Meteorol.

L_t/U_t phase effects have been removed by digital time shifting