Estimating the Random Error in Eddy-Covariance Based Fluxes and ...

Boundary-Layer Meteorol (2012) 144:113–135 DOI 10.1007/s10546-012-9710-0 ARTICLE

Estimating the Random Error in Eddy-Covariance Based Fluxes and Other Turbulence Statistics: The Filtering Method Scott T. Salesky · Marcelo Chamecki · Nelson L. Dias

Received: 13 October 2011 / Accepted: 20 February 2012 / Published online: 16 March 2012 © Springer Science+Business Media B.V. 2012

Abstract A spatially local decomposition of turbulent fluxes based on properties of spatial filters is used to develop a new method of estimating random error in turbulent moments of any order. The proposed error estimation method does not require an estimate of the integral time scale, which can be highly sensitive to the method used to calculate it. The error estimation method is validated using synthetic flux data with a known ensemble mean and intercompared with existing methods using data from the Advection Horizontal Array Turbulence Study (AHATS). Typical errors for a 27.3-min block of data collected at a height of 8 m are found to be approximately 10% for the heat flux and 7–15% for variances. The error in the momentum flux increases rapidly with increasing atmospheric instability, reaching values of 40% or greater for unstable conditions. A new method based on filtering is also proposed to estimate integral time scales of turbulent quantities. Keywords Atmospheric turbulence · Eddy covariance · Filtering · Integral scale · Random error · Turbulent fluxes

1 Introduction The eddy-covariance technique (e.g. Moncrieff et al. 1997; Aubinet et al. 1999; Baldocchi 2003; Lee et al. 2004) is a method for directly calculating exchanges of sensible heat, momentum, water vapour, and trace gases between the Earth’s surface and the atmospheric boundary Electronic supplementary material The online version of this article (doi:10.1007/s10546-012-9710-0) contains supplementary material, which is available to authorized users. S. T. Salesky · M. Chamecki (B) Department of Meteorology, The Pennsylvania State University, University Park, PA 16802, USA e-mail: [email protected] N. L. Dias Laboratory for Environmental Monitoring and Modeling Analysis, Federal University of Paraná, P.O. Box 19100, Curitiba, PR 80011-970, Brazil e-mail: [email protected]

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layer (ABL); it is the basis of global micrometeorological measurement networks (Baldocchi et al. 2001). Eddy-covariance measurements of turbulent fluxes can include contributions from several types of errors, including systematic (i.e. bias) errors (Mahrt 1998; Lenschow et al. 1994) due to systematically undersampling the largest scales that contribute to fluxes, errors due to non-stationarity of the time series (Vickers and Mahrt 1997; Mahrt 1998), and random errors (Lumley and Panofsky 1964; Wyngaard 1973; Sreenivasan et al. 1978; Lenschow and Stankov 1986; Mann and Lenschow 1994; Lenschow et al. 1994). Random errors are defined as the errors due to time averaging over an insufficient period for the time mean to converge to the ensemble mean by the ergodic hypothesis (Lumley and Panofsky 1964; Wyngaard 1973; Sreenivasan et al. 1978; Lenschow and Stankov 1986; Mann and Lenschow 1994; Lenschow et al. 1994). In practice, to avoid strong non-stationarity effects, maximum possible averaging times are limited by the diurnal evolution of the ABL to periods not much larger than 1 h (e.g. Stull 1988). Non-stationarity on shorter time scales can introduce other types of errors that are not considered here. Although random error is used by some authors as a comprehensive category for many kinds of errors, such as those due to limits on sensor response and random noise (Businger 1985; Moncrieff et al. 1996), we focus solely herein on errors due to an insufficient averaging period for the time mean to converge to the ensemble mean. Note that the technique for estimating random errors proposed herein applies in principle to spatial data as well. In the case of spatial data, homogeneity is required for spatial averages to converge to the ensemble mean. However, since most micrometeorological datasets are obtained in time, we present the subject only in terms of stochastic processes in time. In the studies where they have been reported, the magnitude of random errors in turbulent fluxes has been found to be significant. Businger (1985) estimated that random errors in ozone fluxes calculated from the eddy-covariance technique were roughly on the order of 20–30% for several studies (Wesely et al. 1981; Lenschow et al. 1981, 1982). Berger et al. (2001) estimated random errors of 20% for the heat flux and 40–50% for CO2 fluxes in the surface layer for an averaging period of 1 h; these errors increased with z/z i , where z is height, and z i is the height of the ABL. Random errors decrease as the length of the averaging period increases (Moncrieff et al. 1996); they are typically estimated to be approximately 5% for studies of annual budgets (Baldocchi 2008), e.g. quantifying net ecosystem exchange of CO2 . However, for studies of the ABL on shorter time scales, random errors in turbulent fluxes are non-negligible. Although some authors (e.g. Bernardes and Dias 2010) include error bars when reporting measured values of turbulent fluxes, it is not common micrometeorological practice to do so. This is unfortunate, since random errors are, in principle, possible to estimate, and can be large over typical averaging periods (i.e. minutes to hours). Furthermore, when measurement uncertainty is reported, it is not always clear whether it is associated with random errors, systematic errors, or other factors (Luyssaert et al. 2007). This article is organized as follows: existing methods for estimating random error are reviewed briefly in Sect. 2. We introduce a new method of estimating errors in turbulent fluxes in Sect. 4; it is based on a new spatially local decomposition of turbulent fluxes that we present in Sect. 3 and does not require a priori knowledge of the integral time scale. Our new method is validated using time series of synthetic random data with a known ensemble mean in Sect. 5.1, and is intercompared with other methods in Sect. 5.2 using atmospheric surface-layer data from the Advection Horizontal Array Turbulence Study (AHATS). We direct the reader who is primarily interested in applying the new method to the Appendix, where an algorithm giving the steps of the filtering method has been provided. A Matlab function that applies the filtering method may be found online in the Supplemental Material. A schematic view of how the filtering method works is given in Fig. 2. Validation using

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synthetic random data may be found in Sect. 5.1 and using atmospheric surface-layer data in Sect. 5.2.

2 Overview of Methods for Estimating Random Error Estimating the random error of the average f is an elementary result in statistics given by σ 2 = σ 2f /n, where σ 2f is the variance of the random variable f and n is the number of indef pendent samples. When f is a stochastic process with space or time dependence, however, the random error becomes much more difficult to estimate because successive measurements (“samples”) of f are no longer independent. A number of methods have been proposed for estimating the random error Given a random variable f with ensemble mean in this case. f and variance f 2 = [ f − f ]2 , Lumley and Panofsky (1964) expressed the random error in terms of the error variance for an averaging period T : ⎡ ⎤2 t+T /2

2 1 ⎢ ⎥ 2 σ f (T ) = ⎣ (1) f (t )dt − f ⎦ = f − f T t−T /2

where f is the variable of interest, and f is its time average. By assuming that f is statistically stationary and by requiring that an integral scale exists, Lumley and Panofsky (1964) found the error variance could be expressed as 2T f f 2 , (2) T where T f is the integral time scale of f . Note that this type of analysis originated with Liepmann (1952). One can then define a relative error f = σ f /| f | that is given as σ 2f (T ) =

2T f f 2 f = f 2 T

1/2 .

(3)

The above equation holds for any stationary random variable f whose integral scale exists. When (3) is used in practice to estimate the random error, one must replace the ensemble 2 mean and variance f and f 2 by the sample quantities f and s 2f = f − f calculated from time averaging and determine a suitable estimate of the integral time scale T f (i.e. from the spectrum or autocorrelation of f ). Wyngaard (1973) used (3) together with results from the Kansas experiment to make order-of-magnitude estimates of required averaging times for θ θ , w w , u w , and w θ . He approximated the integral scale as T f ≈ z/U for all fluxes in unstable and neutral conditions, and showed that for a given averaging time, relative errors would be higher for the momentum flux than for the heat flux, with errors in the variances being the smallest. He also found that the relative error for a given variable increased rapidly with height. Sreenivasan et al. (1978) used near-neutral marine boundary-layer data collected at a height of z = 5 m and applied (3) to determine required averaging times for moments up to order 4 as well as covariances. They calculated the integral time scales (i.e. T f ) of several second-order moments through numerical integration of the autocorrelation function to its first zero on the time axis and compared them to the approximate integral time scale of w, i.e. Tw = z/U . They found that T f ≈ z/U was a reasonable order-of-magnitude approximation for u w and w θ (i.e. T f = 1.2z/U ), but not as good for w w (T f = 0.56z/U ) and θ θ (T f = 0.73z/U ), showing

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that, although the approximation T f ≈ z/U is of the correct order of magnitude, it should not be interpreted as an equality. Lenschow et al. (1993, 1994) and Mann and Lenschow (1994) used a slightly different definition for the error variance

2 σ 2f (T ) = f − f , (4) which coincides with (1) whenever f is an unbiased estimator of f . By assuming stationarity, the existence of an integral scale, and joint Gaussian behaviour between the vertical velocity w and the scalar of interest c, Lenschow et al. (1994) found f =

2T f T

1/2

2 1 + ρwc 2 ρwc

1/2 (5)

as the expression for the relative error, where ρwc is the correlation coefficient for w and c. A major source of uncertainty in using (3) or (5) to estimate the random error in turbulent flux measurements in the ABL arises from the method used to determine the integral time scale of the flux, T f . By definition, the integral time scale of the flux is given as ∞ Tf =

ρ(τ )dτ,

(6)

[ f (t) − f ][ f (t + τ ) − f ] f 2

(7)

0

where ρ(τ ) =

is the autocorrelation function of f . The autocorrelation function depends on the time lag τ alone if f is a stationary random process. If one wishes to use the definition given in (6) to determine the integral scale, the integration can only be carried out to some arbitrary maximum lag τ ∗ . The choice of τ ∗ can affect the estimate of T f , since ρ(τ ) may fluctuate around zero for large τ (Yaglom 1987). Some authors set τ ∗ to be the first zero of ρ(τ ) (e.g. Sreenivasan et al. 1978; Lenschow and Stankov 1986; Katul and Parlange 1995), while others integrate until a small positive value of ρ(τ ) is reached (Finkelstein and Sims 2001). Still other definitions of τ ∗ are possible, such as when ρ(τ ) reaches e−1 , or the minimum of the negative portion of the autocorrelation curve (Tritton 1988; Theunissen et al. 2008). Another method for estimating the integral time scale of the flux is to fit an exponential function of the form |τ | ρ(τ ) = exp − (8) Tf

to the autocorrelation function that is calculated from the data (Lenschow et al. 1993, 1994; Kaimal and Finnigan 1994; Sullivan et al. 2003). However, the assumption of an exponential autocorrelation function is not always appropriate. Finkelstein and Sims (2001) found |τ |1/2 ρ(τ ) = exp − , (9) b to be a much better fit to their data, where b is a fitted parameter. It must be emphasized that, although fitting an exponential function of the form given in (8) to the measured

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autocorrelation function is common practice, it is done for the sake of convenience rather than being based on fundamental turbulence theory.1 Another option when calculating the autocorrelation function is to use a lag-window, i.e. ρ ∗ (τ ) = a(τ )ρ(τ )

(10)

where a(τ ) is the lag-window function (Dias et al. 2004) that reduces the spurious oscillations that occur for large τ (Yaglom 1987). The lag-window is a weighting function that starts at 1 for τ = 0, and decays monotonically to zero for large τ . Dias et al. (2004) found that when the lag-window was applied, ρ ∗ (τ ) did not always intersect the time axis, leading to the non-existence of some integral scales. Calculated values of the integral time scale T f can vary widely depending on the method used to calculate it. Theunissen et al. (2008) showed that, depending on the length of the record, the maximum time lag τ used to compute the autocorrelation, and the choice of τ ∗ , an estimate of the integral time scale could vary over an order of magnitude. Due to these difficulties of estimating the integral time scale, methods of estimating the random error in turbulent fluxes that do not depend on the integration of the sample autocorrelation function are desirable. One such method is the bootstrap technique, which consists of resampling variables from a set of data in different order that allows one to generate new “surrogate” samples and to estimate statistics that otherwise might be unavailable. Bernardes and Dias (2010) used a form of the bootstrap proposed by Gluhovsky and Agee (1994) for atmospheric measurements to estimate random errors in the momentum flux u w . They found that estimated errors from the bootstrap were in reasonable agreement with the Lumley and Panofsky (1964) method, both when T f was calculated by integration of ρ(τ ) to the first crossing and when a (Parzen) lag-window was applied. In this article, we choose to use a form of the bootstrap called the moving block bootstrap (MBB), which has been validated for turbulence measurements by Garcia et al. (2006). In the MBB technique, one samples in different order blocks of data, which preserves correlation between successive measurements within each sampled block. For a series of length N , one defines blocks of length b, samples N /b blocks at random, and combines them together into a new series. Sampling is done with replacement, so that a block of data can be used more than once in the new series. This procedure is repeated to generate B new series, which then are used as a new “ensemble” to calculate statistics. Politis and White (2004) proposed a method of determining the optimal block length bopt , where 1/3 2H 2 1/3 bopt = N (11) D with D = 4/3h 2 (0), h(0) =

M

λ(k/M) × ρ(kΔt),

(12)

λ(k/M) × |k| × ρ(kΔt),

(13)

k=−M

H =

M k=−M

1 Note that an exponential autocorrelation function corresponds to a κ −2 spectrum (κ is the wavenumber) at

high wavenumbers (Lumley and Panofsky 1964; Sullivan et al. 2003). A more fundamental approach would be to use a series expansion of the Fourier transform of the longitudinal von Kármán spectrum, which yields an autocorrelation function of the form ρ(τ ) = 1 − Cτ 2/3 , where C is a constant.

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and ⎧ ⎪ 0 ≤ |(k/M)| ≤ 0.5 ⎨1, λ(k/M) = 2(1 − |k/M|), 0.5 ≤ |(k/M)| ≤ 1 , ⎪ ⎩ 0, otherwise

(14)

where Δt is the sampling interval, ρ(kΔt) is the autocorrelation function calculated from the data, and M = 2m, where m is the first integer value after which the value of the autocorrelation function is negligible (Garcia et al. 2006). Typically B = 200 repetitions is sufficient to calculate error variances (Efron and Tibshirani 1993); Garcia et al. (2006) found more repetitions (i.e. B = 1, 000) to be necessary for estimating confidence intervals. For the purposes of this article, we refer to the method proposed by Lumley and Panofsky (1964) given in Eq. 3 as the LP method, the method proposed by Lenschow et al. (1994) given in Eq. 5 as the LMK method, and the moving block bootstrap (Garcia et al. 2006) given in (11)–(14) as the MBB method.

3 A Spatially-Local Flux Decomposition Turbulent fluxes may be decomposed in a number of ways. Some methods of decomposing fluxes are local in space, such as the multi-resolution decomposition proposed by Howell and Mahrt (1997). Others are local in scale, such as cospectra or structure functions. Still other methods, such as wavelets (Meneveau 1991; Hagelberg and Gamage 1994; Katul and Parlange 1995; Katul and Vidakovic 1996) have intermediate locality in both space and scale. Because turbulent fluxes are known to contain a large contribution from localized events in space or time that are responsible for a large fraction of the total flux, we here consider a spatially-local decomposition of turbulent fluxes. We define these spatially-local turbulent fluxes in such a way that the time average of the local flux recovers the Reynolds-averaged flux at any scale Δ. The spatially-local flux decomposition may be defined using properties of spatial filters. Filtered turbulent quantities are defined formally in physical space (Leonard 1974) through the convolution of a turbulent signal with a filter kernel, e.g. 1 f (x, t) = G f = Δ

x+Δ/2

f (x , t)dx

(15)

x−Δ/2

where G is a box filter of width Δ. Although it is useful to consider decomposing turbulent fluxes locally in space, we shall refer to local fluxes in time hereafter since we are dealing with time series data. This is done without loss of generality and allows us to avoid the unnecessary application of Taylor’s hypothesis. In order to define local turbulent fluxes, we consider the application of two filters in time to a turbulent signal f . We take the first filter F to be a time average over a single block of data (e.g. T = 30 min) and the second filter G to be a filter in time of width Δt . Note that the time average does indeed satisfy the properties required of a filter, namely the conservation of constants, linearity, and commutation with differentiation (Sagaut 2006). We denote quantities filtered with F with an overbar, i.e. F f = f and quantities filtered with G with a tilde: G f = f . Because the filtering operation (15) is defined in terms of convolution, which is commutative, we have

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u w to the time-averaged flux u w as a function of Fig. 1 Ratio of the time mean of the local filtered flux the time filter width Δt . The error bars denote the standard deviation of the local flux, also normalized by the −1/2 /u w . The dashed line denotes a power-law fit of the form Cuw Δt . The solid time-average flux, i.e. σ u w vertical line at T = 1,638 s denotes the length of the block used for time averaging

f = f, f = F G f =G F f = where the last equality holds true because f is constant. Taking f = flux of a scalar c, we have w c = w c .

(16) w c

to be the vertical (17)

By (17), the time-averaged flux w c for a block of some length T is equivalent to the w c . Furthermore, this holds true for any filter width Δt . time average of the filtered flux A demonstration of the behaviour of the local flux can be found in Fig. 1, where the time average of the local momentum flux u w has been normalized by the time-averaged flux u w and plotted as a function of time filter width Δt for a 27.3-min block of the AHATS data. The error bars correspond to the standard deviation of the local flux, also normalized by the time mean flux, σ /u w . One can see that the time average of the local flux is equivalent u w to the time-averaged flux for any given filter width Δt . The dashed line denotes a power−1/2 law fit of the form Cuw Δt to the normalized standard deviation of the local flux, where −1/2 Cuw is the coefficient to be determined through least squares. We observe that the Δt −1/2 power-law decay of σ /u w is none other than the T decay of the random error in u w the time-averaged flux, as given by (3) and (5), except that now time averages, instead of ensemble averages, are being used. We use this property of local turbulent fluxes to develop a new method of estimating random error in turbulent flux measurements. The new method requires no a priori knowledge of the integral time scale, and is presented in Sect. 4. Note that a local flux decomposition could also be defined by dividing a block of data (e.g. of length T = 30 min) into sub-blocks and calculating the average flux for each sub-block. However, this type of decomposition would have a larger bias error (Lenschow et al. 1993)2 than the local fluxes we have defined above. Although there is bias error in the local flux as defined in Eq. 17, it occurs for averaging period T rather than filter width Δt because the time average of the local flux is identically the time-averaged flux for period T ; this is true 2 i.e. The time average for each sub-block would be systematically smaller than the time average over

period T .

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for any filter width Δt . Thus for a typical block length (i.e. T = 30 min), the bias error will be small. Although the emphasis here is on developing a local decomposition of turbulent fluxes, the equality in (16) holds true for any f and can be used to decompose other quantities such as the mean velocity ui = ui

(18)

u i u j u k = u i u j uk

(19)

or higher order moments, e.g.

as a function of scale Δt .

4 The Filtering Method of Estimating Random Error Our method of estimating random error in turbulent fluxes is based on directly calculating the error variance of a local flux as a function of scale Δt and extrapolating to estimate the error variance for period T . Recall that Lumley and Panofsky (1964) defined the ensemble error variance (1) for the variable of interest f over an averaging period T ; by assuming stationarity and the existence of the integral time scale T f , they found the expression for the error variance given in Eq. 2. We apply their expression in (1) to the flux f = w c ⎡ ⎢ 1 σ 2 (Δt ) = ⎣ f Δt

t+Δ t /2

⎤2 ⎥ f (t )dt − f ⎦

=

f −f

2

(20)

t−Δt /2

for a given time-filter width Δt . Assuming ergodicity we can rewrite Eq. 20 using time averages 2 2 c − w c . σ (Δ ) = w t wc

(21)

Note that (21) is identically the variance of the local flux because of the property of local fluxes that w c = w c . Because (20) is identically the expression given by Lumley and Panofsky (1964) in (1) written for averaging period Δt , we can apply Eq. 2, finding (Δt ) = σ w c

1/2 −1/2 −1/2 2T f (w c − w c )2 Δt = Cwc Δt ,

(22)

where (w c − w c )2 is the variance of the unfiltered flux, and Cwc is a constant coefficient. Equation 22 holds true at any scale Δt . One can see from Figs. 1 to 2 that σ does indeed w c −1/2

decay with a Δt power law. In order to use filtering to estimate random error in turbulent fluxes, we fit a power law to σ of the form given in (22) where Cwc is a coefficient to be w c determined by least squares. The standard deviation of the flux for an averaging period T is then found by evaluating the power-law fit for Δt = T , i.e. (T ) = Cwc T −1/2 , σ w c

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(23)


121

Fig. 2 A demonstration of the filtering method of estimating random errors. The solid black line denotes the relative error uw in the local momentum flux u w as a function of time-filter width Δt . The dashed line is −1/2 a power-law fit of the form Cuw Δt . The vertical dashed lines denote the minimum (Δmin = 10τ f ) and maximum (Δmax = T /10) filter widths used for the fit, and the block length T

and the estimate of the random error is simply the standard deviation of the filtered flux for Δt = T normalized by its mean value: wc =

σ (T ) w c w c

.

(24)

Fitting a power law to the standard deviation of the filtered flux σ therefore allows us to w c estimate the random error in the turbulent fluxes without a priori knowledge of the integral time scale T f . A demonstration of the filtering method of estimating random errors can be found in Fig. 2, where the relative error in the local momentum flux uw is plotted as a function of filter width Δt for the same block of AHATS data as shown in Fig. 1. Note that in Fig. 2 we have normalized σ by the average flux u w and then fitted a power law to the error u w

in u w in order to demonstrate the value of the error obtained through the power-law fit. However, in practice we fit the power law to the standard deviation of the filtered flux σ , w c evaluate the standard deviation of the flux at Δt = T , and then normalize by the mean flux w c . We do this in order to avoid difficulties with the fitting procedure when the mean flux becomes small (e.g. w θ for near-neutral conditions). The solid black line in Fig. 2 denotes uw calculated from the data; the dashed line is the power-law fit given in Eq. 22 to the range of filter widths between Δmin and Δmax . One can see that the calculated error deviates from the power law for Δt < Δmin and Δt > Δmax . For Δt < Δmin , the deviations are due to a lack of statistical convergence since too few integral scales are sampled. We found that Δmin = 10T f was a conservative lower bound for the filter widths included in the power-law fit, although in principle one could use a minimum filter width as small as Δmin = 2T f (i.e. the decorrelation time). An explicit calculation of the integral time scale is not necessary; an estimate or model will be sufficient. Deviations from the fitted function for Δt > Δmax occur when the filter width approaches the block length; σ decreases rapidly in this range because we must have σ → 0 when Δt → T . An w c w c upper bound on the filter ranges included in the power-law fit is therefore also necessary; it should only be a function of the averaging period T . As a conservative estimate of Δmax ,

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we use Δmax = T /10. However, it is evident from Fig. 2 that a larger value of Δmax could be safely used. We found that the average difference between using Δmax = T /10 and a larger value of Δmax = T /4 was never in excess of 2.2% for any of the fluxes. The power-law decay that is exhibited in Fig. 2 is for an ideal case. In practice, if we allow the power-law exponent to remain free, fitting a function of the form σ (Δt ) = Cwc Δbt , w c we do not always obtain b = −1/2. We believe that departures from b = −1/2 are due to non-stationarities in the time series of the flux. Therefore, we always impose b = −1/2 when applying the filtering method. Our method is similar to that of Lumley and Panofsky (1964), except that we calculate the variance of the local flux at scale Δt directly and extrapolate to Δt = T , which allows us to estimate the random error without a priori knowledge of the integral time scale T f . By (22), we find 2 = 2T f (w c − w c )2 Cwc

(25)

as the relationship between the fitted power-law coefficient, the integral time scale, and the variance of the unfiltered flux. One may therefore solve for T f in (25), which gives us a new method of estimating the integral time scale of the flux. This new method for estimating T f is compared with existing methods below.

5 Comparison of Methods for Estimating the Random Error The filtering method of estimating random error in turbulent fluxes is validated and intercompared with other methods in this section, both for random data with a known ensemble mean and for atmospheric surface-layer data from AHATS. Results from random data with a known ensemble mean are presented in Sect. 5.1; an intercomparison of the various methods is given in Sect. 5.2 using surface-layer data from AHATS. 5.1 Synthetic Random Data Although a number of previous studies have intercompared estimates of random errors from several methods (e.g. Finkelstein and Sims 2001; Hollinger and Richardson 2005; Richardson et al. 2006), these studies use ABL data where the true ensemble mean is unknown. As a result, it is not possible to compare the error estimates from a given method to the true ensemble mean. In this section we intercompare error estimation methods using synthetic random flux data with a known ensemble mean. The generation of the synthetic dataset is discussed in Sect. 5.1.1; results are presented in Sect. 5.1.2.

5.1.1 Generation of a Synthetic Dataset In order to evaluate error estimation methods objectively, data for the true ensemble mean of the covariance (i.e. flux) between two random variables are needed. We consider the covariance of two random variables, X and Y that are related via Y = αX + βZ

123

(26)


123

where Z is another random variable and α and β are coefficients to be determined. We assume that X and Z are independent and identically distributed. The mean and variance of Y are μY = αμ X + βμ Z ,

(27)

σY2

(28)

=

α 2 σ X2

+ β 2 σ Z2 .

One can show that the covariance of X and Y is Cov(X, Y ) = E[(X − μ X )(Y − μY )] = ασ X2

(29)

where we have used the fact that Cov(X, Z ) = 0 since X and Z are independent. The correlation between X and Y is ρX Y =

Cov(X, Y ) ασ X = 2 2 . σ X σY (α σ X + β 2 σ Z2 )1/2

We can then use (28) and (30) to solve for α and β, finding that σX α= ρX Y , σY σY (1 − ρ X2 Y )1/2 . β= σZ

(30)

(31) (32)

If we take X and Z to be standard Gaussian random variables and impose σY = 1, we have Y = ρ X Y X + (1 − ρ X2 Y )1/2 Z ,

(33)

and it therefore holds that ρ X Y = Cov(X, Y ), so we can specify the ensemble mean value of the covariance through our choice of ρ X Y in (33). The product X Y has a spectral density with equal power at all wavenumbers (i.e. white noise); its autocorrelation is the delta function. In order to estimate the integral scale T f of X Y that is required for the LP and the LMK methods, we low-pass filter X and Z before generating the series of Y , since low-pass filtered Gaussian noise has an integral scale that exists (Bendat and Piersol 2000). 5.1.2 Results We generated 20,000 blocks of random data of length 32,768 points, equivalent to 27.3-min blocks at 20 Hz. The low-pass filter width was chosen so that the integral time scale of the flux was approximately T f = 2.0 s. The correlation coefficient was fixed as ρ X Y = 0.5, both typical values for the heat flux in the unstable surface layer. The true ensemble mean error was evaluated from the definition of the error variance in Eq. 1. We applied the LP and LMK methods using T f calculated both from integrating ρ(τ ) to the first zero-crossing and from the exponential fit (8), the MBB method, and the filtering method to produce one estimate of the random error for each block of data. The MBB method was applied through generating the time series of the flux X Y for each block of data, then bootstrapping the flux using B = 200 repetitions and the optimal block length estimated from (11) to (14). The filtering method was applied through using 50 filter widths Δt between Δmin = 10T f and Δmax = T /10 spaced evenly on a logarithmic scale. We then calculated probability density functions (PDFs) of the random error estimated from each method; they are displayed in Fig. 3a. Cumulative distribution functions (CDFs) for each method are given in Fig. 3b. Mean values of the random error predicted by each method are given in Table 1. Several conclusions can be drawn from the PDFs and CDFs of random error predicted by each method displayed in Fig. 3. We find that the filtering method is in good agreement

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(a)

(b)

Fig. 3 a PDFs of random error estimates from each method for synthetic random data with a known ensemble mean and b cumulative distribution functions (CDFs) of error estimates from each method. The vertical lines in (a, b) denote the true ensemble error, f = 0.0887 Table 1 Mean values of the estimated random error f from each method for synthetic random data Method

True error

LP, T f

LMK

LP, T f , fit

LMK, fit

Filtering

MBB

Error

0.0887

0.1115

0.1119

0.1018

0.1023

0.1083

0.1057

Table 2 Heights of sonic anemometers on the AHATS profile tower from June 25 to July 17, 2008

Sonic

z (m)

1

1.51

2

3.30

3

4.24

4

5.53

5

7.08

6

8.05

with other methods for the random data; in particular the PDF of the random error predicted by the filtering method is similar to those predicted by both the LP and LMK methods when T f is calculated through integrating ρ(τ ) to the first crossing and by the MBB method. Also notice that the PDFs of the random error are highly sensitive to the method used to estimate the integral scale. For example, when the LP method is applied using T f calculated from the exponential fit, the error has a positively skewed distribution with a mean of f = 0.1018, but when it is applied using the value of T f obtained through integrating ρ(τ ) to the first crossing, the error has a near-Gaussian distribution with a mean of f = 0.1115. Furthermore, it is not surprising that methods that make simple assumptions about the data (e.g. the LMK method, where joint-Gaussian behaviour between the two variables is assumed) perform well for the synthetic random data, where these assumptions hold true. From the tests using synthetic random data, we find that the filtering method is in good agreement with other existing methods, and therefore can be regarded as a reasonable method for estimating random errors. These tests also illustrate the point that many methods of estimating the random error are highly sensitive to the method used to estimate the integral scale; methods where the integral scale need not be known a priori are therefore desirable.

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5.2 Atmospheric Surface-Layer Data 5.2.1 Dataset and Analysis Procedure The ASL data used for our analysis came from the Advection Horizontal Array Turbulence Study (AHATS), collected from June 25 to August 16, 2008 in Kettleman City, California, USA. AHATS is an array turbulence study designed to collect ABL turbulence data that can be spatially filtered to evaluate subfilter-scale models for large-eddy simulation. We used data from the AHATS profile tower during the period from June 25 to July 17. The AHATS profile tower consists of six Campbell Scientific CSAT-3 sonic anemometers mounted on a vertical tower with heights given in Table 2, sampling the three components of the velocity vector and sonic virtual temperature at 60 Hz. The typical procedure of aligning the mean wind to the tower so that V = 0 was performed during preprocessing. The raw 60 Hz data performed were downsampled to 20 Hz and divided in blocks of 32,768 points, i.e. 27.3 min. Blocks of data were rejected if σw /u ∗ exhibited more than a 30% departure from the value predicted by Monin–Obukhov similarity theory (MOST) (Lee et al. 2004, pp. 191–193). We also calculated along-wind (RNu), cross-wind (RNv), and vector-wind (RNS) non-stationarity ratios (Vickers and Mahrt 1997), excluding runs where RNu, RNv, or R N S ≥ 0.5. These non-stationarity runs were removed due to the strong effect of non-stationarity on the behaviour of the autocorrelation function ρ(τ ). Error estimates from the LP and the LMK methods were calculated using T f calculated both from integration of ρ(τ ) to the first zero-crossing and from the exponential fit (8). In addition, the error from the LP method was also calculated using T f ≈ Tw with Tw calculated both from the autocorrelation of w and using the approximation Tw = z/U . The MBB method was applied by calculating the time series of w c for the flux of interest, then bootstrapping the flux time series directly using B = 200 repetitions and bopt determined from (11) to (14), i.e. based on the autocorrelation of w c . The filtering method was applied by fitting the power-law of the form given in (22) with 50 values of Δt evenly spaced on a logarithmic scale such that Δmin < Δt < Δmax , with Δmin = 10τ f and Δmax = T /10 where T = 1638s. Filtered quantities were calculated by applying a 1-D box filter in spectral space, that is, by taking the fast Fourier transform (FFT) of the original time series, multiplying the transformed series by the transfer function of the 1-D box filter, and then taking the inverse FFT. Note, however, that box filtering in time is identical to calculating the running mean of the time series. 5.2.2 Errors in Fluxes Data from the top sonic on the AHATS profile tower at z = 8.05 m were used to intercompare the error estimates from the different methods. In Fig. 4, we show scatterplots comparing the LP, the LMK, and the MBB methods to the filtering method for the relative error in the u-momentum flux, u w . Figure 4d is a comparison of the MBB method to the filtering method. The errors predicted by the two methods agree quite well, lying close to the 1:1 line with little scatter. Furthermore, no assumptions regarding the integral scale are made in either of these two methods. From the agreement with the MBB method, and the accurate prediction of the true ensemble mean by the filtering method that was found in the random data, we conclude that the filtering method is a reasonable estimate of the true error against which to compare other methods. Figure 4a is a comparison of the LP method using Tw using the approximation Tw = z/U and from integration of ρw (τ ) to its first crossing to the filtering method. The LP method

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(a)

(b)

(c)

(d)

Fig. 4 Scatterplots comparing various methods of estimating random error against the filtering method, plotted for the momentum flux u w : a LP method using Tw ; b LP method using T f ; c LMK method using T f ; d MBB method. In a grey circles denote Tw = z/U ; black dots denote Tw calculated from integrating ρw (τ ) up to the first crossing. In b, c grey circles denote an exponential fit to ρuw (τ ); black dots denote integration of ρuw (τ ) up to the first crossing

using Tw from integration has the tendency to predict larger values of uw than the filtering method. Agreement between LP using Tw = z/U and the filtering method is worse still; the two predictions vary by as much as a factor of 2–3. In Fig. 4b, uw predicted by the LP method using T f calculated both from direct integration and the exponential fit is plotted against the prediction from the filtering method. When T f is calculated from direct integration, it has a linear trend relative to the filtering method, although it is slightly above the 1:1 line. This is due to truncating the integration of ρ f (τ ) at its first crossing; if the integration were carried on further, the negative region of ρ f (τ ) included in the integral would lead to smaller values of T f and therefore smaller values of uw . The value of uw predicted by LP using T f from the exponential fit is systematically larger than the error predicted by the filtering method, which indicates that the exponential fit is not a good assumption for atmospheric data. Figure 4c compares the LMK method with the filtering method. As with the LP method using T f , much better agreement is found between the two methods when T f is calculated by integrating the autocorrelation function to the first zero-crossing. From Fig. 4 we conclude that replacing T f with Tw is not a good approximation, regardless of whether Tw is calculated from the autocorrelation function or through the use of a scaling relation. Furthermore, we see that despite the convenience of an exponential fit of the form (8) to estimate T f , this leads to a less accurate prediction of the error than integrating ρ(τ ) directly.

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Table 3 Slope (m) for a linear regression through the origin and R 2 value for comparison of each error estimation method to the filtering method LP, Tw z/U u w w θ uu v v w w θ θ U Θ

LP, T f ρ(τ )

ρ(τ )

LMK, T f Fit

ρ(τ )

MBB Fit

m

1.04

1.23

1.07

1.31

1.06

1.27

1.01

R2

0.860

0.945

0.994

0.945

0.978

0.922

0.970

m

1.27

1.25

1.08

1.41

1.00

1.33

0.98

R2

0.088

0.578

0.977

0.786

0.884

0.691

0.841

m

0.49

0.51

1.12

1.33

1.12

1.33

0.96

R2

0.216

0.064

0.948

0.678

0.807

0.582

0.788

m

0.42

0.45

1.14

1.36

1.16

1.38

0.94

R2

0.055

0.182

0.960

0.684

0.822

0.636

0.811

m

1.08

1.08

1.19

1.75

0.89

1.42

1.12

R2

0.692

0.053

0.947

0.566

0.775

0.571

0.861

m

0.52

0.54

1.15

1.48

1.07

1.33

0.99

R2

0.001

0.008

0.960

0.746

0.796

0.637

0.814

m

–

–

1.24

1.32

–

–

1.01

R2

–

–

0.985

0.894

–

–

0.848

m

–

–

1.42

1.80

–

–

1.14

R2

–

–

0.962

0.892

–

–

0.928

We consider the LP method using the integral time scale of vertical velocity Tw calculated both from integrating ρ(τ ) to its first crossing and the approximation Tw ≈ z/U . We also consider the LP and LMK methods using the integral time scale for the flux of interest T f calculated both from ρ(τ ) and an exponential fit of the form ρ(τ ) = exp(−|τ |/T f )

Statistics for the comparison between errors predicted by filtering method and the other methods can be found for all fluxes in Table 3, where we show the slope for a linear fit through the origin and R 2 values. Table 3 further illustrates that the approximation T f ≈ Tw is not a good one, yielding small R 2 values for most fluxes regardless of whether Tw is calculated through the approximation Tw = z/U or by integration of ρ(τ ). Calculating T f through an exponential fit to the measured autocorrelation function is also a poor assumption for atmospheric data; errors predicted by both the LP and LMK methods using T f from the exponential fit have slopes with larger deviations from unity and smaller R 2 values. Both the LP and LMK methods using T f from integration of ρ(τ ) to the first crossing and the MBB methods are in good agreement with the filtering method. Although the LMK and LP methods have larger R 2 values for most fluxes, the MBB has typical slopes that are closer to unity. This is again due to the overestimation of T f caused by ending the integration of ρ(τ ) at the first zero, which neglects negative values of ρ(τ ). In order to investigate the differences between the filtering method and the others in more detail, a scatterplot comparing the LP method using T f from direct integration of ρ(τ ) and the filtering metals hod is displayed for u u in Fig. 5a. The LP method overpredicts the error relative to our method for many cases. In Fig. 5b, we show the autocorrelation function ρu u (τ ) for a run where the two methods are in poor agreement; Fig. 5c is a plot of the autocorrelation for a case where the two methods are in good agreement. Poor agreement,

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(a)

(b)

(c)

Fig. 5 a Scatterplot comparison of random error in the velocity variance u u between the LP method using T f calculated from direct integration of ρuu (τ ) to the first crossing and the filtering method; b Autocorrelation function of u u for a case where the two methods are in poor agreement; c autocorrelation for a case where the two methods are in good agreement. In b, c the solid line denotes the calculated autocorrelation function; the dashed line is the exponential fit

i.e. a larger prediction from the LP method than the filtering method, occurs when the autocorrelation function does not intersect the time-axis until a large lag τ is reached as shown in Fig. 5b. As a result, the estimate of T f is large, as is the estimate of uw . When ρu u (τ ) intersects the time-axis after a much shorter time lag as in Fig. 5c, the predictions of the two methods are in much better agreement. Note that the exponential autocorrelation function is a poor model for the observed autocorrelation in Fig. 5b, mostly because of the weight of the points for large τ . Attempting to fit the exponential model only to a small range [0, τ0 ] of the autocorrelation function might improve error predictions, but would require an ad hoc specification of τ0 for each block (which would be similar to the lag-window approach of Dias et al. (2004)), and was not tried. Furthermore, the location of the first crossing of ρ(τ ) can be influenced by spurious oscillations, which would lead to larger or smaller calculated values of T f . These plots demonstrate the inherent difficulties of estimating the integral time scale. The LP and LMK methods are highly sensitive to the estimate of T f , whereas the filtering method requires no estimate of the integral scale. The results displayed in Figs. 4, 5 and in Table 3 for the filtering method were obtained by applying a box filter in spectral space, which is equivalent to calculating the running mean of the time series. We also calculated errors from the filtering method using a Gaussian filter and found nearly identical predictions of the errors for all of the fluxes, showing that the filtering method has no dependence on filter type. 5.2.3 Stability-Dependence of Random Errors In this section we consider typical values of errors from the AHATS dataset, and the dependence of random errors on atmospheric stability. Figure 6 is a plot of errors estimated from the filtering method that have been conditionally averaged in bins of the MOST stability variable −u 3 Θ

2

2

∗ 0 z/L where L = is the Obukhov length. Here u ∗ = (u w + v w )1/4 is the friction κg(w θ )0 velocity, Θ0 is the mean potential temperature, κ = 0.4 is the von Kármán constant, and g is gravity. We chose bins such that the resulting lines were relatively smooth functions of z/L. In Fig. 6, we show errors in the fluxes, mean wind, and mean temperature where data from the top five heights from the AHATS profile tower were included together in the bin averaging. Curves of the error vs z/L collapsed neatly for all heights for u u , v v , w w , and u w .

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Fig. 6 Bin-averaged errors in the fluxes, mean wind, and mean temperature from the filtering method as a function of the MOST stability parameter z/L. The top five heights from the AHATS profile tower have been combined for the bin averaging

The bin-averaged relative errors did not collapse for all heights for θ θ and w θ ; we found this occurred because the mean values of temperature variance and heat flux increased with height for a given value of z/L, thus preventing a collapse for all heights in the bin-averaged relative error. Because the average values of w θ and θ θ depended on the measurement height for any given z/L, and because of the large variability within any given bin, we chose to bin average the top five heights together for the heat flux and temperature variance to show typical values of the random error as a function of z/L. In Fig. 6, one can see that u w has a very strong stability dependence for unstable conditions; u w and the variances have a weak stability dependence for stable conditions. The relative error in u w is large for unstable conditions because the surface shear stress vanishes in the free convective limit, i.e. u w → 0 as −z/L → ∞. This increase in uw was found by Bernardes and Dias (2010) to be the main cause of the apparent non-alignment between the Reynolds stress vector and the mean wind vector. Typical errors in u w are on the order of 10% for stable conditions and range from 20% to over 50% in unstable conditions for an averaging period of T = 27.3 min. Random errors in the variances range from approximately 8% for w w to 15% for v v . Typical random errors in w θ were estimated to be approximately 10% for most stabilities. Note that points surrounding z/L = 0 have been excluded from the bin averaging in Fig. 6 for w θ because errors in the heat flux are undefined for neutral stratification (i.e. w θ → 0 as z/L → 0). 5.2.4 Errors in First-Order Moments The filtering method of estimating errors in turbulent moments is general and can be applied to estimate errors in moments of any order. We also considered the magnitude of errors in the mean wind speed U and the mean temperature Θ as a function of the MOST stability variable z/L. In general, the time series of u and θ are more non-stationary than the fluxes. This means that the standard deviations of the filtered quantities u and θ exhibit more departures from −1/2 the Δt power-law; as a result, more care must be taken when selecting the range of filter

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widths to use for the power-law fit. For U (Θ), we use Δmin = 2Tu (2Tθ ) as the lower bound for the power-law fit and Δmax = T /4 as the upper bound for both variables. Note that the lower bound used for the fluxes Δmin = 10T f is a conservative estimate; using a lower filter width of twice the integral scale of the variable of interest means that the series of the filtered flux becomes decorrelated at scale Δt , since 2T f is the decorrelation time. A comparison of the error estimates in U and Θ with the LP and MBB methods is displayed in Table 3. Because the LMK method only applies to central moments, it cannot be used to estimate random errors in U or Θ. We find that the filtering method is in good agreement with the MBB method for estimating errors in U and Θ; the slope for a linear fit through the origin is near unity with R 2 values of 0.848 and 0.928 for U and Θ respectively. Agreement with the LP method is not as good; although the R 2 values are larger than those found with the MBB method, the LP method produced larger estimates of errors in U and Θ than the filtering method. Larger departures from the 1:1 line were found when the integral scale was estimated from the exponential fit to ρ(τ ) than when ρ(τ ) was integrated to its first crossing. Typical values of the errors in U and Θ are displayed in Fig. 6. Once again we have averaged the errors in bins of z/L for the top five sonic anemometers from the AHATS profile tower, including all heights together in the bin average. When the estimated error in U was averaged for each height individually, the errors did not collapse for all heights for the unstable cases. Because it was not clear whether this was due to systematic variability among the sonics at different heights or to insufficient data for the averages to converge, we chose to average errors for all heights together. Typical values of the error in U were found to be approximately 5% for unstable conditions and 2–3% for stable stratification. We found that random errors in Θ (in Kelvin) were less than 0.1% for all stabilities; notice that this corresponds to errors on the order of 0.3 K for typical values of Θ in the surface layer. 5.2.5 Estimation of Integral Scales The filtering method of estimating random error is based on calculating the error variance of the filtered flux directly as a function of scale Δt . When we equate the error estimate from our method and the LP method, we obtain the expression given in (25) that allows us to estimate the integral scale of any variable without the difficulties that exist when estimating the integral scale from the autocorrelation function or from spectra. We used the filtering method to estimate the integral time scales of u, v, w, and θ as a function of stability then converted to length scales through the use of Taylor’s hypothesis, e.g. Λw = U Tw , where Λw is the (Eulerian) integral length scale of w. For u, v, and θ , we observed no systematic trends in the integral length scale with z/L for either stable or unstable cases. This result contradicts the findings of Kaimal et al. (1972), who found from the Kansas data that the peak frequencies of the stable u, v, and θ spectra (related to the integral length scales) varied systematically with z/L. Recent studies (Khanna and Brasseur 1997; Johansson et al. 2001) have shown that additional scales (e.g. z/z i ) need to be included in the scaling for unstable stratification. However, the reason for a lack of systematic variation of Λu , Λv , and Λθ with z/L for stable conditions from our results is unclear. The integral length scale of vertical velocity Λw calculated from the filtering method is compared with the value from integration of ρ(τ ) to the first zero-crossing and to the exponential fit to ρ(τ ) in Fig. 7 as a function of z/L. The empirical fit of Sullivan et al. (2003) to data from the Horizontal Array Turbulence Study (HATS) is also displayed in Fig. 7. Sullivan et al. (2003) used λw , the wavelength associated with the peak in the w spectrum, as a characteristic length scale of turbulence in the surface layer. They used Taylor’s hypothesis, and

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Fig. 7 Dimensionless integral length scale of vertical velocity Λw /z as a function of the MOST stability parameter z/L. Λw was calculated through use of Taylor’s hypothesis by direct integration of ρ(τ ) to the first zero crossing, exponential fit of the form ρ(τ ) = exp(−|τ |/Tw ), and by the filtering method. All points are calculated for z = 8.05 m from the AHATS profile tower. Also shown is the empirical curve of Sullivan et al. (2003)

assumed the autocorrelation of w can be modelled well by an exponential (Eq. 8). Under these assumptions, the relationship between λw and the integral scales is λw = 2πΛw = 2πU Tw . They proposed the empirical relation λw = 2πΛw = z/H(z/L)

(34)

where H(z/L) =

0.17, z/L ≤ −0.2 0.38 + z/L(1.04 − 0.2z/L), −0.2 < z/L < 2.

(35)

Equations 34–35 are the empirical fit to λw estimated from 35 ideal blocks of the HATS data. Sullivan et al. (2003) selected these blocks because of stationarity of mean quantities and fluxes; each block of data was 25 min in length or greater. Sullivan et al. estimated Tw using the exponential fit to the calculated autocorrelation function. Note that the integral scale Λw is plotted in Fig. 7, rather than the wavelength λw related to the peak in the w spectrum. We find that the filtering method produces a prediction of Λw in good agreement with the empirical curve of Sullivan et al. (2003) for unstable and stable stratification. The value of Λw estimated from integration of ρ(τ ) is consistently larger than the prediction from the filtering method; values of Λw from the exponential fit are larger still. When one fits an exponential function to the calculated autocorrelation function, the maximum value of ρ(τ ) included in the fit must be selected arbitrarily. We are fitting the exponential function to ρ(τ ) between ρ(0) and the first zero crossing; a different maximum value used for the fit could lead to a different calculated value of the integral time scale. It is not entirely clear why the value of Λw calculated from the exponential fit to ρ(τ ) is not in better agreement with the empirical curve of Sullivan et al. (2003), since they calculated the integral length scale based on an exponential fit to ρ(τ ); this may be due to a difference in the range ρ(τ ) used for the fit to the exponential function, length of the blocks, or to block selection criteria. The results displayed in Fig. 7 give us further confirmation of the accuracy and robustness of the filtering method for estimating errors in turbulent moments and for estimating integral scales.

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6 Summary and Conclusions Random errors in measured turbulent fluxes in the ABL are due to averaging over time periods too short for time-averaged fluxes to converge sufficiently to the true ensemble mean as required by the ergodic hypothesis. Random errors can be a significant source of error in studies of ABL turbulence over short time scales, i.e. hours to days. Unfortunately, it is not common practice for authors to include an estimate of random error when reporting micrometeorological measurements. Most existing methods for estimating random errors in turbulent fluxes (e.g. Lumley and Panofsky 1964; Wyngaard 1973; Lenschow and Stankov 1986; Mann and Lenschow 1994; Lenschow et al. 1994) require an estimate of the integral time scale, which can be highly sensitive to the method of calculation employed and produce uncertainties in the estimate of the error. A new method for estimating random errors where a priori knowledge of the integral scale is not needed has been proposed based on a local decomposition of turbulent fluxes. This local flux decomposition is based on the commutative property of spatial filters. The time average of the local flux at any filter time scale Δt is identically the time-averaged flux for period T . −1/2 We estimate random error by fitting a power law of the form σ (Δt ) = Cwc Δt to the w c standard deviation of the local flux and extrapolating to a block of length T ; the estimate of the random error is simply σ (T ) normalized by the time-average flux w c . w c −1/2

Note that σ (Δt ) = Cwc Δt behaviour is predicted by statistical theory, but in w c practice if one would fit a power law of the form σ (Δt ) = Cwc Δbt , one would not w c always obtain b = −1/2. We believe that these deviations from b = −1/2 could be due to non-stationarities in the time series of the flux. Difficulties in predicting random errors for non-stationary time series is not unique, however, to the filtering method. As Dias et al. (2004) point out, the autocorrelation function (and therefore the integral time scale) becomes undefined for non-stationary time series. They found that in general, the integral time scale failed to exist more frequently for even-ordered (e.g. u u ) than for odd-ordered (e.g. u u u ) moments. The filtering method of estimating random errors is shown to predict the ensemble mean accurately for synthetic flux data where the ensemble mean is known. Several other methods of estimating random error in turbulent fluxes were compared with the filtering method using ASL data from AHATS. We found that the methods in best agreement with the filtering method are the Lumley and Panofsky (1964) and Lenschow et al. (1994) methods using the integral time scale of the flux calculated from integrating the autocorrelation of the flux to the first zero-crossing and the moving block bootstrap method (Garcia et al. 2006). The assumption T f ≈ Tw resulted in poor agreement with the filtering method, as did estimating T f from an exponential fit to the calculated autocorrelation function. For 27.3-min blocks from the AHATS data, we found typical errors of 10% for the heat flux and 7–15% for variances. Errors in u w were approximately 10–15% for neutral and stable conditions and they increased rapidly with increasing instability, reaching values of 40% or greater for unstable stratification. Errors in the mean quantities U and Θ were also estimated using the filtering method; we found errors in U to be on the order of 5% for unstable conditions and 2–3% for stable conditions. Errors in Θ were found to be on the order of 0.1%, which corresponds to an error of 0.3 K for typical conditions in the surface layer. Because the coefficient Cwc calculated from the power-law fit to the standard deviation of the filtered flux σ is related to the intew c gral time scale, we are able to estimate the integral time scale of any turbulent quantity using filtering. We found that the integral length scales of u, v, and θ exhibited no systematic trends

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with z/L. However, the estimate of the integral length scale of vertical velocity Λw from the filtering method was found to be in good agreement with the empirical function of Sullivan et al. (2003) for all stabilities (−2 ≤ z/L ≤ 1) that we considered. The filtering method was in better agreement with their results than other methods typically employed for estimating the integral scale, i.e. integrating ρ(τ ) directly to the first crossing or estimating Tw through an exponential fit of the form ρ(τ ) = exp(−|τ |/Tw ) to the calculated autocorrelation function. Because random errors can be significant over typical averaging periods used in micrometeorological studies, and are not difficult to estimate, we encourage authors to include an estimate of random errors when reporting measurements. Having an estimate of random error available may allow researchers to determine whether scatter exhibited in measurements is due solely to random error, or whether other dynamical factors in the ABL may also be responsible. Acknowledgments STS and MC gratefully acknowledge support from the National Science Foundation, Grant AGS-0638385. MC is also grateful for support from the NSF, Grant OISE-1061712, which made collaboration with N.L. Dias possible. The AHATS data were collected by NCAR’s Integrated Surface Flux Facility

Appendix: Algorithm for Estimating Random Errors Via Filtering For the reader interested in the practical details of how to apply the filtering method of estimating random errors, an algorithm is outlined below. Suppose one is interested in estimating the random error in w c , the vertical flux of a scalar c, over an averaging period of T = 30 min. The steps required to estimate random error in w c using the filtering method are the following: 1. 2.

3. 4.

5.

Apply data selection criteria and preprocessing algorithms to the data (e.g. despiking, linear interpolation, alignment with mean wind, removal of nonstationary runs). Calculate the time averages W and C for period T and remove from the time series of w(t) and c(t) to generate time series of the fluctuating quantities w (t) = w(t) − W and c (t) = c(t) − C. Multiply each point in the time series of w (t) and c (t) to obtain a time series of the instantaneous flux w c (t). Generate a series of time filter widths Δt between Δmin and Δmax that are spaced evenly on a logarithmic scale. We recommend the conservative values Δmax = T /10 and Δmin = 10T f . Here a rough estimate of the integral scale T f can be used, e.g. using the approximation T f ≈ z/U . Filter the time series of the instantaneous flux w c (t) at each scale Δt and calculate its 2 (Δ ) as a function of filter width (i.e. using the definition given in Eq. 21). variance σ t wc

6.

7. 8.

−1/2

(Δt ) = Cwc Δt (Eq. 22) to the standard deviation Fit a power law of the form σ w c of the local filtered flux as a function of filter width, where Cwc is the coefficient to be determined. Evaluate the power-law fit for averaging period T (Eq. 23) to obtain the estimate of error in the filtered flux. If desired, divide the error estimate from filtering by the mean value of w c for averaging period T to obtain the fractional error in w c (Eq. 24).

A Matlab code that applies steps 3–7 of the above algorithm is available online in the supplemental material.

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