Application of the band‐pass covariance technique to portable flux measurements over the Tibetan Plateau

Abstract

[1] Two versions of the band‐pass covariance technique were applied to the turbulence data collected during daytime with a simple and portable measurement system over the sparse grasslands of the Tibetan Plateau. The coherency spectra between the temperature and the specific humidity, which is a spectral counterpart of the correlation coefficient, were used as a dynamic indicator of the energy‐containing ranges as well as that of the sensor attenuation at higher frequencies. The comparison with independent measurements by the eddy covariance method showed that the original version of the band‐pass covariance technique occasionally fails. This indicates breakdowns of the similarity between the temperature and the water vapor, especially in the lowest‐frequency regions. On the other hand, the latent heat flux computed with the advanced version exhibited adequate agreement with the eddy covariance method. This paper demonstrates that the current implementation of the advanced version with the embedded self‐calibration procedure provides a robust method of frequency extrapolation in scalar flux measurements under unstable conditions.

1. Introduction

[2] Sensors of humidity that are not capable of fully capturing turbulence fluctuations of water vapor, nevertheless, can be used for the measurements of the latent heat flux from the Earth's surface. One possibility is through the band‐pass covariance technique, or the band‐pass covariance method, independently proposed or suggested by Hicks and McMillen [1988] and Högström et al. [1989]. This technique is primarily a spectral extrapolation accompanied by the eddy covariance technique, and calculates the higher‐frequency component of a scalar flux in question that was lost due to the inadequate sensor response. This is done with a guide of a perfectly measured scalar, usually the sonic temperature. This technique facilitates measurements of a scalar flux using a portable instrumentation [see, e.g., Horst and Oncley, 1995; Horst et al., 1997] and a long‐term monitoring of a scalar flux [Watanabe et al., 2000; Yasuda and Watanabe, 2001].

[3] One of the important advantages of the band‐pass covariance technique is that, unlike the other flux correction methods [e.g., Moore, 1986; see also, Massman and Lee, 2002], it does not require a predetermined frequency response function of the sensor, nor does it necessitate assuming a functional form of the flux cospectra. As it is shown in this paper, however, there has been an ambiguity left with the band‐pass covariance technique and an explanation based on an existing theory or an empirical relationship is desired. Surely, this has been a factor that has prevented its potentially wider application, despite strong potential capability of the band‐pass covariance technique.

[4] The aims of this study are to investigate the band‐pass covariance technique from the view point of the surface layer similarity. Particularly, the similarity theory for the two‐point statistics in the unstable atmospheric surface layer (ASL) proposed by Kader and Yaglom [1991] will be used in an effort to give a clear explanation of its physical basis to the technique. Throughout this process, it will be shown that this technique has, in fact, two different versions, each with a different physical background. A plain distinction between these two versions will be made in this paper. This document, then, proposes an improved algorithm for the technique using the coherency spectra as an indicator of the similarity between the sonic temperature and the water vapor. The band‐pass covariance technique with the newly proposed scheme is applied to the latent heat flux measurements obtained over the sparse grasslands of the Tibetan Plateau using a portable flux measurement system called mobile turbulence measurement system (MTMS). There any laboratory calibration results were not expected to be applicable, and any kind of in situ calibration was virtually impossible. This is the most challenging of the flux measurements. Therefore this paper, without using any additional information other than the measured turbulence data itself, fully utilizes and qualifies the capability of the band‐pass covariance technique.

2. Band‐Pass Covariance Technique

[5] Here, two major versions of the band‐pass covariance technique are introduced. For the sake of generality, they are written in terms of a scalar, s. Their application to latent heat flux, E = ρL_e, or the flux of carbon dioxide, F_c = ρ, can be readily derived by substituting s with either specific humidity, q, or carbon dioxide concentration, c, where ρ is the air density, and L_e the heat of vaporization.

2.1. Extension of Monin‐Obukhov Similarity to Cospectra

[6] Cospectra between the vertical velocity component and the scalar are a projection of the vertical transport of s to the Fourier domain. Its integration over the entire frequency domain is the covariance between them.

where S_ws(n) represents the cospectra, z the height above the displacement height, the mean velocity, and n = fz/ the normalized frequency, in which f is the cyclic frequency. Taylor's hypothesis is used to normalize the frequency and other ordinary mathematical notations in the surface hydrology are used in this paper.

[7] In a steady and horizontally homogeneous ASL, S_ws(n) can be written by extending Monin‐Obukhov similarity (MOS) to the Fourier domain, as follows [e.g., Wyngaard and Coté, 1972; Kader and Yaglom, 1991]

where u_* is the friction velocity, s_* ≡ ‒/u_* the scale of s, F_s the universal function of n and ζ ≡ z/L, in which L is the Obukhov length. Within the context of MOS, the similarity between scalars, or shortly the scalar similarity, is defined as an universality of dimensionless moments, independent of the choice of scalars [Hill, 1989; Dias and Brutsaert, 1996]. If this concept can be extended to be applied to the cospectra, it is assumed that F_s is universal, i.e., independent of the choice of s. This is written for the temperature, T, and a scalar, s, as follows

A considerable number of experimental evidences for eq. (3) have been presented using the measurements in ASL [Schmitt et al., 1979; Anderson and Verma, 1985; Ohtaki, 1985].

[8] Plugging (2) into (3) gives the following relationship,

Integrating (4) over a frequency range, X, gives the first version of the band‐pass covariance technique as follows

Equation (5) can be taken as a spectral extension of the Bowen ratio method, and gives an estimate of from and the ratio of the cospectra in a limited frequency band. This is a core concept of the original band‐pass covariance technique proposed by Hicks and McMillen [1988] that was later also used by Verma et al. [1992] for methane flux measurements. Equation (5) has been used often in the measurements of trace gases fluxes, such as, ozone [Jacob et al., 1992], carbon dioxide and methane [Fan et al., 1992] and isoprene [Guenther and Hills, 1998] among others.

[9] Another application of (5) can also be found in the measurements of surface heat fluxes [Blanken et al., 1997; Twine et al., 2000], when the eddy covariance measurements of the sensible heat flux and the latent heat flux, denoted as H_EC and E_EC, do not close the surface energy budget equation. In such circumstances, a new value of H and E that satisfies the energy budget equation is often computed using the Bowen ratio of the measured fluxes, Bo = H_EC/E_EC, and the measurements of the other energy budget components. This procedure implicitly assumes that measurements of H_EC and E_EC both suffer from the same spectral attenuation, and that their ratio, H_EC/E_EC, is equal to the actual value. Henceforth, this can be considered as an implicit application of the original band‐pass covariance technique, i.e., equation (5).

[10] In should be noted that these practical applications of (5), whether explicit or implicit, are usually utilized only for a rough estimate of the flux values, while, in some cases, they are used in seeking more accurate estimates of fluxes.

2.2. Advanced Version of Band‐Pass Covariance Technique

[11] Another, and more advanced, version of the band‐pass covariance technique has been suggested independently by Högström et al. [1989]. They tried to make a practical estimate of lost covariance in with q measured with wet bulb thermometer over a sparse pine forest. They observed that the ratio of the cospectra, S_wT(n)/S_wq(n), remained at a constant value over a middle frequency range. Then, they assumed this to be also constant at the higher frequencies where true fluctuations of q were not available due to the sensor attenuation. With this assumption, they made a rough estimate of underestimation in . The same procedure was also applied in CO₂ flux measurements by Grelle and Lindroth [1996].

[12] Although the initial intention of Högström et al. [1989] appears to be limited to a rough estimate designated to meet practical necessity, their procedure can be used to formulate the advanced version of the band‐pass covariance technique for the flux of the scalar, s, in general. First, it is assumed that the following equation is valid over a certain frequency range.

where n₁ is the lower end of the frequency range, in which (6) is valid, and β₀ is the value of the constant of β(n). With this assumption, can be computed as follows [Yasuda and Watanabe, 2001].

where n₂ is the upper end of the frequency range in which unattenuated measurements of S_ws(n) are available. In practice, β₀ can be determined as an average of β(n) over the frequency range, n₁ < n < n₂.

[13] Equation (7) forms the advanced version of the band‐pass covariance technique. It was used by Horst and Oncley [1995] and Horst et al. [1997] in their algorithm for NCAR (National Center for Atmospheric Research) PAM (Portable Automated Mesonet) III to compute latent heat flux. More recently, Watanabe et al. [2000] utilized (7) with water vapor flux measurements over a deciduous forest where q was measured by a capacitance hygrometer. They compared the results with the eddy covariance measurement and found an overall satisfying performance of (7), except when was small. Yasuda and Watanabe [2001] applied (7) to CO₂ flux measurements with closed path infrared gas analyzer and obtained an adequate comparison with open path measurements.

[14] A basic assumption which allows (7) to compute is that (6) is valid over a frequency range, n₁ < n < ∞. This has been observed by some investigators: As described above, Högström et al. [1989, Table 1] found that β(n) remained at constant values from around the energy‐containing range up to the frequency at which the cospectra drops off due to the slow q sensor. Watanabe et al. [2000, Figure 2] also observed that S_wT(n)/S_wq(n) was roughly constant at the ratio, /, around the energy‐containing range.

[15] Equation (6) can be derived by extending a surface similarity argument of Kader and Yaglom [1991], who applied the directional similarity of Kader and Yaglom [1990] to spectra and cospectra of the velocity components and the temperature in the unstable ASL. The directional similarity assumes that there exist three sublayers in the unstable ASL, namely, (1) the dynamic sublayer (D sublayer, ζ ~ ‒0.04) where the buoyancy does not affect the turbulence field, (2) the dynamic‐convective sublayer (DC sublayer, ‒1.2 ≲ ζ ≲ ‒0.12) where both the buoyancy and the shear stress affect the turbulence but in the different directions, causing the vertical and horizontal motions uncoupled with each other, and (3) the convective sublayer (C sublayer, ζ ≲ ‒2) where the shear stress does not play an important role in generating turbulence. Kader and Yaglom [1991] applied this three‐layered structure of ASL to cospectra and spectra and predicted their formulation as a function of ζ and n. One of the most notable predictions of their results is ‒1 power decay of the longitudinal velocity spectra in the energy‐containing range, and this receives growing attention in the micrometeorological community [see, e.g., Katul and Chu, 1998; McNaughton and Laubach, 2000; Högström et al., 2002].

[16] The theory for cospectra, S_wT(n), by Kader and Yaglom [1991] can be extended to any scalar, s. In the inertial range (1 ≲ n ≲ z/η, where η is Kolmogorov's microscale), Kader and Yaglom [1991] invoked ‒7/3 power decay of the cospectra, first suggested by Lumley [1964, 1967] and later validated with the ASL data for S_wT(n) [e.g., Kaimal et al., 1972; Wyngaard and Coté, 1972; Katul et al., 1998, 2001; Su et al., 2004] and for S_wq(n) and S_wc(n) [e.g., Ohtaki, 1985; Anderson and Verma, 1985; Katul et al., 1998, 2001]. See also Warhaft [2000] for recent developments. With S_ws(n) ~ n^‒7/3 and the aforementioned stability requirements for each of the three sublayers, Kader and Yaglom [1991] gave predictions of S_wT(n), which can be rewritten for S_ws(n) in general, as follows

where C_s⁽¹⁾ and C_s⁽³⁾ are constants to be determined experimentally, and f_s⁽²⁾(ζ) the dimensionless function of ζ.

[17] In the energy‐containing range (n < 1), according to Kader and Yaglom [1991], z is no longer a relevant parameter for the turbulent statistics, and S_ws(n) is proportional to n^‒1. With this assumption and the aforementioned three sublayer structure, S_ws(n) is predicted as follows

where D_s⁽¹⁾, D_s⁽²⁾ and D_s⁽³⁾ are constants that can be determined experimentally. Decay with n^‒1 of these cospectra was also observed by Katul et al. [1998].

[18] Equations (8) and (9) readily yield a simple expression for β(n) in the inertial and the energy‐containing ranges as follows.

where φ_β(ζ) ≡ f_T(ζ)/f_s(ζ) is a function of ζ, and the value of the constant in the second expression can vary in different sublayers and in different frequency ranges (the inertial or the energy‐containing range). In each realization of β(n), φ_β(ζ) is constant, and therefore equation (10) states that β(n), as a function of n, takes a constant value in the energy‐containing and the inertial ranges, irrespective of the stability. Since these two frequency ranges are almost contiguous to each other [Kader and Yaglom, 1991], it can be postulated that β(n) is a constant between the energy‐containing and the dissipation frequencies in all stability classes.

[19] It should be noted that any form of the scalar similarity between T and s defined above as the equity of the dimensionless function, was not assumed in the derivation of (10). In fact, if the scalar similarity between T and s had been assumed, then all of the constants and the functions were identical for T and s, i.e., C_T⁽ⁱ⁾ = C_s⁽ⁱ⁾ (for i = 1, 3), D_T⁽ⁱ⁾ = D_s⁽ⁱ⁾ (for i = 1, 2, 3), and f_T⁽²⁾(ζ) = f_s⁽²⁾(ζ), as suggested by Hill [1989]. This would lead to β(n)∣s_*/T_*∣ = 1, that is equivalent to (4). This is not meant in (10) where the value of the constancy is left unspecified. The only assumption that leads to (10) is that both T and s follow the similarity arguments that result in (8) and (9), that is n^‒7/3 and n^‒1 dependence of the cospectra, respectively. Decay of S_ws(n) with n^‒7/3 in the inertial range can be derived from a dimensional argument where S_ws(n) is a function of ξ, n, ∂/∂z [Lumley, 1967; Wyngaard and Coté, 1972], in which ξ is the average dissipation rate of the turbulent kinetic energy. In the energy containing range, a hypothesis that z is not a relevant parameter leads to its proportionality to n^‒1. Therefore (10) is a consequence of the fact that the both T and s fluctuations share the same flow characteristics between the production and the dissipation frequencies.

[20] It has been shown in this section that the band‐pass covariance technique, in fact, points to the two different flux calculation methods, and that they have different physical bases. The original version formulated as (5) requires that T and s be similar at all frequencies. This is the Fourier version of the scalar similarity compliant to MOS. On the other hand, the advanced version, (7), requires that the ratio of the cospectra, β(n), should be constant in the higher‐frequency range. This is most likely valid between the production and the dissipation frequencies, where well‐defined dynamic forcing on the scalar field exists. It is apparent that the former is more strict than the latter. If (7) fails, (5) should also fail, while (7) works even if (5) fails.

3. Mobile Turbulence Measurements Over Tibetan Plateau

[21] The data subject to analyses in this study were collected with MTMS during GAME‐Tibet (GEWEX Asia Monsoon Experiment‐Tibet) IOP (Intensive Observation Period) in the summer of 1998. GAME‐Tibet is one of the regional subprograms of the GAME, and the main objective of GAME‐Tibet is the better understanding of the energy and water cycle over the Tibetan Plateau. To fulfill this objective, spatially distributed time series of hydrological and meteorological data were acquired during its IOP in the region of approximately 150 km × 150 km in the central Tibetan Plateau. Among all ground‐based measurements, several measurement systems were used to highlight the spatial distribution of surface energy components along the north‐south cross section of the central Tibetan Plateau. These measurement systems include two tower‐based eddy covariance (EC) systems, two PAM III systems, four AWS (Automated Weather Stations) systems, and MTMS. Those except for MTMS are site‐fixed measurements placed along the Qinghai‐Xizang highway. Several publications have already reported these measurement systems and some results from them. These include those by Tanaka et al. [2001, 2003], Ma et al. [2003], Choi et al. [2004], and Yang et al. [2004].

[22] The roles of MTMS during GAME‐Tibet IOP are closely related to the other flux measurement systems. EC and PAMIII are designed to provide independent measurements of the surface energy components. AWS's measured T, q, and wind speed at a single level, and the surface radiative temperature. Therefore, in order to compute surface fluxes out of the data of AWSs, surface roughness for wind speed and temperature need to be determined. MTMS was planned to provide flux data to calculate these parameters. Another major role of MTMS is to provide reference flux data for the cross comparison of the other tower‐based surface flux measurements. The flux measurement systems used in GAME‐Tibet '98 IOP were composed of different sensors and were based on different measurement techniques. Therefore a cross comparison between these systems is essential to derive spatial distribution of the surface energy components out of their measurements. For this cross comparison, rather than gathering all of the measurement systems at one place, a method with a portable flux measurement system was adopted in GAME‐Tibet. If a single flux measurement system can collect flux data at all of the flux sites, site‐to‐site comparisons of the flux measurements are possible using this system as a reference. The second role of MTMS is to serve as a reference for this indirect cross comparison.

[23] MTMS was designated specifically to perform these two roles over the Tibetan Plateau. It was designed and assembled at the Disaster Prevention Research Institute of Kyoto University. Sensors used in MTMS are tabulated in Table 1. Of them all, the two capacitance‐type hygrometers, namely, HumAir (AIR Inc.) and Humicap (Väisälä Inc.), give measurements of specific humidity, q, at a relatively slower response. Hereafter, subscripts _HA and _HC, are used to represent HumAir and Humicap, respectively, when the origins of the measurements need to be specified. Since these sensors are not designed for turbulence measurements, use of the band‐pass covariance technique was planned. Both q sensors were mounted in radiation shields. HumAir has a motored fan for ventilation, whereas Humicap is ventilated with the natural wind to minimize the power consumption of the system. All of the sensors are mounted on the tripod at about 2 meters above ground, and all of the system runs on a shielded battery. Measured data were sampled at 10 Hz and all of the unprocessed data were stored in a custom‐made logger. The whole measurement system is so simple and light weight that it can be transported in a single sport utility vehicle.

Table 1. Sensors Used in the Eddy Correlation System at the BJ Site and Those Used in the Mobile Turbulent Measurement System

Sensors	Manufacture and Type	Measured Variables
Eddy Correlation System at BJ
Sonic anemometer	Campbell Scientific, CSAT‐3	u, v, w, T
Krypton hygrometer	Campbell Scientific, KH‐20	q
Mobile Turbulent Measurement System
Sonic anemometer	Kaijo, DA‐100	u, v, w, T
Capacitance hygrometer	AIR, HumAir	T, RH
Capacitance hygrometer	Väisälä, Humicap	T, RH

[24] Of the measurements collected with MTMS (Table 2), those measured at the BJ site were used when a comparison with eddy covariance measurements is needed. Detailed information regarding the sites and the measurement instruments at the BJ site are available elsewhere [Gao et al., 2004; Choi et al., 2004]. Briefly, the BJ site is located in an open area to the south of the town of Naqu. The ground surface there is almost bare soil during the premonsoon period, while in the middle of the summer monsoon, the land surface is covered with short, sparse grass. The topography is almost flat within 10 km from the site. Therefore the site meets a condition for micrometeorological measurements in the horizontally homogeneous ASL.

Table 2. Sites and Periods of the Mobile Turbulence Measurements

Site	Location	Period,^aa Local time (LT) is Beijing Standard Time. LT
EC Tower Sites
Amdo	4700 m asl, 32°15′N, 91°38′E	23 May 1150 to 24 May 0650
		31 Jul 1450 to 2 Aug 1350
BJ	4580 m asl, 31°22′N, 91°54′E	5 Aug 1710 to 6 Aug 1040
		8 Aug 1140 to 9 Aug 1600
AWS Sites
D110	32°42′N, 91°53′E	1 Jun 1250 to 3 Jun 1040
MS3608	4610 m asl, 31°14′N, 91°47′E	11 Aug 1640 to 12 Aug 1710

• a Local time (LT) is Beijing Standard Time.

[25] Measured 10 Hz data were segmented into 30‐min data runs. After linear trends were removed, the Fourier transform [Frigo and Johnson, 1998] were applied to the data. Only the data under the unstable condition, judged tentatively for this purpose as H > 10 W/m², were subject to the analyses.

4. New Implementation of Band‐Pass Covariance Technique

[26] As mentioned in section 3, the sensors for q used in MTMS are not specialized for the turbulence measurements. The frequency responses of q sensor was investigated with spectra, S_qq(n), and cospectra with w, S_wq(n), and their examples are shown in Figures 1a and 1b. Expected attenuation of q signals at higher frequencies are obvious in Figure 1b. Moreover, q measured with HumAir has a better response at higher frequencies than that with Humicap. This is not only due to the characteristics of the sensors themselves, but also due to their difference in the ventilation described in section 3. The consequences of this difference in the frequency response will be later investigated using the results of the band‐pass covariance technique.

Figure 1

An example of (a) a spectrum, S_qq(n), (b) cospectra, S_wT(n) and S_wq(n), and (c) a coherency spectrum, R_Tq(n), measured with MTMS. The data are from run 300, measured during the operation at the BJ site, on 9 August 1998 from 1210 to 1240 LT.

[27] Application of (7) to the measurements requires a set of predetermined frequencies, n₁ and n₂, so that the value of β₀ can be determined with the data in the frequency range, n₁ < n < n₂. Usually, this frequency range, hereafter called “BP range”, is determined with results of laboratory tests [e.g., Horst and Oncley, 1995; Yasuda and Watanabe, 2001]. Unlike the case with closed path sensors which are often kept in a well‐controlled environment, frequency responses of the capacitance hygrometers used in MTMS can vary with the atmospheric conditions and ventilations. Moreover, sensor characteristics might have changed due to sensor degradation during the four months of the operations over the Tibetan Plateau. In situ calibrations, however, were not practically possible over the Tibetan Plateau, where power supply was not available. In this study, therefore, a more data‐oriented approach was taken, and the BP range was determined from the data themselves. This results in a dynamic determination of the BP range, allowing it to vary run by run, rather than taking it as a constant frequency range. In light of the discussion in section 3 the lower end of the BP range, n₁, is taken as the lower end of the energy‐containing range, while the upper end, n₂, is determined so that the effect of attenuation in q is not significant in the BP range. In this paper, coherency spectra between T and q, defined as [e.g., Priestley, 1981]

were used as an indicator to identify the frequency band, n₁ ≤ n ≤ n₂. Note should be made as to the difference between R_Tq and the coherency function [e.g., Bendat and Piersol, 1971; Ohtaki, 1985].

[28] Coherency spectra is a spectral correlation coefficient between Fourier modes of T and q, and can be interpreted as a phase alignment between them [Yeung, 1996]. This quantity has been used as an indicator of scalar similarity in Fourier space [e.g., Phelps and Pond, 1971; McBean and Miyake, 1972; Tong and Warhaft, 1995; Yeung, 1996; Pope, 2000, p. 247; Ulitsky et al., 2002; I. Tamagawa et al., Dissimilarity between temperature and specific humidity fluctuation observed over Tibetan Pleateau in GAME, manuscript in preparation, 2005]. Recently, J. Asanuma et al. (Spectral similarity between scalars at the lowest frequencies in the unstable atmospheric surface layer over Tibetan Plateau, submitted to Boundary‐Layer Meteorology, 2005, hereinafter referred to as Asanuma et al., submitted manuscript, 2005) investigated the frequency behavior of R_Tq(n) with the current data set. By extending the similarity arguments of Kader and Yaglom [1991], they showed that a procedure similar to the derivation of (10) yields an expression for R_Tq(n), as follows

where φ_R is a function of ζ. The value of the constant in the second expression may be different in the different sublayers and in the different frequency ranges (the inertial or the energy‐containing range). Asanuma et al. (submitted manuscript, 2005) also have found that R_Tq(n) remains almost constant within the energy‐containing range and that, at its lower end, R_Tq(n) levels off sharply toward the lower frequencies.

[29] The frequency, n₂, at which the attenuation of q becomes significant, was also identified with R_Tq(n). An example of R_Tq(n), shown in Figure 1c, exhibits clear drop‐off toward the higher frequencies. Especially, it is clearly shown that R_Tq(n) with q_HC levels off earlier than that of q_HA. This is consistent with the relatively slower response of Humicap, shown in spectra and cospectra in Figures 1a and 1b. These indicates the capability of R_Tq(n) to detect signal attenuation.

[30] With these background, R_Tq(n) was used to detect the BP frequency range in this article. For each 30‐min run, R_Tq(n) was calculated from the corresponding spectra and cospectra that were block averaged over frequency bands equally spaced in the logarithmic frequency. Then, BP range, n₁ < n < n₂, was identified as a frequency range that meets R_Tq(n) > R_cr, where the critical value, R_cr, is a predetermined constant. For the value of R_cr, R_cr = 0.9 is tentatively chosen, and sensitivity of the results on the value of R_cr will be investigated later in this paper.

[31] The vertical bars in Figure 2 indicate the BP range identified for all the data analyzed. Note that the vertical axis in Figure 2 is f = n/z, and it is not normalized. It is noticed that BP ranges identified in June are consistently lower than those of July and August. This can be attributed to the fact that HumAir was not aspirated in June due to a hardware problem. This accidental failure of the hardware happens to suggest that R_Tq(n) successfully detects the sensor response that may vary under the influence of external conditions, in this case, the ventilation.

Figure 2

BP frequency range determined with R_Tq > 0.9 for HumAir (the vertical bars). The circles denote f₀, the characteristic frequency of the q sensor response.

[32] In order to investigate further the validity of R_Tq(n) as an indicator of the attenuation in q, a first‐order response function was assumed for the response of the capacitance hygrometers used in MTMS as, S_wq^m = S_wq(f)/(1 + (f/f₀)²). where the superscript, ^m, denotes the measured value, f₀ the characteristic frequency of the sensor at which the measurements are the half of the actual value [Horst et al., 1997; see also Toda et al., 2002]. Then, (6) can be recast as follows

Using (13), the value of f₀ was computed by linearly fitting f² against β^m(f), and the calculated values were also plotted in Figure 2. Figure 2 exhibits that f₂ = n₂/z determined with R_Tq(n) closely follows f₀ during the course of the seasons. Especially, the difference in the value of f₂ between June and August was explained plainly by that of f₀. This close relationship between f₀ and f₂ is also manifested in Figure 3, where most of the data are shown to fall within the range f₀/2 < f₂ < 2f₀. In Figure 3, it is also shown that f₀ tends to be larger than f₂. These agreements between f₀ and f₂ strongly support the use of R_Tq(n) as an indicator of the sensor attenuation in q and therefore as an identifier of the higher end of the BP frequency range.

Figure 3

Higher end of BP range, f₂, determined as R_Tq > 0.9 plotted against f₀, the characteristic frequency of the sensor response. The lines for y = 2x and y = 0.5x are also shown.

[33] It should also be noted that as the definition of R_Tq(n) suggests, simple attenuation of q does not solely reduce the value of R_Tq(n). The actual cause of the reduction in R_Tq(n) in the higher frequency is the phase shift between T and q fluctuations, accompanied with q attenuation [see Horst, 2000; Watanabe et al., 2000].

[34] The BP range determined in this section will be used in the application of the band‐pass covariance technique, (5) and (7), to the current data set in the next section.

5. Results

[35] By using the BP frequency range identified in section 4 the two versions of the band‐pass covariance technique, namely equations (5) and (7), were applied to the measurements. In this section the resulting latent heat flux, E_BC, was compared with the eddy covariance measurements of the tower at the BJ site, E_EC, in various aspects. For each of the comparisons, relevant statistics, namely the correlation coefficient, r, the slope of the linear regression line forced through the origin with E_BC on abscissa, a, and the root mean squared error, RMSE, were calculated.

5.1. Sensitivity Analysis to R_cr

[36] In section 4, BP frequency ranges were identified as R_Tq(n) > R_cr with the tentative value, R_cr = 0.9. Therefore sensitivity of E_BC computed with (7) to the value of R_cr was investigated by calculating E_BC with different values of R_cr, i.e., R_cr = 0.5, 0.6, 0.7, 0.8 and 0.9. With general shape of R_Tq(n) (see Figure 1c), the higher values of R_cr result in the narrower BP range. Each of the flux values computed from q_HA signals were compared with E_EC, and the relevant statistics were tabulated in Table 3.

Table 3. Statistics Relevant to the Comparison Between the Tower Eddy Covariance Measurements E_EC and the Computations by the Band‐Pass Covariance Techniques E_BC (Equation (7)) With Different Values for R_cr^aa HumAir measurements were used. Here n_d is the number of the data, r is the correlation coefficient, a is the slope of the linear regression line forced through the origin, and RMSE is the root‐mean‐square error in W/m². BC0 in the first column refers to the values computed from the raw covariances of without any application of the band‐pass covariance technique.

Rcr	nd	r	a	RMSE
BC0	25	0.92	0.71 (±0.03)	71.8
0.5	25	0.93	0.91 (±0.03)	40.2
0.6	25	0.93	0.92 (±0.03)	39.5
0.7	25	0.94	0.92 (±0.03)	37.6
0.8	25	0.94	0.94 (±0.03)	35.7
0.9	25	0.94	0.97 (±0.03)	33.2

• a HumAir measurements were used. Here n_d is the number of the data, r is the correlation coefficient, a is the slope of the linear regression line forced through the origin, and RMSE is the root‐mean‐square error in W/m². BC0 in the first column refers to the values computed from the raw covariances of without any application of the band‐pass covariance technique.

[37] The first row in Table 3, denoted as “BC0”, represents the latent heat flux computed using the raw covariance without any application of the band‐pass covariance technique. This revealed that about 30% of the latent heat flux was under measured due to the high‐frequency attenuation of the q sensor. The current results show that more than two thirds of the under measurements were recovered by the band‐pass covariance technique. It is important to emphasize that the current implementation of the band‐pass covariance technique does not utilize any external information other than the measured data.

[38] It can be observed that r and RMSE are almost insensitive to the value of R_cr. A noticeable feature in Table 3 is that the slope, a, slightly yet constantly increases with increasing R_cr. In other words, the smaller values of β₀ in (7) are computed with the narrower BP range associated with the higher value of R_cr. This indicates that attenuated q signals are still contained in the BP range. This is expected from Figure 3, where the higher end of the BP range roughly equals to the half‐attenuated point, f₀.

[39] Table 3 suggests that the higher values of R_cr give superior results. R_cr = 0.9, however, is the highest among the practically possible values. A value of R_cr higher than 0.9 would not give a frequency range for R_Tq(n) > R_cr, as suggested by Figure 1c. One possibility to improve this further is to assume the first‐order sensor response expressed in (13). Applying the data to (13) computes β₀ [Horst, 1997], which, in turn, can be used in (7) to compute the latent heat flux. This was tested with the data in the BP range with R_cr = 0.9. The comparison with E_EC gives the statistics as r = 0.91, a = 1.00(±0.04), RMSE = 47.6 W/m². The slope, a, has thus been improved, while RMSE has been degraded. By imposing a functional form of the sensor response, uncertainty was added to the computed flux values With these considerations, hereafter the threshold value R_cr = 0.9 will be used.

5.2. Comparison Between the Humidity Sensors

[40] Latent heat flux computed by using (7) with the q measurements of the two hygrometers, namely, q_HA and q_HC, are compared in Figure 4. Each of the two panels in Figure 4 shows the latent heat flux from the raw covariances, , without applying the band‐pass covariance technique. Again, these raw covariances are significantly smaller than E_EC and these under measurements were corrected substantially by the band‐pass covariance technique (open circles), with which both of r and the slope, a, are close to unity. This again demonstrates the strong capability of the band‐pass covariance technique in frequency extrapolations. A comparison between the two panels shows slight differences in the performance of the two sensors. HumAir gives a slightly smaller value of RMSE than does Humicap. This is expected from the better frequency response of the former than the latter (Figure 1). A closer look at these figures indicates that the performance of Humicap is worse when the values of latent heat flux are smaller. This may be related to the fact that Humicap was naturally aspirated.

Figure 4

Comparison between the latent heat flux measured with the eddy correlation systems, E_EC, and that computed with equation (7), E_BC (open circles). (top) Measurements with HumAir; (bottom) measurements with Humicap. The solid circles are those computed from the raw covariances plotted against E_EC.

6. Discussion

[41] The comparison between the two versions of the band‐pass covariance technique, namely (5) and (7), can elucidate the validity of the assumptions underlying the methods. This is the main interest of this paper.

[42] As discussed earlier in this document, (5) requires that the dimensionless form of S_wT(n) and S_wq(n) are equal for the range, 0 < n < ∞ (equation (3)), while less strict (7) demands that these cospectra have the same frequency dependency between the energy‐containing and the dissipation frequencies. From the more physical point of view, (5) assumes that T and q fluctuations are similar to each other over the entire Fourier space, whereas (7) only assumes that these scalar fluctuations share the same flow characteristics at their production and during the course of cascading down to their dissipation. Apparently, (7) is based on a less strict assumption than the one (5) is based on.

[43] The first two rows of Table 4 compare the two versions of the band‐pass covariance technique with all runs analyzed. Although the difference is small, it can be observed that (5) considerably underestimates the latent heat flux. These differences between the two methods were further investigated by following Asanuma et al. (submitted manuscript, 2005). All of the runs analyzed were categorized into two groups using the behavior of R_Tq(n) at the lower frequencies. One of them are those that reduce the value of R_Tq(n) below 0.5 in the frequency range, 0 < n < n₁, and the others are those that do not. The specific threshold, 0.5, was chosen so that the two groups have nearly the same population. These two groups are hereafter termed “lower” and “higher correlation” runs. For example, the run shown in Figure 1 is categorized into the former. The comparison statistics were, again, computed for each of these two groups, and they were also tabulated in Table 4. The statistics shown in Table 4 give a clear distinction between the two versions of the band‐pass covariance technique. With the higher correlation runs, both of (5) and (7) works almost equally well. The value of the slope, a, almost equals to unity, and RMSE are almost equal to each other. On the other hand, the performance of (5) with the lower‐correlation runs were worse: RMSE is twice larger and the flux is underestimated by about 15%. Equation (7), in contrast, remains relatively unchanged even with the lower‐correlation runs. This illuminates the robustness of (7).

Table 4. Same Statistics as Table 3 but for Those Relevant to the Comparison Between the Two Versions of the Band‐Pass Covariance, Namely, Equations (5) and (7)

Equation	nd	r	a	RMSE
All Runs
(5)	25	0.93	0.90 (±0.03)	40.5
(7)	25	0.94	0.97 (±0.03)	33.2
Runs With Higher Correlation
(5)	12	0.97	0.99 (±0.04)	26.0
(7)	12	0.97	1.00 (±0.03)	24.1
Runs With Lower Correlation
(5)	13	0.91	0.84 (±0.05)	39.7

[44] These observations can be explained with the findings of Asanuma et al. (submitted manuscript, 2005). By using R_Tq(n) as an indicator, they investigated the similarity between T and q in the Fourier domain with the same data set. By categorizing the analyzed data runs into the two groups as in this study, they found that the lower‐correlation runs have remarkably reduced R_Tq(n) at the lower frequencies, 10^‒4 < n < 10^‒3, while the higher‐correlation runs maintain higher values of R_Tq(n) even at these frequencies. Dissimilarity between T and q occurred at the lower frequencies has been also pointed out by Dias et al. [2004]. The fact that the reduced R_Tq(n) occurs only in the lower‐correlation runs indicates that the dissimilarity is occasional, and its occurrence is limited in time. Asanuma et al. (submitted manuscript, 2005) argued that the dissimilarity between T and q found at the lower frequencies was associated with the “decorrelation events” at the scale of the atmospheric boundary layer (ABL) height. With their longer return period relative to the length of the data run, these events were sensed to be occasional or sporadic. They also mentioned the entrainment at the top of ABL and the convective cloud activities as possible candidates for the decorrelation events.

[45] If we extend their arguments, the failure of (5) found in this study can also be associated with these decorrelation events at the scale of the ABL height. In other words, occasional occurrences of the dissimilarity at the lower frequencies caused occasional failures of (5), which relies on the similarity of T and q over whole Fourier domain. In contrast, equation (7) only relies on the characteristics of the scalar field at the higher frequencies, and remains fairly intact against these decorrelation events. This points to the robustness of (7), as well as the validity of the fundamental assumption of (7) expressed as (6).

[46] A comment should be made on the consistency of the current result with the findings in the literature. Some authors have already presented observational evidences in which T and q become dissimilar at the higher frequencies [e.g., Phelps and Pond, 1971; McBean and Miyake, 1972; Katul and Parlange, 1994]. Katul and Parlange [1994], for example, argued that the active role of T and the relatively passive role of q in the turbulent field become significant at the higher frequencies. The current result that (6) is valid does not contradict this: Since β(n) is the ratio of the cospectra of w and these scalars, the different behavior of T and q associated with their different roles might be filtered out when the cospectra with w are taken. This concept also characterizes the robustness of the assumption underlying (7).

7. Conclusion

[47] The band‐pass covariance technique forms a physically sound method for frequency extrapolation in latent heat flux measurements. The term ‘band‐pass covariance technique,’ in fact, points to two different methods that have a different physical background. Distinctions between these two versions and their different backgrounds are clearly addressed in this paper.

[48] The original version by Hicks and McMillen [1988] requires the similarity between the temperature and the specific humidity, or a scalar in question, over the entire Fourier domain, while the other, more advanced, version [Högström et al., 1989] assumes “similarity” between the two scalars only at the higher frequencies. The word “similarity” for the latter has been used ambiguously, yet not incorrectly, in the literature. In light of the surface similarity arguments, this paper demonstrates that the assumption invoked by the advanced version is different from the similarity between scalars in its strict definition. Instead, it relies only on the fact that the two scalars obey the same flow characteristics at the higher frequencies. The surface similarity arguments also suggest that this is most likely to be valid during the course of the production, the cascading, and the dissipation of the flux covariance in the turbulence field. This requirement for the advanced version turns out to be less strict than that of the original version. Moreover, the flux calculation algorithm, fully described by Watanabe et al. [2000] and Yasuda and Watanabe [2001], has a self‐calibration procedure embedded in itself. These facts make the advanced version more robust and more practical for use in the real atmosphere.

[49] The algorithm of the advanced version [Horst and Oncley, 1995; Horst, 1997; Watanabe et al., 2000; Yasuda and Watanabe, 2001] was further modified and was applied to the daytime measurements of latent heat flux above the sparse grasslands over the Tibetan Plateau. In the modified algorithm, the energy‐containing range was dynamically identified using the coherency spectra between temperature and specific humidity. The coherency spectra were also used to detect attenuation in the measured signals of specific humidity. Though the number of the data is limited, the computed latent heat flux values exhibited adequate agreement with the eddy covariance measurements, indicating strong capability of the current algorithm in frequency extrapolation under unstable conditions. The application of the original version, on the other hand, clearly showed that it fails when the similarity between temperature and humidity is not maintained at the lower frequencies.

[50] The occasional failures of the original version need an special attention. Clearly, the current observations invalidate the frequently invoked “perfect similarity” assumption between the temperature and the specific humidity. Use of this erroneous assumption in the flux measurements of trace gases may cause substantial biases. Instead, direct measurements of the lower‐frequency portions of these fluxes are recommended for better accuracy. In contrast, the background assumption behind the advanced version was shown to be relatively robust.

[51] The modified algorithm of the band‐pass covariance technique proposed in this text gives the band‐pass limit, once the thresholding value of the coherency spectra is chosen. Therefore the algorithm offer a semiautomatic procedure of the scalar flux calculation from measurements of a simple and robust scalar sensor. Hence the algorithm renders long‐term flux measurements, i.e., for months and years, more feasible. For this to be complete, however, applicability of the band‐pass covariance under the stable stratification need to be investigated, which is beyond the coverage of this text.

[52] Finally, the strong extrapolating capability of the band‐pass covariance technique with the proposed algorithm does have a large potential in scalar flux measurements. One possibility is to correct attenuated scalar fluxes due to the separation between the sonic anemometer and the scalar sensor by using a sonic temperature as a guide. The band‐pass covariance technique can be also used as an alternative measurements for gap filling [e.g., Falge et al., 2001] of long‐term data of water vapor, CO₂ and trace gas fluxes with an additional low‐cost scalar sensor as a backup in case of failures of the fast response scalar sensor.

Acknowledgments

[53] The authors are grateful to T. Yasunari at Nagoya University and T. Koike at Tokyo University, without whose efforts and leadership the GAME‐Tibet project could not have been possible. Appreciation is also given to the other participants of the GAME‐Tibet field campaigns for their help and cooperation. Invaluable comments from G. Katul at Duke University, N. Dias at UFPR, Brazil, and T. Watanabe at Forestry and Forest Products Research Institute, Japan, are also appreciated. One of the authors (J.A.) is supported, in part, by Ministry of Education, Culture, Sport, Science and Technology (MEXT), Japan, through a Grant‐in‐Aid for Scientific Research (A) (13308027) and (C)(17560450) and Asahi Breweries Foundation through its academic grant.