the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Air temperature equation derived from sonic temperature and water vapor mixing ratio for turbulent airflow sampled through closed-path eddy-covariance flux systems

### Xinhua Zhou

### Eugene S. Takle

### Xiaojie Zhen

### Andrew E. Suyker

### Tala Awada

### Jane Okalebo

### Jiaojun Zhu

Air temperature (*T*) plays a fundamental role in many
aspects of the flux exchanges between the atmosphere and ecosystems.
Additionally, knowing where (in relation to other essential measurements)
and at what frequency *T* must be measured is critical to accurately describing
such exchanges. In closed-path eddy-covariance (CPEC) flux systems, *T* can be
computed from the sonic temperature (*T*_{s}) and water vapor mixing ratio
that are measured by the fast-response sensors of a three-dimensional sonic
anemometer and infrared CO_{2}–H_{2}O analyzer, respectively. *T* is then
computed by use of either $T={T}_{\mathrm{s}}{\left(\mathrm{1}+\mathrm{0.51}q\right)}^{-\mathrm{1}}$, where *q* is
specific humidity, or $T={T}_{\mathrm{s}}{\left(\mathrm{1}+\mathrm{0.32}e/P\right)}^{-\mathrm{1}}$, where *e* is water
vapor pressure and *P* is atmospheric pressure. Converting *q* and $e/P$ into the same
water vapor mixing ratio analytically reveals the difference between these
two equations. This difference in a CPEC system could reach ±0.18 K,
bringing an uncertainty into the accuracy of *T* from both equations and
raising the question of which equation is better. To clarify the uncertainty
and to answer this question, the derivation of *T* equations in terms of
*T*_{s} and H_{2}O-related variables is thoroughly studied. The two
equations above were developed with approximations; therefore, neither of
their accuracies was evaluated, nor was the question answered. Based on
first principles, this study derives the *T* equation in terms of *T*_{s} and
the water vapor molar mixing ratio (${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$) without any assumption and
approximation. Thus, this equation inherently lacks error, and the accuracy
in *T* from this equation (equation-computed *T*) depends solely on the
measurement accuracies of *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$. Based on current
specifications for *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ in the CPEC300 series, and
given their maximized measurement uncertainties, the accuracy in
equation-computed *T* is specified within ±1.01 K. This
accuracy uncertainty is propagated mainly (±1.00 K) from the
uncertainty in *T*_{s} measurements and a little (±0.02 K) from the
uncertainty in ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ measurements. An improvement in measurement
technologies, particularly for *T*_{s}, would be a key to narrowing this
accuracy range. Under normal sensor and weather conditions, the specified
accuracy range is overestimated, and actual accuracy is better.
Equation-computed *T* has a frequency response equivalent to high-frequency
*T*_{s} and is insensitive to solar contamination during measurements.
Synchronized at a temporal scale of the measurement frequency and matched at a
spatial scale of measurement volume with all aerodynamic and thermodynamic
variables, this *T* has advanced merits in boundary-layer meteorology and
applied meteorology.

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The equation of state, *P*=*ρ**R**T*, is a fundamental equation for
describing all atmospheric flows, where *P* is atmospheric pressure, *ρ* is
moist-air density, *R* is the gas constant for moist air, and *T* is air temperature
(Wallace and Hobbs, 2006). In boundary-layer flow, where turbulence is
nearly always present, accurate representation of the “state” of the
atmosphere at any given “point” and time requires consistent
representation of spatial and temporal scales for all thermodynamic factors
of *P*, *ρ*, and *T* (Panofsky and Dutton, 1984). Additionally, for observing
fluxes describing exchanges of quantities, such as heat and moisture between
the Earth and the atmosphere, it is critical to know all three-dimensional
(3-D) components of wind speed at the same location and temporal scale as
the thermodynamic variables (Laubach and McNaughton, 1998).

In a closed-path eddy-covariance (CPEC) system, the 3-D wind components and
sonic temperature (*T*_{s}) are measured by a 3-D sonic anemometer in the
sonic measurement volume near which air is sampled through the orifice of an
infrared H_{2}O–CO_{2} analyzer (hereafter referred to as the infrared
analyzer) into its closed-path H_{2}O–CO_{2} measurement cuvette, where
air moisture is measured by the analyzer (Fig. 1). The flow pressure inside
the cuvette (*P*_{c}) and the differential (Δ*P*) between *P*_{c} and
ambient flow pressure in the sampling location are also measured (Campbell
Scientific Inc., 2018c). Atmospheric *P* in the sampling volume, therefore, is
a sum of *P*_{c} and Δ*P*. *P*_{c}, along with the internal *T*, is further
used for infrared measurements of air moisture (i.e., *ρ*_{w},
H_{2}O density) to calculate the water mixing ratio (*χ*_{w}) inside
the cuvette that is also equal to *χ*_{w} in the CPEC measurement
volume, including sonic measurement volume and the air sampling location.
Finally, the *T*_{s} and *χ*_{w} from the CPEC measurement volume,
after spatial and temporal synchronization (Horst and Lenschow, 2009), are
used to calculate the *T* inside this volume. Two optional equations (Schotanus
et al., 1983; Kaimal and Gaynor, 1991; see Sect. 2, Background), which
need rigorous evaluation, are available for this *T* calculation. In summary,
the boundary-layer flow measured by a CPEC system has all variables
quantified with consistent representation of spatial and temporal scales for
moist turbulence thermodynamics (i.e., state) if the following are
available: 3-D wind; *P* measured differentially; *T* from an equation; and
*ρ* from *P*, *T*, and *χ*_{w}.

In this paper, the authors (1) derive a *T* equation in terms of *T*_{s} and
*χ*_{w} based on first principles as an alternative to the commonly
used equations that are based on approximations, (2) estimate and verify the
accuracy of the first-principles *T*, (3) assess the expected advantages of the
first-principles *T* as a high-frequency signal insensitive to solar
contamination suffered by conventional *T* sensor measurements (Lin et al.,
2001; Blonquist and Bugbee, 2018), and (4) brief the potential applications
of the derived *T* equation in flux measurements. We first provide a summary of
the moist turbulence thermodynamics of the boundary-layer flows measured by
CPEC flux systems.

A CPEC system is commonly used to measure boundary-layer flows for the
CO_{2}, H_{2}O, heat, and momentum fluxes between ecosystems and the
atmosphere. Such a system is equipped with a 3-D sonic anemometer to measure
the speed of sound in three dimensions in the central open space of the
instrument (hereafter referred to as open space), from which can be
calculated *T*_{s} and 3-D components of wind with a fast response. Integrated
with this sonic anemometer, a fast-response infrared analyzer concurrently
measures CO_{2} and H_{2}O in its cuvette (closed space) of infrared
measurements, through which air is sampled under pump pressure while being
heated (Fig. 1). The analyzer outputs the CO_{2} mixing ratio (i.e., ${\mathit{\chi}}_{{\mathrm{CO}}_{\mathrm{2}}}={\mathit{\rho}}_{{\mathrm{CO}}_{\mathrm{2}}}/{\mathit{\rho}}_{\mathrm{d}}$, where ${\mathit{\rho}}_{{\mathrm{CO}}_{\mathrm{2}}}$ is
CO_{2} density and *ρ*_{d} is dry-air density) and *χ*_{w}
(i.e., ${\mathit{\rho}}_{\mathrm{w}}/{\mathit{\rho}}_{\mathrm{d}}$). Together, these instruments provide
high-frequency (e.g., 10 Hz) measurements from which the fluxes are computed
(Aubinet et al., 2012) at a point represented by the sampling space of
the CPEC system.

These basic high-frequency measurements of 3-D wind speed, *T*_{s}, *χ*_{w}, and ${\mathit{\chi}}_{{\mathrm{CO}}_{\mathrm{2}}}$ provide observations from which mean and
fluctuation properties of air, such as *ρ*_{d}, *ρ*, *ρ*_{w}, and
${\mathit{\rho}}_{{\mathrm{CO}}_{\mathrm{2}}}$, and, hence, fluxes can be determined. For instance, water
vapor flux is calculated from ${\overline{\mathit{\rho}}}_{\mathrm{d}}\stackrel{\mathrm{\u203e}}{{w}^{\prime}{\mathit{\chi}}_{\mathrm{w}}^{\prime}}$, where *w* is the vertical velocity of air, and the prime indicates the
fluctuation of the variable away from its mean as indicated by the overbar
(e.g., ${w}^{\prime}=w-\overline{w}$). Given the measurements of *χ*_{w} and *P*
from CPEC systems and based on the gas laws (Wallace and Hobbs, 2006),
*ρ*_{d} is derived from

where *R*_{d} is the gas constant for dry air and *R*_{v} is the gas constant for
water vapor. In turn, *ρ*_{w} is equal to *ρ*_{d}*χ*_{w}
and *ρ* is a sum of *ρ*_{d} and *ρ*_{w}. All mentioned
physical properties can be derived if *T* in Eq. (1) for *ρ*_{d} is
acquired.

Additionally, equations for ecosystem exchange and flux require ${\overline{\mathit{\rho}}}_{\mathrm{d}}$ (Gu et al., 2012) and $\stackrel{\mathrm{\u203e}}{{\mathit{\rho}}_{\mathrm{d}}w}$ (Foken et al.,
2012). Furthermore, due to accuracy limitations in measurements of *w* from a
modern sonic anemometer, the dry-air flux of $\stackrel{\mathrm{\u203e}}{{\mathit{\rho}}_{\mathrm{d}}w}$ must
be derived from $\stackrel{\mathrm{\u203e}}{{\mathit{\rho}}_{\mathrm{d}}^{\prime}{w}^{\prime}}-{\overline{\mathit{\rho}}}_{\mathrm{d}}\overline{w}$ (Webb et al., 1980; Lee and Massman, 2011). Because of its role
in flux measurements, a high-frequency representation of *ρ*_{d} is
needed. To acquire such a *ρ*_{d} from Eq. (1) for advanced
applications, high-frequency *T* in temporal synchronization with *χ*_{w} and
*P* is needed.

In a modern CPEC system, *P* is measured using a fast-response barometer
suitable for measurements at a high frequency (e.g., 10 Hz; Campbell
Scientific Inc., 2018a), and, as discussed above, *χ*_{w} is a
high-frequency signal from a fast-response infrared analyzer (e.g., commonly
up to 20 Hz). If *T* is measured using a slow-response sensor, the three
independent variables in Eq. (1) do not have equivalent synchronicity in their
frequency response. In terms of frequency response, ${\mathit{\rho}}_{\mathrm{d}}^{\prime}$ cannot
be correctly acquired. ${\overline{\mathit{\rho}}}_{\mathrm{d}}$ derived based on Eq. (1) also has
uncertainty, although it can be approximated from either of the two
following equations:

and

Equation (2) is mathematically valid in averaging rules (Stull, 1988), but the
response of the system to *T* is slower than to *χ*_{w} and even *P*, while
Eq. (3) is invalid under averaging rules, although its three overbar
independent variables can be evaluated over an average interval.
Consequently, neither $\stackrel{\mathrm{\u203e}}{{\mathit{\rho}}_{\mathrm{d}}w}$ nor ${\overline{\mathit{\rho}}}_{\mathrm{d}}$ can
be evaluated strictly in theory.

Measurements of *T* at a high frequency (similarly to those at a low frequency) are
contaminated by solar radiation, even under shields (Lin et al., 2001) and
when aspirated (Campbell Scientific Inc., 2010; R.M. Young Company, 2004;
Apogee Instruments Inc., 2013; Blonquist and Bugbee, 2018). Although a
naturally ventilated or fan-aspirated radiation shield could ensure the
accuracy of a conventional (i.e., slow-response) thermometer often within
±0.2 K at 0 ^{∘}C (Harrison and Burt, 2021) to satisfy
the standard for conventional *T* measurement as required by the World
Meteorological Organization (WMO, 2018), the aspiration shield method cannot
acquire *T* at a high frequency due to the disturbance of an aspiration fan and
the blockage of a shield to natural turbulent flows. Additionally, fine
wires have limited applicability for long-term measurements in rugged field
conditions typically encountered in ecosystem monitoring.

To avoid the issues above in use of either slow- or fast-response *T* sensors
under field conditions, deriving *T* from *T*_{s} and *χ*_{w} (Schotanus
et al., 1983; Kaimal and Gaynor, 1991) is an advantageous alternative to the
applications of *T* in CPEC measurements and is a significant technology for
instrumentation to pursue. In a CPEC system, *T*_{s} is measured at a high
frequency (e.g., 10 Hz) using a fast-response sonic anemometer to detect the
speed of sound in the open space (Munger et al., 2012), provided there is no
evidence of contamination by solar radiation. It is a high-frequency
signal. *χ*_{w} is measured at the same frequency as for *T*_{s} using
an infrared analyzer equivalent to the sonic anemometer with a high-frequency
response time (Ma et al., 2017). *χ*_{w} reported from a CPEC system
is converted from water vapor molar density measured inside the closed-space
cuvette, whose internal pressure and internal temperature are more stable
than *P* and *T* in the open space and can be more accurately measured. Because of
this, solar warming and radiation cooling of the cuvette is irrelevant, as
long as water molar density, pressure, and temperature inside the
closed-space cuvette are more accurately measured. Therefore, it could be
reasonably expected that *T* calculated from *T*_{s} and *χ*_{w} in a CPEC
system should be a high-frequency signal insensitive to solar radiation.

The first of two equations commonly used to compute *T* from *T*_{s} and air-moisture-related variables is given by Schotanus et al. (1983) as

where *q* is specific humidity, defined as a ratio of water vapor to moist-air
density. The second equation is given by Kaimal and Gaynor (1991) as

where *e* is water vapor pressure. Rearranging these two equations gives *T* in
terms of *T*_{s} and *χ*_{w}. Expressing *q* in terms of *ρ*_{d} and
*ρ*_{w}, Eq. (4) becomes

and expressing *e* and *P* using the equation of state, Eq. (5) becomes

The *χ*_{w}-related terms in the denominator inside parentheses in
both equations above clearly reveal that *T* values from the same *T*_{s} and
*χ*_{w} using the two commonly used Eqs. (4) and (5) will not be the
same. The absolute difference in the values (Δ*T*_{e}, i.e., the
difference in *T* between Eqs. 4 and 5) can be analytically expressed as

Given that, in a CPEC system, the sonic anemometer has an operational range
in *T*_{s} of −30 to 57 ^{∘}C (Campbell Scientific Inc., 2018b) and
an infrared analyzer has a measurement range in *χ*_{w} of 0 to 0.045 kgH_{2}O kg^{−1} (Campbell Scientific Inc., 2018a), Δ*T*_{e}
ranges up to 0.177 K, which brings an uncertainty in the accuracy of *T* calculated
from either Eq. (4) or Eq. (5) and raises the question of which equation is
better.

Reviewing the sources of Eq. (4) (Schotanus et al., 1983; Swiatek, 2009; van
Dijk, 2002) and Eq. (5) (Ishii, 1935; Barrett and Suomi, 1949; Kaimal and
Businger, 1963; Kaimal and Gaynor, 1991), it was found that approximation
procedures were used in the derivation of both equations, but the approach to
the derivation of Eq. (4) (Appendix A) is different from that of Eq. (5)
(Appendix B). These different approaches create a disparity between the two
commonly used equations as shown in Eq. (8), and the approximation
procedures lead to the controversy as to which equation is more accurate.
The controversy can be avoided if the *T* equation in terms of *T*_{s} and
*χ*_{w} can be derived from the *T*_{s} equation and first-principles
equations, if possible without an approximation and verified against
precision measurements of *T* with minimized solar contamination.

As discussed above, a sonic anemometer measures the speed of sound (*c*)
concurrently with measurement of the 3-D wind speed (Munger et al., 2012).
The speed of sound in the homogeneous atmospheric boundary layer is defined
by Barrett and Suomi (1949) as

where *γ* is the ratio of moist-air specific heat at constant pressure
(*C*_{p}) to moist-air specific heat at constant volume (*C*_{v}).
Substitution of the equation of state into Eq. (9) gives *T* as a function of
*c*:

This equation reveals the opportunity to use measured *c* for the *T* calculation;
however, both *γ* and *R* depend on air humidity, which is unmeasurable
by sonic anemometry itself; Eq. (10) is, therefore, not applicable for *T*
calculations inside a sonic anemometer. Alternatively, *γ* is replaced
with its counterpart for dry air (*γ*_{d}, 1.4003, i.e., the ratio
of dry-air specific heat at constant pressure (*C*_{pd}, 1004 J K^{−1} kg^{−1}) to dry-air specific heat at constant volume (*C*_{vd}, 717 J K^{−1} kg^{−1})) and *R* is replaced with its counterpart for dry
air (*R*_{d}, 287.06 J K^{−1} kg^{−1}, i.e., the gas constant for dry air).
Both replacements make the right side of Eq. (10) become ${c}^{\mathrm{2}}/{\mathit{\gamma}}_{\mathrm{d}}{R}_{\mathrm{d}}$, which is no longer a measure of *T*. However, *γ*_{d} and
*R*_{d} are close to their respective values of *γ* and *R* in
magnitude, and, after the replacements, the right side of Eq. (10) is
defined as sonic temperature (*T*_{s}), given by (Campbell Scientific Inc.,
2018b)

Comparing this equation to Eq. (10), given *c*, if air is dry, *T* must be equal
to *T*_{s}; therefore, the authors state that “sonic temperature of moist air is the temperature that its dry-air component reaches when moist air has the same enthalpy.” Since both *γ*_{d} and *R*_{d} are constants and *c* is measured by a sonic
anemometer and corrected for the crosswind effect inside the sonic anemometer
based on its 3-D wind measurements (Liu et al., 2001; Zhou et al., 2018),
Eq. (11) is used inside the operating system (OS) of modern sonic anemometers to
report *T*_{s} instead of *T*.

Equations (9) to (11) provide a theoretical basis of first principles to
derive the relationship of *T* to *T*_{s} and *χ*_{w}. In Eq. (9),
*γ* and *ρ* vary with air humidity and *P* is related to *ρ* as
described by the equation of state. Consequently, the derivation of *T* from
*T*_{s} and *χ*_{w} for CPEC systems needs to address the relationship
of *γ*, *ρ*, and *P* to air humidity in terms of *χ*_{w}.

## 3.1 Relationship of *γ* to *χ*_{w}

For moist air, the ratio of specific heat at constant pressure to specific heat at constant volume is

where *C*_{p} varies with air moisture between *C*_{pd} and *C*_{pw} (water
vapor specific heat at constant pressure, 1952 J kg^{−1} K^{−1}). It is
the arithmetical average of *C*_{pd} and *C*_{pw} weighted by the dry air mass and
water vapor mass, respectively, given by (Stull, 1988; Swiatek, 2009)

Based on the same rationale, *C*_{v} is

where *C*_{vw} is the specific heat of water vapor at constant volume (1463 J kg^{−1} K^{−1}). Substituting Eqs. (13) and (14) into Eq. (12)
generates

## 3.2 Relationship of $P/\mathit{\rho}$ to *χ*_{w}

Atmospheric *P* is the sum of *P*_{d} and *e*. Similarly, *ρ* is the sum of
*ρ*_{d} and *ρ*_{w}. Using the equation of state, the ratio of
*P* to *ρ* can be expressed as

In this equation, the ratio of *R*_{v} to *R*_{d} is given by

where *R*^{∗} is the universal gas constant, *M*_{w} is the molecular mass
of water vapor (18.0153 kg kmol^{−1}), and *M*_{d} is the molecular mass of
dry air (28.9645 kg kmol^{−1}). The ratio of *M*_{w} to *M*_{d} is 0.622,
conventionally denoted by *ε*. Substituting Eq. (17), after its
denominator is represented by *ε*, into Eq. (16) leads to

## 3.3 Relationship of *T*_{s} to *T* and *χ*_{w}

Substituting Eqs. (15) and (18) into Eq. (9), *c*^{2} is expressed in terms
of *T* and *χ*_{w} along with atmospheric physics constants:

Further, substituting *c*^{2} into Eq. (11) generates

This Eq. (20) now expresses *T*_{s} in terms of the *T* of interest to this
study, *χ*_{w} measured in CPEC systems, and atmospheric physics
constants (i.e., *ε*, *C*_{pw}, *C*_{pd}, *C*_{vw}, and *C*_{vd}).

## 3.4 Air temperature equation

Rearranging the terms in Eq. (20) results in

This equation shows that *T* is a function of *T*_{s} and *χ*_{w} that are
measured at a high frequency in a CPEC system by a sonic anemometer and an
infrared analyzer.

A CPEC system outputs the water vapor molar mixing ratio (Campbell Scientific
Inc., 2018a) commonly used in the community of eddy-covariance fluxes
(AmeriFlux, 2018). The relation of water vapor mass to the molar mixing ratio
(${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ in molH_{2}O mol^{−1}) is given by

Substituting this relation into Eq. (21) and denoting ${C}_{\mathrm{vw}}/{C}_{\mathrm{vd}}$ with *γ*_{v}
= 2.04045 and ${C}_{\mathrm{pw}}/{C}_{\mathrm{pd}}$ with *γ*_{p}=1.94422, Eq. (21) is expressed as

This is the air temperature equation in terms of *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$
for use in CPEC systems. It is derived from a theoretical basis of first
principles (i.e., Eqs. 9 to 11). In its derivation, except for the use of
the equation of state and Dalton's law, no other assumptions or
approximations are used. Therefore, Eq. (23) is an exact equation of *T* in
terms of *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ for the turbulent airflow sampled
through a CPEC system and thus avoids the controversy in the use of Eqs. (4) and
(5) arising from approximations, as shown in Appendices A and B. Therefore,
*T* computed from this equation (hereafter referred to as equation-computed
*T*) should be accurate, as long as the values of *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ are
exact.

For this study, however, *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ are measured by the CPEC
systems deployed in the field under changing weather conditions through four
seasons. Their measured values must include measurement uncertainty in
*T*_{s}, denoted by Δ*T*_{s}, and in ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ as well, denoted
by $\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$. The uncertainties, Δ*T*_{s} and/or
$\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$, unavoidably propagate to create uncertainty in
equation-computed *T*, denoted by Δ*T*, which makes an exact *T* impossible.
In numerical analysis (Burden and Faires, 1993) or in statistics (Snedecor
and Cochran, 1989), any applicable equation requires the specification of an
uncertainty term. Therefore, the equations for *T* should include a
specification of their respective uncertainty expressed as the bounds (i.e.,
the maximum and minimum limits) specifying the range of the
equation-computed *T* that need to be known for any application. According to
the definition of accuracy that was advanced by the International
Organization for Standardization (2012), this uncertainty range is
equivalent to the “accuracy” of the range contributed by both systematic
errors (trueness) and random variability (precision). Apparently, Δ*T*_{s} is the accuracy of *T*_{s} measurements and $\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$
is the accuracy of ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ measurements. Both should be evaluated
from their respective measurement uncertainties. The accuracy of
equation-computed *T* is Δ*T*. It should be specified through its
relationship to Δ*T*_{s} and $\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$.

## 3.5 Relationship of Δ*T* to Δ*T*_{s} and $\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$

As measurement accuracies, Δ*T*_{s} and $\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ can
be reasonably considered small increments in a calculus sense. As such,
depending on both small increments, Δ*T* is the total differential of
*T* with respect to *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$, given by

The two partial derivatives on the right side of this equation can be derived from Eq. (23). Substituting the two partial derivatives into this equation leads to

This equation indicates that in dry air when *T*=*T*_{s}, Δ*T* is equal
to Δ*T*_{s} if ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ is measured accurately (i.e.,
$\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}=\mathrm{0}$ while ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}=$ 0). However, air in
the atmospheric boundary layer where CPEC systems are used is always moist.
Given this equation, Δ*T* at *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ can be
evaluated by using Δ*T*_{s} and $\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$, both of
which are related to the measurement specifications of sonic anemometers for
*T*_{s} (Campbell Scientific Inc., 2018b) and of infrared analyzers for
${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ (Campbell Scientific Inc., 2018a). Sonic anemometers and
infrared analyzers with different models and brands have different
specifications from their manufacturers. The manufacturer of the anemometer
we studied (Fig. 1) employs carbon fiber with minimized thermo-expansion and
thermo-contraction for sonic strut stability (via personal communication with CSAT
structural designer Antoine Rousseau, 2021); structural design with
optimized sonic volume for less aerodynamic disturbance (Fig. 1); and
advanced proprietary sonic firmware for more accurate measurements (Zhou et
al., 2018), which reduces the variability in *T*_{s} by several kelvins
compared to what has been reported for sonics from other models (Mauder and
Zeeman, 2018). Any combination of sonic and infrared instruments has a
combination of Δ*T*_{s} and $\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$, which are
specified by their manufacturers. In turn, from Eq. (25), the combination
generates Δ*T* of equation-computed *T* for the corresponding combination
of the sonic and infrared instruments with given models and brands.
Therefore, Eqs. (23) and (25) are applicable to any CPEC system beyond the brand of our
study (Fig. 1). The applicability of Eq. (23) for any sonic or infrared
instrument can be assessed based on Δ*T* against the required *T* accuracy
for a specific application.

On the right side of Eq. (25), the first term with Δ*T*_{s} can be
expressed as $\mathrm{\Delta}{T}_{{T}_{\mathrm{s}}}$ (i.e., uncertainty portion of Δ*T* due
to Δ*T*_{s}) and the second term with $\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ can
be expressed as $\mathrm{\Delta}{T}_{{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}$ (i.e., uncertainty portion of
Δ*T* due to $\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$). Using $\mathrm{\Delta}{T}_{{T}_{\mathrm{s}}}$ and
$\mathrm{\Delta}{T}_{{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}$, this equation can be simplified as

Assessment of the accuracy of equation-computed *T* is to evaluate $\mathrm{\Delta}{T}_{{T}_{\mathrm{s}}}$ and $\mathrm{\Delta}{T}_{{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}$ correspondingly from Δ*T*_{s} and $\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$.

The CPEC system for this study is CPEC310 (Campbell Scientific Inc., UT,
USA), whose major components are a CSAT3A sonic anemometer (updated version
in 2016) for a fast response to 3-D wind and *T*_{s} and an EC155 infrared
analyzer for a fast response to H_{2}O along with CO_{2} (Burgon et al.,
2015; Ma et al., 2017). The system operates in a *T* range of −30 to 50 ^{∘}C and measures ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ in a range up to 79 mmolH_{2}O mol^{−1} (i.e., 37 ^{∘}C dew point temperature at 86 kPa in
manufacturer environment); therefore, the accuracy of equation-computed *T*,
depending on Δ*T*_{s} and $\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$, should be defined
and estimated in a domain over both ranges.

## 4.1 Δ*T*_{s} (measurement accuracy in *T*_{s})

As is true for other sonic anemometers (e.g., Gill Instruments, 2004), the
CSAT3A has not been assigned a *T*_{s} measurement performance (Campbell
Scientific Inc., 2018b) because the theories and methodologies of how to
specify this performance, to the best of our knowledge, have not been
clearly defined. The performance of the CSAT series for *T*_{s} is best near the
production temperature at around 20 ^{∘}C and drifts a little away from
this temperature. Within the operational range of a CPEC system in ambient
air temperature, the updated version of CSAT3A has an overall uncertainty of
±1.00 ^{∘}C (i.e., |Δ*T*_{s}|<1.00 K,
via personal communication with CSAT authority Larry Jacobsen through email
in 2017 and in person in 2018).

## 4.2 $\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ (measurement accuracy in ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$)

The accuracy in H_{2}O measurements from infrared analyzers depends upon
analyzer measurement performance. This performance is specified using four
component uncertainties: (1) precision variability (${\mathit{\sigma}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$), (2) maximum zero drift range with ambient air
temperature (*d*_{wz}), (3) maximum gain drift with ambient air temperature
(±${\mathit{\delta}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$, where ${\mathit{\delta}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ is the gain drift percentage), and (4) cross-sensitivity to
CO_{2} (*s*_{c}) (LI-COR Biosciences, 2016; Campbell Scientific Inc.,
2018c). Zhou et al. (2021) composited the four component uncertainties as an
accuracy model formulated as the H_{2}O accuracy equation for CPEC systems
applied in ecosystems, given by

where
*T*_{c} is the ambient air temperature at which an infrared
analyzer was calibrated by the manufacturer to fit its working equation or
zeroed/spanned by a user in the field to adjust the zero/gain drift; subscripts rh and
rl indicate the range of the highest and lowest values, respectively; and
*T*_{rh} and *T*_{rl} are the highest and lowest *T*, respectively, over the
operational range in *T* of CPEC systems. Given the infrared analyzer
specifications – ${\mathit{\sigma}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$, *s*_{c}, *d*_{wz}, ${\mathit{\delta}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}\mathrm{\_}\mathrm{g}}$, *T*_{rl}, and *T*_{rh} – this equation can be used to
estimate $\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ in Eq. (25) and eventually $\mathrm{\Delta}{T}_{{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}$ in Eq. (26) over the domain of *T* and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$.

## 4.3 Δ*T* (accuracy of equation-computed *T*)

The accuracy of equation-computed *T* can be evaluated using Δ*T*_{s} and $\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ (Eq. 25), varying with *T*,
*T*_{s}, and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$. Both *T* and *T*_{s} reflect air temperature,
being associated with each other through ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ (Eq. 23). Given
${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$, *T* can be calculated from *T*_{s}, and vice versa; therefore,
for the figure presentations in this study, it is sufficient to use either
*T* or *T*_{s}, instead of both, to show Δ*T* with air temperature.
Considering *T* to be of interest to this study, *T* will be used. As such,
Δ*T* can be analyzed over a domain of *T* and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ within the
operational range in *T* of CPEC systems from −30 to 50 ^{∘}C across
the analyzer measurement range of ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ from 0 to 0.079 molH_{2}O mol^{−1}.

To visualize the relationship of Δ*T* with *T* and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$,
Δ*T* is presented better as the ordinate along *T* and as the abscissa associated with
${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$. However, due to the positive dependence of air water vapor
saturation on *T* (Wallace and Hobbs, 2006), ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ has a range that
is wider at a higher *T* and narrower at a lower *T*. To present Δ*T* over the
same measure of air moisture, even at different *T* values, the saturation water vapor
pressure is used to scale air moisture to 0, 20, 40, 60, 80, and 100 (i.e.,
RH, relative humidity in %). For each scaled RH value, ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$
can be calculated at different *T* and *P* values (Appendix C) for use in Eq. (25). In this
way, over the range of *T*, the trend of Δ*T* due to each measurement
uncertainty source can be shown along the curves with equal RH as the
measure of air moisture (Fig. 2).

### 4.3.1 $\mathrm{\Delta}{T}_{{T}_{\mathrm{s}}}$ (uncertainty portion of Δ*T* due to Δ*T*_{s})

Given $\mathrm{\Delta}{T}_{\mathrm{s}}=\pm $1.00 K and *T*_{s} from the algorithm in
Appendix C, $\mathrm{\Delta}{T}_{{T}_{\mathrm{s}}}$ in Eq. (26) was calculated over the domain
of *T* and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ (Fig. 2a). Over the whole *T* range, the $\mathrm{\Delta}{T}_{{T}_{\mathrm{s}}}$ limits range ±1.00 K, becoming a little narrower with
${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ increasing due to a decrease, at the same *T*_{s}, in the
magnitude of $T/{T}_{\mathrm{s}}$ in Eq. (25). The narrowest limits of $\mathrm{\Delta}{T}_{{T}_{\mathrm{s}}}$, in an absolute value, vary < 0.01 K over the range of *T* below 20 ^{∘}C, although > 0.01 K but < 0.03 K above 20 ^{∘}C.

### 4.3.2 $\mathrm{\Delta}{T}_{{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}$ (uncertainty portion of Δ*T* due to $\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$)

Given ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ from the algorithm in Appendix C and $\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ from Eq. (27), $\mathrm{\Delta}{T}_{{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}$ was calculated over the
domain of *T* and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ (Fig. 2b). The parameters in Eq. (27) are
given through the specifications of the CPEC300 series (Campbell Scientific
Inc., 2018a, c; ${\mathit{\sigma}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ is 6.0 × 10^{−6} molH_{2}O mol^{−1}, where mol is a unit (moles) for dry air; *d*_{wz}, ±5.0 × 10^{−5} molH_{2}O mol^{−1}; ${\mathit{\delta}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}\mathrm{\_}\mathrm{g}}$, 0.30 %; *s*_{c}, ±5.0 × 10^{−8} molH_{2}O mol^{−1} (µmolCO_{2} mol${}^{-\mathrm{1}}{)}^{-\mathrm{1}}$; *T*_{c}, 20 ^{∘}C as normal temperature – Wright et al., 2003; *T*_{rl}, −30 ^{∘}C; *T*_{rh}, 50 ^{∘}C).

As shown in Fig. 2b, $\mathrm{\Delta}{T}_{{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}$ tends to be smallest at *T*=*T*_{c}. However, away from *T*_{c}, its range nonlinearly becomes wider, very
gradually widening below *T*_{c} but widening more abruptly above it because,
as temperature increases, ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ at the same RH increases
exponentially (Eqs. C1 and C5 in Appendix C), while $\mathrm{\Delta}{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$
increases linearly with ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ in Eq. (27). This nonlinear range
can be summarized as ±0.01 K below 30 ^{∘}C and ±0.02 K above 30 ^{∘}C. Compared to $\mathrm{\Delta}{T}_{{T}_{\mathrm{s}}}$, $\mathrm{\Delta}{T}_{{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}$ is much smaller by 2 orders of magnitude. $\mathrm{\Delta}{T}_{{T}_{\mathrm{s}}}$ is a larger component in Δ*T*.

### 4.3.3 Δ*T* (combined uncertainty as the accuracy in equation-computed *T*)

Equation (26) is used to determine the maximum combined uncertainty in
equation-computed *T* for the same RH grade in Fig. 2 by adding together the
same sign (i.e., ±) curve data of $\mathrm{\Delta}{T}_{{T}_{\mathrm{s}}}$ in Fig. 2a and
$\mathrm{\Delta}{T}_{{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}$ in Fig. 2b. Δ*T* ranges at different RH
grades are shown in Fig. 2c. Figure 2c specifies the accuracy of
equation-computed *T* at 101.325 kPa (i.e., normal atmospheric pressure as used
by Wright et al., 2003) over the ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ measurement range to be
within ±1.01 K. This accuracy for high-frequency *T* is currently the
best in turbulent flux measurement because ±1.00 K is the best in terms of the
accuracy of *T*_{s} from the individual sonic anemometers which are widely
used for sensible heat flux in almost all CPEC systems.

## 4.4 Accuracy of equation-computed *T* from CPEC field measurements

Equation (23) is derived particularly for CPEC systems in which *T*_{s} and
${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ are measured neither at the same volume nor at the same
time. Both variables are measured separately using a sonic anemometer and an
infrared analyzer in a spatial separation between the *T*_{s} measurement
center and the ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ measurement cuvette (e.g., Fig. 1), along
with a temporal lag in the measurement of ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ relative to
*T*_{s} due to the transport time and phase shift (Ibrom et al., 2007) of
turbulent airflows sampled for ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ through the sampling orifice
to the measurement cuvette (Fig. 3).

Fortunately, the spatial separation scale is tens of centimeters, and
the temporal lag scale is tens of milliseconds. In eddy-covariance
flux measurements, such a separation misses some covariance signals at a
higher frequency, which is correctable (Moore, 1986), and such a lag
diminishes the covariance correlation, which is recoverable (Ibrom et al.,
2007). How such a separation along with the lag influences the accuracy of
Eq. (23), as shown in Fig. 2, needs testing against precision measurements
of air temperature. The two advantages of the equation-computed *T* discussed
in the Introduction, namely the fast response to high-frequency signals and
the insensitivity to solar contamination in measurements, were studied and
assessed during testing when a CPEC system was set up in the Campbell
Scientific instrument test field (41.8^{∘} N, 111.9^{∘} W;
1360 m a.s.l.; UT, USA).

## 5.1 Field test station

A CPEC310 system was set as the core of the station in 2018. Beyond its
major components briefly described in Sect. 4, the system also included a
barometer (model MPXAZ6115A, Freescale Semiconductor, TX, USA) for flow
pressure; pump module (SN 1001) for air sampling; valve module (SN 1003)
to control flows for auto-zero/span CO_{2} and H_{2}O; scrub module (SN 1002) to generate zero air (i.e., without CO_{2} and H_{2}O) for the auto-zero procedure; CO_{2} cylinder for CO_{2} span; and EC100
electronic module (SN 1002, OS Rev 07.01) to control and measure a CSAT3A,
EC155, and barometer. In turn, the EC100 was connected to and instructed
by a central CR6 datalogger (SN 2981, OS 04) for sensor measurements,
data processing, and data output. In addition to receiving the data output
from the EC100, the CR6 also controlled the pump, valve, and scrub modules
and measured other micrometeorological sensors in support of this study.

The micrometeorological sensors included an LI-200 pyranometer (SN 18854,
LI-COR Biosciences, NE, USA) to monitor incoming solar radiation, a
precision platinum resistance temperature detector (RTD, model 41342, SN TS25360) inside a fan-aspirated radiation shield (model 43502, R.M. Young
Company, MI, USA) to more accurately measure the *T* considered with minimized
solar contamination due to higher fan-aspiration efficiency, and an HMP155A
temperature and humidity sensor (SN 1073, Vaisala Corporation, Helsinki,
Finland) inside a 14-plate wind-aspirated radiation shield (model 41005) to
measure the *T* under conditions of potentially significant solar contamination
during the day due to low wind-aspiration efficiency. The sensing centers of
all sensors related to *T*_{s}, *T*, and RH were set at a height of 2.57 m above
ground level. The land surface was covered by natural prairie with a grass
height of 5 to 35 cm.

A CR6, supported by EasyFlux DL CR6CP (revised version for this study,
Campbell Scientific Inc., UT, USA), controlled and sampled the EC100 at 20 Hz. For spectral analysis, the EC100 filtered the data of *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ for anti-aliasing using a finite-impulse-response filter with a
0-to-10 Hz (Nyquist folding frequency) passing band (Saramäki, 1993).
The EC155 was zeroed for CO_{2}–H_{2}O and spanned for CO_{2}
automatically every other day and spanned for H_{2}O monthly using an
LI-610 Portable Dew Point Generator (LI-COR Biosciences, NE, USA). The
LI-200, RTD, and HMP155A were sampled at 1 Hz because of their slow response
and the fact that only their measurement means were of interest to this
study.

The purpose of this station was to measure the eddy-covariance fluxes to
determine turbulent transfers in the boundary-layer flows. The air
temperature equation (i.e., Eq. 23) was developed for the *T* of the turbulent airflows sampled through the CPEC systems. Therefore, this equation can be
tested based on how the CPEC310 measures the boundary-layer flows related to
turbulent transfer.

## 5.2 Turbulent transfer and CPEC310 measurement

In atmospheric boundary-layer flows, air constituents along with heat and
momentum (i.e., air properties) are transferred dominantly by individual
turbulent flow eddies with various sizes (Kaimal and Finnigan, 1994). Any
air property is considered to be more homogenous inside each smaller eddy
and more heterogenous among larger eddies (Stull, 1988). Due to this
heterogeneity, an eddy in motion among others transfers air properties
to its surroundings. Therefore, to measure the transfer in amount and
direction, a CPEC system was designed to capture *T*_{s}, ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$,
and 3-D flow speeds from individual eddies. Ideal measurements, although impossible, would be fast
enough to capture all eddies with different sizes
through the measurement volume and sampling orifice of the CPEC system (Fig. 1). To capture more eddies of as many sizes as possible, the CPEC
measurements were set at a high frequency (20 Hz in this study) because,
given 3-D speeds, the smaller the eddy, the shorter time the said eddy takes to
pass the sensor measurement volume.
Ideally, each measurement captures an individual eddy for all variables of
interest so that the measured values are representative of this eddy. So,
for instance, in our effort to compute *T* from a pair of *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ values, the pair simultaneously measured from the same eddy could better
reflect its *T* at the measurement time; however, in a CPEC system, *T*_{s} and
${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ are measured with separation in both space (Fig. 1) and time
(Fig. 3).

If an eddy passing the sonic anemometer is significantly larger than the
dimension of separation between the *T*_{s} measurement volume and the
${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ sampling orifice (Fig. 1), the eddy is instantaneously
measured for its 3-D wind and *T*_{s} in the volume while also sampled in the
orifice for ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ measurements. However, if the eddy is smaller
and flows along the alignment of separation, the sampling takes place either
a little earlier or a little later than the measurement (e.g., earlier if *T*_{s} is
measured later, and vice versa). However, depending on its size, an eddy
flowing beyond the alignment from other directions, although measured by the
sonic anemometer, may be missed by the sampling orifice passed by other
eddies and, in other cases, although sampled by the orifice, may be missed
by the measurement of the sonic anemometer.

Additionally, the airflow sampled for ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ measurements is not
measured at its sampling time on the sampling orifice but instead is
measured, in lag, inside the ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ measurement cuvette (Fig. 3).
The lag depends on the time needed for the sampled flow to travel through
the CPEC sampling system (Fig. 3). Therefore, for the computation of $T,{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ is better synchronized and matched with *T*_{s} as if they were simultaneously
measured from the same eddy.

## 5.3 Temporal synchronization and spatial match for *T*_{s} with ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$

In the CPEC310 system, a pair of *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ values that were
received by the CR6 from the EC100 in one data record (i.e., data row) were
synchronously measured, through the Synchronous Device for Measurement
communication protocol (Campbell Scientific Inc., 2018c), in the *T*_{s}
measurement volume and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ measurement cuvette (Fig. 1).
Accordingly, within one data row of time series received by the CR6, ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ was sampled earlier than *T*_{s} was measured. As discussed above,
*T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ in the same row, although measured at the same time,
might not be measured from the same eddy. If so, the ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$
measurement from the same eddy of this *T*_{s} might occur in another data
row, and vice versa. In any case, a logical procedure for a synchronized
match is first to pair *T*_{s} with ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ programmatically in the CR6, as
the former was measured at the same time as the latter was sampled.

### 5.3.1 Synchronize *T*_{s} measured to ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ sampled at the same time

Among the rows in time series received by the CR6, any two consecutive rows were measured sequentially at a fixed time interval (i.e., measurement interval). Accordingly, anemometer data in any data row can be synchronized with analyzer data in a later row from the eddy sampled by the analyzer sampling orifice at the measurement time of the sonic anemometer. How many rows later depends on the measurement interval and the time length of the analyzer sample from its sampling orifice to the measurement cuvette. The measurement interval commonly is 50 or 100 ms for a 20 or 10 Hz measurement frequency, respectively. The time length is determined by the internal space volume of the sampling system (Fig. 3) and the flow rate of sampled air driven by a diaphragm pump (Campbell Scientific Inc., 2018a).

As shown in Fig. 3, the total internal space is 10.563 mL. The rate of
sampled air through the sampling system nominally is 6.0 L min^{−1}, at
which the sampled air takes 106 ms to travel from the analyzer sampling
orifice to the cuvette exhaust outlet (Fig. 3). Given that the internal
optical volume inside the cuvette is 5.887 mL, the air in the cuvette was
sampled during a period of 47 to 106 ms earlier. Accordingly, anemometer
data in a current row of time series should be synchronized with analyzer
data in the next row for 10 Hz data and, for 20 Hz data, in the row after that.
After synchronization, the CR6 stores anemometer and analyzer data in a
synchronized matrix (variables unrelated to this study were omitted) as a
time series:

where *u* and *v* are horizontal wind speeds orthogonal to each other; *w* is
vertical wind speed; *d*_{s} and *d*_{g} are diagnosis codes for the sonic
anemometer and infrared analyzer, respectively; *s* is the analyzer signal strength
for H_{2}O; *t* is time, and its subscript *i* is its index; and the difference
between *t*_{i} and *t*_{i+1} is a measurement interval ($\mathrm{\Delta}t={t}_{i+\mathrm{1}}$ − *t*_{i}). In any row of the matrix (28) (e.g., the *i*th
row), *t*_{i} for anemometer data is the measurement time plus instrument
lag, and *t*_{i} for analyzer data is the sampling time plus the same lag.
The instrument lag is defined as the number of measurement intervals used
for data processing inside the EC100 after the measurement and subsequent data
communication to the CR6. Regardless of instrument lag, *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ in each row of the synchronization matrix were temporally synchronized
as measured and sampled at the same time.

### 5.3.2 Match *T*_{s} measured to ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ sampled from the same eddy

As discussed in Sect. 5.2, at either the *T*_{s} measurement or the ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$
sampling time, if an eddy is large enough to enclose both the *T*_{s} measurement
volume and the ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ sampling orifice (Fig. 1), *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ in the same row of the synchronization matrix (28) belong to the
same eddy; otherwise, they belong to different eddies. For any eddy size, it
would be ideal if *T*_{s} could be spatially matched with ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ as
a pair for the same eddy; however, this match would not be possible for all
*T*_{s} values simply because, in some cases, an eddy measured by the sonic
anemometer might never be sampled by the ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ sampling orifice,
and vice versa (see Sect. 5.2). Realistically, *T*_{s} may be matched with
${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ overall with the most likelihood to as many pairs as
possible for a period (e.g., an averaging interval).

The match will eventually lag either *T*_{s} or ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$, relatively,
in the synchronization matrix (28). The lag can be counted as an integer
number (*l*_{s}, where subscript s indicates the spatial separation causing lag) in
measurement intervals, where *l*_{s} is positive if an eddy flowed through
the *T*_{s} measurement volume earlier, negative if it flowed through later, or zero if it flowed through
the ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ sampling orifice at the same time. This number is
estimated through the covariance maximization (Irwin, 1979; Moncrieff et
al., 1997; Ibrom et al., 2007; Rebmann et al., 2012). According to *l*_{s}
over an averaging interval, the data columns of the infrared analyzer over
an averaging interval in the synchronization matrix (28) can be moved
together up *l*_{s} rows as positive, down *l*_{s} rows as negative, or
nowhere as zero to form a matched matrix:

For details on how to find *l*_{s}, see EasyFlux DL CR6CP on
https://www.campbellsci.com (last access: 11 December 2021). In the matched matrix (29), over an averaging
interval, a pair of *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ values in the same row can be
assumed to be matched as if they were measured and sampled from the same eddy.

Using Eq. (23), the air temperature can now be computed using

where subscript *l*_{s} for *t* indicates that spatially lagged ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$
is used for computation of *T*. In verification of the accuracy of
equation-computed *T* and in assessments of its expected advantages of
high-frequency signals insensitive to solar contamination in measurements,
${T}_{{l}_{\mathrm{s}}i}$ could minimize the uncertainties due to the spatial separation
in measurements of *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ between the *T*_{s} measurement
volume and the ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ sampling orifice (Fig. 1).

## 6.1 Verification of the accuracy of equation-computed *T*

The accuracy of equation-computed *T* was theoretically specified by Eqs. (25)
to (27) and was estimated in Fig. 2c. This accuracy specifies the range of
equation-computed *T* minus true *T* (i.e., Δ*T*). However, true *T* was not
available in the field, but, as usual, precision measurements could be
considered benchmarks to represent true *T*. In this study, *T* measured by the
RTD inside a fan-aspirated radiation shield (*T*_{RTD}) was the benchmark to
compute Δ*T* (i.e., equation-computed *T* minus *T*_{RTD}). If almost all
Δ*T* values fall within the accuracy-specified range over a measurement
domain of *T* and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$, the accuracy is correctly defined, and the
equation-computed *T* is accurate as specified.

To verify that the accuracy over the domain is as large as possible,
Δ*T* values in the coldest (January) and hottest (July) months were used
as shown in Fig. 4 (−21 ^{∘}C < *T* < 35.5 ^{∘}C, and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ up to 20.78 mmolH_{2}O mol^{−1} in a
30 min mean over both months). Out of 2976 Δ*T* values from both
months, 44 values fell out of the specified accuracy range but were near the
range line within 0.30 K. The Δ*T* values were 0.549 ± 0.281 K in
January and 0.436 ± 0.290 K in July. Although these values were almost
all positively away from the zero line due to either overestimation for
*T*_{s} by the sonic anemometer within ±1.00 K accuracy or
underestimation for *T*_{RTD} by the RTD within ±0.20 K accuracy, the
ranges are significantly narrower than the specified accuracy range of
equation-computed *T* (Figs. 2c and 4).

It is common for sonic anemometers to have a systematic error in *T*_{s} of ±0.5 ^{∘}C or a little greater, which is the reason
that the *T*_{s} accuracy is specified by Larry Jacobsen (anemometer
authority) to be ±1.0 ^{∘}C for the updated CSAT3A. The
fixed deviation in measurements of sonic path lengths is asserted as a
source of bias in *T*_{s} (Zhou et al., 2018). This bias brings an error to
equation-computed *T*. If the *T* equation were not exact as in Eqs. (4) and (5),
there would be an additional equation error. In our study effort, this bias
from fixed deviation is possibly around 0.5 ^{∘}C. With this
bias, the equation-computed *T* is still accurate, as specified by Eqs. (25) to
(27), and even better.

## 6.2 Assessments of the advantages of equation-computed *T*

As previously discussed, the data stream of equation-computed *T* consists of
high-frequency signals insensitive to solar contamination in measurements.
Its frequency response can be assessed against known high-frequency signals
of *T*_{s}, and the insensitivity can be assessed by analyzing the
equation-computed, RTD-measured, and sensor-measured *T*, where the
sensor is an HMP155A inside a wind-aspirated radiation shield.

### 6.2.1 Frequency response

The matched matrix (29) and Eq. (30) were used to compute ${T}_{{l}_{\mathrm{s}}i}$ (i.e.,
equation-computed *T*). Paired power spectra of equation-computed *T* and
*T*_{s} are compared in Fig. 5 for three individual 2 h periods of
atmospheric stratifications, including unstable ($z/L$ is −0.313 to −2.999, where *z* is a dynamic height of measurement minus
displacement height and *L* is the Monin–Obukhov length), near-neutral ($z/L$ is −0.029 to +0.003), and stable ($z/L$ is +0.166 to +0.600). Slower response of equation-computed *T* than
*T*_{s} at a higher frequency (e.g., > 5 Hz) was expected because
equation-computed *T* is derived from two variables (*T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$) measured in a spatial separation, which attenuates the frequency
response of correlation of two measured variables (Laubach and McNaughton,
1998), and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ from a CPEC system has a slower response than
*T*_{s} in terms of frequency (Ibrom et al., 2007). However, the expected slower
response was not found in this study. In unstable and stable atmospheric
stratifications (Fig. 5a and c), each pair of power spectra almost
overlaps. Although they do not overlap in the near-neutral atmospheric
stratification (Fig. 5b), the pairs follow the same trend slightly above or
below one another. In the higher-frequency band of 1 to 10 Hz in Fig. 5a
and b, equation-computed *T* has a little more power than *T*_{s}. The three
pairs of power spectra in Fig. 5 indicate that equation-computed *T* has a
frequency response equivalent to *T*_{s} up to 10 Hz, with a 20 Hz measurement
rate considered to be a high frequency. The equivalent response might be
accounted for by a dominant role of *T*_{s} in the magnitude of
equation-computed *T*.

### 6.2.2 Insensitivity to solar contamination in measurements

The data of equation-computed, sensor-measured, and RTD-measured *T* in July,
during which incoming solar radiation (*R*_{s}) at the site was strongest in
a yearly cycle, are used to assess the insensitivity of equation-computed
*T*. From the data, Δ*T* is considered to be an error in equation-computed
*T*. The error in sensor-measured *T* can be defined as sensor-measured *T* minus
RTD-measured *T*, denoted by Δ*T*_{m}. From Fig. 6, Δ*T* (0.690 ± 0.191 K) is > Δ*T*_{m} (0.037 ± 0.199 K)
when *R*_{s} < 50 W m^{−2} at lower radiation. However, Δ*T* (0.234 ± 0.172 K) is < Δ*T*_{m} (0.438 ± 0.207 K) when *R*_{s}>50 W m^{−2} at higher radiation. This
difference between Δ*T* and Δ*T*_{m} shows a different effect of
*R*_{s} on equation-computed and sensor-measured *T*.

As shown in Fig. 6, Δ*T*_{m} increases sharply with increasing
*R*_{s} for *R*_{s}<250 W m^{−2}, beyond which it asymptotically
approaches 0.40 K. In the range of lower *R*_{s}, atmospheric stratification
was likely stable (Kaimal and Finnigan, 1994), under which the heat exchange
by wind was ineffective between the wind-aspirated radiation shield and
boundary-layer flows. In this case, sensor-measured *T* was expected to
increase with *R*_{s} increase (Lin et al., 2001; Blonquist and Bugbee,
2018). Along with *R*_{s} increase, the atmospheric boundary layer develops
from stable to neutral or unstable conditions (Kaimal and Finnigan, 1994).
During the stability change, the exchange becomes increasingly more
effective, offsetting the further heating from *R*_{s} increase on the
wind-aspirated radiation shield as indicated by the red asymptote portion in
Fig. 6. Compared to the Δ*T*_{m} mean (0.037 K) while *R*_{s}<50 W m^{−2}, the magnitude of the asymptote above the mean is the
overestimation of sensor-measured *T* due to solar contamination.

However, Δ*T* decreases asymptotically from about 0.70 K toward zero
with the increase in *R*_{s} from 50 to 250 W m^{−2} and beyond, with a
more gradual rate of change than Δ*T*_{m} at the lower radiation
range. Lower *R*_{s} (e.g., < 250 W m^{−2}) concurrently occurs
with lower *T*, higher RH, and/or unfavorable weather to *T*_{s} measurements.
Under lower *T* (e.g., below 20 ^{∘}C of normal CSAT3A manufacturing
conditions), the sonic path lengths of CSAT3A (Fig. 1) must become, due to
thermo-contraction of sonic anemometer structure, shorter than those at 20 ^{∘}C. As a result, the sonic anemometer could overestimate the speed of
sound (Zhou et al., 2018) and, hence, *T*_{s} for equation-computed *T*,
resulting in greater Δ*T* with lower *R*_{s}. Under higher RH
conditions, dew may form on the sensing surface of the six CSAT3A sonic
transducers (Fig. 1). The dew, along with unfavorable weather, could
contaminate the *T*_{s} measurements, resulting in Δ*T* greater in
magnitude. Higher *R*_{s} (e.g., > 250 W m^{−2}) concurrently
occurs with weather favorable to *T*_{s} measurements, which is the reason
that Δ*T* slightly decreases rather than increases with *R*_{s} when
*R*_{s} > 250 W m^{−2}.

Again from Fig. 6, the data pattern of Δ*T*>Δ*T*_{m} in the lower *R*_{s} range and Δ*T*<Δ*T*_{m} in
the higher *R*_{s} range shows that equation-computed *T* is not as sensitive to
*R*_{s} as sensor-measured *T*. The decreasing trend of Δ*T* with *R*_{s}
increase shows the insensitivity of equation-computed *T* to *R*_{s}. Although
the purpose of this study is not particularly to eliminate solar radiation
contamination, equation-computed *T* is indeed less contaminated by solar
radiation, as shown in Fig. 6.

## 7.1 Actual accuracy

The range of Δ*T* curves for each RH level in Fig. 2 is the maximum at
that level because the data were evaluated using the maximized measurement
uncertainties from all sources. Accordingly, in field applications under
weather favorable to *T*_{s} measurements, the range of actual accuracy in
equation-computed *T* can be reasonably inferred to be narrower. In our study
case as shown in Figs. 4 and 6, the variability in Δ*T* was narrower
than the accuracy range as specified in Fig. 2. In other words, the actual
accuracy is better.

However, under weather conditions unfavorable to *T*_{s} measurements, such
as dew, rain, snow, or dust storms, the accuracy of *T*_{s} measurements
cannot be easily evaluated. *T*_{s} measurements also possibly have a
systematic error due to the fixed deviation in the measurements of sonic
path lengths for sonic anemometers, although the error should be within the
accuracy specified in Fig. 2. A ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ measurement can also be
erroneous if the infrared analyzer is not periodically zeroed and spanned
for its measurement environment. Therefore, if *T*_{s} is measured under
unfavorable weather conditions and the sonic anemometer produces a
systematic *T*_{s} error and if the infrared analyzer is not zeroed and
spanned as instructed in its manual, then the accuracy of equation-computed
*T* would be unpredictable. Normally, the actual accuracy is better than that
specified in Fig. 2. Additionally, with the improvement in measurement
accuracies of sonic anemometers (e.g., weather-condition-regulated, heated,
3-D sonic anemometers; Mahan et al., 2021) and infrared analyzers, this
accuracy of equation-computed *T* would gradually become better.

For this study, filtering out the *T*_{s} data in the periods of unfavorable
weather could narrow the error range of equation-computed *T*. The unfavorable
weather was suspected of contributing to the stated error. However, although
filtering out unfavorable weather cases could create a lower error estimate,
most field experiments include periods when weather increases a *T*_{s}
error, so including a weather contribution to error would prevent
overstating instrument accuracy under typical (unfiltered) applications.
Therefore, both *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ data in this study were not
programmatically or manually filtered based on weather.

## 7.2 Spatial separation of *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ in measurements

In this study, *T* was successfully computed from *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ as
a high-frequency signal (Fig. 5) with expected accuracy as tested in Figs. 2, 4, and 6, where both were measured separately from two sensors in a
spatial separation. Some open-path eddy-covariance (OPEC) flux systems
(e.g., CSAT3A + EC150 and CSAT3B + LI-7500) measure *T*_{s} and *ρ*_{w}
also using two sensors in a spatial separation. To OPEC systems, although the
air temperature equation (Eq. 23) is not applicable, the algorithms
developed in Sect. 5.3 to temporally synchronize and spatially match
*T*_{s} with ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ for computation of *T* are applicable for computation
of *T* from *T*_{s} and *ρ*_{w} along with *P* in such OPEC systems (Swiatek,
2018).

In Sect. 5.3, programming and computing are needed to pair *T*_{s} measured to
${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ sampled at the same time into the synchronization matrix
(28) as the first step and from the same eddy into matched matrix (29) as
the second step. The second requires complicated programming and much
computing. To test the necessity of this step in specific cases, using Eq. (30), *T*_{0i} was computed from a row of the synchronization matrix,
and ${T}_{{l}_{\mathrm{s}}i}$ was computed from this matrix by lagging ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$
columns up *l*_{s} rows if *l*_{s}>0 and down $\left|{l}_{\mathrm{s}}\right|$ rows if *l*_{s}<0, where *l*_{s} is $-\mathrm{5},\mathrm{\dots},-$1 and $+\mathrm{1},\mathrm{\dots},+$5. From the data of this study, individual ${T}_{{l}_{\mathrm{s}}i}$ values were
different for different subscript *l*_{s} values, but their means for subscript *i*
over an averaging interval $\left({T}_{{l}_{\mathrm{s}}}\right)$ are the same to at least the
fourth digit after the decimal place. Further, the power spectrum of
*T*_{0i} time series was compared to those of ${T}_{{l}_{\mathrm{s}}i}$ time series,
where *l*_{s}≠0. Any pair of power spectra from the same period overlaps
exactly (figures omitted). Therefore, the second step of lag maximization to
match *T*_{s} measured to ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ sampled from the same eddy is not
needed if only the hourly mean and power spectrum of equation-computed *T* are of
interest to computations, for both CPEC and OPEC systems.

## 7.3 Applications

The air temperature equation (Eq. 23) is derived from first principles without
any assumption and approximation. It is an exact equation from which *T* can be
computed in CPEC systems as a high-frequency signal insensitive to solar
radiation. These merits, in additional to its consistent representation of
spatial measurement and temporal synchronization scales with other
thermodynamic variables for boundary-layer turbulent flows, will be
particularly needed for advanced applications. The EasyFlux series is one of
the two most popular field eddy-covariance flux software packages used in
the world, the other being EddyPro (LI-COR Biosciences, 2015). Currently, it
has used equation-computed *T* for *ρ*_{d} in Eq. (1), sensible heat flux
(*H*), and RH as a high-frequency signal in CPEC systems (Campbell Scientific
Inc., 2018a).

### 7.3.1 Dry-air density

As a high-frequency signal insensitive to solar radiation, equation-computed
*T* is more applicable than sensor-measured *T* for calculations of ${\overline{\mathit{\rho}}}_{\mathrm{d}}$ and $\stackrel{\mathrm{\u203e}}{{\mathit{\rho}}_{\mathrm{d}}w}$ for advanced applications (Gu et
al., 2012; Foken et al., 2012). In practice, equation-computed *T* can surely
be used for ${\overline{\mathit{\rho}}}_{\mathrm{d}}$ and $\stackrel{\mathrm{\u203e}}{{\mathit{\rho}}_{\mathrm{d}}w}$ under normal
weather conditions while the sonic anemometer and infrared analyzer are
normally running, which can be judged by their diagnosis codes (Campbell
Scientific Inc., 2018a). Under a weather condition unfavorable to *T*_{s}
measurements, such as dew, rain, snow, and/or ice, equation-computed *T* from
weather-condition-regulated, heated, 3-D sonic anemometers (Mahan et al.,
2021) and infrared analyzers could be an alternative.

Currently, in CO_{2}, H_{2}O, and trace gas flux measurements,
${\overline{\mathit{\rho}}}_{\mathrm{d}}$ for flux calculations is estimated from *T* and RH along
with *P*. *T* and RH are measured mostly by a slow-response *T*–RH probe without
fan aspiration (e.g., HMP155A; Zhu et al., 2021). As shown in Fig. 6,
equation-computed *T* is better than probe-measured *T*. The air moisture measured
by an infrared analyzer in CPEC systems must be more accurate (Eq. 27 and
Fig. 2b) than probe-measured air moisture. The better equation-computed *T*
along with more accurate air moisture has no reason not to improve the
estimation for ${\overline{\mathit{\rho}}}_{\mathrm{d}}$.

### 7.3.2 Sensible heat flux estimated from a CPEC system

Currently, beyond the EasyFlux DL CR6CP series, *H* is derived from $\stackrel{\mathrm{\u203e}}{{T}_{\mathrm{s}}^{\prime}{w}^{\prime}}$ with a humidity correction (van Dijk, 2002). The
correction equations were derived by Schotanus et al. (1983) and van Dijk (2002) in two ways, but both were derived with the approximation from Eq. (4)
(see Appendix A). Using the exact equation from this study, theoretically *H* can be more accurately estimated directly from $\stackrel{\mathrm{\u203e}}{{T}^{\prime}{w}^{\prime}}$,
where *T* is the equation-computed air temperature, although more studies and
tests for this potential application are needed. Without our exact *T*
equation, in any flux software, either Eq. (4) or Eq. (5) must be used for *H*
computation. Both equations are approximate (see Appendices A and B).
Compared to either, our exact equation must be an improvement on the
mathematical representation of *H*. If the equation for sensible heat flux is
approximate, then even a perfect measurement gives only an approximate value
for the flux.

### 7.3.3 RH as a high-frequency signal

Conventionally, RH is measured using a *T*–RH probe, which is unable to track
the high-frequency fluctuations in RH. In a CPEC system, equation-computed
*T*, analyzer-measured ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$, and transducer-measured *P* are able to
catch the fluctuations in these variables at a high frequency, from which RH
can be computed (Sonntag, 1990; also see Appendix C). This method should
provide high-frequency RH, although verification for a frequency response is
needed. Currently, the applications of high-frequency properties in this RH
are unknown in a CPEC system. Regardless, equation-computed *T* provides a
potential opportunity to acquire the high-frequency RH for its application
in the future.

In a CPEC flux system, the air temperature (*T*) of boundary-layer flows
through the space of sonic anemometer measurement and infrared analyzer
sampling (Fig. 1) is desired for a high frequency (e.g., 10 Hz) with
consistent representation of spatial and temporal scales for moist
turbulence thermodynamics characterized by three-dimensional wind from the
sonic anemometer and H_{2}O–CO_{2} and atmospheric pressure from the
infrared analyzer. High-frequency *T* in the space can be measured using
fine-wire thermocouples, but this kind of thermocouple for such an
application is not durable under adverse climate conditions, being easily
contaminated by solar radiation (Campbell, 1969). Nevertheless, the
measurements of sonic temperature (*T*_{s}) and H_{2}O inside a CPEC system
are high-frequency signals. Therefore, high-frequency *T* can be reasonably
expected when computed from *T*_{s} and H_{2}O-related variables. For this
expectation, two equations (i.e., Eqs. 4 and 5) are currently available. In
both equations, converting H_{2}O-related variables into H_{2}O mixing
ratios analytically reveals the difference between the two equations. This
difference in CPEC systems reaches ±0.18 K, bringing an uncertainty
into the accuracy of *T* from either equation and raising the question of which
equation is better. To clarify the uncertainty and answer this question, the
air temperature equations in terms of *T*_{s} and H_{2}O-related variables
are thoroughly reviewed (Sects. 2 and 3, Appendices A and B). The two
currently used equations (i.e., Eqs. 4 and 5) were developed and completed
with approximations (Appendices A and B). Because of the approximations,
neither of their accuracies was evaluated, nor was the question answered.

Using the first-principles equations, the air temperature equation in terms
of *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ (H_{2}O molar mixing ratio) is derived
without any assumption and approximation (Eq. 23); therefore, the equation
derived in this study does not itself have any error, and, as such, the
accuracy in equation-computed *T* depends solely on the measurement accuracies
of *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$. Based on the specifications for *T*_{s} and
${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ in the CPEC300 series, the accuracy of equation-computed *T*
over the *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ measurement ranges can be specified
within ±1.01 K (Fig. 2). This accuracy range is propagated
mainly (±1.00 K) from the uncertainty in *T*_{s} measurements (Fig. 2a) and a little (±0.02 K) from the uncertainty in ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$
measurements (Fig. 2b).

Under normal sensor and weather conditions, the specified accuracy is
verified based on field data as valid, and actual accuracy is better (Figs. 4 and 6). Field data demonstrate that equation-computed *T* values under unstable,
near-neutral, and stable atmospheric stratifications all have frequency
responses equivalent to high-frequency *T*_{s} up to 10 Hz at a 20 Hz
measurement rate (Fig. 5), being insensitive to solar contamination in
measurements (Fig. 6).

The current applications of equation-computed *T* in a CPEC system are to
calculate dry-air density (*ρ*_{d}) for the estimations of CO_{2}
flux (${\overline{\mathit{\rho}}}_{\mathrm{d}}\stackrel{\mathrm{\u203e}}{{\mathit{\chi}}_{{\mathrm{CO}}_{\mathrm{2}}}^{\prime}{w}^{\prime}}$, where
${\mathit{\chi}}_{{\mathrm{CO}}_{\mathrm{2}}}$ is the CO_{2} mixing ratio, *w* is vertical velocity of air, the
prime indicates the fluctuation of the variable away from its mean, and the overbar implies the mean), H_{2}O flux (${\overline{\mathit{\rho}}}_{\mathrm{d}}\stackrel{\mathrm{\u203e}}{{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}^{\prime}{w}^{\prime}}$), and other fluxes. Combined with measurements of ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$, 3-D wind speeds, and *P*, the equation-computed *T* can be applied to
the estimation of ${\overline{\mathit{\rho}}}_{\mathrm{d}}$ and $\stackrel{\mathrm{\u203e}}{{\mathit{\rho}}_{\mathrm{d}}w}$ if
needed (Gu et al., 2012; Foken et al., 2012), to the computation of
high-frequency RH (Sonntag, 1990), and to the derivation of sensible heat
flux (*H*) avoiding the humidity correction as needed for *H* indirectly from
*T*_{s} (Schotanus et al., 1983; van Dijk, 2002).

In a CPEC flux system, although *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ are measured
using two spatially separated sensors of a sonic anemometer and infrared
analyzer, *T* was successfully computed from both measured variables as a
high-frequency signal (Fig. 5) with an expected accuracy (Figs. 2 and 4).
Some open-path eddy-covariance (OPEC) flux systems measure *T*_{s} and water
vapor density (*ρ*_{w}) also using two sensors in a similar way. The
algorithms developed in Sect. 5.3 to temporally synchronize and spatially
match *T*_{s} with ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ for computation of *T* are applicable to such
OPEC systems to compute *T* from *T*_{s} and *ρ*_{w} along with *P*. This *T*
would be a better option than sensor-measured *T* in the systems for the
correction of the spectroscopic effect in measuring CO_{2} fluctuations at
high frequencies (Helbig et al., 2016; Wang et al., 2016). With the
improvements in measurement technologies for *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$,
particularly for *T*_{s}, the *T* from our developed equation will become
increasingly more accurate. Having its accuracy combined with its high
frequency, this *T* with consistent representation of all other thermodynamic
variables for moist air at the spatial and temporal scales in CPEC
measurements has its advanced merits in boundary-layer meteorology and
applied meteorology.

The sonic temperature (*T*_{s}) reported by a three-dimensional sonic
anemometer is internally calculated from its measurements of the speed of
sound in moist air (*c*) after the crosswind correction (Zhou et al., 2018),
using

where subscript d indicates dry air, *γ*_{d} is the specific heat
ratio of dry air between constant pressure and constant volume, and
*R*_{d} is the gas constant for dry air (Campbell Scientific Inc., 2018b).
The speed of sound in the atmospheric boundary layer as in a homogeneous
gaseous medium is well defined in acoustics (Barrett and Suomi, 1949), given
by

where *γ* is the counterpart of *γ*_{d} for moist air, *P* is
atmospheric pressure, and *ρ* is moist-air density. These variables are
related to air temperature and air specific humidity (*q*, i.e., the mass ratio
of water vapor to moist air).

## A1 Moist-air density (*ρ*)

Moist-air density is the sum of dry-air and water vapor densities. Based on
the ideal gas law (Wallace and Hobbs, 2006), dry-air density (*ρ*_{d})
is given by

where *e* is water vapor pressure, and the water vapor density (*ρ*_{w})
is given by

where *R*_{v} is the gas constant for water vapor. Therefore, moist-air
density in Eq. (A2) can be expressed as

Because ${R}_{\mathrm{d}}/{R}_{\mathrm{v}}=\mathit{\epsilon}$ (i.e., 0.622, the molar mass ratio between water vapor and dry air), this equation can be rearranged as

Using Eqs. (A4) and (A6), the air specific humidity can be expressed as

Because $P\gg (\mathrm{1}-\mathit{\epsilon})e$, *q* can be approximated as

Substituting this relation into Eq. (A6) generates

## A2 Specific heat ratio of moist air (*γ*)

The specific heat ratio of moist air is determined by two moist-air
properties: (1) the specific heat at constant pressure (*C*_{p}) and (2) specific heat at constant volume (*C*_{v}). *C*_{p} varies with the air
moisture content between the specific heat of dry air at constant pressure
(*C*_{pd}) and the specific heat of water vapor at constant pressure
(*C*_{pw}). It must be the average of *C*_{pd} and *C*_{pw} that is
arithmetically weighted by the dry air mass and water vapor mass, respectively,
given by (Stull, 1988)

*C*_{v} can be similarly determined:

where *C*_{vd} is the specific heat of dry air at constant volume and
*C*_{vw} is the specific heat of water vapor at constant volume. Denoting
${C}_{\mathrm{pd}}/{C}_{\mathrm{vd}}$ as *γ*_{d}, Eqs. (A10) and (A11) are used to express
*γ* as

## A3 Relation of sonic temperature to air temperature

Substituting Eqs. (A9) and (A12) into Eq. (A2) leads to

Using this equation to replace *c*^{2} in Eq. (A1), *T*_{s} is expressed as

Given *C*_{pw}=1952, *C*_{pd}=1004, *C*_{vw}=1463,
and *C*_{vd}=717 J K^{−1} kg^{−1} (Wallace and Hobbs, 2006), this
equation becomes

Expression of the last two parenthesized terms on the right side of this
equation separately as Taylor series of *q* (Burden and Faires, 1993) by
dropping, due to *q*≪1, the second-or-higher-order terms related
to *q* leads to

On the right side of this equation, the three parenthesized terms can be
expanded into a polynomial of *q* of the third order. Also due to *q*≪1 in this polynomial, the terms of *q* of the second or third order
can be dropped. Further arithmetical manipulations result in

This is Eq. (4) in a different form. In its derivations from Eqs. (A1) and (A2), three approximation procedures were used from Eqs. (A7) to (A8), (A15) to (A16), and (A16) to (A17). The three approximations must bring unspecified errors into the derived equation.

Equation (5) was sourced from Ishii (1935) in which the speed of sound in
moist air (*c*) was expressed in his Eq. (1) as

where all variables in this equation are for moist air, *γ* is the
specific heat ratio of moist air between constant pressure and constant
volume, *P* is moist-air pressure, *ρ* is moist-air density, *α* is the
moist-air expansion coefficient, and *β* is the moist-air pressure
coefficient. Accordingly, the speed of sound in dry air (*c*_{d}) is given
by

where subscript d indicates dry air in which *γ*_{d}, *P*_{d}, *ρ*_{d}, *α*_{d}, and *β*_{d} are the counterparts of *γ*, *P*, *ρ*, *α*, and *β* in moist air. Equations (B1) and (B2) can be combined
as

Experimentally by Ishii (1935), each term inside the three pairs of
parentheses in this equation was linearly related to the ratio of water
vapor pressure (*e*) to dry-air pressure (*P*_{d}). Substituting the relationship into Eq. (B3) leads to

The three parenthesized terms in this equation sequentially
correspond to the three parenthesized terms in Eq. (B3). Dividing
*γ*_{d}*R*_{d}, where *R*_{d} is the gas constant for dry air, over both
sides of Eq. (B4) and referencing Eq. (11), sonic temperature (*T*_{s}) is
expressed in terms of air temperature (*T*), *e*, and *P*_{d} as

Using the relationship of *P*_{d}=*P*–*e*, this equation can be manipulated
as

Dropping the second-order terms due to $e/P\ll \mathrm{1}$ in boundary-layer flows, this equation becomes

Expanding the second parenthesized term into Taylor series and, also due to $e/P\ll \mathrm{1}$, dropping the terms related to $e/P$ of an order of 2 or higher, this equation becomes

Further expanding the two parenthesized terms on the right side of this equation and dropping the second-order term of $e/P$ led to

This is Eq. (5) in a different form. From the experimental source of Eq. (B4), it was derived using three approximations from Eqs. (B4) to (B7), (B7)
to (B8), and (B8) to (B9). The approximations and therefore combined uncertainty in *T* bring unspecified errors into Eq. (5) (i.e., Eq. B9) as an equation error.

For a given air temperature (*T* in ^{∘}C) and atmospheric pressure
(*P* in kPa), air has a limited capacity to hold water vapor (Wallace and
Hobbs, 2006). This limited capacity is described in terms of saturation
water vapor pressure (*e*_{s} in kPa) for moist air, given through the
Clausius–Clapeyron equation (Sonntag, 1990):

where *f*(*P*) is an enhancement factor for moist air, being a function of
atmospheric pressure: $f\left(P\right)=\mathrm{1.0016}+\mathrm{3.15}\times {\mathrm{10}}^{-\mathrm{5}}P-\mathrm{0.0074}{P}^{-\mathrm{1}}$. At relative humidity (RH in %), the water vapor pressure (*e*_{RH}(*T*,*P*) in kPa) is

Given the mole numbers of H_{2}O (*n*_{RH}) and dry air (*n*_{d}) at RH,
the H_{2}O molar mixing ratio at RH (${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}^{\mathrm{RH}})$ is

where *R*^{∗} is the universal gas constant and *P*_{d} is dry-air
pressure in kilopascals. Using this equation and the relation

${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}^{\mathrm{RH}}$ can be expressed as

Using Eq. (23), this ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}^{\mathrm{RH}}$ along with *T* can be used to
calculate sonic temperature (*T*_{s} in K) at RH, given by

where *ε*=0.622 (Eq. 17), *γ*_{v}=2.04045, and *γ*_{p}=1.94422 (Eq. 23). Through Eqs. (C1) and (C2), Eqs. (C5) and (C6)
express ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}^{\mathrm{RH}}$ and ${T}_{\mathrm{s}}(T,{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}^{\mathrm{RH}})$,
respectively, in terms of *T*, RH, and *P*. ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}^{\mathrm{RH}}$ and ${T}_{\mathrm{s}}(T,{\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}^{\mathrm{RH}})$ can be used to replace ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ (H_{2}O
molar mixing ratio) and *T*_{s} in Eq. (25). After replacements, Eq. (25) can
be used to evaluate the uncertainty, due to *T*_{s} and ${\mathit{\chi}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ measurement accuracy uncertainties, in air temperature
computed from Eq. (23) for different RH values over a *T* range.

Code is available at https://www.campbellsci.com/downloads?b=5 (last access: 18 December 2021, Campbell Scientific Inc., 2018a).

Data are available at https://datadryad.org/stash/share/ZiwOBaIBtu85UQ2kFye2LCtkzgp6l_UFg7dMeFi52ww (Zhou and Gao, 2021) via the following files: Data_Fig2a.xlsx, Data_Fig2b.xlsx, Data_Fig2c.xlsx, Data_Fig4.xlsx, and Data_Fig6.xlsx.

XinZ and TG developed the manuscript; ET substantially structured and revised the manuscript; XiaoZ analyzed time series data; AS, TA, and JO made comments on the manuscript; and JZ led the team.

Xinhua Zhou is affiliated with Campbell Scientific Inc., whose products were used in this research. The authors have no other competing interests to declare.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors thank the anonymous reviewers for their professional review, understanding of our study topic, and constructive comments on the manuscript for significant improvement; Brittney Smart for her professional and dedicated proofreading; Rex Burgon for his advice about the technical design of a CPEC sampling system; and Edward Swiatek for his installing of the CPEC system in the Campbell Scientific instrument test field.

This research has been supported by the Bureau of Development and Planning, Chinese Academy of Sciences (grant no. XDA19030204); Research and Development, Campbell Scientific Inc. (project no. 14433); the Bureau of International Co-operation, Chinese Academy of Sciences (grant no. 2020VBA0007); Chinese Academy of Sciences President's International Fellowship Initiative, Chinese Academy of Sciences (grant no. 2020VBA0007); and Long-Term Agroecosystem Research, USDA (award no. 58-3042-9-014).

This paper was edited by Keding Lu and reviewed by two anonymous referees.

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- Abstract
- Introduction
- Background
- Theory
- Accuracy
- Materials and methods
- Results
- Discussion
- Concluding remarks
- Appendix A: Derivation of Eq. (4)
- Appendix B: Derivation of Eq. (5)
- Appendix C: Water vapor mixing ratio and sonic temperature from relative humidity, air temperature, and atmospheric pressure
- Code availability
- Data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References

- Abstract
- Introduction
- Background
- Theory
- Accuracy
- Materials and methods
- Results
- Discussion
- Concluding remarks
- Appendix A: Derivation of Eq. (4)
- Appendix B: Derivation of Eq. (5)
- Appendix C: Water vapor mixing ratio and sonic temperature from relative humidity, air temperature, and atmospheric pressure
- Code availability
- Data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References