the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# New Absolute Cavity Pyrgeometer equation by application of Kirchhoff's law and adding a convection term

### Bruce W. Forgan

### Julian Gröbner

### Ibrahim Reda

An equation for the Absolute Cavity Pyrgeometer (ACP) is
derived from application of Kirchhoff's law and the addition of a convection
term to account for the thermopile being open to the environment, unlike a domed radiometer. The equation is then used to investigate four methods to
characterise key instrumental parameters using laboratory and field
measurements. The first uses solar irradiance to estimate the thermopile
responsivity, the second uses a minimisation method that solves for the thermopile responsivity and transmission of the cavity, and the third and
fourth revisit the Reda et al. (2012) linear least squares calibration
technique. Data were collected between January and November 2020, when the ACP96 and two IRIS radiometers monitoring terrestrial irradiances were
available. The results indicate good agreement with IRIS irradiances using
the new equation. The analysis also indicates that while the thermopile
responsivity, concentrator transmission and emissivity of an ACP can be
determined independently, as an open instrument, the impact of the
convection term is minor in steady-state conditions but significant when the base of the instrument is being subjected to rapid artificial cooling or
heating. Using laboratory characterisation of the transmission and
emissivity, together with use of an estimated solar calibration of the thermopile, generated mean differences of less than 1.5 Wm^{−2} to the two IRIS radiometers. A minimisation method using each IRIS radiometer as the reference also provided similar results, and the derived thermopile
responsivity was within 0.3 µV W^{−1} m^{2} of the solar-calibration-derived infrared responsivity estimate of 10.5 µV W^{−1} m^{2} estimated
using a nominal solar calibration and provide irradiances within ±2 %
of the terrestrial irradiance measured by the reference pyrgeometers
traceable to the International System of Units (SI). The calibration method using linear least squares regression introduced by Reda et al. (2012) that relies on rapid cooling of
the ACP base but utilising the new equation was found to produce consistent results but was dependent on the assumed temperature of the air above the thermopile. This study demonstrates the potential of the ACP as another
independent reference radiometer for terrestrial irradiance once the
magnitude of the convection coefficient and any potential variations in it have been resolved.

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Reda et al. (2012) introduced the Absolute Cavity Pyrgeometer (ACP), its operational equations and its characterisation process. The ACP is an Eppley Laboratory Precision Infrared Radiometer (PIR) with its dome replaced by a symmetrical cavity (called the concentrator) internally coated by polished gold and a cooling and heating system attached to the base of the pyrgeometer to assist in cooling or heating the ACP body.

The Reda et al. (2012) derivation of the ACP equation uses a combination of radiative transfer but without consideration of reflected irradiance components and impacts of convection (Vignola et al., 2012). Blackbody calibration of an ACP has proven difficult, and Reda et al. (2012) proposed a method of characterisation and calibration that included laboratory methods to determine the transmission of the concentrator and the emissivity of polished gold, while the thermopile sensitivity is determined using a linear least squares regression (LSQ) technique in the field at night under stable incoming irradiance conditions. Calibrations provided by Reda et al. (2012) assumed that the responsivity and transmission of the ACP changes over hours and days with variations of the order of several percent. More recently, Gröbner (2021) showed that the selection of points used in the field calibration has a significant influence on the result.

The ACP's body uses an Eppley Laboratory F3 thermopile which is used for Eppley Laboratory pyranometers (PSPs) and pyrgeometers (PIRs). The stability of the F3 thermopile used for solar and infrared irradiance measurements in
domed instruments is well within a percent over several years. Therefore, it
was surprising to see the large variation in the ACP thermopile responsivity
(µV (W^{−1} m^{2})) reported by Reda et al. (2012).

This paper derives a new ACP equation that adheres to Kirchhoff's law of thermal radiation for radiative transfer in vacuum and in-air measurements with a thermopile not protected by a dome and therefore includes an energy transfer term due to convection. The similarity and key differences in the contributing terms of the new in-air and Reda et al. (2012) equations are also examined. Using the new equation, the impact on the laboratory characterisation, night-time calibration compared against two IRIS pyrgeometers, and an application of the linear LSQ methods are investigated.

An ACP without a concentrator is an Eppley Laboratory pyrgeometer without a dome that includes a thermistor to measure the temperature of the body. It has a flat thermopile receiver painted with Parsons Black. In a vacuum there is only radiative transfer between the source and the thermopile receiver, with no possibility of a convection component.

The ACP equation in this instance only involves Kirchhoff's law at the black surface of the thermopile receiver, namely

where *α*_{r} is the fraction absorbed by the receiver, which from Kirchhoff's law is equivalent to emissivity *ε*_{r}, and *ρ*_{r} is the fraction reflected from the receiver as there is no
transmission through the black receiver surface.

The net flux between the incoming and outgoing flux results in a temperature difference between the base and receiver of the thermopile generating a voltage that is proportional to the net flux. That is,

where K is the responsivity of the thermopile (Wm^{−2} µV^{−1}), V is the voltage and *F*↓ and *F*↑ are the downward and upward radiant fluxes. The downward flux is made up of a single component of the irradiance from the source *W*:

The upward flux has two components, the emission from the surface and the reflection of the incoming flux, that is,

where *ε*_{r} is the emissivity of the receiver, and *W*_{r} is
the blackbody irradiance from the receiver. *ρ*_{r} is equal to (1−*ε*_{r}) as there is no transmission through the receiver
surface. Solving the two simultaneous equations, the result is

which gives, for *W*,

where *K*_{r} is the responsivity of the thermopile receiver or ${K}_{\mathrm{r}}=K/{\mathit{\epsilon}}_{\mathrm{r}}$.

*W*_{r} is given by $\mathit{\sigma}{T}_{\mathrm{r}}^{\mathrm{4}}$, where *σ* is the Stefan–Boltzmann constant and *T*_{r} is the temperature at the top of the
receiver. As *T*_{r} cannot be measured directly at time *t*, it is approximated by

where *T*_{b} is the ACP body temperature and *S* is calculated based on known values of the Seebeck coefficient for the thermopile junctions. If *n* is the number of junctions and *ϖ* is the efficiency of the thermopile, then

For the Eppley Laboratory F3 thermopile used in an ACP, with 56
copper–constantan junctions, *S*_{0} is ∼40 µV K^{−1}, and Reda et al. (2012) suggested *ϖ*∼0.65 or 65 %
efficiency. (*T*_{r}−*T*_{b}) is dependent on the net incoming irradiance and
the thermal conductivity of the thermopile, while *S* is a property solely of
the thermopile and impacts directly the thermopile responsivity. *ϖ* may vary due to the manufacturing process. During operation of an ACP, the maximum expected (*T*_{r}−*T*_{b}) is about 0.7 K. Reda et al. (2012) proposed
*S* to be $\mathrm{7.044}\times {\mathrm{10}}^{-\mathrm{4}}$. For a *T*_{b}=273.15 K and steady-state conditions where $\mathrm{V}\sim -\mathrm{800}$ µV (corresponding to the net radiation exchange of the ACP with a cloud-free sky), if *S* is in error by 20 %, the impact on *W*_{r} is about 0.7 Wm^{−2} and increases
proportionally with V and *T*_{b}.

The concentrator is assumed to have symmetrical transmission, absorption and backscatter characteristics. That is,

where *τ* is the transmitted fraction of the incoming irradiance through
the concentrator, *α* is the fraction of the incoming irradiance
absorbed by the concentrator and *β* is the fraction of the incoming
irradiance reflected out of the concentrator. Being a symmetrical cavity,
each component's magnitude will remain the same if irradiance enters either
end of the concentrator.

The concentrator walls coated in gold have an emissivity (or absorptivity)
of *ε*_{c} that is a property of the surface and is independent of the incoming irradiance. The fraction of incoming irradiance absorbed by the
concentrator, *α*, is a consequence of *ε*_{c} and
the multiple reflection of incoming irradiance on the concentrator walls, and hence *α*≥*ε*_{c}.

For an ACP in a vacuum, the incoming flux *F*↓ at the receiver (at
one end of the symmetrical concentrator) has three components, the
transmitted incoming atmospheric irradiance *τ**W*, any emission from the
walls of the concentrator with a blackbody irradiance of *W*_{c},
*ε*_{c}*W*_{c}, and the back reflectance towards the receiver of
the flux from the receiver *β**F*↑, that is,

The outgoing flux from the receiver is made up of two components in Eq. (4); thus,

Solving the two simultaneous equations results in

As a result, the incoming irradiance transmitted by the concentrator is

and the required irradiance is

Note that Eq. (14) would be similar to the domed pyrgeometer equation by
Philipona et al. (1995) if the latter used *T*_{r} instead of the
thermopile base temperature, and the transmission and emission are those of a dome instead of an open cavity.

In air, as the concentrator is open to the atmosphere and convection effects are not minimised by a dome (Robinson, 1966; Kondratyev, 1969; Vignola et al., 2012), a convection term is required. The effective flux input to the receiver by convection is given by

where *γ* is the convection coefficient that is dependent on *T*_{air}, the temperature of the air at the surface of the receiver, water vapour
content, wind speed and air pressure (Vignola et al., 2012). The equivalent
version of Eq. (10) is

The outgoing flux from the receiver is made up of two components, identical to Eq. (4).

Solving the two simultaneous equations results in

and replacing ($\mathrm{1}-\mathit{\beta}(\mathrm{1}-{\mathit{\epsilon}}_{\mathrm{r}}))K/{\mathit{\epsilon}}_{\mathrm{r}}$ with
*K*_{1}, the atmospheric irradiance is

where *W*_{net} represents the non-voltage irradiance components, and

*C* is the effective responsivity of the thermopile receiver – µV (W^{−1} m^{2}). The only difference between Eqs. (14) and (18) is the convection term *F*_{conv}. In a domed radiometer, as the sensor surface and
air under the dome are at near equilibrium, the effects of convection are
minimised, and their inclusion in the flux balance of the thermopile is not used.

As there is no direct measure of the air temperature in the concentrator
near the receiver surface, Reda et al. (2012) averaged the output of six
temperature sensors embedded in the concentrator *T*_{c} to represent
*T*_{air}.

The emissivity of the polished gold-plated concentrator in APC95 was found by National Institute of Standards and Technology (NIST) measurements to be 0.0225. The transmission of the concentrator derived by Zeng et al. (2010) used Eq. (6) for measurements in a vacuum such that

with subscripts o and c representing ACP measurements with the concentrator
removed and with the concentrator in place. They also assumed the emissivity of the concentrator has no impact on the numerator, implying that the emissivity
of the concentrator was 0. *S*_{c} and *S*_{o} are the reference output signals of the irradiance source; the derived value was 0.92. Reda et al. (2012) indicated that the *K*_{1} value used by Zeng et al. (2010) was incorrect
and used a value of *K*_{1}∼0.080 µV Wm^{−2} µV^{−1} (or *C*∼12.5 µV (W ^{−1} m^{2})) from field calibrations to generate a *τ*∼0.993.

As these measurements were conducted in air and the concentrator emissivity is greater than 0, Eq. (18) applies, and hence a convection term and concentrator emission term should have been added to the concentrator emissivity term in the numerator and the convection term in both the numerator and denominator, namely

The laboratory set-up used by Zeng et al. (2010) included a 10 µm laser and its irradiance was higher than the irradiance from the base of the ACP, and hence positive signals resulted from the ACP thermopile and *T*_{r} would
have been higher than *T*_{c}. Setting *β*∼0,
*ε*_{c}=0.0225 and *γ*∼8.5 and assuming that the thermopile-to-air temperature difference was about +0.3 K in steady-state conditions resulted in a 1.5 % reduction in the transmission, giving
∼0.977 when compared to the values used in Reda et al. (2012).
The impact of a zero contribution from the convection term decreased the
derived transmission by less than 0.001.

Reda et al. (2012) utilised Eq. (19) and the results from Zeng et al. (2010)
to derive a value of *τ* for each measurement sequence after updating
*K*_{1} via a linear LSQ calibration run. As a result, *τ* was deemed a
function of *K*_{1} rather than a unique characteristic of the concentrator.

For the remainder of this paper, 0.977 will be used as the transmission of the concentrator.

Using the symbols above, the Reda et al. (2012) equation for incoming irradiance is

with the only additional term being the emissivity of the air in the cavity
*ε*_{cav}, which Reda et al. (2012) set to 1. Rearranging the terms, we have

The first three terms of Eqs. (18) and (22) are identical if the concentrator backscatter *β* is 0. The latter two terms are where significant differences to the new equation exist. The (*W*_{r}−*W*_{c}) term is a
difference between irradiances rather than a difference in temperatures in
Eq. (18). In steady-state conditions with the base of the ACP not subject to artificial cooling or heating, *W*_{r}≤*W*_{c} and $-\mathrm{0.6}\mathit{<}({T}_{\mathrm{r}}-{T}_{\mathrm{c}})\le \mathrm{0.0}$, there is a relatively simple relationship between
the irradiance difference and the temperature difference, namely

where $\mathrm{\Psi}\cong \mathrm{5}\pm \mathrm{2}$ depending on the usual range of
irradiance terms. The magnitude of *γ* from blackbody investigations using ACP96 is *γ*∼8.4 and 6.5, depending on the blackbody configuration, and is higher than Ψ. In essence, the
(*W*_{r}−*W*_{c}) is a lower-magnitude version of the convection term in the new equation. The last term in Eq. (22), namely $-{W}_{\mathrm{r}}{\mathit{\epsilon}}_{\mathrm{c}}/\mathit{\tau}$, adds a negative irradiance contribution due to the concentrator emissivity but sourced from the thermopile irradiance; this is
not consistent with Kirchhoff's law as it adds emission from the
concentrator walls other than due to the concentrator's temperature.

Hence the only differences between Eqs. (22) and (18) are that, for Eq. (22),

- a
the ${\mathit{\epsilon}}_{\mathrm{c}}/\mathit{\tau}$ terms have approximately double the contribution to the derived atmospheric irradiance and

- b
the (

*W*_{r}−*W*_{c}) term could be slightly less in magnitude compared to*γ*(*T*_{r}−*T*_{c}).

The doubling of the ${\mathit{\epsilon}}_{\mathrm{c}}/\mathit{\tau}$ contribution in Eq. (22)
impacts directly any derivation of *K*_{1} as it increases the negative contributions from both the concentrator and the thermopile irradiance emission.
That is, given V is normally negative and as the concentrator emissivity
*ε*_{c} is a constant, the Reda et al. (2012)-derived *K*_{1} will be smaller (and hence *C* is larger) compared to Eq. (18)
derivations by about 8 %.

As the ACP was developed to be an absolute radiometer that did not require
calibration through comparison to another pyrgeometer or blackbody source, Reda et al. (2012) developed an innovative calibration method using linear
LSQ that relies on periods of constant *W*_{atm} together with rapid changes
in the thermopile base temperature. The base and concentrator temperature
provides irradiance traceability to the International System of Units (SI). As the calibration process rapidly
and continuously drops the base temperature of the ACP, the changes in signals and component irradiances are used to generate a linear LSQ
regression solution. Two parameters are derived from the linear LSQ
calibration process, *<**K*_{1}*>* and *<**τ**W*_{atm}*>*. For Reda et al. (2012), *ε*_{c} and *ε*_{cav} coefficients in Eq. (22) are based on laboratory
measurements or assumptions from the literature.

To provide data for the Reda et al. (2012) linear LSQ process, the ACP body is rapidly cooled over a set period. The rapid change in base temperature is
required to minimise the risk that *W*_{atm} changes significantly over the cooling period. The measurements during the rapid heating after a cooling
process are not used.

Using ACP96 data, Gröbner (2021) examined the linear LSQ process using the equation from Reda et al. (2012) and developed procedures to remove the
influence of the initial and final transient values, only using those data where a continuous cooling process is evident. The Gröbner (2021)
processing generated *<**K*_{1}*>* approximately 6 % less than that of the Reda et al. (2012) implementation.

Blackbody methods have been used successfully for decades to calibrate domed pyrgeometers and to solve for Eq. (14) equivalents that have shown high levels of stability over several years (Gröbner and Wacker, 2012). In
blackbody calibrations of a pyrgeometer, the base and dome temperatures of the pyrgeometer and the blackbody output irradiance are changed independently and allowed to stabilise at set values. The data from this process allow a multivariant solution by LSQ optimisation methods. However,
the final determination of *K*_{1} is typically by using non-thermopile
coefficients derived from the blackbody calibration together with a reference irradiance during night-time measurements (Gröbner and Walker,
2012).

Equation (18) assumes *T*_{air}∼*T*_{c}, and this maybe is the reason standard blackbody methodologies (Gröbner and Walker, 2012) have not been successful for calibration of an ACP to date. The differences
between a typical pyrgeometer and ACP are the replacement of a dome with the
open concentrator and the careful matching of the thermistors, with the latter an improvement on normal pyrgeometer thermometry. The blackbody calibration process used for pyrgeometers requires a fixed number of
temperature and blackbody stable temperature points that approximate atmospheric irradiances. Using a standard blackbody pyrgeometer calibration sequence, the ACP thermopile and concentrator cavity are exposed to the air
in the black body, and the black body is cooled to several temperature
points well below the ACP body temperature. As a result, *T*_{c}*<**<**T*_{b}, and it is highly likely that *T*_{air}*<**<**T*_{c}.

## 7.1 The impact of uncertainty in concentrator, thermopile and convection coefficients on *W*_{atm}

Using the new equation, the concentrator properties required are the
concentrator transmission *τ*, its emissivity *ε*_{c}, the concentrator backscatter *β* and the convection term *γ*.

A value for the thermopile emissivity *ε*_{r} is not required
as it is a constant and it is incorporated into *K*_{1} (and *C*). For Parsons Black at terrestrial irradiance wavelength, *ε*_{r} is
∼0.92 and at solar wavelengths ∼0.98.
*ε*_{r} only becomes relevant if *C* is determined at solar
wavelengths (*C*_{solar}) and then converted to a terrestrial irradiance value, as we will see below.

For ACP95, the concentrator emissivity was measured by NIST (Reda et al., 2012) and was found to be 0.0225, which is within 0.0015 of other known values for the emissivity (and hence absorptivity) of polished gold.

The impact of the irradiance backscatter fraction *β* and the receiver emissivity *ε*_{r} is minimal. Using the Zeng et al. (2010)
transmission measurements and the new equation suggests that, for a concentrator transmission *τ* greater than 0.9, and hence ($\mathrm{1}-\mathit{\tau})\ge \mathit{\beta}$ and *ε*_{r}*>*0.9, ($\mathrm{1}-\mathit{\beta}(\mathrm{1}-{\mathit{\epsilon}}_{\mathrm{r}})$) ≥0.99 is essentially constant. Hence, uncertainties in *β* and *ε*_{r} have little impact when
incorporated into *K*_{1}.

The greatest potential impact due to concentrator transmission *τ* and
backscatter *β* is on the *W*_{r} term, where $\mathrm{1.1}\ge (\mathrm{1}-\mathit{\beta})/\mathit{\tau}\mathit{>}\mathrm{1.0}$ when the fraction of incoming irradiance absorbed
by the concentrator *α*_{c} is greater than 0. If there is no absorption of the incoming irradiance by the concentrator (i.e. *α*_{c}=0), then (1−*β*) is equal to *τ*. If *α*_{c}*>**ε*_{c} given the NIST measurements and the
Eq. (20) derivation of *τ*∼0.977 and *ε*_{c}∼0.0225, this necessitates *β*≤0.005 if *α*_{c}=*ε*_{c}. As *β*→0, any error in
concentrator transmission will dominate the error contribution to
*W*_{atm}.

The convection coefficient *γ* is problematic for several reasons. Firstly, at present it needs to be derived assuming the other coefficients
or by approximation. Secondly, it is dependent on the air flow, air temperature, relative humidity and air pressure at the surface of the thermopile receiver. The empirical evidence from blackbody and atmospheric
measurements suggests 6*<**γ**<*10.

The receiver temperature *T*_{r} and hence blackbody irradiance *W*_{r} are
dependent on the estimate of the Seebeck coefficient and the construction of
the thermopile and the measurement of the base temperature *T*_{b}. As the
thermistors in an ACP have been characterised, Reda et al. (2012) estimated
that the standard uncertainty in *W*_{r} and *W*_{c} at about 0.1 Wm^{−2} and the standard uncertainty in the estimation of the Seebeck coefficient for the thermopile provide an additional 0.1 Wm^{−2}
uncertainty contribution to *W*_{r}.

*T*_{r} is calculated using Eq. (7) on the assumption that the
efficiency of the thermopile is as stated in Reda et al. (2012) and that *T*_{b} is equivalent to the thermopile base temperature. *K* is also dependent on the
Seebeck coefficient of the copper and constantan, the efficiency of the thermopile, the emissivity of the receiver surface *ε*_{r} and
the conductivity of the thermopile. Incorrect assignment of the true Seebeck
coefficient *S* in Eq. (7) will impact the two terms in Eqs. (18) and (22). However, *S* has not been derived for an individual ACP, so the Reda et al. (2012) value will be assumed for this paper.

For solar wavelengths the emissivity of Parsons Black changes as the paint discolours over time due to solarisation but has little if any impact on the IR emissivity.

Based on the above assumptions, the components of uncertainty of a single measurement of *W*_{atm} using Eq. (18) are provided in Table 1. The dominant
uncertainty components are *K*_{1} and *ε*_{c} for the
calculation of *τ**W*_{atm}, and the standard uncertainties for both *τ**W*_{atm} and *τ* make similar contributions to the uncertainty of *W*_{atm}.

Four calibration methods will be examined below for ACP96 based at PMOD/WRC in Davos, Switzerland, using either characterisation data or comparison measurements with other reference IRIS pyrgeometers and implementing two versions of linear LSQ.

Based on the likely magnitudes of the uncertainties, assumed values for some parameters are used in all the calibration methods investigated below. The
value of the cavity emissivity is fixed at 0.0225 using the NIST-derived value in Reda et al. (2012) for ACP95. The value for the backscatter from the
concentrator *β* will be assumed to be 0.

Using blackbody investigations, two values of the convection coefficient have been estimated, 8.4 and 6.5. The latter value, 6.5, derived from early blackbody investigations and field measurements, produces convection-based
irradiances closer to the equivalent “air cavity” irradiance values used by
Reda et al. (2012). The logic behind this adjustment is not solely due to the
convection coefficient being different, but rather the approximation
*T*_{c}∼*T*_{air}.

This leaves the sensitivity of the thermopile *K*_{1} and the concentrator
transmission *τ* to be either assumed or provided by a characterisation
methodology.

For the work reported below, the transmission of the concentrator *τ* is assumed to be only due to the construction of the concentrator and is
independent of *K*_{1}. The values of *τ* for ACP95 derived by the
reanalysis of the Zeng et al. (2010) data set but using Eq. (20) will be used when not derived as part of a calibration process.

The fraction of incoming terrestrial irradiance absorbed by the concentrator
*α*_{c} is not independent of the concentrator *ε*_{c}.
Concentrator transmission is a function of the cosine response of the
concentrator, and ray tracing suggests that for most sky zenith angles there will be multiple reflections on its surface, and then *α*_{c}*>**ε*_{c}. For *ε*_{c}=0.0225 derived
by NIST as reported by Reda et al. (2012), the implication is that *τ**<*0.9775.

For the results below when the method requires a fixed value of concentrator
transmission, *τ* is set to 0.977 and a fixed estimate of *ε*_{c}=0.0225 and two values of the convection coefficient *γ*
(8.4 and 6.5).

## 8.1 Data sets

During 2020 there were 242 d of ACP96 data collected at PMOD/WRC, and sometimes coincidentally with days, IRIS4 and or IRIS2 data were collected. Night-time data were available from ACP96 and IRIS2 between 7 January 2020 and 10 December 2020 and between 15 March and 10 December for IRIS4. The data consisted of an average value every 60 s for any IRIS irradiance and a 1 s measurement sequence every 10 s for ACP96. Simultaneous measurements were available in 2020, with 41 d of IRIS2 data and 36 d of IRIS4 that could be compared to ACP96.

Figure 1 shows a box–whisker representation of the differences between the simultaneous measurements of atmospheric terrestrial irradiances
*W*_{IRIS2} and *W*_{IRIS4}. The typical daily range in differences is 1.5 Wm^{−2}, which is within the individual instrument expanded uncertainty (*k*=2) of 2 Wm^{−2} (Gröbner, 2021). Slightly larger differences,
with IRIS2 lower than IRIS4, are observed on two days in August (days of the year 233 and 234), which are still within the combined uncertainties of the two radiometers. There appears to be a trend in the daily mean differences until
day 260 and then a restoration of the early 2020 mean daily differences
after day 300.

While there appears to be a drift between the two data sets, it was decided to use both data sets as a reference or comparison data set. These IRIS data tested the impact of using different reference irradiances and were used to corroborate the results of the methods described below.

## 8.2 Deriving *K*_{1} or *C* from an estimated solar calibration of the thermopile

For this method, either prior to an ACP being assembled or by removing the ACP's concentrator, the concentrator would be replaced with a pyrheliometer
aperture system that conforms to pyrheliometer requirements, with the
closest aperture to the receiver surface being identical to the aperture of
the concentrator. The ACP would be pointed at the Sun and compared to a well-calibrated WRR (or SI) pyrheliometer to produce an estimate of the
thermopile responsivity to solar irradiance. That estimate would then be
converted to the infrared responsivity by assuming the emissivity of the
receiver surface for both solar (*ε*_{rsolar}) and infrared
emission (*ε*_{r}).

Unfortunately, no solar calibration exists for the thermopile of ACP96, so an estimate had to be made, and we will assume that the ACP thermopile responsivity
for solar irradiance, *C*_{solar}, would likely be that of a new F3
thermopile and use the calibrations of new PSP pyranometers that were used to estimate a likely solar calibration for an F3 using an ACP. The data from over
82 individual PSP calibrations sourced from Eppley Laboratory and multiple
national calibration centres in the USA, Canada and Australia indicated that
the mode and mean solar sensitivities of new PSPs manufactured after 2000
were ∼9.3 µV (W^{−1} m^{2}).

An estimate for *C* is the effective responsivity of the thermopile receiver – µV (W^{−1} m^{2}):

As Parsons Black is used to coat the receiver surface, with a typical
receiver solar emissivity *ε*_{rsolar}∼0.98 and
for infrared *ε*_{r}∼0.92, and a PSP has a
double dome with both domes having a nominal transmission at solar
wavelengths of *τ*_{dome}∼0.91, this then gives an estimate of *C*∼10.5 µV (W^{−1} m^{2}).

Using Eq. (18), the atmospheric irradiance *W*_{APC96} was calculated when both IRIS4 and IRIS2 were operating and ACP96 was monitoring in steady-state night-time conditions. This resulted in comparisons over 41 nights
(18 802 measurements) with IRIS2 and 33 nights (14 085 measurements) with
IRIS4. The results are presented in Table 2 using *C*=10.5, *γ*=8.4,
*τ*=0.977 and *ε*_{s}=0.0225; the daily mean
differences (*W*_{IRIS2}−*W*_{ACP96}) and (*W*_{IRIS2}−*W*_{ACP96}) for each
of the days are shown in Fig. 2. Similar statistics are presented in Table 2 and Fig. 3 using *γ*=6.5.

The differences to *W*_{IRIS2} were larger than for *W*_{IRIS4}, and
there appears to be a similar trend in the relationship between IRIS2 and
ACP96, as seen in the comparison between *W*_{IRIS2} and *W*_{IRIS4}. The differences between Table 2 and Table 3 show that the impact of a 22 % change
in *γ* for steady-state conditions is a 0.6 Wm^{−2} APC96 irradiance difference for Δ*γ*=1. Decreasing *γ* by −1.9 shifted all the mean values down by ∼1.2 Wm^{−2} but
increased the range of the *W*_{IRIS2}−*W*_{ACP96}, while the *W*_{IRIS4}−*W*_{ACP96} showed little change.

## 8.3 Outdoor calibration using a reference irradiance

This method also assumes fixed values for the concentrator emissivity
*ε*_{c} and convection coefficient *γ* and finds the minimum difference between the reference irradiance *W*_{IRIS2} or
*W*_{IRIS4} and *W*_{ACP96} using paired values of *K*_{1} and concentrator transmission *τ*. That is, for a set of *n* observations made up of *m* nights, ideally with ranges in *W*_{IRIS} and *W*_{ACP96}, the pair [*C*, *τ*]
is found that provides a mean difference of (*W*_{IRIS}−*W*_{ACP96}) of less
than 0.1 Wm^{−2}. Given the low irradiance impact of concentrator
emissivity and the convection coefficient in steady-state conditions, the convergence to a solution is straightforward.

In the *γ*=8.4 set, the (*W*_{IRIS}−*W*_{ACP96}) statistics for simultaneous measurements with IRIS2 and IRIS4 observations are presented in
Table 4. There are differences of 0.4 (or ∼4 %) between the
*C* values and 0.011 (∼1.2 %) between the resultant
concentrator transmission values. The table also presents the results of
using the average of the two *C* and transmission values derived from IRIS2 and
IRIS4, giving *C*=10.5 and *τ*=0.9764 and deriving the difference statistics to both IRIS2 and IRIS4.

If the three *C* values in Table 4 are converted to equivalent PSP F3 thermopile
solar *C*_{solar} values, this results in values centred on 9.35±0.3.

The process was repeated but using a convection coefficient *γ* of 6.5,
with the results presented in Table 5. The standard deviations and range of differences increase slightly when compared to the values derived using
8.4 for the convection coefficient. The resultant *C* values were reduced by
0.2, while the transmission values are reduced by ∼0.0013.

The results in Tables 4 and 5 indicate that a negative 22 % change in the
convection coefficient reduces *C* by 2 % and increases the transmission by
0.1 % to achieve mean irradiance differences of less than 0.1 Wm^{−2}. These changes are self-consistent given the high correlation between the
components of the ACP equations, either of Reda et al. (2012) or the new equation, and show a 2 Wm^{−2} impact with a change in the convection
component of 1.9. However, for the averaged values of *C* and transmission, the lower convection coefficient provided the averages closest to 0 for both reference irradiances. The transmissions in Tables 4 and 5 from using the mean
of the IRIS2 and IRIS4 results are within 0.002 of the 0.977 value derived
for ACP95 using the new equation and NIST laboratory measurements (Zeng et
al., 2010).

The small differences (*W*_{IRIS2}−*W*_{IRIS4}) for 2 d in August and the
high correlation between components in the new equation demonstrate that uncertainty in the reference irradiance impacts the minimisation method and shows the benefit of having multiple reference irradiances to assess
confidence intervals.

The increase in *C* with an increase in transmission and the magnitude of these
changes is a consequence of the difference in the measured *W*_{IRIS2} and
*W*_{IRIS4}. The two dominant components of *W*_{atm} using the new equation are the thermopile voltage and the thermopile blackbody irradiance
*W*_{r}; the contributions from *W*_{c} and (*T*_{r}−*T*_{c}) are less than
4 %. The magnitude of the irradiance derived from the thermopile signal is
of the order of −80 Wm^{−2}, while *W*_{r} is typically between 300 and 500 Wm^{−2}. Hence, if the minimisation method is to achieve a balance between *K*_{1} and transmission, for a 1 Wm^{−2} change in reference irradiance,
then *K*_{1} changes by the higher percentage as the *W*_{r} is unchanged. If
only *K*_{1} was minimised instead of a (*K*_{1}, *τ*) pair, then a Δ Wm^{−2} difference in *W*_{atm} would result in *K*_{1} changing
by $\mathrm{\Delta}/{W}_{\mathrm{r}}$. Further complications arise if the relationship between
the true *W*_{atm} and *W*_{ref} changes.

## 8.4 Adaption of the Reda et al. (2012) linear LSQ calibration method to the new equation

From Eq. (18), and assuming that the fraction of backscatter of incoming irradiance *β* is 0, we can define the predictand for the linear LSQ
analysis as

with the thermopile voltage V the predictor for the linear LSQ analysis, and hence the equation to solve by linear LSQ is

which results in $\mathit{<}C\mathit{>}=\mathrm{1}/\mathit{<}{K}_{\mathrm{1}}$ and is independent of concentrator transmission. From *<**τ**W*_{atm}*>*, assuming a value for the concentrator transmission results in values for *<**W*_{atm}*>* which could be compared
to a reference irradiance. The inverse would be to prescribe a reference
irradiance and derive a concentrator transmission.

For the linear LSQ process to be successful, *W*_{atm} and *γ* must be
constant during the data collection process and the ACP equation must be
valid. In stable *W*_{atm} conditions, the process for collecting the
required rapid cooling periods results in only small changes in *T*_{c} and
*W*_{c}. As a result, the changes in the concentrator irradiance component *ε*_{c}*W*_{c} are less than 0.1 Wm^{−2} over the entire rapid
cooling process and hence have a minimal impact on *<**K*_{1}*>*.

Given the properties of linear LSQ, using a single predictor, V(*t*), if the
predictand is made up of multiple linear components, one can solve for each component of the predictands independently. The three predictand components from Eq. (18) are

Similarly,

and lastly,

d*T*(*t*) can also be split into three separate components, but that will be left
to the discussion section of this paper on the impact of incorrect estimates
of the Seebeck coefficient and assuming *T*_{c}(*t*) is equivalent to
*T*_{air}(*t*).

Derived *<**K*_{1}*>* and *<**τ**W*_{atm}*>* using the new equation are given by

and

Given that *W*_{c} is almost constant through the ∼7 min cooling of the thermopile and $\left|\mathit{<}{A}_{\mathrm{c}}\mathit{>}\right|\mathit{<}\mathrm{0.005}$, then $\left|{\mathit{\epsilon}}_{\mathrm{c}}\mathit{<}{A}_{\mathrm{c}}\mathit{>}\right|\mathit{<}\mathrm{0.00015}$ and contributes less than
0.1 % to *<**K*_{1}*>*, and hence the concentrator emissivity has minimal impact on deriving *<**K*_{1}*>* using the new
equation. For the intercept terms, *ε*_{c}*<**B*_{c}*>* typically makes a small negative contribution to
*<**τ**W*_{atm}*>* of the order of 2.5 %. *W*_{r} and
(*T*_{r}−*T*_{c}) dominate contributions to both *<**K*_{1}*>* and *<**τ**W*_{atm}*>*.

The concentrator transmission is irrelevant to deriving *<**τ**W*_{atm}*>* or *<**K*_{1}*>* but is essential for
estimating *<**W*_{atm}*>* from *<**τ**W*_{atm}*>*. If *W*_{atm} is known through a reference radiometer
(*W*_{ref}), then the concentrator transmission can be estimated by

For any linear LSQ process there is a key requirement that the process is
linear, and for Eq. (18), *τ**W*_{atm} must be constant. As a result, initial criteria for acceptable conditions were established for a valid
linear LSQ analysis period.

When the base of the ACP is cooled rapidly, the thermopile signal must
continuously become less negative. As the thermopile voltage was measured
every 10 s, a valid time was defined when the following criteria were
met. (a) The difference in consecutive thermopile voltages was more than +3.5 µV. (b) The difference in consecutive
(*T*_{r}(*t*)−*T*_{c}(*t*)) was less than −0.04 K. (c) The total range of the voltage was greater than 200 µV. (d) (${T}_{\mathrm{r}}-{T}_{\mathrm{c}}\left)\right({t}_{i})-({T}_{\mathrm{r}}-{T}_{\mathrm{c}}\left)\right({t}_{i-\mathrm{1}})\mathit{<}\mathrm{0.02}$. These ensured that the cooling was not nearing the new base temperature or that cooling had
stopped.

Out of 266 possible periods during 2020 for ACP96, 244 linear LSQ
calibration periods satisfied the criteria. Figures 4 to 5 show the time series of the individual slopes *<**A*_{c}*>*,
*<**A*_{r}*>* and *<**A*_{dT}*>* and intercepts (*<**B*_{r}*>*, *<**B*_{c}*>* and
*<**B*_{dT}*>*) derived from the valid linear LSQ analyses.
*<**A*_{c}*>* and *<**A*_{r}*>* are stable about a mean value but not the slopes for (*T*_{r}−*T*_{c}) and *<**A*_{dT}*>*; meteorological data for these periods indicate that the dew point temperature was less than 4 K below the ambient temperature and
that thermopile surface temperatures during cooling were close to or less than the dew point. While *<**B*_{dT}*>* is relatively constant
over the year, as expected, *<**B*_{r}*>* and *<**B*_{c}*>* follow the irradiance of the ambient temperature peaking
in summer periods.

The thermopile responsivities *<**C**>* found for 244 linear LSQ calibration periods are shown in Fig. 6. Between days 210 and 260
there is a significant increase in the range of *<**C**>* compared to the rest of the year.

There were 115 periods that were coincident with IRIS2 measurements when the
standard deviation of *W*_{IRIS2} in a cooling sequence was less than 0.4 Wm^{−2}, and 63 were coincident with IRIS4, also with a standard deviation of less than 0.4 Wm^{−2}. *<**C**>* statistics for the 244 linear
calibration periods and irradiance differences for the coincident periods
with *W*_{IRIS2} or *W*_{IRIS4} are presented in Tables 6 and 7 for *γ*=6.5 and *γ*=8.4 respectively.

The differences (*W*_{IRIS2}−*<**W*_{atm}*>*) and
(*W*_{IRIS4}−*<**W*_{atm}*>*) for coincident measurements
using a convection coefficient of 6.5 are shown in Fig. 7. The results
between days 200 and 254 for both *<**C**>* and *<**W*_{atm}*>* appear to be anomalous, with significantly higher values of
*<**C**>* and underestimates of the irradiance differences;
these are during periods when the steady-state base temperature is typically high for the year and within 4 K of the dew point temperature and high
relative humidity of 80 %. The means of pre day 200 and post day 300 are separated by about 2.2 Wm^{−2}. Given that *<**C**>* is likely constant over the two periods, possible reasons for the 2.2 Wm^{−2}
irradiances are that (i) both reference IRIS irradiances' calibrations may have changed by the same amount, (ii) the transmission of the concentrator may have decreased and (iii) the use of a constant convection coefficient over the
entire year is inappropriate.

The mean derived *<**C**>* value in Table 5 using 6.5 as the convection
coefficient is 10.49 µV (W^{−1} m^{2}), which is within 0.3 µV (W^{−1} m^{2}) of the solar and minimisation methods. Table 7 using the higher convection coefficient of 8.4 shows a mean *C* about 18 % lower and
irradiance differences greater than 11 Wm^{−2} between the ACP96 and IRIS2 and IRIS4.

No attempt was made to adjust the concentrator transmission *τ* based on
the derived *<**C**>* (or *K*_{1}), as it is a property of the concentrator, not the thermopile. However, it was possible to estimate *τ* using the
derived *<**τ**W*_{atm}*>* from the linear LSQ intercept, which is independent of any assumed value of *τ* by dividing *<**τ**W*_{atm}*>* by *W*_{IRIS4}. Similarly, the derived *<**τ**W*_{atm}*>* from the Reda et al. (2012) equation could also
produce an estimate of the concentrator transmission *τ*. Figure 8 shows
the results of dividing the *<**τ**W*_{atm}*>* derived
from both LSQ equations by IRIS4 data. Similar results were obtained using
*W*_{IRIS2.} The results using the new equation suggest a concentrator
transmission *τ*∼0.98, while for the Reda et al. (2012) equation a significant majority of periods gave unphysical values of *τ*
greater than 1.

## 8.5 Ensuring the representativeness of *τ**W*_{atm} during a linear LSQ calibration period

The thermopile voltage measurement is a consequence of net irradiance based
on the temperature difference between the base of the thermopile and the top of the thermopile. The blackbody equivalent irradiance of the thermopile receiver is calculated by assuming that the Seebeck coefficient is valid and that the body temperature represents the temperature at the base of the thermopile.
Provided the time constants of the thermopile and thermistors are similar
and the heating and cooling of the body are not too rapid, *C* and the convection coefficient should provide *τ**W*_{atm} for all measurements
(or *K*_{1} in Eq. 22) and ideally produce a near-constant value during both cooling and heating.

Using the data for ACP96 in 2020 and the calculated mean values of *C* given in Table 6, *τ**W*(*t*) was generated for each cooling and heating period.
*τ**W*_{atm}(*t*) was found to maintain some repeatable oscillations that could
not be minimised by changing either the convection coefficient or *C* for the new equation. For the Reda et al. (2012) equation, only *K*_{1} could be varied and resulted in decreases in calculated irradiances over the cooling and
heating period regardless of the *K*_{1} used, with little if any impact on deviations from a presumably constant *τ**W*_{atm}.

The sinusoid shape of the oscillation in the derived *τ**W*_{atm} using
Eq. (18) gave higher values during cooling and lower values during heating,
suggesting that there was a phase difference between the thermopile voltage and the body temperature or that some processes were unaccounted for using the new
equation. If a phase issue, the thermopile voltage at measurement period *p* was lagging the changing body temperature, and hence the temperature of the
body at time *t* was not representing the temperature of the thermopile base
at *t*. Such differences would be tiny in steady-state conditions given the slow rate of change in *T*_{b}.

Linear interpolation in time was used to find a more representative
thermopile voltage that reduced the sinusoidal oscillation in the derived
*τ**W*_{atm} and found that for ACP96 a lag time of about 9 s ± 2 s
was required to reduce the magnitudes of oscillations about the mean when using the new equation's *τ**W*_{atm}. It also reduced the magnitude of the
difference from the constant *τ**W*_{atm} using the Reda et al. (2012)
equation, but a distinct sinusoid always remained with peak deviations of 2 Wm^{−2} or more but 180^{∘} out of phase with the new equation values.

Given that measurements for all quantities were repeated every 10 s, the most representative thermopile voltage for measurement *p* every 10 s, ${\mathrm{V}}_{{p}^{\prime}}$, was

Using this interpolated voltage ${\mathrm{V}}_{{p}^{\prime}}$ to represent the thermopile voltage
at *p* resulted in significantly improved standard errors and confidence intervals for each of the linear LSQ-derived components of *<**K*_{1}*>* and *<**C**>* by factors of 3 to 10
depending on the linear LSQ component and provided statistics for the
variation of *τ**W*_{atm} throughout each cooling and heating period. The
improved linear LSQ fits did not impact significantly the derived *<**K*_{1}*>* or *<**C**>*, only raising
*<**C**>* by less than 0.02, with no significant difference to
the results presented in Sect. 8.4.

Using Eq. (33) to represent the thermopile signal for measurement *p* and setting a maximum standard deviation limit of *τ**W*_{atm} over the
cooling and heating period of 0.6 Wm^{−2} as acceptable when using the new
equation, the results for *<**C**>* derived by linear LSQ in
Sect. 8.4 were re-examined. Figure 10 shows the same *<**C**>* values as in Fig. 7 and those that satisfy the standard deviation of
the *τ**W*_{atm} criterion. Of the 244 original values, only 51 had a larger standard deviation in *τ**W*_{atm} over the cooling period. The main impact of this limit was removal of outliers. It had little impact on the
divergence of results between days 200 and 260 in 2020.

The phase shift showed that both the new and Reda et al. (2012) equations
could represent *τ**W*_{atm} through the cooling and heating with varying
degrees of success. A cumbersome visual method showed that varying the
convection coefficient constant for each cooling and heating cycle further reduces the sizes of the deviations from *τ**W*_{atm} and was not
independent of the estimate for *C*, but this is not the subject of the current paper. Most importantly, automation of the visual method may provide a method of judging whether *τ**W*_{atm} was nominally constant during a linear LSQ calibration period and thus remove the requirement of a reference
radiometer for that purpose.

The four different methods using the new ACP irradiance equation to
calibrate the ACP96 provided irradiances that compared well with the
irradiances from IRIS2 and IRIS4 during 2020. One based on laboratory or blackbody estimates for concentrator emissivity, transmission and the
convection coefficient provided an estimate of *C* based on the modal value
of 80 new F3 thermopile solar calibrations. Another used minimisation of the differences between the ACP and IRIS radiometers for pairs of *C* and
concentrator transmission. The third used the new equation with the linear
LSQ of Reda et al. (2012) but treated every contributor separately. The
fourth used the derived calibrations in the third method to estimate *τ**W*_{atm} from every measurement during a cooling and heating period and
thereby filter the results for stable periods without the need for a
separate pyrgeometer. All the methods produced mean differences from IRIS2 and IRIS4 of less than 1.2 Wm^{−2} and typically ranges of ±3 Wm^{−2} from
the mean difference for IRIS4. The differences in irradiances between IRIS2
and ACP96 were not symmetric about the mean, suggesting an identical trend
in the calibration of either both ACP96 and IRIS4 simultaneously or just IRIS2. As the year progressed, the daily mean differences between IRIS2 and ACP96 became increasingly negative until day 300, when irradiances recovered and equated to IRIS4 as during March and April 2020.

That the pseudo-solar calibration method produced a value very close to the other methods was fortuitous given that it was based on the modal value of
initial PSP calibrations based on 82 instruments. The range of potential
values matched the derived results and suggests that a solar calibration of
the ACP F3 thermopile is both a useful first step in characterising an ACP
thermopile as well as estimating the maximum potential ACP *C* calibration, and the method could be used periodically to check the stability of the
thermopile. An extended solar calibration over ambient temperature ranges
using the method of Pascoe and Forgan (1980) could also confirm the
temperature compensation of the thermopile. However, given the decadal
decrease in responsivity of the F3 thermopiles in PSP radiometers, exposure
of an ACP thermopile to solar exposure should be kept to a minimum to reduce
the impact of solarisation of the Parsons Black paint. Using the solar method as a primary calibration also negates the ACP as an absolute
irradiance reference standard and is based on historical estimates of the emissivity of Parsons Black in both the IR and solar wavelengths.
However, as most World Meteorological Organization regional instrument
centres have ready access to well-maintained reference pyrheliometers but do
not have laboratory facilities to characterise the concentrator, solar calibrations could be a useful verification and monitoring tool. At a
minimum, the solar calibration will provide a lower limit for *K*_{1} (and
hence an upper limit for *C*). That the theoretical value derived from the nominal solar calibration from an ensemble of new PSP F3 thermopiles gave
mean deviations of less than 1.5 Wm^{−2} for over 14 000 measurements with
a standard deviation of ∼1 Wm^{−2} supports this
recommendation.

The second method used an IR reference irradiance using IRIS pyrgeometers to
solve for both *C* and concentrator transmission simultaneously. The reference
pyrgeometers, both IRIS, are not influenced by calibration coefficients
dependent on the spectral transmission and emission of the IR of the domes. However, it was clear from the 2020 comparison data that any reference
radiometer must have an up-to-date calibration, with distinct steps and
trends in the derived relationship between the ACP and IRIS radiometers in
the comparison data. However, irradiance differences are all well within the
current WMO traceability requirement for terrestrial irradiances of 5 Wm^{−2}.

The concentrator transmission derived for ACP95 using the data from Zeng et
al. (2010) but the new equation and the NIST value of concentrator emissivity reported by Reda et al. (2012) were applied to ACP96 and produced
good agreement with the IRIS2 and IRIS4 measurements regardless of the
methods described above. This suggests that these parameters could be used
as a first approximation for any ACP. If an ACP is to be used without
reference to a black body or a reference radiometer, the concentrator emissivity should be obtained independently in the laboratory using the laboratory techniques reported by Reda et al. (2012), and the impact of a significant error in the emissivity for any irradiance calculation by the
new equation would be small. However, as the difference between the true
versus assumed concentrator transmissions will have a directly proportional effect on *W*_{atm}, an alternative method to obtain the concentrator
emissivity would be to repeat the Zeng et al. (2010) methodology for each ACP
using the new equation to generate a concentrator transmission and then
assume the emissivity is (1−*τ*).

By deriving *τ**W*_{atm} from each measurement in a cooling and heating
calibration period, the phase lag between the cooling of the base and the base of the thermopile became clear. The distance between where the base
temperature is measured and the base of the thermopile is about 10 mm, and
during the calibration periods the delay in the response of the thermopile base was found to be about ∼9 s for ACP96. Including that phase
lag in the linear LSQ methods improved the confidence intervals for each
linear LSQ analysis by factors of 3 to over 10 but had little impact on the
derived gradients and intercepts. However, it did improve the measurement
estimate of *τ**W*_{atm} from individual measurements and provided a
method to estimate the variance of *τ**W*_{atm} during a calibration
period without the need for a reference radiometer.

The comparisons between the IRIS and ACP irradiances in the results above
suggest that the ACP thermopile was stable over the year and produced
irradiance ratios to a reference within 2 % over 2020 and with a maximum difference of 5 Wm^{−2}. Reda et al. (2012) stated that the linear LSQ *<**K*_{1}*>* value from a single linear LSQ calibration
period be used as the valid sensitivity for the period between the end of
the heating period that generated the linear LSQ value until the next LSQ
calibration period, usually within 3 h. The results from Reda et al. (2012) and the results presented above for ACP96 suggest that during a single night of linear LSQ calibrations the derived *<**K*_{1}*>*
can vary by more than ±5 %, yet the typical F3 thermopile is found to be stable well within ±2 % over years for both solar and IR
measurements. In other radiometric linear LSQ calibration methods, mean or mode statistics of several linear LSQ calibrations are used to reduce
uncertainty in calibrations on the assumption that *<**C**>* is
a constant. The results above support using a *C* that represents a mean or
mode resulting from more than 20 calibration periods spread over several
nights.

## 9.1 Uncertainty in the Seebeck coefficient using linear LSQ

The equation from Reda et al. (2012) and the new equation are dependent on
the estimate of the Seebeck coefficient *S* in Eq. (7). A fixed value of
$\mathrm{7.044}\times {\mathrm{10}}^{-\mathrm{4}}$ was used in the analysis above. In steady-state conditions when measuring the incoming irradiance, the impact of any offset from the
true value is likely minor provided the other coefficients in the new equation have low uncertainties.

The Seebeck coefficient has a direct influence on both the *W*_{r} term and the (*T*_{r}−*T*_{c}) term of the new equation. For the (*T*_{r}−*T*_{c}) term, the
impact is straightforward given Eq. (7) in that, if the error in the Seebeck coefficient is Δ*S*, then the contributory error is Δ*γ* for *<**K*_{1}*>* and *<**τ**W*_{atm}*>*. The impact of any error in *S* is slightly more
complicated for *W*_{r}, but the ACP96 2020 data suggest similar impacts. This is shown in Fig. 10 plotting the difference in *<**K*_{1}*>* when ignoring *S* in the *<**A*_{r}*>* term.
The difference was calculated by subtracting the receiver slope assuming
*S*=0, that is, base irradiance slope *<**A*_{b}*>*, from the slope derived using *S*. The difference changes through the year inversely to
the magnitude of the base temperature, but on average it is $\sim -\mathrm{0.0033}$ or about −4 % of *<**K*_{1}*>*, which implies that a 25 % error in *S* has an impact of 1 % on *<**K*_{1}*>*.

For the Reda et al. (2012) equation, the impact of the Seebeck coefficient is nearly doubled, as the scaling factor is (2−*ε*_{c}) instead of 1 for the new equation.

## 9.2 Comparing *<**C**>* and *<**τ**W*_{atm}*>* using the Reda et al. (2012) equation and the new equation

Isolating the coefficients that impact the derived *<**K*_{1}*>* via linear LSQ also allows the calculation of the *<**K*_{1}*>* value based on the Reda et al. (2012) equation. Figure 11
shows the two components in Eq. (22) after applying the scaling factors to
generate *<**K*_{1}*>*; in essence, *W*_{r} dominates the calculation, with a small negative contribution from *W*_{c}.

The differences between the derived *<**C**>* for both the new
and Reda et al. (2012) equations by linear LSQ are shown in Fig. 12 and for
*<**W*_{atm}*>* in Fig. 13. The different types of
*<**C**>* are separated by about 2.5 µV (W^{−1} m^{2}), with the Reda et al. (2012) values being higher. The *<**W*_{atm}*>*
differences between IRIS4 and the Reda et al. (2012) equation were between ±12 Wm^{−2}, while the differences to the new equation are bounded by
+4 and −8 Wm^{−2}, about half the range of the Reda et al. (2012) equation results.

## 9.3 Uncertainties in concentrator emissivity and convection coefficient

Three coefficients related to the concentrator are required for Eq. (18) to derive ambient irradiances and use the LSQ method of calibration. Zeng et al. (2010) provided a laboratory method for determining the transmission and an estimate of its uncertainty, but laboratory determinations of the emissivity and convection coefficient have not occurred.

The *W*_{atm} uncertainty estimates in Table 1 indicate that the incorrect
assignment of the convection coefficient *γ* has a minor contribution
to the calculation of *W*_{atm} even if the coefficient's standard uncertainty is 25 % from the true value. However, the emissivity is the
second-largest contributor to uncertainty after the thermopile calibration coefficient in the determination of *W*_{atm}.

Table 8 provides an assessment of the uncertainty of the derived components of the linear LSQ method for the new equation. For this analysis, the uncertainties of the voltage signals are simply the estimate of the signal resolution, and the derived calibration constant incorporates any proportional uncertainty into the true voltage. The uncertainties of the receiver and concentrator irradiance are incorporated into LSQ slope and intercept statistics, as are the uncertainties of the differences between the receiver and the assumed air temperature.

In Table 1 the uncertainty of the convection coefficient was 1.5, but in
Table 8 it is 0.3. Even after reducing the uncertainty component for the
convection coefficient by a factor of 5 from that used in Table 1, this coefficient is the dominant contribution to the standard uncertainty of the derived *<**K*_{1}*>* and is close to the dominant
uncertainty contribution from the receiver irradiance *<**B*_{r}*>* for the estimate of *<**τ**W*_{atm}*>*.

The convection coefficient of air is dependent on the design of the air flow path and temperature, with a higher water vapour content also giving a higher coefficient. Empirical models of convection for the ACP are yet to be developed to determine the non-dimensional Nusslet parameter necessary for assigning and estimating the convection coefficient.

The new equation was developed by applying Kirchhoff's law of radiative transfer for radiative transfer in air. For the solar calibration method and the calibration using a reference irradiance, the ACP is essentially in steady state, while in the linear LSQ method the ACP is in a transient mode. Kirchhoff's law only applies in periods of radiative equilibrium, and this must be considered when modifying the cooling and heating cycle for linear LSQ calibrations.

## 9.4 Future work

While the investigations above demonstrate that the new equation can be used with an ACP for terrestrial irradiance measurements and give good agreement with pyrgeometers with traceability to SI, there are still uncertainties related to the new equation and the characteristics of ACPs to be suitable direct references to SI irradiances. For example, a significant issue is how the Reda et al. (2012) equation and the new Eq. (18) can produce valid terrestrial irradiances but utilise thermopile sensitivities that differ by 25 % or more.

While the uncertainty in the convection coefficient has little impact on calculating outdoor irradiances, it is the dominant uncertainty when using the linear LSQ method for calibration with the new equation. The following are future actions recommended to increase the confidence of ACPs in acting as a primary reference to SI: a method for determining the convection coefficient, determining theoretical approximations of the ACP convection coefficient by developing an appropriate dimensionless Nusslet coefficient for the thermal and air flow characteristics of an ACP, determining whether the ACP can be calibrated in a black body but with a different process to that used with domed pyrgeometers, higher-frequency measurements in cooling and heating cycles to conform to the time offset between the thermopile reacting to a temperature change in the ACP base temperature, investigating whether the heating part of the LSQ data collection process can be used for LSQ analyses, and performing solar calibrations of the thermopile to determine an ACP's maximum possible thermopile responsivity.

The new equation for an ACP derived from the application of Kirchhoff's law and inclusion of a convection term provided irradiances that agreed with measurements from two reference IR radiometers over 11 months in 2020 assuming either a solar-derived calibration or a minimisation method.

The linear LSQ method of Reda et al. (2012) was modified for use with a new equation and developed so that the impact of individual contributors to the linear LSQ process could be assessed. As the only LSQ predictor was the thermopile voltage, the method allowed determination of five linear components independently of the Reda et al. (2012) and new equations. This also provided an estimate of the relative contribution of each component to the calibration values and their uncertainty contribution.

The linear LSQ results indicated that the new equation irradiances were for
most cases consistent with the two reference radiometer irradiances but that consistency was dependent on the value of the convection coefficient. A
method of examining the convection coefficient independently of a reference irradiance was developed by solving for *τ**W*_{atm} during the cooling and heating periods and highlighted the ∼9 s time lag between the
representative voltage for the body and concentrator temperature
measurements. When the lag was incorporated into the linear LSQ method, the confidence intervals for all slope quantities improved significantly, and systematic variations in the derived irradiance during a heating and cooling period were reduced but not eliminated. However, a process to determine the
convection coefficient independently of outdoor or laboratory measurements has yet to be developed.

Via the linear LSQ method, an estimate for the concentrator transmission can also be obtained using a reference terrestrial irradiance. However, the preferred method should be laboratory measurements as performed by Zeng et al. (2010) but using the new equation rather than assuming that the measurements are performed in a vacuum.

A solar calibration of future ACP thermopiles is recommended provided the thermopile has not been subjected to solar irradiance for extended periods over several years. The solar calibration will produce an estimate of the thermopile responsivity that will be close to the maximum possible for the thermopile and thus provide either an independent estimate or a mechanism to assess the long-term stability of the ACP thermopile responsivity.

The dataset and example software code have been published open access in Zenodo: https://doi.org/10.5281/zenodo.7605287 (Forgan and Gröbner, 2023).

The first author derived the new equation, did the data analysis and drafted the paper. The second author collected the data and modified the data collection techniques to assist with the delineation of components of the new equation and contributed to the analysis and interpretation of the results. The last author is the inventor of the absolute cavity pyrgeometer and provided insights in the design criterion and the background to the original paper as well as examples of their analysis techniques using the original equation.

The contact author has declared that none of the authors has any competing interests.

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

This paper was edited by Saulius Nevas and reviewed by Laurent Vuilleumier, Stefan Wacker, and one anonymous referee.

Forgan, B. W. and Gröbner, J.: Data set and pseudo-code for publication https://doi.org/10.5194/amt-2022-250 (1.0), Zenodo [data set], https://doi.org/10.5281/zenodo.7605287, 2023.

Gröbner, J.: A transfer standard radiometer for atmospheric longwave irradiance measurements, Metrologia, 49, 105–111, 2012.

Gröbner, J.: Investigation of ACP96 at PMOD/WRC, Zenodo, https://doi.org/10.5281/zenodo.7047680, 2021.

Gröbner, J. and Wacker, S.: Pyrgeometer calibration procedure at the WRC/PMOD-IRC, WMO, Instrument and Observing Report 120, 13 pp., 2012.

Kondratyev, Y. A.: Radiation in the Atmosphere, Academic Press, International Physics Series, vol. 12, 912 pp., ISBN 13: 978-0124190504, 1969.

Pascoe, D. and Forgan, B. W.: An investigation of the Linke-Feussner pyrheliometer temperature coefficient, Sol. Energy, 25, 191–192, 1980.

Philipona, R., Frohlich, C., and Betz, C.: Characterization of pyrgeometers and the accuracy of atmospheric long-wave radiation measurements, Appl. Opt., 34, 1598–1605, 1995.

Reda, I., Zeng, J., Hanssen, L., Wilthan, B., Myers, D., and Stoffel, T.: An absolute cavity pyrgeometer to measure the absolute outdoor longwave irradiance with traceability to international system of units, SI, J. Atmos. Sol-Terr. Phy., 77, 132–143, 2012.

Robinson, G. D.: Solar Radiation, Elsevier, 269 pp., ISBN 13 978-0444404824, 1966.

Vignola, F., Michalsky, J., and Stoffel, T.: Solar and infrared radiation measurements, CRC Press, ISBN 978-1-4398-5189-0, 2012.

Zeng, J. Hanssen, L., Reda, I., and Scheuch, J.: Preliminary characterization study of a gold-plated concentrator for hemispherical longwave irradiance measurements, Paper no 77920Z, Proceedings of SPIE Conference, 3–5 August 2010, San Diego, vol. 7792, 2010.

- Abstract
- Background
- The steady-state equation for an ACP without a dome or concentrator in a vacuum
- The steady-state equation of an ACP with a symmetrical concentrator in a vacuum
- The steady-state equation of an ACP with a symmetrical concentrator in the atmosphere
- Examining the laboratory-determined coefficients
- Comparing the terms between the original and new ACP equations
- ACP calibration methods to date
- New calibration methods
- Discussion
- Conclusions
- Code and data availability
- Author contributions
- Competing interests
- Disclaimer
- Review statement
- References

- Abstract
- Background
- The steady-state equation for an ACP without a dome or concentrator in a vacuum
- The steady-state equation of an ACP with a symmetrical concentrator in a vacuum
- The steady-state equation of an ACP with a symmetrical concentrator in the atmosphere
- Examining the laboratory-determined coefficients
- Comparing the terms between the original and new ACP equations
- ACP calibration methods to date
- New calibration methods
- Discussion
- Conclusions
- Code and data availability
- Author contributions
- Competing interests
- Disclaimer
- Review statement
- References