New Absolute Cavity Radiometer equation by application of Kirchhoff’s law and adding a convection term
 ^{1}Docklands, Victoria, 3008, Australia
 ^{2}PMOD/WRC, Davos Dorf, 7260, Switzerland
 ^{3}NREL, Golden, Colorado, 804013393, USA
 ^{1}Docklands, Victoria, 3008, Australia
 ^{2}PMOD/WRC, Davos Dorf, 7260, Switzerland
 ^{3}NREL, Golden, Colorado, 804013393, USA
Abstract. 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 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. The minimization method using each IRIS radiometer as the reference also provided similar results, and the derived thermopile responsivity was within 0.3 μV/Wm^{2} of the solar calibration derived infrared responsivity estimate of 10.5 μV/Wm^{2} estimated using a nominal solar calibration and within +/2 % of the terrestrial irradiance measured by the reference pyrgeometers traceable to 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 analogue used for temperature of air above the thermopile. The result of this study demonstrates the potential of the ACP as another independent reference radiometer for terrestrial irradiance once the impact of convection on the ACP has been resolved.
Bruce W. Forgan et al.
Status: open (until 14 Dec 2022)

RC1: 'Comment on amt2022250', Anonymous Referee #1, 29 Nov 2022
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Comments to Editor:
Dear Editor,This manuscript provides a new valuable equation for the Absolute Cavity Pyrgeometer (ACP). The authors present theoretical derivation of the new equation and evaluate the caluculation results by the new equation and the exiting one very carefully. The authors also point out the problems to be solved in the equation in the future, and the proposed new equation is expected quite useful for calculation of the reference terrestrial irradiance from the ACP measurements.
I would recommend it for acceptance after the very minor points listed below.
line 1: The word in the title of "Radiometer" would be better to be changed as "Pyrgeometer".
line 28: Same as above.
line 72: The word "Trb(t)" in the Eq. (7) should be written as "Tb(t)".
line 80: "×" should be added in the description of S value as "7.044×104" and the unit for S should be added as "K/μV" after the value.
line 102: In the numerator of the first term on the right side of the Eq. (13), there is one missing parenthesis of ")" before "K".
line 104: Same as above in the Eq. (14).
line 116: Description of the equation number should have parenthesis as "Eq. (11)" the same as the other equations.
line 304: The ACP thermopile responsivity should be written as "Csolar" instead of "Cs" the same as in the other description.
line 317: "WIRIS2" in the second parenthesis should be written as "WIRIS4".
line 330: The order of parameters description should be the same as that in the Figure 2.
line 484: The word "cooling" is written twice, but either one of them is considered as "heating".
line 581: "×" should be added in the description of S value as "7.044×104".I hope that my comment is very useful for the improvement of the article.
Thank you.Best regards,
NozomuComments to Authors:
Dear Authors,I think this study provides a new valuable equation for the Absolute Cavity Pyrgeometer (ACP). The theoretical derivation of the new equation and evaluation of the caluculation results by the new equation and the exiting one are described very carefully. And the problems to be solved in the equation in the future are precisely pointed out, the proposed new equation is expected quite useful for calculation of the reference terrestrial irradiance from the ACP measurements.
I would recommend the very minor points below before its publication.
line 1: The word in the title of "Radiometer" would be better to be changed as "Pyrgeometer".
line 28: Same as above.
line 72: The word "Trb(t)" in the Eq. (7) should be written as "Tb(t)".
line 80: "×" should be added in the description of S value as "7.044×104" and the unit for S should be added as "K/μV" after the value.
line 102: In the numerator of the first term on the right side of the Eq. (13), there is one missing parenthesis of ")" before "K".
line 104: Same as above in the Eq. (14).
line 116: Description of the equation number should have parenthesis as "Eq. (11)" the same as the other equations.
line 304: The ACP thermopile responsivity should be written as "Csolar" instead of "Cs" the same as in the other description.
line 317: "WIRIS2" in the second parenthesis should be written as "WIRIS4".
line 330: The order of parameters description should be the same as that in the Figure 2.
line 484: The word "cooling" is written twice, but either one of them is considered as "heating".
line 581: "×" should be added in the description of S value as "7.044×104".I hope that my comment is very useful for the improvement of the article. Thank you for your attention.

CC1: 'Reply on RC1', Bruce Forgan, 29 Nov 2022
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Many thanks Nozomu for your comments.
I appreciate the detail in your comments and finding the typographical errors.
I have no issue with changing the title, that is changing 'radiometer' to 'pyrgeometer. That said, I deliberately chose 'radiometer' as a general description as I believe the ACP could also be used for other measurements not only as a pyrgeometer. However, as you implicitly point out using 'pyrgeometer' does provide the continuity to the Reda et. al. (2012), hence I support the suggested change.
The correction of the typographical errors you (and I in the meantime) have identified in the manuscript will be as you recommend.
Best regards, Bruce

CC1: 'Reply on RC1', Bruce Forgan, 29 Nov 2022
reply

RC2: 'Comment on amt2022250', Laurent Vuilleumier, 29 Nov 2022
reply
Overview and general recommendation
This manuscript describes the inference of equations depicting how the electric signal of the Absolute Cavity Pyrgeometer (ACP) can be related to the incoming irradiance at the entrance of the instrument considering the different fluxes within the instrument. The ACP was first described by Reda et al. (2012) and is considered as one of two types of instrument that can be used for providing the World primary reference for terrestrial radiation in the future. The description given by Reda et al. (2012) included assumptions that strongly restricted the conditions when the instrument can be used. In any case, the ACP having no dome, it can only be used during fair weather nights, which is already a restriction, but the further added restrictions by Reda et al. (2012) complicated its use. Thus, this manuscript is an important contribution to the description of an instrument that may participate in the definition of the next primary reference for terrestrial radiation and its scope is appropriate for publication in Atmospheric Measurement Techniques.
I find the manuscript of good quality and most of my comments concern minor points where I think that clarifications are needed or I find some formulation unclear where I usually suggest reformulation that I hope to be clearer. I have two more important comments, the first being that I believe all elements are present in the manuscript for computing a full uncertainty budget, which I recommend to add. The second is that the manuscript is overly long, and I recommend suppressing one part that I find can be omitted. I think the manuscript can be published after my main comments are addressed and the minor points are resolved.
I detail my comments below starting with my two most important comments followed by minor points. Some of the minor points still emphasize questions that need clarification, while some other are just corrections of typos.
Major Comments
 Because the ACP is considered as a potential instrument for a group providing the future primary reference for terrestrial radiation, it is important that its uncertainty budget is well defined. The manuscript provides improved equations for describing the physics of the ACP. With these, it should be possible to better determine the uncertainty budget than was possible for Reda et al. (2012), and the resulting uncertainty for the ACP may also be smaller than the one given by Reda et al. In addition, sections 7.1, 9.1 and 9.3 already contain most of the elements that should be considered for the uncertainty budget. I am aware that the authors recommend further work, especially concerning the influence of the convection term when it varies, for instance, when humidity is high, but I believe the authors can already give estimates. I strongly recommend including a full uncertainty budget computation just before the conclusion. I think this would not be too overwhelming an effort given how valuable it would be.
 While the manuscript is extremely thorough, it is also very long. It is valuable to compare several methods for determining the characteristics of the ACP, especially detailing how the new determinations differ from those of Reda et al. (2012). This partially explains the length of the manuscript. However, I do not find the part describing the calibration using solar irradiance very helpful, especially since generic values had to be used for the thermopile calibration because no solar calibration of an ACP was performed. I would recommend suppressing the parts of the manuscript referring to the calibration using solar irradiance and eventually let it for a further publication when such calibrations could actually be performed.
Minor Points
Note: assuming the authors used \LaTeX to format their manuscript, I used \LaTeX formatting for the minor comments allowing me to use the corresponding mathematical typesetting. I also include the PDF file (available as supplement) obtained by processing the \LaTeX document to facilitate reading the mathematical formulas.
\begin{enumerate}
\item Eq. 5 (line 67) should be $ KV = \varepsilon_r \left( W  W_r \right) $. The different equation in the manuscript is most likely a typo because the equation given here is consistent with both the previous and next equation in the manuscript.
\item Eq. 7 (line 72). The first term of the righthand side should be $ T_{b}(t) $, not $ T_{rb}(t) $.
\item Lines 9091: is the concentrator absorptance a consequence of the scattering on the walls because some radiation is again scattered toward the concentrator wall where it can again be absorbed? Please, clarify.
\item Line 92: Replace “at one end of the symmetrical concentrator” with “at the lower end of the symmetrical concentrator affixed to the body part of the ACP pyrgeometer", so that there is no possible confusion.
\item Lines 98100 and derivation from Eq. (11) to Eq. (12): Eq. (11) is the same as Eq. (4), and it can be suppressed while simply mentioning that Eq. (4) is also valid for the outgoing flux from the receiver when the concentrator is present. It may be more useful to replace Eq. 11 from the manuscript with an equation that results from solving Eq. (4) and Eq. (10):
\begin{equation} F_{\downarrow} = \frac{ \tau W + \varepsilon_c W_c + \beta \varepsilon_r W_r }{ 1  \beta ( 1\varepsilon_r ) } \end{equation}
Which then easily leads to Eq. (12) in the manuscript.
\item Line 112: Replace “… dependent on several factors such as water vapur content” with “… dependent on several factors such as water vapor content."
\item Line 130, citation of Jinan et al. (2010): I think Jinan is the given name and Zeng is the surname, which is a rather common Chinese surname. Thus, I believe this reference should be Zeng et al. (2010) and not Jinang et al., (2010). This also applies at lines 134, 139, 144, 215, 265, 548, 555556, 675, and at line 691 where the reference is given that should be “Zeng, J., Hanssen L., Reda I., Scheuch J.” and not “Jinan, Z., Hanssen L., Reda I., Scheuch J.”
\item Eqs. 19 and 20 (line 131138): The derivation by Zeng et al. (2010) does not use the same terminology as this manuscript. I believe it is assumed in the manuscript that Zeng et al. described the relationship between the thermopile signal and a reference irradiance signal $ S $ when the concentrator is not present as $ K_1 V_0 = S_0  W_{r0} $, while the relationship with the concentrator present was taken as $ K_1 V_c = \tau S_c  W_{rc} $. Dividing the latter by the former allows obtaining Eq. 19. If these assumptions were made, it would help the reader stating them in the manuscript. I also believe this manuscript includes corrections for convection and emission by the concentrator wall in these two relationships as $ K_1 V_0 = S_0  W_{r0} + \gamma ( T_{air0}  T_{r0} )$ (without concentrator) and $ K_1 V_c = \tau S_c  W_{rc} + \varepsilon_c W_{cc} + \gamma ( T_{airc}  T_{rc} ) $. Again, dividing the latter by the former allows obtaining Eq. 20. It would also help stating it in the manuscript.
\item Eqs. 19 and 20 (line 131138): I wonder if other subscripts than “0” and “c” could be used for the situation with and without concentrator. Subscript 'c' is also used for describing characteristics of the concentrator itself such as its emissivity, temperature, thermal emission, etc. Also using it to denote measurements taken with and without concentrator makes it confusing for the reader: in this case, $W_{rc}$ is the thermal emission from the receiver (not the concentrator), when the concentrator is in place.
\item Lines 183184: I am not sure what the authors meant with “Using Eq. (22) these results implied data selection generates...” The authors may have meant “Using Eq. (22), the data selection implied by these results generates $ \langle K_1 \rangle $ that are approximately 6\% less than the Reda et al. (2012) implementation.”
\item Line 185: Replace “… the difference between the results from Reda et al., 2012 and the use Eq. (18) could be…” with “… the difference between the results from Reda et al. (2012) and results using Eq. (18) could be…”
\item Line 190: Replace “the base temperature of the pyrgeometer, and its dome, and the blackbody output irradiance are changed…” with “the base and dome temperatures of the pyrgeometer, and the blackbody output irradiance are changed…”
\item Lines 206207: I am not sure how to understand the end of the sentence. With “$\alpha_c$ is not required but given Kirchhoff’s law indicates it would not be independent of $\varepsilon_c$” did the authors mean “Kirchhoff’s law allows relating $\alpha_c$ to $\varepsilon_c$, reducing the number of concentrator properties to determine”?
\item Line 211: “For ACP95 the concentrator emissivity … was found to be 0.0225…” is already mentioned at the beginning of section 5.
\item Line 225: A range for $ \gamma $ is given at the end of the paragraph. In addition, there is an estimation of the impact of $\gamma$ uncertainty on $ W_{atm} $ in the third paragraph of section 8 (starting at line 250). I would suggest to move this estimation here after line 225 (section 7.1 is about impact of uncertainties on $ W_{atm} $) and refer to it later in section 8.
\item Lines 247249: “but $\alpha_c$ the absorption fraction…” It is unclear to me if $\alpha_c$ is different from $\alpha$ in eq. 9. By analogy with $\epsilon_c$ (that I understand as the emissivity of the concentrator walls), $\alpha_c$ may be the absorptivity of the walls and not the concentrator as a whole. If they are the same, this sentence only expresses Kirchhoff’s law in words or is a rewording of the fact that the backscatter is assumed insignificant. Please clarify.
\item Lines 252253: Replace “convection coefficient $\gamma$ divided by concentrator transmission of…” with “convection coefficient divided by concentrator transmission ($ \gamma / \tau $) of…”
\item Lines 276278: Replace “… and for ACP96 a 1 s measurement sequence every 10 seconds” with “… and a 1 s measurement sequence every 10 seconds for ACP96.”
\item Line 282: Replace “… instrument expanded (k=2) of 2 Wm$^{2}$…” with “… instrument expanded uncertainty (k=2) of 2 Wm$^{2}$…”
\item Line 303: Replace “One we will assume…” with “We will assume…”
\item Line 314: “Using the new equation…” Do the authors mean “Using Eq. 18 with $\tau$ estimated with Eq. 20? Please be more specific.
\item Line 355: Replace “… the average differences were much smaller in magnitude” with “… the average differences were smaller in magnitude when using the data from both IRIS instruments to derive single $C$ and $\tau$ values.”
\item Line 371: “in the new equation…” Again, I think the authors are referring to Eq. 18. Please refer to it by its number.
\item Eq. 26 (line 383): To strictly follow Eq. 18, the term $ \langle K_1 \rangle V(t) $ should be preceded by a minus sign.
\item Lines 413421: It is not clear whether there are three or four conditions. One condition states that a cooling sequence does not start before $ \left(T_r  T_c \right) \left( t_i \right)  \left(T_r  T_c \right) \left( t_{i+1} \right) < 0.04$ or that it ends when this condition is not anymore fulfilled. On the other hand, there is also a condition that $ \left(T_r  T_c \right) \left( t_i \right)  \left(T_r  T_c \right) \left( t_{i+1} \right) < 0.02$. Does this mean that, within a cooling sequence, only measurements satisfying the latter condition are kept? Is there also a condition on the minimum number of valid point in a cooling sequence? Please clarify. I also think it will help the reader if the paragraph started with a general statement on the conditions then listed the conditions explicitly. In case the authors have three conditions (adapting to four conditions is straightforward), I suggest introducing the paragraph as follows. “Three conditions were used to select measurements sequences acceptable for the LSQ calibration, limiting the voltage difference between two successive measurements, the change in the temperature difference between the receiver and the concentrator between two successive measurements, and the total voltage increase over a full cooling sequence. Explicitly, the conditions were:” After this introduction, the authors can include a numbered list with the conditions explicitly detailed.
\item Lines 415416: Replace “… the voltage difference between two consecutive voltage was least than or equal to +3.5 μV” with “… the voltage difference between two consecutive voltage was less than or equal to +3.5 μV.”
\item Lines 425426: “… the slopes for $ ( T_r  T_c ) $, and $ \langle A_{dT} \rangle $, shows an upwards shift between days 200 and 255 that recovers when data collection recommenced on day 312.” It is difficult to see such an upwards shift on Figure 4. There are three groups of points between days 200 and 255. For these groups, the green points have extremely similar values, except maybe for the last day in the period where there is some dispersion, but it is not larger than the dispersion in the group of point between day 0 and 30. I am not sure this statement can be upheld and I would suppress it.
\item Line 433: Only a single short sentence mentions Figure 6, just saying what parameter is displayed on the figure. If this figure is not discussed, it should be suppressed as well as this sentence. I think Tables 5 and 6 are sufficient for the discussion.
\item Table 5 caption (lines 441443): The caption of Table 5 is confusing for me. I think the authors meant, and should indicate as caption “ACP $ \langle C \rangle $ determined using linear LSQ calibrations and using IRIS measurement as reference irradiance (section 8.3). LSQ calibrations use all 244 periods regardless of the stability of $ W_{atm} $. When IRIS measurement are used as reference, values of $ \epsilon_c = 0.0225 $, $ \gamma = 6.5 $ and $ \tau = 0.977 $ are chosen and only periods with a standard deviation of $ W_{atm} $ from the IRIS less than 0.4 Wm${2}$ are used, including 115 periods for IRIS2 and 63 periods for IRIS4”.
\item Table 6 caption (lines 445447): In relation to the change suggested for Table 5 caption, I suggest to replace the caption of Table 6 with “Same as Table 5 for the determination with IRIS as reference, but with values of $ \epsilon_c = 0.0225 $, $ \gamma = 8.4 $ and $ \tau = 0.977 $”.
\item Lines 449450: Replace “… are shown in Figure 7 with the convection coefficient used value is 6.5” with “… are shown in Figure 7 (computations use a convection coefficient $ \gamma = 6.5 $”
\item Line 453: I do not understand what the authors meant with "give consistent < C > values about a mean".
\item Lines 474475: Replace “… is a consequence of net irradiance based on the temperature difference of the base of the thermopile to the top of the thermopile” with “… is a consequence of net irradiance inducing a temperature difference between the base and the top of the thermopile.”
\item Lines 505506: Replace “… over the cooling and heating period was below 0.6 Wm${2}$” with “… over the cooling and heating period to be below 0.6 Wm${2}$.”
\item Lines 514515: Replace “… provided irradiances compared well” with “… provided irradiances that compared well.”
\item Line 580: Replace “The Reda et al., 2012 and the new equation are dependent…” with “The equation from Reda et al., 2012 and the new equation are dependent…”
\item Line 582: Replace “… provided the other coefficients in new equation are known” with “… provided the other coefficients in the new equation are known."
\item Line 615: Replace “… this is complicated by needing to divide by the transmission” with “… this is complicated by the need to divide by the transmission.”
\item Lines 653654: Replace “Further work is required to ensure that confirms the ACP…” with “Further work is required to confirm the ACP…”
\item Lines 668669: Replace “… were reduced but not eliminate” with “… were reduced but not eliminated.”
\end{enumerate}

CC4: 'Reply on RC2', Bruce Forgan, 30 Nov 2022
reply
Hi Laurent
Many thanks for your comments  I sent a long reply but it seems to have been rejected.
I noted your indications of typographical comments  and will be incorporating those.
As to your main issues:
1. Uncertainty analysis: based on suggestion I will put in two (a) in the solar section, and (b) a new subsection after the last LSQ method. Both will be short so not add to the paper.
2. Re deleting the solar section: I have a number of reasons for putting in this section and am loath to remove it. I will reword this subsection to highlight why it is important  even as a guess.
Sorry my long reply did not make it.
Cheers, Bruce

CC2: 'Comment on amt2022250', Ibrahim Reda, 29 Nov 2022
reply
it would be much informative and good idea to compare the irradiance results between the proposed method and my method, note that based on 6 IRIS&ACP95F3 comparisons the avergae difference is less than 2 w/m2 using my method as shown in Table 2 below.
Reda
Table 2. U_{95 }of the irradiance measured by ACP95F3 with respect to SI units.
ACP95F3IRIS
Difference, W/m^{2}
Standard Deviation , W/m^{2}
U_{95}, W/m^{2}
PMOD February 2013
0.10
0.83
1.66
PMOD IPgCII October 2015
0.57
0.31
0.85
SGP phase 1 October 2017
0.86
0.78
1.79
SGP phase 2 November 2017
1.05
0.85
2.00
PMOD July 2019
0.75
1.11
2.34
PMOD IPgCIII October 2021
0.10
1.20
2.40
Average ^{ }
0.03
0.85
1.84

CC3: 'Reply additional paper', Bruce Forgan, 30 Nov 2022
reply
Hi Reda
There is already some disquiet on the length of the current paper and international comparisons isn't the focus of the paper.
The current paper is about introducing a new equation like the foundational Reda et al (2021) paper introducing the ACP and the LSQ method. It's just a different equation and 3+ calibration method suggestions with examples.
The idea of comparison type papers is a very good one. As there is for example now plenty (4 years) of data at Davos with ACP96 and IRIS. And as there is now evidence that a good calibration of an ACP is possible using a static BB at room temperature some other work should be in our work pipeline.
As we have discussed previously there are several ways ahead and all very interesting  but off line from this review process  I will be emailing you, Julian, Stefan and Nozomu on some potential ways ahead.
Cheers, Bruce

CC5: 'Reply on CC3', Ibrahim Reda, 30 Nov 2022
reply
agree with potential characterization which is a good idea to explore how the ACP is used in different applications, yet the Beauty of using my method to run the ACP is its independance from any Blackbody charcterization, since it is using the source which is the atmospheric longwave irradiance which we are trying to measure, note that the atmoseric emittance is ~ 0.7; +/ 0.15 based on atmosheric contents, which is not a blackbody with emittance 0f > 0.999 and to establish a world reference the two radiometers must be independant and have different methods of charcterization/measurementmethods, hope this helps clear my thoughts about the imortance of showing the difference in the measured irradiance using all methods. which would add value to the article.

CC5: 'Reply on CC3', Ibrahim Reda, 30 Nov 2022
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CC3: 'Reply additional paper', Bruce Forgan, 30 Nov 2022
reply
Bruce W. Forgan et al.
Bruce W. Forgan et al.
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