Optimizing the precision of infrared measurements using the Eppley Laboratory, Inc. model PIR pyrgeometer

Michalsky, Joseph J.; Augustine, John A.; Hall, Emiel; Sheffer, Benjamin R.

doi:https://doi.org/10.5194/egusphere-2025-3787

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https://doi.org/10.5194/egusphere-2025-3787

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05 Sep 2025

| 05 Sep 2025

Status: this preprint is open for discussion and under review for Atmospheric Measurement Techniques (AMT).

Optimizing the precision of infrared measurements using the Eppley Laboratory, Inc. model PIR pyrgeometer

Joseph J. Michalsky, John A. Augustine, Emiel Hall, and Benjamin R. Sheffer

Abstract. The Eppley Model PIR is widely used for thermal infrared wavelength (3.5–50 μm) measurements of the downwelling and upwelling radiation from the atmosphere and surface, respectively. The field of view of the instrument is 2π steradians with a receiver that has an approximate cosine response. In this paper we examine four equations in the literature that have been used to transfer calibration from standards to field units that are used for network operations. After the introduction we discuss various equations used to convert the resistance of the YSI 44031 thermistors used in PIRs for temperature measurements of the body, aka case, and dome that are used in the derivation of incoming irradiance. We then use the four related, but distinct, equations for the transfer of the calibration from standards to field instruments. A clear choice for the preferred equation to use for calibration and transfer of calibration to field PIRs emerges from this study.

Received: 04 Aug 2025 – Discussion started: 05 Sep 2025

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Joseph J. Michalsky, John A. Augustine, Emiel Hall, and Benjamin R. Sheffer

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RC1: 'Comment on egusphere-2025-3787', Christopher Cox, 10 Sep 2025 reply

“Optimizing the precision of infrared measurements using the Eppley Laboratory, Inc. model PIR pyrgeometer” by Michalsky et al. evaluates existing approaches for transferring calibrations from PIR standards to sensors used for field operations. The authors conduct a careful and transparent analysis that provides useful results for long term networks with a historical investment in the PIR sensor. The study is a good fit for AMT. I have some comments that should be addressed in a revision before publication.
General comments:
(1) What equation is used to produce calibrated fluxes in the WISG sensors? If the WISG also uses eq (3), then wouldn’t the results shown here be interpreted firstly as an indication of consistency, but not necessarily an indication of accuracy?
(2) Figure 7: Some additional analysis here is warranted. The most obvious candidate for explaining the difference between the first and second half of the cal period is the cloud fraction (the repeatability of pyrgeometer measurements is much worse under clear skies than stratiform clouds; e.g., https://doi.org/10.5194/amt-14-1205-2021), but perhaps mean temperature or precipitable water vapor could also explain it. Figures 6 and 7 could mean that the small differences reported earlier (from any equation) are overly optimistic compensation between opposing errors given fortuitous proportions of conditions within the cal period.
(3) It would be good to apply t-tests to determine which means are different from one another, or from zero, where appropriate. The analyzed differences are small enough that they may not be significant.
(4) L320-322: Regarding conclusions, what about the fact that the differences amongst transfer equations is so much smaller than either the WISG uncertainty or (speculatively, see 2 above) the uncertainty caused by the sampling of conditions during outdoor calibrations?
Specific comments:
L14: For clarity, “broadband thermal IR…”
L40: Maybe clarify that the dome is designed to partly transmit only in the range of 3.5-50 um.
Figure 1: A few suggestions to improve the communication in this figure: (a) Label “dome” in the picture as you have done with the thermopile so that it is not interpreted as schematic of example paths in the sky (as I did at first). (b) In the caption after the word “rays” clarify that these are the numbered vectors in the picture. (c) Tb is not actually at the base of the thermopile, but is potted in the bronze casing nearby, so it would be helpful to depict the upper part of the case to show that Tb and Tr are not measuring the same thing. (d) Label Td and Tb as being thermistor measurements to distinguish from Tr, which is estimated (see also my comment at L92, which could also refer back to this figure).
L88: Since it is not clear from this text what Reda et al.’s justification was for including k0, it is also not clear what the present study’s justification is for dropping it.
L90: I think this paragraph would benefit from a leading statement expressing the problem this paper is solving. That statement might be supported by another that explains the reason prior studies modified the original Albrecht and Cox approach. As is, the text presumes too much insider knowledge on the historical context and current gap in understanding.
L92: The fact that YSI44031s are used to measure the temperature, and which temperatures are measured this way, should be included in Figure 1.
L112: Eq. 7 is odd. Can you write “c” instead of 0 in the equation to be more consistent with the Section 2 analysis/figure and then clarify in the text that in the classical form, S-H set c = 0?
L119: I’m confused about the use of the quadratic term. It looks like c = 0 for all lines in Figure 2. Where in the figure is the full cubic found? If it is the dashed blue line, it seems to be defined differently, as there is a minus sign in both the legend and the y-axis (Is the dashed blue line actually comparable to the other lines?) Also, what is c when it isn’t 0, and when it is not 0, are a, b, and d the same or do they also change?
L120: “Interestingly…” I don’t understand this statement. It seems like it would be much more surprising that changing the units yields a different result. The paper is not very long. Perhaps the appendix can be returned to the main text.
Figure 2: An error of 0.01 C in the thermistor will produce an error < 0.05 Wm2 at 0 C, which is negligible compared to other uncertainties (similar, in fact, to the error produced by the conventional, though incorrect, assumption that sigma is 5.6700e-8). Isn’t it true that the most relevant problem attributable to the YSI44031 is not the calibration method, but instead either the representativeness of its placement in the sensor in the case or the variance amongst individual thermistors in conforming to the calibration coefficients? So, I’m left not being entirely sure what the purpose of this exercise is. Is the take-away message that the YSI calibration isn’t the problem with the flux calibration? If so, make that clear. [Returning to this point after reading the conclusion, I appreciate the point you made at L299-304, though it might be worth commenting on the other issues with the thermistor in the conclusion. At very least, I suggest making the purpose of the thermistor section clearer in Section 2.]
L164: When you say “using these standards”, do you mean that the average of the standards was used for the calibration?
L247, 266: I think Figures 6 and 7, which show larger differences than Figures 3 and 4, suggest that the conditions under which outdoor calibrations (clarify somewhere that these are indeed outdoor?) are carried out are responsible for larger calibration uncertainty than the choice of equation. Yet, I think the community has historically been more focused on methodology. Maybe a recommendation to be made there?
L313: “…are small.” Specifically, the differences are an order of magnitude smaller in the transfer of relative calibrations than the reported uncertainty of the WISG.
L360, 364: Are these equation references supposed to be to A#?
L417: Is this Grobner (2025) from the main text?

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Citation: https://doi.org/10.5194/egusphere-2025-3787-RC1
CC1: 'Comment on egusphere-2025-3787', Julian Gröbner, 10 Oct 2025 reply

I have done a similar study a few years ago, which I showed at the BSRN meeting in 2008 (see attached excerpt of slides). I had used 78 pyrgeometers at the time that were characterised at PMOD/WRC in our blackbody. It also included two types available at the time, Eppley PIR and Kipp&Zonen CG4. The conclusions were similar, that the simple Albrecht equation with k1=0 and k2=1 produces larger residuals than the extended Albrecht equation with k1, k2, and k3. By now we have characterised a few hundred pyrgeometers from different manufacturers which could be included in such an analysis.
Note that high-end pyrgeometers have a temperature compensation circuitry attached to the signal cables of the thermopile to reduce the temperature sensitivity of the pyrgeometer, and thereby achieve a k1 very close to zero. So for those, the simple Albrecht equastion would work quite well. However I know of one manufacturer who does not use such a temperature compensation circuitry, and in this case the resulting k1 are of the order of -0.13, so significantly different from zero, and clearly not compatible with the simple Albrecht equation.

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Citation: https://doi.org/10.5194/egusphere-2025-3787-CC1
RC2:
'Comment on egusphere-2025-3787', Anonymous Referee #2, 16 Oct 2025 reply
In “Optimizing the precision of infrared measurements using the Eppley Laboratory, Inc. model PIR pyrgeometer”, Michalsky et al. assess the performance of four different equations that are used to convert raw voltage measurements of pyrgeometers into irradiances during transfer calibrations. The authors conclude that the method by Philipona et al. (1995) is the preferred option for this purpose. Additionally, some analysis on the conversion of thermistor-measured resistance to temperature and on the consistency among different calibration events are provided. The topic is well suitable for publication in ATM. However, I think, the study has more potential and should first receive a careful revision considering the comments below.

General comments
The introduction could be improved. First, a general motivation on why accurate thermal infrared radiation measurements are required is missing. Although this might be obvious, it might help putting the later results into context. Second, please add a short explanation on the measurement principle of the pyrgeometers and why the instrument temperature needs to be corrected for. Thereby, especially readers less familiar with such instruments will better understand the components in Fig. 1 and Eq. (1) and the difference between T_B and T_R

What were the meteorological conditions during the instrument calibration? Was the calibration performed according to WRC standards (calibration coefficient C from outdoor measurements in clear-sky night-time conditions and coefficients k_i from lab experiments)? L 166-167 only mentions the exclusion of outliers and periods of precipitation. Please provide more information on the measurement site and the conditions during calibration (temperature, humidity, clouds). Perhaps, meteorological data could complement the irradiance time series shown in Fig. 3. I think, the study can largely benefit from such data. Regarding Fig. 7, this data could be used to more accurately filter for specific conditions, such as cloudiness, temperature or humidity regimes, and assess, how different conditions affect the transfer calibration of the pyrgeometers. I would look forward to an extended analysis on this problem in Sect. 4 as a second focus of the study.

I would suggest testing the consistency of the standard PIRs’ calibration (Fig. 6) prior to calibration transfer to get an estimate of the impact of potential instrument instabilities on the transfer calibration. Fig. 6 seems to reveal that PIR 32909 is less stable than the other ones, especially regarding the 2024 calibration. This probably causes the underestimation in measured irradiance that is visible in Fig. 3 also for Philipona’s coefficients. However, due to the significantly stronger underestimation for Albrecht’s coefficients, I assume that Fig. 6 would also show larger differences for Albrecht. Consider showing Fig. 6 for both Albrecht and Philipona in conjunction with Fig. 3 (see also more detailed comment on Fig. 3 below). The results might imply a possible exaggeration of instrument instabilities by Albrecht, which would further substantiate the preference for Philipona’s method.

Based on the general comments above, consider modifying the general structure of the manuscript: My suggestion would be the following:
Motivation: Different equations for conversion of raw signal (voltages) to irradiance, all depending on body and dome temperature (measured by thermistor)

Sect. 2: evaluate different methods to convert thermistor resistance to temperature → no significant impact on irradiance

Sect. 3: evaluate equations to convert voltages to irradiances
3 regularly calibrated “standard” PIRs: first assess instrument stability and consistency of calibration events (Fig. 6) → PIRs generally stable, but 32909 least stable, impact of instability larger for Albrecht → preference Philipona (see general comment 3)

Transfer calibration standard PIRs → test PIRs (Figs. 4, 5) → preference Philipona

Modified Sect. 4 “Impact of meteorological conditions on precision of transfer calibration” (see general comment 2)

Furthermore, I would like to encourage the authors to carefully revise the text. The wording frequently sounds too colloquial, is imprecise, or lacks clarity. Sometimes, the language could be more concise. Some comments are given below, but there is room for more improvement.

Specific comments
Title: Consider changing to something like “Optimizing the precision of infrared radiation measurements by Eppley PIR pyrgeometers”

L 16-17: Is the cosine response important for the study? Rather mention more important instrument characteristics, such as the thermistors that are used for temperature measurements and referred to in L 20.

L 18: This sentence lacks clarity. Do “standards” and “field units” refer to the standard pyrgeometers calibrated at WRC and the physical unit of irradiance, respectively? Please clarify.

L 87-88: Please justify why k_0 is dropped.

L 91-92: Avoid the term Steinhart-Hart equation here since it was not defined yet. Better “various versions of the Steinhart-Hart equation that have been used to convert …” → “various methods used to convert …”.

L 93: “three test PIRs” – later, the text mentions 6 instruments. Please clarify.

Eq. (7): I think, the notation of the exponentiated logarithms is improper. I suggest using either (ln R)² or ln²(R). Furthermore, I would suggest including the quadratic term with coefficient c in Eq. (7) and set to 0 when necessary.

L 117: Is my understanding right that the regression (Eq. 7) to derive the coefficients a, b, (c), d is based on tabulated data of resistance and temperature provided by the manufacturer? At least, this statement is made in the conclusions section. If so, what is the benefit of performing the fit instead of simply interpolating between the tabulated values? Please add important information and be more accurate in describing what was done here and why.

L 118-123: Consider altering the argumentation here and sticking to ohms when including the quadratic term into the regression. Although kiloohms could be skipped completely from this analysis, it might be worth mentioning the difference when the quadratic term is omitted in the following sentence. I like the idea of drawing attention to this discrepancy arising from the choice of the unit here and explain it mathematically. However, since this explanation is not the main scope of the study, I agree that the appendix is the right place to do that.

L 125: Is the reference Gröber (2025) accessible to the public? It would be good to know which coefficients are used for the corresponding fit. Maybe, a table listing the coefficients for all fits considered in Fig. 2 can be added.

L 126: “… but does well over the entire range.” Please specify the temperature range referred to at the beginning of the section. Furthermore, “entire range” is imprecise here and should be changed to something like “entire temperature range analyzed here”.

L 129-130: Could you justify your choice quantitatively? What does the temperature deviation mean in terms of irradiance?

L 148: Was the calibration of the field/test PIRs only done for period shown in Fig. 3? According to the text, Fig. 3 is just an example. Please be more accurate in describing the calibration procedure and mention, which instrument was calibrated when.

Fig. 3: To put the irradiance measurements into context with the concurrent meteorological conditions, my suggestion is to add time series of relevant meteorological quantities to the time series of the irradiances, if available. Furthermore, only show the mean of the three instruments in the time series. The difference between the individual instruments (e.g. with respect to the mean) could be plotted in an additional histogram (or boxplots), from which the magnitude of potential over- or underestimation can be better quantified.

L 163: How was “the mean IR irradiance of the three standards” obtained? How were the three WRC calibration (2018, 2022, 2024) combined to calculate the standards’ irradiance? Please provide more information.

L 188-189: I think, it’s fine to only plot the results of two instruments. However, maybe summarize statistics for all instruments in a table and mention some key numbers in the text to corroborate the conclusions.

L 218-219: Is it fair to argue that the results are better or worse in Fig. 5 compared to Fig. 4? I mean, you are comparing to different references (standards calculated with Albrecht vs. Philipona) but none of these references really represents the truth. I think the comparison only shows that calibrating the test PIRs with Albrecht’s equation can fit better to either the Albrecht- or the Philipona-calibrated standard and that the consistent application of Albrecht’s method does not assure a minimized spread.

Fig. 7: How consistent are the results if you randomly split the data into equally sized subsets for calibration and validation?

Appendix: Equation numbers 8–14 in the text → A1–A7

Text improvements
throughout: the use of written-out “infrared” and abbreviated “IR” is inconsistent. I suggest using the acronym “TIR” for thermal infrared, since the infrared also covers parts of the solar spectrum where the pyrgeometer is not sensitive.

throughout: “degrees K” → “Kelvin”

throughout: “standards” → “standard PIRs”

L 14: “The Eppley Model PIR is widely used …” → “The Eppley Precision Infrared Radiometer (PIR) is a widely used pyrgeometer …”

L 17: “equations in the literature” → “equations suggested by the literature”

L19-20: “… used to convert the resistance of the YSI 44021 thermistors used in PIRs for temperature measurements …” → “… used to convert the resistance measurements of the thermistors to temperatures …”

L 20, 31: skip “aka case”

L 29: “The Eppley model PIR pyrgeometer was developed …” → “The Eppley Precision Infrared Radiometer (PIR) is a pyrgeometer developed …”

L 29: skip “longwave”

L 30: “sky” → “atmosphere”

L 46: skip “L is the external incoming infrared irradiance” since L was defined above

Caption Fig. 1: “rays” → “radiation components on the thermopile surface” and “incoming infrared” → “incident atmospheric TIR irradiance”.

L 112: “… where T is in degrees K and R is in ohms or kiloohms” → “… where T is the temperature in Kelvin and R is the measured resistance in Ohms or Kiloohms.”

L 118: “The least-squares fit to Eq. (7) … is the red line …” → “The least-squares fit to Eq. (7) … is indicated by the red line …”

L 118-119 and 129-130: “full cubic” → “full cubic relationship”

L 139: “we apply Eq. (2), (3), (4), and (6) to examine how well each performs …” → “we examine the performance of Eqs. (2), (3), (4), and (6) …”

L 141: “Our three PIRs …” → “The three standard PIRs …”

L 141-142 and throughout: be consistent with terms “PMOD” and “WRC”.

L 229-235: Most information in this paragraph is redundant, because it is already known. Suggestion to simplify: “The regular calibration of the standard PIRs at the WRC leads to different calibration results. Here, the consistency and repeatability of those calibration events is assessed.” Also consider changing the structure (see general comments).

L 306-307: “The three standard PIRs … are sent biennially to be calibrated …” → “The three standard PIRs … are biennially calibrated …”

L 328: “In this paper, the World Infrared Standard Group (WISG) …” → “The WISG …”

L 345-348: “The source of the difference …” → “This difference is due to numerical reasons, which are explained in the following.”

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Citation: https://doi.org/10.5194/egusphere-2025-3787-RC2

Joseph J. Michalsky, John A. Augustine, Emiel Hall, and Benjamin R. Sheffer

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Short summary

We examine four equations for calculating infrared radiation (3–50 mm) measured with a Eppley PIR pyrgeometer. These equations are used to transfer calibrations from the World Infrared Standard Group at the World Radiation Center in Davos, Switzerland, to the three PIR pyrgeometers we use as standards. A clear choice in terms of the most precise method to follow emerges from this study. Furthermore, we evaluate the stability of the Eppley PIR, necessary for long-term trend analysis.


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