NREL is a national laboratory of the U.S. Department of Energy, Office of Energy 

Efficiency & Renewable Energy, operated by the Alliance for Sustainable Energy, LLC.

Contract No. DE-AC36-08GO28308

Method to Calculate 

Uncertainties in Measuring 

Shortwave Solar Irradiance Using 

Thermopile and Semiconductor 

Solar Radiometers

Ibrahim Reda

Technical Report

NREL/TP-3B10-52194

July 2011
NREL is a national laboratory of the U.S. Department of Energy, Office of Energy 

Efficiency & Renewable Energy, operated by the Alliance for Sustainable Energy, LLC.

National Renewable Energy Laboratory

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303-275-3000 • www.nrel.gov

Contract No. DE-AC36-08GO28308

Method to Calculate 

Uncertainties in Measuring 

Shortwave Solar Irradiance Using 

Thermopile and Semiconductor 

Solar Radiometers

Ibrahim Reda

Prepared under Task No. 3B10.3000

Technical Report

NREL/TP-3B10-52194

July 2011
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1 

Acknowledgments

Many thanks to Bev Kay and Preston Morse for their continued assistance and persistence during 

the radiometer calibrations and for making sure that all instruments were properly calibrated and 

within tolerance. Many thanks also for NREL’s Metrology Laboratory, PV, Resource Assessment, 

and Atmospheric System Research programs for their continuous improvement in the field of 

radiometry, and for providing the support and funds for this effort.
2 

Abstract

The uncertainty of measuring solar irradiance is fundamentally important for solar energy and 

atmospheric science applications. Without an uncertainty statement, the quality of a result, model, 

or testing method cannot be quantified, the chain of traceability is broken, and confidence cannot 

be maintained in the measurement. Measurement results are incomplete and meaningless without a 

statement of the estimated uncertainty with traceability to the International System of Units (SI) or 

to another internationally recognized standard. 

This report explains how to use International Guidelines of Uncertainty in Measurement (GUM) to 

calculate such uncertainty. The report also shows that without appropriate corrections to solar 

measuring instruments (solar radiometers), the uncertainty of measuring shortwave solar 

irradiance can exceed 4% using present state-of-the-art pyranometers and 2.7% using present 

state-of-the-art pyrheliometers. Finally, the report demonstrates that by applying the appropriate 

corrections, uncertainties may be reduced by at least 50%. The uncertainties, with or without the 

appropriate corrections might not be compatible with the needs of solar energy and atmospheric 

science applications; yet, this report may shed some light on the sources of uncertainties and the 

means to reduce overall uncertainty in measuring solar irradiance.
3 

1 Introduction

The uncertainty of measuring solar irradiance is fundamentally important for solar energy and 

atmospheric science applications. Without an uncertainty statement, the quality of a result, model, 

or testing method cannot be quantified, the chain of traceability is broken, and confidence cannot 

be maintained in the measurement. Measurement results are incomplete and meaningless without a 

statement of the estimated uncertainty with traceability to the International System of Units (SI) or 

to another internationally recognized standard. 

The International Guidelines of Uncertainty in Measurement GUM (JCGM/WG, 2008) are used 

for calibrating solar radiometers at the National Renewable Energy Laboratory (NREL) (Reda et. 

al 2008). This report describes a method for calculating estimated measurement uncertainties of 

solar irradiance data obtained with properly calibrated pyranometers and pyrheliometers. It also 

outlines GUM guidelines used to calculate the estimated uncertainty for field measurement. 

Uncertainty in field measurement is a result of radiometer calibration, equipment installation, data 

acquisition, system maintenance methods, and the environmental conditions at the site where the 

radiometer is deployed. 

2 GUM Guidelines

The GUM process can be summarized in four steps:

1.1 Determine the measurement equation.

1.2 List or estimate the standard uncertainty analogous to a standard deviation associated 

with each variable in the measurement equation (e.g. voltage, irradiance) and for each 

component that might introduce uncertainty to the calibration process (e.g. interpolation, 

environmental conditions). 

1.3 Calculate the combined standard uncertainty using the root-sum-of-squares method for 

all standard uncertainties in step 1.2. 

1.4 Calculate the expanded uncertainty by multiplying the combined standard uncertainty by 

the coverage factor, k (typically known as Student’s “t”), or prescribed coverage factors 

for known distributions of measurements representing the single value of the quantity to 

be measured (e.g. Gaussian, triangular, rectangular). 
4 

3 Measurement Equation

The following equation is used to calculate global solar irradiance using a thermopile pyranometer 

at each data point:

 

G

V R W

R

nt nt = − *

 (1) 

where:

 G is the calculated global solar irradiance in watts per square meter (Wm-2) 

 V is the pyranometer's thermopile output voltage in microvolts (μV) 

 Rnt is the pyranometer's net longwave responsivity estimated or determined by 

Blackbody characterization in μV/(Wm-2) 

 Wnt is the net longwave irradiance measured by a collocated pyrgeometer in W/m-2

(Pyrgeometers are radiometers that measure atmospheric longwave irradiance) 

R is the pyranometer's responsivity determined by calibration in μV/(Wm-2). 

Either a single responsivity value or the responsivity as a function of solar zenith angle can be 

uniquely determined for an individual pyranometer from calibration and used to compute global 

irradiance data. The uncertainty in the responsivity value can be reduced by as much as 50% if the 

responsivity as a function of solar zenith angle is used (Reda, 1998 and Reda et al., 2008). 

For other types of solar radiometers, the same equation is used by removing or adding variables 

relevant to the type of radiometer. For example, in the case of pyrheliometers and semiconductor 

pyranometers with no thermal offset response, Rnt equals zero; therefore, G equals V divided by R.

Note that all the variables in Equation 1 are measured or calculated using independent methods; 

therefore, they are assumed to be independent with no correlation. 

4 Standard Uncertainties

A standard uncertainty (u) can be estimated using statistical methods (Type A), non-statistical 

methods (Type B), or both. For example, a standard deviation of a number of data points is a Type 

A uncertainty. A manufacturer's specification of radiometer performance characteristics or an 

uncertainty reported by a calibration vendor is a Type B uncertainty (Taylor and Kuyatt 1994). A 

Type B uncertainty that changes as a function of one or more variables may be a combination of 

Type A and Type B. 
5 

Standard uncertainty calculations depend on the distribution of sources of uncertainty. A 

requirement of the GUM approach is to state the type of statistical distribution, either measured or 

assumed, for uncertainty calculations. Table 1 shows how to estimate the standard uncertainty for 

some common distributions using the expanded uncertainty. Throughout this report, standard

uncertainties are denoted by “u” and expanded uncertainties by “U”.

Table 1. Standard uncertainty estimation for common distributions

Distribution Source Standard uncertainty

Normal: 

Standard deviation = σ

Number of readings = n

or 

Expanded uncertainty = U 

and coverage factor = k

Statistical

(Type A) 

u

n

OR u

U

k = = σ

Rectangular Non-statistical

(Type B) 

u

U = 3

Triangular Non-statistical

(Type B) 

u

U = 6

Normal 

k = 2 (or t)

Non-statistical

(Type B) 

u

U

OR u

U

t = = 2

The following steps explain how to calculate the combined standard uncertainty and its effective 

degrees of freedom for Type B uncertainties first, then for Type A uncertainties. 

To calculate the combined standard uncertainty and its effective degrees of freedom for Type B 

uncertainties: 

4.1 List the estimated expanded uncertainty (U95) for each variable in Equation 1. The 

expanded uncertainty is obtained as follows:

A. The uncertainty of measuring the radiometer’s output voltage (V) is calculated with 

a calibrated data acquisition system. This uncertainty is typically listed in the 

manufacturer specifications in the user’s manual. Table 2 displays the expanded

uncertainty calculated using a Campbell Scientific, Inc. Model CR3000 data 

logger1

. 

 

1

 NREL makes no product endorsement, this product was chosen as an example. 
6 

Table 2. Expanded uncertainty listed in manufacturer specifications

Model Range Resolution

(Differential)

Temperature 

Range

Offset U95

(%)

Campbell 

CR3000 

20 

millivolts 

(mV) 

0.67 uV -25 C to 50 C 3*resolution+2

uV 

0.07% reading + 

offset

B. The uncertainty of the net responsivity (Rnt) is listed in a solar radiometer’s 

calibration certificate or manufacturer specifications. From experience, an 

estimated uncertainty of 20% is used in this report.

C. The uncertainty of the net longwave irradiance (Wnt) is listed in a pyrgeometer’s 

calibration certificate. It may also be estimated by users based on experience.

D. The uncertainty of the solar radiometer Responsivity (R) is listed in the 

radiometer’s calibration certificate. Other sources of uncertainties must be included 

in the uncertainty of the responsivity. Such sources are a result of responses to 

change in solar zenith angle, solar azimuth angle, spectral irradiance distribution, 

ambient temperature, etc. These uncertainties are listed in the manufacturer 

specifications or may be estimated based on prior experience.

Table 3 is an example of uncertainty estimates for four types of solar radiometers: a

thermopile pyranometer, a semiconductor pyranometer, a thermopile 

pyrheliometer, and a semiconductor pyrheliometer (Semiconductor pyrheliometers 

are not frequently available to solar data users, but are used here to illustrate the 

effects of spectral irradiance distributions on measurements). These estimates are 

for current state-of- the-art radiometers and have been determined via calibrations

such as NREL’s (NREL-Annual), data collection and experience with hundreds of 

radiometers over many years. Actual values may be listed on manufacturers’ web 

sites or in radiometers specifications. Additional uncertainties may be added 

depending on the instrument type and instrument set-up onsite. 
7 

Table 3. Example of uncertainty estimates for solar radiometer responsivity (R) 

Uncertainty 

Source

Thermopile 

pyranometer

Semiconductor 

pyranometer

Thermopile 

pyrheliometer

Semiconductor 

pyrheliometer

Calibrationa 3% 5% 2% 3% 

Zenith responseb 2% 2% 0.5% 1% 

Azimuth response 1% 1% 0% 0% 

Spectral response 1% 5% 1.5% 8% 

Tiltc 0.2% 0.2% 0% 0% 

Nonlinearity 0.5% 1% 0.5% 1% 

Temperature 

response

1% 1% 1% 1% 

Aging per year 0.2% 0.5% 0.1% 0.5% 

Total U = Sum 8.9% 15.7% 5.6% 14.5% 

Total U = RSS 4.1% 8.0% 2.7% 8.9% a Includes zenith angle response from 30° to 60° b Includes zenith angle response from 0° to 30° and 60° to 90° c This uncertainty is set to zero for untilted radiometers

The percentage contribution of each uncertainty source listed in Table 3 is calculated to 

show which source contributes the most to the total uncertainty. This contribution is 

calculated by dividing the value of each source listed in the table by the sum of all 

uncertainties for each radiometer type. For example, for the calibration value for 

thermopile pyranometers, 3% divided by 8.9% multiplied by 100 = 33.7%. Table 4 is a 

list of the percentage contributions for the values in Table 3. 

In Table 4, the greatest contributors to the uncertainty of the responsivity are the 

uncertainties resulting from calibration. For pyranometers, most of the calibration 

uncertainty is from the radiometers zenith angle response; see Figure 1 for an example of 

zenith angle response during pyranometer calibration. Therefore, reducing the overall 

uncertainty can be accomplished by reducing the uncertainty resulting from the 

calibration process. One method, used at NREL, is to report the uncertainty of the 

responsivity after being corrected for the zenith angle response. NREL’s Broadband 

Outdoor Radiometer Calibration (BORCAL) reports (NREL-Annual) show that this is 

accomplished by fitting the responsivity to an AM and PM spline interpolation functions. 

Under clear-sky conditions, this may reduce the overall uncertainty by at least 30%. 

Other fitting methods may be used to fit the responsivity versus zenith angle, yet users 
8 

must calculate the Type A standard uncertainty resulting from the fitting (uA), where uA

equals the standard deviation of the calculated residuals resulting from the fitting.

Table 4. Uncertainty sources contributing to the total uncertainty of the responsivity, R

Uncertainty Source Thermopile 

pyranometer

Semiconductor 

pyranometer

Thermopile 

pyrheliometer

Semiconductor 

pyrheliometer

Calibration 33.7% 31.9% 35.7% 20.7% 

Zenith response 22.5% 12.7% 8.9% 6.9% 

Azimuth response 11.2% 6.4% 0% 0% 

Spectral response 11.2% 31.9% 26.8% 55.2% 

Tilt 2.3% 1.3% 0% 0% 

Nonlinearity 5.6% 6.4% 8.9% 6.9% 

Temperature 

response

11.2% 6.4% 17.9% 6.9% 

Aging per year 2.3% 3.2% 1.8% 3.5% 

Total U = Sum 100% 100% 100% 100% 
9 

Figure 1. Responsivity versus zenith angle for PSP-33852F3 from outdoor calibration

Models can also be used to correct for other sources of uncertainty (e.g. spectral 

irradiance distribution, ambient temperature, etc.). The uncertainties due to using such 

models must be added as Type A and/or Type B standard uncertainties when computing 

expanded uncertainty. 

4.2 Using Table 1 for the most common distribution, list the distribution, degrees of freedom, 

and coverage factor for each variable in Equation 1. When the distribution of the 

expanded uncertainty is not known, it is common to assume a rectangular distribution 

with infinite (∞) degrees of freedom (Taylor and Kuyatt 1994). Table 5 lists the 

distribution, degrees of freedom, and coverage factor for each variable. 

7.00

7.10

7.20

7.30

7.40

7.50

7.60

7.70

15 20 25 30 35 40 45 50 55 60 65 70 75 80

Responsivity (uVW-2)

Solar zenith angle (°)

AM Responsivity (uV/W/m2) PM Responsivity (uV/W/m2)
10 

Table 5. Distribution, degrees of freedom, and coverage factor for the variables in the measurement 

equation 

Variable Distribution Degrees of freedom Coverage Factor

V Rectangular 

(from manufacturer)

∞ 3

Rnt Rectangular 

(from estimate)

∞ 3

Wnt Normal

(from calibration)

n>500 data points 1.96 

R Normal

(from calibration)

n>500 data points 1.96 

4.3 Calculate the estimated standard uncertainty (u) for each variable in Equation 1. For 

example, a calibrated voltmeter is used to measure the output voltage of a solar 

radiometer. Using the manufacturer specifications for the voltmeter, the expanded 

uncertainty of the voltage variable (V) in Equation 1 equals UV, 95 with 95 % confidence 

level, and the distribution is assumed to be rectangular. Therefore, the standard 

uncertainty for the voltage measurement is uV , and equals UV, 95 divided by the coverage 

factor listed in Table 5: 

 

u

U

V

V

= ,95

3 (2) 

Table 6 is a list of standard uncertainties for the variable in the measurement equation. 

The standard uncertainties listed in the table are based on the listed values in the “Value” 

column. The coverage factors are obtained from Table 5. Note that the standard 

uncertainties are calculated at each data point, except for Rnt , which is constant. Also, R 

can be considered constant, unless it is corrected for zenith angle dependence (i.e. using 

it as a function of zenith angle as mentioned in 4.1.1 above). 
11 

Table 6. Standard uncertainty for each variable in the measurement equation 

Variable Value %U95 U95

(Absolute Value)

u 

(Absolute Value)

V (uV) 5083.5 N/A 7.57 4.37 

Rnt [uV/(Wm-2)] 0.61 20% 0.122 0.07 

Wnt (Wm-2) -174.2 5% 8.711 4.44 

R [uV/( Wm-2)] 7.4 4% 0.296 0.15 

4.4 Calculate the sensitivity factor (cj) for the jth variable in the measurement equation 

(Equation 1) by using the partial derivative of the global solar irradiance (G) with respect 

to each variable as follows (Taylor and Kuyatt 1994): 

 

c

G

V R V = = ∂

∂

1

 (3) 

 

c

G

R

W

R R

nt

nt

nt = = ∂ −

∂ (4) 

 

c

G

W

R

R W

nt

nt

nt = = ∂ −

∂ (5) 

 

c

G

R

V R W

R R

nt nt = = ∂ − −

∂

( * ) 2

 (6) 

Table 7 lists the calculated value of cj at a specific data point using the values listed in Table 6.

Table 7. Sensitivity factors for variables in the measurement equation at a specific data point

cV cRnt cWnt cR

Sensitivity 

Factor 

0.135 

(Wm-2/uV)

23.557 

[(Wm-2)

2

/uV)

-0.0825 

(Unity)

-94.88 

[(Wm-2)

2

/uV]

4.5 Calculate the combined standard uncertainty for Type B, uB (Taylor and Kuyatt 1994):

 

u c u B j j

j

= =

−

∑( * ) 2

0

 1

 (7) 

where j is the jth variable and l is the number of variables. As an example, Table 8 is a list of 

cj * uj and the calculated uB for the above data point. 
12 

Table 8. Combined standard uncertainty for Type B uncertainties

cV*uV

(Wm2

)

cRnt*uRnt

(Wm2)

cWnt*uWnt

(Wm2)

cR*uR

(Wm2)

Combined Standard Uncertainty, uB

(Wm2

)

0.591 1.659 0.367 -14.321 14.433 

The standard uncertainty uB is calculated, and the contribution of each variable in the 

measurement equation to the total Type B uncertainty of measuring the solar irradiance 

is also calculated. The contribution is used to decide which variable contributes the 

most to the total standard uncertainty, and consequently to the expanded uncertainty. 

This contribution is calculated at each data point by dividing each cj * uj by the sum of 

all cj * uj. Figure 2 shows the percentage contribution for each variable versus zenith 

angle. In the figure, the uncertainty in the responsivity value is the greatest contributor 

to the overall uncertainty. Therefore, improving the uncertainty in the responsivity 

would have the greatest effect on the overall uncertainty in measuring solar irradiance.

Figure 2. Contribution of each variable in the measurement equation to the total uncertainty versus 

zenith angle

0

10

20

30

40

50

60

70

80

90

100

15 25 35 45 55 65 75

Contribution (%)

Zenith angle (°)

V Rnt Wnt R
13 

4.6 Calculate the effective degrees of freedom for Type B DFB-eff where dfj is the degrees of 

freedom of the jth variable in Equation 1 (Taylor and Kuyatt 1994): 

 

DF

u

c u

df

B eff

B

j j

j j

,

( )

( * ) =

=

−

∑

4

4

0

 1

 (8) 

Note that the value of dfj equals ∞ for most variables; therefore DFB, eff equals ∞.

4.7 Calculate Type A standard uncertainty (uA). This uncertainty is calculated as the standard 

deviation of a data set or of a set of measured irradiance.

4.8 Calculate the combined standard uncertainty (uc): 

 u u u c = + A B

2 2

 (9) 

Then calculate the effective degrees of freedom (DFeff) using Equation 8 by replacing uB

with uc and cj * uj with uA and uB . As stated in 4.1.6, DFeff equals infinity because all 

other values are infinity. Note that in this report, the measured irradiance is calculated 

instantaneously; therefore, uA equals zero.

Since uc will change at each data point, it is plotted against the zenith angle. Therefore, a 

conservative value for the standard uncertainty equals the maximum value of uc = uc, max. 

Figure 3 shows the uc for PSP 33852F3, where uc, max = 2.1 % 
14 

Figure 3. Standard Uncertainty versus zenith angle for PSP-33852F3

5 The Expanded Uncertainty and Reporting

The expanded uncertainty with 95% confidence level (U95) is calculated as follows: 

 U k u 95 = c *

 (10) 

where k is the coverage factor, calculated using student “t” tables. For DFeff equals infinity, k 

equals 1.96. 

The reporting of uncertainty must include Type A and Type B standard uncertainties, coverage 

factors, and expanded uncertainties with 95% confidence level. Table 9 is a list of the reported 

results for PSP-33852F3. In the table, there are two lists, one using one value for the responsivity 

(R) and the other using R as a function of zenith [F(z)]. 
15 

Table 9. The reported uncertainty for PSP-33852F3

Responsivity = 

constant Responsivity = F(z)

Type A standard uncertainty (uA) 0% 0.3 % = standard deviation of 

fitting residuals

Type B standard uncertainty (uB) 2.1% 1.34% 

Combined standard uncertainty (uc) 2.1% 1.37% 

Coverage factor (k) 1.96 1.96 

Expanded uncertainty (U95) 4.1% 2.69% 

Table 10 is a list of expanded uncertainties for current state-of-the-art solar radiometers. Using the 

radiometers without any corrections and using responsivity as a constant will yield estimated 

expanded uncertainties of 4.4% and 2.7% for thermopile pyranometers and pyrheliometers, and 

8% and 8.9% for semiconductor pyranometers and pyrheliometers. These uncertainties are 

consistent those previously reported (Myers et al. 2004) (King 1997). For each radiometer, the 

expanded uncertainty is calculated by using the responsivity as a constant value and as a function 

of zenith angle. In the list, note that improvement in the expanded uncertainty exceeds 29%. 

Correcting for other uncertainty contributors might result in reduction in the uncertainty that is at 

least 20% greater. Therefore, when correcting for other sources, the zenith angle correction may

result in a reduction of at least 50% for clear-sky conditions similar to those during the 

pyranometer calibration. 

Table 10. Expanded uncertainties for solar radiometers

Thermopile 

pyranometer

Semiconductor 

pyranometer

Thermopile 

pyrheliometer

Semiconductor 

pyrheliometer

U95 

Using R = 

Constant

4.1% 8.0% 2.7% 8.9% 

U95 using R = 

F(z) 2.6% 4.0% 1.9% 4.7% 

U95

Improvement 

due to F(z)

38% 50% 29% 47% 
16 

6 Conclusion

Measurement uncertainties must be reported with any solar irradiance data. The described 

procedure in this report is consistent with GUM guidelines and may be used for calculating 

uncertainty estimates for field measurements of solar irradiance. It is evident from the results 

shown in this report that the uncertainty in measuring solar irradiance using current state-of-the-art 

radiometers is greater than 4% for pyranometers and greater than 2.7% for pyrheliometers under 

clear-sky conditions similar to those during the radiometer calibration. 

Since the radiometer’s inception in the late 1800s, there have been major improvements in the 

manufacturing process, but very little improvement in measurement accuracy. Many methods and 

models have been implemented to improve the uncertainty of solar irradiance data (Reda et al. 

2008) (King and Myers 1997) (King et al. 1997). Implementing such methods and models can 

reduce the uncertainty of measuring solar irradiance by at least 50% (as discussed in Section 5);

therefore, the best achievable uncertainty is 2% for pyranometers and 1.35% for pyrheliometers 

under clear-sky conditions. 

The goal of this report is to raise user awareness of the measurement uncertainties associated with 

solar radiometer data and inform users that these uncertainties might not meet the requirements for 

solar energy and atmospheric science applications. This report may also encourage manufacturers 

and interested researchers to improve the manufacturing process and develop improved methods 

for characterizing radiometers so that better corrections can be developed for the sources of 

uncertainty. Increased demand for accurate solar irradiance measurement may result in an

Expanded Uncertainty (U95) of less than ±1% in the near future. 
17 

References

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King, D.L.; Boyson, W.E.; Hansen, B.R. (1997). Improved accuracy for low-cost solar irradiance 

sensors. SAN-97-3175C, CONF-980735. Sandia National Laboratory. 

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