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Contract No. DE-AC36-08GO28308

Evaluation of Clear Sky Models 

for Satellite-Based Irradiance

Estimates

Manajit Sengupta and Peter Gotseff

National Renewable Energy Laboratory

Technical Report

NREL/TP-5D00-60735

December 2013
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

This report is available at no cost from the National Renewable Energy

Laboratory (NREL) at www.nrel.gov/publications.

Contract No. DE-AC36-08GO28308

National Renewable Energy Laboratory

15013 Denver West Parkway

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303-275-3000 • www.nrel.gov

Evaluation of Clear Sky Models 

for Satellite-Based Irradiance

Estimates

Manajit Sengupta and Peter Gotseff

National Renewable Energy Laboratory 

Prepared under Task No. SS13.8041

Technical Report

NREL/TP-5D00-60735

December 2013
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Table of Contents

Abstract........................................................................................................................................................ 5

1 Background ........................................................................................................................................... 5

2 Clear Sky Model Conversion to MATLAB .......................................................................................... 6

2.1 Model Conversion............................................................................................................................ 6

2.2 Model Validation ............................................................................................................................. 7

3 Model Sensitivity ................................................................................................................................ 12

4 Clear Sky Model Performance........................................................................................................... 13

4.1 Model Inputs.................................................................................................................................. 13

4.1.1 Surface Measurement Inputs........................................................................................... 13

4.1.2 Satellite-Based Model Inputs.......................................................................................... 15

4.2 Model Results................................................................................................................................ 15

4.2.1 Clear Sky Results............................................................................................................ 15

4.2.2 Model Run Times............................................................................................................ 19

5 Model Selection and Conclusions .................................................................................................... 20

References................................................................................................................................................. 21

Appendix A ................................................................................................................................................ 22

Optical Depth Effect on Irradiance....................................................................................................... 22

Determination of Angstrom Turbidity Coefficients for Model Validation .......................................... 23

Appendix B ................................................................................................................................................ 26

Appendix C ................................................................................................................................................ 27

Model Improvement Due to Use of Daily AOD and Half-Hour Water Column ................................. 27

Appendix D ................................................................................................................................................ 29

GHI Clear Sky Model Comparisons for All Sites................................................................................ 29

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5 

Abstract

This report describes an intercomparison of three popular broadband clear sky solar irradiance 

model results with measured data, as well as satellite-based model clear sky results compared to 

measured clear sky data. The authors conclude that one of the popular clear sky models (the Bird 

clear sky model developed by Richard Bird and Roland Hulstrom) could serve as a more 

accurate replacement for current satellite-model clear sky estimations. Additionally, the analysis 

of the model results with respect to model input parameters indicates that rather than 

climatological, annual, or monthly mean input data, higher-time-resolution input parameters 

improve the general clear sky model performance. 

1 Background

The existing Satellite Algorithm for the Shortwave Radiation Budget (SASRAB) Geostationary 

Operational Environmental Satellite (GOES)-based irradiance model for solar resource 

assessment has been shown to underestimate direct normal irradiance (DNI) and global 

horizontal irradiance (GHI) under clear sky conditions, and especially during low solar zenith 

angle conditions. The reason for this underestimate is that the SASRAB algorithm requires a 

background reflectance field to calculate solar insolation for the current image. This is generated 

by recording the second-darkest value for each image pixel over the previous 28 days. This is a 

visible channel measurement that has been shown to be adversely affected by the earth’s surface 

reflectivity. Thus, desert environments, snow, or any high-albedo conditions that existed at the 

time of the background calibration force the model to assume it was actually due to high 

atmospheric aerosols. The result is that the SASRAB clear sky shortwave radiation results will 

often be underestimated compared to the actual surface radiation. At the time of this algorithm’s 

development in the early 1980s, aerosol content over land via satellite imagery did not exist. 

However, with satellite aerosol data now available, the opportunity exists to correct the current 

and historical SASRAB data.

The work presented here is the first part of an effort to implement an improved clear sky 

algorithm for calculating shortwave GHI and DNI during times when retrievals from the satellite 

imagery indicate clear skies. In this work, the GHI and DNI from three industry-accepted clear 

sky models are computed and quantitatively compared to the current implementation of the 

satellite-based predicted radiation. Satellite image surface locations have been chosen to coincide 

with the National Oceanic and Atmospheric Administration (NOAA) Surface Radiation Budget 

Network (SURFRAD) ground stations to allow validation with measured radiation at the earth’s 

surface.

The three models chosen for comparison are the Simplified Solis Broadband (SolisBB) clear sky 

model developed by Pierre Ineichen [1], the Bird clear sky model developed by Richard Bird and 

Roland Hulstrom [2], and the REST2 model developed by Chris Gueymard [3]. Each model 

required conversion from its native format to the Mathworks/MATLAB scripting language and 

was validated against the results from existing Microsoft Excel spreadsheets or FORTRAN code 

before their final intercomparison for this work. 

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6 

2 Clear Sky Model Conversion to MATLAB

2.1 Model Conversion 

The clear sky models were converted to a vectorized MATLAB form using the sources listed in 

Table 1 as a starting point. These sources include the original model in Excel or FORTRAN

format and the respective original literature describing the development of each model. 

Table 1. Clear Sky Model Validation Sources

Clear Sky 

Model 

Author(s) Validation Source MATLAB File Name Reference

Bird R. Bird, R. Hulstrom BIRD_10_04_2011.

xls*

pvl_clearsky_bird.m [1]

SolisBB P. Ineichen solis-2008-tool.xls* pvl_clearsky_solisbb.m [2]

REST2 C. Gueymard REST2_v8.5_PC.f* pvl_clearsky_rest2.m [3]

*Obtained via private communication

In order to build a common input data structure for each model, the input format was designed 

around Sandia National Laboratories’ PV_LIB Toolbox for MATLAB [4]. PV_LIB is a 

collection of subroutines used for photovoltaic (PV) performance modeling that also includes the 

solar position and translation algorithms that are used throughout this work. All converted 

models use the PV_LIB Time (year, month, day, hour, minute, second) and Location (latitude, 

longitude, elevation) structure. However, each model uses a slightly different subset of input 

atmospheric parameters. In order to make the models consistent as callable subroutines, all the 

atmospheric parameters are passed in as a named structure. Each model then uses its respective 

parameters. Each parameter in this structure is a vector of the same length as the Time structure. 

Key parameters and important assumptions used by each of three clear sky models are 

summarized in Table 2. Some important notes regarding these inputs include: a) The solar 

constant used by each model is listed because accepted values have changed over the years and 

are reflected in each model’s development; b) The annual variation in extraterrestrial DNI is 

calculated using the PV_LIB function pvl_extraradiation for the Bird and REST2 models, but 

SolisBB uses an internal calculation. The result is less than +-0.8 W/m2 extraterrestrial DNI 

difference over the course of a year between the two methods; c) The solar zenith angle 

calculation used in the model may be either geometric or apparent. The geometric sun position 

angle is based purely on earth-sun geometry, whereas the apparent sun position angle accounts 

for atmospheric diffraction and can introduce corrections from 0.3° to 1.5° at very high 

geometric zenith angles from 85° to 92°. The apparent and geometric sun positions are discussed

in more detail in Section 2.2. 

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7 

Table 2. Key Model Input Requirements

Model Property Bird SolisBB REST2

Solar Constant, En

[W/m2]

1367 (corresponds to 

mult = 0.9751; see [1])

1367 1361.2

Extraterrestrial 

DNI Annual 

Variation

pvl_extraradiation Internal Modified 

pvl_extraradiation to 

match En = 1361.2

Zenith Angle

Calculation

Apparent Geometric Apparent

Surface Pressure Required by 

pvl_ephemeris, 

pvl_absoluteairmass

Required by 

pvl_ephemeris

Required by 

pvl_ephemeris

Surface 

Temperature

Required by 

pvl_ephemeris

Required by 

pvl_ephemeris

Required by 

pvl_ephemeris

Surface Albedo Required Assumed fixed 0.20 Required

Aerosol Optical 

Depth

380 nm, 500 nm required 700 nm required Uses α1, α2, β below

Ang Turb Coeff 

(<7 μm), α1

N/A N/A Required (typical 1.3)

Ang Turb Coeff 

(>7 um), α2

N/A N/A Required (= α1 if no other 

info available) (typical

1.3)

β (AOD Intercept 

at 1 μm)

N/A N/A Required

H2O Vapor [cm] Required (typical 0.5–5) Required (typical 0.5–5) Required (typical 0.5–5)

O3 [cm] Required (typical 0.3) Assumed 0.340 Required (typical 0.3)

NO2 [cm] Optional (default 2E-4)

% Forward Scatter 

Ba

Optional (default 0.84) Required (default 0.84) Uses scatter albedo 

below

Single Scatter 

Albedo (<7 μm), 

ω1

N/A N/A Required (typical 0.92)

Single Scatter 

Albedo (>7 μm), 

ω2

N/A N/A Required (typical 0.84)

2.2 Model Validation

Once all the models were converted to MATLAB, they were cross checked against the validation 

sources listed in Table 1. As a final check, each validation source (e.g., Excel worksheet or 

FORTRAN program) was modified to represent a full year of hourly daytime (zenith angle < 

85°) clear sky results. The same input datasets, including time, location (Table Mountain, 

Colorado, was arbitrarily chosen), zenith angle, elevation, air mass, extraterrestrial radiation, and 

the atmospheric parameters listed in Table 3, were input to each of the three models. The 

exception was the REST2 model written in FORTRAN: here, the solar position was calculated 

internally by the SunPSA subroutine [5], which closely matches the PV_LIB pvl_ephemeris

results. 

The literature for all the clear sky models does not specify whether they are designed for 

geometric solar position or refracted solar position. Review of Gueymard’s REST2 FORTRAN

code shows it uses a refracted sun angle obtained from the independently developed SunPSA 

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8 

subroutine [5]. Ineichen reports that the SolisBB model is designed around geometric sun 

elevation. The Bird model uses geometric solar position, but it should be noted that calculation 

of air mass uses the Kasten air mass correction formula, which produces refraction-corrected 

(apparent) air mass. In the end, all three models correct for the sun’s refraction through the 

atmosphere at some point, which becomes more important at zenith angles greater than 85°. 

Properly calculating air mass determines the thickness of the atmosphere for a given solar 

position. All the atmospheric properties (i.e., aerosol, H2O, O3, etc.) are then scaled by the air 

mass and used to determine the transmittance through each component. 

Table 3 shows the input parameters used for the model conversion validations. The aerosol 

optical depth (AOD) values as well as the Angstrom turbidity coefficients are all internally 

consistent with the annual climatological values at the SURFRAD Table Mountain site. AOD 

calculations and their relationship to the Angstrom coefficients, α and β, are discussed in Section 

4.1.1. 

Table 3. Model Input Parameters for Conversion Validation

Latitude/Longitude/Elevation 40.125°/-105.237°/1689 m

Surface Pressure/Temperature 831.31 mPa/15.67°C

Surface Albedo 0.2

H2O Vapor 0.5 atm-cm

O3 0.3 atm-cm

NO2 0.0002 atm-cm

% Forward Scatter Ba 0.84

AOD380/AOD500/AOD700 0.1306/0.0985/0.0706

α1, α2, β 0.9174/0.9174/0.0502

ω1, ω2 0.92/0.84

The model differences in W/m2 between the new MATLAB models and the original source 

models are shown in Figure 1 through Figure 6. All show relatively good agreement. The 

SolisBB conversion shows nearly perfect agreement with no observable bias beyond what are 

likely numerical precision differences between MATLAB and Excel. The Bird conversion shows 

a strong zenith response for clear sky DNI with differences up to +0.045 W/m2 and to a lesser 

extent for GHI differences up to +0.021 W/m2

. Finally, the REST2 model shows the largest 

differences, +2 W/m2 for DNI and +0.7 W/m2 for GHI. This has been tracked to the differences 

in the apparent sun position calculations performed in the solar position algorithms used by 

REST2 (SunPSA) and PV_LIB pvl_ephermeris, which is an implementation of an 

undocumented solar position algorithm. The differences between the model validation source 

and MATLAB version were deemed acceptable for all three models.

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9 

Figure 1. SolisBB chronological model error [W/m2] between MATLAB and original source. 

Figure 2. SolisBB model error [W/m2] between MATLAB and original source vs. zenith angle. 

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10 

Figure 3. Bird chronological model error [W/m2] between MATLAB and original source vs. time.

Figure 4. Bird model error [W/m2] between MATLAB and original source vs. zenith angle. 

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11 

Figure 5. REST2 model error [W/m2] between MATLAB and original source vs. time.

Figure 6. REST2 model error [W/m2] between MATLAB and original source vs. zenith angle. 

In the following sections, the model sensitivities with atmospheric input parameters are 

presented, and then each model’s respective performance is analyzed with respect to surface 

measurements.

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12 

3 Model Sensitivity

This section provides a general idea of how the clear sky models vary with atmospheric 

conditions. In general, all three models vary in the same manner for a given set of model inputs. 

For this analysis the Bird model is chosen to show model response, in order of decreasing 

sensitivity, to: AOD, water column, ozone, forward scattering ratio, and albedo at a fixed air 

mass of 1.5 (zenith angle = 48.2°) and site location equivalent to Desert Rock, Nevada (latitude

36.63°, longitude -116.02°, elevation 1,007 m).

Figure 7 clearly shows that for the typical ranges encountered for most sites, AOD is the 

strongest driver of clear sky determination, followed by water column. Thus, to calculate

accurate clear skies, the AOD (or alternatively the Angstrom turbidity coefficients) and water 

column are the two most important parameters to specify correctly. This is especially relevant for 

the DNI radiation estimate. For example, over the expected range of AOD at 380 nm the 

modeled GHI can vary up to 80 W/m2. However, the modeled DNI can vary up to 225 W/m2, the 

physical reason being that DNI transmission extinction is strongly driven by aerosol absorption 

and scattering. Appendix A contains more information on the rate of extinction of radiation for 

the most relevant atmospheric properties, such as Raleigh scattering, AOD, water vapor, and 

ozone. 

Figure 7. Sensitivity of Bird Model GHI and DNI to atmospheric properties.

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13 

4 Clear Sky Model Performance

The relative improvement in performance of the three newly converted clear sky models over the 

current SASRAB clear sky algorithm is quantified in this section. For this work, seven

continental NOAA SURFRAD ground measurement sites were used to quantify the performance 

of each of the clear sky models over the SASRAB clear sky estimates for the year 2009. These 

seven sites represent varying mid-latitude continental atmospheric sky conditions from the desert 

southwest (Desert Rock, Nevada) to the high-humidity south (Goodwin Creek, Mississippi). 

NOAA also has nine Integrated Surface Irradiance Study (ISIS) measurement stations that are 

well distributed throughout the United States, but their use here was rejected due to a lack of 

ground-based AOD measurements required for clear sky model input. The complete list of

SURFRAD sites is given in Section 4.1.1. The data flow for this work is detailed in Appendix B. 

4.1 Model Inputs

4.1.1 Surface Measurement Inputs

The seven NOAA SURFRAD sites supplying broadband radiation (GHI and DNI) at one-minute 

intervals are used for quantifying the clear sky model performance. In addition, the co-located 

surface meteorological measurements (surface pressure, temperature, relative humidity, wind 

speed, and direction) as well as AOD measurements are used to create more accurate sitespecific model inputs to the clear sky models. This allows the models to be evaluated with the 

best available surface data for determining local atmospheric conditions as model input. 

The SURFRAD sites used for analysis are shown in Table 4 along with the latitude and longitude 

of the center of the assigned 4-km SASRAB grid point and the GOES source for the imagery. 

Table 4. SURFRAD Sites Used for Model Performance Analysis

SURFRAD Site

(abbreviated name)

SURFRAD

Latitude/Longitude

SASRAB

Latitude/Longitude

GOES Source

Table Mountain, CO (tbl) 40.125/-105.237 40.109°/-105.257° West

Desert Rock, NV (dra) 36.62/-116.02 36.643°/-116.030° West

Goodwin Creek, MS (gwn) 34.25/-89.87 34.267°/-89.870° East

Penn State, PA (psu) 40.72/-77.93 40.700°/-77.900° East

Bondville, IL (bnd) 40.05/-88.37 40.051°/-88.327° East

Fort Peck, MT (fpk) 48.31/-105.1 48.335°/-105.065° West

Sioux Falls, SD (sxf) 43.73/-96.62 43.732°/-96.600° East

The SURFRAD GHI and DNI solar irradiance are preprocessed to align properly with the 

instantaneous GOES images that are taken twice hourly at :00 and :30 past the hour for GOES 

West and at :15 and :45 past the hour for GOES East. That is, the one-minute SURFRAD

irradiance is 30-minute time-averaged on the center of the corresponding GOES image 

timestamp.

AOD measurements are determined from the co-located Multi-Filter Rotating Shadowband 

Radiometer (MFRSR) (Harrison, Michalsky, Berndt, 1994). Because each clear sky model 

requires AOD at different wavelengths, the following approach was used for this analysis. The 

MFRSR measures AOD wavelengths at: 415 nm, 501 nm, 616 nm, 672 nm, and 868 nm. As 

stated earlier, the Bird model uses AOD at 380 nm and 500 nm, SolisBB uses AOD at 700 nm,

and REST2 uses the Angstrom Law coefficients β, the Angstrom turbidity coefficient, and α, the 

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14 

Angstrom exponent, to describe atmospheric turbidity. For this analysis the Angstrom 

coefficients were calculated for the REST2 model, and using those values, the AODs were back 

calculated for Bird (380 nm, 500 nm) and SolisBB (700 nm) to ensure that all AODs were 

internally consistent and described the same atmospheric turbidity conditions. A more detailed 

background on atmospheric turbidity and the Angstrom coefficient calculations is given in 

Appendix A. 

Additionally, co-located temperature and relative humidity are also used to determine 

atmospheric water column based on the method by Gueymard [6]. This is preprocessed on the 

same time scale as the irradiance data to coincide with the satellite image.

The variation of surface-measured AOD and half-hour estimated water column over the 2009 

evaluation period for each SURFRAD site is shown in Figure 8 with the mean value shown in 

blue, standard deviation shown as asterisks, and maximum and minimum values shown as red 

dots. This figure provides a visual clue to how much these two atmospheric parameters can vary 

over a year for a site.

Figure 8. Atmospheric input parameter statistics for AOD and water column. 

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15 

4.1.2 Satellite-Based Model Inputs

The GOES-based PATMOS-X/SASRAB product makes over 50 measured atmospheric 

parameters available in self-contained Hierarchical Data Format (HDF) files. However, for this 

analysis only the calculated insolation (GHI), insolation_diffuse (DHI), and cloud_type are used. 

Furthermore, this analysis is restricted to clear skies as determined by SASRAB. This is 

equivalent to those satellite image pixels where cloud_type is identified as “0.” 

The nearest GOES East or GOES West satellite pixel corresponding to a SURFRAD location is 

used to extract calculated insolation and cloud type for analysis. The SURFRAD site and 

corresponding SASRAB pixel (identified by latitude and longitude) used in this analysis are 

shown in Table 4. 

The satellite cloud_type is the primary filter for identifying clear skies with a secondary 

requirement that the satellite reported zenith angle be less than 85°. This threshold is intended to 

eliminate the highest zenith angles (sunrise and sunset). The times of all half-hour SASRAB data 

that pass these criteria are then intersected with the corresponding half-hour averaged 

SURFRAD data for subsequent processing of the 2009 analysis year. 

4.2 Model Results

4.2.1 Clear Sky Results

This section illustrates the performance differences between the three clear sky models compared 

to the SURFRAD measurements for clear sky conditions as identified by the SASRAB 

cloud_type index for the 2009 analysis year. Model performance is quantified for each of three 

models for two input parameter cases: 1) annual climatic values for the AOD and water column 

model inputs and 2) average daily values for AOD and half-hour water column estimates using 

the method by Gueymard. 

The model sensitivities shown in Section 3 would imply that more accurate AOD and water 

column would result in the largest clear sky model improvements. The scatter plots in Figure 9

and Figure 10 graphically demonstrate this for the Desert Rock SURFRAD site. This 

improvement can be seen at all sites and shows the important fact that providing higherresolution model inputs (here, daily and half-hourly rather than annual climatic values) is critical 

in accurately estimating clear sky irradiance. For the Desert Rock location, this results in mild 

improvement in modeled GHI from all three models—improving the average RMSE to 28.0 

W/m2 from 32.5 W/m2

. DNI improvement is striking—improving the model average value to 

63.3 W/m2 from 89.5 W/m2. Appendix C demonstrates how the clear sky models improve 

statistically as annual fixed climatic values are replaced with more refined daily AOD or halfhour water column estimates, and Appendix D includes individual scatter plots for all 

SURFRAD sites. The figures show the modeled vs. measured GHI and DNI for: 1) annual 

climatic AOD values and water column and 2) daily AOD and half-hour averaged water 

column—with both parameters derived from the SURFRAD datasets. 

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16 

Figure 9. GHI clear sky model comparisons for Desert Rock, Nevada: a) climatic AOD, fixed water 

column; b) daily AOD, half-hour water column.

Figure 10. DNI clear sky model comparisons for Desert Rock, Nevada: a) climatic AOD, fixed water

column; b) daily AOD, half-hour water column. 

The relative performance of each clear sky model is ranked based on summary statistics for the 

2009 analysis year using the refined daily model inputs. Figure 11 and Figure 12 bar charts show 

the statistical performance of each model with respect to the others for each SURFRAD site. 

Note the different axis limits for both the GHI and DNI statistics, which show that GHI has 

similar errors irrespective of the model, whereas DNI is more strongly influenced by AOD and 

water column, and its degree of “correctness” is more strongly dependent on the model being 

used. This same data is presented in Table 5. Furthermore, Table 6 presents average summary 

statistics for all SURFRAD sites combined, which are used in final model selection as described 

in Section 5. 

In general, all three clear sky models compare well, with REST2 slightly outperforming Bird and 

SolisBB for most sites. For most sites, the REST2 root mean square error (RMSE) is typically 

less than the Bird RMSE by 3 W/m2

 to 10 W/m2 for GHI, but not significantly different for DNI, 

with typical site-to-site differences of less than 2 W/m2. 

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17 

Figure 11. GHI model error comparisons for daily AOD and half-hour water column.

Figure 12. DNI model error comparisons for daily AOD and half-hour water column. 

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18 

Table 5. Summary Statistics for Individual SURFRAD Sites for Daily AOD, Half-Hour Water Column

SURFRAD Site GHI [W/m2] DNI [W/m2]

SASRAB SolisBB Bird REST2 SASRAB SolisBB Bird REST2

TblMtn, CO RMSE 102.85 36.35 37.74 39.27 353.90 94.94 97.51 96.36

MBE -71.28 8.46 -6.31 -14.76 -266.41 6.05 -13.51 -14.94

R 0.95 0.99 0.99 0.99 0.08 0.75 0.73 0.74

DesertRock, 

NV RMSE 148.08 30.21 28.42 25.36 435.08 67.50 61.34 61.09

MBE -109.17 16.30 2.97 0.63 -374.77 28.69 10.84 9.41

R 0.94 1.00 1.00 1.00 0.04 0.88 0.87 0.87

GoodwinCreek, 

MS RMSE 58.88 42.31 42.05 37.69 150.63 113.90 109.20 111.12

MBE -7.63 18.88 13.58 -1.55 -11.50 48.44 24.83 34.41

R 0.97 0.99 0.99 0.99 0.66 0.80 0.78 0.79

PennState, PA RMSE 86.97 41.69 39.44 35.29 233.88 138.02 127.41 130.33

MBE -46.00 21.06 9.78 1.03 -112.35 78.65 37.57 59.49

R 0.96 0.99 0.99 0.99 0.41 0.73 0.68 0.71

Bondville, IL RMSE 72.41 38.30 37.19 35.56 199.93 109.98 105.70 107.45

MBE -33.42 16.35 8.45 -7.41 -95.67 39.82 14.13 22.92

R 0.97 0.99 0.99 0.99 0.54 0.77 0.77 0.76

FortPeck, MT RMSE 54.66 37.20 31.46 29.25 186.46 78.48 75.43 75.12

MBE -20.07 24.42 11.86 3.16 -112.45 28.63 -7.80 8.70

R 0.98 0.99 0.99 0.99 0.59 0.83 0.83 0.83

SiouxFalls, SD RMSE 60.57 44.68 39.21 30.78 193.14 102.34 98.36 98.29

MBE -13.07 33.77 25.37 9.46 -81.13 48.08 30.46 31.68

R 0.97 0.99 0.99 0.99 0.51 0.79 0.78 0.78

Table 6. Summary Statistics for All SURFRAD Sites Combined

GHI [W/m2] DNI [W/m2]

SolisBB Bird REST2 SolisBB Bird REST2

RMSE 38.68 36.50 33.31 100.74 96.42 97.11

MBE 19.89 9.39 -1.35 39.76 13.79 21.67

R 0.992 0.991 0.991 0.791 0.777 0.784

The larger DNI RMSE in the statistics and additional scatter in the DNI comparison figures 

when compared to GHI also reveal an artifact of the analysis method as well as one of the issues 

with using satellite data. The satellite value is calculated based on an instantaneous pixel image 

corresponding to a 4-km by 4-km surface, whereas the SURFRAD ground measurements in this 

study are half-hour averages centered on the time of the satellite image and represent a point 

measurement. The cloud_type index may indicate a clear sky within that pixel at the instant the 

image was captured, but the measurement average over the half hour may be partly cloudy over 

that time, lowering the overall average. This affects the GHI, which is a combination of both 

direct and diffuse irradiance components—but more so the DNI. During partly cloudy 

conditions, DNI surface measurements can be anywhere between zero and clear sky irradiance 

within the 30-minute period. As with the GHI, the satellite instantaneous DNI irradiance 

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19 

estimates are based on some value in that range not necessarily reflecting the half-hour average. 

Furthermore, there may be no satellite pixel center point directly overhead of the surface 

measurement point. 

4.2.2 Model Run Times

All three clear sky models were evaluated for runtime performance. Results for per-point runtime 

are listed in Table 7. To clarify, because the models are vectorized in MATLAB, vectors of time, 

location, and model parameters are passed to each clear sky model. The per-point time is the 

total function execution time divided by the number of elements in the vectors, excluding the 

first call to the function, which has been found to have considerable overhead in terms of 

execution time for all the models. In practice it has been found that the lengths of the input 

parameter vectors passed to the clear sky models make negligible differences in per-point timing. 

Thus, this level of performance should be expected for a production version of these models, 

which will be used to correct the SASRAB clear sky GHI and DNI values. 

Table 7. Model Per-Point Timing

Model Per-Point Time [μsec]

SolisBB 3.95

Bird 4.12

REST2 6.23

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5 Model Selection and Conclusions

The summary statistics in Table 6 point to Bird and REST2 as the most promising of the three 

models for replacing the SASRAB clear sky estimator, with overall RMSE for GHI of 36.50 

W/m2 and 33.31 W/m2

,respectively, and DNI errors of 96.42 W/m2 and 97.11 W/m2, 

respectively. In final production, each SASRAB clear sky satellite pixel value for GHI and DNI 

will be replaced with one of these models’ clear sky values. Thus, runtime will be critical—

which leads to the final selection of the Bird model, which runs in two-thirds of the time of 

REST2. This selection should not preclude the use of REST2 in other applications. This project 

is solely focused on broadband GHI and DNI values (i.e., wavelengths from .3–3 μm). However, 

REST2 is spectrally selective and specifically designed to estimate broadband, illuminance, and 

photosynthetically active radiation (PAR) values that can be used in other disciplines. 

Furthermore, with some modification by its author, REST2 could provide irradiance values that 

coincide with particular PV materials’ wavelengths (e.g., 0.3–1.1 μm for silicon), which could 

potentially be useful to more accurately estimate power production from these materials.

Overall conclusions from this work are as follows: 

• Any of the three clear sky models used in this analysis is superior to the current SASRAB 

clear sky model. 

• For this analysis, the SASRAB cloud_type parameter has been found to be an accurate 

clear sky identifier.

• High-temporal-resolution model input data parameters such as AOD and water vapor 

significantly improve clear sky model accuracy.

• The Bird clear sky model combines relatively good performance coupled with 

computational efficiency.

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References

[1] Ineichen, P. “A Broadband Simplified Version of the Solis Clear Sky Model.” Solar Energy

(82:8), 2008; pp. 758–762. 

[2] Bird, R.E.; Hulstrom, R.L. A Simplified Clear Sky Model for Direct and Diffuse Insolation on 

Horizontal Surfaces. SERI/TR-642-761. Golden, CO: Solar Energy Research Institute, 1981. 

[3] Gueymard, C. “Clear-sky irradiance predictions for solar resource mapping and large-scale 

applications: Improved validation methodology and detailed performance analysis of 18 

broadband radiative models.” Solar Energy (86:8), 2012; pp. 2145–2169. 

[4] PV_LIB MATLAB Framework. Available online: http://pvpmc.org/pv-lib/. 

[5] Blanco-Muriel, M., et al. “Computing the solar vector.” Solar Energy (70:5), 2001; pp. 431–

441. Available online: http://www.psa.es/sdg/sunpos.htm 

[6] Gueymard, C. “Analysis of monthly average atmospheric precipitable water and turbidity in 

Canada and northern United States.” Solar Energy (53:1), 1994; pp. 57–71. 

[7] Gueymard, C. “SMARTS2, simple model of the atmospheric radiative transfer of sunshine: 

Algorithms and performance assessment.” FSEC-PF-270-95. Cocoa, FL: Florida Solar 

Energy Center, 1995.

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22 

Appendix A

Optical Depth Effect on Irradiance

All three clear sky models require a measure of the atmospheric optical depth (AOD) to 

determine the attenuation or “extinction” of extraterrestrial radiation as it passes through the 

atmosphere. The exponential law of solar radiation extinction takes the form 

I(λ) = I0(λ)e

− τ(λ)

cos(θ) (1) 

where I0(λ) is the extraterrestrial flux at wavelength(λ), I(λ) is the flux reaching the ground, θ is 

the solar zenith angle, and τ(λ) is the optical depth of the vertical column of atmosphere over the 

observation location at that wavelength. The total optical depth is typically comprised of three 

components: Raleigh scattering, aerosols, and atmospheric gas absorption, typically shown as

τ(λ) = τR(λ) + τAOD(λ) + τg(λ) (2) 

Raleigh optical depth, τR, is the scattering of light due to individual molecules in the atmosphere 

and is a very strong driver of solar radiation extinction. In general, τR varies as λ-4 and can be 

relatively easily estimated from atmospheric pressure. Although τR is an important extinction 

term, it doesn’t figure into model sensitivities because it is so easily and accurately determined 

for any site. 

Aerosol optical depth, τAOD, is a measure of the scattering and absorption of light by dust and 

haze and is the second most important contributor to radiation extinction after Raleigh scattering. 

Its extinction percentage over the whole spectrum is often the same order of magnitude as 

Raleigh scattering. It is the most complex atmospheric parameter to accurately estimate and 

strongly affects clear sky calculations, as aerosol extinction is typically large in the middle of the 

visible spectrum where the solar irradiance is highest. AOD is generally a smooth extinction 

function dependent on wavelength 

τ(λ) = βλ−α (3) 

where α and β are called the Angstrom exponent and Angstrom turbidity coefficient, 

respectively, and can be determined empirically from satellite data (e.g., NASA MODIS data 

product) or from surface-based narrow-band radiometers, such as the MFRSR or sun photometer 

found in the NOAA SURFRAD or NASA AERONET (AErosol RObotic NETwork) systems, 

respectively. 

Finally, gaseous optical depth, τg, is radiation extinction due to absorption by atmospheric gases 

(H2O, ozone, N2). The most important of these is water column, which has strong absorption 

bands in the near-infrared. Water column is site dependent and can be estimated using the 

method by Gueymard [6] using surface measurements of temperature and relative humidity. 

Alternatively, water column can be estimated from satellite measurements (e.g., NASA MODIS 

data products). Ozone and nitrogen are two other contributors, but for these clear sky models, 

fixed values as recommended by the authors have provided sufficient accuracy. 

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The spectral distribution of the extinction of direct radiation by each of these constituents is 

shown graphically in the figure below. The data for this figure is taken from the NREL 

SMARTS model developed by Gueymard [7] using typical atmospheric conditions. Originally 

detailed in Section 3, the reason for the particular model sensitivities is now more evident. 

Aerosols drive the radiation extinction, followed by Rayleigh scattering, water column, and 

ozone absorption. Other atmospheric gases (N2, O2, NO2) also contribute to this attenuation, but 

to a much lesser degree. Most clear sky models neglect or simply combine the contribution of 

these less important constituents.

Figure A1. Direct solar transmission of major components of the clear sky atmosphere 

superimposed on spectral DNI for the same conditions. 

Determination of Angstrom Turbidity Coefficients for Model Validation 

For this model validation work, the clear sky models require inputs of either τAOD at specific 

wavelengths (Bird, SolisBB) or the Angstrom exponent and turbidity coefficient α and β

(REST2). Using the relationship in Equation (3), the natural log of each side can be taken, 

resulting in: 

ln(τλ) = ln(β) − α ∙ ln(λ) (4)

This equation has the form of a linear equation for λ measured in microns. In a stable 

atmosphere, this results in a straight line with an intercept of β = τ (λ = 1 μm) and slope α. Once 

α and β are determined, the AOD at any specified wavelength can be calculated from Equation 

(3).

Surface-measured τAOD is required for clear sky model validation. The narrow-band radiometer 

that performs this measurement is the SURFRAD MFRSR and its associated data stream. The 

MFRSR makes simultaneous measurements of the solar irradiances at six wavelengths 

(nominally 415 nm, 500 nm, 615 nm, 673 nm, 868 nm, and 940 nm) at one-minute intervals 

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throughout the day. Time series of direct solar beam extinctions and horizontal diffuse fluxes are 

derived from these measurements, as well as AOD at each of these wavelengths. Only the 500-

nm and 868-nm channels are used to calculate the Angstrom turbidity coefficients. The choice of 

the 500-nm and 868-nm MFRSR channels for computation of α and β is explained by the 

instrument co-developer Joseph Michalsky of NOAA. He states that “those are the most trusted 

channels as they have little to no ozone correction, and, more importantly, they do not typically 

show light leakage that the two 600’s have in the past. ... Also, 415 is typically noisier because 

of the response of the silicon detector.”

As an example, a single one-minute measurement from the MFRSR at Desert Rock, Nevada, for 

May 20, 2009, at 9:00 a.m. is shown below on a ln(λi) vs. ln(AODi) scale for the following data: 

T.able A1. Desert Rock τAOD Values from the MFRSR

Wavelength [μm] Measured AOD

0.415 0.1

0.501 0.089

0.616 0.061

0.672 0.059

0.868 0.051

Figure A2. Determination of Angstrom turbidity coefficients. 

Calculating the line between AOD at 500 nm and 868 nm gives a slope of -1.013 and y-intercept 

at -3.12. Angstrom exponent α is defined as the negative of the slope. Hence, α = 0.986. The yintercept is lnβ = -3.12. Hence, the Angstrom turbidity coefficient β = e-3.12 = 0.044. Gueymard 

makes the further distinction that α and β be determined separately above and below 7 μm for 

REST2. However, for this analysis it was assumed the slope and intercept were the same for 

AOD above and below 7 μm, which Gueymard recommends when additional spectral data is not 

available. 

A final screening measure recommended by Gueymard was used to limit potentially bad 

measured data. Here, α was limited to within 0.3 and 2.5 and β limited to within 0.001 and 0.5. 

Any α, β pairs outside this range were replaced with annual climatic values.

y = -1.0131x - 3.1194

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

-1 -0.8 -0.6 -0.4 -0.2 0

ln(AOD) 

ln(wavelength(um)) 

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The SURFRAD AOD data was processed to create both a daily and an annual climatic dataset 

for all the SURFRAD stations. The daily value dataset is the daily average of the one-minute 

AODs for the entire 2009 analysis year, excluding any data flagged as bad, which includes any 

cloudy periods. Any missing days are replaced with the annual average. The annual dataset

consists of the annual mean climatic values for all flagged good measurements in 2009. These 

datasets were the basis of the AOD data used for the clear sky model evaluations. 

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Appendix B

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Appendix C

Model Improvement Due to Use of Daily AOD and Half-Hour Water Column 

GHI

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DNI

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Appendix D

GHI Clear Sky Model Comparisons for All Sites

Table Mountain GHI

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Table Mountain DNI

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Desert Rock GHI

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Desert Rock DNI

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Goodwin Creek GHI

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Goodwin Creek DNI

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Penn State GHI

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Penn State DNI

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Bondville GHI

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Bondville DNI

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Fort Peck GHI

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Fort Peck DNI

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Sioux Falls GHI

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Sioux Falls DNI

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