Modeling Solar Radiation at the Earth’s Surface

Geometric, Environmental and Equipment Uncertainty

Additional contributors to the uncertainty of all radiometers include: temperature coefficients, linearity, thermal electromotive forces, and electromagnetic interfer­ence. Moreover, the field of view of many pyrheliometers differs from that of the ref­erence ACR, which results in slight differences in the circumsolar radiation sensed during calibration. For pyranometers, inaccuracies in the zenith angle calculation and in the sensor’s cosine response must be considered. The latter issue is usually significant, particularly when measuring global irradiance under clear skies, because of the predominance of the direct beam. This issue is further discussed below and in Sect. 7. Finally, the specifications and performance of the data logging equipment (resolution, precision, and accuracy) must also be considered.

A pyranometer’s departure from perfect Lambertian response is often called “co­sine error”. This has been documented in various publications (e. g., Michalsky et al. 1995; Wardle et al. 1996), with the result that improved calibration techniques using variable responsivity coefficients (rather than the conventional fixed single calibration number) became the recommended procedure (Lester and Myers 2006). What follows is an overview of the most advanced method currently used to cali­brate pyranometers at research-class sites.

This method must be performed during a whole clear summer day, with z reach­ing values as close to 0 as possible. The responsivity for each zenith angle, Rs(z), is calculated as before. The calibration data for the morning and afternoon are sepa­rately segregated into a number of zenith angle intervals. These data points are then fitted to a high-order polynomial in the form of Eq. (1.6):

rs (z)am/pm = E a cos'(z) (16)


where the ai are n + 1 coefficients for each morning and afternoon set of z. Thus there are two n-degree polynomials in cos(z) mapping the responsivity curve of each pyranometer. This method can be used with various z intervals. The original version used 5-degree intervals (Reda 1998). The current version uses 2-degree intervals, with n = 48.

With this sophisticated approach, uncertainties of no more than ±2.1% in mea­sured pyranometer data can be achieved. This is a significant improvement over the conventional method of using a single value, Rs (z0), which may induce errors of up to ±10% at zenith angles largely separated from z0.

Example data for various calibration results for a single pyranometer, report­ing responsivity as a function of zenith angle in bins of 2°, and 9°, as well as derived coefficients for a fit to Eq. (1.6), may be found on the CD-ROM, in the folder CM22_all_Rs_NREL2006_02. In the folder NREL2006_02COEFF are the re­sults of coefficient fits to several models of pyranometer (Kipp and Zonen CM-22 and CM6b, Eppley PSP, and Li-Cor LI200SB). Also included in that folder is a spreadsheet file, ‘RCC-Function_RsCalculator. xls’, which implements calculation of responsivities as a function of zenith angle using the coefficient files. The calcu­lated uncertainties published in the ‘2006-02_NREL_SRRL _BMS. pdf’ and ‘2006- 02_ARM_SGP_Full. pdf’ reports on the CD-ROM are based upon the techniques specified in the GUM and current knowledge of the sources of uncertainty, and their estimated magnitudes, during outdoor calibrations.

Basic calibration uncertainties of about 2.1% for pyranometers, and 1.8% for pyrheliometers, at “full scale” (i. e., 1000Wm~2 or “1 sun”) are the very best that can be expected with present instrumentation. This is equivalent to an uncertainty of 21W m~2 for global solar radiation and 18W m~2 for direct normal radiation.

When radiometers are deployed to the field, further sources of uncertainty arise, such as differing (usually lower resolution) data logging, cleanliness, and even at­mospheric conditions, which must be considered in addition to the basic calibration uncertainty. Field measurements under varying, sometimes harsh, environmental conditions can easily double or triple the basic uncertainties (Myers 2005; Myers et al. 2004).

Solar radiation model developers must be aware that random and bias errors in models represent how well the model reproduces the measured data, and not nec­essarily the absolute accuracy of the radiation component. The particular case of empirical radiation models is worth discussing further. Such models are not based on algorithms that attempt to describe the physics of the various extinction pro­cesses in the atmosphere, but on simple relationships using some correlations be­tween different phenomena and the observed irradiance. For instance, it has long been known that monthly-average global irradiation was roughly linearly correlated with sunshine duration. Because such models use irradiance observations for their development, any systematic or random error in irradiance measurement is embed­ded in the model. If the model is based on irradiance data that have been measured at site X with one set of instruments, and is used to compare its predictions to irra - diance measured at site Y with a different set of instruments, a part of the apparent prediction errors at site Y will be due to a mismatch between the instrument char­acteristics. This insidious problem is usually overlooked, but is one possible reason why such models are rarely found of “universal” applicability.

Modeling Solar Radiation at the Earth’s Surface

Quality Assessment Based Upon Comparison with Models

Many models based on the physics of radiation transfer through the clear atmo­sphere have been developed (Lacis and Hansen 1974; Atwater and Ball 1978; Hoyt 1978; Bird and Hulstrom 1981a, …

Solar Horizontal Diffuse and Beam Irradiation on Clear Days

There exist a number of models to determine the solar horizontal diffuse irradia­tion on a clear day (Kondratyev 1969) but they are complex and have very stringent conditions. Similarly, there …


Reading the twenty chapters of this book caused me mixed reactions, though all were positive. My responses were shaped by several factors. Although I have main­tained a “watching brief’ on …

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