Astronomy

Measured values of the solar irradiance at other values than 1 AU?

Measured values of the solar irradiance at other values than 1 AU?


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It is straight forward to estimate the total solar irradiance of a planet using the Earth's solar constant, and scaling it according to the (mean) distance of the planets. But for which planets have measurements been made? A quick search of the usual suspects likearxiv.orgorscholar.google.comdid not yield any conclusive or easy to interpret results.

  • Mercury: $12,121 { m W}/{ m m}^2$ according to Wolframalpha which use the simple scaling. However, there are other sources, one of which claiming to cite a reference handbook's value of $9,937 { m W}/{ m m}^2$ which is a rather large deviation.
  • Venus: $2,601.3 { m W}/{ m m}^2$, according to NASA's Venus fact sheet, which is reasonable close to $2,569 { m W}/{ m m}^2$ according to Wolframalpha
  • Earth: $1,361^{+1}_{-0} { m W}/{ m m}^2$ according to Wikipedia on the solar constant
  • Mars: around $560.7 { m W}/{ m m}^2$, this is probably measured more exactly
  • Jupiter: probably around $52.7 { m W}/{ m m}^2$

How accurate are the values? What are the error bars?


Total Solar Irradiance (TSI) datasets: An overview

The solar radiation arriving at Earth (once known as the “solar constant”, now usually referred to as Total Solar Irradiance (TSI)), is the most fundamental of climate parameters as it indicates the totality of the energy driving the climate system. All climate models need to prescribe a value for it, either explicitly or implicitly, but its measurement with the precision and stability needed for climate studies has proved challenging. -From Expert Guidance by Drs. Joanna Haigh and William Ball please see the "Expert Guidance" tab for more.

TSI datasets generally fall under two categories, historical reconstructions and satellite-based radiometric measurements:

Historical reconstructions: The two key datasets for long, historical records of TSI are (1) SATIRE and (2) NRLTSI. The NRLTSI data are being produced by NOAA as a Climate Data Record, available in a useful netCDF format with data for 1882-present.

For CMIP6 historical experiments, the recommended solar forcing data (see "Get Data" tab for access) are based on a mean of these historical reconstructions.

Contemporary radiometric measurements from satellites: Recently, TSI has been measured by the Total Irradiance Monitor (TIM) two versions of this instrument have flown on the SORCE spacecraft (providing TSI measurements since 2003) and the TCTE platform (providing TSI measurements since 2013). SORCE and TCTE data are available through the University of Colorado's Laboratory for Atmospheric and Space Physics.

A joint European Space Agency-NASA experiment known as VIRGO, which encompasses 3 instruments for TSI on the SOHO spacecraft, has provided TSI data from the mid-1990s to present.

Earlier satellite-era TSI measurements came from the ERB/HF, ACRIM, and ERBS missions. Data from these earlier missions have been combined with contemporary VIRGO measurements to form a composite satellite-era TSI record known as PMOD (named for the institute in Davos, Switzerland that produces the data), which spans 1978-present.

Please see the "Expert Guidance" tab for more background on these datasets, and the "Get Data" tab for links to access the datasets.


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2. Space-Based Measurements of Total Solar Irradiance

[9] All TSI instruments employ a common measurement approach, that of active cavity electrical substitution radiometry, in which an absorptive blackened cavity is maintained in thermal equilibrium by application of measured electrical heater power while incident radiant solar power passing through a defining precision aperture of calibrated area is modulated via a shutter [ Butler et al., 2008 ]. Once on orbit, radiometric calibrations drift for multiple reasons including solar degradation of the absorptive cavity's interior surfaces, electronic degradation affecting the measured heater power, surface degradation of the precision aperture, and varying surface emissivities and temperatures that alter instrument thermal backgrounds. Instruments utilize various approaches to minimize and quantify known sources of on-orbit instabilities.

2.1. The 32-Year Space-Based Record

[10] The space-based TSI record comprises measurements from more than ten radiometers spanning three solar activity cycles. As Figure 1a illustrates, the SORCE/TIM TSI values are lower than prior measurements made by the Earth Radiometer Budget Experiment (ERBE) on the Earth Radiation Budget Satellite (ERBS) [ Lee et al., 1995 ], VIRGO on the Solar Heliospheric Observatory (SoHO) [ Fröhlich and Lean, 2004 Fröhlich, 2009 ], and the ACRIM instruments on the Solar Maximum Mission (SMM), Upper Atmosphere Research Satellite (UARS), and ACRIMSat [ Willson and Mordvinov, 2003 ]. Ground calibrations of these flight radiometers relied on component rather than system level measurements since irradiance standards prior to their launches lacked desired absolute accuracies.

[11] Because uncertainties of individual irradiance observations exceed solar irradiance variations (∼0.1%), instrument stability and measurement continuity are relied upon to discern real solar irradiance variations in the database. Instrument stability is achieved primarily by exposing redundant radiometer cavities to different accumulations of solar radiation to quantify exposure-dependent degradation effects that are then corrected in reported solar signals. Overlap of sequential radiometer observations permits corrections for absolute offsets and, if sufficiently long, validation of instrumental drifts.

[12] Although ACRIM I, ACRIM II, ACRIM III, VIRGO, and TIM all track degradation with redundant cavities, notable and unexplained differences are evident in Figure 3a among their reported irradiance variations and in Figure 3b with the modeled influences of sunspots and faculae (shown in Figure 1c and used to estimate climate responses in Figure 2). Features not easily attributable to solar activity include an annual cycle that is nearly in phase with the Sun-Earth distance in ACRIM III data, and 90-day spikes in the VIRGO data coincident with SoHO spacecraft maneuvers that are most apparent during the 2008 solar minimum. Disagreement among overlapping observations, as apparent in Figure 3, indicates undetected drifts that suggest the TSI record is not sufficiently stable to discern solar changes on decadal time scales. For example, Figure 3b shows that only the ACRIM composite shows irradiance increasing by ∼1 W m −2 between 1986 and 1996 if real, the solar origins of this increase are ambiguous since it is also absent in the model.

[13] The sunspot and faculae model accounts for 92% of the irradiance variance that TIM observes. The correlation of TIM observations and the model is 0.96 (for 2481 daily mean values from 2003 to 2009), the 1σ standard deviation of their differences is 0.09 W m −2 , and the trend in the differences is −0.009 W m −2 (−0.00066%) per year, well within the TIM stability. The differing and lower correlations between the model and each of the three irradiance composites (0.91 for PMOD, 0.81 for ACRIM, and 0.92 for RMIB) over their entire records suggest the presence of variations in these composite time series, as apparent in Figure 3b, that cannot be explained by our current understanding of the sunspot and facular influences on total solar irradiance.

2.2. Resolving Instrumental Differences

[14] Specific technological advances designed to improve measurement accuracy and stability distinguish the TIM from prior space-based radiometers. These include the forward placement of the defining aperture relative to the view-limiting aperture (rather than the inverse) to reduce stray light (see Figure 4a), phase-sensitive (rather than time domain) signal detection to reduce sensitivity to thermal drifts and signals not in phase with the shuttered sunlight, and etched metal black (instead of painted) cavity interiors to minimize degradation from solar exposure. The TIM also employs a digital signal processor controlled servo system with feed forward that anticipates radiometer heater changes to nearly eliminate thermal fluctuations as sunlight is modulated [ Kopp and Lawrence, 2005 ]. Furthermore, only the TIM acquires multiple daily measurements of dark (∼4K) space to directly determine and correct for instrument thermal background signals.

[15] To better understand the causes of instrument offsets, NIST and NASA hosted a workshop in 2005 to discuss instrument uncertainties and stabilities in detail. Recommendations to resolve the instrument discrepancies [ Butler et al., 2008 ] include: validating optical power measurement accuracy by comparing ground-based versions of the instruments to laboratory references, such as those at NIST NIST validation of aperture area calibrations using flight spare apertures from each instrument and applying corrections for diffraction from the view-limiting aperture (a sizeable correction not applied by all instruments).

[16] Notably, for the ACRIM instrument NIST determined that diffraction from the view-limiting aperture contributes a 0.13% signal not accounted for in data from the three ACRIM instruments [ Butler et al., 2008 ]. This correction lowers the reported ACRIM values, resolving part of ACRIM's difference with TIM. In ACRIM and all other instruments, the precision aperture used to define the measured solar beam is deep inside the instrument with a larger view-limiting aperture at the front, which, depending on edge imperfections, in addition to diffraction can directly scatter light into the absorbing cavity. Additionally, this design allows into the instrument interior two to three times the amount of light intended to be measured if not completely absorbed or scattered back out, this additional light produces erroneously high signals. In contrast, the TIM's optical design places the precision aperture at the front so only light intended to be measured enters (Figure 4a).


Improved measurements of sun to advance understanding of climate change

Scientists have taken a major step toward accurately determining the amount of energy that the sun provides to Earth, and how variations in that energy may contribute to climate change.

In a new study of laboratory and satellite data, researchers report a lower value of that energy, known as total solar irradiance, than previously measured and demonstrate that the satellite instrument that made the measurement -- which has a new optical design and was calibrated in a new way -- has significantly improved the accuracy and consistency of such measurements.

The new findings give confidence, the researchers say, that other, newer satellites expected to launch starting early this year will measure total solar irradiance with adequate repeatability -- and with little enough uncertainty -- to help resolve the long-standing question of how significant a contributor solar fluctuations are to the rising average global temperature of the planet.

"Improved accuracies and stabilities in the long-term total solar irradiance record mean improved estimates of the sun's influence on Earth's climate," said Greg Kopp of the Laboratory for Atmospheric and Space Physics (LASP) of the University of Colorado Boulder.

Kopp, who led the study, and Judith Lean of the Naval Research Laboratory, in Washington, D.C., published their findings January 14 in Geophysical Research Letters, a journal of the American Geophysical Union.

The new work will help advance scientists' ability to understand the contribution of natural versus anthropogenic causes of climate change, the scientists said. That's because the research improves the accuracy of the continuous, 32-year record of total solar irradiance, or TSI. Energy from the sun is the primary energy input driving Earth's climate, which scientific consensus indicates has been warming since the Industrial Revolution.

Lean specializes in the effects of the sun on climate and space weather. She said, "Scientists estimating Earth's climate sensitivities need accurate and stable solar irradiance records to know exactly how much warming to attribute to changes in the sun's output, versus anthropogenic or other natural forcings."

The new, lower TSI value was measured by the LASP-built Total Irradiance Monitor (TIM) instrument on the NASA Solar Radiation and Climate Experiment (SORCE) spacecraft. Tests at a new calibration facility at LASP verify the lower TSI value. The ground-based calibration facility enables scientists to validate their instruments under on-orbit conditions against a reference standard calibrated by the National Institute of Standards and Technology (NIST). Before the development of the calibration facility, solar irradiance instruments would frequently return different measurements from each other, depending on their calibration. To maintain a long-term record of the sun's output through time, scientists had to rely on overlapping measurements that allowed them to intercalibrate among instruments.

Kopp said, "The calibration facility indicates that the TIM is producing the most accurate total solar irradiance results to date, providing a baseline value that allows us to make the entire 32-year record more accurate. This baseline value will also help ensure that we can maintain this important climate data record for years into the future, reducing the risks from a potential gap in spacecraft measurements."

Lean said, "We are eager to see how this lower irradiance value affects global climate models, which use various parameters to reproduce current climate: incoming solar radiation is a decisive factor. An improved and extended solar data record will make it easier for us to understand how fluctuations in the sun's energy output over time affect temperatures, and how Earth's climate responds to radiative forcing."

Lean's model, which is now adjusted to the new lower absolute TSI values, reproduces with high fidelity the TSI variations that TIM observes and indicates that solar irradiance levels during the recent prolonged solar minimum period were likely comparable to levels in past solar minima. Using this model, Lean estimates that solar variability produces about 0.1o Celsius (0.18o Fahrenheit) global warming during the 11-year solar cycle, but is likely not the main cause of global warming in the past three decades.


Understanding Radiance (Brightness), Irradiance and Radiant Flux

Evaluation of performance of a radiation source has to involve radiometry – the measurement of quantities associated with radiation. To those new to the field, the units and terms, such as Radiance, Irradiance and Radiant Flux, may be unfamiliar. In addition, non-standard terms such as brightness, radiant power, flux, and intensity are often used casually without explanations. Finally, photometry terms such as luminance are often misused when discussing radiometry situations.

This Application Note attempts to explain radiometry terms and units, to differentiate them from photometry terms, and to clarify the non-standard terms commonly heard. In addition we will illustrate how the radiometric terms help in selecting an appropriate light source for a particular application.

Definitions of radiometry and photometry

  • Radiometry is the science of measuring radiation energy in any portion of the electromagnetic spectrum. In practice, the term is usually limited to the measurement of ultraviolet (UV), visible (VIS), and infrared (IR) radiation using optical instruments.
  • Photometry is the science of measuring visible radiation, light, in units that are weighted according to the sensitivity of the human eye. It is a quantitative science based on a statistical model of the human visual perception of light (eye sensitivity curve) under carefully controlled conditions.

Definition of the frequently used Radiometry Units

The SI System (Système International d’unités) defines six radiometric units, of which three are most commonly used for describing the effectiveness of radiation coupling between a light source and an optical system. These most commonly used units are: (1) Radiance (2) Irradiance and (3) Radiant Flux. Radiance is often casually called “brightness”, a term also used in photometry to describe the perception of human eyes looking at a light source. An example of brightness perception will be given in the following section.

Radiance

Figure 1. Radiance (R) of source is the Power (P) emitted from the source emitting Area (A) and propagated in the Solid Angle (Ω).

Figure 2. Steradian [sr] is a unit for measuring solid angles (Ω) defined by the solid angle that projects on the surface of a sphere, with a radius of r, having an area of A = r2 (Ω = A/r2 = r2/r2 = 1 [sr]).

The radiance of a source is increased by increasing its emitted power, by making the emitting area of the source smaller or by emitting the radiation into a smaller solid angle. Strictly speaking, radiance is defined at every point on the emitting surface, as a function of position, and as a function of the angle of observation. Often, as in the example above we use radiance of a source to mean the radiance averaged over a finite sized aperture and over some solid angle of interest.

Radiance is a conserved quantity in an optical system so that radiance measured as watts per unit area per unit solid angle incident on a detector will not exceed the radiance at the emitter. In practice, for any bundle of rays mapping an emitter to a detector, the radiance seen at the detector will be diminished by the light which is absorbed along the way or scattered out of the solid angle of the bundle of rays reaching the detector.

Let us consider an example. Suppose one observes with the eye a 35W Xenon (Xe) short-arc lamp, and then a 60W straight tube fluorescent lamp, both at a similar distance of a few meters. (As background information, the 35W arc lamp emits significantly less visible power than the 60W fluorescent tube.) Which light source is perceived to be brighter, or in radiometric terms, has higher radiance? The Xe short-arc lamp is perceived to be much brighter, although the 35W arc lamp emits less power than the 60W fluorescent lamp. This is as a result of the much smaller emitting area (A) of the short-arc lamp compared to the very large emitting area of the fluorescent lamp, while the eye is receiving the radiation at more or less the same solid angle (Ω) when the distance between the eye and the source is the same. The eye’s lens forms a bright image of the Xe arc on a very small area of the retina and the eye does not feel comfortable. The larger area fluorescent lamp will form an image over a much larger area on the retina, which the eye can tolerate more comfortably. The arc-lamp has a much higher radiance than the fluorescent lamp, even though it emits less power.

By way of a further example, imagine using the Xe and fluorescent lamps to illuminate a small area such as the end of a 200 μm diameter optical fiber. As a result of the higher source radiance the radiation from the 35W Xe arc-lamp can be much more efficiently collected and focused into the fiber. In contrast, the low radiance 60W fluorescent lamp will be ineffective in coupling its radiation energy into the fiber, no matter what type of focusing optic is used.

Energetiq’s Laser-Driven Light Sources have ultrahigh radiance from their small emitting area (

100 μm diameter). Radiation from such a high radiance and small emitting area source can be even more efficiently coupled into the 200 μm diameter optical fiber described above. This is also true for other optical systems with small apertures and a limited accepting solid angle - optical systems with small ‘étendue’ - such as the narrow slits of a monochromator. (For further discussion of étendue, see Application Note #002-2-14-2011, Etendue and Optical Throughput Calculations.)

Irradiance

Irradiance is the radiometry term for the power per unit area of electromagnetic radiation incident on a surface. The SI unit for irradiance is watts per square meter [W/m2], or milliwatts per square millimeter [mW/mm2]. (Irradiance is sometimes called intensity, but this usage leads to confusion with another standard, but infrequently used, radiometry unit —Radiant Intensity — which is measured in watts per steradian.)

If a point radiation source emits radiation uniformly in all directions and there is no absorption, then the irradiance drops off in proportion to the distance squared from the source, since the total power is constant and it is spread over an area that increases with the distance squared from the radiation source. To compare the irradiance of different sources, one must take into account the distance from the source. A 50 cm distance is often used for such measurements.

Irradiance is a useful measure for applications where power must be delivered to large areas. For example, illuminating a classroom or a football field is primarily a question of delivering a certain number of watts per square meter. This can be achieved by using a single high power source. However, since irradiance does not depend on solid angle, multiple sources can be combined, illuminating the walls or the field from different angles.

The irradiance of a source is not the most useful measure when designing an efficient optical coupling system that collects radiation from a source, and then delivers the radiation into an optical instrument. Such optical instruments will have a limited entrance aperture and a limited acceptance solid angle. In such cases it is the radiance of the source (its ‘brightness’) that is most useful.

Radiant Flux

Radiant flux is radiant energy per unit time, also called radiant power [W, mW or μW]. Radiant flux is often used to describe the radiation power output of a radiation source, or the radiation power received by an optical instrument. Examples of radiant flux are: the radiation power passing through a pinhole the radiation power emerging from the optical fiber of a fiber-coupled laser the radiation power received by a power detector.

The units of Radiant Flux do not include area or solid angle, and are therefore not helpful in determining whether a particular light source with a particular radiant flux will be useful in delivering its power to an optical instrument. In our earlier example, the 60W fluorescent tube emits a greater radiant flux (power) than the 35W Xe arc-lamp. But, with an appropriate focusing optic, the arc lamp will deliver a higher radiant flux to the 200μm diameter optical fiber. A Laser-Driven Light Source, such as Energetiq’s EQ-99, may have a lower radiant flux emitted than the 35W arc-lamp, but its higher radiance allows it to deliver even higher radiant flux to the 200 μm diameter optical fiber than the 35W arc-lamp.

Spectral Radiance, Spectral Irradiance, and Spectral Radiant Flux

The three terms discussed above are quantities used to characterize radiation within a certain wavelength band, (UV, VIS and/or IR). It is also common to consider those values for unit wavelength (per nm) in the spectrum. For radiation power per unit of wavelength, spectral radiant flux is used with SI units of watts per meter [W/m], or more commonly milliwatts per nanometer [mW/nm]. For radiation incident on a surface, the term spectral irradiance is used, and has the SI unit of [W/m3], or more commonly units of [mW/mm2-nm]. For radiation power within in a unit solid angle from a unit emitting area and unit wavelength, the term is spectral radiance, most commonly with units of [mW/mm2-nm-sr].

Spectral radiance is a key measure when selecting a source for an application. In general, most radiation sources exhibit variations in spectral radiance across their spectrum of emission. In Figure 3, the spectral radiance is shown for a 30W deuterium lamp (D2), a 75W high-brightness Xe arc-lamp, and for two versions of Energetiq’s Laser-Driven Light Source, the EQ-99 and the EQ-1500.

Figure 3: Spectral radiance of EQ-99X LDLS, EQ-77 LDLS, EQ-400, LDLS, 75W short-arc Xe Lamp,
Tungsten Lamp and D2 lamp.

For our earlier example of illuminating a 200 μm optical fiber, let us assume that we wish to compare the four light sources in Figure 3 at delivering 200 nm wavelength radiation into the fiber. Since the key parameter is the spectral radiance of the sources at 200 nm, we can see from Figure 3 that the Xe lamp’s spectral radiance is about one order of magnitude higher (‘brighter’) than the D2 lamp and the LDLS sources are a further order of magnitude higher than the Xe lamp. With the same focusing optic used to couple the light from each source into the 200 μm fiber, the radiant flux delivered into the fiber would similarly vary by the same orders of magnitude.

Conclusions

In the design of optical instruments, scientists and engineers choosing light sources will be exposed to a variety of source specifications and radiometric terms. It is important to understand the nature of the specifications and to couch them in radiometric terms that will enable appropriate design decisions. In general, for typical optical instrument applications, such as spectroscopy and imaging, it is the radiance and spectral radiance of the light source that most needs to be understood. For an instrument with limiting apertures and solid angles, it is the radiance of the source that determines how much radiation passes through the instrument. By carefully matching the instrument with a source of appropriate radiance, an optimum system can be designed.


A new, lower value of total solar irradiance: Evidence and climate significance

[1] The most accurate value of total solar irradiance during the 2008 solar minimum period is 1360.8 ± 0.5 W m −2 according to measurements from the Total Irradiance Monitor (TIM) on NASA's Solar Radiation and Climate Experiment (SORCE) and a series of new radiometric laboratory tests. This value is significantly lower than the canonical value of 1365.4 ± 1.3 W m −2 established in the 1990s, which energy balance calculations and climate models currently use. Scattered light is a primary cause of the higher irradiance values measured by the earlier generation of solar radiometers in which the precision aperture defining the measured solar beam is located behind a larger, view-limiting aperture. In the TIM, the opposite order of these apertures precludes this spurious signal by limiting the light entering the instrument. We assess the accuracy and stability of irradiance measurements made since 1978 and the implications of instrument uncertainties and instabilities for climate research in comparison with the new TIM data. TIM's lower solar irradiance value is not a change in the Sun's output, whose variations it detects with stability comparable or superior to prior measurements instead, its significance is in advancing the capability of monitoring solar irradiance variations on climate-relevant time scales and in improving estimates of Earth energy balance, which the Sun initiates.

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Dep. of Geography, Dickens Hall, Kansas State University, Manhattan, KS, 66506-0801 USA

Dep. of Geography, Dickens Hall, Kansas State University, Manhattan, KS, 66506-0801 USA

Weather Data Library and Dep. of Communications, Kansas State University, Manhattan, KS, 66506 USA

Weather Data Library and Dep. of Communications, Kansas State University, Manhattan, KS, 66506 USA

Dep. of Geography, Dickens Hall, Kansas State University, Manhattan, KS, 66506-0801 USA

Dep. of Geography, Dickens Hall, Kansas State University, Manhattan, KS, 66506-0801 USA

Weather Data Library and Dep. of Communications, Kansas State University, Manhattan, KS, 66506 USA

Weather Data Library and Dep. of Communications, Kansas State University, Manhattan, KS, 66506 USA

Contribution no. 97-165-J, Kansas Agric. Exp. Stn., Manhattan, KS.

Abstract

Crop growth models require solar irradiance as input data, yet there are few places where such data are routinely measured. For locations where measured values are not available, solar irradiance can be estimated using empirical models such as the Bristow–Campbell (B–C) model. This study was conducted to assess the spatial and seasonal accuracy of the B–C model for midcontinental locations in Kansas. A 30-year data set from Manhattan, KS, was used to calibrate and evaluate unmodified and modified forms of the B–C model. The effect of seasonality was investigated by subdividing the yearly data into two subsets, a high noontime solar elevation angle period, ranging from DOY 121 to 273, and a low noontime elevation angle period comprising the remainder of the year. The B–C model was also evaluated without seasonal division of the year. The calibrated models were then tested against measured solar irradiance values for 10 sites distributed across the state of Kansas. Results indicate that, for the calibration site at Manhattan, irradiance was more accurately estimated using a modified form of the B–C model. For the yearly data, root mean square error (RMSE) was 3.9 MJ m −2 d −1 (25% error), compared with 5.2 MJ m −2 d −1 (24% error) for the high solar elevation angle period and 3.6 MJ m −2 d −1 (32% error) for the low solar elevation angle period. The RMSE for the 10 test sites ranged from 2.0 to 6.2 MJ m −2 d −1 percentage error ranged from 26 to 47%. Neither latitude nor distance from the calibration site significantly affected the accuracy of irradiance estimates at the evaluation sites. Results suggest that the modified B–C model provides reasonably accurate estimates of irradiance at noninstrumented sites and that the model can successfully be used at sites away from the calibration site. Seasonal subdivision of the data adds little to the accuracy of estimates.


Estimating Solar Irradiance for Crop Modeling Using Daily Air Temperature Data

Contribution no. 97-165-J, Kansas Agric. Exp. Stn., Manhattan, KS.

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Abstract

Crop growth models require solar irradiance as input data, yet there are few places where such data are routinely measured. For locations where measured values are not available, solar irradiance can be estimated using empirical models such as the Bristow–Campbell (B–C) model. This study was conducted to assess the spatial and seasonal accuracy of the B–C model for midcontinental locations in Kansas. A 30-year data set from Manhattan, KS, was used to calibrate and evaluate unmodified and modified forms of the B–C model. The effect of seasonality was investigated by subdividing the yearly data into two subsets, a high noontime solar elevation angle period, ranging from DOY 121 to 273, and a low noontime elevation angle period comprising the remainder of the year. The B–C model was also evaluated without seasonal division of the year. The calibrated models were then tested against measured solar irradiance values for 10 sites distributed across the state of Kansas. Results indicate that, for the calibration site at Manhattan, irradiance was more accurately estimated using a modified form of the B–C model. For the yearly data, root mean square error (RMSE) was 3.9 MJ m −2 d −1 (25% error), compared with 5.2 MJ m −2 d −1 (24% error) for the high solar elevation angle period and 3.6 MJ m −2 d −1 (32% error) for the low solar elevation angle period. The RMSE for the 10 test sites ranged from 2.0 to 6.2 MJ m −2 d −1 percentage error ranged from 26 to 47%. Neither latitude nor distance from the calibration site significantly affected the accuracy of irradiance estimates at the evaluation sites. Results suggest that the modified B–C model provides reasonably accurate estimates of irradiance at noninstrumented sites and that the model can successfully be used at sites away from the calibration site. Seasonal subdivision of the data adds little to the accuracy of estimates.


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