* Manuscript
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Comparison of estimates of daily solar radiation from air temperature
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range for application in crop simulations
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M.G. Abraha and M.J. Savage
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Soil-Plant-Atmosphere Continuum Research Unit, Agrometeorology Discipline, School of
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Environmental Sciences, University of KwaZulu-Natal, Pietermaritzburg, South Africa
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Address paper for correspondence:
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Ms J Manickum, Agrometeorology, School of Environmental Sciences,
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University of KwaZulu-Natal, P Bag X01, Scottsville, 3209 South Africa
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FAX 27 33 2605514 or 27 33 2605426
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Tele 27 33 2605510
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E-mail [email protected]
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Comparison of estimates of daily solar radiation from air temperature
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range for application in crop simulations
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M.G. Abraha and M.J. Savage*
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Soil-Plant-Atmosphere Continuum Research Unit, Agrometeorology Discipline, School of
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Environmental Sciences, University of KwaZulu-Natal, Pietermaritzburg, South Africa
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Abstract
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Daily solar radiation is an input required by most crop growth, development and
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yield simulation models. It is, however, not observed at many locations, preventing the
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application of such models. The objective of this work was to (i) evaluate several existing
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models estimating solar radiation based on daily minimum and maximum air temperature
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and/or precipitation for seven sites in the world, and (ii) investigate the impact of the
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estimated solar radiation on grass reference evapotranspiration (ETo) and total plant dry
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biomass simulations for maize. Comparisons of the solar radiation models was made
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based on a single modular indicator, Irad, computed using a fuzzy expert system that
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aggregated several statistical indices, and distribution of mean daily errors over a year.
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The model estimations were also evaluated according to their ability to simulate ETo and
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total dry biomass that matches simulations from the observed solar radiation. According
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to the Irad indicator, there was no solar radiation model which consistently outperformed
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all the other models at all the sites tested, but Irad indicated models that relatively
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underperformed at all the sites. The graphical presentation of the mean fluctuation of
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errors over a year gave a good assessment of the solar radiation estimation models in
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investigating the temporal behaviour and magnitude of the residuals. Performance of the
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models according to Irad and simulations of grass ETo and total dry biomass agreed better
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for models that relatively underperformed. Ranking of the models according to the root
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mean square error (RMSE) in solar radiation estimation and the RMSE in the grass ETo
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simulations agreed very well. Comparison of the ranking of the models using the Irad (or
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the individual statistical indices thereof) and total biomass simulation was difficult
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Corresponding author. Tel. and fax: 27 33 2605514.
E-mail address: [email protected] (M.J. Savage).
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because of the difference in the time scale used in calculation of the statistical indices.
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The difference in simulations of total dry biomass accumulated over the years, however,
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qualitatively agreed with the graphs of the mean fluctuation of errors over a year. In
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general, the Irad indicator demonstrated which solar radiation estimation models should be
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used for crop simulation modelling.
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Keywords: Daily solar radiation; Air temperature range; Radiation modelling; Model
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evaluation; Crop yield modelling
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________________________________________________________________________
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1. Introduction
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Crop simulation models have been successfully used to provide simulations of
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growth, development and yield of crops (Jones and Ritchie, 1990). Most crop simulation
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models require daily solar radiation (Is), maximum and minimum air temperatures (Tx and
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Tn), and precipitation (PP) (Whisler et al., 1986; Ritchie, 1991; Hunt and Boote, 1998).
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Solar radiation is the primary input for estimations of reference evaporation and plant
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biomass accumulation in most crop simulation models. However, solar radiation data is
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not as readily available as air temperature and precipitation data (Liu and Scott, 2001;
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Weiss and Hays, 2004). Even at stations where solar radiation is observed there could be
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many days when solar radiation data are missing or lie outside the expected range due to
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equipment failure and other problems (Hunt et al., 1998). These problems could be one of
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many: calibration problems, problems with dirt on the sensor, accumulated water,
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shading of the sensor by masts, etc. The lack of solar radiation data restricts the
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application of crop simulation models (Hook and McClendon, 1992) at locations where
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records on past crop experiments and other weather variables are available. This has led
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researchers to develop a number of methods for estimating solar radiation. Some of these
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methods include estimating solar radiation from other available meteorological
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observations (e.g., Ångström, 1924; Cengiz et al., 1981; Bristow and Campbell, 1984;
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Hargreaves et al., 1985; Hunt et al., 1998; Thornton and Running, 1999; Liu and Scott,
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2001; Mahmood and Hubbard, 2002), substitution of data from nearby stations (e.g.,
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Hunt et al., 1998; Trnka et al., 2005; Rivington et al., 2006), linear interpolation (e.g.,
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Soltani et al., 2004), interpolation in neural networks (e.g., Elizondo et al., 1994; Reddy
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and Ranjan, 2003), satellite-based methods (e.g., Pinker et al., 1995) and generation from
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stochastic weather models (e.g., Richardson and Wright, 1984; Hansen, 1999).
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For accurate crop simulations accurate inputs of the weather variables, including
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solar radiation, are required. Therefore, in the absence of solar radiation data, accurate
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estimation techniques have considerable significance. The above techniques for
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estimating solar radiation have varying degree of complexity, input requirements and
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accuracy of outputs. Use of solar radiation data from nearby stations is not always the
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best option (e.g., Rivington et al., 2006) and their accuracy decreases with increase in
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distance (Hunt et al., 1998; Trnka et al., 2005) as they are dependent on climate and/or
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topography (e.g., Weiss et al., 2001; Rivington et al., 2006). Linear interpolation requires
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solar radiation data at the site of interest and often fails to reproduce the actual day-to-day
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variation (e.g., Soltani et al., 2004). Training neural networks usually requires large data
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sets and the resulting model may not be applicable to other locations (Weiss and Hays,
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2004). The low sampling frequency and coarse spatial resolution of satellite-based
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methods (Pinker et al., 1995) renders them inadequate for site specific application.
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Satellite-based methods are also relatively new and may not provide long-term historical
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weather data. Generated weather data may be used for creating possible scenarios, but
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cannot be used for calibration and validation of crop simulation models for a particular
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period of time (Hook and McClendon, 1992; Liu and Scott, 2001).
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Solar radiation estimation methods from other standard meteorological
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observations, however, have the advantage that the variables used for estimation are
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commonly observed and available within the site of interest. The most common approach
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of estimating daily solar radiation using these methods is to determine the daily
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extraterrestrial solar radiation (Iex) for the site and modify it using the daily atmospheric
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transmission coefficient (tti). The inter-relationship between tti and other meteorological
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observations such as air temperature, atmospheric vapour pressure, precipitation and
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sunshine duration is exploited to estimate daily solar radiation.
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Solar radiation can be easily estimated from sunshine duration measurements
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using several equations with varying degree of complexity following the classic work of
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Ångström (1924). In fact, models that estimate solar radiation from sunshine duration
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fared better than models involving other standard meteorological observations involving
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air temperature and precipitation (e.g., Podestá et al., 2004; Rivington et al., 2005; Trnka
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et al., 2005). However, sunshine duration is not commonly observed in standard
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meteorological stations as are air temperature and precipitation. In this context, solar
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radiation estimation models based on daily air temperature range and/or precipitation are
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attractive and viable options. These models are very simplistic, but allow widespread
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application because air temperature and precipitation are observed practically in all
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meteorological stations.
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The daily solar radiation that is received at the earth’s surface strongly affects the
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thermal conditions at the surface and the immediate atmosphere which in turn can be
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used as an index of cloudiness and solar radiation load (Mahmood and Hubbard, 2002).
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Solar radiation estimation models that utilize these thermal conditions are based on the
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assumptions that - (i) clear skies will increase the daily maximum air temperature because
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of the greater short wave radiation input while resulting in decreased minimum air
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temperature due to reduced long wave emission from the atmosphere; and (ii) cloudy
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conditions will decrease the daily maximum air temperature due to reduced transmissivity
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while resulting in increased minimum air temperature due to increased long wave
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emission from the clouds (Donatelli and Campbell, 1998). Bristow and Campbell (1984)
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using this relationship estimated daily solar radiation using an exponential function of
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daily air temperature range (ΔT) and precipitation (see Appendix A for details). They
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were able to account for 70 to 90 % of the variation in incoming daily solar radiation data
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at three northwestern sites in the USA. The Bristow and Campbell (hereafter called BC)
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model has been modified by others for specific applications. For example, Donatelli and
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Marletto (1994) and Donatelli and Campbell (1998) included a summer night air
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temperature factor to improve underestimation of predicted values during the summer;
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Goodin et al. (1999) refined the equation by adding an extra Iex term that is meant to act
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as a scaling factor allowing ΔT to accommodate a greater range of solar radiation values,
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although Mahmood and Hubbard (2002) found it to perform worse than the original
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model for the Northern Great Plains; Thornton and Running (1999) introduced
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atmospheric water vapour pressure to the equation in an attempt to eliminate the need for
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site-specific calibration of coefficients; and Donatelli and Belocchi (2001) attempted to
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account for the effect of seasonal variation by introducing a trigonometric function.
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Hargreaves et al. (1985) also developed a simple linear relationship between daily air
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temperature range and tti. Hunt et al. (1998), based on the evaluation of five solar
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radiation models, found best estimates in a model with multiple-linear relationship
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between daily incoming solar radiation, and air temperature and precipitation. Mahmood
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and Hubbard (2002) also found more stable estimations of daily incoming solar radiation
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from clear-sky solar radiation and daily air temperature range compared with the BC
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model. Most of these models require some observed solar radiation data for derivation of
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coefficients, which may inhibit their application at sites where solar radiation has not
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been observed before.
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Previous work in this field has concentrated on evaluating the uncertainty of daily
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solar radiation estimation models over a localized area (e.g., Hunt et al., 1998; Liu and
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Scott, 2001; Mahmood and Hubbard, 2002; Rivington et al., 2005). Evaluation of most of
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the existing solar radiation estimation models over a wide range of geography and climate
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in the world is lacking. Therefore, the objective of this study was to (i) compare various
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models that estimate daily global solar radiation from daily maximum and minimum air
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temperatures and/or precipitation over several locations in the world, and (ii) investigate
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the impact of the estimated solar radiation on simulations of grass reference
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evapotranspiration and total maize dry biomass using a cropping systems simulation
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model (CropSyst).
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2. Materials and methods
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2.1. Data
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Meteorological data were obtained from seven stations with a range of latitude,
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longitude and elevation around the world. Information on the sites and period of recorded
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data is presented in Table 1. The sites included at least daily maximum and minimum air
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temperature (Tx and Tn), precipitation (PP) and solar radiation (Is) data. The data were
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checked for outliers for each weather variable. For Davis, CA missing data were replaced
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by data from another weather station at the same latitude and longitude. For the other
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sites, if one or more weather variable was missing, then all the weather variables for that
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day were replaced by that from another year (a year or two of data were set aside for such
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purposes for each site) of the same site and day. Precipitation occurrences in that day and
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two previous days were taken into account when replacing missing data as this may affect
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the estimations. This avoids replacement of data through optimization or other
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relationships and ensures that the data are still from the same site. A year with more than
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30 days of missing or faulty data was discarded (e.g., the year 1996 for Cortez,
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Colorado).
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2.2. Solar radiation estimation formulae
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For each site and year of available data, Is was estimated using six solar radiation
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estimation models presented in Table 2. These models were chosen as representative of
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the existing models that utilize Iex and readily available weather data of Tx, Tn and/or PP.
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Further information on the models is given in Appendix A.
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2.3. Coefficients
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All the models require Is (MJ m-2) for derivation of the coefficients for estimation
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of solar radiation. The BC, CD and DB models are contained within the software RadEst
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tool (beta v3.00) (SIPEAA, 2006) and iterative optimization utilities are provided within
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the software for determining the coefficients. The RadEst tool has been used for
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estimation of solar radiation in previous works (e.g., Bellocchi et al., 2002; Rivington et
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al., 2006). Derivation of the coefficients for the Hgvs model involved simple linear
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regression, and for the HKS and MH models multiple linear regression with natural log
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transformations for the latter (Genstat, 2006). For each site, three years of daily data of
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Tx, Tn, PP, and Is were used for derivation of the coefficients. The data sets used for
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derivation of coefficients were from consecutive years with no or little missing data. The
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derived coefficients were then used to estimate solar radiation for all the years excluding
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the ones used for derivation of the coefficients.
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2.4. Statistical evaluation
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Most studies evaluating the performance of solar radiation models have
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traditionally used coefficient of determination (R2), mean square error (RMSE) and/or
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model bias to assess model suitability and comparison (e.g., Hunt et al., 1998; Lui and
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Scott, 2001; Mahmood and Hubbard, 2002; Ball et al., 2004; Chen et al., 2004; Weiss and
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Hays, 2004).
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particular single or separate multiple statistics not organized in a systematic manner may
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be inadequate as each statistic evaluates a particular aspect of the model (Bellocchi et al.,
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2002; Rivington et al., 2005). Bellocchi et al. (2002) argued that a model may be deemed
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unsuitable according to one statistic evaluating certain aspects of the model but other
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features of the model may still be desirable. On the other hand, a model may be judged
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suitable according to one statistic but it may be deficient according to another statistic. To
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obviate such problems, Bellocchi et al. (2002) used a fuzzy-logic based system that
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simultaneously considers several statistical indices. The system allows aggregation of
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several statistical indices into a single module by assigning an expert weight according to
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the relative importance of the particular statistical index. The weight assigned to each
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statistical index was on the basis of the authors’ experience. The modules are also
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aggregated, in the same manner as for the indices, into a single indicator. The indices and
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modules considered are presented in Table 3. The indices relative root mean square error
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(RRMSE), model efficiency (EF) and two-tailed paired t-test P(t) are aggregated into
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module Accuracy; correlation coefficient (R) into module Correlation; and pattern of the
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residuals against independent variables of day of year (PIdoy) and daily minimum air
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temperature (PITn) are aggregated into module Pattern. Computation of the pattern
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indices, PIdoy and PITn, involves dividing the residuals into four groups according to the
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independent variables and then calculating pairwise differences between the four average
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residuals. Further information on patterns of residuals is documented by Donatelli et al.
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(2004). The modules Accuracy, Correlation and Pattern are then aggregated into a single
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modular indicator, Irad, which enabled ranking of the models (Fig. 1).
Evaluation of the performance of the solar radiation models using a
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Three membership classes (subsets) are defined for all indices: favourable (F),
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unfavourable (U) and partial (fuzzy) membership. These membership classes, along with
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decision rules, are used to calculate a dimensionless module whose value ranges between
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0 (best model performance) and 1 (worst model performance). Membership functions that
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are S-shaped in transition interval were used (Bellocchi et al., 2002). The relative
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importance of the indices, modules and the weights assigned to them are presented in Fig.
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1. Detailed illustration of aggregation of indices into modules and modules into a single
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indicator is presented by Bellocchi et al. (2002).
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This study makes use of such an indicator to evaluate the solar radiation models.
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The aggregation of indices into modules and modules into an indicator was performed
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following the logic in the IRENE_DLL system for model implementation (Fila et al.,
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2003). Besides, a simple graphical presentation of the mean difference between estimated
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and observed solar radiation across all years for all the sites was considered for
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assessment of the solar radiation models.
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2.5. ETo and biomass simulations
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Observed and estimated solar radiation along with Tx, Tn and PP, maximum and
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minimum relative humidity (RHx and RHn) or vapour pressure and wind speed were used
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to simulate grass reference evapotranspiration (ETo) and total dry biomass of maize at
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two sites for which the solar radiation estimation was best and worst according to Irad as
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computed for all the models. The only variable that kept changing with each simulation
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was the solar radiation as observed and estimated from the different models. Daily ETo
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was calculated according to the FAO Penman-Monteith procedure as recommended by
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Allen et al. (1998). A soil and plant growth simulator, CropSyst - a multi-year multi-crop
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simulation model developed to study the effect of cropping systems management on
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productivity and environment (Stöckle et al., 2003), was used for biomass simulation
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purposes. The model has been tested and validated for a wide range of management
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conditions and a variety of crops and cropping systems in a range of locations over the
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world (Stöckle, 1996). Default crop parameters for maize were used. Planting date at
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each location was set to that locally practiced. The simulation was run for all the number
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of available continuous years in rotation along with fallow conditions.
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The effect of estimated solar radiation on simulated total dry biomass of
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maize was analyzed using the difference between cumulative and mean of total dry
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biomass simulated from the observed and estimated solar radiation inputs, seasonal
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means of absolute differences of total dry biomass, R2 during the simulation period,
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RMSE for individual seasonal simulations and maximum error between the seasonal dry
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biomass simulations from the observed and estimated solar radiation inputs. The
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calculated statistical measures were used in ranking the performance of the solar radiation
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models on their ability to simulate total dry biomass that matches that simulated using the
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observed solar radiation.
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3. Results
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All the solar radiation models were calibrated using three years data for the
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specific locations under study. The coefficients derived for each location and model are
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presented in Table 4. Not many works report coefficients for models. Bellocchi et al.
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(2002) reported model coefficients for the BC, CD and DB models for several locations
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over the world. For these models, the coefficients derived in our study fall within the
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range of coefficients they reported. Bechini et al. (2000) also reported model coefficients
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for the CD model for 29 locations in northern Italy. Their Tnc parameter appears to be
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smaller than presented in this study. This could be because of the localized location they
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used for their study.
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The Hgvs and HKS models predicted quite a few negative solar radiation values
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at low daily air temperature range records, especially in the temperate sites during winter.
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The estimated solar radiation for such days was set to zero. The HKS model, because it
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uses a polynomial function of daily precipitation amount for estimation of solar radiation,
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also predicted an unreasonably high solar radiation at high precipitation records, in two
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occasions for Pretoria and Griffith. Assuming these days were overcast, the value of
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estimated solar radiation was corrected by multiplying the Iex by 0.25 for such days
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(Gates, 1980).
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3.1. Model evaluation according to the indicator Irad
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Mean values of indices and modules are calculated and presented in Table 5 for
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each site. Table 6 also presents the mean indices and modules for each model over all the
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sites. It is apparent from Table 5 that ranking of the solar radiation models based on an
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individual statistic or index is difficult. A model may perform well according to one
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index but may not do so according to another. Overall, according to Irad, the DB (0.2061)
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model closely followed by the CD (0.2089) model was ranked top of the group. The DB
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model had in general best results for PIdoy while the CD model was best according to
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RRMSE, EF and PITn. The correlation coefficient, R, had little effect on the integrated
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index, Irad, as the R value for most of the sites and years was close to or greater than 0.9
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except for Pretoria. The MH model was ranked last of the group with an overall Irad value
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of 0.3655, by far larger (less desirable) than the other models. The MH model also was
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often ranked last of the group according to most of the other indices for each site. The
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mean values of indices and modules over all years for all the sites may be indicative of
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model performance for that index and module but it should not mean that the model
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ranked first based on that overall mean index or module is best for all sites. It should also
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be noted that the aggregation of the indices was performed on a yearly basis and not after
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averaging across all years. In evaluating the models, not only the rank but also the
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difference between the scores of each rank for each model should be considered.
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The BC model, according to Irad, appeared to perform well for higher elevation
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(Cortez (0.1759), Rothamsted (0.1838), Pretoria (0.2143) and Griffith (0.1945)) than for
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lower elevation (Padova (0.2880), Davis (0.2738) and Wageningen (0.3063)) sites (Table
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5). The best result of Irad achieved by this model was for Cortez. This was also the best
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Irad value scored for that site. The BC model had the best overall results for the paired t-
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test and in general, it produced small residuals compared to the other models with overall
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mean RMSE of 3.27 MJ m-2 (Table 6). This enabled it to achieve first rank according to
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the overall mean Accuracy (Table 6). It also produced the best Accuracy result at one site
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and second to best at five other sites. Otherwise it produced large PIdoy and PITn values
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which resulted in large Pattern (less desirable) values compared to the other models,
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especially for the lower elevation sites. The model was also not consistent in the ranks it
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achieved according to the module Pattern for the individual sites.
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The CD model was ranked second with a slightly greater Irad value than the DB
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model (Table 6). The CD model was strong in producing small PITn values. According to
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this index, it was the best model at four sites (Cortez, Davis, Padova and Wageningen),
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and this led to best Pattern result at three of these sites (Cortez, Davis and Wageningen).
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But it produced larger residuals which saw it ranked in the middle of the group according
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to the module Accuracy (Table 6). It had particularly poor results for the paired t-test,
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P(t), for which it was ranked last at two sites (Cortez and Davis). Otherwise it produced
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overall best results for the indices RRMSE (RMSE 3.23 MJ m-2) and EF (0.82) (Table 6).
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The DB model was the best of all the models according to Irad (Table 6). The rank
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of the model according to most of the individual indices was about average at all the sites.
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It resulted in two best Accuracies (Davis and Rothamsted) and two best Patterns
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(Rothamsted and Pretoria). Its overall RMSE value was 3.27 MJ m-2 (Table 6). The DB
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model generally produced larger PITn at most sites but it produced best results of PIdoy at
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two sites (Rothamsted and Pretoria) that finally enabled it to achieve the best Pattern
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value at those sites.
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The Hgvs model was ranked third in the overall ranking according to Irad (Table
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6). The model did not show consistency in the ranks it achieved according to most
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modules and indices. For example, it resulted in best Accuracy at three sites (Cortez,
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Wageningen and Griffith) but then it was at the tail of the ranks for all the other sites for
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the same module. Consequently, it was ranked fifth in the overall ranking for all sites
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according to the module Accuracy. The model was ranked second according to the
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module Pattern. This was mainly because it produced about the smallest PIdoy at two sites
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(Padova and Wageningen) compared to the other models.
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The HKS model was ranked fifth in the overall ranking according to Irad (Table 6).
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This model also did not show consistency in the ranks it achieved according to Irad as it
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scored different ranks at different sites, from first to last (Table 5). Its overall ranking
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according to the modules Accuracy and Pattern was fourth and third respectively. It
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scored the best result for module Pattern at Griffith jointly with the BC model. This was
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mainly because of the best result in PITn (see Table 5). At the other sites, it produced best
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results of PIdoy (Cortez, Davis and Rothamsted), however, the corresponding PITn values
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were not as good to result in best Pattern values. The HKS model appeared not to perform
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well for the semi-arid regions where there occurred infrequent but high rainfall amounts.
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This was reflected in the module Correlation at Pretoria and Griffith, where the HKS
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model was ranked last and second to last respectively according to this module.
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The MH model, which was ranked last according to Irad (Table 6), was almost
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consistently ranked last of the models according to all the indices and modules (Table 5).
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The only time that it came out of the last ranks was when the models were ranked
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according to the paired t-test, P(t), in which there was no clear pattern of ranks, and PITn.
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3.2. Patterns of observed versus estimated daily mean errors
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A graphical distribution of mean daily errors over a year for the sites considered
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could indicate a systematic behaviour of the model and serve as a means of quick visual
15
evaluation of model performance (Rivington et al. 2005). A graph of the mean daily
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errors against day of year (Fig. 2) revealed the existence of temporal and spatial pattern
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for most of the models. The patterns appeared to be different for the northern and
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southern hemisphere sites, but similar within the respective hemispheres for each model
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implying that solar radiation estimations are dependent on latitude and season. Cortez,
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with the highest elevation of all the sites in this study, showed a pattern which is less
21
clear or different from the other sites for some of the models. This also suggests that
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estimations may be dependent on elevation as well. In general, Davis and Padova showed
23
the smallest and largest fluctuations of mean daily errors respectively for all sites.
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For convenience, the seasons winter, spring, summer and autumn for the northern
25
hemisphere are defined roughly as the months from December to February, March to
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May, June to August and September to November respectively; and for the southern
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hemisphere June to August, September to November, December to February and March
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to May respectively.
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The BC model showed similar patterns at Davis and Padova with an
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overestimation of solar radiation in winter, early spring, late summer and autumn and
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underestimation of solar radiation in late spring and early summer. A slight tendency of
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overestimation was observed at both Cortez and Rothamsted in summer. For the southern
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hemisphere sites, the mean daily errors were well-distributed around the observed mean
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at Pretoria from autumn through spring but an overestimation was observed at Griffith for
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similar seasons. The best Irad value for BC was observed at Cortez (0.1759).
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The CD model, for the sites in the northern hemisphere, overestimated (Cortez)
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and underestimated (Rothamsted and Wageningen) in winter. At Davis, underestimation
7
in winter, spring and summer and overestimation in autumn was observed. At Padova,
8
errors were evenly distributed with a tendency of underestimation in summer. In the
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southern hemisphere the model overestimated solar radiation from late autumn up to late
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spring but resulted in an even distribution of errors in spring and summer. The best Irad
11
value for the CD model was observed at Davis (0.1326) where the fluctuations of the
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mean daily errors were small.
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The DB model, for the sites in the northern hemisphere, appeared to overestimate
14
solar radiation in winter, early spring and autumn but underestimate it in late autumn and
15
summer (Cortez, Davis and Padova). The errors were evenly distributed for Rothamsted
16
with a tendency to underestimate at Wageningen. For the sites in the southern
17
hemisphere, the DB model overestimated solar radiation for most of the year. The best
18
Irad value for the DB model was for Pretoria (0.1349) although the model seemed to
19
overestimate from autumn to late spring at this site.
20
The Hgvs model showed the tendency to overestimate solar radiation at all of the
21
northern hemisphere sites except at Cortez where errors were evenly distributed and at
22
Padova where the model underestimated during winter and early summer. For the
23
southern hemisphere sites, in spring, the model tended to overestimate at Pretoria and
24
underestimate at Griffith. The best Irad value observed for the model was at Davis
25
(0.1517) where the fluctuations of the mean daily errors were small.
26
For the HKS model, there was no clear pattern of errors for the northern
27
hemisphere sites. At Wageningen, solar radiation was underestimated almost throughout
28
the year. At Rothamsted, the errors seem to be evenly distributed. At Cortez, they appear
29
to be evenly distributed in spring and summer with underestimations in winter and
30
autumn. At Davis, the errors fluctuated about the mean with a slight tendency to
31
underestimate in spring and summer. Large underestimations were observed at Padova in
13
1
the summer. For the southern hemisphere, the HKS model overestimated in all the
2
seasons except in summer. The best Irad value for the HKS model was observed at Padova
3
(0.1359). But the model appeared to underestimate solar radiation in summer and also the
4
largest fluctuations of errors for all models were observed at this site.
5
The existence of patterns was best illustrated by the MH model where all the sites
6
produced similar shapes of fluctuation of mean daily errors for the respective
7
hemispheres. For the first half of the year, the sites in the northern hemisphere showed a
8
“V” shaped pattern in which there was consistent underestimation of solar radiation. This
9
was followed by an even fluctuation of errors in the second half of the year. For the sites
10
in the southern hemisphere, the model produced errors with a “W” shape in which an
11
overestimation occurred in the middle of the year. The best Irad value for the MH model
12
was observed at Griffith where the temporal distribution of the daily mean errors
13
fluctuated around the mean throughout the year except in autumn.
14
15
3.3. Grass ETo and biomass simulation
16
17
Grass ETo and total dry biomass were simulated for Davis and Wageningen for
18
which solar radiation estimation models collectively produced the smallest and largest Irad
19
values respectively (Table 5). For the grass ETo simulation, the full weather data set
20
including Tx, Tn, PP, RHx and RHn or water vapour pressure and wind speed along with
21
the observed and estimated solar radiation were used. For total dry biomass simulation,
22
all the available years with observed and estimated solar radiation at both sites were used.
23
For both Davis and Wageningen, the grass ETo simulated from observed and
24
estimated solar radiation appear to be well distributed along the 1:1 line (Fig. 3 and 4)
25
with a slight overestimation of lower and underestimation of higher values of grass ETo.
26
Slope, intercept, R2 and RMSE, presented in the graph, were used to rank the models,
27
although not as rigorously as Irad. Wageningen had a lower slope and larger RMSE
28
compared with Davis implying that the solar radiation was not well estimated at
29
Wageningen as it was at Davis. The ranking of the models according to Irad was reflected
30
in the ranking of the models according to the grass ETo simulation at Davis but not at
31
Wageningen, with only the first and last ranks matching at Wageningen. Nevertheless,
14
1
the RMSE achieved from the grass ETo simulations for both sites confirmed the ranking
2
of the solar radiation models for Davis and Wageningen according to the index RRMSE
3
(Table 5).
4
The total dry biomass simulated using the observed solar radiation is taken as a
5
baseline and all simulations from model-estimated solar radiation were compared against
6
it (Table 7). The smallest and largest mean differences (magnitude) between the baseline
7
and model-simulated solar radiation simulations were, in tons ha-1, 0.02 and 0.71 (n =
8
18), and 0.31 and 1.53 (n = 19) for Davis and Wageningen respectively. These suggest
9
that the simulations of total dry biomass from model-estimated solar radiation reasonably
10
matched the baseline simulations. The agreement between the baseline and model-
11
estimated solar radiation simulations of total dry biomass was better for Davis compared
12
to Wageningen (Fig. 5), as was the solar radiation estimation from all the models for
13
these sites. The total and mean differences presented in Table 7 indicated that, for both
14
Davis and Wageningen, the simulated total dry biomass from all model-estimated solar
15
radiation was underestimated except for an overestimation from the DB model at Davis
16
and the Hgvs model at Wageningen. But the magnitude of the underestimations was
17
greater at Wageningen than at Davis. Fig. 5 illustrates the extent of the under- and
18
overestimations of total dry biomass for each season by all models for both sites.
19
The total biomass simulation for the BC model at Davis, in general, resulted in a
20
closest match to the baseline simulations compared with simulations from the other
21
models. The largest seasonal under- and overestimations by this model were (tons ha-1)
22
-0.73 and 0.59 respectively (only the larger of the two in magnitude is presented in Table
23
7). These indicated that there was compensation of errors from the under- and
24
overestimations in the total biomass simulations over all the years. As a result, the
25
absolute difference for this model was by far larger than that indicated by the mean
26
difference (Table 7). The total and mean differences also indicated that the CD model
27
under-estimated the total and mean biomass simulations (Table 7). The CD model under-
28
estimated the seasonal total biomass simulations in almost all years leaving little room for
29
error compensation in the total dry biomass accumulated over all the years (Fig. 5). The
30
largest under- and overestimations by this model were (tons ha-1) -0.81 and 0.49
31
respectively. In contrast, the total and mean biomass simulations were overestimated by
15
1
the DB model. The DB model was the only model that overestimated the total dry
2
biomass simulations at Davis. The largest under- and overestimations by this model were
3
-0.48 tons ha-1 and 1.77 tons ha-1 respectively. The largest overestimation that occurred
4
with the simulations using the DB model-estimated solar radiation was an isolated single
5
incidence that happened in a year when the DB model over-estimated the solar radiation
6
for most of the days during the growing season. Otherwise, the next largest
7
overestimation by the model was 0.67 tons ha-1. The total and mean difference of total
8
biomass simulations from the Hgvs and HKS models produced second and third closest
9
match to the baseline simulations respectively. These models also had well distributed
10
under- and overestimations, and simulations with close to zero errors (Fig. 5). The
11
smallest absolute maximum error in the biomass simulations (0.66 tons ha-1) was
12
observed from the Hgvs model. The largest under- and overestimations for Hgvs and
13
HKS models were (tons ha-1) -0.66 and 0.62, and -0.70 and 0.63 respectively. The
14
absolute differences indicated that these two models and the BC model simulated
15
seasonal dry biomass that matched well to the baseline compared with the other models
16
(Table 7). The MH model consistently underestimated the total biomass simulation for all
17
seasons except once implying there was no room for error compensation in the total dry
18
biomass accumulation. Consequently, it was the worst according to the measures of total,
19
mean and absolute differences. The largest under- and overestimations by this model
20
were (tons ha-1) -1.63 and 0.06 respectively.
21
At Wageningen, the picture was different with most of the models
22
underestimating the total dry biomass over all the years and seasons. The BC model had
23
three, the CD and DB- four, the HKS and MH- two seasons in which there was an
24
overestimation otherwise the total dry biomass simulations were underestimated in all the
25
seasons. The overestimations also appeared to be smaller in magnitude than the
26
underestimations. This indicated that there was little room for compensation of errors in
27
the total dry biomass simulations accumulated over all the years. The largest under- and
28
overestimations by these models were (tons ha-1) -1.77 and 0.73, -1.82 and 0.59, -1.57
29
and 1.02, -2.98 and 0.97, and -2.67 and 0.62 for the BC, CD, DB, HKS and MH models
30
respectively. The Hgvs model, however, overestimated the total dry biomass in more than
31
half of the total simulation years with underestimation in the rest of the years. The
16
1
overestimations were larger than the underestimations. The largest under- and
2
overestimations by this model were (tons ha-1) -0.54 and 1.76 respectively. The largest
3
and smallest absolute maximum errors observed in the simulations were by the HKS
4
(2.98 tons ha-1) and DB (1.57 tons ha-1) models respectively. The under- and
5
overestimations observed in the total biomass simulation at this site were in accordance
6
with the mean daily error graphs of model-estimated solar radiation for Wageningen (Fig.
7
2).
8
According to most of the statistics (Table 7) used in the biomass simulation the
9
MH model was ranked last at Davis, as it was in the solar radiation estimation according
10
to Irad. The ranking of the CD and BC models seems to have swapped ranks according to
11
the statistics used in the biomass simulation compared with the ranking according to Irad.
12
The CD and BC models were ranked first and fifth according to Irad respectively, but the
13
BC model was ranked first and the CD fourth several times according to the statistics
14
used in the biomass simulation. But it should be noted that the difference in score among
15
the four models ranked first to fourth using the statistics in the biomass simulation was
16
small. Most of the statistics used in the biomass simulation were measures of residuals,
17
but the CD model was ranked first at Davis mainly because it produced the best result for
18
the module Pattern otherwise it produced similar residuals compared to some of the other
19
models. The ranking of the rest of the models according to the statistics used in the
20
biomass simulation at this site was similar to the ranking of the models according to Irad.
21
At Wageningen, the HKS and MH models were ranked last and second to last
22
respectively according to Irad, and so were they in their ranking according to the statistics
23
used in the biomass simulation. There was little difference in the score of Irad for the other
24
four models and this was reflected in the ranking according to the statistics used in the
25
biomass simulation. But, overall, the Hgvs model appeared to be ranked first by most of
26
the statistics used in the biomass simulations followed by the DB, CD and BC models.
27
This ranking did not match the ranking of the solar radiation models according to the Irad
28
scores, or for that matter any of the modules or conventionally-used statistical indices
29
thereof.
30
31
17
1
4. Discussion
2
3
The results demonstrated that most of the models tested were, in general, able to
4
adequately estimate daily solar radiation from daily air temperature range and/or
5
precipitation. The ranking of the models, according the indicator, Irad, revealed that there
6
was no model as such which consistently outperformed all of the other models at all sites.
7
It, however, indicated that the performance of the MH model was consistently lower
8
(largest Irad value) at all sites compared to the other models.
9
The values of the indices and modules achieved for each model from the
10
calculation indicate the strength and weaknesses of the model in dealing with the
11
respective index (Table 5) (Bellocchi et al., 2002). The BC model produced better
12
residuals (RRMSE, EF and P(t)) but poor patterns (PIdoy and PITn). The CD model was
13
good in handling the indices RRMSE, EF and PITn but was poor in P(t). The DB model
14
was particularly good in producing better P(t) and PIdoy. The Hgvs model appears to
15
perform well in all the indices but PIdoy. The performance of the HKS model was about
16
an average in the overall results of most of the indices but lacked consistency at the
17
individual sites. The MH model showed some strength for the index PITn only but
18
otherwise it was poor according to all indices and modules.
19
The models contained within the RadEst tool (BC, CD and DB) generally
20
performed very well. The BC model was superior in producing smaller residuals than the
21
CD and DB models at most of the sites. But it was inconsistent in the patterns of residuals
22
it produced. This may be expected as the improvement of the CD and DB models over
23
the BC model was in these aspects. Although correlation had little effect in the sites
24
tested, the CD model appeared to produce better correlations. The DB model had the
25
overall best Irad value, but the CD model relatively produced better individual indices at
26
most of the sites. The overall ranks that the DB and CD models achieved and the
27
consistency that they showed in the scores of the individual indices makes them better
28
options for estimating solar radiation from daily air temperature range and precipitation.
29
The Hgvs model, considering the simplicity of the formulae and relative ease of
30
deriving the coefficients for each geographic area compared to the other models,
31
performed well to be ranked third in the overall ranking. But the inconsistency in the
18
1
results of the statistical indices that it had showed makes it less desirable as a better
2
option for estimating solar radiation. The largest fluctuations that were observed from this
3
model were during the rainy season implying the inclusion of precipitation in solar
4
radiation models is appropriate.
5
The HKS model includes the precipitation quantity as PP and (PP)2 in its
6
formulae. The coefficients of these two variables were negative and positive respectively
7
for all sites. In the case of high rainfall amounts, the estimated solar radiation could be by
8
far greater than that observed. It may be easy to detect and correct such estimations when
9
they lie outside the expected range of solar radiation but not when they lie within the
10
expected range. The latter can cause undesirable model performance. Moreover, the
11
coefficients derived for such a model may be erroneous if derived using a calibration data
12
set with abnormally high precipitation values. Liu and Scott (2001) found that models
13
that use precipitation as a binary input (present or absent) to perform better than the ones
14
that use precipitation quantity such as HKS. In general, the HKS model may have less
15
application for semi-arid climates where infrequent and high rainfall amounts may be
16
observed.
17
The MH model showed clear seasonal and spatial patterns as was illustrated for
18
the northern and southern hemisphere sites irrespective of latitude, altitude and distance
19
to the coast. Mahmood and Hubbard (2002) found similar results when they compared
20
the MH with the BC model, leading them to conclude that the MH model was more
21
stable. But they found the BC model to slightly outperform the MH model based on
22
RMSE, d index of agreement (Willmott, 1981) and relative error for the Northern Great
23
Plains. Identification of the cause of the patterns the MH model showed could help in
24
modification of the model to better estimate solar radiation.
25
Of particular interest is also the fluctuation of errors at Padova. Padova showed
26
consistently the largest temporal fluctuations of errors compared to the other sites for all
27
models. This could be due to, as Padova is located near the Adriatic Sea, a maritime
28
climate influence. Rivington et al. (2005) have noted similar observations for sites
29
located near the coast in the UK.
30
Rivington et al. (2005) illustrated fluctuations of mean daily errors for sites in the
31
UK using the CD and DB models. These were similar to the fluctuations observed at
19
1
Rothamsted in this study, but the values of Irad and other indices calculated for
2
Rothamsted in this study were better than theirs. The difference in number and type of
3
data sets used for calibration and validation of the models might be responsible for these
4
discrepancies. In general, they achieved Irad values ranging from 0.301 to 0.642 and 0.322
5
to 0.633 using the CD and DB models respectively for several locations in the UK. In
6
other studies, involving data from ten locations around the world, Irad values ranging
7
between 0.0086 to 0.5518, 0.0385 to 0.5040, and 0.0044 to 0.4704 were reported for the
8
BC, CD and DB models respectively (Bellocchi et al., 2002). The Irad values calculated in
9
this study for all the sites fall well between the ranges reported in both the above studies.
10
The RMSE has been widely used for evaluation of daily solar radiation models and mean
11
or range of this value (in MJ m-2) reported by other authors include: for the BC model 3
12
(Bristow and Campbell, 1984), 4.7 (Hunt et al., 1998), 3.53 to 4.78 (Mahmood and
13
Hubbard, 2002); for the CD model 2.37 to 4.26 (Donatelli and Campbell, 1998); for the
14
DB model 2.3 to 3.9 (Bechini et al., 2000); for the Hgvs model 4.2 to 4.7 (Hunt et al.,
15
1998); for the HKS model 3.4 to 4.1 (Hunt et al., 1998); and for the MH model 3.90 to
16
4.93 (Mahmood and Hubbard, 2002). The range of RMSE achieved in this study for all
17
models at the individual locations was (in MJ m-2) between 2.56 and 4.11 (data not
18
shown) which falls well within the ranges reported in the other studies.
19
The statistical results (Table 7) and graphical presentations (Figs 3, 4, and 5)
20
demonstrated that estimates of solar radiation from daily air temperature range and/or
21
precipitation could be successfully used for grass ETo and total dry biomass simulations.
22
But care should be exercised in choosing the best model to represent the actual solar
23
radiation. At Davis, estimations from all the models but the MH model could be
24
successfully used in place of measured data for biomass simulation purposes. At
25
Wageningen, the systematic errors in model-estimated solar radiation may cause a
26
systematic bias in total plant dry biomass simulations. But still, the Hgvs, DB, CD and
27
BC models could be used to give a representative total dry biomass simulation at this site.
28
The choice of model-estimations may also depend on particular interest of application.
29
For example, at Wageningen the Hgvs model should be used if the interest is the smallest
30
absolute difference and the DB if the smallest maximum error is required. The statistical
31
results used in the grass ETo using observed and estimated inputs of solar radiation
20
1
reflected the ranking of the solar radiation models according to the overall means of
2
RMSE and to some extent Irad. The ranking of the models according to the statistics used
3
in the biomass simulations and Irad, however, agreed well only for models that did not
4
relatively perform well. The rank of the models that performed well with respect to Irad
5
was not reflected in the statistics used in the biomass simulation, but the difference in the
6
score of the statistics was small. Rivington et al. (2005) suggested that the yearly
7
individual indices of RRMSE, EF, R and pattern indices may be more indicative than the
8
overall mean of Irad since the accuracy and precision of daily weather values becomes
9
more important. But the yearly values of the above indices gave little indication of the
10
corresponding errors in the total dry biomass simulations. This could be because the
11
statistical indices used in the evaluation of the solar radiation models were calculated for
12
the whole year but the statistics in the biomass simulations were calculated only from part
13
of the year which was involved in growing the plant. The response of the solar radiation
14
model during the growing season may be different compared to the whole year. Besides
15
crop growth models contain non-linear functions in which a certain change in input may
16
affect the output differently. In general, the models with small Irad produced better
17
simulations (small residuals) and the models with large Irad produced worse simulations
18
(large residuals) although the exact rank of the models was not reflected in the biomass
19
simulations for the models that performed relatively well. The pattern of the mean daily
20
errors (Fig. 2), however, agreed well with the under- and overestimations of total dry
21
biomass simulations, especially at Wageningen. Graphical presentation of the distribution
22
of the mean daily errors, along with the Irad, makes the evaluation of the solar radiation
23
models easier. In general, Irad as a means of evaluation of solar radiation models for
24
application in crop simulation models was better in discriminating models that did not
25
relatively perform well. It was also helpful in indicating sites for which the performance
26
of the solar radiation models was good or poor.
27
28
29
30
31
21
1
5. Conclusions
2
3
In the absence of solar radiation measurement data, reliable estimates can be made
4
from easily available meteorological observations of air temperature and/or precipitation
5
along with extraterrestrial radiation using several existing models. Comparison of the
6
performance of the models using individual statistical indices proved to be difficult
7
because of the number of statistical indices considered and the contrasting results they
8
give. A model could be good according to one statistical index but poor according to
9
another index. Aggregation of the statistical indices into a single modular indicator, Irad,
10
using the fuzzy-logic expert system enabled evaluation and ranking of the solar radiation
11
models performance. According to the Irad indicator, there was no model which
12
consistently outperformed the other models as some models made good estimates at one
13
site and poor at other sites. But the Irad was good in discriminating models that relatively
14
underperformed at all sites. Overall, the DB and CD models were found to be best
15
estimators of solar radiation and the MH model the worst. Graphical presentation of the
16
mean fluctuation of the errors over a year also demonstrated the temporal behaviour of
17
the models in estimating solar radiation. Sites for which the solar radiation models
18
resulted in relatively good Irad values produced better simulations of grass ETo and total
19
dry biomass that matched the simulations from using inputs of observed solar radiation.
20
Ranking of the models according to the Irad and the grass ETo and total dry biomass
21
simulations agreed better for models which relatively did not perform well. The ranking
22
of the models according to the RMSE in the grass ETo simulations also agreed very well
23
with the RRMSE (or RMSE) in the solar radiation estimation. Comparison of the solar
24
radiation estimation models according to statistical indices used in the solar radiation
25
estimation and the total dry biomass simulations was difficult because of the difference in
26
the timescale used in the indices calculation (overall mean or yearly in the former case
27
and seasonal in the latter). Although the ranks of the relatively good performing models
28
according to the Irad was not reflected in the grass ETo and total dry biomass simulation
29
exactly, the difference between the statistical indices used in the grass ETo and total dry
30
biomass simulation was small. The graphical presentation of the mean fluctuation of the
31
errors also roughly indicated the direction of the errors in grass ETo or total dry biomass
22
1
simulation. In general, Irad was able to indicate which solar radiation estimation model
2
should be used for application in crop simulation models.
3
4
5
Acknowledgements
6
7
Weather data for Cortez from COlorado AGricultural Meteorological nETwork
8
(COAGMET); for Davis from University of California - Integrated Pest Management
9
Program (UC IPM); for Padova from University of Basilicata Viale Ateneo Lucano,
10
Potenza - Italy; for Rothamsted from Rothamsted Research, UK; for Wageningen from
11
Wageningen University and Research Center - Meteorology and Air Quality section,
12
Wageningen, The Netherlands; for Pretoria from Agricultural Research Council, Institute
13
of Soil, Climate and Water, Pretoria, South Africa; and for Griffith from CSIRO, Land
14
and Water, Griffith, Australia is gratefully acknowledged. The paper benefited from the
15
comments of the anonymous reviewers.
23
1
Appendix A. Models for estimating daily solar radiation from daily air temperature
2
range and/or precipitation
3
4
5
The BC, CD and DB models estimate daily solar radiation, Is (MJ m-2), at the
earth’s surface as:
6
7
I s  tti  I ex
[A1]
8
9
where tti is the daily atmospheric transmission coefficient and Iex (MJ m-2) is the daily
10
extraterrestrial solar radiation which is calculated based purely on solar geometry and the
11
solar constant (e.g., Swift 1976; Campbell and Norman, 1998). The tti part of Eq. [A1] is
12
estimated by the BC, CD and DB models as follows:
13
14
BC model (Bristow and Campbell, 1984):
15
16

 b  Ti c
tti    1  exp 
 Tm


 ;

[A2]
17
18
CD model (Donatelli and Campbell, 1998):
19
20


 T  
tti    1  exp  b  (0.017  exp (exp (0.053  Tavg ( i ) )))  Ti 2  exp  n(i)    ;
 Tnc   


[A3]
21
22
DB model (Donatelli and Bellocchi, 2001):
23
24
 b  Ti 2  

 i 
  i 
 
tti    1  c1  sin  r
 c2    r
f  c2     1  exp 
 ;
 180
  180
 

 Tw  
[A4]
25
26
27
f  c2   1  1.90  c2  int  c2    3.83  c2  int  c2  
2
[A5]
24
1
2
where  is clear-sky atmospheric transmission coefficient, i is day of year, b and c are the
3
daily air temperature range coefficients, Tx(i) and Tn(i) (oC) are the daily maximum and
4
minimum air temperatures respectively, Ti (oC) = Tx(i) – (Tn(i) + Tn(i+1))/2, Tm (oC) is
5
the fixed monthly mean T, Tnc is the summer night air temperature factor, Tavg(i) (oC) =
6
(Tx(i) + Tn(i))/2, ir is a reverse option (ir = 1 for no reverse; ir =361- i for reverse), c1 and c2
7
are general seasonality factors, int(c2) is the integer of c2 and Tw (oC) is the mobile
8
weekly mean T. On rainy days Ti is reduced by 25%, and if T on the day before rain
9
occurred, T(i-1), was less than T(i-2) by 2oC it was also reduced by 25% assuming that
10
cloudy conditions began on day (i -1) (Bristow and Campbell, 1984). Extended rain
11
periods enable equilibration between incoming solar radiation and T and do not require
12
adjustments (Bristow and Campbell, 1984). The Tnc factor is meant to prevent
13
underestimation of solar radiation prediction during summer that may be introduced due
14
to higher Tn. The BC, CD and DB models are contained within the software RadEst tool
15
which is freely available via the website http://www.sipeaa.it/tools.
16
17
Hgvs model (Hargreaves et al., 1985):
18
19
I s  b1  b2 DT(i )  I ex
[A5]
20
21
where b1 and b2 are empirical coefficients and DT(i)= Tx(i) - Tn(i).
22
23
HKS model (Hunt et al., 1998):
24
25
I s  b0  b1  I ex  DT( i ) 0.5  b2Tx( i )  b3  PPi  b4  PPi 2
[A6]
26
27
where b0, b1, b2, b3 and b4 are empirical coefficients, and PP (mm) is the daily total
28
precipitation.
29
30
MH model (Mahmood and Hubbard, 2002):
25
1
2
The MH model estimates daily incoming solar radiation based on clear-sky solar
3
radiation (Icc) - calculated from day of year, maximum daylength for the year for a given
4
latitude following Cengiz et al. (1981):
5
6
I s  b0  DT(i ) b1  I ccb2
[A7]
7
8
where b0, b1 and b2 are empirical coefficients.
26
1
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10
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12
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13
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14
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18
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19
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20
21
22
23
24
25
26
27
28
29
30
31
30
1
List of Figures
2
3
Fig. 1. The statistical indices, modules and the indicator used for statistical evaluation
4
along with the decision rules and their systematic aggregation (Indices: RRMSE =
5
relative root mean square error, EF = model efficiency, P(t) = two-tailed paired t-test, R
6
= correlation coefficient, PIdoy and PITn pattern of the residuals by day of year and by
7
minimum air temperature respectively); Modules: Accuracy, Correlation and Pattern, Irad
8
= single modular indicator, F = Favourable, U = Unfavourable).
9
10
Fig. 2. Difference between estimated and observed mean daily solar radiation against day
11
of year for all models and sites.
12
13
Fig. 3. Grass ETo simulated from observed and model-estimated solar radiation for
14
Davis, California.
15
16
Fig. 4. Grass ETo simulated from observed and model-estimated solar radiation for
17
Wageningen, The Netherlands.
18
19
Fig. 5. Total dry biomass (tons ha-1) simulations from using observed and estimated solar
20
radiation from the different models (a) for Davis, California and (b) Wageningen, The
21
Netherlands.
22
23
24
List of Tables
25
26
Table 1
27
Sites of meteorological stations and period of data records used for estimation of solar
28
radiation
29
30
31
31
1
Table 2
2
Models used for estimation of solar radiation
3
4
5
Table 3
6
Statistical indices and modules used in evaluation of solar radiation models
7
8
9
10
11
Table 4
Calibrated model coefficients for all locations. See Appendix A for details of the models
12
13
Table 5
14
Performance of the solar radiation models by site according to mean of the indices
15
RRMSE (relative root mean square error), EF (model efficiency), P(t) (two-tailed paired
16
t-test), R (correlation coefficient), and PIdoy and PITn (Pattern of residuals by day of year
17
and minimum air temperature respectively); and the modules Accuracy, Correlation and
18
Pattern, and the indicator, Irad
19
20
21
Table 6
22
Performance of each model for the indices RMSE (root mean square error), RRMSE
23
(relative root mean square error), EF (model efficiency), P(t) (two-tailed paired t-test), R
24
(correlation coefficient) and PIdoy and PITn (Pattern of residuals by day of year and
25
minimum air temperature respectively); and the modules Accuracy, Correlation and
26
Pattern, and the indicator, Irad as averaged over all the sites
27
28
29
Table 7
30
Statistical comparison of total dry biomass (tons ha-1) simulations for maize at Davis,
31
California and Wageningen, The Netherlands using observed and estimated solar
32
radiation
32
Correlation
Pattern
F Partial U F Partial U
=0
=1
F
F
U
U
F
F
U
U
=0
=1
F
U
F
U
F
U
F
U
rad
Modular
indicator
I
Modules
Expert Accuracy
weight F Partial U
=0
=1
F
0.00
F
0.30
F
0.15
F
0.45
U
0.55
U
0.85
U
0.70
U
1.00
1
2
Fig. 1. The statistical indices, modules and the indicator used for statistical evaluation
3
along with the decision rules and their systematic aggregation (Indices: RRMSE =
4
relative root mean square error, EF = model efficiency, P(t) = two-tailed paired t-test, R
5
= correlation coefficient, PIdoy and PITn pattern of the residuals by day of year and by
6
minimum air temperature respectively); Modules: Accuracy, Correlation and Pattern, Irad
7
= single modular indicator, F = Favourable, U = Unfavourable).
33
Indices
-2
Expert RRMSE (%)
EF
P(t)
Expert
R
(MJ m )
weight F Partial U F Partial U F Partial U weight F Partial U
PIdoy
PITn
< 20
> 40 > 0.9 < 0.4 >0.1 < 0.05
> 0.9 < 0.7 Expert
F Partial U
F Partial U
weight
F
0.00
0.00
F
F
F
<1
> 2.5 < 1
> 2.5
F
1.00
0.20
F
U
U
0.00
F
F
U
0.40
F
F
0.50
U
F
U
0.60
F
U
0.50
F
U
F
0.40
U
F
1.00
U
U
F
0.60
U
U
U
0.80
U
F
U
1.00
U
U
1
2
Fig. 2. Difference between estimated and observed mean daily solar radiation against day
3
of year for all models and sites.
34
1
2
Fig. 3. Grass ETo simulated from observed and model-estimated solar radiation for
3
Davis, California.
35
1
2
Fig. 4. Grass ETo simulated from observed and model-estimated solar radiation for
3
Wageningen, The Netherlands.
36
1
2
Fig. 5. Total dry biomass (tons ha-1) simulations from using observed and estimated solar
3
radiation from the different models (a) for Davis, California and (b) Wageningen, The
4
Netherlands.
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
37
1
Table 1
2
Sites of meteorological stations and period of data records used for estimation of solar
3
radiation
Location
Latitude
Longitude
Elevation (m)
Period
Cortez, Colorado, USA
37º14′ (N)
108º41′ (W)
1833
1992-2005
Davis, California, USA
38º32′ (N)
121º47′ (W)
18
1985-2005
Padova, Italy
44º58′ (N)
12º11′ (W)
0
1990-2003
Rothamsted, UK
51º48′ (N)
0º24′ (E)
128
1980-2000
Wageningen, The Netherlands 51º58′ (N)
5º38′ (E)
7
1985-2005
Pretoria, South Africa
25º45′ (S)
28º11′ (E)
1308
1993-2003
Griffith, Australia
34º17′ (S)
146º3′ (E)
125
1986-2005
4
5
6
7
8
9
10
11
Table 2
12
Models used for estimation of solar radiation
Authors
13
14
Model abbreviation
Model requirements
Bristow and Campbell (1984)
BC
Iex, Tx, Tn and PP*
Donatelli and Campbell (1998)
CD
Iex, Tx, Tn and PP
Donatelli and Bellocchi (2001)
DB
Iex, Tx, Tn and PP
Hargreaves et al. (1985)
Hgvs
Iex, Tx and Tn
Hunt et al. (1998)
HKS
Iex, Tx, Tn and PP
Mahmood and Hubbard (2002)
MH
Iex, Tx and Tn
*
The BC, CD and DB models require presence or absence of PP while the HKS model requires the amount
(see Appendix A)
38
1
Table 3
2
Statistical indices and modules used in evaluation of solar radiation models
Index
Notation Range
Best
value
Relative root mean square
error (%)
Model efficiency
Paired Student t-test
probability of equal means
Correlation coefficient
Pattern index by day of year
(MJ m-2)
Pattern index by minimum air
temperature (MJ m-2)
RRMSE
0 to ∞
0
EF
1 to -∞
1
P(t)
0 to 1
1
R
-1 to 1
1
PIdoy
0 to ∞
0
PITn
0 to ∞
0
Module
Accuracy
(amount of residuals)
Correlation
Pattern (state of
pattern in residuals)
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
39
1
2
3
4
Table 4
Calibrated model coefficients for all locations. See Appendix A for details of the models
Location
Model
BC
Parameter
b
c
CD
b
Tnc
DB
b
c1
c2
Hgvs b1
b2
HKS b0
b1
b2
b3
b4
MH b0
b1
b2
Cortez,
Colorado,
USA
0.107
2
0.203
104.1
0.112
0.034
1.410
-1.172
0.155
-1.112
0.161
-0.032
-0.355
0.004
0.077
0.758
1.129
Davis,
California,
USA
0.137
2
0.262
44.5
0.122
-0.026
1.410
-0.784
0.162
0.301
0.164
-0.050
-0.264
0.004
0.067
0.796
1.172
Padova,
Italy
Rothamsted,
UK
0.141
2
0.396
42.1
0.131
-0.040
0.008
-2.041
0.190
-1.420
0.186
0.003
-0.302
0.003
0.076
0.897
1.122
0.110
2
0.345
106.1
0.106
-0.011
1.183
-0.610
0.141
0.022
0.147
-0.019
-0.356
0.008
0.146
0.488
1.151
Wageningen,
The
Netherlands
0.100
2
0.331
64.1
0.099
-0.026
0.215
-0.728
0.140
-0.235
0.132
0.039
-0.305
0.008
0.099
0.721
1.088
Pretoria,
South
Africa
0.126
2
0.39
84.8
0.134
-0.053
0.041
-0.258
0.173
0.149
0.169
0.021
-0.395
0.006
0.263
0.791
0.775
Griffith,
Australia
0.123
2
0.282
100.0
0.119
0.015
1.137
-0.330
0.163
2.197
0.169
-0.112
-0.760
0.019
0.374
0.745
0.677
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
40
1
2
3
4
5
6
7
Table 5
Performance of the solar radiation models by site according to mean of the indices
RRMSE (relative root mean square error), EF (model efficiency), P(t) (two-tailed paired
t-test), R (correlation coefficient), and PIdoy and PITn (Pattern of residuals by day of year
and minimum air temperature respectively); and the modules Accuracy, Correlation and
Pattern, and the indicator, Irad
Location
Cortez,
Colorado,
USA
Davis,
California,
USA
Padova, Italy
Rothamsted,
UK
Wageningen,
The
Netherlands
Pretoria,
South Africa
Griffith,
Australia
Model
BC
CD
DB
Hgvs
HKS
MH
BC
CD
DB
Hgvs
HKS
MH
BC
CD
DB
Hgvs
HKS
MH
BC
CD
DB
Hgvs
HKS
MH
BC
CD
DB
Hgvs
HKS
MH
BC
CD
DB
Hgvs
HKS
MH
BC
CD
DB
Hgvs
HKS
MH
RRMSE
(%)
17.50
17.77
18.42
18.34
16.88
19.32
15.23
14.34
14.68
15.87
15.30
20.28
26.34
25.72
26.32
25.42
23.54
28.73
31.51
31.69
31.66
34.30
32.26
37.43
33.96
33.27
34.24
33.26
36.33
35.11
15.81
15.87
16.47
17.62
16.23
17.24
21.77
21.53
21.62
21.38
20.74
22.74
EF
P(t)
R
0.84
0.83
0.82
0.83
0.85
0.80
0.91
0.92
0.91
0.90
0.91
0.83
0.82
0.83
0.82
0.84
0.86
0.79
0.83
0.82
0.82
0.79
0.82
0.75
0.81
0.81
0.80
0.81
0.78
0.79
0.74
0.74
0.72
0.69
0.71
0.69
0.78
0.79
0.79
0.79
0.80
0.77
0.14
0.00
0.03
0.17
0.07
0.04
0.08
0.07
0.17
0.08
0.08
0.08
0.24
0.03
0.08
0.00
0.00
0.00
0.24
0.19
0.31
0.04
0.10
0.00
0.14
0.07
0.11
0.07
0.00
0.00
0.06
0.08
0.04
0.01
0.05
0.04
0.06
0.12
0.06
0.23
0.25
0.12
0.92
0.92
0.92
0.91
0.93
0.91
0.96
0.96
0.96
0.95
0.96
0.93
0.91
0.92
0.91
0.92
0.93
0.91
0.91
0.92
0.91
0.90
0.91
0.89
0.90
0.91
0.90
0.91
0.92
0.91
0.88
0.88
0.86
0.85
0.74
0.84
0.89
0.90
0.89
0.90
0.90
0.88
PIdoy PITn
(MJ m-2)
1.78 1.55
1.74 1.35
1.83 1.66
1.89 2.08
1.69 1.59
2.94 1.79
2.44 2.02
1.82 1.06
1.92 1.50
1.80 1.27
1.77 1.51
3.72 1.96
2.23 1.86
1.70 1.13
1.90 1.51
1.15 1.34
1.30 1.30
2.76 1.13
0.92 0.70
0.99 0.79
0.81 0.68
0.88 0.84
0.81 1.08
2.4 0.86
1.36 1.10
1.16 0.85
1.24 1.13
1.17 1.09
1.82 1.44
2.55 1.12
1.66 1.44
1.68 0.99
1.11 0.59
1.76 1.25
1.39 0.84
1.71 1.94
1.37 1.60
1.82 1.69
1.25 2.15
1.55 1.64
1.92 1.33
2.19 1.36
Accuracy
Correlation
Pattern
Irad
0.1848
0.2286
0.2186
0.1676
0.1905
0.2123
0.1445
0.1449
0.1225
0.1675
0.1796
0.1975
0.2396
0.2141
0.2674
0.2971
0.2463
0.4255
0.3739
0.4280
0.3579
0.5401
0.4491
0.6461
0.4960
0.4968
0.5046
0.4924
0.6164
0.5838
0.2669
0.2605
0.2698
0.3377
0.2937
0.3217
0.2301
0.2105
0.2117
0.1553
0.1644
0.2502
0.0006
0.0006
0.0016
0.0022
0.0000
0.0008
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0008
0.0004
0.0004
0.0000
0.0000
0.0012
0.0003
0.0002
0.0008
0.0028
0.0007
0.0097
0.0057
0.0021
0.0066
0.0011
0.0001
0.0006
0.0676
0.0642
0.0954
0.1591
0.8476
0.2021
0.0131
0.0116
0.0101
0.0093
0.0197
0.0289
0.3799
0.3741
0.4084
0.5606
0.4297
0.7147
0.7416
0.3241
0.4578
0.3616
0.4018
0.7545
0.6632
0.2970
0.4925
0.1628
0.2096
0.5154
0.0557
0.0555
0.0368
0.0650
0.0850
0.4428
0.1412
0.1250
0.1277
0.1325
0.4095
0.4692
0.3422
0.2399
0.0929
0.3036
0.1195
0.5181
0.3721
0.4511
0.4582
0.3969
0.3721
0.5286
0.1759
0.1962
0.1902
0.2227
0.1970
0.2823
0.2738
0.1326
0.1719
0.1517
0.1700
0.3018
0.2880
0.2126
0.2542
0.1541
0.1359
0.3787
0.1838
0.2313
0.1611
0.3307
0.2470
0.5131
0.3063
0.3036
0.3128
0.3111
0.4870
0.4708
0.2143
0.1703
0.1349
0.2670
0.2887
0.3422
0.1945
0.2157
0.2172
0.1637
0.1699
0.2682
8
9
41
1
Table 6
2
Performance of each model for the indices RMSE (root mean square error), RRMSE
3
(relative root mean square error), EF (model efficiency), P(t) (two-tailed paired t-test), R
4
(correlation coefficient) and PIdoy and PITn (Pattern of residuals by day of year and
5
minimum air temperature respectively); and the modules Accuracy, Correlation and
6
Pattern, and the indicator, Irad as averaged over all the sites
Model
BC
CD
DB
Hgvs
HKS
MH
RMSE
(MJ m-2)
3.27
3.23
3.30
3.36
3.24
3.66
RRMSE
(%)
23.16
22.89
23.34
23.74
23.31
25.83
EF
P(t)
R
0.82
0.82
0.81
0.81
0.82
0.78
0.14
0.08
0.11
0.09
0.08
0.04
0.91
0.91
0.91
0.91
0.90
0.90
PIdoy PITn
(MJ m-2)
1.68 1.47
1.53 1.12
1.44 1.32
1.46 1.36
1.53 1.30
2.61 1.45
Accuracy
Correlation
Pattern
Irad
0.2765
0.2948
0.2789
0.3082
0.3053
0.3767
0.0126
0.0113
0.0164
0.0249
0.1240
0.0348
0.3851
0.2667
0.2963
0.2833
0.2896
0.5633
0.2338
0.2089
0.2061
0.2288
0.2422
0.3653
7
8
9
10
Table 7
11
Statistical comparison of total dry biomass (tons ha-1) simulations for maize at Davis,
12
California and Wageningen, The Netherlands using observed and estimated solar
13
radiation
Davis, California (n = 18)
14
15
16
17
18
Observed
BC
CD
DB
Hgvs
HKS
MH
Acc. total dry biomass (tons ha-1)
177.26
176.88 174.34 180.49 176.74 176.22
164.53
Total difference (tons ha-1)
-(0.38) -(2.92)
7.51
-(0.52) -(1.04) -(12.74)
9.85
9.83
9.69
10.03
9.82
9.79
9.14
Mean total dry biomass (tons ha-1)
Mean difference (tons ha-1)
-(0.02) -(0.16)
0.18
-(0.03) -(0.06)
-(0.71)
Maximum error (tons ha-1)
-(0.73) -(0.81)
1.77
-(0.66) -(0.70)
-(1.63)
mean absolute error (tons ha-1)
0.25
0.29
0.38
0.24
0.25
0.72
RMSE (tons ha-1)
0.321
0.355
0.531
0.319
0.330
0.848
R2
0.99
0.99
0.98
0.99
0.99
0.99
Wageningen, The Netherlands (n = 19)
Acc. total dry biomass (tons ha-1)
345.33
331.95 336.15 338.55 351.13 316.33
320.04
Total difference (tons ha-1)
-(13.38) -(9.19) -(6.78)
5.80
-(29.00) -(23.50)
Mean total dry biomass (tons ha-1)
18.18
17.47
17.69
17.82
18.48
16.65
16.84
Mean difference (tons ha-1)
-(0.70) -(0.48) -(0.36)
0.31
-(1.53)
-(1.33)
Maximum error (tons ha-1)
-(1.77) -(1.82) -(1.57)
1.76
-(2.98)
-(2.67)
mean absolute error (tons ha-1)
0.87
0.61
0.64
0.54
1.66
1.45
RMSE (tons ha-1)
0.981
0.790
0.743
0.697
1.841
1.610
R2
0.90
0.92
0.91
0.91
0.82
0.86
†
Acc. dry biomass - the total dry biomass accumulated during the simulation period; Total and Mean
difference - are the absolute differences in total and mean dry biomass simulated from estimated and
observed solar radiation; Maximum error - is the largest absolute difference in simulated total dry biomass
(negative values indicate the direction of change); RMSE - root mean square error; R2 – coefficient of
determination
42