Arvid Skartveit

A model for the diffuse fraction of hourly global radiation

Abstract Hourly values of diffuse and beam irradiance are often required in cases where, at best, only global irradiance is available. For use in the evaluation of the climatological potential for solar energy utilization in Norway, an analytical model is proposed expressing the hourly diffuse fraction of global irradiance in terms of hourly solar elevation and clearness index. This model, developed for average snow-free close to sea level conditions in Norway, fairly well fits the average picture of an extensive independent data base.

Modelling slope irradiance at high latitudes

Abstract Irradiance on slopes is often required in cases where it is not directly measured and, consequently, has to be modelled from alternative data. In this paper three models for the derivation of hourly slope irradiances from hourly horizontal beam and diffuse sky irradiances are tested against 5 years of recorded slope irradiances at Bergen (60°24′N, 5°19′E). The test demonstrated the importance of accounting for sky radiance anisotropy both at overcast and cloudless weather. Moreover, a stepwise procedure for the derivation of long-term mean slope irradiance from various kinds of input data is outlined. Testing against observations demonstrated that, except for steeply south-facing slopes at low mid-winter sun, long-term slope irradiance is derived with reasonably accuracy from mean cloudiness. Moreover, observed horizontal irradiances, whether global or beam and diffuse, as input data yield satisfactory accuracy throughout the year on all slopes.

An hourly diffuse fraction model with correction for variability and surface albedo

Abstract The paper presents an improved version of a previously published model for the diffuse fraction of hourly global irradiance. In addition to hourly solar elevation and clearness index, an hour-to-hour variability index and regional surface albedo are included among the input parameters. Moreover, to prevent excessively high normal incidence beam irradiances at very low solar elevations, the model does not allow a solar elevation dependent maximum beam transmittance to be exceeded. This new model is tuned to 32 years of data from Bergen, Norway. Moreover, a test against independent data from four European stations showed that the model performs better than the models of Erbs et al. (1982) , Maxwell (1987) and Perez et al. (1992) .

The probability density and autocorrelation of short-term global and beam irradiance

Abstract For individual hours, a characteristic bimodal pattern of short-term global and beam irradiance is frequently observed, with modes at high and low irradiances and with low probabilities near the hourly averages. For such hours, averaging over the hour will imply smoothing of quite significant variations within the hour. Models for the probability density distributions of short-term (5 min or less) irradiances are presented in this paper. These distributions are not unique functions of the hourly averages, but depend heavily also on the irradiance variability within the hour. This intrahour variability is found to depend on the averaging time and also on the interhour variability among three hourly averages, namely, the hour in question, the preceding and the deceding hour. The distribution differences between 5 min averages and instantaneous values are, however, negligible for most practical purposes. The lag one autocorrelation is evaluated as a function of averaging time, and a first order autoregressive model is presented. With hourly averages as the only input the probability density and autoregressive model in combination produce time series of short-term intrahour averages having realistic distributions and autocorrelation structure.

A probability density function for daily insolation within the temperate storm belts

Abstract Long-term insolation data are frequently obtainable either from recordings or from various empirical formulae. Many radiation-driven processes are, however, highly non-linear. The ability to model the correct long-term frequency distribution of a sequence of radiation events having a specified long-term mean insolation, is, therefore, valuable. This paper emphasizes that the “U-shaped” cloud cover frequency distribution, characteristic of the temperate storm belts, is “mapped” into the irradiation domain in a way that augments the number of “poor” and “excellent” days of sunshine at the expense of “average” days. An analytical and adjustable model is proposed, correlating the probability density of daily clearness index with the monthly average clearness index. This model is tuned to the two longest records of solar radiation in Norway, and it is verified against independent observations from the northern temperate storm belt.

Observed and modelled hourly luminous efficacies under arbitrary cloudiness

Abstract The luminous efficacy of solar irradiance under cloudless sky is evaluated by a spectral radiative transfer model. Based on model runs with input data from Bergen, Norway, the beam and diffuse luminous efficacies are parameterized for the cloudless case in terms of solar elevation and day number of the year. The luminous efficacy under an unbroken cloud cover is evaluated by a combination of the cloudless model and an overcast radiative transfer model, and then parameterized in terms of solar elevation and clearness index. Based on a physical argument, an overall model is finally presented, yielding the luminous efficacy at arbitrary cloud conditions by a tuned interpolation between the efficacies for the overcast and the cloudless case. This model yields deviations from observations that are small relative to the luminous efficacy variations caused by variations in solar elevation and cloudiness.

A probability density model for hourly total and beam irradiance on arbitrarily orientated planes

Abstract An analytical and adjustable model is proposed, by which the probability density of hourly global irradiance may be derived from long-term mean global irradiation. The model explicitly accounts for the bimodal character of the probability density of solar irradiance, and it is tuned to a developmental data sample and verified against an independent data base. In combination with previously published deterministic models this probability density model even forms a composite model which yields the probability density of hourly irradiance (total, beam or diffuse) on arbitrarily orientated planes. This composite model is also verified against the independent data base.

Luminous efficacy models and their application for calculation of photosynthetically active radiation

From a spectral radiative transfer model, an algorithm is developed for the conversion of illuminance to different measures of Photosynthetically Active Radiation (PAR) (in W m−2 or in μEm−2s−1). This illuminance to PAR conversion may even be used in combination with a luminous efficacy model and, thus, form a photosynthetic efficacy model. In this work, two luminous efficacy models are chosen, one empirical and one derived from the above radiative transfer model. Observed PAR energy flux and PAR photon flux from seven Nordic stations (56–70°N) and illuminance from one U.S. station (43°N) are, together with observed all-wave solar irradiance from all stations, used for verification. Observed and modelled luminous efficacies agree on the average within 1% at high solar elevation under cloudless sky, while it is indicated that the illuminance, PAR energy flux, and PAR photon flux radiometers are mutually inconsistent by some 6–16%. Even differences in cosine response between radiometer types are apparent at low solar elevation under cloudless sky. In the present climates, the global radiation efficacy is 10–12% higher under an average cloudless atmosphere than it is outside the atmosphere. By introducing an average cloud deck in this cloudless atmosphere, a further efficacy increase, slightly exceeding these 10–12%, is observed. However, observations indicate that the cloud transmittance algorithm used in the radiative transfer model significantly overestimates the global radiation efficacy increase caused by horizontally inhomogeneous cloud decks.

Spatial distribution of photosynthetically active radiation over complex topography

Abstract The paper presents a technique for spatially continuous mapping of local screening effects on diffuse and direct beam fluxes of short-wave radiation (0.3–3 μm) or photosynthetically active radiation (PAR: 0.4–0.7 μm). The method requires a digitized topographical map, and radiation data either from a site with an unobscured horizon or from radiative transfer calculations requiring minimum input data. Within one selected complex topography, screening/reflection effects yield substantial local spatial variations in the PAR field. These variations depend on albedo and cloudiness, and they are (on a percentage basis) most pronounced at high latitudes.

Characteristics of hourly global irradiance modelled from cloud data

A model yielding hourly global irradiance from cloud observations is presented. Tested against independent observations of hourly irradiance, the model yields a correlation coefficient 0.92. The corresponding correlation coefficients for daily and monthly values are 0.975 and 0.998, respectively. The mean bias and root-mean-square errors for hourly data are less than, respectively, 3% and 30% of the average. Moreover, the frequency distributions and the lag one autocorrelations of modelled hourly averages are close to those of the observed data.