Modelling monthly diffuse solar radiation fraction and its validity over the Indian sub-tropics

Authors

  • Jyotsna Singh,

    Corresponding author
    1. Centre of Excellence in Climatology, Birla Institute of Technology, Mesra-835215, Ranchi, Jharkhand, India
    • Centre of Excellence in Climatology, Birla Institute of Technology, Mesra-835215, Ranchi, Jharkhand.
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  • Bimal K. Bhattacharya,

    1. Agriculture, Terrestrial Biosphere and Hydrology Group (ABHG), Space Applications Centre (ISRO), Ahmedabad-380015, Gujarat, India
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  • Manoj Kumar,

    1. Centre of Excellence in Climatology, Birla Institute of Technology, Mesra-835215, Ranchi, Jharkhand, India
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  • Kaniska Mallick

    1. Water and Carbon Cycles Group, NASA, Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109
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Abstract

A three-parameter sigmoidal ‘local’ model (climate-specific) and a ‘regional’ model (common for all climates) have been developed for measuring the monthly average diffuse solar radiation fraction from atmospheric transmissivity by using large, 21-year datasets (1973–1993) over four stations (Jodhpur, New Delhi, Nagpur and Kolkata). These stations represent the four ‘prime’ climates (arid, semi-arid, sub-humid and humid) over the Indian sub-tropics. The models have been validated with the longer time-series (10 years) of independent datasets (1994–2003) of 4 ‘prime’ climates as well as datasets from 16 stations using one-year datasets (termed as secondary stations) of the Indian region. The monthly diffuse fraction estimates were also compared with seven globally existing models. The ‘regional’ model showed more accurate estimates than ‘local’ models over three (semi-arid, sub-humid and humid) of the four ‘prime’ climates. However, the different error statistics showed that the ‘regional’ model outperformed the globally existing models which failed to capture diffuse fraction variability over the Indian sub-tropics. The extendibility of the ‘regional’ model over ‘secondary’ stations in India showed an overall good performance with R2: 0.78–0.96 and RMSE: 0.017–0.125, except for two stations. These models are unique for Indian sub-tropics and can undoubtedly be used for predicting future diffuse solar radiation fraction from transmissivity datasets of climate simulations, and also for other meteorological, climatological, solar energy-based applications. Copyright © 2011 Royal Meteorological Society

1. Introduction

Global insolation (RG) and diffuse solar radiation fraction (KD) reaching the surface of the Earth is altered by atmospheric clarity (based on aerosol concentration) or cloudiness that determines atmospheric transmissivity (KT). Changes in cloud properties, fog, atmospheric aerosol loadings including dust, volcanic or anthropogenic emissions, alter both the KT and KD which also affect plant productivity, and the land carbon sink globally (Mercado et al., 2009). Recent observational findings over diverse plant functional types also established the role of KD on modulating canopy gas exchange processes (Niyogi et al., 2004; Knohl and Baldocchi, 2008; Still et al., 2009; Jing et al., 2010), vegetation, light and water use efficiencies (Rocha et al., 2004; Chen et al., 2009). Studies have also shown the sensitivity of vegetation productivity to the fluctuations in RD and it is more efficiently used by the plants than the direct component (Roderick et al., 2001). Because of its omnidirectional nature (i.e. incident from multiple angles), the diffuse component has the greater capacity to penetrate the light-limited layers of the dense forest canopies, thus stimulating photosynthesis and productivity (Gu et al., 2002; Still et al., 2009).

The main concern of today's world is sustainable development which is directly linked to the utilisation of energy resources. For attaining sustainable development, we need to harness sustainable energy sources, and the use of renewable energy such as ‘solar energy’ will promote sustainability (Dincer and Rosen, 1999). Values of RD and KD are also required for building the solar energy systems. In addition to these, long-term records of RD and KD have significant roles in constructing quantitative information on atmospheric turbidity, aerosol and cloudiness. This, in turn, can be used to study the response-feedback mechanisms between Earth and atmosphere (Barth et al., 2005; Carslaw et al., 2010), for example, atmospherically driven changes in global hydrology and carbon cycles, and the impact of these cycles on the atmospheric properties. Different atmospheric conditions (turbidity and transparency), airmass, content of water vapour in the atmosphere and the cloud cover distribution influence the insolation by absorption, scattering and re?ection (Okogbue et al., 2009). The knowledge of KD can be useful to get an idea of concentration of atmospheric load indirectly, where a low KD will indicate clear sky and more pristine atmosphere, and vice versa. In a developing country like India, RD measurement is sparse because it is expensive and tedious (Veeran and Kumar, 1993; Gopinathan and Soler, 1995; Pandey and Katiyar, 2009). Therefore, a common alternative is to formulate robust correlation models of RD or KD from the observational networks. Many empirical models of KD have been developed in some advanced countries such as Europe and North America (Reindl et al., 1990), US (Erbs et al., 1982), Australia (Spencer, 1982), Canada (Orgill and Hollands, 1977), Italy (Barbora et al., 1981; Jain, 1990), as well as in some developing countries such as Thailand (Janjai et al., 1996), Turkey (Ulgen and Hepbasli, 2009) and Saudi Arabia (Elhadidy and Abdel-Nabi, 1991). These models cannot be straightway extrapolated to a sub-tropical country like India because of differences in the radiative forcing patterns. Besides this, the models of the developed countries are valid for the higher latitudes (above 40°) only. Though Gopinathan and Soler (1995) have developed a KD versus KT relationship taking observations from different parts of the world including two stations of India, yet that relationship suffers from limited climatological information, and the model was not validated over other independent Indian stations. A comprehensive list of correlation models between monthly KD and KT developed earlier is given in Table I (Liu and Jordan, 1960; Erbs et al., 1982; Ulgen and Hepbasli, 2009; and others). No such robust model of KD is available so far for India. This paper aims at developing a monthly model of KD from the time-series measurements of RG and RD. The objectives of the present study are: (1) development of local as well as regional models for monthly average KD using long time-series RG and RD observations over ‘prime’ climates (Section 2) over the Indian sub-tropics; (2) comparison of newly developed KD model outputs with the estimates from existing global models; and (3) validation of the KD model with independent datasets and shorter time series data from 16 different radiometric stations of India.

Table I. Globally existing monthly average models for diffuse solar radiation fraction (KD) based on transmissivity (KT)
StudyModelsLocation
Page (1961)KD = 1.00 − 1.13(KT)40°N-40°S
Liu and Jordan (1960)KD = 1.39 − 4.027(KT) + 5.531(KT)2 − 3.108(KT)3USA
Erbs et al. (1982)KD = 1.317 − 3.023(KT) + 3.372(KT)2 − 1.769(KT)3USA
Barbaro et al. (1981)KD = 1.0492 − 1.3246(KT)Italy
Elhadidy and Abdel-Nabi (1991)KD = 1.039 − 1.741(KT)2Saudi Arabia
Ulgen and Hepbasli (2009)KD = 0.981 − 1.9028(KT) + 1.9319(KT)2 − 0.6809(KT)3Turkey
Gopinathan and Soler (1995)KD = 0.91138 − 0.96225(KT)30°S-60°N

2. Study region and datasets

Time series measurements on daily RG and RD were obtained for 4 stations in the Indian sub-tropics namely, Jodhpur (26°23′N, 73°08′E), New Delhi (28°37′N, 77°13′E), Nagpur (21°08′N, 79°10′E) and Kolkata (22°36′N, 88°24′E), representing ‘prime’ climates - arid, semiarid, sub-humid and humid, respectively. These stations are referred to here as primary stations where consistently longer time series datasets on RG and RD were available for a period of 31 years (1973–2003). In Jodhpur, the transport of sand and dust through the Thar Desert especially in pre-monsoon months, and clouds from western disturbances during winter months modulate the KT and KD. The persistent fog and clouds from western disturbances in winter months are responsible for controlling both the variables in New Delhi. Shortwave radiation regime in Kolkata is mostly controlled by cloud dynamics associated with the Bay of Bengal branch of the southwest monsoon, cyclones, mist and maritime aerosols. In Nagpur, the clouds through the Arabian Sea branch off, and continental aerosols control RG and RD. The four climates represented by these stations thus dominate the atmosphere on KD over the Indian sub-tropics. Hence, longer time series datasets from these four stations are suitable for the development of semi-empirical regression models.

A shorter period of datasets of about one year (June 1998–May 1999) were available for other 16 stations across India. These are referred to here as secondary stations. The distribution of ‘primary’ and ‘secondary’ stations is shown in the Figure 1. The RG and RD data were recorded at hourly intervals at the IMD (India Meteorological Department) stations using a pyranometer and a shading ring pyranometer, respectively, and accumulated over a day in megajoules per square meter (MJ m−2). The pyranometers were calibrated from time to time with respect to World Radiometric Reference (WRR), and maximum ± 5% uncertainty was found in the measured data (Singh and Tiwari, 2005). The calibration results were found satisfactory. The data over a period of 31 years (1973–2003) were used for developing and validating the KD model. The model developed over the ‘primary’ stations were used to evaluate the performance over the ‘secondary’ stations.

Figure 1.

Four primary (used for monthly average diffuse fraction (KD) model formulation) and sixteen secondary (used for validation of the KD model) stations of India with their latitudes and longitudes. Primary stations are represented with symbol (.) in upper case letters and secondary stations with symbol (*) in numeric

3. Methodology

3.1. Computation of KD and KT

The development of the KD model requires basic observations of KT and KD. Since these ratios cannot be measured directly, the daily KT (also called clearness index) and KD were computed from the measurements of RG and RD as follows:

equation image(1)

KT was computed as the ratio of global radiation to extraterrestrial daily radiation (Ra) received at the top of atmosphere.

equation image(2)

The ‘Ra’ was computed after Burman and Pochop (1994)

equation image(3)

S0 = solar constant megajoule per meter per day (MJ m−1 day−1), L = latitude (degree), equation image is the Sun–Earth distance ratio:

equation image(4)

Where, D = calendar day of the year.

ωhs = sunset hour angle (degree) which was calculated as follows:

equation image(5)

δ = daily solar declination angle (degree) which was computed as follows:

equation image(6)

3.2. Model development

We have tried to develop a new climate-specific as well as a common model for KD from its observational relationship with KT for the Indian sub-tropics. All the datasets were divided into two parts. Out of a total of 31 years (1973–2003), 21 years' (1973–1993) datasets were used for model formulation in the ‘primary’ stations. The rest of the period (1994–2003) was used for validation of the developed model. The climate-specific model is hereafter referred to as local model. A common model is developed by pooling the data from four climate types and referred to as regional model hereafter. A three-parameter sigmoidal model is fitted on KT−1 and KD data in all different climatic types (section 4.1 gives the details). According to the Lambert–Beer Law of radiation extinction, transmissivity of the atmosphere can be given as (Houghton, 2002)

equation image(7)

Where τa, τg, τmath image, τω, τmath image, and τr are optical depths of aerosol, gases, nitrous oxide, water, ozone and Rayleigh scattering from oxygen and nitrogen. m is airmass factor and we can write ‘τ’ for expressing the total atmospheric optical depth

equation image(8)
equation image(9)

Transmissivity of the atmosphere is also known as clearness index KT (KT = T) (Lopez et al., 2010). Thus, we can write

equation image(10)
equation image(11)

Exponential relationship between KT−1 and optical depth indicates that when atmospheric optical depth increases, KT−1 increases exponentially. So, KT−1 is a direct indicator of the atmospheric conditions and optical depth.

3.3. Statistical indicators for model evaluation

Different statistical indicators such as MBE (Mean Bias Error), RMSE (Root Mean Square Error), MAPE (Mean Absolute Percentage Error), R2 (Coefficient of Determination) and AIC (Akaike's Information Criterion) (Droulia et al., 2009) were used to evaluate the performance of ‘local’ and ‘regional’ models. AIC is a tool for selecting the best model among different models, and the model having the lowest AIC value is considered to be the best.

equation image(12)
equation image(13)
equation image(14)
equation image(15)

Where, n = number of data points, ci = ith calculated value, mi = ith measured value and k = number of parameter plus one.

4. Results and discussions

4.1. Seasonal behaviour of KT and KD over the four ‘prime’ climates of the Indian sub-tropics

The monthly averages of KT and KD from 1973 to 2003 are plotted in Figure 2. Values of KT and KD for each year is connected by a line (step horizontal). The data points are connected perpendicularly through this line with an initial horizontal line. A denser line means more years having the same average KT and KD. The annual spectrums of both KT and KD showed prominent seasonal variation in each of the four climates with KT being the highest during winter months and lowest in monsoon months. The KD followed just the opposite trend. It showed maxima during the monsoon months and minima in winter months. The aerosol load, cloud mass, its types and persistent fog largely determine KD (Krakauer and Randerson, 2003, Mercado et al., 2009). Both the spectrums showed significant intra-seasonal variability during the entire year. The KT was found to be relatively lower (KT < 0.7) in case of humid maritime climate (Figure 2(d), Kolkata) as compared to semi-arid (Figure 2(b) New Delhi) and arid (Figure 2(a) Jodhpur) climates where the upper limit of KT is more than 0.7. In sub-humid continental climate, the upper limit of KT rarely crossed 0.75. Interestingly, the lower and upper limits of KT were higher in New Delhi for the months of December, February, March and April as compared to Jodhpur. Being situated in the western part of India, Jodhpur experiences large (intense) events of western disturbances which originate from the Mediterranean regions. Frontal characteristics of western disturbances are lost when it moved towards eastward side of India across Afghanistan and Pakistan (Hatwar et al., 2005) although New Delhi experiences the effects of intense western disturbances and may cause small levels of nebulous atmosphere as compared to Jodhpur. In addition to this, Jodhpur also experiences sand and dust storm effects from the nearby deserts that lead to scattering of dust particles thus reducing atmospheric clearness.

Figure 2.

Monthly mean of transmissivity (KT) and diffuse solar radiation fraction (KD) for primary stations of India for the period (1973–2003). Horizontal lines represent different years. KT of (a) Jodhpur, (b) New Delhi, (c) Nagpur and (d) Kolkata have shown similar trend with least values in monsoon and high values during pre-monsoon. Their respective variation in KD is shown in (e)–(h) with higher values at the time of monsoon

The higher humidity levels associated with higher cloud dynamics coupled with norwesters (premonsoon) cause the upper limit of transmissivity to be lower in sub-humid and maritime climates than those in drier climates.

The lower limit of KD spectrum never went below 0.2 in semi-arid (Figure 2(f)) and humid maritime climates (Figure 2(h)). For continental regimes of arid (Figure 2(e)) and humid (Figure 2(g)) climates, KD was below 0.2 during the post-monsoon and winter parts of the year. During the southwest monsoon (July–August) and post-monsoon (November–December) months, the upper limit of KD falls within 0.8–0.9 in all the four ‘prime’ climates. Relatively higher KD was observed in almost all climatic regions during winter months (December, January and February) as compared to pre-monsoon months (April and May). It was associated with the burning of crop residues in the winter months (Sharma et al., 2010) and occurrence of fog. In India, fog persistence during the daytime over the Indo-Gangetic belt is very much coupled with increase in aerosol load (Gautam et al., 2007). Moisture regime over the Indo-Gangetic plain was controlled by the River Ganges, and the irrigation events in wheat fields contribute to dense fog development.

The KT and KD spectrums on monthly scale could well capture the differences in climate types due to the differences in atmospheric constituents and cloud covers. The visibility of foggy weather drastically changes throughout the day in the winter season. This must have a significant role leading to unique KT−1KD relation over India as compared to the rest of the world.

4.2. KT−1KD scatter plots and monthly model parameters

The two-dimensional scatter plots of monthly KT−1 and KD are shown for the four ‘prime’ climates in Figure 3. Here, a quasi-linear pattern was evident between the two variables. In all the four climate types, linearity in relationship was found only up to KT−1 = 2.0, beyond which a nonlinear shape exists for KT−1 > 2.0. Combining both the behaviours for two different KT−1 thresholds, a sigmoidal relationship was apparent in the KT−1KD plots. On the basis of these KT−1KD scatter plots, a three-parameter sigmoidal function was fitted between them. The model statistics with its coefficients are given in Table II. The following nonlinear three-parameter sigmoidal function is proposed for both ‘local’ and ‘regional’ models

equation image(16)
Figure 3.

Scatter plots of inverse of clearness index (KT−1) and diffuse fraction (KD) for the period (1973–2003) on monthly scale of four primary stations

Table II. Coefficients (Coeff), standard errors (SE) and R2 for ‘local’ and ‘regional’ models (N = number of datasets)
CoeffLocal modelsRegional model
 Arid continental (R2 = 0.91, N = 204)Semi-arid continental (R2 = 0.91, N = 240)Sub-humid continental (R2 = 0.91, N = 252)Humid (R2 = 0.90, N = 204)All climates (R2 = 0.89, N = 720)
 ValueSEValueSEValueSEValueSEValueSE
a0.89080.0350.75930.0160.85340.0120.77490.0170.82950.012
b0.44590.0280.34180.0180.38880.0130.39360.0290.43190.015
c1.82650.0411.56060.0151.78080.0151.65990.0161.72900.014

Where a, b and c are model parameters. These three parameters of the function are based on the KT−1KD plots from monthly data over the four ‘prime’ climate types.

The KT−1KD correlations for ‘local’ models were high with R2 = 0.91 (N = 204), 0.91 (N = 240), 0.91 (N = 252), and 0.90 (N = 204) for arid, semi-arid, sub-humid continental and monsoon maritime climates, respectively. The coefficients were found to vary from 0.7593 to 0.8908 for ‘a’, 0.3418 to 0.4459 for ‘b’ and 1.5606 to 1.8265 for ‘c’. The ‘regional’ model was found to produce high correlation with R2 = 0.89 (N = 336). The parameters were a = 0.8295, b = 0.4319 and c = 1.729, respectively.

4.3. Model validation and comparison

The performance of ‘local’ and ‘regional’ models for monthly KD estimates was evaluated with independent KT−1KD paired datasets from 1994 to 2003. These were also compared to the estimates from the existing seven monthly models reported by Page (1961), Barbaro et al. (1981), Elhadidy and Abdel-Nabi (1991), Liu and Jordan (1960), Erbs et al. (1982), Gopinathan and Soler (1995), and Ulgen and Hepbasli (2009). Comparisons between ‘regional’ models and globally existing models are shown in Figure 4 for all the four stations. Figure 4 clearly indicated that Elhadidy and Abdel-Nabi (1991) overestimated, while all other models underestimated the KD value. The error statistics such as MBE, RMSE, MAPE, AIC and correlation coefficient (R2) are summarized in Table III. In arid continental climate (e.g. Jodhpur), we found the best results with the ‘local’ model as compared to the ‘regional’ model, with positive bias having MBE = 0.021, RMSE = 0.043, MAPE = 8.251, R2 = 0.93 and lowest AIC (−669.18). However, semi-arid continental climate (e.g. New Delhi) showed MBE = 0.025, RMSE = 0.045, MAPE = 8.12, R2 = 0.90 and AIC = − 660.111 for the ‘local’ model. The errors were less for the ‘regional’ model with better correlation (R2 = 0.91) and lowest AIC (−678.286).

Figure 4.

Comparison of seasonality of ten years (1994–2003) monthly average KD from ‘local’, ‘regional’ plots and ‘existing’ models—Page (1961), Barbaro et al. (1981), Elhadidy and Abdel-Nabi (1991), Liu and Jordan (1960), Erbs et al. (1982), Gopinathan and Soler (1995), and Ulgen and Hepbasli (2009)

Table III. Validation statistics (MBE, RMSE, MAPE, R2 and AIC) for ‘local’, ‘regional’ and ‘globally existing’ models
Climate typesLocal modelRegional modelPageBarbaro etal.Elhadidy and Abdel-NabiLiu and JordanErbs etal.Gopinathan and SolerUlgen and Hepbasli
Arid         
MBE0.0210.032− 0.056− 0.1140.070− 0.101− 0.052− 0.053− 0.026
RMSE0.0430.0540.0870.1270.0850.1320.0910.0930.113
MAPE8.25111.12410.78123.16517.50718.03810.88610.79816.067
R20.930.930.910.910.880.920.920.910.93
AIC− 669.180− 623.386− 521.372− 440.150− 525.881− 428.120− 506.681− 507.083− 461.910
Semi-arid         
MBE0.025− 0.019− 0.113− 0.1680.017− 0.160− 0.110− 0.111− 0.087
RMSE0.0450.0420.1210.1730.0630.1700.1210.1220.121
MAPE8.1207.00622.19634.66110.03631.04921.08921.25317.215
R20.900.910.890.890.860.890.900.890.90
AIC− 660.111− 678.286− 450.897− 373.146− 592.384− 372.496− 447.076− 448.809− 445.492
Sub-humid         
MBE0.0120.017− 0.075− 0.1300.051− 0.118− 0.069− 0.073− 0.047
RMSE0.0530.0520.0990.1410.0800.1440.1000.1060.125
MAPE9.3079.58616.13328.88414.99323.26414.76215.33617.951
R20.910.910.900.900.880.900.900.900.89
AIC− 628.064− 632.051− 493.311− 416.630− 540.583− 408.691− 487.825− 478.642− 439.280
Humid         
MBE0.0140.018− 0.101− 0.1430.075− 0.166− 0.107− 0.110− 0.121
RMSE0.0420.0390.1130.1490.0830.1770.1200.1250.147
MAPE5.6625.32715.00422.42712.70025.28715.87116.14017.168
R20.910.920.910.920.910.910.920.920.91
AIC− 676.851− 688.050− 465.480− 404.717− 531.577− 363.976− 447.932− 442.493− 404.055

In sub-humid continental climate (e.g. Nagpur), the MBE from the ‘local’ model was found to be the lowest as compared to other climates. The MAPE was also less for the ‘local’ model, but the RMSE was higher (0.053) with R2 = 0.91 and AIC = − 628.064. Here also, the RMSE of the ‘regional’ model was nearly equal to ‘local’ (0.052), but AIC was lowest − 632.051. This confirmed the better performance of the ‘regional model’. For monsoon maritime climate (e.g. Kolkata), though the MBE was higher (0.018) for the ‘regional’ model as compared to the ‘local’ model, the RMSE = 0.039 and MAPE = 5.327 were less. The R2 and AIC were better for the ‘regional’ model (0.92, − 688.50) as compared to the ‘local’ model. In general, ‘regional’ models produced lower errors as compared to ‘local’ models and also had lowest AIC except in arid continental climate. Climatic conditions of Jodhpur are greatly influenced by the Thar Desert which contributes to increased aerosol load and dust storm events. The long-distance intercontinental dust transport through Iran, Afghanistan and Pakistan puts an additional dust aerosol burden on Jodhpur (Santra et al., 2010). This feature makes this station unique from the rest of the other stations. These could have resulted in better performances of ‘local’ than ‘regional’ models in the arid climate of Jodhpur.

The R2 and AIC were better for both ‘local’ and ‘regional’ models as compared to the seven existing models (Table III). The seasonality of average of monthly estimates of KD from ‘local’ and ‘regional’ models were compared with those from ‘existing’ models with the average of monthly KD over ten years (1994–2003). The intra-seasonal variation of monthly KD was better for ‘local’ and ‘regional’ models than most of the ‘existing’ models except for the Page (1961) model which showed similar seasonality, although the model statistics were poorer. The functional form of ‘existing’ models is mostly linear or nonlinear even up to fourth-order polynomial. The models developed in this study, i.e. three-parameter sigmoidal, are quite different from other models. The existing models are generally developed and validated with a limited period datasets, that too with 1–3 stations; but, the new models in the present study were developed with 21 years' datasets of 4 different stations in India representing the 4 different ‘prime’ climates.

4.4. Extendibility of ‘regional’ model over secondary' stations in India

In order to check the applicability of the ‘regional’ model across the country, we compared the KD estimates with available monthly KD datasets for one year from the so-called secondary stations. We found satisfactory performances of ‘regional’ models in terms of MBE, RMSE, MAPE and R2 (Table IV). In most of the stations such as Jaipur, Ahmedabad, Ranchi, Mumbai, Pune, Hyderabad, Vishakhapatnam, Panjim, Port Blair and Chennai, the R2 were > = 0.9. It was 0.89 in Bhopal and Thiruvanathapuram, and 0.85 in Patiala. The R2 was 0.78 for Patna and 0.65 for Shillong. The R2 was found to be the lowest in Bangalore (R2 = 0.59). The RMSE was found to be < = 0.05 in four stations (Patiala, Hyderabad, Panjim and Thiruvanathapuram). It was less than 0.09 for Jaipur, Ahmedabad, Bhopal, Ranchi, Mumbai, Pune, Vishakhapatnam and Panjim, but it was higher (0.1) for Bangalore and Patna, and was highest in Shillong (0.17). The MBE was found to be < = 0.07 in all stations with the lowest for Patiala. MAPE was less than 10% in Mumbai, Hyderabad and Thiruvanathapuram, around 10% in Panjim, Patiala and Chennai, less than 15% in Jaipur, Ranchi and Vishakahapatnam, and within 20% in Ahmedabad, Bangalore and Port Blair. The MAPE was high in Pune and Patna and highest in Shillong and Bhopal. The station, Shillong, showed the highest MBE, RMSE and MAPE (0.126, 0.174 and 33.47) among all the stations. Generally, the MBE, MAPE and RMSE of the ‘secondary’ stations were little higher than the validation statistics over ‘prime’ climate types represented by ‘primary’ stations.

Table IV. Validation statistics (MBE, RMSE, MAPE and R2) of ‘regional’ model over ‘secondary’ stations (N = no. of datasets)
Station no.StationData availabilityMBERMSEMAPER2
1JaipurJune 1998–May19990.0470.06513.520.94
2PanjimJune 1998–May19990.0290.04810.080.96
3MumbaiJune 1998–May19990.010.068.070.91
4PuneJune 1998–May19990.0550.08322.340.93
5PatialaJune 1998–May19990.0010.01710.040.85
6PatnaJune 1998–Mar19990.1070.12522.260.78
7ShillongJune 1998–May19990.1260.17333.470.65
8AhmedabadJune 1998–May19990.0590.07617.630.93
9BhopalJune 1998–May19990.0750.09332.510.89
10RanchiJune 1998–Mar19990.0290.07513.820.96
11HyderabadJune 1998–May1999− 0.0160.0468.810.91
12VishakhapatnamJune 1998–May19990.070.09413.360.94
13ChennaiAug 1998–May19990.0520.07410.480.9
14BangaloreAug 1998–May19990.0570.12617.230.59
15Port BlairJune 1998–May19990.0970.1118.230.92
16ThiruvanathapuramJune 1998–May19990.0120.055.740.89

The validation datasets of monthly KD over ‘prime’ climates are substantially larger than ‘secondary’ ones. This could have resulted in differences in observed model statistics. The higher error in KD estimates was observed over Shillong and Bangalore as compared to other stations. Shillong is a hill station with elevation of 1598 m having fragmented topography. The extraterrain reflected radiation component and skyview form a major part in diffuse radiation. For Bangalore, it could be due to other influencing factors that need further detailed study. Our present study primarily focuses on development of regional model purely based on atmospheric transmissivity to be applicable at climate scale for plain-to-plateau region. This resulted in neglecting the role of fragmented topography on KT−1KD modelling. A separate model needs to be developed in the future to obtain more accurate KD estimates over such hilly topography within the Indian sub-tropics.

5. Conclusions

The major conclusions of the present study are as follows:

  • 1.Both ‘local’ and ‘regional’ models performed better than globally existing models. The ‘regional’ models were found to produce more accurate estimates than the ‘local’ models over three of four ‘prime’ climates. In arid climate, ‘local’ model statistics were better than those of the ‘regional’ model.
  • 2.The extendibility of the ‘regional’ model over ‘secondary’ radiation measuring stations in India showed good results in 14 stations out of 16 secondary stations on the basis of R2. It failed in two provinces Bangalore (MBE = 0.057, RMSE = 0.126, MAPE = 17.23 and R2 = 0.59) and Shillong (MBE = 0.126, RMSE = 0.173, MAPE = 33.47, R2 = 0.65). In a hilly station (Shillong) it is attributed to lack of accounting for topographical factors.
  • 3.We can use the regional model KD = 0.8295/{1 + exp[−(KT−1 − 1.7290)/0.4319]} for predicting monthly average diffuse solar radiation fraction for Indian region except hilly areas.
  • 4.These models are unique for the Indian sub-tropics and can undoubtedly be used for predicting future KD from KT datasets of climate simulations to characterize future productivity, other meteorological and climatological applications. For devising different ecofriendly equipment (solar-based systems like photovoltaics, solar collectors, etc.), these can be used as input parameters.

Acknowledgements

This study was carried out under an ISRO-GBP project titled Energy and Mass Exchange in Vegetative Systems. The authors are grateful to the Space Applications Centre (ISRO) for funding the study as well as for guidance during the study. We wish to acknowledge India Meteorological Department, Pune, for providing the required datasets. We are also very thankful to Prof. N. C. Mahanti, Head, Department of Applied Mathematics, Birla Institute of Technology, Mesra, Ranchi, for his support and encouragement.

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