modelling monthly diffuse solar radiation fraction and its validity over the indian sub-tropics

$: Modelling monthly diffuse solar radiation fraction and its validity over the Indian sub-tropics$
INTERNATIONAL JOURNAL OF CLIMATOLOGYInt. J. Climatol. (2011)Published online in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/joc.3408

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

Jyotsna Singh,a* Bimal K. Bhattacharya,b Manoj Kumara and Kaniska Mallickc

a Centre of Excellence in Climatology, Birla Institute of Technology, Mesra-835215, Ranchi, Jharkhand, Indiab Agriculture, Terrestrial Biosphere and Hydrology Group (ABHG), Space Applications Centre (ISRO), Ahmedabad-380015, Gujarat, India

c Water and Carbon Cycles Group, NASA, Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA91109

ABSTRACT: A three-parameter sigmoidal ‘local’ model (climate-specific) and a ‘regional’ model (common for allclimates) have been developed for measuring the monthly average diffuse solar radiation fraction from atmospherictransmissivity 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. Themonthly diffuse fraction estimates were also compared with seven globally existing models. The ‘regional’ model showedmore 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 whichfailed 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 fortwo stations. These models are unique for Indian sub-tropics and can undoubtedly be used for predicting future diffuse solarradiation fraction from transmissivity datasets of climate simulations, and also for other meteorological, climatological,solar energy-based applications. Copyright 2011 Royal Meteorological Society

KEY WORDS atmospheric transmissivity; diffuse solar radiation fraction; India; modelling

Received 14 June 2011; Revised 25 September 2011; Accepted 25 October 2011

1. Introduction

Global insolation (RG) and diffuse solar radiation fraction(KD) reaching the surface of the Earth is altered by atmo-spheric clarity (based on aerosol concentration) or cloudi-ness that determines atmospheric transmissivity (KT ).Changes in cloud properties, fog, atmospheric aerosolloadings including dust, volcanic or anthropogenic emis-sions, alter both the KT and KD which also affect plantproductivity, and the land carbon sink globally (Mercadoet al., 2009). Recent observational findings over diverseplant functional types also established the role of KD

on modulating canopy gas exchange processes (Niyogiet al., 2004; Knohl and Baldocchi, 2008; Still et al.,2009; Jing et al., 2010), vegetation, light and water useefficiencies (Rocha et al., 2004; Chen et al., 2009). Stud-ies have also shown the sensitivity of vegetation produc-tivity to the fluctuations in RD and it is more efficientlyused by the plants than the direct component (Rodericket al., 2001). Because of its omnidirectional nature (i.e.incident from multiple angles), the diffuse component hasthe greater capacity to penetrate the light-limited layers of

∗ Correspondence to: J. Singh, Centre of Excellence in Climatology,Birla Institute of Technology, Mesra-835215, Ranchi, Jharkhand.E-mail: [email protected]

the dense forest canopies, thus stimulating photosynthesisand productivity (Gu et al., 2002; Still et al., 2009).

The main concern of today’s world is sustainabledevelopment which is directly linked to the utilisation ofenergy resources. For attaining sustainable development,we need to harness sustainable energy sources, and theuse of renewable energy such as ‘solar energy’ will pro-mote sustainability (Dincer and Rosen, 1999). Values ofRD and KD are also required for building the solar energysystems. In addition to these, long-term records of RD

and KD have significant roles in constructing quantitativeinformation on atmospheric turbidity, aerosol and cloudi-ness. 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 andcarbon cycles, and the impact of these cycles on theatmospheric properties. Different atmospheric conditions(turbidity and transparency), airmass, content of watervapour in the atmosphere and the cloud cover distribu-tion influence the insolation by absorption, scattering andre?ection (Okogbue et al., 2009). The knowledge of KD

can be useful to get an idea of concentration of atmo-spheric load indirectly, where a low KD will indicateclear sky and more pristine atmosphere, and vice versa.

Copyright 2011 Royal Meteorological Society

J. SINGH et al.

In a developing country like India, RD measurement issparse because it is expensive and tedious (Veeran andKumar, 1993; Gopinathan and Soler, 1995; Pandey andKatiyar, 2009). Therefore, a common alternative is toformulate robust correlation models of RD or KD fromthe observational networks. Many empirical models ofKD have been developed in some advanced countriessuch 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 suchas Thailand (Janjai et al., 1996), Turkey (Ulgen and Hep-basli, 2009) and Saudi Arabia (Elhadidy and Abdel-Nabi,1991). These models cannot be straightway extrapolatedto a sub-tropical country like India because of differencesin the radiative forcing patterns. Besides this, the modelsof the developed countries are valid for the higher lat-itudes (above 40°) only. Though Gopinathan and Soler(1995) have developed a KD versus KT relationship tak-ing observations from different parts of the world includ-ing two stations of India, yet that relationship suffers fromlimited climatological information, and the model was notvalidated over other independent Indian stations. A com-prehensive list of correlation models between monthlyKD and KT developed earlier is given in Table I (Liuand Jordan, 1960; Erbs et al., 1982; Ulgen and Hep-basli, 2009; and others). No such robust model of KD isavailable so far for India. This paper aims at developinga monthly model of KD from the time-series measure-ments of RG and RD . The objectives of the present studyare: (1) development of local as well as regional modelsfor monthly average KD using long time-series RG andRD observations over ‘prime’ climates (Section 2) overthe Indian sub-tropics; (2) comparison of newly devel-oped KD model outputs with the estimates from existingglobal models; and (3) validation of the KD model withindependent datasets and shorter time series data from 16different radiometric stations of India.

2. Study region and datasets

Time series measurements on daily RG and RD wereobtained 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 sta-tions are referred to here as primary stations where con-sistently longer time series datasets on RG and RD wereavailable for a period of 31 years (1973–2003). In Jodh-pur, the transport of sand and dust through the TharDesert especially in pre-monsoon months, and cloudsfrom western disturbances during winter months modu-late the KT and KD . The persistent fog and clouds fromwestern disturbances in winter months are responsiblefor controlling both the variables in New Delhi. Short-wave radiation regime in Kolkata is mostly controlled bycloud dynamics associated with the Bay of Bengal branchof the southwest monsoon, cyclones, mist and maritimeaerosols. In Nagpur, the clouds through the Arabian Seabranch off, and continental aerosols control RG and RD .The four climates represented by these stations thus dom-inate the atmosphere on KD over the Indian sub-tropics.Hence, longer time series datasets from these four sta-tions are suitable for the development of semi-empiricalregression models.

A shorter period of datasets of about one year (June1998–May 1999) were available for other 16 stationsacross India. These are referred to here as secondarystations. The distribution of ‘primary’ and ‘secondary’stations is shown in the Figure 1. The RG and RD datawere recorded at hourly intervals at the IMD (IndiaMeteorological Department) stations using a pyranome-ter and a shading ring pyranometer, respectively, andaccumulated over a day in megajoules per square meter(MJ m−2). The pyranometers were calibrated from timeto time with respect to World Radiometric Reference(WRR), and maximum ±5% uncertainty was found in themeasured data (Singh and Tiwari, 2005). The calibrationresults were found satisfactory. The data over a periodof 31 years (1973–2003) were used for developing andvalidating the KD model. The model developed over the‘primary’ stations were used to evaluate the performanceover the ‘secondary’ stations.

3. Methodology

3.1. Computation of KD and KT

The development of the KD model requires basic obser-vations of KT and KD . Since these ratios cannot bemeasured directly, the daily KT (also called clearnessindex) and KD were computed from the measurements

Table I. Globally existing monthly average models for diffuse solar radiation fraction (KD) based on transmissivity (KT ).

Study Models Location

Page (1961) KD = 1.00 − 1.13(KT ) 40 °N-40 °SLiu and Jordan (1960) KD = 1.39 − 4.027(KT ) + 5.531(KT )2 − 3.108(KT )3 USAErbs et al. (1982) KD = 1.317 − 3.023(KT ) + 3.372(KT )2 − 1.769(KT )3 USABarbaro et al. (1981) KD = 1.0492 − 1.3246(KT ) ItalyElhadidy and Abdel-Nabi (1991) KD = 1.039 − 1.741(KT )2 Saudi ArabiaUlgen and Hepbasli (2009) KD = 0.981 − 1.9028(KT ) + 1.9319(KT )2 − 0.6809(KT )3 TurkeyGopinathan and Soler (1995) KD = 0.91138 − 0.96225(KT ) 30 °S-60 °N

Copyright 2011 Royal Meteorological Society Int. J. Climatol. (2011)

DIFFUSE RADIATION FRACTION MODEL AND ITS VALIDATION OVER INDIA

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 secondarystations with symbol (*) in numeric. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

of RG and RD as follows:

KD = RD/RG (1)

KT was computed as the ratio of global radiation toextraterrestrial daily radiation (Ra) received at the top ofatmosphere.

KT = RG/Ra (2)

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

Ra = (S0/π)(d/d

)2[ωhs(π/180) sin L sin δ

+ cos L cos δ sin ωhs] (3)

S0 = solar constant megajoule per meter per day(MJ m−1 day−1), L = latitude (degree),

(d/d

)2is the

Sun–Earth distance ratio:(d/d

)2 = 1 + 0.034 cos(360D/365.25) (4)

Where, D = calendar day of the year.ωhs = sunset hour angle (degree) which was calculated

as follows:

ωhs = cos−1(− tan δ tan L) (5)

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

δ = 23.5 sin{[360(D + 284)]/365} (6)

3.2. Model development

We have tried to develop a new climate-specific aswell as a common model for KD from its observational

relationship with KT for the Indian sub-tropics. Allthe datasets were divided into two parts. Out of atotal 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 forvalidation of the developed model. The climate-specificmodel is hereafter referred to as local model. A commonmodel is developed by pooling the data from four climatetypes 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 thedetails). According to the Lambert–Beer Law of radiationextinction, transmissivity of the atmosphere can be givenas (Houghton, 2002)

T = exp{−m

(τa + τg + τNO2 + τω + τO3 + τr

)}(7)

Where τa , τg , τNO2 , τω, τo3 , and τr are optical depths ofaerosol, gases, nitrous oxide, water, ozone and Rayleighscattering from oxygen and nitrogen. m is airmass factorand we can write ‘τ ’ for expressing the total atmosphericoptical depth

T = exp(−mτ) (8)

1/T = exp(mτ) (9)

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

1/T = KT−1 (10)

KT−1 = exp(mτ) (11)

Exponential relationship between KT−1 and optical

depth indicates that when atmospheric optical depth


J. SINGH et al.

increases, KT−1 increases exponentially. So, KT

−1 is adirect indicator of the atmospheric conditions and opticaldepth.

3.3. Statistical indicators for model evaluation

Different statistical indicators such as MBE (Mean BiasError), RMSE (Root Mean Square Error), MAPE (MeanAbsolute Percentage Error), R2 (Coefficient of Determi-nation) and AIC (Akaike’s Information Criterion) (Drou-lia et al., 2009) were used to evaluate the performance of‘local’ and ‘regional’ models. AIC is a tool for selectingthe best model among different models, and the modelhaving the lowest AIC value is considered to be the best.

MBE = 1

n

n∑i=1

(ci − mi) (12)

RMSE =√√√√1

n

n∑i=1

(ci − mi)2 (13)

MAPE = 1

n

n∑i=1

∣∣∣∣(

ci − mi

mi

)× 100

∣∣∣∣ (14)

AIC = n. ln

(1

n

n∑i=1

(ci − mi)2

)+ 2k (15)

Where, n = number of data points, ci = ith calculatedvalue, mi = ith measured value and k = number ofparameter 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 to2003 are plotted in Figure 2. Values of KT and KD foreach year is connected by a line (step horizontal). Thedata points are connected perpendicularly through thisline with an initial horizontal line. A denser line meansmore years having the same average KT and KD . The

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 leastvalues 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. This figure is available in colour online at wileyonlinelibrary.com/journal/joc



annual spectrums of both KT and KD showed prominentseasonal variation in each of the four climates with KT

being the highest during winter months and lowest inmonsoon months. The KD followed just the oppositetrend. It showed maxima during the monsoon monthsand minima in winter months. The aerosol load, cloudmass, its types and persistent fog largely determine KD

(Krakauer and Randerson, 2003, Mercado et al., 2009).Both the spectrums showed significant intra-seasonalvariability during the entire year. The KT was found tobe relatively lower (KT < 0.7) in case of humid maritimeclimate (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 upperlimits of KT were higher in New Delhi for the monthsof December, February, March and April as comparedto Jodhpur. Being situated in the western part of India,Jodhpur experiences large (intense) events of westerndisturbances which originate from the Mediterraneanregions. Frontal characteristics of western disturbancesare lost when it moved towards eastward side of Indiaacross Afghanistan and Pakistan (Hatwar et al., 2005)although New Delhi experiences the effects of intensewestern disturbances and may cause small levels ofnebulous atmosphere as compared to Jodhpur. In additionto this, Jodhpur also experiences sand and dust stormeffects from the nearby deserts that lead to scattering ofdust particles thus reducing atmospheric clearness.

The higher humidity levels associated with highercloud dynamics coupled with norwesters (pre-monsoon) cause the upper limit of transmissivity to belower in sub-humid and maritime climates than those indrier climates.

The lower limit of KD spectrum never went below0.2 in semi-arid (Figure 2(f)) and humid maritime cli-mates (Figure 2(h)). For continental regimes of arid(Figure 2(e)) and humid (Figure 2(g)) climates, KD wasbelow 0.2 during the post-monsoon and winter parts ofthe year. During the southwest monsoon (July–August)and post-monsoon (November–December) months, theupper limit of KD falls within 0.8–0.9 in all the four‘prime’ climates. Relatively higher KD was observed inalmost all climatic regions during winter months (Decem-ber, January and February) as compared to pre-monsoonmonths (April and May). It was associated with the burn-ing of crop residues in the winter months (Sharma et al.,2010) and occurrence of fog. In India, fog persistenceduring the daytime over the Indo-Gangetic belt is verymuch coupled with increase in aerosol load (Gautamet al., 2007). Moisture regime over the Indo-Gangeticplain was controlled by the River Ganges, and the irri-gation events in wheat fields contribute to dense fogdevelopment.

The KT and KD spectrums on monthly scale couldwell capture the differences in climate types due to thedifferences in atmospheric constituents and cloud cov-ers. The visibility of foggy weather drastically changes

throughout the day in the winter season. This must havea significant role leading to unique KT

−1 − KD relationover India as compared to the rest of the world.

4.2. KT−1 − KD 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 thetwo variables. In all the four climate types, linear-ity 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 theKT

−1 − KD plots. On the basis of these KT−1 − KD

scatter plots, a three-parameter sigmoidal function wasfitted between them. The model statistics with its coef-ficients are given in Table II. The following nonlinearthree-parameter sigmoidal function is proposed for both‘local’ and ‘regional’ models

KD = a/{1 + exp[−(KT−1 − c)/b]} (16)

Where a, b and c are model parameters. These threeparameters of the function are based on the KT

−1 − KD

plots from monthly data over the four ‘prime’ climatetypes.

The KT−1 − KD 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 from0.7593 to 0.8908 for ‘a’, 0.3418 to 0.4459 for ‘b’ and1.5606 to 1.8265 for ‘c’. The ‘regional’ model was foundto 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 formonthly KD estimates was evaluated with independentKT

−1 − KD paired datasets from 1994 to 2003. Thesewere also compared to the estimates from the existingseven monthly models reported by Page (1961), Bar-baro et al. (1981), Elhadidy and Abdel-Nabi (1991), Liuand Jordan (1960), Erbs et al. (1982), Gopinathan andSoler (1995), and Ulgen and Hepbasli (2009). Compar-isons between ‘regional’ models and globally existingmodels 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 underesti-mated the KD value. The error statistics such as MBE,RMSE, MAPE, AIC and correlation coefficient (R2) aresummarized 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 positivebias having MBE = 0.021, RMSE = 0.043, MAPE =8.251, R2 = 0.93 and lowest AIC (−669.18). However,


J. SINGH et al.

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).

Coeff Local models Regional 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)

Value SE Value SE Value SE Value SE Value SE

a 0.8908 0.035 0.7593 0.016 0.8534 0.012 0.7749 0.017 0.8295 0.012b 0.4459 0.028 0.3418 0.018 0.3888 0.013 0.3936 0.029 0.4319 0.015c 1.8265 0.041 1.5606 0.015 1.7808 0.015 1.6599 0.016 1.7290 0.014

semi-arid continental climate (e.g. New Delhi) showedMBE = 0.025, RMSE = 0.045, MAPE = 8.12, R2 =0.90 and AIC = −660.111 for the ‘local’ model. Theerrors were less for the ‘regional’ model with better cor-relation (R2 = 0.91) and lowest AIC (−678.286).

In sub-humid continental climate (e.g. Nagpur), theMBE from the ‘local’ model was found to be the lowestas compared to other climates. The MAPE was alsoless for the ‘local’ model, but the RMSE was higher(0.053) with R2 = 0.91 and AIC = −628.064. Herealso, the RMSE of the ‘regional’ model was nearly equalto ‘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), thoughthe MBE was higher (0.018) for the ‘regional’ model ascompared to the ‘local’ model, the RMSE = 0.039 andMAPE = 5.327 were less. The R2 and AIC were betterfor the ‘regional’ model (0.92, −688.50) as compared tothe ‘local’ model. In general, ‘regional’ models producedlower errors as compared to ‘local’ models and also hadlowest AIC except in arid continental climate. Climaticconditions of Jodhpur are greatly influenced by the TharDesert which contributes to increased aerosol load anddust storm events. The long-distance intercontinental dust



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).

transport through Iran, Afghanistan and Pakistan puts anadditional dust aerosol burden on Jodhpur (Santra et al.,2010). This feature makes this station unique from therest of the other stations. These could have resulted inbetter performances of ‘local’ than ‘regional’ models inthe arid climate of Jodhpur.

The R2 and AIC were better for both ‘local’ and‘regional’ models as compared to the seven existingmodels (Table III). The seasonality of average of monthlyestimates of KD from ‘local’ and ‘regional’ models werecompared with those from ‘existing’ models with theaverage of monthly KD over ten years (1994–2003).The intra-seasonal variation of monthly KD was betterfor ‘local’ and ‘regional’ models than most of the‘existing’ models except for the Page (1961) model whichshowed similar seasonality, although the model statisticswere poorer. The functional form of ‘existing’ modelsis mostly linear or nonlinear even up to fourth-orderpolynomial. The models developed in this study, i.e.three-parameter sigmoidal, are quite different from othermodels. The existing models are generally developed andvalidated with a limited period datasets, that too with 1–3stations; but, the new models in the present study were

developed with 21 years’ datasets of 4 different stationsin 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 estimateswith available monthly KD datasets for one year fromthe so-called secondary stations. We found satisfactoryperformances of ‘regional’ models in terms of MBE,RMSE, MAPE and R2 (Table IV). In most of the stationssuch as Jaipur, Ahmedabad, Ranchi, Mumbai, Pune,Hyderabad, Vishakhapatnam, Panjim, Port Blair andChennai, the R2 were >= 0.9. It was 0.89 in Bhopaland Thiruvanathapuram, and 0.85 in Patiala. The R2 was0.78 for Patna and 0.65 for Shillong. The R2 was found tobe the lowest in Bangalore (R2 = 0.59). The RMSE wasfound to be <= 0.05 in four stations (Patiala, Hyderabad,Panjim and Thiruvanathapuram). It was less than 0.09for 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 allstations with the lowest for Patiala. MAPE was less than


J. SINGH et al.

Table III. Validation statistics (MBE, RMSE, MAPE, R2 and AIC) for ‘local’, ‘regional’ and ‘globally existing’ models.

Climatetypes

Localmodel

Regionalmodel

Page Barbaroet al.

Elhadidy andAbdel-Nabi

Liu andJordan

Erbset al.

Gopinathanand Soler

Ulgen andHepbasli

AridMBE 0.021 0.032 −0.056 −0.114 0.070 −0.101 −0.052 −0.053 −0.026RMSE 0.043 0.054 0.087 0.127 0.085 0.132 0.091 0.093 0.113MAPE 8.251 11.124 10.781 23.165 17.507 18.038 10.886 10.798 16.067R2 0.93 0.93 0.91 0.91 0.88 0.92 0.92 0.91 0.93AIC −669.180 −623.386 −521.372 −440.150 −525.881 −428.120 −506.681 −507.083 −461.910Semi-aridMBE 0.025 −0.019 −0.113 −0.168 0.017 −0.160 −0.110 −0.111 −0.087RMSE 0.045 0.042 0.121 0.173 0.063 0.170 0.121 0.122 0.121MAPE 8.120 7.006 22.196 34.661 10.036 31.049 21.089 21.253 17.215R2 0.90 0.91 0.89 0.89 0.86 0.89 0.90 0.89 0.90AIC −660.111 −678.286 −450.897 −373.146 −592.384 −372.496 −447.076 −448.809 −445.492Sub-humidMBE 0.012 0.017 −0.075 −0.130 0.051 −0.118 −0.069 −0.073 −0.047RMSE 0.053 0.052 0.099 0.141 0.080 0.144 0.100 0.106 0.125MAPE 9.307 9.586 16.133 28.884 14.993 23.264 14.762 15.336 17.951R2 0.91 0.91 0.90 0.90 0.88 0.90 0.90 0.90 0.89AIC −628.064 −632.051 −493.311 −416.630 −540.583 −408.691 −487.825 −478.642 −439.280HumidMBE 0.014 0.018 −0.101 −0.143 0.075 −0.166 −0.107 −0.110 −0.121RMSE 0.042 0.039 0.113 0.149 0.083 0.177 0.120 0.125 0.147MAPE 5.662 5.327 15.004 22.427 12.700 25.287 15.871 16.140 17.168R2 0.91 0.92 0.91 0.92 0.91 0.91 0.92 0.92 0.91AIC −676.851 −688.050 −465.480 −404.717 −531.577 −363.976 −447.932 −442.493 −404.055

Table IV. Validation statistics (MBE, RMSE, MAPE and R2) of ‘regional’ model over ‘secondary’ stations (N = no. of datasets).

Station no. Station Data availability MBE RMSE MAPE R2

1 Jaipur June 1998–May1999 0.047 0.065 13.52 0.942 Panjim June 1998–May1999 0.029 0.048 10.08 0.963 Mumbai June 1998–May1999 0.01 0.06 8.07 0.914 Pune June 1998–May1999 0.055 0.083 22.34 0.935 Patiala June 1998–May1999 0.001 0.017 10.04 0.856 Patna June 1998–Mar1999 0.107 0.125 22.26 0.787 Shillong June 1998–May1999 0.126 0.173 33.47 0.658 Ahmedabad June 1998–May1999 0.059 0.076 17.63 0.939 Bhopal June 1998–May1999 0.075 0.093 32.51 0.8910 Ranchi June 1998–Mar1999 0.029 0.075 13.82 0.9611 Hyderabad June 1998–May1999 −0.016 0.046 8.81 0.9112 Vishakhapatnam June 1998–May1999 0.07 0.094 13.36 0.9413 Chennai Aug 1998–May1999 0.052 0.074 10.48 0.914 Bangalore Aug 1998–May1999 0.057 0.126 17.23 0.5915 Port Blair June 1998–May1999 0.097 0.11 18.23 0.9216 Thiruvanathapuram June 1998–May1999 0.012 0.05 5.74 0.89

10% in Mumbai, Hyderabad and Thiruvanathapuram,around 10% in Panjim, Patiala and Chennai, less than15% in Jaipur, Ranchi and Vishakahapatnam, and within20% in Ahmedabad, Bangalore and Port Blair. TheMAPE was high in Pune and Patna and highest inShillong and Bhopal. The station, Shillong, showed thehighest MBE, RMSE and MAPE (0.126, 0.174 and 33.47)among all the stations. Generally, the MBE, MAPE andRMSE of the ‘secondary’ stations were little higher

than the validation statistics over ‘prime’ climate typesrepresented by ‘primary’ stations.

The validation datasets of monthly KD over ‘prime’climates are substantially larger than ‘secondary’ ones.This could have resulted in differences in observed modelstatistics. The higher error in KD estimates was observedover Shillong and Bangalore as compared to other sta-tions. Shillong is a hill station with elevation of 1598 mhaving fragmented topography. The extraterrain reflectedradiation component and skyview form a major part



in diffuse radiation. For Bangalore, it could be dueto other influencing factors that need further detailedstudy. Our present study primarily focuses on devel-opment of regional model purely based on atmospherictransmissivity to be applicable at climate scale for plain-to-plateau region. This resulted in neglecting the roleof fragmented topography on KT

−1 − KD modelling. Aseparate model needs to be developed in the future toobtain more accurate KD estimates over such hilly topog-raphy within the Indian sub-tropics.

5. Conclusions

The major conclusions of the present study are as follows:

1. Both ‘local’ and ‘regional’ models performed betterthan globally existing models. The ‘regional’ modelswere found to produce more accurate estimates thanthe ‘local’ models over three of four ‘prime’ climates.In arid climate, ‘local’ model statistics were better thanthose of the ‘regional’ model.

2. The extendibility of the ‘regional’ model over ‘sec-ondary’ radiation measuring stations in India showedgood results in 14 stations out of 16 secondary stationson the basis of R2. It failed in two provinces Ban-galore (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 ahilly station (Shillong) it is attributed to lack ofaccounting for topographical factors.

3. We can use the regional model KD = 0.8295/{1 +exp[−(KT

−1 − 1.7290)/0.4319]} for predictingmonthly average diffuse solar radiation fraction forIndian region except hilly areas.

4. These models are unique for the Indian sub-tropicsand can undoubtedly be used for predicting futureKD from KT datasets of climate simulations tocharacterize future productivity, other meteorologicaland climatological applications. For devising differentecofriendly equipment (solar-based systems like pho-tovoltaics, solar collectors, etc.), these can be used asinput parameters.

Acknowledgements

This study was carried out under an ISRO-GBP projecttitled Energy and Mass Exchange in Vegetative Systems.The authors are grateful to the Space ApplicationsCentre (ISRO) for funding the study as well as forguidance during the study. We wish to acknowledgeIndia Meteorological Department, Pune, for providing therequired datasets. We are also very thankful to Prof. N.C. Mahanti, Head, Department of Applied Mathematics,Birla Institute of Technology, Mesra, Ranchi, for hissupport and encouragement.

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modelling monthly diffuse solar radiation fraction and its validity over the indian sub-tropics

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