Spatial interpolation of climatic variables using land ...

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Spatial interpolation of climatic variables using land surface temperature and modified inverse distance weighting a

a

b

Emre Ozelkan , Serdar Bagis , Ertunga Cem Ozelkan , Burak Berk a

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Ustundag , Meric Yucel & Cankut Ormeci a

Agricultural & Environmental Informatics Research and Application Centre, Istanbul Technical University, Istanbul, Turkey b

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Systems Engineering & Engineering Management, The University of North Carolina at Charlotte, Charlotte, NC, USA c

National Software Certification Centre, Istanbul Technical University, Istanbul, Turkey d

Department of Geomatics Engineering, Faculty of Civil Engineering, Istanbul Technical University, Istanbul, Turkey Published online: 18 Feb 2015.

To cite this article: Emre Ozelkan, Serdar Bagis, Ertunga Cem Ozelkan, Burak Berk Ustundag, Meric Yucel & Cankut Ormeci (2015) Spatial interpolation of climatic variables using land surface temperature and modified inverse distance weighting, International Journal of Remote Sensing, 36:4, 1000-1025 To link to this article: http://dx.doi.org/10.1080/01431161.2015.1007248

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International Journal of Remote Sensing, 2015 Vol. 36, No. 4, 1000–1025, http://dx.doi.org/10.1080/01431161.2015.1007248


Spatial interpolation of climatic variables using land surface temperature and modified inverse distance weighting Emre Ozelkana*, Serdar Bagisa, Ertunga Cem Ozelkanb, Burak Berk Ustundaga, Meric Yucelc, and Cankut Ormecid a Agricultural & Environmental Informatics Research and Application Centre, Istanbul Technical University, Istanbul, Turkey; bSystems Engineering & Engineering Management, The University of North Carolina at Charlotte, Charlotte, NC, USA; cNational Software Certification Centre, Istanbul Technical University, Istanbul, Turkey; dDepartment of Geomatics Engineering, Faculty of Civil Engineering, Istanbul Technical University, Istanbul, Turkey

(Received 26 August 2013; accepted 7 November 2014) Accurate spatial interpolation (SI) of climate data is vital for the management and supervision of natural resources and agriculture. Owing to the lack of an adequate number of meteorological stations, meteorological-station-data-based SI methods may not always reflect the real climatic conditions of an interpolated point. Land surface temperature (LST) data obtained from satellite sensors enable the characterization of meteorological conditions of areas without meteorological stations. The aim of this article is to present a new modified inverse distance weighting (M-IDW) SI method for air temperature (Ta), total precipitation (Pt), and relative humidity (RH) by integrating Landsat LST data with meteorological station data for the interpolation process. The M-IDW approach is based on the correlation relationship between the climate data and LST at each meteorological station, which is incorporated into the traditional IDW to improve the estimation of the climate data at an interpolation location of interest. The proposed method, M-IDW, is applied for the interpolation of long years’ (i.e. long term) monthly average (LYMA) Ta, Pt, and RH climate data from meteorological stations in the Eastern Thrace region, which is 23,764 km2, located in southeast Europe. The LYMA of the Ta, Pt, and RH has been constructed using data obtained from 27 meteorological stations that had functioned at least 10 years between 2000 and 2012 and from the corresponding satellite data. The outputs of the interpolation are in the form of LYMA, so are the analysed climate data. The spatial resolution of the predicted surface was taken as 30 m, similar to the original data presented by United States Geological Survey. The results were compared with those of the standard IDW, ordinary kriging (OK), and ordinary cokriging (OCK) methods to analyse the performance and accuracy of the proposed method. The results show that the proposed M-IDW method has the potential for SI of climate data, if enough number of images and cloudless pixels are incorporated in the LYMA LST computation. The proposed method, in general, yields better results than standard IDW and OK methods, especially during spring, summer, and partially in autumn for the interpolation of Ta (with 0.72°C, 0.53°C, and 0.66°C root mean square error (RMSE) values, respectively) and Pt (with 11.07 mm, 7.64 mm, and 4.85 mm RMSE values, respectively). OCK and M-IDW results were comparable in spring, summer, and autumn where M-IDW was slightly better for Ta in autumn and spring and was slightly better for Pt in summer. For the RH interpolation, although M-IDW results were found to be close to the results of IDW, OK, and OCK in spring, summer, and autumn, for the overall seasonal interpretation, the RMSE values of M-IDW were worse than the others. In general, M-IDW yields worse results for the winter months, which in turn is related to cloudiness and availability of satellite images. *Corresponding author. Email: [email protected] © 2015 Taylor & Francis

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1. Introduction A good knowledge about an area’s climatic condition is necessary for optimal management of agriculture, natural resources, disaster recovery, urbanization, transportation, and many other activities. To assess an area’s climatic condition, a typical approach is to rely on the meteorological data coming from in situ stations. Since the number of in situ stations is limited, spatial interpolation (SI) of measurements from various meteorological stations is widely used to approximate the actual meteorological parameters (Hartkamp et al. 1999). SI techniques are classified as deterministic and geostatistical techniques (see e.g. Hartkamp et al. 1999; Apaydin, Sonmez, and Yildirim 2004; Dobesch, Dumolard, and Dyras 2007; Li and Heap 2008; Sluiter 2009). Unlike the deterministic techniques, geostatistical techniques consider estimation error. Some of the well-known deterministic methods used in climate data interpolation are nearest neighbour and triangulation, inverse distance weighting (IDW), polynomial functions (splines), linear regressions, and artificial neural networks (Hartkamp et al. 1999; Apaydin, Sonmez, and Yildirim 2004; Dobesch, Dumolard, and Dyras 2007; Li and Heap 2008; Sluiter 2009). Geostatistical methods used in climate data interpolation include optimum interpolation, ordinary kriging (OK), simple kriging, universal kriging, residual kriging, indicator kriging, probability kriging, disjunctive kriging, and stratified kriging and cokriging of all kriging methods (such as ordinary cokriging (OCK)) with ancillary data (Hartkamp et al. 1999; Apaydin, Sonmez, and Yildirim 2004; Dobesch, Dumolard, and Dyras 2007; Li and Heap 2008; Sluiter 2009). Deterministic methods such as IDW intend to predict values at unknown points by using known measurements (Johnston et al. 2001; Chen and Liu 2012). Geostatistical methods such as OK estimate values at points without measurement using an evaluated spatial autocorrelation (Johnston et al. 2001; Lloyd 2005; Ustuntas 2006). Unlike cokriging, a drawback of pure deterministic IDW and geostatistical OK methods is that they do not take into account the structural characteristics of the interpolation area by using ancillary data (such as satellite data, in situ measurement data, digital elevation model, etc. (Hartkamp et al. 1999; Li and Heap 2008; Sluiter 2009). In this study, we created a modified version of IDW (M-IDW), which uses ancillary data, and compared it with IDW, OK, and OCK. Satellite remote-sensing data (RSD) reflect instant characteristics and spatial variations within the study area (Voogt and Oke 2003). In addition, using multi-temporal satellite images of a certain area can express temporal variations (Goncalves et al. 2011). RSD is of important utility for contemporary meteorological models as it allows a wide range of analysis and modelling opportunity at the same time (McVicar and Jupp 1999). RSD enables tracking the motion of large atmospheric systems in high spatial resolution (Wan, Wang, and Li 2004) compared with the simulated meteorological models. The relationship between meteorological parameters and derived remote-sensing (RS) parameters such as land surface temperature (LST) allows application opportunities for various study areas over wide regions (Dash et al. 2002; Inamdar et al. 2008; Duan et al. 2012). LST, which indicates upward thermal radiation (Freitas et al. 2013), is an important parameter in the land–atmosphere energy exchange and the global hydrological cycle which is commonly associated with climate change (Wan and Li 1997; Duan et al. 2012). Since LST is affected by different meteorological, topographic, and landcover parameters (Karnieli et al. 2010; Duan et al. 2012), it has been used as a pixel-based input parameter in several environmental models of the energy water cycle, numerical weather forecast, global ocean cycle, and climate change (Dash et al. 2002). LST is


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closely related to meteorological parameters such as air temperature (Ta), precipitation (Pt), and relative humidity (RH), presenting an opportunity to estimate meteorological data and to perform meteorological analysis in places without meteorological stations (Tian et al. 2012). It has been shown that LST is positively correlated with Ta and negatively correlated with Pt and RH (note that RH is indirectly proportional to vapour pressure deficit (Singh 2010), which positively correlates with LST (Hashimoto et al. 2008)) (Wan, Wang, and Li 2004; Karnieli et al. 2010; Recondo et al. 2013). Hay and Lennon (1999) used RS and SI parameters to obtain monthly prediction of Ta, atmospheric moisture, and Pt of Africa for the year 1990 to control vector-borne diseases. Ta and atmospheric moisture were predicted using National Oceanic and Atmospheric Administration – Advanced Very High Resolution Radiometer (AVHRR) data. Pt was obtained using METEOSAT-high resolution radiometer cold cloud duration data. Ground measurement data were obtained from the World Meteorological Organization meteorological stations. Ta predictions were obtained using a thin-plate spline SI method applied to the stations’ data, which were shown to be superior to the RS-based prediction. On the other hand, RS-based atmospheric moisture and Pt predictions were found to be superior to the SI results. Goetz, Prince, and Small (2000) analysed the RS of environmental variation for epidemiological applications in Africa. Ta and humidity water vapour pressure were analysed using the AVHRR instrument. Their results showed an 83% correlation between RS temperature data and the temperature data from the stations. Similarly, an 89% correlation was found between RS temperature data and radiometric measured temperature data from the stations, and 45% correlation was found between RS atmospheric water vapour data and the stations’ measurements. In their study, surface temperature and soil moisture were determined with an accuracy of 66%. Angelo Colombi et al. (2007) used LST data from the Moderate Resolution Imaging Spectroradiometer satellite to obtain Ta for the Italian Alpine region to integrate hydrological and environmental models. This study was performed on cloudless areas of 12 different satellite images taken between January and June 2003. Ta obtained using LST data and IDW interpolation applied to meteorological stations’ data had a root mean square error (RMSE) of 1.89°C and 2.23°C, respectively. Cristóbal, Ninyerola, and Pons (2008) predicted instantaneous, minimum, mean, and maximum daily Ta for the period between 2002 and 2004, and monthly and yearly minimum, mean, and maximum Ta for the period between 2000 and 2005 for Spain– Catalonia using RS and geographical information systems. Geographical (such as altitude, latitude, continentality, and solar radiation) and RS variables (LST and normalized difference vegetation index data sets obtained from Landsat-5 (TM), Landsat-7 (ETM+), National Oceanic and Atmospheric Administration – AVHRR, and Terra (Moderate Resolution Imaging Spectroradiometer)) were analysed by means of multiple regression analysis and SI techniques. When RS variables were combined with geographical variables, the best results were obtained for daily mean Ta (R2 = 0.60 and RMSE = 1.75°C) and for monthly and annual mean Ta (R2 = 0.86 and RMSE = 1.00°C). Finally, this combined method was found to be superior to the methods that used only RS or geographical variables. LST and normalized difference vegetation index were identified as the most powerful RS predictors for Ta. The literature review showed that there are studies that used meteorological station data independently using simple interpolation models such as IDW or incorporate LST as ancillary data along with the meteorological station data in a relatively complex fashion using interpolation techniques such as cokriging. Therefore, our objective here is to



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propose a new relatively simple M-IDW SI method by directly integrating Landsat LST data with meteorological station data for the interpolation process. In addition, while there are studies in the literature which analyse the relationship of instantaneous Landsat LST data and a station’s instantaneous meteorological data, there seems to be a gap in the analysis of long-term LST data to generate long years’ (i.e. long term) monthly average (LYMA) LST to associate with the regional climate. As illustrated in the subsequent sections, the results indicate that the proposed method provides higher spatial resolution (through the use of Landsat images) and a more precise climate analysis if a large number of thermal images and amount of cloudless pixels are inputted for the LST computations. The proposed M-IDW method is compared with several known SI techniques: a deterministic method standard IDW, a basic geostatistical method OK, and the OCK (which can be considered as a modified version of OK using ancillary data).

2. Study area The study region, Eastern Thrace, extends over 23,764 km2 and is located in northeast Turkey between 26° 0ʹ–29° 2ʹ eastern meridians and 40° 0ʹ–42° 2ʹ northern parallels. In the north, the Strandzha Mountains, with a highest elevation of 1031 m, are located parallel to the Black Sea coast. In the South, the Ganos and Koru mountains have lower elevations up to 945 m and 726 m. The remaining area is generally covered with low hills (TDA 2012). The area is encircled by the Black Sea in the north, the Aegean Sea in the west, and the Marmara Sea in the south. Containing a variety of geographical conditions, the Eastern Thrace region has three different regional climates: continental, Black Sea, and Marmara (Sensoy et al. 2008) (Figure 1). Long-term average meteorological data for the three regional climates are summarized in Table 1. The continental climate in Eastern Thrace is described by hot summers and relatively cold winters. The natural vegetation of the continental climate is generally in the form of ‘dry’ forests and steppes (Sensoy et al. 2008). The Marmara climate reflects a transition between the continental, Black Sea, and Mediterranean climates: winters are not as warm as the Mediterranean climate and not as cold as the continental climate, and summers are not as rainy as the Black Sea climate and not as dry as the continental climate (Sensoy et al. 2008). According to this, the general vegetation of the Marmara climate is plants of Mediterranean origin such as scrub at the lower altitudes, and plants of Black Sea origin such as moist forests in north-facing slopes at higher altitudes (Sensoy et al. 2008). The Black Sea climate is observed mostly in the northern slopes of the mountains and the Black Sea coast of the Marmara region. The Ta difference between summers and winters is lower compared with other climates and all seasons are rainy. Summers are chilly, winters are warm in coastlines, and at higher elevations, it is snowy and cold (Sensoy et al. 2008). Natural vegetation of the Black Sea climate is moist broadleaf forests in coastlines and coniferous forests at higher elevations, which can grow in cold and moist climate conditions. According to the Turkish Sate Meteorological Service, Pt can be more than 1000 mm at the western coast (Sensoy et al. 2008; TSMS 2003). Every season is rainy in the Black Sea climate, but precipitation occurs the most in winters and the least in summers similar to the other climates in the region. All presented meteorological data about regional climates were obtained from the data archives of the Turkish State Meteorological Service (TSMS 2003).


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Figure 1. The climate zones in the study area, distribution of the meteorological stations, and Landsat coverage in the Eastern Thrace region (16 July 2011 dated Landsat-5 (TM)).

Table 1.

Comparison of continental, Marmara, and Black Sea climates (TSMS 2003). Regional climates

Climatic variables LYMA Ta (°C) coldest month – January LYMA Ta (°C) hottest month – July The long years’ annual average Ta (°C) The long years’ annual average Pt (mm) The long years’ average Pt percentage of summer (%) The long years’ annual average RH (%)

Continental

Marmara

Black Sea

2.8 23.9 13.2 559.7 17.6 69.6

4.9 23.7 14.0 595.2 11.7 73.0

4.2 22.1 13.0 842.6 19.4 71.0


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3. Data sets


Data from meteorological stations and Landsat satellite images were used in the study. All data sets were transformed into the Universal Transverse Mercator (UTM) projection system with WGS 84 datum (UTM-WGS84) and zone 35 North.

3.1. Meteorological station data Out of the 27 stations used in this study, all are located in the Eastern Thrace region and of which 13 are still actively functioning. Only the inactive stations that have at least 10 years of LYMA Ta, Pt, and RH data were included in this study to reflect the longyears’ average in the best possible way. Figure 1 shows the distribution of the used meteorological stations in the Eastern Thrace region.

3.2. Remote-sensing data In this study, blue and thermal bands of Landsat-5 (TM) and Landsat-7 (ETM+) satellite images with a spatial resolution of 30–120 m and 30–60 m, respectively, were used. The Eastern Thrace region is covered by two Landsat frames lying from south to north (Figure 1). The study area is encompassed by 7973 × 7517 pixels with 30 m spatial resolution each. The analysis was performed using 432 satellite images obtained from the Landsat-5 (TM) and Landsat-7 (ETM+) satellites taken between January 2000 and January 2012. The number of images from the Landsat-5 (TM) and Landsat-7 (ETM+) satellites was 202 and 230, respectively. The thermal bands, which are the sixth bands of the Landsat-5 (TM) and Landsat-7 (ETM+) satellite images, were used for the generation of LST data (USGS 2013). Moreover, for the correction of radiometric distortions and cloud effects, thermal and blue bands of Landsat images were used (Martinuzzi, Gould, and González 2007). LST images were resampled to 30 m spatial resolution as recommended by the United States Geological Survey (USGS) (USGS 2013).

4. Methodology In this section, we present the detailed methodology about the proposed M-IDW interpolation method. The adopted method can be generally summarized into five steps: (1) selection of the RSD processing methods; (2) association of generated LST with meteorological data and determination of the most correlated filter size; (3) conducting SI; (4) generation of filters for correction of sensor-based distortions and cloud effects; and (5) creation of an interpolation procedure that comprises the four steps mentioned above.

4.1. Remote-sensing data processing Radiometric correction processes were performed for the thermal infrared band of the Landsat satellite images (6th and 6.1th bands of Landsat-5 (TM) and Landsat-7 (ETM+), respectively). To generate LST images, each thermal band was (1) converted from digital number (DN) to sensor radiance using the spectral radiance scaling method, (2) processed using the radiative transfer equation as an atmospheric corrector to convert the sensor radiance to surface radiance, and (3) converted from radiance to radiant temperature (kelvin) using the Planck function (see e.g. the following references for a similar approach: Qin, Karnieli, Berliner 2001; Gangopadhyay, Lahiri-Dutt, and Saha 2006; Chander,


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Markham, and Helder 2009; Coll et al. 2010). Note that if steps 1, 2, and 3 are applied in order, the computations would yield the surface temperature, but if only steps 1 and 3 are applied in order, the result would yield the sensor temperature. We would like to remark that in this study, due to the large amount of satellite data (432) which were processed to produce general long-term climate data of each pixel using M-IDW, a constant emissivity (ε) value was used. The ε value was taken here as 0.95, which is close to most of the land-cover classes’ ε values for thermal wavelength range (Arnfield 1982; Voogt 2000; Hewison 2001; Gangopadhyay, Lahiri-Dutt, and Saha 2006; Jin and Liang 2006; YCEO 2010). Finally, after performing the LST generation steps, the values (in kelvin) were converted to degrees Celsius by subtracting 273.15. Note that here we did not incorporate atmospheric correction. As reported in Song et al. (2001), atmospheric correction may not always be necessary and does not necessarily provide more accurate results if the spectral signatures defining the essential classes correlations are obtained from the images. Song et al. (2001) recommended atmospheric correction especially for land-use classification and change detection to keep the multi-temporal data in the same scale (the reader can also refer to Goslee [2011] for a similar approach). Moreover, atmospheric effects (upwelling and downwelling radiances) are managed by aerosols, O3, CO2, and water vapour (Dash et al. 2002; Sobrino, JiménezMuñoz, and Paolini 2004), which are decisive in climate formation (NOAA 2013; EPA 2013) and are removed using atmospheric correction methods (Barsi, Barker, Schott 2003; Weng, Lu, and Schubring 2004). To verify whether atmospheric correction would have a significant effect on this study, a preliminary data analysis was conducted using atmospherically corrected (steps 1, 2, and 3) and uncorrected (steps 1 and 3) LST of 10 Landsat-5 (TM) satellite images. The 10 selected images corresponded to the following dates: 19 February 2010, 7 March 2010, 24 April 2010, 26 May 2010, 11 June 2010, 13 July 2010, 29 July 2010, 14 August 2010, 15 September 2010, and 1 October 2010. There was no specific reason to select 2010 other than that the 13 stations, which are numbered 0, 3, 4, 5, 18, 19, 20, 21, 22, 23, 24, 25, and 26, were active. These images were selected since they do not contain stripes and thick clouds (that prevent monitoring of land surface and have lower temperature values compared with land surface because of the structure and altitude of the clouds) on the corresponding 13 active meteorological stations where Ta was observed. It was found that atmospherically corrected LST was correlated with Ta with the coefficient of determination (R2) value varying between 0.53 and 0.98 among the 13 stations, with an average of 0.76. The same correlation was observed as 0.57 to 0.97 (with an average value of 0.79) for the atmospherically uncorrected LST. Significance probabilities (p) varied between approximately 0 and 0.016 (average p = 0.004) for the atmospherically corrected LST, and between approximately 0 and 0.012 (average p = 0.003) for the atmospherically uncorrected LST. In addition, R2 and the p-values between the atmospherically corrected and uncorrected LST data set were found to be 0.99 and approximately 0, respectively. These preliminary results aligned with the literature cited earlier and indicated that in our case, atmospherically corrected and uncorrected LST are close to each other. As explained further in Section 4.4, here instead of atmospheric correction, a simple filter consisting of different threshold values was defined to identify cloudy pixels using the first and the sixth bands of Landsat satellite images (Martinuzzi, Gould, and González 2007; Goslee 2011). Also same filter was used to determine the dead or striped pixels. A benefit of excluding atmospheric correction was a decrease in processing times due to the exclusion of the related processing steps.


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4.2. Association of LST with meteorological data and determination of the filter size LST pixel-based (shortly referred as LST(1 × 1)) values corresponding to the coordinates of the meteorological stations were computed from each Landsat frame. In addition, 3 × 3, 5 × 5, 7 × 7, 9 × 9, 11 × 11, 13 × 13, and 15 × 15 mean filters were applied around the coordinates of the meteorological stations in each frame, and LYMA LST values were computed (shortly referred as LST(M × M), where M = 1, 3, 5, 7, 9, 11, 13, and 15). The relationship between climate parameters and LST (M × M) was analysed to find the highest correlated ones. According to the results, the correlation relationship between LST(1 × 1) and Ta, Pt, and RH was found to be unreliable across stations: for example, a high correlation was found in some stations (e.g. average of 0.95, 0.74, and 0.91 coefficient of determination R2 for Ta, Pt, and RH, respectively, at station 21, and 0.87, 0.83, and 0.88 R2 for Ta, Pt, and RH, respectively, at station 24) and conversely a low correlation was found in some others (e.g. 0.22, 0.24, and 0.23 R2 for Ta, Pt, and RH, respectively, at station 14 and 0.78, 0.53, and 0.69 R2 for Ta, Pt, and RH, respectively, at station 19). While not shown here to keep the presentation concise, the unreliable correlation behaviour was observed for LST(1 × 1) across all stations. LST(M × M) with M ≥ 3 mean filters was found to be more reliable and remained relatively close to each other; however, analysis of all 27 stations showed that LST(M × M) mean filters with M ≥ 5 generated slightly better results than LST(3 × 3) for LYMA Ta, Pt, and RH (see Figure 2). Again, the results of all filters seemed close to each other; since M ≥ 5 yields slightly better results and the enlargement of the filter size after M = 5 did not give significant improvements (on the contrary, they resulted in a decrease of the spatial resolution of the defined/investigated region), LST(5 × 5) mean filter was selected for further LYMA LST computations. Figure 3 shows sample analysis results of RSD and climate data for meteorological station 14. The results presented in the simple linear regression plots show that the variance explained (R2) for the LYMA LST(5 × 5) with regard to LYMA Ta, Pt, and RH is over 0.74 for station 14. In general, R2 was relatively high for Ta with minimum and maximum R2 being 0.85 and 0.98 at stations 21 and 20, respectively. Similarly, for Pt with minimum and maximum R2 being 0.58 and 0.83 at stations 8 and 26, respectively, and for

Figure 2. R2 between LST(M × M) and Ta, Pt, and RH, and average of R2 for each filter size (correlation labels correspond to the average line).


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Figure 3.

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LYMA LST and Ta, Pt, and RH relation for station 14 using simple linear regression.

RH with minimum and maximum R2 being 0.61 and 0.95 at stations 26 and 25, respectively. The other stations, which are not shown here, behave in a similar way. Based on the analysis results, LYMA LST and LYMA Ta were found to be directly proportional and in contrast, Pt and RH values of the two locations were found to be indirectly proportional.

4.3. Spatial interpolation In this study, a new M-IDW is proposed and the results are compared with some of the well-known SI techniques such as IDW, OK, and OCK. In this section, we will first briefly summarize the existing SI techniques used here, and then discuss M-IDW and the results in more detail. IDW and OK methods are widely used in practice due to their relative simplicity and can be respectively considered as ‘pure’ deterministic and geostatistic data interpolation methods since they only require the meteorological data of the respective stations. On the other hand, OCK is a more complex or sophisticated method that can use ancillary data. As shown below, M-IDW aims to incorporate ancillary data into the interpolation process as well, but in a more direct and relatively simple way.

4.3.1. Inverse distance weighting (IDW) IDW is a type of deterministic method for multivariate interpolation based on the principle that the estimated point’s value is more correlated with a closer known point value than a far known one. As the distance between queried and measured points (stations’ locations) increases, the efficacy of the measured point decreases (Hartkamp et al. 1999). The effect can be also noted as the weights that are taken here are inversely proportional to the square of the distance. The IDW equation is given as follows (Chen and Liu 2012).


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N P

Zðui Þwi i¼1 ^ ; Zðu0 Þ ¼ N P wi

(1)


i¼1

where u0 denotes the predicted location and ui, i = 1 … N denote the measurement ^ 0 Þ is a weighted average of N measured values Z(ui) locations. The predicted value Zðu on different locations, i = 1… N. wi ¼ di0r refers to the weight or influence of each of the measured data, di0 is the Euclidian distance between the estimated location (0) and measurement. Note that as discussed in Lloyd (2005), a typical setting for r (exponential power parameter) is 2, denoting squared distances, which is used in this study. r = 2 magnifies the large distances further, thus reducing the weighting of distant locations in the interpolation computations.

4.3.2. Ordinary kriging Kriging is a geostatistic interpolation method. It is similar to IDW since both of them use a weighted mean of measured values (Lloyd 2005). When weights are computed, kriging takes into account not only the distance, but also the spatial dispersion of measurement locations, which reflects the autocorrelation of these measurements (Johnston et al. 2001). As in IDW, observed values are kept unchanged after interpolation (Hartkamp et al. 1999; Cheng, Yeh, and Tsai 2000). Goovaerts (1997, 1998, 2000) indicates that all kriging estimators are variants of the basic linear regression estimators (Wang 2011; Mutua and Kuria 2012) and defines the required computational relationship as: Z ðuÞ mðuÞ ¼

N X

λi ½Zðui Þ mðui Þ;

(2)

i¼1

where Z*(u) P is the value to be estimated, λi is the weights assigned to measurements such that Ni¼ 1 λi ¼ 1, Z(ui) is the ith measurement at location u, and m(u) and m(ui) are the means of Z(u) and Z(ui), respectively. N is the number of measurements. Kriging methods differ in their treatments of the trend component m(u). In OK, it is assumed that m(ui) = m(u) is known and constant in the local neighbourhood of each estimation point (Garška and Krūminiene 2004; Wang 2011), which further simplifies Equation (2), yielding the well-known OK formula as follows. ZOK ðuÞ ¼

with

N X i¼1

N X i¼1

λOK i ðuÞZðui Þ

λOK i ðuÞ

(3) ¼ 1:

The weights in Equations (2) and (3) are computed so that they minimize the variance given σ 2E ðuÞ ¼ VarfZ ðuÞ ZðuÞg under the unbiasedness constraint EfZ ðuÞ ZðuÞg ¼ 0 (Goovaerts 2000; Oliver, Webster, and Slocum 2000; Hu 2010). An empirical semivariogram (the spherical model used in our study) may

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be generated which in turn may be used to obtain the weights (λi) solving Equation (4) (Hartkamp et al. 1999; Cheng, Yeh, and Tsai 2000; Ustuntas 2006; Wang, Li, and Christakos 2009).


γi;j ¼ 0:5E½ðZðui Þ Zðuj ÞÞ2 ;

(4)

where γi,j is the semivariance and covariance between i and j data points. For more information about semivariogram models, the reader may refer to Bastin and Gevers (1985), Ishida and Ando (1999), Goovaerts (2000), Wu et al. (2009), Garška and Krūminiene (2004), Sertel, Kaya, and Curran (2007).

4.3.3. Ordinary cokriging (OCK) Cokriging is an extension of the kriging technique described above in Section 4.3.2, which uses information from one or more correlated secondary variables (covariates) (Hartkamp et al. 1999; Yang, Wang, and August 2004; Sluiter 2009). It considers the spatial cross-correlation between primary and secondary variables to increase the accuracy of the interpolation and is often used when the primary variable (i.e. climate data in this study) is under-sampled (Zhang, Li, and Travis 2009). The OCK formula with a secondary variable (e.g. LST for Ta, Pt, and RH) is presented below. ðuÞ ¼ ZOCK

with

N1 X i1 ¼1

N1 X i1 ¼1

λOCK ðuÞZðui1 Þþ i1

λOCK ðuÞ i1

¼ 1; and

N2 X i2 ¼1

λOCK ðuÞZðui2 Þ i2

N2 X i2 ¼1

(5) λOCK ðuÞ i2

¼ 0:

Note that in this study, ArcGIS software was used to conduct kriging and cokriging. For the kriging parameters, a trial and error approach was taken to identify the best parameters manually, but it was observed that the default parameters identified by the ArcGIS software yielded better results. Therefore, we have used the values as identified by ArcGIS in this study.

4.3.4. Modified inverse distance weighting (M-IDW) In Section 4.2., the relationship between LYMA LST and LYMA climate data was analysed with a correlation and regression analysis. It was found that LYMA LST and LYMA climate data of Ta, Pt, and RH are significantly correlated as explained in Section 4.2. These results indicated that LYMA LST, which reflects the characteristics of the study area, reflects the general climate conditions of corresponding points. Based on these findings, the IDW formula (Equation (1)) was modified as follows to incorporate LST as ancillary data: N P

CDðu0 Þ ¼

CDðu0;i Þ wi

i¼1 N P i¼1

; wi

(6)


1011

where


CDðu0;i Þ ¼

LSTðu0 ÞCDðui Þ ; i ¼ 1; . . . ; N : LSTðui Þ

● The notation in Equation (6) is described below: u0 denotes the estimated location, which in this case corresponds to each pixel in the Landsat images of the corresponding month, LST(u0) is the estimated location’s LYMA LST computed from Landsat images of the corresponding month, LST(ui) is the measurement location’s LYMA LST computed from Landsat images of the corresponding month, CD is the climate data that is input from meteorological stations and output of MIDW, CD(u0,i) is the individual contribution of station i on the estimated location’s LYMA climate data, CD(ui) are the measurement location’s LYMA climate data (i.e. (z(ui) in IDW formula Equation (1)), CD(u0) are the estimated location’s LYMA climate data, ● N is the number of meteorological stations, ● wi is the weight of the corresponding measurement location, ● di0 is the Euclidian distance between the estimated location (0) and measurement location (i). r is an exponential power parameter and set as 2. As seen in Equation (6), instead of using the climate parameter z(ui) obtained from the meteorological stations directly as in the standard IDW, here new terms CD(u0,i) enhanced with LST values are used. These terms estimate each stations’ contribution to the LYMA climate parameter estimation using a simple linear interpolation. In other words, these terms simply compute the value of the climate data at point u0 given that LST at point u0 (LST(u0)) is known, and by using the available meteorological station-based information data (i.e. LST(ui), CD(ui)). Note that a linear interpolation (cross-multiplication of the terms) results in the (LST(u0) CD(ui))/LST(ui), which yields CD(u0,i). We would like to remark once again here that a linear relationship assumption is made here based on our findings in Section 4.2, which showed a strong correlation between LST and the climate variables (Ta, Pt, and RH). Figure 4 presents a simple example to illustrate the M-IDW methodology. In this example, the climate data and the LST values are available for the three sample stations at corresponding pixels. For other pixels, while the LST values are available, the climate data CD(u0) are to be estimated. The distance between three sample stations and the estimated pixel, which are shown as d10, d20, and d30, respectively, are used for weighting the influence of each station. The numerical values are presented in the figure for illustration purposes.

4.4. Correction of sensor-based distortions and cloud effect Here, we will elaborate on the two digital filters integrated into the system. The first filter is designed for determining distortions of radiometric and atmospheric effects of the sixth band before the LST computation steps. This filter assures that if a pixel is outside the threshold values, then the corresponding pixel value is set as ‘no data’. The second filter is an adaptive enlarging mean filter designed to calculate the mean


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E. Ozelkan et al.

Figure 4. Illustration of M-IDW computations, where pink pixels correspond to the meteorological stations and the others represent pixels to be estimated. A sample computation is shown for one cell.

LST values starting from a 5 × 5 window and growing if not enough data were found. The distortion caused by sensors in Landsat-5 (TM) and Landsat-7 (ETM) is well known (Chander, Markham, and Helder 2009). Generally, in Landsat-5 (TM) images, dead pixels may occur and Landsat-7 (ETM) images may have stripes. Additionally, except for sensor distortions, natural distortions such as cloudy pixels can occur. Here, initially, the threshold values were determined for the filter design. The ranges for cloudy pixels were defined as 120–250 DN for band 1 and as 102–128 DN for band 6 (Martinuzzi, Gould, and González 2007). Cloud’s reflection interval in the blue region is defined with band 1. Since, within this interval, reflection of rocks, cities, etc. can also occur, clouds were distinguished from other structures by considering the sixth band threshold range, which also includes densely forested areas (see e.g. Ackerman et al. [1998] and Martinuzzi, Gould, and González [2007] for a related approach). Since shaded areas reflect the real LST condition of the cloudy days, the shaded areas were not masked in the current study presented. As a result, a cloud was determined by evaluating the intersection set of the first and sixth bands and the corresponding pixels were eliminated. In addition, to eliminate no data (dead


1013

pixels and stripes), a DN > 0 threshold value was set for all bands using Equation (7) below.


MF IMðx; yÞ ¼

8 > > > > > > < > > > > > > :

DN

No data

if 0 < IM Band1ðx; yÞ < 120 and 0 < IM Band6ðx; yÞ < 102 or IM Band6ðx; yÞ > 128 and DN > 0

(7)

else;

where IM Bands 1 and 6 are the DNs for the first and sixth bands of the Landsat sensor and (x, y) shows the coordinate of each pixel in the working frame. In effect, MFIM, which is a masking filter image, is used to detect pixel-wise noise and distortion caused by atmospheric and radiometric effects as well as in image pixel values. The second filter is a two-dimensional adaptive mean filter and was applied on LST resulting images for noise and distortion reduction by taking the average of neighbouring pixels (NPs) (see e.g. Kundu, Mitra, and Vaidyanathan 1984; Vernon 1991; Yagou, Ohtake, and Balyaev 2002). As discussed in the previous sections, a preliminary analysis indicated that the 5 × 5 mean filter is fairly adequate to reflect the climate data. On the other hand, when the number of NPs is not sufficient, the filter size needs to be enlarged. Hence this is the reason for using an adaptive filter size. In our case, the minimum number of NPs was selected using NP = M – 3, where M is the filter size which was initialized as M = 5 (i.e. a 5 × 5 filter) and was enlarged to M = 25 when needed.

4.5. Building the interpolation procedure The study area was chosen to be between the UTM coordinates defined in the upper left corner (42° 9′ 34″ N, 25° 55′ 17″ E) and in the lower right corner (39° 59′ 48″ N, 28° 36′ 2″ E). The area is covered by two Landsat frames and consists of 7973 × 7517 pixels of resampled 30 m spatial resolution pixels. The proposed method was implemented at each pixel of the study area. A total number of 432 (202 Landsat-5 (TM) and 230 Landsat-7 (ETM)) images recorded between 2000 and 2012 were used to build the system. The system was designed based on the UTM WGS 84 zone 35 north map projection similar to that which USGS Landsat uses. Note that the spatial resolution of the system output was set as 30 m, similar to the original data presented by USGS. The overall procedure for the interpolation system is summarized in Figure 5. There are mainly two phases: in phase 1, the LYMA LST values were computed by using the Landsat images and by applying filters for correction, and in phase 2, the M-IDW method was applied to compute LYMA Ta, Pt, and RH values at each pixel using the LST and meteorological data from stations. The interpolation procedure for M-IDW was implemented using the C# programming language on a MS Windows platform. The reason for using C# was related to the huge amount of data that had to be processed. Using the speed of processing, less memory usage and parallelization capability of C# (Fourment and Gillings 2008) results could be obtained relatively fast. Four hundred and thirty-two first and 432 sixth band Landsat satellite images were processed in ~2 h. The computations were run on a 64-bit Windows 7 Professional platform with an Intel® Xeon® CPU, 2.67 GHz processor, and 4 GB RAM.


1014

Figure 5.

E. Ozelkan et al.

Flow chart of interpolation procedure for an individual month.

5. Results In this part of the study, the proposed M-IDW – as well as the standard IDW, OK, and OCK – were applied to the study area and the results were compared. As indicated earlier, the study area has 27 meteorological stations. In this study, 21 of these stations were used for interpolation model building, and six of these stations (8, 9, 15, 19, 20, and 21) corresponding to about 22% of the overall data were selected for validation or for testing the accuracy of the interpolation model. These test stations were chosen to represent the three different climates in the Eastern Thrace region. An RMSE between the estimated and observed values was used to measure the accuracy of test data results obtained through the IDW and OK techniques (Zhang, Wu, and Zhou 2011). The RMSE results for the M-IDW, IDW, OK, and OCK-based interpolations are given in Table 2. Some of the major observations can be summarized as follows. Monthly analysis indicates that M-IDW gives better or comparable results in some months compared with the other techniques. More specifically, for Ta computations, M-IDW was better than IDW for 8 months (between March and October). M-IDW was better than OK for 6 months (March, April, June, July, September, and October). M-IDW was better than OCK for 5 months (March, April, September, October, and November). On the other hand, for the winter months (January, February, and December months), the Ta results of M-IDW seem consistently inferior compared with the other techniques. For Pt interpolation, M-IDW generated a higher accuracy in comparison to IDW between March and August and for October. M-IDW generated a higher accuracy compared to OK between April and August and for October. M-IDW generated a higher accuracy than OCK for the April, June, July, and October months. Again, for the winter months, the Pt results of M-IDW seem consistently inferior compared with the other techniques. For RH interpolation, M-IDW generated more accurate results than IDW for May and August. M-IDW generated more accurate results than OK for the May, August, and September months. M-IDW generated more accurate results than OCK for the May month. The RMSE differences between M-IDW and the other methods were low between March and September months, again M-IDW yielding worse results for the winter months similar to the Ta and Pt results.

M-IDW 2.83 1.19 0.27 0.19 1.13 0.66 0.60 0.73 0.71 0.58 0.87 29.59 3.28 11.20 0.72 0.53 0.66 0.64

Month

9 1 17 2 33 3 40 4 45 5 56 6 71 7 59 8 47 9 35 10 10 11 10 12 Grand average Winter [12-1-2] Autumn [9-10-11] Spring [3-4-5] Summer [6-7-8] Grand average without winter [3-11]

Number of images 0.57 0.37 0.37 0.98 1.23 0.82 0.81 0.75 0.79 0.75 0.75 0.53 0.73 0.49 0.76 0.86 0.79 0.81

IDW 0.77 0.51 0.44 1.25 0.78 0.60 0.55 0.59 0.79 0.61 0.91 0.58 0.70 0.62 0.77 0.82 0.58 0.73

OCK

Ta RMSE (°C)

0.79 0.47 0.33 0.73 1.08 0.75 0.63 0.70 0.97 0.91 0.85 0.63 0.74 0.63 0.91 0.71 0.69 0.77

OK 43.25 13.09 8.95 5.69 8.28 4.60 5.26 4.70 6.78 7.98 18.47 381.72 42.40 146.02 11.07 7.64 4.85 7.86

M-IDW 9.69 5.38 9.51 6.11 9.37 5.75 5.60 5.71 6.73 8.73 13.58 9.70 7.99 8.26 9.68 8.33 5.69 7.90

IDW 10.64 5.36 8.64 5.88 5.52 5.14 6.48 4.55 6.00 10.85 13.11 10.48 7.72 8.83 9.99 6.68 5.39 7.35

OCK

Pt RMSE (mm)

10.55 5.35 8.44 6.50 10.05 6.79 6.21 5.07 6.12 10.82 13.65 11.30 8.40 9.07 10.20 8.33 6.02 8.18

OK

30.46 22.25 5.08 6.39 3.59 5.72 5.09 4.29 4.22 6.18 7.88 391.81 41.08 148.17 6.09 5.02 5.04 5.38

M-IDW

2.78 2.67 2.58 2.81 3.60 3.23 4.07 4.45 4.19 3.16 3.28 3.21 3.34 2.89 3.54 3.00 3.92 3.49

IDW

3.14 2.87 2.86 2.65 2.79 2.46 5.50 4.05 4.17 3.86 3.52 3.49 3.45 3.17 3.85 2.77 4.01 3.54

OCK

RH RMSE (%)

2.93 3.53 3.37 3.21 3.80 3.77 3.64 4.48 4.66 3.80 3.76 3.73 3.72 3.39 4.07 3.46 3.96 3.83

OK

Table 2. Analysis of monthly and seasonal RMSE values for predicting Ta, Pt, and RH using M-IDW, IDW, OCK, and OK (bold reflects the most accurate method with smallest RMSE and numbers in brackets indicate the months of the year (i.e. January to December is 1 to 12).


International Journal of Remote Sensing 1015


1016

E. Ozelkan et al.

When seasonally analysed, Ta interpolation results using M-IDW were more accurate than IDW and OK except for winter. M-IDW was more accurate than OCK for autumn and spring, and in summer OCK and M-IDW were very close. Pt interpolation using M-IDW generated higher accuracy than IDW and OK for spring and summer. Owing to the high RMSE of M-IDW for November, IDW and OK results were superior to M-IDW for autumn. For winter, IDW and OK results were also superior to M-IDW. M-IDW generated higher accuracy than OCK for summer. In spring and autumn, M-IDW and OCK RMSE values were found to be relatively close. For RH interpolation results, winter seems to be again the season when the accuracy of IDW, OK, and OCK was much more superior to M-IDW. For the other seasons, the performances of the M-IDW and the other techniques (IDW, OK, and OCK) were close. We would like to briefly elaborate on the statistical significance of these presented results. The significance of results (i.e. significance F (SF)) was analysed using analysis of variance. As shown in Table 3, for Ta, most results were statistically significant at a 90% confidence level. For Pt, the significance of the results was worse than the Ta results, in general, for all SI techniques. For RH, while the general significance results were better than the ones for Pt (with the exception of OK), overall they were also worse than the Ta results for all SI techniques. The distribution of available images per month is given in Table 2. Note that the number of images is negatively influenced by cloudy weather preventing optical RSD from being recorded or having large regions covered by cloud. Further analysis indicates that the M-IDW accuracy seems to drop with the drop in the number of images, thus the accuracy is negatively influenced by weather conditions that cause insufficient ancillary data. On the other hand, OCK, which is a much more sophisticated method, seems to be more robust and was not affected by insufficient ancillary data as M-IDW. Long-years’ average daily (LYAD) insolation hours and LYMA rainy days of Eastern Thrace are shown in Table 4. According to these data, it can be said that winter months have more Pt and less sunshine, indicating a general cloudy weather characteristic of the study area. Especially in winter months, because of frequent cloudy weather and lower number of useful satellite images, statistical M-IDW accuracy over all study field decreases. While in January and December months, there are the lowest insolation hours and the highest rainy days, December has also the least number of useful images due to cloudiness, which influences M-IDW accuracy negatively. The correlation of the number of satellite images (Table 2) with the climate parameters of LYAD insolation hours and LYMA rainy days (Table 4) is shown in Figure 6. The number of images is correlated to LYAD insolation hours with r = 0.98 (R2 = 0.95) and LYMA rainy days with r = –0.87 (R2 = 0.76). The figure shows that when insolation hours increase (which typically corresponds to a decrease in rainy days (i.e. cloudiness)), the number of images increases. During the non-winter months when more cloudless images of stations are available, the proposed M-IDW method seems to provide more accurate results than the standard IDW and OK, also again, excluding the winter, M-IDW seems to provide similar or slightly better results than OCK. 6. Summary, discussions, and conclusions In this study, we proposed a new M-IDW technique to integrate the LYMA LST data into the SI process. M-IDW was compared with standard IDW, OK, and OCK. In general, the results show that there is no single method that is the best for all months and seasons. The results also indicate that M-IDW has some potential for a reliable SI application and

Month

9 1 17 2 33 3 40 4 45 5 56 6 71 7 59 8 47 9 35 10 10 11 10 12 Grand average Winter [12-1-2] Autumn [9-10-11] Spring [3-4-5] Summer [6-7-8] Grand average without winter [3-11]

Number of images 0.19 0.12 0.02 0.00 0.13 0.20 0.05 0.08 0.03 0.08 0.27 0.01 0.10 0.11 0.13 0.05 0.11 0.10

M-IDW 0.01 0.00 0.00 0.01 0.04 0.03 0.01 0.01 0.03 0.19 0.02 0.04 0.03 0.02 0.08 0.02 0.02 0.04

IDW

Ta (SF)

0.02 0.03 0.01 0.01 0.01 0.02 0.00 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.01 0.01 0.01

OCK 0.02 0.04 0.00 0.00 0.02 0.01 0.00 0.12 0.03 0.01 0.10 0.10 0.04 0.05 0.05 0.01 0.04 0.03

OK 0.55 0.41 0.78 0.66 0.26 0.04 0.17 0.08 0.91 0.25 0.38 0.75 0.44 0.57 0.51 0.57 0.10 0.39

M-IDW 0.58 0.93 0.18 0.78 0.48 0.30 0.21 0.14 0.28 0.02 0.65 0.50 0.42 0.67 0.32 0.48 0.22 0.34

IDW

Pt (SF)

0.44 0.89 0.76 0.64 0.23 0.00 0.37 0.12 0.54 0.17 0.74 0.51 0.45 0.61 0.48 0.54 0.17 0.40

OCK 0.31 0.91 0.48 0.90 0.64 0.00 0.21 0.40 0.01 0.03 0.86 0.60 0.45 0.61 0.30 0.67 0.20 0.39

OK 0.85 0.42 0.32 0.61 0.52 0.49 0.29 0.08 0.21 0.72 0.20 0.16 0.40 0.48 0.38 0.48 0.29 0.38

M-IDW

0.04 0.02 0.04 0.33 0.22 0.89 0.23 0.16 0.43 0.16 0.24 0.05 0.23 0.04 0.28 0.20 0.42 0.30

IDW

0.12 0.04 0.18 0.17 0.19 0.06 0.13 0.14 0.28 0.64 0.36 0.14 0.21 0.10 0.43 0.18 0.11 0.24

OCK

RH (SF)

0.02 0.59 0.98 0.99 0.78 0.88 0.13 0.55 0.53 0.73 0.14 0.10 0.53 0.24 0.47 0.92 0.52 0.63

OK

Table 3. Analysis of monthly and seasonal SF values of the test stations for predicting Ta, Pt, and RH using M-IDW, IDW, OCK, and OK (bold reflects the most significant results, and numbers in the brackets indicate the months of the year (i.e. January to December is 1 to 12).


International Journal of Remote Sensing 1017

1018 Table 4.

E. Ozelkan et al. LYAD insolation times (hour) and LYMA rainy days of Eastern Thrace.

Parameter/month


LYAD insolation hour LYMA rainy day

Figure 6.

1

2

3

4

5

6

7

8

9

10

11

12

2.4 12.3

3.4 10.6

4.5 10.3

6.1 10.4

8.1 8.8

9.3 7.3

10.1 4.6

9.6 3.9

7.6 5.0

5.1 8.3

3.3 10.0

2.2 12.7

Correlation of number of images with LYAD insolation hours and LYMA rainy days.

produce comparable results, especially in non-winter months. Some of the major findings, discussions, and conclusions are presented below. While in the standard IDW and OK, the users obtain a synthetic spatial resolution depending on the gridding ability of the program, the proposed M-IDW as well as the OCK methods on the other hand yield high spatial resolution results since they give best use of the LST characteristics of each pixel. Beyond the interpolation of meteorological parameters of terrestrial stations only, M-IDW and OCK also provide a possibility to examine images pixel by pixel and to interpolate according to the characteristics of each pixel individually. In other words, unlike the IDW, OK, and similar methods, here a point is interpolated by not only considering station data and distances but also by considering the pixel-based LST values. The difference of M-IDW from OCK is its relative simplicity, M-IDW uses LST as a direct parameter in its formula, whereas OCK uses LST as an ancillary data as a covariate through a relatively more sophisticated semivariogram approach. The simplicity of the M-IDW procedure also reflects in machine processing times, which are significantly lower for M-IDW compared with OCK. For the interpolation of the whole area, between the start of data acquisition until the end of the process, M-IDW took 46 s on average as opposed to OCK, which took 423 s on average. Basic techniques such as IDW and OK took only 2–3 s, which was expected since these techniques did not incorporate processing of the ancillary data. Note that these are not CPU times but since the same computer setup was used for each method, the times should be reflective of the relative computational requirements. Long years’ April average Ta, Pt, and RH results produced by the M-IDW, IDW, OCK, and OK for Eastern Thrace region are shown in Figure 7. The figures of M-IDW indicate an especially high variation according to the natural background (vegetation,

1019



Figure 7. The M-IDW, IDW, OCK, and OK long years’ April average (LYMA – 4) for (a) Ta, (b) Pt, and (c) RH.


1020

E. Ozelkan et al.

topography, etc.). IDW, OCK, and OK do not show structural variations, not reflecting geographic distribution. Thus, it is possible to say that if the station data are statistically different from the general average, standard IDW, OCK, and OK, interpolations may generate an abnormal neighbouring region around the corresponding station, an effect that is called a ‘Bull’s Eye’ (Johnston et al. 2001). As almost half of the stations that supply meteorological data are not active currently and the reliability of these data is not well known, even if incompatible and anomaly data are incorporated for correction, it may not be possible to obtain the actual data. The spatial resolution of the proposed M-IDW method helps minimize the negative effects of missing data. SI methods such as the standard IDW, OK, and OCK yield values within a controlled range as dictated by the minimum and maximum of the input data to be interpolated or the variogram. Owing to the incorporation of LST along with distances, the M-IDW method is not constrained with the input data range when it comes to predictions. Hence, when the LST data are sufficient, M-IDW enables the estimation of extreme or anomaly climatic conditions. For example, consider a case where all meteorological stations are positioned at lower altitudes, say,

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