a. Regression relation of temperature with auxiliary variables
Table 1 reports the correlation results of actual temperature data with auxiliary variables of latitude, longitude, altitude, and satellite temperature products. As shown, although the annual data is somewhat correlated with latitude so that temperature decreases from north to south, the temperature is not significantly correlated with latitude in the studied month. However, increasing altitude in most months has a significant negative correlation with temperature in Mazandaran Province. Moreover, temperature data in Mazandaran Province are insignificantly correlated with longitude. According to the table, satellite temperature product (TRMM column) has a significant positive correlation with annual data in most months.
Table 1
correlation results of monthly and annual Temperature with covariate variables
TRMM
|
Z
|
Y
|
X
|
|
|
r*
|
r**
|
r*
|
r
|
Year
|
Month
|
0.58
|
-0.92
|
0.52
|
0.12
|
2012
|
Jan
|
0.61
|
-0.91
|
0.44
|
0.24
|
2013
|
0.2
|
-0.43
|
0.23
|
0.24
|
2012
|
Feb
|
0.6
|
-0.92
|
0.52
|
0.12
|
2013
|
0.42
|
-0.64
|
0.33
|
0.42
|
2012
|
Mar
|
0.51
|
-0.82
|
0.32
|
0.24
|
2013
|
0.58
|
-0.92
|
0.32
|
0.33
|
2012
|
Apr
|
0.57
|
-0.93
|
0.52
|
0.23
|
2013
|
0.59
|
-0.81
|
0.44
|
0.24
|
2012
|
May
|
0.58
|
-0.94
|
0.53
|
0.34
|
2013
|
0.63
|
-0.92
|
0.42
|
0.24
|
2012
|
June
|
0.61
|
-0.93
|
0.52
|
0.38
|
2013
|
0.63
|
-0.92
|
0.53
|
0.23
|
2012
|
July
|
0.62
|
-0.94
|
0.54
|
0.24
|
2013
|
0.59
|
-0.82
|
0.43
|
0.21
|
2012
|
Agu
|
0.63
|
-0.93
|
0.52
|
0.33
|
2013
|
0.66
|
-0.94
|
0.51
|
0.32
|
2012
|
Sep
|
0.62
|
-0.93
|
0.52
|
0.23
|
2013
|
0.63
|
-0.92
|
0.51
|
0.23
|
2012
|
Oct
|
0.6
|
-0.91
|
0.53
|
0.34
|
2013
|
0.61
|
-0.91
|
0.54
|
0.23
|
2012
|
Nov
|
0.59
|
-0.92
|
0.44
|
0.32
|
2013
|
0.57
|
-0.93
|
0.53
|
0.13
|
2012
|
Dec
|
0.56
|
-0.92
|
0.42
|
0.32
|
2013
|
0.65
|
-0.93
|
0.51
|
0.22
|
2012
|
Annual
|
0.63
|
-0.94
|
0.53
|
0.32
|
2013
|
*: significant at 95% **: significant at 99%
b. Relationship between temperature and TRMM satellite images
TRMM satellite images were used auxiliary variables along with latitude and altitude in the regression equation to estimate temperature and plot isothermal maps. As shown in Table 2, the coefficient of determination of regression equations is above 0.8 except for a few cases. The coefficient of determination of annual data is also greater than 0.9.
Table 2
Temperature regression equation of Mazandaran province
year
|
month
|
Regression Equation of Regres
|
R2
|
2012
|
jan
|
t = − 63.3 + 1.54 Y − 0.00294 Z − 0.033 trmm
|
0.79
|
2013
|
t = 3.8 − 0.73 Y − 0.00359 Z + 0.117 trmm
|
0.82
|
2012
|
feb
|
t = − 192 + 3.95 Y − 0.00129 Z − 0.187 trmm
|
0.40
|
2013
|
t = − 27.3 + 0.55 Y − 0.00317 Z + 0.048 trmm
|
0.82
|
2012
|
mar
|
t = − 156 + 2.67 Y − 0.00159 Z − 0.113 trmm
|
0.49
|
2013
|
t = − 2.4–0.52 Y − 0.00293 Z + 0.050 trmm
|
0.70
|
2012
|
apr
|
t = 16.8–1.02 Y − 0.00333 Z + 0.121 trmm
|
0.81
|
2013
|
t = − 89.5 + 1.87 Y − 0.00245 Z − 0.082 trmm
|
0.84
|
2012
|
may
|
t = 26–1.02 Y − 0.00441 Z + 0.186 trmm
|
0.84
|
2013
|
t = − 137 + 2.80 Y − 0.00302 Z − 0.119 trmm
|
0.86
|
2012
|
June
|
t = − 53.3 + 0.85 Y − 0.00421 Z + 0.097 trmm
|
0.83
|
2013
|
t = − 134 + 2.80 Y − 0.00321 Z − 0.087 trmm
|
0.89
|
2012
|
July
|
t = − 51.3 + 0.96 Y − 0.00421 Z + 0.127 trmm
|
0.84
|
2013
|
t = − 91.9 + 2.05 Y − 0.00329 Z + 0.023 trmm
|
0.87
|
2012
|
agu
|
t = 18–0.64 Y − 0.00457 Z + 0.190 trmm
|
0.72
|
2013
|
t = − 123 + 2.64 Y − 0.00311 Z − 0.069 trmm
|
0.91
|
2012
|
sep
|
t = − 126 + 2.49 Y − 0.00332 Z − 0.002 trmm
|
0.92
|
2013
|
t = − 71.7 + 1.45 Y − 0.00339 Z + 0.012 trmm
|
0.86
|
2012
|
oct
|
t = − 84.1 + 1.60 Y − 0.00370 Z + 0.028 trmm
|
0.92
|
2013
|
t = − 107 + 1.98 Y − 0.00352 Z − 0.066 trmm
|
0.95
|
2012
|
Nov
|
t = − 96.2 + 1.86 Y − 0.00365 Z − 0.019 trmm
|
0.93
|
2013
|
t = − 76.9 + 1.08 Y − 0.00389 Z − 0.001 trmm
|
0.9
|
2012
|
Dec
|
t = − 45.1 + 0.92 Y − 0.00324 Z − 0.023 trmm
|
0.9
|
2013
|
t = − 114 + 1.87 Y − 0.00313 Z − 0.037 trmm
|
0.82
|
2012
|
Annual
|
t = − 69.8 + 1.24 Y − 0.00338 Z + 0.015 trmm
|
0.93
|
2013
|
t = − 88.9 + 1.67 Y − 0.00326 Z − 0.030 trmm
|
0.95
|
c. Selecting the best temperature estimation method
Tables 3 and 4 show the RMSE and MBE of estimated precipitation in 2012 and 2013, respectively. The highest temperature estimation error was observed in January, February, May, and August 2012 and in January, November, and December 2013. The lowest error was observed in March, April, and December 2012 and 2013. According to Tables 3 and 4, the highest and lowest temperature estimation errors were observed in winter and autumn.
Comparing interpolation methods shows that CK and IDW methods have the highest temperature estimation errors. Despite good correlation with actual data, the TRMM satellite cannot properly estimate temperature alone. Given the negative MBE in all months, the temperature products of this satellite have a lower estimation error. Temperature estimation by the regression method in 2013 reduced error by 55% relative to CK and IDW methods and by 68% relative to the TRMM satellite. Error reduction in 2012 by the above methods was 65 and 70%, respectively.
Comparing interpolation methods in different months showed that among the IDW, CK, regression, and TRMM satellite network, the regression method, as a combination of interpolation and TRMM satellite network data, has the lowest temperature estimation error in all months, and the TRMM satellite network alone shows a relatively large error. Analyzing the MBE of temperature estimation methods shows the highest error for the TRMM satellite, ranging from 2 to 5̊. The regression method showed the lowest MBE of zero among the studied methods. According to the error analysis results of methods used in this study (Tables 3 and 4), the regression method is the best temperature estimation method in Mazandaran Province. It can be concluded that using the auxiliary variable of the TRMM satellite increases the precision of temperature estimation methods while decreasing errors.
Table 3
RMSE and MBE values of interpolation method for monthly and annual average temperature data for 2012
TRMM
|
Cokrg
|
IDW
|
Regres
|
Month
|
RMSE
|
MBE
|
RMSE
|
MBE
|
RMSE
|
MBE
|
RMSE
|
MBE
|
4.7
|
-3.1
|
2.5
|
-0.09
|
2.8
|
-0.03
|
1.3
|
0
|
Jan
|
7.4
|
-5.5
|
3.8
|
-0.47
|
3.2
|
0.01
|
2.7
|
0
|
Feb
|
5.3
|
-3.9
|
2.7
|
-0.18
|
2.4
|
0.04
|
1.8
|
0
|
Mar
|
3.5
|
-1.2
|
3.1
|
-0.03
|
3.3
|
0.09
|
1.4
|
0
|
Apr
|
3.8
|
-0.2
|
4.4
|
-0.06
|
4.7
|
0.05
|
2.3
|
0
|
May
|
3.7
|
-1.5
|
3.9
|
0.08
|
4.3
|
0
|
1.7
|
0
|
June
|
3.4
|
-1.2
|
3.8
|
0.04
|
4.3
|
-0.40
|
1.7
|
0
|
July
|
3.9
|
-0.4
|
4.7
|
0.08
|
5.0
|
0
|
2.5
|
0
|
Agu
|
3.4
|
-2.1
|
3.0
|
0.05
|
3.4
|
-0.03
|
0.9
|
0
|
Sep
|
3.7
|
-1.8
|
3.4
|
0.06
|
3.7
|
-0.01
|
1.0
|
0
|
Oct
|
4.2
|
-2.6
|
3.1
|
0.01
|
3.5
|
0.01
|
0.9
|
0
|
Nov
|
4.6
|
-2.8
|
2.6
|
-0.03
|
2.9
|
-0.64
|
0.9
|
0
|
Dec
|
3.6
|
-2.2
|
2.8
|
0.02
|
3.1
|
0.01
|
0.8
|
0
|
annual
|
Table 4
RMSE and MBE values of interpolation method for monthly and annual average temperature data for 2013
TRMM
|
Cokrg
|
IDW
|
Regres
|
Month
|
RMSE
|
MBE
|
RMSE
|
MBE
|
RMSE
|
MBE
|
RMSE
|
MBE
|
4.4
|
-1.5
|
3.4
|
0.02
|
3.6
|
0.06
|
1.5
|
0
|
Jan
|
3.4
|
-1.4
|
3.2
|
-0.4
|
3.4
|
-1
|
1.3
|
0
|
Feb
|
3.6
|
-1.1
|
3.3
|
-0.3
|
3.2
|
-0.6
|
1.6
|
0
|
Mar
|
3.3
|
-2.0
|
2.2
|
-0.09
|
2.4
|
-0.2
|
0.9
|
0
|
Apr
|
4.2
|
-3.0
|
2.4
|
-0.1
|
2.8
|
-0.2
|
1.1
|
0
|
May
|
3.7
|
-2.7
|
2.6
|
-0.06
|
3.1
|
-0.2
|
1.1
|
0
|
June
|
2.9
|
-1.4
|
2.8
|
-0.05
|
3.3
|
-0.2
|
1.2
|
0
|
July
|
3.5
|
-2.6
|
2.7
|
-0.07
|
3
|
-0.2
|
0.9
|
0
|
Agu
|
2.9
|
-1.1
|
2.9
|
-0.05
|
3.3
|
-0.2
|
1.2
|
0
|
Sep
|
3.7
|
-2.1
|
2.7
|
-0.1
|
3.1
|
-0.2
|
0.7
|
0
|
Oct
|
4.1
|
-2.2
|
3.3
|
-0.1
|
3.7
|
-0.4
|
1.1
|
0
|
Nov
|
4.8
|
-3.2
|
3.1
|
0.04
|
3.2
|
0.04
|
1.3
|
0
|
Dec
|
3.4
|
-2.0
|
2.6
|
-0.1
|
3.01
|
-0.27
|
0.7
|
0
|
annual
|
d. Regression analysis of temperature estimation methods
Diagrams were plotted for 2012 and 2013 for the regression analysis of the methods used in this study. Similar results were obtained. To this end, the diagram for 2013 in Fig. 2 was used. In this diagram, the actual and estimated temperatures were plotted, and the line equation was fitted on data for different interpolation methods. According to the equations and resulting diagram, the regression method gives the best line of y = x as the closest line equation than other methods. The analysis results revealed the regression method as the top method.
e. Analysis of isothermal maps
Annual isothermal maps of 2012 and 2013 were plotted in ArcGIS to better compare and understand the performance of the regression method in correcting the TRMM satellite for estimating the annual temperature of Mazandaran Province. Equations in Table 2 were used to plot the regression map in Fig. 3.
Analyzing the map plotted by the IDW method shows that this method only gives a correct estimation in the vicinity of meteorological stations. Due to not using the auxiliary variable, a suitable isothermal map was not obtained compared to other methods, not showing a correct spatial distribution of temperature, particularly in Mazandaran highlands. Despite better temperature distribution in coasts and highlands from the isothermal map obtained from the CK method compared to the IDW method, it gives no correct understanding of temperature distribution of coastal regions, not properly showing the temperature distribution of the western regions of Mazandaran province because of its complex topography. The isothermal map plotted by the regression method properly displays the temperature range of Alborz highlands. This isothermal map also properly shows the temperature algorithm of Caspian coasts so that temperature increases from eastern coasts to Nowshahr. Maps generated by this method consider a very good spatial distribution, especially in the west of Mazandaran Province, showing a good temperature distribution. These two maps show that spatial temperature variations are almost the same, and temperature increases in the eastern part of Mazandaran Province. In total, maps obtained from different methods showed the higher precision and accuracy of the regression method than other methods, displaying the temperature distribution of Mazandaran Province more properly. Accordingly, the regression method is the best method for spatial temperature estimation and map plotting, indicating increased precision of the methods using the auxiliary variable of the satellite image than other methods.