In this research, the monthly mean of geopotential height [in meters]for the past 13 years, obtained from Atmospheric Infrared Sounder [AIRS]level 3, was used. AIRS was the instrument on AQUA satellite, which launched in May 2002.

This satellite has very high spectral resolutions: e.g., it captures climate data through nearly 2382 bands in the electromagnetic spectrum and its geopotential height product is very high resolution 0.5˚ × 0.5˚ grid cell. Version 6 of its product contains fewer biases in geopotential height [5]. Besides good quality of climate data, GESDISC¹ provides geopotential height data for the whole global.

Randomly 30% of the 324 samples were missing data which were then estimated from the 70% known data using different interpolation techniques like IDW, NN, BI and NI [2][7]. Robeson [21]; Price, McKenney [22]; Perry and Hollis [7]; Yozgatligil, Aslan [3]considered these performance parameters like, Mean Absolute Error [MAE], Root Mean Square Error [RMSE], Coefficient of Determinations [R²]and Correlation Coefficient [Corr], to find out the best interpolation technique for missing climatic data set.

1) INVERSE DISTANCE WEIGHTING

This imputation resembles to Tobler’s first law of geography in which the weight of the known samples will be determined based on the distances from the imputed sample [Robeson, 1994]. More will be the distance of neighbors from a predicted sample less will be their weight in interpolation. Ferrari and Ozaki [9]used Equation (1) which is given below:

z i j = ∑ a = 1 n z o a d a   j − r ∑ a = 1 n d a   j − r (1)

where d a j − r is the weighting factor of distance between the a^th original neighbor sample z o i , z i j is j^th the point to be estimated, n is the total number of the sample used, and r weighting factor. Langella [23], formula for IDW was used in the missing data imputations.

2) NEAREST NEIGHBORS INTERPOLATION [NN]

Missing values were directly imputed with a most suitable neighbor around the missing sample [24][25]in this interpolation technique.

3) BILINEAR INTERPOLATION [BI]

Junninen et al. [2004]used Equations (2) and (3) for Bilinear Interpolations

z i = z i 1   + m ( z o + z o 1 ) (2)

m = z i 2 − z i 1 z o 2 − z o 1 z o 1 < z o < z o 2     and     z i 1   < z i < z i 2 (3)

It was a linear equation with ( z o 1 ,   z I 1 ) and ( z o 2 ,   z i 2 ) sample values, m being a gradient of this line.

4) NATURAL NEIGHBORS INTERPOLATION [NI]

This spatial interpolation gives the nearest neighbor value of the sample to the missing geopotential height. D. and Boissonnat and Cazals [25]explain the selection of such natural neighbors for randomly missing data being on Delaunay triangulation.

These following performance parameters have been frequently used by Robeson [21]; Price, McKenney [22]; Junninen, Niska [2]; Perry and Hollis [7]; Stahl, Moore [24]; Norazian [4]; Ferrari and Ozaki [9]; Saleem and Ahmed [18]for imputation of missing climate data set.

1) ROOT MEAN SQUARE ERROR [RMSE]

Root Mean Square was calculated by dividing the sum of the square of the difference between imputed geopotential heights and actual value with the total number of samples, and then finally taking the square root of this term [4]. Smaller values indicate a perfect estimation of missing data set. Equation (4) was its mathematical formula used in this research.

R M S E = ( 1 n ∑ a = 1 n [ z o a   − z i a ] 2 ) 1 2 (4)

This parameter calculates the total difference [±]between original and interpolated geopotential height.

2) MEAN ABSOLUTE ERROR [MAE]

This provides more information about the residual error as compared with RMSE. Junninen, Niska [2]and Norazian [4]provided Equation (5) for MAE.

M A E = 1 n ∑ a = 1 n | z a i   − z i a | (5)

MAE value range from 0 to ∞ . Its value close to 1 indicates more accurate and perfect imputation of missing data set.

3) CORRELATION COEFFICIENT [Corr]

Its value of +1 indicates very strong correlation and near to 0 signifies a bad correlation between actual and predicted geopotential height. Equation (6) was used for the correlation coefficient in this research.

c o r r   =     cov ( z i   , z o ) ∂ z i ∂ z O (6)

In Equation (6) nominator represents covariance while denominator represents the product of their standard deviations in the data set.

4) COEFFICIENT OF DETERMINATION [R²]

This parameter provides a degree of correlation between the actual and predicted sample geopotential height [1]which varies between 0 and 1. Noor, Abdullah [4], suggested values closer to 1 indicate a perfect fit for the data set. Rahman and Islam, [2011]used the following formula for R².

R 2 = [ 1 n   ∑ a = 1 n ( z i a − A i ) ( z o a − A o ) ∂ z i ∂ z o ] (7)

In Equation (7), A_i was the average of predicted samples and A_o is the average of sample values before prediction.

On all pressure level IDW showed very biased results. IDW produced highest RMSE ± 14.45 m over 1 hPa while lowest value of this error was ±3.66 m at 925 hPa. Actual and predicted values indicating low quality of interpolation for missing values of geopotential height with IDW as correlation coefficient was very low (Table 1).

RMSE value remains between ±4.925 and ±11.369 m with Nearest Neighbor Interpolations. Such a large RMSE, poor correlation, and poor fit to the surface indicated bad refilling of data with this interpolation technique (Table 2).

Bilinear Interpolation appeared to be relatively better as compared to the above mentioned two interpolations. RMSE was ±2.461 to ±5.241 m in refilling of gaps in data up to 1000 hPa. MAE remains less than 1 and strong correlation (0.98) was found in the imputation of geopotential height. Coefficient of Determination was close to 0.98 for imputation over 1, 1.5, 2, 3, 5, 7, 10, 15, 70, 100, 150, 200, 250, 300 hPa (Table 3).

Reasonable low RMSE come in refilling of geopotential height over 2, 3, 5, 7, 30, 50, 70, 200, 250, 400, 500, 600 hPa. Largest RMSE was ±5.10 m at 10 hPa and lowest RMSE ±2.2 m for refilling of gaps in data at 850, 925, 1000 hPa. A good correlation coefficient [near to 0.99]was come in the refilling of geopotential height. R² was near to 1 concluding a good line of fit between actual and predicted data set (Table 4).