3.2.3 Inverse Distance Weighted
IDW was first proposed by Shephard (1961), he stated this method to be a deterministic. This method assumes that observation points farther away from the estimation points have less impact over the estimated value while the observation points nearer to estimated points have more impact on the estimated value. Hence it can be seen that the estimated value is inversely proportional to the distance of observed value. IDW function uses a parameter defined as weight (Wj), which assigns the impact of an observed station to any other point of measurement. The parameter (Wj) uses a parameter p, which act as a decay function. As the value of p increases the weight of the stations which are farther decreases. For p=0, the predicted value will be the arithmetic mean of the stations in the search neighbourhood because the weight of all stations become equal for p=0. For very high value of p, the values are estimated taking into effect only the immediate surrounding stations. In this paper the values are estimated using the p=2, 3 and 4.
If r (x, y) is any arbitrary point within the region of interpolation then interpolated value is given by:
\(P\left(r\right)=\sum_{j=1}^{N}{W(r_{j})(f(r_{j})}\) 2
where P (r) = interpolated value at
W (rj)= \(\frac{d_{j}r^{-p}}{\sum_{j=1}^{N}{d_{j}r^{-p}}}\) 3
Where dj(r) =\(\sqrt{{(x-x_{j})}^{2}+{(y-y_{j})}^{2}}\) distance betweenr and rj; p = decay determining parameter.
A comprehensive discussion has been made by Kravchenko and Bullock (1999) for the choice of parameter p, the effect of weight parameter for IDW interpolation is discussed by Cecilio and Pruski (2003), a similar discussion has been made by Vicento-Serrano (2003) over the importance of weight parameter in IDW function for prediction model.