Question

The table provides the X- and Y-coordinates of the points, measured in row and column of a raster with a cell size of 1 meter, and their known values. Using inverse distance weighted (IDW) interpolation method and Euclidean distance, the interpolated value at Point 0 is ................. (Rounded to 2 decimal places). A constant rate of change in value between points should be assumed. 

 

Point	$X$	$Y$	Value
1	69	76	27
2	59	67	10
3	74	79	13
0	69	67	?

 

Hint:

If the power $p$ is not specified for IDW, use $p=1$ (pure inverse distance). The formula is \(\displaystyle \hat z(\mathbf{x})=\frac{\sum_i \frac{z_i}{d_i^p}}{\sum_i \frac{1}{d_i^p}}\). Always compute distances first; normalization by $\sum w_i$ avoids unit issues.

Answer 1

Step 1: Distances from point 0 $(69,67)$ (Euclidean). \[ \begin{aligned} d_1 &= \sqrt{(69-69)^2+(76-67)^2} = 9, \\ d_2 &= \sqrt{(59-69)^2+(67-67)^2} = 10, \\ d_3 &= \sqrt{(74-69)^2+(79-67)^2} = 13. \end{aligned} \] Step 2: IDW weights (power $p=1$, i.e. “inverse distance”). \[ w_i = \frac{1}{d_i}, \quad w_1 = \tfrac{1}{9}, \quad w_2 = \tfrac{1}{10}, \quad w_3 = \tfrac{1}{13}. \] Step 3: IDW estimate. \[ \hat{z}_0 = \frac{\sum w_i z_i}{\sum w_i} = \frac{\tfrac{27}{9} + \tfrac{10}{10} + \tfrac{13}{13}}{\tfrac{1}{9} + \tfrac{1}{10} + \tfrac{1}{13}} = \frac{3 + 1 + 1}{0.11111 + 0.1 + 0.07692} = \frac{5}{0.28803} = 17.359 \approx \boxed{17.36}. \]