Question$-1(50$ points$)$

Consider the figure given below:

$00000000000$

$z_1=20$

$z_2=10$

$10$ points are distributed in space with equal spacing. There is a distance of $1$ units between each consecutive point. The solid points on the very left and on the very right represent data points. The data are given in the figure. Perform the following kriging operations with a spherical variogram that has a sill of $10$ and a correlation range of $2$ units.

a$-)(10$ points$)$ Determine the ordinary kriging estimates and the ordinary kriging estimation variances for the $8$ points given in the figure $($for the empty circles$).$

b$-)(10$ points$)$ Determine the simple kriging estimates and the simple kriging estimation variances for the points. Use a mean of $40$ when you are performing simple kriging.

c$-)(10$ points$)$ Determine the simple kriging estimates and the simple kriging estimation variances for the points. Use the mean you obtain from the data when you are performing the simple kriging.

d$-)(20$ points$)$ Comment on the results for parts a$-$c$.$

Question$-2(50$ points$)$

a$-)(25$ points$)$ Perform simple kriging using the data given in the file "data.dat" using the GSLIB program $\text{'}$kt$3$d$\text{'}$ or any other geostatistical software. You should be performing the simple kriging for the $50501$ grid we have been working on so far. Please perform the simple kriging using a spherical variogram with no nugget contribution, a sill that is equal to the variance of the data and a correlation range of $10$ blocks. Assume that the variogram is isotropic.

b$-)(25$ points$)$ Compute the variogram for the kriging estimate that you have obtained in part$-$a$.$ You can accomplish this by using the GSLIB program "gam" or any other geostatistical software that you like. Comment on the results.