3.13 Repeat Exercise 3.12 with the same test function ????(h), but this time using the linear intrinsic covariance function ????I(h)=−|h|, where ????I(h)

is defined up to an additive constant. As before, define the regularized intrinsic covariance function ????I,????(h) by (3.67).

(a) Show that

????I,????(h) = {1 3

|h|

3 − h2 − 1 3

, 0 ≤ |h| ≤ 1,

−|h|, |h| ≥ 1.

(b) The semivariogram for the original and regularized random fields are defined by ????(h) = ????I(0) − ????I(h) and ????????(h) = ????I,????(0) − ????I,????(h), respectively. Note they do not depend on the arbitrary additive constant in the definition of ????I(h). Show that ????(h) ∼ |h| and ????????(h) ∼ h2 as |h| → 0.

Hence, conclude that ????????(h) is smoother at the origin than ????(h).