  Given a zero-mean Gaussian random field with a covariance function that
belongs to a parametric family of covariance functions, we introduce a new
notion of likelihood approximations, termed truncated-likelihood functions.
Truncated-likelihood functions are based on direct functional approximations of
the presumed family of covariance functions. For compactly supported covariance
functions, within an increasing-domain asymptotic framework, we provide
sufficient conditions under which consistency and asymptotic normality of
estimators based on truncated-likelihood functions are preserved. We apply our
result to the family of generalized Wendland covariance functions and discuss
several examples of Wendland approximations. For families of covariance
functions that are not compactly supported, we combine our results with the
covariance tapering approach and show that ML estimators, based on
truncated-tapered likelihood functions, asymptotically minimize the
Kullback-Leibler divergence, when the taper range is fixed.
