  Spatial process models popular in geostatistics often represent the observed
data as the sum of a smooth underlying process and white noise. The variation
in the white noise is attributed to measurement error, or micro-scale
variability, and is called the "nugget". We formally establish results on the
identifiability and consistency of the nugget in spatial models based upon the
Gaussian process within the framework of in-fill asymptotics, i.e. the sample
size increases within a sampling domain that is bounded. Our work extends
results in fixed domain asymptotics for spatial models without the nugget. More
specifically, we establish the identifiability of parameters in the Mat\'ern
covariance function and the consistency of their maximum likelihood estimators
in the presence of discontinuities due to the nugget. We also present
simulation studies to demonstrate the role of the identifiable quantities in
spatial interpolation.
