  Model calibration or data inversion is one of fundamental tasks in
uncertainty quantification. In this work, we study the theoretical properties
of the scaled Gaussian stochastic process (S-GaSP), to model the discrepancy
between reality and imperfect mathematical models. We establish the explicit
connection between Gaussian stochastic process (GaSP) and S-GaSP through the
orthogonal series representation. The predictive mean estimator in the S-GaSP
calibration model converges to the reality at the same rate as the GaSP with a
suitable choice of the regularization and scaling parameters. We also show the
calibrated mathematical model in the S-GaSP calibration converges to the one
that minimizes the $L_2$ loss between the reality and mathematical model,
whereas the GaSP model with other widely used covariance functions does not
have this property. Numerical examples confirm the excellent finite sample
performance of our approaches compared to a few recent approaches.
