  Calibration of expensive computer models with high-dimensional output fields
can be approached via history matching. If the entire output field is matched,
with patterns or correlations between locations or time points represented,
calculating the distance metric between observational data and model output for
a single input setting requires a time intensive inversion of a
high-dimensional matrix. By using a low-dimensional basis representation rather
than emulating each output individually, we define a metric in the reduced
space that allows the implausibility for the field to be calculated
efficiently, with only small matrix inversions required, using projection that
is consistent with the variance specifications in the implausibility. We show
that projection using the $L_2$ norm can result in different conclusions, with
the ordering of points not maintained on the basis, with implications for both
history matching and probabilistic methods. We demonstrate the scalability of
our method through history matching of the Canadian atmosphere model, CanAM4,
comparing basis methods to emulation of each output individually, showing that
the basis approach can be more accurate, whilst also being more efficient.
