  This paper introduces variational design methods that are novel to
Geophysics, and discusses their benefits and limitations in the context of
geophysical applications and more established design methods. Variational
methods rely on functional approximations to probability distributions and
model-data relationships. They can be used to design experiments that best
resolve either all model parameters, or the answer to specific questions about
the system to be interrogated. The methods are tested in three schematic
geophysical applications: (i) estimating a source location given arrival times
at sensor locations, and (ii) estimating the contrast in seismic wavefield
velocity across a stratal interface given measurements of the amplitudes of
seismic wavefield reflections from that interface, and (iii) designing a survey
to best constrain CO2 saturation in a subsurface storage scenario. Variational
methods allow the value of an experiment to be calculated and optimised
simultaneously, which results in substantial savings in computational cost. In
the context of designing a survey to best constrain CO2 in a subsurface storage
scenario, we show that optimal designs may change substantially depending on
the questions of interest. Overall, this work indicates that optimal design
methods should be used more widely in Geophysics, as they are in other
scientifically advanced fields.
