Glacier surface elevation responds to a variety of localized processes
occurring beneath the ice. Subglacial-lake volume change in particular
is inherently time-dependent, producing time-varying perturbations in
ice-surface elevation. Here, we introduce inverse methods for
quantifying time-varying subglacial perturbations from altimetry data
and, when available, horizontal surface velocity data. The forward model
is based on a small-perturbation approximation of the Stokes equations
that is solved efficiently with Fourier transform methods. The inverse
methods are derived from variational least-squares optimization problems
and the associated normal equations are solved with the conjugate
gradient method. We conduct synthetic tests for reconstructing
time-varying basal vertical velocity and drag perturbations that are
motivated by subglacial-lake activity and slippery spots beneath
Antarctic ice streams. We show that incorporation of horizontal surface
velocity data as additional constraints can refine altimetry-based
inversions or facilitate reconstruction of multiple fields, depending on
whether the data are spatially discrete or continuous. We further
validate the method by showing that it can reconstruct basal
perturbations from synthetic elevation data that are produced by a
nonlinear subglacial lake model. With the advent of high spatial and
temporal resolution altimetry data from NASA's ICESat-2 mission, these
inverse methods will facilitate further assessment of the relation
between ice-sheet flow and subglacial processes.