  We develop direct inversion methods for inferring the rheology of a fluid
from observations of its shallow flow. First, the evolution equation for the
free-surface flow of an inertia-less current with general constitutive law is
derived. The relationship between the volume flux of fluid and the basal
stress, $\tau_b$ is encapsulated by a single function $F(\tau_b)$, which
depends only on the constitutive law. The inversion method consists of (i)
determining the flux and basal stress from the free-surface evolution, (ii)
comparing the flux to the basal stress to constrain $F$ and (iii) inferring the
constitutive law from $F$. Examples are presented for both steady and transient
free-surface flows demonstrating that a wide range of constitutive laws can be
directly obtained. For flows in which the free-surface velocity is known, we
derive a different method, which circumvents the need to calculate the flux.
