  Finite volume schemes for hyperbolic balance laws require a piecewise
polynomial reconstruction of the cell averaged values, and a reconstruction is
termed `well-balanced' if it is able to simulate steady states at higher order
than time evolving states. For the shallow water system this involves
reconstructing in surface elevation, to which modifications must be made as the
fluid depth becomes small to ensure positivity, and for many reconstruction
schemes a modification of the inertial field is also required to ensure the
velocities are bounded.
  We propose here a reconstruction based on a convex combination of surface and
depth reconstructions which ensures that the depth increases with the cell
average depth. We also discuss how, for cells that are much shallower than
their neighbours, reducing the variation in the reconstructed flux yields
bounds on the velocities. This approach is generalisable to high order schemes,
problems in multiple spacial dimensions, and to more complicated systems of
equations. We present reconstructions and associated technical results for
three systems, the standard shallow water equations, shallow water in a channel
of varying width, and a shallow water model of a particle driven current.
Positivity preserving time stepping is also discussed.
