  Topological simplification of scalar and vector fields is well-established as
an effective method for analysing and visualising complex data sets. For
multi-field data, topological analysis requires simultaneous advances both
mathematically and computationally. We propose a robust multivariate topology
simplification method based on ``lip''-pruning from the Reeb Space.
Mathematically, we show that the projection of the Jacobi Set of multivariate
data into the Reeb Space produces a Jacobi Structure that separates the Reeb
Space into simple components. We also show that the dual graph of these
components gives rise to a Reeb Skeleton that has properties similar to the
scalar contour tree and Reeb Graph, for topologically simple domains. We then
introduce a range measure to give a scaling-invariant total ordering of the
components or features that can be used for simplification. Computationally, we
show how to compute Jacobi Structure, Reeb Skeleton, Range and Geometric
Measures in the Joint Contour Net (an approximation of the Reeb Space) and that
these can be used for visualisation similar to the contour tree or Reeb Graph.
