We suggest a sequential, or “tracking,”algorithm
for estimating a smooth fault line in a response surface. The method starts
with an approximation to a point on the line, and from there the line is
tracked as it meanders through the plane. The technique differs from recent
approaches in that it does not require a large part of the plane to be searched
for evidence of a fault line. This offers potential computational savings, and
produces a method that is invariant under rotations of coordinate axes (except
insofar as a rotation might affect the estimated starting point, and the
relative orientation of the grid on which calculations are done). That feature
is important if design points are not located on a regular grid. We investigate
properties of the method under very general conditions on the design, allowing
Poisson cluster processes, jiggled grid processes and deterministic, regular
lattices. Uniform rates of convergence are derived in all these settings, for
the case of noisy data, and shown to be within logarithmic factors of optimal
pointwise convergence rates in the no-noise setting.