  We consider the scalar conservation law in one space dimension with a
genuinely nonlinear flux. We assume that an appropriate velocity function
depending on the entropy solution of the conservation law is given for the
comprising particles, and study their corresponding trajectories under the
flow. The differential equation that each of these trajectories satisfies
depends on the entropy solution of the conservation law which is typically
discontinuous in both time and space variables. The existence and uniqueness of
these trajectories are guaranteed by the Filippov theory of differential
equations. We show that such a Filippov solution is compatible with the front
tracking and vanishing viscosity approximations in the sense that the
approximate trajectories given by either of these methods converge uniformly to
the trajectories corresponding to the entropy solution of the scalar
conservation law. For certain classes of flux functions, illustrated by traffic
flow, we prove the H\"older continuity of the particle trajectories with
respect to the initial field or the flux function. We then consider the inverse
problem of recovering the initial field or the flux function of the scalar
conservation law from discrete pointwise measurements of the particle
trajectories. We show that the above continuity properties translate to the
stability of the Bayesian regularised solutions of these inverse problems with
respect to appropriate approximations of the forward map. We also discuss the
limitations of the situation where the same inverse problems are considered
with pointwise observations made from the entropy solution itself.
