  We propose an interdependent random geometric graph (RGG) model for
interdependent networks. Based on this model, we study the robustness of two
interdependent spatially embedded networks where interdependence exists between
geographically nearby nodes in the two networks. We study the emergence of the
giant mutual component in two interdependent RGGs as node densities increase,
and define the percolation threshold as a pair of node densities above which
the giant mutual component first appears. In contrast to the case for a single
RGG, where the percolation threshold is a unique scalar for a given connection
distance, for two interdependent RGGs, multiple pairs of percolation thresholds
may exist, given that a smaller node density in one RGG may increase the
minimum node density in the other RGG in order for a giant mutual component to
exist. We derive analytical upper bounds on the percolation thresholds of two
interdependent RGGs by discretization, and obtain $99\%$ confidence intervals
for the percolation thresholds by simulation. Based on these results, we derive
conditions for the interdependent RGGs to be robust under random failures and
geographical attacks.
