  Avalanche statistics of various threshold activated dynamical systems are
known to depend on the magnitude of the drive, or stress, on the system. Such
dependencies exist for earthquake size distributions, in sheared granular
avalanches, laboratory scale fracture and also in the outage statistics of
power grids. In this work we model threshold-activated avalanche dynamics and
investigate the time required to detect local variations in the ability of
model elements to bear stress. We show that the detection time follows a
scaling law where the scaling exponents depend on whether the feature that is
sought is either weaker, or stronger, than its surroundings. We then look at
earthquake data from Sumatra and California, demonstrate the trade-off between
the spatial resolution of a map of earthquake exponents i.e. the $b$-values of
the Gutenberg-Richter law) and the accuracy of those exponents, and suggest a
means to maximise both.
