  We investigate the regional gradient observability of fractional
sub-diffusion equations involving the Caputo derivative. The problem consists
of describing a method to find and recover the initial gradient vector in the
desired region, which is contained in the spacial domain. After giving
necessary notions and definitions, we prove some useful characterizations for
exact and approximate regional gradient observability. An example of a
fractional system that is not (globally) gradient observable but it is
regionally gradient observable is given, showing the importance of regional
analysis. Our characterization of the notion of regional gradient observability
is given for two types of strategic sensors. The recovery of the initial
gradient is carried out using an expansion of the Hilbert Uniqueness Method.
Two illustrative examples are given to show the application of the developed
approach. The numerical simulations confirm that the proposed algorithm is
effective in terms of the reconstruction error.
