Generalised model-independent characterisation of strong gravitational
  lenses VII: impact of source properties and higher-order lens properties on
  the local lens reconstruction
  We investigate the impact of higher-order gravitational lens properties and
properties of the background source on our approach to directly infer local
lens properties from observables in multiple images of strong gravitationally
lensed extended, static background sources developed in papers I to VI. As the
degeneracy between local lens and source properties only allows to determine
relative local lens properties between the multiple image positions, we cannot
distinguish common scalings and distortions caused by lensing from intrinsic
source characteristics. The consequences of this degeneracy for lens modelling
and our approach and ways to break it are detailed here. We also set up
quantitative measures around the critical curve to find clear limits on the
validity of the approximation that source properties are negligible to infer
local lens properties at critical points. The impact of the source on the local
lens properties depends on the reduced shear at the image position and the
amplitude and orientation of the source ellipticity, as we derive in this
paper. Similarly, we investigate the role of third-order lens properties
(flexion), in two galaxy-cluster simulations and in the Lenstool-reconstruction
of the galaxy-cluster lens CL0024. In all three cases, we find that flexion is
negligible in over 90% of all pixels of the lensing region for our current
imprecision of local lens properties of about 10%. Decreasing the imprecision
to 2%, higher-order terms start to play a role, especially in regions with
shear components close to zero.
