  We present new tests to identify stationary position-dependent additive shear
biases in weak gravitational lensing data sets. These tests are important
diagnostics for currently ongoing and planned cosmic shear surveys, as such
biases induce coherent shear patterns that can mimic and potentially bias the
cosmic shear signal. The central idea of these tests is to determine the
average ellipticity of all galaxies with shape measurements in a grid in the
pixel plane. The distribution of the absolute values of these averaged
ellipticities can be compared to randomized catalogues; a difference points to
systematics in the data. In addition, we introduce a method to quantify the
spatial correlation of the additive bias, which suppresses the contribution
from cosmic shear and therefore eases the identification of a
position-dependent additive shear bias in the data. We apply these tests to the
publicly available shear catalogues from the Canada-France-Hawaii Telescope
Lensing Survey (CFHTLenS) and the Kilo Degree Survey (KiDS) and find evidence
for a small but non-negligible residual additive bias at small scales. As this
residual bias is smaller than the error on the shear correlation signal at
those scales, it is highly unlikely that it causes a significant bias in the
published cosmic shear results of CFHTLenS. In CFHTLenS, the amplitude of this
systematic signal is consistent with zero in fields where the number of stars
used to model the PSF is higher than average, suggesting that the
position-dependent additive shear bias originates from undersampled PSF
variations across the image.
