•Derivation of a robust Likelihood Ratio (LR) in case of detection with a small noise mismatch between the hypotheses.•Derivation of a robust Adaptive Matched Filter (AMF).•Pointing out differences with standard detection schemes using Matched Filter Residual (MFR) plots.•Performance improvement based on a hyperspectral imaging detection benchmark. One of the main issue in detecting a target from an hyperspectral image relies on properly identifying the background. Many assumptions about its distribution can be advocated, even if the Gaussian hypothesis prevails. Nevertheless, the huge majority of the resulting detection schemes assume that the background distribution remains the same whether the target is present or not. In practice, because of the spectral variability of the target and the non-linear mixing with the background radiance, this hypothesis is not strictly true. In this paper, we consider that an unknown background mismatch exists between the two hypotheses. Under the assumption that this mismatch is small, we derive an approximation of the Likelihood Ratio for the problem at hand. This general formulation is then applied to the case of Gaussian distributed background, leading to a robust Adaptive Matched Filter. The behaviour of this new detector is analysed and compared to popular detectors. Numerical simulations, based on real data, show the possible improvement in case of target signature mismatch.