Unresolved Doppler velocity measurements are not homogenous across the solar disc (Brookes et al. 1978). We consider one cause of the inhomogeneity that originates from the BiSON instrumentation itself: the intensity of light observed from a region on the solar disc is dependent on the distance between that region on the image of the solar disc formed in the instrument and the detector. The non-uniform weighting affects the realization of the solar noise and the amplitudes of the solar oscillations observed by a detector. An offset velocity, which varies with time, is observed in BiSON data and has consequences for the long-term stability of observations. We have attempted to model, in terms of the inhomogeneous weighting, the average observed offset velocity.