Shadowing of an imaging aperture occurs when ultrasound beams are partially obstructed by an acoustically hard tissue, e.g., bone tissue. This effect leads to reduced resolution and, in some cases, geometrical distortion. In this paper, we initially introduce a binary apodization model to simulate effects of the shadowing on the point scatterers located close to a bone structure. Further, in a simulation study and an in vitro experiment, the minimum variance (MV) beamforming method is employed to image scatterers partly located in the shadow of bone. We show that the MV beamformer can result in a distorted image when the imaging aperture is highly obstructed by the bone structure. This distortion can be seen as an apparent lateral shift of the point spread function and a decrease in the sensitivity. Based on the signal power across the aperture, we adaptively determine the shadowed elements and discard their corresponding data from the covariance matrix to improve the MV beamformer performance. This modified MV beamformer can retain the resolution and compensate for the apparent lateral shifting and signal attenuation for the shadowed point scatterers.