The goal of image reconstruction is obtaining an image of the object of interest from indirectly measured, and typically noisy, data. Mathematically this is formulated as an inverse problem, where the forward operator models data acquisition. In practice, not only the data are noisy, but also the forward operator is often not perfectly known as it may involve imperfect calibration measurements or simplified models. We present an approach to inverse problems with imperfect forward models that relies on partially ordered functional spaces - Banach lattices. A typical example of such an inverse problem is image deblurring, where the blurring kernel is not perfectly known. Failure to acknowledge the errors in the blurring kernel may lead to reconstruction artifacts that can be eliminated by modifying the reconstruction algorithms to account for errors in the operator.