## 4 Discussion  
From the experimental results of the proposed model verification, it can be seen that as long as the watermarking algorithm is consistent with the fundamental principles, the algorithmic robustness against image cropping attack can be determined by the proposed model theoretically. According to the properties of the proposed model, the calculation process has no relationship to experimental data, and avoids the computational resource cost of experiments. Hence, it provides a theoretical method to evaluate the robustness against image cropping attack. In specific applications of watermarking remote sensing images, the model helps the copyright owner consider how much of the image may be cropped and decide the parameters of watermarking algorithms. Especially for the massive data providers, the theoretical calculation saves much energy and cost compared to practical experiments.  
Further applicability improvements have also been made by the deduction experiments based on the proposed model. The relationships among robustness index, watermark information length and watermark repeat times have been investigated and revealed by these experiments. Thus, these enlightenments can be used to facilitate the design of watermarking algorithms which obey the proposed watermarking principles. For example, considering the fidelity requirement of remote sensing images, the total embedded watermark bits, which equals \(L\times N\), are strictly restricted. To increase the robustness against image cropping for existing watermarking algorithms, it is significant to increase \(N\) and decrease \(L\) without causing prominent distortions at the same time. Another example may be when watermarking remote sensing images in the form of big data. In this situation, larger \(L\), which is necessary for copyright identification, will increase the complexity and the time of robustness computation. From the experimental results, smaller \(L\) can be used to reduce the computation time and provide the efficient method to accurately estimate the robustness.  
Another potential application of the proposed model may be the promotion on research on robustness computation models against other kinds of attacks. The analysis of other image attacks needs further and detailed study, hence some views discussed here are incomplete. Generally, image attacks can be divided into two categories: content/pixel attacks and geometric transformation attacks. For example, the attacks of contents include filtering, JPEG compression (a standard created by the Joint Photographic Experts Group), noise addition and so on, while the attacks of geometric transformation include rotation, scaling, other affine transformations and so on. We note that the process of geometric attacks, as the compound of image geometric transformation and image contents resampling, seem to be more complicated in robustness analysis comparing with attacks purely on contents. Among the attacks mentioned above, the analysis principles of image cropping attack are simpler as watermark synchronization of the remaining part keep correct. Further study may be conducted on the basis of principles proposed in this paper and combined with the characteristics of various attacks.