## 1 Introduction  
Currently, remote sensing images are widely used in scientific research and various industries, including agriculture, vegetation monitoring, ecology and so on [1; 2; 3]. However, security issues inhibit the convenience of remote sensing image sharing and transmission [4], especially in the big data era [5]. Robust watermarking technology has become a promising solution, which could be a viable form of protection regarding issues such as user tracking, forgery prevention and many other illegal uses [6; 7; 8; 9; 10; 11]. The major property of these watermarking algorithms, named robustness, represents the resistance towards different types of attack [12; 13; 14]. As the image cropping operation is the common processing method towards remote sensing images, the watermark algorithmic robustness against image cropping attack is evaluated in almost every watermarking algorithm study [10; 11; 15; 16]. In fact, this type of robustness determines whether the copyright can be successfully identified after cropping a certain amount of remote sensing image. It is common to verify the watermarking algorithmic robustness against image cropping attacks by experimentation, both for common images and remote sensing images. From the perspective of computational complexity, the experimental methods can be divided into three types: robustness verification, algorithm comparisons, and criteria systems. The robustness verification experiments are mainly conducted after the watermarking algorithms have been proposed by performing some attacks [8; 10; 11; 17; 18; 19; 20; 21; 22; 23]. Normalized correlation (\(NC\)) [10; 17; 23] and bit error rate (\(BER\)) [17; 24] have been used as the evaluation indexes, which are calculated by comparing the extracted copyright information with the original. Thus the watermarking algorithmic robustness is determined conveniently and directly in this method. To demonstrate the robustness in a more reliably persuasive way, algorithm comparisons verify the robustness result of those proposed watermarking algorithms and others [8; 18; 21; 22; 23]. In addition, the robustness criteria systems and platforms, which defined a variety of sample data, attacking types (including image cropping attack), attacking strength and other parameters, have been studied and published. These criteria systems, including \"Stirmark\" [25; 26], Watermark Evaluation Testbed (WET) [27] and Information Hiding and its Criteria for Evaluation (IHC) [28], strive to extend the range of experimental parameters and conditions to more accurately estimate the robustness of the watermarking algorithms. For example, resistance to image cropping attack is considered as the most basic and low-level type of robustness in the Stirmark Benchmark evaluation system [26], and image databases with strategies of repeated tests have been adopted [29]. Unfortunately, as indicated above, it is obvious that the experimental method of evaluating robustness consumes many computational resources and much time. With the different parameters and experimental conditions, the computational resources and time used to evaluate the robustness will thus, increase further, especially for the remote sensing images in the big data era. Further, the experiments cannot cover all kinds of conditions as the parameters related to sample data, attacking type, attacking strength and so on. Thus, the robustness of the experimental results can only be used to approximate robustness ability and cannot, in theory, deduce the robustness index value. The significance to theoretically analyze and calculate the robustness of watermarking algorithms against image cropping attack is believed to be important and therefore necessary for further study. Compared to research on watermark embedding and extraction strategies, not much attention has been paid to robustness analysis and its calculation methods.