## 5 Conclusions  
In this study, we have described the underdetermined problem that arises when inverting the inner part values using a linear spectral mixing model. Through simulation studies and a case study with a class map, we have provided a detailed illustration of this problem. We have concluded that the main reason for encountering the most underdetermined cases is the imperfect approximation of land surface using current regular window-based methods. To address this issue, we have proposed an iterative approximation strategy that focuses on local mixed pixels and their adjacent regions. This strategy aims to provide a solution to the underdetermined problem by gradually approaching the solution instead of attempting to solve it in a single step when dealing with mixed pixels and their neighborhoods. As an example of this strategy, we have developed an elastic-window-based algorithm that can handle mixed pixels that may be involved in linear systems to prevent these systems from encountering the underdetermined problem.  
Based on a statistical analysis of sixty land cover maps from around the world, the results have shown that the underdetermined problem exists at a relatively low level (ranging from dozens to hundreds in every ten thousand mixed pixels) in all of these areas. However, the severity of the problem tends to increase to some extent with the increase in mixed scales. A comparison between the single-window-based method and the proposed algorithm has demonstrated that the proposed algorithm can reduce the proportions of mixed pixels that encounter the underdetermined problem. However, it should be noted that the proposed algorithm requires more time compared to the single window-based method, mainly due to the iterative approximation of local mixed pixels. Ensuring that equations do not result in an underdetermined problem is crucial for successfully inverting linear systems.  
The presented strategy and algorithm can have potential applications in various areas that require the inversion of inner part values directly--for example, downscaling high-temporal-resolution images or time series of derived indices (such as NDVI) to obtain high-spatial-resolution time sequences based on linear spectral unmixing. Additionally, it can provide an alternative approach for parameter inversions related to local land surface approximation, with mixed pixels as the computational unit. However, it is important to note that although the experiment has been tested with ten mixed scales based on 30 m spatial resolution land cover maps, further studies are needed considering various land cover maps with different spatial resolutions and classification systems. Future research should focus on improving the accuracy of unmixing in these scenarios.  
Conceptualization, B.L. and Y.Z.; Methodology, B.L.; Writing--Original Draft Preparation, B.L.; Funding Acquisition, Y.Z. All authors have read and agreed to the published version of the manuscript. This research was supported by Fundamental Research Program of Shanxi Province, grant number 202203021212496; the Shanxi Province Higher Education Science and Technology Innovation Plan Project, grant number 201910249; the National Natural Science Foundation of China (NSFC), grant number U21A20107. The data presented in this study are available on request from the corresponding author. The data are not publicly available due to the restrictions of the project term.  
The authors would like to thank the National Geomatics Center of China for providing the Globeland30 dataset. The authors declare no conflict of interest.