## 2 Related Work
### Neural Radiance Fields for Satellite Photogrammetry  
Reconstructing 3D geometry from satellite images has become a key research topic in the fields of remote sensing and geographic information science. Since satellite images are almost always taken at an oblique angle, providing a unique multi-view perspective of the scene that cannot be provided by a single-view image, the reconstruction of a 3D model of the Earth's surface from a collection of satellite imagery taken from different angles plays a vital role in the fields of urban planning, environmental monitoring and protection, and disaster assessment and management.  
Neural radiance field technology has been successfully applied to reconstruct the Earth's surface from multi-view satellite imagery, demonstrating its potential in the domain of 3D reconstruction for satellite imagery. Refs. [11; 16; 37; 38; 39] have explored satellite 3D reconstruction through the neural radiance field method. [PERSON] et al. [11] took the lead in exploring remote sensing 3D reconstruction based on neural radiance fields and proposed S-NeRF, which is a significant advancement in the discipline of remote sensing 3D reconstruction based on neural radiance fields. A basic assumption of the NeRF is that the scene is static and captured at the same time. However, unlike stereo or tri-stereo images, multi-view satellite images are often captured at different times, resulting in obvious appearance and color differences between different images. S-NeRF adapts to multi-view satellite images with different illumination conditions by modeling radiance as the product of irradiance and albedo. In addition, Sat-NeRF [16] introduces the RPC camera model to more accurately calculate the origin and direction of rays for ray marching in the NeRF, and it solves the problem of transient phenomena that are not easily explained by the sun's position through the shadow-aware irradiance model and uncertainty weights. Sat-Mesh [39], on the other hand, uses the MLP to learn the sign distance function (SDF) and integrates it within the volume-rendering framework to achieve multi-view satellite reconstruction. Season-NeRF [37] enables the NeRF model to learn and render seasonal features by including time as an additional input variable. SpS-NeRF [38] uses the low-resolution dense depth generated by the traditional multi-view stereo (MVS) for supervision, which enables the network to learn geometric structures even with few satellite image inputs.  
Although the above works have achieved good results in remote sensing 3D reconstruction, they usually require 8-10 h or even longer for training and optimization, as shown in Table 1, which significantly limits further research and applications in practical scenarios. Therefore, one of the goals of our research work is to accelerate neural radiance field methods for multi-view satellite imagery.  
In this study, we thoroughly explore and extend Instant-NGP and Sat-NeRF. By adopting hash grid encoding in Instant-NGP, we significantly improve the efficiency of model training, which provides new possibilities for processing large-scale datasets. At the same time, we improved the irradiance model of Sat-NeRF to cope with illumination changes and transient objects (cars, shadows, etc.) in multi-date images. In addition, we use the random structural similarity loss function to capture the rich structural information in remotely sensed images and improve the model's ability to reconstruct geometric details in complex scenes.