## 1 Introduction  
Urbanization is an inevitable trend of global social and economic development. The expansion speed of urban construction is of great significance to studying the urban growth model [1, 2, 3], as well as identifying urbanization processes [4]. Arbitrary expansion of urban land use has brought great challenges to groundwater exploitation, ground bearing capacity [5, 6], air quality [7], and even leads to many harmful chain reactions [8, 9, 10, 11]. The authorities need reliable and rapid statistics on the three-dimensional expansion or expansion velocity of buildings in the jurisdiction in order to plan urban development and formulate development policy [12, 13]. However, it is difficult to achieve rapid and large-scale statistics due to a large amount of construction volume and intricate demolition information [14, 15].  
Remote sensing technology provides a convenient solution for statistics and comparison in this task. At present, many methods and cases have been presented in the literature.  
For example, urban land-use change detection based on remote sensing images has been used in many case studies [16; 17; 18; 19; 20; 21; 22; 23]. In recent years, night-time light data are also used to study urban expansion [24]. This method is very effective for the horizontal expansion of buildings, but it is not sensitive to the vertical expansion of the city.  
Megacities usually begin as horizontal expansion initially, and then grow vertically in the already developed areas due to the shortage of land resources [25]. Therefore, capturing the height change of urban buildings in time is an effective way to dynamically monitor the urban development process. Nowadays, building height can be retrieved from synthetic aperture radar (SAR) imagery [26; 27]. Furthermore, high-precision building height information can be extracted from photon-counting LiDAR data [28; 29]. Other predecessors focus on generating a digital surface model from optical stereo image pairs or SAR images, and then compare the height differences of different DSMs to identify newly constructed buildings, as well as demolitions in dense residential areas [30; 31; 32; 33; 34]. Nevertheless, more details, such as the exclusion of non-construction factors [35; 36; 37; 38], are still worth further study. Moreover, openly accessible information at high spatial resolution is still missing for complete countries or regions [39].  
Multi-temporal DSMs released for free worldwide, including those already opened and those to be opened in the future, have the advantage of rapidly detecting building height changes in large areas, and even on a global scale. However, due to many uncertain factors such as data errors and mixed non-building changes, they are less used for building expansion statistics [40].  
Our research mainly focuses on using AW3D30 DSM and SRTM DSM as an example to calculate building height expansion. By registering DSMs data, verifying the change of sample building height, and excluding the non-building height shift, the three-dimensional expansion of the building is calculated. The result can be converted to the average completed building floor space per unit area (ACBFS) to compensate for statistical deficiencies.