## 6 Expansion Rate of the Dam
This randomness is crucial for validating the decomposition model; structured residuals would indicate missing components or inappropriate model selection [64]. Figure 16 presents a linear regression model with noise results, showcasing the expansion rate (in millimeters/month) of a dam's soil in the downstream area from January 2010 to December 2020. This comprehensive dataset offers valuable insights into the temporal variations in soil expansion, allowing for a thorough analysis of seasonal patterns and long-term trends. Upon examining the data, a distinct seasonal pattern emerges in the soil expansion rates. The highest expansion rates consistently occur during the summer months (June, July, and August) across all years. For instance, in July 2010, the expansion rate peaked at 17.48 mm/month, while in August 2011 and July 2012, it reached 18.03 mm/month and 18.07 mm/month, respectively. This seasonal trend persists throughout the entire dataset, with summer months exhibiting expansion rates typically ranging from 10 to 15 mm/month. Conversely, winter months (November, December, January, and February) display significantly lower expansion rates. For instance, the expansion was slight in December 2010 (4 mm/month). This stark contrast between summer and winter suggests a strong correlation between soil expansion and temperature, a well-documented phenomenon in geotechnical literature [13]. Interestingly, the data also reveal a slight increase in peak expansion rates over the years. For example, the highest expansion rate in 2010 was 17.62 mm/month (August), while in 2018 and 2020, it reached 18.00 mm/month (August) and 19.21 mm/month (August), respectively. This gradual increase could indicate long-term changes in soil properties or environmental factors, such as climate change's effects on soil moisture content and temperature. Our findings align with several vital studies. [PERSON] et al. reported similar seasonal variations in expansive soils, with peak expansion rates during summer months due to increased evapotranspiration and soil suction. Their research on a highway embankment in Texas showed maximum heave rates of 12-16 mm/month in July and August, closely matching our observed range [66]. Furthermore, [PERSON] et al. investigated the downstream soil expansion of the Three Gorges Dam in China using InSAR technology. They found seasonal displacements of 5-20 mm/month, primarily during summer, attributed to temperature-induced expansion and reservoir level fluctuations [67]. [PERSON] and [PERSON], in their study of soil expansion in Northern Nigeria, noted a gradual increase in soil expansion rates over five years, attributing it to changing rainfall patterns and increased irrigation activities. However, their study's expansion magnitude was 20-40 mm/month [68]. Our data fall within this range, supporting the generalizability of these findings across different dam sites. The expansion rate is always positive regarding the cumulative soil volume increase, indicating a consistent increase in soil volume over time. However, it is essential to note that this does not necessarily translate to a simple, monotonic increase in cumulative soil volume. Expansive soils often experience cycles of swelling and shrinkage, particularly in response to seasonal changes in moisture content. While our data show net positive expansion, there may be periods of slight contraction not captured at our monthly sampling interval. Additionally, factors such as consolidation under the dam's weight could partially offset some of the expansion over time. The novelty of our study lies in its comprehensive temporal coverage and granularity. Most existing research focuses on short-term monitoring (1-3 years) or uses coarser temporal resolutions (quarterly or bi-annual measurements). In contrast, our dataset spans from 2010 to 2020 with monthly measurements, providing an unprecedented view of short-term (seasonal) and long-term (inter-annual) variations in soil expansion rates.