## 1 Introduction  
The existence and stability of snow and ice cover on Earth are integral components of the global climate system. Glacier fluctuation due to climatic change is a natural occurrence that has been taking place throughout the history of the Earth. Over the past few decades, there has been a discernible shift in global climate patterns, exerting a pronounced influence on the high mountain glacial environment. Glaciers and discontinuous permafrost are exceptionally responsive to alterations in atmospheric conditions and changes in land cover, leading to changes in albedo. The Hindu Kush-Himalaya (HKH) region, which spans over 4,192,000 km\({}^{2}\) in eight countries, is one of the most dynamic, fragile, and complex mountain systems. Renowned as the 'Third Pole', it holds the highest concentration of snow and glaciers outside the high-latitude polar regions [1]. The glacier meltwater of the HKH region sustains the ten largest river systems in South Asia, including the Amu Darya, Brahmaputra, Ganges, Indus, Irrawaddy, Mekong, Salween, Tarim Interior, Yangtze, and Yellow rivers. These rivers collectively provide vital resources to more than 1.3 billion people residing in the downstream basin regions [2]. The HKH region has already experienced discernible impacts attributed to changing climatic conditions. Numerous documented observations have highlighted the effects of climate change within the HKH region. Studies have reported a concerning trend of heterogeneous mass loss in the HKH region for the past few decades. A thorough study of HKH glacier extent changes [3] reveals a slow and consistent retreat trend of Hindu Kush glaciers, especially in China, India, Nepal, and Pakistan, and a rapid retreat trend of more than the global average in Himalayan glaciers. However, over the Karakoram range, glaciers exhibit a more diverse response with comparatively less change and more stable conditions. A multi-temporal analysis and comparison study from 1991 to 2014 shows that the glaciers of the Ladakh range, Jammu and Kashmir, India, have shrunk by \(-\)12.8% (\(-\)45.3 km\({}^{2}\)) [4]. From 1968 to 2019, the Swetvarn and Thelu glaciers in the Rakvarn glacial valley of the Bhagiriathi basin, Garhwal Himalaya, India, retreated by about 16.60% and 24.50%, respectively [5]. The largest glacier in the Sikkim Himalaya, Zemu, shrank by 30.67 \(\pm\) 2.87% at a rate of 0.35% per year, and its snout retreated by \(\sim\)797 \(\pm\) 19.7 m at a rate of 9.1 ma\({}^{-1}\) during the last 87 years between 1931 and 2018. However, the retreat rate has increased to \(\sim\)20 ma\({}^{-1}\) lately, from 2014 to 2018. A total of 147 glaciers from the Kashmir Himalaya show a notable loss of 28.82% (29.32 \(\pm\) 12.09 km\({}^{2}\)) of covered area during the last 38 years. These glaciers covered an area of around 101 \(\pm\) 16.79 km\({}^{2}\) in 1980, which shrank to 72.41 \(\pm\) 4.7 km\({}^{2}\) in 2018. This recession is synchronous with the temperature increase during the period from 1980 to 2018 [6]. According to reports [7; 8], 65 glaciers situated in the five mountain ranges of the North-Western Himalaya (the Karakoram range, the Zanaskar range, the Ladakh range, the Shamawari range, and the Greater Himalaya) have shrunk to 1073 \(\pm\) 24.6 km\({}^{2}\) area in 2014 from 1106 \(\pm\) 33.6 km\({}^{2}\) in 1990, and the glacier volume has reduced to 211 \(\pm\) 4.85 km\({}^{3}\) from 219 \(\pm\) 6.66 km\({}^{3}\) during this period. Based on the glacier inventory prepared by the International Centre for Integrated Mountain Development (ICIMOD), Kathmandu, Nepal, 885 glaciers of the Bhutan Himalaya show around 23.3 \(\pm\) 0.9% glacial cover fall-off between \(\sim\)1980 and 2010, with a minimum fall-off of around 6.7 \(\pm\) 0.1% (2000-2010) and a maximum of around 11.6 \(\pm\) 1.2% (\(-\)1980-1990) [9]. Between 2000 and 2012, almost 12,243 glaciers with a 19,727 \(\pm\) 1054 km\({}^{2}\) area in the Upper Indus Basin of the Jammu and Kashmir Himalaya thinned by around \(-\)0.35 \(\pm\) 0.33 ma\({}^{-1}\).