## 1 Introduction  
Soil liquefaction represents a prominent phenomenon triggered by seismic motion, particularly relevant in regions where loose, water-saturated sandy layers are confined within impermeable strata close to earthquake epicenters. The ground motion during earthquakes can induce excess pore pressures and temporary losses of strength in these saturated sediments, causing them to behave akin to fluids. Frequently observed in alluvial and coastal plains due to distinct subsurface stratigraphy, liquefaction events have historically caused significant damage to critical infrastructures, agricultural areas, and properties, as exemplified by notable seismic sequences such as the 1810-1811 New Madrid earthquakes [1], the 1964 Alaska and Niigata earthquakes [2, 3], the events in Kobe (1995), Chi-Chi (1999), and Izmit (1999) [4, 5, 6], the Canterbury earthquake of 2010-2011 in NewZealand [7], the Emilia earthquake sequence of 2012 in the Po Plain, Northern Italy [8], and the 2018 Palu earthquake in Indonesia [9]. Liquefaction primarily stems from the generation of excess pore pressures in loose sand deposits subjected to cyclic loading during earthquakes. These deposits may liquefy under seismic stress, resulting in significant losses of stiffness and strength [10]. Structures built atop such deposits face considerable risk of damage from coseismic shear stresses and subsequent post-liquefaction reconsolidation, during which, excess pore pressures gradually dissipate [11; 12; 13]. In cases where the sand deposit allows drainage only from the base (singly draining), pore fluid flows upward, prolonging the liquefied state of sandy layers at depth and extending the pore pressure dissipation time, thus causing further delayed subsidence. The damage induced by liquefaction arises from intricate interactions among various mechanisms, with post-liquefaction reconsolidation playing a significant role in sand deformation [14; 15]. Measuring the extent, amplitude, and temporal evolution of liquefaction-induced ground subsidence is, then, crucial to understanding this phenomenon [12] and contributes to the hazard assessment of liquefaction-prone areas. Unfortunately, such measurements are often difficult to perform because of the spatial and temporal variability of the observed phenomenon. Nowadays, the Interferometric Synthetic Aperture Radar (InSAR) technique is extensively adopted to map regional- to local-scale surface deformations caused by earthquakes, volcanoes, lava flows, ground subsidence, and landslides [16; 17; 18; 19; 20], providing a synoptic and precise measurement of the deformations at the ground with unprecedented spatial coverage over both space and time. InSAR displacement time series have been successfully applied to monitor pre- and post-earthquake ground movements of tectonic origin [21; 22], but there has been limited exploration of using SAR for remotely sensing liquefaction phenomena and mapping local liquefaction-induced effects at the ground, i.e., sand boils and water leaks from cracks. Primarily, complex coherence, which measures how much a scene has changed between two acquisitions, enabled the detection of differential compaction- or liquefaction-induced failure at the scale of building blocks, along with the identification of expelled sands and water. A few case studies have exploited the InSAR technique to detect the local ground subsidence caused by sand ejecta at the surface [13; 23; 24; 25]. However, none of these studies provided insights into the spatial and temporal evolution of post-liquefaction consolidation subsidence, which could last for a long time, depending on the geomorphological setting of the subsoil, further threatening infrastructures at the surface. This research study utilized high-resolution X-band SAR data collected by COSMO-SkyMed satellites to detect and monitor, for the first time, post-liquefaction ground subsidence during the 2012 Emilia seismic sequence in northern Italy [26].