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Slip inversion is essential to understand source mechanics of earthquakes. Problems of inverting for static/kinematic rupture from surface measurements have been almost always stated to be inherently non-unique in the seismological literature. Given the geometry of a fault and the displacement functions on the surface of the semi-infinite elastic medium, the slip function on the fault is defined by Fredholm integral equation of the first kind. We prove that Fredholm integral equation of the first kind for slip inversion is mathematically of a unique solution, theoretically implying that earthquake slip/rupture can be properly reconstructed. The inherent non-uniqueness issue of slip inversion in the seismological literature is not mathematically true but is practically caused due to lack of measurements. We propose a new inequality-constrained regularized inversion of slip distributions on multiple faults, which implements physically more general inequality constraints to accept more complex dislocation models and/or a combination of complex dislocation models and can be applied to a large earthquake involving multiple faults with different rake angles. It is a natural extension of positivity and the 45 degrees constraints by Olson/Apsel and Hartzell/Heaton. We apply the proposed method to the 2016 Kumamoto Mw7.0 earthquake with GEONET GNSS measurements. The inversion results of slips show that the 2016 Kumamoto earthquake is only of magnitude Mw6.6-6.7 and severe rupture takes place very shallowly on Hinagu and Futagawa faults, with the maximum slips of 4.81m on Hinagu fault and 7.89m on Futagawa fault, respectively, which may well explain the largest damage in Mashiki town., The 28th IUGG General Assembly (IUGG2023) (Berlin 2023)

Multi-GNSS now routinely measures positions precisely at a high sampling rate, up to mm-level of precision within a few minutes or cm-level in the long term of hours and days. A high sampling rate becomes critical in GNSS determination of velocity and acceleration, since difference methods are always used to determine velocity and acceleration from GNSS precise positioning on the implicit assumption of a low sampling rate of GNSS measurements, because the random errors of GNSS-based velocity and acceleration are proportional to the sampling rate in the case of velocity or square-proportional to the sampling rate in the case of acceleration. There are a number of critical technological problems with GNSS accelerometry: (i) with the increase of sampling rate, measurement errors are significantly amplified such that computed velocity and acceleration are completely immersed into noise and of no physical significance. Thus, GNSS precise positioning with a sampling rate higher than a few Hz has to be first down-sampled to, say 1 or 2 Hz; (ii) down-sampling very high-rate GNSS precise positioning measurements directly results in a lower resolution of velocity and acceleration, a larger aliasing effect and signal distortion, the extent of which depend on that specific rate of sampling; and (iii) a low sampling rate also implies impossibility to provide instantaneous velocity and acceleration. We present a regularization method to compute velocity and acceleration and design its apparatus, which solves the mentioned critical technological problems and lead to the technological invention of accelerometers., The 28th IUGG General Assembly (IUGG2023) (Berlin 2023)