GNGTS 2024 - Atti del 42° Convegno Nazionale

Session 3.3 GNGTS 2024 Discrete Cosine Transform (DCT) reparameterizaton. This approach is applied to a real dataset, acquired in the framework of the InterPACIFIC project at the test site of Grenoble (France, Garofalo et al. 2016). Method The method we employed is the same described in Bert et al. (2023) and applied to solve the acoustc FWI, but in the present study the method has been extended to the elastc case and validated on real data. For the sake of brevity, the theoretcal descripton of the method is not included here, and we refer the reader to Bert et al. (2023) for more details. In essence, our implemented MCMC method defnes the proposal distributon as a localized approximaton of the PPD by leveraging informaton derived from the local gradient and Hessian of the negatve log posterior computed around the current state of the chain. This signifcantly reduces the tme requested by the algorithm to reach the steady state. The drawback of this procedure is that derivatves need to be evaluated for each sampled model, posing computatonal challenges when dealing with extensive model and data spaces. Therefore, a convenient strategy to reduce the computatonal complexity of this inverse problem is to compress the model and data spaces through appropriate reparameterizaton techniques, such as the DCT. The DCT of a signal reveals the energy distributon of the signal in the frequency domain spectrum. Typically, the majority of the signal’s energy is expressed by low-order DCT coefcients and consequently, this mathematcal transformaton serves as a tool for model and data compression, achieved by setng the coefcients of the base functon terms beyond a certain threshold equal to zero. The estmaton of the optmal number of DCT coefcients needed to approximate the model and data spaces is a critcal step of our inversion framework. For the seismic data, we have analyzed how the relatve percentage error, calculated as the rato between the L2 norm diference of the observed and compressed data and the L2 norm of the observed data, varies using diferent combinatons of DCT coefcients; for the model space instead, we have used the available borehole data, investgatng how the variability of the model, calculated as the rato between the variance of the compressed and uncompressed models (Aleardi, 2021), changes with the number of retained DCT coefcients. Results To validate our proposed methodology, we applied the approach to a feld dataset, acquired in Grenoble, France, as part of the InterPACIFIC project. Three in-line boreholes, spaced at 4.5m intervals, were drilled up to 50m depth. These available well-log data were used to validate the results obtained in our work. The dataset consists of three shot gathers, of which one is split- spread and two are of-ends, recorded by 48 vertcal geophones with a spacing of 1 m and a natural frequency of 4.5 Hz. Pre-processing steps, including trace-by-trace amplitude normalizaton and a zero-phase band-pass flter (3-30Hz), were applied to enhance data quality. Then, a 3D to 2D correcton is needed to compensate for the geometrical spreading between the real case point source and the 2D forward modelling where line sources are implicitly used in the simulatons. For generatng our predicted data, we have constructed a grid with a size of 276(nx 0 ) x 150(nz 0 ), where nx 0 and nz 0 are the number of grid points in the horizontal and vertcal directon. The grid spacing is set to 0.25m in both directons, to avoid numerical dispersion in the fnite diference modelling. The tme sampling is 0.1ms for the forward modelling and the registraton tme is 0.5s. Both predicted and observed data were resampled to a 2ms tme interval. The simulaton of the shots is performed using SOFI2D (Bohlen, 2002), a viscoelastc forward modelling code that solves the pure elastc or viscoelastc wave equaton by a fnite diference scheme in the tme domain. The model parameters to be estmated are the Vs and Vp values, and we are considering a homogeneous