GNGTS 2022 - Atti del 40° Convegno Nazionale

GNGTS 2022 Sessione 3.3 481 The major computational requirement of the implemented approach is the computation of the Jacobian matrix associated with each accepted model and, consequently, the manipulation of the Hessian matrix and the gradient vector. Using a finite difference scheme to evaluate the Jacobian, the number of forward evaluations needed for its computation increases linearly with the number of model parameters. In this work we have used the DCT to compress our model and data spaces. The DCT of a signal indicates the energy distribution of the signal in the frequency domain spectrum. Usually, most of the energy of the signal is expressed by low-order DCT coefficients and consequently, this mathematical transformation can be used for model compression, which is accomplished by setting the coefficients of the base function terms beyond a certain threshold equal to zero. For the estimation of the optimal number of DCT coefficients needed to approximate the model and data spaces, we quantified how the variability of the model and data changes as the number of DCT coefficients varies: we have computed the variability as the ratio between the variance of the approximated model and data and the one of the uncompressed model and data (Aleardi, 2021). Synthetic inversion. we have applied the proposed GB-MCMC FWI method to the 2D Marmousi V p model. The grid size for generating the observed data is 251( nx 0 ) x 101( nz 0 ), with a grid spacing of 0.02 x 0.02 km , so 25351 parameters form the full model space. A Ricker wavelet with the peak frequency at 5 Hz is used as the source. We have simulated 5 shots equally spaced along the horizontal axis and each shot is recorded by 126 receivers with a 0.04 km receiver interval. Noncorrelated Gaussian white noise is added to the observed data. The simulation of the shots has been performed using Devito , a newdomain-specific language for implementing high performance finite difference partial differential equation solvers. By embedding this kind of language within Python , it’s possible to develop finite difference simulators quickly using a syntax that strongly resembles mathematics (Louboutin, 2019). In order to reduce the computational cost of the calculation of the Jacobian matrix using the finite difference approach, we have used the Multiprocessing Python package, which allows to parallelize the execution of a function across multiple input values, distributing the input data across processes (data parallelism). The data and models must be projected onto the DCT space, where the MCMC sampling runs. We have observed that less than 20 coefficients along the two DCT spatial dimensions explain more than 98% of variability of the original model. This means that the compression allows for a reduction of the 25351-D model space to a 9 x 18 = 162-D domain. A similar analysis has been carried out on our seismic data, and in this case 90 x 100 = 9000 retained coefficients in the data space explain almost the total variability of the uncompressed seismogram. Therefore, the full 751 x 126 = 94626-D data space has been reduced to a 9000-D domain. We can see that all the matrices that we are using in our inversion procedure are a lot more tractable in the compressed domain, for example the 25351 x 25351 Hessian in the full domain has been reduced to a 162 x 162 matrix in the compressed space. Notice that, although the MCMC inversions run in the reduced DCT space, the sampled models must be projected back into the full domain just before the forward modelling phase performed with Devito, generates the predicted data needed to compute the likelihood value. We have used six cores on an Intel® Core™ i7-8700 CPU @ 3.20GHz to run the numerical tests. Each iteration, including computing the Jacobian, the gradient, the Hessian matrix and drawing a sample, takes approximately 45 s wall clock time. We have run one single chain with 10.000 iterations, and it finished running within 70 h . The chain starts from a 1D gradient velocity model, using as a prior information the mean of each row of the original velocity model, and we can observe in Fig .2 that the mean model