GNGTS 2018 - 37° Convegno Nazionale

GNGTS 2018 S essione 3.3 713 (8) where k is the total number of inverted subsurface properties (1 for the post-stack inversion and 3 for the pre-stack inversion). Note that equation 8 expresses the prior probability for a multiparameter inversion as well. We consider a likelihood value based on the L2 norm difference between observed and predicted data under the assumption of Gaussian distributed noise: (9) Where ϕ (m) = | | d – d pre | | 2 2 and σ 2 d is the noise variance. Our approach implements the rjMCMC by defining four updating strategies for each iteration: • Velocity move (for even iteration numbers): Randomly pick one layer and perturb the model properties (i.e. acoustic impedance in the post-stack inversion or Vp , Vs and density in the pre-stack inversion). We consider a Gaussian proposal with zero mean and a variance value properly sets to obtain an acceptance ratio between 0.2-0.4; • For odd iteration numbers chose one of the following updating strategies with equal probability: a. Thickness move: Randomly pick one layer and perturb the layer thickness. A uniform proposal with zero mean is used in this case. b. Birth move: Randomly choose a vertical location and create a new layer; determine the layer thicknesses and the model properties. These are assigned by perturbing the properties pertaining to the unperturbed model at the selected vertical location; c. Death move: Randomly select one layer and remove it. For the velocity and thickness moves the number of model parameters does not change, then the proposal ratio is equal to one and the resulting acceptance probabilities are given by: (10) For the birth move we have: (11) where σ 2 e i is the variance of the proposal Gaussian distribution for the i -th parameter, whereas e´ i and e i are the elastic properties in the picked layer for the perturbed and unperturbed model, respectively. For the death move we get: (12) Note that equations 11 and 12 extend the acceptance probability tomultiparameter inversions. The entire procedure is summarized in Fig. 1. Note that we collect models form different chains in order to increase the confidence on the final computed posterior density. In the following examples we use 20 chains, we collect 40000 models for each chain, and we consider a burn-in period of 20000 samples.