Model reduction in geostatistical seismic inversion with functional data analysis
In subsurface modelling and characterization, predicting the spatial distribution of subsurface elastic properties is commonly achieved by seismic inversion. Stochastic seismic inversion methods, such as iterative geostatistical seismic inversion, are widely applied to this end. Global iterative geostatistical seismic inversion methods are computationally expensive as they require, at a given iteration, the stochastic sequential simulation of the entire inversion grid at once multiple times. Functional data analysis is a well-established statistical method suited to model long-term and noisy temporal series. This method allows to summarize spatiotemporal series in a set of analytical functions with a low-dimension representation. Functional data analysis has been recently extended to problems related to geosciences, but its application to geophysics is still limited. We propose the use functional data analysis as a model reduction technique during the model perturbation step in global iterative geostatistical seismic inversion. Functional data analysis is used to collapse the vertical dimension of the inversion grid. We illustrate the proposed hybrid inversion method with its application to three-dimensional synthetic and real data sets. The results show the ability of the proposed inversion methodology to predict smooth inverted subsurface models that match the observed data at a similar convergence as obtained by a global iterative geostatistical seismic inversion, but with a considerable decrease in the computational cost. While the resolution of the inverted models might not be enough for a detailed subsurface characterization, the inverted models can be used as starting point of global iterative geostatistical seismic inversion to speed-up the inversion or to test alternative geological scenarios by changing the inversion parameterization and obtaining inverted models in a relatively short time.