Optimal experimental design for full waveform inversion using a wavenumber sampling criterion—part 1 : 2-D to methodological development

SUMMARY Due to the complexity of seismic surveys, different strategies have been investigated to optimize characteristics of the acquisition in order to fulfil various criteria, most intuitively those related to the operational and economical aspects of the survey. However the question can also be approached from an imaging perspective. Andrzej Kikjo’s seminal work on the optimal design of seismic stations for earthquake localization, dating back to the seventies, has encouraged several works related to optimal acquisition in exploration geophysics. A common choice for the optimality criterion is the conditioning of the Hessian operator. In our work, we have chosen our optimality criterion to be the regularity of the wavenumber sampling at a specified target: using the theory of diffraction tomography we can estimate the wavenumber sampling at a target point and we note that for a regularly spaced acquisition the corresponding sampling is not regular. In our target-oriented approach, we want to find the best positions of sources and receivers to improve the regularity of this sampling. We develop our work in the context of full waveform inversion, although our approach is more general. Using a simple formula for the estimation of the wavenumber points we are able to get a geometric characterization of the accessible wavenumber space that helps us define its envelope. We describe the problem as finding the geometry for which the sampling points are regular in terms of spacing inside this envelope and to that end we incorporate in our workflow the notion of centroidal Voronoi tessellation, a tool from computational geometry that links the regularity of the sampling to the uniformity of the domain tessellation. To assess the uniformity of the tessellation we utilize an energy function the minimal values of which provide regular samplings. By combining this energy function with a map that estimates the wavenumber content of an acquisition we define a new objective function that assess the quality of such acquisition with regards to our regularity criterion. We then mathematically express our problematic as a minimization problem which we solve using local optimization strategies. In this article we focus on the theoretical aspect of our method and validate it through numerical examples of varying complexities in the 2-D cases: when comparing optimized acquisitions with regular ones the results show a consistent local improvement of the quality of the inversion with the optimized geometries at the target and motivates the exploitation of the method for target-oriented applications such as seismic monitoring. The subsequent extension to 3-D and application to more realistic industrial examples will be discussed in a companion paper.