Inversion of the Earth’s free oscillations using dense array observations and wavefield gradiometry

Sjoerd A.L. de Ridder1

1Geoscience Research Centre, Total E&P UK, Aberdeen, UK (

With the advent of large and dense seismic arrays the information comprised by the observations as a whole is greater than the sum of its parts.

Recently, novel imaging and inversion methods have been developed to exploit the information captured by stations in close proximity to each other. Processing array data using wavefield gradiometry explicitly exploit the constraint on near-surface surface wave velocities from the spatial wavefield gradients of the wavefield. In contrast to cross-correlation based processing, this strategy does not require energy equipartitioning, as long as local acting sources can be characterized or neglected. Several field data examples using ambient seismic recordings dominated by surface waves show that reliable phase velocity maps can be obtained by inverting a Helmholtz equation directly.

Beyond wavefield gradiometry, inverting the ambient seismic field can be posed as a full wavefield inversion problem jointly estimating a reconstructed wavefield and the medium parameters. The PDE system governing the wave propagation is imposed as a soft constraint and the boundary conditions of the wavefield can remain unknown. An example using synthetic elastic data generated over a model containing a subsurface step-function will illustrate how the inversion is able to characterise the step-function in a surface-wave phase velocity image.

The principle of wavefield gradiometry suggest that obtaining subsurface properties from closely spaced stations requires recordings spaced below the spatial Nyquist criteria. Yet recent work suggests that reconstructing surface waves over a larger bandwidth allows including significantly aliased energy at the higher end of the frequency bandwidth. This means that dense-array methods can be used on not-so-dense arrays.