Adaptive Lp Inversion

Traditional geophysical inversion methods can only recover either smooth features by
applying L2 model norm in the model objective function to be minimized, or blocky
features by applying L1 model norm. In reality, however, both smooth and blocky
features can be present in the subsurface physical property or interfaces to be recovered.
We develop a new method to adaptively recover both smooth and blocky features in the
reconstructed model from inversions of geophysical datasets. This method first detects
the smoothness or blockiness of different regions of a model based on a sequence of
inversions and then adaptively applies appropriate Lp model norm with different p to
complete the final inversion. We use two synthetic examples from basement inversion
using gravity data and cross-hole seismic travel-time tomography to illustrate the new
method.

Fig 1_1 shows the synthetic faulted basement model and the simulated noisy gravity data.

Fig 1_2 shows the smoothness change at each location of the model as different threshold
values are used in Ekblom norm measure which is used as the measure of length of
vectors in this study because of its flexibility and differentiability. By examining which
part of the model changes its smoothness most rapidly, we are able to identify the
approximate locations of faults in this basement model.

Fig 1_3 shows the reconstructed basement model by applying adaptive Lp inversion
method. This model is much more interpretable than traditional L1 or L2 inversion result.

Fig 2_1 shows the synthetic cross-hole seismic survey over a 2D area that is 1600 meters
long by 800 meters deep. Positions of the transmitters and receivers are marked by red
triangles and circles, respectively. There are two velocity anomalies with high-velocity
anomaly represented by a 2D Gaussian function and low-velocity anomaly represented
by a block.

Fig 2_2 shows there exist two regions of high change in smoothness in the x direction
corresponding to the vertical boundaries of the blocky anomaly.

Fig 2_3 shows a similar result in the z direction and outlines the horizontal boundaries of
the blocky anomaly.

Fig 2_4 clearly indicates the overall boundary of the blocky feature by applying the
Laplacian operator.

Fig 2_5 shows the recovered velocity distribution by adaptive Lp inversion.