Full-waveform inversion with wavefield gradients

<p>Wavefield gradient instruments, such as rotational sensors and DAS systems, are becoming more and more accessible in seismology. Their usage for Full Waveform Inversion (FWI) is in sight. Nevertheless, local small-scale heterogeneities, like geological inhomogeneities, surface topographies, and cavities are known to affect wavefield gradients. This effect is in fact measurable with current instruments. For example, the agreement between data and synthetics computed in a tomographic model is often not as good for rotation as it is for displacement.</p><p>The theory of homogenization can help us understand why small-scale heterogeneities strongly affect wavefield gradients, but not the wavefield itself. It tells us that at any receiver measuring wavefield gradient, small-scale heterogeneities cause the wavefield gradient to couple with strain through a coupling tensor <strong>J</strong>. Furthermore, this <strong>J</strong> is 1) independent of source, 2) independent of time, but 3) only dependent on the receiver location. Consequently, we can invert for <strong>J</strong> based on an effective model for which synthetics fit displacement data reasonably well. Once inverted, <strong>J</strong> can be used to correct all other wavefield gradients at that receiver.</p><p>Here, we aim to understand the benefits and drawbacks of wavefield gradient sensors in a FWI context. We show that FWIs performed with rotations and strains are equivalent to that performed with displacements provided that 1) the number of data is sufficient, and 2) the receivers are placed far away from heterogeneities. In the case that receivers are placed near heterogeneities, we find that due to the effect of these heterogeneities, an incorrect model is recovered from inversion. In this case therefore, the coupling tensor <strong>J</strong> needs to be taken into account for each receiver to get rid of the effect.</p>

Correcting wavefield gradients for the effects of local small-scale heterogeneities

Summary Local small-scale heterogeneities, like geological inhomogeneities, surface topographies and cavities are known to affect seismic wavefield gradients. This effect is in fact measurable with current instruments. For example, the agreement between data and synthetics computed in a tomographic model is often not as good for rotation as it is for displacement. Here, we use the theory of homogenization to explain why small-scale heterogeneities strongly affect wavefield gradients, but not the wavefield itself. We show that at any receiver measuring wavefield gradient, small-scale heterogeneities cause the wavefield gradient to couple with strain through a coupling tensor J. Furthermore, we show that this J is (1) independent of source, (2) independent of time, but (3) only dependent on the receiver location. Consequently, we can invert for J based on an effective model for which synthetics fit displacement data reasonably well. Once inverted, J can be used to correct rotations at that receiver for other sources. Results of the correction are shown for synthetic rotational receivers and for the ring laser located in Wettzell, Germany. We find that compared to synthetics, the synthetics corrected through J are a better fit to the rotation data. Although results here are derived for rotations, they can be extended to receivers measuring any wavefield gradient.

The tetragonal elastic dipole Cr5+ in SrTiO3

Tetragonal Cr⁵⁺ impurity centre in single crystals of the perovskite SrTiO₃. After careful analyses we came to the conclusion that this centre is an off-centre system and no Jahn-Teller impurity. From the previous stress experiments in EPR we calculated the linear stress coupling tensor as β = 3.56×10^-30 m³, which is in agreement with other stress coupling tensors of off-centre systems in SrTiO₃ as well as BaTiO₃.

The tetragonal elastic dipole Cr5+ in SrTiO3

Tetragonal Cr⁵⁺ impurity centre in single crystals of the perovskite SrTiO₃. After careful analyses we came to the conclusion that this centre is an off-centre system and no Jahn-Teller impurity. From the previous stress experiments in EPR we calculated the linear stress coupling tensor as β = 3.56×10^-30 m³, which is in agreement with other stress coupling tensors of off-centre systems in SrTiO₃ as well as BaTiO₃.

The tetragonal elastic dipole Cr5+ in SrTiO3

Tetragonal Cr⁵⁺ impurity centre in single crystals of the perovskite SrTiO₃. After careful analyses we came to the conclusion that this centre is an off-centre system and no Jahn-Teller impurity. From the previous stress experiments in EPR we calculated the linear stress coupling tensor as β = 3.56×10^-30 m³, which is in agreement with other stress coupling tensors of off-centre systems in SrTiO₃ as well as BaTiO₃.