To quantitatively image fractures with high resolution, we have developed an elastic least-squares migration (LSM) algorithm coupled with linear-slip theory, which accurately addresses seismic wave interaction with thin structures. We derive a linearized waveform inversion using the Born approximation to the boundary integral equation for scattered waves, including linear-slip interfaces for P-SV and SH wavefields. Numerical modeling tests assuming a laboratory-scale fracture where a 20 cm long fracture is illuminated by waves with a 50 kHz center frequency show that our LSM successfully estimates fracture compliances. Furthermore, due to the presence of coupling compliances at the fracture, the results using our LSM show better images than those using the conventional LSM estimating the Lamé constants. We also numerically illustrate that our LSM can be successfully applied to dipole acoustic borehole logging data with 3 kHz center frequency for single-well reflection imaging of a 10 m long, dipping fracture embedded in a random background. Finally, we apply LSM to laboratory experimental data, measuring PP reflections from a fluid-filled fracture. We confirm that the estimated fracture compliances correspond well to those estimated by earlier amplitude variation with offset inversion. Furthermore, the LSM resolves the spatially varying fracture compliances due to local filling of water in the fracture. Because the linear-slip theory can be applied to thin structures in a wide range of scales, high-resolution imaging results and estimated fracture compliance distributions will be crucial to further address small-scale properties at fractures, joints, and geologic faults.
Elastic least-squares migration for quantitative reflection imaging of fracture compliances
