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.

中文翻译：

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弹性最小二乘偏移用于定量反映裂缝顺应性

为了对高分辨率的裂缝进行定量成像，我们开发了一种结合线性滑移理论的弹性最小二乘迁移（LSM）算法，该算法可精确解决地震波与薄结构的相互作用。我们使用Born逼近到散射波的边界积分方程，包括P-SV和SH波场的线性滑移界面，得出了线性化的波形反演。数值模拟测试假设一个实验室规模的裂缝，其中20厘米长的裂缝被中心频率为50 kHz的波照亮，表明我们的LSM成功地估算了裂缝的顺应性。此外，由于在裂缝处存在耦合顺应性，因此使用LSM的结果显示出比使用传统LSM估计Lamé常数的结果更好的图像。我们还用数值方法说明了我们的LSM可以成功地应用于中心频率为3 kHz的偶极声波测井数据，以进行单井反射成像，将10 m长的浸入裂缝嵌入随机背景中。最后，我们将LSM应用于实验室实验数据，测量充满流体的裂缝中的PP反射。我们确认估计的裂缝顺应性与通过偏移反演的早期振幅变化估算的裂缝顺应性很好。此外，由于裂缝中局部注水，LSM解决了空间变化的裂缝顺应性问题。由于线性滑移理论可以应用于各种规模的薄结构，因此高分辨率成像结果和估计的裂缝顺应性分布对于进一步解决裂缝的小规模特性至关重要，