We have analyzed the angle-dependent reflectivity of microseismic wavefields at a hydraulic fracture, which we modeled as an ideal thin fluid layer embedded in an elastic, isotropic solid rock. We derived full analytical solutions for the reflections of an incident P-wave, the P-P and P-S reflection coefficients, as well as for an incident S-wave, and the S-S and S-P reflection coefficients. The rather complex analytical solutions were then approximated and we found that these zero-thickness limit approximations are in good agreement with the linear slip model, representing a fracture at slip contact. We compared the analytical solutions for the P-P reflections with synthetic data that were derived using finite-difference modeling and found that the modeling confirmed our theoretical results. For typical parameters of microseismic monitoring by hydraulic fracturing, e.g., a layer thickness of [Formula: see text] and frequencies of [Formula: see text], the reflection coefficients depend on the Poisson’s ratio. Furthermore, the reflection coefficients of an incident S-wave are remarkably high. Theoretical results suggested that it is feasible to image hydraulic fractures using microseismic events as a source and to solve the inverse problem, that is, to interpret reflection coefficients extracted from microseismic data in terms of reservoir properties.

SUMMARYWe consider the problem of anelastic full waveform inversion in a multi-layered, isotropic viscoelastic medium from microseismic and cross-hole perforation shots. Usually, the source wavelet is not well known, so we focus on Q estimation techniques that can handle this problem. We revisit the spectral ratio method (SRM) and discuss its domain of applicability. We conclude that SRM can be used for interval Q estimation if the sources and receivers are contained in the same homogeneous layer. To overcome the restrictions of SRM, we consider the more sophisticated source-independent full waveform inversion (SIFWI). We compare three different source-independent misfit functions on synthetic data and show that they can be used for Q estimation in a viscoelastic medium. We use 3-D complex ray theory to implement SIFWI in a layered viscoelastic medium.

SUMMARY It has been increasingly popular to use shallow-seismic full-waveform inversion (FWI) to reconstruct near-surface structures. Conventional FWI tries to resolve the earth model by minimizing the difference between observed and synthetic seismic data using a certain criterion (conventionally, l2-norm of waveform difference). In this paper, we propose a multi-objective waveform inversion (MOWI) in which the similarity of data is quantified and minimized using multiple criteria simultaneously. By doing so, we expand the dimensionality of objective space as well as the mapping from data space to objective space, which provides MOWI higher freedom in exploring the model space compared to single-objective FWI. We combine three different scalar-valued objective functions into a vector-valued multi-objective function which measures the similarity of the waveform, the waveform envelope, and the amplitude spectra of the data, respectively. This multi-objective function takes not only trace-based waveform and wave packet similarity but also the dispersion characteristics of surface waves into account. Furthermore, the uncertainty in the inversion result could be estimated and analysed quantitatively by the variance of the optimal models. We propose a modified ϵ-constraint algorithm to solve the multi-objective optimization problem. Two synthetic examples are used to show the advantages of using MOWI compared to single-objective FWI. We also test the efficiency of MOWI by using two synthetic shallow-seismic examples, which confirm that MOWI can converge to a better result compared to the conventional single-objective FWI.

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