Particle relabelling transformations in elastodynamics

The motion of a self-gravitating hyperelastic body is described through a time-dependent mapping from a reference body into physical space, and its material properties are determined by a referential density and strain-energy function defined relative to the reference body. Points within the reference body do not have a direct physical meaning, but instead act as particle labels that could be assigned in different ways. We use Hamilton’s principle to determine how the referential density and strain-energy functions transform when the particle labels are changed, and describe an associated “particle relabelling symmetry”. We apply these results to linearised elastic wave propagation, and discuss their implications for seismological inverse problems. In particular, we show that the effects of boundary topography on elastic wave propagation can be mapped exactly into volumetric heterogeneity while preserving the form of the equations of motion. Several numerical calculations are presented to illustrate our results.O.C. is supported through a NERC PhD studentship.This is the final version of the article. It first appeared from Oxford University Press via http://dx.doi.org/10.1093/gji/ggw03

An extended ice-Age sea-level equation: Incorporating water flux across sills

Summary We present a generalized theory governing gravitationally self-consistent, spatio-temporal sea-level changes within an ocean-plus-lake system that is intermittently connected by water mass flux across a sill. Our expressions for the change in sea level (defined as the difference in height of the sea surface equipotential relative to the solid surface) hold for any Earth model, and easily incorporate effects of viscoelastic deformation of the solid Earth and perturbations in both the gravitational field and rotation vector (as is now standard in ice-age sea-level calculations). In its most general form, the theory also includes an exact treatment of the evolving shoreline position in both water bodies. Our formalism involves three cases: (1) one global ocean, in which mass transfer may occur between ice sheets and the global ocean; (2) an ocean and lake separated by an exposed sill, in which mass transfer may occur between ice sheets and the global ocean, and between the ocean and lake via evaporative flux; and (3) transitional phases between these two states, when the ocean surface reaches the height of the sill from below (i.e., the sill is breached) or above (the sill is exposed). We illustrate the new theory using examples from the Black Sea flooding during the last deglacial phase (∼10 ka) and sea-level fall in the Mediterranean Sea during the Messinian Salinity Crisis (5.96-5.33 Ma). These examples demonstrate the importance of including the geophysical feedbacks associated with sea-level change in an isolated basin in the dynamics of flooding and desiccation.</jats:p

Forward and inverse modelling of post-seismic deformation

We consider a new approach to both the forward and inverse problems in post-seismic deformation. We present a method for forward modelling post-seismic deformation in a self-gravitating, heterogeneous and compressible earth with a variety of linear and nonlinear rheologies. We further demonstrate how the adjoint method can be applied to the inverse problem both to invert for rheological structure and to calculate the sensitivity of a given surface measurement to changes in rheology or time-dependence of the source. Both the forward and inverse aspects are illustrated with several numerical examples implemented in a spherically symmetric earth model.Natural Environment Research Council, British Antarctic Surve

Intrinsic non-uniqueness of the acoustic full waveform inverse problem

SUMMARY In the context of seismic imaging, full waveform inversion (FWI) is increasingly popular. Because of its lower numerical cost, the acoustic approximation is often used, especially at the exploration geophysics scale, both for tests and for real data. Moreover, some research domains such as helioseismology face true acoustic media for which FWI can be useful. In this work, an argument that combines particle relabelling and homogenization is used to show that the general acoustic inverse problem based on band-limited data is intrinsically non-unique. It follows that the results of such inversions should be interpreted with caution. To illustrate these ideas, we consider 2-D numerical FWI examples based on a Gauss–Newton iterative inversion scheme and demonstrate effects of this non-uniqueness in the local optimization context.</jats:p

Sensitivity kernels for body tides on laterally heterogeneous planets based on adjoint methods

SUMMARY We apply the adjoint method to efficiently calculate the linearized sensitivity of body tide observations to perturbations in density, elastic/anelastic moduli and boundary topography. This theory is implemented practically within the context of normal mode coupling calculations, with an advantage of this approach being that much of the necessary technical machinery is present in existing coupling codes. A range of example sensitivity kernels are calculated relative to both spherically symmetric and laterally heterogeneous background models. These results reaffirm the conclusions of earlier studies that the M2 body tide is strongly sensitive to spherical harmonic degree-2 density variations at the base of the mantle. Moreover, it is found that the sensitivity kernels are only weakly dependent on the background model, and hence linearized methods are likely to be effective within inversions of body tide observations.NSF grant EAR-192386

Anelasticity across seismic to tidal timescales: a self-consistent approach

In a pioneering study, Wahr & Bergen developed the widely adopted, pseudo-normal mode framework for predicting the impact of anelastic effects on the Earth's body tides. Lau $\textit{et al.}$ have recently derived an extended normal mode treatment of the problem (as well as a minor variant of the theory known as the direct solution method) that makes full use of theoretical developments in free oscillation seismology spanning the last quarter century and that avoids a series of assumptions and approximations adopted in the traditional theory for predicting anelastic effects. There are two noteworthy differences between these two theories: (1) the traditional theory only considers perturbations to the eigenmodes of an elastic Earth, whereas the new theory augments this set of modes to include the relaxation modes that arise in anelastic behaviour; and (2) the traditional theory approximates the complex perturbation to the tidal Love number as a scaled version of the complex perturbation to the elastic moduli, whereas the new theory computes the full complex perturbation to each eigenmode. In this study, we highlight the above differences using a series of synthetic calculations, and demonstrate that the traditional theory can introduce significant error in predictions of the complex perturbation to the Love numbers due to anelasticity and the related predictions of tidal lag angles. For the simplified Earth models we adopt, the computed lag angles differ by ∼20 per cent. The assumptions in the traditional theory have important implications for previous studies that use model predictions to correct observables for body tide signals or that analyse observations of body tide deformation to infer mantle anelastic structure. Finally, we also highlight the fundamental difference between apparent attenuation (i.e. attenuation inferred from observations or predicted using the above theories) and intrinsic attenuation (i.e. the material property investigated through experiments), where both are often expressed in terms of lag angles or $\textit{Q}$$^{-1}$. In particular, we demonstrate the potentially significant (factor of two or more) bias introduced in estimates of $\textit{Q}$$^{-1}$ and its frequency dependence in studies that have treated $\textit{Q}$$^{-1}$ determined from tidal phase lags or measured experimentally as being equal. The observed or theoretically predicted lag angle (or apparent $\textit{Q}$$^{-1}$) differs from the intrinsic, material property due to inertia, self-gravity and effects associated with the energy budget. By accounting for these differences we derive, for a special case, an expression that accurately maps apparent attenuation predicted using the extended normal mode formalism of Lau $\textit{et al.}$ into intrinsic attenuation. The theory allows for more generalized mappings which may be used to robustly connect observations and predictions of tidal lag angles to results from laboratory experiments of mantle materials.This work was supported by NSF EAR-1464024, NSF EAR-1215061, and Harvard University

Exact free oscillation spectra, splitting functions and the resolvability of earth's density structure

Seismic free oscillations, or normal modes, provide a convenient tool to calculate low-frequency seismograms in heterogeneous Earth models. A procedure called ‘full mode coupling’ allows the seismic response of the Earth to be computed. However, in order to be theoretically exact, such calculations must involve an infinite set of modes. In practice, only a finite subset of modes can be used, introducing an error into the seismograms. By systematically increasing the number of modes beyond the highest frequency of interest in the seismograms, we investigate the convergence of full-coupling calculations. As a rule-of-thumb, it is necessary to couple modes 1–2 mHz above the highest frequency of interest, although results depend upon the details of the Earth model. This is significantly higher than has previously been assumed. Observations of free oscillations also provide important constraints on the heterogeneous structure of the Earth. Historically, this inference problem has been addressed by the measurement and interpretation of splitting functions. These can be seen as secondary data extracted from low frequency seismograms. The measurement step necessitates the calculation of synthetic seismograms, but current implementations rely on approximations referred to as self- or group-coupling and do not use fully accurate seismograms. We therefore also investigate whether a systematic error might be present in currently published splitting functions. We find no evidence for any systematic bias, but published uncertainties must be doubled to properly account for the errors due to theoretical omissions and regularization in the measurement process. Correspondingly, uncertainties in results derived from splitting functions must also be increased. As is well known, density has only a weak signal in low-frequency seismograms. Our results suggest this signal is of similar scale to the true uncertainties associated with currently published splitting functions. Thus, it seems that great care must be taken in any attempt to robustly infer details of Earth's density structure using current splitting functions.The research leading to these results has received funding from the European Research Council (ERC) under the European Union’s Seventh Framework Programme (FP/2007-2013) grant agreement number 320639 (iGEO) and under the European Union’s Horizon 2020 research and innovation programme grant agreement number 681535 (ATUNE)

Silent onset of postmenopausal endometriosis in a woman with renal failure in hormone replacement therapy: a case report

<p>Abstract</p> <p>Introduction</p> <p>Postmenopausal endometriosis is a rare form of a common disease, since the absence of estrogenic hormone production should halt disease progression.</p> <p>Case presentation</p> <p>We present the case of a 54-year-old Italian Caucasian woman in surgical menopause with a history of ovarian endometriosis, who underwent voluntary hormone replacement therapy for seven years. She developed postrenal renal failure due to bilateral compression of the pelvic ureteral tract caused by two large, deeply infiltrating endometriotic nodules with no pelvic pain. She underwent operative laparoscopy with adhesiolysis of enteroenteric adhesions and excision of the endometriotic nodules encompassing the juxtavesical tract of the ureters, without obtaining improvement of renal failure.</p> <p>Conclusion</p> <p>Postmenopausal endometriosis can manifest itself in an unpredictable and potentially very serious manner. It is therefore important to carefully evaluate the risks and benefits of administering hormone replacement therapy to patients with previous endometriosis.</p