Complex Rays and Wave Packets for Decaying Signals in Inhomogeneous,Anisotropic and Anelastic Media

Diffraction and anelasticity problems involving decaying, “evanescent” or “inhomogeneous” waves can be studied and modelled using the notion of “complex rays”. The wavefront or “eikonal” equation for such waves is in general complex and leads to rays in complex position-slowness space. Initial conditions must be specified in that domain: for example, even for a wave originating in a perfectly elastic region, the ray to a real receiver in a neighbouring anelastic region generally departs from a complex point on the initial-values surface. Complex ray theory is the formal extension of the usual Hamilton equations to complex domains. Liouville's phase-space-incompressibility theorem and Fermat's stationary-time principle are formally unchanged. However, an infinity of paths exists between two fixed points in complex space all of which give the same final slowness, travel time, amplitude, etc. This does not contradict the fact that for a given receiver position there is a unique point on the initial-values surface from which this infinite complex ray family emanates.In perfectly elastic media complex rays are associated with, for example, evanescent waves in the shadow of a caustic. More generally, caustics in anelastic media may lie just outside the real coordinate subspace and one must trace complex rays around the complex caustic in order to obtain accurate waveforms nearby or the turning waves at greater distances into the lit region. The complex extension of the Maslov method for computing such waveforms is described. It uses the complex extension of the Legendre transformation and the extra freedom of complex rays makes pseudocaustics avoidable. There is no need to introduce a Maslov/KMAH index to account for caustics in the geometrical ray approximation, the complex amplitude being generally continuous. Other singular ray problems, such as the strong coupling around acoustic axes in anisotropic media, may also be addressed using complex rays.Complex rays are insightful and practical for simple models (e.g. homogeneous layers). For more complicated numerical work, though, it would be desirable to confine attention to real position coordinates. Furthermore, anelasticity implies dispersion so that complex rays are generally frequency dependent. The concept of group velocity as the velocity of a spatial or temporal maximum of a narrow-band wave packet does lead to real ray/Hamilton equations. However, envelope-maximum tracking does not itself yield enough information to compute synthetic seismogramsFor anelasticity which is weak in certain precise senses, one can set up a theory of real, dispersive wave-packet tracking suitable for synthetic seismogram calculations in linearly visco-elastic media. The seismologically-accepiable constant-Q rheology of Liu et al. (1976), for example, satisfies the requirements of this wave-packet theory, which is adapted from electromagnetics and presented as a reasonable physical and mathematical basis for ray modelling in inhomogeneous, anisotropic, anelastic media. Dispersion means that one may need to do more work than for elastic media. However, one can envisage perturbation analyses based on the ray theory presented here, as well as extensions like Maslov's which are based on the Hamiltonian properties.     

Scattering of Seismic Waves by Cracks with the Boundary Integral Method

— We develop a new scheme to compute 2-D SH seismograms for media with many flat cracks, based on the boundary integral method. A dry or traction-free boundary condition is applied to crack surfaces although other kinds of cracks such as wet or fluid-saturated cracks can be treated simply by assigning different boundary conditions. While body forces are distributed for cavities or inclusions to express scattered wave, dislocations (or displacement discontinuities between the top and the bottom surfaces of each crack) are used as fictitious sources along crack surfaces. With these dislocations as unknown coefficients, the scattered wave is expressed by the normal derivative of Green's function along the crack surface, which is called “double-layer potentials” in the boundary integral method, while we used “single-layer potentials” for cavities or inclusions. These unknowns are determined so that boundary conditions or crack surfaces are satisfied in the least-squared sense, for example, traction-free for dry cracks. Seismograms with plane-wave incidence are synthesized for homogeneous media with many cracks. First, we check the accuracy of our scheme for a medium with one long crack. All the predicted phases such as reflected wave, diffraction from a crack tip and shadow behind the crack are simulated quite accurately, under the same criterion as in the case for cavities or inclusions. Next, we compute seismograms for 50 randomly distributed cracks and compare them with those for circular cavities. When cracks are randomly oriented, waveforms and the strength of scattering attenuation are similar to the cavity case in a frequency range higher than k d $\simeq$ 2 where the size of scatterers d (i.e., crack length or cavity diameter) is comparable with the wavelength considered (k is the wavenumber). On the other hand, the scattering attenuation for cracks becomes much smaller in a lower frequency range (k d<2) because only the volume but not detail geometry of scatterers becomes important with wavelength much longer than each scatterer. When all the cracks are oriented in a fixed direction, the scattering attenuation depends strongly on the incident angle to the crack surface as frequency increases (k d>2): scattering becomes weak for cracks oriented parallel to the direction of the incident wave, while it gets close to the cavity case for cracks aligned perpendicular to the incident wave.     

Scattering diagrams in seismic imaging: More insights into the construction of virtual events and internal multiples

The key processes in marine seismic imaging include (i) removing from seismic data all seismic events (free-surface multiples and ghosts) which contain at least one reflection at the sea surface in their wave-propagation path, and leaving those with no reflection at the free surface (internal multiples and primaries), (ii) removing events with at least two reflections in the subsurface (internal multiples), and leaving events with only one reflection in the subsurface (primaries), and then (iii) locating the scattering points and reflectors inside the subsurface which are the sources of primaries and internal multiple events. All these processes are here explained, derived, and optimized via scattering diagrams (diagrammatica) in a way similar to the way the quantum field theory is often explained via Feynman diagrams. Our discussion of the removal of events with free-surface reflections from the data will be brief, as the diagrammatica of these events are now well understood.The main focus of this paper is the diagrammatica of internal multiples and primaries. Although these events do not contain any reflection at the sea surface, it is important to reconstruct them with scattering points near the sea surface, where seismic data are recorded. So our diagrammatica of primaries and internal multiples include events which are not directly recorded in seismic data but which can be constructed from seismic data. These events have allowed us to construct scattering diagrams of primaries and internal multiples with scattering points near the sea surface. Furthermore, these new diagrammatica of internal multiples and primaries can be used to remove internal multiples from the data.     

Seismic Precursors to Space Shuttle Shock Fronts

—?Seismic precursors to space shuttle re-entry shock fronts are detected at TXAR in Southwest Texas when the ground track of the orbiter vehicle passes within ～150–200?km of the observatory. These precursors have been termed “shuttle-quakes” because their seismograms superficially mimic the seismograms of small earthquakes from shallow sources. Analysis of the “shuttle-quake” seismograms, however, reveals one important difference. Unlike ordinary earthquakes, the propagation azimuths and horizontal phase velocities of the individual phases of the “shuttle-quakes” are functionally related. From a theoretical model developed to account for the origin of these precursors it is found that the seismic phases of “shuttle-quakes” are “bow” waves. A “bow” wave originates at the advancing tip of the shock front trace (i.e., intersection of the re-entry shock front with the surface of the earth) when the ground speed of the orbiter vehicle exceeds the horizontal phase velocity of a particular seismic phase. “Bow” waves are shown to differ in two important respects from the ordinary seismic phases. They vanish ahead of the advancing tip of the shock front trace and their propagation azimuths and horizontal phase velocities are functionally related. The ground speed of the orbiter vehicle exceeds the horizontal phase velocities of crustal seismic phase over much of the re-entry flight profile. As a result, P,S, and R _g “bow” waves will be seen as precursors to the re-entry shock front at stations located within a few hundred km of its ground track.     

NUMERICAL VLF MODELING*

The VLF response of laterally inhomogeneous and anisotropic models is calculated numerically using the finite element method. Some results are presented for a slab model in terms both of the polarization parameters, i.e., the tilt angle and ellipticity of the magnetic polarization ellipse, and the amplitude ratio |H_z/H_x|. On the basis of both the ellipticity and the tilt angle, it is possible to discriminate between a poor conductor and a good one. The direction of the dip can be determined from the anomaly profiles of all diagnostic parameters. The effect of the conductive overburden is most noticeable on the ellipticity profile: one observes attenuation for a poor conductor and “negative attenuation” for a good conductor. The anomaly profiles for anisotropic cases are consistent with the ones of the isotropic cases.     

Migration methods for imaging different-order multiples

Multiples contain valuable information about the subsurface, and if properly migrated can provide a wider illumination of the subsurface compared to imaging with VSP primary reflections. In this paper we review three different methods for migrating multiples. The first method is model-based, and it is more sensitive to velocity errors than primary migration; the second method uses a semi-natural Green's function for migrating multiples, where part of the traveltimes are computed from the velocity model, and part of the traveltimes (i.e., natural traveltimes) are picked from the data to construct the imaging condition for multiples; the third method uses cross-correlation of traces. The last two methods are preferred in the sense that they are significantly less sensitive to velocity errors and statics because they use “natural data” to construct part of the migration imaging conditions. Compared with the interferometric (i.e., crosscorrelation) imaging method the semi-natural Green's function method is more computationally efficient and is sometimes less prone to migration artifacts. Numerical tests with 2-D and 3-D VSP data show that a wider subsurface coverage, higher-fold and more balanced illumination of the subsurface can be achieved with multiple migration compared with migration of primary reflections only. However, there can be strong interference from multiples with different orders or primaries when multiples of high order are migrated. One possible solution is to filter primaries and different orders of multiples before migration, and another possible solution is least squares migration of all events. A limitation of multiple migration is encountered for subsalt imaging. Here, the multiples must pass through the salt body more than twice, which amplifies the distortion of the image.     

Stochastic Modeling of Groundwater Fluoride Contamination: Introducing Lazy Learners

While it remains the primary source of safe drinking and irrigation water in northwest Iran's Maku Plain, the region's groundwater is prone to fluoride contamination. Accordingly, modeling techniques to accurately predict groundwater fluoride concentration are required. The current paper advances several novel data mining algorithms including Lazy learners [instance-based K-nearest neighbors (IBK); locally weighted learning (LWL); and KStar], a tree-based algorithm (M5P), and a meta classifier algorithm [regression by discretization (RBD)] to predict groundwater fluoride concentration. Drawing on several groundwater quality variables (e.g., concentrations), measured in each of 143 samples collected between 2004 and 2008, several models predicting groundwater fluoride concentrations were developed. The full dataset was divided into two subsets: 70% for model training (calibration) and 30% for model evaluation (validation). Models were validated using several statistical evaluation criteria and three visual evaluation approaches (i.e., scatter plots, Taylor and Violin diagrams). Although Na⁺ and Ca²⁺ showed the greatest positive and negative correlations with fluoride (r = 0.59 and −0.39, respectively), they were insufficient to reliably predict fluoride levels; therefore, other water quality variables, including those weakly correlated with fluoride, should be considered as inputs for fluoride prediction. The IBK model outperformed other models in fluoride contamination prediction, followed by KStar, RBD, M5P, and LWL. The RBD and M5P models were the least accurate in terms of predicting peaks in fluoride concentration values. Results of the current study can be used to support practical and sustainable management of water and groundwater resources.     

Complex active regions as the main source of extreme and large solar proton events

A study of solar proton sources indicated that solar flare events responsible for ≥2000 pfu proton fluxes mostly occur in complex active regions (CARs), i.e., in transition structures between active regions and activity complexes. Different classes of similar structures and their relation to solar proton events (SPEs) and evolution, depending on the origination conditions, are considered. Arguments in favor of the fact that sunspot groups with extreme dimensions are CARs are presented. An analysis of the flare activity in a CAR resulted in the detection of “physical” boundaries, which separate magnetic structures of the same polarity and are responsible for the independent development of each structure.     

Analysis of Thomsen parameters for finely layered VTI media

Since the work of Postma and Backus, much has been learned about elastic constants in vertical transversely isotropic (VTI) media when the anisotropy is due to fine layering of isotropic elastic materials. Nevertheless, there has continued to be some uncertainty about the possible range of Thomsen's anisotropy parameters ε and δ for such media. We use both Monte Carlo studies and detailed analysis of Backus' equations for both two- and three-component layered media to establish the results presented. We show that ε lies in the range ?3/8 ε ½[〈v²_p〉〈v^?2_p〉?1], for finely layered media; smaller positive and all negative values of ε occur for media with large fluctuations in the Lamé parameter λ in the component layers. We show that δ can also be either positive or negative, and that for constant density media, sign (δ) = sign (〈v^?2_p〉 ? 〈v^?2_s〉〈v²_s/v²_p〉). Monte Carlo simulations show that among all theoretically possible random media, positive and negative δ are equally likely in finely layered media. (Of course, the δs associated with real earth materials may span some smaller subset of those that are theoretically possible, but answering this important question is beyond our present scope.) Layered media having large fluctuations in λ are those most likely to have positive δ. This is somewhat surprising since ε is often negative or a small positive number for such media, and we have the general constraint that ε ? δ > 0 for layered VTI media. Since Gassmann's results for fluid-saturated porous media show that the mechanical effects of fluids influence only the Lamé parameter λ, not the shear modulus μ, these results suggest that small positive δ occurring together with small positive ε (but somewhat larger than δ) may be indicative of changing fluid content in a layered earth.     

Global Change and Health — The Good, The Bad and The Evidence

Debates over the merits and demerits of globalisation for health are increasingly polarised. Conclusions range from globalisation being essentially positive for health, albeit with a need to smooth out some rough edges, to one of utter condemnation, with adverse effects on the majority of the world's population. Anyone wading into this debate is immediately confronted by seemingly irreconcilable differences in ideology, opinion and interests. Both camps agree that global changes are occurring, and with them many of the determinants of population health status. While some skepticism persists about whether “globalisation” has value beyond being a fashionable buzzword, most agree that we need better understanding of these changes. Two difficult questions arise: (i) What are the health impacts of these changes; and (ii) how can we respond more effectively to them? To move beyond the stand-offs that have already formed within the health community, this paper reviews the main empirical evidence that currently exists, summarises key points of debate that remain, and suggests some ways forward for the research and policy communities. In particular, there is need for an informed and inclusive debate about the positive and negative health consequences of globalisation.     

Constrained Regional Recovery of Continental Water Mass Time-variations from GRACE-based Geopotential Anomalies over South America

We propose a “constrained” least-squares approach to estimate regional maps of equivalent-water heights by inverting GRACE-based potential anomalies at satellite altitude. According to the energy integral method, the anomalies of difference of geopotential between the two GRACE vehicles are derived from along-track K-Band Range-Rate (KBRR) residuals that correspond mainly to the continental water storage changes, once a priori known accelerations (i.e. static field, polar movements, atmosphere and ocean masses including tides) are removed during the orbit adjustment process. Newton's first law merely enables the Difference of Potential Anomalies from accurate KBRR data and the equivalent-water heights to be recovered. Spatial constraints versus spherical distance between elementary surface tiles are introduced to stabilize the linear system to cancel the effects of the north-south striping. Unlike the “mascons” approach, no basis of orthogonal functions (e.g., spherical harmonics) is used, so that the proposed regional method does not suffer from drawbacks related to any spectrum truncation. Time series of 10-day regional maps over South America for 2006–2009 also prove to be consistent with independent data sets, namely the outputs of hydrological models, “mascons” and global GRACE solutions.     

Intraseasonal predictability of the duration of flooding above National Weather Service flood warning levels across the U.S. Midwest

Unpreparedness is often the main cause of the economic and social damages caused by floods. To mitigate these impacts, short-term forecasting has been the focus of several studies during the past decades; however, less effort has been paid to flood predictions at longer lead times. Here, we use forecasts by six models from the North American Multi-Model Ensemble project with a lead time from 0.5 to 9.5 months to predict the seasonal duration of floods above four National Weather Service flood categories (“action,” “flood,” “moderate” and “major”). We focus on 202 U.S. Geological Survey gage stations across the U.S. Midwest and use a statistical framework which considers precipitation, temperature, and antecedent wetness conditions as predictors. We find that the prediction skill of the duration of floods for the “action” and “flood” categories is overall low, largely because of the low accuracy of the climate forecasts rather than of the errors introduced by the statistical models. The prediction skill slightly improves when considering the shortest lead times (i.e., from 0.5 to 2.5 months) during spring in the Northern Great Plains, where antecedent wetness conditions play an important role in influencing the generation of floods. It is very difficult to draw strong conclusions with respect to the “moderate” and “major” flood categories because of the limited number of available events.     

Models and Design of Aerobic Biological Treatment Processes

Various models, applied to the design of aeration tanks, trickling filters and rotating discs, are compared. The adequacy of the models is established from the minimal mean square deviation of the measured and calculated values of the effluent concentration of a pollutant. It is shown that at the oxidation of simple substrates of the glucose type Moser's model is an adequate model and Monod's model is a particular case of this model. If a single substrate has a large concentration, activated sludge or biofilm become “saturated” and the oxidation kinetics follows the zero order. As the processing of the results of some experiments showed, the classical Monod model and its extreme case—the first-order equation of a reaction—are not adequate to the processes of the oxidation of multicomponent municipal sewage water. Equations for reactions of higher orders are true for them. The rate constant of the first-order reaction correlates well with the effluent concentration-influent concentration ratio, i.e., it is variable and depends on the treatment conditions and, above all, it is inversely proportional to the level of treatment.     

Global desertification: Drivers and feedbacks

Desertification is a change in soil properties, vegetation or climate, which results in a persistent loss of ecosystem services that are fundamental to sustaining life. Desertification affects large dryland areas around the world and is a major cause of stress in human societies. Here we review recent research on the drivers, feedbacks, and impacts of desertification. A multidisciplinary approach to understanding the drivers and feedbacks of global desertification is motivated by our increasing need to improve global food production and to sustainably manage ecosystems in the context of climate change. Classic desertification theories look at this process as a transition between stable states in bistable ecosystem dynamics. Climate change (i.e., aridification) and land use dynamics are the major drivers of an ecosystem shift to a “desertified” (or “degraded”) state. This shift is typically sustained by positive feedbacks, which stabilize the system in the new state. Desertification feedbacks may involve land degradation processes (e.g., nutrient loss or salinization), changes in rainfall regime resulting from land-atmosphere interactions (e.g., precipitation recycling, dust emissions), or changes in plant community composition (e.g., shrub encroachment, decrease in vegetation cover). We analyze each of these feedback mechanisms and discuss their possible enhancement by interactions with socio-economic drivers. Large scale effects of desertification include the emigration of “environmental refugees” displaced from degraded areas, climatic changes, and the alteration of global biogeochemical cycles resulting from the emission and long-range transport of fine mineral dust. Recent research has identified some possible early warning signs of desertification, which can be used as indicators of resilience loss and imminent shift to desert-like conditions. We conclude with a brief discussion on some desertification control strategies implemented in different regions around the world.     

Heavy tailed probability distributions for non-Gaussian simulations with higher-order cumulant parameters predicted from sample data

Forecasting of extreme events and phenomena that respond to non-Gaussian heavy-tailed distributions (e.g., extreme environmental events, rock permeability, rock fracture intensity, earthquake magnitudes) is essential to environmental and geoscience risk analysis. In this paper, new parametric heavy-tailed distributions are devised starting from the exponential power probability density function (pdf) which is modified by explicitly including higher-order “cumulant parameters” into the pdf. Instead of dealing with whole power random variables, novel “residual” random variables are proposed to reconstruct the cumulant generating function. The expected value of a residual random variable with the corresponding pdf for order G, gives the input higher-order cumulant parameter. Thus, each parametric pdf is used to simulate a random variable containing residuals that yield, in average, the expected cumulant parameter. The cumulant parameters allow the formulation of heavy-tailed skewed pdfs beyond the lognormal to handle extreme events. Monte Carlo simulation of heavy-tailed distributions with higher-order parameters is demonstrated with a simple example for permeability.     

山西地区震源新参数间关系初探

利用“十五”数字化后山西地震台网记录的612次 ML ≥2．0地震,使用中国地震局地震预测研究所推广的中小地震新参数计算软件,计算其中148次事件的震源新参数,对地震矩、矩震级、震源破裂尺度、应力降、拐角频率等参数间的定标关系做初步研究,得出 ML 震级与地震矩、矩震级间存在较好的线性正相关;震级与震源尺度间存在多项式关系;与应力降的对数呈显著正相关,当大于 ML 4．0后,随着震级的增大,释放的应力降略有变缓;与拐角频率的对数呈负相关。     

Burial stress and elastic strain of carbonate rocks

Burial stress on a sediment or sedimentary rock is relevant for predicting compaction or failure caused by changes in, e.g., pore pressure in the subsurface. For this purpose, the stress is conventionally expressed in terms of its effect: “the effective stress” defined as the consequent elastic strain multiplied by the rock frame modulus. We cannot measure the strain directly in the subsurface, but from the data on bulk density and P‐wave velocity, we can estimate the rock frame modulus and Biot's coefficient and then calculate the “effective vertical stress” as the total vertical stress minus the product of pore pressure and Biot's coefficient. We can now calculate the elastic strain by dividing “effective stress” with the rock frame modulus. By this procedure, the degree of elastic deformation at a given time and depth can be directly expressed. This facilitates the discussion of the deformation mechanisms. The principle is illustrated by comparing carbonate sediments and sedimentary rocks from the North Sea Basin and three oceanic settings: a relatively shallow water setting dominated by coarse carbonate packstones and grainstones and two deep water settings dominated by fine‐grained carbonate mudstones and wackestones.     

General-planetary character of three-dimensional planetary waves propagation in lower and upper Earth atmosphere

A new branch of planetary Rossby waves is found taking into account latitudinal gradient of horizontal component of the earth rotation's angular velocity. Frequencies of a new branch and ordinary Rossby waves numerically coincide but strongly differ on character of dispersion. The general condition naturally “filtering” planetary waves in the long-wave approximation and internal inertial waves in the short-wave approximation was established.     

Applications of Directional Wavefield Decomposition, Phase Space, and Path Integral Methods to Seismic Wave Propagation and Inversion

— Recently, de Hoop and coworkers developed an asymptotic, seismic inversion formula for application in complex environments supporting multi-pathed and multi-mode wave propagation (de Hoop et al., 1999; de Hoop and Brandsberg-Dahl, 2000; Stolk and de Hoop, 2000). This inversion is based on the Born/Kirchhoff approximation, and employs the global, uniform asymptotic extension of the geometrical method of “tracing rays” to account for caustic phenomena. While this approach has successfully inverted the multicomponent, ocean-bottom data from the Valhall field in Norway, accounting for severe focusing effects (de Hoop and Brandsberg-Dahl, 2000), it is not able to account properly for wave phenomena neglected in the “high-frequency” limit (i.e., diffraction effects) and strong scattering effects. To proceed further and incorporate wave effects in a nonlinear inversion scheme, the theory of directional wavefield decomposition and the construction of the generalized Bremmer coupling series are combined with the application of modern phase space and path (functional) integral methods to, ultimately, suggest an inversion algorithm which can be interpreted as a method of “tracing waves.” This paper is intended to provide the seismic community with an introduction to these approaches to direct and inverse wave propagation and scattering, intertwining some of the most recent new results with the basic outline of the theory, and culminating in an outline of the extended, asymptotic, seismic inversion algorithm. Modeling at the level of the fixed-frequency (elliptic), scalar Helmholtz equation, exact and uniform asymptotic constructions of the well-known, and fundamentally important, square-root Helmholtz operator (symbol) provide the most important results.     

Note on Coriolis-Stokes force and energy

In this study, we consider the origin of the Coriolis-Stokes (CS) force in the wave-averaged momentum and energy equations and make a short analysis of possible energy input to the ocean circulation (i.e., Eulerian mean velocity) from the CS force. Essentially, we find that the CS force appears naturally when considering vertically integrated quantities and that the CS force will not provide any energy input into the system for this case. However, by including the “Hasselmann force”, we show some inconsistencies regarding the vertical structure of the CS force in the Eulerian framework and find that there is a distinct vertical structure of the energy input and that the net input strongly depends on whether the wave zone is included in the analysis or not. We therefore question the introduction of the “Hasselmann force” into the system of equations, as the CS force appears naturally in the vertically integrated equations or when Lagrangian vertical coordinates are used.