There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Thank you for reading principles of optics electromagnetic theory of propagation interference and diffraction of light. As you may know, people have search hundreds times for their favorite novels like this principles of optics electromagnetic theory of propagation interference and diffraction of light, but end up in harmful downloads. Rather than enjoying a good book with a cup of coffee in the afternoon, instead they are facing with some infectious bugs inside their computer.

Principles Of Optics Electromagnetic Theory Of Propagation Interference And Diffraction Of Light

The geology of northern and central Africa is characterized by broad plateaux, narrower swells and volcanism occurring from ∼45 Myr ago to the present. The greatest magma volumes occur on the >1,000-km-wide Ethiopian and east African plateaux, which are transected by the Red Sea, Gulf of Aden and east African rift systems, active since the late Oligocene epoch. Evidence for one or more mantle plumes having impinged beneath the plateaux comes from the dynamic compensation inferred from gravity studies, the generally small degrees of extension observed and the geochemistry of voluminous eruptive products1,2,3,4. Here we present a model of a single large plume impinging beneath the Ethiopian plateau that takes into account lateral flow and ponding of plume material in pre-existing zones of lithospheric thinning5. We show that this single plume can explain the distribution and timing of magmatism and uplift throughout east Africa. The thin lithosphere beneath the Mesozoic–Palaeogene rifts and passive margins of Africa and Arabia guides the lateral flow of plume material west to the Cameroon volcanic line and south to the Comoros Islands. Our results demonstrate the strong control that the lithosphere exerts on the spatial distribution of plume-related melting and magmatism.

Cenozoic magmatism throughout east Africa resulting from impact of a single plume

It is well accepted that the lithosphere may exhibit nonzero mechanical strength over geological time and space scales, associated with the existence of non-lithostatic (deviatoric) stress. The parameter that characterizes the apparent strength of the lithosphere is the flexural rigidity D, which is commonly expressed through the effective elastic thickness (Te) of the litho-sphere. Estimates of Te for oceanic lithosphere approximately follow the depth to a specific iso-therm (-600°C), which marks the base of the mechanical lithosphere. The physical meaning and significance of the effective elastic thickness for continents are still enigmatic, because for continental lithosphere estimates of Te bear little relation to specific geological or physical boundaries. Although high observed values of Te (70-90 km for cratons) can be partly explained by the present day temperature gradients, the low values (10-20 km), in general, cannot. In addition, the elastic plate models are self-inconsistent in that they mostly predict intraplate stresses high enough to lead to inelastic (brittle or ductile) deformation, according to data of rock mechanics. To provide a basis for a physically consistent unified interpretation of the observed variations of Te for continental and oceanic lithosphere, we developed an analytical and numerical approach that allows direct treatment of Te in terms of the lithospheric rheology, thermal structure, and strain/stress distribution. Our technique is based on finding true inelastic and equivalent (effective) elastic solutions for the problem of deformation of the lithosphere with realistic brittle-elasto-ductile rheology. We show that the thermal state (thermotectonic age) of the lithosphere is only one of at least three equally important properties that determine apparent values of Te. These other properties are the state of the crust-mantle interface (decoupling of crust and mantle), the thickness and proportions of the mechanically competent crust and mantle, and the local curvature of the plate, which is directly related to the bending stresses. The thickness of the mechanically competent crust and the degree of coupling or decoupling is generally controlled by composition of the upper and lower crust, total thickness of the crust, and by the crustal geotherm. If decoupling takes place, it permits as much as 50% decrease of Te, compared with Te implied from conventional thermal profiles. Comparison of the theoretically predicted Te with inferred values for different regions suggests that the lower crust of most continental plates has a low-temperature activation rheology (such as quartz) which permits crust and mantle decoupling. The curvature of the plate depends on the theological structure and on the distribution of external loads applied to the plate (e.g., surface topography, sediment fill, and plate-boundary forces). Bending stresses created by major mountain belts are large enough to cause inelastic deformation (brittle failure and a ductile flow) in the underlying plate, which, in turn, leads to a 30 to 80% decrease of Te beneath such belts and less beneath the adjacent regions. The boundary forces and moments (e.g., due to the slab pull, etc.) lead to more localized but even stronger reductions in Te (e.g., plate necking in subduction zones). Our approach provides a feedback between the "observed" Te and rheology, allowing to constrain the lithospheric structure from estimates of Te.

/pdf/the-effective-elastic-thickness-te-of-continental-43ks5eo3ne.pdf

The effective elastic thickness (Te) of continental lithosphere: What does it really mean?

Normal faulting in the upper continental crust: observations from regions of active extension

The broad topographic domes, or plateaus, of East Africa and Afar are characterized by long-wavelength negative Bouguer gravity anomalies and volcanically active rift valleys. Gravity and topography data from the East African and Afar plateaus and data from the stable cratonic regions to the west were subdivided into 17 smaller regions to study the variation of elastic plate thickness within part of the African continent, its relation to rifting processes within these intracontinental plateau regions, and compensation mechanisms for the broad uplifts. Assuming that loads at the surface, within, and beneath the base of a thin elastic plate contribute to the observed Bouguer gravity anomalies, the wavelength dependence of the coherence between gravity and topography was used to determine the effective elastic plate thickness in each of the subregions. Estimates of elastic plate thickness were found to be 64–90+ km in the stable cratonic areas, provided that surface and subsurface loads are uncorrelated. Lower estimates (43–49 km) were obtained in the largely unfaulted regions encompassing the broad uplifted plateaus and the narrower Darfur dome to the west of the Afar plateau. Estimates of 21–36 km correspond to regions that include the severely faulted and commonly volcanically active Kenya, Western, and Ethiopian rift valleys as well as unfaulted regions adjacent to the rift valleys. We attribute the smallest estimates of elastic plate thickness (21–36 km) to averaging unfaulted topography with mechanically weakened topography within the severely faulted Kenya, Western, and Ethiopian rifts. The linear transfer function between gravity and topography within the uplifted East African plateau region at wavelengths longer than 1000 km can be explained by a dynamic uplift mechanism and associated heating of the thermal lithosphere above a convecting region within the asthenosphere. These isostatic and dynamical compensation mechanisms are consistent with existing geological and geophysical data and with constraints on the timing of volcanism and uplift within the East African plateau region.

Effective elastic plate thickness beneath the East African and Afar plateaus and dynamic compensation of the uplifts

A technique for estimating flexural rigidity that is not limited to sedimentary basins is used here to map variations in the effective elastic thickness of the North American lithosphere. The effective elastic thickness ranges from a minimum of about 4 km in the Basin and Range Province to more than 100 km in the Precambrian core of the continent. This finding supports the idea that flexural rigidity has increased with time since the last thermal event.

Variations in effective elastic thickness of the North American lithosphere

The isostatic compensation of Australia is investigated using an isostatic model for the Australian lithosphere that assumes regional compensation of an elastic plate which undergoes flexure in response to surface and subsurface loading. Using the coherence between Bouguer gravity and topography and two separate gravity/topography data sets, it was found that, for the continent as a whole, loads with wavelengths above 1500 km are locally compensated. Loads with wavelengths in the range 600-1500 km are partially supported by regional stresses, and loads with wavelengths less than 600 km are almost entirely supported by the strength of the lithosphere. It was found that the predicted coherence for a flexural model of a continuous elastic plate does not provide a good fit to the observed coherence of central Australia. The disagreement between model and observations is explained.

/pdf/effective-elastic-thicknesses-of-the-lithosphere-and-2aygrdtlgp.pdf

Effective Elastic Thicknesses of the Lithosphere and Mechanisms of Isostatic Compensation in Australia

Summary

The topography and gravity anomalies in the area surrounding the East African Rift in Kenya can be modelled as the sum of the effects of surface and subsurface loading of an elastic plate. Assuming surface and subsurface loading are independent processes, the observed coherence between the 2-D Fourier transforms of Bouguer gravity and topography provides a constraint on the effective elastic thickness or flexural rigidity of the plate. Distinct linear segments of the log gravity power spectrum suggest that components of the gravity field with wavelengths of 250–1000 km are generated predominantly by a density contrast at a depth of about 32 km. Most shorter wavelength gravity anomalies are probably associated with source depths of less than 1 km and indicate variations in thickness of low density sedimentary or volcanic layers or lateral variations in density of the surface rocks. A simple density model based on these estimates and the geology of Kenya consists of a cover layer averaging 0.5 km thick with density 2300 kg m−3, a layer 32 km thick with density 2800 kg m−3, and an underlying half-space with density 3200 kg m−3. Using this density model and assuming loading due to relief on all three density interfaces, the elastic thickness that best predicts the observed coherence in the least squares sense is 25 km (Flexural rigidity of 1.4 × 1023Nm). Amplitudes of loads on the density interfaces can be calculated based on the model response. Topography with wavelength greater than 650 km is locally compensated making surface and subsurface loading indistinguishable. At shorter wavelengths, rift volcanics and volcanic cones including Mts Kenya, Elgon and Kilimanjaro can be identified as surface loads. The largest amplitude subsurface load is an upward directed, regionally supported load beneath the Kenya Dome that may correspond to a region of hot, low density mantle recognized by other geophysical studies. In creating the topography for which surface and sub-surface loading are distinguishable, surface loading is much more important than doming due to subsurface loading.

/pdf/mechanisms-of-isostatic-compensation-in-the-vicinity-of-the-40ldcujl04.pdf

Mechanisms of isostatic compensation in the vicinity of the East African Rift, Kenya

Holographic subsurface radars (HSR) are not in common usage now; possibly because of the historical view amongst radar practitioners that high attenuation of electromagnetic waves in most media of interest will not allow sufficient depth of penetration. It is true that the fundamental physics of HSR prevent the possibility to change receiver amplification with time (i.e., depth) to adapt to lossy media (as is possible with impulse subsurface radar or ISR). However, use of HSR for surveying of shallow subsurface objects, defects, or inhomogeneities is an increasingly proven area of application. In this case, HSR can record images with higher resolution than is possible for ISR images. The RASCAN family of holographic radars is presented along with technical specifications and typical case histories. Among the applications considered are civil and historic building surveys, nondestructive testing of dielectric materials, security applications, and humanitarian demining. Each application area is illustrated by relevant data acquired in laboratory experiments or field tests. This paper presents experiments with RASCAN imaging in media with different degrees of attenuation, and illustrates the principle of HSR through an optical analogy.

Timothy Bechtel

Papers

Effective elastic plate thickness beneath the East African and Afar plateaus and dynamic compensation of the uplifts

Variations in effective elastic thickness of the North American lithosphere

Effective Elastic Thicknesses of the Lithosphere and Mechanisms of Isostatic Compensation in Australia

Mechanisms of isostatic compensation in the vicinity of the East African Rift, Kenya

Holographic Subsurface Radar of RASCAN Type: Development and Applications