Real and Complex Analysis

Compare your culture to one of the cultures discussed in this unit. On a sheet of paper, list the cultures you are comparing and make one column titled “similarities,” and a second column titled “differences.” Now, list as many similarities and differences between the two as you can think of. Are there more similarities or differences between the two cultures you selected? Have you ever met anyone from this culture? How can you use this information to build greater respect between cultures?

Similarities and Differences

WITH the ever-widening scope of modern mathematical analysis and its many ramifications, it is quite impossible to include, in a single volume of reasonable size, an adequate and exhaustive discussion of the calculus in its more advanced stages. It therefore becomes necessary, in planning a thoroughly sound course in the subject, to consider several important aspects of the vast field confronting a modern writer. The limitation of space renders the selection of subject-matter fundamentally dependent upon the aim of the course, which may or may not be related to the content of specific examination syllabuses. Logical development, too, may lead to the inclusion of many topics which, at present, may only be of academic interest, while others, of greater practical value, may have to be omitted. The experience and training of the writer may also have, more or less, a bearing on both these considerations.Advanced CalculusBy Dr. C. A. Stewart. Pp. xviii + 523. (London: Methuen and Co., Ltd., 1940.) 25s.

Advanced Calculus I

Volatiles that are transported by subducting lithospheric plates to depths greater than 100 km are thought to induce partial melting in the overlying mantle wedge, resulting in arc magmatism and the addition of significant quantities of material to the overlying lithosphere. Asthenospheric flow and upwelling within the wedge produce increased lithospheric temperatures in this back-arc region, but the forearc mantle (in the corner of the wedge) is thought to be significantly cooler. Here we explore the structure of the mantle wedge in the southern Cascadia subduction zone using scattered teleseismic waves recorded on a dense portable array of broadband seismometers. We find very low shear-wave velocities in the cold forearc mantle indicated by the exceptional occurrence of an 'inverted' continental Moho, which reverts to normal polarity seaward of the Cascade arc. This observation provides compelling evidence for a highly hydrated and serpentinized forearc region, consistent with thermal and petrological models of the forearc mantle wedge. This serpentinized material is thought to have low strength and may therefore control the down-dip rupture limit of great thrust earthquakes, as well as the nature of large-scale flow in the mantle wedge.

/pdf/an-inverted-continental-moho-and-serpentinization-of-the-qck5yka1nh.pdf

An inverted continental Moho and serpentinization of the forearc mantle

Problem Given the number of times in which an unknown event has happened and failed: Required the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named. SECTION 1 Definition 1. Several events are inconsistent, when if one of them happens, none of the rest can. 2. Two events are contrary when one, or other of them must; and both together cannot happen. 3. An event is said to fail, when it cannot happen; or, which comes to the same thing, when its contrary has happened. 4. An event is said to be determined when it has either happened or failed. 5. The probability of any event is the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the value of the thing expected upon it’s 2 happening.

An essay towards solving a problem in the doctrine of chances. [Facsimil]

This is the first paper in a three-part series that examines formal inversion of the teleseismic P wave coda for discontinuous variations in elastic properties beneath dense, three-component, seismic arrays In this paper, we develop the theoretical framework for a migration method that draws upon the tenets of inverse scattering theory and is amenable to practical implementation The forward problem is formulated for two-dimensional (2-D) heterogeneity in observance of formal sampling requirements and currently accessible instrumentation A ray theoretic Green's function, corresponding to a line source with axial component of forcing, is employed within the 2-D Born approximation to accommodate planar, incident wave fields at arbitrary back azimuths Both the forward scattered response generated by the upgoing incident wave field and the backscattered response created by its reflection at the free surface are included within the formulation In accordance with the high-frequency and single-scattering approximations employed in the forward problem the inverse problem is cast as a generalized Radon transform The resulting back projection operator is well suited to the teleseismic context in several respects It is tolerant of irregularities in array geometry and source distribution and allows a full complement of global seismicity to be utilized through its accommodation of oblique incidence By permitting both independent and simultaneous treatment of different scattering modes (reflections, transmissions, conversions) the inversion formula facilitates a direct appraisal of individual mode contributions to the recovery of structure In particular, it becomes evident that incorporation of backscattered modes leads to (1) a better localization of structure than possible using forward scattered energy and (2) the imposition of complementary constraints on elastic properties

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2001JB000330

Multiparameter two-dimensional inversion of scattered teleseismic body waves 1. Theory for oblique incidence

This is the third paper in a three-part series that examines formal inversion of the teleseismic P wave coda for discontinuous, two-dimensional (2-D) variations in elastic properties beneath dense arrays of three-component, broadband seismometers. In this paper, the method is applied to data from the Incorporated Research Institutions for Seismology-Program for Array Seismic Studies of the Continental Lithosphere (IRIS-PASSCAL) Cascadia 1993 experiment undertaken across central Oregon. Two major features are imaged in the resulting model. The continental Moho becomes evident � 150 km from the coast beneath the Western Cascades and extends through the eastern end of the profile at 35-40 km depth. In the western portion of the model, oceanic crust of the subducting Juan de Fuca plate dips shallowly (12� ) at the coast and more steeply (27� ) below the Willamette Valley and is evident to depths of >100 km beneath the High Cascades. The abrupt increase in plate dip at � 40 km depth coincides with an apparent thickening of the oceanic crust followed by a diminution in its signature. Building on previous work, we argue that these results are consistent with the consequences of prograde metamorphic reactions occurring within the oceanic crust. Progressive dehydration at lower-grade facies conditions culminates in the transformation to eclogite, producing a pronounced increase in the seismic velocity, density and dip of the subducting plate, and structural complexity in the overlying wedge.

/pdf/multiparameter-two-dimensional-inversion-of-scattered-2w7we0g7gb.pdf

Multiparameter two-dimensional inversion of scattered teleseismic body waves 3. Application to the Cascadia 1993 data set

Efficiently modeling seismic data sets in complex 3D anisotropic media by solving the 3D elastic wave equation is an important challenge in computational geophysics. Using a stress-stiffness formulation on a regular grid, we tested a 3D finite-difference time-domain solver using a second-order temporal and eighth-order spatial accuracy stencil that leverages the massively parallel architecture of graphics processing units (GPUs) to accelerate the computation of key kernels. The relatively small memory of an individual GPU limits the model domain sizes that can be computed on a single device. To circumvent this constraint and move toward modeling industry-sized 3D anisotropic elastic data sets, we parallelized computation across multiple GPU devices by using domain decomposition and, for each time step, employing an interdevice communication protocol to exchange data values falling within interior boundaries of each subdomain. For two or more GPU devices within a single compute node, we use direct peer-to-peer (i.e., GPU-to-GPU) communication, whereas for networked nodes we employed message-passing interface directives to route data over the network. Our 2D GPU-based anisotropic elastic modeling tests achieved a 10× speedup relative to an OpenMP CPU implementation run on an eight-core machine, whereas our 3D tests using dual-GPU devices produced up to a 28× speedup. The performance boost afforded by the GPU architecture allowed us to model seismic data for 3D anisotropic elastic models at lower hardware cost and in less time than has been previously possible.

Solving 3D anisotropic elastic wave equations on parallel GPU devices

In this paper, we investigate the formal inversion of synthetic teleseismic P coda waves for subsurface elastic properties using a ray theoretic approach which assumes single scattering [Bostock et al., this issue]. We consider a model comprising an idealized lithospheric suture zone whose geometrical configuration is drawn from previous deep crustal seismic studies. Two-dimensional, pseudospectral synthetic seismograms representing plane waves propagating through this model are preprocessed to extract an estimate of the scattered wave field generated by short-wavelength structure. These data are employed in a series of numerical simulations which examine the dependence of multiparameter inversion results on a range of input parameters. In particular, we demonstrate (1) the contrasting sensitivity which forward and backscattered waves display to structural recovery, (2) the diminution of the problem null-space accompanied by increased source coverage, (3) improvements in model reconstruction achieved through simultaneous treatment of multiple scattering modes, and (4) the robustness of the method for data sets with noise levels and receiver geometries that approach those of field experiments.

http://sepwww.stanford.edu/sep/jeff/BRS2.pdf

Multiparameter two‐dimensional inversion of scattered teleseismic body waves 2. Numerical examples

Migration of seismic data from topography using methods based on finite-difference (FD) approximation to acoustic wave propagation commonly suffers from a number of imaging drawbacks due to the difficulty of applying FD stencils to irregular computational meshes. Altering the computational geometry from Cartesian to a topographic coordinate system conformal to the data acquisition surface can circumvent many of these issues. The coordinate transformation approach allows for acoustic wave propagation and the crosscorrelation and inverse-scattering imaging conditions to be posed and computed directly in topographic coordinates. Resulting reverse time migration (RTM) images may then be interpolated back to the Cartesian domain using the known inverse mapping. Orthogonal 2D topographic coordinates can be developed using known conformal mapping transforms and serve as the computational mesh for performing migration from topography. Impulse response tests demonstrate the accuracy of the 2D generalized a...

Jeffrey Shragge

Papers

Multiparameter two-dimensional inversion of scattered teleseismic body waves 1. Theory for oblique incidence

Multiparameter two-dimensional inversion of scattered teleseismic body waves 3. Application to the Cascadia 1993 data set

Solving 3D anisotropic elastic wave equations on parallel GPU devices

Multiparameter two‐dimensional inversion of scattered teleseismic body waves 2. Numerical examples

Reverse time migration from topography