Attenuation tomography. Modelling regional love waves: Imperial Valley to Pasadena

Abstract to Part I

The inverse problem of seismic wave attenuation is solved by an iterative back-projection method. The seismic wave quality factor, Q, can be estimated approximately by inverting the S-to-P amplitude ratios. Effects of various uncertain ties in the method are tested and the attenuation tomography is shown to be useful in solving for the spatial variations in attenuation structure and in estimating the effective seismic quality factor of attenuating anomalies.

Back-projection attenuation tomography is applied to two cases in southern California: Imperial Valley and the Coso-Indian Wells region. In the Coso-Indian Wells region, a highly attenuating body (S-wave quality factor (Q_β ≈ 30) coincides with a slow P-wave anomaly mapped by Walck and Clayton (1987). This coincidence suggests the presence of a magmatic or hydrothermal body 3 to 5 km deep in the Indian Wells region. In the Imperial Valley, slow P-wave travel-time anomalies and highly attenuating S-wave anomalies were found in the Brawley seismic zone at a depth of 8 to 12 km. The effective S-wave quality factor is very low (Q_β ≈ 20) and the P-wave velocity is 10% slower than the surrounding areas. These results suggest either magmatic or hydrothermal intrusions, or fractures at depth, possibly related to active shear in the Brawley seismic zone.

No-block inversion is a generalized tomographic method utilizing the continuous form of an inverse problem. The inverse problem of attenuation can be posed in a continuous form , and the no-block inversion technique is applied to the same data set used in the back-projection tomography. A relatively small data set with little redundancy enables us to apply both techniques to a similar degree of resolution. The results obtained by the two methods are very similar. By applying the two methods to the same data set, formal errors and resolution can be directly computed for the final model, and the objectivity of the final result can be enhanced.

Both methods of attenuation tomography are applied to a data set of local earthquakes in Kilauea, Hawaii, to solve for the attenuation structure under Kilauea and the East Rift Zone. The shallow Kilauea magma chamber, East Rift Zone and the Mauna Loa magma chamber are delineated as attenuating anomalies. Detailed inversion reveals shallow secondary magma reservoirs at Mauna Ulu and Puu Oo, the present sites of volcanic eruptions. The Hilina Fault zone is highly attenuating, dominating the attenuating anomalies at shallow depths. The magma conduit system along the summit and the East Rift Zone of Kilauea shows up as a continuous supply channel extending down to a depth of approximately 6 km. The Southwest Rift Zone, on the other hand, is not delineated by attenuating anomalies, except at a depth of 8-12 km, where an attenuating anomaly is imaged west of Puu Kou. The Ylauna Loa chamber is seated at a deeper level (about 6-10 km) than the Kilauea magma chamber. Resolution in the Mauna Loa area is not as good as in the Kilauea area, and there is a trade-off between the depth extent of the magma chamber imaged under Mauna Loa and the error that is due to poor ray coverage. Kilauea magma chamber, on the other hand, is well resolved, according to a resolution test done at the location of the magma chamber.

Abstract to Part II

Long period seismograms recorded at Pasadena of earthquakes occurring along a profile to Imperial Valley are studied in terms of source phenomena (e.g., source mechanisms and depths) versus path effects. Some of the events have known source parameters, determined by teleseismic or near-field studies, and are used as master events in a forward modeling exercise to derive the Green's functions (SH displacements at Pasadena that are due to a pure strike-slip or dip-slip mechanism) that describe the propagation effects along the profile. Both timing and waveforms of records are matched by synthetics calculated from 2-dimensional velocity models. The best 2-dimensional section begins at Imperial Valley with a thin crust containing the basin structure and thickens towards Pasadena. The detailed nature of the transition zone at the base of the crust controls the early arriving shorter periods (strong motions), while the edge of the basin controls the scattered longer period surface waves. From the waveform characteristics alone, shallow events in the basin are easily distinguished from deep events, and the amount of strike-slip versus dip-slip motion is also easily determined. Those events rupturing the sediments, such as the 1979 Imperial Valley earthquake, can be recognized easily by a late-arriving scattered Love wave that has been delayed by the very slow path across the shallow valley structure.

Steady capillary-gravity waves on deep water

The properties of capillary-gravity waves of permanent form on deep water are studied. Two different formulations to the problem are given. The theory of simple bifurcation is reviewed. For small amplitude waves a formal perturbation series is used. The Wilton ripple phenomenon is reexamined and shown to be associated with a bifurcation in which a wave of permanent form can double its period. It is shown further that Wilton's ripples are a special case of a more general phenomenon in which bifurcation into subharmonics and factorial higher harmonics can occur. Numerical procedures for the calculation of waves of finite amplitude are developed. Bifurcation and limit lines are calculated. Pure and combination waves are continued to maximum amplitude. It is found that the height is limited in all cases by the surface enclosing one or more bubbles. Results for the shape of gravity waves are obtained by solving an integra-differential equation. It is found that the family of solutions giving the waveheight or equivalent parameter has bifurcation points. Two bifurcation points and the branches emanating from them are found specifically, corresponding to a doubling and tripling of the wavelength. Solutions on the new branches are calculated.

Some problems of edge waves and standing waves on beaches

Some problems of edge waves and standing waves on beaches are examined.

The nonlinear interaction of a wave normally incident on a sloping beach with a subharmonic edge wave is studied. A two-timing expansion is used in the full nonlinear theory to obtain the modulation equations which describe the evolution of the waves. It is shown how large amplitude edge waves are produced; and the results of the theory are compared with some recent laboratory experiments.

Traveling edge waves are considered in two situations. First, the full linear theory is examined to find the finite depth effect on the edge waves produced by a moving pressure disturbance. In the second situation, a Stokes' expansion is used to discuss the nonlinear effects in shallow water edge waves traveling over a bottom of arbitrary shape. The results are compared with the ones of the full theory for a uniformly sloping bottom.

The finite amplitude effects for waves incident on a sloping beach, with perfect reflection, are considered. A Stokes' expansion is used in the full nonlinear theory to find the corrections to the dispersion relation for the cases of normal and oblique incidence.

Finally, an abstract formulation of the linear water waves problem is given in terms of a self adjoint but nonlocal operator. The appropriate spectral representations are developed for two particular cases.

Topics in linear and nonlinear dispersive waves

The various singularities and instabilities which arise in the modulation theory of dispersive wavetrains are studied. Primary interest is in the theory of nonlinear waves, but a study of associated questions in linear theory provides background information and is of independent interest.

The full modulation theory is developed in general terms. In the first approximation for slow modulations, the modulation equations are solved. In both the linear and nonlinear theories, singularities and regions of multivalued modulations are predicted. Higher order effects are considered to evaluate this first order theory. An improved approximation is presented which gives the true behavior in the singular regions. For the linear case, the end result can be interpreted as the overlap of elementary wavetrains. In the nonlinear case, it is found that a sufficiently strong nonlinearity prevents this overlap. Transition zones with a predictable structure replace the singular regions.

For linear problems, exact solutions are found by Fourier integrals and other superposition techniques. These show the true behavior when breaking modulations are predicted.

A numerical study is made for the anharmonic lattice to assess the nonlinear theory. This confirms the theoretical predictions of nonlinear group velocities, group splitting, and wavetrain instability, as well as higher order effects in the singular regions.

Nonlinear dispersive waves in nonlinear optics

A study is made of solutions of the macroscopic Maxwell equations in nonlinear media. Both nonlinear and dispersive terms are responsible for effects that are not taken into account in the geometrical optics approximation. The nonlinear terms can, depending on the nature of the nonlinearity, cause plane waves to focus when the amplitude varies across the wavefront. The dispersive terms prevent the singularities that nonlinearity alone would produce. Solutions are found which de scribe periodic plane waves in fully nonlinear media. Equations describing the evolution of the amplitude, frequency and wave number are generated by means of averaged Lagrangian techniques. The equations are solved for near linear media to produce the form of focusing waves which develop a singularity at the focal point. When higher dispersion is included nonlinear and dispersive effects can balance and one finds amplitude profiles that propagate with straight rays.

Nonexistence of looping trajectories in hydromagnetic waves of finite amplitude. Breaking of waves in a cold collision-free plasma in a magnetic field. On stabilities of periodic waves in a cold collision-free plasma in a magnetic field.

This dissertation consists of three parts. In Part I, it is shown that looping trajectories cannot exist in finite amplitude stationary hydromagnetic waves propagating across a magnetic field in a quasi-neutral cold collision-free plasma. In Part II, time-dependent solutions in series expansion are presented for the magnetic piston problem, which describes waves propagating into a quasi-neutral cold collision-free plasma, ensuing from magnetic disturbances on the boundary of the plasma. The expansion is equivalent to Picard's successive approximations. It is then shown that orbit crossings of plasma particles occur on the boundary for strong disturbances and inside the plasma for weak disturbances. In Part III, the existence of periodic waves propagating at an arbitrary angle to the magnetic field in a plasma is demonstrated by Stokes expansions in amplitude. Then stability analysis is made for such periodic waves with respect to side-band frequency disturbances. It is shown that waves of slow mode are unstable whereas waves of fast mode are stable if the frequency is below the cutoff frequency. The cutoff frequency depends on the propagation angle. For longitudinal propagation the cutoff frequency is equal to one-fourth of the electron's gyrofrequency. For transverse propagation the cutoff frequency is so high that waves of all frequencies are stable.

Part I. On the breaking of nonlinear dispersive waves. Part II. Variational principles in continuum mechanics

A model equation for water waves has been suggested by Whitham to study, qualitatively at least, the different kinds of breaking. This is an integro-differential equation which combines a typical nonlinear convection term with an integral for the dispersive effects and is of independent mathematical interest. For an approximate kernel of the form e^(-b|x|) it is shown first that solitary waves have a maximum height with sharp crests and secondly that waves which are sufficiently asymmetric break into "bores." The second part applies to a wide class of bounded kernels, but the kernel giving the correct dispersion effects of water waves has a square root singularity and the present argument does not go through. Nevertheless the possibility of the two kinds of breaking in such integro-differential equations is demonstrated.

Difficulties arise in finding variational principles for continuum mechanics problems in the Eulerian (field) description. The reason is found to be that continuum equations in the original field variables lack a mathematical "self-adjointness" property which is necessary for Euler equations. This is a feature of the Eulerian description and occurs in non-dissipative problems which have variational principles for their Lagrangian description. To overcome this difficulty a "potential representation" approach is used which consists of transforming to new (Eulerian) variables whose equations are self-adjoint. The transformations to the velocity potential or stream function in fluids or the scaler and vector potentials in electromagnetism often lead to variational principles in this way. As yet no general procedure is available for finding suitable transformations. Existing variational principles for the inviscid fluid equations in the Eulerian description are reviewed and some ideas on the form of the appropriate transformations and Lagrangians for fluid problems are obtained. These ideas are developed in a series of examples which include finding variational principles for Rossby waves and for the internal waves of a stratified fluid.

I. Singular perturbation problems involving singular points and turning points. II. On the averaged Lagrangian technique for nonlinear dispersive waves

In Part I a class of linear boundary value problems is considered which is a simple model of boundary layer theory. The effect of zeros and singularities of the coefficients of the equations at the point where the boundary layer occurs is considered. The usual boundary layer techniques are still applicable in some cases and are used to derive uniform asymptotic expansions. In other cases it is shown that the inner and outer expansions do not overlap due to the presence of a turning point outside the boundary layer. The region near the turning point is described by a two-variable expansion. In these cases a related initial value problem is solved and then used to show formally that for the boundary value problem either a solution exists, except for a discrete set of eigenvalues, whose asymptotic behaviour is found, or the solution is non-unique. A proof is given of the validity of the two-variable expansion; in a special case this proof also demonstrates the validity of the inner and outer expansions.

Nonlinear dispersive wave equations which are governed by variational principles are considered in Part II. It is shown that the averaged Lagrangian variational principle is in fact exact. This result is used to construct perturbation schemes to enable higher order terms in the equations for the slowly varying quantities to be calculated. A simple scheme applicable to linear or near-linear equations is first derived. The specific form of the first order correction terms is derived for several examples. The stability of constant solutions to these equations is considered and it is shown that the correction terms lead to the instability cut-off found by Benjamin. A general stability criterion is given which explicitly demonstrates the conditions under which this cut-off occurs. The corrected set of equations are nonlinear dispersive equations and their stationary solutions are investigated. A more sophisticated scheme is developed for fully nonlinear equations by using an extension of the Hamiltonian formalism recently introduced by Whitham. Finally the averaged Lagrangian technique is extended to treat slowly varying multiply-periodic solutions. The adiabatic invariants for a separable mechanical system are derived by this method.

Non-linear dispersive waves with a small dissipation

The general theory of Whitham for slowly-varying non-linear wavetrains is extended to the case where some of the defining partial differential equations cannot be put into conservation form. Typical examples are considered in plasma dynamics and water waves in which the lack of a conservation form is due to dissipation; an additional non-conservative element, the presence of an external force, is treated for the plasma dynamics example. Certain numerical solutions of the water waves problem (the Korteweg-de Vries equation with dissipation) are considered and compared with perturbation expansions about the linearized solution; it is found that the first correction term in the perturbation expansion is an excellent qualitative indicator of the deviation of the dissipative decay rate from linearity.

A method for deriving necessary and sufficient conditions for the existence of a general uniform wavetrain solution is presented and illustrated in the plasma dynamics problem. Peaking of the plasma wave is demonstrated, and it is shown that the necessary and sufficient existence conditions are essentially equivalent to the statement that no wave may have an amplitude larger than the peaked wave.

A new type of fully non-linear stability criterion is developed for the plasma uniform wavetrain. It is shown explicitly that this wavetrain is stable in the near-linear limit. The nature of this new type of stability is discussed.

Steady shock solutions are also considered. By a quite general method, it is demonstrated that the plasma equations studied here have no steady shock solutions whatsoever. A special type of steady shock is proposed, in which a uniform wavetrain joins across a jump discontinuity to a constant state. Such shocks may indeed exist for the Korteweg-de Vries equation, but are barred from the plasma problem because entropy would decrease across the shock front.

Finally, a way of including the Landau damping mechanism in the plasma equations is given. It involves putting in a dissipation term of convolution integral form, and parallels a similar approach of Whitham in water wave theory. An important application of this would be towards resolving long-standing difficulties about the "collisionless" shock.

Liquid silicate equation of state : using shock waves to understand the properties of the deep Earth

The equations of state (EOS) of several geologically important silicate liquids have been constrained via preheated shock wave techniques. Results on molten Fe₂SiO₄ (fayalite), Mg₂SiO₄ (forsterite), CaFeSi₂O₆ (hedenbergite), an equimolar mixture of CaAl₂Si₂O₈-CaFeSi₂O₆ (anorthite-hedenbergite), and an equimolar mixture of CaAl₂Si₂O₈-CaFeSi₂O₆-CaMgSi₂O₆(anorthite-hedenbergite-diopside) are presented. This work represents the first ever direct EOS measurements of an iron-bearing liquid or of a forsterite liquid at pressures relevant to the deep Earth (> 135 GPa). Additionally, revised EOS for molten CaMgSi₂O₆ (diopside), CaAl₂Si₂O₈ (anorthite), and MgSiO₃ (enstatite), which were previously determined by shock wave methods, are also presented.

The liquid EOS are incorporated into a model, which employs linear mixing of volumes to determine the density of compositionally intermediate liquids in the CaO-MgO-Al₂O₃-SiO₂-FeO major element space. Liquid volumes are calculated for temperature and pressure conditions that are currently present at the core-mantle boundary or that may have occurred during differentiation of a fully molten mantle magma ocean.

The most significant implications of our results include: (1) a magma ocean of either chondrite or peridotite composition is less dense than its first crystallizing solid, which is not conducive to the formation of a basal mantle magma ocean, (2) the ambient mantle cannot produce a partial melt and an equilibrium residue sufficiently dense to form an ultralow velocity zone mush, and (3) due to the compositional dependence of Fe2+ coordination, there is a threshold of Fe concentration (molar X_Fe ≤ 0.06) permitted in a liquid for which its density can still be approximated by linear mixing of end-member volumes.

Seismic structure above and below the core-mantle boundary

Seismic structure above and below the core-mantle boundary (CMB) has been studied through use of travel time and waveform analyses of several different seismic wave groups. Anomalous systematic trends in observables document mantle heterogeneity on both large and small scales. Analog and digital data has been utilized, and in many cases the analog data has been optically scanned and digitized prior to analysis.

Differential travel times of S - SKS are shown to be an excellent diagnostic of anomalous lower mantle shear velocity (V s) structure. Wavepath geometries beneath the central Pacific exhibit large S- SKS travel time residuals (up to 10 sec), and are consistent with a large scale 0(1000 km) slower than average V_s region (≥3%). S - SKS times for paths traversing this region exhibit smaller scale patterns and trends 0(100 km) indicating V_s perturbations on many scale lengths. These times are compared to predictions of three tomographically derived aspherical models: MDLSH of Tanimoto [1990], model SH12_WM13 of Suet al. [1992], and model SH.10c.17 of Masters et al. [1992]. Qualitative agreement between the tomographic model predictions and observations is encouraging, varying from fair to good. However, inconsistencies are present and suggest anomalies in the lower mantle of scale length smaller than the present 2000+ km scale resolution of tomographic models. 2-D wave propagation experiments show the importance of inhomogeneous raypaths when considering lateral heterogeneities in the lowermost mantle.

A dataset of waveforms and differential travel times of S, ScS, and the arrival from the D" layer, Scd, provides evidence for a laterally varying V_s velocity discontinuity at the base of the mantle. Two different localized D" regions beneath the central Pacific have been investigated. Predictions from a model having a V_s discontinuity 180 km above the CMB agree well with observations for an eastern mid-Pacific CMB region. This thickness differs from V_s discontinuity thicknesses found in other regions, such as a localized region beneath the western Pacific, which average near 280 km. The "sharpness" of the V_s jump at the top of D", i.e., the depth range over which the V_s increase occurs, is not resolved by our data, and our data can in fact may be modeled equally well by a lower mantle with the increase in V_s at the top of D" occurring over a 100 krn depth range. It is difficult at present to correlate D" thicknesses from this study to overall lower mantle heterogeneity, due to uncertainties in the 3-D models, as well as poor coverage in maps of D" discontinuity thicknesses.

P-wave velocity structure (V_p) at the base of the mantle is explored using the seismic phases SKS and SPdKS. SPdKS is formed when SKS waves at distances around 107° are incident upon the CMB with a slowness that allows for coupling with diffracted P-waves at the base of the mantle. The P-wave diffraction occurs at both the SKS entrance and exit locations of the outer core. SP_dKS arrives slightly later in time than SKS, having a wave path through the mantle and core very close to SKS. The difference time between SKS and SP_dKS strongly depends on V_p at the base of the mantle near SK Score entrance and exit points. Observations from deep focus Fiji-Tonga events recorded by North American stations, and South American events recorded by European and Eurasian stations exhibit anomalously large SP_dKS - SKS difference times. SKS and the later arriving SP_dKS phase are separated by several seconds more than predictions made by 1-D reference models, such as the global average PREM [Dziewonski and Anderson, 1981] model. Models having a pronounced low-velocity zone (5%) in V_p in the bottom 50-100 km of the mantle predict the size of the observed SP_dK S-SKS anomalies. Raypath perturbations from lower mantle V_s structure may also be contributing to the observed anomalies.

Outer core structure is investigated using the family of SmKS (m=2,3,4) seismic waves. SmKS are waves that travel as S-waves in the mantle, P-waves in the core, and reflect (m-1) times on the underside of the CMB, and are well-suited for constraining outermost core V_p structure. This is due to closeness of the mantle paths and also the shallow depth range these waves travel in the outermost core. S3KS - S2KS and S4KS - S3KS differential travel times were measured using the cross-correlation method and compared to those from reflectivity synthetics created from core models of past studies. High quality recordings from a deep focus Java Sea event which sample the outer core beneath the northern Pacific, the Arctic, and northwestern North America (spanning 1/8th of the core's surface area), have SmKS wavepaths that traverse regions where lower mantle heterogeneity is pre- dieted small, and are well-modeled by the PREM core model, with possibly a small V_p decrease (1.5%) in the outermost 50 km of the core. Such a reduction implies chemical stratification in this 50 km zone, though this model feature is not uniquely resolved. Data having wave paths through areas of known D" heterogeneity (±2% and greater), such as the source-side of SmKS lower mantle paths from Fiji-Tonga to Eurasia and Africa, exhibit systematic SmKS differential time anomalies of up to several seconds. 2-D wave propagation experiments demonstrate how large scale lower mantle velocity perturbations can explain long wavelength behavior of such anomalous SmKS times. When improperly accounted for, lower mantle heterogeneity maps directly into core structure. Raypaths departing from homogeneity play an important role in producing SmKS anomalies. The existence of outermost core heterogeneity is difficult to resolve at present due to uncertainties in global lower mantle structure. Resolving a one-dimensional chemically stratified outermost core also remains difficult due to the same uncertainties. Restricting study to higher multiples of SmKS (m=2,3,4) can help reduce the affect of mantle heterogeneity due to the closeness of the mantle legs of the wavepaths. SmKS waves are ideal in providing additional information on the details of lower mantle heterogeneity.

Structure of the earth's core and lowermost mantle from seismic PKP waves

This thesis addresses the fine structure, both radial and lateral, of compressional wave velocity and attenuation of the Earth's core and the lowermost mantle using waveforms, differential travel times and amplitudes of PKP waves, which penetrate the Earth's core.

The structure near the inner core boundary (ICB) is studied by analyzing waveforms of a regional sample. The waveform modeling approach is demonstrated to be an effective tool for constrainning the ICB structure. The best model features a sharp velocity jump of 0.78km/s at the ICB and a low velocity gradient at the lowermost outer core (indicating possible inhomogeneity) and high attenuation at the top of the inner core.

A spherically symmetric P-wave model of the core, is proposed from PKP differential times, waveforms and amplitudes. The ICB remains sharp with a velocity jump of 0. 78km/ s. A very low velocity gradient at the base of the fluid core is demonstrated to be a robust feature, indicating inhomogeneity is practically inevitable. The model also indicates that the attenuation in the inner core decreases with depth. The velocity at D" is smaller than PREM.

The inner core is confirmed to be very anisotropic, possessing a cylindrical symmetry around the Earth spin axis with the N-S direction 3% faster than the E-W direction. All of the N-S rays through the inner core were found to be faster than the E-W rays by 1.5 to 3.5s. Exhaustive data selection and efforts in insolating contributions from the region above ensure that this is an inner core feature.

The anisotropy at the very top of the inner core is found to be distinctly different from the deeper part. The top 60km of the inner core is not anisotropic. From 60km to 150km, there appears to be a transition from isotropy to anisotropy.

PKP differential travel times are used to study the P velocity structure in D". Systematic regional variations of up to 2s in AB-DF times were observed, attributed primarily to heterogeneities in the lower 500km of the mantle. However, direct comparisons with tomographic models are not successful.

Characterization and developmental regulation of a gene expressed specifically in the skeletogenic lineage of the sea urchin embryo

The sea urchin embryonic skeleton, or spicule, is deposited by mesenchymal progeny of four precursor cells, the micromeres, which are determined to the skeletogenic pathway by a process known as cytoplasmic localization. A gene encoding one of the major products of the skeletogenic mesenchyme, a prominent 50 kD protein of the spicule matrix, has been characterized in detail. cDNA clones were first isolated by antibody screening of a phage expression library, followed by isolation of homologous genomic clones. The gene, known as SM50, is single copy in the sea urchin genome, is divided into two exons of 213 and 1682 bp, and is expressed only in skeletogenic cells. Transcripts are first detectable at the 120 cell stage, shortly after the segregation of the skeletogenic precursors from the rest of the embryo. The SM50 open reading frame begins within the first exon, is 450 amino acids in length, and contains a loosely repeated 13 amino acid motif rich in acidic residues which accounts for 45% of the protein and which is possibly involved in interaction with the mineral phase of the spicule.

The important cis-acting regions of the SM50 gene necessary for proper regulation of expression were identified by gene transfer experiments. A 562 bp promoter fragment, containing 438 bp of 5' promoter sequence and 124 bp of the SM50 first exon (including the SM50 initiation codon), was both necessary and sufficient to direct high levels of expression of the bacterial chloramphenicol acetyltransferase (CAT) reporter gene specifically in the skeletogenic cells. Removal of promoter sequences between positions -2200 and -438, and of transcribed regions downstream of +124 (including the SM50 intron), had no effect on the spatial or transcriptional activity of the transgenes.

Regulatory proteins that interact with the SM50 promoter were identified by the gel retardation assay, using bulk embryo mesenchyme blastula stage nuclear proteins. Five protein binding sites were identified and mapped to various degrees of resolution. Two sites are homologous, may be enhancer elements, and at least one is required for expression. Two additional sites are also present in the promoter of the aboral ectoderm specific cytoskeletal actin gene CyIIIa; one of these is a CCAA T element, the other a putative repressor element. The fifth site overlaps the binding site of the putative repressor and may function as a positive regulator by interfering with binding of the repressor. All of the proteins are detectable in nuclear extracts prepared from 64 cell stage embryos, a stage just before expression of SM50 is initiated, as well as from blastula and gastrula stage; the putative enhancer binding protein may be maternal as well.

Singularities and horizons in the collisions of gravitational waves

This thesis presents a study of the dynamical, nonlinear interaction of colliding gravitational waves, as described by classical general relativity. It is focused mainly on two fundamental questions: First, what is the general structure of the singularities and Killing-Cauchy horizons produced in the collisions of exactly plane-symmetric gravitational waves? Second, under what conditions will the collisions of almost-plane gravitational waves (waves with large but finite transverse sizes) produce singularities?

In the work on the collisions of exactly-plane waves, it is shown that Killing horizons in any plane-symmetric spacetime are unstable against small plane-symmetric perturbations. It is thus concluded that the Killing-Cauchy horizons produced by the collisions of some exactly plane gravitational waves are nongeneric, and that generic initial data for the colliding plane waves always produce "pure" spacetime singularities without such horizons. This conclusion is later proved rigorously (using the full nonlinear theory rather than perturbation theory), in connection with an analysis of the asymptotic singularity structure of a general colliding plane-wave spacetime. This analysis also proves that asymptotically the singularities created by colliding plane waves are of inhomogeneous-Kasner type; the asymptotic Kasner axes and exponents of these singularities in general depend on the spatial coordinate that runs tangentially to the singularity in the non-plane-symmetric direction.

In the work on collisions of almost-plane gravitational waves, first some general properties of single almost-plane gravitational-wave spacetimes are explored. It is shown that, by contrast with an exact plane wave, an almost-plane gravitational wave cannot have a propagation direction that is Killing; i.e., it must diffract and disperse as it propagates. It is also shown that an almost-plane wave cannot be precisely sandwiched between two null wavefronts; i.e., it must leave behind tails in the spacetime region through which it passes. Next, the occurrence of spacetime singularities in the collisions of almost-plane waves is investigated. It is proved that if two colliding, almost-plane gravitational waves are initially exactly plane-symmetric across a central region of sufficiently large but finite transverse dimensions, then their collision produces a spacetime singularity with the same local structure as in the exact-plane-wave collision. Finally, it is shown that a singularity still forms when the central regions are only approximately plane-symmetric initially. Stated more precisely, it is proved that if the colliding almost-plane waves are initially sufficiently close to being exactly plane-symmetric across a bounded central region of sufficiently large transverse dimensions, then their collision necessarily produces spacetime singularities. In this case, nothing is now known about the local and global structures of the singularities.

Techniques for strength measurement at high pressures and strain-rates using transverse waves

The study of the strength of a material is relevant to a variety of applications including automobile collisions, armor penetration and inertial confinement fusion. Although dynamic behavior of materials at high pressures and strain-rates has been studied extensively using plate impact experiments, the results provide measurements in one direction only. Material behavior that is dependent on strength is unaccounted for. The research in this study proposes two novel configurations to mitigate this problem.

The first configuration introduced is the oblique wedge experiment, which is comprised of a driver material, an angled target of interest and a backing material used to measure in-situ velocities. Upon impact, a shock wave is generated in the driver material. As the shock encounters the angled target, it is reflected back into the driver and transmitted into the target. Due to the angle of obliquity of the incident wave, a transverse wave is generated that allows the target to be subjected to shear while being compressed by the initial longitudinal shock such that the material does not slip. Using numerical simulations, this study shows that a variety of oblique wedge configurations can be used to study the shear response of materials and this can be extended to strength measurement as well. Experiments were performed on an oblique wedge setup with a copper impactor, polymethylmethacrylate driver, aluminum 6061-t6 target, and a lithium fluoride window. Particle velocities were measured using laser interferometry and results agree well with the simulations.

The second novel configuration is the y-cut quartz sandwich design, which uses the anisotropic properties of y-cut quartz to generate a shear wave that is transmitted into a thin sample. By using an anvil material to back the thin sample, particle velocities measured at the rear surface of the backing plate can be implemented to calculate the shear stress in the material and subsequently the strength. Numerical simulations were conducted to show that this configuration has the ability to measure the strength for a variety of materials.