Critical Ray Research Articles

Slowness‐domain kinematical characteristics for horizontally layered orthorhombic media. Part II: Pre‐critical slowness match

ABSTRACTPart II of this paper is a direct continuation of Part I, where we consider the same types of orthorhombic layered media and the same types of pure‐mode and converted waves. Like in Part I, the approximations for the slowness‐domain kinematical characteristics are obtained by combining power series coefficients in the vicinity of both the normal‐incidence ray and an additional wide‐angle ray. In Part I, the wide‐angle ray was set to be the critical ray (‘critical slowness match’), whereas in Part II we consider a finite long offset associated with a given pre‐critical ray (‘pre‐critical slowness match’). Unlike the critical slowness match, the approximations in the pre‐critical slowness match are valid only within the bounded slowness range; however, the accuracy within the defined range is higher. Moreover, for the pre‐critical slowness match, there is no need to distinguish between the high‐velocity layer and the other, low‐velocity layers. The form of the approximations in both critical and pre‐critical slowness matches is the same, where only the wide‐angle power series coefficients are different. Comparing the approximated kinematical characteristics with those obtained by exact numerical ray tracing, we demonstrate high accuracy. Furthermore, we show that for all wave types, the accuracy of the pre‐critical slowness match is essentially higher than that of the critical slowness match, even for matching slowness values close to the critical slowness. Both approaches can be valuable for implementation, depending on the target offset range and the nature of the subsurface model. The pre‐critical slowness match is more accurate for simulating reflection data with conventional offsets. The critical slowness match can be attractive for models with a dominant high‐velocity layer, for simulating, for example, refraction events with very long offsets.

Influence of heterogeneous refractivity on optical wave propagation in coastal environments

Spatial variations of refractivity significantly dictate the characteristics of optical wave propagation through the atmosphere. Consequently, the ability to simulate such propagation is highly dependent upon the accurate characterization of refractivity along the propagation path. Unfortunately, the scarcity of high spatiotemporal resolution observational data has forced many past studies of optical wave propagation to assume horizontally homogeneous (HH) atmospheric conditions. However, the (adverse) impact of such an assumption has not been quantified in the literature. In this paper, we attempt to fill this void by utilizing a mesoscale modeling-based approach to explicitly simulate atmospheric refraction. We then compare the differences of the HH refractivity fields to the mesoscale model-derived refractivity fields by means of a realistic atmospheric event and through ray tracing simulations. In this study, we model a coastal low-level jet, a common coastal atmospheric phenomenon which is associated with heterogeneous thermal and refractivity fields. Observational data from a radiosonde and a radar wind profiler near the northeastern region of the United States are used for model validation. The observed characteristics of low-level jet (e.g., evolution, intensity, location) and associated temperature inversion are found to be reasonably well captured by the mesoscale model. The simulated nighttime refractivity gradient field manifests significant spatial heterogeneity; over land, the refractivity gradient is much stronger and amplified near the ground, whereas it becomes much weaker over the ocean. We quantify the effect of this heterogeneity on optical ray trajectories by simulating a suite of rays and documenting the variability of their altitudes at certain propagation ranges. It is found that the altitude of optical rays may vary tens of meters during a diurnal cycle, and at nighttime the rays may bend downward by more than 150 m at a range of 100 km. We run additional ray tracing simulations using refractivity profiles from a single location and assuming HH refractivity along the propagation path. It is observed that the HH approach yields instantaneous ray bending magnitudes up to 30 % less than the ray bending based on the refractivity simulated by the mesoscale model. At the same time, it is found that the mesoscale model-based refractivity fields may have uncertainty introduced by different factors associated with the model configuration. Of these factors, turbulence parameterization is explored in-depth and found to be responsible for more uncertainty than spatial grid resolution. To be more specific, different turbulence parameterizations are found to produce significantly varying temperature inversion parameters (e.g., height, magnitude), which are critical factors influencing ray trajectories. Collectively, these results highlight the potential advantages and disadvantages of utilizing a mesoscale model to simulate refractivity in coastal areas as opposed to assuming HH refractivity.

Development of a buccal mucoadhesive film for fast dissolution: mathematical rationale, production and physicochemical characterization

The validity of a mathematical rationale for preparation of a fast-dissolving buccal mucoadhesive was tested. A buccal mucoadhesive biopolymeric formulation has been developed having pullulan as the main component. The formulation was duly evaluated physicochemically, via assays for intrinsic viscosity (resulting in 71.61 cm3 g−1), differential scanning calorimetry analysis (resulting in a Tg = 63 °C), thermogravimetric analysis (244–341 °C), moisture content determinations (14%, w/w), dissolution timeframe (41.6 s), mucoadhesion force (40 kg/cm2), scanning electron microscopy analyses (critical ray under 1.0 μm), mechanic strength (tensile strength = 58 N/mm2, deformation = 4.4%). The mucoadhesive formulation exhibited important characteristics for a drug carrier, that is, a 6 cm2 area, a fast dissolution timeframe, an adequate mucoadhesivity, resistance to both oxygen and water vapor penetration, increased viscosity in solution (ranging from 33.2 cm3/g to 71.61 cm3/g), easy molding, suitable water solubility and transparency.

The effect of interface curvature on AVO inversion of near-critical and postcritical PP-reflections

Widely exploited in the industry, amplitude-variation-with-offset (AVO) inversion techniques are based on weak-contrast approximations of the plane-wave reflection coefficients. These approximations are valid for plane waves reflected at almost flat interfaces with weak contrasts in seismic parameters and for reflection angles below the critical angle. Regardless of the underlying assumptions, linearized coefficients provide a simple and physically adequate tool to accurately invert AVO data for seismic parameters at precritical angles. However, the accuracy of linearized coefficients drastically decreases with increasing incidence angle. Limitations occur around and beyond the critical ray, where the effect of wavefront curvature becomes prominent and thus can no more be neglected. The effective reflection coefficients generalize the plane-wave reflection coefficients for waves generated by point sources and reflected at curved interfaces. They account for the wavefront curvature and are adequate at any incidence angle. Our previous studies have shown that including the reflections around and beyond the critical angle in the AVO inversion significantly improves the accuracy of estimated parameters. However, the interface curvature also must have its contribution to the long-offset AVO inversion. We find that the interface curvature affects the energy propagation along the ray tube and the energy diffusion across the ray tube. The energy propagation along the tube is characterized by the geometrical spreading, which is strongly affected by interface curvature. The transverse diffusion is captured by the effective reflection coefficients which are less influenced by interface curvature. The long-offset AVO inversion is thus sensitive to interface curvature through a combination of several wave propagation factors.

A PARABOLIC EQUATION AND A PEARCEY-TYPE INTEGRAL FOR THE WAVE FIELD IN THE VICINITY OF THE CRITICAL RAY

The high-frequency wave field of a point source situated near the interface of two homogeneous media and placed in a faster medium is considered. A description of the wave field in a neighborhood of the critical ray in a slower medium is found by the parabolic-equation method. The result involves a Pearcey-type integral. The approach is not based on the Fourier method and can be generalized to inhomogeneous media with curved interfaces. Bibliography: 14 titles.

Corrections for critical rays in 2-D seismic modelling

The usual high-frequency (ω) ray method of modelling reflections from smooth boundaries does not account for interference arising near critical angles of incidence. When the reflector is effectively convex towards the incident-wave side, the ray method fails (i) in a transition zone of width O(ω-1/2) separating regions of partial and total ray reflection, (ii) for the transmission in a zone along the reflector having thickness O(ω-1/2) and where a whispering-gallery is formed by multiply-reflected turning waves, and (iii) for turning rays refracted back into the first medium at distances from the critical point less than O(mω-1/4), where m is the number of turns. The interference is a local effect and approximate analytic waveforms can be found for laterally varying media. The 2-D acoustic case is described here. The fields near the point of critical incidence and about the critically reflected ray are found in a way similar to the grazing ray solution described in an earlier paper. However, through the critical ray problem the first two terms of the asymptotic expansion must be considered. The whispering-gallery is obtained by exploiting work on modes by Buldyrev, Lewis, Ludwig and others. The modal solution is then matched with the solution near the critical point to determine the initial modal amplitudes. The individual ray contributions separate from the modes as the waves progress along the boundary. The modal form of solution is needed, even if one wishes to demonstrate only the matching with ray theory for the primary refracted/turning wave near the critical point. This is because the transition zone for the refraction is much wider than that for the reflected wave. Though it is more complicated than the earlier grazing ray solution, the critical-ray diffracted wavefield is still continued way from the boundary using ray coordinates. Hence, it is not difficult to provide ray tracing programs with corrections to ray synthetics. Some preliminary numerical examples are presented to indicate the waveforms and potential significance of these corrections. They are small in comparison with the total reflection, but they are very significant in comparison with the weaker refracted waves. Even where ray theory is (just) valid, for practical reasons one may wish to avoid tracing multiple reflections close to a numerically specified boundary. The boundary-layer formulae provide a more ‘natural’ solution to both the theoretical and numerical problems for the refracted part of the field.

Saddle point analysis of the reflected acoustic field

The field due to reflection of spherical sound waves at the plane interface between two homogeneous fluids is examined in terms of saddle points of an integrand. The resulting new expression for the field is valid at all ranges including the vicinity of the critical ray and the caustic due to beam displacement. A simple interpretation of the field in terms of wave fronts and eigenrays, which is valid at all ranges, is obtained. At some ranges, the wavenumber for eigenrays is complex. The new result provides an alternative to the usual specular ray and lateral wave interpretation.

A note on the zeros of solutions [wacute]+P(z)w= 0 where P is a polynomial

For the class of equations described in the title, there is a classical result which states that there are (deg P) + 2 critical rays, arg z = θj, such that for any e > 0, all but finitely many zeros of any solution f(z) ≢ 0 must lie in the union of the sectors |arg z - θj| < e. We prove that any infinite set of zeros in such a sector (for sufficiently small e) must actually approach a definite ray, which is a translate ofthe critical ray (and which can be explicitely calculated). In addition, we estimate the rate at which the zeros approach the ray, thus obtaining new information on the exact location of the zeros. The class of equations treated here contains equations which arise in applications in other areas, such as Airy's equation and Titchmarsh's equation

A new asymptotic method for the modeling of near-field accelerograms

AbstractWe study high-frequency radiation from a dislocation model of rupture propagation at the earthquake source. We demonstrate that in this case all the radiation emanates from the rupture front and, by a change of variables, that at any instant of time the high-frequency waves reaching an observer come from a line on the fault plane that we call isochrone. An asymptotic approximation to near-source velocity and acceleration is obtained that involves a simple integration along the isochrones for every time step. It is shown that wave front discontinuities (critical or stopping phases) are radiated every time an isochrone becomes tangent to a barrier. This leads to what we call the critical ray approximation which is given in a closed form. The previous results are compared with discrete wavenumber synthetics obtained by Bouchon (1982) for the Gilroy 6 recording of the Coyote Lake earthquake of 1980. The fit between the asymptotic and full numerical method is extremely good. The critical ray approximation permits the identification of different phases in Bouchon's synthetics and the prediction of the behavior of the signal in the vicinity of their arrival time.

Critical ray-like propagation modes along graded-index planar optical waveguides

Graded-index planar optical waveguides and surface compression layers were formed simultaneously in the surface layer of glass plates by ion exchange. A change in the refractometer patterns was observed. Before ion exchange, only a critical ray fringe was observed, but with ion exchange a guided wave fringe appeared on the high effective refractive index side of the ‘critical ray’ fringe, and the number of guided wave fringes increased. The guided wave fringe or fringes were birefringent, whereas the ‘critical ray’ fringe was kept nonbirefringent. It was concluded that the ‘critical ray’ propagated along the bottom of a waveguide, ie at the foot of the refractive index distribution.

Surface Stress Measurement of Curved Tempered Glasses by Critical Ray

Surface Stress Measurement of Curved Tempered Glasses by Critical Ray

Thermally tempered glass surface stress measurement by critical ray

A critical ray was excited along the glass surface by laser beam incidence. Part of the critical ray was refracted during propagation. The angle of refraction depended on the refractive index of the glass sirface. When the glass surface was birefringent due to the photoelastic effect, the angles of refraction differed from each other between two linearly polarized beams. This difference allows surface stress determination.

Reflection from a variable-index half space: Uniform and nonuniform representations

Consider the problem of a point harmonic source in a homogeneous half space over a second half space in which the sound speed is larger than that in the source half space and for which the speed in the second half space decreases as a function of distance along the interface. The usual geometrical acoustics ansatz is used to obtain the reflected and transmitted field as well as the lateral wave field. The lateral wave field displays some new properties not present in the case when the higher-speed medium is homogeneous. It shows modifications to both the phase function and the 3/2 range decay law for the portion of the lateral wave path along the interface. In addition, it shows that the square-root decay with range is modified by an additional decay which is proportional to the range multiplied by the derivative of the index. This is a consequence of the divergence of the lateral wave rays as they leave the interface. As in the case of the homogeneous higher-speed medium, the lateral wave expansion derived by the usual geometrical acoustic ansatz becomes singular on the critical ray. It is shown how the reflected and lateral waves can be used to obtain a uniform expansion for the scattered field which is valid for all observation points beyond the so-called critical ray. The ansatz for the uniform expansion is provided by the uniform expression for the scattered field for the case when the higher speed medium is isospeed and it involves the parabolic cylinder functions. The ansatz does not require that the source medium be homogeneous, nor does it require that the interface be planar. Away from the critical ray this expansion reduces to the usual results of geometrical acoustics.

Reflected and lateral waves for the Sommerfeld model

The Sommerfeld model, a constant-density isospeed half space over a second constant-density isospeed half space, has offered extensive insight into the role of the ocean bottom in acoustic propagation by exploiting properties of the ground or lateral wave and the specularly reflected waves. This paper presents some new results for this model. A uniform asymptotic expansion is obtained for the source and receiver on the interface, which reduces to the exact solution as either the density ratio or the sound-speed ratio approaches unity. In addition, this result shows the ground or lateral wave decays initially—not like 1/kr2, the classical result, but like 1/r, and as the index ratio approaches unity this region increases. For the source and receiver above the interface, the reflected fields are expressed in a uniform expansion involving the parabolic cylinder functions. Away from the critical ray these expressions reduce to the usual specular and lateral fields. The expressions reduce to the interface results and are not singular as the sound-speed index approaches unity, and confirm the above decay predictions for the lateral wave. Subject Classification: [43]30.20, [43]30.30; [43]20.15.

Refractive attenuation formulas for an unstable critical ray in a circularly symmetric medium

Through application of the basic equation of geometrical optics, the eikonal equation, we have derived an expression for the refractive defocusing of a critical ray in a medium which has circular symmetry. The equation of the ray itself is simply a circle, since by definition its radius of curvature is exactly the same as its distance from the center of symmetry. This mathematical development has a practical significance in that it permits for the first time the calculation of the strength of a signal which travels in the atmosphere of a planet at a fixed critical height. Our specific interest in this matter was due to the probable existence of such a critical ray about 33 km high in the atmosphere of Venus; at this altitude the pressure and temperature are reasonably Earth‐like and, as a result, some advanced space exploration studies have considered telecommunication by means of the critical ray (J. E. Blamont, personal communication, 1974). Through the application of the result given here, the strength of such a propagation mode can be determined. The result also has potential applications in geocentric studies and in other media which possess either spherical or cylindrical symmetry.

Uniform asymptotic representation of a field of reflected and head waves

A uniform asymptotic expansion for integrals of the type ʃ +∞ -∞ u ½ F ( u ) exp {i kψ ( u )}d u has been obtained in terms of generalized Airy functions, which are the solutions of the equation V"( z )+ z 2 V ( z ) = 0. This result is applied to the construction of a uniform asymptotic representation of a solution of the wave equation in the case of reflexion of a spherical wave from a plane boundary in a region including a critical ray. This asymptotic series may be divided into two series corresponding to reflected and head waves respectively, which are transformed into the ray series for these waves far from the critical ray and reduced to the expressions given by Brekhovskikh (1960) in the vicinity of the critical ray.

On the radio occultation method for studying planetary atmospheres

The problem of determining the refractivity profile of a planetary atmosphere from optical or radio occultation data is identical in principle to the problem of determining the variation of seismic velocities in the earth from the observed travel times of seismic body waves. In either case, a complete set of data can be inverted uniquely, the only constraints being those fundamental to geometric optics. Expressions are given for converting observed Doppler shifts to the index of refraction as a function of depth in the atmosphere. The effect of various approximations on the analysis is discussed; it is found that a ‘thin atmosphere’ approximation simplifies the mathematics and preserves the singularity at the critical ray curvature.

Abnormal Audibility Zones in Long Distance Propagation through the Atmosphere

Five thousand tons of high explosives detonated on Helgoland, April 18, 1947, created air pressure perturbations recorded on microbarographs between 66 and 1000 km SSE from the blast. Instruments responded to frequencies 0.05–5 c.p.s. Arrival times of abnormal signals at six stations more distant than 220 km, supplemented by high altitude meteorological data and the assumption of negligible winds above 30 km, permit upper atmosphere temperature calculations. Temperatures agree with N.A.C.A. values up to 42 km, but show a reduced gradient above that altitude, and a maximum value 294°K in the temperature hump between 30 and 70 km. This temperature maximum establishes critical ray which is refracted to infinity. A new explanation for observed outer boundaries of abnormal zones is therefore proposed, and substantiated by recorded evidence of dispersion near the temperature maximum. In the signal received near the abnormal zone outer boundary, high frequency content predominates.

On the Accuracy Attainable with Critical Angle Refractometers

The three factors controlling the determination of a refractive index by means of a critical angle refractometer are, so far as the prism system is concerned, (i.) the angle of the prism, (ii.) its refractive index, and (iii.) the angle of emergence of the critical ray from the second prism face. Expressions are obtained for the variation of the required refractive index with each of these factors separately and curves are given connecting these variations with the angle of emergence from the second prism face for various prism angles.