Variations In Velocity Structure Research Articles

Numerical Simulation of Wave Equation Using the Finite Difference Method with Variable Refined Grid in Complex Near-surface Condition

The horizontal layered refined grid is adopted in the conventional finite difference method (FDM) of variable refined grid for the numerical simulation of wave equation. But, the method cannot well adapt to the characteristics of undulated topography and the variation of velocity structure in near-surface region. To solve this problem, a numerical simulation of wave equation using FDM with undulated variable refined grid is proposed in this paper. The grids are generated according to the undulated topography and the spatial distribution of velocity in the near-surface region from low-velocity to the high-velocity layer. For ensuring the accuracy while taking into account the computational efficiency, the FDM with variable coefficient in above grids is used to discretize the acoustic wave equation. The numerical evaluation and example show that the same accuracy as the method using fine grid can be obtained by the new method. However, its computational time is only 43.6% of the one spent by the fine grid. Furthermore, the new method can also stably adapt to the real complex loess plateau near-surface model in a working area of Ordos Basins.

Acoustic full waveform inversion for 2-D ambient noise source imaging

SUMMARYWe present a method for estimating seismic ambient noise sources by acoustic full waveform inversion (FWI) of interstation cross-correlations. The method is valid at local scales for laterally heterogeneous media, and ambient noise sources confined to the Earth’s surface. Synthetic tests performed using an actual field array geometry, are used to illustrate three unique aspects of our work. First: the method is able to recover noise sources of arbitrary spatial distribution, both within and outside the receiver array, with high fidelity. This holds true for complex velocity models and does not require a good initial guess for inversion, thereby addressing an outstanding issue in the existing research literature. Second: we analyse the extent of biases in source inversion that arise due to inaccurate velocity models. Our findings indicate that source inversion using simplified (e.g. homogeneous) velocity models may work reliably when lateral variations in velocity structure are limited to 5 or 10 per cent in magnitude, but is vitiated by strong variations of 20 per cent or higher, wherein the effect of strong scattering and/or phase distortions become significant. Finally, our technique is implemented without the adjoint method, which is usually inextricably linked to FWI. Inversions are performed using source kernels computed for each receiver pair, and this approach is computationally tractable for real-world problems with small aperture seismic arrays.

Upper mantle and transition zone structure beneath Leizhou–Hainan region: Seismic evidence for a lower-mantle origin of the Hainan plume

The origin of the widespread volcanism at the Leizhou–Hainan (Leiqiong) region in the Southern China remains obscure. We take advantage of the highly active seismicity and dense seismic networks surrounding this region to investigate its upper mantle and Mantle Transition Zone (MTZ) structure. Over 5000 P-wave waveforms whose raypaths bottom at depths around the MTZ are collected, and traveltimes of their first arrivals are hand-picked. By matching the traveltime curve variation over the epicentral distance range from 10° to 35°, we first construct a 1-D upper mantle and MTZ velocity structure for the region. This initial model is then refined by forward modeling, in which the observed triplicated waveforms from selected earthquakes are compared with the synthetic seismograms with varying velocity structure. In our preferred model for Leiqiong, the P-wave velocities deeper than 200km at the upper mantle are 0.8–1.2% lower than the IASP91, and 0.6% slower in the MTZ, while the top and bottom boundaries of the MTZ depresses 12km and slightly uplifted, respectively, compared to the global averages. This model provides independent constraints on the structure beneath Leiqiong, suggesting a thermal anomaly within the MTZ and a lower mantle origin for the volcanism seen in this region.

An upper-mantle S-wave velocity model for East Asia from Rayleigh wave tomography

We present a new shear velocity model of the upper mantle beneath the East Asia region derived by inverting Rayleigh wave group velocity measurements between 10 and 145 s combined with previously published Rayleigh wave phase velocity measurements between 150 and 250 s. Rayleigh wave group velocity dispersion curves along more than 9500 paths were measured and combined to produce 2D dispersion maps for 10–145 s periods. The group velocity maps benefit from the inclusion of new data recorded by the China National Seismic Network and surrounding global stations. The increase in available data has resulted in enhanced resolution compared with previously published group velocity maps; the horizontal resolution across the region is about 3° for the periods used in this study. The new shear-wave velocity models indicate varying velocity structure beneath eastern China, which yields estimates of a lithosphere–asthenosphere boundary depth from around 160 km beneath the Yangtze block to approximately 140 km beneath the western part of the North China Craton (NCC), up to depths of 70–100 km beneath the eastern NCC, Northeast China, and the Cathaysia block. The models reveal the subduction of two opposite-facing continental plates under the southern and northern margin of Tibet. An obvious low-velocity anomaly appears in the top 200 km of the upper mantle beneath northern Tibet, which is inconsistent with the presence of subducted Asian or Indian mantle lithosphere beneath northern Tibet. The Cenozoic volcanism fields in the Mongolian plateau are characterized by an obvious upper mantle negative anomaly, but no signature of deep-seated plume was observed.

Surface wave tomography: Imaging of the lithosphere–asthenosphere boundary beneath central and southern Africa?

The lithosphere–asthenosphere boundary (LAB) remains a controversial subject in Earth sciences, and beneath cratonic regions appears to be a particularly difficult boundary to consistently image. Seismic methods give different indicators on the velocity structure of the upper mantle: tomographic models provide estimates of the velocity variations at a variety of lateral scales, but have limited vertical resolution; receiver function techniques provide good indication of the depth to seismic discontinuities, but less information on the absolute velocities. This study assesses whether the different methods give consistent estimates for the depth of the LAB in southern Africa. Using a surface wave dataset with nearly 12,000 paths in the African region, new tomographic models of central and southern Africa are calculated. To show the non-unique nature of tomography, results are presented for two different parameterisations. The models indicate varying velocity structure beneath the cratonic regions of central and southern Africa, which yield estimates of the LAB depth from around 150 km depth in Tanzania, to approximately 200 km depth beneath the Kalahari Craton, down to depths of 225–250 km beneath parts of the Congo. At a broad-scale these depth estimates are compatible with geothermometry from kimberlite xenoliths. In regions such as Tanzania, the kimberlite magmatism is observed to occur along strong horizontal gradients in upper mantle seismic velocity structure — potentially edge features in lithospheric structure. In contrast, a detailed comparison beneath the Kalahari Craton indicates that in this region, and given the present resolution of the tomography, there is no clear link between the kimberlites and velocity gradients. However, in general the kimberlites do not sample the regions of fastest seismic velocities. The relationship between LAB depth estimates from the tomographic modelling and those estimates from receiver functions is not clear. Results from receiver function techniques beneath southern Africa tend to place discontinuities at either shallower (100–150 km), or deeper (300–350 km), depths than the thermally defined LAB estimates. As such, particular care should be taken in automatically associating a discontinuity from fast to slow seismic velocities as the same location as a thermally defined lithosphere–asthenosphere boundary.

Regional estimation of Q from seismic coda observations by the Gauribidanur seismic array (southern India)

Attenuation properties of the lithosphere in southern India are estimated from 1219 vertical-component, short-period observations of microearthquake codas recorded by the Gauribidanur seismic array. The magnitudes of the earthquakes range from 0.3 to 3.7 and have focal depths less than 10 km. Coda-wave attenuation ( Q c −1) is estimated by means of a single isotropic scattering method and a multiple lapse time window analysis based on the hypothesis of multiple isotropic scattering and uniform distribution of scatterers is used to estimate the contribution of intrinsic absorption ( Q i −1) and scattering ( Q s −1) to total attenuation ( Q t −1). All the attenuation parameters are estimated, as a function of frequency for hypocentral distances up to 255 km. Results show a frequency dependent relation of the Q c −1 values in the range 1–10 Hz that fit the power law Q −1(f)=Q 0 −1(f/f 0) ηA Q 0 −1 value of 0.014 and a decrease of f −1.2 have been found using data from the whole region. On the other hand, scattering attenuation is found to be greater than intrinsic absorption for all the frequency bands. A high value of the seismic albedo (which ranges from 0.68 to 1) is found which indicates that scattering is the dominant effect in the study region. Nevertheless, the attenuation parameters estimated are much lower than the obtained for other regions in the world. On the other hand, the observed energy at 0–15 s from the S-wave arrival time bends significantly downward with decreasing distance. In order to clarify this phenomenon, there is a need to take into account the vertical varying velocity structure in the theoretical model.

3-D Velocity Structure beneath the Chia-Nan Area, Taiwan

In this study, we adopt a damping least-square inversion method to investigate the Vp structures and Vp/Vs ratios of the crust and upper mantle beneath the Chia-Nan area, Taiwan. Previous studies have shown that, velocity structure can be used as an indicator of the geometry of a fault and the general aspects of tectonics. Therefore, the first goal of this research is to analyze the degree of correlation between the velocity structure and the seismic characteristics with respect to the tectonic implications of the area. The second intention is to study the relationship between the Vp/Vs ratio and the pressure change occurring with crack opening in the deeper crust. Finally, the distribution of Vp/Vs ratios and its association with fault activities is also investigated. Our results indicate that the variations in velocity structure beneath the Chia-Nan area is caused by local geological structures, fault crossing and the existence of the Pei-Kang High Area. We also find that most earthquakes occur in areas that have Vp/Vs gradients varying rapidly. In addition, according to the distribution of the earthquakes, there seems to exist a westdipping fault west of the Chukou fault; however, this prospect needs to be investigated further in a future study.

Three-dimensional crustal velocity structure beneath the Strait of Georgia, British Columbia

Summary The Strait of Georgia is a topographic depression straddling the boundary between the Insular and Coast belts in southwestern British Columbia. Two shallow earthquakes located within the strait (M = 4.6 in 1997 and M = 5.0 in 1975) and felt throughout the Vancouver area illustrate the seismic potential of this region. As part of the 1998 Seismic Hazards Investigation of Puget Sound (SHIPS) experiment, seismic instruments were placed in and around the Strait of Georgia to record shots from a marine source within the strait. We apply a tomographic inversion procedure to first-arrival traveltime data to derive a minimum-structure 3-D P-wave velocity model for the upper crust to about 13 km depth. We also present a 2-D velocity model for a profile orientated across the Strait of Georgia derived using a minimum-parameter traveltime inversion approach. This paper represents the first detailed look at crustal velocity variations within the major Cretaceous to Cenozoic Georgia Basin, which underlies the Strait of Georgia. The 3-D velocity model clearly delineates the structure of the Georgia Basin. Taking the 6 km s−1 isovelocity contour to represent the top of the underlying basement, the basin thickens from between 2 and 4 km in the northwestern half of the strait to between 8 and 9 km at the southeastern end of the study region. Basin velocities in the northeastern half are 4.5–6 km s−1 and primarily represent the Upper Cretaceous Nanaimo Group. Velocities to the south are lower (3–6 km s−1) because of the additional presence of the overlying Tertiary Huntingdon Formation and more recent sediments, including glacial and modern Fraser River deposits. In contrast to the relatively smoothly varying velocity structure of the basin, velocities of the basement rocks, which comprise primarily Palaeozoic to Jurassic rocks of the Wrangellia Terrane and possibly Jurassic to mid-Cretaceous granitic rocks of the Coast Belt, show significantly more structure, probably an indication of the varying basement rock lithologies. The 2-D velocity model more clearly reveals the velocity layering associated with the recent sediments, Huntingdon Formation and Nanaimo Group of the southern Georgia Basin, as well as the underlying basement. We interpret lateral variation in sub-basin velocities of the 2-D model as a transition from Wrangellian to Coast Belt basement rocks. The effect of the narrow, onshore–offshore recording geometry of the seismic experiment on model resolution was tested to allow a critical assessment of the validity of the 3-D velocity model. Lateral resolution throughout the model to a depth of 3–5 km below the top of the basement is generally 10–20 km.

Deep crustal structure in the southwestern Yilgarn Craton, Western Australia

The Archaean rocks of the Yilgarn Craton comprise the largest geological entity in Western Australia and one of the largest areas of Archaean crust anywhere in the world. However, poor outcrop has prevented a thorough understanding of the structure of the crust in the region, a subject of particular interest being the relationship between high-grade metamorphic rocks in the west, and lower-grade granitoid-greenstone terrains further east. Interpretation of previously unpublished seismic refraction data, combined with the re-interpretation of published data, allows crustal velocity structure in the southwest of the Craton to be resolved. Overall, the structure is similar to shield areas elsewhere in the world, the data defining a two-layered crust with an average thickness of about 35 km. However, significant lateral variations in velocity structure are also observed, and these coincide with a terrane boundary previously postulated on the basis of geological observations and seismic reflection data. The terrane boundary dips to the east, extends through the entire crust, and is notable for an area of anomalously high seismic velocity in the lower crust. Two possible interpretations of the high-velocity zone are proposed. Firstly, it is due to the presence of mafic to ultramafic intrusions. Secondly, and the preferred interpretation, the high-velocity zone is a fault-bounded mega-sliver, which may be a suspect terrane in its own right, perhaps with oceanic affinities. The results of the seismic experiment strongly support tectonic models for Yilgarn Craton based on terrane tectonics. The possibility that the terrane boundary is being reactivated by contemporary stresses also provides a plausible explanation for the local intra-plate seismicity.

Three-dimensional crustal structure beneath the TOR array and effects on teleseismic wavefronts

The temporary seismic station array (TOR) was designed to study the lithosphere–asthenosphere system across the northwestern part of the Trans-European Suture Zone (TESZ) by teleseismic tomography. Teleseismic wavefronts, when propagating through complex crustal structure, undergo severe distortion that may result in travel time residual anomalies of significant amplitude. The inversion of teleseismic travel time residuals for deep structures without accounting for such crustal-related anomalies may erroneously map these travel time anomalies into features at greater depth. In this study we apply a three-dimensional (3-D) technique to estimate effects of a priori known 3-D crustal structure on travel times of teleseismic waves observed at the TOR seismic array across the TESZ to correct for these effects in future tomographic studies.A uniform 3-D crustal model is developed by use of published two-dimensional crustal models from previous active seismic surveys. The parameterization of this 3-D crustal model is designed to adequately represent those crustal structures that mostly influence the propagation of teleseismic wavefronts. The 3-D model includes lateral variation in velocity structure, Moho topography, and large and deep sedimentary basins. The teleseismic forward problem for this local 3-D model is solved by calculation of travel times to the base of the model using a standard whole Earth model and by subsequent propagation of spherical wavefronts using finite difference methods. Travel time calculations for an event near Japan reveal significant lateral variations in the range between −0.3s and +0.5s due to crustal structures. Being able to obtain the full travel time field at the surface of the model has the additional advantage of improving the identification and timing of seismic phases observed at the TOR seismic array.

TheOleanderProject: Monitoring the Variability of the Gulf Stream and Adjacent Waters between New Jersey and Bermuda

Abstract An overview of the first 4.5 years of operation of a program to monitor the structure and variability of the Gulf Stream (GS) is presented. A container vessel that operates on a weekly schedule between Port Elizabeth, New Jersey, and Hamilton, Bermuda, is equipped with a 150-kHz narrowband acoustic Doppler current profiler to measure currents from the surface to ∼300 m depth. A major objective of the multiyear program is to study the annual cycle and interannual variations in velocity structure and transport by the GS. In this survey the focus is on the transport and lateral structure of the current at 52-m depth. The velocity maximum is constant at 2.07 ± 0.24 m s−1 (4 kt) with a seasonal range of ∼0.1 m s−1. Seasonal and interannual variations in total transport are observed but appear to be limited to the edges of the current, apparently reflecting low-frequency variations in the intensity of the recirculating waters adjacent to the stream. The transport by the central core of the current, def...

Slant stacks of refraction data from maine: Effects of lateral variations in velocity structure

Abstract We discuss some of the advantages and pitfalls associated with τ( p ) analysis of wide-angle profiles recorded over terranes characterized by significant lateral heterogeneity. We construct composite slant stacks for a pair of reversed shots recorded along the axis of the Merrimack Synclinorium in southwest Maine and invert picks for bounds on averaged one-dimensional slowness-depth structure. Extremal bounds on total crustal thickness derived from shot point 12 are 40 to 43 km. Average crustal velocities for individual slowness models corresponding to the extremal depths range from 6.5 to 6.6 km/sec. Predicted normal-incidence two-way travel times for the Moho lie between 12.2 and 13.2 sec. The utility of one-dimensional inversion of τ( p ) data is largely in estimating the maximum resolving power of the data set; this is information that forward modeling approaches do not provide. Comparisons of slant stacks of reversed shot gathers with intercept times predicted for a two-dimensional model derived by ray tracing highlight some of the pitfalls associated with interpretation of τ( p ) data in terms of one-dimensional structure. In some cases, the principal arrival may become multi-valued, resulting in misidentification of phases and compounding biases in estimates of average crustal parameters. Slant stacks can be highly useful, however, in refining two-dimensional models derived from travel times because of the sensitivity of ray parameters to dip and lateral velocity gradients. We use slant stacks in conjunction with the original shot gathers to refine our estimate of the configuration of the Moho beneath southwest Maine. Our final model shows a step in the Moho that is consistent with variations in normal two-way travel times inferred from analysis of postcritical fan shots. This abrupt change in crustal thickness may mark the northeast extent of significant underthrusting of Avalon crust beneath Medial New England basement during the Acadian Orogeny. The preservation of offsets in Moho that were formed during Paleozoic compression is consistent with the limited amount of rifting experienced by this portion of the orogen during the opening of the Atlantic.

Shear-wave splitting near Guam

Polarities of shear waves from intermediate-focus events underneath Guam are studied. For records from a group of ten events, shear-wave splitting with faster-arriving E-W components are observed. This event group occurred within, or above, one geographic portion of the Wadati-Benioff zone, with depths ranging between 57 and 148 km. Ray tracing calculations were performed for 3-D and 1-D velocity models constructed for the region to determine expected S-wave polarities and ray patterns, as well as their sensitivities to variations in velocity structure. These were used to infer the probable existence of intrinsic anisotropy at depth and to determine the location and magnitude of anisotropy which can explain the observed shear-wave splitting. The most probable location of the anisotropy is beneath the crust and above, or partially within, the subducting slab. Assuming a maximum depth range of 10–120 km for the location of the anisotropy, its amount is about 1%, which may be viewed as a lower bound. Plausible causes of the anisotropy include mantle flow and thin, sheet-like channels filled with lava, or water vapor migrating upward from the subducting slab.

A 3-D raytracer for teleseismic body-wave arrival times

Abstract In this article, we present a method for determining the minimum time ray path from a planar wavefront at depth to a fixed point at the surface through a 3-D heterogeneous velocity structure. The program is based on the two point raytracer of Prothero et al. (1988), with a modification to use odd quarter-cycles of sinusoids to distort the rays. These functions produce zero displacement at the fixed receiver but non zero displacement at the wavefront. This allows the raytracer to explore a wide variety of ray paths starting at different points on the wavefront, while the rays leaving the wavefront remain normal to it. To find the minimum time path, amplitudes of these functions are systematically perturbed using the simplex algorithm. The approach is analogous to using a Fourier series to fit a curve with specific properties at each end. Development of this raytracer was motivated by the desire to model our observations of large perturbations in the bearings and phase velocities of teleseismic P waves recorded in Long Valley caldera, California. These observations clearly show the inadequacy of 1-D raytracing in regions having complex, three-dimensionally varying velocity structure. This raytracer provides a basic tool with which to study the effects of ray bending on tomographic results from these kinds of environments.

Kinematic source modelling of normal-faulting earthquakes using the finite element method

Near-field full elastic wave synthetic seismograms were computed for simulations of propagating normal faults using a 2-D, finite element method. The kinematics of faulting were simulated using a modified application of the ‘split-node’ technique of Melosh & Raefsky. Fault simulations using ‘split-nodes’ require only a modification of the load vector and correctly models relative fault slip as a double-couple force distribution without moment. Synthetic seismograms were computed for a suite of dip-slip, normal fault models designed to illustrate waveform variations due to temporal and spatial changes of the relative slip and rupture velocity along the fault, in addition, to fault slip at the free surface. Results from these models show that distinct P- and S-wave phases, observed in the near-field, are produced by the initiation and termination of faulting. Primarily observed in the hanging wall, these phases are identified as the “starting” and “stopping” phases. Their general amplitude behaviour can be explained by appropriately orientated double-couple point sources. For fault models with rapidly changing source-property variations in the slip distribution, rupture velocity, and fault orientation, notable S-wave phases were produced and observed in the hanging wall. Comparison of seismograms show that changes in rupture velocity produce larger amplitude variations in this phase than corresponding changes in slip distribution. The amplitudes of the associated P-wave phase were generally too small, compared with the S-wave, to be observed on the seismograms. Development of the ‘split-node’ technique for use in synthetic seismogram calculations represents an important new method permitting simple and accurate modelling of elastic waves due to fault propagation in a laterally varying velocity structure.

Simultaneous inversion for hypocenters and lateral velocity variations: An iterative solution with a layered model

An iterative inversion technique has been developed that uses the direct P and S wave arrival times from local earthquakes to compute simultaneously a three‐dimensional velocity structure and relocated hypocenters. Crustal structure is modeled by subdividing flat layers into rectangular blocks. An interpolation function is used to smoothly vary velocities between blocks, allowing ray trace calculations of travel times in a three‐dimensional medium. Tests using synthetic data from known models show that solutions are reasonably independent of block size and spatial distribution but are sensitive to the choice of layer thicknesses. Application of the technique to observed earthquake data from north‐central Utah shows the following: (1) lateral velocity variations in the crust as large as 7% occur over 30‐km distances, (2) earthquake epicenters computed with the three‐dimensional velocity structure were shifted an average of 3.0 km from locations determined assuming homogeneous flat layered models, and (3) the laterally varying velocity structure correlates with anomalous variations in the local gravity and aeromagnetic fields, suggesting that the new velocity information can be valuable in acquiring a better understanding of crustal structure.

Horizontal Velocity Structure in Waterspouts

We have measured the spatial variation of a single horizontal component of the velocity in a number of waterspouts using an airborne infrared Doppler lidar. In 21 data sets, maximum velocities range from 4.2 to 33.6 m s−1 and visible funnel diameters from 6.6 to 90 m. Data were taken at altitudes between 675 m, near cloud base, and 95 m above the surface. The sequences show time development of the velocity as a function of radius at a fixed altitude and the velocity structure at different altitudes and sequential times with a horizontal resolution of ∼0.75 m between data points. The variation in velocity structure between waterspouts is large, with some showing marked azimuthal asymmetry and mixing with the ambient flow, and others showing multiple concentric vortex shells.

Lateral variation in the structure of the upper mantle beneath Eurasia

Summary. A travel-time curve for seismic rays bottoming within the upper mantle beneath Western Russia and recorded at the NORSAR array is sub- stantially different from a curve for rays bottoming beneath Southern and Central Europe, recorded at the NORSAR and Eskdalemuir arrays. This difference is interpreted in terms of a laterally varying velocity structure to a depth of approximately 500 km beneath the two regions. Introduction Pronounced variations in the structure of the upper 200 km of the Earth have been clearly defined by surface wave, travel-time and teleseismic phase residual studies. In contrast, travel-time data which reflect deeper mantle structure in various regions have often been interpreted as indicating essentially similar velocity distributions. Hales (1972) concludes that the upper mantle beneath the whole of the US includes two sharp velocity increases of about 10 per cent near depths of 400 km and 650 km. More recently, Wiggins & Helmberger (1973) have shown that below a depth of 300km, one model satisfies all their observational data for rays bottoming beneath the Western and Central US.

Reflection, refraction and mode conversion of long-period surface waves and the measurement of Q−1 for free oscillations

abstract Filtered time series of Rayleigh waves from 4 events recorded on the quartz accelerometer at the IGPP, Camp Elliott Station, were analyzed. Attenuation (Q−1) was computed for spheroidal fundamental modes (oS19 to oS24) from several sequences of time-lapsed records for each event. A five-fold variation in measured Q−1 (and some variations in peak frequency) was assumed to be the result of lateral inhomogeneities in earth structure. Utilizing the duality between Rayleigh waves and fundamental-mode spheroidal oscillations, model power spectra were computed by summing the simulated Fourier transforms of dispersed wave trains. The effect of lateral variations in earth structure resulting in reflection, refraction and mode conversion of fundamental-mode surface waves was simulated by changes in amplitude, phase angle, and group and component travel times. Assuming an anelastic 10,000/Q of 33.3 (Q = 300), the observed range of measured Q−1 (and peak frequency) variations was duplicated by models with up to 5 per cent of the fundamental-mode Rayleigh-wave energy being “scattered”, i.e., reflected, refracted or converted to higher modes. In the real Earth, this would call for lateral variations in velocity structure well below the upper few hundred kilometers of the mantle. Recent seismological investigations have suggested lateral variations at these depths.

Two-Timing in Underwater Acoustics

Correlation studies of high-frequency signals over long periods of time must take into account the slowly varying velocity structure of the medium. Experimental studies have pointed out that amplitude and phase of a CW signal are modulated by the fluctuation frequency of the channel stratification. A theory based upon the Method of Multiple Scales as expounded by Van Dyke is applied to the wave equation. The technique appears to be successful, but complicated. An extension of Lighthill's rule, which was originally developed for the Method of Strained Coordinates, is presented.