Travel-time Curves Research Articles

Modelling and parameterization errors in body wave seismology

Summary. When body wave seismic data are analysed under ray theory assumptions it is common to transform the observations so that p , the horizontal slowness, is the independent variable to allow use of a linear formulation of the inverse problem. Modelling errors, such as the presence of inhomogeneities in the real Earth but not in the model, introduce correlated errors in distance X(p) and time T(p), as does the uncertainty in estimating p, the derivative of the travel-time curve. The propagation of errors into the model space is most easily done if the covariance matrix of the observations is diagonalized. For the error sources listed above, the transformation producing the desired result is a transformation to the variables ~(p) = T-pX and

Travel times for Pacific earthquakes

Pacific earthquakes studied by Gogna, also three important explosions in the Tuamotu archipelago, are rediscussed. The results are very consistent, but those from Tuamotu are later than Gogna's by about 1 s in the times of P about 60°. Both sets of data give PKP residuals about −5 s about 140° – 142°, indicating that the observations there referred to the neighbourhood of the cusp of the travel-time curve but the ISS had compared them with the DEF branch. The corresponding difference in the 1940 tables is about 2 s. Analysis at intervals of 1° indicated that the cusp of PKP is about 141° instead of 143° as in the 1940 tables and the difference between it and the DEF branch at these distances is about — 5 s. Travel times of S under the Pacific were found but need more data, especially at distances under 10°. Times of PcP reported by Kogan and Carder were compared with those calculated from P in Gogan's explosions, and indicated a radius of the core of 3479.8 ± 1.8 km.

Attenuation measurement with sonobuoys

Sonobuoy data in a number of shelf areas have yielded travel time curves which are relatively smooth and which result in velocity–depth profiles that are nearly straight lines to depths of several kilometers. Moreover, many of the records contain several multiples of the first refracted arrival which are well separated from the first arrival and free of excessive noise. Such data are ideal for comparing the amplitude of the first arrival and the amplitude at the corresponding point on the multiple at double distance and double time. By taking the ratio of these amplitudes, the effects of geometrical spreading and the angular dependence of the source are removed so that estimates of total attenuation as a function of depth and frequency are much more easily studied. In this paper we present an example of such a study using sonobuoy data from the Sarawak basin which is located in the South China Sea (4 °N–110 °E). Measurements made in the frequency range of 10 to 60 Hz suggest that the logarithmic decrement δ or the quality factor Q are not constants but depend rather strongly on frequency in the diving waves that were studied.

Combined estimate of the mechanical properties of a rock mass at great depth at the deposit-surveying stage

1. At the surveying stage for deep deposits, in conjunction with the drilling of exploratory wells, it is possible to obtain reliable data for calculating the coefficient of structural weakening. 2. Seismoacoustic well-logging and fractionation (break-up) of the core provides a complete picture of the state of the rock mass near the exploratory well. Based on the travel-time curve for longitudinal elastic wave velocity, the fractured (karst) zones can be determined with the necessary accuracy, as well as the acoustic coefficient of weakening in the rock mass due to the scale factor A⋆ = λsf, the acoustic fracturing coefficient Ai, or the coefficient η1, based on which the intensity of fracturing, the size of the structural block, the coefficient of structural weakening λ, and the modulus of elasticity E can be determined in the rock mass. 3. The method proposed permits the coefficient of structural weakening and modulus of elasticity to be most operatively determined, the values for which can be used at the planning stage for mining operations.

Seismic refraction survey at a dam construction site

The paper deals with seismic refraction survey as a method forin situ pre-investigation for the Rwegura hydro-electric power plant at Burundi. After a short description and discussion of theoretical relations the performance of the measurements is reported. The presentation of the results includes travel-time curves, comparison with sounding results and discussion of wave velocities. Some cases of negative apparent wave velocity are reported and analysed.

Inverting travel times with a triplication

abstractA new technique has been developed to overcome difficulties which arise when body-wave data in the form of T-delta is reduced to the tau-p form in the presence of cusps. The technique uses sophisticated constrained spline fits to allow for constraints on the derivatives and thus eliminates the possibility of points of inflexion occurring as well as making sure that the curvature is in the right direction. Having arrived at a good fit to the T-delta data with a single spline, the tau-p curve can be derived which has taken the cusp properly into consideration. The tau-p curve is then inverted with a new linear inversion method, which again takes into consideration whether a particular point lies on a prograde or retrograde branch.The technique is applied to the lunar seismic data, where at least one triplication is presumed to occur for the P-wave travel-time curve. It is shown that the present analysis is not accurate enough to address itself to the problem of a possible lunar core, since there are insufficient data points for events close to the antipode of the center of the lunar network.

Inversion for sediment structure using multichannel data

The problem of inferring a velocity or impedance structure for a layered sediment has traditionally been posed in terms of a restricted data set, such as a normal incidence reflection seismogram or a refraction travel time curve. Data acquisition systems used for petroleum exploration record 96 or more channels in a linear array, and thus obtain a virtually unaliased representation of the wavefield. The data contains reflections from all angles of incidence, including postcritical angles, all normal modes, leaky modes, and head waves. A key tool in dealing with this kind of data is the Radon transform, which transforms the data set into a plane-wave representation. A raw data set u(x,t) is transformed into v(p,τ), where p, which parameterizes the plane waves, is the x component of slowness, and τ is a reduced (intercept) time. The ensemble of plane-wave seismograms is the basis for inversion of the data to obtain a one-dimensional velocity/impedance model. Modeling the lossy elastic transient plane-wave response is easily done by one of the modifications of the Thomson-Haskell matrix method. The algorithm can be vectorized to run very rapidly on a machine with an array processor. The object of the inversion is to obtain a model whose plane-wave decomposition agrees exactly with that of the observed wavefield. In the absence of noise, one plane-wave seismogram suffices to determine the model. An ensemble for a range of wave parameters overdetermines the model. In the presence of noise, the problem has the character of an ill-posed, overdetermined least squares problem. In order to carry through the solution by means of least squares, it is necessary to go through a construction procedure to make the trial model linearly close to the correct one in some parameter subspace. Several construction procedures proceed by stripping from the top down, although none have been reduced to a fully automatic alogrithm, and require some human intervention in the iterations.

Earthquake location in island arcs

A comprehensive data set of selected teleseismic P-wave arrivals and local-network P- and S-wave arrivals from large earthquakes occurring at all depths within a small section of the central Aleutians is used to examine the general problem of earthquake location in island arcs. Reference hypocenters for this special data set are determined for shallow earthquakes from local-network data and for deep earthquakes from combined local and teleseismic data by joint inversion for structure and location. The high-velocity lithospheric slab beneath the central Aleutians may displace hypocenters that are located using spherically symmetric Earth models; the amount of displacement depends on the position of the earthquakes with respect to the slab and on whether local or teleseismic data are used to locate the earthquakes. Hypocenters for trench and intermediate-depth events appear to be minimally biased by the effects of slab structure on rays to teleseismic stations. However, locations of intermediate-depth events based on only local data are systematically displaced southwards, the magnitude of the displacement being proportional to depth. Shallow-focus events along the main thrust zone, although well located using only local-network data, are severely shifted northwards and deeper, with displacements as large as 50 km, by slab effects on teleseismic travel times. Hypocenters determined by a method that utilizes seismic ray tracing through a three-dimensional velocity model of the subduction zone, derived by thermal modeling, are compared to results obtained by the method of joint hypocenter determination (JHD) that formally assumes a laterally homogeneous velocity model over the source region and treats all raypath anomalies as constant station corrections to the travel-time curve. The ray-tracing method has the theoretical advantage that it accounts for variations in travel-time anomalies within a group of events distributed over a sizable region of a dipping, high-velocity lithospheric slab. In application, JHD has the practical advantage that it does not require the specification of a theoretical velocity model for the slab. Considering earthquakes within a 260 km long by 60 km wide section of the Aleutian main thrust zone, our results suggest that the theoretical velocity structure of the slab is presently not sufficiently well known that accurate locations can be obtained independently of locally recorded data. Using a locally recorded earthquake as a calibration event, JHD gave excellent results over the entire section of the main thrust zone here studied, without showing a strong effect that might be attributed to spatially varying source-station anomalies. We also calibrated the ray-tracing method using locally recorded data and obtained results generally similar to those obtained by JHD.

Topological approach to remote sensing

Summary. The topological problem underlying remote sensing is analysed by determining the geometric singularities that an unknown surface or structure generically impresses on a sensing wavefield. It is shown that the analytical singularities observed in scattering amplitudes and echograms are produced by the topological singularities of the scattering system. Imposing the principle of structural stability on the inverse scattering problem, the singularities that generically occur in recorded signals, travel-time curves, surface contour maps and Fresnel-zone topographies can, together with the associated high-intensity diffraction patterns, be classified into a few universal standard forms described by catastrophe polynomials. As the source-receiver positions vary, the patterns change their morphologies in terms of specific bifurcation sets. By applying singularity and bifurcation theory to allow the effects of caustics (both in ray and wave theory) to be incorporated into three-dimensional techniques for reconstructing surfaces and subsurface structures from their echoes, the interpretation process is considerably simplified and permits an on-site 3D survey. Universal power laws for singularity-dominated echo amplitude variations with the source frequency are deduced. The shape of a scattering surface is reconstructed using the high-frequency regime alone. Discontinuities in the surface, edges and faults, are discussed in terms of constraint catastrophes and the patterns they produce in echograms are classified.

Analysis of earthquakes in the distance range 40–70° and inferred lower mantle structure

About fifty earthquakes in the distance range 40–70° and azimuthal range 45–120° from the Celebes, Philippines, Mariana and Kurile Island regions, and recorded at Gauribidanur seismic array in southern India, were used in the present study. Measurements on slowness and apparent azimuths were made on the first 30 s of the short period P-wave trains using an adaptive processing technique. Analysis of this data set has revealed no strong evidence for any triplications in the travel-time curve over the ranges in question. The P-wave velocity increases continuously with an almost uniform gradient below 1000 km depth range and is in very close agreement with the JB model. Almost all the observed slowness values of the events were anomalously low and consistent suggesting that they are caused by some azimuthal dependent structure near the array.

Crustal structure near the Arctic Mid‐Ocean Ridge

Seismic reflection and crustal refraction studies were carried out at the FRAM I ice station near the Arctic Mid‐Ocean Ridge crest. This ridge is spreading at the very slow rate of 5 mm yr−1. Upper mantle Pn arrivals with apparent velocities averaging 7.9 km/s are observed at distance ranges less than 15 km, with corresponding intercept times of typically 1.1 s of which 0.5 s can be attributed to the travel path through a sedimentary layer. Anisotropy of about 6% appears to be present. These early Pn arrivals are observed on seven of the eight refraction lines studied, but crustal velocities are not well constrained. Modeling was done by computing travel time curves for a number of possible velocity structures for comparison with the data and suggests a crustal thickness between 2 and 3 km. This thin crust is associated with low‐amplitude magnetic anomalies. The remaining refraction line was shot within a region of enhanced magnetic anomaly amplitudes the Yermak H zone, where the data interpretation indicates a crustal thickness of about 8 km. This thicker crust and associated strong magnetic anomalies may be due to the ‘Yermak hot spot,’ a region of high magmatic activity. The generally thin crust is probably due to low basaltic productivity at the ridge crest, as a result of increased cooling at the very slow spreading rate.

Crustal low-velocity zone south of Shatsky Rise, northwest Pacific Ocean

The detailed seismic refraction investigation of the oceanic crust south of Shatsky Rise in the Northwestern Pacific revealed a low velocity zone (LVZ) with an average compressional wave velocity of 6.3 km/s within layer 3. This conclusion is based on the shadow zone for refractions on the travel time curves in their first arrivals from the M discontinuity. The LVZ may be composed of oceanic plagiogranites because serpentinization of peridotites would probably lead to an increase in crustal block volume with a concomitent decrease in density and thereby thickening and upwelling at the place of now “overdeepened” ocean would be expected.

VELOCITY ANALYSIS FOR WIDE APERTURE SEISMIC DATA*

AbstractCommon midpoint data are now being collected with ever increasing source‐receiver offsets. For wide aperture seismic data classical methods of interpretation fail, since velocity analyses and signal‐to‐noise enhancement methods based on hyperbolic traveltime curves are no longer appropriate. Therefore, the goals of increased velocity resolution and signal enhancement, which motivate the increase in offset, are not achieved. Approximate methods, involving higher order traveltime curves or extrapolations, have been developed for velocity analysis but these are ineffective in the presence of refracted arrivals, and lack a physical basis. These problems can be minimized by transforming the observational data to the domain of intercept or vertical delay time τ and horizontal ray parameter p. In this domain headwave refractions are collapsed into points and both near vertical and wide angle reflections can be analyzed simultaneously to derive velocity‐depth information, even in the presence of velocity gradients or low velocity zones.

Seismic travel-time anomalies from events in the western Soviet Union

abstract An iterative technique and the joint epicenter determination technique are utilized to compute improved locations for 33 presumed underground explosions in the western Soviet Union and station corrections to the Herrin et al. (1968) travel-time curves for 98 stations. Comparison of the results of the two techniques indicates that the iterative procedure produces improved location accuracy relative to the joint epicenter determination technique. Analysis of the station corrections determined by this study indicate that epicenters located on a worldwide basis with these corrections are as reliable, or slightly better than those determined using corrections from previous studies. For events in the western Soviet Union, the station corrections are superior to previous estimates. Comparison of the travel-time corrections with magnitude residuals previously determined by North (1977) fails to indicate a strong worldwide correlation of early arrivals with increased amplitudes and late arrivals with decreased amplitudes.

Travel-time curves for a simple sea floor model

This paper reviews a simple technique for interpreting the velocity structure of upper oceanic crust from travel-time data of sonobuoy and ocean bottom receiver refraction experiments. The technique does not involve sophisticated digital processing or synthetic seismogram analysis. Interpretations can be carried out with a pencil, paper and slide rule.

Slowness techniques in seismic interpretation

Wave propagation in a horizontally stratified medium may be characterized by the slowness p of the waves. In ray theory the travel time curves are parameterized by the slowness p, and this dependence is exploited in modern methods of tau inversion for travel times. The complete response of a stratified medium may also be represented as a superposition of cylindrical waves of varying slowness. This representation leads to the possibility of presenting the response of the stratification in a slowness time map containing both travel time and amplitude information. Such an approach requires copious good quality data but offers advantages in interpretation. Even in the presence of lateral variations of structure, slowness methods can provide useful information.

Inversion of the travel-time curve of a reflected wave

Summary. A general theory of inversion of the reflected travel-time curve is developed. The properties of the travel-time curve and its analytical continuation are examined. The inversion formulae are presented and the structure of the set of velocity—depth functions is studied in detail. The inverse problem or a finite number of reflectors and the problem of separation of the reflected and multiple travel-time curves are both considered.

Seismic investigation of crust-mantle transition in continental rift systems - Jordan-Dead Sea rift and Rhinegraben

The recent acquisition of high-quality seismic refraction data in the Jordan—Dead Sea rift and adjacent areas has made possible the investigation of the dynamic properties of seismic P-waves refracted and reflected at the crust—upper mantle boundary. These waves cause high-amplitude arrivals near the outer cusp of the travel-time curve which are followed by an abrupt decrease in amplitudes at increasing distances beyond the cusp. It has been shown that such amplitude distributions can only be the result of a smooth rapid increase of velocity with depth. In the case of the Jordan—Dead Sea rift the amplitude distribution indicates the presence of a transition zone between the lower crust and upper mantle in which the velocity increases smoothly. The interpretation of seismic refraction data in the Rhinegraben indicates the existence of a similar transition zone. In both rifts the crust—mantle boundary outside the rift is represented by sharp velocity discontinuity. The comparison of the velocity structure of the crust—upper mantle boundary suggests that a smooth transition zone at the base of the lower crust is a characteristic property of continental rifts which could be interpreted in terms of crust—mantle interaction.

Crustal structure near Palmdale, California, from borehole-determined ray parameters

Abstract Seismic refraction profiles across and along the San Andreas fault zone near Palmadale, California, were used to establish propagation paths for several phases whose travel times are being monitored for earthquake precursors. The profiles were generated using an airgun submerged in Bouquet Reservoir, together with surface and borehole geophone receivers. Source signal stacking, coupled with the noise reduction provided by the downhole receivers, provided data of sufficient quality to measure the ray parameters of the phases. Using the ray parameters and surface travel-time curves, velocity structures parallel and transverse to the fault zone have been constructed. Two of the observed phases are confined to the upper portions of the crustal structure, which was found to differ across the fault. A third phase is consistent with a reflection from a depth of approximately 10 km. Two later arrivals are suggestive of reflections from depths of 21 and 27 km.

Composite travel x of seismic waves and general velocity models of the Voronezh Shield crust and upper mantle

Summary. Extensive explosion seismology investigations using industrial quarry blasts have been undertaken across the Voronezh Shield. Various phases of P- and S-waves, including crustal refractions (Pg and Sa, upper mantle refractions (P, and S,), intercrustal Conrad reflections (Pc and Sc), Moho reflections (PM and SM) and reflections from boundaries within the upper mantle (PM1 and SM1, PM2 and SM2) have all been recognized. Independent and compatible velocity models V, (z) and V, (z) of the lithosphere have been derived from the composite travel-time curves and have been used to determine the variation of Poisson's ratio (6) and the seismic parameter (@) with depth. These quantities significantly advance the knowledge of the layered structure and composition of the lithosphere in the Voronezh Shield region. The models for this area are compared with those for the Canadian Shield.