Geiger Method Research Articles

Accurate Location of Synthetic Acoustic Emissions and Location Sensitivity to Relocation Methods, Velocity Perturbations, and Seismic Anisotropy

Acoustic emission (AE) monitoring is a non-invasive method of monitoring fracturing both in situ, and in experimental rock deformation studies. Until recently, the major impediment for imaging brittle failure within a rock mass is the accuracy at which the hypocenters may be located. However, recent advances in the location of regional scale earthquakes have successfully reduced hypocentral uncertainties by an order of magnitude. The least-squares Geiger, master event relocation, and double difference methods have been considered in a series of synthetic experiments which investigate their ability to resolve AE hypocentral locations. The effect of AE hypocenter location accuracy due to seismic velocity perturbations, uncertainty in the first arrival pick, array geometry and the inversion of a seismically anisotropic structure with an isotropic velocity model were tested. Hypocenters determined using the Geiger procedure for a homogeneous, isotropic sample with a known velocity model gave a RMS error for the hypocenter locations of 2.6 mm; in contrast the double difference method is capable of reducing the location error of these hypocenters by an order of magnitude. We test uncertainties in velocity model of up to ±10% and show that the double difference method can attain the same RMS error as using the standard Geiger procedure with a known velocity model. The double difference method is also capable of precise locations even in a 40% anisotropic velocity structure using an isotropic model for location and attains a RMS mislocation error of 2.6 mm that is comparable to a RMS mislocation error produced with an isotropic known velocity model using the Geiger approach. We test the effect of sensor geometry on location accuracy and find that, even when sensors are missing, the double difference method is capable of a 1.43 mm total RMS mislocation compared to 4.58 mm for the Geiger method. The accuracy of automatic picking algorithms used for AE studies is ±0.5 µs (1 time sample when the sampling rate is 0.2 µs). We investigate how AE locations are effected by the accuracy of first arrival picking by randomly delaying the actual first arrival by up to 5 time samples. We find that even when noise levels are set to 5 time samples the double difference method successfully relocates the synthetic AE.

A statistical technique for validating velocity models

Abstract This study investigates the use of a station influence statistic to identify velocity model shortcomings in the earthquake hypocenter location problem. Two groups of microearthquake events are examined. The first is a group of 81 events from the Mount St. Helens region that occurred between November 1987 and September 1991; the second, 110 well-located events from the 1992 Joshua Tree after-shock sequence. We describe a method for validating a postulated earth model. Let λ denote the hypocenter estimates that Geiger's method obtains. Systematically remove each station observation from the location problem, and recompute the location estimate. Call this estimate λ(i) when the ith station is removed. For a single event, define a station's influence (SI) as a weighted difference between λ and λ(i). Distributional summaries of SI statistics across events are used to identify model shortcomings: Given a specified velocity model, SI distributions that are not homogeneous across stations provide evidence of model inadequacies and/or failures in the weighting scheme. We show that velocity model shortcomings detected using SI statistics for the Mount St. Helens sequence under a one-dimensional model appear to correlate with known physical anomalies; while SI distributions evaluated under a three-dimensional model are more homogeneous and reflect a modest improvement over the one-dimensional model. SI distributions provide evidence of model failure for the Joshua Tree sequence under a one-dimensional model, but no evidence of failure under a three-dimensional model. Finally, the weighting scheme's validity is verified for the Joshua Tree sequence under the three-dimensional model.

A new regional hypocenter location method

AbstractA new regional hypocenter location method is presented in this article to address location problems associated with sparse station distributions and lack of velocity information. For an arbitrary slowness structure, an unknown parameter, slowness deviation, is introduced to construct a more precise slowness model than a homogeneous slowness model. By incorporating a genetic algorithm, the new method yields reasonable solutions for epicenters and origin times both inside and outside the seismic network. The synthetic data tests indicate that outside the network the new method gives excellent results for epicenter locations compared with Geiger's method, even when Geiger's method is used with the true velocity structure. The relocation of the Loma Prieta mainshock and 26 aftershocks occurring within the first 24 h after the mainshock by using the new hypocenter location method places the aftershocks an average of 2.29 km to the southwest of the locations published by the Northern California Earthquake Data Center. The new locations are in better agreement with the geodetic measurements.

Lithospheric structure models applied for locating the Romanian seismic events

The paper presents our attempts made for improving the locations obtained for local seismic events, using refined lithospheric structure models. The location program (based on Geiger method) supposes a known model. The program is run for some seismic sequences which occurred in different regions, on the Romanian territory, using for each of the sequences three velocity models: 1) 7 layers of constant velocity of seismic waves, as an average structure of the lithosphere for the whole territory; 2) site dependent structure (below each station), based on geophysical and geological information on the crust; 3) curves deseribing the dependence of propagation velocities with depth in the lithosphere, characterizing the 7 structural units delineated on the Romanian territory. The results obtained using the different velocity models are compared. Station corrections are computed for each data set. Finally, the locations determined for some quarry blasts are compared with the real ones.

Teleseismic epicentre locations from arrival times at regional networks

SUMMARY Arrival time reportings from local and regional networks can be used for preliminary epicentre location of teleseismic events via slowness and azimuth estimation. Robust techniques for doing so are presented, and their relative performance tested on ISC listed P arrival times of Fennoscandian stations (aperture ca. 10°). For the 10 events analysed, the most efficient technique, a variant of Geiger's method for fitting arrival times to traveltime tables, gave average location errors of less than 1° for distances from 15° to 85°.

Approximative ray tracing and the accuracy of earthquake localization in media approximated by block structures

Localization errors made by approximating a given medium by a homogeneous rectangular block model and due to the approximative solution of two-point ray tracing are estimated for four different structures. The separate influence of the individual characteristics of the medium is discussed. The iterative Geiger method supplemented by a generalized inversion of the damped least-squares and Marquardt algorithm is used for localizating five hypothetical hypocentres using 9 stations. Travel times in the original structures, used as input data for localization, are calculated using a special ray tracing system.

Analysis methods for kinematic data from local earthquakes

A review of the techniques underlying the basic analysis of local and regional earthquakes is presented, describing modern methods for earthquake location, velocity inversion, and fault plane solution. These three topics require a method for ray tracing, determining the seismic ray path connecting the earthquake source to the observing station. A basic introduction to common ray tracing techniques is presented. Earthquake location based on Geiger's method remains the most common in use. Our understanding of this fundamental problem has been enhanced by a variety of studies, and by progress in geophysical inverse theory, but difficulties with the method remain. Alternatives to Geiger's method are reviewed to emphasize the continuing need for study of earthquake location techniques. The more complex problem of simultaneous inversion is receiving increasing attention. Algorithms have been developed to deal with a wide range of velocity structures, including layered, continuous one‐dimensional, and three‐dimensional models. Modern techniques for manipulating and transforming large matrix equations have been critical to the success of simultaneous inversion. Finally, methods for determining fault plane solutions are reviewed, such as first motion, amplitude ratio, and polarization, and techniques are discussed for inferring stress orientation from sets of focal mechanisms.

Nonlinear earthquake location: Theory and examples

AbstractA fundamental modification to Geiger's method of earthquake location for local earthquakes is described which incorporates nonlinear behavior of travel time as a function of source position. The use of Newton's method rather than the usual Gauss-Newton method allows the inclusion of second-order partial derivatives of travel time with respect to source coordinates in the location algorithm. These second-order derivatives can be calculated quite easily for half-space and layered crustal models. Expected benefits are improved convergence and stability, as demonstrated in a series of examples, and more realistic assessment of solution uncertainty.

Rapid solution of ray tracing problems in heterogeneous media

abstract The determination of local earthquake hypocenters and orgin times from first-P-arrival times by Geiger's method requires a technique for finding the minimum travel time (and derivatives) between the source and the station. Sophisticated ray tracing techniques have been developed for this purpose for use in complex velocity structures. Unfortunately, the two common techniques, shooting and bending, are generally prohibitively expensive for routine use in data analysis. The bending method is also particularly vulnerable to the problem of local minima in travel time. A method has been developed known as the ray initializer, which can be used to circumvent these problems in many cases. First, the technique can find a reasonable estimate of the minimum-time ray path in a quick and efficient manner. The velocity in a region local to the source and receiver is laterally averaged to yield an approximate layered velocity model. One-dimensional ray tracing techniques are used to find the minimum-time path for this layered structure. The ray path estimate can then be used as the starting path in a bending routine, a procedure resulting in more rapid convergence and the avoidance of local minima. Second, the travel time found by numerical integration along the estimated ray path is an excellent approximation to the actual travel time. Thus, in many cases, the ray initializer can be substituted for a three-dimensional ray tracing routine with a tremendous increase in efficiency and only a small loss in accuracy. It is found that the location of an explosion, derived using the ray initializer, is nearly identical to a complete ray tracing solution, even for a highly complex velocity structure.

Determination of three-dimensional velocity anomalies under a seismic array using first P arrival times from local earthquakes: 1. A homogeneous initial model

Geiger's method of locating local earthquakes has been extended to include the effect of P velocity variation along the ray paths in three dimensions. The crustal structure was modeled by rectangular blocks, and a parameter was assigned to each block describing the perturbation of P wave slowness in the block. On the basis of an initial model, a set of linear equations for the observed first P arrival times was formulated in terms of the source and medium parameters. The source parameters for all the earthquakes in the data set and medium parameters for all the blocks penetrated by the seismic rays were then determined simultaneously by the damped least squares method. A computer program has been written for the case of a homogeneous initial medium model with constant P velocity. It has been tested with artificial data and applied to a set of local earthquakes recorded by a dense seismic array in Bear Valley, California. The resultant velocity distribution in the top 5 km shows a narrow low-velocity zone of about 5 km/s in the San Andreas fault zone sandwiched between high-velocity regions of about 6 km/s.

The mechanics of locating earthquakes

abstract A complete reexamination of Geiger's method in the light of modern numerical analysis indicates that numerical stability can be insured by use of the QR algorithm and the convergence domain considerably enlarged by the introduction of step-length damping. In order to make the maximum use of all data, the method is developed assuming a priori estimates of the statistics of the random errors at each station. Numerical experiments indicate that the bulk of the joint probability density of the location parameters is in the linear region allowing simple estimates of the standard errors of the parameters. The location parameters are found to be distributed as one minus chi squared with m degrees of freedom, where m is the number of parameters, allowing the simple construction of confidence levels. The use of the chi-squared test with n-m degrees of freedom, where n is the number of data, is introduced as a means of qualitatively evaluating the correctness of the earth model.

Earthquake location based on the gradient method and on minima of beams of directions

abstract Earthquake location is usually made by means of the Gauss-Newton iterative method, known in seismology as the Geiger method. As this method may fail to be efficient in some cases, attention is turned to the possibilities offered to the problem by the gradient method, which is always convergent, and, when properly used, may lead to the vicinity of the solution after a few iterations. But in the vicinity of the solution, the gradient method becomes slowly convergent. Therefore, in addition, another convergent method based on the minimum value, g, of the sum of the squared residuals taken on a beam of directions, is presented. In it the sum g is approximated by a polynomial of Laplace spherical harmonics. The new method includes the gradient method as a particular case; it is well suited in the vicinity of the solution and may lead quickly to it.

Effect of the beta spectrum on self-absorption of beta radiation from 14C

By integration of the beta spectrum for 14C, and application of a formula for the absorption of monoenergetic beta rays, self-absorption was calculated for thin layers of carbohydrate. The absorption coefficient was shown to increase from 0·2 at 6 mg/cm 2 to over 1·05 at near zero-thickness. Absorption rather than scattering was shown to be the prime source of error in detection at near-zero thickness. It was recommended that the thickness range of 0–0·5 mg/cm 2 be avoided in making counts by the Geiger method, or, if this is not possible, that the 0–30 keV electrons be excluded from the count.

Seismicity in Utah, 1850 through June 1965

abstractDuring 1850 through June 1965, at least 609 earthquakes occurred in Utah, of which at least 38 were damaging. Pertinent data of these earthquakes are listed, and an earthquake epicenter map of Utah is presented. The epicenter determinations of about 344 earthquakes (all felt) in Utah during 1850 through 1949 (Williams and Tapper, 1953) were non-instrumental. The epicenters and magnitudes and depths of focus (when determined) of the earthquakes during 1950 through June 1962 (77) were generally instrumental determinations taken from the Coast and Geodetic Survey, U. S. Department of Commerce (CGS). The epicenters of the earthquakes during July 1962 through June 1965 (188) were all instrumental and were obtained as follows: 23 were obtained from already-published CGS data; 165 were obtained by the University of Utah, of which 68 epicentral determinations were also made by the CGS. The comparisons of epicenter locations were generally good. Our epicenters were determined using a slightly modified Geiger's method programmed on an IBM 7044 digital computer at the University of Utah. The computer program used unpublished travel-time curves (B. H. Atwell, 1967) computed from the earth crustal model of Berg and others (1960) for the Utah region and included depths-of-focus data at 10-km-depth intervals from 0 to 60 km. The data show recent earthquake activity in the Ephraim (1961), Cache Valley (1962), and Levan (1963) areas, and swarm activity in the lehman Caves (1963) area. During 1962-1964 Utah averaged about 60 earthquakes per year for which instrumental epicenter determinations were made.

The Alaskan earthquake of July 22, 1937*

SummaryThe study of the Alaskan earthquake of July 22, 1937, is based on the examination of original seismograms and photographic copies from seismological observatories throughout the world.The arrival times of P at 71 stations were used in locating the epicenter. By Geiger's method and the use of Jeffreys' travel times, the position of the epicenter was found to be: geographical latitude, 64.67±.04° N, longitude, 146.58±.12° W, and the time of occurrence to be 17h 9m 30.0±.25s, U.T. The epicenter lies in the Yukon-Tanana upland in central Alaska, which is not a region of frequent major earthquakes.The disagreement caused by the apparently early arrivals at College and Sitka was reduced by replacing the standard travel-time curve of P by a linear travel-time curve in the interval of epicentral distance 0° to 16° and by interpreting the first arrival at College as P.It was possible to determine the direction of the first motion of P for 51 stations. The observed distribution of first motion and the geological trends in the region of the epicenter are consistent with the earthquake's having been caused by movement along a fault with strike between N 30° E and N 37° E, and dip between 64° and 71° to the southeast, in which the southeast side of the fault was displaced relatively northeastward with the line of movement pitching between 12° and 16° northeast.A wave designated F (for “false S”) was found to precede S on the records by 20 to 55 seconds, depending on the epicentral distance. The wave is longitudinal in type and the arrival times define a linear travel-time curve. It is suggested that this wave may be a longitudinal surface wave, of the type proposed by Nakano, produced at the surface of the earth by the arrival of a transverse wave which has been reflected at a surface of discontinuity within the earth.The records show two impulses near the time when S is expected. The average time interval between the two impulses is 11.3 sec. The first, called S1, has a plane of vibration intermediate in direction between the plane of propagation and the normal thereto. The second impulse, called S2, is nearly pure SH movement.The writer wishes to express his indebtedness to Professor Perry Byerly for invaluable suggestions and criticism during the course of the investigation.

On the differential coefficients in Geiger's method of locating epicenters*

On the differential coefficients in Geiger's method of locating epicenters*

On the question of dispersion in the first preliminary seismic waves1

AbstractSummaryBy use of the Byerly-Jeffreys travel-time curve for P, and Geiger's method of least-squares adjustment, the epicenter of the Alaskan earthquake of October 24, 1927, was placed at5 7 ° 26 ' ± 5 0 ' N . 13 7 ° 03 ' ± 1 9 ' W .and the time of occurrence was placed at 15h 59m 55s ± 2s, G.M.C.T.A second solution was obtained using Mohorovičić's multiple travel-time curves for P. The co-ordinates of the epicenter were the same as those given above, but the time of occurrence was found to be 16h00m, G.M.C.T.It has been held by some seismologists that anomalous dispersion can be observed in the first preliminary waves; i.e., that shorter periods travel faster than long ones. Investigations of periods were made with a view to testing this hypothesis, with the following results:The general conclusion is that observation of periods gives no evidence for dispersion in waves of longitudinal type.It is shown that, if dispersion did exist, the travel time of the beginning would be a continuous function of epicentral distance, and, therefore, Mohorovičić's curves are not evidence for dispersion. The observations of the epicentral distances at which P2, P1, and Pn are most frequently recorded first are contrary to dispersion.In the Alaskan earthquake the distribution of first motion (condensation or rarefaction) is very complicated. Dispersion offers no explanation for this fact, and it is believed that complex movements at the source are responsible for the observed distribution.