On the question of dispersion in the first preliminary seismic waves1

1931 ◽  
Vol 21 (2) ◽  
pp. 87-158 ◽  
Author(s):  
H. Henrietta Sommer
Keyword(s):  
Continuous Function ◽  
Least Squares ◽  
Travel Time ◽  
Epicentral Distance ◽  
Anomalous Dispersion ◽  
Method Of Least Squares ◽  
Time Curve ◽  
Least Squares Adjustment ◽  
First Motion

Abstract Summary By 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 at 5 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.

The Alaskan earthquake of July 22, 1937*

1940 ◽  
Vol 30 (4) ◽  
pp. 353-376
Author(s):  
John N. Adkins
Keyword(s):  
Travel Time ◽  
Epicentral Distance ◽  
Time Interval ◽  
Time Curve ◽  
Arrival Times ◽  
The Earth ◽  
Geographical Latitude ◽  
The World ◽  
First Motion ◽  
First Arrival

Summary The 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.

The Texas earthquake of August 16, 1931*

1934 ◽  
Vol 24 (2) ◽  
pp. 81-99
Author(s):  
Perry Byerly
Keyword(s):  
Travel Time ◽  
Travel Times ◽  
Time Curve ◽  
P Waves ◽  
First Order ◽  
First Motion ◽  
Order Discontinuity

Summary The travel-time curve of P for the Texas earthquake of August 16, 1931, shows that there is a definite break in the travel-time curve near Δ = 16°. This is interpreted as indicating a first-order discontinuity at a depth of about 300 kilometers. Another break in the travel-time curve at Δ = 25° is strongly suggested. Beyond Δ = 75° the curve has two branches, the lower following most existing curves, the upper following the Montana curve which latter seems to be a usual one for American earthquakes. This part of the curve is interpreted as indicating that the discontinuity at depth about 2,400 kilometers is a first-order one at which the speed of P waves drops discontinuously. From the direction of first motion on the records it is concluded that a sufficient source would have been motion on a fault of strike about N 35° W, the movement being up on the easterly side and down on the westerly side. The travel times of all waves read on the records are plotted on graphs. The scattering of all waves after P is marked.

Classical Concepts of Least Squares Adjustment

1965 ◽  
Vol 19 (1) ◽  
pp. 16-26
Author(s):  
Gottfried Konecny
Keyword(s):  
Statistical Analysis ◽  
Least Squares ◽  
Standard Error ◽  
Statistical Tests ◽  
Arithmetic Mean ◽  
Method Of Least Squares ◽  
Correlated Observations ◽  
Symmetrical Distribution ◽  
Least Squares Adjustment ◽  
Normally Distributed

A review of the fundamental classical concepts of least squares adjustment, based on the development by Gauss is presented. The method of least squares leads to a most probable value of the adjusted quantity under the assumption that the arithmetic mean yields the most probable result and that the uncorrelated observations are normally distributed. It is shown further that the method of least squares leads to an adjusted value with a resulting smallest standard error for any symmetrical distribution. It is indicated that this is also true for correlated observations and that various statistical tests make least squares adjustment a powerful tool for statistical analysis.

On focal points of SKS*

1938 ◽  
Vol 28 (3) ◽  
pp. 197-200
Author(s):  
B. Gutenberg
Keyword(s):  
Travel Time ◽  
Extreme Values ◽  
Focal Point ◽  
Epicentral Distance ◽  
Outer Part ◽  
Focal Points ◽  
Time Curve ◽  
Longitudinal Waves ◽  
The Core ◽  
Preliminary Study

Summary The travel-time curve of the first section of SKS depends much on the velocity in the outer part of the core. It begins on the travel-time curve of ScS at an epicentral distance somewhere between 65° and 90°, depending on the velocity of longitudinal waves just below the surface of the core. The extreme values correspond to velocities there of approximately 8.0 and 7.4 km/sec., respectively. If the distance where SKS begins is relatively large, its first section extends to decreasing distances and is convex towards the axis of distance, and SKS must have an odd number of cusps with focal points (at least one) where it reverses in direction and changes from convex to concave or vice versa. If SKS begins at a relatively short distance, its first segment extends to increasing distances and is concave towards the axis of distance; in this case the number of cusps is even (possibly zero). In an intermediate case, SKS begins with a focal point. In any case, the first segment of the travel-time curve of SKS is below the travel-time curve of ScS. Similar conclusions are correct for SKS. A preliminary study of the observations seems to indicate a focal point of SKS at a distance between 70° and 80°. More detailed investigations which are under way may be used to draw conclusions respecting the velocity of longitudinal waves in the outer part of the core.

The Montana earthquake of June 28, 1925, G.M.C.T.

1926 ◽  
Vol 16 (4) ◽  
pp. 209-265 ◽  
Author(s):  
Perry Byerly
Keyword(s):  
Epicentral Distance ◽  
Maximum Amplitude ◽  
Average Period ◽  
Time Curve ◽  
Natural Period ◽  
The North ◽  
The Pacific ◽  
First Motion ◽  
Instrumental Period ◽  
The Waves

Summary 1. The epicenter of the great Montana earthquake of June 28, 1925, G.M.C.T., is located at: ϕ = 46 ° 24 ′ ± 05 ′ N . λ = 111 ° 14 ′ .5 ± 06 ′ W . The time of occurrence is placed at: O = 1 h 21 m 05 s ± 01 s . G.M.C.T. 2. The travel-time curve for the first preliminary wave is drawn and compared with previous curves. Two abrupt changes of slope are present at epicentral distances, corresponding to wave paths penetrating to depths of 400 m and 1,700 km. The only possibilities of an early first preliminary type, P1, are at Cartuja, where the arrival is five seconds before the time indicated by the new curve, and at Tacubaya, from which only a bulletin was received. 3. Examination of the nature of the first wave indicates that the first motion was a compression within a sector of angular magnitude of between 61° and 105° toward the north, and a dilatation in other directions. The earth amplitude of the first motion appears to change very little with Δ, the second amplitude showing greater loss with increased Δ. 4. The periods of the preliminary groups do not appear to depend on epicentral distance. The periods of the reflections P do not differ greatly from those of PR1, but there seems to be a possible tendency for the periods of SR as registered to exceed those of S. 5. The position of the maximum in the group is measured, where possible, for P and S. In the P group, there appears no tendency for the position of the maximum to vary regularly with epicentral distance. But in the S group, the maximum tends to advance toward the beginning of the group. There appears to be a tendency for the S group to vibrate in the plane of propagation. 6. There appear to be four groups of surface waves registered, (1) the long wave of Gutenberg, v = 4.35 km/sec. (ca.), but with a period of less than one minute; (2) the normal L, v=3.8 to 3.9 (ca.); (3) a new phase called X, which carries the maximum amplitude on Pacific paths, and less energy on other paths, v=3.6 (ca.); (4) the normal M, which carries the waves of maximum amplitude and regular character over non-Pacific paths, v=3.3 (ca.). 7. The periods of the first six minutes of the maximum group are investigated. It is found that for the Pacific Coast stations, 10° <Δ<15°, the average period is nine to twelve seconds, regardless of the natural period of the seismograph. For eastern America, the period registered was a function of the natural period of the instrument. These periods were five to seven, nine to eleven, fifteen to twenty. At stations of great epicentral distances, periods of fifteen to twenty-four seconds were registered regardless of the instrumental period. It appears that the long period is present in the first minute at nearer stations, but does not dominate because of the presence of shorter periods. At the more distant stations, the short periods have been lost. 8. The periods of the L group were measured. The average period for southwestern America is twenty seconds, for eastern America, eight seconds; for Europe, thirty seconds; for Honolulu and Apia, ten seconds. It is concluded that there is evidence of a thinner surface layer in eastern America than in western America. 9. The heavy aftershock, which took place about forty-five minutes after the main shock, originated at a focus probably to the south and west of that of the first shock.

The Hebgen Lake, Montana, earthquake of August 18, 1959: P waves

1962 ◽  
Vol 52 (2) ◽  
pp. 235-271
Author(s):  
Alan Ryall
Keyword(s):  
Travel Time ◽  
Sierra Nevada ◽  
Fault Plane ◽  
Time Curve ◽  
P Waves ◽  
Arrival Times ◽  
The Pacific ◽  
First Motion ◽  
Plane Solution ◽  
The Pacific Ocean

ABSTRACT The instrumental epicenter of the Hebgen Lake earthquake is found to lie within the region of surface faulting. The depth of focus had a maximum value of 25 kilometers. Times of P are studied in detail for epicentral distances less than 13 degrees. The apparent scatter of arrival times from 700 to 1400 kilometers can be explained by variations of the velocity of Pn between the physiographic provinces of the western United States. A comparison of observations for the Hebgen Lake earthquake with published times for blasts in Nevada and Utah indicates that the velocity of Pn in the central and eastern Basin and Range is about 7.5 km/sec, and that the crust in that region thickens toward the east and thins toward the south. A comparison of apparent velocities in northern California, in directions parallel and transverse to the structure, indicates that the crust thins by about 19 kilometers, from the edge of the Sierra Nevada to the Pacific Ocean. A discontinuity is observed in the travel-time curve at a distance of 24–25 degrees. Arrivals of P waves in the range 65–128 degrees fall on two parallel travel-time branches; this multiplicity in the travel-time curve may be related to repeated motion at the source. Travel-times of PKIKP appear to deviate from published curves. The fault-plane solution for the Hebgen Lake earthquake, together with a consideration of the first motion at Bozeman, Montana, indicates a focal mechanism of the dipole, or fault, type. The strike and dip of the instrumental fault plane agree well with observed ruptures at the surface.

scholarly journals Longitudinal waves through the Earth's Core*

1952 ◽  
Vol 42 (2) ◽  
pp. 119-134
Author(s):  
M. E. Denson
Keyword(s):  
Velocity Distribution ◽  
Travel Time ◽  
Focal Point ◽  
Epicentral Distance ◽  
Outer Core ◽  
Travel Times ◽  
Time Curve ◽  
Longitudinal Waves ◽  
The Core ◽  
The Waves

Abstract Amplitudes, periods, and travel times of the longitudinal P′ or PKP core waves have been investigated. Results indicate that the epicentral distance of the main focal point and the travel time of P′ phases vary with the periods of the waves. This variation would seem reasonably explained in terms of dispersion. The point of reversal in the travel-time curve of the waves through the outer core is believed to lie near 157°. Data suggest a discontinuity between 120° and 125° rather than 110°. Anomalies existing in energy, period, and travel-time relationships of the P′ phases indicate that current concepts of velocity distribution and of propagation paths within the core are in need of modification.

Appendix II. Notes on the phase determination from anomalous dispersion observations

1971 ◽  
Vol 323 (1555) ◽  
pp. 484-487
Keyword(s):  
Least Squares ◽  
Royal Society ◽  
Complex Form ◽  
Anomalous Dispersion ◽  
Phase Determination ◽  
Structure Factors ◽  
Least Squares Adjustment ◽  
Independent Observations ◽  
Cobyric Acid

At the final stages of refinement of the structure of cobyric acid full allowance was made for the effects of anomalous dispersion on the measured structure amplitudes. Refinement was effected by least squares adjustment of calculated structure factors to the 6188 corrected observed F values. The cobalt scattering factors were given the complex form, f Co + Δ f' Co + i f'' Co as in the International tables , vol. 3, and the observed F ( hkl ) and F ( hkl ) values were treated as independent observations. The program used derived from one written by Gantzel, Sparks and Trueblood, modified for the Atlas computer. Table 8 provides a small extract from the final table 8 of observed and calculated F ( hkl ) and terms, which is deposited with the Royal Society (see footnote on page 466).

The limit state of concrete and reinforced concrete rods at complex and longitudinal-transverse bending

2020 ◽  
pp. 60-73
Author(s):  
Yu V Nemirovskii ◽  
S V Tikhonov
Keyword(s):  
General Solution ◽  
Least Squares ◽  
Nonlinear Differential Equations ◽  
Limit State ◽  
Algebraic Equations ◽  
Method Of Least Squares ◽  
Elasticity Limit ◽  
Limit Values ◽  
Symbolic Calculations ◽  
Deformation Diagrams

The work considers rods with a constant cross-section. The deformation law of each layer of the rod is adopted as an approximation by a polynomial of the second order. The method of determining the coefficients of the indicated polynomial and the limit deformations under compression and tension of the material of each layer is described with the presence of three traditional characteristics: modulus of elasticity, limit stresses at compression and tension. On the basis of deformation diagrams of the concrete grades B10, B30, B50 under tension and compression, these coefficients are determined by the method of least squares. The deformation diagrams of these concrete grades are compared on the basis of the approximations obtained by the limit values and the method of least squares, and it is found that these diagrams approximate quite well the real deformation diagrams at deformations close to the limit. The main problem in this work is to determine if the rod is able withstand the applied loads, before intensive cracking processes in concrete. So as a criterion of the conditional limit state this work adopts the maximum permissible deformation value under tension or compression corresponding to the points of transition to a falling branch on the deformation diagram level in one or more layers of the rod. The Kirchhoff-Lyav classical kinematic hypotheses are assumed to be valid for the rod deformation. The cases of statically determinable and statically indeterminable problems of bend of the rod are considered. It is shown that in the case of statically determinable loadings, the general solution of the problem comes to solving a system of three nonlinear algebraic equations which roots can be obtained with the necessary accuracy using the well-developed methods of computational mathematics. The general solution of the problem for statically indeterminable problems is reduced to obtaining a solution to a system of three nonlinear differential equations for three functions - deformation and curvatures. The Bubnov-Galerkin method is used to approximate the solution of this equation on the segment along the length of the rod, and specific examples of its application to the Maple system of symbolic calculations are considered.

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