Local earthquake location with an electronic computer

1960 ◽  
Vol 50 (3) ◽  
pp. 467-470
Author(s):  
E. A. Flinn
Keyword(s):  
Least Squares ◽  
Iterative Procedure ◽  
Electronic Computer ◽  
Probable Error ◽  
Earthquake Location ◽  
Depth Of Focus ◽  
Origin Time ◽  
Local Earthquakes ◽  
National University ◽  
Local Earthquake

ABSTRACT A straightforward least-squares iterative procedure for locating local earthquakes using only the direct waves Pg and Sg is now in use at the Australian National University. An IBM 650 electronic computer is used for all calculations, including estimates of the probable error of epicentral coordinates, depth of focus, and origin time.

scholarly journals The Rann of Cutch Earthquake of 21 July 1956

MAUSAM ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 137-146
Author(s):  
A. N. TANDON
Keyword(s):  
Least Squares ◽  
Depth Of Focus ◽  
Method Of Least Squares ◽  
Origin Time

A seismometric study of the earthquake of 21 July 1956 in the Rann of Cutch, which caused destruction to life and property at Anjar, has been made. The epicentre and origin time have been determined by the method of least squares and are Lat. 23°20'N, Long 70000' E and 15h 32m 26S GMT respectively. The shock had a magnitude of 7 and a depth of focus of nearly 13 to 18 km.

scholarly journals A special-purpose program for earthquake location with an electronic computer

1962 ◽  
Vol 52 (2) ◽  
pp. 431-437
Author(s):  
John M. Nordquist
Keyword(s):  
Arrival Time ◽  
Electronic Computer ◽  
Method Of Least Squares ◽  
Origin Time ◽  
P Waves ◽  
Electronic Digital Computer ◽  
Network Output ◽  
Local Earthquake ◽  
The Difference ◽  
The Times

ABSTRACT An electronic digital computer program has been developed for determining the source and origin time of a local earthquake by the method of least squares, using the times of arrival of direct and refracted P waves at stations in the Pasadena network. Output includes the geographic coordinates and depth of the source, the origin time, direct distances from the source to each station, and the difference between observed and computed arrival time of P at each station. Limitations to the applicability of the program are discussed.

Short-term tilt anomalies preceding local earthquakes near San Jose, California

1980 ◽  
Vol 70 (6) ◽  
pp. 2221-2228
Author(s):  
C. E. Mortensen ◽  
E. Y. Iwatsubo
Keyword(s):  
Partial Recovery ◽  
San Jose ◽  
Short Term ◽  
Origin Time ◽  
Central California ◽  
Local Earthquakes ◽  
The North ◽  
Surface Creep ◽  
The Times ◽  
Black Mountain

abstract A tilt anomaly preceded a pair of earthquakes (ML = 4.2, origin time 0014 UTC, and ML = 3.9, origin time 0018 UTC, both on 29 August 1978) on the Calaveras Fault near San Jose, California. These earthquakes occurred at hypocentral depths of 8.5 and 9.0 km, respectively, and were located 6.7 and 5.2 km northwest of the Mt. Hamilton tiltmeter site. The anomaly is similar in shape and time scale to signals observed on other tiltmeters at the times of recorded surface creep events. The anomaly began approximately 40 hr before the earthquake pair and consisted of gradual down-to-the-east tilting followed by rapid tilting down-to-the-north-northeast at a rate of 12 μrad/hr. This was followed by 1 hr of rapid down-to-the-east tilting amounting to 1.5 μrad. The maximum peak tilt of 10.6 μrad down-to-the-northeast was followed by gradual decelerating tilting down-to-the-southwest constituting partial recovery. An anomaly of nearly identical form, but smaller in amplitude and duration, preceded an ML = 2.2 aftershock on 5 September 1978. Other nearby earthquakes as large as ML = 4.7 have occurred without accompanying creep-like signals. A similar, but a much smaller (0.74 μrad) creep-event-like signal preceded an ML = 3.5 earthquake with epicenter 3 km east of the Black Mountain tiltmeter site. In general, however, short-term tilt anomalies such as these are not observed to precede local earthquakes within the central California tiltmeter network. The tilt signal preceding the 29 August earthquake pair may be interpreted in terms of a model of a propagating creep event, at depth, associated with seismic failure at a “stuck” patch on the fault. However, the data are not adequate to constrain the model sufficiently to constitute a test of the hypothesis.

Near earthquakes in Central California*

1939 ◽  
Vol 29 (3) ◽  
pp. 427-462 ◽  
Author(s):  
Perry Byerly
Keyword(s):  
Least Squares ◽  
Travel Time ◽  
Sierra Nevada ◽  
Accurate Determination ◽  
Depth Of Focus ◽  
Surface Layers ◽  
P Waves ◽  
Central California ◽  
The Waves

Summary Least-squares adjustments of observations of waves of the P groups at central and southern California stations are used to obtain the speeds of various waves. Only observations made to tenths of a second are used. It is assumed that the waves have a common velocity for all earthquakes. But the time intercepts of the travel-time curves are allowed to be different for different shocks. The speed of P̄ is found to be 5.61 km/sec.±0.05. The speed for S̄ (founded on fewer data) is 3.26 km/sec. ± 0.09. There are slight differences in the epicenters located by the use of P̄ and S̄ which may or may not be significant. It is suggested that P̄ and S̄ may be released from different foci. The speed of Pn, the wave in the top of the mantle, is 8.02 km/sec. ± 0.05. Intermediate P waves of speeds 6.72 km/sec. ± 0.02 and 7.24 km/sec. ± 0.04 are observed. Only the former has a time intercept which allows a consistent computation of structure when considered a layer wave. For the Berkeley earthquake of March 8, 1937, the accurate determination of depth of focus was possible. This enabled a determination of layering of the earth's crust. The result was about 9 km. of granite over 23 km. of a medium of speed 6.72 km/sec. Underneath these two layers is the mantle of speed 8.02 km/sec. The data from other shocks centering south of Berkeley would not fit this structure, but an assumption of the thickening of the granite southerly brought all into agreement. The earthquakes discussed show a lag of Pn as it passes under the Sierra Nevada. This has been observed before. A reconsideration of the Pn data of the Nevada earthquake of December 20, 1932, together with the data mentioned above, leads to the conclusion that the root of the mountain mass projects into the mantle beneath the surface layers by an amount between 6 and 41 km.

On locating local earthquakes using small networks

1969 ◽  
Vol 59 (3) ◽  
pp. 1201-1212
Author(s):  
David E. James ◽  
I. Selwyn Sacks ◽  
Eduardo Lazo L. ◽  
Pablo Aparicio G.
Keyword(s):  
Least Squares ◽  
Focal Depth ◽  
P Wave ◽  
Time Interval ◽  
Travel Times ◽  
Origin Time ◽  
Average Value ◽  
Fundamental Properties ◽  
P Wave Velocity ◽  
Wave Travel

abstract Mathematical instability in four-parameter least squares hypocenter solutions arises primarily from the fact that the four computed variables—origin time (T0), focal depth (h), latitude (θ), and longitude (λ)—are not strictly independent. Specifically, T0 exhibits a non-independent relationship with the geometric parameters. For small networks (< 10–15 stations), the lack of independence between T0 and the other variables results in unstable least-squares solutions. This instability is manifest most clearly by the fact that different station subsets of the observational network produce significantly different solutions for the same earthquake. The instability can be eliminated by computing T0 independently for each station using the formula ( T 0 ) i = ( T p ) i − V k ( T s − p ) i V p , where Tp = P-wave arrival time, Vk = S-P velocity, Vp = P-wave velocity, and Ts-p = time interval between P and S arrivals. An average value of T0 can be obtained from the individually calculated origin times and the P-wave travel times calculated. The variables ϕ, λ and z are then computed by the usual least-squares procedure using P-wave travel times only. The method is iterative and an average T0 is recalculated in the course of each iteration. Fundamental properties of travel times within the Earth impose definite limitations upon the accuracy of the locations. Low values of the derivative dTp/dh at epicentral distances of a few degrees introduce a large uncertainty in focal depth, particularly for shallow (0–60 km) earthquakes. There is normally little error in epicenter, however, even for solutions in which depth is poorly determined. The dimensions and geometric configuration of the network in relation to the epicenter and the proximity of the epicenter to any one station are controlling factors in predicting the minimum uncertainty for any given hypocenter solution.

Accuracy in phase determination of LP electromagnetic seismographs based on transient calibration pulse

1976 ◽  
Vol 66 (1) ◽  
pp. 97-104 ◽  
Author(s):  
W. Mitronovas
Keyword(s):  
Least Squares ◽  
Large Error ◽  
Phase Response ◽  
Additional Parameter ◽  
Origin Time ◽  
Pulse Inversion ◽  
Transient Calibration ◽  
A Minor ◽  
Typical Solution

abstract A modification of the Jarosch and Curtis (1973) formulation for the calibration pulse inversion is suggested in an effort to improve its accuracy. The modification involves the introduction of the origin time of the pulse (time of the step in the calibration current) as an additional parameter to be solved in the least-squares inversion. This approach is found to be not completely satisfactory in general, but does improve the solution when a large error in origin time (> ±0.5 sec) is introduced in digitizing the pulse. A typical solution, when the origin time of the pulse has to be determined by least-squares, is accurate to within ± 1 sec in the phase response of a system for periods up to 500 sec. A more satisfactory approach is to record the origin time of calibration pulses on seismograms. Only a minor modification of the existing calibration circuit is necessary to provide a simultaneous signal to a time relay and calibration coil through a double pole switch. For pulses, where the origin time is known accurately, the existing techniques give results accurate to within ± 0.3 sec in phase response.

Unbiased Nonlinear Least Squares Estimations of Short-distance Equations

2017 ◽  
Vol 70 (4) ◽  
pp. 810-828 ◽  
Author(s):  
Shuqiang Xue ◽  
Yuanxi Yang
Keyword(s):  
Least Squares ◽  
Short Distance ◽  
Iterative Procedure ◽  
Nonlinear Least Squares ◽  
Long Distance ◽  
Geometrical Condition ◽  
Least Squares Solution ◽  
The Monte Carlo Method ◽  
Linearization Error ◽  
Geometric Dilution Of Precision

Nonlinear least squares estimations have been widely applied in positioning. However, nonlinear least squares estimations are generally biased. As the Gauss-Newton method has been widely applied to obtain a nonlinear least squares solution, we propose an iterative procedure for obtaining unbiased estimations with this method. The characteristics of the linearization error are discussed and a systematic error source of the linearization error needs to be removed to guarantee the unbiasedness. Both the geometrical condition and the statistical condition for unbiased nonlinear least squares estimations are revealed. It is shown that for long-distance observations of high precision, or for a positioning configuration with the lowest Geometric Dilution Of Precision (GDOP), the nonlinear least squares estimations tend to be unbiased; but for short-distance cases, the bias in the nonlinear least squares solution should be estimated to obtain unbiased values by removing the bias from the nonlinear least squares solution. The proposed results are verified by the Monte Carlo method and this shows that the bias in nonlinear least squares solution of short-distance distances cannot be ignored.

scholarly journals A local earthquake coda magnitude and its relation to duration, moment Mo, and local Richter magnitude ML

1979 ◽  
Vol 69 (2) ◽  
pp. 353-368
Author(s):  
Anne M. Suteau ◽  
James H. Whitcomb
Keyword(s):  
Epicentral Distance ◽  
Total Duration ◽  
Correlation Coefficients ◽  
Origin Time ◽  
Theoretical Relationship ◽  
Statistical Confidence ◽  
Geometric Spreading ◽  
Local Earthquake ◽  
Linear Fit ◽  
Shallow Crust

abstract A relationship is found between the seismic moment, Mo, of shallow local earthquakes, coda amplitudes, and the total duration of the signal, t, in seconds, measured from the earthquake origin time. Following Aki, we assume that the end of the coda is composed of backscattering surface waves due to lateral heterogeneity in the shallow crust. Using the linear relationship between the logarithm of Mo and the local Richter magnitude ML, we obtain a relationship between ML and t, of the form: ML = a0 + a1 log t + a2t1/3 + f(t), where a0, a1, a2 are constants depending on an attenuation parameter (effective Q) and geometric spreading; and f(t) is a function of the instrument response and a (weak) function of the scattering process. This relationship is different from the empirical one generally used ML = a0 + a1 log τ + a2(log τ)2 + a3Δ, where τ is the duration measured from the first P arrival time and Δ is epicentral distance in kilometers. In the theoretical relationship, the dependence on epicentral distance is implicit in t. The theoretical relationship is used to calculate a coda magnitude MC that is compared to ML for southern California earthquakes which occurred during the period from 1972 to 1975. This comparison is made independently at six stations of the CIT network. At all stations, a good linear fit (ML = C0 + C1MC) is obtained. The standard errors range from 0.2 to 0.3 and the correlation coefficients from 0.80 to 0.90. Once station gain is accounted for, station correction terms are less than 0.17 magnitude unit when comparing ML and Mc. Mc calculation is not limited to a duration measurement but can utilize the entire earthquake coda in order to increase by many times the statistical confidence in an estimate of an earthquake's magnitude.

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