Gumbel Method Research Articles

Improving the Gumbel analysis by using M-th highest extremes

Improvements to the Gumbel method of extreme value analysis of wind data made over the last two decades are reviewed and illustrated using sample data for Jersey. A new procedure for extending the Gumbel method to include M-th highest annual extremes is shown to be less effective than the standard method, but leads to a method for calibrating peak-over-threshold methods against the standard Gumbel approach. Peak-over-threshold methods that include at least the 3rd highest annual extremes, specifically the modified Jensen and Franck method and the Method of independent storms are shown to give the best estimates of extremes from observations.

Comparison of earthquake recurrence-rate estimates using the Gumbel and Gutenberg-Richter formulae

AbstractOne thousand earthquakes distributed according to Gutenberg and Richter's formula, log N = 6 − M, were assumed to occur during a 50-yr period. Various fractions were assumed to be overlooked, and randomly distributed errors were added to their magnitudes. These data were used to calculate the expected recurrence rates of large earthquakes using methods developed by Gumbel and by Gutenberg and Richter. Although the method of Gutenberg and Richter leads to a more accurate estimation of recurrence rates with ideal data, when the data are incomplete and reported magnitudes contain errors, Gumbel's method sometimes leads to a more accurate estimate.

A Program of Frequency Analysis Using Gumbel's Method

AbstractA program (GUMBEL) has been written in QBASIC language to carry out the mathematical procedure for preparing frequency curves using Gumbel's extreme value distribution. The program can be used to analyze hydrologie and meteorologie events so as to establish a relationship between the magnitude of extreme events and their frequency through the use of probabilistic distributions.Six different plotting positions formulae are included in the program so as to select the most reliable formula of these graphical fitting methods in analyzing the observed hydrologie data. A sample input data file of annual rainfall data from Jordan has been tested, in order to show the applicability and usefulness of the program.

On the variation of seismic hazard along the San Andreas fault

AbstractRecurrence probabilities of earthquakes were evaluated at 33 locations along the San Andreas fault using Gumbel's method. Two minima in the mode of the largest annual earthquakes occur, one north of San Francisco centered around 38.75°N latitude and the other centered in the Carizzo Plain area at 35.25°N latitude. The latter is offset by an unusually large ratio of large to small earthquakes, so that only one minimum (around 38.75°N) is observed in the expected 100-yr earthquake size. This broad minimum closely coincides with where the San Andreas fault broke in 1906, and may be only temporary.There appears to be a slight, systematic tendency for the observed largest 50-year earthquake to become larger than the expected 50-yr earthquake as the size of the area being studied is reduced. The preferred explanation for this is that the pattern of size as a function of frequency becomes nonlinear when the size of the area studied is less than the focal area of the largest earthquakes.

Earthquakes and seismic hazard in Finland

In this paper the seismicity of Finland has been re-evaluated on the basis of a new regional magnitude scale, which gives somewhat lower magnitude values than the previously used macroseismic scale. Macroseismic magnitudes for Finnish earthquakes have been calculated by using the value 4.0 for the coefficient of attenuation, and the value 2.0 for the lower limit of intensity at the boundary of the felt area. Using these parameters, we obtained macroseismic magnitudes comparable to those of the instrumental M L-scale. The distribution of seismicity in Finland is presented in an epicentral map and maximum-intensity maps (1880–1980). These maps indicate some regional variation of seismic activity. The seismicity was also considered as a function of time using strain release curves. The b-value 0.59 was evaluated from the log N(M) relationship for the last hundred years. The time variation of b-values was also studied. Estimation of magnitude and intensity extremes by Gumbel's method yielded the value 5.0 for the upper limit of magnitude and VI-VII for the upper limit of intensity. The return periods for different magnitudes and intensities were calculated as well as the probabilities that they are exceeded.

Towards better estimation of extreme winds

The use of dynamic pressure in place of wind speed as the variable when fitting extreme values to the Fisher—Tippett (Type 1) model is discussed. A procedure is proposed by which estimates of extreme winds may be obtained that are superior to those from the standard Gumbel method. The procedure is extended to allow the estimation of extreme winds according to direction, in sectors 30° wide. Adoption of the sectorial estimates leads to substantial reductions in design wind loads over a large range of wind directions in the U.K.

Maximum one-day point rainfall estimation for North Indian plains using district average rainfall ratios

A quick and simple procedure has been developed for evaluating maximum point rainfall for different return periods for any location in the plains of north India. According to this procedure, 2-year one-day rainfall of a location is estimated from the 2-year generalized chart of the region. Average district rainfall ratios for higher return periods of 5, 10, 25 and 50 years to 2-year return period are obtained with the help of (i) district average ratio map of 100/2 and (ii) frequency interpolation nomogram. The magnitudes of 5, 10, 25, 50 and 100-year rainfall are then obtained by multiplying the 2-year value by the corresponding district average ratios pertaining to different return periods. The estimates of point rainfall obtained by this procedure are quite comparable with those obtained directly by the Gumbel method. By using the procedure given in this study, a design engineer or a hydrologist can estimate point rainfall of different return periods for any station in north Indian plains without undertaking elaborate statistical calculations.

Time series analyses on the maximum depth of snow cover in Akita city

The objective of this study is to clarify the time series characteristics on the maximum depth of snow cover (Smax) at Akita city located in the northwest of Japan, from view-points of periodic variation and winter climatic variation. For this purpose, mehtod of moving average, correlogram analysis, Fourier analysis, cross-correlation analysis, and probability analysis by Gumbel method were applied. On the basis of these analyses, numerical evaluations were made for the secular variation of past 90 years record on Smax.(1) The long-periodic variation on Smax was found by Fourier analysis. It seems to be significant as an indicator of the trend of winter climatic variation in the sense of statistics.(2) The present time Now is presumed as a transitional age in progress of warm winter to cold winter.(3) There are upward type and downward type between the peaks of Smax in similar intervals. The latter one tends to appear in recent years. This penomenon is probable in response to the general fluctuation of the amount of annual sunspot numbers.(4) The basic periodicity of 11 years in heavy snow fall is discerned by correlogram analysis.(5) The amount of Smax accompanied by this periodicity is estimated about 80 cm larger, according to the probability of Gumbel method.

A simple version of Gumbel's method for flood estimation / Version simplifiée de la méthode de Gumbel pour l'estimation des crues

Abstract Gumbel's method has been simplified in such a manner that one can obtain the magnitude of a given return period flood without recourse to looking at a table and working out the value of the coefficient of variation of the given data. The results obtained by the simplified version are compared with those obtained from using both the original approach and those from Powell's modification of Gumbel's method.

On the prediction of extreme wind speeds from the parent distribution

This paper examines an alternative approach from the traditional Gumbel method to the estimation of extreme mean wind speeds. The method is based on Rice's upcrossing rate expression and only requires a statistical description of the parent population. Initially, the long-term power spectrum of wind speeds is developed in an attempt to evaluate some necessary parameters and the results are found to be similar to those obtained by other researchers. Favourable comparisons are found between Rice's expression and observed crossing rates of low and intermediate mean speeds. Rice's expression is then extended to yield return periods of extreme speeds, enabling a comparison to be made with Gumbel's method applied to eighteen annual maxima. Again, the agreement is satisfactory. Finally, the data requirements for the application of this method are examined and a relationship is derived between the length of data employed and the coefficient of variation of predicted extreme speeds.

The application of extreme-value theory to M-ary detection systems

An application of Gumbel's extreme-value theory to M-ary detection systems is described. Application of Gumbel's method for binary detection requires a “fixed threshold”. Since a fixed threshold does not exist for M-ary processes, a method is developed whereby an “artificial threshold” is created. The method developed is used to analyze the signal detection portion of an advanced digital computing system. The results obtained show a general agreement with the previously presented theoretical results. The savings obtained by using Gumbel's method, as opposed to simple monitoring of errors, is large.

HIGH TEMPERATURE EXTREMES IN AUSTRALIA∗

ABSTRACT Magnitude-frequency analysis by the Gumbel method of temperature records at 166 Australian stations produced data for maps of extreme temperatures at recurrence intervals of 1.58, 2.33, five, ten, twenty-five, fifty, and 100 years on the annual series, plus a generalized map of actual values for the 100-year interval. Whereas the extremes for recurrence intervals up to about twenty-five years conform well to the Theory of Extreme Values, those for greater recurrence intervals produce convex-upward graphs, suggesting a similarity to Gumbel's Type III probability. In some parts of the continent ranges between the 1.58-year and the 100-year extremes are small. Graphs for some very exposed coastal stations flatten abruptly in their upper ranges. Sample occurrences of absolute extremes at selected stations involved unusual transport of cTW air from the interior at low levels across the coastlands in association with patterns of cyclonicity and anticyclonicity that for the most part contrast with most ...

The Application of Extreme-Value Theory to Error-Free Cornrnunication

This article gives an application of Gumbel's extreme-value theory to the problem of the estimation of low error probabilities in certain types of communications receivers, for the purpose of rejecting receivers having unacceptably high error rates. The systems considered all involve threshold detectors in a binary system. Thus, the receiver calls a received symbol a “0” if and only if a certain detected voltage falls below a certain value; otherwise, the symbol is called a “1”. The maximum of a run of a large number of successive independent samples of voltage is taken, and these maxima have approximately a Gumbel distribution. A large number of these maxima themselves are taken, and the two parameters of the Gumbel distribution are thereby estimated using large sample theory. These two parameters in turn allow the estimation of error probability. The variance of the estimate is found, as well as a confidence interval. In addition, the savings to be obtained using the Gumbel method, as opposed to the sim...

Discussion of paper by D. M. Hershfield and M. A. Kohler, ‘An empirical appraisal of the Gumbel Extreme-Value Procedure’

In Tables 1, 2, and 3 Hershfield and Kohler [1960] show a comparison of the estimated number of events that have a return period of 2, 5, 10, ... years with the corresponding computed number of events based upon the Gumbel method of curve fitting to extreme-value data. They point to a bias in the results of Table 1 using the dependent data and to an apparent lack of bias in Tables 2 and 3 when they test their results on independent data. Ordinarily one expects that an equation of estimate will perform better upon the data sample from which it is derived than it would upon an independent sample. But the authors apparently suggest that the reverse is true.

An empirical appraisal of the Gumbel extreme-value procedure

In this study the predictive value of the widely used Fisher-Tippett type I distribution, when fitted to rainfall data by the Gumbel method, is appraised. Thousands of station years of rainfall data have been analyzed in several ways in evaluating the Gumbel procedure. The results provide evidence of the acceptability of the Gumbel procedure for predicting the probability of occurrence of the externe values of rainfall.