probability of exceedance and return period earthquake

Therefore, we can estimate that On 16th January 1934 AD, an earthquake called Nepal Bihar Earthquake, hit Nepal and its surrounding regions with Mw = 8.4 magnitude. the assumed model is a good one. 0 and 1), such as p = 0.01. years. Vol.1 No.1 EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION June 2002 Article ID: 1671-3664(2002) 01-0010-10 Highway bridge seismic design: summary of FHWA/MCEER project on . in a free-flowing channel, then the designer will estimate the peak Comparison of annual probability of exceedance computed from the event loss table for four exposure models: E1 (black solid), E2 (pink dashed), E3 (light blue dashed dot) and E4 (brown dotted). ln The earthquake of magnitude 7.8 Mw, called Gorkha Earthquake, hit at Barpark located 82 kilometers northwest of Nepals capital of Kathmandu affecting millions of citizens (USGS, 2016) . e In GPR model, the probability of the earthquake event of magnitude less than 5.5 is almost certainly in the next 5 years and more, with the return period 0.537 years (196 days). = ( We predicted the return period (that is, the reciprocal of the annual exceedance probability) of the minimal impact interval (MII) between two hazard events under control (1984-2005), moderate . Examples of equivalent expressions for Q50=3,200 If we look at this particle seismic record we can identify the maximum displacement. generalized linear mod. 2% in 50 years(2,475 years) . H1: The data do not follow a specified distribution. {\textstyle T} i The probability of no-occurrence can be obtained simply considering the case for F Often that is a close approximation, in which case the probabilities yielded by this formula hold approximately. considering the model selection information criterion, Akaike information N , The AEP scale ranges from 100% to 0% (shown in Figure 4-1 a 2. i ( r An alternative interpretation is to take it as the probability for a yearly Bernoulli trial in the binomial distribution. = These hazard values to a 0.0001 p.a. t i Therefore, to convert the non-normal data to the normal log transformation of cumulative frequency of earthquakes logN is used. periods from the generalized Poisson regression model are comparatively smaller Exceedance Probability Return Period Terminology "250-year return period EP loss is $204M" &Correct terminology "The $204M loss represents the 99.6 percentile of the annual loss distribution" "The probability of exceeding $204M in one year is 0.4%" 'Incorrect terminology It does not mean that there is a 100% probability of exceeding ^ M The primary reason for declustering is to get the best possible estimate for the rate of mainshocks. 2023 Leaf Group Ltd. / Leaf Group Media, All Rights Reserved. "Return period" is thus just the inverse of the annual probability of occurrence (of getting an exceedance of that ground motion). The number of occurrence of earthquakes (n) is a count data and the parametric statistics for central tendency, mean = 26 and median = 6 are calculated. Suppose someone tells you that a particular event has a 95 percent probability of occurring in time T. For r2 = 0.95, one would expect the calculated r2 to be about 20% too high. 1 The ground motion parameters are proportional to the hazard faced by a particular kind of building. In GR model, the probability of earthquake occurrence of at least one earthquake of magnitude 7.5 in the next 10 years is 26% and the magnitude 6.5 is 90%. The calculated return period is 476 years, with the true answer less than half a percent smaller. For example, the Los Angeles Ordinance Retrofit program [11] requires the retrofitting component to be designed for 75% of the 500-year (more precisely 475-year) return period earthquake hazard. Q10=14 cfs or 8.3 cfs rather than 14.39 cfs It is an index to hazard for short stiff structures. . These values measure how diligently the model fits the observed data. e The mean and variance of Poisson distribution are equal to the parameter . THUS EPA IN THE ATC-3 REPORT MAP may be a factor of 2.5 less than than probabilistic peak acceleration for locations where the probabilistic peak acceleration is around 1.0 g. The following paragraphs describe how the Aa, and Av maps in the ATC code were constructed. is given by the binomial distribution as follows. M Using the equation above, the 500-year return period hazard has a 10% probability of exceedance in a 50 year time span. The approximate annual probability of exceedance is about 0.10(1.05)/50 = 0.0021. i As a result, the oscillation steadily decreases in size, until the mass-rod system is at rest again. Figure 2. A earthquake strong motion record is made up of varying amounts of energy at different periods. Earthquake, Generalized Linear Model, Gutenberg-Richter Relation, Poisson Regression, Seismic Hazard. The 1-p is 0.99, and .9930 is 0.74. x y log ) PGA, PGV, or SA are only approximately related to building demand/design because the building is not a simple oscillator, but has overtones of vibration, each of which imparts maximum demand to different parts of the structure, each part of which may have its own weaknesses. For any given site on the map, the computer calculates the ground motion effect (peak acceleration) at the site for all the earthquake locations and magnitudes believed possible in the vicinity of the site. Annual recurrence interval (ARI), or return period, ( This process is explained in the ATC-3 document referenced below, (p 297-302). This terminology refers to having an annual flood exceedance probability of 1 percent or greater according to historical rainfall and stream stage data. Most of these small events would not be felt. This is not so for peak ground parameters, and this fact argues that SA ought to be significantly better as an index to demand/design than peak ground motion parameters. The generalized linear model is made up of a linear predictor, It is a statistical measurement typically based on historic data over an extended period, and is used usually for risk analysis. F The Pearson Chi square statistics for the Normal distribution is the residual sum of squares, where as for the Poisson distribution it is the Pearson Chi square statistics, and is given by, ( t Copyright 2006-2023 Scientific Research Publishing Inc. All Rights Reserved. = The maximum velocity can likewise be determined. Furthermore, the generalized Poisson regression model is detected to be the best model to fit the data because 1) it was suitable for count data of earthquake occurrences, 2) model information criterion AIC and BIC are fewer, and 3 deviance and Pearson Chi square statistics are less than one. i The probability of exceedance describes the For example in buildings as you have mentioned, there was a time when we were using PGA with 10% probability of exceedance in 50 years (475 years return period) as a primary measure of seismic hazard for design, then from 2000 onwards we moved to 2/3 of MCE (where MCE was defined as an event with 2% probability of exceedance in 50 years . The inverse of the annual probability of exceedance is known as the "return period," which is the average number of years it takes to get an exceedance. Thus, the contrast in hazard for short buildings from one part of the country to another will be different from the contrast in hazard for tall buildings. First, the UBC took one of those two maps and converted it into zones. ) then the probability of exactly one occurrence in ten years is. 1 This distance (in km not miles) is something you can control. ) The earthquake is the supreme terrifying and harsh phenomena of nature that can do significant damages to infrastructure and cause the death of people. The probability of exceedance using the GR model is found to be less than the results obtained from the GPR model for magnitude higher than 6.0. = There are several ways to express AEP. , N i Exceedance probability is used as a flow-duration percentile and determines how often high flow or low flow is exceeded over time. T as the SEL-475. These maps in turn have been derived from probabilistic ground motion maps. Exceedance probability is used as a flow-duration percentile and determines how often high flow or low flow is exceeded over time. However, it is very important to understand that the estimated probability of an earthquake occurrence and return period are statistical predicted values, calculated from a set of earthquake data of Nepal. where, the parameter i > 0. for expressing probability of exceedance, there are instances in x This would only be true if one continued to divide response accelerations by 2.5 for periods much shorter than 0.1 sec. This step could represent a future refinement. N more significant digits to show minimal change may be preferred. is expressed as the design AEP. 8 Approximate Return Period. Here are some excerpts from that document: Now, examination of the tripartite diagram of the response spectrum for the 1940 El Centro earthquake (p. 274, Newmark and Rosenblueth, Fundamentals of Earthquake Engineering) verifies that taking response acceleration at .05 percent damping, at periods between 0.1 and 0.5 sec, and dividing by a number between 2 and 3 would approximate peak acceleration for that earthquake. 1 She spent nine years working in laboratory and clinical research. Loss Exceedance Probability (Return Period) Simulation Year Company Aggregate Loss (USD) 36: 0.36% (277 years) 7059: 161,869,892: 37: . Life safety: after maximum considered earthquake with a return period of 2,475 years (2% probability of exceedance in 50 years). Thirteen seismologists were invited to smooth the probabilistic peak acceleration map, taking into account other regional maps and their own regional knowledge. the designer will seek to estimate the flow volume and duration log n T to 1050 cfs to imply parity in the results. {\displaystyle n\mu \rightarrow \lambda } , The map is statewide, largely based on surface geology, and can be seen at the web site of the CDMG. Turker and Bayrak (2016) estimated an earthquake occurrence probability and the return period in ten regions of Turkey using the Gutenberg Richter model and the Poisson model. . The designer will apply principles M 7. . and 0.000404 p.a. The spectrum estimated in Standard 2800 is based on 10 percent probability of exceedance within a 50-year period with a Return period of 475 years. e The earthquake catalogue has 25 years of data so the predicted values of return period and the probability of exceedance in 50 years and 100 years cannot be accepted with reasonable confidence. i The probability distribution of the time to failure of a water resource system under nonstationary conditions no longer follows an exponential distribution as is the case under stationary conditions, with a mean return period equal to the inverse of the exceedance probability T o = 1/p. The relation between magnitude and frequency is characterized using the Gutenberg Richter function. This data is key for water managers and planners in designing reservoirs and bridges, and determining water quality of streams and habitat requirements. or (12), where, The approximate annual probability of exceedance is about 0.10 (1.05)/50 = 0.0021. i When the damping is small, the oscillation takes a long time to damp out. Care should be taken to not allow rounding Seasonal variation of the 1%, 10%, 50%, and 99% exceedance probability levels. event. Maps for Aa and Av were derived by ATC project staff from a draft of the Algermissen and Perkins (1976) probabilistic peak acceleration map (and other maps) in order to provide for design ground motions for use in model building codes. The loss amount that has a 1 percent probability of being equaled or exceeded in any given year. acceptable levels of protection against severe low-probability earthquakes. The return periods commonly used are 72-year, 475-year, and 975-year periods. The exceedance probability may be formulated simply as the inverse of the return period. t 0 = To get an approximate value of the return period, RP, given the exposure time, T, and exceedance probability, r = 1 - non-exceedance probability, NEP, (expressed as a decimal, rather than a percent), calculate: RP = T / r* Where r* = r(1 + 0.5r).r* is an approximation to the value -loge ( NEP ).In the above case, where r = 0.10, r* = 0.105 which is approximately = -loge ( 0.90 ) = 0.10536Thus, approximately, when r = 0.10, RP = T / 0.105. The estimated parameters of the Gutenberg Richter relationship are demonstrated in Table 5. is the return period and , In GPR model, the return period for 7.5, 7 and 6 magnitudes are 31.78 years, 11.46 years, and 1.49 years respectively. The true answer is about ten percent smaller, 0.63.For r2* less than 1.0 the approximation gets much better quickly. = Peak Acceleration (%g) for a M6.2 earthquake located northwest of Memphis, on a fault at the closest end of the southern linear zone of modern . Probabilistic ground motion maps have been included in the seismic provisions of the most recent U.S. model building codes, such as the new "International Building code," and in national standards such as "Minimum Design Loads for Buildings and Other Structures," prepared by the American Society of Civil Engineers. 0 through the design flow as it rises and falls. Any potential inclusion of foreshocks and aftershocks into the earthquake probability forecast ought to make clear that they occur in a brief time window near the mainshock, and do not affect the earthquake-free periods except trivially. If m is fixed and t , then P{N(t) 1} 1. Uniform Hazard Response Spectrum 0.0 0.5 . Another example where distance metric can be important is at sites over dipping faults. ] [ Figure 4 provides an overview of the estimated EEWS-related reduction in injury and fatality exceedance by return period for each of 11 large Swiss municipalities . Gutenberg and Richter (1954) have suggested an expression for the magnitude and frequency of earthquake events larger than magnitude (M). The model selection information criteria that are based on likelihood functions and applications to the parametric model based problems are 1) Akaike information criterion (AIC): AIC procedure is generally considered to select the model that minimizes AIC = 2LL + 2d, where LL is the maximized log likelihood of the model given n observation, d is the dimension of a model. ) The other assumption about the error structure is that there is, a single error term in the model. F This does not mean that a 100-year flood will happen regularly every 100 years, or only once in 100 years. n i 2 ) , it is tempting to assume that the 1% exceedance probability loss for a portfolio exposed to both the hurricane and earthquake perils is simply the sum of the 1% EP loss for hurricane and the 1% EP loss . It includes epicenter, latitude, longitude, stations, reporting time, and date. Currently, the 1% AEP event is designated as having an 'acceptable' risk for planning purposes nearly everywhere in Australia. Given the spectrum, a design value at a given spectral period other than the map periods can be obtained. {\displaystyle r} Probability of Exceedance for Different. This probability gives the chance of occurrence of such hazards at a given level or higher. The theoretical return period between occurrences is the inverse of the average frequency of occurrence. ^ n ) The constant of proportionality (for a 5 percent damping spectrum) is set at a standard value of 2.5 in both cases. The dependent variable yi is a count (number of earthquake occurrence), such that n=30 and we see from the table, p=0.01 . corresponding to the design AEP. Despite the connotations of the name "return period". How to . (6), The probability of occurrence of at least one earthquake of magnitude M in the next t years is, P If the observed variability is significantly smaller than the predicted variance or under dispersion, Gamma models are more appropriate. PDF | Risk-based catastrophe bonds require the estimation of losses from the convolution of hazard, exposure and vulnerability models. The probability of exceedance in a time period t, described by a Poisson distribution, is given by the relationship: An event having a 1 in 100 chance The maps come in three different probability levels and four different ground motion parameters, peak acceleration and spectral acceleration at 0.2, 0.3, and 1.0 sec. S Definition. M Similarly, in GPR model, the probability of earthquake occurrence of at least one earthquake of magnitude 7.5 in the next 10 years is 27% and the magnitude 6.5 is 91%. If you are interested only in very close earthquakes, you could make this a small number like 10 or 20 km. T On the average, these roughly correlate, with a factor that depends on period.While PGA may reflect what a person might feel standing on the ground in an earthquake, I don't believe it is correct to state that SA reflects what one might "feel" if one is in a building. Table 4. b , , Rather, they are building code constructs, adopted by the staff that produced the Applied Technology Council (1978) (ATC-3) seismic provisions. T They would have to perform detailed investigations of the local earthquakes and nearby earthquake sources and/or faults in order to better determine the very low probability hazard for the site. If the probability assessment used a cutoff distance of 50 km, for example, and used hypocentral distance rather than epicentral, these deep Puget Sound earthquakes would be omitted, thereby yielding a much lower value for the probability forecast. A 5-year return interval is the average number of years between Q10), plot axes generated by statistical where, F is the theoretical cumulative distribution of the distribution being tested. t % ^ The procedures of model fitting are 1) model selection 2) parameter estimation and 3) prediction of future values (McCullagh & Nelder, 1989; Kokonendji, 2014) . R i Earthquake Parameters. [4]:12[5][failed verification]. ( = , Frequencies of such sources are included in the map if they are within 50 km epicentral distance. Note that for any event with return period Photo by Jean-Daniel Calame on Unsplash. It states that the logarithm of the frequency is linearly dependent on the magnitude of the earthquake. n ) This probability is called probability of exceedance and is related to return periods as 1/p where p is return period. Thus the maps are not actually probability maps, but rather ground motion hazard maps at a given level of probability.In the future we are likely to post maps which are probability maps. i In our question about response acceleration, we used a simple physical modela particle mass on a mass-less vertical rod to explain natural period. and 2) a variance function that describes how the variance, Var(Y) depends on the mean, Var(Y) = V(i), where the dispersion parameter is a constant (McCullagh & Nelder, 1989; Dobson & Barnett, 2008) . ) Then, through the years, the UBC has allowed revision of zone boundaries by petition from various western states, e.g., elimination of zone 2 in central California, removal of zone 1 in eastern Washington and Oregon, addition of a zone 3 in western Washington and Oregon, addition of a zone 2 in southern Arizona, and trimming of a zone in central Idaho. , Time HorizonReturn period in years Time horizon must be between 0 and 10,000 years. The probability function of a Poisson distribution is given by, f A typical seismic hazard map may have the title, "Ground motions having 90 percent probability of not being exceeded in 50 years." Example: "The New Madrid Seismic Zone.". ( a = 6.532, b = 0.887, a' = a log(bln10) = 6.22, a1= a log(t) = 5.13, and Periods much shorter than the natural period of the building or much longer than the natural period do not have much capability of damaging the building. Solving for r2*, and letting T1=50 and T2=500,r2* = r1*(500/50) = .0021(500) = 1.05.Take half this value = 0.525. r2 = 1.05/(1.525) = 0.69.Stop now. Predictors: (Constant), M. Dependent Variable: logN. That is disfavoured because each year does not represent an independent Bernoulli trial but is an arbitrary measure of time. Note that, in practice, the Aa and Av maps were obtained from a PGA map and NOT by applying the 2.5 factors to response spectra. In these cases, reporting = Answer:Let r = 0.10. is the counting rate. The probability that the event will not occur for an exposure time of x years is: (1-1/MRI)x For a 100-year mean recurrence interval, and if one is interested in the risk over an exposure It can also be noticed that the return period of the earthquake is larger for the higher magnitudes. But EPA is only defined for periods longer than 0.1 sec. t AEP i design engineer should consider a reasonable number of significant In taller buildings, short period ground motions are felt only weakly, and long-period motions tend not to be felt as forces, but rather disorientation and dizziness. y ( e The most important factors affecting the seismic hazard in this region are taken into account such as frequency, magnitude, probability of exceedance, and return period of earthquake (Sebastiano, 2012) . the exposure period, the number of years that the site of interest (and the construction on it) will be exposed to the risk of earthquakes. produce a linear predictor This is consistent with the observation that chopping off the spectrum computed from that motion, except at periods much shorter than those of interest in ordinary building practice has very little effect upon the response spectrum computed from that motion, except at periods much shorter than those of interest in ordinary building practice. to occur at least once within the time period of interest) is. 1-30 Seismic Rehabilitation Prestandard FEMA 356 Chapter 1: Rehabilitation Requirements where: and the mean return period, P R, at the desired exceedance probability shall be calculated from Equation (1-2): (1-2) where P EY is the probability of exceedance (expressed as a decimal) in time Y (years) for the desired earthquake hazard level. ( Nepal is one of the paramount catastrophe prone countries in the world. ) i .For purposes of computing the lateral force coefficient in Sec. In this table, the exceedance probability is constant for different exposure times. . The relation is generally fitted to the data that are available for any region of the globe. Annual Exceedance Probability and Return Period. The GR relation is logN(M) = 6.532 0.887M. / n An EP curve marked to show a 1% probability of having losses of USD 100 million or greater each year. The other significant measure of discrepancy is the generalized Pearson Chi Square statistics, which is given by, These models are. Let r = 0.10, 0.05, or 0.02, respectively. Over the past 20 years, frequency and severity of costly catastrophic events have increased with major consequences for businesses and the communities in which they operate. as AEP decreases. The frequency magnitude relationship of the earthquake data of Nepal modelled with the Gutenberg Richter (GR) model is logN= 6.532 0.887M and with generalized Poisson regression (GPR) model is lnN = 15.06 2.04M. probability of an earthquake incident of magnitude less than 6 is almost certainly in the next 10 years and more, with the return period 1.54 years. N The probability of exceedance in 10 years with magnitude 7.6 for GR and GPR models is 22% and 23% and the return periods are 40.47 years and 38.99 years respectively.