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The Frechet distribution as an alternative model of extreme value data Shahriari, Shahriar 1987

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THE FRECHET DISTRIBUTION AS AN ALTERNATIVE MODEL OF EXTREME VALUE DATA by Shahriar Shahriari B.Sc. The University of Southampton, 1985 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering We accept the thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA APRIL 1987 e Shahriar Shahriari, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. SHAHRIAR SHAHRIARI Department of M E G H A N T H A T , K N T C T N T T O T N C , The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date APRIL 1987  DE-6(3/81) ABSTRACT The Frechet distribution was applied to a set of earthquake data in order to test i t s validi t y as a practical alternative distribution for extreme value data. It was concluded that the Frechet distribution was the best model representing that data set. Also, a Poisson model of occurrence could not be rejected for that data set. The combination of these two models resulted in a closed form unconditional extreme value distribution which was developed analytically. The appropriate s t a t i s t i c a l tests and sensitivity analyses were performed on the obtained model. i i ACKNOWLEDGEMENT I w o u l d l i k e t o t a k e t h i s o p p o r t u n i t y t o e x p r e s s my s i n c e r e t h a n k s a n d a p p r e c i a t i o n t o P r o f e s s o r K. V. B u r y f o r h i s i n v a l u a b l e h e l p a n d g u i d a n c e t h r o u g h o u t t h e c o u r s e o f t h i s t h e s i s . The f i n a n c i a l s u p p o r t o f my f a t h e r d u r i n g t h e c o u r s e o f t h i s t h e s i s i s a l s o g r e a t l y a p p r e c i a t e d . i i i CONTENTS Abstract i i Acknowledgements i i i Contents i v L i s t of figures v i i L i s t of tables ix 1. Introduction 1 1.1 The need for research 1 1.2 The data base . 1 2. L i t e r a t u r e review 3 2.1 Geological background 3 2.1.1 Geological source modelling . . . 3 2.1.2 Magnitude scales 4 2.1.3 Swarms, foreshocks, aftershocks . . 7 2.1.4 C y c l i c e f f e c t s 8 2.1.5 Attenuation 8 2.2 Stochastic models 10 2.2.1 Occurrence modelling 10 2.2.2 Data a v a i l a b i l i t y 13 2.2.3 Magnitude modelling 14 2.2.4 Geographical source modelling . . . 15 2.2.5 The Cornell method of PGA estimation . 16 2.2.6 Bayesian models 16 2.2.7 Extreme PGA models 17 2.2.8 Seismic hazard maps 19 iv 2.2.9 Uncertainties 19 2.3 Other features 21 2.3.1 Duration and frequency of earthquakes 21 2.3.2 S o i l composition 22 2.3.3 Peak ground v e l o c i t y and displacement 2 3 2.3.4 Seismic energy propagation . . . . 23 2.3.5 Topography 24 3. Proposed Analysis 2 5 3.1 Assumptions 2 5 3.2 Proposed analysis . 26 4. Details of the proposed analysis 3 0 4.1 Data 3 0 4.2 Earthquake magnitude 31 4.3 Occurrence modelling 32 4.3.1 Graphical examination 3 2 4.3.2 Main event determination . . . . 36 4.3.3 Other occurrence models 37 4.4 Scatter p l o t 4 0 4.5 PGA modelling 42 4.5.1 Attenuation 4 2 4.5.1.1 OASES attenuation law . . . 4 2 4.5.1.2 Nearest distance c a l c u l a t i o n . 43 4.5.2 S t a t i s t i c a l modelling . . . . . . 45 4.5.2.1 Graphical examination . . . 45 4.5.2.2 The choice between Frechet and Log-Normal d i s t r i b u t i o n s . . . . 49 v 4.5.2.3 Summary 54 4.6 Extreme value modelling . . . . . . . 54 4.7 S t a t i s t i c a l tests 56 5. Summary of Results and Conclusions . . . . 62 5.1 Summary of results 62 5.2 Conclusions 63 5.3 Recommendations for future work . . . . 64 Bibliography . . . 65 Appendices' I Extreme value model derivation 69 1 II Maximum likelihood estimation for Frechet distribution parameters of a censored sample 71 III Main computer program 73 IV data 85 V Table of bias for maximum likelihood estimation of Frechet distribution shape parameter . . 9 6 VI Extreme Value estimation for Log-normal and Frechet as parent distributions 97 VII Hazard plots for time of occurrence modelling 98 VIII Probability plots for rejected distributions 99 IX Probability plots for various site locations and orientations 1 0 4 v i List of Figures 4.1 - Probability Mass Function of Richter magnitude 31 4.2 - Time between occurrences of earthquakes on Weibull Hazard paper 33 4.3 - Time between occurrences of earthquakes for yearly data on Weibull Hazard paper . . 38 4 . 4 - 'Time between occurrences of earthquakes for large PGA on Weibull Hazard paper . . . 39 4.5 - Geographic dispersion of earthquakes . . 41 4.6 - Spherical wave propagation 44 4.7 - Plot of PGA on Frechet and Log-Normal probabi-l i t y paper . . . . 47 4.8 - Plot of maximum yearly PGA on Gumbel and Frechet probability paper 4 8 4.9 - Comparison of Frechet and Log-Normal models 50 4.10 - Extreme value distribution for two specified time spans 55 4.11 - Model sensitivity to attenuation parameters 61 VII. l - Time between occurrences of earthquakes 9 8 V I I I . l - Plot of PGA on Extreme Value type I probability paper . 9 9 VIII.2 - Plot of PGA on Weibull probability paper . tOO V I I I . 3 - Plot of PGA on Gamma probability paper 101 VIII.4 - Plot of PGA on Gamma probability paper 102 / v i i VIII. 5 - Plot of PGA on Normal probability paper 103 IX. 1- Comparison of Frechet and Log-Normal distributions for a different sit e at 13 N, 90 W and 0 and 90 degrees W of N 1 0 4 IX.2- Comparison of Frechet and Log-Normal distributions for a different sit e at 15 N, 89 W and 0 and 90 degrees W of N 1^ 5 v m List of Tables Determination of main events . . . . . 3 5 Table of OASES parameters [17] . . . . 43 Comparison of projected and recorded extreme value for PGA 52 Effects of censorship on vertical PGA estimates 57 Sensitivity to attenuation laws (Normal PGA only) 59 Sensitivity to physical displacement of the site (Normal PGA only) 59 Sensitivity to attenuation law parameters 60 ix CHAPTER 1  INTRODUCTION 1.1 The need for research For engineering purposes, information i s necessary. This information may be in many forms. Scientific theories and data are two such forms. The data normally have errors and s t a t i s t i c a l distributions associated with them. For most engineering purposes, extreme values are important. For example earthquake engineers are normally concerned with extreme earthquake loads applied to structures. A common distributional assumption for large extremes in an analysis i s the Gumbel distribution. In many cases this i s a valid assumption, however, at other times other models may f i t the data better. In this thesis, a different extreme value model, namely the Frechet model w i l l be investigated. One characteristic of this distribution is the long t a i l associated with i t . Naturally, this longer t a i l may result in radically different decisions of engineering sig-nificance. 1.2 The data base For the purposes of this investigation, the earth-quake phenomenon was chosen as the one of engineering interest. For this reason, a real data set was chosen 1 which included sufficient information for a s t a t i s t i c a l analysis. The earthquake data are i n i t i a l l y attenuated to a chosen location and then the Peak Ground Acceleration (PGA) values are s t a t i s t i c a l l y modelled. Sta t i s t i c a l models are tested for this data set and the best f i t model is determined and further investigated. 2 CHAPTER 2  LITERATURE REVIEW The purpose of this chapter is to provide the reader with some background information on the practical context from which the data for the thesis investigation were drawn. 2.1 Geological background 2.1.1 Geological source modelling A number of geological models have been developed to explain the causes of earthquake occurrence. The simplest and the best known of these models i s the "elastic rebound" model proposed by Reid in 1910 [19]. This model simply states that the tectonic plates of the earth move relative to each other. When they come into contact with one another, there w i l l be a stress build-up because of the pressure they exert on each other. When this stress reaches a certain level, depending on a number of factors such as the rock structure of the tectonic plates or the type of contact between them, the stress w i l l be relieved in the form of an energy release, giving rise to an "earthquake". On a much larger scale, the tectonic plates movement i s said to have caused the well known "continent-al d r i f t " . This model i s a simple one and explains the exist-ence of fault lines adequately. However, i t cannot be considered as a universal model since i t does not explain 3 the occurrence of earthquakes in locations other than along fault lines. There are some areas which have high seismicity, yet cannot be classified as fault lines. These are horizontal or inclined planes which, when projected on a map, w i l l indicate seismically active areas. Some theories explain these as the collection of a number of fault lines, others describe them as an area of weak geological structure the build up of stress within which may result in an earthquake at the weakest part of that plane. Furthermore, some earthquakes have occurred in the middle of tectonic plates which again cannot be explained by either of the above models. Two of the largest ever recorded earthquakes have occurred in different parts of China, in the middle of a tectonic plate. The 1557 earthquake of central China was one of them and 112 years later, another earthquake of approximately the same magnitude occurred at a different part of the same tectonic plate [19]. No other earthquakes have occurred since in either of those locations. This kind of occur-rence has not been satisfactorily explained yet. 2.1.2 Magnitude scales The variation in the level of stress build-up and the tolerance of different locations for differing amounts of energy storage, imply different magnitudes of energy release during the occurrence of each earthquake. There 4 have been attempts to quantify this value. However, what can easily be observed is the energy intensity on the ground where the observer or the observation mechanism is situated. One of the earlier earthquake magnitude scales made use of this fact. Rather than trying to quantify the magnitude of the release of energy, the Modified Mercalli Intensity (MMI) scale used observations to give a magnit-ude of ground shaking intensity. However, these observat-ions were done rather crudely. The scale used visual observations of the damage incurred by the earthquake to quantify the intensity of the ground motion in a discrete fashion. For example a magnitude XII on the MMI scale indicates total destruction of a l l existing structures; whereas magnitude XI refers to major damage to the majority of the existing structures. The usefulness of this scale i s rather limited due to the discrete nature of the scale; i t i s clear that the subjectivity of the scale adds to i t s uncertainty. It i s however useful in classify-ing histo r i c a l earthquakes. The f i r s t person who proposed the use of a continu-ous scale was Charles Richter. In 1935 [28], he suggested the use of his proposed magnitude scale. The Richter scale makes use of instrumental observations of Peak Ground Acceleration (PGA) at the observation site and standardis-es i t with respect to a standard PGA at a certain distance 5 from the epicentre of the earthquake. Richter in his paper defined the magnitude of an earthquake as follows. "The magnitude of any shock is taken as the logarithm of maximum trace amplitude, expressed in microns, with which the standard short-period torsion seismometer (T D = 0.8 sec. v=2800 h=0.8), would register at an epicentral distance of 100 kilometers." The improvements over the previous models were great. Primarily, the Richter scale uses an instrumentally recorded observation which elimi-nates much of the subjectivity involved in previous scales; secondly, i t is a continuous scale which implies less uncertainty involved in evaluation of earthquake magnitudes. The Richter scale however, s t i l l uses some subjective calculations such as in comparing the earth-quake magnitude with that of a standard one i t must use some geologically based technique which i s not necessarily objectively defined. The significant characteristic of this scale i s that by the virtue of i t s logarithmic nature, i t includes a wide range of energy levels. An earthquake of magnitude 6 has ten times the energy of one of magnitude 5. It is interesting to note that a significant proportion of the earth's seismic energy i s released by the occurrence of the larger earthquakes; while the number of the larger shocks i s small compared to the total number of earth-quakes [28] . 6 2.1.3 Swarms, foreshocks, aftershocks r191 Swarms are sequences that occur i n regions of active tensional tectonics, or i n high volcanic or hydrothermal a c t i v i t y regions. By t h e i r nature, swarms have a shallow f o c a l point, and for t h i s reason they have a high destruc-t i v e p o t e n t i a l . However, because of the low capacity of these shallow a c t i v i t i e s for stress accumulation, earth-quakes of magnitude above 6.5 on Richter scale are not known to have occurred i n swarms. Foreshocks are shocks of smaller magnitude that occur p r i o r to the main event. One theory i s that the occurrence of the foreshocks s t a r t s a chain reaction of earthquakes [19]. In seismology, foreshocks are looked upon as earthquake predictors that act as warnings of a larger magnitude event. By d e f i n i t i o n , each foreshock can be considered as a main event u n t i l a larger one occurs l a t e r i n that sequence. Another way of looking at the same d e f i n i t i o n i s to consider each shock as a foreshock, u n t i l a smaller magnitude sequence of events takes place. Aftershocks are shocks of smaller magnitude that follow the main event. Normally, the main event t r i g g e r s a seri e s of smaller shocks that occur u n t i l the seismic energy i s s i g n i f i c a n t l y released. One further addition to the above i s that some major foreshocks may have t h e i r own sequence of aftershocks following them. The 21 May 1960 South Chile earthquake . 7 (magnitude 7.5) was considered to be the main event, until a larger earthquake followed 33 hours later. There were six shocks of magnitude 7-7.5 in the eleven hours prior to the main shock [19]. 2.1.4 Cyclic effects Earthquakes have a cyclic characteristic. In some parts of the world, the existing records support this. For example in eastern Turkey, in 1939, an earthquake of magnitude 7 occurred which started a sequence of shocks. This was previously considered to be a quiet region. However, in talking to the farmers along that fault, many family stories which were handed-down, repeatedly related similar occurrences. A l l these stories referred to similar earthquakes which dated back by 100-200 years. Earlier hist o r i c a l documents point out the occurrence of similar series of shocks between 967 and 1050 AD; and s t i l l further back, i t i s known that same kind of acti v i t i e s occurred in the 3rc*, 5*-*1, and 7t*1 centuries [19]. 2.1.5 Attenuation The seismic energy stored during the stress build-up is released in the form of an earthquake. This energy release i s dissipated in the form of seismic waves. The energy intensity of these waves decreases with distance due to two main factors. Like any other wave front, the 8 amplitude of ground shaking decreases as the distance from the focal point increases due to the wave front expansion. Also, as the wave front travels i t looses energy due to f r i c t i o n a l effects. There are a number of attenuation laws, the most prominent of which i s the Esteva law [12]. PGA - C em R_q (2.1) Where C, K, q are empirical constants, M i s the earthquake magnitude, and R is the focal distance. Most of these laws have a similar exponential form; however, for each region of the world, a particular law with i t s appropriate parameters i s in use. The parameters of these laws are determined empirically [12]. A large number of modific-ations to the law, are appropriate for particular regions of the world [5], and are also widely accepted. Most of the analyses using attenuation laws assume some particular type of wave propagation, either implicit-ly or otherwise [9, 19, 24]. The simplest form of this assumption i s that of a spherical wave propagation pattern. This assuntrrption would hold i f the s o i l composit-ion was homogeneous, energy intensity reduction was independent of depth, distance, frequency of shaking, and a number of other minor reasons. In reality however, this i s not the case; the wave propagation i s either e l l i p s o i d -9 a l , or i t w i l l not have a uniform or symmetrical pattern. For the scope of this thesis however, the spherical wave propagation assumption w i l l suffice. Attenuation laws are generally different for different depths. They also d i f f e r depending on the epicentral distance. Higher frequency vibrations generally attenuate faster than the lower frequency ones. There are three general categories of attenuation laws. Those applicable' to short distances, usually less than 2 0 kilometers from the source; those for long distance attenuation, above 200 kilometers; and those for medium length distances [32]. The last type is the more important one, since the chances of having an event close to a site i s small unless the site i s located on or very near a fault line. Also, the chances are small of having such large magnitudes for distant events that extensive damage would result. 2.2 Stochastic models  2.2.1 Occurrence modelling Earthquakes are generally agreed to be discrete events. The duration of each earthquake compared to the time between the occurrence of two successive earthquakes is so small that i t can be neglected, and hence the earthquake can be considered as a discrete event. 10 Another property of the time of occurrence i s i t s random nature. This property enables the modelling of the number of occurrences of the earthquakes i n a given time as a s t a t i s t i c a l event. A number of stochastic models have been proposed for t h i s phenomenon. Some of the more prominent ones are mentioned below. The Poisson Model [8, 9, 12, 17, 19, 22, 24]; The simplest model for the number of occurrences of earthquakes i n a given time i s the Poisson model. P p <n|x) - <-*-)nexp i-^-}^- (2.2) Where x i s the time i n t e r v a l , n i s the number of occur-rences during that i n t e r v a l , a i s the mean i n t e r - a r r i v a l time of events and Pp i s the p r o b a b i l i t y mass function (pmf). This model has the d i s t i n c t property of being memory-less. What t h i s property implies i s that the occur-rence of each earthquake i n the time domain i s independent of any other events that have occurred previously. One shortcoming of t h i s model i s that i t contradicts the occurrence of fore-shocks, after-shocks, and swarms (Sect. 2.1.3) and i t i s also i n disagreement with the c y c l i c nature of earthquakes (Sect. 2.1.4). The Compound Poisson Model [2, 30]; This model i s simply an extension of the above Poisson model. I t divides the events into two parts: one part 11 comprises the main shocks that have a Poisson model of number of occurrences; the second part of the model suggests the existence of a different Poisson model for the number of occurrence of fore and aftershocks around the main event. Although this i s a better and at the same time more complex model, i t s t i l l does not model the cyclic nature of the earthquakes. Frequency Of Occurrence [3, 17, 22, 24]; This model approaches the same phenomenon differently. Rather than modelling the number of occurrences in a given time, this model chooses a duration of time and models the number of occurrences related to magnitude, in that duration. It proposes a logarithmic relationship between the number of earthquakes in that length of time exceeding a certain magnitude versus that magnitude. The simple relationship Log N = a - bM (2.3) was proposed where a and b are empirically derived const-ants, M i s the Richter magnitude, and N i s the number of earthquakes exceeding that magnitude in one year. The simplest form of this approach proposes a linear relationship as suggested by the above equation. Other models propose a truncated bi-linear or in some cases a parabolic relationship [17, 22, 24]. 12 2.2.2 Data availability An earthquake i s a geological phenomenon; and as such, i t can only be comprehensively considered on a large time scale basis. Since the availability of data on a geological time basis i s beyond the reach of today's records and technology, the sparse available information must be u t i l i z e d to i t s utmost. Historical records often refer to earthquakes and note the resulting destruction. Scientists can use these records to estimate the magnitude of these occurrences. However, there are two major problems associated with this. F i r s t l y , the estimation contains a large and often indeterminate uncertainty associated with i t since i t would be the subjective opinion of one or more experts. Secondly, the historical records only note the major earthquakes. This causes a censorship of the smaller shocks and a bias towards large magnitude events. A reasonable record of earthquake occurrences is only available since the late 19 t n century. These records have the advantage of being documented, but they are s t i l l non-standard and lacking in accuracy. Accurate recorded data are only available from the early 1960's onward. This limits the a v a i l a b i l i t y from a desired geological time scale to about a quarter of a century of existing records. The size of the available 13 data base i s obviously very l i m i t e d even i f that of the twentieth century up to the present i s considered. Of a l l the available data approximately 71% of earthquakes i n t o t a l h i s t o r i c a l records were reported i n the l a s t 40 years (1930-1969) , 23% i n the previous 40 years, and 6% were recorded p r i o r to that time [31]. This demonstrates some of the problems associated with data recording. Another problem associated with the data recording i s the bias of the recording equipment not to record small magnitude shocks. There i s a threshold value for each seismometer below which the ground shaking cannot be recorded. These factors w i l l add to the incompleteness and inhomogeneity of the available data. 2.2.3 Magnitude modelling A number of researchers have developed s t a t i s t i c a l models for the occurrence of earthquake magnitudes. The models vary from the proposal of a Gamma d i s t r i b u t i o n for the magnitude [12] to the assumption of B e r n o u l l i events [24], to truncated models of the magnitude [22]. The l a t t e r was subsequently extended to B i - l i n e a r [17] and Quadratic laws for the occurrence of these magnitudes [22]. The a b i l i t y to predict the p r o b a b i l i t y of magnitude of forthcoming earthquakes i s valuable. However, t h i s 14 thesis has not considered magnitude modelling, and has concentrated on the analysis and evaluation of PGA at the site. 2.2.4 Geographical source modelling Earthquake sources are mainly modelled in three ways. The source may be assumed to be a point source [19,24]. This simplifies the earthquake to a release of a f i n i t e amount of energy by some mechanism at a point. The second model is a simplified representation of a fault line. This model assumes that the seismic energy is released along a line [19, 24]. The focal point i s considered to be either in the middle of that line or at a point on the line, nearest point to the site. A more sophisticated model represents the release of seismic energy along a plane [24] which may be horizontal, ver t i c a l , or inclined. This i s a better model of what actually takes place; However, the focal point of the earthquake i s assumed to be the centroid of that plane. The attenuation laws are usually applied from the focal point. More refined models introduce the concept of "significant distance". This i s defined as the shortest distance from the ruptured length to the site [24]. This model assumes that the effect of energy release nearer the site may be more significant than that of the focal point, 15 and therefore applies the attenuation law by using the significant distance rather than epicentral or hypocentral distances. Other models use numerical analysis to evaluate the effect at the site from an event that has uniform or weighted probability of occurrence anywhere within a given seismically active plane. 2.2.5 The Cornell method of PGA estimation [9] The Cornell method is one of the more established design methods available for the prediction of maximum PGA. This methodology recommends the prediction of maximum earthquake magnitude for a given probability of exceedence in the v i c i n i t y of the site, and conversion of that magnitude to design PGA through the use of an appropriate attenuation law. 2.2.6 Bayesian models Some researchers argue that, because of the limited data a v a i l a b i l i t y and the significant advances made in the fields of geology and earthquake engineering, the addi-tional knowledge from these fields should be used to a larger extent. These researchers advise the adoption of a Bayesian approach which enables the use of much of the additional information already existing, as well as a means for updating and re-evaluating the results [12, 17, 19, 24, 29]. The thesis work uses the classical statis-t i c a l approach to uncertainty modelling, and no further reference to the Bayesian approach w i l l be made. 2.2.7 Extreme PGA models In this thesis load inputs to a static design procedure are considered only. That i s , design c r i t e r i a other than peak amplitude of ground acceleration are excluded from analysis. The reason for this particular focus i s that the resonant frequencies of common struc-tures are quite separate from the high energy frequencies of a typical earthquake forcing function [10, 25, also Sect. 2.3.1]. There have been many versions of extreme value models related to earthquakes. Some researchers have attributed a Gumbel distribution to earthquake accelerat-ions [5]: fjUjii.o) - 4- exp {-^J_U_ e x p {--JL^JL} } (2.4) Where x= independent variable, u • the location parameter, o -the scale parameter of the Gumbel distribution, and f= probability density function (pdf). Others have chosen the annual maximum values of PGA. The annual maximum intensity of energy has been given a Gumbel distribution [9]. Some have suggested a Gumbel distribut-§ ion for annual maximum PGA [21]. Modified Mercalli 17 Intensity extreme value has been given a type I (Gumbel) d i s t r i b u t i o n , while the maximum s i t e displacement was thought to be type II [8, 30]: Where x= the independent variable, o » the scale parameter, and * - the shape parameter of the Frechet d i s t r i b u t i o n . Most researchers predict the p r o b a b i l i t y of earth-quake magnitudes and t h e i r l o c a t i o n . Then they attenuate the magnitude to the design s i t e to estimate the PGA for that p r o b a b i l i t y [5, 9, 30], Others evaluate the proba-b i l i t y of maximum yearly magnitude and i t s location, and thus evaluate the design PGA [19]. Yet some prefer to model the p r o b a b i l i t y of maximum yearly PGA and then choose the design PGA for a given l i f e of the design structure by se l e c t i n g an appropriate p r o b a b i l i t y of exceedance [8, 9]. The present work assumes a data set of earthquake magnitude, and obtains d i r e c t l y from the attenuated data set a model f o r the site-PGA, using established s t a t i s -t i c a l p r i n c i p l e s . This model, together with a model of earthquake occurrence, forms the proposed basis of extreme value p r e d i c t i o n . f l i ( x ; C , A ) - - i - (_£_) X+l \ exp {- (-£-) } (2.5) 18 2.2.8 Seismic hazard maps [191 An extension of the Cornell method is to develop seismic maps. A relevant area map i s divided into a grid. Each node of that grid represents a site for which an extreme PGA for a given return period i s calculated from earthquake data of the region. These grid values allow the plotting of iso-acceleration lines. The points on these lines predict equal extreme PGA values for that return period. Seismic maps have been developed for most regions of the world. They can act as good references when quick and approximate values are required. They are also useful for the purpose of comparison. However, when sites between iso-acceleration lines are evaluated, only approximate values are obtained. Thus they become inappropriate when used for detailed design purposes due to their approximate nature. 2.2.9 Uncertainties Uncertainty in an analysis i s due to a number of factors. Generally, there are three types of uncertainties [27]. 1- Data uncertainty. This i s the type of un-certainty associated with the shortcomings of the data. The recording instruments may have some systematic and some random error which contribute to this uncertainty. 19 The data incompleteness and inhomogeneity are also other contributing factors. The short time span for which earthquake data are available causes the incompleteness of that data. The recordability of some smaller shocks and omission of others cause the inhomogeneity of the data. 2- Model uncertainty. The deviation between the actual event and the simplified laws and models represen-ting the event are the cause of this uncertainty. The nature of the attenuation laws has already been mentioned (Sect. 2.1.5). The spherical wave propagation model is another example of this deviation. These uncertainties are dealt with in hazard analysis with varying degrees of conservatism [27]. 3- Methodology uncertainty. This i s a "man made" uncertainty. Different designers, when presented with the same data and same design c r i t e r i a , produce different results. This i s due to a subjective weighting of the different parts of the methodology, and the varying degree of conservatism applied to the design work. It is d i f f i c u l t to quantify this kind of uncertainty in s t a t i s t i c a l terms. Because of the absence of a quantita-tive value for methodology uncertainty, i t i s impractical to derive a value for total uncertainty, which i t s e l f would have associated uncertainties. 20 2.3 Other features There are a number of other features of earthquakes and their effects on different sites that must be consi-dered during a design process. The following outlines some of the more important ones. 2.3.1 Duration and frequency of earthquakes Duration and frequency of vibration during an earthquake'are very important [19, 25]. An earthquake with a long duration of vibrations w i l l impart a much greater amount of energy than a similar one that only lasts for a short length of time. In the same way, i f the frequency spectrum of an earthquake has a peak corresponding to the natural frequency of the structure that i s being shaken, then the imparting of energy w i l l be much greater than otherwise. The Mexico City earthquake of September 19, 1985 resulted in extensive damage to buildings resonating at, or near, the shaking frequencies. However, many smaller buildings with a much lower natural frequency were unaffected; furthermore, the 43 story Latino Americano Tower with a natural period of 3.5 seconds also suffered only superficial damage. Most of the structures with a natural frequency between the two groups mentioned suffered extensive damage or total destruction [16]. Another related problem is that the energy distribu-tion spectra of strong shocks and weak ones di f f e r [19]. 21 This implies that strong shocks w i l l affect a different range of structures from the weak ones. In a more detailed design procedure, this fact must also be considered. 2.3.2 Soil composition T16, 19. 32] The s o i l composition and conditions w i l l affect the PGA at a site. Generally, soft s o i l composition magnifies the ground shaking, whereas structures built on hard rocks are subjected to smaller magnitude of ground shaking for otherwise identical conditions. In the above mentioned Mexico City earthquake, in areas of firm ground, the maximum PGA was about 4% of gravity. In the centre of the city which i s built on a drained lake bed, the maximum PGA was as high as 2 0% of gravity [16]. Furthermore, the amount of water present in the s o i l w i l l also affect the magnitude of PGA [19]. A soft dry s o i l has a high damping associated with the capacity of the s o i l to push out the air and i t s a b i l i t y to become more compact. If the same s o i l i s saturated with water, then i t s a b i l i t y to push out water is much less than that applied to a i r . This makes the s o i l act as a more solid composition and reduces the damping effects greatly. The effects of s o i l conditions and composition on the seismic energy transfer w i l l not be discussed further here. 22 2.3.3 Peak ground velocity and displacement Other peak ground motion parameters also play an important role in determining the fi n a l damage to the structure [9, 14, 15, 19]. Earthquake engineers are mainly concerned with the PGA in their design work. There are other parameters such as the Peak Ground Displacement and the Peak Ground Velocity which must be looked at separate-ly. An earthquake may produce a very small PGA, while at the same time cause a large ground displacement at the site. Although larger accelerations may not be experienced by the structures, the ground displacement may simply tear them apart. In a similar way, the Peak Ground Velocity may be the major cause of damage in certain cases or struc-tures . 2.3.4 Seismic energy propagation The energy propagation i s an interesting phenomenon. The simplest assumptions represent this phenomenon by a spherical wave propagation pattern [9, 19, 24]. As already mentioned (Sect. 2.1.5) in reality this i s not so. Not only the propagation pattern i s different and non-homo-geneous, but also effects such as reflection off the free surface of the earth crust are normally ignored. In earth-quake engineering, the energy propagation pattern is thought to be in the form of Body waves and Surface waves simultaneously. The former is a three dimensional pheno-23 menon, whereas the latter i s only two dimensional. The dimensional properties of these waves imply that they attenuate at different rates [32]. The body waves attenu-ate at a rate inversely proportional to the epicentral distance while the surface wave attenuation i s inversely proportional to the square root of the epicentral dis-tance. 2.3.5 Topography The topography of the region in question can also affect the magnitude of PGA. Due to the presence of some particular shapes of h i l l s , or around very sharp edges of valleys or mountains, the PGA can be magnified by as much as a factor of 2 [6, 7]. Furthermore, The structure i t s e l f can affect the PGA value at that site. The measurements obtained in the basement of large buildings or at the foot of dams varied slightly from those obtained at nearby sites without the existence of large structures, when corrected for other factors [7]. 24 CHAPTER 3  PROPOSED ANALYSIS T h i s c h a p t e r w i l l c o n s i s t o f a s t a t e m e n t o f t h e a s s u m p t i o n s o f t h i s t h e s i s w o r k f o l l o w e d b y a b r i e f e x p l a n a t i o n o f t h e a p p r o a c h t a k e n i n t h e a n a l y s i s . 3.1 A s s u m p t i o n s T h r o u g h o u t t h i s t h e s i s , t h e f o l l o w i n g a s s u m p t i o n s a r e made: E a c h e a r t h q u a k e i s c o n s i d e r e d t o b e a p o i n t s o u r c e . T h u s t h e n e a r e s t d i s t a n c e i n e a c h c a s e i s c a l c u l a t e d a s t h e h y p o c e n t r a l d i s t a n c e ( F i g . 4 . 6 ) . F o r t h e p u r p o s e s o f c o m p a r i s o n a n d e a s e o f c o m p u t a t i o n , i t i s a s s u m e d t h a t s e i s m i c w a v e s f o l l o w a s p h e r i c a l p r o p a g a t i o n p a t t e r n a n d , when a t t e n u a t e d t o t h e s i t e , c a n be d e c o m p o s e d i n t o t h r e e o r t h o g o n a l c o m p o n e n t s . F u r t h e r -m ore, e a c h o f t h e s e c o m p o n e n t s c a n be m o d e l e d s e p a r a t e l y . L a t e r o n i n t h e a n a l y s i s , t h r e e c o m p o n e n t s may b e c o m b i n e d t o y i e l d a g e n e r a l PGA v a l u e . F o r c o m p u t a t i o n a l p u r p o s e s , a n a r b i t r a r y h o r i z o n t a l d i r e c t i o n was c h o s e n a n d PGA c o m p o n e n t s t h a t a r e v e r t i c a l a n d h o r i z o n t a l l y n o r m a l t o t h a t d i r e c t i o n a r e e v a l u a t e d a s t h e t w o o r t h o g o n a l d i r e c t i o n s . 25 3.2 Proposed analysis The proposed analysis e s s e n t i a l l y consists of f i v e steps. St a r t i n g with a s p e c i f i e d s i t e , and given earth-quake data for the region, as well as a time horizon, a value f o r maximum PGA with a given p r o b a b i l i t y of exceed-ance i s estimated. Step 1: The i n i t i a l stage of the proposed analysis i s to derive a model for the number of earthquake occur-rences, i n a chosen time period, from a v a i l a b l e earthquake data of the region. I t i s generally accepted that i f only the main events are considered, the number of earthquake occurrences i n a chosen time period can be modelled as a Poisson phenomenon (2.2, also Sect. 2.2.1). This has already been discussed i n section 2.2.1, and w i l l be tested i n section 4.3, where some r e s u l t s w i l l be shown. With the assumption of a Poisson model for the number of earthquake occurrences, one can proceed to estimate the parameter of that model for the s i t e of the design structure on the basis of the avail a b l e data. Step 2: Having evaluated the occurrence model of earthquakes, the next step i s to use the avail a b l e earth-quake data of the magnitude, location and depth of each i n d i v i d u a l h i s t o r i c a l event and to attenuate each event to the design s i t e . Obviously, the choice of the appropriate attenuation law i s c r i t i c a l . This choice must be made on the basis of geological and empirical information a v a i l -26 a b l e . A t t h e e n d o f t h i s s t a g e , a new s e t o f d a t a , e q u i v a l e n t t o t h e o r i g i n a l d a t a s e t , w i l l b e a v a i l a b l e t o t h e a n a l y s t , c o n s i s t i n g o f i n d i v i d u a l PGA's v a l u e s w i t h t h e i r r e s p e c t i v e d i r e c t i o n s f o r e a c h e v e n t . P e a k G r o u n d A c c e l e r a t i o n , a s i t s name s u g g e s t s , r e f e r s t o t h e maximum g r o u n d a c c e l e r a t i o n o b t a i n e d a t t h e s i t e d u r i n g a n e a r t h q u a k e e v e n t . The n a t u r e o f t h i s u n i t s u g g e s t s a n e x t r e m e v a l u e phenomenon w i t h r e s p e c t t o many g r o u n d a c c e l e r a t i o n v a l u e s o c c u r r i n g d u r i n g a n e v e n t . T h e r e f o r e i t i s r e a s o n a b l e t o e x p e c t a n e x t r e m e v a l u e m o d e l o t h e r t h a n a W e i b u l l one t o r e p r e s e n t PGA. I t i s e v i d e n t t h a t t h e W e i b u l l m o d e l i s n o t j u s t i f i e d a s i t i m p l i e s a n e x t r e m e v a l u e phenomenon whose v a l u e i s s t r i c t l y l i m i t e d . S t e p 3: The t h i r d s t a g e i s t o d ecompose t h e PGA d a t a o b t a i n e d i n t o t h r e e p a r t s , n a m e l y t h r e e o r t h o g o n a l d i r e c t i o n s . The r e s u l t s o f s e c t i o n 4.5 s u g g e s t t h a t f o r t h e d a t a a n a l y z e d a n a p p r o p r i a t e s t a t i s t i c a l m o d e l f o r e a c h o f t h e s e o r t h o g o n a l c o m p o n e n t s i s t h e F r e c h e t m o d e l ( 2 . 5 ) . Once t h e m o d e l h a s b e e n v e r i f i e d o n a g r a p h i c a l b a s i s [ 4 ] , t h e p a r a m e t e r s o f t h e F r e c h e t m o d e l c a n be e v a l u a t e d u s i n g maximum l i k e l i h o o d e s t i m a t i o n ( 4 . 2 , 4 . 3 ) . S t a t i s t i c a l t e s t s s u c h a s t h e K o l m o g o r o v t e s t a n d t h e C h i s q u a r e t e s t s h o u l d t h e n be u s e d t o c o n f i r m t h e n o n - r e j e c t -i o n o f t h e s e s p e c i f i c m o d e l s . 27 At t h i s point, there i s an estimated Poisson model of the number of earthquake occurrences for a chosen time period, and an estimated Frechet model of PGA magnitude i n each of the three s p e c i f i e d orthogonal d i r e c t i o n s at the disposal of the analyst. Furthermore, s t a t i s t i c a l t e sts should not have rejected any of the estimated models. Step 4: The next stage i s to combine the Poisson occurrence model with the Frechet model for the PGA magnitude i n each orthogonal d i r e c t i o n , to obtain a single unconditional d i s t r i b u t i o n of the maximum expected PGA i n a chosen time span for each orthogonal d i r e c t i o n . This can be done a n a l y t i c a l l y (Append. I) r e s u l t i n g i n a closed form sol u t i o n . This f i n a l model, see equations (4.9, 4.10), i s the extreme value d i s t r i b u t i o n f o r the PGA component whose parameters are the scale and shape parameters of the Frechet d i s t r i b u t i o n , the scale para-meter of the Poisson occurrence model, and the chosen time span. The chosen time span would t y p i c a l l y correspond to that of the underlying data set, or may be of longer duration. Step 5: The f i n a l stage makes use of the obtained model to cal c u l a t e a PGA value for a given p r o b a b i l i t y of exceedance for the s i t e i n each orthogonal d i r e c t i o n . Furthermore, t h i s stage should include a s e n s i t i v i t y analysis of the model parameters and, i f possible, 28 an uncertainty evaluation of the calculated PGA value (not dealt with in this thesis, see section 5.3). 29 CHAPTER 4  DETAILS OF THE PROPOSED ANALYSIS This chapter presents some of the t e s t s performed to derive and q u a l i f y the proposed analysis. The graphs and tables shown are only a selected representation of what was done, observed, and used i n making decisions regarding the d e t a i l s of the proposed analysis. The presented figures and tables are believed to be s u f f i c i e n t i n suggesting the q u a l i f i c a t i o n of the derived conclusions. 4•1 Data In order to t e s t the proposed methodology, a data set was required. There are two sources of data. The f i r s t i s to simulate data on the basis of assumptions with respect to the earthquake phenomenon. The second source i s an e x i s t i n g set of actual observations. In t h i s t h esis, a data set was chosen which was based on seismic a c t i v i t i e s i n the Honduran region [18]. To eliminate inhomogeneities and inaccuracies of the data set, only the time period with comprehensive and accurate recording was considered (Sect. 2.6). This reduced the o r i g i n a l data set of recorded earthquake events to 609 events between l a t e 1963 and early 1978; i n other words only f i f t e e n years of seismic data were considered. The data set included information such as the time of occur-rence of the events, t h e i r geographical location, depth and Richter magnitude. This information was s u f f i c i e n t to analyze the PGA at an a r b i t r a r y s i t e i n a general manner. 4.2 Earthquake magnitude As was mentioned i n section 2.2.3, the modelling of earthquake magnitude was not necessary for the purpose of analysis and evaluation of PGA within t h i s thesis work. However, a PMF p l o t of Richter magnitude i s shown i n Fig. 4.1 to give a general idea of the nature of the d i s t r i b u t -ion of earthquake magnitude. G.12-1 0.10-0.08-0.06-0.04 -0.02-0.00 I I I I 3 i 5 6 IL 2 7 6 MAGNITUDE IN RICHTER F i g . 4.1 - P r o b a b i l i t y Mass Function of Richter magnitude 4.3 Occurrence modelling I t i s evident that the proposed model should either consider the occurrences of events or i t s cumulative value from a given time o r i g i n . The l i t e r a t u r e available generally agrees that the number of occurrences of earth-quakes may be considered a Poisson event i n a s p e c i f i e d time period. V » | t ) - (-T-) n e x P <- 4"> 4r - t 4 - 1 ) p p where t i s the s p e c i f i e d time period, n i s the number of occurrences i n that time span, and ° p i s the mean i n t e r - a r r i v a l time. This model implies that the occurrence of each event i s independent of past occurrences. There-fore, given that an event has taken place, the length of time to the occurrence of the next event follows an Exponential d i s t r i b u t i o n . The time between two events i s referred to as the i n t e r - a r r i v a l time or time between occurrences. 4.3.1 Graphical examination The next l o g i c a l step was to process the data to ca l c u l a t e the i n t e r - a r r i v a l times of successive events. With the a v a i l a b i l i t y of t h i s new data set, the i n t e r -a r r i v a l times were to be modelled on a s t a t i s t i c a l basis. I n i t i a l l y the data were examined graphically. Hazard Plotting [4] of the data was performed for a l l types of Extreme value s t a t i s t i c a l distributions. Extreme value types I and II were rejected on a graphical basis (Append. VII). The weibull distribution was not rejected (Fig. 4.2) , ^ - • ' » = i ( f ) ' - ' « p ( - ( f ) ' j Fig. 4.2 Time between occurrences of earthquakes on Weibull Hazard paper This was not surprising since the Weibull distribution is one of the parent distributions (when the shape parameter is 1) of the Exponential distribution which i t s e l f models a Poisson event. The determination of parameters on a graphical basis [4] yielded a shape parameter value of approximately 0.7. This was considered to be too far from a value of 1.0 to consider the inter-arrival time to be Exponential. Maximum Likelihood estimation later confirmed these results ([4], also Table 4.1). The main events were then separated from the raw data (see discussion later in this section) and the time between occurrences of those events were then calculated and analyzed in the same manner. This approach was considered to be relevant so long as simple damage to the structure was modelled. If cumulative damage was to be modelled, then the elimination of fore and aftershocks would become unjustifiable. The determination of the shape parameter on a graphical basis yielded a value much nearer 1.0 this time (Table 4.1). Maximum likelihood estimation of both scale and shape parameters was done at this stage [4], using the following equation with scale parameter o as the unknown: (4.2) 34 The shape parameter X was then obtained simply as: The conclusion made at this stage was that i f the main shocks were to be considered only, then the inter-arrival time could be considered to follow an Exponential di s t r i b -ution. NUMBER OF DATA EVENT CLUSTER TIME LIMIT ± t HOURS WEIBULL SHAPE PARAMETER 589 0.0 0.739 569 1.0 0.809 560 2.0 • 0.837 545 5.0 0.880 520 10.0 0.942 510 15.0 0.966 497 20.0 0.994 445 50.0 1.100 Table 4.1 - Determination of main events. Latitude enclosure * 2 degrees. Longitude enclosure - 4 degrees. 35 4.3.2 Main event determination To separate the main events from their fore and aftershocks a number of approaches may be considered. Obviously, one approach i s to only input the relevant data for those events that are believed to be main events by seismic experts and omit the rest from each cluster of events. The shortcoming of this method is that an after-shock may occur that has a smaller magnitude; however, since i t s focal point may be much nearer the design site, i t may produce a larger PGA than the main shock i t s e l f . Therefore i t i s seen that the problem is the definition of "the main event". It was decided that, for purposes of analysis, the main event be defined as the event within a cluster that produces the largest PGA at the chosen site. The new definition required the attenuation of a l l shocks to the site and the modification of the data to include the main events only. For computational ease and saving of much time, i t was decided to determine the main events somewhat arb i t r a r i l y in the following manner. The Honduran region has most of i t s faults in an east-west direction. The length of these faults do not exceed 4 degrees of longitude and 2 degrees of latitude. Therefore, for each event, i t s epicentre was examined and compared to the previous event's. If i t lay within 4 degrees of longitude and 2 degrees of latitude of the epicentre of the previous event, also i f i t occurred within a given time period from the occurrence of the previous event, i t was considered to be i n the same c l u s t e r of events as the previous event. The PGA's at the s i t e were then compared and the one which produced the larger PGA was determined to be the main event. This process continued u n t i l a c l u s t e r of events was considered to be i d e n t i f i e d . The given time period for a c l u s t e r of events was chosen a r b i t r a r i l y . A c l u s t e r of approximately +2 0 hours about the main event was found to y i e l d the shape parameter value that was nearest 1.0 (Table 4.1) for the i n t e r - a r r i v a l time model. The data were then reduced to 497 "main" events. These 497 data were used throughout the computations. Obviously, for a more accurate analysis the data should be divided by a seismic expert to many sets of event c l u s t e r s . Each event should then be attenua-ted to the s i t e , and the event that produces the largest PGA within each c l u s t e r should then form part of the screened data set of main events to be analyzed. 4.3.3 Other occurrence models Other approaches were also examined. The time between occurrences of events within each year also suggested a Weibull d i s t r i b u t i o n (Fig. 4.3) which also gave a shape parameter value near 1.0 when main events were considered only. However, t h i s approach was not 37 thought to be of any r e a l advantage and was not pursued further. 10: 1968 o z o CC tvl CC X 0 . 1 : 0 . 0 1 -0 .1 1 0 : z o o cr. cc CC X _J 0 . H CD UJ TTl 10 100 T I M E I N HOURS 1973 1000 10000 0.01-0.1 T | I 1 10 100 T I M E I N HOURS 1000 F i g . 4.3 Time between occurrences of earthquakes for yearly data on Weibull Hazard paper 38 Normal PGA above 0.015 g ">3 Lam =0.52 0.H 0.01 too • I I 1000 10000 TIME IN HOURS V e r t i c a l PGA above 0.0011 g Lam = 0.69 1000 10000 TIME IN HOURS 100000 100000 F i g . 4.4 Time between occurrences of earthquakes for large PGA on Weibull Hazard paper 39 Another alternative that was also considered was the selection of large PGA's only. The time between occur-rences of those events did not yield an Exponential distribution (Fig. 4 .4) . That i s , the estimated shape parameter was very different from 1. This was expected since many main events which are of small magnitude or which produce a small PGA at the site are ignored. Furthermore, many large magnitude shocks also have large magnitude fore and aftershocks which a l l may produce large PGA's at the site. Another disadvantage of this approach is the discarding of much information at such an early stage in the design procedure. In conclusion i t was believed that main events, defined as those events that produce the largest PGA at the site within a cluster of related events, could be considered as Poisson events in time. 4.4 Scatter plot In order to obtain a better visual understanding of the effects of geography and magnitude on PGA at the site, consider Figures 4 .5 A, B, C, and D. Earthquake events from the original data set, of varying magnitude, are plotted on latitude Vs. longitude graphs (maps) and their relative positions to the site are shown. This site corresponds to an actual proposed dam site. Obviously, most shocks of large magnitude are also distant and Magnitude above 6.0 Magnitude above 5.5 - i r--92 -90 -88 -86 -84 LONGITUDE -82 20 n o 12 A 10 » » « I I — - » 2 -90 -88 - 8 « -84 LONGITUDE -82 Magnitude above 5 A l l recorded earthquakes 20-1 to • I . -92 I -90 -88 -86 LONGITUDE -84 - 1 — -82 20 - i •M tz M - B 8 -86 LONGITUDE -84 - 8 2 y - source 8 - s i t e F i g . 4.5 - Geographic dispersion of earthquakes therefore, when attenuated to the s i t e , may have a less dramatic e f f e c t than expected. However, some shocks of smaller magnitude with t h e i r epicenters nearer the s i t e may have a more c r i t i c a l and damaging e f f e c t . In short, some smaller magnitude shocks nearer the s i t e may produce a larger PGA at the s i t e than more distant yet larger shocks. 4.5 PGA modelling This section i s divided into two general sub-sections. One i s a b r i e f discussion of attenuation, and the other i s a description of the PGA modelling analysis. 4.5.1 Attenuation 4.5.1.1 OASES attenuation law The appropriate attenuation law was investigated by Shah et a l . [17]. For the Honduras region the OASES (Off-shore Alaska SeisTnic Exposure Studies) attenuation law was chosen. This attenuation law has the following form b, P G A = b x exp (b 2 M } ( R + 0 . 8 6 4 exp { 0 . 4 6 3 M } ) ( 4 * 4 ) where PGA i s the Peak Ground Acceleration i n cm/s2, R i s the nearest distance from the rupture s i t e or the distance from the f o c a l point to the design s i t e i n km, M i s the Richter magnitude, and b 1 ; b 2, and b 3 are constants that vary with the focal depth, h. The following table shows this depth dependent variation of these constants. Table of OASES parameters [17] FOCAL DEPTH (km) *>1 *>2 "3 h > 19 190 0.823 -1.561 b < 19 285 0.589 -1.050 4.5.1.2 Nearest distance calculation To obtain the nearest distance during the computat-ions, the geographic location of the design site was compared with the epicentral location for each "main" event (Sect. 4.3.2). This difference was then translated into kilc-meters of latitude and longitude. The nearest distance from the focal point to the design site was calculated as the vector sum of the epicentral distance and the focal depth (Fig. 4.6). This assumption neglects the effects of the curvature of the earth. This was considered to be reasonable as the distances involved were small compared to the radius of the earth. Furthermore, i t i s assumed that the nearest distance corresponds to the hypocentral distance. Appendix III shows the computer program used to calculate a l l the r e s u l t s . Horz . F i g . 4 .6 - Spherical wave propagation Thus the knowledge of the l o c a t i o n of the fo c a l point of each shock together with i t s depth, and the magnitude of that shock can determine the PGA at the chosen s i t e through the use of the attenuation law. Furthermore, the assumption of spherical wave propagation allowed the decomposition of PGA into three desired orthogonal components for computational and comparative purposes. A l l the shocks were thus attenuated and a f i n a l data set with the following d e t a i l s became av a i l a b l e . The data included for each event the time of occurrence, time between i t s occurrence and the previous event's, location 44 of the epicentre i n l a t i t u d e and longitude, the focal depth, PGA at the s i t e due to that event i n three ortho-gonal components. 4.5.2 S t a t i s t i c a l modelling  4.5.2.1 Graphical examination At t h i s point the data were avail a b l e for the s t a t i s t i c a l analysis. I t was decided to examine two orthogonal components only since the t h i r d component could be treated i n the same manner. Again, i n i t i a l l y a graphic-a l approach was thought to be best. This time p r o b a b i l i t y p l o t s [4] of the two PGA components were drawn. Appendix VIII includes some of the rejected s t a t i s t i c a l d i s t r i b u -t i o n s . P r o b a b i l i t y p l o t t i n g (Append. VIII) rejected Gumbel d i s t r i b u t i o n s (2.4), and Weibull d i s t r i b u t i o n s . The l a t t e r pdf i s defined as follows: fw(x;o,A) - -L. [JL.) exp {- (-£-) } (4.5) where f= the dependent variable, x= the independent variable, o - the scale parameter, and A - the shape parameter of the Weibull d i s t r i b u t i o n . 45 The data were also used to estimate the parameters of a Gamma d i s t r i b u t i o n . The cdf of the Gamma d i s t r i b u t i o n was then plo t t e d and compared with the cmf of the data. The Gamma pdf i s defined by: . X - 1 fG<x;°'X> * o r(X) HH exp {--£-> (4.6) Where x= the independent variable, o - the scale parameter, and X - the shape parameter of the Gamma d i s t r i b u t i o n . Further to a v i s u a l r e j e c t i o n of the d i s t r i b u t i o n , Kolmogorov t e s t s [4] also rejected t h i s d i s t r i b u t i o n . The data were then transformed to be plotted on Normal p r o b a b i l i t y paper (Append. VIII). This model was also rejected. The graphical examination rejected neither the Frechet d i s t r i b u t i o n (2.5) nor the Log-Normal d i s t r i b u t i o n (Fig. 4.7). For the purpose of completion and consistency with the occurrence modelling (Sect. 4.3), the maximum yearly PGA was also plotted on Gumbel p r o b a b i l i t y paper [3] and Frechet p r o b a b i l i t y paper. This was done since some designers assume the maximum yearly PGA follows a Gumbel d i s t r i b u t i o n due to i t s Extreme value nature. Both models were rejected graphically (Fig. 4.8). 46 Log-Normal F r e c h e t Normal PGA Log-Normal F r e c h e t V e r t i c a l PGA 0.8-0.6-2 O O z 3 0.1 • 0.2-o.oi o. i i io ACCELERATION IN 0.001 g 2 2" 100 0.001 0.01 0.1 1 10 ACCELERATION IN 0.001 g Fig. 4.7 - Plot of PGA on Frechet and Log-Normal probability paper 47 Gumbel F r e c h e t Normal PGA 3 -10 20 30 40 50 60 A C C E L E R A T I ON I N 0.001 g ACCELERATION IN 0.001 Gumbel V e r t i c a l PGA F r e c h e t 10 15 20 25 10 A C C E L E R A T I O N IN 0.001 g 3S 40 0.001 0.01 0.1 I A C C E L E R A T I O N IN 0.001 .g. 4.8 - P l o t o f maximum y e a r l y PGA on Gumbel F r e c h e t p r o b a b i l i t y paper 48 4.5.2.2 The choice between Frechet and Log-Normal d i s t r i - butions The graphs in figure 4.7 show that Log-Normal and Frechet distributions appear plausible. However, they posed two questions: Which model i s more appropriate, and Should there be two parameters or three. Finally, i s the whole process one phenomenon, or two phenomena consisting of one occurrence mechanism for lower values of PGA and another for higher ones? If the process comprises two phenomena then should the data set be truncated or censored? It should be noted that the choice between Log-Normal and Frechet distributions may be c r i t i c a l since the natures of these distributions and in particular their t a i l s are very different and can have a large effect on the prediction of design PGA for a given l i f e of a structure (Fig. 4.9). 49 4. Legend evil LOG-NORHBL -> 50 a) For Vertical PGA Legend tvn LOE-NQRWSL ACCELERATION IN 0.001 g b) Magnification of the lower part of the above graph Fig. 4.9 - Comparison of Frechet and Log-Normal models 3-2 -1 -IO 20 30 10 ACCELERATION IN 0.001 g 50 Different site locations and orientations were evaluated and again similar results to those obtained at the original site were obtained (Append. IX) . The next step was to assume that the model also had a location parameter. The result was a better f i t of the Frechet distribution to the data. At the same time, the Log-Normal distribution s t i l l appeared plausible. To decide upon an appropriate model, extreme value models were developed for 600 occurrences corresponding to the number of data points used for both the Frechet and Log-Normal distributions. The Frechet distribution has a reproductive property which means i t s extreme value also has a Frechet form (Append. VI). f I T ( x ; o , A ) - f n ( x ; o n 1 / X , X ) (4.7) where n= the number of events for which the extreme value is being determined. The maximum PGA obtained from the data was within the middle 90 percent of that extreme value distribution. The extreme value model had a larger dispersion than i t s parent Frechet distribution. For the case of the Log-Normal model also an extreme value distribution was calculated. This was a Gumbel type I distribution (2.4, and Append. VI) with a very small dispersion. The middle 99 percent of that extreme value 51 model underestimated the maximum recorded PGA of the data by a factor of approximately 2.5 or 5 (Table 4.2). DIRECTION OF PGA PARENT DISTRIBUTION PERCENTILE 10Z 5 0 Z 9 0 Z Normal Frechet 2 0 . 4 4 5 . 4 1 5 9 . 4 Normal Log-Normal 2 1 . 6 2 2 . 3 2 3 . 3 Vertical Frechet 2 4 . 8 7 9 . 5 4 9 5 . 2 Vertical Log-Normal 1 7 . 6 1 8 . 2 1 9 . 2 Table 4 . 2 - Comparison of projected and recorded extreme value for PGA. Maximum obtained PGA: Normal 1 2 4 . 6 , Vertical 4 5 . 6 . Projected extreme PGA for 497 occurrences (al l units in 0 . 0 0 1 g). On t h i s basis the Frechet d i s t r i b u t i o n was chosen despite i t s large dispersion. In other words, t h i s model was considered more representative of the actual phenomenon than the Log-Normal d i s t r i b u t i o n . The next decision considered the choice between a two parameter or three parameter version of the Frechet model. I t had already been observed that by the appro-p r i a t e choice of a.location parameter a better f i t to the data was obtained. However the a n a l y t i c a l complications that r e s u l t from the in c l u s i o n of a lo c a t i o n parameter are 52 considerable. It was noticed that due to the logarithmic nature of the probability paper, the deviations from a straight line trend were at the lower end of the scale. This meant that to improve the straightness of the plot a very small location parameter was required. Furthermore, the effect of such a small parameter at the upper and more c r i t i c a l t a i l was negligible. Therefore i t was decided that the omission of the location parameter would not affect the fin a l outcome noticeably. The scale and shape parameters of the Frechet model had been estimated graphically. These parameters were then estimated using the Maximum Likelihood approach. The equations that were used for iteration are shown in Appendix II. The use of the Maximum Likelihood approach to the parameter estimat-ion implies that the maximum amount of relevant informa-tion i s extracted from the data to obtain a Poisson model for the occurrences of events and a Frechet model for the components of PGA. The pdf of the Frechet model i s : f n ( x ; o.X) - - i . ( J L ) A + l exp { - <-£-) X } (4.8) Where x= the independent variable, ° m the scale parameter, X- the shape parameter of the Frechet distribution. 53 4.5.2.3 Summary At this stage two s t a t i s t i c a l distributions were estimated from the "main" event data. One i s the Frechet distribution of the magnitude of PGA in two chosen directions at the site, given that an earthquake has occurred, This i s a conditional distribution. The second model is a Poisson model of occurrence of earthquakes. 4.6 Extreme value modelling To determine an Extreme Value model for the design PGA components, the occurrence and PGA models can be used. The Poisson occurrence model can be combined analytically with the Frechet PGA model. The details of this analysis are presented in appendix I. The result i s : fEv<"lT> " - H - Hr>* «p i - (-f )* - (4-> i 1 - «p <- t -r) X > J) ( 4.9) F E V (x |T) - exp {- <-L_) + (_L_) e x p <- (_£_)*) } ( 4 > i o ) Where x= PGA in O.OOlg units T= Design l i f e op = Poisson scale parameter 0 = Frechet scale parameter A = Frechet shape parameter Thus an Extreme value model for the analysis of PGA corresponding to a chosen time span i s derived. The 54 parameters of that model are the scale parameter of the Exponential distribution for the time between occurrences, the scale and shape parameters of the Frechet distribution for the magnitude of PGA, and the chosen time span. Figure 4.10 compares the Probability density functions of this distribution for two different specified times. Thus the basic result of the analysis i s the unconditional d i s t r i b -ution of the extreme value for site-PGA, in two specified directions, corresponding to a specified time span. Thus any appropriate percentile can be determined once the distribution i s obtained. Furthermore, the time span can be changed as a parameter of the Extreme Value model. 0.02S 0.020 0.015 L. a a. 0.010-0.005-0.000 EXTRDC T l * 100 Fig. 4.10 - Extreme value distribution for two specified time spans The time span of the data i s small compared to the design l i f e of a structure. This i s an inherent problem of 55 this analysis and. similar analyses. One way to overcome this problem is to increase the time base of the data set. In other words, use a data set which is based on a longer time period. This would imply including earlier, less reliable, data after verification and adjustment by seismic experts [31]. The extrapolation of these short time spans to longer ones obviously adds to the uncertainties of prediction'. Thus an uncertainty analysis w i l l provide the analyst with insight on the extent of deviation caused by this extrapolation. 4.7 S t a t i s t i c a l tests To examine the validity of the s t a t i s t i c a l models a number of tests were performed. Several s t a t i s t i c a l distributions were rejected visually on a graphical basis, clearly being unlikely candidates for representing the reality of the earthquake phenomenon. However, the va l i d i t y of the graphically accepted models needed to be established more rigorously. Therefore the following s t a t i s t i c a l tests were performed on the Frechet models of the PGA components. Kolmogorov tests [4] were performed on both the complete sample models and the models for censored data (Table 4.3). Even for the case of highest discrepancy 56 b e t w e e n m o d e l a n d d a t a , t h e F r e c h e t d i s t r i b u t i o n c o u l d n o t be r e j e c t e d a t s i g n i f i c a n c e l e v e l s b e t w e e n 0 . 0 1 a n d 0 . 2 0 . The maximum l i k e l i h o o d e s t i m a t i o n o f p a r a m e t e r s o f t h e F r e c h e t d i s t r i b u t i o n was c o n s i d e r a b l y a f f e c t e d b y t h e l a r g e number o f s m a l l m a g n i t u d e PGA d a t a . One a p p r o a c h t o p u t g r e a t e r e m p h a s i s on t h e more i m p o r t a n t l a r g e m a g n i t u d e d a t a was t o c e n s o r t h e d a t a t o t h e l e f t ( T a b l e 4.3). The NUMBER OF DATA POINTS USED 15 YEARS PERCENTILES 50Z 90Z KOLM0GOR0V SET ACCEPTANCE LEVEL 497 458 368 107 37 3 85.7 80.0 57.1 14.5 17.5 37.5 541.8 495.0 316.4 49.3 66.0 245.5 0.0217 0.0214 0.0152 0.00898 0.00838 0.00175 0.01 - 0.20 0.01 - 0.20 0.01 - 0.20 0.01 - 0.20 0.01 - 0.20 0.01 - 0.20 Table 4.3 - Eff e c t of censorship on v e r t i c a l PGA estimates. Maximum recorded v e r t i c a l PGA - 45.6. c a s e shown i s f o r v e r t i c a l PGA m o d e l w h i c h showed t h e l a r g e s t d i s a g r e e m e n t s w i t h t h e p r o c e s s e d d a t a . A g a i n , t h e p r o b l e m o f l o s s o f i n f o r m a t i o n was e n c o u n t e r e d . F u r t h e r -m ore, t h e d e t e r m i n a t i o n o f t h e p o i n t o f c e n s o r i n g i t s e l f was s u b j e c t i v e . I f t o o many d a t a p o i n t s a r e c e n s o r e d , t h e n t h e r e s u l t i s b i a s e d t o w a r d s t h e l a r g e r m a g n i t u d e s ; i f t o o few a r e c e n s o r e d t h e f i n a l r e s u l t may b e a n o v e r / u n d e r -57 estimation of reality. Another problem is the bias produced by the small number of large data during maximum likelihood estimation (Append. V). It was decided not to discard any information rather than risk any kind of potential deviation. In other words, a l l the "main" event data were used in the computation. The Chi-square test only applies to the uncensored sample cases since a large number of the points were eliminated'in the censoring process. One problem with this test was that due to the large number of the small magnitude PGA's most of the testing emphasis was on those data points rather than the few large PGA's that determine the t a i l of the distribution and hence i t s extreme value characteristics. In other words, the model f i t to the small magnitude data was tested rather than the f i t of the upper t a i l to the available data. With a chi-square value of 15.7, this model corresponding to the vertical PGA, could not be rejected at the 5% significance level, with 17 degrees of freedom. In addition to these s t a t i s t i c a l tests, the Extreme value model i t s e l f was also investigated further. The substitution of the Esteva attenuation law for the OASES attenuation law showed the high sensitivity of the final results to the choice of attenuation law. The results were affected by as much as an order of magnitude when the 58 attenuation law was changed (Table 4.4). Only results for normal PGA are shown. ATTENUATION LAW 15 YEARS PERCENTILES 50% 90% OASES 124.6 159.1 ESTEVA 684.3 1498.9 Table 4.4 - S e n s i t i v i t y to a t t e n u a t i o n laws. Normal PGA only. The model was tested for variation in location and orientation of the chosen site and was shown to be sensitive in both cases (Table 4.5). Any change in the input values produced a noticeable change in the estimated PGA values. LATITUDE LONGITUDE ORIENTATION MAXIMUM 15 YEARS PERCENTILES (degrees) (degrees) (degrees W of N) PGA 502 90Z * 15.0 87.7 18 124.6 45.4 159.1 14.8 87.7 18 121.1 52.8 189.8 15.0 87.5 18 111.3 42.6 147.6 15.0 17.7 28 98.5 37.3 126.5 Table 4.5 - Sensitivity to physical displacement of the site. Normal PGA only. * site location and orientation 59 The OASES attenuation law was then assumed with various parameter values (Table 4.6). I t was shown that the most s e n s i t i v e parameter of t h i s law was b3, while the le a s t s e n s i t i v e was b^. Further to t h i s , a s e n s i t i v i t y analysis was performed on these parameters. Figure 4.11 shows the s e n s i t i v i t y of the re s u l t s to the attenuation law parameters i n non-dimensional terms. t 5Z C H A N G E I N A T T E N U A T I O N P A R A M E T E R Z C H A N G E I N 50 Y E A R S P E R C E N T I L E 50Z 90Z »1 + 8 + 9 - 11 - 12 b 2 + 50 + 58 - 21 - 33 b 3 - 27 - 25 +103 +115 Table 4 . 6 - Sensitivity to attenuation law parameters 60 0-| 1 1 1 1 1 0.90 0.95 I 1.05 1.10 1.15 NON-D1MENS IONflL OASES PARAMETER F i g . 4.11 - Model s e n s i t i v i t y to attenuation para-meters 61 CHAPTER 5  SUMMARY OF RESULTS AND CONCLUSIONS A summary of the results obtained and the deduced conclusions are outlined in this chapter. Furthermore, some suggestions are made for the direction of future research work. 5.1 Summary of results A analysis of PGA magnitude at a given location was proposed and tested for the Honduran region. A Poisson model of occurrence was obtained with scale parameter 0p= 255.7. V n I t ) • ( - H n « p {-4->4r p p The PGA magnitude at the given location was calculated in two chosen orthogonal directions for a l l events: Vertical and horizontal. A Frechet distribution provided a better f i t to the given data than competing other models: f i i ( x ; o , X ) - - L _ ( J L ) e x p {_ ( J L . } } Neither Chi-square tests nor Kolmogorov tests rejected these distributions at the 5% significance level. The PGA magnitude and occurrence models were combined to yield an unconditional extreme value distribu-t i o n for PGA. The cumulative d i s t r i b u t i o n function (cdf) of t h i s unconditional extreme value model i s F E V ( x | T ) - e x p {- [ l - e x p { - ( - £ - f \ ] } Furthermore, the r e s u l t s showed that the proposed analysis was s e n s i t i v e to the choice of the attenuation law, and changes i n location and orient a t i o n of the chosen s i t e . A s e n s i t i v i t y analysis of the OASES attenuation parameters showed that the model was most s e n s i t i v e to parameter b 3 . A -5% change i n b 3 resulted i n an increase of over 100% i n the maximum PGA value for 50 years. The model was le a s t s e n s i t i v e to the parameter b]_. A change of +5% i n b^' caused an increase of 8% i n the estimated maximum PGA value for 50 years. 5.2 Conclusions 1- The proposed analysis produces s t a t i s t i c a l l y acceptable extreme value PGA d i s t r i b u t i o n s . 2- Since the Frechet Extreme Value model proved to represent these given data more c l o s e l y than other models, i n p a r t i c u l a r the Gumbel model,a tentative conclusion of t h i s t h e s i s i s that the Frechet model should be seriously considered when s t a t i s t i c a l l y analyzing data from extreme value phenomena such as earthquakes. 63 3- The a p p l i c a t i o n o f t h e F r e c h e t m o d e l t o E x t r e m e V a l u e d a t a i s r e m a r k a b l y s i m p l e w i t h t h e a i d o f t h e u n c o n d i t i o n a l m a g n i t u d e m o d e l d e r i v e d i n t h i s t h e s i s . 5.3 R e c o m m e n d a t i o n s f o r f u t u r e w o r k I n o r d e r t o s t r e n g t h e n t h e c o n c l u s i o n s o f t h i s t h e s i s , i t w o u l d be d e s i r a b l e t o t e s t t h e a n a l y s i s b y a p p l y i n g i t t o d a t a s e t s f r o m v a r i o u s s o u r c e s . 64 BIBLIOGRAPHY 1. 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" S t a t i s t i c a l analysis of vertical ground motion characteristics", Proc. of the 6*-n World Conference on earthquake engineering, New Delhi, 1977, Vol. I, pp. 954 34. Whitman R. V. et a l . "Seismic design decision analysis", J. of Structural Division ASCE, May 1975 68 APPENDIX I EXTREME VALUE MODEL DERIVATION Frechet model of magnitude: Extreme value of magnitude given that n events have occurred: : I I ( x | n ) = f n f T T __ (x; a n l A , A | n) , „ 1/A A+l n 1/A A ^ ( J L 2 _ _ ) exp { - } x on . A+l A f l I ( x | n ) = - S j L . ( J L . ) exp { - n (-£-) } Poisson model of occurrence f o r a s p e c i f i e d time T: P (n|T) = ) exp { - <-£-_) } J L _ where n - 0, 1, 2, P P P The u n c o n d i t i o n a l extreme value model i n a s p e c i f i e d time T: f E V ( x | T ) = I f I X ( x | n ) P„(n|T) n=o I _!LS_ e,P ( -„ (^-) X ) • C - i - ) " exp I - (-|_)) - L . P P n=o 4. <^ _/+1 eltp t . (4_„ I ^ [<-£_> e x p < - <-2->> )]" p p K n=o v 69 - 4 - HHX + 1 e x? {- <4->} e x p t<4->exp {- HHX}> LEV V 1 ' O v x ' — r x O ' — r l N 0 P P f E V ( x | T ) - (4)X exp | - (4)1 - <-§-> [1 - e*P {- (4)X } J} P P I t s cummulative density f u n c t i o n i s X F E V ( x | T ) = exp {- (-|-) + (4~) **P <- } J P P 70 APPENDIX I I MAXIMUM LIKELIHOOD ESTIMATION FOR FRECHET PARAMETERS OF A CENSORED SAMPLE L i k e l i h o o d f u n c t i o n : •a r n L ( x ; 8 ) = [ / f ( x ; 6 ) dx ] II f ( x ; 6) i=r+l where f ( x ; 6 ) i s censored to the l e f t at a and r = number of events to the l e f t of a For the Frechet d i s t r i b u t i o n r n L ( x ; 6 ) = [? (a;o,X)] II f (x ;o,X) i=r+l L ( x ; 6 ) = [exp { - (-i-)V X n " r o X< n" r> I xT*"1 exp {- ( J L ) * } i=r+l 1 L o g - L i k e l i h o o d f u n c t i o n i s n Ln L(x ; 9 ) = - r o X a ~ X + (n - r ) Ln X + A ( n - r ) Ln o - (X + 1) I Ln x ± i=r+l n X r -X i=r+l For ML estimates — | Q - (Ln L ( x ; 6 ) ) = 0 For a n X -X , , . X r -X - r o a + (n - r) - o I i=r+l n X , 1 . -X . v -X - _ 1 Let T = o i=r + l Hence o = T1^ For X - r a" a " Ln o + r o" a " Ln a + X -X . _ . _X -X , (n - r ) , , N T \ r - r — — + (n - r ) Ln o - I Ln x. 1 i=r+l n n X . v -X . X r -X - o Ln o I x~ + o I x ~ Ln x± = 0 i = r + l . i=r+l £ Ln x i = / ( r a ^ Ln a + I Ln x i } + ^ n ^ r ^ + (n - r ) Ln a i=r+l i=r+l n - a Ln a {r a" + I x~ } i=r+l But T -Hence n n ,-1 — + {r a~* Ln a + I x ^ Ln x j (r a ^ + I x ^ } = J Ln x i i=r+l i=r+l i=r+l 72 A P P E N D I X I I I M A I N COMPUTER PROGRAM THIS PROGRAM INITIALLY CONVERTS ALL THE DATA TO THE DESIRED DATA FORM. IT CALCULATES THE INTER-ARRIVAL TIMES, AND ATTENUATES ALL EVENTS TO THE SITE, IN THE REQUIRED DIRECTIONAL COMPONENTS. IT THEN EVALUATES THE SCALE PARAMETER OF THE POISSON MODEL OF OCCURRENCES, THEN EVALUATES THE SHAPE AND SCALE PARAMETERS OF THE FRECHET MODEL IN EACH DIRECTION. IF REQUIREb, IT PERFORMS STATiSTiCAL TESTS SUCH AS THE KOLMOGOROV TESTS AND THE CHI-SQUARE TESTS. IT HAS THE ABILITY TO DISCARD EVENTS OF LOWER THAN ANY DESIRED MAGNITUDE, OR DIRECTIONAL PGA. IT ALSO SELECTS MAIN EVENTS FROM THE GIVEN DATA ON ANY DESIRED BASIS. FINALLY THE PROGRAM COMPUTES THE ESTIMATED 50 AND 90 PERCENT-ILE VALUES OF MAXIMUM PGA FOR A LIFE OF 15 AND 50 YEARS. C PROGRAM A L L PARAMETER E S T I M A T I O N BY MAX L I K E L I H O O D C C INTEGER DAY(IOOO),. MONTH( 1 0 0 0 ) , YEAR( 1 0 0 0 ) , HOUR( 1 0 0 0 ) , D( 1 0 0 0 ) INTEGER MINUTE( 1 0 06 j , C( 1 6 6 0), F R E Q ( 1 6 6 ) , RUB, I , ESTEVA, KOLMO INTEGER TOT, RUB, DOFN, DOFV, C H I T E S INTEGER N, J , K, INPUT, OUTPUT, AFTER, CENSOR, MAIN, N I , ANSWER REAL L A T ( 1 0 0 0 ) , LONG( 1 06 6 ) , DEPTH( 1 0 0 0 ) , MAG( 1 0 0 0 ) , H Y P D I S ( 9 9 9 ) REAL A V E R ( 1 0 0 0 ) , A N O R M ( 1 0 0 0 ) , D I F ( l O O O ) , A V A R ( 1 0 0 0 ) , T I M E ( 1 0 0 0 ) REAL R1 ( 1 0 0 0 ) , E ( 1 0 0 0 , 1 0 ) , EP.I DI S (.1 0.00.)., AH.O.RZ..(. 1.0.00.L ACC(iOQO.).. REAL A L F A ( 1 0 0 0 ) , B E T A ( 1 0 0 0 ) REAL MODAY, DAYHR, MINHR, R U B B I S , TOTAL, AVARM, ORIENT, T I M , B1 REAL LONGS, LSTCOS, E P S I L O , L O L I M , L A L I M , T I L I M , D1, D2, D3, T I REAL MU, S I G , TOP, BOT, LAM, TRY2, C 3 , T 5 0 , L A T S , LARGE, LO, L I REAL LAMW, SIGW, S I P , LAMN, S I G N , LAMV, S I G V , B 2 , B 3 , C 1 , C 2 , F REAL AMINV, AMINN, GNMAX, GVMAX, TRY 1, T 5 0 , GNEE5, GNEE9, GNTY5 REAL GNTY9, GVEE5, G V E E 9 , GVTY5, GVTY9, S T E P , R, P I , CMF, SN, S REAL S T E P , P R E S , C H I , C H I N , C H I V , L N , T O P 1 , BOT1, TRY, LOG, SUM REAL SUM1, SUM2, SUM3,X, X T , RUN C C c C THE NUMBER OF USEFUL DATA NI= 5 8 9 C MIN A C C E P T A B L E PGA V A L U E AVARM= 6 . 6 6 6 l C L I M I T S ON GEOGRAPHIC P O S I T I O N AND T I M E OF EARTHQUAKES TO C DETERMINE WHAT I S THE M A I N SHOCK L A L I M = 2 . 6 LOLIM= 4.0 T I L I M = 2 0 . 0 ... C THE S I T E L A T I T U D E AND LONGITUDE AND O R I E N T A T I O N ARE LATS =15.0 LONGS=87.7 6 R I E N T = 1 8 . 6 C C ACCEPTABLE ACCURACY ...FOR ..PARAMETER...E.ST ACCUR=6.601 c C^  S T ARTING V A L U E OF LAM-STEP :  LAMSTA= -0.9999 C OTHER CONSTANTS c INPUT*3 OUTPUT*7 NO. OF DATA TO BE READ IS N c TOTAL*0.0 LARGE=9999999999.0 DATA FOR INTER=ARRIVAL TIME DETERMINATION M0DAY=31.0 DAYHR=24.0 MI NHR=1 .0/60.0 . .. C CONSTANTS FOR TRUNCATION ARE AMINV=0.0 AMINN=0.0 C C CONSTANTS FOR EXTREME VALUE TIME ARE HOURS IN 15 YYEARS T15=133320.0 C C HOURS IN 50 YEARS T50=438288.0 • THE CONSTANTS ARE: C PI=3.1415926 THE EQUATORIAL RADIUS OF THE EARTH R =6377.3 ORIENT=ORIENT*PI/18 0.0 WRITE(OUTPUT,910) WRITE(OUTPUT,911) C WRITE(OUTPUT,912) WRITE(0UTPUT,913) NOW READ THE DATA! 1 1 DO 10 1=1,NI READdNPUT, 1 00)DAY(I ) , MONTH ( I ) , YEAR ( I ) , HOUR ( I ) ,MINUTE ( I ) , & L A T ( I ) , LONG{ I ) jDEPTH(I), MAG(I) 10 I F ( MAG(I) .LE. 0.0) GOTO 11 MAG(I) = MAG(I)/100.0 CONTINUE c c c WORK OUT THE INTER-ARRIVAL TIME VALUES FROM THE DATA DO 20 1=1,NI TIM= (FLOAT( MONTH(I) -1 )) * MODAY IF ( MONTH(I) .GT. 2 ) TIM=TIM-3.0 I F ( MONTH(I) .GT. 4 ) TIM=TIM-1.0 IF ( MONTH(I) .GT. 6 ) TIM=TIM-1.0 I F ( MONTH(I) .GT. 9 ) TIM=TIM-1.0 IF ( MONTH(I) .GT. 11) TIM=TIM-1.0 RUBBIS = (FLOAT(YEAR(I)))/4.0 IF (RUBBIS .NE. (AINT(RUBBIS) ) ) GOTO 50 1 C 50 I F ( MONTH(I) .GT. 2 ) TIM=TIM+1.0 FOR PREVIOUS YEARS RUBBIS = FLOAT(YEAR(I) -64) C TIM = TIM + RUBBIS * 365.0 FOR LEAP YEARS RUBBIS= AINT((RUBBIS +3.0)/4.0) c TIM = TIM +RUBBIS TO ALLOW FOR THE LAST THREE MONTHS OF 1963 TIM = TIM + 80.0 20 TIM= (TIM + (FLOAT (DAY (I j - 1 j ) j * DAYHR + FLOAT (HOUR(D) TIME(I) = TIM + (FLOAT(MINUTE(I)))* MINHR CONTINUE 74 c c c DECIDE UPON THE LOCATION & ORIENTATION OF DAM WRITE(6,997) READ(5,995) ANSWER IF (ANSWER .NE. 1) GOTO 1 19 WRITE(6,996) READ(5,994) LATS, LONGS, ORIENT ORIENT* 0RIENT*PI/180.0 1 c CONTINUE C C WORK OUT THE DISTANCES AND IMPLEMENT THE ATTENUATION LAW c TO DETERMINE THE SITE GROUND ACCELERATIONS. V , C ORIGINAL OASES ATTENUATION CONSTATNTS ARE B1=190.0 B2=0.823 B3=-1.561 EPSILO=1.0 D1=285.0 D2=0.589 C D3=-1.050 ORIGINAL ESTEVA ATTENUATION PARAMETERS ARE C1=200.0 C C2=0.8 C3=-1.0 c DETERMINE THE PARAMETERS L 22 WRITE(6,993) READ (5,995) ESTEVA WRITE(6,992) READ(5,995) ANSWER IF (ANSWER .NE. 1 ) GOTO 12 IF(ESTEVA .EQ. 1) GOTO 6 WRITE(6,991) READ (5,990) D1, D2, D3, EPSILO WRITE(6,989) READ (5.988) B1. B2.B3 6 GOTO 12 WRITE(6,977) READ (5,988) C1, C2., C3 12 C C CONTINUE C C NOW WORK OUT SOME GEOMETRIC PARAMETERS LATCOS = COS( LATS*PI /180.0) C N=NI DO 51 1=1,N BETA(I) = LONG(I) -LONGS RUBBIS = (LAT(I) - LATS)**2 EPIDIS(I)=SQRT(((COS(LAT(l)*PI/180.0))*BETA(I))** 2 + RUBBIS) EPIDISd)=EPIDIS(I)+ (SQRT( (LATCOS*BETA(I))**2 EPIDISd )=EPIDIS(I) * PI* R/360.0 HYPDIS(I)=SQRT( (EPIDIS(I))**2 + (DEPTH(I))**2 +RUBBIS)) ) C C NOW FIND SOME ANGLES ALFA(I) = ACOS( EPIDISd ) / HYPDIS(D) 75 B E T A ( I ) = B E T A ( I ) * P I * R * ( C O S ( L A T ( I ) * P I / 1 8 0 . 0 ) + L A T C O S ) B E T A ( I ) = A S I N ( B E T A ( I ) / ( 3 6 0 . 0 * E P I D I S ( I ) ) ) I F ( ( L A T ( I ) - L A T S ) . G E . 0.0) B E T A ( I ) = B E T A ( I ) - O R I E N T I F ( ( L A T ( I ) - L A T S ) . L T . O . 6 ) B E T A ( I j = P I - B E T A ( I ) + O R I E N T C C A T T E N U A T I O N L A W S C E S T E V A I F ( E S T E V A . N E . 1) G O T O 13 R 1 ( I ) = S Q R T ( 400.0 + ( H Y P D I S ( I ) ) * * 2 ) A C C ( I ) = C l * ( E X P ( C 2 * M A G ( I j j ) * ( R 1 ( I ) * * C 3 j G O T O 61 C O A S E S 13 I F ( D E P T H ( I j . L T . 19 . 6) G O T O 60 R 1 ( I ) = ( H Y P D I S ( I ) + 0 . 8 6 4 * E X P ( 0 . 4 6 3 * M A G ( I ) ) )**B3 A C C ( I ) = B1* ( E X P . ( B 2 * M A G ( I ) ).)* R 1 ( I . ) . * E P S I L O G O T O 61 60 R 1 ( I ) = ( H Y P D I S ( I ) + 0 . 8 6 4 * E X P ( 0 . 4 6 3 * M A G ( D ) )**D3 A C C ( l ) = D1* ( E X P ( D 2 * M A G ( I ) ) ) * R 1 ( I ) * E P S I L O C C N O W D E T E R M I N E T H E C O M P O N E N T S O F A C C E L E R A T I O N C 61 A V E R ( I ) = A C C ( i ) * ( S I N ( A L F A ( I ) ) ) A H O R Z ( l ) = A C C ( I ) * ( C O S ( A L F A ( I ) ) ) A N O R M ( I ) = A B S . ( A H O R Z ( I ) .. * ( C Q S . . ( B E T A ( I ) ) ) . ) 51 C O N T I N U E C D O 7 I = 1 , N E ( I , 1 j = T I M E ( I ) E ( I , 2 ) = L A T ( I ) E ( I j 3 ) = L O N G ( I ) E ( i , 4 ) = M A G ( l j E ( I , 5 ) = A V E R ( I ) E ( I J 6 ) = A N O R M ( I ) 7 C O N T I N U E C C C H A N G E T H E S H O C K C L U S T E R L I M I T S ? W R I T E ( 6 , 9 8 7 ) R E A D (5,995) A N S W E R I F ( A N S W E R . N E . 1) G O T O 14 3 W R I T E ( 6 , 9 9 8 ) R E A D ( 5 , 9 9 9 ) L A L I M , L O L I M , T I L I M I F ( L A L I M . L T . 0.0) G O T O 4 C N = N I D O 8 1=1,N T I M E ( i ) = E ( i , i ) L A T ( I ) = E ( I , 2 ) L O N G ( I ) =£(1,3) M A G ( I ) = E ( I , 4 ) A V E R ( I ) = E ( I , 5 ) A N O R M ( I ) = E ( I , 6 ) 8 C O N T I N U E C C E L I M I N A T E F O R E & A F T E R S H O C K S C C C D E T E R M I N E T H E L I M I T S F O R A F T E R S H O C K S C 14 W R I T E ( 6 , 9 8 6 ) R E A D (5.995) M A I N  76 AFTER=0 J= 1 D ( i ) = J C C C N=NI DO 89 I=2,N TI = ABS(TIME(Ij - T i M E ( b ( j ) j j IF ( TI .GE. TILIM ) GOTO 80 LA= ABS(LAT(I) - L A T ( D ( J ) ) ) IF ( LA .GE. LALIM j GOTO 80 LO= ABS(LONG(l)-LONG(D(J))) IF .(. LO .GE. LOLIM ) GOTO..80 AFTER = AFTER +1 IF (MAIN .NE. 0) GOTO 16 IF ( MAG(I) .LE. MAG(D(J))) GOTO .89 GOTO 18 16 IF (MAIN .NE. 1) GOTO 17 IF (AVER(I) .LE. AVER(D(J)) ) GOTO 89 GOTO 18 17 I F (ANORM(I) .LE. ANORM(D(J))) GOTO 89 18 D(J) = I GOTO 89 80 J=J+1 P(J). =..I 89 CONTINUE C C CONFIRM MINIMUM ACCEPTABLE PGA WRITE(6,984) READ(5,995) ANSWER I F ( ANSWER .NE. 1) GOTO 68 WRITE(6,9831 j READ (5,982) AMINV, AMINN WRITE(6,982) AMINV, AMINN 68 CONTINUE CENSOR=0 C C CENSOR TO THE LEFT C N= N - AFTER k=i DO 64 1=1,N J r D l l ) IF ((AVER(J) .GT. AMINV) .AND. (ANORM(J) .GT. AMINN))GOTO 66 CENSOR=CENSOR+1 GOTO 64 66 C ( k j = J K=K+1 64 CONTINUE C C C NOW THE DATA IS REDUCED TO MAIN SHOCKS ONLY C DIF(1)=TIME(1) IF .(.DIF(I) .GT. 1000.0) DIF ( l)=500.0 DO 82 i=2,N D I F ( I ) = T I M E ( D ( I ) ) - TIME(D(I-1)) 82 CONTINUE  * 77 c c THE POISSON PARAMETER SIP= (TIME(D(N)) - TI M E ( D ( 1 ) ) ) / ( F L 0 A T ( N ) ) c c c STEP SIZE FOR WEIBULL ITTERATION PROCESSES c STEP=0.10 STARTING VALUE OF LAM-STEP LAM=-0.09999 c TOTAL=0.0 DO 21 1=1,N 21 IF ( D I F ( I ) .LT.0.0001) GOTO 21 TOTAL=TOTAL + A L O G ( D I F ( l ) ) CONTINUE OOO; r> TOTAL= TOTAL ./ FLOAT(N) C c NOW DO THE ITTERATION PROCESS c 5 TRY2=LARGE LAM=LAM +STEP TRY1=TRY2 BOT=0.0 TOP=0.0 DO 34 1=1,N . RUBBIS= D I F ( I ) * * L A M IF ( D I F ( I ) .LT. 0.0001) GOTO 34 34 TOP=TOP + ALOG(DIF(I)) *RUBBIS BOT=BOT + RUBBIS CONTINUE TRY2=-1.0/LAM +TOP/BOT - TOTAL IF (ABS(TRY2) .LT. ABS(TRY1) ) GOTO 5 IF (STEP .LT. ACCUR) GOTO 35 TRY2=LARGE LAM=LAM-STEP*1.7 STEP=STEP/10.0 C C GOTO 5 NOW EVALUATE OTHER PARAMETER C 35 RUBBIS=0.0 DO 40 1=1 (N * 40 RUBBIS=RUBBIS+(DIF(I)**LAM) CONTINUE RUBBIS=( RUBBIS/FLOAT(N)) SIG=RUBBIS**(1.0/LAM) LAMW=LAM SIGW=SIG C TOTAL=0.0 TRY2=LARGE C STEP=EVSTEP LAM= LAMSTA NOW READ THE DATA! C C SUM =0.0 78 DO 63 1= 1 , N AVAR(I) = ANORM(I) SUM = SUM + AVAR(I) 63 CONTINUE C LAM = 1.000 X = FLOAT ( CENSOR) IF (AMINN .GT. 0.0) LN= ALOG(AMINN) IF (AMINN .GT. 0.0 ) GOTO 15 LN = 6.0 15 SUM1 = 0.0 SUM2 =...0,0 SUM3 = 6.6 I F (AMINN .GT. 0.0) RUBBIS = X/(AMINN**LAM) IF (AMINN .GT. 0.0) GOTO 70 RUBBIS = 6.6 70 XT = RUBBIS * LN DO 90 I = 1 t N IF (AVAR(I) .LE. 6 .6) GOTO 90 LOG = ALOG(AVAR(I)) RUN = 1.0 /(AVAR(I)**LAM) SUM1 = SUM1 + RUN SUM2 = SUM2 + RUN * LOG SUM3 = SUM3 + RUN * LOG * LOG 90 CONTINUE TOPI = (XT + SUM2) / (RUBBIS + SUM1) BOT1 = (SUM3 + XT*LN) / (RUBBIS....+ SUM 1 ) TOP = TOP1 - SUM + 1.6/LAM BOT = BOT1 - TOP1**2 + 1.0/(LAM**2) TRY = TOP /BOT IF (ABS(TRYj .LT. ACCUR) GOTO 31 LAM = LAM+ TRY GOTO 15 C C C NOW EVALUATE OTHER PARAMETER C 31 RUBBIS=0.0 DO 41 1=1,N IF (AVAR(I) .LT. 6.661) GOTO 41 RUBBIS=RUBBIS+1.0/(AVAR(I)**LAM) 41 CONTINUE IF ( AMINN .LT. 6.6661) GOTO 53 RUBBIS=(RUBBIS + X /(AMINN**(LAM))) 53 RUBBIS=FLOAT(N) / RUBBIS SIG=RUBBiS**(1.6/LAM) LAMN=LAM SIGN=SIG . ... C C CHI SQUARE TEST C WRITE(6,973) READ (5,995) CHITES IF (CHITES .NE. 1) GOTO 55 STEP=6.65 PRES=STEP BOT=AMINN J=1 IF (BOT .EQ. 0.0) GOTO 37 PRES = STEP + (EXP(-(SIG/BOT)**LAM))  -79 c 37 FREQ(J)=0 TOP=SIG/((ALOG(1.0/PRES))**(1,0/LAM)) DO 38 1= 1 , N IF ((AVAR(I) .LT. BOT) .OR. (AVAR(I) .GE. FREQ(J)= FREQ(J) +1 TOP)) GOTO 38 38 CONTINUE PRES=PRES+STEP J=J+1 IF (PRES .GE. 0.99) GOTO 39 BOT=TOP GOTO 37 39 RUB=J-1 TOT= FREQ(1) DO 42 I=2,RUB 42 TOT =TOT+ FREQ(I) CONTINUE FREQ(J)=N-TOT C TOTAL =0.0 DO 43 1=1.RUB 43 CHI=(FLOAT(FREQ(I)))**2 /STEP TOTAL=TOTAL + CHI CONTINUE TOTAL=TOTAL+ ( ( ( F L O A T ( F R E Q ( J ) ) ) * * 2 ) /(1.0 CHIN=TOTAL/FLOAT(N) -FLOAT(N) DOFN=J-2 + STEP - PRES)) OOOj KOLMOGOROV TEST 55 WRITE(6,974) READ (5,995) KOLMO IF (KOLMO .NE. 1) GOTO 26 SN=0.0 DO 2 1=1,N RUB=I+1 IF (RUB .GT. N) GOTO 45 DO 23 J=RUB,N IF (AVAR(J) .GE. AVAR(I}) GOTO 23 RUBBIS =AVAR(Ij AVAR(I)=AVAR(J) AVAR(J)= RUBBIS 23 45 CONTINUE CMF=FLOAT(CENSOR+I)/FLOAT(CENSOR+N) F= EXP(-((SIG/AVAR(I))**LAM)) ? RUBBIS=ABS(F-CMF) IF (RUBBIS .GT. SN) SN=RUBBIS CONTINUE O O Oj NOW DO THE SAME FOR VERTICAL PGA C SUM = 0.0 DO 63 1= 1, N 63 AVAR(I) = AVER(I) SUM = SUM + AVAR(I) CONTINUE C LAM = 1.000 X = FLOAT(CENSOR) 80 IF (AMINN . G T . 0.0) LN= ALOG(AMINN) IF (AMINN . G T . 0.0) GOTO 15 LN = 0.0 15 SUM1 = 0 . 0 SUM2 = 0.0 SUM3 = 0.0 IF (AMINN . G T . 0.0) RUBBIS = X/(AMINN**LAM) IF (AMINN . G T . 0.0) GOTO 70 RUBBIS = 0 . 0 70 XT = RUBBIS * LN DO 90 I = 1 , N IF (AVAR(I) . L E . 0.0) GOTO 90 LOG = ALOG(AVAR(I)) RUN = 1.0 / (AVAR( I ) * *LAM) SUM1 = SUM1 + RUN 90 SUM2 = SUM2 + RUN * LOG SUM3 = SUM3 + RUN * LOG * LOG CONTINUE TOP1 = (XT + SUM2) / (RUBBIS + SUM1) BOT1 = (SUM3 * XT*LN) / (RUBBIS + SUM1) TOP = TOP1 - SUM + 1.0/LAM BOT • BOT1 - TOPI* *2 + 1 .0 / (LAM**2 ) TRY = TOP /BOT IF (ABS(TRY) . L T . ACCUR) GOTO 31 C LAM = LAM+ TRY GOTO 15 C NOW EVALUATE OTHER PARAMETER c 32 RUBBIS=0.0 DO 44 1=1,N IF (AVAR(I) . L T . 0.001) GOTO 44 RUBBIS=RUBBIS+1.0/(AVAR(I)**LAM) 44 CONTINUE IF (AMINV . L T . 0.0001) GOTO 52 RUBBIS= (RUBBIS + X / (AMINV** (LAM)) ) 52 RUBBIS=FLOAT(N) / RUBBIS SIG=RUBBIS**(1.0/LAM) LAMV=LAM SIGV=SIG c. c p CHI SQUARE TEST IF (CHITES . N E . 1) GOTO 56 STEP=0.05 PRES=STEP BOT=AMINV J=1 IF (BOT . E Q . 0.0) GOTO 36 PRES=STEP+ ( E X P ( - ( S I G / B O T ) * * L A M ) ) C 36 FREQ(J)=0 TOP=S IG/ ((ALOG ( 1 . 0/PRES) ) * * (1 . 0/LAM).) DO 48 1= 1, N IF ((AVAR(I) . L T . BOT) .OR. (AVAR(I) . G E . TOP)) GOTO 48 FRE.Q(J)= FREQ(J) +1 48 CONTINUE PRES=PRES+STEP J=J+1  8t • ' I F ( P R E S . G E . 0 . 9 9 ) G O T O 4 9 B O T = T O P G O T O 3 6 4 9 R U B = J - 1 T O T = F R E Q ( 1 ) D O 47 1 = 2 , R U B T O T = T6T + F R E Q ( I j 4 7 C O N T I N U E F R E Q ( J ) = N - T 0 T . C T O T A L = 0 . 0 D O 5 4 1 = 1 , R U B C H I = ( F L O A T ( F R E Q ( I j j j * * 2 / S T E P T O T A L = T O T A L + C H I 5 4 C O N T I N U E I T 6 T A L = T 6 T A L + ( ( ( F L O A T ( F R E Q ( J ) ) ) * * 2 j / ( i . 6 + S T E P - P R E S ) ) C H I V = T O T A L / F L O A T ( N ) - F L O A T ( N ) D O F V = J - 2 C C K O L M O G O R O V T E S T C 5 6 I F ( K O L M O . N E . 1 ) G O T O 2 7 S = 0 . 0 D O 9 1 = 1 , N R U B = I + 1 I F ( R U B . G T . N ) G O T O 4 6 D O 2 8 J = R U B , N I F ( A V A R ( J ) . G E . A V A R ( I ) j G O T O 2 8 R U B B I S = A V A R ( I ) A V A R ( I ) . = A V A R ( J ) A V A R ( J ) = R U B B I S 2 8 C O N T I N U E 4 6 C M F = F L O A T ( C E N S O R + I ) / F L O A T ( C E N S O R + N ) F = E X P ( - ( ( S I G / A V A R ( I j ) * * L A M j j R U B B I S = A B S ( F - C M F ) » I F ( R U B B I S . G T . S ) S = R U B B I S 9 C O N T I N U E C C P R I N T T H E R E S U L T S C R U B B I S = O R I E N T * 1 8 0 . 0 / P I 2 7 I F ( E S T E V A . E Q . 1 ) G O T O 9 9 W R I T E ( O U T P U T , 9 0 0 ) L A T S , L O N G S , R U B B I S W R I T E ( O U T P U T , 9 0 3 ) B 1 , B 2 , B 3 , E P S I L O , D 1 , D 2 , D 3 , L A L I M , L O L I M , T I L I M , & N , C E N S O R G O T O 9 8 9 9 W R I T E ( O U T P U T , 9 0 1 ) L A T S , L O N G S , R U B B I S W R I T E ( O U T P U T , 9 0 2 ) C 1 , C 2 , C 3 , L A L I M J L O L I M , T I L I M , N , C E N S O R . 9 8 W R I T E ( O U T P U T , 9 0 4 ) S I P , L A M W , S I G W , L A M N , S I G N , L A M V , S I G V I F ( M A I N . N E . 0 ) G O T O 9 2 W R I T E ( O U T P U T , 9 0 5 ) G O T O 9 4 9 2 I F ( M A I N . N E . 1) G O T O 9 3 W R I T E ( O U T P U T , 9 0 6 ) G O T O 9 4 9 3 I F ( M A I N . N E . 2 ) G O T O 9 4 W R I T E ( O U T P U T , 9 0 7 ) 9 4 C O N T I N U E C C I F N O C H I - S Q U A R E T E S T S E T V A L U E S T O Z E R O  82 c I F ( C H I T E S . E Q . 1 ) G O T O C H I N = 0 . 0 5 7 C H I V = 0 . 0 D O F N = 0 D O F V = 0 5 7 n C O N T I N U E c H O W A B O T E X T R E M E V A L U E P R E D I C T I O N ? c W R I T E ( 6 , 9 8 1 ) R E A D ( 5 , 9 9 5 ) A N S W E R I F ( A N S W E R . N E . 1 ) G O T O F = 0 . 5 G N T Y 5 = E X T R E ( F , S I P , T 5 0 , 3 0 L A M N , S I G N ) G N E E 5 = E X T R E ( F , S I P , T 1 5 , G V T Y 5 = E X T R E ( F , S I P , T 5 0 , G V E E 5 = E X T R E ( F , S I P , T 1 5 , L A M N , S I G N ) L A M V , S I G V ) L A M V , S I G V ) F = 0 . 9 G N T Y 9 = E X T R E ( F , S I P , T 5 0 , G N E E 9 = E X T R E ( F , S I P , T15, L A M N , S I G N ) L A M N , S I G N ) G V T Y 9 = E X T R E ( F , S I P , T 5 0 , G V E E 9 = E X T R E ( F , S I P R T 1 5 , W R I T E ( O U T P U T , 9 8 0 ) L A M V , S I G V ) L A M V , S I G V ) 3 0 W R I T E ( O U T P U T , 9 7 9 ) G N M A X , G N E E 5 , G N E E 9 , G N T Y 5 , G N T Y 9 , A M I N N I F ( K O L M O . N E . 1 ) G O T O 2 9 W R I T E ( O U T P U T , 9 7 5 ) S N 2 9 W R I T E ( O U T P U T , 9 0 7 ) I F ( C H I T E S . N E . 1 ) G O T O W R I T E ( O U T P U T . 9 7 2 ) C H I N , 5 8 D O F N . ... 5 8 3 3 I F ( A N S W E R . N E . 1 ) G O T O 3 3 W R I T E ( O U T P U T , 9 7 8 ) G V M A X , G V E E 5 , G V E E 9 , G V T Y 5 , G V T Y 9 , A M I N V I F ( K O L M O . N E . 1 ) G O T O 5 9 5 9 W R I T E ( O U T P U T , 9 7 5 ) S W R I T E ( O U T P U T , 9 0 6 ) I F ( C H I T E S . N E . 1 ) G O T O 6 7 W R I T E ( O U T P U T , 9 7 1 ) C H I V , D O F V ; L C R E P E A T T H E P R O C E D U R E F R O M A N Y S T E S ? c 6 7 W R I T E ( 6 , 9 8 5 ) R E A D ( 5 , 9 9 5 ) A N S W E R I F ( A N S W E R . N E . 1 ) G O T O W R I T E ( 6 , 9 9 7 ) R E A D ( 5 , 9 9 5 ) A N S W E R 4 I F ( A N S W E R . E Q . 1 ) G O T O W R I T E ( 6 , 9 7 6 ) R E A D ( 5 , 9 9 5 ) A N S W E R 1 9 I F ( A N S W E R . E Q . 1 ) G O T O W R I T E ( 6 , 9 8 7 ) R E A D ( 5 , 9 9 5 ) A N S W E R 2 2 4 I F ( A N S W E R . E Q . 1 ) G O T O G O T O 1 4 C O N T I N U E 3 C C 1 0 0 F O R M A T ( 5 ( I 2 , 1 X ) , 2 ( 2 X , F 6 . 3 ) , 2 ( 1 X , F 4 . 0 ) ) 9 9 9 9 9 8 8 8 8 F O R M A T ( 3 ( F 4 . 1 , 1 X ) ) F O R M A T ( ' I N P U T L A L I M , L O L I M , T I L I M ; F O R M A T I S F 4 . 1 ' ) F O R M A T ( 5 ( I 2 . 1 X ) . 2 ( 2 X . F 6 . 3 ) . 2 X . F 4 . 0 . 2 X . F 4 . 2 . 2 X . F 8 . 2 . 2 ( 2 X . F 8 . 4 ) ) 83 997 995 996 FORMAT('TO CHANGE DEFAULT/LAST LOCATION, ORIENTATION TYPE 1*) FORMAT(11) FORMAT('NEW SITE LAT, LONG, & ORIENTATION W of N 3(F7.3,1X)') 994 993 992 FORMAT(3(F7.3,1X)) FORMAT('CHOOSE ONE OF 0=OASES OR 1=ESTEVA') FORMAT('DO YOU WANT TO CHANGE THE CONSTANTS? YES=1') 991 990 989 FORMAT('FOR OASES, DEPTH<19Km INPT D1.. EPSILON 3(F7.3,X)F3.1') FORMAT(3(F7.3 ,1X),F3.l) FORMAT('OASES DEPTH>19Km; INPUT B1...B3 3(F7.3,1X)') 988 987 986 FORMAT(3(F7.3,1X)) FORMAT('CHANGE THE DEFAULT/LAST LALIM ETC? YES=1') FORMAT('MAIN EVENT IS ACCORDING TO MAG; TO CHANGE 1=AVER,', 985 900 & ' 2=ANORM') FORMAT(' DO YOU WANTTO TRY MORE?') FORMAT (///,'OASES WITH LAT' tF8 . 3 , ' LONG',F8.3,' & ORIENT', 901 & & F8.3, ' W of N') FORMAT(///,'ESTEVA WITH LAT',F8.3,' LONG',F8.3,' & ORIENT', F8.3,1 W of N') 902 903 S. FORMAT('MAIN EVENT' ,2X,3(F7.3,1X),30X,3(F4.1,1X),2(I 3,1X)) FORMAT('MAIN EVENT',2X,3(F7.3,1X),F5.3,3(1X,F7.3), 3(1X,F4.1),2(1X,I3)) 904 905 906 FORMAT('CHOSEN BY ',F7.3,1X,3(1X,F5.3,2X,F7.3)) FORMAT('MAGNITUDE',/) FORMAT('VERTICAL PGA',/) 907 910 & FORMAT('NORMAL PGA',/) FORMAT('COMMENTS',18X,'OASES/ESTEVA PARAMETERS',14X, 'MAIN SHOCK LIMITS') 91 1 912 & FORMAT(14X,'B1',6X,'B2',6X,'B3',3X,'EPSILON',2X,'DT,6X,'D2', 6X,*D3',4X,'LAT LONGTIME N CEN') FORMAT(/,12X,'POISSON WEIBULL TIME' ,6X,'NORMAL ', 8X,'VERTICAL') 913 984 983 FORMAT(13X,'SIGMA',3(4X,'LAM',5X,'SIG'),/) FORMAT('ARE THE DEFALT PGA MIN ACCEPTABLE? NO=1') FORMAT('INPUT AMINV, AMINN 2(F6.3,1X)') 982 981 980 FORMAT(2(F6.3,1X)) FORMAT('HOW ABOT EXTREME VALES? YES=1') FORMAT('DIRECTION',6X,'MAX PGA 15yr .5 15yr .9 50yr .5*, 979 978 & 50yr .9 MIN PGA') FORMAT('NORMAL',6X,5(3X,F7.1),4X,F6.3) FORMAT('VERTICAL',4X,5(3X,F7.1),4X,F6.3) 977 976 975 FORMAT('ESTEVA INPUT B1, B2, B3 3(F7.3,1X)') FORMAT('TO CHANGE ATTENUATION LAW FROM LAST ONE,TYPE 1') FORMAT('KOLMOGOROV TEST, LARGEST DIFFERENCE=',3X,F7.5,' FOR') 974 973 972 FORMAT('DO YOU WANT A KOLMOGOROV TEST? YES=1') FORMAT ('DO YOU WANT A CHI - SQUARE TEST? YES=D FORMAT('NORMAL PGA CHI-SQUARE IS'tF9.3,3X,'WITH',I 5,' DOF',/) 971 C FORMAT('VERTICAL PGA CHI-SQUARE ',F9.3,3X,'WITH',I 5 D O F ' , / ) END C c REAL FUNCTION EXTRE(F, SIP, T, LAM, SIG) c c REAL F, SIP, T, LAM, SIG, RB RB=1.0/(1.0+((SIP*ALOG(F))/T)) EXTRE=SIG*((ALOG(RB))**(-1.0/LAM)) RETURN END 84 APPENDIX IV DATA [18] D M y H M L L D M A 0 E 0 I A 0 E A y N A U N T N P G T R R U I G T N H T T I H I U T T D U U E D D E E 13 10 63 10 21 12.500 87.100 30. 560. 19 10 63 21 22 15.200 89.200 33. 296. 28 10 63 2 25 14.800 90.800 33. 296. 1 1 1 63 1 1 34 13.600 90.100 128. 379. 6 1 1 63 21 49 12.200 88.000 37. 329. 10 1 1 63 7 18 12.900 88.500 45. 345. 23 1 1 63 2 59 12.000 87.200 30. 650. 16 12 63 6 23 12.200 88.400 34. 362. 16 12 63 10 49 13.900 90.900 59. 362. 20 12 63 22 28 13.200 88.000 64. 362. 21 12 63 5 5 13.400 87.200 30. 560. 27 12 63 3 57 14.500 90.700 33. 296. 31 12 63 14 22 12.400 87.900 77. 362. 9 1 64 18 36 14.900 87.900 33. 428. 18 1 64 21 57 16.100 89.300 33. 279. 21 1 64 15 43 16.600 86.400 33. 279. 25 1 64 17 35 17.000 86.900 33. 296. 25 1 64 22 46 17.000 86.600 33. 296. 27 1 64 1 1 21 17.000 87.000 33. 279. 3 2 64 1 46 13.000 88.000 61 . 329. 9 2 64 10 49 12.300 90.600 33. 296. 18 2 64 16 12 13.000 87.500 30. 560. 21 2 64 7 24 12.800 87.900 62. 296. 24 2 64 2 29 13.000 87.200 83. 296. 2 3 64 16 9 12.500 88.000 63. 379. 2 3 64 18 13 13.500 90.500 73. 246. 5 3 64 14 30 12.100 88.100 53. 362. 6 3 64 2 55 12.400 87.600 100. 296. 1 1 3 64 8 39 12.500 87.100 30. 560. 13 3 64 11 54 12.900 90.400 128. 462. I 13 3 64 18 51 13.700 90.700 64. 379. 13 3 64 21 8 14.500 90.900 33. 445. 16 3 64 6 6 13.500 88.300 92. 296. 20 3 64 6 35 12.900 89.900 125. 345. 2 4 64 3 48 12.500 87.800 32. 345. 4 4 64 6 43 12.500 87.700 41 . 362. 9 4 64 4 7 13.600 90.200 53. 263. 9 4 64 4 15 13.500 89.900 89. 478. 16 4 64 1 1 20 14.000 90.000 209. 263. 24 4 64 14 40 13.300 88.800 158. 580. 29 4 64 8 8 12.100 88.400 33. 296. 29 4 64 20 24 12.300 87.400 30. 530. 30 4 64 7 37 16.600 85.100 33. 279. 15 5 64 2 13 12.800 87.500 51 . 263. 15 5 64 10 17 12.400 89.000 53. 379. 85 10 6 64 16 25 12.000 87.900 55. 345. 14 6 64 5 42 12.000 89.200 33. 329. 15 6 64 9 41 12.600 88.300 56. 329. 15 6 64 11 13 12.600 88.000 14. 329. 15 6 64 11 27 12.100 89.300 33. 246. 17 6 64 9 1 1 12.000 87.200 33. 329. 19 6 64 17 31 12.400 88.100 53. 312. 27 6 64 8 50 16.500 85.700 28. 410. 1 7 64 0 42 12.000 86.700 108. 362. 3 7 64 2 33 13.000 90.900 8. 296. 3 7 64 1 1 3 15.000 88.200 60. 279. 6 7 64 9 31 13.500 90.400 33. 312. 9 7 64 10 48 13.000 90.200 56. 246. 20 7 64 8 27 13.600 87.900 87. 279. 21 7 64 7 1 13.100 88.400 68. 428. 26 7 64 3 58 12.700 89.600 65. 445. 31 7 64 12 44 14.400 88.900 68. 279. 8 8 64 15 45 12.500 87.800 63. 611 . 10 8 64 14 0 12.400 88.300 71 . 312. 15 8 64 14 43 12.000 88.000 39. 312. 16 8 64 7 30 13.300 87.600 60. 296. 16 8 64 12 34 12.000 88.600 33. 362. 17 8 64 19 5 12.400 86.800 1 16. 395. 25 8 64 5 3 12.100 87.700 72. 312. 30 8 64 15 20 12.600 88.500 54. 412. 3 9 64 1 55 12.200 87.000 108. 312. 12 9 64 21 19 12.500 89.100 55. 345. 13 9 64 7 27 12.400 89.200 51 . 362. 14 9 64 15 45 15.500 90.800 38. 445. 15 9 64 5 9 15.100 91.000 33. 0. 15 9 64 7 51 15.400 90.600 33. 296. 15 9 64 21 56 15.300 90.800 74. 379. 22 9 64 13 27 13.300 89.700 78. 445. 25 9 64 1 54 13.300 89.400 94. 312. 28 9 64 3 20 13.300 90.100 72. 279. 28 9 64 5 25 13.600 90.000 33. 362. i 1 10 64 18 19 13.600 90.100 62. 379. 18 10 64 10 7 12.100 89.600 33. 345. 18 10 64 10 45 12.200 89.300 33. 428. 29 10 64 12 21 13.200 88.500 33. 312. 7 11 64 1 36 12.000 88.000 33. 379. 7 11 64 20 7 13.100 89.900 82. 362. 7 11 64 20 21 13.000 90.100 73. 329. 10 11 64 19 13 12.500 90.300 33. 345. 13 11 64 8 2 13.000 88.600 86. 462. 17 11 64 16 17 13.200 89.600 54. 395. 27 11 64 10 55 13.400 88.700 33. 329. 27 11 64 23 50 15.800 89.400 33. 312. 3 12 64 2 0 .13.800 89.000 41 . 345. 3 12 64 14 51 12.300 88.500 33. 329. 9 12 64 14 22 13.200 87.200 200. 296. 14 12 64 21 29 13.900 90.500 33. 379. 5 1 65 6 24 13.300 90.800 33. 362. 1 1 1 65 4 10 14.000 89.500 144. 478. 12 1 65 3 19 12.300 88.900 39. 362. 21 1 65 20 43 12.300 86.700 138. 379. 1 2 65 15 43 13.100 89.600 62. 345. 2 2 65 2 46 14.300 90.400 94. 412. 2 2 65 3 37 14.000 91.000 33. 0. 3 2 65 18 35 16.100 90.300 33. 329. 86. 6 2 65 8 14 13.800 89.300 16. 362. 10 2 65 3 14 14.100 91 .000 33. 0. 10 2 65 19 30 13.000 89.500 55. 312. 28 2 65 6 56 12.800 89.700 97. 312. 22 3 65 7 53 13.900 90.900 60. 395. 23 3 65 7 58 12.300 88.200 33. 329. 30 3 65 14 1 13.800 90.800 33. 362. 2 4 65 1 13 13.800 90.800 33. 296. 24 4 65 13 25 12.700 82.000 33. 530. 3 5 65 10 1 13.650 89.150 23. 650. 15 5 65 5 6 12.400 89.500 33. 312. 26 5 65 4 58 13.700 90.600 39. 511. 6 6 65 3 37 12.700 90.100 30. 312. 1 1 1 6 65 22 33 13.200 91.000 33. 0. 12 6 65 16 35 13.700 91.000 33. 0 19 6 65 1 39 13.100 90.400 32. 412. 27 6 65 13 8 12.600 87.900 33. 329. 1 7 65 16 57 12.100 87.500 176. 362. 17 7 65 13 59 12.200 87.900 56. 379. 26 7 65 13 6 14.900 90.900 133. 279. 28 7 65 13 20 14.500 90.300 220. 362. 30 7 65 6 33 12.000 88.500 33. 345. 5 8 65 19 5 14.800 91 .000 59. 0. 29 8 65 1 45 14.000 90.600 85. 495. 29 8 65 2 32 14.000 90.900 28. 379. 27 9 65 17 37 12.900 88.400 33. 345. 2 10 65 23 3 13.700 90.700 69. 296. 14 10 65 8 30 13.200 90.800 106. 263. 14 10 65 10 17 12.500 88.300 96. 279. 20 10 65 23 54 12.500 87.300 72. 545. 20 10 65 23 54 12.700 87.000 91 . 578. 21 11 65 2 19 12.100 88.900 33. 362. 27 11 65 19 50 17.000 88.100 34. 312. 6 12 65 4 39 12.500 87.300 50. 395. 6 12 65 13 19 12.400 87.400 44. 296. PO 12 65 17 59 12.400 87.300 33. 345. 1 1 12 65 15 30 14.200 90.800 33. 345. 11 12 65 19 20 14.100 90.900 33. 329. 1 1 12 65 23 52 13.400 89.300 57. 329. 14 12 65 14 21 14.500 89.700 273. 462. 16 12 65 5 23 12.600 88.800 152. 379. 24 12 65 22 57 12.600 89.000 118. 395. 25 12 65 4 13 13.900 89.400 33. 362. 24 1 66 3 46 12.300 88.400 46. 395. 9 2 66 6 22 12.700 87.800 98. 379. 13 3 66 21 46 14.200 88.400 36. 345. 18 3 66 21 55 13.300 90.900 40. 362. 22 3 66 3 41 13.100 90.000 20. 445. 23 3 66 5 1 1 16.800 85.900 33. 462. 4 4 66 19 50 13.800 89.700 102. 495. 12 4 66 3 34 12.600 87.600 104. 296. 12 4 66 17 30 12.600 88.000 33. 379. 16 4 66 13 21 12.300 88.400 33. 379. 21 4 66 6 9 14.000 90.800 66. 312. 24 4 66 6 3 12.900 89.200 47. 362. 24 4 66 15 41 13.700 88.300 33. 296. 4 5 66 18 13 12.500 87.700 68. 495. 1 1 5 66 5 57 12.300 86.800 33. 312. 4 6 66 14 12 13.200 90.100 55. 379. 23 6 66 9 1 13.400 90.300 33. 312. 87 26 6 66 5 52 12.300 88.700 33. 279. 1 7 66 20 17 13.700 88.400 201. 412. 5 7 66 4 5 13.700 90.800 53. 312. 8 7 66 13 22 12.400 88.900 43. 362. 13 7 66 8 20 12.600 87.800 56. 528. 24 7 66 6 1 1 12.200 88.300 61 . 329. 24 7 66 18 50 12.200 88.700 33. 329. 25 7 66 0 42 13.200 87.800 210. 312. 31 7 66 19 10 12.400 88.100 63. 345. 7 8 66 5 33 13.400 89.900 74. 379. 12 8 66 14 40 14.300 89.900 77. 279. 23 8 66 6 7 12.400 88.200 39. 312. 25 8 66 12 4 12.600 89.300 73. 379. 27 8 66 12 12 16.100 86.100 36. 312. 29 8 66 17 35 13.300 90.900 33. 279. 18 9 66 10 51 13.200 89.600 66. 329. 20 9 66 3 53 13.100 88.200 81. 312. 20 9 66 9 51 13.800 89.000 8. 379. 23 9 66 19 1 12.900 89.500 65. 379. 20 10 66 3 40 13.100 89.700 67. 329. 26 10 66 20 43 15.200 90.600 20. 379. 29 10 66 14 21 12.900 89.300 68. 329. 13 11 66 6 4 13.500 89.600 69. 412. 5 12 66 16 1 1 14.400 90.500 53. 345. 12 12 66 12 35 14.300 91.000 52. 0. 23 12 66 2 2 12.900 88.600 88. 395. 23 12 66 22 10 12.600 87.700 32. 395. 25 12 66 2 20 13.500 90.800 63. 263. 27 12 66 21 22 13.300 88.800 78. 550. 3 1 67 17 1 1 13.546 90.264 51 . 362. 4 1 67 0 28 13.100 88.625 72. 329. 9 1 67 22 31 13.138 88.981 78. 329. 19 1 67 2 19 12.561 86.613 200. 395. 19 1 67 19 43 16.998 85.740 34. 428. 30 1 67 9 30 12.196 87.596 8. 329. 5 2 67 5 24 13.232 90.893 33. 362. 24 2 67 9 16 13.437 90.117 135. 362. 3 3 67 19 39 13.078 90.392 65. 312. 6 3 67 23 31 12.389 87.127 159. 329. 8 3 67 15 52 13.049 88.127 75. 296. 8 3 67 23 7 15.101 88.782 76. 345. 13 3 67 8 37 13.726 88.069 82. 279. 20 3 67 8 42 13.433 90.048 8. 263. 21 3 67 9 55 13.924 90.955 112. 312. 22 3 67 9 15 12.653 87.823 70. 379. 25 3 67 7 26 12.406 87.876 76. 312. 27 3 67 14 40 12.221 87.712 54. 345. 30 3 67 5 30 13.873 89.532 33. 312. 9 4 67 5 28 13.059 89.243 69. 329. 1 1 4 67 7 49 12.259 89.800 33. 379. 12 4 67 4 56 12.217 88.102 33. 445. 13 4 67 6 10 12.778 89.141 84. 379. 15 4 67 22 10 12.743 88.163 33. 362. 17 4 67 4 27 15.196 90.023 205. 395. 18 4 67 13 5 12.765 87.943 84. 296. 30 4 67 3 10 12.632 86.852 115. 345. 8 5 67 14 40 12.974 88.073 33. 412. 16 5 67 12 58 13.480 90.639 92. 570. 16 5 67 15 1 1 13.275 90.939 75. 329. 19 5 67 7 52 13.241 89.518 89. 495. 88 20 5 67 7 48 13.640 90.555 89. 428. 20 5 67 7 58 14.001 90.420 94. 329. 21 5 67 3 12 13.283 89.694 71. 395. 21 5 67 5 42 13.844 89.614 62. 362. 23 5 67 11 37 13.889 90.528 76. 279. 23 5 67 19 12 13.340 90.721 68. 395. 28 5 67 14 37 14.194 90.872 33. 263. 29 5 67 1 43 13.791 89.686 123. 345. 17 6 67 17 43 14.100 90.000 103. 445. 23 6 67 13 30 12.900 87.800 11. 345. 14 7 67 18 2 13.500 88.800 147. 412. 15 8 67 2 56 14.700 90.700 45. 395. 27 6 67 6 32 14.200 89.600 65. 362. 27 8 67 13 8 12.300 86.200 183. 51 1 . 10 9 67 15 51 13.200 89.500 69. 279. 4 10 67 0 12 15.700 88.600 33. 379. 8 10 67 21 29 13.600 90.800 53. 345. 12 10 67 3 17 14.400 90.400 190. 296. 4 11 67 22 29 14.200 89.700 24. 312. 6 11 67 18 49 13.500 88.000 33. 362. 8 11 67 3 10 16.800 85.900 28. 545. 15 11 67 22 2 14.300 90.400 26. 412. 18 11 67 12 16 13.400 89.100 78. 495. 18 11 67 12 26 13.300 89.300 70. 428. 18 11 67 12 53 12.900 89.700 50. 395. 25 11 67 4 13 14.100 90.500 96. 345. 26 11 67 15 14 13.100 90.100 58. 379. 6 12 67 2 53 12.500 87.200 87. 528. 4 1 68 10 3 12.100 86.300 5. 412. 1 1 1 68 20 48 14.300 89.900 197. 379. 26 1 68 10 37 12.700 88.800 60. 296. 28 1 68 8 47 12.600 87.900 75. 379. 28 1 68 18 20 12.100 86.900 152. 379. 14 3 68 10 9 12.000 86.800 25. 395. 24 3 68 17 13 12.500 86.500 79. 495. 1 4 68 7 28 12.400 88.500 33. 379. 3 4 68 0 6 12.200 88.300 33. 362. 18 4 68 4 40 12.500 88.600 54. 462. 6 5 68 5 41 13.218 88.605 78. 279. 6 5 68 14 37 14.641 90.821 123. 495. 6 5 68 21 23 13.021 87.734 56. 412. 7 5 68 23 39 13.043 89.703 91 . 412. 24 5 68 17 13 12.500 86.500 79. 510. 1 1 6 68 5 52 13.941 88.761 199. 528. 29 6 68 6 21 13.607 90.220 97. 428. 7 7 68 19 49 14.678 88.939 134. 395. 14 7 68 3 55 15.237 88.843 14. 395. 7 8 68 5 17 12.665 89.720 37. 345. 20 8 68 0 34 13.105 87.833 157. 362. 2 9 68 4 53 13.108 90.365 33. 329. 10 9 68 14 35 12.432 89.174 63. 329. 15 9 68 4 32 13.466 87.243 164. 296. 19 9 68 0 37 12.143 88.714 33. 379. 28 9 68 19 21 12.232 89.061 51 . 362. 30 9 68 1 1 26 13.632 89.626 94. 379. 2 10 68 2 45 14.207 90.989 99. 395. 14 10 68 21 12 13.903 89.719 90. 345. 16 10 68 6 8 13.893 88.250 16. 362. 22 10 68 6 42 13.208 88.247 64. 428. 31 10 68 0 18 14.007 90.762 81. 312. 89 28 11 68 13 44 12.451 88.006 49. 462. 8 12 68 3 12 13.924 90.594 116. 329. 10 12 68 18 7 13.214 89.597 69. 362. 16 12 68 9 18 13.317 90.077 156. 395. 11 1 69 6 4 13.586 90.780 73. 279. 21 1 69 14 48 13.526 90.186 79. 329. 8 2 69 4 56 13.624 88.271 33. 362. 25 2 69 2 1 15.274 87.544 13. 445. 25 2 69 7 39 15.231 87.466 15. 500. 9 3 69 7 12 13.101 88.528 75. 379. 10 3 69 8 15 12.349 87.455 62. 600. 14 3 69 8 47 12.916 86.755 178. 580. 1 4 69 21 19 12.677 88.233 71 . 462. 4 4 69 8 43 13.534 88.981 105. 362. 8 4 69 19 41 12.670 88.340 76. 445. 21 4 69 2 19 14.100 91.000 82. 625. 22 4 69 0 29 14.078 90.964 89. 445. 25 4 69 16 46 13.311 89.174 70. 362. 3 5 69 3 19 12.987 88.449 77. 395. 9 5 69 10 47 12.889 89.190 68. 329. 1 3 5 69 16 39 1 2.447 87.830 33. 345. 23 5 69 9 18 13.685 89.808 94. 345. 29 5 69 20 25 14.722 90.595 218. 445. 6 6 69 16 16 12.467 88.000 48. 450. 15 6 69 19 29 12.865 90.135 128. 362. 23 6 69 20 46 12.410 88.052 33. 428. 27 6 69 13 20 13.321 89.529 63. 362. 28 6 69 4 34 12.818 89.217 69. 513. 7 7 69 4 20 13.082 89.014 64. 379. 17 7 69 13 24 12.590 87.006 175. 379. 26 7 69 7 21 12.625 87.798 84. 445. 3 8 69 21 17 13.342 88.342 52. 362. 9 8 69 1 19 13.202 90.892 33. 312. 29 8 69 13 53 14.465 90.989 118. 362. J 9 69 6 25 13.216 88.429 53. 412. 24 9 69 11 40 13.184 90.934 63. 362. 27 9 69 10 43 16.978 85.565 33. 362. 1 10 69 8 41 12.084 88.358 33. 379. 9 10 69 9 6 13.301 89.778 67. 362. 10 10 69 2 32 13.467 89.751 65. 329. 10 10 69 4 6 13.199 89.717 71 . 412. 12 10 69 22 48 13.211 90.587 68. 312. 15 10 69 1 1 54 14.320 90.994 105. 379. 12 11 69 3 48 13.888 89.984 111. 312. 15 11 69 16 40 14.575 90.405 121 . 296. 16 11 69 10 30 13.351 89.650 79. 488. 3 1 70 5 39 12.388 89.797 33. 362. 8 1 70 16 18 15.335 90.507 25. 379. 18 1 70 23 43 14.165 90.929 106. 362. 31 1 70 22 23 15.727 88.750 46. 450. 8 2 70 16 19 14.670 90.237 33. 412. 12 2 70 17 12 12.139 86.616 44. 379. 15 2 70 7 0 12.861 90.219 70. 395. 24 3 70 8 24 14.443 90.324 197. 345. 27 3 70 18 39 12.631 86.823 182. 495. 21 4 70 5 37 12.947 87.880 79. 51 1 . 25 5 70 4 56 15.873 89.319 35. 329. 16 6 70 1 56 13.641 90.678 87. 412. 5 7 70 5 42 12.561 87.446 33. 395. 7 8 70 13 3 12.325 87.791 59. 395. 90 12 8 70 9 24 12.015 86.542 33. 630. 12 8 70 10 24 12.051 86.539 33. 550. 1 9 70 1 3 13.077 89.350 80. 345. 6 9 70 20 44 12.783 88.595 75. 395. 23 9 70 22 48 13.514 90.086 80. 428. 23 9 70 22 56 13.521 90.078 83. 412. 23 9 70 23 57 12.021 87.446 70. 395. 4 10 70 13 28 12.687 90.318 33. 345. 6 10 70 15 18 13.561 88.312 192. 412. 13 10 70 23 21 13.764 90.046 100. 445. 8 11 70 11 49 13.553 90.665 96. 395. 16 11 70 6 17 14.277 90.188 204. 412. 19 2 71 16 59 13.562 88.795 176. 511 . 25 2 71 4 15 12.159 87.489 52. 528. 26 2 71 2 0 12.555 89.925 71 . 312. 14 4 71 3 9 14.125 90.133 124. 445. 19 4 71 5 7 13.907 90.540 92. 478. 1 5 71 14 32 13.259 88.449 93. 545. 28 5 71 3 9 13.049 89.963 86. 362. 28 5 71 20 19 1.3.732 89.753 84. 379. 29 5 71 0 0 13.663 90.016 72. 395. 11 6 71 11 51 12.647 86.967 148. 395. 18 6 71 13 36 14.678 87.567 7. 478. 6 8 71 19 52 14.298 90.834 87. 412. 12 8 71 5 59 12.877 87.341 211 . 428. 18 9 71 17 11 13.526 90.026 82. 445. 8 1 1 71 4 26 12.472 88.165 68. 462. 16 11 71 0 26 13.412 90.729 115. 412. 23 1 1 71 10 24 14.412 90.171 190. 412. 25 11 71 22 6 12.759 87.458 62. 462. 26 1 1 71 4 6 12.148 87.626 47. 379. 26 11 71 14 1 13.305 89.140 79. 379. 3 1 72 4 31 12.311 86.300 33. 329. 5 1 72 11 52 12.286 86.236 33. 345. 15 1 72 19 21 15.958 90.988 33. 428. 22 1 72 13 8 14.000 91.000 102. 600. 29 1 72 12 49 12.434 90.687 33. 412. 17 2 72 9 19 13.269 88.691 80. 428. 4 3 72 22 4 12.933 88.743 64. 428. 1 1 3 72 1 41 13.203 89.825 157. 296. 1 1 3 72 3 50 13.368 90.042 85. 560. 22 3 72 7 16 16.038 90.948 16. 330. 29 3 72 15 8 14.417 90.777 82. 345. 9 4 72 15 26 13.528 90.464 85. 445. 11 4 72 7 33 13.523 89.135 93. 395. 17 4 72 19 25 13.614 89.590 88. 362. 30 4 72 10 55 12.896 88.668 77. 379. 28 5 72 3 2 14.130 85.837 33. 445. 5 6 72 6 22 13.356 90.278 33. 428. 11 6 72 18 21 13.463 90.584 52. 495. 1 1 6 72 19 9 13.511 90.521 79. 379. 25 6 72 1 52 12.668 89.062 65. 379. 27 6 72 14 45 12.182 86.588 104. 412. 2 7 72 22 37 12.149 89.709 33. 296. 5 7 72 13 0 13.951 90.526 109. 445. 27 7 72 20 3 13.317 90.131 78. 395. 27 7 72 20 44 13.276 90.049 77. 395. 27 7 72 21 16 13.210 90.004 91 . 395. 28 7 72 13 53 13.151 87.753 209. 379. 6 8 72 23 6 13.478 90.590 83. 412. 21 8 72 23 33 1 3 . 9 5 7 9 0 . 8 1 6 1 1 9 . 4 4 5 . 31 8 72 2 24 1 3 . 2 7 2 8 8 . 5 1 3 5 0 . 4 8 0 . 26 9 72 3 20 12 .231 8 7 . 4 1 2 3 7 . 3 8 0 . 26 9 72 8 25 1 2 . 1 9 9 9 0 . 1 9 2 3 3 . 3 6 0 . 1 10 72 0 21 1 2 . 2 7 5 8 9 . 1 9 7 3 3 . 3 6 2 . 4 10 72 4 5 1 2 . 2 0 7 9 0 . 1 9 8 3 3 . 4 2 8 . 19 11 72 4 35 1 2 . 9 7 3 8 8 . 5 2 4 7 3 . 5 7 8 . 12 12 72 0 32 1 2 . 5 5 2 8 9 . 0 4 8 4 5 . 3 9 5 . 23 12 72 6 29 1 2 . 3 5 7 8 6 . 1 1 9 5 . 6 2 0 . 23 12 72 7 17 1 2 . 0 5 0 8 6 . 3 7 8 5 . 4 7 8 . 23 12 72 7 19 1 2 . 0 0 0 8 6 . 1 8 9 5 . 51 1 . 25 12 72 0 26 1 5 . 2 1 5 8 8 . 9 4 3 3 4 . 4 1 2 . 25 1 73 22 25 1 3 . 7 0 0 8 9 . 3 2 4 100 . 3 7 9 . 22 2 73 7 38 1 4 . 5 0 4 9 0 . 9 7 7 100 . 4 1 2 . 7 3 73 13 0 1 2 . 7 0 5 8 8 . 0 2 7 6 4 . 4 1 2 . 9 3 73 18 19 13.611 9 0 . 6 8 7 3 3 . 3 7 9 . 29 3 73 6 47 1 4 . 0 2 8 9 0 . 2 2 6 9 6 . 3 4 5 . 9 4 73 3 17 1 2 . 2 6 7 8 6 . 5 5 3 167 . 4 2 8 . 20 4 73 9 19 1 3 . 4 8 2 9 0 . 0 2 8 8 8 . 4 4 5 . 25 4 73 10 45 12.. 454 9 0 . 0 8 2 3 3 . 2 9 6 . 30 4 73 7 15 1 2 . 3 2 2 8 9 . 5 3 9 5 0 . 3 7 9 . 7 5 73 1 1 54 12.344 8 7 . 9 4 9 3 3 . 4 0 0 . 31 5 73 5 39 13.941 9 0 . 8 7 7 9 9 . 5 2 0 . 26 6 73 7 24 1 4 . 2 9 0 9 0 . 7 0 6 3 8 . 3 7 0 . 30 6 73 8 8 1 3 . 7 6 3 9 0 . 9 3 5 7 8 . 4 9 5 . 27 7 73 8 51 1 2 . 4 1 5 8 7 . 4 0 7 3 3 . 3 9 5 . 27 7 73 19 42 1 2 . 8 1 3 8 6 . 6 7 4 1 9 9 . 5 2 8 . 28 7 73 20 48 1 2 . 3 3 3 8 7 . 0 8 4 7 9 . 5 4 5 . 4 8 73 11 54 1 2 . 1 4 0 8 8 . 9 8 3 3 3 . 3 7 9 . 19 8 73 22 17 12.651 8 8 . 5 0 7 6 8 . 4 9 5 . 21 8 73 12 42 1 3 . 2 5 2 9 0 . 5 6 4 7 0 . 2 9 6 . 5 9 73 6 15 1 3 . 0 0 0 9 0 . 6 0 0 3 3 . 3 4 5 . 8 9 73 15 59 1 2 . 8 0 0 8 9 . 8 0 0 6 3 . 3 2 9 . 19 9 73 23 42 1 2 . 7 0 0 8 8 . 0 0 0 5 5 . 4 7 8 . 11 10 73 8 40 1 3 . 3 0 0 8 9 . 8 0 0 6 8 . 3 9 5 . 2 11 73 3 6 1 3 . 2 0 0 8 8 . 0 0 0 118. 3 9 5 . 9 1 1 73 2 36 1 4 . 0 0 0 8 9 . 6 0 0 1 4 9 . 4 6 2 . 10 1 1 73 2 50 1 2 . 4 1 7 8 8 . 3 0 7 3 3 . 0 . 10 1 1 73 19 46 1 4 . 6 0 0 9 0 . 2 0 0 2 4 3 . 4 1 2 . 28 1 1 73 6 45 1 4 . 5 0 0 9 0 . 0 0 0 2 3 0 . 4 7 8 . 8 12 73 0 55 1 3 . 3 0 0 8 9 . 9 0 0 8 8 . 4 6 2 . 27 12 73 17 29 1 6 . 7 4 4 8 6 . 5 0 2 3 3 . 0 . 12 1 74 4 47 1 3 . 6 5 3 9 0 . 9 5 8 6 4 . 0 . 17 1 74 1 1 16 1 3 . 2 0 0 8 8 . 3 0 0 1 0 3 . 4 1 2 . 4 3 74 14 54 1 3 . 6 0 0 9 0 . 6 0 0 8 9 . 3 9 5 . 4 3 74 14 56 1 3 . 6 0 0 9 0 . 6 0 0 7 9 . 4 1 2 . 5 3 74 15 10 1 3 . 2 0 0 9 0 . 8 0 0 8 4 . 3 6 2 . 6 3 74 1 40 1 2 . 3 0 0 8 6 . 4 0 0 1 1 0 . 6 1 0 . 13 3 74 1 1 35 1 3 . 1 5 0 9 0 . 0 1 0 8 5 . 0 . 21 3 74 22 1 1 3 . 9 0 0 9 0 . 1 0 0 9 9 . 3 6 2 . 1 4 74 2 40 1 2 . 2 0 0 8 7 . 9 0 0 156 . 3 6 2 . I 2 4 74 20 12 1 3 . 6 0 0 9 0 . 3 0 0 91 . 4 1 2 . 1 20 4 74 7 41 1 3 . 0 0 0 9 0 . 0 0 0 2 3 . 4 2 8 . 30 4 74 11 43 1 6 . 9 1 8 8 5 . 4 8 0 3 3 . 0 . 2 5 74 3 12 1 3 . 9 0 0 9 0 . 7 0 0 7 6 . 4 2 8 . 2 5 74 3 51 1 3 . 8 0 0 9 0 . 5 0 0 6 5 . 4 1 2 . 4 5 74 17 49 1 3 . 9 0 0 9 0 . 5 0 0 1 15 . 3 9 5 . 7 6 74 3 1 1 2 . 2 0 0 6 8 . 2 0 0 3 3 . 4 1 2 . 21 6 74 15 29 1 4 . 2 0 0 9 0 . 0 0 0 6 5 . 3 4 5 . 19 8 74 19 54 1 2 . 3 0 0 8 8 . 9 0 0 6 7 . 511 . 92 30 8 74 7 45 12.800 87.400 62. 462. 1 9 74 13 49 13.300 88.900 111. 462. 2 9 74 22 4 13.200 90.800 41 . 395. 12 9 74 20 14 13.600 89.900 85. 478. 20 10 74 13 22 12.000 86.700 307. 345. 23 10 74 13 36 14.700 90.600 231 . 412. 4 11 74 12 34 13.000 90.100 33. 362. 4 11 74 21 23 12.700 90.100 33. 428. 3 12 74 4 42 12.400 89.900 33. 345. 17 12 74 15 31 13.200 88.500 50. 478. 24 12 74 2 10 14.300 90.100 155. 545. 10 2 75 3 59 13.100 88.600 179. 412. 1 3 75 12 37 12.600 89.600 33. 379. 5 3 75 17 25 12.300 87.900 104. 445. 13 3 75 23 40 16.300 87.000 33. 51 1 . 24 3 75 6 59 13.600 90.800 59. 462. 24 3 75 7 15 15.100 90.800 33. 395. 24 3 75 1 1 51 13.000 89.300 93. 412. 25 3 75 2 31 13.600 90.700 33. 511. 29 3 75 9 30 12.900 88.400 33. 395. 4 4 75 16 54 12.400 88.600 1 13. 395. 14 4 75 23 17 13.600 90.200 107. 362. 1 1 5 75 8 55 12.136 88.984 33. 0. 21 5 75 16 41 13.300 89.800 33. 412. 22 5 75 12 51 12.700 87.900 33. 478. 4 6 75 19 36 12.700 88.500 33. 412. 19 6 75 16 12 13.400 89.300 93. 478. 24 6 75 3 31 12.000 86.200 157. 428. 26 6 75 6 45 14.000 90.800 76. 445. 17 7 75 17 52 13.600 86.100 33. 395. 13 8 75 23 7 14.500 89.900 256. 428. 31 8 75 13 19 14.100 90.600 88. 395. 4 9 75 19 14 12.900 90.700 33. 428. 12 9 75 8 56 13.600 90.900 43. 362. 13 9 75 19 31 16.800 85.900 33. 412. 1* 9 75 17 38 12.400 88.400 33. 445. 15 10 75 20 55 13.453 88.045 33. 428. 21 10 75 21 19 12.200 88.000 65. 296. 30 10 75 22 23 13.398 84.127 33. 0. 27 11 75 8 43 12.897 87.368 191. 362. 4 1 76 11 34 13.037 88.666 33. 478. 25 1 76 10 59 13.932 90.883 103. 478. 26 1 76 7 28 13.692 90.848 33. 445. 26 1 76 9 31 14.146 90.518 50. 310. 30 1 76 10 41 13.796 88.418 38. 370. 31 1 76 4 55 13.703 88.354 123. 395. 4 2 76 9 1 15.324 89.101 5. 750. 4 2 76 9 30 14.943 90.562 5. 545. 4 2 76 10 3 14.172 90.729 5. 478. 4 2 76 11 24 14.696 90.879 5. 428. 4 2 76 12 57 15.421 89.223 5. 362. 4 2 76 16 40 15.361 89.184 5. 380. 4 2 76 17 20 15.856 88.286 5. 390. 5 2 76 4 9 15.232 90.626 5. 390. 6 2 76 1 57 15.363 90.278 5. 390. 6 2 76 18 1 1 14.316 90.432 5. 478. 6 2 76 18 19 14.764 90.610 5, 530. 7 2 76 2 28 14.335 90.886 5. 310. 8 2 76 4 38 14.392 90.515 5. 296. 8 2 76 8 13 15.571 88.466 5. 560. 93 9 2 76 1 1 44 15.319 89.073 5. 470. 10 2 76 2 28 14.921 90.778 5. 340. 10 2 76 6 17 14.833 89.798 5. 380. 10 2 76 17 35 14.515 88.964 5. 300. 11 2 76 10 18 12.313 89.080 33. 350. 14 2 76 16 2 13.298 89.715 33. 340. 19 2 76 22 56 14.300 90.335 61. 400. 24 2 76 6 25 14.221 90.418 5. 379. 7 3 76 3 15 14.808 90.889 5. 500. 9 3 76 7 42 14.869 90.943 5. 430. 3 4 76 10 2 12.991 89.772 68. 362. 4 4 76 17 55 14.258 87.636 45. 440. 25 4 76 14 - 0 13.182 89.812 90. 362. 26 4 76 15 24 15.016 89.607 5. 330. 28 4 76 4 39 13.931 90.654 69. 445. 28 4 76 5 15 15.883 86.126 36. 312. 4 5 76 8 43 13.075 87.568 33. 528. 15 5 76 12 56 14.273 90.859 83. 379. 19 5 76 17 43 16.868 85.529 43. 450. 22 5 76 9 0 12.637 88.454 77. 379. 23 5 76 2 50 12.574 88.607 61 . 379. 23 5 76 9 4 12.428 88.315 96. 279. 8 6 76 11 14 12.646 89.448 33. 379. 12 6 76 3 29 13.407 89.941 94. 412. 15 6 76 15 19 13.318 89.425 71 . 395. 18 6 76 7 26 12.610 87.797 31 . 0. 21 6 76 1 51 13.132 89.480 77. 345. 22 6 76 1 1 58 13.944 90.777 81 . 462. 24 6 76 21 15 12.712 88.459 65. 412. 27 6 76 7 41 13.117 89.432 95. 362. 28 6 76 14 33 13.929 90.705 100. 445. 2 7 76 5 35 12.143 86.645 148. 462. 14 7 76 9 23 15.571 88.042 33. 412. 29 7 76 16 3 12.647 88.786 33. 330. 5 8 76 18 48 12.745 88.201 70. 51 1 . 9 8 76 17 1 12.671 88.175 82. 51 1 . 15 8 76 18 21 13.580 89.472 69. 379. 23 8 76 3 41 14.775 89.649 88. 395. 1 9 76 22 34 13.512 88.111 73. 310. 2 9 76 4 15 15.634 88.487 43. 350. 2 9 76 10 20 13.259 89.989 81 . 478. 14 9 76 23 47 13.990 90.990 103. 412. 15 9 76 20 52 12.544 87.797 86. 462. 22 10 76 4 4 12.553 87.852 59. 462. 22 10 76 5 53 13.075 88.397 83. 445. 6 11 76 12 37 13.619 90.568 86. 462. 7 11 76 6 41 13.672 90.421 85. 412. 1 12 76 17 44 13.531 89.251 85. 462. 8 12 76 5 20 15.054 89.328 101 . 0. 9 12 76 4 24 14.356 90.214 199. 428. 9 12 76 12 50 12.338 87.866 77. 445. 11 12 76 17 20 13.596 88.895 79. 478. 14 1 77 17 5 13.144 87.755 33. 263. 17 1 77 0 55 12.417 88.601 33. 362. 22 1 77 5 14 13.055 89.133 85. 395. 28 2 77 0 44 13.931 90.892 61 . 428. 15 3 77 21 28 13.353 89.668 95. 428. 22 3 77 19 29 12.972 88.536 76. 478. 1 1 4 77 10 1 13.154 88.134 10. 430. 17 4 77 14 41 13.629 88.24J 208. 445. 94 30 4 77 2 42 13.587 90.592 59. 379. 20 5 77 7 32 12.851 89.144 33. 330. 13 6 77 5 28 13.787 90.418 111. 379. 16 6 77 15 50 13.245 89.957 62. 329. 17 7 77 5 10 13.225 90.874 39. 296. 30 7 77 7 56 12.978 88.771 81. 445. 4 8 77 13 20 12.346 87.294 33. 500. 12 8 77 14 10 13.214 89.849 98. 362. 15 8 77 5 54 14.399 90.963 121. 362. 20 8 77 2 46 16.612 86.852 14. 6. 20 8 77 3 51 16.700 86.606 36. 610. 20 8 77 12 21 16.952 86.430 33. 428. 21 8 77 11 33 13.788 90.057 84. 445. 3 9 77 22 23 12.540 87.532 79. 528. 12 9 77 21 35 12.400 87.854 118. 362. 18 9 77 5 44 13.558 89.954 108. 478. 23 10 77 13 44 13.231 87.761 193. 462. 30 10 77 22 28 13.653 90.117 96. 312. 31 10 77 6 53 12.983 89.640 84. 495. 4 11 77 12 49 13.862 90.588 104. 478. 20 11 77 3 17 13.361 87.333 36. 0. 12 4 78 16 51 12.590 87.790 100. 428. THE DATA IS AN EXTRACT OF THE RECORDED DATA FOR THE.HO.NDTO 1.978, 95 A P P E N D I X V T a b l e o f b i a s f o r maximum l i k e l i h o o d e s t i m a t i o n o f F r e c h e t d i s t r i b u t i o n shape p a r a m e t e r [4] II ' bin) n n bin) n bin) 5 0 . 6 6 9 18 0 . 9 2 3 4 2 0 . 9 6 8 6 6 0 . 9 8 0 6 0 . 7 5 2 2 0 0.931 4 4 0 . 9 7 0 6 8 0.981 7 0 . 7 9 2 22 0 . 9 3 8 4 6 0.971 7 0 0.981 8 0 . 8 2 0 2 4 0 . 9 4 3 4 8 0 . 9 7 2 72 0 . 9 8 2 9 0 . 8 4 2 26 0 . 9 4 7 5 0 0 . 9 7 3 74 0 . 9 8 2 10 0 . 8 5 9 28 0.951 52 0 . 9 7 4 7 6 0 . 9 8 3 II 0 . 8 7 2 3 0 0 . 9 5 5 54 0 . 9 7 5 78 0 . 9 8 3 12 0 . 8 8 3 32 0 . 9 5 8 56 0 . 9 7 6 8 0 0 . 9 8 4 13 0 . 8 9 3 34 0 . 9 6 0 58 0.977 85 0 . 9 8 5 14 0.901 36 0 . 9 6 2 6 0 0 . 9 7 8 9 0 0.9R6 15 0 . 9 0 8 38 0 . 9 6 4 62 0 . 9 7 9 100 0 . 9 8 7 16 0 . 9 1 4 4 0 0 . 9 6 6 64 0 . 9 R 0 120 0 . 9 9 0 T h i s t a b l e a p p l i e s t o t h e W e i b u l l d i s t r i b u t i o n , i s a l s o v a l i d f o r t h e F r e c h e t d i s t r i b u t i o n . 96 APPENDIX VI EXTREME VALUE ESTIMATION FOR LOG-NORMAL AND FRECHET AS PARENT DISTRIBUTIONS Frechet has a reproductive property [4] * n(x;a,X) = F J J (x;a,X) = F n ( x ; a n l A , X) Hence a = a n 1 / X n X n Log-Normal Gumbel d i s t r i b u t i o n models the extreme value w i t h the f o l l o w i n g parameters; u = Quantile of order (1 — ) e n o 1 " fLN ( P e ? yLN» °LN>  6 " fLN ( V \ N ' °LN> 97 APPENDIX VII Hazard plots for time of occurrence modelling Gumbel Hazard Paper 10 O D.l-i cc cr M cr x • • LJ 0.0JJ 0.C01 500 — I — 1000 1500 2000 2S00 T I M E I N HOURS 3000 3500 VII. 1 - Time between occurrences of earthquakes 98 APPENDIX VIII Probability plots for rejected distributions Gumbel Probability Paper Vertical PGA o z - 1 — 40 — I — 60 BO —r— 100 — i 120 ACCELERATION IN 0.001 g + z o o z Normal PGA — i — 50 100 150 200 2S0 ACCELERATION IN 0.001 g i 300 VIII.l - Plot of PGA on Extreme Value type I probability paper 99 Weibull Probability Paper Vertical PGA 6H 8-0.001 TT, I ' ;•' 0.01 0.1 1 10 100 ACCELERATION IN 0.001 g 1000 Normal PGA 10 100 ACCELERATION IN 0.001 g 1000 Plot of PGA on Weibull probability paper 100 Gamma P r o b a b i l i t y Paper V e r t i c a l PGA Lam =1.0 e o.) ».* o.t o.i i i . i ruNCTiON or A C C E L E R A T I O N I N O . O O I G Lam = 2.0 '1 0 0 . 1 0 . 4 O.t 0 . 0 I 1.1 fUNCTlON OT SCCCLfflTlON IN 0.001 g . .. Lam =5.0 e o.i e.« o.« e.i i 1.1 ruNCTiON or A C C E L E R A T I O N I N O . O O I 9 P l o t o f PGA on Gamma p r o b a b i l i t y paper 101 Gamma Distribution Estimated Vs. Sample data Vertical PGA Lam = 0.28, Sig =3.04 0.2-CflLCULATCQ E S T I M A T E D 1 1 1 1 5 10 15 20 flCCELERRTION IN 0.001 g Normal PGA Lam = 0.28, Sig = 17.0 0.2-CflLCUL«7ED E S T I M A T E D —I I 1 1 I I 1 20 40 60 80 100 120 M0 RCCELERRTI0N IN 0.001 g VIII.4 - Plot of PGA on Gamma probability paper 102 Normal Probability Paper Vertical PGA i n 0.4 -0.2-20 Normal PGA I 0.2-o H 1 1 1 1 1 i 1 0 20 40 60 60 100 120 MO ACCELERATION IN 0.001 g VIII.5 - Plot of PGA on Normal probability paper - i 1 1— 5 10 15 ACCELERATION IN 0.001 g o.eH CC 0.6 Ti o >^  0.4 -I 103 APPENDIX IX Probability plots for various site locations and orientations Log-Normal Probability Paper Frechet Probability Paper Vertical PGA fe i £ 0 . 4 5 * 0.001 0 .01 0 .1 I 10 100 1000 ACCaCRATJCM IN 0.001 g 0 .Q0I 0 .01 0 .1 I 10 100 1000 flCCOOWTlON IN 0.001 g Normal PGA 5 — 0 . » b E £ 0 . 4 0 .01 0 .1 I flCCOXRATION IN 0.001 g 0.01 0.1 I ACCELERATION IN 0.001 g IS S C o.«-Tangential PGA o.ai o.oi o.i i io too ion %mC*ATI0N IN 0.001 g 0.001 0 .01 0 .1 I 10 100 1000 HCCEUEWT10N IN 0.001 g IX. 1- Comparison of Frechet and Log-Normal distributions for a different site at 13 N, 90 W and 0 and 90 degrees W of N 104 Log-Normal P r o b a b i l i t y Paper F r e c h e t P r o b a b i l i t y Paper V e r t i c a l PGA " o..H fe S 0.001 0 .01 0 .1 1 ftCCCLPmTlON IN 0.001 g 0.01 0 .1 I flCCCLERATlON IN 0.001 g fe 5 o.« Normal PGA 0 .001 0 .01 O.I I 10 100 iooo ACCCLCRATION IN 0.001 g 0.001 0 .01 0.1 I 10 100 IOOO R C C C L D W I O N I N 0.001 g fe S £ 0.4 T a n g e n t i a l PGA 10 I X 1000 ACCDXRATION IN 0.001 g 5 5 5 100 1000 CCOXRRTION IN 0.001 g I X . 2 - Comparison o f F r e c h e t and Log-Normal d i s t r i b u t i o n s f o r a d i f f e r e n t s i t e a t 15 N, 89 W and 0 and 90 de g r e e s W o f N ."t05 

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