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Working Paper C Characterization of Earthquake Hazards for Loss Studies Technically, earthquake hazard or seismic hazard refers to the di- rect impact of an earthquake on the earth, including ground shaking, ground failures (liquefaction, surface faulting, landslides, and settle- ment), and the water-related phenomena of tsunamis and seiches. In this usage, the characterization of earthquake or seismic hazard does not include effects on the humanly constructed environment. Thus, threats such as collapsing buildings, overturning shelving, or breaking gas lines which in common language are often caned earth- quake hazards, or which are encompassed in the National Earthquake Hazards Reduction Program, are not topics in this working paper. This paper describes earthquake hazards and how they can be quantified, reviews current practice in the specification of hazards for loss studies, and describes a range of hazard specifications that might be used ~ future loss studies. TYPES OF EARTHQUAKE HAZARDS The primary and most pervasive hazard associated with earth- quakes is the shaking of the ground. This causes direct damage to structures as well as physical phenomena (seiches, liquefaction, and landslides) that can result in significant damage and loss. Ground shaking is generally caused by the release of crustal energy by rup- ture along a fault surface. Sometimes the rupture reaches the earth's 113

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114 surface and is evident after the event, but often the rupture is buried beneath surficial sediments and rocks. Voicanic earthquakes are less common, tend to be limited to moderate magnitudes, and are pri- mariTy caused by thermal rather than mechanical action. The energy released by a rupture propagates in the form of com- pressional waves and shear waves. The character of ground motion from these waves is a function of the source characteristics of energy release, the attenuation (clamping) characteristics of the earth's crust along the wave travel path, the near-surface geologic characteristics that may modify the frequency content of the motion (amplifying certain frequencies and damping others), and the interaction of the body waves with the earth's surface to form surface waves. Collateral earthquake hazards caused by ground shaking include seiches, liquefaction, and landslides. These hazards can lead to com- plete destruction of structures. For example, the Niigata earthquake of 1964 led to the liquefaction of soil in a large area, causing the loss of foundation strength for many apartment buildings. As a result, buildings tilted by as much as 70 degrees. A recent report of the National Research Council (1986) summarizes the state of the art of estimating the hazard from liquefaction. Landslides constitute a similar, shaking-induced hazard. In the 1964 Alaska earthquake, landslides in Anchorage caused the destruction and total loss of many residences and buildings. Landslides into bodies of water, or on the bottom of harbors and bays, can produce water waves that may cause very serious losses. Another hazard associated with earthquakes is rupture of the earth's surface caused by displacement of a fault. In the United States this phenomenon is generally observed only in the western states and Alaska. The fault movements associated with great earthquakes may be on the order of 5-10 meters. Such deformations are difficult or impossible to design for, and the best policy may be to avoid fault locations entirely. For some civil engineering works (e.g., pipelines, transmission lines, highways, railroacis, and aqueducts) it is not possible to avoid fault crossings. In these cases the hazard to the facility can be identified and quantified, and the eject on the system's function can be evaluated. If the system damage and its probability of occurrence are not acceptable, alternative system designs are usually possible to alleviate damage to components crossing faults (e.g., designing redundant links in the system, designing the fault crossing to be easily

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115 repairable, or devising an earthquake response plan that reduces the functional loss). Still another earthquake hazard is a tsunami, which is a wave generated in the open ocean as a result of tectonic movement of the floor of the ocean. Where such waves come ashore, they can rise to significant heights and cause considerable damage. Tsunamis are a potentially severe problem for Alaska, and tsunamis generated by Alaskan earthquakes have also caused damage to the West Coast states. Tsunamis generated both in Alaska and Chile have caused great destruction and fatalities in Hawaii. Apparently tsunamis are not a problem along the Atlantic and Gulf coasts, but may be a threat in Puerto Rico and the Virgin Islands. SCOPE OF EARTHQUAKE HAZARD ASSESSMENT The purpose of assessing earthquake hazards is to identify and quantify the severity of the various hazards in the geographic area of interest for the loss estimation study, given a scenario earthquake. In most cases an estimate of the frequency in tune (or probability of occurrence over a given period of time) of the hazard is also necessary. The results of the hazard assessment are combined with the ground- motion and damage relationships and the inventory of facilities in the study region to produce estimates of losses. Assessing earthquake hazards and specifying the other inputs to a loss study are related but independent activities that can be undertaken by different investigators at different times. One advan- tage of this independence is that various parts of the analysis can be updated (e.g., more recent data from a U.S. census can be incorpo- rated) without having to reinvestigate other inputs to the analysis. Also, studies of facility inventories and vulnerability relations can be locally or regionally undertaken, while nationally developed seismic hazard data (e.g., studies by the U.S. Geological Survey) can be pro- duced separately. The converse division of labor is also possible, as when national inventory data (e.g., census data on housing) is com- bined with seisrn~c hazard studies locally undertaken by state or local government geological agencies or local geotechnical consultants. Earthquake hazards and other inputs to loss estimation are re- lated in that the hazard must be specified in terms that are meaning- ful to the vulnerability analysis. For example, if the seismic hazard is specified in terms of the peak acceleration during the earthquake,

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116 the vulnerability functions cannot be given using a qualitative inten- sity scale such as MMI. If either the hazard or vulnerability analyses must be translated to make it compatible with the other, care must be taken in the translation, and the conversions used must be fully documented. Simple empirical correlations based on observed statistics often lead to incorrect results in particular applications. Simple repre- sentations of the shaking hazard, for example, those using peak acceleration or Modified Mercalli Intensity (MMI), are attractive be- cause hazard analyses are frequently available for these parameters, and vulnerability functions are available to estimate losses for them for many types of structures. The price for this simplicity is a wide range of uncertainty in the damage and loss, because simple rep- resentations of seismic shaking or earthquake-caused ground failure cannot capture the details of the underlying phenomena. More complex representations of ground shaking, for example, through a filtered "effective peak motion, a single-degree-of-freedom linear response spectrum, a nonlinear spectrum, a time history of motion, and the duration of strong shaking, have the ability to be more accurate predictors of damage and loss. There is less agreement, however, on how to estimate these functions for a future earthquake, how to quantify the single- or multidimensional hazard associated with them, and how to derive an accurate predictor of damage from them. CHARACTERIZATION OF GROUND SHAKING For historical and pragmatic reasons, MMI has been used as the ground-shaking measure in most earthquake loss studies conducted in the past, and likely will remam the standard for studies in the near future. This procedure was popular in early loss studies because multiple, instrumental records of ground shaking were not available to correlate motion levels to damage. Even today, records of strong shaking at the site of buildings damaged during earthquakes are rare, whereas assessments of the MMI level at that site can always be made by an experienced investigator. The MM! assessment denotes the severity of earthquake shaking at a particular location in terms of the effects on people, on construction, and on the earth's natural features. The MMI level depends on seismic, geologic, engineering, and human factors. The assigned MM! value for a particular earthquake and

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117 location is a qualitative measure of the integrated response of the natural and man-made environments to earthquake energy. As Richter (1958) also notes, at the higher intensity levels (MMI _ X, XT, XIT) the scale refers primarily to ground failure rather than directly to ground vibration. The primary fault trace phenomena and secondary ground effects depend on the type of faulting motion (vertical versus horizontals, the duration of the faulting movement, and the nature of the ground in the immediate vicinity of the fault. Ground failures such as liquefaction, faulting, and landslides are not good measures of ground shaking since they can occur at low as well as high levels and durations of ground motion. Contrary to conventional seismological practice and to previous applications in the loss estunation field (e.g., ATC-13), it is desirable to use MMI intensity as follows: First, ground failure phenomena should be treated separately from ground shaking, and second, only intensities below MMI X! should be used to describe the severity of ground vibration. This does not unply that quantitative measures of ground vibration (e.g., peak ground acceleration or velocity) are limited to a maximum value that would correspond, according to one of the various MMI-acceleration or MMI-velocity relationships proposed, to MM! X. In other words, MMI X is not necessarily the maximum severity of ground shaking that could occur on earth. As a general rule, the estimation of MM} can rarely be refined beyond a 1-unit range. When making loss estimates, the ejects of soil conditions on intensity of shaking must be considered. One means of accomplishing this, if ground motion is used as an intermediate variable to estimate losses, is to modify the estimates of ground motion (e.g., MMI) to reflect the expected effects of soils at locations in the region from the hypothesized earthquakes. Table C-1 presents one set of correction factors that has been proposed for southern California and that attempts to account for both the type of rock that might underlie a site, ~d the depth to the water table (Evernden and Thomson, 1985~. Other correlations may have equal or greater justification, depending on the data base and the region of study. The use of MMI in loss estimates is a gross simplification that is justifiable only if more precise methods are not available. It is known, for example, that modifications of earthquake ground motions by soils are frequency dependent. Therefore, an accurate modification of ground motion should consider the frequency characteristics of the structures for which losses are to be estunated. Similarly, explicit

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118 TABLE C-1 Ground-Motion Correction Factors for Southern California Geologic Condition Change in Intensity Quaternary alluvium (water table > 100 ft) Quaternary alluvium (water table 30-100 ft) Quaternary alluvium (water table < 30 ft) Sedimentry rock Volcanic, granitic, and metamorphic rock 0.0 +1.0 +1.5 0.0 to -1.6 -1.7 to 2.0 SOURCE: E,rernden and Thomson (1985~. quantification of the effects of earthquake magnitude, distance, and duration are ignored when MMI is used. For example, an MMI VII observed at the epicenter of a magnitude 5 earthquake does not unply the same ground motion as an MMI VII observed (for the same soil conditions) 100 km from a magnitude 7 (Murphy and O'Brien, 1977~. For these reasons, use of MMI (and similar intensity scales) should be recognized as a less-than-perfect representation of earthquake ground shaking to be used only until more precise parameters and methods are available. HAZARD AND LOSS ESTIMATION PROCEDURES Scenario earthquakes have been deterrn~ned following rationales that in different ways express the need to compromise between the likelihood or credibility of the event and its destructive potential. Once the scenario event has been selected, all existing large-scale loss estunation procedures use deterministic relationships to calculate the level of ground motion at each site, the resulting damage, and the losses. Occasionally, some of the relationships (most notably the rela- tionship between ground-motion intensity and damage) include un- certainty, but in no study has uncertainty been propagated through the analysis to produce probabilistic estimates of loss. For the treat- ment of uncertainty, one can therefore regard all previous large-scale, general-purpose loss est~rnation methods as Deterministic proce- dures under scenario earthquakes. This type of analysis reflects

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119 TABLE C-2 Applications of Loss Estimation Methods Applications Appropriate Methods Advocacy Response planning Preparedness planning Mitigation strategies Relocation/reco~rery Risk evaluation Economic impact analysis Insurance National security A,B A,B,D A,8 A,B,C B,C B A,B,C,D,E B,C,D . KEY: Current practice: A = extrapolation from historical data; and B = scenario analyses and accuracy estimates. Emerging methods: C = lose-frequency analysis; and D = simulation of actual events with variability. E = cumulative over time. the heritage of early loss estimation efforts. Resistance to change has resulted from the difficulty of more complete representations of uncertainty and by computational constraints. Deterministic scenario-type analyses are easy to interpret and will likely remain the basic type of earthquake loss calculation in the near future. However, they are not ideal for all purposes because they are not capable of fully representing uncertainty and because of the lack of a clear rationale for selecting scenario earthquakes. For cer- tain uses, deterministic scenario analyses are actuaby inappropriate, and alternative methods, not yet fully available, must be developed. A listing and brief description of loss estunation procedures (some existing, others in need of development) and an indication of their potential uses follow. Appropriate applications of each are shown in Table C-2. Extrapolation from Historic Losses Loss information from historical events in the region of inter- est can be adjusted to reflect recent changes in the inventory and differences between characteristics of the scenario earthquakes and those of the historic events. This analysis is simple and inexpensive, but typically less accurate and less general than analyses based on models of ground motion, damage, and loss. Such a technique is

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120 appropriate when a historic event, not much different from the sce- nario earthquake of interest, has occurred in the recent past, as in the case of the largest of the six scenario earthquakes used in the National Oceanic and Atmospheric Adm~nistration's San Francisco study (AIgermissen et al., 1972~. This condition, however, seldom applies. Selecting a historic event as the basis for a loss stiffly does not avoid the problem of uncertainty, although the nontechnical audience is less likely to bring up the issue of uncertainty because of the intuitively convincing nature of events that have happened before. Results of analyses of this type are illustrated in Figure C-la. Scenario Analysis with a Statement of Accuracy This type of analysis is most often used today, although often without the accompanying statement of accuracy. One or more earth- quakes are selected, based on criteria reviewed earlier herein, and single-value estimates of the resulting losses are produced. Accom- panying these estimates with even sunple but objective statements of certainty would prove useful to the users. For example, one might use sensitivity analysis: each major input parameter or relationship is modified in turn and the analysis is repeated to calculate the effect of the change on the calculated losses. The amount by which each input parameter is modified should reflect the degree of uncertainty on that parameter. The results of such an analysis are similar to the "type A" seismic hazard analysis (Figure C-la), but a description of uncertainty in the estimates for the scenario event may be included. Historic Maximum E:arthqu~ce Historic earthquakes, when judged to be suitable and perhaps with adjusted magnitude, intensity distribution, or location, can be used in loss estimation studies. These earthquakes are convincing to both the users and the general public. Recurrence Bate Earthquake This can be an appropriate selection technique when a known seismic source zone or fault dominates the problem at hand, for ex- ample, the Was atch Fault at Salt Lake City, Utah. In such a case, an earthquake magnitude may be selected on the basis of its recurrence rate, that is, from the magnitude-frequency law flog N = a - bM). N

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121 L = Best estimate loss (a) HISTORIC LOSSES OR SCENARIO EARTHQUAKE 100% ,'_ m Cl: m o a: J ~ m LL' -1 ~ ~ 10 m an O ~ a: Cal Q ~ 10-2 ~ X Z 1 0-3 0% L = Loss (b) PROBABILISTIC 1W _ ~ ,. \ Existing Hazard - With Hazard ~ Reduction L= Loss (c) LOSS-FREQUENCY ANALYSIS FIGURE C-1 Form of results of different loss estimation techniques.

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122 is the number per unit tune of earthquakes exceeding magnitude M, and a and b are coefficients typically estimated by statistical analy- sis. For example, the magnitude of earthquake expected to occur on the average once in every 100 or perhaps 500 years or more Knight be selected. The location of the earthquake can be defined to maximize damage or loss. This method is more difficult to apply In an area of diffuse seismicity in that the locationts) of the seismogenic tectonic structures are not well constrained, as in much of the eastern United States. However, even in those areas, it is possible that the recurrence rate method could be a viable approach for some problems. Geologically Defined Characteristics This method is also most applicable to regions where the tec- tonic regime and the seismogenic tectonic structures are well known. Earthquake magnitude and location are determined on the basis of geologic and seismologic parameters that have been specifically asso- ciated with a given fault or area, for example, slip rate, stress drop, and typical fracture directions and lengths. Locations where this criterion might be applied are Pallet Creek and Anza, California, because of the ability of geologists to define the characteristic fault behavior at these places. Forecasting Based on past seism~city as well as relevant physical premonitory considerations, specific earthquake forecasts can sometimes be made. Such forecasted events (earthquake predictions) might then be used as scenario earthquakes. Regions of the United States where earth- quake prediction efforts are under way include the Aleutian Islands and central California (Parkfield). Maximum Credible Earthquake This and similar undefined criteria have sometimes been used in the past, but such concepts are vague and should be avoided in favor of more objective criteria. [os~Erequency Analysis This Toss estimation procedure is very different from the previous two methods (extrapolation of historical losses and scenario analysis).

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123 Its objective is not to estimate earthquake losses under a single postulated event, but to calculate the frequency with which various levels of loss are exceeded in the region of study. It is important to realize that a particular loss may result from earthquakes of different characteristics, for example, from a nearby earthquake of moderate size or from a distant earthquake of larger magnitude. This procedure sums up the contributions to the frequency with which any given loss is exceeded from all possible earthquakes, large and small, near and far away. The final result is a plot of the annual probability of exceedance versus loss (Ioss-frequency curve), as shown in Figure C-lc. From an operational point of view, the analysis proceeds as follows: discrete ranges of possible magnitudes and locations are se- lected and a frequency of occurrence is assigned to each magnitude- location combination. The loss produced by each combination is then estimated using a procedure signaler to scenario-earthquake analysis and the results of all such loss calculations are summa- rized through the loss-frequency curve. (A mathematical formu- lation of this method is given later.) The major departure from existing scenario-type analyses is that, in loss-frequency calculation, one would typically regard the loss from a given magnitude-Iocation combination as a random variable, due to the uncertainties of pre- dicting ground-motion intensity, physical damage, and economic and human losses. The mathematical procedure for this type of analysis follows closely the method of probabilistic seismic hazard analysis described in a report authored by the Committee on Seismology, National Research Council (1987~. Described here is the specific application of these probabilistic concepts to derive estimates of losses and their frequencies of occurrence for a region or metropolitan area. Procedures are well established for estimating the probabilities of seismic ground motion at a point. Three types of input are required: (~) a designation of faults or sources that generate earthquakes and the distribution of earthquake locations on the faults or sources, (2) a description of the distributions of earthquake sizes (magnitude*) and tones of occurrence for each fault or source, and (3) a function that estimates the intensity of ground motion at the site, for earthquakes of specified magnitudes and locations on the faults or sources. With *It is essential to be clear as to which magnitude scale is used.

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124 these three inputs, probabilities that a specified amplitude of ground motion a wiD be exceeded at the site per unit time can be calculated. The total probability theorem is used for this calculation: PI`A > a)`Gime ~ As, Hi / ~ PtA > Am, is fM,Rtm, r`J&m~r. (~) ~ m r in equation 1, Pt.] indicates probability, the vertical bar (~) indicates conditions, f(.) is probability density, the summation is over all sources, i, that might produce ground motions affecting the site, pi represents the expected number of earthquakes per unit time in source i, and m and r are general descriptors of earthquake size (e.g., magnitude) and location tenth respect to the site (e.g., distance). The approximation in equation ~ results from using the expected number of earthquakes rather than calculating probabilities of multiple occurrences and from neglecting the effects of multiple exceedances of amplitude a. For the usual annual probabilities of interest this approximation is very accurate. The formulation of seismic hazard considers (and integrates over) all earthquakes that can affect the site. The resulting annual probability is calculated, in effect, by weighting all these earthquakes by the ground motion that they may produce at the site. Practical applications of seismic hazard analysis are accom- plished in several steps, which lead to the calculation of equation 1. An illustration ~ given in Figure C-2. First, earthquake faults or zones of seismicity must be delineated; from these, the distribu- tion of distance fR(r) between earthquakes and the site is obtained. Next the probability distribution of earthquake magnitudes fat (m) is derived for each source or fault, often by analysis of historical earth- quakes that are spatially associated with that feature. The product of these two distributions is fM,R(m,r) in equation 1. The third specification concerns the ground motion occurring at a site, which is a distribution of ground-motion levels conditional on earthquake magnitude, distance, and local geology. This distribution allows calculation of the probability PEA > a~m,r] in equation I. The equation that calculates a mean or median ground-motion level as a function of m and ~ is often termed an attenuation equation. The final step in the process consists of integrating over all earthquake magnitudes and distances, in the manner of equation 1, to calculate hazard results for various ground-motion amplitudes, as illustrated in Figure C-2d by a typical hazard curve.

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A. Seismic Source i (Earthquake locations in space lead to a distribution of epicentral distances fR (rl m) - / fR(r~ m) ,' RuDture _,_ Site a L. I Distance r C. Ground motion estimation: Alm,r~ cut cut c' a, a - Fault i m =~\ GA|m,r( Distance (log scale) 125 B. Magnitude distribution and rate of occurrence for Source i: f M (m), Hi f M (m) Lo, mO mmax Magnitude m D. Probability analysis: P[ A ~ a in time t] /t ~ Zi V! !|GA~mr(a*) f M (m) f R (r ~ m) dmdr - a, c' 0 ce a, 0 _ CL At_ , ok\ \  Ground Motion Level a. (log scale) FIGURE C-2 Graphs indicating probabilistic seismic hazard analysis steps. Source: McGuire (1987~.

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126 A similar procedure can be applied to calculate annual prob- abilities of earthquake losses from earthquake ground shaking in a region, but additional information is needed. First, a scalar variable needs to be chosen to represent the potential losses (e.g., dollar loss or number of deaths). Second, the correlation of ground motion at different sites must be taken into account. This correlation results because separate sites may be affected simultaneously by the same earthquake, by similar focusing effects of the source, by similar travel paths, and by similar geologic conditions at the site. Just as the un- certainty in ground motion is important in site hazard calculations, PtA > a~m,r] in equation 1, correlation of ground motion at multiple sites is important to regional risk estunates. The estimation of an- nual probability of exceeding a loss $' for the region can then proceed by an enumeration of all earthquakes that might affect the region: P[$ > $ ~ ~ Eui / ,/ Pt$ > $'m' r] fM,R(m, r`)dmdr, {~2~) m r which is similar to equation 1 except that the summation is over all earthquakes that may affect the region of interest, and R represents earthquake location (without reference to a particular site). The conditional probability in equation 2 is evaluated as: P[$ > $' ~ m,r] = Pt(~$j~xj,yj)) > $' ~ m,r), (3) .j where $j (I, y, ) is the loss at location I, yj and the summation is over all locations in the region. The correlation of ground motion enters into this calculation of total loss over all locations in the region for an earthquake of specified size and location. In practice the region is divided into convenient units (e.g., census tracts, statistical areas, or blocks) for this enumeration. The available format for the facility inventory or census information obviously plays a role in choosing the appropriate size of subdivisions for estimating total losses. Several simplifying assumptions are usually made in applying equation 2 to estimate earthquake losses. Often the uncertainty in earthquake losses during a hypothesized earthquake is ignored, leading to a great simplification of equation 3. These uncertainties may result from variabilities in the ground motion generated at the site (whether or not this is used as an intermediate variable), the effect of local geology on ground motion, the loss that might be generated in specified facilities for a given ground motion, and the

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127 number and type of facilities at a given location (e.g., uncertainty in the facility inventory). Ignoring these uncertainties constitutes a simplification that may be justified if, for example, only a best estimate of losses versus annual probability Is desired, but in this case the statistical mean of all relations (rather than, for example, the median) should be usecI. Even then the results only approximate the mean loss, and they are likely to underestimate it, perhaps substantially. Other sunplifying approximations are appropriate under certain conditions. If the region considered for the loss estunate Is rather small (several tens of square kilometers), the integration over loca- tions in the region can be avoided by assuming that the entire region is subjected to the same ground motion. Then, an accurate hazard analysis can be performed for one point (e.g., the geographical center of the region), and the seismic hazard results can be translated to loss estimates. In effect, this assumes that the region Is small enough that the same ground motion occurs over its entirety. In some parts of the United States, the earthquake hazard re- sults from specific fault zones or sources that are small relative to the size of the area that could be affected, for example, the New Madrid fault zone. In these cases the earthquakes occur in an area that is small relative to the region that may be examined for loss calculations, for example, the Mississippi Valley. The loss (more specifically, the range of losses) calculated for a specific magnitude earthquake occurring in the fault zone can then be associated with the annual probability of that event, using the recurrence relation for earthquake magnitudes in that source. In erect, one avoids the integration over location in equation 2; the 1,000~year earthquake is used to estimate the 1,00~year loss. Note that this circumstance does not by itself justify ignoring uncertainty in the resulting losses; as a minimum, mean values (rather than medians) should be used in aD relations to calculate the resulting losses. Simulation of Actual Events In this typical analysis, all of the uncertain input parameters, ex- cept possibly for earthquake magnitude and location, are numerically simulated. Hence, for each given earthquake magnitude and location, different loss scenarios are generated in different simulations. The results are patterns of damage and Toss, which are more realistic

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128 than those produced by bes~estimate scenario analysis. Figure C-lb illustrates results of this type. Cumulative [ose Over Time For applications that depend on the total loss that may be ex- pected in a certain geographical region over a given time period, one should sum losses due to future earthquake occurrences in the region. One use of this result would be to compare earthquake risks with risks from other natural phenomena. For certain uses, one may need only the expected (actualized) cumulative loss, whereas for others it may be important to calculate the entire probability distribution of the cumulative loss. SU~RY This description of loss estimation methods is neither exhaus- tive nor exclusive, meaning that certain applications may require the development of specialized procedures not included in the present list or the combined use of several methods. Generally speaking, deterministic scenario analyses can be made at a level of detail and spatial resolution that is ~rnpractical in probabilistic risk calcula- tions, because of the large number of calculations required by the latter methods. Hence, deterministic analyses (methods A and B as described in Table C-2) might be ideal tools for use in disaster exercises and for the detailed evaluation and improvement of loss reduction strategies. On the other hand, public safety policies (such as the selection of suitable building code provisions) and economic decisions would be best made considering the integrated results of risk studies (method C). The usefulness of scenario type analyses may vary geographically. For example, in regions where events of size close to the magnum possible magnitude occur frequently, a single-event analysis using one such event may be all one needs to make informed decisions. By contrast, in regions where seismicity is low and the maximum earthquake size is unknown, the earthquake threat may be dominated by events in an intermediate magnitude range. ~ the latter case, one should make decisions based on the projected loss from a variety of earthquakes, considering the frequency with which each type of event occurs in the area. For many uses, a fully probabilistic risk calculation by method C is the ideal type of analysis. For example, knowledge of the risk curve

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129 would allow quantitative assessments of public safety with respect to earthquakes and comparisons with other risk sources. Insurance and financial institutions would find risk curves appropriate for the evaluation of expected (Iong-term average) profits as well as for the evaluation of the frequency of catastrophic losses. Another example is the comparison of risk reduction options: different risk reduction or preventive actions might have different effects depending on the earthquake size and the amount of damage. The effectiveness of each proposed action could then be represented in terms of the downward shift that a particular action produces on the original risk function, as shown by the dotted curve of Figure C-Ic. Method D (repeated simulation) is perhaps most appropriate to plan emergency response, when one needs to evaluate the adequacy of response strategies in the context of certain damage and loss scenarios.