## 176 Characterization of Seismicity

The previous section described the global distribution of seismicity, in qualitative terms. This section describes how that seismicity may be mathematically characterized, in terms of magnitude-frequency and other relations.

The term magnitude-frequency relation was first characterized by Gutenberg and Richter (1954) as log N (m) = aN — bNm (17.27)

where N(m) is the number of earthquake events equal to or greater than magnitude m occurring on a seismic source per unit time, and aN and bN are regional constants (10aN is equal to the total number of earthquakes with magnitude >0, and bN is the rate of seismicity; bN is typically 1±0.3). Gutenberg and Richter's examination of the seismicity record for many portions of the earth indicated this relation was valid for selected magnitude ranges. The Gutenberg-Richter relation can be normalized to

where F(m) is the cumulative distribution function (CDF) of magnitude, BM is a regional constant, and M0 is a small enough magnitude such that lesser events can be ignored. Combining this with a Poisson distribution to model large earthquake occurrence (Esteva 1976) leads to the CDF of earthquake magnitude per unit time:

which has the form of a Gumbel (1958) extreme value type I (largest values) distribution (denoted EXi l), which is an unbounded distribution (i.e., the variate can assume any value). The parameters aM and mM can be evaluated by a least squares regression on historical seismicity data, although the probability of very large earthquakes tends to be overestimated. Several attempts have been made to account for this (e.g., Cornell and Merz 1973). Yegulalp and Kuo (1974) have used Gumbel's Type III (largest value, denoted EXIII L) to successfully account for this deficiency. This distribution

has the advantage that w is the largest possible value of the variate (i.e., earthquake magnitude), thus permitting (when w, u, and k are estimated by regression on historical data) an estimate of the source's largest possible magnitude. It can be shown (Yegulalp and Kuo 1974) that estimators of w, u, and k can

7Rake is a continuous variable representing the angle between the direction of slip on the fault plane and the strike or the orientation of the fault on the Earth's surface.

' — m\ k be obtained by satisfying Kuhn-Tucker conditions although, if the data are too incomplete, the EXnI,L parameters approach those of the EXIjL:

Determination of these parameters requires careful analysis of historical seismicity data (which is highly complex and something of an art; Donovan and Bornstein 1978), and the merging of the resulting statistics with estimates of maximum magnitude and seismicity made on the basis of geological evidence (i.e., as discussed above, maximum magnitude can be estimated from fault length, fault displacement data, time since last event and other evidence, and seismicity can be estimated from fault slippage rates combined with time since last event, see Schwartz, 1988, for an excellent discussion of these aspects). In a full probabilistic seismic hazard analysis, many of these aspects are treated fully or partially probabilistically, including the attenuation, magnitude-frequency relation, upper and lower bound magnitudes for each source zone, geographical bounds of source zones, fault rupture length, and many other aspects. The full treatment requires complex specialized computer codes, which incorporate uncertainty via use of multiple alternative source zonations, attenuation relations, and other parameters (EPRI 1986; Bernreuter et al. 1989) often using a logic tree format. A number of codes have been developed using the public domain FRISK (Fault Risk) code first developed by McGuire (1978).

Several topics are worth noting briefly:

• While analysis of the seismicity of a number of regions indicates that the Gutenberg-Richter relation log N(M) = a — bM is a good overall model for the magnitude-frequency or probability of occurrence relation, studies of late Quaternary faults during the 1980s indicated that the exponential model is not appropriate for expressing earthquake recurrence on individual faults or fault segments (Schwartz 1988). Rather, it was found that many individual faults tend to generate essentially the same size or characteristic earthquake (Schwartz and Coppersmith 1984), having a relatively narrow range of magnitudes at or near the maximum that can be produced by the geometry, mechanical properties, and state of stress of the fault. This implies that, relative to the Gutenberg-Richter magnitude-frequency relation, faults exhibiting characteristic earthquake behavior will have relatively less seismicity (i.e., higher b value) at low and moderate magnitudes, and more near the characteristic earthquake magnitude (i.e., lower b value).

• Most probabilistic seismic hazard analysis models assume the Gutenberg-Richter exponential distribution of earthquake magnitude, and that earthquakes follow a Poisson process, occurring on a seismic source zone randomly in time and space. This implies that the time between earthquake occurrences is exponentially distributed and that the time of occurrence of the next earthquake is independent of the elapsed time since the prior earthquake.8 The CDF for the exponential distribution is

Note that this forms the basis for many modern building codes, in that the probabilistic seismic hazard analysis results are selected such that the seismic hazard parameter (e.g., PGA) has a '' 10% probability of exceedance in 50 years'' (UBC 1994) — if t = 50 years and F(t) = 0.1 (i.e., only 10% probability that the event has occurred in t years), then l = 0.0021 per year, or 1 per 475 years. A number of more sophisticated models of earthquake occurrence have been investigated, including time-predictable models (Anagnos and Kiremidjian 1984), renewal models (Kameda and Takagi 1981; Nishenko and Buland 1987), and time-dependent models (Ellsworth et al. 1999). The latter have formed the basis for state-of-the-art estimation of seismic hazard for the San Francisco Bay Area by the U.S. Geological Survey, but can only be used when sufficient data are available.

8For this aspect, the Poisson model is often termed a memoryless model.

• Construction of response spectra is usually performed in one of two ways:

A. Using probabilistic seismic hazard analysis to obtain an estimate of the PGA, and using this to scale a normalized response spectral shape. Alternatively, estimating PGA and PSV (also perhaps PSD) and using these to fit a normalized response spectral shape, for each portion of the spectrum. Since probabilistic response spectra are a composite of the contributions of varying earthquake magnitudes at varying distances, the ground motions of which attenuate differently at different periods, this method has the drawback that the resulting spectra have varying (and unknown) probabilities of exceedance at different periods. Because of this drawback, this method is less favored at present, but still offers the advantage of economy of effort.

B. An alternative method results in the development of uniform hazard spectra (Anderson and Trifunac 1977), and consists of performing the probabilistic seismic hazard analysis for a number of different periods, with attenuation equations appropriate for each period (e.g., those of Boore, Joyner, and Fumal). This method is currently preferred, as the additional effort is not prohibitive, and the resulting response spectra has the attribute that the probability of exceedance is independent of frequency.

The reader is referred to Chen and Scawthorn (2002) for a more extensive discussion of this topic.

Glossary

Attenuation — The rate at which earthquake ground motion decreases with distance.

Benioff zone — A narrow zone, defined by earthquake foci, that is tens of kilometers thick dipping from the surface under the earth's crust to depths of 700 km (also termed Wadat-Benioff zone).

Body waves — Vibrational waves transmitted through the body of the earth, and are of two types: (1) P waves (transmitting energy via dilatational or push-pull motion) and (2) slower S waves (transmitting energy via shear action at right angles to the direction of motion).

Characteristic, earthquake — A relatively narrow range of magnitudes at or near the maximum that can be produced by the geometry, mechanical properties, and state of stress of a fault (Schwartz and Coppersmith 1984).

Completeness — Homogeneity of the seismicity record.

Corner frequency, f0 — The frequency above which earthquake radiation spectra vary with —-3 below f0, the spectra are proportional to seismic moment.

Cripple wall — A carpenter's term indicating a wood frame wall of less than full height T, usually built without bracing.

Critical damping — The value of damping such that free vibration of a structure will cease after one cycle (ccrit — 2mo). Damping represents the force or energy lost in the process of material deformation (damping coefficient c — force per velocity).

Dip — The angle between a plane, such as a fault, and the earth's surface.

Dip slip — Motion at right angles to the strike, up- or down-slip.

Ductile detailing — Special requirements such as, for reinforced concrete and masonry, close spacing of lateral reinforcement to attain confinement of a concrete core, appropriate relative dimensioning of beams and columns, 135° hooks on lateral reinforcement, hooks on main beam reinforcement within the column, etc.

Ductile frames — Frames required to furnish satisfactory load-carrying performance under large deflections (i.e., ductility). In reinforced concrete and masonry this is achieved by ductile detailing.

Ductility factor — The ratio of the total displacement (elastic plus inelastic) to the elastic (i.e., yield) displacement.

Epicenter — The projection on the surface of the earth directly above the hypocenter.

Far-field — Beyond near-field, also termed teleseismic.

Fault — A zone of the earth's crust within which the two sides have moved — faults may be hundreds of miles long, from one to over one hundred miles deep, and not readily apparent on the ground surface.

Focal mechanism — Refers to the direction of slip in an earthquake, and the orientation of the fault on which it occurs.

Fragility — The probability of having a specific level of damage given a specified level of hazard.

Hypocenter — The location of initial radiation of seismic waves (i.e., the first location of dynamic rupture).

Intensity — A metric of the effect, or the strength, of an earthquake hazard at a specific location, commonly measured on qualitative scales such as MMI, MSK, and JMA.

Lateral force resisting system — A structural system for resisting horizontal forces, due, for example, to earthquake or wind (as opposed to the vertical force resisting system, which provides support against gravity).

Liquefaction — A process resulting in a soil's loss of shear strength, due to a transient excess of pore water pressure.

Magnitude — A unique measure of an individual earthquake's release of strain energy, measured on a variety of scales, of which the moment magnitude MW (derived from seismic moment) is preferred.

Magnitude-frequency relation — The probability of occurrence of a selected magnitude — the commonest is log10 n(m) = a — bm (Gutenberg and Richter 1954).

Meizoseismal — The area of strong shaking and damage.

Near-field — Within one source dimension of the epicenter, where source dimension refers to the length or width of faulting, whichever is less.

Nonductile frames — Frames lacking ducility or energy absorption capacity due to lack of ductile detailing — ultimate load is sustained over a smaller deflection (relative to ductile frames), and for fewer cycles.

Normal fault — A fault that exhibits dip-slip motion, where the two sides are in tension and move away from each other.

Peak ground acceleration (PGA) — The maximum amplitude of recorded acceleration (also termed the ZPA, or zero period acceleration).

Pounding — The collision of adjacent buildings during an earthquake due to insufficient lateral clearance.

Response spectrum — A plot of maximum amplitudes (acceleration, velocity, or displacement) of a single-degree-of-freedom (SDOF) oscillator as the natural period of the SDOF is varied across a spectrum of engineering interest (typically, for natural periods from 0.03 to 3 or more seconds or frequencies of 0.3 to 30+ Hz).

Reverse fault — A fault that exhibits dip-slip motion, where the two sides are in compression and move away toward each other.

Ring of fire — A zone of major global seismicity due to the interaction (collision and subduction) of the Pacific plate with several other plates.

Sand boils or mud volcanoes — Ejecta of solids (i.e., sand, silt) carried to the surface by water, due to liquefaction.

Seismic gap — A portion of a fault or seismogenic zone that can be deduced to be likely to rupture in the near term, based on patterns of seismicity and geological evidence.

Seismic hazards — The phenomena and/or expectation of an earthquake-related agent of damage, such as fault rupture, vibratory ground motion (i.e., shaking), inundation (e.g., tsunami, seiche, dam failure), various kinds of permanent ground failure (e.g., liquefaction), fire, or hazardous materials release.

Seismic moment — The moment generated by the forces generated on an earthquake fault during slip.

Seismic risk — The product of the hazard and the vulnerability (i.e., the expected damage or loss, or the full probability distribution).

Seismotectonic model — A mathematical model representing the seismicity, attenuation, and related environment.

Soft story — A story of a building signifiantly less stiff than adjacent stories (i.e., the lateral stiffness is 70% or less than that in the story above, or less than 80% of the average stiffness of the three stories above (BSSC 1194).

Spectrum amplification factor — The ratio of a response spectral parameter to the ground motion parameter (where parameter indicates acceleration, velocity, or displacement).

Strike — The intersection of a fault and the surface of the earth, usually measured from north (e.g., the fault strike is N60° W).

Subduction — Refers to the plunging of a tectonic plate (e.g., the Pacific) beneath another (e.g., the North American) down into the mantle, due to convergent motion.

Surface waves — Vibrational waves transmitted within the surficial layer of the earth, and are of two types: horizontally oscillating Love waves (analogous to S body waves) and vertically oscillating Rayleigh waves.

Tectonic — Relating to, causing, or resulting from structural deformation of the earth's crust, (from Greek tektonikos, from tektn, builder).

Thrust fault — Low-angle reverse faulting (blind thrust faults are faults at depth occurring under anticlinal folds — they have only subtle surface expression).

Trans-alpide belt — A zone of major global seismicity, extending from the Mediterranean through the Middle East, Himalayas, and Indonesian archipelago, resulting from the collision of several major tectonic plates.

Transform or strike-slip fault — A fault where relative fault motion occurs in the horizontal plane, parallel to the strike of the fault.

Uniform hazard spectra — Response spectra with the attribute that the probability of exceedance is independent of frequency.

Vulnerability — The expected damage given a specified value of a hazard parameter.

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### Further Reading

There is a plethora of good references on earthquakes. Chen and Scawthorn (2002) provides an extensive reference. The reader is recommended to Earthquakes by B.A. Bolt (1993, Freeman, San Francisco) for an excellent and readable introduction to the subject; to The Mechanics of Earthquakes and Faulting by C.A. Scholz (1990, Cambridge University Press, New York) for an erudite treatment of seismogenesis; to The Geology of Earthquakes by R.S. Yeats, K. Sieh, and C.R. Allen (1997, Oxford University Press, New York) for an exhaustive review of faulting around the world; and to Modern Global Seismology by T. Lay and T.C. Wallace (1995, Academic Press, New York) for a readable theoretical text on seismology (a very rare thing).

Mark Yashinsky

Division of Structures Design California Department of Transportation, Sacramento, CA

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