APPENDIX A-RECURRENCE MODELS

The earthquake probability estimates in this report are based on current stochastic recurrence models of characteristic earthquakes, including explicit consideration of the uncertainty in the values of the parameters of these models. By "characteristic earthquakes" we mean the relatively narrow range of large events associated with successive "complete" ruptures of a specific segment (for example, Schwartz and Coppersmith, 1984). The frequency of occurrence of such events is not necessarily predicted by extrapolation of the conventional (Gutenberg-Richter) linear (log) frequency-versus-magnitude relationship. Further, because characteristic earthquakes are associated with a "cycle" of major stress drop and stress recovery, it is believed that the interevent, or recurrence, times of these events may follow a temporal pattern associated with a relatively narrow probability distribution (relative in this case to the exponential distribution associated with the reference case, a Poissonian recurrence model). In contrast to the Gutenberg-Richter magnitude-frequency distribution and the familiar Poisson recurrence model, the two general characteristics of this report's models, that is, a relatively narrow magnitude (or slip per event) range and a relatively narrow recurrence time distribution, are consistent with the notions of nearconstant strain rate and a nearly deterministic characteristic earthquake cycle. Furthermore, in this mechanical context, these two characteristics are also consistent with each other. As long as some degree of proportionality exists between actual successive earthquake slips and recurrence times-for example, that proportionality associated with a constant slip rate1-this narrowness of one distribution will imply the narrowness of the other.

There are many probabilistic models that display these two basic characteristics. The discussion here is limited, first, to the simplest, the renewal model, because it has been widely studied, and second, to the time-predictable model favored by the current Working Group. The developments for the first are easily extended to the second. In both cases the recurrence time, T, follows a probability distribution, f,(t), with (marginal) median value, Î, and variability or dispersion measure, o. In this report the dispersion measure is defined to be the standard deviation of the (natural) logarithm of T. In the range of our interest, this parameter is approximately equal to the coefficient of variation, that is, the standard deviation divided by the mean of the recurrence times. Various distribution types have been used in the literature for f,(t), including normal, lognormal, Weibull, and gamma. For any single segment there is insufficient data to distinguish among these distribution types; fortunately the majority of forecasts in this report are insensitive to the choice. The Working Group's general policy has been to retain the assumptions of the 1988 Working Group unless more recent evidence compels us to do otherwise. Therefore the lognormal distribution has been used again in this report. Research by Nishenko and Buland (1987) supports this assumption.

RENEWAL MODEL

The renewal model for characteristic events on a segment is based on the assumption of (probabilistic) independence among the sequence of recurrence times (T1, T2, ...) and the sequence of slips per event (D1, D2, ...). Probability forecasts are based on conditional probability statements, the condition being that no event has occurred between the previous event and the day of the forecast, that is, that a time, T., has elapsed since the last event. For the renewal model, the forecast for the next 30-year interval is written

1

Note, however, that there is always proportionality between the mean (or median) of the slips per event and the mean (or median) of the recurrence times (by definition of the slip rate), even if, for example, the characteristic earthquakes occur in a Poisson fashion.

in which the cumulative distribution function, Fr(t), is related to the density function by

Fr(t) = P(T ≤ 1) = [* 1, (u)du.

(A-1)

(A-2) A graphical interpretation of equation (A-1) is given in figure A-1. Equation (A-1) is equivalent to equation 2 in the main body of the report. Typical plots of the function C versus T. (for given parameter values, Î and σ) are shown in figure A-2. The value of a dictates the sensitivity of the forecast to the elapsed time; for o≈ 1.0, the probability is virtually independent of the elapsed time. (More precisely, for an exponential recurrence distribution, that is, for a Poissonian recurrence model, which has a coefficient of variation of 1.0, C is independent of T.. In other words, the Poisson process has no memory. Note that, in general, when T, is about two-thirds of Î, the hazard is approximately equivalent to that of the Poisson model no matter what the value of a. See fig. A-2.)

PARAMETRIC UNCERTAINTY

In practice it is difficult to know with precision the numerical values of parameters in this model for a specific fault. Following the practice of the 1988 Working Group (and practice in the engineering seismic hazard community), we treat the uncertain parameters in turn as random variables. The simplest model of parametric uncertainty considers only Î, the median, as uncertain and ignores the uncertainty in the dispersion measure. For reasons that will become clear below, this Working Group concurs with the 1988 Working Group in adopting this parametric uncertainty model and, further, in using a common value of 0.21 for the measure of variability of recurrence times. (The basis for this particular numerical value is the report by Nishenko and Buland (1987), who found it to be a representative value for circum-Pacific segments.) It will be seen below that the precise value of this parameter estimate is not critical provided it is less than about 0.3. Again like the 1988 Working Group, we assume that the parametric uncertainty in the median, ↑, can be represented by a lognormal (prior) distribution with a specified best-estimate (median) value and a specified parametric uncertainty measure, denoted σ,, which is the standard deviation of the (natural) log of the uncertain median. In this report, σ, reflects the combined uncertainty in the slip per event and slip rate whose ratio is used to estimate the median ↑. Typical values are about 0.4 (tables 3 and 7).

The simplest way to deal with parametric uncertainty is to "fold it in" with the intrinsic, obtaining what is called a "predictive distribution" on the recurrence T. For the assumptions here it can be shown (for example, Benjamin and Cornell, 1970, Chapter 6) that the predictive distribution of T is again lognormal with median ↑ and net uncertainty parameter, ON:

in which σ, is now used to denote the "intrinsic," random, or event-to-event recurrence-time variability observed on a given segment, the parameter which was set equal to 0.21 in the discussion above. With this approach, one can again use equation (A-1) to calculate "the" conditional probability of an event in the next 30 years given an elapsed time interval of T, years. The result is "the" value of this probability in that it has effectively considered each possible value of ↑ and its relative likelihood (multiplied times the 30-year probability associated with that value of †). As we shall see below, the result can also be interpreted as a mean estimate of this conditional probability,

CONDITIONAL PROBABILITY, C

Figure A-1. Graphical interpretation of C. the conditional probability of T. ≤T≤ T. +30 given T > T.. See text for explanation of variables.

σ = 0.2

σ= 0.4

σ = 1.0

ELAPSED TIME, T.

Figure A-2. Conditional probability. C. of an earthquake in the next 30 years given an elapsed time of T, since the last event, for several values of σ. the degree of dispersion in the recurrence-time distribution. (Assumption: Recurrence interval. Î, is much greater than 30 years.)

C. Equation (A-3) explains why these estimates of C are insensitive to the intrinsic variability o, if the parametric uncertainty, σ,, is approximately 0.4 or more, then the net uncertainty, is insensitive to σ,, provided σ, is less than about 0.3.

It has been found effective when dealing with technical probability assessments to report more than simply a best estimate; we can also make explicit the degree of uncertainty in the estimates. Here the parametric uncertainty in the median (represented by the value of σ) induces uncertainty in C. If we rewrite equation (A-1) as

CT:(T) =

Fr (Te + 30; Î) — Fr(Te; Î) ̧
1- FT(T.; T)

(A-4)

it emphasizes the fact that C is a function of the uncertain parameter Î. (It is understood in this paragraph that the distribution Fr(t; Î) has for its dispersion level only the intrinsic value,

that is, 0.21 in these calculations.) Assuming that C(T) is monotonically decreasing in T in the range of interest, we can find the fractile, c', of CT by calculating the probability that ↑ is less than the corresponding value of the median, t'. (For a given value of c', the corresponding value of t' is found by solving equation (A-4) for T.) To calculate this probability we must use the distribution on the uncertain parameter ↑. This computation is complicated somewhat by the fact that the distribution on ↑ must be "updated" to reflect the information that this particular, current recurrence time is greater than T., the elapsed time since the last event (see, for example, Davis and others, 1989). The updating uses Bayes theorum:

f*(t | T > T.) = k f1(t) P[T > T. | Î = t].

(4-5) In this equation f1(t) is the “prior" distribution on the uncertain median (here lognormal with dispersion σ,), while f(t|T>T.) is the "posterior" distribution (given the observation that T > T.). Note that f(t) is modified by the "likelihood function" (that is, the likelihood of the observation given that the true median, Î, has value t), which here is P[T > T. | Î = t]. This probability is obtained from the (intrinsic; σ = o,) distribution on the recurrence time, T, but as a function of its median, ↑. In this application P[T>T,|Î=t] varies from zero to one as t increases; for example, if the true median is very small, it is unlikely that one would have observed a recurrence interval as large as T.. Therefore, such small values of Î are “downweighted." Finally, the coefficient k in equation (A-4) is a normalizing factor that "ensures" that the posterior distribution on ↑ has unit area. In practice these computations are conducted by numerical integration (or simulation). Making the calculations at a set of values, c', defines the probability distribution on the uncertain forecast Cinduced by the uncertainty in the parameter T. From this distribution one can read specified fractiles, for example, quantiles corresponding to probabilities of 0.25, 0.5, and 0.75. Results of such calculations appear in appendix C. In addition to fractiles, one can calculate the mean of the distribution of C; it can be shown that it is equivalent to "the" probability calculated from equation (A-1) using the predictive distribution on T, that is, using the total uncertainty o (equation (A-3)). Therefore this result, which was also used by the 1988 Working Group, implicitly includes the updating of the distribution on the median due to the "open interval" information, T > T..

TIME-PREDICTABLE MODEL

In contrast to the renewal model, the time-predictable model of characteristic earthquake recurrence is based on the assumption that there is positive correlation between the slip, D., in a particular event on a segment and the subsequent recurrence time, T., to the next event. Further, some form of proportionality is assumed between the recurrence time and the slip. In this report

in which D, is the (random) slip in the ith characteristic earthquake in a sequence, T; is the subsequent recurrence time to the next event, V is a constant (the constant slip rate), and €, is an (independent) random deviation term (with unit median value). Then, as discussed above, the (marginal) median of T is equal to the (marginal) median of D (that is, the median slip per event) divided by the slip rate, V. Conditional on knowing that the slip D, was, say, d, the conditional median of T; is d/V. Further, noting that In T, = − In V + In D, + In e, we see that σ, the marginal standard deviation of the log of T, is o+o, in which σ is the marginal (event-to-event) standard deviation of log D and σ, is the standard deviation of log E. In contrast, the conditional standard deviation of In T (given D.) is only σ.. (We retain the somewhat unusual notation of a for standard deviation of the log of the variable.)

We need not repeat the results (equations (A-1) through (A-5)) for the time-predictable model. All the analysis developed above for the renewal model applies equally well to the timepredictable model, provided one interprets those distributions, parameters, and probabilities as conditional on the slip in the last event. For example, Î and σ in equation (A-1) are now the conditional median and (log) standard deviation given the slip. The probability distribution functions Fr and fr in equation (A-1) and (A-2) are those of the conditional distribution of T given D, and so on.

As stated, the Working Groups utilized the time-predictable model, and therefore the adoption of the lognormal type of distribution and the value σ, = 0.21 are both strictly applicable to the conditional distribution on T given the past slip. For notational and editorial simplicity in the main body of the report, the notion that all is conditional on D d is normally deleted in the presentation. It is implicit. Note, as is clear in the model above, that the conditional (log) standard deviation of T is less than (or equal to) the marginal value. Hence, using 0.21 for the conditional value may be an upper bound because the Nishenko and Buland (1987) analysis, upon which the value is based, was conducted on marginal distributions. In fact there is as yet little evidence to establish the relative values of the marginal and conditional values of these dispersion measures, or equivalently the correlation coefficient between In D and the successive In T. Preliminary investigations show negligible estimated correlation between (estimated) characteristic magnitudes and logs of the succeeding recurrence times on a given segment, but the implied measurement noise (in relation to log slips and log times) is severe.

In the current application of these models to San Francisco Bay region forecasts there is little possibility to distinguish between the renewal and time-predictable model in any case. For virtually every segment there is only one past known earthquake. Therefore the best current estimate of the median slip per event, Ô, is simply the slip in the last event, D. In this case the current estimate of the marginal median of T (that is, D/V) is numerically equal to the conditional median of T given the past slip3 (that is, D/V). The former is used in the renewal model and the latter in the time-predictable model. Provided one continues to use 0.21 for both the marginal and conditional variability measure, the two models will then produce the same forecast probability. As more information becomes available it will be possible to distinguish between the two.

2

For the model in equation (A-6), the correlation coefficient between In T and In D is 03/(03 +03), that is, /02. The renewal model, incidently, is obtained by replacing D; by its median, Ô, in equation (A-6).

3 The slip in the last event, like the "constant" slip rate V, can only be estimated, of course, but that is a separate parameter estimation problem discussed above and in the body of the report.