Estimating Surface Rupture Length and Magnitude of Paleoearthquakes. From Point Measurements of Rupture Displacement

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1 Estimating Surface Rupture Length and Magnitude of Paleoearthquakes From Point Measurements of Rupture Displacement Glenn P. Biasi and Ray J. Weldon II August 16, 24 Abstract. We present a method to estimate paleomagnitude and rupture extent from measurements of displacement at a single point on a fault. The variability of historic ruptures is summarized in a histogram of normalized slip, then scaled to give the probability of finding a given displacement within a rupture for any magnitude considered. The histogram can be inverted to yield probability density functions of magnitude and rupture length for any given displacement measurement. The probability by magnitude of an earthquake actually producing ground rupture are included in the inversion. Probability distributions can be improved if the distribution of earthquake magnitudes for the fault is known. The Gutenberg-Richter magnitude distribution predicts shorter rupture lengths and smaller magnitudes than does a uniform distribution where any magnitude earthquake is considered equally likely. Longer ruptures and larger magnitudes than the uniform model are predicted by an alternative magnitude distribution designed to return site average displacement. This model is a generalization of the characteristic earthquake model, and reasonably describes paleoseismic findings on the southern San Andreas fault, where slip is accommodated by infrequent ruptures with large average 1

2 displacements. Our results should increase the value of paleoseismic displacement measurements for hazard assessment. In particular, they quantify probability estimates of earthquake magnitude and rupture length where point observations of rupture displacement are available, and so can contribute to probabilistic seismic hazard analyses. Introduction Paleoseismic investigations have had good success in locating and dating preinstrumental earthquake ruptures, but have been more limited in their ability to estimate paleoearthquake magnitude or rupture length. For example, long, well dated event chronologies at Pallett Creek and Wrightwood on the San Andreas fault in California (Sieh et al., 1989; Fumal et al. 1993, 22a) constrain recurrence rate and suggest patterns in underlying fault behavior (e.g., Sieh et al., 1989; Biasi et al., 22; Weldon et al., 24). However, slip estimates for individual ruptures (e.g., Sieh, 1984; Salyards et al., 1992; Grant and Sieh, 1994; Weldon et al., 22, 24; Liu, et al., 24) have thus far contributed only a general understanding of paleoearthquake magnitude. Because these measurements are, by their nature, point estimates of slip and rupture behavior, it is impossible to say whether the observed slip is representative of the average over the entire rupture, or whether it happens to be more or less than average. Rupture length is even less constrained than average displacement by studies at individual sites. Lengths have been estimated primarily from by speculative correlation of events using overlap their age ranges (Sieh et al., 1989; Weldon et al., 24). Since seismic moment estimates depend directly on rupture length and average displacement (M o = µdlw, where M o is seismic moment, L = rupture length, d=average slip, W = rupture width, and µ = the rock 2

3 shear modulus), detailed chronologies recovered from the best paleoseismic sites provide the frequency of ground rupture, but only weakly constrain the socially more important frequency of earthquakes of known sizes. Hemphill-Haley and Weldon (1999) proposed a method of estimating average displacement from point displacement measurements. By a Monte Carlo method they showed that a reasonably precise estimate of average slip could be made if five to ten measurements of slip, preferably well distributed along of the fault, were available. The difficulty in applying their method is that even on a relatively well studied fault such as the southern San Andreas, very few slip estimates are available before the most recent earthquake. Furthermore, earthquake dates are never precise enough to show that the same rupture is being observed at multiple sites. Chang and Smith (22) used an alternative means to estimate average displacement from measurements in paleoseismic excavations. They assume that rupture profiles are elliptical in shape, with the lengths of the ruptures taken from a segmentation model and the heights taken from trenching displacement measurements and trench position within the segment. Average displacement and magnitude are estimated from the resulting ellipses. This approach would not be suitable where the segmentation model is unknown or disputed, or where the elliptical rupture shape cannot be assumed. To develop a probabilistic estimate of magnitude given a point measure of slip, we begin by examining the natural variability of slip along strike for ground-rupturing earthquakes. Slip distributions of historic earthquakes have some common features that allow them to 3

4 be summarized in the form of an empirical probability distribution. This empirical probability distribution can be scaled to give the forward probability of finding a surface displacement given earthquake magnitude. We then invert this relationship for the probability of earthquake magnitude and rupture length given a measured displacement. Average slip and rupture length given magnitude Wells and Coppersmith (1994) obtained relationships of average surface displacement and rupture length to magnitude from published reports of surface ruptures. Using data from all types of faults they found magnitude M and surface rupture length L related by: Equation 1: M = *log(L) To relate magnitude and average displacement, d ave, we combined the forward and inverse relationships in Wells and Coppersmith (1994) for M given d ave and d ave given M: Equation 2: M = *log(d ave ) Equation 2 balances the misfit of M and d ave data and gives a reversible relationship for M and d ave. Equations 1 and 2 were derived from observations of, respectively, 77 and 56 mapped surface ruptures ranging in magnitude from 5.6 to

5 Incorporating slip variability Hemphill-Haley and Weldon (1999) investigated the variability of surface slip for a selection of well documented earthquakes. Figure 1 illustrates slip variability as a function of position along several mapped fault ruptures. As may be seen, slip at a point commonly can be a factor of two and the maximum a factor of three larger than the average slip along the rupture. To characterize surface rupture variability, each rupture profile was sampled at 1% intervals and presented in the form of a histogram (Figure 1, insets). The histograms retain the variability of the mapped surface slip, but summarize it by showing only the frequency of occurrence of a given slip without specifying the spatial patterns of individual measurements. This summarization is useful because in the paleoseismic context, trenching at random within the rupture and drawing a displacement at random from the histogram are equivalent procedures. While individual slip distributions can be quite variable, Hemphill-Haley and Weldon (1999) noted two important features about them. First, slips tend to be as variable for small earthquakes as for large ones. That is, one could not tell from the shape of the slip distribution alone whether a large or small earthquake was plotted. Second, all distributions necessarily have ends and tend to taper to small slip offsets as the ends are approached. This suggests an approximate shape upon which the variability is expressed. Hemphill-Haley and Weldon (1999) used these qualities as a means of combining slip distributions of small and large earthquakes. They normalized each observed slip distribution by the average displacement for that event, and the length by the total 5

6 observed length. From these, two stacked distributions were developed (Figure 2). When the normalized rupture profiles themselves are stacked (Figure 2a), the average rupture profile shape includes a nearly flat central part amounting to approximately one third of the total slip length and tapers on each end. The average shape is far less variable than any contributing rupture because of the tendency for above average normalized displacement in one event to be matched by another that is below average. The histogram drawn from averaging the ruptures (Figure 2b) has far less variability than actual ruptures (Figure 1) demonstrating that much of the slip variability has been averaged out. On the other hand, when the individual histograms themselves are stacked (Figure 2c), variability is preserved, including extreme normalized values (e.g., d/d ave =3), which are retained but weighted by their relative frequency. For this reason the averaged histogram is useful for inverting rupture variability while the histogram of the average rupture profile (Figure 2b) is not. Probability of Slip Given Magnitude A key to the Bayesian inversion of slip observations for earthquake magnitude and rupture length is the observation that, if given a unit area, the histograms of individual earthquake ruptures (Figure 1) may be interpreted as probability density functions for slip during that earthquake. Probability of slip in a given range around a given displacement (e.g., meters) can be found by integrating the appropriate portion of the histogram. This interpretation also applies to the average histogram (Figure 2c). If Equation 2 is then used to scale the histogram in Figure 2c it becomes a probability density function for 6

7 surface displacement given magnitude, p(d obs M(d ave )). Individual probability density functions p(d M) are shown in Figure 3 for example magnitudes. The histogram reflects the expectation that, given M, both smaller and larger values of displacement than the mean from Equation 2 are expected within the rupture. Bayesian inverse for magnitude given slip. Bayes Theorem allows the slip variability p(d M) to be inverted for the probability distribution of earthquake magnitude given an observed displacement, p(m d obs ). Bayes Theorem may be stated as, Equation 3 p(m d obs ) = p(m)[p(d obs M) /P(d obs )]. Briefly, Equation 3 revises the a priori distribution of earthquake of magnitude M, p(m), by its likelihood in light of a paleoseismic displacement observation. The distribution of magnitudes is unknown, but has basic limits. For the moment, we assume that any magnitude in the range M6.6 to M8.1 in.1 magnitude unit increments is equally likely to produce ground rupture and thus p(m) is uniformly distributed on this range. We explore the effects of the range and shape of p(m) in later sections. To estimate the relative weighting of individual magnitudes in Equation 3, the slip histogram from Figure 2c is scaled for each by the average displacement predicted by 7

8 Equation 2 (Figure 3). The probability that magnitude M i caused observed slip d obs is the ratio of the area in the i th histogram near d obs, P(d:d-δ <=d obs <=d+δ M i ) to the total area in all the histograms in this displacement range. The relative likelihoods among candidate magnitudes is given by the area of each as their fraction of the total. P(d obs ) in Equation 3 is that total area. The shaded vertical bar in Figure 3 corresponds to an example slip of d=2 +/-.25 meters. For each magnitude, prior distribution p(m) then multiplies the fraction to yield p(m d obs ). Figure 4 shows the inverse probability distribution functions p(m d obs ) for d obs = 1, 2, and 4 meters. For the example d obs =2m, Figure 4b shows that it would be an unlikely random displacement measurement from ruptures of M<7. earthquakes. For M7.4 the fractional weight P(d:d-δ <=d obs <=d+δ M) is near its maximum. Larger earthquakes have some part of their length around 2 meters, but their fraction of all expected displacements begins to decline. That is, for M>7.4 events, two meter displacements would be somewhat less likely because larger displacements are expected. Probable Rupture Length To estimate probabilities of rupture length, we use p(m d) just calculated, and the scaling of M to L in Equation 1. Figure 5 shows the result in the form of cumulative probabilities instead of the density functions in Figure 4. As an example, if a 2 meter observed displacement is drawn at random, Figure 5 predicts a 75% likelihood of a total rupture length of at least 5 km. 8

9 The Role of Limits on the Magnitude Range Thus far we have assumed that earthquake magnitudes are uniformly distributed in the range 6.6<=M<=8.1. However, smaller earthquakes can produce ground rupture, and for some faults, larger earthquakes may occur. Qualitatively the effect of the lower limit can be seen in Figure 3 and Equation 3. Magnitudes whose upper displacement range is less than a given d obs contribute zeros to P(d M) and P(d) in Equation 3. However, for smaller displacements the lower limit is relevant. Figure 6 (dashed) shows p(m d) where the lower limit of magnitudes has been decreased to M 6.. Decreasing the minimum magnitude limit adds ways that a 1 meter displacement could be explained, namely by events of M<6.6. This slightly decreases the probability that it was caused by a M>=6.6 event. For d obs of 2 meters or more, the lower bound has no effect. The upper magnitude limit of p(m) does have an effect on p(m d), since unlike smaller earthquakes, the largest magnitude earthquakes include displacements of all sizes. Figure 6 (dotted lines) illustrates the effect for P(M) uniformly distributed as 6.6<= M <=8.4. The relative weights of, say, M 7.2 vs. M 7.4 changes little, but the absolute probabilities would decrease slightly. The difference is largest for large observed displacements. We adopted a maximum case of M 8.1 for the balance of this paper because this is the limit of data in the Wells and Coppersmith (1994) regressions, but Figure 6 shows that larger or smaller magnitude limits are readily accommodated. 9

10 The Role of Magnitude Distribution Thus far we have assumed that earthquakes in a large magnitude range are equally likely to have caused the observed surface displacement. This assumption corresponds to the least informative prior distribution p(m) in Equation 3. Sometimes, however, information is available with which to further shape p(m). We consider two alternative magnitude distribution models. One is a modified Gutenberg-Richter relation and the other is an average displacement earthquake model that assumes that a fault produces a narrower range of earthquakes and larger individual ruptures. We also incorporate the probability that an earthquake will rupture the surface. This becomes particularly important when P(M) includes smaller magnitude ranges. It is well known that most small earthquakes, and many moderate ones, do not rupture the ground surface. Thus, one cannot directly compare a distribution of magnitudes for a fault to a record of paleoearthquakes. While rupture is known for events as small as M 3 (Bonilla, 1988) very few earthquakes smaller than M 5 rupture the surface, and for crustal faults that reach the Earth s surface (that paleoseismic investigations could be carried out on), it is rare to not to see rupture for events greater than M 7 (McCalpin, 1996). To estimate the probability of rupture we replotted the data in Wells and Coppersmith (1993) in a form that gives the fraction of earthquakes that produce rupture as a function of magnitude (Figure 7). For their compilation they regard as negligible the probability of ground rupture for earthquakes smaller than M 5. The resulting curve is consistent with the observation of Bonilla (1982) that rupture becomes likely (passes 5% likelihood) 1

11 at about M 6 in the western U. S. Our goal here is not to argue strongly for this distribution, but to demonstrate how the known decline in rupture probability with decreasing magnitude may affect our results. To account for the likelihood that an earthquake in our magnitude distribution will be recorded at the paleoseismic site as a surface rupture we add a term to Equation 3: Equation 4 p(m d obs ) = [P(rupture M)*p(M)][P(d obs M) /P(d obs )]. P(dobs) remains the sum of possible outcomes of the numerator, but is numerically different from its value in Equation 3. Because of the increasing improbability of rupture with decreasing magnitude, a side effect of including the P(rupture M) term is that it decreases the importance of the lower limit in P(M) (Figure 6a). Figure 8 shows the how the probability of rupture modifies the three probability distributions of magnitude we consider. All span the distribution range 5. M 8.1. The modified uniform distribution (Figure 8a, solid line) is, in effect, a plot of the probability of rupture itself. In the Gutenberg-Richter (GR) model the number of earthquakes of size M is given by N(M) = a +blog 1 (M), where a relates to fault productivity and b is typically about 1.. This model predicts 1 times more earthquakes of M 6. than M 7.. The exponential increase in the number of earthquakes with decreasing magnitude is checked by the declining probability of rupture, so that the probability of magnitude peaks at about M5.4 and decreases to zero by M5 (Figure 8b, solid line). 11

12 The GR model is seen to exert a strong influence on the calculated magnitudes and rupture length; compared to a uniform model, using the Gutenberg-Richter model in Equation 4 raises the relative probability that a given displacement observation came from an above average slip point of a smaller magnitude earthquake The average displacement p(m) model is intended to model the common paleoseismic case where ground rupture is taken up in a few earthquakes with a large average displacement. The size of any given earthquake is not seriously constrained, but the average displacement over many events is set to the geologic or geodetic average for the fault at the location of interest. The characteristic earthquake model (Schwartz and Coppersmith, 1984) is an extreme form of this model. In the characteristic earthquake distribution there is little variability about the mean displacement for an individual fault (or fault segment if the fault is long enough to rupture in multiple pieces). Because the fault or segment length is essentially fixed and displacement varies little between events, the magnitude range of the earthquakes produced is limited. To construct the average displacement model, an average displacement is selected with a standard deviation consistent with the width of the scatter in the data of Wells and Coppersmith (1994) for that magnitude. We then convert this displacement distribution into a magnitude distribution using the displacement-magnitude scaling relationship in Equation 2. (This application motivated our modification of Equation 2 from the original result of Wells and Coppersmith, 1994). 12

13 To illustrate the average displacement model we use a mean displacement of 4.3 meters (Figure 8c). This choice is motivated by a typical San Andreas fault offset, so that we can compare the results to a familiar fault and dataset. The exact number, 4.3 m, is based on an average recurrence interval of 135 years derived from Pallett Creek paleoseismic site (Sieh et al., 1989; Salyards et al, 1992; Biasi et al., 22), and a slip rate of ~32 mm/yr. The upper range was truncated at M 8.1 to facilitate direct comparison with the other p(m) models. Our intention with this model is to illustrate the consequences of paleoseismic data indicating that slip occurs in a few large ruptures, and not to argue that our implementation is necessarily unique or even optimal. From a practical point of view one could construct such a distribution from the mean value of a population of observed offsets on a fault, were a sufficient number of ruptures to be available, or from the slip rate and recurrence interval (mean displacement=recurrence interval*slip rate). The results for the three p(m) models are shown in Figure 9. Each p(m) model has different properties. In application, the GR p(m) model implies that one should rarely expect a slip of one meter to have been caused by an earthquake above the low M 7 range. The difference decreases compared to the uniform p(m) model with increasing values of d obs, but for all values the mean earthquake size given d obs is smaller than the uniform case. (Figure 9b-f). The average displacement p(m) model also exerts a strong influence on consequent p(m d) and p(l d) distributions (Figure 9, dashed). For small d obs, the rarity of smaller rupture-producing earthquakes in p(m) means that even small observed displacements are attributed to mid-m 7 and larger earthquakes. It is opposite in that respect to the up-weighting of the frequency of small to moderate earthquakes by 13

14 the GR model. For the largest observed displacements the opposite occurs; because the size of possible earthquakes is limited to a range about the average offset, offsets significantly larger than the mean are, in effect, interpreted as points of unusually large slip within an approximately average rupture. As a stricter form of the average displacement model, the characteristic earthquake model would produce even narrower p(m d) distributions. Discussion Hemphill-Haley and Weldon (1999) inferred that surface displacements are usefully similar when normalized by average displacement and surface rupture length. This underlies the scaling employed in Figure 3 and all subsequent results. The variability of distinct magnitude levels might be tested if a much larger ground rupture set were consulted. Histograms developed by magnitude might be constructed instead of the scaling method of Figure 3. Another obvious refinement to our results would be to separate a larger data set by tectonic setting or fault type. The average histogram we used includes earthquake ruptures from all the principal tectonic styles, and reasonably represents the individuals from which it was compiled. Differences among strike-slip, reverse, and normal faults suggest that factors such as the continuity of rupture and the ratio of the average displacement to the maximum may vary systematically. For example, McCalpin and Slemmons (1998) found differences among fault types when maximum displacement was used to scale ruptures. It remains to be seen, however, whether likely differences between fault types or environments are sufficient to be worth 14

15 implementing because the large uncertainties in the a priori magnitude distribution model and the huge variability in offset along individual ruptures. The inversion for p(m d) and p(l d) assume that the observed displacement is drawn at random from within the rupture. This assumption may suit the case where little else is known about the rupture. In some cases, however, the observed displacement may be known to be exceptional. Exceptionally large displacement observations compared to the rupture average correlate with higher probabilities of preservation and discovery in the field. Small offsets in a strike-slip environment are harder to detect and less likely to be preserved and studied. Biased sampling of slip observations can also occur at splays and step-overs where rupture trails off on one trace and is taken up on a parallel trace some distance away (e.g., Hog Lake, Rockwell et al., 23; Pallett Creek, Sieh, 1984; Frazier Park, Lindvall et al., 22). As shown by the average displacement model above, if the recurrence interval and fault slip rate are available, per-event slip estimates may not be crucial for seismic hazard estimation purposes. Deciding that an observed displacement is greater or less than average for a rupture is, ultimately, a professional judgment that will affect probability estimates of magnitude and rupture length. Figure 1 suggests another way, in principle, by which to find a biased observation of displacement. If it is known that the observation comes from one or other end of a rupture, then smaller than average displacements may be expected. Chang and Smith (22) applied this idea in their study of the Wasatch Frontal fault system to estimate 15

16 paleomagnitudes from trench offsets. If the segment boundaries are known, and ruptures honor the boundaries, adjustment of the observed displacements might be entertained. Bayesian inverse models characteristically depend on the prior model to be shaped by the data. The uniform prior p(m) model is usually considered to be the least informative prior upon which to apply p(d M). Mathematically the information in the uniform p(m) model is in where one sets the ends. By including the probability of rupture (Figure 7), the lower magnitude bound can be arbitrarily small. The upper magnitude bound (Figure 8a) has little effect on the relative weight among smaller events. For a fault about which little is known, or where one is reluctant to assert additional information, a uniform prior case model that spans the range of conceivable magnitudes is perhaps the best choice. The Gutenberg-Richter model was developed to characterize seismicity in a region, and includes an exponential increase in the number of smaller earthquakes. Even when the probability of rupture is included, the resulting p(m) still predicts that most ground ruptures are caused by the relatively smaller end of the magnitude range (Figure 8b). In the field assuming this model will have the effect of preferentially attributing rupture to relatively smaller earthquakes, just because a greater fraction of all ruptures one might find are in this magnitude range. The applicability of the model to individual faults has been debated (Wesnousky, 1994; Scholz, 22; Stirling et al., 1996), but nevertheless the GR p(m) prior model might be a good choice when studying a region about which little is known. 16

17 The average displacement p(m) model (Figure 8c) also strongly shapes p(m d) and p(l d). Applied to the San Andreas fault, however, we would argue that it is informed, at least, by paleoseismic data taken from the fault. The features of a paleoseismic record most important for the average displacement model are its completeness of event detection and the approximate date of the oldest event in the complete section. From the Carrizo Plains to Indio (Grant and Sieh, 1994; Liu et al., 24; Sieh et al. 1989; Fumal et al. 22a; Seitz et al. 1997; Yule and Howland, 21; Fumal et al., 22b; Sieh, 1986) records interpreted to be fairly complete require average slips of a few meters per event if one accepts the reported recurrence intervals the geodetic slip rates. This requires some sort of enforcement of an average slip and thus a shape on p(m). Figure 8c shows that a strong penalty against M<7.2 events is needed to achieve average displacements of 4 meters or more. Better ideas on the precise shape of the average displacement model may emerge, but even in the form of Figure 8c it has at least this, that the long paleoseismic records on the San Andreas fault favor it. The average displacements implied by the p(m) models in Figure 8 contribute to which may be preferred in a given situation. Average displacement is computed by summing the probability of each magnitude times the displacement predicted from Equation 2. Average displacements for the models of Figure 8, after the correction for probability of rupture, are 2.65,.23, and 4.3 meters for the uniform, GR, and average displacement models, respectively. The average displacement for the uniform p(m) model is most affected by the upper magnitude bound. The Gutenberg-Richter average displacement is least affected by the choice of a maximum magnitude because while rupture displacement 17

18 increases exponentially with magnitude (under the regression), the frequency of the largest events decreases exponentially, so that the average displacement increases only very slowly with the maximum magnitude. The average displacement models are built to match site average displacement, and thus have to be evaluated by other criteria. Some modification of the probability of rupture (Figure 7) might be argued on the grounds that small displacements are less likely to be detected with paleoseismic methods, but reasonable modifications are unlikely to change the general properties of the p(m) models. An immediate application for the p(m d) and p(l d) relationships in Figures 4 and 5 is to help quantify certain elements of probabilistic seismic hazard analysis. Logic tree assessments typically recognize a range of magnitudes. Where rupture displacement information is available, the present results may be applied directly or with suggested adjustments to quantify branch weights for magnitude and rupture length. Conclusions We show that probabilities of magnitude and rupture length can be developed given a displacement measurement from a paleoseismic excavation. Rupture variability, an historical hindrance to interpreting individual displacement measurements, is summarized in a histogram that captures the variability without prescribing how it is distributed within a rupture. A fact making the inversion possible is that sampling a histogram of slip measurements at random is equivalent to sampling in a random location within the 18

19 corresponding rupture. With the results presented here, paleoseismic displacement measurements from ground ruptures become a new resource for estimating paleomagnitude and length. The average histogram of variability can be interpreted as the probability of displacement given magnitude, p(d obs M) once it is scaled using a regression of average displacement versus magnitude. Bayes Theorem allows us to invert p(d obs M) for probabilities of earthquake magnitude and rupture length given point observations of displacement, p(m d) and p(l d), respectively. These distributions are less than the explicit answer to the question, How big was it?, but they do quantify the probability of any magnitude range or length estimate given a rupture displacement measurement. These are common input parameters to probabilistic seismic hazard analysis. The inverse probabilities for magnitude and length do depend on the magnitude distribution model assumed as an input. We analyze three, the uniform, Gutenberg- Richter, and average displacement models. We modify of each to account for the decreasing probability of rupture with decreasing magnitude. The least restrictive form of p(m), the uniform distribution, assumes any magnitude is as likely as another. This model might be used when one does not wish to assume in advance any magnitude distribution. The Gutenberg-Richter magnitude distribution is a very strong prior assumption about p(m), and it strongly biases probabilities given a displacement observation toward smaller magnitudes and lengths. Average displacement for the model depends little on the maximum magnitude chosen because the presumed exponential 19

20 increase in displacement is offset by an exponential decrease in probability of occurrence. The Gutenberg-Richter model might be suitable to studies of regions where little is known about individual faults. Paleoseismic findings of large average slip per event, such as occur on the southern San Andreas fault, motivated the third model. Called the average displacement model, it can be viewed as a generalization of the characteristic earthquake model. It is formed using a range and probability of magnitudes designed such that sampling from it returns the average slip known from the recurrence interval and geologic or geodetic slip rate on the fault. Slip-per-event is not otherwise constrained, distinguishing it in that respect from the characteristic earthquake model. For faults with large average displacements, this model predicts that even modest rupture displacement observations of a meter or two likely correspond to ruptures over 1 km in length. Virtually all paleoseismic studies of the southern San Andreas fault indicate that slip is accommodated in relatively infrequent ruptures with a few meters average slip. Thus, among the models considered, the average displacement model is preferred for the southern San Andreas fault. Captions Figure 1. Rupture profiles for seven events used in Hemphill-Haley and Weldon (1999). Profiles have been normalized to unit length. Histograms are normalized to mean slip. Rupture variability is seen at all scales. For most earthquakes most slip is less than twice the average, but with small probability, slips up to three times the average are observed. 2

21 Figure 2a. If the rupture profiles themselves are normalized and stacked, realistic slip variability is removed and a fairly smooth semi-elliptical rupture profile results. (b) The histogram of the average rupture profile in (a) is too peaked around the maximum slip (~1.3 times the average) to approximate any of the contributing profiles. (c) Averaging the histograms yields a useful average rupture variability because histograms specify variation without constraining its pattern. Figure 3. Average histogram from Figure 2c scaled using Equation 2 for four example magnitudes. Constant bin widths of.5 meters are used, giving the appearance of different underlying data. The original observed data and not the histograms are used for actual inversions. Vertical bars on an example observed displacement of 2±.3 m suggest the concept of the Bayesian inversion. For a given magnitude, the area within the bars as its fraction of the individual histogram area is the probability of finding a displacement in that range. The probability that the i th magnitude is the correct one is the fraction its area between the bars to the total area over all magnitudes. This is the contribution of the i th magnitude to P(d obs M) /P(d obs ) in Equation 3. Figure 4. (a) Probability distribution function for earthquake magnitude given an observed displacement of 1 meter. P(M d) is peaked around M 6.7 for a 1 meter displacement because smaller earthquakes are unlikely to produce so large a slip, and less likely for much larger earthquakes because they should produce larger displacements. (b) Same as (a), but for d obs = 2 m. Magnitudes from M 7.1 to M7.3 are most likely to have caused d obs, but as in (a), smaller and larger earthquakes are possible. (c) Same as (a) and 21

22 (b), but for d obs =4 meters. Irregularities in the probabilities are caused by the fine details in the shape of the average histogram (Figure 2c). Figure 5. Probability of surface rupture length given 1 d obs 6 m. In this figure earthquakes of any magnitude are considered equally probable. Figure 6. Parametric study of the effect of the width of p(m) on p(m d obs ) when the lower limit of earthquake magnitudes contributing to ground rupture is decreased to M 6. (dashed) or the upper magnitude limit is extended to M 8.4 (dots on solid line). At M 8.4 Equation 1 predicts a rupture length of over 7 km, long enough to rupture the entire southern San Andreas fault. For d obs = 2 and 4 m, decreasing the lower limit to M 6. has no effect, and the dashed line is not visible. Increasing M max causes peak probabilities to decrease and spreads the probability into larger magnitudes. Figure 7. Probability of ground rupture given magnitude. Data (solid line) are from Wells and Coppersmith (1993). The probability of rupture for earthquakes with M<5. is neglected. The dashed line is the fit to the data used for P(rupture M) in Equation 4. Figure 8. Three p(m) models. Dashed is the original, solid line is after adjustment for the probability of rupture. (a) Uniform (any event magnitude in a range is as likely as another), (b) Gutenberg-Richter, truncated at the same maximum magnitude, and (c) an average displacement model, which, by way of Equation 2, approximately enforces a 4.3 meter average displacement. This average displacement model is patterned to apply to 22

23 the paleoseismic record at Pallett Creek, California (Sieh et al., 1989) on the San Andreas fault. The average displacement p(m) model allows smaller and larger events in a paleoseismic event sequence, but probabilities of magnitude are adjusted so that on average sampling recovers the average displacement. The average displacement model will be different for sites with different recurrence and slip rate data. Figure 9. P(M d obs ) and P(L d obs ) for Gutenberg-Richter (dots on solid line) and 4.3 meter average displacement (dashed) p(m) models from Figure 8 for d obs from 1 to 6 meters. For the average displacement model, d obs =1 m is below the distribution average, so probabilities of magnitude and length are biased larger. In effect, given the magnitude distribution, a 1 meter observation is reweighted as an anomaly from a larger earthquake. For displacements larger than the mean the situation reverses, and p(m) causes larger d obs to be interpreted as large outliers of a slightly smaller magnitude and rupture length. Acknowledgements This work was supported by the National Earthquake Hazards Reduction Program, Cooperative Agreements 1HQAG9 and 4HQAG4, and the Southern California Earthquake Center. SCEC is funded by NSF Cooperative Agreement EAR and the USGS Cooperative Agreement 2HQAG8. The SCEC contribution number for this paper is

24 References Biasi, G.P., R. J. Weldon II, T. E. Fumal, and G. G. Seitz (22). Paleoseismic event dating and the conditional probability of large earthquakes on the southern San Andreas fault, California, Bull. Seism. Soc. Am. 92, no. 7, Bonilla, M. G. (1982). Evaluation of potential surface faulting and other tectonic deformation, U.S. Geol. Surv. Open File Rept , 91 p. Bonilla, M. G. (1988). Minimum earthquake magnitude associated with coseismic surface faulting, Bull.Assoc. Eng. Geol., 25, Chang, W. L. and R. B. Smith (22). Integrated seismic-hazard analysis of the Wasatch Front, Utah, Bull. Seism. Soc. Am. 92, no. 5, Fumal, T. E., Weldon, R. J., G. P. Biasi, T. E. Dawson, G. G. Seitz, W. T. Frost, and D. P. Schwartz (22a). Evidence for large earthquakes on the San Andreas fault at the Wrightwood, California, paleoseismic site: A.D. 5 to Present, Bull. Seism. Soc. Am. 92, no. 7, Fumal, T. E., M. J. Rymer, and G. G. Seitz (22b). Timing of large earthquakes since A.D. 8 on the Mission Creek strand of the San Andreas fault zone at Thousand Palms Oasis, near Palm Springs, California, Bull. Seism. Soc. Am. 92, no. 7, Fumal, T.E., S. K. Pezzopane, R. J. Weldon II, and D. P. Schwartz (1993). A 1-year average recurrence interval for the San Andreas fault at Wrightwood, California, Science, 259,

25 Grant, L. B. and K. Sieh (1994). Paleoseismic evidence of clustered earthquakes on the San Andreas fault in the Carrizo Plain, California, J. Geophys. Res, 99, Hemphill-Haley, M. A. and R. J. Weldon II (1999). Estimating prehistoric earthquake magnitude from point measurements of surface rupture, Bull. Seism. Soc. Am. 89, Lindvall, S. C., T. K. Rockwell, T. E. Dawson, J. G. Helms, and K. W. Bowman, (22). Evidence for two surface ruptures in the past 5 years on the San Andreas fault at Frazier Mountain, California, Bull. Seism. Soc. Am. 92, no. 7, Liu, J., Y. Klinger, K.Sieh, and C. Rubin (24). Six similar sequential ruptures of the San Andreas fault, Carrizo Plain, California, Geology, 32, McCalpin, J. (1996). Paleoseismology, Academic Press, 588 pp. McCalpin, J. and D. B. Slemmons (1998). Statistics of paleoseismic data, Final Technical Report, Contract 1434-HQ-96-GR-2752, U.S.G.S. National Earthquake Hazards Reduction Program, 62 pp. Rockwell, T. K., J. Young, G. Seitz, A. Meltzner, D. Verdugo, F. Khatib, D. Ragona, O. Altangerel, and J. West (23). 3, years of ground-rupturing earthquakes in the Anza Seismic Gap, San jacinto fault, southern California: Time to shake it up?, Seismol. Res. Lett., 74, 236. Scholz, C. H. (22). The Mechanics of Earthquakes and Faulting, Cambridge University Press, Cambridge, 471 pp. Schwartz, D. P. and K. J. Coppersmith (1984). Fault behavior and characteristic earthquakes Examples from the Wasatch and San Andreas fault zones, J. Geophys. Res. 89,

26 Salyards, S. L., K. E. Sieh, and J. L. Kirschvink (1992). Paleomagnetic measurement of non-brittle coseismic deformation across the San Andreas fault at Pallett Creek, J. Geophys. Res. 97, Seitz, G. G., R. J. Weldon II, and Biasi, G. P. (1997). The Pitman Canyon paleoseismic record: a re-evaluation of southern San Andreas fault segmentation, J. Geodynam. 24, Sieh, K. (1978b). Slip along the San Andreas fault associated with the great 1857 earthquake, Bull. Seismol. Soc. Am. 68, Sieh, K. E. (1984). Lateral offsets and revised dates of large earthquakes at Pallett Creek, California, J. Geophys. Res. 89, Sieh, K. E. (1986) Slip rate across the San Andreas fault and prehistoric earthquakes at Indio, California, EOS Transactions, 67, 12. Sieh, K., M. Stuiver, and D. Brillinger (1989). A more precise chronology of earthquakes produced by the San Andreas fault in southern California, J. Geophys. Res., 94, Stirling, M. W., S. G. Wesnousky, and K. Shimazaki (1996). Fault trace complexity, cumulative slip, and the shape of the magnitude-frequency distribution for strike-slip faults: A global survey, Geophys. J. Int., 124, Weldon, R., K. Scharer, T. Fumal, and G. Biasi (24). Wrightwood and the earthquake cycle: what a long recurrence record tells us about how faults work, GSA Today, 14, in press. 26

27 Weldon, R. J. II, T. E. Fumal, T. J. Powers, S. K. Pezzopane, K. M. Scharer, and J. C. Hamilton (22). Structure and earthquake offsets on the San Andreas fault at the Wrightwood, California paleoseismic site, Bull. Seism. Soc. Am. 92, no. 7, Wells, D. L., and K. J. Coppersmith (1993). Likelihood of surface rupture as a function of magnitude, Seismological Research Letters, 64, 54. Wells, D. L., and K. J. Coppersmith (1994). New empirical relationships among magnitude, rupture length, rupture width, rupture area, and surface displacement, Bull. Seism. Soc. Am. 84, Wesnousky, S. (1994). The Gutenberg-Richter or characteristic earthquake distribution, which is it?, Bull. Seism. Soc. Am. 84, Yule, D. and C. Howland (21). A revised chronology of earthquakes produced by the San Andreas fault at Burro Flats, near Banning, California, in SCEC Annual Meeting, Proceedings and Abstracts, Southern California Earthquake Center, Los Angeles, 121. University of Nevada Reno Seismological Laboratory, MS-174 Reno, NV glenn@seismo.unr.edu (G.P.B.) University of Oregon Department of Geological Science 1272 Eugene, OR

28 (R.J.W.) 28

29 Normalized Dis placement Normalized Dis placement S an Andreas, Normalized Length E rzincan.5 1 Normalized Length D/D ave 2 4 D/D ave 2 4 Normalized Dis placement T osya.5 1 Normalized Length.4.2 D/D ave 2 4 Normalized Dis placement Normalized Dis placement Normalized Length E dgecumbe B olugerde.5 1 Normalized Length D/D ave D/D ave 2 4 Normalized Dis placement Normalized Dis placement Normalized Length K ern F u yun.5 1 Normalized Length F igure D/D ave D/D ave 2 4

30 D/D ave S tacked R upture P rofiles 1% S ampling (a) Normalized Length F requency Histogram of S tacked R uptures (b) Normalized Dis placement F requency Histogram of All 1% d/dbar (c) Normalized Dis placement F igure 2

31 .6.4 S caled Histograms of Displacement Mw = Mw = Mw = Mw = Dis placement (m) F igure 3

32 P (M d) vs. Magnitude, d obs 1. to 4. m, prior: 1 P robability.15 dobs = 1. m.1.5 (a) P robability P robability dobs = 2. m.1.5 (b) dobs = 4. m.1 (c) Moment Magnitude F igure 4

33 1 P (L d) vs. R upture Length, d obs 1. to 6. m, prior: 1.8 d obs = 6 P robability d obs = S urface R upture Length (km) F igure 5

34 p(m d) for T hree Widths of Uniform p(m) P rior d obs 1. to 4. m P robability.15 dobs = 1. m.1.5 (a) P robability.15 dobs = 2. m.1.5 (b) P robability.15 dobs = 4. m.1.5 (c) Moment Magnitude F igure 6

35 1 F raction P roducing R upture, from W&C F raction Magnitude F igure 7

36 Uniform P (M) Model.15 (a) P robability G utenbergr ichter P (M) Model.15 (b) P robability C onstrained Mean Displacement P (M) Model.15 (c) P robability Moment Mag F igure 8

37 P robability P robability.3 dobs = 1. m.2.1 p(m d) vs. Magnitude dobs = 2. m p(l d) vs. S urface R upture Length dobs = 1. m dobs = 2. m P robability P robability.3 dobs = 4. m dobs = 6. m dobs = 4. m dobs = 6. m.5 P robability P robability dobs = 8. m dobs = 1. m Moment Magnitude dobs = 8. m dobs = 1. m S urface R upture Length F igure 9