Probabilistic prediction of rupture length, slip and seismic ground motions for an ongoing rupture: implications for early warning for large earthquakes

stochastic of heterogeneous slip. large slip amplitudes the probability for the continuation of rupture and the evolution into a the on spatially smooth fault earthquakes recognition the causative fault real-time faults, the in the fault exceeds a few metres, because the probability for a large earthquake is high and strong shaking is expected to occur in large areas around the fault. A more sophisticated EEW system could use the present slip on the fault to estimate the future slip evolution and ﬁnal rupture dimensions, and (using this information) could provide probabilistic predictions of seismic ground motions along the evolving rupture. The decision on whether an EEW system should be realized in the ﬁrst or in the second way (or in a combination of both) is user-speciﬁc.


I N T RO D U C T I O N
Earthquake Early Warning (EEW) requires fast and robust predictions of earthquake source and ground motion parameters shortly after the initiation of an earthquake. If given in a timely manner, warnings can be used to trigger and execute automatic measures to reduce expected damage at distant sites (e.g. Goltz 2002;Allen et al. 2009). Several algorithms for EEW have been proposed and tested during recent years (e.g. Nakamura 1988;Allen & Kanamori 2003;Kanamori 2005;Böse 2006;Böse et al. 2007;Cua & Heaton 2007;Wu et al. 2007;Hoshiba et al. 2008;Böse et al. 2009a,b). The majority of these approaches use the amplitude and frequency content of seismic waves at one or more seismic sensors in the epicentral area for a rapid assessment of earthquake source and ground motion parameters. With a few exceptions (e.g. Yamada et al. 2007;Yamada & Heaton 2008), the majority of the current EEW approaches consider the earthquake as a point source, that is, neglect rupture finiteness.
Large magnitude earthquakes (M > 7.0), with rupture lengths of up to hundreds of kilometres, cause damaging ground shaking in much larger areas than moderate-to-strong events. Despite their rare occurrence, many more people and users could benefit from an EEW system during large earthquakes (Heaton 1985;Allen 2006). Due to the low propagation speed of seismic ruptures (∼80 per cent of the seismic S-wave speed), warning times to heavily shaken areas may exceed more than 1 min, while users who are strongly shaken Early warning for large earthquakes 1015 in smaller earthquakes are likely to be near the epicentre and will generally receive warnings of a few seconds only.
A key to EEW for large magnitude earthquakes is the rapid prediction of the potential final dimensions of the ongoing rupture. This is because the level and distribution of seismic ground motions, such as peak ground velocity (PGV) or seismic intensity, are usually predicted from the earthquake magnitude and the rupture-to-site distance, which both can be estimated from the expected rupture dimensions. If the rupture is still propagating, these predictions can only be given in a probabilistic manner. Appropriate decision making is necessarily based on probabilistic estimates of future shaking.
The rupture of a large (strike-slip) earthquake propagates mainly in the horizontal direction along a fault, that is, rapid prediction of the rupture length is especially important for EEW. An empirical relationship between the subsurface rupture length L(km) and the (moment) magnitude M for strike-slip earthquakes with M 4.8 to M 8.1, for example, has been proposed by Wells & Coppersmith (1994) with [see Working Group on California Earthquake Probabilities (1999) for discussions on this relation].
A simple way to predict L (without any further information on the rupture) is by the usage of frequency-size statistical distributions of earthquakes, such as produced by the Gutenberg-and-Richter model (G-R; Gutenberg & Richter 1944) or by the Characteristic Earthquake model (CE; Schwartz & Coppersmith 1984).
The G-R model describes the frequency-size distribution of earthquakes by the power-law where N(M) is the number of earthquakes exceeding magnitude M, and a and b are constants (Gutenberg & Richter 1944). The b-value in eq. (2) is usually observed to be about 1.0, although there are regional variations (e.g. Wiemer & Wyss 2002;Schorlemmer et al. 2004); b also depends on the particular definition of magnitude chosen (e.g. M l , M S and M w ). The probability of occurrence of an earthquake exceeding a certain magnitude M in the range [M min , M max ] can be described by the probability density function (PDF) with β = b ln[10] and ∫ Mmax Mmin prob G-R (M)dM = 1 (Fig. 1a). The probability of exceedance (POE) of a present magnitude M p is (4) If, for example, we knew that the present magnitude of an ongoing earthquake was M p 5.5 (corresponding to a present rupture length of L p ≈ 6 km) and b = 1, then the G-R model predicts that there is only a 1 per cent chance that the event will grow to be larger than M 7.5 (Fig. 1b). An EEW system based on the G-R model will always conclude that there is a very low probability that such an earthquake will grow to be a 'Big One'; only 1 out of 100 earthquakes with a present magnitude of M p 5.5 will eventually grow to be larger than M 7.5. Assuming higher b-values in the G-R model will increase the probabilities of earthquake growth (Iwata et al. 2005).
The CE model, in contrast, assumes that single faults tend to produce earthquakes of similar size. The magnitudes of these 'characteristic' earthquakes mainly depend on the dimension of the fault zones or fault segments (Schwartz & Coppersmith 1994). It is typically supposed that characteristic earthquakes are accompanied by foreshocks, aftershocks and low-level background seismicity along the considered fault or fault zone, which are distributed according to the G-R model. Youngs & Coppersmith (1985) and Convertito et al. (2006) give a PDF of the earthquake occurrence that is appropriate for this model with prob CE where the constant c is given by The probability for the occurrence of a large earthquake is much lower in the Gutenberg-and-Richter (G-R) than in the Characteristic Earthquake (CE) model: only 1 out of 100 earthquakes will exceed M 7.5. An EEW system based on the G-R model will likely never predict a 'Big One'. The CE model, on the other hand, is very poorly constrained (here: b = 1.0, M 1 = 1.75 and M 2 = 0.5).

M. Böse and T. H. Heaton
The parameters M 1 and M 2 describe two intervals below and above the characteristic magnitude M c = M max -M 2 (Fig. 1a).
The POE for the CE model, p CE , can be determined analogous to eq. (4). Due to the occurrence of characteristic earthquakes, p CE takes generally higher values than p G-R . For example with M 1 = 1.75 and M 2 = 0.5, the CE model predicts that there is a 40 per cent chance for an ongoing event with M p 5.5 to grow to be larger than M 7.5, compared to 1 per cent predicted by the G-R model (Fig. 1b).
Spatially smooth ruptures along so-called 'mature' faults usually travel over larger distances than heterogeneous ruptures along 'immature' or 'generic' faults (e.g. Kanamori & Allen 1986;Wesnousky 1988;Manighetti et al. 2005Manighetti et al. , 2009. For this study we assume that the frequency-size statistics of earthquakes along mature faults can be described by the CE model, while the G-R model seems to be more appropriate to characterize the seismicity along generic faults. There is an ongoing debate among seismologists about which model might be more appropriate to characterize the seismic activity along individual faults (e.g. Wesnousky 1994;Kagan 1996;Page et al. in preparation). We want to avoid these discussions here. Even if we knew that the G-R or CE models are adequate to describe the seismicity in a certain region or along a specific fault, it seems clear that there is insufficient knowledge to assign appropriate parameters to the wide range of possible rupture scenarios.
A more flexible approach to study the rupture probabilities along faults is by the application of stochastic slip models that can be tuned from G-R-to CE-like behaviour. These models can be used to generate series of spatially variable slip (Liu-Zeng et al. 2005). Slip heterogeneities can be caused, for example, by the frictional properties and the spatial heterogeneities in the material characteristics on the fault, or by the complexities of the fault geometry (Rice 1993;Andrews 1994;Aki 1995;Aagaard & Heaton 2008). In this study, we use 1-D stochastic slip models to simulate large suites of possible rupture scenarios for a systematic statistical analysis of the relationship between seismic slip and rupture length.
We find that the larger the present slip amplitude D p , the higher is the probability for the continuation of a rupture and the evolution into a 'Big One', that is, a long remaining rupture length L r (Yamada 2007). We also find that the D p -L r relationship shows a significant amount of scatter and is strongly controlled by the degree of assumed spatial heterogeneity of slip on the fault.
In this paper, we are not addressing the origin of spatially heterogeneous slip. We simply note that if slip changes rapidly, then it is more difficult to predict the future evolution of the rupture (in space and time) than if the slip is spatially smooth. In principle, we could also try to estimate the smoothness of an ongoing rupture in real time from the variability of slip amplitudes relative to the mean evolution of slip. This idea will be briefly discussed in the next section, but shall not be a major topic in this study.
The real-time application of our probabilistic approach to EEW requires the continuous monitoring of seismic slip along faults, for example, by a global positioning system (GPS; e.g. Crowell et al. 2009) or by the backprojection of seismic displacement data onto the fault line (Yamada 2007). Again, to understand the implications of a small or large slip amplitude, it is essential to consider the respective 'smoothness' of the fault along which a detected earthquake propagates. This means that an EEW system for large earthquakes needs some mechanism for the rapid recognition of the rupturing fault and the capability to consider the respective statistical features. This procedure requires that the rupture is automatically linked to a (hopefully) known geological structure, for which a 'smoothness measure' from an earlier assessment was determined and stored in a database.
If rupture is occurring on a smooth fault, for example, on the San Andreas Fault (SAF), an EEW system could be implemented in two ways: the system could issue a warning, whenever slip on the fault exceeds a few metres, because the probability for a large earthquake is high and strong shaking is expected in large areas around the fault. A more sophisticated EEW system could use the present slip on the fault to estimate the final rupture dimensions and future slip evolution, and (using this information) could provide probabilistic predictions of seismic ground motions along the evolving rupture. We will demonstrate this concept of an EEW system on the example of the southern Californian SAF.

1-D stochastic slip models
We assume that seismic ruptures consist of slip pulses that propagate along a fault (Heaton 1990). The ultimate slip D(x) at the spatial point x on the fault is achieved quickly after the passage of the rupture front. That is, we suppose that the current distribution of slip up to a present rupture length L p is approximately the final (static) distribution of slip along this length, and no further slip accumulation occurs after the passage of the rupture front.
Although earthquakes actually occur on rupture surfaces, we are constructing here a simple simulation of long ruptures and assume a 1-D line source. We assume that slip pulses vary (perhaps chaotically) as they propagate along the fault. We also assume that the event is over, once the amplitude of the slip pulse drops to zero.
We simulate the propagation of slip pulses from simple 1-D stochastic slip models as proposed by Liu-Zeng et al. (2005), in which the slip amplitudes at adjacent points, x and (x + x), change by some random amount in either positive or negative direction. We model this change in the amplitude by a low-pass filtered Gaussian white noise series R(x) with zero mean and standard deviation one, whereR(k) is the Fourier transform of R(x), k is the wavenumber, FT −1 the inverse Fourier transform and D a scaling factor. The filtering exponent μ determines the spatial roughness of slip: the smaller μ, the rougher is the slip. Each point x, at which D(x) = 0, defines the end of an individual rupture. We take the absolute amplitudes of these 'parent series' to obtain contiguous positive slip functions (Fig. 2). Note that the slip models described by eq. (7) are purely geometric; they only assume that slip is spatially contiguous and that it has the scale-invariant properties of a self-affine fractal (Turcotte 1997). Liu-Zeng et al. (2005) found that models with a rupture roughness of up to μ = 1.5 produce slip series that have similar statistical features like real earthquakes (see below for further discussions). We use the stochastic slip models in eq. (7) to generate a set of 500 parent series of 65 536 points each for μ = 1.1, μ = 1.3 and μ = 1.5. A total of 22 541 individual slip events, which are identified by determining adjacent zero crossings of the parent series, are catalogued: 15 006 for μ = 1.1, 5121 for μ = 1.3 and 2414 for μ = 1.5.
Models with small rupture roughness μ, that is, high slip heterogeneity, tend to generate more small events, that is, shorter ruptures, than models with large μ (Fig. 3a), because rougher slip distributions are more likely to terminate rupture (Fig. 2). The frequencysize statistics of these events are close to the power-law distributions Figure 2. Unscaled parent series of slip generated from eq. (7) for three levels of rupture roughness μ. The larger the slip heterogeneity, the larger is the probability to cross the zero line, which defines the end of an individual rupture event. generated by the G-R model; slip series generated from models with large μ, on the other hand, produce smooth ruptures with a higher number of large events, that is, the frequency-size statistics of these series are more similar to the distributions produced by the CE model (Fig. 3a). This means that we can use the roughness parameter μ to tune the stochastic models from G-R-to CE-like behaviour. This is in good agreement with, for example, Hillers et al. (2007), who found from quasi-dynamic rupture simulations that smooth continuum faults usually tend to have narrow event distributions, and that strong spatial heterogeneity is required to produce spatiotemporal non-uniform seismic behaviour.
The calibration of the 1-D slip functions is driven by two desires: (1) we want to reproduce realistic average slip-to-rupture length ratios as observed during earthquakes, and (2) we want to avoid maximum slip amplitudes that seem unrealistic high (although we are not really sure what is unrealistic). In this study, we set in eq. (7) D = 3 m. We assume that the 1-D slip amplitudes correspond to the average slip taken over the fault width.
The slip amplitudes D in our models follow one-sided Gaussian distributions (Fig. 3b) as a consequence of the assumed Gaussian distribution of noise amplitudes in eq. (7). If desired, we could replace the Gaussian by any other distribution, for example, by the logarithmic distribution. The slip amplitudes in the stochastic models increase with increasing rupture roughness, that is, decreasing μ (Fig. 3b). The largest slip amplitudes in our models are less than 30 m for μ = 1.3 and μ = 1.5, and, in a very few cases, grow up to 65 m for μ = 1.1. The range of simulated slip amplitudes seems to be fairly realistic if compared, for example, with the slip models in the Finite-Source Rupture Model Database (SRCMOD version 7, http://www.seismo.ethz.ch/srcmod/). Note, however, that the inverted slip distributions in the SRCMOD are usually strongly smoothed and the SRCMOD does not contain a sufficient number of large events to allow for a statistical analysis as aimed for in our studies.
We generally find a good agreement between the ratios of the average slip D (taken over the entire rupture length L) to the rupture length L in the simulated slip functions and earthquakes which usually have D/L values within the wedge-shaped area in Fig. 4: the D/L ratios, which are related to the seismic stress drops, increase among small earthquakes compared to large events, if we consider the combined effect of different levels of rupture roughness ( Fig. 4). Ruptures with rougher slip distributions (smaller μ) tend to produce higher slip-length ratios than smooth ruptures. Generally, the ratios decrease with increasing rupture length L, except for smooth ruptures (μ = 1.5) that hardly show any variation in D/L.
For rupture lengths that are large relative to the seismogenic width of the fault (L 30 km), constant slip versus length scaling (of smooth ruptures) implies constant stress drops with L. This behaviour is compatible with the 'L-model' proposed by Scholz (1982). Note, however, that the stress drop is not really a fundamental frictional parameter in our 1-D stochastic slip models: the end of a rupture is simply defined by the location at which the amplitude of a slip pulse dies to zero. That is, the stress drop in our models is determined by the details of spatially variable slip pulses (Liu-Zeng et al. 2005).  The corresponding values of real earthquakes are typically located in the wedge-edged area between the two lines. The scaling of the parent series in eq. (7) is driven by the compromise between avoiding too large slip amplitudes and the desire to simulate ruptures with fairly realistic D/L ratios.
Generally, the D/L ratios obtained from the simulated slip series tend to be smaller than the ratios observed during real earthquakes (Fig. 4, wedge-edged area). In principle, we could improve the fit by increasing the scaling factor D in eq. (7), but this would lead to single slip amplitudes exceeding several tens of metres, which seems too large for a real earthquake. The decision to set in this study D = 3 m is driven by the compromise between avoiding slip amplitudes that are too large and the desire to simulate ruptures with realistic D/L ratios.
Another way to evaluate the suitability of the simulated slip functions is by comparing their shape with observations made during real earthquakes. The mean shape of the 1-D slip functions can be characterized by three parameters (Ward 2004): the (total) rupture length L, the peak average slip E peak and a skewness parameter q, where q controls whether a slip function has more off-centre (q ≈ 2.0) or more elliptical shape (q ≈ 0.5; Ward 1997). We determine for each simulated slip series of length L parameters E peak and q from fitting the model in eq. (8) to the generated 1-D slip functions (Figs 5a-d).
The majority of the slip series tend to have elliptical shapes, independent from the level of slip heterogeneity (Fig. 5c). This symmetry is not surprising since the probability for slip increase or decrease in the stochastic slip models is constant at all rupture lengths l, that is, on average we should obtain the same shape, independent from whether the rupture starts at l = 0 or at l = L. Manighetti et al. (2005) found that the slip functions of real earthquakes usually tend to be roughly triangular both along strike and dip, and most of them (70-80 per cent) are asymmetric. The reason for this phenomenon is not well understood, and cannot be reproduced by our simple static 1-D models.
We have seen that the 1-D stochastic slip models as used in this study imitate important statistical features of earthquakes, such as reproducing realistic slip amplitudes and average slip versus rupture length ratios (Fig. 4). On the other hand, the models fail to reproduce asymmetric shapes of the slip series as often are observed during real earthquakes. We think that employing the 1-D models for our purposes, however, is justified, because we are mainly interested in statistical features such as fractal behaviour of slip, which is well reproduced by the 1-D functions (see Discussions and Conclusions).
What is the relationship between the simulated slip series D in eq. (7) and the mean slip function E in eq. (8)? Neither the shape factor q nor the peak slip value E peak show a clear dependency on μ (Figs 5c and d). However, the variability of the simulated slip amplitudes D relative to E increases strongly with increasing roughness of the slip series, that is, the stronger the slip heterogeneity, the larger are the differences of single slip amplitudes and the mean slip function, and the larger the standard deviation σ of the residuals (E-D) (Fig. 6).
From our stochastic 1-D models we find that σ ≤ 1.5 m for μ = 1.5, and σ ≤ 2.25 m for μ = 1.1 and μ = 1.3 (Fig. 6). The variability increases with increasing mean slip E. We could in principle use the variability of the slip amplitudes along an ongoing rupture to estimate the rupture roughness in real-time. This procedure, however, requires densely sampled slip profiles. In the following, we assume that μ is a fault-characteristic parameter, which has been determined from previous geological and geophysical studies (e.g., Sagy et al. 2007) and is known when the rupture nucleates.

Bayesian approach
The statistical analyses of seismic slip amplitudes D and rupture length L in this paper are based on the Bayesian theorem. Probabilistic methods, such as provided by the Bayesian framework, are reasonable approaches to deal with the large variability in the observed seismic source and ground motion characteristics and the inherent uncertainties in predicting these parameters.
In this study, we determine the probabilities of remaining rupture length L r of an ongoing earthquake rupture using a presently observed seismic slip amplitude D p . We are not only interested in determining the most probable, but also the less likely, however, still possible solutions. In other words: we want to determine the entire PDF for L r . Both slip amplitudes D and rupture length L span several orders of magnitudes. They usually follow power-law distributions, and we therefore use their logarithmic values log(D) and log(L) in the following analyses.
We describe the rupture length-slip relationship by the 'conditional probability' of L r for a given D p , also known as the 'posterior probability' p(log(L r )|log(D p )). The Bayesian theorem states that the posterior probability can be determined from the product of the likelihood function p(log(D p )|log(L r )) and the a priori probability p(log(L r )), that is, where the normalization factor in the denominator, p(log(D p )) = n i=1 p(log(D p ) i | log(L r )) p(log(L r )), ensures that the sum over all probabilities gives 1.
The likelihood function p(log(D p )|log(L r )) in eq. (9) describes, how likely it is to observe a certain log(D p )-log(L r ) combination. ; E depends on the peak mean value E peak , shape factor q and rupture length L (see eq. (8)). (c) and (d): Relative frequency of parameters q and E peak for all 22 541 simulated ruptures. The majority of slip functions has a symmetric shape (q = 0.5) as a consequence of equal probabilities of slip increase and decrease at all length l, that is, in a statistical sense the slip series should not be different if the rupture starts at l = 0 or l = L. Figure 6. Standard deviation σ of residuals (mean slip E-slip D) for three levels of rupture roughness μ. The smaller the slip heterogeneity, the smaller is σ , that is, the better is the fit between E and D.
The a priori probability p(log(L r )) tells, how often we will observe a given log(L r ) value. Obviously, p(log(L r )) depends on both the probability of occurrence of a rupture with length L and on how far the considered rupture already has propagated, that is, p(log(L r )) = p(log(L-L p )), where L p is the present rupture length, at which slip D p is observed. In principle, we can use the log-linear relationship between the rupture length and magnitude in eq. (1) and describe the a priori probability p(M) by the earlier discussed frequency-size statistics of earthquakes as produced, for example, by the G-R or the CE model.
The likelihood function p(log(D p )|log(L r )) in eq. (9) can be determined from the joint probability function p(log(D p ),log(L r )), which describes the probability of the joint occurrence of log(D p ) and log(L r ). The likelihood function p(log(D p )|log(L r )) can be calculated from where we determine the joint probabilities from the 2-D kernelbased estimation method (e.g. Bishop 1995) using Gaussian kernels where the standard deviation σ in the denominator controls the width of the kernels, and N is the number of data points. Kernelbased methods belong to the group of 'non-parametric' density estimation techniques that do not require assumptions about the forms of the PDFs (except that they are smooth). They therewith are highly flexible and well suited for many applications (e.g. Bishop 1995).

Probabilistic prediction of rupture length L and magnitude M
We go along each stochastically simulated 1-D slip series and determine for each slip amplitude D p the PDF of remaining rupture length L r , that is, the posterior probability p(log(L r )|log(D p )). We consider two levels of slip heterogeneity in the stochastic models: μ = 1.1 (rough rupture) and μ = 1.5 (smooth rupture) representing 'generic' and 'mature' fault conditions, respectively. Note that the slip and length values as specified in the following should be understood in a qualitative rather than a quantitative way, because they depend on the calibration of the 1-D slip models as described in the previous section. Fig. 7 shows the probabilities of remaining rupture length L r for slip amplitudes of D p = 2.0 m, D p = 5.0 m and continuous values of D p ≤ 20.0 m. The dashed lines mark the median values of the PDFs, that is, 50 per cent of the simulated ruptures propagate over a shorter and 50 per cent over a larger distance than specified by this value. Note the strong asymmetry of the PDFs, leading to significant differences between the median values and the maxima of the PDFs. The greyish colours mark the 38 per cent, 68 per cent and 95 per cent confidence intervals of the PDFs, corresponding to 0.5σ , 1σ and 2σ of a Gaussian distribution.
Our model estimates a 50 per cent chance that a rough rupture (μ = 1.1) keeps growing for L r > 6 km if D p = 2.0 m, and for L r > 20 km if D p = 5.0 m (Fig. 7, left-hand side). Using eq. (1) these lengths correspond to magnitudes of M > 5 and M > 6, respectively. Note that a more accurate specification of M requires the knowledge of the present rupture length L p . The probabilities for rupture continuation increase substantially, if the rupture travels along a smooth (mature) fault with little slip heterogeneity (μ = 1.5; Fig. 7, right-hand side): 50 per cent of the ruptures keep growing for L r > 45 km (M ∼ 7.0) if D p = 2.0 m, and for L r > 70 km (M > 7) if D p = 5.0 m. Our model thus predicts that the potential sizes of ruptures on heterogeneous (generic) and smooth (mature) faults may differ by more than one full magnitude unit for the same slip amplitude.
On a rough fault, L r scales almost linearly with D p (Fig. 7c, lefthand side), while the confidence intervals become broader, that is, the uncertainties in the prediction increase with increasing slip. Note that even for large slip amplitudes (e.g. D p = 6.0 m) every second rupture does not exceed 25 km and therewith M < 6.5 (provided that D p is observed at L p ≈ 0 km, i.e. close to the initiation point of the rupture); for 38 per cent of the earthquakes ruptures we observe L r < 50 km, for 68 per cent L r < 75 km and for 95 per cent L r < 125 km. If the rupture, however, occurs along a smooth (mature) fault (Fig. 7c, right-hand side), we observe a strong increase of L r for D p ≤ 5.0 m. The PDFs hardly change for larger D p amplitudes, that is, the remaining rupture length is largely independent from the present slip and every second rupture will grow for L r > 65 km, that is, M > 7.0. Fig. 8 shows a comparison of the predicted probabilities of remaining rupture length L r and the relative frequency of L r determined from the slip models in the Finite-Source Rupture Model Database (SRCMOD version 7, http://www.seismo.ethz.ch/ srcmod/) for slip amplitudes of D p = 1-5 m. Note that these plots show observational slip data from both heterogeneous and smooth ruptures. There is qualitative agreement between these data and the probabilities derived from the stochastic models with slip heterogeneities of μ = 1.1 and μ = 1.3. The stochastic models capture essential features of the SRCMOD. A more detailed comparison, however, required a classification of the SRCMOD ruptures with respect to their levels of slip heterogeneity, which needs to be done in the future.

Prediction of mean slip function
Although the prediction of single slip amplitudes D along the evolving rupture may likely remain an elusive goal, we can try to estimate the mean slip function E using eq. (8). This involves some knowledge or the prediction of the total rupture length L, the peak mean slip E peak and shape factor q. The estimation of these parameters requires some information on the evolution of slip. For example, if we knew the entire slip series up to rupture length L p , we could estimate L = L p + L r from eq. (9) and then determine E peak and q from curve fitting of function E in eq. (8) to the observed slip amplitudes D. The shape of the slip function, however, remains elusive, if E peak and q cannot be reasonably well resolved.
We test the robustness of the curve-fitting procedure to determine E peak and q if the number of slip amplitudes is reduced by downsampling of data points in the simulated slip series. We use the slip amplitudes along the full rupture, that is, from l = 0 to l = L, and assume that the total rupture length L of each event is known. Fig. 9 shows the rms errors for both parameters as a function of data completeness C. A completeness of C = 0.1, for example, means that only every 10th slip amplitude of the original slip series is used.
We can easily recognize a rapid decrease of the rms errors in Fig. 9 with increasing data completeness C, that is, the more details (slip amplitudes) of the slip series are known, the more reliable are the estimates of q and E peak . Rough ruptures (with small μ) that have a large variability in the slip amplitudes require more information on the evolution of slip than smooth series. Ruptures with slowly varying slip maintain their memory of past rupture over relatively longer lengths, that is, slip variations occur over larger length scales. We call this behaviour the (spatial) 'memory effect' of smooth ruptures.
In the previous example we assumed that the rupture is already terminated and slip amplitudes (or subsets of amplitudes) along this rupture as well as the total rupture length L are known, before q and E peak are determined from curve fitting. In the case of an ongoing rupture, we will know the slip amplitudes only of the initial portion of the rupture (if at all). The results in Fig. 9, however, indicate that due to the (spatial) 'memory effect' of smooth ruptures the prediction of future slip does not require detailed information on , as well as the 38 per cent, 68 per cent and 95 per cent confidence intervals, corresponding to 0.5σ , 1σ and 2σ of a Gaussian distribution. Note that the probabilities are plotted on a linear length scale, while determined on a logarithmic scale using eq. (9). Rough ruptures with a high level of slip heterogeneity (left-hand side) tend to propagate over shorter distances than smooth ruptures (right-hand side). The median values of the PDFs increase almost linearly with increasing slip D p for a rough rupture (c, left-hand side), while staying almost constant after a rapid increase up to D p = 5.0 m for a smooth rupture (c, right-hand side). the evolution of slip if a smooth (mature) fault is considered. We will come back to this when demonstrating the probabilistic prediction for a scenario earthquake along the SAF.

Implications for EEW
EEW requires a rapid and robust determination of seismic source and ground motion parameters, such as magnitude and PGV, shortly after the initiation of a moderate-to-large earthquake. Most chal-lenging in EEW for large earthquakes is the real-time determination of the probability that the rupture of an ongoing earthquake with a present length L p (corresponding to a present magnitude M p ) will continue and exceed a certain magnitude level. We have seen earlier, that in the G-R model the POE for a large magnitude earthquake is extremely small: only 1 out of 100 earthquakes with M p 5.5 (L p ≈ 6 km) will exceed M 7.5 (Fig. 1). The output of the CE model, on the other hand, depends very much on the model parameters chosen and thus is not well constrained. Figure 8. Comparison of the predicted probabilities of remaining rupture length L r for two levels of slip heterogeneity μ with the relative frequency of L r determined from the slip models in the Finite-Source Rupture Model Database (SRCMOD, http://www.seismo.ethz.ch/srcmod/); slip amplitudes vary from D p = 1-5 m from left to right. Note that these plots show observational slip data from both heterogeneous and smooth ruptures. There is good agreement between these data and the probabilities derived from the stochastic slip models with μ = 1.1 and 1.3. A more detailed comparison required a classification of the SRCMOD ruptures with respect to their levels of slip heterogeneity. Figure 9. Robustness of the curve-fitting procedure to determine (a) the shape parameter q and (b) the peak average slip E peak of the mean slip function E in eq. (8) if only a subset of slip amplitudes along the rupture is used. The rms errors decrease rapidly with increasing data completeness up to C = 0.2, which means that every fifth slip amplitude is assumed to be known. The larger the slip heterogeneity, the larger are the errors, that is, the more important is the knowledge of past slip for predicting future amplitudes. Ruptures with slowly varying slip (higher μ) maintain their memory of past rupture over relatively longer lengths. This is the so-called (spatial) 'memory effect' of smooth ruptures.
For the stochastic slip models the POE of a given magnitude tends to increase with increasing slip amplitude D p (Fig. 10). For example, for M p 5.5 and a slip amplitude of D p = 0.5 m it is three times more likely that the rupture on a rough fault (μ = 1.1) will become M > 7.0 and 10 times more likely to become M > 7.5 than in the G-R model ( Table 1). The probabilities increase by an additional factor of two, if the slip increases to D p = 5.0 m. A further increase of the probabilities by an additional factor of two to three is observed, if the same slip occurs along a smooth fault (μ = 1.5). This observation demonstrates that although the present slip amplitudes D p affect the POE, the smoothness of the causative fault has an even larger impact.
When should an EEW system issue a warning that an earthquake may become large? We have seen in Fig. 7c that L r scales almost linearly with D p on a rough rupture (left-hand side) and even for large slip amplitudes (e.g. D p = 6.0 m) only every second rupture exceeds L r = 25 km. If the rupture, however, occurs along a smooth (mature) fault, we observe a very strong increase of L r with increasing D p ≤ 5.0 m (Fig. 7c, right-hand side). The PDFs hardly change for D p > 5.0 m, suggesting that the remaining rupture length is largely independent from the present slip and every second rupture will grow L r > 65 km, that is, M > 7.0.
These findings imply that we might have a critical slip amplitude (here D p,critical ∼ 5.0 m) on a smooth (mature) fault, for which we have an increased probability for the occurrence of a large earthquake for which EEW is needed. For the ruptures along heterogeneous (generic) faults, however, we cannot identify a critical slip value (Fig. 7c, left-hand side). Note again, that D p,critical as specified here depends on the scaling of the 1-D slip models, and might not be equal to the critical slip amplitude that we have to expect for a real earthquake. A more accurate quantification of D p,critical will require further statistical analyses of earthquake data and modelling of earthquake ruptures that is beyond the scope of this paper.
The statistical analyses of slip and rupture length in this paper demonstrate that a large slip amplitude D p does not necessarily mean that we have a high probability for a large magnitude earthquake. Or, a small-to -moderate slip amplitude does not necessarily mean that the probability for a large event is low. We need to consider the respective characteristics of the underlying fault. We conclude from this finding, that an EEW system for large earthquakes requires a mechanism for the rapid recognition of the causative fault and consideration of its smoothness, that is, slip heterogeneity.
The majority of the algorithms, which currently are applied to EEW, use a fixed time window of a couple of seconds of the seismic Figure 10. Probability of exceedance (POE) of different magnitude levels M for an ongoing rupture with a present magnitude of M p 5.5 and present slip amplitude D p . The probabilities are derived from the 1-D stochastic slip models for rough (left-hand side) and smooth ruptures (right-hand side), respectively. The POE is controlled by the slip amplitudes, D p , the smoothness of the causative fault, however, has an even larger impact. The probabilities for a large event are much higher than in the G-R model (Fig. 1).  P wave for a rapid estimation of the earthquake magnitude, using, for example, the average or the predominant period of shaking (e.g. Nakamura 1988;Allen & Kanamori 2003;Kanamori 2005;Lewis & Ben-Zion 2007;Wu et al. 2007;Lewis & Ben-Zion 2008;Böse et al. 2009a,b). This approach is heavily disputed, since the rupture process of earthquakes with M > 6 takes much longer than 3 s and it appears questionable, why the eventual rupture length and therewith the magnitude of an earthquake should be pre-determined at this early stage of the rupture process (e.g. Rydelek & Horiuchu 2006;Rydelek et al. 2007). Only if a large amount of the seismic energy of the earthquake is released at its initial stage, that is, only if seismic ruptures start in patches of large seismic slip (as proposed, e.g. by Mai et al. 2005or Manighetti et al. 2005, a fast and robust magnitude prediction might become feasible. If this is not the case, the magnitude will likely remain underestimated, if we use a limited time window of a few seconds only. Our analyses might shed some light onto these discussions: it seems appropriate to assign to a smooth (mature) fault, such as the SAF, a higher probability that a rupture will continue and finally evolve into a 'Big One', than to a heterogeneous (generic) fault. This implies that the observation of a (apparent) moderate earthquake (judging from the first few seconds of waveform data) on a smooth (mature) fault should result in a stronger warning.
Once the probabilities of the total rupture length L and the eventual magnitude M of an ongoing earthquake are estimated (provided that L p or M p , respectively, are known), we can predict in a probabilistic manner the level and distribution of ground shaking amplitudes, such as PGV, along the evolving rupture using empirical magnitude/distance relationships (see, e.g. Next generation attenuation-special issue by Earthquake Spectra, 2008). The consideration of rupture dimensions allows for much more realistic assessments of the expected ground shaking than could be delivered by a point source approximation, in which the ground motion amplitudes are assumed to decay with increasing epicentral rather than rupture-to-site distance.
Peak ground displacement (PGD) [or inferred from this the response spectral acceleration (SA) at long periods], on the other hand, shows a good correlation with the static slip on the fault (e.g. Aagaard et al. 2001). This means that if we were able to predict the future slip of the evolving rupture, we could use this information for a significant improvement of the estimates of parameters PGD and SA for EEW.

Demonstration for an M 7.8 scenario earthquake along the southern SAF
We are going to demonstrate the probabilistic prediction of seismic ground motion and future slip for an M 7.8 scenario earthquake along the southern SAF in California (Fig. 11a, left-hand side). The SAF is a right-lateral strike-slip fault with a total length of about 1200 km. The average time interval between significant ruptures along the SAF varies from about 25 yr in the Cholame Valley to about 300 yr on segments with very large slips. There is a 50 per cent chance of a magnitude M 7.5 or greater earthquake in California in the next 30 yr (www.scec.org/ucerf). The possible impact of an M 7.8 in southern California has been analysed and described by Jones et al. (2008). The warning time for a large earthquake on the SAF could be more than 1 min for large cities like Los Angeles that will experience a high level of shaking and damage during such an event.
Similar to the 'ShakeOut' scenario earthquake (Jones et al. 2008), the rupture of our scenario event starts on the southern portion of the SAF, close to Bombay Beach, and propagates unidirectionally to the northwest over a distance of L ∼ 240 km (Fig. 11a, left-hand side). We assume a constant rupture speed of v r = 2.8 km s −1 , that is, the total rupture duration is ∼85 s. We simulate a 1-D seismic slip function of this scenario earthquake by usage of the earlier described stochastic slip models; that is, from our large collection of stochastic slip models, we choose several models with spatial distributions of slip that are qualitatively similar to the hypothesized 'ShakeOut' event. The SAF might be considered as a smooth fault with little slip heterogeneity. Thus, setting μ = 1.5 in eq. (7) might be adequate. For comparison, however, we also model and analyse a scenario rupture of the same length L and similar slip D along a heterogeneous (generic) fault (μ = 1.1). The maximum slip amplitudes are in the order of 15 m (Fig. 11a, right-hand side). By modelling purely unidirectional rupture propagation, we assume that the scenario earthquake starts with (almost) zero slip. The ruptures of real earthquakes often appear to propagate bidirectional and often start in patches of significant slip (e.g. Mai et al. 2005). EEW for unidirectional ruptures, as assumed in this scenario, is usually considered to be more difficult because of the small slip amplitudes at the beginning of the rupture. On the other hand, unidirectional rupture scenarios offer the largest possible warning times for EEW.
We use the earlier determined posterior probabilities p(log(L r )|log(D p )) in eq. (9) to resolve for each slip amplitude D p in Fig. 11a (right-hand side) the PDFs of remaining and total rupture length, L r and L = L p + L r , as a function of time after the rupture nucleation of the scenario earthquake. We then apply eq. (1) to estimate the corresponding magnitude M.
Obviously, the convergence rate of the estimated magnitude towards the true magnitude (M 7.8) is controlled by the slip heterogeneity on the causative fault: the median value of PDF is initially underestimated by 1.5 magnitude units, if slip occurs along a rough fault (Fig. 11b, left-hand side). Since on a heterogeneous (generic) fault, a short rupture is always more likely to occur than a large one, the predictions converge only slowly towards M 7.8 with increasing time and rupture growth. The median values are only slightly higher than the present magnitude M p derived from the present rupture length L p . If we assume, in contrast, that slip occurs along a smooth fault (μ = 1.5), the median values of the magnitude PDFs are initially underestimated by less than one magnitude unit only (Fig. 11b, right-hand side), and show a fairly rapid convergence towards M 7.8.
In the next step, we are going to estimate the PGV in four cities along the SAF as potential recipients of EEWs: Borrego Springs, Palm Springs, Riverside and Los Angeles (Fig. 11a, left-hand side). At each time step after the rupture nucleation of the scenario earthquake, we determine PGV (for rock condition) from an empirical attenuation relation for earthquakes in southern California (Cua & Heaton 2008, submitted) using the predicted magnitudes in Fig. 11b and the respective distances of each user site to the present and predicted rupture along the SAF. A more detailed analysis could include the variability of PGV predictions due to the use of different attenuation relations into the Bayesian framework (see Discussions and Conclusions).
The scenario earthquake produces damaging shaking in large areas of southern California (Fig. 11c). At two out of four sites, Borrego Springs and Palm Springs, PGVs exceed 0.3 m s −1 . The other two sites, Riverside and Los Angeles, still experience PGVs of around 0.2 m s −1 . Similar to the magnitude M (Fig. 11b), PGV is initially underestimated for both rough and smooth rupture conditions (Fig. 11c). For the rough rupture (Fig. 11c, left-hand side), the predictions are once again only slightly better than if we base the prediction on the present rupture length L p (rather than predicting L = L p + L r ). The convergence is faster, if we assume that slip occurs along a smooth fault (Fig. 11c, right-hand side).
As said earlier, certain (mid-and long-period) ground motion parameters, such as PGD or SA, show a stronger correlation with the static slip on the fault than with earthquake magnitude. If the eventual length L of the rupture is known (or predicted), we can fit the mean slip function E in eq. (8) to the observed slip amplitudes D. We have seen in Fig. 9 that the required information on previous slip for predicting E decreases with increasing rupture smoothness due to the spatial 'memory effect' of smooth faults. In the following analyses we will assume the idealized case that the entire slip history up to the present rupture length L p is known. Fig. 12 shows the probabilistic prediction of the mean slip function E for the scenario M 7.8 earthquake at three time steps after the rupture nucleation (from top to bottom). The shape and peak parameters of the mean slip functions, q and E peak , are determined from curve fitting of the 'known' slip amplitudes (in the grey area) to eq. (8) using the probabilistically predicted remaining rupture length L r . The red line shows the mean slip function for the median value of the PDF of predicted L r , the green and blue lines refer to the predicted rupture length within the 68 per cent and 95 per cent confidence intervals, respectively.
During the first two time steps (Fig. 12, figures in the top and middle panels), the peak mean slip E peak is underestimated by almost a factor of two (5 m rather than 10 m) due to the fairly small slip amplitudes at the beginning of the rupture. The estimates of E improve, once the half of the rupture is finished. Note again that the predicted values of L r are much smaller on the rough fault (Fig. 12, left-hand panels) than on the smooth fault (Fig. 12, righthand panels), which has a strong impact on the shape parameter q: for the rough rupture it is q < 0.5, that is, E peak is predicted to be close to l = 0. For the smooth (mature) fault, in contrast, we observe q ≥ 0.5, that is, the slip function is assumed to be either symmetric or E peak is predicted to be close to l = L (elliptical to skewed shape). The smoother the rupture, the smaller are the differences between the predicted mean slip functions and the observed slip amplitudes.
Since we assume that the SAF can be characterized as a 'smooth' fault, the results shown on the right-hand side of Figs 11 and 12 seem to be more appropriate for this scenario. We conclude that a system for EEW for large earthquakes on the SAF (or on any other smooth fault with little slip heterogeneity) can be implemented in two ways: (1) The system can either provide real-time predictions of magnitude, slip and seismic ground motions with the corresponding uncertainties, whereby these estimates are steadily updated using observations of the present rupture length and present slip (Figs 11 and 12). Or, (2) The mere observation of slip in the order of a few metres in amplitude can trigger the declaration of a warning, since large slip on a smooth fault suggests a high probability of a large earthquake and strong shaking in wide areas around the fault (Fig. 7c). In contrast to the first implementation, the second EEW system will not provide detailed estimates of expected ground motions.

D I S C U S S I O N S A N D C O N C L U S I O N S
An EEW system that estimates the potential source dimensions of a currently ongoing earthquake rupture using the present magnitude M p together with the G-R frequency-size statistics will likely never predict a large magnitude earthquake, because of the rare occurrence of 'extreme events'. For an ongoing rupture with a present length Figure 12. Probabilistic prediction of the mean slip function E for the scenario M 7.8 earthquake in Fig. 11 at three time steps after the rupture nucleation (from top to bottom), assuming a rough (left-hand side) and a smooth rupture (right-hand side) with high and low slip heterogeneity, respectively. The shape and peak parameters of the mean slip functions, q and E peak , are determined from curve fitting of the 'known' slip amplitudes (in the grey area) to eq. (8) using the probabilistically predicted remaining rupture length L r . The red line shows the mean slip function for the median values of the PDF of predicted L r , the green and blue lines refer to the predicted rupture length within the 68 per cent and 95 per cent confidence intervals, respectively. The smoother the rupture, the smaller are the differences between E and the observed slip amplitudes. corresponding to, for example, M p 5.5, the probability to exceed M 7.5 is only 1 per cent. However, it seems to be reasonable to assume that the evolution into a 'Big One' is supported by large slip amplitudes and a high level of 'smoothness' on the causative fault.
To investigate the relationship between the slip and the eventual size of an ongoing rupture, we simulated in this study suites of evolving ruptures using 1-D stochastic models of spatially heterogeneous slip. We found that while large slip amplitudes D p increase the probability for the continuation of a rupture and the possible evolution into a large earthquake, the recognition that rupture is occurring on a spatially smooth fault with low slip heterogeneity has an even stronger impact. The higher the variability in the slip amplitudes along the evolving rupture, the harder it is to predict whether the trend of declining slip amplitudes suggests that a rupture is likely to cease or whether this decrease might be followed by another increase in slip, caused, for example, by the rupture of a further asperity.
We conclude that an EEW system for large earthquakes needs some mechanism for the rapid recognition of the causative fault (e.g. from a GPS) and for consideration of its 'smoothness'. For earthquake ruptures occurring on a smooth fault, for example, on the SAF, an EEW system could be implemented in two ways: the system could issue a warning, whenever slip on the fault exceeds a few metres, because the probability for a large earthquake is high (Table 1) and strong shaking is expected to occur in large areas around the fault. A more sophisticated EEW system could use the present slip on the fault to estimate the final rupture dimensions and future slip evolution, and (using this information) could provide probabilistic predictions of seismic ground motions along the evolving rupture. The decision on whether an EEW system should be realized in the first or in the second way (or in a combination of both) is clearly user-specific, and depends on (1) the vulnerability of the considered facility, and (2) the costs in the case of under-or overestimated ground shaking (Cua & Heaton 2007;Grasso et al. 2007;. The possible integration of our probabilistic approach into an EEW system for large earthquakes is shown in Fig. 13. A regional monitoring (GPS and/or seismic sensor network) and processing system provides real-time measurements of slip along the rupturing fault (e.g. Crowell et al. 2009). From a regional fault model, the system identifies the fault or fault segment along which the rupture likely propagates. Depending on the corresponding fault characteristics (dimensions, generic/mature fault, probability of multisegment propagation, etc.) and on the current slip amplitude D p , the processing system evaluates the probability of remaining rupture length L r . Combined with the present rupture length L p , a probabilistic distribution of the expected total length L = (L p + L r ) is computed, and from this, using, for example, eq. (1), a probabilistic distribution of the magnitude M. Length and magnitude estimates are combined with empirical attenuation relationships for a probabilistic prediction of the seismic ground motions along the evolving rupture. As time progresses, additional data will become available and allow for updating these estimates.
The probabilistic real-time prediction of rupture expansions can improve EEW for large earthquakes by more realistic estimates of the levels and distributions of ground shaking along the earthquake rupture compared to point source approximations. Uncertainties are expressed by the specification of PDFs. The real-time prediction of the final rupture dimensions, slip and seismic ground motions along the ongoing rupture becomes feasible for EEW, if the rupture duration exceeds the processing time of the warning system. If we assume a constant rupture velocity of v r = 2.8 km s −1 , it takes around 11 s for a 30-km-long rupture (∼ M 6.5) to cease, 22 s for a 60-km-long rupture (∼M 7.0), ∼50 s for a 140-km-long rupture (∼M 7.5) and around 100 s for a 300-km-long rupture (∼M 8.0). This implies that the probabilistic rupture and ground motion prediction as outlined in Fig. 13 might become feasible for EEW for earthquakes with M > 6.5, which usually are considered to be most destructive.
Our findings might also shed some light onto the discussions of whether the eventual size of an earthquake is pre-determined at the initial stage of the rupture process, that is, of whether seismic ruptures are 'deterministic' (e.g. Olson & Allen 2005;Rydelek & Horiuchu 2006;Rydelek et al. 2007). This is indirectly assumed by many EEW algorithms when deriving seismic information on magnitude from the initial portion of the seismic P wave only (Nakamura 1988;Allen & Kanamori 2003;Kanamori 2005;Böse et al. 2007;Wu et al. 2007). Our results imply that the observation of a (apparent) moderate earthquake (judging from the first few seconds of seismic waveform data, e.g. from the predominant or the average period of shaking) on a smooth (mature) fault should result in a stronger warning than if observed on a heterogeneous (generic) fault, because of a higher probability of future strong shaking at sites far from the epicentre. Hillers et al. (2006) simulated evolving ruptures in quasidynamic continuum models of a 2-D strike-slip fault in an elastic solid with different levels of geometrical heterogeneity tuned by the spatial variations of the critical slip distance L c . They found that the nucleation phases of small and large events differ in models with heterogeneous L c distributions [see also Ellsworth & Beroza (1995) and Beroza & Ellsworth (1996)]. Even if we do not analyse the seismic nucleation phase here, it seems that our results suggest that if ruptures occur on faults with very high heterogeneity, small and large events cannot be distinguished from each other at the early stage of the rupture. This observation is in good agreement with the model of cascading subevents on heterogeneous faults, in which the event size is influenced strongly by the highly variable stress-strength conditions along the fault (e.g. Ben-Zion et al. 2003).
The 1-D stochastic slip models used in this study were chosen for their conceptual simplicity, and there are indications that some of their features are incompatible with observations at real earthquakes. However, the 1-D models may capture several basic features of slip-pulse type models of dynamic rupture (Liu-Zeng et al. 2005). Other, more complex (dynamic) models might be better suited to reproduce time-dependent features of earthquakes, but they require the specification of a large number of free parameters (such as the pre-stress conditions on the fault) that are largely unknown. We therefore think that the usage of our static 1-D models in the presented study is justified by the fact that we describe the general features of slip, such as the dependency of rupture propagation on slip heterogeneity, in a statistical sense, rather than providing a quantitative analysis.
The root physical cause of heterogeneous slip is not well understood. It seems likely that geometric complexities in fault geometry and in material properties play a role (Rice 1993;Andrews 1994;Aki 1995). However, dynamic chaos that occurs when sliding friction includes strong slip velocity weakening may also be a source of slip heterogeneity (Aagaard & Heaton 2008). Regardless of the cause of the heterogeneity, our findings point to the desirability of characterizing spatial heterogeneity on recognized faults. Manighetti et al. (2007) suggest that the variability in the slipto-rupture length ratios arises from earthquakes breaking a variable number of fault segments with variable frictional strength that depends on both their long-term slip history ('structural maternity') and on their geometry (large-scale segmentation). This interpretation could provide a possible link between the concepts of slip roughness μ and fault maturity.
The classification of faults with respect to their level of maturity is not trivial. We hypothesize that a mature fault like the SAF could be identified from its seismicity (e.g. very few small earthquakes), its geological persistence, the total offset on the fault and its geometric simplicity (e.g. Aviles et al. 1987). Clearly, the number of mature faults worldwide is small, and we therefore suggest installing GPS monitoring systems along these few faults to build very simple and effective EEW systems.
In this study, we based the probabilistic prediction of L r on the present slip amplitude D p . We might possibly expect that also the average slip D taken between the spatial start point and the present rupture length L p = L -L r might be suited for the prediction of L r of an ongoing rupture. We found, however, that the correlation is smaller for D[R(D, L r ) ≈ 0.6] than for D p [R(D p , L r ) ≈ 0.8], implying that D p is more suited for the prediction of L r than D. If the slip is very heterogeneous, then the rupture quickly loses its memory of past slip as it propagates along the fault. Ruptures with slowly varying slip maintain their memory of past rupture over relatively longer lengths (spatial 'memory effect' of smooth faults).
Even if we demonstrate our concept in this paper for a scenario earthquake with unidirectional rupture propagation, our probabilistic method is directly applicable to earthquakes with both uni-and bidirectional rupture propagation. Because of the symmetry of the problem, the variability in p(log(L r )|log(D p )) does not increase in the bidirectional case: assuming that the underlying fault is infinite, the estimates of remaining rupture length L r to both directions are independent from each other and can be determined in the same manner as in the unidirectional case. However, the convergence to the true magnitude will be much faster in the bidirectional case and the distribution of ground motions might likely differ significantly compared to the unidirectional case. Bidirectional ruptures clearly complicate the problem of ground motion prediction and remain a field of future research.
Our probabilistic approach allows specifying inherent uncertainties in the estimated source and ground motion parameters. A complete assessment would require the additional consideration of, for example, the uncertainties in the rupture length-magnitude relation in eq. (1), uncertainties in the empirical attenuation laws, site and directivity effects. Yet another important issue in the context of rupture prediction clearly is the consideration of rupture propagation along different fault segments (possibly with different rupture roughness), as studied, for example, by Jackson et al. (1995) and Field et al. (2007). Including these effects in the probabilistic models as presented here, will lead to a further improvement of EEW for large magnitude earthquakes in the future.

A C K N O W L E D G M E N T S
This work is funded through contract G09AC00258 from USGS/ANSS to the California Institute of Technology (Caltech). This is contribution #10039 of the Seismological Laboratory, Geological and Planetary Sciences at Caltech. Calculations and most figures in this paper were generated from Matlab 7.8. The map in Fig. 11a (left-hand side) was made using the Generic Mapping Tools version 4.2.1 (www.soest.hawaii.edu/gmt; Wessel & Smith 1998). We would like to thank Egill Hauksson for proofreading. We are grateful for the helpful comments of P. Martin Mai, two anonymous reviewers and the editor Yehuda Ben-Zion that helped us to improve an earlier version of this manuscript.