UCERF3 Task R2- Evaluate Magnitude-Scaling Relationships and Depth of Rupture: Proposed Solutions

UCERF3 Task R2- Evaluate Magnitude-Scaling Relationships and Depth of Rupture: Proposed Solutions Bruce E. Shaw Lamont Doherty Earth Observatory, Columbia University Statement of the Problem In UCERF2 [WGCEP, 2008] Magnitude-Area scaling relations contributed one of the main sources of uncertainty in the final hazard estimates. In this task, we seek ways of finding additional constraints or alternative methods of using scaling which will reduce these uncertainties. Summary of Recommendations A thorough study was made of the two main uses of magnitude-area relations: one, to figure out the sizes of events, and two, to figure out the rates of events through moment or slip-rate balancing. The accompanying paper [Shaw, 2011b] discusses the issues and results more comprehensively. Here, we summarize the recommendations for UCERF3. With respect to moment or slip-rate balancing, qualitative consistency was found between the implied slip-length from magnitude-area scaling, and surface slip-length observations. However, some quantitative difference, of order 30%, remained, with the implied slip from magnitude-area being higher. This could be due to the deep coseismic slip being mapped onto shallower seismogenic layers, so that magnitude-area relations are overestimating average slip. Alternatively, it could be due to the surface slip measurements underestimating slip, perhaps due to distributed plastic deformation in unconsolidated layers or other such mechanisms. This quantitative discrepancy remains an unresolved epistemic uncertainty. The qualitative agreement, however, does give useful information in distinguishing different scaling laws. The accompanying paper discusses this more thoroughly. Given the epistemic uncertainties, we suggest two main branches. The difference in the two branches relates to how the slip-rate balancing is done. In all cases, we suggest continuing to use magnitude-area relations to estimate the sizes of events. Whatever the epistemic uncertainties associated with the depth dependence of slip, magnitude-area relations give good size estimates, and seismogenic area is a good proxy, and self-consistent when understood as such, for these relations. Three magnitude-area scaling relations are considered as most viable for estimating sizes of events: Ellsworth-B [WGCEP, 2003], Hanks-Bakun [Hanks and Bakun, 2002, 2008], and Shaw09 [Shaw, 2009]. The Shaw09 relation generalizes the Hanks and Bakun relation from two to three regimes, adding a third regime for the largest events with slip saturating with width W scaling. The relations are Ellsworth-B: M = log A +.2 (1) This has the virtue of being extremely simple, being a one parameter fit, and being a good empirical fit to the large earthquake (M > 6.5) data it was developed to fit. Hanks and Bakun [2002] developed a two regime saling: { log A +3.98 A 537 km 2 ; M = 3 log A +3.07 A>537 km2. This has the virtue of doing a better job fitting the data at smaller magnitudes. Shaw [2009] developed a generalization of the Hanks and Bakun model, which extends the bilinear Hanks-Bakun two-regime scaling to a three-regime scaling which has a third asymptotic regime valid for very long ruptures L W whereby S approaches W scaling asymptotically. This is done at the price of one additional (2) 1

scaling parameter (the scale at which the transition to the third regime occurs). The Shaw [2009] scaling relation is parameterized as follows. M = log 10 A + 2 A 3 log max(1, ) H 2 10 + const. (3) A (1 + max(1, ))/2 H 2 β Here H is the seismogenic depth, and β is a fitting parameter which gives a crossover scale length to the asymptotic W scaling. In the limit of β it reduces to the Hanks-Bakun scaling relation, and in this sense is a one parameter extension of it. Note also that in the limit as L it scales asymptotically as M 2 3 log 10A which gives S W (using W H), as the Knopoff [1958] solution suggests. Fits to the Hanks-Bakun data give best fitting parameters of values H = 16.1 km and β =6.6. Based on the whole magnitude range in the Hanks and Bakun database for magnitude-area of strike-slip events, Shaw09 slightly outperforms Hanks and Bakun on an AIC test, though they perform comparably, and both outperform Ellsworth-B. Based on M > 6.5, all three scaling relations perform comparably. This is based on misfits in magnitude-log area space, so all events regardless of size are weighted equally. Figure 1 shows the fit of the three scaling relations to the Hanks-Bakun data. Table 1 summarizes these results of the fits. Looking in linear weighted space, as is relevant for moment or slip-rate balancing, the picture changes substantially. There, the Hanks-Bakun relation is seen to be a poor fit of the largest events, with Shaw09 slightly outperforming Ellsworth-B, and both very much outperforming Hanks and Bakun. Figure 2a shows these results, plotting the magnitude-area data transformed into slip-length data, and plotted on a linear scale relevant for slip-rate balancing. Table 2 summarizes these results of the fits. Weights for sizes of events branches For UCERF3 purposes, all three magnitude-area scaling relations perform sufficiently for estimating sizes of events. Shaw09 performed well on both the log-weighting (across all magnitudes) and linear weighting (weighted towards largest magnitudes), while Hanks and Bakun did not perform well on the linear weighting and Ellsworth-B did not perform as well on the log weighting (though questions have been raised about the area estimates for the smaller magnitude events). Given these results, viable branch weights could be replacing the UCERF2 Hanks and Bakun weight with the Shaw09 relation which generalizes it. Alternatively, some weight on the Hanks and Bakun branch could be retained for continuity. Weights for Moment or Slip-Rate Balancing Two main branches are suggested. One branch continues to use magnitude-area scaling to estimate average slip, as was done in UCERF2. For this branch, we suggest using the Ellsworth-B and Shaw09 relations, since the Hanks and Bakun relation performed poorly on the linear weighting relevant to rate balancing. (See Table 2). For the second branch, slip rate would be balanced using slip-length scaling relations. Linear scaling with L out to the largest events was found to be a poor fit to the data. The best fits were for a constant stress drop model, which transitions from asymptotic L scaling to asymptotic W scaling, and for a square root L 1/2 model. The equations for these two models are as follows. For the square root scaling consistent with the implied Ellsworth-B scaling, S A 1/2 = C (LW ) 1/2 () 2

with C a fitting constant. For the constant stress drop model [Shaw, 2011b] consistent with the Shaw09 scaling S = σ 1 (5) µ 7 3L + 1 2W with µ the shear modulus and σ the stress drop. This relationship comes from a more careful treatment of how slip would be expected to transition from circular ruptures just beginning to break the surface to long rectangular ruptures. Best fit values of the constants C and σ were found for fits to the Wesnousky [2008] data in the associated paper Shaw [2011b], with C =.91 10 5,and assuming µ =30GP a abestfit σ =3.91 MPa. These constants are expected to change slightly when fit to the updated UCERF3 large event dataset. The constant stress drop model somewhat outperformed the L 1/2 scaling, but they both performed well in fitting the data. Figure 2b shows the fits of these slip-length functions to surface slip data. Table 3 summarizes the results of these fits. Based on the fits, and a desire to span epistemic uncertainty, we suggest equal weighting to these two models. W scaling For use with variable downdip width W, the dependence on W has been included in the slip-length scaling relations, Equations () and (5). (See paper for details). Note that the scalings imply different W dependence. Looking at the largest events to try to distinguish between the candidate scalings, the sparse noisy data favored W scaling over W 1/2 scaling (See Figure 6 in paper). New datasets Data sets examined thus far are the Hanks and Bakun [2008] magnitude-area dataset and the Wesnousky [2008] surface slip dataset. The updated dataset for large earthquakes being compiled for UCERF3 would be very useful in further refining weightings for scaling relations. The basic framework is not expected to change, but the relative performance in fitting the data might, and this could impact recommended branch weights. Epistemic Uncertainty Currently, to fits for the two types of slip-length relations, those derived from implied magnitude-area scaling laws, and those derived from fits to surface slip data, span the range of epistemic uncertainty. This epistimic undertainty is illustrated in Figure 3, which shows superposed from Figure 2a the implied magnitude area scaling laws with dashed lines, and from Figure 2b the surface slip scaling laws with solid lines. Newer datasets are expected to raise somewhat the amplitude of the surface slip measurement scaling laws. Cybershake Cybershake [Graves et al., 2010] is somewhat outside the scope of UCERF3, but it is, nevertheless an interested user, and potentially can develop into a useful constraint. Some of the branches in UCERF3 will be more compatible with cybershake than others. Since cybershake itself has many underlying assumptions (such as the degree of coherence in the ruptures), sensitivities in cybershake need to be explored further before it is used as a constraint on UCERF relations. Nevertheless, it is a goal to have compatibility developed, and thus exploring the implications of the various relations in this context is an important task. An unresolved epistemic uncertainty in the depth dependence of slip concerns how much coseismic slip may be occurring below the seismogenic layer. Dynamic models suggest a significant fraction of 3

slip may be occurring below the seismogenic layer up to of order 1/3 the total moment [Shaw and Wesnousky, 2008]. This slip would be occurring mainly as long period motion, with a dearth of high frequencies [Shaw and Wesnousky, 2008]. An exponential decay or a linear decay with depth below the seismogenic layer are both reasonable parameterizations of what is seen in the dynamic models [Shaw and Wesnousky, 2008]. The scale length of the decay depends on the degree of velocity strengthening; the degree of strengthening is not well constrained, and thus the scale length remains an epistemic uncertainty. The distributions of slip, and rupture velocities, are important factors which warrant further study in cybershake. Correlations of slip at the surface have gaussian, or even faster than gaussian falloff in the tails [Shaw, 2011a], so cybershake simulations positing extreme slip values in the distributions (e.g. 50m slip in great strike-slip earthquake patches) should be reexamined. Given the implied high stress drops at moderate magnitudes in the Ellsworth-B relations, and the implied high stress drops at large magnitudes in the Hanks-Bakun relations, it would be interesting to see if constant stress drop models can be seen to be compatible with cybershake. Is, for example, the Shaw09 magnitude-area scaling relation, combined with a 30% subseismogenic high-frequency deficient moment, compatible with more high variance rupture velocity low variance slip patch versions of cybershake? References Graves, R., et al., Cybershake: A physics-based seismic hazard model for Southern California, Pageoph, 2010. Hanks, T. C., and W. H. Bakun, A blinear source-scaling model for M-log A observations of continental earthquakes, Bull. Seismol. Soc. Am., 92, 181, 2002. Hanks, T. C., and W. H. Bakun, M-log A observations of recent large earthquakes, Bull. Seismol. Soc. Am., 98, 90, 2008. Knopoff, L., Energy release in earthquakes, Geophys. J. R. A. S., 1,, 1958. Shaw, B. E., Constant stress drop from small to great earthquakes in magnitude-area scaling, Bull. Seismol. Soc. Am., 99, 871, 2009. Shaw, B. E., Surface slip gradients of large earthquakes, Bull. Seismol. Soc. Am., 101, 792, 2011a. Shaw, B. E., An alternative pathway for seismic hazard estimates based on slip-length scaling, submitted, 2011b. Shaw, B. E., and S. G. Wesnousky, Slip-length scaling in large earthquakes: The role of deep penetrating slip below the seismogenic layer, Bull. Seismol. Soc. Am., 98, 1633, 2008. Wesnousky, S. G., Displacement and geometrical characteristics of earthquake surface ruptures: Issues and implications for seismic-hazard analysis and the process of earthquake rupture, Bull. Seismol. Soc. Am., 98, 1609, 2008. WGCEP, Earthquake probabilities in the San Francisco Bay Region: 2002 to 2031, U.S. Geol. Surv. Open File Rep., 03-21, 2003. WGCEP, The Uniform California Earthquake Rupture Forecast, version 2 (ucerf2), U.S. Geol. Surv. Open File Rep., 2007-137, 2008.

Tables of Fits of Scalings to Data Magnitude-Area Scaling Table 1: Magnitude-Area Scaling Fits Scaling Best fits Parameters Name Eqn Std. dev. AIC No. Best fit values S09 (3).12-86.58 3 C 0 =3.98 W =16.1 km β =6.6 HB (2).1-86.55 1 C 0 =3.98 A c = 537 km 2 EB (1).19-36.07 2 C 0 =.20 The standard deviation in the above table measures the difference in magnitudes of the data from the predicted curve. The data comes from the Hanks-Bakun database [Hanks and Bakun, 2008], which is reproduced in the electronic supplement S1. The number of free parameters is given by No., with the best fitted parameter values following that. The parameter C 0 is the constant in the magnitudearea scaling relations. Ranking in this table, and those that follow, is by AIC, which rewards reduced standard deviation but penalizes additional parameters. Slip-Length Scaling from Implied Magnitude Area Data Table 2: Implied Slip-Length from Magnitude-Area Scaling Best fits Parameters Name Eqn Std. dev. AIC No. Values S09 (3) 1.180 281.77 2 C 0 =3.98 Wβ = 106.3 km EB (1) 1.211 282.17 1 C 0 =.20 HB (2) 1.59 316.62 2 C 0 =3.98 A c = 537 km 2 The standard deviation in the above table measures the difference in slip of the data from the predicted curve. The data comes from the Hanks-Bakun database [Hanks and Bakun, 2008], which is transformed from magnitude-area to slip-length as follows. To get slip we convert magnitude to moment and divide by area and modulus to get slip. To get length we divide area by width, assuming width is the lesser of the length compared with the downdip seismogenic width. The table shows a comparison of the transformed data with scaling relations derived from transformed magnitude area to implied slip-length. scaling relations by a constant factor ρ. 5

Slip-Length Scaling from Surface Slip Data Table 3: Slip-Length from Surface Slip Data Scaling Best fits Parameters Name Eqn Std. dev. AIC No. Best fit values S11 (5).760 50.05 1 σ =3.91 MPa L 1/2 ().80 52.3 1 C =.91 10 5 L 1.17 68.3 1 C 5 =7.10 10 7 km 1 The standard deviation in the above table measures the difference in slip of the data from the predicted curve. The data is derived from the Wesnousky database for surface slip in large Strike-Slip events [Wesnousky, 2008]. The table shows a comparison of the surface slip data with scaling relations proposed for surface slip-length scaling. The best overall fit to the data, in terms of AIC, is S11, the constant stress drop model. 6

9 8 M 7 6 5 1 2 3 5 log A [km 2 ] 10 Figure 1: Magnitude area relations for large strike-slip events. Red circles denote magnitude and area of events from Hanks and Bakun [2008] database. Solid black line is linear Ellsworth-B [WGCEP, 2003] magnitude-area relation, Equation (1). Dashed green line is Hanks and Bakun [2002] bilinear relation, Equation (2). Blue line is new generalized scaling relation, Equation (3) [Shaw, 2009]. 7

10 5 8 S[m] 6 S[m] 3 2 2 1 (a) 0 0 100 200 300 00 500 L[km] (b) 0 0 100 200 300 00 500 L[km] Figure 2: Fits of slip-length scaling. (a) Slip-length scaling implied by magnitude-area data. Data (light blue circles) derived from Hanks and Bakun [2008] mag-area data by rescaling as follows. Magnitude is converted to moment, then divided by area and modulus to get slip. Area is converted to length by dividing area by width assuming seismogenic depth H=15 km. Different color lines represent different magnitude-area scaling laws rescaled the same way the data has been. The curves are the implied slip-length for: Ellsworth-B [WGCEP, 2003] (red), Hanks-Bakun [Hanks and Bakun, 2008] (green), and Shaw [2009] (black). (b) Fits of surface sliplength scaling laws to surface slip observations of average slip versus length. Color indicates focal mechanism: strike-slip (blue), normal (cyan), thrust (magenta). In order of best fit to least good fit: black line shows Equation (5) S11 scaling; red line shows Equation () L 1/2 scaling. Note change in vertical axis scale in (a) versus (b), reflecting overall systematic difference, and fundamental epistemic uncertainty, in average slip estimates from magnitude-area versus surface slip observations. Some of this difference reflects different events in the two different databases, but some of it, a difference of around 30%, remains when we restrict to looking at the same events. 10 8 S[m] 6 2 0 0 100 200 300 00 500 L[km] Figure 3: Slip length scaling relations for large strike-slip events. Dashed lines are implied slip-length scaling relations derived from Magnitude-Area scaling relations. Solid lines are slip-length scaling relations derived from surface slip observations. Note that the dashed lines are systematically above the solid lines. Dashed cyan line is implied slip-length from Ellsworth-B [WGCEP, 2003] magnitude-area relation, Equation (1). Dashed green line is implied slip-length from Hanks and Bakun [2002] bilinear relation, Equation (2). Dashed blue line is implied slip-length from [Shaw, 2009], Equation (3). Solid red line is L 1/2 scaling, Equation (). Solid black line is constant stress drop scaling [Shaw, 2011b], Equation (5). The dashed blue line best fits the slip-length data derived from the magnitude-area data. The solid black line best fits the surface slip-length data. 8