UCERF3 Task R- Evaluate Magnitude-Scaling Relationships and Depth of Rupture: Proposed Solutions Bruce E. Shaw Lamont Doherty Earth Observatory, Columbia University Statement of the Problem In UCERF 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. Proposed Solution In UCERF, Magnitude-Area scaling was used in two different ways. One way was to estimate the sizes of events that occurred on given sections of faults and the associated shaking through empirical attenuation relations. A second way was through deriving effective average slip in events to do moment or slip-rate balancing which then set overall rates. While this second use is a valid methodology, it introduces a few difficulties which are sources of significant uncertainty in the final hazard estimates. Prominently, differences in proposed empirical magnitude-area scaling laws imply substantial differences in hazard estimates, mainly due to the fact that the moment sum is dominated by the largest events, and differences in magnitude estimates of the largest events in the different empirical relationships end up affecting rates across the whole spectrum of sizes of events. An additional complication in this pathway is the recently raised question of whether or not a significant fraction of the moment may be coming from deeper slip below the seismogenic layer, thereby complicating ways of moment-balancing to ensure long term slip rates are matched. In UCERF3, we propose to use an alternative complementary scaling law, slip versus length scaling, in combination with magnitude-are scaling. The slip-length scaling would be used in two different ways. First, and most significantly, we propose to use slip length scaling instead of magnitude-area scaling to do the slip-rate (or moment ) balancing on faults. This alternative pathway has a number of advantages. Importantly, because it is observable at the surface, it substitutes a directly observable relation for one which requires assumptions about how slip is distributed in depth. In this way, it reduces epistemic uncertainties. Moreover, since it feeds back on the rate of smaller events through the rate balancing, it contributes to reducing what is probably the largest contribution to uncertainty associated with the magnitude-area scaling relations. The second proposed use of slip-length scaling observations is as an additional constraint on magnitude-area scaling in determining the sizes of events. While the down-dip distribution of slip matters in trying to use magnitude-area scaling to balance slip rates, it does not matter in trying to use them to determine the size of events and the associated shaking when using empirical attenuation relationships. In this latter case, there is a self-consistency in how seismogenic area is defined so that the resulting magnitudes and inferred shaking do not depend on uncertain depth dependencies of slip. At the same time, however, we can use the implied slip-length scaling inherent in definitions of area to see whether these implied scalings are consistent with the direct observations. In this second use, we project the magnitude-area scaling onto slip-length scaling, as was done in UCERF to do moment or slip rate balancing, but now use this to see whether the empirical scaling laws in that projection look consistent with the direct slip-length scaling observations. Figure 1 shows this approach. We learn a few things by doing this. First, this transformation to slip-length viewed in linear-linear space, as opposed to the log-log space where the empirical fits were originally made, is a better reference frame to try to do slip-rate balancing from since the slips will be added linearly in the sum. Transforming to linear-linear space emphasizes trying to fit the largest events, whereas log-log space emphasizes trying to fit the widest 1
8 1 1 1 1 6 S [m] 8 6 S [m] 8 6 S [m] (a) 1 3 5 6 L [km] (b) 1 3 5 6 L [km] (c) 1 3 5 6 L [km] Figure 1: Slip-length scaling. (a) Data (red circles) derived from Hanks and Bakun [8] mag-area scaled by assumed seismogenic depth H=15km. Different color lines represent different magnitude-area scaling laws rescaled the same way the data has been; Ellsworth-B [WGCEP, 3] (black), Wells-Coppersmith [Wells and Coppersmith, 199] (yellow), Hanks-Bakun [Hanks and Bakun, 8] (green), Shaw [9] (blue). (b) Surface slip data from Wesnousky [8]. Colored lines same as in (a). (c) Same data as (b), but lines rescaled by factor of.6. scale range of events. To insure proper weighting of different information contained in the different events, and given the structure of the uncertainties in the data, it is still the case that it makes sense to do the fitting of the curves in log-log space; but it is important to make sure that the resulting fits do a good job in linear-linear space, since the largest events are most important to the sum, and where the most ambiguity lies. Second, the scaling laws developed for magnitude-area seem fairly reasonable compared with the magnitude-area data rescaled to implied slip-length values (here assuming a constant seismogenic thickness of H = 15km for all events, and constant down-dip average slip). However, when compared with a database of geologically observed surface slip-length data there are a number of significant issues which arise. One issue, seen in Figure 1b, is that the scaling laws appear to overpredict the observations of geological surface slip. Part of the discrepancy here is due to the different events in the two databases; this is an issue we will return to. Another issue is we see one of the scaling laws used in UCERF, the Hanks-Bakun scaling law [Hanks and Bakun, 8], just does not appear viable even when rescaled in amplitude as all of the curves are done in Figure 1c. What are the origins of the differences and how might we deal with it? The Wells and Coppersmith [199] empirical scaling relations for large earthquakes have played a central role in established thinking about the scaling of large earthquakes. They were, however, developed as empirical relations, and the limitations of these kinds of relations can become apparent when they are pushed beyond the range over which there is sufficiently constraining data. In particular, for the largest events where data is pretty sparse, some problems arise. One fundamental issue is that, for large aspect ratio events, where the length is large compared to the width of the rupture, the scaling relations are not self-consistent. This is reflected in the epistemic uncertainty of the two leading magnitude-area scaling laws used in UCERF, the Ellsworth-B [WGCEP, 3] scaling relation and the Hanks-Bakun one [Hanks and Bakun,, 8]. Leaving aside small differences in the exponents and a constant amplitude offset, the Ellsworth-B scaling relation connects to the moment-area scaling relation of Wells and Coppersmith [199], whereas the Hanks-Bakun scaling relation connects for the largest events with the slip-length scaling relation of Wells and Coppersmith [199]. From this, we can see the inconsistency. For the Ellsworth-B relation based on the scaling of moment M with area A, withm A 3/, expressed in terms of slip S scaling with length L for very long ruptures where A = LW with width W = const we get S M/A A 1/ L 1/ or S L 1/. In contrast, for the Hanks-Bakun for very large aspect ratio events M A which gives S M/A A or S L connecting to the nearly linear scaling relation of Wells and Coppersmith [199] for slip versus length. All of these scaling relations were developed in log-log space. If we look in linear-linear space, however, we find much less support for the linear increase of slip with length out to the largest events.
Proposed Magnitude-Area scaling relations In this task we propose that Area A continue to be defined as it was in UCERF, as the seismogenic depth estimated from depth of seismicity and the along-strike length. As in UCERF, we propose to use two magnitude-area scaling laws to span the epistemic uncertainty. Because of its simplicity, and consistency with both the magnitude-area data (at least for M>6.5) and slip-length data, we propose to continue using the Ellsworth-B relation M = log A +. (1) as one scaling relationship for magnitude-area. For the second scaling relation, we propose to replace the Hanks-Bakun scaling relation with a generalization which matches better the largest events. For the second scaling relation we propose to use the scaling relation of Shaw [9] 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 scaling parameter (the scale at which the transition to the third regime occurs). It has been shown, however, to be a better fit to the Hanks-Bakun magnitude-area data, and to be a sufficiently better fit from an AIC and Fisher F test to justify the additional parameter [Shaw, 9]. The Shaw [9] scaling relation is parameterized as follows. M = log 1 A + 3 log 1 max(1, A H ) + const. () A (1 + max(1, ))/ H 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 3 log 1A 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 =15.6km and β =6.9 [Shaw, 9]. Figure shows the fit of the various scaling relations to the Hanks-Bakun data. Some important epistemic uncertainties remain with this proposed solution, particularly at the lower magnitudes where the different magnitude-area scaling laws diverge (Ellsworth-B on the one hand, [Wells and Coppersmith, 199; Hanks and Bakun, 8; Shaw, 9] on the other, though the later are all based on the same data and are thus not independently determined). Finite source inversions and precise relative relocations of aftershocks are two potential ways of going after these uncertainties. Research in these areas is encouraged, and would be incorporated if it sufficiently advances in the time period of UCERF3, but it may require waiting for future UCERF iterations. Proposed Slip-Length scaling relations For Slip-Length scaling relations we propose to use two scaling relations to span the epistemic uncertainty. One is the effective scaling law implied by the Ellsworth-B scaling relation and supported empirically by the Wesnousky [8] fit to the surface slip data, that S = αl 1/ (3) where α is a constant amplitude still to be fit. This scaling has the advantage of a minimal parameterization, simplicity, empirical support [Wesnousky, 8], and consistency with other scaling laws used in other regimes (Ellsworth-B magnitude-area). As a result of the March 11 Menlo Park UCERF3 workshop on the Distribution of Slip in Large Earthquakes, the database of events which will be fit will be supplemented with additional events, in particular a number of Mongolian and Chinese events, as well as an enveloping procedure on the slip, which is expected to change the amplitude of the scaling. The amplitude α will be revised after the database of surface slip events is reworked. 3
9 8 M 7 6 5 1 3 5 log 1 A [km ] Figure : Magnitude area relations for large strike-slip events. Red circles denote magnitude and area of events from Hanks and Bakun [8] database. Solid black line is linear Ellsworth-B [WGCEP, 3] magnitude-area relation. Dashed green line is Hanks and Bakun [] bilinear relation. Blue line is new proposed scaling relation, Equation () [Shaw, 9]. As a second scaling relationship, we propose to use a generalization of a scaling law which connects L scaling at small aspect ratio events to W scaling at large aspect ratio events [Shaw and Scholz, 1; Manighetti et al., 7; Shaw and Wesnousky, 8]. An advance we have developed is a more careful treatment of how this scaling connects in the two regimes, which finds, importantly, that a constant stress-drop model can match the data well. First, let us discuss the data, shown in Figure 3. The circles show individual data points for average surface slip as a function of surface rupture length, with strike-slip events colored blue, normal faulting events cyan, and thrust events magenta. The data without error bars is taken from a database compiled by Wesnousky [8]. I have supplemented this with the slip from the 8 M7.8 Wenchuan earthquake using data from [Xu et al., 1], shown with error bars given their estimate of 3 m of average slip at the surface. We see confirmed a number of things noted previously, in particular the clear trend towards saturation of the slip at large lengthscales and the large crossover lengthscale in the saturation. We also see something new by combining the Wenchuan data: evidence that large aspect ratio events with other focal mechanisms besides strike-slip also saturate in slip at large lengths, and scale similarly as the strike-slip events. The dotted and solid lines on the figure represent major progress in understanding the data. Recently, in an effort to find an origin of the large apparent lengthscale of the saturation, I studied 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. Interestingly, if one assumes a circular rupture with constant stress drop on the interior, and looks at the intersection of the slip profile with a line representing the surface (neglecting free surface effects at this stage), the average slip S as a function of the line length L scales linearly with the line interesection length independently of the radius of the circle, with average slip S σ 3 µ 7L where σ is the stress drop and µ is the elastic shear modulus. At the other asymptotic limit, for rupture lengths L much greater than the seismogenic width W, slip on a constant stress drop long rectangular rupture saturates and scales with W : S σ µ W [Knopoff, 1958; Scholz, ]. The two scaling limits are shown with dashed lines, using an assumed value for constant stress drop of σ =MPa and seismogenic width W =15km to
illustrate an example scaling. Right away we find something very interesting: the crossover lengthscale L c, where the two scalings intersect, is a large multiple of W, L c = 1 3 W. Taking W =15km gives a crossover lengthscale of 7 km, which is very similar to the crossover lengthscale found seismologically [Romanowicz, 199, 199]. Combining these two asymptotic limits in parallel gives a new proposed scaling [Shaw, 11]: S = σ µ 1 7 3L + 1 W shown in the figure with a solid line. This scaling is similar qualitatively to previous proposed scalings [Shaw and Scholz, 1], but differing quantitatively in the coefficients on the lengthscales, and thereby resolving the puzzling aspects associated with the large inferred crossover lengthscale [Shaw and Scholz, 1; Manighetti et al., 7; Shaw and Wesnousky, 8]. The fit to the data is strikingly good. () 5 S[m] 3 1 1 3 5 L [km] Figure 3: Geological surface slip observations of average slip versus length. Color indicates focal mechanism: strike-slip (blue), normal (cyan), thrust (magenta). Dashed lines show asymptotic scaling limits for circular and long rectangular ruptures. Solid line shows scaling combining these two limits, Equation (). Parameters on lines are: constant stress drop σ =MPa, seismogenic width W =15km. Figure shows unoptimized fits of both proposed scaling relations, Equation (3) and Equation () to the data. One further feature shown in the plots is the addition of slip-length data of various focal mechanisms, shown as circles of different colors. While the data is quite sparse for the other focal mechanisms, we suggest that the simplest hypothesis, that they do not differ, is the best approach for now. Note that, due to the downdip width W dependence, we would expect much larger slip values on shallow dipping subduction zone events, which are not included in the available data. Here, differences in W for the subareal crustal events do not appear as strongly. A few further comments on this proposed scaling relation. First, we did relax the constraint that the stress drops for the two asymptotic regimes matched, but found best fit values of stress drop which differed little at the small end and large end, and thus did not justify the introduction of an additional parameter. Second, the best fitting values are expected to change with the updated surface slip database. It is interesting to note that the range in stress drop variations about the mean seems to be relatively narrow at the large events of order a factor of or less on either side. This contrasts 5
5 S[m] 3 1 1 3 5 L [km] Figure : Geological surface slip observations of average slip versus length. Color indicates focal mechanism: strike-slip (blue), normal (cyan), thrust (magenta). Solid red line shows Equation (3) scaling. Solid black line shows Equation () scaling. Neither line have optimized parameters yet; shown here to give feel for fit of functional form. with more like a factor of 1 variation on either side of the mean values at small events [Hanks and Kanamori, 1979]. It could be measurement uncertainties in corner frequencies at small events are affecting the scatter there, or it could be a real effect. Nevertheless, the at most small difference in the mean values of stress drops for the great events relative to the small events is remarkable, and gives support for the constant stress drop approach. Discussion Other magnitude-area scaling laws Given the support of the constant stress drop slip-length scaling, one might wonder why there is not an analogously proposed scaling for the magnitude-area scalings. We did explore this avenue, but found such scalings to have a more limited range in variation of magnitude versus area in the moderate magnitude regime (M6-7), and thus did not span the epistemic uncertainty as much as the two proposed scaling laws do. Since spanning the epistemic uncertainty was considered the most important feature, we did not at this stage propose such a logic-tree branch. Other magnitude-area scaling laws considered but not ultimately recommended include the Somerville [6] scaling relation used in cybershake and a scaling proposed by Jackson [1, WGCEP discussions]. In the case of the Somerville scaling, the procedure used to estimate area based on trimmed source inversions was deemed non-robust, being sensitive to details of the inversion algorithms and unconstrained smoothing assumptions. Non-robustness and untransparent reproducibility were considered undesirable features for UCERF purposes. In the case of the Jackson [1] proposed scaling relations, the linear increase of slip with length out to the largest magnitudes, in common with the Hanks-Bakun scaling, was deemed an insufficiently good fit to the large aspect ratio events. 6
area as given area H=15km SurfaceSlip / (m/aµ) 1.5 1.5 6.5 7 7.5 8 M Figure 5: Exploring discrepancy between implied average slip values from magnitude-area data and surface slip data. Ratio of data in Figure 1(b) to data in Figure 1(a) for same events. Red circles use area as given. Blue circles assume seismogenic depth H = 15km as in Figure 1(a). Surface slip measurements average around 3% lower than magnitude area estimates. Reconciling differences in magnitude-area and surface estimates of slip As noted earlier in Figure 1a and 1b, magnitude-area estimates of average slip based on the Hanks- Bakun data [Hanks and Bakun, 8] were found to be larger than surface slip estimates in the Wesnousky [8] data. Attempting to reconcile the estimates, we note one difference is that different events are considered in the two databases. To control for this, we first look at events which are common to both datasets. Examining the ratio of the surface slip estimate to the magnitude-area estimate, shown in Figure 5, we find the surface slip estimate is on average roughly 3% less. We find, as well, no obvious magnitude dependence to this difference. Two possibilities arise in explaining this difference. One is that the surface slip measurements are systematically low, missing some slip. The other is that the magnitude-area estimates are systematically high, mapping some deep coseismic slip below the seismogenic layer onto the seismogenic layer. Both of these possibilities appear to be valid concerns, and are a source of epistemic uncertainty. As a result of discussions of this topic at the March 11 UCERF workshop, an effort to reexamine the surface slip data is being undertaken, with particular attention to the issue of enveloping the surface slip data in a more consistent way. It is expected that at least some events (e.g. Hector Mine) will have a net increase in their estimated mean surface slip as a result of this reexamination. Thus some of this difference is expected to narrow. An additional source of differences between the magnitude-area estimates and the surface slip estimates come from differences in the event populations, with, in particular, high slip Mongolian and Chinese events in the magnitude-area database not being in the surface slip database. The March 11 UCERF workshop also discussed this issue and concluded there were not clear reasons to exclude the Mongolian and Chinese events, and thus supplemental events from this population are to be added in the reevaluation of the surface slip data. This is also expected to decrease the differences between the Figure 1a and 1b mean behaviors. Remaining differences are proposed to be treated as epistemic uncertainties. Branch Weights The choices between the two alternative scalings for each case Equation (1) vs. Equation () for magnitude area and Equation (3) vs. Equation () for slip vs length are not correlated with eachother in the current simplest proposal, though, as noted below, one could choose branch weightings to trim the uncorrelated case to pick up a naturally correlated physics underlying them 7
One logic tree branch would use magnitude area as in ucerf, for size and for slip-rate balancing. This logic tree would have two branches, one for Equation (1) and one for Equation (). Essentially, thought of in terms of an equation used for size and an equation used for rate, you use: size rate (1) & (1) () & () for the two branches Another logic tree branch would use slip vs length for slip-rate balancing and magnitude-area for size. This logic tree would have four branches for the uncorrelated combining: size rate (1) & (3) (1) & () () & (3) () & () Alternatively, we could trim this second approach to size rate (1) & (3) () & () since they are, at some level, based on related perspectives. In addition to the functional form epistemic uncertainties, there are parametric epistemic uncertainty in the values of the parameters α and σ in Equation (3) and (), respectively. If it were desireable, further parametric epistemic uncertainties could be considered in the constant in Equation (1) and the constant and the parameters H and β in Equation (). Deep slip The existence or not of significant amounts of coseismic slip below the seismogenic layer in the deeper stably sliding fault remains an open question not currently resolved by observations. Dynamic modeling results suggest it is a real possibility, but parameter uncertainties and most especially observational resolution uncertainties make this difficult to constrain. As such, it remains an epistemic uncertainty. Modeling results suggest an exponential falloff in slip with depth is a good parameterized fit for slip propagating into a logarithmically velocity strengthening a ln V layer, where V is the slip rate. Numerical results find the scale of the exponential falloff scaling linearly with the strengthening coefficient a. Figure 6 and Figure 7 illustrates these modeling results. Approximating the exponential falloff as a linear falloff is also a reasonable parameterization [Shaw and Wesnousky, 8]. Once refined surface slip enveloping efforts are completed, we anticipate quite modest amounts of deep slip (< few tens of percent) will enable the magnitude-area slip estimates to match the surface-slip estimates. Based on the lack of obvious magnitude dependence in Figure 5, a one parameter family could be used to represent the difference. Aseismicity Factor For aseismicity factor an area reduction rather than slip rate reduction is recommended for seismogenic depths. Below the seismogenic zone, for purposes such as cybershake and other modeling, a slip rate reduction would be recommended as a more appropriate representation. Other uses of scaling laws Finally, independent of the slip-rate balancing branch based on slip-length scaling, we also need slip-length scaling laws to incorporate the Hecker et al. [11] proposed constraint on magnitude frequency distribution based on CV data for slip derived from compilations of paleoseismic slip data. We recommend that the analyses performed by Hecker et al. [11] using the slip-length scaling implied by the Hanks and Bakun [, 8] relations be redone using the two slip-length scaling relations proposed here. 8
1 1 1 1 1 1 1 1 1 1 slip 1 slip 1 1 3 1 3 1 1 (a) 1 5..5 1. 1.5..5 3. 3.5. depth/h (b) 1 5..5 1. 1.5..5 3. 3.5. depth/h Figure 6: Average slip as a function of depth in 3D dynamic model. (a) Less velocity strengthening on a ln v term: a =.3 (b) More velocity strengthening: a =.1. In both cases, note exponential falloff of slip with depth for large events, with differing slope depending on a. But also note slip at surface, at depth=, is similar for both cases. mean slope. 3.5 3..5. 1.5 1..5.....6.8.1.1 a Figure 7: Scale length of exponential falloff as a function of velocity strengthening parameter a in dynamic model. Note scale length depends approximately linearly on the amplitude of the logarithmic velocity strengthening term. 9
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