A hierarchy of tremor migration patterns induced by the interaction of brittle asperities mediated by aseismic slip transients J.-P. Ampuero (Caltech Seismolab), H. Perfettini (IRD), H. Houston and B. Delbridge (U. Washington) Coupled slow slip and tectonic tremor phenomena offer an exceptional opportunity to investigate the behavior of faults at depth. The possible relations between the spatio-temporal organization of tremor activity and large earthquakes open interesting perspectives in the study of earthquake physics with implications on the assessment of seismic hazard, including the along-strike segmentation and the potential depth extent of seismic moment release. Tectonic tremors may act as natural creepmeters to monitor aseismic slip with a resolution that cannot be afforded by geodetic and strainmeter networks, which could unveil precursory slow slip associated with the nucleation of large earthquakes. Tremor phenomena might provide unique high resolution observations to elucidate the role of frictional heterogeneities and fault zone fluids on the rupture process of natural faults. The following hierarchy of tremor migration patterns has been observed in Cascadia and (partly) in Japan: 1. Large scale tremor migration along-strike at 10 km/day (Obara, Kao, etc), consistent with the propagating front of slow slip events (SSE), see Figure 1 2. Rapid tremor reversals : tremor swarms that propagate in the opposite direction at speeds of order 100 km/day (Houston et al, 2010), see Figure 1 3. Fast tremor swarms propagating along-dip on narrow streaks at speeds of order 1000 km/day (Shelly et al, 2007; Ghosh et al, 2010), see Figure 2 While the first migration pattern can be naturally associated to triggering of tremor by a propagating slow slip pulse, the two other patterns are currently unexplained. The purpose of this paper is to provide a unifying framework to understand these three patterns. Our model focuses on the tremor generation process driven by an ongoing SSE, but relies only on generic assumptions about the underlying SSE. The proposed model leads to predictions that can be tested by currently affordable observations.
Figure 1. Two along-strike migration patterns of tectonic tremor observed in Cascadia: large scale, along-strike slow migration at 10 km/day and rapid tremor reversals, in the opposite along-strike direction, at 100 km/day (after Houston et al 2010). Figure 2: Two examples of fast tremor swarms observed in Cascadia, migrating along-dip over narrow streaks at speeds of order 1000 km/day (after Ghosh et al 2010).
Building blocks of the model: Seismic failure of deep asperities embedded in a more stable fault zone: In a prevailing view, tremor is generated by dynamic failure of a multitude of potentially brittle asperities present on the deep, mainly ductile portions of the fault zone. Although the origin of such deep asperities (spatial heterogeneities of fault rheology) remains unknown, there is some independent evidence of their existence. For instance, Rolandonne et al (2004) observed that the M7.3 Landers earthquake triggered seismicity deeper than the usual seismogenic depth. In rate-and-state models an asperity driven by surrounding creep generates regular and spontaneous seismic events if its size exceeds a critical size (intrinsically unstable asperity). If its size is sub-critical (in the sense of linear stability) seismic slip on the asperity can still be triggered if the creep rate suddenly exceeds a threshold (conditionally stable asperity with the slip law ). In our model the regions surrounding the asperities are not necessarily velocity strengthening, they could also be velocity-weakening zones that are relatively more stable due to larger Dc or smaller effective normal stress. Seismic activity correlated to loading rate: The slip budget combined with the most basic Coulomb failure model implies that the recurrence time of an asperity is inversely proportional to loading rate. This suggests that seismic activity rate on deep asperities is correlated with loading rate. Perfettini and Avouac (200x) proposed that the temporal decay of aftershock rates are controlled by the decaying loading rate induced by postseismic slip. Moreover, Peng et al (200x) observed that repeating microearthquakes had larger magnitude and shorter recurrence time following a nearby large earthquake than before the mainshock, so the degree of coupling in each asperity failure might also correlate with loading rate. In rate-and-state models a transient increase in the surrounding creep rate induces a transient load on the asperity that decreases its recurrence time (if intrinsically unstable) or pushes it to failure (if conditionally stable). Secondary creep waves: The breaking of an asperity transfers stresses to the surrounding creeping fault. These stresses are initially concentrated at the edge of the asperity and induce a post-seismic slip perturbation that propagates outward, a creep wave. The expansion of aftershock area observed after some large earthquakes (e.g. Peng et al, 2010) provides indirect evidence of the post-seismic slip migration process (Kato, 2007). Figure 3 shows an example from a typical rate-and-state numerical simulation. The propagation speed V prop of the creep wave is proportional to its peak slip velocity V max (Ampuero and Rubin, 2008; Perfettini and Ampuero, 2008), a relation valid for any cohesive rupture model: V prop = shear modulus / strength drop * V max In any fault zone rheology containing a viscous component (e.g. the direct effect in rate-and-state friction) V max correlates with the amplitude of the triggering stress perturbation and with the background slip rate V bg. The stress increase beyond the edge of a broken asperity scales with the stress drop in the asperity which, by analogy to regular earthquakes, can be assumed to be constant. A correlation remains between V max and V bg. In the special case of rate-and-state friction (with a logarithmic viscosity term, the direct effect ) V max is proportional to V bg :
V max = exp( [stress perturbation]/a) *V bg Combination of the two relations above predicts that the propagation speed of a creep wave is correlated to the background slip rate of the surrounding creeping fault region. In rate-and-state friction these quantities are proportional, but the existence of a correlation does not depend on the details of the friction or fault rheology model. Figure 3: An asperity breaks and triggers a propagating slow slip perturbation in the surrounding creeping fault. (This example is for a semi-critical aging-law asperity that does not generate fast seismic slip, but the creep wave would look similar for a seismic asperity.) Tremor swarms as a cascade process mediated by creep waves: The secondary creep wave generated by the breaking of an asperity transfers stresses to neighboring asperities, triggering further seismic slip on them. This process produces a self-sustained cascade. Ariyoshi et al (2010) showed some examples of this cascade process in rate-and-state 3D simulations. The migration speed of the resulting tremor swarm is controlled by the propagation speed of the secondary creep waves. From the previous discussion, the tremor migration speed correlates with the background slip rate of the creeping fault regions that surround the asperities. The spatial distribution of slip rate in a slow slip pulse: Although we consider that slow slip is the main driver of the tremor patterns we are studying, we don t address here the origin and detailed mechanics of the slow slip transient. Several models of slow slip have been recently proposed based on rate-andstate friction laws and interaction between slip and fault zone fluids (Rubin, Segall, Shibazaki, etc). We
nevertheless consider some general aspects of the spatial distribution of slip rate, as typically found in rupture pulse models (dynamic or quasi-static) that assume energy-neutral healing. The slip rate is characterized by its skewness along the propagation direction: slip rate is large near the leading front and decays towards the healing front (see Figure 4). The depth distribution is naturally tapered at the top and bottom edges. This applies to the slip rate inside a slow slip pulse, which is the background creep rate that drives the tremor asperities. In Cascadia, SSE source models inferred from geodetic observations imply that the peak slip velocity in the SSE pulse is at least of order 100 times larger than the long-term tectonic slip rate, its pulse size is of order 50km along-strike and along-dip. Figure 4: Typical spatial distribution of slip velocity inside a rupture pulse, inspired by dynamic rupture models. Putting the pieces of the model together: The first migration pattern is naturally associated to triggering of tremor by the ongoing slow slip pulse. Tremor asperities located on the fault plane are triggered within the slip pulse: stress drops almost everywhere inside the pulse except on the asperities left unbroken, accelerated aseismic slip increases the loading rate on the asperities, which increases their seismicity rate. This triggering is much more efficient near the leading edge of the slow slip pulse, where the slip rate is larger, which is confirmed by recent high-resolution tremor locations (Figure 5): the along-strike density of tremor is larger close to the leading front. The breaking of asperities generate secondary, small scale creep pulses that propagate in all directions and trigger other asperities on their way. These in turn generate new secondary pulses, producing a cascade process of tremor swarms. Their migration speed is controlled by the propagation speed of the secondary creep waves, which scales with the background SSE slip rate. The tremor swarms propagate
o o o fast along the rim of the SSE pulse front, where background slip is faster: this explains the fast along-dip tremor migration pattern more slowly towards the interior of the pulse, where background slip is slower: this explains the rapid tremor reversal at intermediate migration speed too slowly in the direction of propagation of the SSE to separate from the SSE. Figure 5: Spatial distribution of tremors along-strike, relative to the tremor front, stacked from observations of an ETS sequence in Cascadia recorded by the Big Skiddar array. The skewed spatial distribution of tremor activity is consistent with the expected spatial distribution of slip rate inside a slow slip pulse.
Figure 6: conceptual description of the model for the three tremor migration patterns and their relation to the spatial distribution of slip rate within a slow slip pulse Figure 7: Tremor distribution along depth in Cascadia. Blue: all tremor swarms. Red: only RTR swarms. The widths reported at the bottom left are standard deviations measured along-dip.
Implications on scaling laws: Slow slip and tremor swarms follow a duration-moment scaling relation very different than that of regular earthquakes (Ide et al, 2007): We explore here the possibility that the three migrating processes studied here (slow slip, rapid tremor reversals and along-dip tremor streaks) lie on the same scaling law. The three processes have very elongated source areas, hence their overall seismic moment release can be represented by a pulse-like source, a slow version of Haskell s source model. The seismic moment of an elongated source of length L and width W (the shortest dimension) is: where D is average slip and μ is shear modulus. The source duration T and its longest dimension L define its average rupture speed V r : Combining these two equations: Assuming typical observed values for SSEs in Cascadia (μ = 30 GPa, D = 2 cm, W = 50 km, Vr = 10 km/day) equation (4) is consistent with the empirical scaling law, equation (1). Moreover, from linear elasticity: where Δτ is the static stress drop. Note that equation (5) involves the shortest dimension W. In their discussion Ide et al (2007) assumed L=W (or typical relations for circular cracks), which is not representative of the elongated shape of the source areas of slow slip and tremor swarms. Combining equations (4) and (5) leads to: This implies proportionality between moment and duration for all the three processes, which is qualitatively consistent with the empirical scaling relation for slow earthquakes. However, nothing a priori guarantees that the three processes share a similar value of the ratio, or average moment rate. Considering equations (1) and (6), this ratio should be: (1) (2) (3) (4) (5) (6) (7)
If we assume a continuum of possible length scales W, equation (7) is a very puzzling relation for which we see no evident mechanical model. However, only three separate scales are relevant here: the alongdip extent of slow slip (W SSE 50 km), the along-dip size of rapid tremor reversals (W RTR ) and the alongstrike width of tremor streaks (W streaks ). If we assume that the stress drop is similar for slow slip and tremors and Δτ 10 kpa, a typical value for SSEs in Cascadia, equation (8) implies: This suggests that the width of the fast tremor swarms which migrate along-dip at V r 100 km/h in Cascadia is W streaks 5 km. Although the width of tremor streaks is hard to resolve in current observations, this value seems to provide a reasonable upper bound. Narrower widths are possible in the model if larger stress drops are assumed. The same equation suggests that the width of the rapid tremor reversals, which migrate at V r 100 km/day, is W RTR 15 km. This value is consistent with the observed depth distribution of tremors in RTRs as summarized in Figure 7. (8) Discussion The formation of structural streaks in faults with large cumulative offset is suggested by field observations on exhumed faults (Sagy and Brodsky, 2009) and by laboratory experiments (Voisin et al, 2008). In that view W streaks is the characteristic scale of the fault plane corrugations. In the model proposed here W streak is not a persistent geometric feature of the fault zone, but is instead controlled by the along-strike extent of the region of rapid slip at the front of the SSE pulse. The width of this SSE process zone is conceivably a small fraction (1/10) of the total SSE pulse width, which is consistent with our estimates above. Tremor locations in Cascadia (Figure 5) show a process zone width (the distance from low to peak tremor rate at the leading edge) of order 20km, larger than the value W streak 5km required by our model, but this difference might be attributed to the spatial smearing in the tremor catalog due to location uncertainties. Assuming the distribution of asperities is statistically uniform, our model implies a quite constant W streak and provides a mechanical link between W SSE and W streak. However, this model needs to be combined with structural features that can modulate the spatial distribution of tremors, such as fault plane undulations elongated in the slip direction as supported by observations that tremor streaks are parallel to the local and past directions of plate convergence (Ide 2010, Ghosh et al 2010). Our model predicts that a linear scaling between M 0 and T exists separately for SSE, RTRs and LFE streaks but, because the ratio M 0 /T (equation (7)) is very similar for the three processes, they can lie on the same scaling law. This universality is not fortuitous but arises from the relation between the length scales W SSE, W RTR and W streak implied by the properties of the SSE pulse, and from the assumption of similar stress drops.
The length scale W RTR might be related to the along-dip distribution of slip rate in the SSE pulse, which is expected to taper off smoothly towards the top and bottom limits of the SSE source region: W RTR might be the width of the region of relatively high SSE slip rate at intermediate depth. The slip pulse model (the classical Haskell s model) assumed for tremor streak swarms predicts that their moment rate spectrum decays as 1/f at intermediate periods between their duration (T) and their rise time. Typical streak swarm durations (T=L/Vr with Vr=1000km/day, L=few 10km) are of order 100s. VLFs are observed down to 0.01Hz so they might be missing this corner frequency (must compare to Ide s stochastic model which suggest a typical duration T=?). The rise time must be shorter than 0.1s. In a dynamic pulse model with healing controlled by geometrical constraints (W) the rise time is a small fraction of W/Vs, which would correspond to a corner frequency hardly above 10 Hz. However, in the cascade swarm model the rise time is rather controlled by the duration of individual LFEs (is that longer than 0.1s?). Figure 8: predicted patterns of RTR migration in the radial direction (green arrows) and fast tremor swarms in the azimuthal direction along the front (yellow arrows) during the initial semi-circular propagation stage of a SSE (cyan arrows, the blue circle is a slow slip zone that has nucleated at the bottom)
Notes for discussion: Equation (8) implies that the peak SSE slip rate is proportional to 1/(process zone size)^2. Is this mechanically consistent? How do LFEs fit in this scaling law discussion? If the slip velocity in the pulse decays smoothly towards its tail should we expect a slowing down of the migration speed of RTR? This slow down is not observed. Does that indicate that the tail is quite flat (slip rate rather constant over a large distance behind the front)? Such flat tails are observed in recent laboratory experiments of dynamic rupture by Jay Fineberg. Our model concerns the tremor produced by shear slip sources on or near the subduction interface or fault zone. There might be other types of tremor located further above the fault interface that can be triggered by the SSE but it is not clear they could induce secondary creep waves. It has been suggested that tremor sources are located further down-dip than slow slip (in Japan, Cascadia, Mexico). However, given the large uncertainties in tremor locations and in slow slip location (geodetic inversions assume uniform half-space, shallow compliant layers bias the source location. Moreover the bias due to the presence of the slab has not been addressed), the data could be consistent with co-located tremor and slow slip. Our model assumes permanent asperities, localized rheological heterogeneities, on the fault. Persistent apparent asperities emerge also in some fault models without assumed structural heterogeneities, due to the non-linear dynamics (Horowitz and Ruina, 198x). [Show example figure.] Allan Rubin suggested that tremor can be the signature of localized slip acceleration transients induced at the intersection between multiple creep pulses propagating in different directions along the fault. The potential of such models to reproduce the full range of tremor phenomena remains to be explored. Mike Brudzinsky (IRIS workshop, 2010) showed examples of space jumps in tremor migration (over an area where a swarm had occurred before) and examples of stopping and reactivation of migration: how do these fit in the picture? Recent observations of depth dependence in Cascadia and Japan (A. Wech and K. Obara, IRIS workshop, 2010): amplitude and recurrence time of tremor swarms decreases with depth, continuously. Suggests stress drop decreases with depth, due to slower healing or faster loading rate at depth (both imply smaller static friction at failure time, assuming non linear, logarithmic healing). This seems inconsistent with the observation by Peng et al (2004): larger magnitude if faster loading rate (shorter recurrence time is ok). If tremors are a cascade of asperity ruptures mediated by secondary creep waves, why tremors triggered by surface waves shut down soon after the passage of the wave? Shouldn t the cascade persist some time after the dynamic load? I don t think so. In contrast to the load induced by a passing slow slip pulse, the dynamic load generated by passing surface waves is temporally symmetric: it contains positive and negative swings. The latter turn off the tremor activity.
If during the early days of a large SSE, before slip has saturated the available down-dip width, the slip propagates mainly as a circular crack, our model predicts that there should be RTRs propagating inwards in the radial direction and fast tremor swarms propagating in the azimuthal direction along the circular front of the SSE (Figure 8). One might be able to test this prediction with forthcoming data from the array of arrays in Cascadia. What simulations should we do? To illustrate that the propagation speed of postseismic creep correlates strongly with the loading slip rate: do cycle simulations of an isolated asperity surrounded by creep with different values of the remote loading velocity. Actually this can be borrowed from simulations done by Chen and Lapusta (200x). To illustrate the rapid tremor reversals: do 2.5D simulations of slow slip on a velocityweakening fault with slightly super-critical width W or with a transition from weakening to strengthening at large slip rate (a la Shibazaki), introduce small tremor asperities in the simulation. The 2.5D simulation can be done with a modified kernel in a 2D code or with a 3D code using only one row of elements of along-dip size equal to W. The asperities can be one cell large, conditionally stable (slightly smaller than nucleation size, using the slip law and a very small Dc) The complete model: as previous but in 3D.