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THICK- AND THIN-LITHOSPHERE MODELS [c7, p195-201]

The most extreme features of locked fault behavior are currently observed on the two San Andreas fault segments where great earthquakes have occurred in historical time, in 1857 and 1906 (see Fig. 5.11 for locations and coseismic-slip distributions). On these segments, no aseismic slip is observed at the Earth's surface, the two faces of the fault are in locked frictional contact to depths of 10 to 15 km, and interearthquake slip is either extremely small or absent. At greater depths, the mechanics of fault movement is uncertain, but two models bound the range of expected behavior ( Fig. 7.9). In the first, the thick-lithosphere model, the depth D of coseismic faulting is much less than the thickness H of elastically strong lithosphere. Interearthquake deformation then predominantly results from episodic or steady aseismic slip on the downward extension of the seismogenic fault zone, and any effects of the underlying weak asthenosphere can be safely neglected. In the second, the thin-lithosphere model ( Fig. 7.9) coseismic faulting depth is comparable to elastic-plate thickness. In this model, transient postseismic and steady interseismic flow in the asthenosphere provide the dominant mechanism for interearthquake strain accumulation.

Note that in the context of these two models, the terms "lithosphere" and "asthenosphere" are linked to mechanical properties of the Earth's crust and upper mantle: The lithosphere is the strong elastic layer near the Earth's surface, and the asthenosphere is the region of ductile flow that lies beneath. Their boundary may thus lie well above the thermal boundary layer that separates the moving plates from the convecting mantle. If so, then at least the upper part of the "asthenosphere" forms part of the tectonic plate and moves with it.

Thus considered, the boundary between "lithosphere" and "asthenosphere" defines the zone of decoupling between surface tectonic processes and those that occur in the ductile region beneath. The location of this boundary is thus of central importance to the broad-scale tectonics of the San Andreas fault, the nature of the earthquake-generation process and its thermomechanical implications (see chap. 9), and the relation between shallow structural features and those inferred at depth (see chap. 8). I explore below the influence of this boundary location on cyclic earthquake-related deformation at the currently locked transform fault zones in the San Andreas, illustrating the contrasting mechanical behavior of the thick- and thin-lithosphere models.

All of the models considered here are two dimensional, and so neither slip nor mechanical properties vary along fault strike. Each model consists of only a single planar, vertical strike-slip fault. However, because the medium properties are linear elastic and (or) viscoelastic, the effects of multistranded fault zones can be obtained by simply superposing the deformation due to slip on individual fault segments. Furthermore, all of the two-dimensional models discussed here have also been considered in three dimensions, and so complexities arising from changes in fault strike, variations in slip along strike, and the finite extent of faulting can be incorporated straightforwardly as necessary. Similarly, except for the transition from elastic lithosphere to viscoelastic asthenosphere, depth variations in material properties are not considered, although, again, solutions have been obtained for faulting in plane-layered elastic and viscoelastic media. Although fault end effects and changes in slip and geometry along fault strike can be locally important, these effects, as well as those due to depth-varying material properties, are generally second order relative to the deformation features described here. More important are the effects of the several subparallel strands that compose much of the San Andreas fault system along its two currently locked sections. In these sections, the interseismic deformation due to each major fault strand contributes significantly to the observed displacement pattern, and as a rule the effects of faults lying off the San Andreas proper cannot be safely ignored in matching models to data across the entire San Andreas boundary zone.

The simplest form of the thick lithosphere model, first proposed by Savage and Burford (1970), is illustrated in Figure 7.10 . In this idealized model, interearthquake strains accumulate uniformly throughout the deformation cycle and have precisely the same spatial pattern as coseismic strains, except that the sense of movement is reversed. The cycle consists of coseismic slip delta u extending from the surface to depth D and steady interearthquake aseismic slip at a constant rate u-dot (= delta u/T, where T is the earthquake recurrence interval) beginning at z = D and extending to great depth. For this model of interearthquake deformation, simple expressions relate surface-displacement rates u-dotx(y) and shear-strain rates e-dot xy(y) to the fault parameters u-dot and D and the distance y from the fault trace:

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A principal utility of this model is the ease with which approximate values of displacement and strain rate can be computed, commonly as a preliminary step to more detailed computations that employ complex models which nonetheless show many of the same general features. For example, using typical San Andreas values of delta u=4 m, T=200 yr, and D= 15 km, then u-dot=20 mm/yr, and the engIneering shear-strain rate (twice the tensor strain rate e-dotxy) at the fault trace (y=0) is about 0.8 mu rad/yr, a value close to some of the peak strain rates plotted in Figure 7.2 and Figure 7.3. Furthermore, the bell-shaped distribution of secular strain across the model fault (middle right, Figure 7.10) generally accords with observations (Fig. 7.3), and the width of the profile is a direct measure of the fault-locking depth D. (Note, however, that the observations summarized in Figure 7.3 include strain rates determined from multistranded segments of the San Andreas fault system, and so they are not directly comparable to the model calculations for a single fault strand illustrated in Figure 7.10. Recalling the observations discussed in the section above entitled "Observations of Crustal Deformation," the wider zone of secular strain across the southern section of the San Andreas can be rationalized if the depth of seismic slip and,thus, the locking depth of the fault are simply greater in southern than in northern California. As it stands, this model has no transient effects and so is too simple to explain the postearthquake strain changes plotted in Figure 7.4. However, introducing a rather straightforward modification remedies this defect while accounting for the observed difference in strain-rate profiles between northern and southern California. Surprisingly, these same features are, for different reasons, natural consequences of the thin-lithosphere model.

The two models are illustrated in Figure 7.11. In the thick-lithosphere model, postseismic effects are introduced by specifying transient postearthquake slip on a segment of the fault immediately beneath the coseismic rupture. Its time history is constrained by an exponentially decreasing slip rate (time constant alpha), and its magnitude by the requirement that the cumulative slip sum to the coseismic offset delta u by the end of the cycle. In the thin-lithosphere model, the transient deformation results from flow in the asthenosphere due to stress relaxation after seismic faulting in the lithosphere. Its time scale is controlled by the asthenosphere-relaxation time tau = 2 eta/mu when eta is the effective viscosity of the asthenosphere and mu is the average shear modulus of lithosphere and asthenosphere, here taken to be equal. In both models, the transient motions are superimposed on a steady component of deformation that is due to relative plate motion.

Detailed computations show that the two models produce surface deformations that with suitable choices of model parameters are observationally indistinguishable (see Thatcher, 1983). Here, the discussion is restricted to qualitative features, as summarized in Figure 7.12. Near the fault, shear-strain rates monotonically decrease over time and gradually approach a constant (Fig. 7. 12B), while the deformation profile broadens and strains diffuse into the interiors of the adjacent plates as the cycle progresses (Fig. 7.12A). It is easy to match the observed temporal decline in strain rate with either model; the particular parameter combinations are themselves not unique, and a range of choices can provide equally good agreement. All satisfactory thin-lithosphere models, however, require an elastic plate only 10 to 15 km thick, the maximum depth of coseismic slip in the 1906 earthquake (Thatcher, 1975). Both models predict a broadening of the zone of deformation that depends on the time interval since the latest great earthquake, and so the greater width of the strain-rate profile in southern California can be accounted for. For example, data from the northern, locked section of the San Andreas fault may correspond to times t1 to t3 in Figure 7.12, whereas those from the southern section may correspond to times t4 and t5.

More complex models that combine features of both the thick- and thin-lithosphere models are also consistent with available data (for example, Li and Rice, 1987). Furthermore, coseismic and interearthquake fault slip undoubtedly vary as a function of depth, rather than abruptly terminating at some specified fault depth. Although this gradationality of the slip distribution modifies the detailed patterns of surface strain and displacement from those illustrated in Figure 7.10, for example, the same qualitative features are preserved, and it will be difficult to distinguish between differing slip-depth distributions on the basis of surface-deformation measurements alone.

In summary, at transform plate boundaries, available data are consistent with both thick- and thin-lithosphere models but do not strongly constrain either. The most geophysically interesting feature of both models is the predicted postearthquake diffusion-like spread of strain from the plate-bounding fault into the interiors of the adjacent plates. Postearthquake surveys, however, are sufficiently infrequent and areal coverage sufficiently limited that these effects, if they indeed occur, have not been directly observed. Details of the temporal decline in deformation rate near the fault are also absent.