Scaling Laws. σ 1. σ = mean stress, which is needed to compute σ 0. η = percent strain energy released in eq. Introduction.

Transcription

1 Scaling Laws Introduction Scaling Laws or Relationships are the result of empirical observation. They describe how one physical parameter varies as a function of another physical parameter within a system. Such relationships may be used to place physical bounds and checks on physics-based theoretical parameters and, similarly, lead to constraints on other model parameters that are not so simply observed. In the absence of theory, scaling relationships can be important in the estimation or prediction of one parameter from another. A simple example of the latter would be the estimation of earthquake magnitude from the measure of a faults mapped length. In this case, through the examination of numerous historical earthquakes, one might observe a systematic relationship between the length of an earthquake rupture and the seismologists measure of magnitude. Armed with such observation, one might then use geologists measures of fault length to estimate the expected size of future earthquakes. Here we ll first consider the various parameters that may be used to describe the earthquake source. Then in the context of expressions that result from elastic dislocation theory, we ll examine the implications of various empirical scaling relationships to the mechanical behavior of earthquakes. Fault Parameters Parameters that describe characteristics may be static in nature, such as the length, fault area, slip or stress drop of an earthquake, or dynamic parameters such as dynamic stress drop or rupture velocity. The static parameters address the net result of an earthquake displacement while the dynamic parameters describe variable related to the rupture process. It is the static parameters that may be independently addressed by seismologists and geologists alike. From viewpoint of statics, the important parameters relating to faulting have been conveniently laid out by Kanamori and Anderson [1975] and listed here. L = fault dimension! # D = average offset " µ = material property # $ which together can be used to estimate M o. σ = mean stress, which is needed to compute σ 0 + σ 1 2 or Δσ = stress drop = σ 0 σ 1 where σ 0 = initial stress and σ 1 = final stress Δw = strain energy change, of which a fraction η = percent strain energy released in eq. 1 of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

2 Scaling Laws Arising From Dislocation Theorey The earthquake cycle is described by the accumulation of stress during the interseismic period and the release of stress at the time of the earthquake. A fundamental parameter of interest is the stress drop, a measure of the amount of the accumulated stress that is released during the earthquake. The general expression for stress drop Δσ from elastic dislocation theory is written where µ = rigidity # Δσ = Cµ % D & $ L ˆ ' C = non - dimensional shape factor D Δe = strain change or strain drop L A measure of the work done during faulting or, analogously, the drop in strain energy Δω during an earthquake is found Δw = SD σ, where S = fault area Note that the product does have the units of work or energy. Now rearranging the expression for stress drop to D = L Cµ Δσ and then inserting it into the expression for strain energy one arrives at an expression for strain energy drop in terms of the stress at the commencement of faulting (initial) and that at the completion of the rupture process (final). " L Δw = $ % ' SΔσσ # Cµ & Δw = L 2Cµ S σ ( σ 1 ), where σ 0 = initial stress σ 1 = final stress Not all energy is radiated as seismic waves. Some is also taken up, for example, by frictional heating during slip. So the seismic energy is more correctly written Ες = ηδw, where η <1 It is useful to note that it generally requires in the above expressions that three source parameters be independently determined to predict the fourth and, hence, fully describe the statics of faulting. L, D and σ or L, M 0, and Δw 2 of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

3 Seismic Moment versus Fault Area. The seismologist has the tools to measure the seismic moment and fault area for an earthquake. Geologists may do the same independently, except for the measure of fault width for which they must allude to the measures of seismologists. Here we follow the development of Kanamori and Anderson [1975] to make certain predictions about how seismic moment should scale to fault area in light of classical dislocation theory. Then we can compare them with observation. For a circular fault where S = Πa 2 expression for moment is written M 0 = µsd = µπa 2 D and the expression for stress drop is Δσ = 7Π 16 µ D a is the fault area, the where we have now included the value of C = 7π/16 for a circular fault. Reearranging terms in the stress drop expression to define D and inserting into the preceding expression for seismic moment M 0 yields an explicit expression for seismic moment in terms of stress drop and the fault area. M 0 = µ ( Πa 2 ) 16aΔσ 7Πµ = 16 ( ) 3 7 Δσa3 and because a 3 = Πa2 2 $ = 16Δσ ' & ) S 3 2 % ( or 7Π 3 2 LogM 0 = 3 2 log S + log $ 16Δσ 7Π3 2' % ( Π 3 2 = S3 2 From this, we now have an expression that relates seismic moment to fault area and stress drop. If we assume that stress drop is constant for all earthquakes, then seismic moment should scale as a function of fault area to the 3/2 power. That is to say, if we are to make a plot of LogMo versus fault area S for the assumption of any given stress drop, the points should mark a slope of 3/2 on the plot. The following Figure 1 from Kanamori and Anderson [1975] is such a plot with the addition that they have plotted up measurements of seismic moment and fault area for a global set of earthquakes. They observed that most of the earthquakes occurred between the lines of 10 bar and 100 bar stress drop and marked a slope of 3/2. From this, they concluded that the stress drop during earthquakes was to a first approximation constant and about 30 bars. Yet, they noticed something else too. The data suggested that static stress drops for earthquakes in intraplate environments appeared to be on average higher than those in intraplate environments. Π of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

4 Figure 1. Plot of Log of seismic moment versus fault area for a global set of earthquakes. Lines of constant stress drop derived from elastic dislocation theory. Open and closed symbols represent earthquakes in interplate and intraplate environments, respectively (plot taken from Kanamori and Anderson [1975]. Moment Magnitude Mw It was the examination of the scaling between seismic moment Mo and minimum strain energy drop Wo during earthquakes that led Kanamori [1977] to develop the moment magnitude scale Mw. His development was as follows The difference in elastic strain energy W before and after an earthquake is W = σ D S where σ = average stress during faulting If the stress drop is complete (the final stress is zero), the stress drop for an earthquake is Δσ = 2σ or σ = Δσ 2 4 of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

5 and the expression for strain energy may be rewritten W = W 0 = 1 2 Δ σ D S = Δσ 2µ M 0 and because M 0 = µ S D Drawing upon the results of Kanamori and Anderson (1975) in Figure 1, Kanamori (1977) then assumed Δσ = constant = bars = 2 6x 10 7 dyn cm 2 and further noted that the rigidity of the crust is equal to µ = 3 6x dyne cm 2 from which the ratio of stress drop to rigidity may be approximated and the expression for strain energy drop in terms of seismic moment then simplifies to W 0 ~ M 0 2 x10 4 Combining this latter relationship with Gutenberg Richter Relationship for Energy which we derived earlier in class ( LogΕ =1.5M +11.8) brings one to this expression. or LogM 0 = 1.5M M w = 2 3 Log M When magnitude is determined in this manner from direct measurements of seismic moment, it is called the moment magnitude. Today this is perhaps the most widely used measure of earthquake size. Incomplete Stress Drop The assumption of whether final stress drop is zero is not excepted by all. Indeed, it is quite a source of argument whether stress drop is total or partial during an earthquake. It is very important because it is this uncertainty that prevents us from clearly obtaining a measure of absolute stress in the earth s crust. The following development illustrates that our estimate of strain energy drop is only a minimum if stress drop is incomplete and is thus a minimum estimate of wave energy. 5 of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

6 Let σ 0 and σ 1 be the initial and final stress. Then # W = σ D S = Δσ $ 2 % & DS + σ 1 D S = W 0 + σ 1 D S and we see that estimate of Wo in preceding section is a minimum. Also, we may let σ F be the frictional stress during faulting which may be rewritten W = Η + Ε, where Η = σ + D S the frictional loss Ε = σ DS σ F DS % = Δσ & 2 and Ε = wave energy ' ( D S + D S ( σ 1 σ F ) = W 0 + D S( σ 1 σ F ) which = W 0 if σ 1 = σ F, which means it is also an estimate of minimum wave energy Earthquake Repeat Time Versus Average Stress Drop or Stress Drop as a Function of Tectonic Environment. The repeat time between large earthquakes on any particular fault is a function of the rate of strain accumulation. The higher the strain rate, the more frequent the occurrence of large earthquakes. We have observed that the return time between large earthquakes on the major strike-slip faults of California may be measured in hundreds to thousands of years and in Nevada in thousands to tens of thousands of years. That is because the rates of crustal strain are much less in the Great Basin than along the San Andreas. The observation has led investigators to divide earthquakes in terms of tectonic environment which, in turn, is directly related to the average rates of strain accumulation across an area. The two extreme members of such a description are the interplate and intraplate. Interplate refers to those regions that sit astride major plate boundaries where crustal strain rates are high and intraplate those regions that sit far from plate boundaries and crustal strain rates are extremely low. There are regions in between as well. The Great Basin is an example. While clearly accommodating relative plate motion between the Pacific and North America, the region is far from the plate boundary. As well, strain rates sit somewhere between interplate and intraplate environments. These are often referred to as plate boundary related 6 of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

7 environments. Scholz et al [1986] introduced the following descriptive scheme. Type Description Slip Rate Recurrence Time I Interplate V > 1 cm/yr 10 2 II Intraplate (plate boundary related) 0.01 < V < III Intraplate (mid-plate) V < 0.01 > 10 4 The question has arisen whether or not stress drop is related to average return time of the faults on which earthquakes occur. We ll initially follow the approach outlined by Kanamori and Allen [1986] Define the average stress drop as the ratio of the following integrals: where Δσ = s ΔσDdS Δσ = stress drop s DdS D = dislocation on fault plane S = fault area Recall from the preceding discussion of Kanamori [1977] that the expression in numerator may also be written (for complete stress drop) Δ σ Dd S = 2Ε s ( = 2W ) s where Ε s = seismic radiated energy of the Gutenberg-Richter energy formula LogΕ s =1.5M s Combining these latter two expressions with the preceding for average stress drop yields Δσ = 2µΕ s M0 = 2 x10 1.5M s LWD or where M 0 = seismic M 0 LWD= 2 x10 1.5M s Δσ where W = width of fault L = fault length which are scaling relationships to which we can compare to observation. To do so, a number of different cases can be considered. 7 of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

8 If (case 1) both W and D are proportional to L, then log L 1.5 # ( 3)M s 1 $ % 3& log Δσ If (case 2) both W and D are fixed, then log L ( 1.5)M s logδσ If (case 3) W is fixed and D L, then log L ( 1.5 2)M s # 1 % log Δσ $ 2& 1.37logl = M s Plots of Log L vs Ms and Log L vs Mw with lines drawn from case 3 with the assumption of different constant stress drops are shown in Figure 2. The earthquakes thought to have the shortest return time (fastest repeat time) tend to have smaller stress drops than earthquakes with longer return times. Figure 2. Relationship between moment magnitude Mw and surfacewave magnitude Ms and fault length L. The solid lines are the trend for constant stress drop for case 3. Symbols are observed value for earthquakes and different symbols are used to indicate the estimated averge return time of such earthquakes on the faults on which they occurred. Figures from Kanamori and Allen [1986]. 8 of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

9 Before continuing, note that the relationship log L ( 1.5 2)M s may also be found by manipulation of the expressions for seismic moment M0 and moment-magnitude Mw The expression for seismic moment M 0 = µ LW D The expression for moment-magnitude Combining fields we obtain Again again assuming LogM 0 = 1.5M w log µ + log L + log W + log D = 1.5M L DandW is fixed, we find 2 log L 1.5M or log L M Scholz et al [1986] simultaneously addressed the same question but followed a slightly different approach than Kanamori and Allen [1986]. They plotted seismic moment Mo directly versus rupture length L for a global set of intraplate and plate boundary earthquakes (Figure 3). They observed a systematic relationship between the two variables Figure 3. Log fault length versus log moment for large interplate and intraplate earthquakes. Figure from Scholz et al [1986]. but also a distinct offset between intraplate versus interplate earthquakes. The lines drawn through the data have slopes of about 1/2 which indicates that Mo is proportional to rupture length squared L 2, which is equivalent to displacement being proportional to rupture length. Mo = µlwd 9 of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

10 if D is proportional to L, then Mo µl 2 W The lines drawn through the interplate and intraplate data are equivalent to values of slope α equal to 6x10-5 and 10-4 in the relationship D=αL for intraplate and interplate earthquakes, respectively. The result indicates that intraplate earthquakes have about 6 times more slip than interplate earthquakes with the same rupture length. Given that stress drop is proportional to slip per unit area, the result also implied higher stress drops for intraplate earthquakes. In contrast, there is no separation of data due to fault mechanism. This would argue that return time or, analogously, tectonic environment plays a stronger role in controlling the static stress drop of an earthquake. There are a number of factors that can lead to differences between the character of interplate and intraplate events: 1) Intraplate fault slip rates are 1-2 orders of magnitude slower than interplate. 2) Total slip is generally 1-10 km as opposed to hundreds of kilometers for plate boundaries. 3) Intraplate events have finite lengths and are not continuous features. Explanations: 1) Lab studies show frictional strength has negative dependence on sliding velocity and increases with time of stationary contact, but this latter effect in lab is much less than in nature. Other processes such as chemical healing may play a role. 2) Crustal deformation is strain softening process since progressive deformation tends to be concentrated in limited, narrow zones. Thus, increased offset may result in weakening of faults. 3) Absence of any correlation of stress drop with focal mechanism type is perhaps surprising in light of Sibson [1974] 4) Net result is that faults are slip weakening or velocity weakening (or both). Thus, deformation tends to concentrate on a few master faults as opposed to being evenly distributed over a broad zone. This may be the mechanism which yields plate tectonics. Large Versus Small Earthquakes and Complications in Scaling Laws Earthquakes generally nucleate and are confined between the surface and some depth H, the depth of the seismogenic layer, which in turn is a function of tectonic environment. The maximum depth to which a rupture of dip δ may extend is H/sin(δ) whereas no similar bound exists to limit the length of earthquake ruptures. Thus, at that point where a growing earthquake rupture reaches the bounds of the seismogenic layer, it is limited to growing in the length L dimension. 10 of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

11 Scholz [1982] defined small earthquakes as those bounded by the seismogenic layer and large earthquakes those with widths greater than H/sin(δ). He further noted that while a circular fault model may be applicable to describing small earthquakes, it is a less than satifactory description of the dimensions of a large earthquake rupture. And therein resides a question of whether or not it is correct to accept the implications for large earthquake scaling laws based on a circular, probably incorrect, fault model. In an earlier paper, Scholz [1982] took the approach of collecting published empirical data showing the correlation between mean slip u, in a large earthquake, and the length L, and width W of rupture for large earthquakes. In this case, he subdivided the data into those occurring along major subduction thrusts and strike-slip earthquakes, with the reasoning that the seismogenic thickness is greater in subduction zones than along strike-slip faults. It was observed that slip correlates with length D = L but the constant of proportionality differs between thrust and strikeslip faults (Figure 4). One can examine the consequences of the observations in light of elastic dislocation models. Figure 4. Plots of average coseismic displacement versus rupture length for large strike-slip and thrust earthquakes. Figure taken from Scholz [1982] Returning to the definition of seismic moment and recalling M 0 = µ LW D # Δσ = Cµ D & % ( or D = ΔσW $ W ' Cµ the expression for moment may be rewrittn and shows Mo scales to LW 2. M 0 = Δσ C LW2 11 of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

12 Log(M0) vs Log(LW 2 ) for the expression is plotted in Figure 5. Data for historical earthquakes with lines of constant stress drop for reference are plotted in Figure 5. Figure 5. Plot of Log(Mo) versus Log(LW 2 ) for historical strike-slip and thrust earthquakes. Lines of constant stress drop are provided for convenience. In this case, the trend of data do not follow lines of constant stress drop but, rather, indicate a trend toward increasing stress drop with earthquake size. The step between the trend of strike-slip and thrust earthquakes reflects the difference in seismogenic thickness or fault length for the two data sets. The trends of increasing stress drop with increasing seismic moment was explained by Scholz [1982] as a consequence of coseismic displacement being proportional to fault length. D = L By placing the preceding expression into the standard expression for stress drop, one obtains # Δσ = Cµ % $ L & ( W ' 12 of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

13 which shows stress drop should increase as a rupture length increases. The step between strike-slip and thrust in above figure is because the seismogenic width is smaller for strike-slip earthquakes than for thrust, so for the an equivalent moment the aspect ratio of strike-slip is greater than thrust for events of same moment. And if the data of Figure 5 are replotted on a graph of Log Mo versus Log L 2 W, a linear dependence between the two variables may be observed in Figure 6. The dependence is explained by substituting the empirically observed proportionality of displacement D to rupture length L (D ~ αl) into the expression for moment M 0 = µ LW D = µ L 2 W Figure 6. Plot of Log Mo versus Log L 2 W for large strike-slip versus thrust earthquakes. Figure taken from Scholz [1982] The question arises why Kanamori and Anderson (1975) observed a dependence of seismic moment Mo on fault Area A 3/2 whereas Scholz [1982] sees a dependence on L 2 W. Scholz [1982] explained it in the following manner. Drawing on the observation that D~αL, then as at the top of the page, Mo must be proportional to L 2 W. Then note that L 2 W may be reexpressed in terms of fault Area A as follows. 13 of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

14 L 2 W Α 3 2# L % $ W & 1 2 Kanamori and Anderson (1975) may have observed a correlation to fault area A 3/2 only because L/W for the earthquakes considered only ranges by a factor of 20 and results of were plotted in logarithmic coordinates. However, when considered in framework of their circular fault model, there interpretations of correlation to mean constant stress drop wold not be correct. Implications on the Dynamics of Rupture The different conclusions and implications led Scholz [1982] to pose the thought-problem two alternate fault models Figure 7. The L- model of fault behavior was put forth to explain the T apparent linear relationship between fault length. In this case, it is assumed that the base of the seismogenic layer is soft and absorbs displacement and goes to zero at the ends of the earthquake rupure. The W-model, in contrasts implies that the rupture is pinned to zero displacement at the base of the seismogenic layer. This model is consistent with standard elastic models. In both models, any point on the fault that has ruptured will not know the fault has stopped until it has reached the limits of L in the L-model or W in the W-model. Assuming that any patch of rupture continues to slip until it receives information back from where the rupture front has stopped, the two models predict very different characteristics of the dynamics of fault rupture (Figure 8). Figure 7. Boundary conditions that may control earthquake rupture. (A) The W-model asserts that displacement at the base of the seismogenic layer is zero. (B) The L-model considers that the base of the seismogenic layer is not pinned but, rather may absorb displacement during earthquake rupture. Displacement is pinned to zero at the ends of the earthquake rupture Figure from Scholz [1982]. 14 of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

15 Figure 8. Given two earthquakes with the same displacement, (a) the W- model requires that slip be accommodated relatively quickly by propagating patches of slip whereas the (b) L-model would predict slip at any given point to be achieved over a longer period. Thus, middle depicts amount of slip at any time on fault and right shows rise-time needed for complete slip to occur at any point. The Problem Revisited Subsequent to the early 1980 studies that first outlined the implication of scaling laws, seismologic measurements of the moment and length of large earthquakes continued to accumulate. Romanowicz [1992] revisited the issue of scaling of fault length and seismic moment with the accumulated data. Her plot of the data are reproduced in Figure of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

16 Figure 9. Plot of seismic moment versus rupture length for large strike-slip earthquakes in California and elsewhere around the globe. On the plot, she places lines of slope n= 1, 2, and 3 and notes that earthquakes smaller than 1 x N-m follow a slope of 3 whereas larger events follow the slope of n=1. Thus, for smaller earthquakes LogM 0 = log L = 3 or M 0 = L because within the expression for seismic Moment, it is assumed that displacement increases as a function of fault radius (effectively, W) and "large" earthquakes, M 0 = µ LW D µ Π R 2 D LogM 0 = LogL =1 because, the interpretation goes, W is fixed, D is fixed, and Mo is proportional to L. So in this analysis, we return to the implication that coseismic slip does not necessarily increase with earthquake size. Surprising are the wheels of science. Scholz [1994a] subsequently pointed out that the cross-over point in her analysis was treated as a free-variable but that there are physical reasons that the controlling factor for the cross-over is physically related to the km depth of the seismogenic layer. If the cross-over is fixed at 15km, it is observed M0 scales with L 2 (Figure 10). 16 of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

17 Figure 10. At this juncture, there is a general bias that the L-model is incorrect. The evidence against the L-model actually arises from dynamic simulations of the slip history of historical earthquakes which show slip to occur by distinct slip patches that migrate along the fault plane. But this also leaves an interesting conundrum. If in fact displacement scales according to A 3/2, then why do we see slip scale with length? Figure 11 is an example of the progression of slip during rupture of the 1999 Koaceli, Turkey earthquake constructed by Sekiguchi and Iwata [2002]. The Figure illustrates the time-sequence of displacement beginning with the initial shock at the hypocenter. The last panel is the cumulative or total slip. The analysis argues that slip is accomplished by the progagation of slip patches along the fault plane. In this case, the rupture is bilateral. There are also a number of other references that follow up these ideas as well as provide a different view [Bodin and Brune, 1996; Romanowicz, 1994; Scholz, 1994a; Scholz, 1994b] 17 of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

18 Figure 10. References cited Bodin, P., and J.N. Brune, On the scaling of slip with rupture length for shallow strike-slip earthquakes: Quasi-static models and dynamics of rupture propagation, Bulletin of the Seismological Society of America, 86, , Kanamori, H., The energy release in great earthquakes, J. Geophys. Res., 84, , Kanamori, H., and C.R. Allen, Earthquake repeat time and average stress drop, in Earthquake Source Mechanics, edited by S. Das, J. Boatwright, and C. Scholz, pp , American Geophysical Union, Washington, D.C., Kanamori, H., and D.L. Anderson, Theoretical basis of some empirical relations in seismology, Bulletin of the Seismological Society of America, 65, , Romanowicz, B., Strike-slip earthquakes on quasi-vertical transcurrent faults: inferences for general scaling relations, Geophysical Research Letters, 19 (5), , Romanowicz, B., Comment on "A reappraisal of large earthquake scaling" by C. Scholz, Bulletin of the Seismological Society of America, 84, , of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws

19 Scholz, C.H., Scaling laws for large earthquakes: consequences for physical models, Bulletin of the Seismological Society of America, 72, 1-14, Scholz, C.H., A reappraisal of large earthquake scaling, Bulletin of the Seismoloical Society of America, 84, , 1994a. Scholz, C.H., Reply to Comment on "A reappraisal of large earthquake scaling" by C. H. Scholz, Bulletin of the Seismological Society of America, 84 (5), , 1994b. Scholz, C.H., C.A. Aviles, and S.G. Wesnousky, Scaling differences between large intraplate and interplate earthquakes, Bulletin of the Seismological Society of America, 76, 65-70, Sekiguchi, H., and T. Iwata, Rupture process of the 199 Kocaelli,Turkey, earthquake estimated from strong-motion waveforms, Bulletin of Seismological Society of America, 92 (1), , Sibson, R.H., Frictional constraints on thrust, wrench, and normal faults, in Nature, of 19 Wesnousky Course Notes Geology 736 Spring 2018 ScalingLaws