Utility of elastic models in predicting fault displacement

Transcription

1 JOURNAL OF GEOPHYSCAL RESEARCH, VOL 103, NO B1, PAGES , JANUARY 10, 1998 Utility of elastic models in predicting fault displacement fields Anupma Gupta and Christopher H Scholz Lamont-Doherty Earth Observatory of Columbia University, Palisades, New York Abstract Although elastic models have long been used to model earthquake deformation, their application to fault problems is questionable, as accumulated fault strain is higher and the relevant timescales are longer We test the utility of using elastic models to predict fault displacement fields by independently measuring the three-dimensional slip (offset) distribution and displacement field of small normal faults The displacement field is constrained from the topography of the deformed bedding planes; the slip distribution constrained from stratal offsets in multiple sections of fault-normal saw-cuts Using the observed slip distribution, we calculate both one- and three-dimensional elastic displacement fields We find that the large strain associated with fault growth can be accommodated with linear elastic models Much of the remaining misfit between the data and the model may result from elastic interaction with other nearby faults, the inelastic zone around surrounding fault tips, and prefaulting irregularities in the measured bedding plane surface 1 ntroduction strain and longer timescales involved in fault growth make the application of elastic models to fault problems open to Elastic models have long been used to model earthquakes question and their displacement field [eg, Chinnery, 1961; Savage and n addition, study of fault displacement fields is complicated Hastie, 1966; Nur and Mavko, 1974; Savage and Prescott, by other geological processes such as isostasy, syntectonic 1978; Thatcher and Rundle, 1979; Stein and Barrientos, 1985; and posttectonic erosion and deposition This can make direct Savage and Gu, 1985] Some of these workers use geodetic and comparison of observed structures and simple elastic model other seismological data in conjunction with an elastic model solutions difficult Another complicating factor is that crustal to constrain the slip distribution at depth; others attempt to scale fault characteristics are usually difficult to measure, model the earthquake cycle assuming a slip distribution An especially in three dimensions Consequently, workers inherent assumption in most elastic models is that the strains compare any available large-scale fault geometries to models considered are small and smoothly varying [Timoshenko and which may incorporate different crustal structure, erosion, Goodier, 1951] n addition, the inelastic or ductile region deposition, and isostasy However, they all start with the surrounding the fault tip must be small compared to the fault assumption that displacement fields can be modeled within an length [Kanninen and Popelar, 1985] (See Figure 1 for elastic upper crust and other influences modify the geometry terminology used in this paper) f strains or inelastic zones over different timescales However, as noted above, the high are too large, a significant fraction of the deformation can be strains and long timescales associated with faulting make this inelastic or nonrecoverable n these cases, more assumption tenuous Because large faults are difficult to study sophisticated models are needed to approximate the and observed structures are complicated by other processes, so displacement field far it has not been possible to test the assumption that fault Elastic models are ideal for the study of earthquake displacement fields can be modeled as elastic response to slip deformation because the strains are quite small n addition, Small normal faults of length 10 '2-102 cm present a over the short timescale of an earthquake the crust can behave remarkable opportunity to test this assumption They are free elastically [eg, Turcotte and Schubert, 1982] Several of influence from isostasy, sedimentation, and they may be workers have also used elastic models to simulate fault-related studied in three-dimensional detail We take advantage of this basins or mountain ranges [eg, Rundle, 1982; Stein et al, opportunity and compare observations of small faults to one- 1988; King et al, 1988; King and Ellis, 1991; Armijo et al, and three-dimensional elastic solutions 1996; Contreras et al, 1997] However, faults grow over While we and other workers employ an elastic model to millions of years Furthermore, fault strains are much larger simulate the fault displacement field, we are not suggesting than earthquake strains The strain field scales with the slipthat fault formation and growth should be modeled wholly as length ratio, which for faults is -10'2[eg, Cowie and Scholz, an elastic process Cowie and Scholz [1992a, b] have shown 1992b; Dawers et al, 1993]; by comparison, it is for from theoretical considerations and observations [see also earthquakes [Scholz, 1982; Scholz et al, 1986] The higher Dawers et al, 1993; Peacock and Sanderson, 1994; Schlische et al, 1996] that the slip-length profile is not elliptical as Copyright 1998 by the American Geophysical Union Paper number 97JB /98/97JB simple linear elastic fracture mechanics would predict, but rather is "bell-shaped"(figure 1) The theory suggests that the slip-length profile is tapered because the rock cannot support infinite stress [Cowie and Scholz, 1992a] Consequently,

2 824 GUPTA AND SCHOLZ: UTLTY OF ELASTC MODELS a: Fault plane and slip:length profiles b: Deformed bedding surface height tapered slip profile slip directi on _ '[' L'N [ length ld i bell-shaped or tapered ds= 0 length elliptical ds _ oo displacement field length Figure 1 Terminology used throughouthis paper (a) Fault length is the maximum dimension of the fault plane perpendicular to the direction of slip; fault height is the dimension parallel to the direction of slip "Bell-shaped" and elliptical slip-length profiles are illustrated (b) We use slip distribution to describe the displacement variation on the fault plane, whereas displacement field means deformation away from and including the fault plane Most workers use the term displacement to describe offset on the fault plane, we will reserve the terms slip and dislocation to describe offset on the fault plane; eg, what workers often call displacement-length profiles, we call slip-length profiles plastic yielding could occur at fault tips However, some workers find that a linear elastic model with varying frictional strength on the fault plane can produce a "tapered" profile [B irgmann et al, 1994] We account for plastic yielding or differences in strength at tips by using observed, rather than elliptical, slip-length profiles as boundary conditions Given an observed, "bell-shaped" slip-length distribution, we show that an elastic dislocation model can be used to simulate observed fault displacement fields 2 Data and Methods The faults occur at high angles to finely layered lacustrine strata which preferentially break along bedding planes Assuming that bedding planes were originally horizontal, we can attribute changes in bedding surface topography to faulting The fine layering enables us to measure the slip distribution accurately along fault-normal saw-cuts Because the faults studied are only centimeters long, we used a modified profilometer to measure the deformation field along the exposed bedding planes of two faults The original profilometer, designed and built by S Brown and T Koczynski, is described by Brown and Scholz [1985] and has since been automated The precision of the profilimeter is 10 '6 The faults studied are taken from a population of rift-related m normal faults from the Solite quarry near Eden, North Carolina All samples were profiled in detail For example, the The 10' cm long faults formed within laminated siltstones bedding plane surface shown in Figure 2b was recorded with of the Triassic Danville rift basin The small faults follow the 6100 data points (063 mm x 28 mm sampling interval) n same scaling laws larger faults obey: powerlaw frequency- addition to profiling surface data, we set both faults in epoxy length and linear slip-length distributions [Schlische et al, and made serial sections perpendicular to fault length in order 1996] Other features that they have in common with large- to measure slip distributions (Figure 2, white lines) We scale faults include bell-shaped slip profiles, pull-apart structures, horse-tail fan terminations, mineralized, multiple direction slickensides along fault planes, fault linkage, "relay ramp" structures, footwall uplift, hanging wall subsidence, and digitally scanned polished sections and measured slip variation along fault width (Figure 3) Section data are accurate to within a few pixels (02 mm) as determined by several replicated measurements high strains (~10'2) Their small size and similarity to largescale structures make them ideal for study For more information about the fault population and the regional 3 Observations geology, see Schlische et al [1996] From this population, we selected two relatively isolated 31 Displacement Field normal faults for detailed study: one 16 cm long (referred to as SA-16), the other 2 cm long (referred to as SA-02) (Figure 2) Detailed descriptions of large fault displacement fields are not available, but we have obtained this information for small

3 GUFFA AND SCHOLZ: UTLTY OF ELASTC MODELS 825 Figure 2 Map view of normal faults studied FW, footwall; HW, hanging wall White lines show where samples have been sectioned The surfaces are profiled and plotted in Figures 4 and 5 (a) 2 cm long normal fault (SA-02) and (b) 16 cm long normal fault (SA-16) faults By measuring faulted bedding surfaces we provide clear representations of fault displacement fields (Figures 4 and 5) Tapered slip-length profiles (Figures 4a and 5a) are indistinguishable from published slip-length profiles of larger faults [eg, Muraoka and Kamata, 1983; Barnett et al, 1987; Walsh and Watterson, 1989; Dawers et al, 1993; Peacock and Sanderson, 1994; Schlische et al, 1996] However, these small faults differ from large scale faults in that their footwall uplift and hanging wall subsidence are symmetrical about the fault plane (Figures 4b and 5b) than footwall uplift [eg, Stein and Barrientos, 1985; Armijo et al, 1996] The coseismic asymmetry is usually attributed to the thult dip and free surfac effect Over a longer timescale, as faults grow, sediment loading of the hanging wall basin, erosion of the footwall, and isostatic compensation change the displacement field (Figure 6a) [eg, King et al, 1988; Stein et al, 1988; Contreras et al, 1997] Some workers have attempted to disentangle the elastic, viscous flow, erosional and depositional components of fault structure and growth [King et al, 1988; Stein et al, 1988; Ma and Kusznir, 1994; (Footwall uplift and hanging wall subsidence are consequences Armijo et al, 1996; Contreras et al, 1997] Their studies of flexure, not drag, but some workers use the term "reverse drag" because of the appearance of deformed beds) However, observed large-scale normal faults and earthquake ruptures can suggest that fault displacement fields become more, but not completely, symmetrical as isostatic compensation occurs Our interpretation for small, buried faults is necessarily display as much as 10 times greater hanging wall subsidence different n our small faults there is no isostatic 0 } S cm, ß -" :', : : : : v:c:, ::<: : :": "'::: : : 5:' w" "-,: ::: : : : :½G::' ; ': ::: ::a? :,"*:::: :':% : : :': ' x" ' ' ': ""'% : ' "??: ::½:** :% -'":-,::;:;: ;:':' ": :' :': 7::v ::; ; :-': '' :-'- '"":--: :: '" sq? ' "'!, ; : '4 :?"T:" ' : ;T ¾7::':?] :47 : "*?;:: " "' :: ' ::-' 4 :,7' : ;:,"e' ½:" ::5': ")-:e' : : %-'":: :::::::: ::*- ::q:½: :,:-:: :? :-w:?:*:->:? ::: 7%?'-"-' ---:- : -' : : :6- -::;:':--:-': ] ] ]' :? ]: : 2: : ' : :': ½:: m" :: ; i;::', ' ':" :::%6: ; ;a;5: 3; : -a : ' :' :e i : ½:?E??½ : ½' ' ½'P' ½ "* '?; ;/ :: ½?:47' '?%: --:: - --:v: r :,? ; : :;, : -: ::j i j ß : : :: :::' :::' " - % : ::: O:&' ' : 4:% :' : V ':::: :: : :' :': 7 ':':: :::':::':"'":* ":::½ 7' ½ ::g :": ': E; : :-? " '-'"'""' ' : :% ';:: :':: :':: :':-: :;t-'"'"" :: '::' " ":'"'"::" - :": ' ' ß ' - :--- : :- :-:<' :; : :; : ::: : :;: : e:::::::::,: ::::--: :, ;:- ',:---?------:---:$, rw,::-: : &? :: : : : : :: : : : c-- : :, -- :- - -,: ::: : "-- -: : :;::: ::' ', ½ :'*'*: ' ' ---:- : ½: : --,: :, :8 $3 - : :,::,: % 4:' * :s ::?: ::? :v ' :? :,: : :;::½'""'""?*' : g : :-:,-:,: :: ½:* z?-:$ : : : : ---:;?:: -::: '--$ : ; :: :: ::m ½ :: ::*: ;: : :::: : :': %::;: : :::::----,- :---- ::: ::: ::-:: ::& '-' "-;';;;3:::: :::::::::::::::::::::: ;::/:-: '- 5:":"-' : 4:-- ': ;---:-: --::v '-";' ''-': " -'':' : - -:L--;:!: :c ':" :': g: Figure 3 Cross sections through the center of each sample Fault plane slip distribution was found by measuring bedding offset on several similar cross sections (a) Cross section through the center of SA-16 The deflected bed, outlined in black, may be influenced by deformatio near the tip of SA-16 (b) Cross section through SA-02 Note the bend in the fault plane (marked by an arrow) where two faults may have linked

4 826 GUPTA AND SCHOLZ: UTLTY OF ELASTC MODELS A' 3-D mesh plot 6b) Thus we can directly compare elastic solutions to the observed displacement fields 16 f 32 Slip Distribution We measured slip distribution for two normal faults using several fault normal serial sections Results for the larger fault (SA-16) are plotted and contoured with 1 mm slip contours in Figure 7 The slip is generally high near the center and tapers near the tips to zero, in agreement with other observations of slip distribution [eg, Rippon, 1985; Barnett al, 1987] n 12 oo A: 3-D mesh plot 2 y:1472 ia c y:1221 B: Profiles normal to fault 2 y: y= o y: o - 0 i E i ' y--4) y=263 Centimeters i o -- Figure 4 Displacement field data for 16 cm long normal fault (SA-16) (a) Mesh plot of fault (maximum displacement is shown) Note the fault tapers at the ends (b) Fault normal profiles Coordinates are the same as in Figure 4a -0s 05 y= Centimeters compensation, nor are there erosion and sedimentation The Figure 5 Displacement field data for 2 cm long normal fault small fault problem starts as a symmetric one because the same (SA-02) (a) Mesh plot of fault (maximum displacement lithostatic pressure and lithology surround the fault for many shown) Note fault tapers at the ends (b) Fault normal fault lengths on all sides: there is no free-surface effect (Figure profiles Coordinates are the same as in Figure 5a

5 GUPTA AND SCHOLZ: UTLTY OF ELASTC MODELS 827 seismogenic upper crust lower and mantle fault studied Figure 6 Schematic illustration of tectonic setting and boundary conditions associated with small normal faults (a) Block diagram showing Triassic rift basin and approximate location of small faults within the basin The large basin bounding fault has free surface on one end and mantle on the other Because the fault is dipping, more hanging wall area (light shading) is in contact with the free surface, and more footwall area (dark shading) is in contact with mantle Hence the boundary conditions this crustal scale fault are asymmetrical, which means that hanging wall subsidence will be greater than footwall uplift For thrust faults, hanging wall deformation is also greater sostasy and sediment loading only modify this initial asymmetry See King et al [1988], Stein et al [1988], and Contreras et al [1997] for more details (b) Small-scale faults The free surface and lower crust are many fault lengths away from these small faults This means that the top and bottom of the fault experience similar boundary conditions; ie, the small fault effectively occurs in an infinite elastic material, rather than in a thin elastic plate addition, the fault is still growing by linkage A small fault has linked to SA-16 (lower right-hand corner of Figure 7a) and makes the slip distribution irregular n Figure 7b we attempted to remove the slip contribution from the "recently" linked smaller fault by excluding slip data along the smaller fault in the plot This contour diagram (Figure 7b) looks more idealized, and the fault plane approximates an ellipse, assuming the unsectioned portion of the fault plane is also simple Sectioning of fault SA-02 revealed that the fault consists of two nearly equal size faults that have linked to form one continuous fault plane (Figure 8) Note the fault normal offset of the two segments shown in Figure 3b (marked by solid arrow) However, for the two fault segments in SA-02, it appears that the faults did not link until late in their slip history (Figures 8 and 3b) The outlines of two separate planes can still be clearly seen, and the slip distribution still has characteristics of two separate, but interacting faults Each fault appeared to be a simple, isolated structure until sectioning revealed the three-dimensional complexity 4 Elastic Models and Results 41 Elastic Models n order to test whether an elastic model can accurately approximate fault geometry for large strains and long timescales, we used the independent measurements of slip distribution described above to compute elastic displacement fields We compared transects perpendicular to fault planes to one-dimensional elastic solutions for flexure caused by single edge dislocations The solution for displacement along a line normal to the fault plane is presented below [eg, Maruyama, 1964]: where w/is the deflection, u is the slip (constant for the length of the dislocation), p is the distance from the dislocation perpendicular to the fault plane, and d is the along dip width This equation does not include any asymmetrical deflection, necessary for large dipping faults; the equation assumes a symmetrical problem Consequently, the deflection solution for large, crustal scale faults is more involved [eg, Contreras et al, 1997] However, solutions to equation (1) can be compared to fault-normal transects near the center of small faults To compare the entire observed displacement field to an elastic solution, we used a three-dimensional boundary element model with the observed slip distribution in SA-16 as boundary conditions Available analytic solutions, which are usually calculated for point or rectangular sources, do not allow the freedom to incorporate the observed fault shape and slip distribution in the problem However, boundary element models allow us to apply slip mea,qnrements as internal boundary conditions on the fault surface as well as to the entire volume Another advantage of boundary element models is that we do not need to model the entire volume with finite elements Because of the complexity of SA-02's observed slip distribution (Figure 8), we did not compute its threedimensional displacement field With the observed slip distribution in SA-16 (Figure 7b), we calculated an expected elastic displacement field using a commercial boundary element modeling program, BEASY (Computational Mechanics nc, 1993, version 50) By using BEASY rather than other methods we had the flexibility to solve a 3-D problem and to input any fault shape and slip distribution (1)

6 828 GUPTA AND SCHOLZ: UTLTY OF ELASTC MODELS A ooo aooo ooo oooo eooo millimeters millimeters Figure 7 Slip distribution on the fault plane (SA-16) All distances are in millimeters; shade contour line interval is 05 mm White contour lines are marked (a) Unaltered data, including linking fault (b) Displacement contribution from small linking fault removed The star is the point of displacement through which the one-dimensional profile in Figure 10a passes Before giving the slip information to BEASY, we needed to simplify it We tried gridding the data using several different contouring methods, but because of the localized nature of sampling, contouring in this way produced unrealistic results We resorted to contouring by hand, and our contouring results and the data used to draw contours are presented in Figure 9a We tried to keep slip gradients as accurate as possible to preserve the bell-shaped nature of the slip profile However, given computational constraints, we solved the problem as a twofold symmetrical one (half plane), which necessitated some simplification of the observed geometry Once contoured, the fault plane was discretized into polygonal elements, and slip at nodes (solid dots) was given to BEASY The program interpolated between contour nodes to obtain a continuouslip distribution We also constrained our elastic problem by setting to zero the displacement at infinity (infinite space is approximated by a cube with side length equal to 10 fault lengths) (Figure 9b) Given such a slip distribution and boundary conditions, BEASY calculated the resulting displacement field Because the model assumes a homogeneous elastic material and we use slip boundary conditions, our displacement field results do not depend on elastic constants However, stress field estimates would depend on elastic constants and rheology BEASY enabled us to use complicated slip boundary conditions and to obtain three-dimensional displacement

7 GUPTA AND SCHOLZ: UTLTY OF ELASTC MODELS os : -- }:: -- ::: : - - :: - :: ' '" ' ':':'::'--" ' ::'' :' ':' :? 'a, s''" :':-: ' ' : :2& ;:' :;?" : ::,::½?'L d ' ::' ::: -- ' " : :, : ' '',- :: ' ' ::"'" ':" ' '" : : :,:::,": " ::-d"' ',- ' '""*'"'" '""'%:-:: ;, :':' 5 ::: ß ' ' : ',' d;': " : - :: ß :: :: :: ß ' :, :: : %, :?: : ':::: ß ' - '::' 0::_ "" -:,, ' : ":: : ' ' : "- ' - - -" ' ;* " ' - :: ::: ::t ß ' ::::' '" ' 'x:'"'" ': :' ': ';' :: " : " ]_j ' ' : : : ' : ZO 25 3 '0 ' '3 5' centimeters Figure 8 Slip distribution on the fault plane (SA-02) Contour interval is 02 mm The now continuous fault plane appears to have grown from at least two smaller faults The star is the point of displacement through which the one-dimensional profile in Figure 10b passes information for any slice through the volume that we needed We chose a slice in the same orientation as the deformed bedding plane from which we obtained displacement field measurements 42 Comparison of Elastic and Observational Results Figure 10 shows data from fault normal transects near the center of faults and one-dimensional displacement predictions based on equation (1) Different displacement curves represent different fault heights, or depths n each case, the curve which most closely matches the shape of the data represents a half height of about one quarter of the total fault length, ie, 40 and 5 mm This agrees with the observed fault aspect ratio The transect across SA-16 (Figure 10a) is consistent with a dislocation solution, but with increased distance from the fault plane, the correlation decreases, especially in the footwall The observed footwall deflection in SA-02 (Figure 10b) shows better agreement with a dislocation solution However, in the hanging wall about 4 mm away from the fault plane, the observed deflection is substantially larger than elastic curves We also show data for faults not chosen for detailed 3-D study because of their proximity to other faults (Figure 11) We find that least squares best fit profiles to equation (1) are usually good approximations to the observed profiles n order to compare the 3-D data and model quantitatively, we calculated their correlation on a common grid The model displacement field is estimated by 515 data points, and the observations (SA-16) are defined by many more data points (6100) Because the density of observations for SA-16 is so high, we simply used a nearest-neighbor gridding technique; for the model, we used an inverse distance gridding method As the grid spacing size changed so did the correlation coefficient (covariance normalized to standard deviations and number of data points) (Figure 12) The correlation coefficient varies between-10 and 10; zero means no correlation, and 10 is a perfect positive correlation When all the information in the data sets was utilized, the correlation coefficient stabilized at -091 Once this stable point was reached, a finer grid spacing did not increase or decrease the coefficient, suggesting that 091 is the true correlation coefficient The regridded three-dimensional observations and model predictions for SA-16 are compared in Figures 13a and 13b, respectively We plot the residual (observed minus model) in Figure 13c The predicted generally matches the observe data; areas of high residual are confined to the lower fight-hand and upper left-hand corners of the domain The areas of high residual are localized, both in 1-D and 3-D comparisons; this suggests that the elastic model is not generally wrong, but its failure in certain areas is due to neglected localized phenomenon Nevertheless, an elastic model can at least reproduce the basic shape of the observed deformation field (Figures 10, 11, and 13) One could, of course, find an equivalent viscoelastic model to produce the same displacement field The difference in the theology affects only the predicted stresses, which do not concern us here For the case we study, the elastic stresses have been completely relaxed since the formation of the faults We study the locked in displacement fields that remain 43 Sources of Misfit The observed displacement field for both faults is consistent with elastic models However, there are various sources of misfit between the observations and models Other nearby faults and their associated elastic displacement fields may account for much of the localized mismatch Because these small faults all dip in the same direction, the effects of other fault displacement fields are simple and predictable

8 830 GUPTA AND SCHOLZ: UTLTY OF ELASTC MODELS a) 000 :z:=-' ----'4" ' ' / oo ,00 _ - 0,00, zooo millilmeters F, initially horizontal reference plane or bedding surface fault plane (polygons represent elements)! / / / / mm each element is given displacement boundary conditions in each comer Figure 9 Three-dimensional boundary element model Using an idealized version of the observed slip distribution (Figure 9a), we calculate the expected displacement field using the boundary element modeling program, BEASY (Computational Mechanics nc) The problem is solved as a twofold symmetrical one, and the fault half plane is given displacement boundary conditions based on observations Note that the cube is not to scale; sides of the cube are 10 times longer than fault length n addition, the cube of elastic space is held fixed on all sides

9 GUPTA AND SCHOLZ: UTLTY OF ELASTC MODELS 831 _ SA-16 y = i bedding plane variation observation l-d, d=20 mm l-d, d=40 mm l-d, d=60 mm 3-D model distance (mm) SA-02 y = 23 ' ' 02 F: nearby fault 9 0 ' distance (mm) observation l-d, d=2 mm l-d, d=5 mm --- l-d, d=10 mm Figure 10 Observed deflection and one-dimensional elastic dislocation solutions Different elastic curves represent different fault plane heights FW, footwall; HW, hanging wall The y coordinates correspond to spatial coordinates in Figures 4 and 5 (a) The 16 cm long normal fault (SA-16) (b) The 2 cm long normal fault (SA-02) Bedding should be deflected above predicted values in the hanging wall and below in the footwall (Figure 14) The below predicted curves in the hanging wall and above in the footwall, is not likely for this population For example, another fault, 4 cm from SA-02 in its hanging wall, suggests that the sharp departure of the observed curve from predicted is due to elastic interaction with another fault (Figures 10b and 14a) n the lower right-hand corner of Figure 13c, another fault is also responsible for the large residual there At the same time, elastic interaction is ruled out for some of the mismatch in SA-16 (Figures 10a and 13c, upper left-hand corner) because these small faults do not dip in opposite directions One possible explanation for the mismatch is that the inelastic tip of another larger fault is modifying the bedding above that predicted in the footwall An example of this "inelastic" interaction is shown in Figure 3a A small bed (outlined in black) is also deflected upward in the footwall of a small fault in the center of the sample (Figure 3a), similar to the deflection in SA-16's footwall (Figure 10a) We see that "inelastic" deformation near the tip of a larger fault is influencing the deflection in Figure 3a A large fault tip may,, h,,n ;,,,, m e,-,,-,,,4n,,;,,, ;, c ^ 76 (Figures 10a and 13c, upper left-hand corner) Additional unexplained misfit may be due to idealization of the model, prefaulting irregularities, and/or plastic deformation 5 Summary and Conclusions We have observed a remarkable population of small riftrelated normal faults These small faults allow accurate and independent measurements of slip distribution and displacement field, which is nearly impossible for large crustal scale faults Elastic models are useful in predicting the displacement fields associated with these faults even though fault strain and timescales involved in fault growth are much larger than for earthquakes The displacement fields and slip distributions in our population show general agreement with available large-scale

10 GUPTA AND SCHOLZ: UTLTY OF ELASTC MODELS a) 40-10x vertical exaggeration 20 O- (:D - E ' _, c -20 o -40 ' 60x u = 0079 d= 118 ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' centimeters b) 40x c) 010 a) 0-',--, (1) _ E ß = e" - (1) ' - u=015 d = 158 [ ] [ [, centimeters d) u = 025 d = 368 e) 02-- [ ] centimeters u =026 d= 183 [ [ centimeters d = -081 ' '''''''1''''''' '''''''1'''''''1 ' centimeters Figure 11 Observedeflection (solid) and best fit elastic profiles (dashed) for several normal faults Vertical exaggeration is 5x unless otherwise noted Best fit parameters u and d are as in equation (1)(a) Nearby fault affecting the deflection profile; 10x vertical exaggeration (b) and (c) Examples of fit very close to the fault plane (d) Deflection of larger fault, which does not change until smaller fault is reached, unlike deflection in Figure 10a (e) An imperfect fit because the fault profile is chipped or eroded, ie, it is difficult to estimate true slip

11 , ß GUPTA AND SCHOLZ: UTLTY OF ELASTC MODELS o i i ß ß ß 6 SA points 3-D model L, 515 points : ß : '''l'''l''' '''l'''l grid size (10 means a 10x 10 grid) Figure 12 Variation in displacement field correlation coefficient between observations of SA-16 and 3-D model: As the grid becomes finer, more information from each data set is utilized As all information is used, the coefficient stabilizes at -,-091 data Slip-length profiles are tapered, and slip distributions are approximately elliptical when slip contributions from linking faults are removed Footwall uplift and hanging wall subsidence, a striking z, feature of observed displacement fields in our small faults, show that they are consequences of flexure, not isostasy This suggests flexure also causes footwall uplift and hanging wall subsidence for large faults However, large fault footwall uplift and hanging wall subsidence are asymmetrical, whereas for o small faults the displacement field is symmetrical This may be because small buried faults do not feel the free surface effect which large faults do Displacement field transects normal to fault planes generally agree with elastic solutions for flexure caused by single screw dislocations A three-dimensional model for an isolated normal fault is also consistent with the observed displacement field Most of the misfit between the models and data may be due to elastic fault interaction, and some may be , 2, due to inelastic deformationear fault tips However, a simple elastic model is sufficient to simulate most of the observed 3 deformation field despite the large strains (~ 10 '2) associated with faulting However, the results presented in this paper do not imply z that fault growth is an elastic process or that an elastic model describes all features of the deformation field We have shown 1 examples where inelastic deformation must exist at fault tips n addition, since the deformation is now permanent, there was obviously some viscous relaxation of the stresses The timescale of relaxation does not concern us Our point is that most of the deformation can be described by an elastic model This does not imply that a viscoelastic model is wrong or that Figure 13 Observed displacement field, 3-D model it cannot producequivalent displacement results, only that it geometry, and comparison All units and contours are in may be unnecessary to resort to complicated lithologies for centimeters (a) Measured SA- 16 displacement field (b) basin and fault problems despite the large strains and long Three-dimensional model results for equivalent displacement timescales field (c) Difference between data and model

12 834 GUPTA AND SCHOLZ: UTLTY OF ELASTC MODELS King, G C P, R S Stein, and J B Rundle, The growth of geological structures by repeated earthquakes, 1, Conceptual framework, J Geophys Res, 93, 13,307-13,318, 1988 Ma, X Q, and NJ Kusznir, Effects of rigidity layering, gravity and stress relaxation on 3-D subsurface fault displacement fields, Geophys J nt, 118, , 1994 Mansinha, L, and D E Smylie, The displacement fields of inclined faults, Bull Seismol Soc Am, 61, , 1971 Maruyama, T, Statical elastic dislocations in an infinite and semi-infinite medium, Bull Earthquake Res nst Univ Tokyo, 42, , 1964 Muraoka, H, and H Kamata, Displacement distribution along minor Figure 14 Effect of nearby faults on deflection profiles fault traces, J Struct Geol, 5, , 1983 Nur, A, and G Mavko, Postseismic viscoelastic rebound, Science, 183, All faults in the population dip in the same direction, so Figure , a is most probable (a) Bedding deflected above predicted Peacock, DC, Displacements and segment linkage in strike-slip fault curves (dashed) in the hanging wall and below in the footwall zones, J Struct Geol, 13, , 1991 (b) nteraction when faults dip in opposite directions Peacock, DC P, and D J Sanderson, Displacements, segment linkage and relay ramps in normal fault zones, J Struct Geol, 13, , 1991 Acknowledgments This work is supported by NSF funding to CHS and M Anders (EAR ) and an AAPG gran to AG We thank C H Gover and the Virginia Solitc Corporation for their support and access to the quarry T Koczynski designed and led AG through rebuilding of the profilimeter S Wills provided assistance with and R Buck access to BEASY Reviews by R Schlische, D Sparks, M Evans, G Lyzenga, an anonymous reviewer, and the Associate Editor greatly improved the quality of the manuscript J Contreras, N Dawers, J Vermilye, R Schlische, R Ackermann, G King, T Dubin, and R Gangloft provided field support and/or discussion L-DEO contribution 5747 Peacock, DC P, and D J Sanderson, Geometry and 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