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RESOLUTION OF FAULT SLIP ALONG THE 470-KM-LONG RUPTURE OF THE GREAT 1906 SAN FRANCISCO EARTHQUAKE AND ITS IMPLICATIONS

Abstract

Data from all available triangulation networks affected by the 1906 earthquake have been combined to assess the trade-off between slip resolution and its uncertainty and to construct a conservative image of coseismic slip along the rupture. Because of varying network aperture and station density, slip resolution is very uneven. Although slip is determined within uncertainties of ±1.0 m along 60 % of the fault, constraints are poor on the remaining, mostly offshore portions of the rupture. Slip decreases from maxima of 8.6 and 7.5 m at Shelter Cove and Tomales Bay to 4.5 m near Mt. Tamalpais and 2.7 m at Loma Prieta. The geodetically-derived slip distribution is in poor agreement with estimates based on analysis of S-wave seismograms, probably because these waves register only 20-30% of the total seismic moment obtained from longer period surface waves. Consideration of a range of fault geometries for 1906 slip near Loma Prieta indicates right-lateral motions lie between 2.3 and 3.1 m. These values are considerably greater than the 1.5 m of measured surface slip on which several assessments of high earthquake hazard for this fault segment were based. This factor, along with the absence of 1989 slippage where 1906 surface slip was used to make the forecasts, casts doubt on some claims of success in predicting the 1989 M=6.9 Loma Prieta earthquake.

1. INTRODUCTION

Ninety years after its occurrence the M=7.7 1906 San Francisco earthquake remains one of the most significant events in the seismic record. It was the first large earthquake to have been intensively documented with seismic, geologic and geodetic observations, and the resulting report on these investigations [Lawson, 1908] is still frequently cited. The extent of fault slippage was unusually great for a continental earthquake, with over 300 km of surface faulting and a zone of high seismic intensity 600 km long and 100 km or more wide (Figure 1). The intense urbanization that has subsequently occurred in the San Francisco bay area, adjacent to the southern 120 km of the surface rupture, make the effects of the 1906 event of continuing importance for seismic zonation studies and longterm earthquake hazard assessment. In particular, the distribution of seismic slip in 1906 has a critical influence on the assignment of longterm earthquake probabilities to specific segments of the northern San Andreas fault [Working Group on California Earthquake Probabilities, 1988, 1990].

In this paper we reassess the geodetic constraints on 1906 slip, updating and extending previous studies by two of us [Thatcher, 1975; Thatcher and Lisowski, 1987]. Several factors have motivated us to revisit this topic. First, we determined that previously unexamined regional geodetic networks located inland from Fort Ross, Point Arena, and Shelter Cove (see Figure 1) were distorted by the 1906 earthquake, providing information on slippage beyond the region of surface faulting. Second, as Figure 1 shows, the data distribution along the fault is very uneven, so that slippage is well-constrained on some fault segments and known poorly or not at all on others. This variability was not accounted for in the earlier work and is important both for site-specific studies such as paleoseismic investigations (e.g., Niemi and Hall, 1992) and for the segment-by-segment assignments of earthquake probabilities that rely on quantitative estimates of slip in the last major event [Working Group on California Earthquake Probabilities, 1988, 1990]. Third, the occurence of the 1989 M=6.9 Loma Prieta earthquake near the southern end of 1906 rupture suggests a further look at the geodetic constraints on 1906 slip there would be worthwhile. Finally, a recent seismological evaluation of the sources of strong seismic radiation along the 1906 rupture [Wald et al., 1993] motivates us to compare these results with the resolvable features of the geodetically-estimated slip distribution.

This paper is divided into eight sections. We first describe the geodetic data used here and how they are analysed (section 2), then discuss how we handle their uneven spatial distribution in estimating the magnitude of fault slip and its uncertainties (section 3). Next we present our derived 1906 slip distribution and compare it with seismic estimates and surface fault offsets (section 4). The possible values of 1906 fault slip near the site of the 1989 Loma Prieta earthquake are then determined for several possible fault geometries (section 5) and the recurrence implications are discussed (section 6). The paper concludes with a discussion (section 7) and a summary of our main results (section 8).

2. DATA AND ANALYSIS METHOD

The measurements used here come from triangulation surveys, repeated measurements of the angular separation of permanent survey marks. The angle changes provide a measure of crustal deformation between surveys. Their errors are estimated for each network by summing the angle changes for each triangle and determining the rms average for the net.

The networks are shown in Figure 1. For detailed configurations of the four small local nets, see Map 25 in Lawson [1908]. Those used by Thatcher [1975] and Thatcher and Lisowski [1987] are described there. Two additional nets are included here. The large aperture northern region network, was established between 1872 and 1880 and resurveyed during 1925-1942. The local Colma net covers the interval 1899-1907. Names, geographic coordinates, and station numbers assigned here are given in Table 1. Observed angle changes are listed in Table 2 along with parameters of the model fits to be described in section 4.

Essential features of the analysis method we use to estimate fault slip from triangulation data are illustrated schematically in Figure 2. For a vertical strike-slip fault like the San Andreas, the geometry is usually well constrained, and along the 1906 rupture we fix it and determine only purely horizontal slippage on specified segments. The elements of the model are thus vertical fault segments imbedded in a homogeneous elastic half-space extending from the surface to a specified depth D (Figure 2b). Faults lying close to, but off the main San Andreas trend are often inclined and have reverse dip-slip motion. It is difficult to generalize about the influence of such faults on our slip estimates, but in section 5 we illustrate how possible 1906 slip on the fault that ruptured in the 1989 Loma Prieta earthquake may have influenced our results.

Results obtained here are relatively insensitive to changes in the assumed fault depth that lie within the range of possible values. Here we have chosen D=10 km, corrresponding to the estimated average depth of 1906 coseismic slip [Thatcher, 1975, pp. 4865-4866] and the thickness of the seismogenic upper crust in California. Shallower assumed depths would in general increase the slip over values estimated here and taking D>10 km would result in smaller values. However, this effect is a second order one: for D in the range 5 to 20 km the slip estimates change by 20% or less. For a specific calculation that illustrates this effect, see Table 2 in Thatcher and Lisowski [1987].

Note that our slip estimates are average values from the surface to depth D but are weighted towards depths shallower than D because of the decreasing sensitivity of surface measurements to slip at depth (see Thatcher, 1975, Figure 4, p. 4866). Thus, our data (and all geodetic measurements) are insensitive to small changes in fault slip at depth. Hence, for example, the 1906 triangulation data from Point Arena analysed by Matthews and Segall [1993] neither preclude nor require the slippage of 2 m or less below 10 km shown in the fault slip distribution obtained by them.

The assumption of spatially homogeneous elastic properties may introduce uncertainties into the derived slip distribution that are difficult to quantify. However, if these effects are significant they are likely to be consistently of one sign, leading to underestimates of the true slip. This follows because the likeliest variations in elastic modulii are an increase with depth and/or a decrease within the San Andreas fault zone. Both of these variations tend to concentrate straining due to fault slip closer to the fault than would be the case if elastic properties were spatially uniform [Rybicki and Kasahara, 1977]. Geodetic stations would then be displaced less than the amount computed using a unform half-space elastic model, resulting in an underestimate of fault slip.

Most of the survey data used here spans a significant time interval before and/or after 1906, and the inclusion of pre- or post-earthquake straining introduces another bias into calculation of coseismic slip. This can be estimated to first order by using recent geodetic measurements to establish the expected rates of inter-earthquake straining, and we have used such data to compute the magnitude of this effect and correct the observations where appropriate. The bias is again consistently of one sign, and neglect of interseismic straining leads to underestimation of coseismic slip. However, the effect is small, generally amounting to at most a few percent of the coseismic slip, and we have ignored it in most of our calculations. However, data from the northern region network span intervals as long as 60 years and we have explicitly corrected them for interseismic effects. Still, the correction is relatively small and the original 1906 slip estimate has been increased by less than 10%.

Then, with fault geometry fixed, given a set of observations di, (i=1, n) and model parameters mj, (j=1, m), the relation between model and data can be expressed as

di=Aij mj (1)

where Aij is an n x m matrix of known elements that depend upon fault geometry and network configuration. Surface displacements are obtained from expressions given in Okada [1985], and the form of the Aij terms that apply for triangulation data are derived in Thatcher [1979]. For dislocations in a homogeneous, isotropic elastic half-space with Poisson's ratio of 0.25 the expressions are independent of elastic modulii. Equation 1 can be solved by standard methods of linear inversion, yielding estimates of the unknowns mj [Jackson, 1972; Menke, 1984].

However, commonly not all model parameters are well-determined and the solutions to (1) may be unstable. In this case an important goal of inversion is to learn where fault slip is well-constrained (resolution) and within what limits it is determined (uncertainty) (Figure 2d). Slip uncertainty, the standard deviation for each model parameter, is directly obtained from (1) as a mapping of the data error onto the model. Model parameter constraint is measured via the resolution matrix.

The model resolution matrix R (Menke, 1984, p. 64) provides a useful means of characterizing the degree to which slip is constrained along fault strike. Each row of the resolution matrix corresponds to a single model parameter. The degree to which the elements of the row resemble a delta function provides a measure of the resolution of the corresponding model parameter. Thus, in the ideal case, each row element would be zero except for that element lying on the diagonal of the matrix, which would have a value of unity. Values of this diagonal element less than one indicate less-than-perfect resolution. The diagonal elements of R thus provide a convenient single-parameter characterization of model parameter resolution and we use them below for this purpose.

In ideal situations, each model parameter would be uniquely determined within modest uncertainties (Figure 2d, left). More commonly, as here, some parameters are resolved, others are not, and uncertainty varies widely from segment to segment (Figure 2d, right). Generally, the more parameters we wish to resolve, the larger is the uncertainty in each individual parameter (resolution-uncertainty trade-off). Deriving a satisfactory slip distribution thus involves making judgements about acceptable levels of slip uncertainty and examining the spatial resolution of slip that can be obtained under this constraint.

3. RESOLUTION OF 1906 EARTHQUAKE SLIP

In order to characterize the unique features of the 1906 earthquake slip distribution, we determine a range of solutions to (1) for successively greater numbers of independent parameters (eigenvalues) using the method of singular-value decomposition [Lanczos, 1961]. For each solution we examine the trade-off between resolution and uncertainty and monitor the resulting misfit between observed and model-computed data values. Based on the rate at which this misfit decreases with increasing eigenvalue we determined how many independent parameters should be estimated. Using the resolution-uncertainty analysis, we decided where fault slip is unknown and where it could be determined, finally resegmenting the model fault to determine slip on the optimum number of resolvable segments. Since this final step involves a least-squares inversion, each of the model parameters in this resegmented model is uniquely resolved.

To explore the resolution-uncertainty tradeoff we divided the San Andreas fault into 48 10-km-long segments (shown in Figure 1, listed in Table 3), following the mapped location of the fault in the region of the 1906 earthquake [Jennings, 1992]. To measure resolution we plot the diagonal elements of the resolution matrix for each solution versus the number of eigenvalues included in the solution (1 to 48). This is shown in color-coded format in Figure 3a. Figure 3b is a corresponding plot of slip uncertainty versus eigenvalue.

The color coding in Figure 3 has been chosen so that the red end of the spectrum corresponds to good resolution and low slip uncertainty. Poor resolution and large uncertainty are denoted by bluish colors. Note, however, that in Figure 3b the reddish colors at small eigenvalue (left third of graph) mostly reflect the small uncertainties associated with fault segments not resolved in the inversion. Such model parameters will be adjusted very little in the inversion process and are therefore assigned low uncertainties. When enough eigenvalues are included in the solution to resolve these parameters their uncertainties rise accordingly.

The misfit versus eigenvalue plot (Figure 4) defines a range of independent parameters that may be estimated but specifies no clearly preferred value. The misfit declines sharply to 14 eigenvalues, decreases more gradually between 14 and 27 eigenvalues, and is nearly level beyond. This suggests that 14 to 27 model parameters may profitably be estimated.

The gross pattern of constrained and unresolved segments is clearly discernible in Figure 3a. Between 14 and 27 eigenvalues, resolution is particularly good in the vicinity of the local networks at Point Arena (16-18), Fort Ross (22-24), Tomales Bay (27-30), and, to a lesser extent, Colma (36). For the same range in eigenvalues, resolution is quite poor on segments 1, 7-14, 20-21, 31, 41-42, and 46-47.

Precisely how many model parameters are inverted and which they should be depends on the preferred trade-off between resolution and uncertainty. Based on previous 1906 slip estimates [Thatcher, 1975; Thatcher and Lisowski, 1987] and on preliminary inversions carried out here, we expect slip to vary between about 2 and 8 m along fault strike. On this basis we decided that slip uncertainties up to about 1 m were acceptable but larger values were not. We were then able to use Figures 3a, b to select those fault segments on which slippage would be uniquely resolved, exclude segments with very poor resolution and/or large uncertainties, and combine several 10-km-long segments together to determine slippage within the prescribed 1 m limit.

The detailed rationale for our segment selection follows, listing resolved and unresloved segments and special cases, in each instance starting at the north end of the fault. Reference to the appropriate portions of the network/fault segment map (Figure 1) and the resolution/uncertainty plots (Figure 3) will complement the discussion included here. Resolved segments include: 2-6 (Shelter Cove); 16, 17, 18 (Pt. Arena); 22, 23, 24 (Ft. Ross); 27, 28, 29, 30 (Tomales Bay); 32-33; 36; 38, 39, 40 (Sierra Morena); 43-44; 45-48. Unresolved segments are 1, 7-15, 19-21, 25, 26, 31, 34, 35, 37, 41, 42. Segments 2-6 are combined into a single 50-km-long segment to increase resolution and decrease uncertainty below the 1 m limit. While finer subdivisions could still produce a nominally acceptable resolution/uncertainty tradeoff, the solutions themselves were oscillatory because of slip covariance between adjacent segments. Because one station of the Fort Ross net (Chaparral, see Map 25 in Lawson [1908]) was unstable, data from it were excluded and as a result segment 25 was not resolved. Segment 26 was excluded; while it was resolved uniquely beyond about 20 eigenvalues, its uncertainty lies just outside the 1 m limit. Segments 32 and 33 are resolved with less than 1 m uncertainty if combined into a single 20-km-long segment. For segment 35, slip uncertainty exceeds 1 m even for the 27 eigenvalue solution. Slip uncertainty on segment 40 lies just within the 1 m limit. Segments 43 and 44 are combined into a single 20-km-long segment and resolved. Segments 45-48 can be combined into a 40-km-long segment that is just barely resolved within 1 m; most of the resolution is contained in segments 45 and 48. There are a total of 18 resolved segments; together they comprise 60 % of the 1906 rupture.

4. PREFERRED MODEL OF 1906 EARTHQUAKE SLIP

The model construction proceeded in several steps. We started by separately determining slip on the resolved segments of each of the four local networks. Fixing slip on these 10 segments, we then used the data from the northern and southern regional nets to determine slip on the 8 remaining resolved segments. Slip on the unresolved segments was assigned by interpolating linearly between segments on which slip is known, initially using slip values from the local nets, then iterating the solution using values determined for the newly-resolved segments until convergence was achieved. The derived slip distribution is quite insensitive to the exact values of slip assumed on the unresolved segments, and variations of several meters have negligible effects on the results.

Statistics of model fits for each network are listed in Table 4. In none of them are data fit within their assigned errors, since reduced chi-squared values range from 1.3 to 4.1 (a perfect fit corresponds to a reduced chi-square of 1.0; statistically acceptable fits can have somewhat larger values that depend upon the number of data and degrees of freedom in the least squares procedure). Nonetheless, except for the Colma net, a very high percentage (89-99 %) of the angle change signal has been matched. The reason for the poor improvement in the reduced chi-squared value over data signal-to-noise for the Colma net is unknown; we do note, however, that only two closed triangles are available to estimate data uncertainties for this net and it is possible that our value is underestimated.

Table 2 lists individual angle changes computed for the final model and the normalized misfit for each observation. Also included is the 'data importance' of each angle change, which measures the degree to which individual data influence the derived model parameters (see Menke, 1984, pp. 62-64). The cumulative sum of these individual data importance values is equal to the number of unknowns solved for, 18 in the least squares inversion for the preferred slip model.

Computed angle changes misfit by more than two standard deviations are listed in Table 5. Most of the misfit in the local nets comes from angles with large changes (20-350 arc seconds), significant data importance, and high signal-to-noise ratio. While most of this signal can be explained, it is typical that 10-20% cannot be matched by the simple elastic half space dislocation models applied here (e.g., Thatcher, 1979). A second class of misfit occurs for angle changes that are quite insensitive to the chosen model parameters; their data importance is close to zero and fits cannot be improved by model parameter adjustments that are consistent with more important data.

The preferred model is shown in Figure 5a and slip values are listed in Table 3. Since the best-fitting model still did not match data within its standard errors, we accounted for this shortcoming by recomputing slip uncertainties, multiplying each by the appropriate square root of the reduced chi-squared value. It is these larger uncertainties that are plotted in Figure 5a and listed in Table 3. In general, model slip values have greater spatial resolution than our previously published estimates [Thatcher, 1975; Thatcher and Lisowski, 1987] but agree within uncertainties in areas of overlap.

Geodetic estimates of moment based on our slip distribution are consistent, within uncertainties, with values obtained previously using long-period surface waves. Summing slip on the resolved segments of the model of Figure 5a yields a geodetic moment of 4.0 x 10 20 N-m; including the interpolated segments increases the value to 7.5 x 10 20 N-m. The geodetic moment is directly proportional to the assumed fault depth, fixed at 10 km in our model, and uncertainty in this depth and probable along-strike variations in it introduce an additional uncertainty of 10-20% into the moment estimate. Thatcher [1975] and Wald et al. [1993] used 1906 surface-wave amplitudes to obtain seismic moments of 4.0 and 4.7 x 10 20 N-m respectively. Because the 1906 rupture length was more than 450 km it is possible that even the 100 second waves used in these two studies are not long period enough to measure the total seismic moment. The entire range of moment estimates corresponds to moment magnitudes between 7.7 and 7.9.

The high slip value of 8.6 m for the segment centered on Shelter Cove is newly determined. As Figure 3a shows, slip resolution is rather evenly distributed along this 50-km-long segment. Slip on the segment is constrained by the first 9 angle changes listed in Table 2. Because these angle changes cover a considerable interval spanning 1906 (1882-1928/1942) we used modern (post-1981) trilateration and GPS observations at these same stations [M. Lisowski, unpublished data, 1994] to compute angle changes due to interseismic strain accumulation during this 46-60 year time interval. In all cases these effects were small (<3 arc sec), and although we corrected for them (and list these corrected values in Table 2), their inclusion has only a small effect on the 1906 slip estimate. Several meters of postseismic buried slip occurred on the San Andreas fault in the several decades following 1906 [Thatcher, 1975]. However, the effects of such motions on the northern region network would also be small compared to coseismic angle changes and have not been included in our calculations (see Thatcher, 1975, Figure 4, for an illustrative example). The 5 data that are most sensitive to 1906 slip (Data Importance>0.10) have high signal-to-noise ratios (7-13) and are fit well by the model, with only one of the 5 calculated angle changes disagreeing by more than one sigma.

Extensive ground deformation was observed at several localities northwest of Shelter Cove in 1906 but no surface fault offsets were reported [Lawson, 1908, pp. 55-58 and Plate 31A]. Brown [1995] shows that several of these sites are located on the currently active trace of the San Andreas fault and argues persuasively that they represent primary surface faulting. However, if fault slip at seismogenic depths indeed exceeded 8 m, it is perhaps surprising that no measurements of surface offset were obtained. It is possible that 1906 slippage also occurred on a more westerly, offshore strand of the San Andreas fault in this region. However, if this were the case even more cumulative 1906 slip would be needed to match the triangulation data.

Figure 5b shows a comparison between our preferred 1906 slip distribution and that estimated by Wald et al. [1993] using historical body-wave seismograms of the earthquake. Given the large error bars of the seismologic estimates, the two independent slip distributions are not greatly inconsistent within their region of overlap. However, it does appear that the seismically-estimated values do not track the along-strike slip variation determined here. The match might be improved by shifting the seismologic profile southeast by 30-50 km, permissible because its exact position is tied to the instrumental (but rather uncertain) epicenter of the 1906 earthquake just offshore of San Francisco [Bolt, 1968; Boore, 1977]. However, the 1906 seismogram analysis did not predict any significant slip northwest of Point Arena, where the intensity of ground shaking in 1906 was high (see Figure 1) and our geodetic slip model shows large amounts of coseismic slip near Shelter Cove.

The precise cause of this discrepancy is uncertain. However, because the S-waves analysed by Wald et al. [1993] contain only 20-30% of the measured surface wave moment, it is clear that a major component of the static slip will be missed. Furthermore, Wald et al. observed there were only two significant pulses on the seismograms and found they could be matched with two events, one located at the epicenter, where motions initiated, and a second, larger event 38 seconds later. Assuming a rupture propagation velocity of 2.7 km/sec, no significant motion more than about 100 km from the epicenter was needed to match the S-wave data. With the epicenter fixed near San Francisco it is thus easy to see why Wald et al. required no slip northwest of Pt. Arena to fit their data.

This comparison illustrates the strengths and shortcomings of each dataset in defining earthquake features. The geodetic data are quite sensitive to the cumulative fault slippage but provide no information on rupture dynamics. In contrast, seismic body-wave amplitudes are more sensitive to acceleration and deceleration of rupture than to the final static slip values. It may be that the rise time of dynamic slippage on the 150-km-long fault segment northwest of Pt. Arena was slow enough that radiated seismic body-waves were too long period to be recorded above ground noise levels within the pass-band of the early seismographs used by Wald et al. [1993]. If this is so, we should not invariably expect even qualitative agreement between these two slip estimates, and the seismic and geodetic data may be providing complementary rather than contradictory information on the 1906 rupture.

The geodetic slip profile is compared with 1906 surface fault offsets in Figure 5c. In this case there is general agreement in the gross character of the along-strike varaitions in fault slip, although the surface offset data show considerable local scatter and tend to be 20-50% less than the geodetic estimates on the same fault segments. The underestimate is particularly marked on the southern 90 km of the 1906 rupture. On only one of the resolved segments (#34, near 350 km on Figure 5c) does the surface fault offset exceed the geodetically-estimated slip. As Thatcher and Bonilla [1989] showed, this feature is not peculiar to the 1906 earthquake. For 9 events with both geodetic and surface slip estimates, in no case did surface slip exceed geodetic values. Although the comparisons showed considerable scatter, the ratio of surface offset to geodetic slip averaged 80%, similar to that shown in Figure 5c.

5. 1906 EARTHQUAKE SLIP NEAR 1989 LOMA PRIETA EARTHQUAKE SOURCE REGION

The occurrence of the 1989 M=6.9 Loma Prieta earthquake near the southern end of the 1906 rupture raises important questions about earthquake recurrence and long-term forecasting that have not yet been resolved. Prior to 1989 several investigators estimated a high seismic potential for a M=6.5 to 7 event during the subsequent several decades on the southern-most 80 km of the 1906 rupture [Lindh et al., 1982; Lindh, 1983; Sykes and Nishenko, 1984; Scholz, 1985]. Thatcher and Lisowski [1987], on the other hand, re-examined the 1906 earthquake slip data on which the forecasts depended and suggested the seismic potential was considerably lower. The occurrence of the 1989 earthquake may well have confirmed the forecasts of high long-term probability (e.g., U. S. Geological Survey Staff, 1990) and refuted the lower estimates. However, some doubts remain. First, in order to be sure the 1989 event was successfully forecast the matter of the disparate 1906 earthquake slip estimates remains to be accounted for. Furthermore, differences between the 1906 and 1989 events in the vicinity of Loma Prieta [Kanamori and Satake, 1990; Segall and Lisowski, 1990] and consideration of the relation between long-term vertical movements and 1989 coseismic uplift [Anderson, 1990; Valensise and Ward, 1991] have raised new questions that bear on the long-term earthquake forecasts. These factors suggest that new investigations are needed to better understand the relation between the 1906 and 1989 earthquakes. In this section we begin this process by examining the available constraints on 1906 slip and fault geometry.

Figure 6 and Table 6 show the range of fault geometries and corresponding values for 1906 slip that are consistent with our triangulation data in the vicinity of Loma Prieta Mountain. The conjectured geometry of faults is shown schematically in both map view and cross-section in the top left panel of Figure 6 and this format is repeated for the 9 models shown in subsequent panels. Two faults are considered, a vertical strike-slip fault whose surface intersection is the mapped trace of the San Andreas fault [Sarna-Wojcicki et al, 1975], and a second fault, here termed the Loma Prieta fault, that is inclined 70° southwest and corresponds (at depths of 5-18 km) to the rupture plane of the 1989 earthquake [Dietz and Ellsworth, 1990]. The surface trace of this fault lies close to the Sargent fault, which is largely strike-slip in this region.

The models are progressively more complex and test contrasting scenarios for 1906 earthquake slip near the 1989 source region. In each, the faults are constrained to extend to 20 km depth; if slippage were confined to shallower depths its magnitude would invariably be greater (see for example Table 2 of Thatcher and Lisowski, 1987). Models 1 and 2 have single strike-slip segments 80 km long; comparing the two results shows that in order to satisfy the same data, faults with southwest dip require 20% more slippage than vertical faults. All subsequent models (3 -9) fix slip on the northern and southern 20 km to 3.5 and 2.0 m respectively, values close to those determined by inversion for the preferred model of Figure 5. Even several meter changes in these fixed values have only a very minor effect on the inverted slip on the central 40-km-long segment. Models 3 and 4 are segmented versions of 1 and 2 and show similar effects. Model 5 simulates slippage on a shallow (0-5 km depth) San Andreas fault segment coupled to a deeper (5 - 18 km) Loma Prieta fault segment. Model 6 confines slip to only the 5 - 18 km Loma Prieta fault, and very large amounts of strike-slip motion (12.1 m) are required to satisfy the data. Model 7 is similar to Model 4 but dip-slip motion is permitted on the inclined fault. Models 8 and 9 test whether slippage on our 'Sargent fault' might satisfy the data, with slip fixed to 0 m (Model 8) and 1.5 m (Model 9) on the shallow portion (0 - 5 km) of the San Andreas. Because the surface trace of the Sargent fault lies closer to the Loma Prieta triangulation station than does the San Andreas, about 25% less strike-slip motion is required to satisfy the data (compare Model 7 and Model 8). The results of the Model 9 calculation show that if San Andreas slip at depth is fixed to the 1.5 m of surface slip inferred by Lawson [1908] at Wrights Tunnel, the cumulative right-lateral slip across the San Andreas and Sargent faults could have been as low as 2.3 m.

The results of these tests indicate that 1906 strike-slip motion at depth on the San Andreas fault zone near Loma Prieta lies between 2.3 and 3.1 m, and this conclusion follows whether the fault is vertical (Models 1,3), inclined (2, 4, 7), or a mix of the two (5). If strike-slip were confined entirely to the Sargent fault (Model 9) the cumulative right-lateral component could be as low as 2.2 m. These conclusions depend entirely on the motion of the triangulation station at Loma Prieta Mountain. However, as Figure 1 shows, the position of this station is controlled by sighting from many surrounding stations, and 22 independent angle changes constrain its coseismic motion. As shown by Hayford and Baldwin [1908] the 1906 motion of Loma Prieta derived by a network adjustment of all data indicates displacement of 1.0 m nearly parallel to the local trend of the San Andreas fault. This was reconfirmed by Segall and Lisowski [1990], who also showed that in 1989 this same station was displaced by 0.2 m in a direction oriented 40° clockwise from the strike of the San Andreas fault. Inversion of a large suite of leveling, trilateration and Global Positioning System data indicate 1989 slippage occurred between 5 and 18 km depth on a 70° southwest-dipping fault plane about 40 km long, with 1.6 m of strike-slip and 1.2 m of reverse dip-slip motion [Lisowski et al., 1990; Marshall et al., 1991], and variable slip models concentrate slip below 10 km depth [Arnadottir and Segall, 1994].

In the analysis of Segall and Lisowski [1990] the 1906 geodetic data were used to invert for displacements of 10 stations rather than slippage on one or a few faults. As a result, the derived 1906 station displacements (17 independent components) each have rather large uncertainties. Sykes (1996, p. 3736) noted this and argued that the geodetic data are therefore too inaccurate to apply in estimating 1906 slip and consequent longterm hazard along the San Andreas fault near Loma Prieta. However this is not the case, because to obtain slip uncertainties the triangulation data must be applied directly to slip estimation without first referring to derived station displacements or their uncertainties. Following this procedure, as we have done here, yields the approproiate slip uncertainties, typically ±0.3 m or better.

The comparisons between 1906 and 1989 station displacements [Segall and Lisowski, 1990] and corresponding fault models highlights the undoubted differences in fault geometry and slip between the two events. A 1989-type event occurring in 1906, or in 1865, when a poorly-documented M~6 earthquake occurred nearby [Toppazada et al., 1981], might well have gone undetected. Such an event would have produced angle changes too small to have been clearly observed at Loma Prieta (compare Model 6, Figure 6). However, the 1906 data clearly require right-lateral strike-slip motion in excess of the 1.5-1.8 m offsets observed in Wrights Tunnel, and this slippage must have occurred on faults that did not rupture in 1989.

6. RECURRENCE IMPLICATIONS OF 1906 EARTHQUAKE SLIP DISTRIBUTION

The matter of the long-term forecast of the Loma Prieta earthquake remains controversial, with both advocates for its success [U. S. Geological Survey Staff, 1990; Working Group on California Earthquake Probabilities, 1990; Ellsworth, 1991; Sykes, 1996] and skeptics [Kanamori and Satake, 1990; Segall and Lisowski, 1990; Savage, 1991]. While Lindh et al. [1982] clearly showed that pre-1989 seismicity defined a conspicuous gap, and this gap was subsequently filled by the 1989 event, the seismicity data alone provide no information on the timing of a gap-filling event. In our view, all of the event timing estimates are suspect in two ways. First, we doubt that it is valid to use earthquake slippage on one fault (1906 slip on the San Andreas at Wrights Tunnel) to forecast an event on another (the inclined oblique slip fault that ruptured in the 1989 earthquake). While the geodetic data cannot preclude the occurrence of a 1989-type event in 1906, we note that right-lateral surface slip occurred on the San Andreas fault in 1906, this slip was the basis for the forecasts of high earthquake potential, and this fault did not rupture in 1989. Second, even if one were to accept the validity of this methodology, the 1906 slip at seismogenic depths on the San Andreas determined from our analysis significantly exceeds the surface slip at Wrights Tunnel upon which the estimates of high seismic potential were based.

Several of these issues are illustrated in Figure 7. It shows that a number of characteristics of the 1989 earthquake were unexpected and differed from those features of the 1906 earthquake upon which the long-term forecasts were based. In 1906, right-lateral fault offsets were recorded at Wrights Tunnel and several other localities on the San Andreas fault to the southeast [Lawson, 1908; Prentice and Schwartz, 1991; Prentice and Ponti, 1996]. Primary right-lateral surface faulting was absent in 1989, although a number of features of ground failure observed in 1989 mimicked those documented in the same localities after the 1906 earthquake [Ponti and Wells, 1991]. Models based on geodetic survey measurements indicate fault slippage came no closer than 4 km to the earth's surface in 1989 [Lisowski et al, 1990; Marshall et al, 1991], with significant slip being concentrated below about 10 km depth [Wald et al., 1991; Arnadottir and Segall, 1994]. The 1989 fault plane also dips 60°-70° to the southwest and movements have a significant component of reverse dip-slip motion, resulting in substantial coseismic uplift of the southern Santa Cruz Mountains [Marshall et al., 1991]. Comparison of the uplift pattern of deformed 80 Kyr marine terraces with the 1989 vertical movements in the same vicinity has suggested that reverse dip-slip events like the 1989 earthquake are comparatively rare, recurring on average every ~800 years [Anderson, 1990; Valensise and Ward, 1991].

Figure 7 also shows that the maximum surface offset in 1906 (1.5 m) is significantly less than the geodetically-estimated slippage at seismogenic depths, 2.4 m or more [Thatcher and Lisowski, 1987]. Although other factors contribute, 1906 fault slip near Loma Prieta is the single most important determinant of long-term hazard on this fault segment: estimates of high seismic potential [Lindh et al., 1982; Lindh, 1983; Sykes and Nishenko, 1984; Scholz, 1985] relied on a 1.5 m fault offset of Wrights Tunnel (recently revised upwards to 1.7-1.8 m by Prentice and Ponti, 1996) while the lower estimate used the geodetically-derived 1906 slip [Thatcher and Lisowski, 1987].

Following the Loma Prieta earthquake, a second group of scientists was assembled to review and if necessary revise the findings of the 1988 Working Group Report relevant to the San Francisco Bay region [Working Group on California Earthquake Probabilities, 1990]. Athough they endorsed the success of the earlier working group (and, implicitly, the assessments of Lindh, Sykes and Nishenko, and Scholz) in forecasting the 1989 earthquake, they provided no detailed rationale for this endorsement nor did they address either the differences between 1906 and 1989 events or the discrepancy between 1906 surface offset and geodetically-estmated slip. Rather, the Working Group argued that the 1906 and 1989 events were sufficiently nearby that their causative faults would experience similar inter-earthquake stress accumulation and similar stress release when either slipped, making any event-to-event differences unimportant for longterm forecasting. We disagree.

Minor faults lying close to but off the main trace of the San Andreas fault are a common feature throughout its active length, particularly where there are local changes in fault strike like those occurring in the Santa Cruz Mountains and in the Transverse Ranges of southern California [Jennings, 1992]. These faults are rarely vertical, purely strike-slip, or precisely parallel to the main San Andreas trace. Their location and geometry often indicate that, like the Loma Prieta fault, they have a strong dip-slip component and originated in order to relieve stresses different from those released by fault slippage on the main trace. Because of their differing orientation, location, and sense of slip, the rate of strain accumulation across these faults will also differ from that across the main trace. The stress state across the minor faults is certainly affected by major events on the main fault (and vice versa). However, because of the proximity of the faults, such stress transfer will be very dependent on the exact geometric relation of major and minor faults and the detailed slip distribution of the perturbing earthquake. The result of such differences will be a very complex relation between the earthquake cycle on the main San Andreas and the minor faults that lie off its trend, and slip history on one fault will be a poor guide in forecasting slip history on another.

However, the main focus of the 1990 Working Group was not on the validity of the previous forecasts but on the future earthquake potential of San Francisco Bay area faults. They used a logic tree approach and a poll of working group members to determine weights for three distinct recurrence scenarios and to assign 1906 fault slip values to four Bay area San Andreas fault segments. On the southern 120 km of the 1906 rupture, these values are 70-75% of the geodetically-estimated slip and their assigned standard errors are about twice the geodetic slip uncertainties. The polling method does not address the scientific issues themselves but is an accepted practice for reaching concensus judgements when opinions differ. While respecting the concensus opinion of the Working Group we disagree with the results--it seems to us that the geodetic slip values are correct or they are not, and cannot be , say, 70% right and 30% wrong. The Working Group also provided no separate scientific rationale for its choices of 1906 slip values and their assigned standard errors. Strict usage of the geodetic slip values and their standard errors would decreases the 30-year probabilities for the two central Bay area San Andreas segments by 5-10% over those reported in the 1990 Working Group report.

Several factors suggest that current seismic risk may be significant on the upper ~10 km of the San Andreas fault near to and southeast of Loma Prieta Mountain. Analyses of seismic [Wald et al., 1991] and geodetic [Arnadottir and Segall, 1994] data indicate there was no significant right-lateral strike-slip fault motion above ~10 km in 1989. However, right-lateral slip of at least 1.6 m below this depth would have incremented stresses at shallower depths on the San Andreas fault near Loma Prieta, advancing the occurrence time of the next strike-slip event (a similar point has been made by Segall and Lisowski, 1990, p. 1243). Slip rate estimates for the Bay area segments of the San Andreas fault are very uncertain, with credible values ranging from 15 to 25 mm/yr [Working Group on California Earthquake Probabilities, 1990, pp. 14-15]. However, if local fault slip rates are as high as the 24±3 mm/yr value obtained recently by Niemi and Hall [1992] near Tomales Bay, much of the 1906 slip deficit near Loma Prieta would now have been recovered. Farther southeast the 1906 slip estimate is quite uncertain (2.3±1.1 m) but the seismic risk there could be comparably high. Such conjectures of risk are very dependent on poorly-constrained parameters as well as on matters which remain controversial. Thus, while insufficient to form a basis for formal assignments of risk or public policy decisions, these considerations suggest a potential hazard and underline the need for better constraints on the factors controlling earthquake recurrence on the southern ~80 km of the 1906 rupture.

7. DISCUSSION

Slip Inversion Methodology

The 1906 earthquake slip distribution obtained here purposely emphasizes its best resolved characteristics and suppresses unresolved or doubtful features. We determined along-strike variations only when they were well-constrained by observations and made no attempt to estimate slip variability as a function of depth. This approach runs counter to recent trends in applying both geodetic and seismologic data to estimating earthquake slip, where continuous distributions are determined over the entire coseismic fault plane (e. g., Archuleta, 1984; Hartzell and Heaton, 1983; Segall and Harris, 1987; Wald and Heaton, 1994).

Although modern geodetic data are often clearly superior to the classical measurements used here to characterize 1906 fault slip, we suggest that our minimalist approach may find useful application in modern data inversions as well. However accurate and complete geodetic survey networks may be, measurements are necessarily confined to the earth's surface and static strains decay as the inverse cube of distance from the deformation source. As a result, slip resolution decreases dramatically as a function of depth (see Thatcher, 1975, Figure 4). Furthermore, even for recent earthquakes in well-monitored regions, geodetic coverage is not areally uniform and gaps are inevitable (e.g., Freymueller et al., 1994), producing spatial variations in slip resolution similar to those documented here. Inversion methods which generate smoothed, continuous slip distributions fail to account for such variations in resolution, producing misleadingly complete images of the coseismic slip pattern. In addition, such methods estimate many more slip values than can be uniquely resolved and thus provide no useful estimates of uncertainties in the magnitude of the derived slip [Segall and Harris, 1987; Wald and Heaton, 1994; Freymueller et al., 1994].

Of course no single slip distribution derived from geophysical data can be truly unique or free from subjective bias. For example, although the least squares solution we obtain is nominally unique, the choice of resolved, combined, and unresolved segments depends upon acceptable levels of slip uncertainty and other subjective factors. However, our approach has the advantage of providing a compact visual representation of the resolution/uncertainty tradeoff (Figure 3) which facilitates the construction of a conservative image of the slip distribution (Figure 5a).

Seismic Body-Wave, Geodetic, and True Slip Distributions

Comparison of the geodetically-estimated slip distribution with that obtained from 1906 earthquake seismograms (Figure 5b) shows that each data type is insensitive to coseismic slip over significant portions of the 1906 rupture. We have estimated slip on only 60% of the fault. However, given the absence of geodetic stations near a 90-km-long fault segment between Shelter Cove and Point Arena as well as adjacent to several shorter fault segments farther southeast, the existence of these blind spots is not too surprising. Less expected are the large reaches of the 1906 rupture over which seismic body-waves provide no information on either dynamic or static fault slip. As Wald et al. [1993] emphasize, the 1906 seismograms are much simpler than might have been expected for a 470-km-long rupture. The main shear-wave arrivals consist of two pulses separated in time by about 40 seconds. Together, these S-waves account for only 20-30% of the total seismic moment measured independently using long-period surface wave amplitudes. This moment deficiency, along with the geodetic-seismic slip comparison shown in Figure 5b, indicates that about 80% of the 1906 fault is unsampled by the S-waves, including several localized zones of high static slip pinpointed by the geodetic inversions.

It is unclear whether disagreements between the body-wave estimates of fault slip and the true slip distribution are fairly common or comparatively rare. For several recent M=6-7 California earthquakes there was substantial agreement among slip distributions (and seismic moments) obtained using distant body-wave seismograms, local strong motion seismograms, and geodetic survey measurements [Hartzell and Heaton, 1983; Wald et al., 1991; Wald and Heaton, 1994]. On the other hand for great earthquakes, discrepancies between seismic moments obtained from body- and surface-waves like that seen for the 1906 event are the rule rather than the exception [Kikuchi and Fukao, 1987]. Although geodetic or other independent slip estimates are lacking for the earthquakes studied by Kikuchi and Fukao, the body-wave / surface-wave discrepancy indicates that much of the rupture process in these events occurs without generating the impulsive arrivals recorded on body-wave seismograms.

The geodetic/seismic comparison for the 1906 earthquake shows that neither the amplitudes nor the along-strike variations in slip can be recovered from the body-wave seismograms. It seems unlikely that this disagreement is due to a deficiency peculiar to the early seismographs that recorded the 1906 event. Although the magnification of these instruments is much less than that of modern systems, the 1906 S-waves were recorded with high signal amplitudes, and the frequency response of these instruments is comparable to that of the long period seismographs of the World Wide Standard Seismographic Network (WWSSN) deployed in the early 1960's and used to study many of the great earthquakes included in the Kikuchi and Fukao [1987] study.

8. SUMMARY

Standard linear inversion methods have been applied to examine the trade-off between spatial resolution of 1906 coseismic fault slip and uncertainty in its magnitude. Angle changes from repeated triangulation surveys carried out from as early as 1853 and as recently as 1942 have been used, correcting for interseismic deformation where necessary. The networks, designed for geodetic control rather than fault zone monitoring, are of varying aperture and station density, and are not uniformly distributed along the fault, posing special problems for application of slip inversion methodologies.

Slip resolution is very uneven along the 1906 rupture. It is excellent on 30-40 km-long sections near Point Arena, Fort Ross, and Tomales Bay, where dense local nets are located. Two large-scale regional networks cover the northern 170 km and southern 230 km of the 1906 rupture with varying resolution, and slip magnitude is determined within uncertainties of ±1.0 m along 60 % of the fault. 1906 coseismic slip is essentially unknown along the remaining 40 % of rupture. In particular, blind spots exist offshore southeast of Shelter Cove (90-km-long section); between Point Arena and Fort Ross (30 km); offshore of San Francisco (20 km); and on a 20-km-long segment on the southern San Francisco Peninsula. Although the procedure adopted here produces an incomplete image of the slip distribution, the result may more accurately reflect the true slip constraints on the fault plane than do smoothed, continuous slip distributions that fail to account for the necessarily limited resolution of all geodetic data.

Notable features of the derived slip distribution include maxima of 8.6 m near Shelter Cove and 7.5 m in Tomales Bay and a general decrease in slippage on the southern 150 km of rupture, from 4.5 m near Mt. Tamalpais to about 2.7 m in the vicinity of Loma Prieta. Application of a range of vertical and inclined fault geometries to the Loma Prieta region indicates 1906 right-lateral strike-slip motion at seismogenic depths on the San Andreas fault lies between 2.3 and 3.1 m, significantly greater than the 1.5 m of surface slip measured nearby upon which several long-term earthquake forecasts were based. This factor, along with the absence of 1989 slippage where 1906 surface slip was used to make the forecasts, casts doubt on some claims of success in predicting the 1989 M=6.9 Loma Prieta earthquake. However, right-lateral slip below ~10 km on the inclined fault that ruptured in 1989 may have incremented stresses on the overlying San Andreas, increasing the seismic potential on this fault as well as on the adjacent 30-km-long segment of the San Andreas to the southeast.

Interpolating linearly across the unresolved segments in the geodetically-estimated 1906 slip distribution and summing slip yields a geodetic moment of 7.5 x 10 20 N-m. This value agrees acceptably with seismologic moment estimates and corresponds to a moment magnitude of 7.9. However, body-wave estimates of slip distribution by Wald et al. disagree significantly with the geodetic values obtained here. This disagreement is at least in part due to the fact that the S-waves sample only 20-30% of the total seismic moment and thus cannot faithfully register the entire slip distribution. However, the short duration of the S-wave seismograms is surprising for such a long rupture, indicating that impulsive body-waves were not radiated to teleseismic distances from a significant portion of the 1906 fault. This comparison suggests that the geodetic and seismic data are complementary, with seismic data sampling dynamic features of the earthquake rupture while the geodetic measurements constrain static fault offsets.

Acknowledgments: Many colleagues provided stimulating discussion on this work and its possible implications. Careful and constructive reviews of the manuscript were provided by J. H. Dieterich, D. J. Wald, R. M. Russo, P.R. Lundgren, and P. M. Davis.

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Figure 1: Region of the April 18, 1906 San Francisco earthquake. 1906 rupture and 10-km-long segments of the San Andreas fault used in fault slip modeling are indicated by the continuous open-barred line. Geodetic networks used in the slip estimation are shown by connected line segments, while detailed local nets at Point Arena, Fort Ross, Tomales Bay, and Colma are denoted by shading. The zone within which ground shaking exceeded Modified Mercali Intensity VII is bounded by heavy dashed lines, taken from Lawson [1908]. SM=Sierra Morena, LP=Loma Prieta, SJB=San Juan Bautista.

Figure 2: Schematic illustration of main elements of analysis to determine fault slip (uj) using geodetic data.

Figure 3: Graphical representation of (a) model parameter resolution and (b) slip uncertainty for the 48 fault segments and the geodetic survey networks shown in Figure 1. The vertical axis of each plot shows position along the 1906 rupture (or, equivalently, 10-km-long fault segment number) from northwest (top) to southeast (bottom). The horizontal axis shows the number of linearly independent model parameters (eigenvalues) used in the calculation of (a) resolution and (b) slip uncertainty, which are shown color-coded in each graph.

Figure 4: Cumulative rms misfit between observed and model-predicted angle changes plotted against the number of independent parameters (eignvalues) used in each inversion. Model elements and data are, respectively, the 48 fault segments and the 181 angle changes from the geodetic networks shown in Figure 1. As discussed in the text, the discrepancy at high eigenvalue between the cumulative data variance and total data error results primarily from the 10-20% misfit of a small number of data with very high signal-to-noise ratio. These observations are listed in Table 5.

Figure 5: (a) Geodetically-estimated 1906 coseismic slip distribution, compared with (b) slip estimated from teleseismic S-wave seismogram analysis [Wald et al., 1993] and (c) surface fault offsets [Lawson, 1908].

Figure 6: Models of 1906 slip in the vicinity of Loma Prieta determined by a subset (22 angle changes) of the triangulation data. Top left-hand frame shows cross-section and map views of region near Loma Prieta Mountain (triangle), with star at 1989 hypocenter and solid line denoting 1989 aftershock zone. Frames numbered 1 to 9 schematically show corresponding cross-section and map view model slip distributions that satisfy the 1906 triangulation data. Models 1 and 2 determine slip on a single 80-km-long segment extending from Black Mountain (west of Palo Alto) to San Juan Bautista. All other models fix slip at 3.5 m and 2.0 m on two ~20-km-long segments adjacent to Loma Prieta and determine slip on a ~40-km-long segment centered on Loma Prieta. One standard deviation uncertainties in slip determinations are shown next to the map view fault segments. Exact slip values and further model details are listed in Table 6.

Figure 7: Schematic cross-section parallel to the San Andreas fault showing 1906 and 1989 earthquake slip zones. Depth of slippage in 1906 is uncertain but 2.4 m estimate assumes slip extends from surface to 20 km depth (Model 1, Figure 7). Star locates hypocenter of 1989 event at 18 km depth and oblique slip of 2.0 m is that estimated by Lisowski et al. [1990]. 1906 slippage of 1.5 m at Wrights Tunnel is value used prior to 1989 by several investigators to estimate high probability for a M=6.5-7.0 earthquake on this segment of the San Andreas fault.

Table 1: Triangulation stations for networks used in 1906 earthquake slip estimation.

Table 2: Angle change data used in 1906 earthquake slip estimation. Each observed angle is defined by three triangulation stations, numbered as in Table 1, with the second number listed corresponding to the station from which the angle was turned, clockwise from the first station number to the third. Calculated angle changes and data importance are for the preferred slip model listed in Table 3 and illustrated in Figure 5.

Table 3: 1906 coseismic slip model parameters. Geographic co-ordinates of northern and southern segment ends are listed sequentially (the northern end of one segment is the southern one of the next, except at the extremities of the rupture). Inverted slip values and standard deviations for preferred least squares model are listed. Segments where slip is fixed in the inversion are denoted by slip uncertainties of zero and the designation F in the Network/Character column. Inverted segments are denoted by I, with an asterisk (*) to indicate where adjacent 10-km-long segments have been combined into a single longer segment for the least squares inversion (each of these combined 10 km segments thus has the same slip and standard deviation). Networks constraining slip on each segment are indicated in the last column, and names corrspond to those shown in Figure 1 and listed in Table 4.

Table 4: Model and network statistics.

Table 5: Angle change data misfit by more than two standard deviations by the preferred 1906 slip model.

Table 6: Alternative 1906 earthquake slip models in the vicinity of Loma Prieta Mountain. See Figure 7 for visual presentation of these same results.