Source parameters for the 1952 Kern County earthquake, California : A joint inversion of leveling and triangulation observations

Coseismic leveling and triangulation observations are used to determine the faulting geometry and slip distribution of the July 21, 1952, Mw 7.3 Kern County earthquake on the White Wolf fault. A singular value decomposition inversion is used to assess the ability of the geodetic network to resolve slip along a multisegment fault and shows that the network is sufficient to resolve slip along the surface rupture to a depth of 10 km. Below 10 km, the network can only resolve dip slip near the fault ends. The preferred source model is a two-segment right-stepping fault with a strike of 51° and a dip of 75° SW. The epicentral patch has deep (6–27 km) left-lateral oblique slip, while the northeastern patch has shallow (1–12.5 km) reverse slip. There is nearly uniform reverse slip (epicentral, 1.6 m; northeast, 1.9 m), with 3.6 m of left-lateral strike slip limited to the epicentral patch. The seismic moment is M0 = 9.2 ± 0.5 × 1019 N m (Mw = 7.2). The signal-to-noise ratio of the leveling and triangulation data is reduced by 96% and 49%, respectively. The slip distribution from the preferred model matches regional geomorphic features and may provide a driving mechanism for regional shortening across the Comanche thrust and structural continuity with the Scodie seismic lineament to the northeast.


Introduction
The Kern County earthquake was one of the largest earthquakes in California during the twentieth century (M,• 7.7, Mw 7.3 [Richter, 1955;Ben-Menahern, 1978]), second only to the great 1906 San Francisco earthquake. The 1952 event ruptured 60 km of the White Wolf fault, north of the junction of the San Andreas and Garlock faults, and near a restraining bend in the San Andreas fault (Figure 1). Even though this earthquake was one of the most well-studied events in southern California at the time [Oakeshott, 1955], nearly 30 years elapsed until the first rigorous geodetic source models were published [Dunbar et al., 1980;Stein and Thatcher, 1981]. Unfortunately, differences in the geometry and slip distribution between these models provide conflicting interpretations on the relationship between the White Wolf fault and a number of other active faults and structures throughout the region.
I present a simple source model for the Kern County earthquake based on a comprehensive analysis of the coseismic triangulation and leveling observations. This study differs from previous coseismic analysis by including a larger set of triangulation observations and by directly inverting the geodetic observations rather than using forward modeling techniques to match shear strain estimates and uplift patterns. Furthermore, I examine where the geodetic observations can resolve slip and, more importantly, where the resolution is poor and slip cannot be determined. I first constrain the geometry of the fault model by combining the geodetic data with aftershock locations and then invert the observations, in a least squares sense, to estimate the This paper is not subject to U.S. copyfight. Published in 2001 by the American Geophysical Union.

Paper number 2000JB900315.
slip distribution along the White Wolf fault. Finally, I compare the results with previous studies and discuss the implications in a regional context.

Triangulation
This study uses coseismic angle changes for the triangulation array that spans the White Wolf fault (Figure 2 and Table l a). The U.S. Coast and Geodetic Survey (USCGS) collected over four decades  of triangulation data throughout the Big Bend region of the San Andreas fault [Thatcher, 1979;Dunbar et al., 1980;Thatcher, 1981' King andSavage, 1984;Eberhart-Phillips et al., 1990;Snay et al., 1996;Bawden et al., 1997], including two surveys that bracket the July 21, 1952 mainshock. Two months prior to the earthquake the USCGS completed a comprehensive, 6-month-long survey of the triangulation array that spans the southern half of the White Wolf fault; this array was again reoccupied beginning 2 months (September 1952 to January 1953) after the mainshock. Both the preseismic and postseismic surveys were conducted to first-order tolerances, where the standard error, o, assigned to each measurement was based on the consistency of the angles turned during each setup and the consistency among different setups at the same station. The standard error values used in this study are <0.8 arc sec.
The 143 coseismic angle changes were calculated by differencing 654 preseismic and postseismic angles (Table lb and Figure 2). If repeated measurements were made in either epoch (1951-1952 or 1952-1953), then the angle was determined by averaging each observation, weighted by its standard error. To assess the quality of the data, I evaluated the misclosures for monument combinations that formed a closed triangle, a total of 16 closed triangle ( The signal available for modeling is best described by the signal-to-noise ratio (S/N). The signal represented here is the coseismic angle change, and the standard deviation of each angle change is set to 1.18 arc sec. The standard deviation is based on the weighted average of the triangle closures [King and Thatcher, 1998] and is consistent with first-order uncertainties (0.8 x/•) typically assigned to triangulation observations [Gergen, 1975]. The signal-to-noise ratio can be expressed as S/N = 1 N-1 where N is the number of observations, Oi is the ith observation, and o'• is the ith standard error. The S/N for the triangulation data is 3.33. About 27% of the angle changes are equal to or smaller than the observation uncertainties. The low S/N is likely a function of the geometry of the network: the majority of the triangulation observations are within the upthrown block and do not span the 1952 surface rupture. Furthermore, the orientations of many of the observed triangles are not optimal for resolving deformation along the fault.

Leveling .
The leveling data used in this study were collected by the USCGS [Whitten, 1955] and evaluated by Stein and Thatcher [1981] for slope-dependent and misclosure errors (Table 2a) [Lofgren, 1975;Stein and Thatcher, 1981]. I use the elevation changes between successive benchmark pairs as the leveling observable, so that the observations are independent of elevation changes at the endp_?ints (Table 2b).
The model resolution matrix R quantifies how well the slip components are resolved along the fault segments [Menke, 1989].
Each row of the resolution matrix corresponds to one slip combined with neighboring segments, or constrained by additional data. Once these poorly resolved segments have been addressed, the final step involves a least squares inversion where each of the model parameters will be uniquely resolved. The single-value decomposition method of treating the geodetic data differs from previous coseismic geodetic studies [Dunbar et al., 1980;Stein and Thatcher, 1981] by directly modeling each angle change as a discrete observation, instead of forward modeling the derived components of shear strain by groups of 3-4 monuments. I use direct observation: no points need to be fixed or constrained to calculate the coseismic angle changes. Thus assumptions that apply to shear strain calculations, such as uniform shear within a network, no network dilation, and no network rotation, are avoided. Furthermore, by modeling the leveling data as relative elevation changes between adjacent monuments, errors associated with datum offsets between the preseismic and postseismic surveys are averted.
However, since the covariance between successive monument pairs is not utilized, then the S/N for the leveling data may be overestimated. Another difference between this study and previous analyses is that I include additional triangulation data both north and south of the surface rupture.

Coseismic Fault Model
Determination of the coseismic fault model proceeded in a multistep process. First, an overparameterized fault model was constructed to examine where slip could be resolved both along the length of the fault and at depth. Second, the dip of the overparameterized fault model was varied over a wide range of values to evaluate how well the network can resolve variations in dip. Third, based on where the geodetic network could resolve slip, the fault model was reparameterized into a four-segment fault. Model parameters were then incrementally varied to determine the fault geometry (endpoint location, dips, and fault depths). Fourth, the model resolution was then reevaluated, resulting in a further reparameterization of the fault into two segments. I again used a grid search on the two-segment model to determine the geometry of the preferred model. Finally, the data were inverted to determine the coseismic slip distribution.

Model Resolution
To evaluate the resolution of the triangulation and leveling data on the fault plane, a fault plane with a dip of 75 ø, the steepest dip suggested for the White Wolf fault, was divided into 30 patches (ten 6.6-km patches along strike and three 5-km patches down dip) (Figure 3a). This overparameterized fault model was not used to estimate slip; rather, it was used to examine where on the modeled fault surface the data could resolve slip and, more importantly, where the resolution is poor.   Zoback, 1995], focal mechanism studies find dips between 60 ø and 66 ø [Gutenberg, 1955]

Fault Geometry
To determine the geometry and fault model parameters to be used in the preferred inversion, I used a grid search method that incrementally varied each parameter while minimizing the reduced chi square (Z 2) in a joint inversion of the leveling and triangulation observations. Each iteration solved for both the  Since the surface rupture was poorly expressed and the aftershock locations have large uncertainties, the dip of the rupture plane is uncertain. Previous geodetic models of coseismic slip have found a variety of dips for the rupture plane(s) that range from 20 ø to 75 ø [Dunbar et al., 1980;Stein and Thatcher, 1981], aftershock and seismicity studies find dips between 50 ø and 75 ø [Cisternas, 1963;Ross, 1986

Coseismic Slip Distribution
The coseismic slip distribution was determined by first inverting the triangulation data to obtain the coseismic strike-slip displacements and then fixing these values while inverting the leveling data for the dip-slip components. This approach was necessary because the low S/N of the triangulation relative to the leveling data placed an inordinate weight on leveling observations during the joint inversion. This resulted in slip models inconsistent with observed coseismic offset patterns ( Table 4). The preferred model, which is, of course, not the only possible model, fits the leveling data well along all three profiles (Figure 7). The offset of the Wheeler Ridge spur line to the west from the Highway 99 line (Figures 7a and b) provided needed constraints on the geometry of the fault model, because minor changes in the geometry would result in large reduced chi square values in these lines. Similarly, lateral variations in the monument spacing along the Caliente leveling line, which followed a winding road somewhat perpendicular to the surface rupture (Figure 1), resulted in an "apparent scatter" of the data (Figure 7c, e.g., between 10 and 25 kin) or unusual uplift patterns (Figure 7d, e.g., between 40 and 50 kin). This apparent scatter is in part due to the lateral vertical deformation gradient and was useful in determining the fault geometry for the northeastern fault segment. This preferred model places 3.6 m of left-lateral strike slip and 1.6 m of reverse slip on the southwestern segment (Table  4 and Table 4 and are specific to the fault geometry of the preferred model. Given that the leveling observations have a high signal-tonoise ratio and that the locations of the three leveling lines cross both of the fault ends, the leveling data alone were inverted to estimate slip (Table 4). The distribution of the coseismic slip in the southwest is similar to the joint inversion at the 2(; confidence level with 3.8 m of left-lateral strike slip and 1.6 m of reverse slip. However, the models differ for the northeastern segment. The leveling-only model produced 4 times (0.9 m) the amount of left-lateral strike slip than the combined triangulation/leveling inversion, with only minor increase in the reverse slip (2.0 m) (Table 4).  Is there a significant difference in the reduction of residuals between the inversion of leveling only and the joint inversion? To address this question, I compared the misfits of the two models with an F test:

Rooks
where r is the residual (observed minus calculated); o is the data uncertainty; and v •, v2 are the number of degrees of freedom for models 1 and 2, respectively. The joint inversion produced F = 2.56, which is significant at the 99% confidence level. Thus the combined leveling and triangulation inversion provides a better fit to the data, even though the model reduced the triangulation signal by only 49%.
The geodetic moment was calculated for the Kern County earthquake from the preferred model (Table 5)  Alternatively, the modeled misfit may be a product of nontectonic or secondary deformation that locally disturbed the network or, may reflect complexities associated with obliquereverse fault earthquakes. The most notable source of nontectonic deformation is from groundwater-related subsidence in the southern San Joaquin valley [Lofgren, 1975;Stein and Thatcher, 1981;Bawden et al., 1997]. To minimize this source of contamination, leveling monuments selected were located in the hanging wall of the White Wolf fault, placed in bedrock, had shallow depths to bedrock, or were surveyed close in time to the mainshock (Figures 1 and 2). Since

Comparison With Other Studies
The geometry and the slip distribution from the preferred model differ significantly from previous geodetic analysis that used forward modeling techniques (Figure 9 and Table 5). One of the most apparent differences among the three geodetic models is the spatial geometry of the fault planes. The Dunbar et al. [1980] model is a straight fault that approximates the endpoints of the primary surface rupture and is divided into two shallow and two deep patches (Figure 9b and Table 5 (Table 5). The inversion found that a uniform dip of 75 ø to the southeast explains the leveling data best. More gently dipping (<60 ø ) fault planes failed to match the elevation changes along the northern leveling line and were rejected. It is unclear why the dip for the northeastern fault segment varies so much between the Stein and Thatcher [ 1981 ] model and the preferred model. I omitted leveling data that may have been subjected to subsidence associated with groundwater or hydrocarbon pumping north of the White Wolf fault and monuments with questionable stability near Tehachapi (Figure 9). The exclusion of these data provided a more robust solution with fewer potential sources of nontectionic contamination. These data omissions may be one of the reasons that the preferred  (Table 5). Left-lateral strike-slip displacements for this study were up to 1.5 m greater than the other studies in the southwest and were significantly lower along the northeastern segment. The reason for the disagreement among the different models is likely from the geometry and placement of the fault models. Observed surface displacements from the Kern County earthquake are small and often contradictory [Buwalda and St. Amand, 1955] and so provide little information to discriminate among the models. Additionally, the spatial distributions of aftershock focal mechanisms also provide little insight with a mixed pattern of reverse, strike-slip, and oblique mechanisms throughout the region [Dreger and Savage, 1999]. Where the geodetic studies agree is that most of the slip occurred below 5 km for the southwestern half of the rupture and the upper and lower depth of the fault is shallower along the northeastern portion of the fault (Table 5).
The geodetic moment that I calculated is at the lower end of the range of both seismic and geodetic moments previously determined for the Kern County earthquake (Table 5). Observed surface breaks from the mainshock extend 12 km northeast of the preferred fault model. Since the geodetic network could not resolve slip beyond this fault patch, then the moment calculated in this study can be taken as a lower bound for the Kern County earthquake.

Implications
Surface displacements and geometry associated with the preferred model are consistent with the regional topography and the current tectonic structures along the White Wolf fault (Plate 1). The southwestern half of the fault has no discernible topographic relief, with the exception of Wheeler Ridge at the southwestern end of the fault. Conversely, the northeastern half of the fault has elevated topography in the hanging wall block with elevations as high as 2100 m, while the footwall block, for the most part, remains at an elevation of 200 m (Figure 1). The region with the greatest coseismic uplift and sharpest deformation gradient agrees well with the present-day topography (Plate 1). Even though the area of model maximum uplift does not directly correspond to the highest topography, it does include Comanche Point, a site of contemporary folding and thrust faulting ( Figure 6 and Plate 1) [Goodman and Malin, 1992]. The preferred slip model has high left-lateral strike slip in the southwest (3.6 m) and minimal strike slip in the northeast (0.2 m). If this slip distribution pattern were typical of earthquakes along the White Wolf fault, then some structure would need to accommodate the differential strike slip between the two fault patches. The position and orientation of the Comanche thrust system is consistent with the regional shortening that would be expected with the strike slip differential that the preferred model produced. Aftershocks for the Kern County earthquake are compatible with northeast shortening across the Comanche thrust system [Dreger and Savage, 1999].
The steeply dipping fault patch along the northeastern portion of the fault is consistent with the present-day seismicity and provides structural continuity between the White Wolf fault and the Scodie lineament, a newly forming strike-slip fault that extends northeast from the White Wolf fault (Figure 1)