Open-File Report 96-517
Our approach was first to find a model of the earthquake source that is most consistent with the displacement of the leveling BMs and GPS monuments and with independent seismological data, and then to identify the bench marks. that depart the most from the model (Figure 8). Bench marks with elevation changes that differed from the model by more than 3 cm (1.2 inches) were then visited for a preliminary site inspection by a Caltrans survey team.
We treat the earth as a homogeneous elastic halfspace, a medium with a flat upper surface that extends to infinite depth, and assign elastic modulii (or stiffness) that approximate observed earth behavior. Into this medium we embed an elastic dislocation--a rectangular cut or fault--at a given location with a specified length, width, inclination, and orientation. We then calculate how slip (shear displacement) of the fault will deform the earth's surface. This is comparable to making a cut inside a stiff block of rubber, displacing the two sides of the cut past each other a fixed amount, and measuring how the surface of the rubber block has deformed as a result of the internal cut. At every point where we have a geodetic observation of the earthquake deformation (vertical or horizontal), we calculate the predicted deformation per meter of slip on the fault. We then find the value of fault slip which minimizes the difference between the predicted and observed surface displacements. The observations are weighted such that the most precise observations carry the most weight, following Marshall [1991].
This method works well principally because the earthquake displacement field is predominantly long wavelength (smooth), whereas the displacements associated with structural damage or ground failure tend to be localized (Figure 8). In addition to the GPS and leveling observations, interferometric synthetic aperture radar satellite images confirm the simplicity of the overall spatial pattern of deformation [Murakami et al., 1995; Massonnet et al., 1993]. Because the Northridge earthquake struck on a `blind' thrust fault [SCEC & USGS, 1994], in which the main fault slip does not reach the ground surface, the deformation field is relatively smooth everywhere .
The 617 leveling observations and 66 GPS-derived displacement vectors were simultaneously inverted to estimate the fault slip that occurred during the Northridge earthquake, which best satisfied the geodetic data [Hudnut et al., 1996]. Since each GPS observation consists of three components of displacement there are thus 815 total observations. This modeling approach is based on elastic dislocation theory, which can be used to compute the displacements at a given point on the ground surface from a slip distribution model [Okada, 1992].
The modeled slip on the fault that caused the Northridge earthquake is shown in Figure 9 and the location of the fault, identical to that used by Hudnut [1996], is shown in Figure 10. This study differs from its predecessors [Hudnut et al., 1996], in that leveling observations were first corrected for subsidence and then introduced into the inversion as section elevation changes weighted by their uncertainty. The benefit in modeling section elevation changes is that no datum offset between the pre- and postseismic deformation need be identified. The uncertainty ascribed to each observation reflected the quality of the survey and error in any subsidence correction that had been applied to the data. The spatial coverage and density of the leveling data, in addition to the large number of sampled leveling sections, contribute substantial new information to the source inversion.
To isolate and remove leveling sections with large residuals, we first inverted all of the leveling and GPS data together, and then removed from the inversion leveling sections with residuals greater than 40 mm. The typical uncertainty of a section elevation observation (in other words, the uncertainty of the height change of one BM compared to an adjacent BM about 1 km away) was 10-15 mm, so 40 mm is about the 3[sigma] or 95% confidence level of the section elevation change. We then re-ran the inversion. Any new 40 mm outliers were then removed from the revised model until the final model incorporated no residuals greater than 40 mm. Finally, this process was repeated with a 30 mm criteria, which represents the 90% confidence level.
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U.S. Geological Survey
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