Open-File Report 96-517
To estimate the slip distribution on an assumed fault plane, we applied Singular Value Decomposition or SVD. The displacement at a point on the Earth's surface caused by slip on a number of faults can be written as
u = Gm (5)
where G is a partial derivative matrix of functions that describes the fault geometry and elastic properties of the crust. The geodetic data, u, is a column vector that contains the measured coseismic displacements, and the matrix, m, is a column vector of slip values that must be estimated. To solve for m, both sides of equation (5) are multiplied by a matrix G-g such that:
= G-gu = G-gGm (6)
where is the solution matrix, i.e., the coseismic slip. G-g is the `inverse' of G and SVD is used to estimate parameters in the G-g matrix.
All geodetic inversions share the deficiency, that while displacement observations are made at the Earth's surface, we seek to resolve the fault geometry and slip at depth. The SVD matrix decomposition technique is used to estimate parameters when the data is insufficient to yield information and slip on certain parts of the fault . When the number of singular values in the matrix G-g equals the number of unknown parameters (in this case, slip on the fault), then the technique is equivalent to least squares inversion of the data. Using a small number of singular values means that only those parts of the model that are well mapped by the data will be estimated.
We inverted the leveling section changes and their errors, along with the GPS estimates of displacement vector components and their associated errors (from Table 1 of [Hudnut, et al., 1996]) to estimate the thrust and strike-slip components of slip on each subfault [Larsen, 1991]. The number of model parameters estimated was 260, and the number of data was 815 (617 section changes from leveling, plus 198 data from GPS, since each GPS displacement vector contains 3 components).
Rupture occurred on a plane 15 km long and 20 km down-dip. Slip was concentrated between 6 and 10 km depth where it was almost purely up-dip (Figure 9). The peak slip was about 3 m on the fault. We did not employ positivity or slip constraints in the inversions; in other words, slip was permitted to be positive or negative at all sites on the fault surface. In the SVD method, when the eigenvalue matrix is truncated, a form of smoothing is implemented; no other smoothing was implemented [Larsen, 1991]. We retained 29 singular values, fitting the geodetic data well, producing a solution in accord with seismic evidence, and yielding a total seismic moment of 1.5 x 1019 N-m (equivalent to a moment-magnitude M=6.75). Other solutions yielded qualitatively similar slip distributions in the range 26 < k < 31, where k is the number of retained singular values. Within this stable range, the seismic moment increased from 1.54 to 1.74 x 1019 N-m, and the normalized-root-mean-square (nrms) residuals decreased slightly. Although a more rigorous selection of the optimal result from this approach is possible using an F-test [Parker, 1977] we made our selection by inspection, in an attempt to fit the largest number of BMs as closely as possible while avoiding mapping errors or anomalous BMs into the slip values. This minimized the number of BMs falsely identified as anomalous.
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U.S. Geological Survey
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