All movies presented show models of finite deformation in southern California through time. These models are generated using geologic and geodetic observations and contain predictions of both horizontal and vertical changes through time. Following is a brief description of how the models are generated. We use the fault slip rate data for southern California, which was compiled by Shen-Tu et al. , with selected rates along the San Andreas Fault from the Big Bend to the San Bernardino Mountains chosen following Meade and Hager . Our faults are digitized from a fault activity map by Jennings , with additional faults (Ozena, Cuyama, Nacimiento) digitized from a USGS/CA Division of Mines and Geology geologic map of California (1966). Geodetic data comes from GPS stations in the Southern California Integrated GPS Network (SCIGN), as well as the EarthScope Plate Boundary Observatory (PBO) initiative. The methods of Haines and Holt [1993; as detailed in Holt et al., 2000 and Kreemer et al., 2000; Beavan and Haines, 2000] are used to generate horizontal strain rate fields for our region. Geologic and geodetic rates are input, and the deformation field is parameterized as a continuous horizontal velocity [Holt et al., 2000],
W(r)=R(W(r) x r) (1)
W(r) above is a vector rotation function, R is Earth’s radius, r is a radial unit vector, and the operation involves a cross product.. W(r) is expanded by bi-cubic Bessel interpolation on a curvilinear grid [Beavan and Haines, 2000]. From these calculations, the horizontal strain rates can be extracted [Holt et al., 2000]. The horizontal strain rates on the surface of a sphere will involve the spatial derivatives of W(r) [Holt et al., 2000; Beavan and Haines, 2000].
In addition, Kostrov  moment-rate-tensor summation is used to infer strain rates within areas, in order to match the strain-rate distribution with the bi-cubic functions [Holt et al., 2000]. To determine the fit of strain rates to the continuous function, a least-squares inversion is used, in which an objective function representing the sum of misfit between observed values of the strain-rate tensor at each point and values of the model strain rate tensors calculated by the above methods, is minimized [Holt et al., 2000].
In addition, strain values at each point on the grid may be smoothed over a radius of a certain number of grid squares – that is, the value for strain is distributed throughout this area on the grid, instead of values at discrete points being considered. We used smoothing radii of 1 and of 0 for trials in our experiment (to be discussed later), determining that the radius of one grid square yielded the most useful results.
Once our horizontal dilatational-strain-rate field is calculated, we assume Airy isostasy and zero crustal-volume change in order to determine rates of uplift and subsidence from rates of compressional and extensional strain, respectively. In the absence of information on erosion rates in southern California, we set erosion rate (removal of volume per unit area per unit time) as one half of the instantaneous, tectonic induced uplift rate for any given time interval. The final uplift rate is adjusted to account for the isostatic response to both the crustal thickening change and the erosion. The assumptions used to generate uplift and erosion rates do not allow for changing rates of erosion relative to uplift over time (e.g., due to changing climatic conditions), and may or may not reflect the reality of conditions in southern California over the past 3 million years (especially given faunal evidence of a more moist climate in this region at ~1.3 – 1.4 Ma [Albright, 1999]). However, it is possible that outside information on changing erosion rates in our study area could be used at a later date to refine this step in the procedure.
In order to map our strain-rate fields, their associated uplift and subsidence, and the topography resulting from finite strain, we use the Generic Mapping Tools (GMT) program [Wessel and Smith, 1995]. The x, y, and z positions of points are calculated at each time step by moving points using their instantaneous velocity and uplift rate (based on both the tectonic or dilatational model strain rate and on the assumed erosion rate) with time steps of 50,000 years. Given these instantaneous motions, the new positions of faults are predicted, or extrapolated, into either the past or the future. With the new fault positions, a new model continuous-strain-rate field is produced using the methods described above. Pacific-North America velocity boundary conditions are used at each time step in the generation of the predicted model velocity field. The process of obtaining new x, y, and z positions, new fault orientations, and new model strain rate fields is repeated until 3 million years of plate boundary evolution into the past is obtained. A shortcoming in the modeling is that we do not change the expected slip rates on the sets of faults as their orientations change. However, model strain rates are obtained through the inversion procedure described above, with emplaced velocity boundary conditions. As faults rotate into new positions, it is possible that the model strain rates predicted for such regions may differ significantly from the input fault slip rates, owing to the requirement that the entire integrated model strain rate field satisfies total Pacific-North America velocity boundary conditions for all time steps. In addition, we extrapolate forward in time by the same method to 3 million years into the future, creating visual tools that can be used in Earth science education. GPS velocities are also used and the inversion procedure can simultaneously match strain rates and GPS velocities. When GPS velocities are used, the station positions at the next time step are obtained by using the instantaneous model value at the particular station location. This creates a new, synthetic GPS set of ‘observations’ that are matched in the inversion procedure at each time step. The movies that you can view were produced by two summer REU undergraduate interns, Barbara Birkes (2004) and Beth Ann Bell (2006). In models created by summer REU scholar Barbara Birkes, GPS data were used. However, in models generated by REU scholar Beth Ann Bell, GPS were not used in final models because they produce strain rate models that fail to match inferred uplift rates in the San Gabriel and San Bernardino mountain regions.
Albright, B., 1999, Magnetostratigraphy and biochronology of the San Timoteo Badlands, southern California, with implications for local Pliocene-Plestocene tecotnic and depositional patterns, GSA Bulletin, v. 111, 1265 – 1293.
Beavan, J. & Haines, A.J. (2001), Contemporary horizontal velocity and strain rate fields of the Pacific-Australian plate boundary zone through New Zealand, J. Geophys. Res., 106, 741-770.
Birkes, B., and Holt, B., 2004, Quantification of tectonic rates using space geodetic and geologic observations, Mineral Physics Institute Summer Scholars Program final report, SUNY at Stony Brook Geoscience Department, Stony Brook, NY.
Haines, J., and Holt., W., 1993, A procedure for obtaining the complete horizontal motions within zones of distributed deformation form the inversion of strain rate data, Journal of Geophysical Research, v. 98, 12,057 – 12,082.
Holt, W., Shen-Tu, B., Haines, J., and Jackson, J., 2000, On the determination of self-consistent strain rate fields within zones of distributed continental deformation, Geophysical Monograph 121, 113 – 141.
Kostrov, V. V., 1974, Seismic moment, energy of earthquakes, and the seismic flow of rock, Izv. Acad. Sci. USSR Phys. Solid Earth, Engl. Transl., 10, 23-44.
Kreemer, C., Holt, W., Goes, S., and Govers, R., 2000, Active deformation in eastern Indonesia and the Philippines from GPS and seismicity data, Journal of Geophysical Research, v. 105, no. B1, 663 – 680.
Meade, B. J., and Hager, B. H., 2005, Block models of crustal motion in southern California constrained by GPS measurements, Journal of Geophysical Research, v. 110, BO3403
Molnar, P., and Gipson, J. M., Very long baseline interferometry and active rotations of curstal blocks in the Western Transverse Ranges, California, GSA Bulletin, v. 106, 594 – 606.
Onderdonk, N. W., 2005, Structures that accommodated differential vertical axis rotation of the western Transverse Ranges, California, Tectonics, v. 24.
Shen-Tu, B., Holt., W. E., and Haines, A. J., 1999, Deformation kinematics in the western United States determined from Quaternary fault slip rates and recent geodetic data, Journal of Geophysical Research, v. 104, no. B12, 28,927 – 28,955.
U.S. Geological Survey and California Division of Mines and Geology, 1966, Geologic Map of California.
Wessel, P, and Smith, W., 1995, New version of the Generic mapping tools released,
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Modified September 27, 2007