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Professor William Holt (State University of New York at Stony Brook) offered a presentation at the Fall meeting of AGU, 1998, in San Francisco on the kinematics and dynamics of the western US inferred from quaternary fault data, GPS and VLBI velocities, and topography and geoid data. While there is no web page for this research, what follows is an excerpt from an email he sent, describing the project.
This research was published in Science as:
Flesch, L.M., W.E. Holt, A.J. Haines and B. Shen-Tu, 2000: Dynamics of the Pacific-North American Plate Boundary in the Western United States, Science, v. 287, pp. 834-836.
From Professor Holt's e-mail:
"We do not have a web page with results on this particular project yet, but it is possible that we can have that soon. Much of the modeling was performed by Lucy Flesch, a PhD student in our department. We got the code tested and working this summer, obtained results using just topography data in late summer. We only obtained the latest result with geoid - which I was very happy to see worked out rather well - in November. Lucy actually presented results using the same modeling method for Asia on Tuesday, but we did not use the geoid for that project.
"We got the idea to use the geoid to infer gravitational potential energy differences from Jones, Unruh and Sonder Nature, 381, 37-41, 1996. One meter of height is equivalent to 2.281x10^11 N/m [Coblentz et al., Tectonics, 13, 929-945, 1994]. We made some approximating assumptions that are not completely justified and we need to correct for this before submitting the paper. That is, we assume that the anomalies arise within the lithosphere or uppermost mantle, which is clearly incorrect for total geoid. What Jones et al. [1996] did was apply a filter - removing terms below degree and order 7 (with a cosine taper to degree and order 11) to the GEOID93. They did this in an attempt to remove the presence of geoid anomalies associated with features well below the lithosphere. After doing this they showed that potential energy inferred from the geoid correlated very well with independent estimates of potential energy inferred from crustal thickness and density profiles obtained from seismic studies.
"We also showed a fairly close agreement between the two independent methods, but again our estimates of potential energy inferred assumming local compensation of long wavelength topography is slightly more simplified that what Jones et al did, since we did not use seismic data to show possible lateral variations in crustal density and indicate where higher elevations might be compensated by lower density mantle. In regard to the geoid data we simply did not have the time to search out a filtering method before the AGU, so we hoped that the long wavelength component would impose only a small error onto the calculation of deviatoric stress field assoicated with gravitational potential energy differences. Given that the greatest influence on the deviatoric stress field is in local variations in geoid height, and not the absolute value, this is probably OK for now. Our next step, however, is to filter as Jones et al. did. If you could suggest anything for this, I would appreciate it.
"What we did in using your new geoid model is run the software, which allows one to query what the geoid height is for an average area (I think this is the case, as Lucy brought the data off). The heigt was determined for a large number of areas - about 600 I think - in the western US. The method we employ that is new is to solve force balance equations with potential energy values as input. That is, we minimize the local rate of doing work subject to the constraint of solving force balance equations. It turns out that the minimization of a functional containing work rate as proportional to (TijTij + Tii^2) plus lagrange multiplier times force balance equations yields a form for deviatoric stress of
Tij = 0.5*(dLi/dxj + dLj/dxi)
where Li is the lagrange multiplier for the differential equation constraints. If this is plugged back into the force balance equations, then it can be shown that minimization of a second functional that looks just like a least-squares norm, satisfies the force balance equations for Tij of the form above, which is a minimum stress associated with potential energy differences. This second functional contains terms that look like
(Tij - Sigma(zz))^2
where Sigma(zz) is the vertically averaged vertical stress or potential energy. So sigma(zz), which is inferred from the geoid or topography is the only input and after the inversion (or minimization of the second functional) Tij will correspond with a self-consistent (minimum) deviatoric stress associated with the given estimates of potential energy.
"We are confident that grid geometry (as long as it extends far enough away from area of interest) does not influence results profoundly. The second phase is to seek a stress boundary condition that when added to contribution from potential energy, gives a total stress field that looks like the present day stress field. What was encouraging here is that the boundary stress was very long wavelength or slowly varying. That is, a very simple stress field boundary condition added to the one calculated above (which is not slowly varying and not simple) yields quite a good match to what is happening on a broad scale in the western US. We showed that both contributions are important, but potential energy alone will certainly not give all of observed deformation in the Great Basin. On the other hand we feel that the potential energy differences are certainly responsible for stresses being at a high angle to the strike of the PA plate boundary along the San Andreas fault (60 - 65 degrees).
"The question I have is whether it is possible to read the data in digitally in a fairly continous fasion and then apply appropriate filter. If you have any suggestions for this I would appreciate it."
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