I am very grateful to the entire faculty of the Division for allowing me the latitude to investigate the problems of my choice. I would also like to thank Olin for the fun time we spent collaborating on the method that is the basis of the first part of this thesis. In the second part we develop a physical model of the source region of subduction zone volcanism.
Their undeniable connection to one of the key features of the plate tectonic system suggests that understanding their generation may lead to an improved understanding of subduction in general. Therefore, a better understanding of them would lead not only to an understanding of the evolution of the mantle over time, but also to an understanding of the production of the continental crust. Only by developing a coherent understanding of the many different phenomena due to mantle convection will we develop a well-constrained dynamical model to explain plate tectonics.
PART 1
Seismology gave us only limited information about the dynamic state of the Earth's interior, for example the inhomogeneity parameter [Masters, 1979]. Seismic tomographs face many challenges in their attempts to develop the best possible images of the Earth. Many employees use huge datasets (for example, the International Seismological Society (ISC) catalog) that make it impractical to personally examine all the original data (because of its size (millions of choices) and its inaccessibility); therefore it is difficult to make a detailed assessment of the errors in the data set.
These surface wave surveys have produced the only global images of Earth's upper mantle. The method is more robust because it concentrates on obtaining a simpler description of the velocity fields, i.e. the spectrum. This is the first presentation of the results for the shear rates and the comparison between the compression and shear wave results.
Inversion
In the sense that x = (A 1 Af1A 1 d 1s least squares solution of d = Ax a n d x=(A1A+a2I)A1d is the damped least squares solution, (where I is the identity matrix and a is the damping parameter) Damped least squares mm1m1ze a linear sum of the prediction error and the L2 norm of the model For the inversion of the P-wave data, it was impossible to examine the trade-off by repeating the inversion for an interval of
A coarse parameterization was undertaken so that the global damping resulted in lower errors than in the fine parameterization of the lower mantle. In the P-wave inversion, the trade-off was investigated with singular values less than one thousandth of the largest singular value set to zero. Due to the poor resolution of the layers in the upper mantle, they are preferentially damped, especially the two regions discussed above.
In Figure 1.18(c) we present the short-scale effect, the part of the total effect due to structure with harmonic degree l > 50. There is very little small-scale effect m the lower mantle; while some of the values (eg the third depth bin) become negative. In Figure 1.19 (a) we present the half-width of the fine-depth parameterization in the P-wave inversion.
In figure 1.21 we represent total power in the fine P-wave model and the coarse S-wave model. It is also within a short distance of the mm1ma m the exchange surface (see figure 1.12) but note here that the half-width in the deepest mantle is only 5OO-6OOkm. This is as expected, given the constant half-width and the presence of the peak at the same deeper level in the S-wave long-wavelength power as discussed above.
In Figures 1.24 and 1.25, we present the data predictions from the model and the original data (minus their intercept estimates) along with their error bars. After considering the total error in the data used in the inversion, it is seen that the data predicted by the P-wave model will also be within _ the error bars of the original data. In the P-wave study, we re-evaluated the intercept estimates to provide a better fit to the data after the model was evaluated.
Discussion
Some of the scatter is considered to be due to uncertainty with the earthquake hypocenter parameters, but most of this scatter is scale invariant and is not inverted to produce spurious features in the model. In surface wave studies modeling minor and main arc paths, one needs not only the location and origin time, but also the source time function of the earthquake. At the very largest scales we will be considering many orientations, therefore part of the scattering at this scale may be due to azimuthal anisotropy.
The estimates of the intercepts for the P and S wave studies are understandable when compared to the complexity of the travel time curves. This is shown in the data as a sharper drop in variance as we approach the origin of the shallower curves. If there is heterogeneity on the scale length of the path difference, then it will increase.
This may be a function of the source and also the complexity of the nearby source structure. If we assume that the depth variation of the intercept is primarily a function of small-scale structure and finite binning, then it is reasonable to take the deep source bin intercept estimate as an estimate of the non-structural signal in the data. The most striking result of this study is the concentration of force in the upper mantle compared to the lower mantle and the similarity in the results for the shear and compression wave structure.
Given the fact that for teleseismic distances the length of the lower path of the mantle is significantly greater than the length of the upper path of the mantle; The contribution of the lower mantle to the traveltime residuals will be greater than the total low power (the product of the half-width and the root-mean-square amplitude of the slowness variations suggests [Zhou et al., 1988]). If the heterogeneity scale lengths were the same in both regions, then the amplitude of the slowness fluctuated. A comparison of the P-wave results in the lower mantle is made with the statistics of previous deterministic studies of the lower P-wave mantle in Figure 1.29.
This additional signal may be the result of aliasing of higher spectral power to the lower harmonics, spurious power due to inversion noise to poorly resolved regions of the model, or leakage to lower mantle due to poor depth resolution in the deterministic inversions.
CME 0
The only previous analysis of lower mantle global shear body waves was by Sengupta and Toksoz [1976]. The value at the top of the lower mantle is greater than 4, but the P-wave studies of the lower mantle suffer from leakage from the upper mantle. They are constant throughout the upper half of the mantle; while potentially increasing in the lower half of the mantle.
The S-wave studies similarly show half-widths of km in the upper half of the mantle, rising to 3,000 km in the lower mantle. As described earlier, the lower cladding value of the S-wave is not robust to variations in the value of the damping parameter. Mantle cooling may also be due to very narrow downwells, with large temperature differences.
Therefore, we propose that most of the Earth's heat flow is due to internal heating and secular cooling. It is conceivable that different parts of the spectra of seismic heterogeneity are controlled by different factors. The limitations of the above simulations serve to remind us that Earth is more complex.
This is a key link in the following argument, and direct experimental confirmation of the above result would be important. He found that the velocity variance decreases more rapidly as a function of the harmonic degree and is a weaker function of the Rayleigh number. The consistency of the spectral shape for the four cases from 60-1400 km gives us a confi.
We have developed a method to image the spectrum of Earth's heterogeneity as a function of depth. We found that Earth's heterogeneity 1 is concentrated in the _upper 400 km of the mantle. In the lowermost mantle, it is hypothesized that this is a natural consequence of the compressive effect on the materials [Anderson, 1987a].
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A simultaneous solution would also give us a quantitative measure of the trade-off between model and intercept.
![Figure 1. 7 P-wave linear intercept estimates. From Gudmundsson et al. (1989]](https://thumb-ap.123doks.com/thumbv2/123dok/10402434.0/37.1155.164.1031.59.406/figure-p-wave-linear-intercept-estimates-gudmundsson-1989.webp)
