Global earth physics : a handbook of physical constants

Rotation of the Earth as a triaxial rigid body
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Earth Sciences: Background Information A guide to the reference sources and indexes to the journal literature in earth sciences. Fairbridge, eds. Kresge Ref. Goudie ed. Kresge GB Office of Naval Research and never published. D44 v. D44 Dana's New Mineralogy, 8th ed. D23 view record for print Recommended websites: Mineralogy Database Mindat. H23 Soil Genesis and Classification 6th ed. Stanley W. Buol, R. Southard, R. Graham, P. The most authoritative source of soil survey information. In Fig. The general level of fit is very good but there are about 50 models which are significantly better than the rest.

The shape of the misfit patterns as a function of ranked model is characteristic of most of the cases we have considered; there are a few models with small misfits and a few with poor performance and a smooth progression between. However, the scaling of the misfits varies between different segments of the Earth. An informative way of looking at the results of the model generation process is to look at the pattern of models as a function of depth and at the way in which the misfit maps into the density distributions.

This is illustrated in Fig. In the left-hand panel of Fig. A broad sampling of the allowed density bounds has been achieved in the outer core, but a more concentrated pattern emerges in the inner core, which arises from the broad range of allowed gradients. In the right-hand panel of Fig. We note immediately that the best fits can appear at the fringes of the sampled zone.

Display of sampling and projected model misfit for a set of models of density structure in the core derived from the same random seed. The model with the least L 2 misfit is superimposed in white. The deviations in both slope and density value from the reference model are not large and are somewhat exaggerated by the presentation in Fig. The most noticeable feature is the trend to lower densities and gradients in the inner core, which helps to improve the fit to the free-oscillation information. The 50 density models for the core region with the least L 1 misfit, compared to PREM reference model shown as a chain-dashed line.

We have allowed significant departures from this behaviour in order to provide a sampling of a broad region around the reference model. The imposition of tighter slope bounds tends to restrict the zone of acceptable models but has to be used judiciously to avoid artificial restrictions on connectivity between successive levels in the model.

So far we have considered the models in isolation, but we can reimpose some degree of smoothness by looking at the ensemble properties. The chain-dotted lines indicate the estimate of the variance associated with this estimate. In each case the jump at the inner-core boundary is enhanced. Ensemble average structures for density and its variation in the core, using an exponential weighting with misfit.

The ensemble results suggest that in the outer core down to km a simple cubic in radius is adequate. But, in the parametrization of density in the core it would be advantageous to allow a separate representation of the zones on the two sides of the inner-core boundary rather than force a single low-order polynomial in each of the inner and outer cores. Alternatively, a higher-order Chebyshev polynomial could be used in the outer core to provide a flexible representation without undue instability.

We now retain the PREM structure in the core and look at the possible variation of density in the lower mantle, which we have taken as the region from km deep to the core-mantle boundary. This zone is represented in the PREM model with a cubic polynomial in radius from km to km depth, a cubic polynomial from km to km at the top of D", and a further cubic to the core-mantle boundary; continuity of density is imposed at km and km depth. We have used a rather different representation, with a set of linear gradients in radius, and in consequence we can test the appropriateness of the PREM parametrization.

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Once a discontinuity is allowed, the probability is overwhelmingly in favour of it developing some contrast, and so if just a discontinuity in gradient is expected it is necesary to impose continuity in the density itself. Thus, at km depth we have normally imposed a continuity condition. However, at the top of D" we have considered cases in which we have forced continuity and also where we have allowed a discontinuity to develop.

For this lower-mantle section, we have found that the mass and moment-of-inertia constraints are of major importance, and that for a single random seed between 30 and models would need to be generated to produce which satisfied the primary constraints within our prescribed tolerances and which then could be compared with the free-oscillation information.

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The mass and moment-of-inertia constraints, when applied to this segment of the Earth, have led to a characteristic banding in the model sampling, which is well illustrated in the left-hand panel of Fig. A set of models derived from a single random seed is displayed to illustrate the patterns of model sampling nearly 60 models were rejected on the basis of the primary constraints and misfit.

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The best-fitting models tend to have somewhat lower density gradients than the general trend of the cluster. For the lower mantle it was difficult to find models that fitted as well as PREM unless a discontinuity was allowed at the top of D". The overall span of misfits was somewhat larger than in the core, with misfit measures reaching about 1.

Display of sampling and projected model misfit for a set of models of density structure in the lower mantle derived from the same random seed. When a broad span of models is allowed grey lines , the behaviour follows closely the trend seen in the model sampling in Fig. However, when attention is focused on the models with a better fit to the free-oscillation data black lines there is a reduced gradient between and km depth, with a mean departure of 0.

We note in Fig. Ensemble average structures for density and its variation in the lower mantle using an exponentional weighting with misfit. When discontinuities were allowed at km and km, relatively minor jumps developed at km but large jumps were common at km. Away from the immediate vicinity of the discontinuities the behaviour was very close to that illustrated in Figs 6 and 7.

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We can expect there to be some edge effects associated with the isolation of only a portion of the Earth, but the patterns of behaviour in Figs 6 and 7 suggest that it would be desirable to allow at least two cubic polynomial segments in any future parametrization of the density distribution in the lower mantle. In this case we have retained the PREM structure for depths below km but have allowed the density to vary through the upper mantle and have looked at a range of different continuity conditions.

The PREM parametrization in the upper mantle is in terms of linear gradient segments, so that there is a closer correspondence with the model representation in this region than for the two deeper zones. The size of the sample population varies somewhat with different continuity conditions, but is not less than 10 27 potential models. The influence of the mass and moment-of-inertia constraints is relatively weak for this portion of the Earth because only a small portion of the mass is involved and the conditions can be fitted by adjustment of density between different levels.

The deviations in density which we have allowed in the upper mantle extend well outside any plausible linearization conditions. The span of the misfits over the model ensemble for each class of trial is substantially larger than for the other cases but a few models approach the fit achieved with PREM or even improve on it. In these cases we have found it most effective to use the Bose-Einstein representation eq.

As shown in Fig. The estimated variance is significant and is reflected in the behaviour of the 50 best models on the L 1 criterion, which are illustrated in Fig. Ensemble average structures for density and its variation in the upper mantle using a Bose-Einstein weighting which emphasizes the properties of the best-fitting models.

The 50 density models for the upper mantle region with the least L 1 misfit, compared to the PREM reference model shown as a heavy black line. The level of misfit can be reduced by the introduction of further discontinuities so that in this depth range the prior specification of potential jumps in density is important. For example, the PREM model includes a discontinuity at km which is of major significance for velocity, yet the presence of such a global feature is certainly a matter of dispute.

The sequence of model tests described in this paper has been undertaken to try to get an assessment of the constraints available on the density distribution without invoking linearization about a reference model. The representation of the models has been deliberately chosen to differ significantly from the PREM model with which comparisons have been made, so that an assessment can be made of the adequacy of the polynomial parametrization employed in PREM. In addition to employing results from groups of models derived from a single random seed, we have used ranked model information across an ensemble of models.

We have also looked at ensemble properties for density and its variance as a function of depth, using both an exponential and a Bose-Einstein weighting in terms of individual model misfit. The ensemble properties allow the reinstatement of smoothness, which is missing in the individual piecewise linear representations. The many trials demonstrate the quality of the PREM density model, particularly as a smooth model within the limits of its parametrization.

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However, the constraints imposed on the polynomials for the different depth intervals in PREM are based on mathematical convenience rather than an attempt to allow for different physical processes. For both the lower mantle and the core, it would be desirable for any future reference models to have more degrees of freedom than in PREM.

A single cubic polynomial in radius is barely adequate to represent the full range of behaviour across a major zone within the Earth, and imposes a very strong constraint on the nature of possible gradients. In particular, it is desirable that special provision is made for the treatment of structure near major boundaries. In future representations of the spherically averaged density models, we must be wary of forcing excessively heavy constraints on our view of the Earth by working with a very limited number of parameters, which is no longer required with recent advances in computational power.

The details of the density models would be modified if we adopted a dynamic density model with a modified moment of inertia, rather than the hydrostatic case as considered above, but this would not affect our conclusions. I would like to thank Guy Masters for the version of the MINOS program used in calculating the free-oscillation frequencies used in this work and for general discussions.

I would also like to thank Malcolm Sambridge, who created the original routines for uniform sampling with function and gradient bounds and who has been very helpful in discussions on extraction of model properties from ensembles. Malcolm Kennett helped with the development of the inversion scheme.

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Why Do Universal Constants Show Up and Why Can We Get Rid of Them?

Advanced Search. Article Navigation. Close mobile search navigation Article Navigation. Volume Article Contents. On the density distribution within the Earth B. Oxford Academic. Google Scholar. Cite Citation. Permissions Icon Permissions. Abstract The distribution of density as a function of position within the Earth is much less well constrained than the seismic velocities. Apart from sampling at the surface, there is very little possibility of direct observation of density.

The major constraints come from the mass and the mean moment of inertia of the Earth about the polar axis, which are moments of the density distribution. The mass is the second moment of the radial density distribution,.