Density Distribution In Earth Research Articles

Neutrino tomography of the Earth with ORCA detector

Using PREM as a reference model for the Earth density distribution we investigate the sensitivity of ORCA detector to deviations of the Earth (i) outer core (OC) density, (ii) inner core (IC) density, (iii) total core density, and (iv) mantle density, from their respective PREM densities. The analysis is performed by studying the effects of the Earth matter on the oscillations of atmospheric nu _{mu }, nu _e, {bar{nu }}_mu and {bar{nu }}_e. We present results which illustrate the dependence of the ORCA sensitivity to the OC, IC, core and mantle densities on the type of systematic uncertainties used in the analysis, on the value of the atmospheric neutrino mixing angle theta _{23}, on whether the Earth mass constraint is implemented or not, and on the way it is implemented, and on the type – with normal ordering (NO) or inverted ordering (IO) – of the light neutrino mass spectrum. We show, in particular, that in the “most favorable” NO case of implemented Earth mass constraint, “minimal” systematic errors and sin ^2theta _{23}=0.58, ORCA can determine, e.g., the OC (mantle) density at 3sigma C.L. after 10 years of operation with an uncertainty of (− 18%)/+ 15% (of (− 6%)/+ 8%). In the “most disfavorable” NO case of “conservative” systematic errors and sin ^2theta _{23}=0.42, the uncertainty on OC (mantle) density reads (− 43%)/+ 39% ((− 17%/+ 20%), while for for sin ^2theta _{23} = 0.50 and 0.58 it is noticeably smaller: (− 37)%/+ 30% and (− 30%)/+ 24% ((− 13%)/+ 16% and (− 11%/+ 14%)). We find also that the sensitivity of ORCA to the OC, core and mantle densities is significantly worse for IO neutrino mass spectrum.

Gravity in Earth’s Interior

The acceleration of gravity in Earth’s interior is determined by the density distribution in Earth. A remarkable result is that the acceleration is approximately constant all over the mantle, which amounts to about 84% of Earth’s volume. This result can be explained by a simple two-layer model of Earth, showing that the constancy of the acceleration in the mantle is a consequence of the particular size and density of Earth’s core with respect to the size and density of the whole Earth. In other planets, with different mass distributions, the dependence of acceleration on depth could be very different.

Analytical treatment of long-term observations of the day-night asymmetry for solar neutrinos

The Earth's density distribution can be approximately considered piecewise continuous at the scale of two-flavor oscillations of typical solar neutrinos, such as the beryllium-7 and boron-8 neutrinos. This quite general assumption appears to be enough to analytically calculate the day-night asymmetry factor for such neutrinos. Using the explicit time averaging procedure, we show that, within the leading-order approximation, this factor is determined by the electron density within about one oscillation length under the detector, namely, in the Earth's crust (and upper mantle for high-energy neutrinos). We also evaluate the effect of the inner Earth's structure on the observed asymmetry and show that it is suppressed and mainly comes from the neutrinos observed near the winter and summer solstices. As a result, we arrive at the strict interval constraint on the asymmetry, which is valid within quite a wide class of Earth models.

Semidiurnal signal in UT1 due to the influence of tidal gravitation on the triaxial structure of the Earth

AbstractThe axial component of Earth rotation, which is conventionally expressed by Universal Time (UT1), contains small physical signals with diurnal and subdiurnal periods. This part of the spectrum is dominated by the tidal effects which are regular and predictable. The largest components express the influence of the gravitationally forced ocean tides with diurnal and semidiurnal periods and amplitudes up to 0.02 milliseconds (ms) in UT1 corresponding to an angular displacement of 0.30 milliarcseconds (mas); see Table 8.3 of the IERS Conventions (IERS, 2003). There are also smaller subdiurnal components (amplitudes up to 0.03 mas), designated as “spin libration” by Chao et al. (1991), due to direct influence of the tidal gravitation on those features of the Earth's density distribution which are expressed by the non-zonal terms of the geopotential. These components are not included in the models recommended by the IERS Conventions, in contrast to the corresponding effect in polar motion (ibid., Table 5.1).Here we consider in detail the subdiurnal libration in UT1. We derive an analytical solution for the structural model of the Earth consisting of an elastic mantle and a liquid core which are not coupled to each other. The reference solution for the rigid Earth is computed by using the satellite-determined coefficients of geopotential and the recent developments of the tide generating potential (TGP). We arrived to the conclusion that the set of terms with amplitudes exceeding the truncation level of 0.005 mas consists of 11 semidiurnal harmonics due to the influence of the TGP term u22 on the equatorial flattening of the Earth expressed by the Stokes coefficients C22, S22. There is an excellent agreement between our estimates for the rigid Earth and the amplitudes derived by Wünsch (1991). The only important difference is the term with the tidal code ν2, which seems to be overlooked in the development of Wünsch. Our amplitudes computed for an elastic Earth with liquid core appear to be in reasonable agreement with those derived by Chao et al. (1991), but the latter model was not complete. The estimated effect is superimposed on the ocean tide influences having the same frequencies but 9 to 11 times larger amplitudes. Nevertheless, its maximum peak-to-peak size is about 0.105 mas, hence definitely above the current uncertainty of UT1 determinations. Comparison with the corresponding model of prograde diurnal polar motion associated with the Earth's triaxiality (IERS Conventions, Table 5.1) shows that: 1) the two effects are of similar size, 2) there is consistency between the underlying dynamical models, parameters employed, etc. In conclusion, we recommend adding the model developed here to the set of procedures provided by the IERS Conventions.

Harmonic spline–wavelets on the 3–dimensional ball and their application to the reconstruction of the Earth's density distribution from gravitational data at arbitrarily shaped satellite orbits

We introduce splines for the approximation of harmonic functions on a 3-dimensional ball. Those splines are combined with a multiresolution concept. More precisely, at each step of improving the approximation we add more data and, at the same time, reduce the hat-width of the used spline basis functions. Finally, a convergence theorem is proved. One possible application, that is discussed in detail, is the reconstruction of the Earth´s density distribution from gravitational data obtained at a satellite orbit. This is an exponentially ill-posed problem where only the harmonic part of the density can be recovered since its orthogonal complement has the potential 0. Whereas classical approaches use a truncated singular value decomposition (TSVD) with the well-known disadvantages like the non-localizing character of the used spherical harmonics and the bandlimitedness of the solution, modern regularization techniques use wavelets allowing a localized reconstruction via convolutions with kernels that are only essentially large in the region of interest. The essential remaining drawback of a TSVD and the wavelet approaches is that the integrals (i.e. the inner product in case of a TSVD and the convolution in case of wavelets) are calculated on a spherical orbit, which is not given in reality. Thus, simplifying modelling assumptions, that certainly include a modelling error, have to be made. The splines introduced here have the important advantage, that the given data need not be located on a sphere but may be (almost) arbitrarily distributed in the outer space of the Earth. This includes, in particular, the possibility to mix data from different satellite missions (different orbits, different derivatives of the gravitational potential) in the calculation of the Earth´s density distribution. Moreover, the approximating splines can be calculated at varying resolution scales, where the differences for increasing the resolution can be computed with the introduced spline-wavelet technique.

Regularized wavelet-based multiresolution recovery of the harmonic mass density distribution from data of the Earth's gravitational field at satellite height

The inverse problem of recovering the Earth's density distribution from data of the first or second derivative of the gravitational potential at satellite orbit height is discussed for a ball-shaped Earth. This problem is exponentially ill-posed. In this paper, a multiscale regularization technique using scaling functions and wavelets constructed for the corresponding integro-differential equations is introduced and its numerical applications are discussed. In the numerical part, the second radial derivative of the gravitational potential at 200 km orbit height is calculated on a point grid out of the NASA/GSFC/NIMA Earth Geopotential Model (EGM96). Those simulated derived data out of SGG (satellite gravity gradiometry) satellite measurements are taken for convolutions with the introduced scaling functions yielding a multiresolution analysis of harmonic density variations in the Earth's crust. Moreover, the noise sensitivity of the regularization technique is analysed numerically.

Scale Continuous, Scale Discretized, and Scale Discrete Harmonic Wavelets for the Outer and the Inner Space of a Sphere and Their Application to an Inverse Problem in Geomathematics

In this paper, we construct a multiscale solution method for the gravimetry problem, which is concerned with the determination of the earth's density distribution from gravitational measurements. For this purpose, isotropic scale continuous wavelets for harmonic functions on a ball and on a bounded outer space of a ball, respectively, are constructed. The scales are discretized and the results of numerical calculations based on regularization wavelets are presented. The obtained solutions yield topographical structures of the earth's surface at different levels of localization ranging from continental boundaries to local structures such as Ayer's Rock and the Amazonas area.

Neutrino tomography of the Earth

Neutrino oscillations inside matter are very sensitive to the density of the medium traversed by the neutrinos. If the neutrino mixing angle (so far unknown) is not very small, then the neutrino oscillation length is of the size of the Earth, and we can use the phenomenon of neutrino matter oscillations to study the Earth's density distribution. Although neutrino attenuation data have already been proposed as a means to infer the density structure of the Earth, we show that neutrino oscillation data could also be used to make a density tomograph of the Earth. A numerical experiment in which neutrino beams from an accelerator are sent across the Earth, and are detected at six different locations on the Earth's surface, shows the feasibility of using neutrino oscillation data, with the advantage of using only currently available neutrino energies.

Application of internal gravitational field equations to geophysical measurement of G

We present the spherical harmonic equation for the gravitational potential inside the Earth. The equation satisfies Poisson's equation and converges uniformly. It obviates the need for downward continuation of the exterior potential with its attendant convergence difficulties but of course requires some knowledge of the Earth's density distribution. Equipped with the equation, we derive the general expression for the geophysical measurement of the gravitational constant G made inside the Earth, such as in boreholes and mine shafts. We also present numerical evidence that the long‐ to intermediate‐wavelength gravity anomalies can masquerade as the “fifth force” if not properly corrected for.

Density Modelling by Monte Carlo Inversion--I Methodology

Summary Monte Carlo inversion of geophysical data provides a method of specifying the Earth's density distribution within the uncertainties present in the data. It is possible to estimate the reliability of non-uniqueness bounds defined by a family of randomly generated models by means of a statistical procedure proposed by Anderssen & Seneta. We have devised a technique by which physically realistic models are generated without violating the condition of randomness, to counter the criticism that the method is inherently biased towards complex models. We have also examined a further criticism that the use of variational parameters to calculate theoretical eigenperiods favours the acceptance of -near (as opposed to -far) solutions. The objection is valid in principle, but may be circumvented by the adoption of suitably conservative acceptability criteria for the eigenperiod residuals. Our Monte Carlo technique was used to invert the seismic data from which the model HB2 is derived. The suite of models thus obtained provide a measure of the non-uniqueness inherent in the HB2 data, whilst confirming that the parametric constraints imposed on the density gradient by Bullen & Haddon are entirely compatible with their data. We conclude that major discrepancies between the density models of Bullen & Haddon and those of Press cannot be attributed to methodological differences in the derivation of these models.