Statistics and Data Analysis in Geology

The robust estimation of geodetic datum transformation is discussed. The basic principle of robust estimation is introduced. The error influence functions of the robust estimators, together with those of least-squares estimators, are given. Particular attention is given to the robust initial estimates of the transformation parameters, which should have a high breakdown point in order to provide reliable residuals for the following estimation. The median method is applied to solve for robust initial estimates of transformation parameters since it has the highest breakdown point. A smooth weight function is then used to improve the efficiency of the parameter estimates in successive iterative computations. A numerical example is given on a datum transformation between a global positioning system network and the corresponding geodetic network in China. The results show that when the coordinates are contaminated by outliers, the proposed method can still give reasonable results.

Robust estimation of geodetic datum transformation

Major unconformities/termination of extension events and associated surfaces in the South China Seas: Review and implications for tectonic development

The mechanisms driving plate motion and the Earth's geodynamics are still not entirely clarified. Lithospheric volumes recycled at subduction zones or emerging at rift zones testify mantle convection. The cooling of the planet and the related density gradients are invoked to explain mantle convection either driven from the hot interior or from the cooler outer boundary layer. In this paper we summarize a number of evidence supporting generalized asymmetries along the plate boundaries that point to a polarization of plate tectonics. W-directed slabs provide two to three times larger volumes to the mantle with respect to the opposite E- or NE-directed subduction zones. W-directed slabs are deeper and steeper, usually characterized by down-dip compression. Moreover, they show a shallow decollement and low elevated accretionary prism, a steep regional monocline with a deep trench or foredeep, a backarc basin with high heat flow and positive gravity anomaly. Conversely directed subduction zones show antithetic signatures and no similar backarc basin. Rift zones also show an asymmetry, e.g., faster Vs in the western lithosphere and a slightly deeper bathymetry with respect to the eastern flank. These evidences can be linked to the westward drift of the lithosphere relative to the underlying mantle and may explain the differences among subduction and rift zones as a function of their geographic polarity with respect to the “tectonic equator.” Therefore also mantle convection and plate motion should be polarized. All this supports a general tuning of the Earth's geodynamics and mantle convection by astronomical forces.

Polarized Plate Tectonics

Computing Helmert transformations

The regularized solution of the external sphericalStokes boundary value problem as being used for computations of geoid undulations and deflections of the vertical is based upon theGreen functions S
1(Λ0, Φ0, Λ, Φ) ofBox 0.1 (R = R
0) andV
1(Λ0, Φ0, Λ, Φ) ofBox 0.2 (R = R
0) which depend on theevaluation point {Λ0, Φ0} ∈ S
 R0
 2
 and thesampling point {Λ, Φ} ∈ S
 R0
 2
 ofgravity anomalies Δ
 γ
 (Λ, Φ) with respect to a normal gravitational field of typegm/R (”free air anomaly”). If the evaluation point is taken as the meta-north pole of theStokes reference sphere S
 R0
 2
 , theStokes function, and theVening-Meinesz function, respectively, takes the formS(Ψ) ofBox 0.1, andV
2(Ψ) ofBox 0.2, respectively, as soon as we introduce {meta-longitude (azimuth), meta-colatitude (spherical distance)}, namely {A, Ψ} ofBox 0.5. In order to deriveStokes functions andVening-Meinesz functions as well as their integrals, theStokes andVening-Meinesz functionals, in aconvolutive form we map the sampling point {Λ, Φ} onto the tangent plane T0S
 R0
 2
 at {Λ0, Φ0} by means ofoblique map projections of type(i) equidistant (Riemann polar/normal coordinates),(ii) conformal and(iii) equiareal.Box 2.1.–2.4. andBox 3.1.– 3.4. are collections of the rigorously transformedconvolutive Stokes functions andStokes integrals andconvolutive Vening-Meinesz functions andVening-Meinesz integrals. The graphs of the correspondingStokes functions S
2(Ψ),S
3(r),⋯,S
6(r) as well as the correspondingStokes-Helmert functions H
2(Ψ),H
3(r),⋯,H
6(r) are given byFigure 4.1–4.5. In contrast, the graphs ofFigure 4.6–4.10 illustrate the correspondingVening-Meinesz functions V
2(Ψ),V
3(r),⋯,V
6(r) as well as the correspondingVening-Meinesz-Helmert functions Q
2(Ψ),Q
3(r),⋯,Q
6(r). The difference between theStokes functions / Vening-Meinesz functions andtheir first term (only used in the Flat Fourier Transforms of type FAST and FASZ), namelyS
2(Ψ) − (sin Ψ/2)−1,S
3(r) − (sinr/2R
0)−1,⋯,S
6(r) − 2R
0/r andV
2(Ψ) + (cos Ψ/2)/2(sin2 Ψ/2),V
3(r) + (cosr/2R
0)/2(sin2
r/2R
0),⋯,
 
$$V_6 (r) + {{(R_0 \sqrt {4R_0^2 - r^2 } )} \mathord{\left/ {\vphantom {{(R_0 \sqrt {4R_0^2 - r^2 } )} {r^2 }}} \right. \kern-\nulldelimiterspace} {r^2 }}$$

 illustrate the systematic errors in the”flat” Stokes function 2/Ψ or ”flat”Vening-Meinesz function −2/Ψ2. The newly derivedStokes functions S
3(r),⋯,S
6(r) ofBox 2.1–2.3, ofStokes integrals ofBox 2.4, as well asVening-Meinesz functionsV
3(r),⋯,V
6(r) ofBox 3.1–3.3, ofVening-Meinesz integrals ofBox 3.4 — all of convolutive type — pave the way for the rigorousFast Fourier Transform and the rigorousWavelet Transform of theStokes integral / theVening-Meinesz integral of type ”equidistant”, ”conformal” and ”equiareal”.

The Stokes and Vening-Meinesz functionals in a moving tangent space

The global earthquake catalog of seismic events with M W ≥ 7.0, for the time interval from 1950 to 2007, shows that the depth distribution of earthquake energy release is not uniform. The 90% of the total earthquake energy budget is dissipated in the first ~30km, whereas most of the residual budget is radiated at the lower boundary of the transition zone (410 km - 660 km), above the upper-lower mantle boundary. The upper border of the transition zone at around 410 km of depth is not marked by significant seismic energy release. This points for a non-dominant role of the slabs in the energy budged of plate tectonics. Earthquake number and energy release, although not well correlated, when analysed with respect to the latitude, show a decrease toward the polar areas. Moreover, the radiated energy has the highest peak close to (±5°) the so-called tectonic equator defined by Crespi et al. (2007), which is inclined about 30° with respect to the geographic equator. At the same time the presence of a clear axial co-ordination of the radiated seismic energy is demonstrated with maxima at latitudes close to critical (±45°). This speaks about the presence of external forces that influence seismicity and it is consistent with the fact that Gutenberg-Richter law is linear, for events with

http://users.ictp.it/~pub_off/preprints-sources/2010/IC2010055P.pdf

Earthquake energy distribution along the earth surface and radius

The upward-downward continuation of a harmonic function like the gravitational potential is conventionally based on the direct-inverse Abel-Poisson integral with respect to a sphere of reference. Here we aim at an error estimation of the “planar approximation” of the Abel-Poisson kernel, which is often used due to its convolution form. Such a convolution form is a prerequisite to applying fast Fourier transformation techniques. By means of an oblique azimuthal map projection / projection onto the local tangent plane at an evaluation point of the reference sphere of type “equiareal” we arrive at a rigorous transformation of the Abel-Poisson kernel/Abel-Poisson integral in a convolution form. As soon as we expand the “equiareal” Abel-Poisson kernel/Abel-Poisson integral we gain the “planar approximation”. The differences between the exact Abel-Poisson kernel of type “equiareal” and the “planar approximation” are plotted and tabulated. Six configurations are studied in detail in order to document the error budget, which varies from 0.1% for points at a spherical height H=10km above the terrestrial reference sphere up to 98% for points at a spherical height H = 6.3×106km.

The Abel-Poisson kernel and the Abel-Poisson integral in a moving tangent space

In present, contribution paleogeographical maps for the time interval 0.6 Ga BP to present are analyzed in terms of (a) the ratio between continental to oceanic crust areas in order to estimate the speed of continental growth and (b) the surface motion of continental plates under the influence of global forces of tidal friction and Eotvos force (“pole-fleeing”). It is concluded that the area of the continents during the Phanerozoic was growing and it exhibited a rate ~0.5 km3/year. It is also found that beside the westward-oriented tidal frictional forces the Eotvos force can play also a role in tectonical processes. It is shown that the continental plates on average tend to find a position close to the equator during the whole investigated 600 Ma time interval.

/pdf/evolution-of-the-oceanic-and-continental-crust-during-neo-15tkv4rdl9.pdf

Evolution of the oceanic and continental crust during Neo-Proterozoic and Phanerozoic

Criterion matrices/ideal variance — covariance matrices are constructed from continuous networks being observed by signal derivatives of first and second order. The method of constrained least- squares leads to differential equations up to fourth order, to characteristic boundary values and constraints according to the datum choice. Here only onedimensional networks on a line and on a circle are discussed. Their characteristic (modified) Green function is constructed and used for the computation of the variance — covariance function of adjusted mean signal functions. Finally numerical aspects originating fm the discrete nature of real observational series are discussed. In detail, the transformation of a criterion matrix into a network datum and its comparison with the variance — covariance matrix of an ideally configurated network is presented.

https://link.springer.com/content/pdf/10.1007%2F978-3-642-70659-2_13.pdf

F. Krumm

Papers

The Stokes and Vening-Meinesz functionals in a moving tangent space

Earthquake energy distribution along the earth surface and radius

The Abel-Poisson kernel and the Abel-Poisson integral in a moving tangent space

Evolution of the oceanic and continental crust during Neo-Proterozoic and Phanerozoic

Continuous Networks I