Unification of vertical datums by GPS and gravimetric geoid models using modified stokes formula

In general, neighboring vertical datums can be compared directly at one or more common points on the border between the datums. This direct method requires leveling and gravity measurements. Such a direct connection is not possible if the datums are separated by an ocean or another body of water. Then a rigorous mathematical model, an indirect approach, may be useful. In order to connect regional vertical datums, a rigorous mathematical model is derived based on a method by Rummel and Teunissen. In this study, two vertical datums are connected indirectly by means of a combination of precise geocentric positions of tide gauge sites and their geoid heights in one geocentric coordinate system and their height values in the respective height datums. This method is used to connect the Swedish and the Finnish height systems. The difference between Swedish RH70 and the Finnish N60 height systems is estimated to —12.1±2.7 cm. The results are mostly in good agreement with those of the direct approach by Sjöberg and by Ekman and the indirect approach by Pan and Sjöberg.     

Design methods for stopes and sill pillars with application to the Zinkgruvan Mine,central Sweden

Summary The design of pillars in cut-and-fill mining and open stoping in vertical and subvertical orebodies is of vital importance in optimizing mining operations. The primary requirement for a good and reliable design technique is the ability to represent the actual physical behaviour of the pillar. In this paper, a new methodology for stope roof and sill pillar design is proposed for the Zinkgruvan Mine in Sweden. Studies of failure modes, local geology and rock mass characteristics were carried out to correlate failure modes to different geomechanical environments. For preliminary design, crude and ready-to-use stress level criteria were extracted from simple linear elastic modelling. More detailed modelling was used to simulate observed failures in a mechanically realistic manner and, at the same time, to help identify the fundamental failure mechanisms. Once the correct models and input parameters were identified, a set of guidelines on choice of model and parameter values were produced. The models can be used for design of new mining areas at Zinkgruvan, and the methodology could also be applied to other mines with similar geomechanical conditions.     

Structures and stabilities of Cd(II) and Cd(II)-phthalate complexes at the goethite/water interface

The complexation of Cd(II) and Cd(II)-phthalate at the goethite/water interface were investigated by EXAFS and IR spectroscopy, by batch adsorption experiments and by potentiometric titrations at 298.15 K. The EXAFS spectra showed Cd(II) to form only inner-sphere corner-sharing complexes with the goethite surface sites in the presence and absence of phthalate. EXAFS spectra also showed the presence of Cd(II)-chloride complexes in 0.1 mol/L NaCl. IR spectra also showed phthalate to form (1) an inner-sphere complex with adsorbed corner-sharing Cd(II) surface complexes in the pH 3.5 to 9.5 and (2) an outer-sphere complex with the same type of corner-sharing Cd(II) complex however at pH > 6, in addition to the inner- and outer-sphere complexes of phthalate reported in a previous study. The potentiometric titration and the batch adsorption data were used to constrain the formation constants of the different Cd(II)-phthalate surface complexes on the dominant {110} and the {001} planes of the goethite. The models were carried out with the Charge Distribution Multisite Complexation model coupled to the Three Plane Model and can predict the molecular-scale speciation of cadmium and phthalate in the presence of goethite. Cd(II) adsorption models calibrated on a 90 m²/g goethite also could accurately predict experimental data for a 37 m²/g goethite of slightly different basic charging properties.     

Quasigeoid-to-geoid determination by EGM08

We present a method to estimate the difference between quasigeoid and geoid heights globally from the Earth Gravitational Model EGM08 and a related topographic model. The numerical computations with the standard topographic density of 2.67 g/cm³ show that the maximum and minimum of the separations are estimated to 5.47 m and −0.11 m on the Tibet plateau and in the Indian Ocean, respectively. These estimates do not consider possible topographic density variations, which result in topographic bias changes proportional to the topographic elevation squared. Assuming such density changes of 10% from the standard value, the separation may change up to 5 dm.     

Åland GPS levelling campaign in 1987

In October 1987 a four day satellite GPS campaign was performed over the Åland archipelago to test the possibility of connecting the Swedish and Finnish national height systems. This paper summarizes the gained experiences using 5 WM 101 GPS receivers and the PoPS software.The computing results for the connection between the two height systems are considerably dependent on the choice of geoidal undulation model and systematic error parameter model. Using the NKG Scandinavian geoid 1989, which is probably the most accurate geoid available for the region, and a bias and tilt parameter model the difference between the Swedish RH70 system and the Finnish N60 system is estimated to 11.4 ± 4.0 cm. An independent check is provided by two connecting border bench marks in northern Scandinavia yielding the difference 19.2 ± 4.2 cm. In view of that merely single frequency GPS receivers were used together with the PoPS software, we consider this result most satisfactory.     

Phase Transformations and Proton Promoted Dissolution of Hydrous Manganite (γ-MnOOH)

The objective of this study was to describe the proton promoted disproportion of synthetic manganite (γ-MnOOH) and to characterise the resulting phase transformations. The solution and remaining solid phase after disproportionation was analysed by techniques including atomic absorbance spectroscopy, X-ray diffraction (XRD), atomic force microscopy (AFM) and scanning electron microscopy (SEM). In suspensions with pH between 5 and 7, −log[H⁺] was monitored for 17 months and equilibrium constants were determined at 9, 12 and 17 months of reaction time for the following reaction (25 °C, 0.1 M (Na)NO₃):

A general model for modifying Stokes’ formula and its least-squares solution

Today the combination of Stokes formula and an Earth gravity model (EGM) for geoid determination is a standard procedure. However, the method of modifying Stokes formula varies from author to author, and numerous methods of modification exist. Most methods modify Stokes kernel, but the most widely applied method, the remove compute restore technique, removes the EGM from the gravity anomaly to attain a residual gravity anomaly under Stokes integral, and at least one known method modifies both Stokes kernel and the gravity anomaly. A general model for modifying Stokes formula is presented; it includes most of the well-known techniques of modification as special cases. By assuming that the error spectra of the gravity anomalies and the EGM are known, the optimum model of modification is derived based on the least-squares principle. This solution minimizes the expected mean square error (MSE) of all possible solutions of the general geoid model. A practical formula for estimating the MSE is also presented. The power of the optimum method is demonstrated in two special cases. AcknowledgementsThis paper was partly written whilst the author was a visiting scientist at The University of New South Wales, Sydney, Australia. He is indebted to Professor W. Kearsley and his colleagues, and their hospitality is acknowledged.     

Mechanism of pyroxene and amphibole weathering—I. Experimental studies of iron-free minerals

The short term (2–40 days) dissolution of enstatite, diopside, and tremolite in aqueous solution at low temperatures (20–60°C) and pH 1–6 has been studied in the laboratory by means of chemical analyses of reacting solutions for Ca²⁺, Mg²⁺, and Si(OH)₄ and by the use of X-ray photoelectron spectroscopy (XPS) for detecting changes in surface chemistry of the minerals. All three minerals were found to release silica at a constant rate (linear kinetics) providing that ultrafine particles, produced by grinding, were removed initially by HF treatment. All three also underwent incongruent dissolution with preferential release of Ca and/or Mg relative to Si from their outermost surfaces. The preferential release of Ca, but not Mg for diopside at pH 6 was found by both XPS and solution chemistry verifying the theoretical prediction of greater mobility of cations located in M₂ structural sites. Loss mainly from M₂ sites also explains the degree of preferential loss of Mg from enstatite at pH 6; similar structural arguments apply to the loss of Ca and Mg from the surface of tremolite. In the case of diopside and tremolite initial incongruency was followed by essentially congruent cation-plus-silica dissolution indicating rapid formation of a constant-thickness, cation-depleted surface layer. Cation depletion at elevated temperature and low pH (~ 1) for enstatite and diopside was much greater than at low temperature and neutral pH, and continued reaction resulted in the formation of a surface precipitate of pure silica as indicated by solubility calculations, XPS analyses, and scanning electron microscopy.From XPS results at pH 6, model calculations indicate a cation-depleted altered surface layer of only a few atoms thickness in all three minerals. Also, lack of shifts in XPS peak energies for Si, Ca, and Mg, along with undersaturation of solutions with respect to all known Mg and Ca silicate minerals, suggest that cation depletion results from the substitution of hydrogen ion for Ca²⁺ and/or Mg²⁺ in a modified silicate structure and not from the precipitation of a new, radically different surface phase. These results, combined with findings of high activation energies for dissolution, a non-linear dependence on a_H⁺ for silica release from enstatite and diopside, and the occurrence of etch pitting, all point to surface chemical reaction and not bulk diffusion (either in solution or through altered surface layers) as the rate controlling mechanism of iron-free pyroxene and amphibole dissolution at earth surface temperatures.     

A solubility and surface complexation study of a non-stoichiometric hydroxyapatite

The dissolution and surface complexation of a non-stoichiometric hydroxyapatite (Ca_8.4(HPO₄)_1.6(PO₄)_4.4(OH)_0.4), (HAP) was studied in the pH range 3.5-10.5, at 25 °C in 0.1 M Na(Cl). The results from well-equilibrated batch experiments, potentiometric titrations, and zeta-potential measurements were combined with information provided by Attenuated Total Reflectance Fourier Transform Infrared (ATR-FTIR) spectroscopy and X-ray Photoelectron Spectroscopy (XPS). The information from the analyses was used to design an equilibration model that takes into account dissolution, surface potential, solution and surface complexation, as well as possible phase transformations. The results from the XPS measurements clearly show that the surface of the mineral has a different composition than the bulk and that the Ca/P ratio of the surface layer is 1.4 ± 0.1. This ratio was also found in solution in the batches equilibrated at low pH where the dominating reaction is dissolution. In the batches equilibrated at near neutral pH values, however, the Ca/P ratio in solution attains values as high as 25, which is due to re-adsorption of phosphate ions to the HAP surface. The total concentration of protons as well as the total concentration of dissolved calcium and phosphate in solution were used to calculate a model for the dissolution and surface complexation of HAP. The constant capacitance model was applied in designing the following surface complexation model:

     

A solution to the downward continuation effect on the geoid determined by Stokes' formula

 The analytical continuation of the surface gravity anomaly to sea level is a necessary correction in the application of Stokes' formula for geoid estimation. This process is frequently performed by the inversion of Poisson's integral formula for a sphere. Unfortunately, this integral equation corresponds to an improperly posed problem, and the solution is both numerically unstable, unless it is well smoothed, and tedious to compute. A solution that avoids the intermediate step of downward continuation of the gravity anomaly is presented. Instead the effect on the geoid as provided by Stokes' formula is studied directly. The practical solution is partly presented in terms of a truncated Taylor series and partly as a truncated series of spherical harmonics. Some simple numerical estimates show that the solution mostly meets the requests of a 1-cm geoid model, but the truncation error of the far zone must be studied more precisely for high altitudes of the computation point. In addition, it should be emphasized that the derived solution is more computer efficient than the detour by Poisson's integral. Received: 6 February 2002 / Accepted: 18 November 2002 Acknowledgements. Jonas ?gren carried out the numerical calculations and gave some critical and constructive remarks on a draft version of the paper. This support is cordially acknowledged. Also, the thorough work performed by one unknown reviewer is very much appreciated.