Some modifications of Stokes' formula that account for truncation and potential coefficient errors

 Stokes' formula from 1849 is still the basis for the gravimetric determination of the geoid. The modification of the formula, originating with Molodensky, aims at reducing the truncation error outside a spherical cap of integration. This goal is still prevalent among various modifications. In contrast to these approaches, some least-squares types of modification that aim at reducing the truncation error, as well as the error stemming from the potential coefficients, are demonstrated. The least-squares estimators are provided in the two cases that (1) Stokes' kernel is a priori modified (e.g. according to Molodensky's approach) and (2) Stokes' kernel is optimally modified to minimize the global mean square error. Meissl-type modifications are also studied. In addition, the use of a higher than second-degree reference field versus the original (Pizzetti-type) reference field is discussed, and it is concluded that the former choice of reference field implies increased computer labour to achieve the same result as with the original reference field. Received: 14 December 1998 / Accepted: 4 October 1999     

Precise geoid determination over Sweden using the Stokes–Helmert method and improved topographic corrections

 Four different implementations of Stokes' formula are employed for the estimation of geoid heights over Sweden: the Vincent and Marsh (1974) model with the high-degree reference gravity field but no kernel modifications; modified Wong and Gore (1969) and Molodenskii et al. (1962) models, which use a high-degree reference gravity field and modification of Stokes' kernel; and a least-squares (LS) spectral weighting proposed by Sj?berg (1991). Classical topographic correction formulae are improved to consider long-wavelength contributions. The effect of a Bouguer shell is also included in the formulae, which is neglected in classical formulae due to planar approximation. The gravimetric geoid is compared with global positioning system (GPS)-levelling-derived geoid heights at 23 Swedish Permanent GPS Network SWEPOS stations distributed over Sweden. The LS method is in best agreement, with a 10.1-cm mean and ±5.5-cm standard deviation in the differences between gravimetric and GPS geoid heights. The gravimetric geoid was also fitted to the GPS-levelling-derived geoid using a four-parameter transformation model. The results after fitting also show the best consistency for the LS method, with the standard deviation of differences reduced to ±1.1 cm. For comparison, the NKG96 geoid yields a 17-cm mean and ±8-cm standard deviation of agreement with the same SWEPOS stations. After four-parameter fitting to the GPS stations, the standard deviation reduces to ±6.1 cm for the NKG96 geoid. It is concluded that the new corrections in this study improve the accuracy of the geoid. The final geoid heights range from 17.22 to 43.62 m with a mean value of 29.01 m. The standard errors of the computed geoid heights, through a simple error propagation of standard errors of mean anomalies, are also computed. They range from ±7.02 to ±13.05 cm. The global root-mean-square error of the LS model is the other estimation of the accuracy of the final geoid, and is computed to be ±28.6 cm. Received: 15 September 1999 / Accepted: 6 November 2000     

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.     

Equilibrium and structural studies of silicon(iv) and aluminium(iii) in aqueous solution. V. Acidity constants of silicic acid and the ionic product of water in the medium range 0.05–2.0 M Na(CI) at 25°C

The apparent ionization constants for silicic acid, k₁ and k₂, and the ionic product of water, k_w, have been determined in 0.05, 0.1, 0.2, 0.4 and 2.0 M Na(CI) media at 25°C. The medium dependence of these constants was found to fit equations of the form

$\text{log}\text{k}{}_{\mathrm{i}}\text{=}\text{log}\text{K}{}_{\mathrm{i}}\text{+a}{}_{\mathrm{i}}\text{I}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(1+I}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)+b}{}_{\mathrm{i}}\text{I}$

where K₁ is the ionization constant in pure water, α_i and b_i are parameters of which b_i has been adjusted to present data. The following results were obtained (α_i, b_i): pK₁ = 9.84, (1.022, ?0.11); pK₂ = 13.43, (2.044, ?0.20); and pK_w = 14.01 (1.022, ?0.22). k_i values are collected in Tables I and II. Attempts have been made to explain the medium dependence of k₁ and k₂ with weak sodium silicate complexing according to the equilibria

$\text{Na}{}^{\mathrm{+}}\text{+}\text{SiO(OH)}{}^{\mathrm{?}}{}_3\text{?}\text{NaSiO(OH)}{}_3\text{;k}{}_{11}$

$\text{Na}{}^{\mathrm{+}}\text{+}\text{SiO}{}_2\text{(OH)}{}^2{}_2\text{?}\text{NaSiO}{}_2\text{(HO)}{}^{\mathrm{?}}{}_2\text{; k}{}_{21}$

giving k₁₁ = 0.37M^?1 and k₂₁= 3.0M^?1. However, these weak interactions cannot be interpreted unambiguously from potentiometric data at different 1-levels. Probably the medium dependence could equally well be expressed by variations in the activity coefficients.The measurements were performed as potentiometric titrations using a hydrogen electrode. The average number of OH^- reacted per Si(OH)₄, Z, has been varied within the limits 0 ? Z ? 1.1 and B₁, the total concentration of Si(OH)₄, between 0.001 M and 0.008 M. k₁ was evaluated from experimental data with B ? 0.003 M, and k₂ with B ? 0.008 M and Z ? 0.95.     

The changing social relations of a community-based mangrove forest project in Zanzibar

Coastal areas in East Africa are experiencing rapid economic, resource management, demographic and technological shifts. In response diverse Community-based Natural Resource Management (CBNRM) applications have been embraced to provide mutual conservation and use benefits. These initiatives have met with mixed success in practice. Reflecting on the limitations of past research using common pool resources theory theory to study CBNRM we use insights from actor oriented theory combined with satellite image analysis to describe and discuss the forces dynamically influencing institutional and mangrove forest cover change at Kisakasaka, Zanzibar focussing on the formal CBNRM project period between 1996 and 2001, but also considering the period before and after this. We examine the shifting social relations that affected the performance and viability of the formal CBNRM arrangements. An integrated approach was taken to the presentation and discussion of results where it was possible to enrich and expand explanations of socio-environmental change, which was driven by a lack of government support, the undermining effects of party political divisions, and the lack of institutional adaptive capacity. We conclude that this was a useful approach to explain CBNRM intervention events at Kisakasaka.     

Baltic Sea Level project with GPS

In order to study the Baltic Sea Level change and to unify national height systems a two week GPS campaign was performed in the region in Autumn 1990. Parties from Denmark, Finland, Germany, Poland and Sweden carried out GPS measurements at 26 tide gauges along the Baltic sea and 8 VLBI and SLR fiducial stations with baseline lengths ranging from 230 km to 1600 km. The observations were processed in the network mode with the Bernese version 3.3 software using orbit improvement techniques. To get rid of the scale error introduced by the ionospheric refraction from single-frequency data, the local models of the ionosphere were estimated using L4 observations. The tropospheric zenith corrections were also considered. The preliminary results show average root mean square (RMS) errors of about ±3 cm in the horizontal position and ±7 cm in the vertical position relative to the Potsdam SLR station in ITRF89 system. After transformation of the GPS results to geoid heights using the levelled heights, an absolute comparison with gravimetric geoid heights using the least squares modification of Stokes' formula (LSMS), the modified Molodensky and the NKG Scandinavian geoid 1989 (NGK-89) models gives a standard deviation of the difference of ±7cm to ±9cm for the NKG-89 model and of ±9cm to ±30cm for the LSMS and the modified Molodensky model. The Swedish height system is found to be about 8-37cm higher than those of the other Baltic countries for NKG-89 model.     

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.     

A recurrence relation for the βn-function

A recurrence relation is presented for the smoothing function, β_n, which is used in geodesy to relate spherical harmonics to their mean values over circular areas (caps). The proposed formula does not require the computation of the Legendre's polynomials. Moreover, it is numerically more stable than the formulas ofPellinen (1969) andMeissl (1971).     

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.