Stokes Integral Research Articles

Modification methods of the Stokes’ kernel for determining the (quasi-) geoid with the Remove-Compute-Restore technique

The geoid and quasi-geoid serve as the reference surfaces of the orthometric and normal height systems, respectively. In order to improve the accuracy of the (quasi-) geoid determined by the Stokes integral with use of the Remove-Compute-Restore (RCR) technique, various modification methods for the spherical Stokes’ kernels, including the spheroidal, cosine-, power-, and Molodensky-modified kernels, are studied in this paper. In addition to the traditional Molodensky-modified Stokes’ kernel, a more effective Molodensky-modified Stokes’ kernel is put forward. A general formula for spectral decomposition of the Stokes integral in the RCR mode is derived, followed by the spectral analysis to reveal the transfer principles of gravity data when using different Stokes’ kernels. The spheroidal and modified Stokes integrals can cause spectral leakage phenomenon, and a method to eliminate spectral leakage is presented based on spectral analysis. The research indicates the low truncation degree of the spheroidal Stokes’ kernel and the low modification degrees of the modified Stokes’ kernel affect the accuracy of the (quasi-) geoid significantly. Quantitative methods for estimating the empirical values of the parameters of the low-degree spheroidal and modified Stokes’ kernels are proposed and the effectiveness of the methods is validated through numerical tests.

An assessment of the implications of gravity reduction methods on local geoid modelling over Ado Ekiti

There are several gravity reduction approaches that are available for use in geodetic applications. More so, with the rapid rate at which gravimetric geoid models are being incorporated into Global Navigation Satellite System (GNSS) solutions, the need for precise regional models is becoming increasingly relevant. The two chosen gravity reduction approaches (Bouguer and Residual Terrain Model (RTM)) were used to reduce gravity anomalies over Ado town. The reduced anomalies were then used to compute a local geoid for the study area using the conventional Stokes integral in the Remove Compute Restore (RCR) technique. Comparison of the computed geoid by both methods with GNSS Levelling data at selected locations show that the RTM reduced anomalies produce a better geoid model than the Bouguer reduced anomalies over the study area with a RMSE of 83.2cm and 83.5cm respectively.

Integration of satellite geodetic observations for regional geoid modeling using remove-compute-restore technique

Advanced satellite geodetic systems have contributed to improving knowledge on changes in global gravity fields in recent years. These systems offer profound opportunities in the determination of a high-precision, high-resolution gravimetric geoid models in data deficient regions. However, due to the absence of research competence in some regions, these conventional datasets have not been implemented to update the obsolete reference frames which have been in use in these regions. This study introduces the well-known Remove-Compute-Restore (RCR) technique in modeling a gravimetric geoid model for a large data deficient region in West Africa (Nigeria) using two sets of long and short wavelength data (a) EGM2008 (long) + Airborne gravimetric observation (AGO) dataset (short) (b) EGM2008 (long) + Terrestrial gravimetric undulation (TGU) dataset (short). This therefore resulted in obtaining two sets of resultant RCR-gravimetric details whose statistics comparative assessments are produced in the body of this work. The RCR-determined gravimetric geoid model showed a strong relationship with the observed terrestrial data. The goodness of fit for the orthometric height correlation between the computed gravimetric geoid and the terrestrial data is 97.87 %. Strong relationships between the computed model and the other height models determined from the primary data are also observed. The difference between the Flury and rummel geoid-based model and the Heiskanen and moritz model in the geoid-quasi geoid separation phase shows the importance of implementing the ‘rigorous modeling technique’ for geoid quasi-geoid separation instead of the approximation method. However, the discrepancy between both models can be overlooked in a case of low accuracy geoidal operations. The RCR-based Fast Fourier Transform (FFT), and the least squares modified stokes (LSMS) was used in the quasi-geoid determination of the study region. Using the 1-parameter fit, the modified Stokes integral (FFT(b)) outperformed the LSMS model with a root mean square (rms) difference of 0.6 cm, while the LSMS outperformed both the unmodified (FFT(a)) and the modified (FFT(b)) Stokes integral using the 4-parameter fit with a rms difference of 0.5 cm. The use of both models in this study created a better perception of the quasi-geoid differential analysis, and a more refined understanding of their interaction with the satellite based quasi geoid.

Unbiased least-squares modification of Stokes\u2019 formula

As the KTH method for geoid determination by combining Stokes integration of gravity data in a spherical cap around the computation point and a series of spherical harmonics suffers from a bias due to truncation of the data sets, this method is based on minimizing the global mean square error (MSE) of the estimator. However, if the harmonic series is increased to a sufficiently high degree, the truncation error can be considered as negligible, and the optimization based on the local variance of the geoid estimator makes fair sense. Such unbiased types of estimators, derived in this article, have the advantage to the MSE solutions not to rely on the imperfectly known gravity signal degree variances, but only the local error covariance matrices of the observables come to play. Obviously, the geoid solution defined by the local least variance is generally superior to the solution based on the global MSE. It is also shown, at least theoretically, that the unbiased geoid solutions based on the KTH method and remove–compute–restore technique with modification of Stokes formula are the same.

Application of the one-step integration method for determination of the regional gravimetric geoid

The regional gravimetric geoid solved using boundary-value problems of the potential theory is usually determined in two computational steps: (1) downward continuing ground gravity data onto the geoid using inverse Poisson’s integral equation in a mass-free space and (2) evaluating geoidal heights by applying Stokes integral to downward continued gravity. In this contribution, the two integration steps are combined in one step and the so-called one-step integration method in spherical approximation is implemented to compute the regional gravimetric geoid model. Advantages of using the one-step integration method instead of the two integration steps include less computational cost, more stable numerical computation and better utilization of input ground gravity data (reduced in each integration step to avoid edge effects). A discrete form of the one-step integral equation is used to convert mean values of ground gravity anomalies into mean values of geoidal heights. To evaluate mean values of the integral kernel in the vicinity of the computation point, a fast and numerically accurate analytical formula is proposed using planar approximation. The proposed formula is tested to determine the regional gravimetric geoid of the Auvergne test area, France. Results show a good agreement of the estimated geoid with geoidal heights estimated at GNSS-levelling reference points, with the standard deviation for the difference of 3.3 cm. Considering the uncertainty of geoidal heights derived at the GNSS/levelling reference points, one can conclude the geoid models computed by the one-step and two-step integration methods have negligible differences. Thus, the one-step method can be recommended for regional geoid modelling with its methodological and numerical advantages.

A high-resolution gravimetric quasigeoid model for Vietnam

A high-resolution gravimetric quasigeoid model for Vietnam and its surrounding areas was determined based on new gravity data. A set of 29,121 land gravity measurements was used in combination with fill-in data where no gravity data existed. Global Gravity field Models plus Residual Terrain Model effects and gravity field derived from altimetry satellites were used to provide the fill-in information over land and marine areas. A mixed model up to degree/order 719 was used for the removal of the long and medium wavelengths and the calculation of the quasigeoid restore effects. The residual height anomalies have been determined employing the Stokes integral using the Fast Fourier Transform approach and deterministic kernel modification proposed by Wong–Gore, as well as by means of Least-Squares Collocation. The accuracy of the resulting quasigeoid models was evaluated by comparing with height anomalies derived from 812 co-located GNSS/levelling points. Results are very similar; both local quasigeoid models have a standard deviation of 9.7 cm and 50 cm in mean bias when compared to the GNSS/levelling points. This new local quasigeoid model for Vietnam represents a significant improvement over the global models EIGEN-6C4 and EGM2008, which have standard deviations of 19.2 and 29.1 cm, respectively, for this region.

Comparison of Three Gravimetric-Geometric Geoid Models for Best Local Geoid Model of Benin City, Nigeria

The conversion of geometric as well as ellipsoidal heights from GNSS observations to practical heights for engineering constructions has necessitated the determination of the local geoid model of areas. Benin City is a developing area which requires a local geoid model for conversion of geometric heights to orthometric heights for physical developments in the area. This paper is on the best local geoid model of Benin City, Nigeria by comparing three gravimetric-geometric geoid models of the study area. GNSS and gravimetric observations were carried out on 49 points to respectively obtain their coordinates and absolute gravity values. The theoretical gravity values of the points were computed on the Clarke 1880 ellipsoid, subtracted from the absolute gravity values and corrected for the air (free air) to obtain the free air gravity anomalies of the points. The computed free air gravity anomalies were applied to compute the geoid heights of the points using the integration of the modified Stokes integral. Three geometric geoid surfaces (plane, second degree and third degree surfaces) were fitted to the computed gravimetric geoid heights using the least squares technique to obtain the gravimetric-geometric geoid models of the study area. The RMSE of the three gravimetric-geometric geoid models were computed to determine their (the models) accuracy. The three gravimetric-geometric geoid models were compared using their accuracy to obtain the most suitable geoid model of the study area. The results of the comparison showed that the third degree gravimetric-geometric geoid model is most suitable for application in the study area. It is recommended that ellipsoidal heights obtained from GNSS observation within Benin City, Nigeria should be converted to orthometric heights using the third degree geoid model.

The New Zealand gravimetric quasigeoid model 2017 that incorporates nationwide airborne gravimetry

A one arc-minute resolution gravimetric quasigeoid model has been computed for New Zealand, covering the region $$25^{\circ }\hbox {S}$$ – $$60^{\circ }\hbox {S}$$ and $$160^{\circ }\hbox {E}$$ – $$170^{\circ }\hbox {W}$$ . It was calculated by Wong and Gore modified Stokes integration using the remove–compute–restore technique with the EIGEN-6C4 global gravity model as the reference field. The gridded gravity data used for the computation consisted of 40,677 land gravity observations, satellite altimetry-derived marine gravity anomalies, historical shipborne marine gravity observations and, importantly, approximately one million new airborne gravity observations. The airborne data were collected with the specific intention of reinforcing the shortcomings of the existing data in areas of rough topography inaccessible to land gravimetry and in coastal areas where shipborne gravimetry cannot be collected and altimeter-derived gravity anomalies are generally poor. The new quasigeoid has a nominal precision of $$\pm \,48\,\hbox {mm}$$ on comparison with GPS-levelling data, which is approximately $$14\,\hbox {mm}$$ less than its predecessor NZGeoid09.

The first Australian gravimetric quasigeoid model with location-specific uncertainty estimates

We describe the computation of the first Australian quasigeoid model to include error estimates as a function of location that have been propagated from uncertainties in the EGM2008 global model, land and altimeter-derived gravity anomalies and terrain corrections. The model has been extended to include Australia’s offshore territories and maritime boundaries using newer datasets comprising an additional {sim }280,000 land gravity observations, a newer altimeter-derived marine gravity anomaly grid, and terrain corrections at 1^{prime prime }times 1^{prime prime } resolution. The error propagation uses a remove–restore approach, where the EGM2008 quasigeoid and gravity anomaly error grids are augmented by errors propagated through a modified Stokes integral from the errors in the altimeter gravity anomalies, land gravity observations and terrain corrections. The gravimetric quasigeoid errors (one sigma) are 50–60 mm across most of the Australian landmass, increasing to {sim }100 mm in regions of steep horizontal gravity gradients or the mountains, and are commensurate with external estimates.

On the equivalence of spherical splines with least-squares collocation and Stokes\u2019s formula for regional geoid computation

This work is an investigation of three methods for regional geoid computation: Stokes’s formula, least-squares collocation (LSC), and spherical radial base functions (RBFs) using the spline kernel (SK). It is a first attempt to compare the three methods theoretically and numerically in a unified framework. While Stokes integration and LSC may be regarded as classic methods for regional geoid computation, RBFs may still be regarded as a modern approach. All methods are theoretically equal when applied globally, and we therefore expect them to give comparable results in regional applications. However, it has been shown by de Min (Bull Géod 69:223–232, 1995. doi:10.1007/BF00806734) that the equivalence of Stokes’s formula and LSC does not hold in regional applications without modifying the cross-covariance function. In order to make all methods comparable in regional applications, the corresponding modification has been introduced also in the SK. Ultimately, we present numerical examples comparing Stokes’s formula, LSC, and SKs in a closed-loop environment using synthetic noise-free data, to verify their equivalence. All agree on the millimeter level.

Boundary integral method for the flow of vesicles with viscosity contrast in three dimensions

We propose numerical algorithms for the simulation of the dynamics of three-dimensional vesicles suspended in viscous Stokesian fluid. Our method is an extension of our previous work (S.K. Veerapaneni et al., 2011) [37] to flows with viscosity contrast. This generalization requires a change in the boundary integral formulation of the solution, in which a double-layer Stokes integral is introduced, and leads to changes in the fluid dynamics due to the viscosity contrast of the vesicles, which can no longer be efficiently resolved with existing algorithms.In this paper we describe the algorithms needed to handle flows with viscosity contrast accurately and efficiently. We show that a globally semi-implicit method does not have any time-step stability constraint for flows with single and multiple vesicles with moderate viscosity contrast and the computational cost per simulation unit time is comparable to or less than that of an explicit scheme. Automatic oversampling adaptation enables us to achieve high accuracy with very low spectral resolution. We conduct numerical experiments to investigate the stability, accuracy, and the computational cost of the algorithms. Overall, our method achieves several orders of magnitude speed-up compared to the standard explicit schemes.

Canadian gravimetric geoid model 2010

A new gravimetric geoid model, Canadian Gravimetric Geoid 2010 (CGG2010), has been developed to upgrade the previous geoid model CGG2005. CGG2010 represents the separation between the reference ellipsoid of GRS80 and the Earth’s equipotential surface of $$W_0=62{,}636{,}855.69~\mathrm{m}^2\mathrm{s}^{-2}$$ . The Stokes–Helmert method has been re-formulated for the determination of CGG2010 by a new Stokes kernel modification. It reduces the effect of the systematic error in the Canadian terrestrial gravity data on the geoid to the level below 2 cm from about 20 cm using other existing modification techniques, and renders a smooth spectral combination of the satellite and terrestrial gravity data. The long wavelength components of CGG2010 include the GOCE contribution contained in a combined GRACE and GOCE geopotential model: GOCO01S, which ranges from $$-20.1$$ to 16.7 cm with an RMS of 2.9 cm. Improvement has been also achieved through the refinement of geoid modelling procedure and the use of new data. (1) The downward continuation effect has been accounted accurately ranging from $$-22.1$$ to 16.5 cm with an RMS of 0.9 cm. (2) The geoid residual from the Stokes integral is reduced to 4 cm in RMS by the use of an ultra-high degree spherical harmonic representation of global elevation model for deriving the reference Helmert field in conjunction with a derived global geopotential model. (3) The Canadian gravimetric geoid model is published for the first time with associated error estimates. In addition, CGG2010 includes the new marine gravity data, ArcGP gravity grids, and the new Canadian Digital Elevation Data (CDED) 1:50K. CGG2010 is compared to GPS-levelling data in Canada. The standard deviations are estimated to vary from 2 to 10 cm with the largest error in the mountainous areas of western Canada. We demonstrate its improvement over the previous models CGG2005 and EGM2008.

Introduction to macro-econophysics and finance

Closed integrals in physics lead to equations for sources and vortices in fluid mechanics, electrodynamics and thermodynamics. In economics, the Stokes integral of economic circuits leads to new fundamental equations of macro-econophysics. These equations differ significantly from the laws of neoclassical theory. Entropy of markets replaces of the economic Cobb Douglas function and leads to stochastic processes and micro-econophysics of financial markets.

THE EVALUATION OF THE NEW ZEALAND'S GEOID MODEL USING THE KTH METHOD / NAUJOSIOS ZELANDIJOS GEOIDO MODELIO VERTINIMAS KTH METODU / ОЦЕНКА МОДЕЛИ ГЕОИДА НОВОЙ ЗЕЛАНДИИ С ПРИМЕНЕНИЕМ МЕТОДА КТН

We compile a new geoid model at the computation area of New Zealand and its continental shelf using the method developed at the Royal Institute of Technology (KTH) in Stockholm. This method utilizes the least-squares modification of the Stokes integral for the biased, unbiased, and optimum stochastic solutions. The modified Bruns-Stokes integral combines the regional terrestrial gravity data with a global geopotential model (GGM). Four additive corrections are calculated and applied to the approximate geoid heights in order to obtain the gravimetric geoid. These four additive corrections account for the combined direct and indirect effects of topography and atmosphere, the contribution of the downward continuation reduction, and the formulation of the Stokes problem in the spherical approximation. The gravimetric geoid model is computed using two heterogonous gravity data sets: the altimetry-derived gravity anomalies from the DNSC08 marine gravity database (offshore) and the ground gravity measurements from the GNS Science gravity database (onshore). The GGM coefficients are taken from EIGEN-GRACE02S complete to degree 65 of spherical harmonics. The topographic heights are generated from the 1×1 arc-sec detailed digital terrain model (DTM) of New Zealand and from the 30×30 arc-sec global elevation data of SRTM30_PLUS V5.0. The least-squares analysis is applied to combine the gravity and GPS-levelling data using a 7-parameter model. The fit of the KTH geoid model with GPS-levelling data in New Zealand is 7 cm in terms of the standard deviation (STD) of differences. This STD fit is the same as the STD fit of the NZGeoid2009, which is the currently adopted official quasigeoid model for New Zealand. Santrauka Stokholmo Karališkajame technologijos institute (KTH) sukurtu metodu apskaičiuotas naujas Naujosios Zelandijos ir kontinentinio šelfo geoido modelis. Taikoma Stokso integralo mažiausiųjų kvadratų modifikacija, įvertinant paklaidas ir jų nevertinant bei ieškant optimalių stochastinių sprendinių. Modifikuotas Bruno ir Stokso integralas sieja regioninius žemyninius gravimetrinius duomenis su globaliuoju geopotencialo modeliu (GGM). Gravimetriniam geoidui gauti skaičiuojamos keturios papildomos pataisos: topografinės situacijos ir atmosferos tiesioginės ir netiesioginės įtakos, redukcijos įtakos ir Stokso integralo taikymo sferiniam paviršiui. Gravimetrinis geoido modelis apskaičiuotas pagal du duomenų rinkinius: DNSC08 jūrinių gravimetrinių duomenų bazėje (šelfas) esančias altimetriniu metodu nustatytas sunkio pagreičio anomalijas ir žemyninės dalies gravimetrinių matavimų duomenis iš GNS gravimetrinės duomenų bazės (pakrantė). GGM koeficientai imti iš EIGEN-GRACE02S modelio sferinių iki 65 laipsnio harmonikų. Topografiniai aukščiai sugeneruoti iš Naujosios Zelandijos 1×1 sekundės detaliojo skaitmeninio reljefo modelio ir iš 30×30 sekundžių globaliojo aukščių modelio SRTM30_PLUS V5.0. Gravimetriniams ir GPS niveliacijos duomenims sujungti taikytas mažiausiųjų kvadratų 7 parametrų metodas. KTH metodu sudaryto geoido modelio vidutinė kvadratinė paklaida 7 cm. Tai sutampa su NZGeoid 2009 geoido modelio, taikomo Naujoje Zelandijoje, tikslumu. Резюме Модель геоида континентального шельфа Новой Зеландии построена с применением метода, созданного в Королевском технологическом институте Стокгольма. Данный метод основан на модификации решения интеграла Стокса методом наименьших квадратов с оценкой или без оценки погрешностей и поиском оптимальных статистических решений. Модифицированный интеграл БрунаСтокса объединяет региональные надземные гравиметрические данные с глобальной геопотенциальной моделью (GGM). Для определения гравиметрического геоида вычисляются дополнительные поправки прямого и косвенного влияния топографии и атмосферы, редукции и применения проблемы Стокса для сферической поверхности. Гравиметрическая модель геоида вычисляется на основе двух баз данных: альтиметрическим методом определенных аномалий силы тяжести в базе морских гравиметрических данных DNSC08 (шельф) и надземной части гравиметрических измерений из базы данных GNS. Коэффициенты GGM взяты из сферических гармоник до 65 степени модели EIGENGRACEO2S. Топографические высоты сгенерированы из детальной цифровой модели рельефа Новой Зеландии с сеткой 1×1 секунду и из глобальной модели высот SRTM30_PLUSv5.0 с сеткой 30×30 секунд. Для объединения гравиметрических и GPSнивелирных данных применялся метод наименьших квадратов с 7 параметрами. Среднеквадратическая погрешность модели геоида, созданной по методу КТН, равна 7 см. Точность аналогична точности применяемой в Новой Зеландии модели геоида NZGeoid2009.

Mean kernels to improve gravimetric geoid determination based on modified Stokes's integration

Gravimetric geoid computation is often based on modified Stokes's integration, where Stokes's integral is evaluated with some stochastic or deterministic kernel modification. Accurate numerical evaluation of Stokes's integral requires the modified kernel to be integrated across the area of each discretised grid cell (mean kernel). Evaluating the modified kernel at the center of the cell (point kernel) is an approximation, which may result in larger numerical integration errors near the computation point, where the modified kernel exhibits a strongly nonlinear behavior. The present study deals with the computation of whole-of-the-cell mean values of modified kernels, exemplified here with the Featherstone–Evans–Olliver (1998) kernel modification [Featherstone, W.E., Evans, J.D., Olliver, J.G., 1998. A Meissl-modified Vaníček and Kleusberg kernel to reduce the truncation error in gravimetric geoid computations. Journal of Geodesy 72(3), 154–160]. We investigate two approaches (analytical and numerical integration), which are capable of providing accurate mean kernels. The analytical integration approach is based on kernel weighting factors which are used for the conversion of point to mean kernels. For the efficient numerical integration, Gauss–Legendre quadrature is applied. The comparison of mean kernels from both approaches shows a satisfactory mutual agreement at the level of 10 −4 and better, which is considered to be sufficient for practical geoid computation requirements. Closed-loop tests based on the EGM2008 geopotential model demonstrate that using mean instead of point kernels reduces numerical integration errors by ∼65%. The use of mean kernels is recommended in remove–compute–restore geoid determination with the Featherstone–Evans–Olliver (1998) kernel or any other kernel modification under the condition that the kernel changes rapidly across the cells in the neighborhood of the computation point.

The high-resolution gravimetric geoid of Italy: ITG2009

A new geoid computed with high-resolution for Italy, is presented in this paper. The computation of this gravimetric geoid is based on the most recent geopotential model: EGM2008 (Earth Gravity Model released in 2008). The method used in the computation of this new gravimetric geoid has been the Stokes integral in convolution form. The terrain correction has been applied to the gridded gravity anomalies, to obtain the corresponding reduced anomalies. Also the indirect effect has been taken into account. Thus, a new geoid model has been calculated and it is provided as a data grid in the Geodetic Reference System of 1980 (GRS80), distributed for the study area from 37° to 48° of latitude and 6° to 19° of longitude, on a 441 × 521 regular grid with a mesh size of 1.5′ × 1.5′. This new high-resolution geoid and the global geoid EGM2008; are compared with the geoid undulations measured for 11 points of the EUropean Vertical Network (EUVN) on Italy. The new geoid shows an improvement in precision and reliability, fitting the geoidal heights of these EUVN points with more accuracy than the global geoid. Moreover, this new geoid has a smaller standard deviation (12.5 cm) than that obtained by any previous geoid developed for Italy (and the surrounding area) up to date ( Pavlis et al., 2008). This new model will be useful for orthometric height determination by GPS over this study area, because it will allow orthometric height determination in the mountains and remote areas, in which levelling has many logistic problems. This new model can be also interesting for other geophysical purposes, other than geodesy and the height measurements, because it can provide a constraint for the density distribution and the thermal state of lithosphere and the viscosity in the mantle. Such details can be inferred from a geoid model and the seismic velocity structure.

Use of EGM08 model and shuttle radar topography mission data for geoid computation in the State of Rio de Janeiro, Brazil: A case study with Voronoi/Delaunay discretisations

The EGM08 geopotential model complete to degree and order 2159 was used in a remove-compute-restore (RCR) method for the geoid computation in the State of Rio de Janeiro, Brazil. Terrain and indirect effect corrections were computed using a 6-arcsec resolution DTE, derived from the TOPODATA Project (Shuttle Radar Topography Mission data) raised by the National Institute for Space Research. INPE, Brazil. We applied Voronoi/Delaunay discretisations for discrete Stokes integration. In these schemes, target area is partitioned into polygons/triangles, respectively, and the computation is carried out by point-wise numerical integration and no gridding is mandatory. For both procedures, the cells were produced using either observed gravity data combined with gridded Bouguer derived information. Particularly in Delaunay scheme, as the gravity anomalies are interpolated into the triangular cells, and geoid undulations are computed for their vertices, Stokes function singularity was gone. Externally estimated errors resulting from a comparison with GPS/leveling data were presented for both the schemes and classical ones, as well as for the EGM08 undulations. They yielded RMS differences equal to 0.105 m, 0.110 m, 0.110 m, 0.115 m and 0.228 m, respectively, for Voronoi, Delaunay, Voronoi/Delaunay with gridded-data alone and EGM08, computed between 32 GPS/leveling points.

Stokes integral of economic growth: Calculus and the Solow model

Economic growth depends on capital and labor and two-dimensional calculus has been applied to economic theory. This leads to Riemann and Stokes integrals and to the first and second laws of production and growth. The mathematical structure is the same as in thermodynamics, economic properties may be related to physical terms: capital to energy, production to physical work, GDP per capita to temperature, production function to entropy. This is called econophysics. Production, trade and banking may be compared to motors, heat pumps or refrigerators. The Carnot process of the first law creates two levels in each system: cold and hot in physics; buyer and seller, investor and saver, rich and poor in economics. The efficiency rises with the income difference of rich and poor. The results of econophysics are compared to neoclassical theory.

The high‐resolution gravimetric geoid of North Iberia: NIBGEO

AbstractThe gravimetric geoid computed in the northern part of Iberia, is presented in this paper. This computation has been performed considering two study windows fitted to the areas with higher density of gravity data, to reduce the computation errors associated to the scarcity of gravity data, as much as possible. The bad influence of a bathymetry with poorer resolution than the topography is also reduced considering the smallest marine area possible. Moreover, the computation of this gravimetric model is based on the most recent geopotential model: EIGEN‐GL04C (obtained in 2006). The method used in the computation of the new gravimetric geoid has been the Stokes integral in convolution form. The terrain correction has been applied to the gridded gravity anomalies, to obtain the corresponding reduced anomalies. Also the indirect effect has been taken into account. Thus, a new geoid model has been calculated and it is provided as a data grid in the Geodetic Reference System of 1980, distributed for the northern part of Iberia from 40 to 44 degrees of latitude and −10 to 4 degrees of longitude, on a 161 × 561 regular grid with a mesh size of 1.5′ × 1.5′. This new geoid and the previous geoid Iberian Gravimetric Geoid 2005, are compared with the geoid undulations measured for eight points of the European Vertical Reference Network (EUVN) on Iberia. The new geoid shows an improvement in precision and reliability, fitting the geoidal heights of these EUVN points with more accuracy than the previous geoid. Moreover, this new geoid has a smaller standard deviation (12.6 cm) than that obtained by any previous geoid developed for the Iberian area up to date. This geoid obtained for the northern part of Iberia will complement the previously obtained geoid for South Spain and the Gibraltar Strait area; both geoids jointly will give a complete picture of the geoid for Spain and the Gibraltar Strait area. This new model will be useful for orthometric height determination by GPS over this study area, because it will allow orthometric height determination in the mountains and remote areas, in which levelling has many logistic problems. This new model contributes to our knowledge of the geoid, but the surrounding areas must be better known to constrain the lithospheric and mantle models.

Assessment of systematic errors in the surface gravity anomalies over North America using the GRACE gravity model

SUMMARY The surface gravity data collected via traditional techniques such as ground-based, shipboard and airborne gravimetry describe precisely the local gravity field, but they are often biased by systematic errors. On the other hand, the spherical harmonic gravity models determined from satellite missions, in particular, recent models from CHAMP and GRACE, homogenously and accurately describe the low-degree components of the Earth’s gravity field. However, they are subject to large omission errors. The surface and satellite gravity data are therefore complementary in terms of spectral composition. In this paper, we aim to assess the systematic errors of low spherical harmonic degrees in the surface gravity anomalies over North America using a GRACE gravity model. A prerequisite is the extraction of the low-degree components from the surface data to make them compatible with GRACE data. Three types of methods are tested using synthetic data: low-pass filtering, the inverse Stokes integral, and spherical harmonic analysis. The results demonstrate that the spherical harmonic analysis works best. Eighty-five per cent of difference between the synthetic gravity anomalies generated from EGM96 and GGM02S from degrees 2 to 90 can be modelled for a region covering North America and neighbouring areas. Assuming EGM96 is developed solely from the surface gravity data with the same accuracy and GGM02S errorless, one way to understand the 85 per cent difference is that it represents the systematic error from the region of study, while the remaining 15 per cent originates from the data outside of the region. To estimate systematic errors in the surface gravity data, Helmert gravity anomalies are generated from both surface and GRACE data on the geoid. Their differences are expanded into surface spherical harmonics. The results show that the systematic errors for degrees 2 to 90 range from about −6 to 13 mGal with a RMS value of 1.4 mGal over North America. A few significant data gaps can be identified from the resulting error map. The errors over oceans appear to be related to the sea surface topography. These systematic errors must be taken into consideration when the surface gravity data are used to validate future satellite gravity missions.