Polar motion for an elastic earth model using a canonical theory

The purpose of this paper is to develop a canonical formulation of the rotational motion for an elastic Earth model. We have obtained the canonical equations for the precession and nutation motion in an inertial frame, and from this we have deduced the equations in an Earth-fixed frame. The linearized equations deduced for polar motion are equivalent to those obtained using Liouville's equations.     

Polar motion for an elastic Earth model with a homogeneous liquid core using a canonical theory

The rotational motion for an elastic Earth model with a homogeneous liquid core has been obtained using Hamilton's equations. From the canonical equations for the precessional and nutational motions in an inertial frame, the corresponding equations in an Earth fixed frame are deduced. The linearized equations obtained for polar motion and liquid core motion are equivalent to the Sasao-Okubo-Saito equations.     

Polar motion for an elastic Earth model with a homogeneous liquid core using a canonical theory

The rotational motion for an elastic Earth model with a homogeneous liquid core has been obtained using Hamilton's equations. From the canonical equations for the precessional and nutational motions in an inertial frame, the corresponding equations in an Earth fixed frame are deduced. The linearized equations obtained for polar motion and liquid core motion are equivalent to the Sasao-Okubo-Saito equations.     

同时顾及章动和极移的地球自转方程

通过引进章动坐标系相对惯性参照系的转动角速度随时间的变化,导出了一个可同时解出章动和极移的地球自转方程,用这个方程可同时研究地球的强迫和自由转动。与现行研究地球自转的惯用方法相比,该方法综合性强,易于理解。     

Transformation of amplitudes and frequencies of precession and nutation of the earth’s rotation vector to amplitudes and frequencies of diurnal polar motion

The temporal change of the rotation vector of a rotating body is, in the first order, identical in a space-fixed system and in a body-fixed system. Therefore, if the motion of the rotation axis of the earth relative to a space-fixed system is given as a function of time, it should be possible to compute its motion relative to an earth-fixed system, and vice versa. This paper presents such a transformation. Two models of motion of the rotation axis in the space-fixed system are considered: one consisting only of a regular (i.e., strictly conical) precession and one extended by circular nutation components, which are superimposed upon the regular precession. The Euler angles describing the orientation of the earth-fixed system with respect to the space-fixed system are derived by an analytical solution of the kinematical Eulerian differential equations. In the first case (precession only), this is directly possible, and in the second case (precession and nutation), a solution is achieved by a perturbation approach, where the result of the first case serves as an approximation and nutation is regarded as a small perturbation, which is treated in a linearized form. The transformation by means of these Euler angles shows that the rotation axis performs in the earth-fixed system retrograde conical revolutions with small amplitudes, namely one revolution with a period of one sidereal day corresponding to precession and one revolution with a period which is slightly smaller or larger than one sidereal day corresponding to each (prograde or retrograde) circular nutation component. The peculiar feature of the derivation presented here is the analytical solution of the Eulerian differential equations.     

Uniquely and overdetermined geodetic boundary value problems by least squares

Least-squares by observation equations is applied to the solution of geodetic boundary value problems (g.b.v.p.). The procedure is explained solving the vectorial Stokes problem in spherical and constant radius approximation. The results are Stokes and Vening-Meinesz integrals and, in addition, the respective a posteriori variance-covariances. Employing the same procedure the overdeterminedg.b.v.p. has been solved for observable functions potential, scalar gravity, astronomical latitude and longitude, gravity gradients Г_xz, Г_yz, and Г_zz and three-dimensional geocentric positions. The solutions of a large variety of uniquely and overdeterminedg.b.v.p.'s can be obtained from it by specializing weights. Interesting is that the anomalous potential can be determined—up to a constant—from astronomical latitude and longitude in combination with either {Г_xz,Г_yz} or horizontal coordinate corrections Δx and Δy, or both. Dual to the formulation in terms of observation equations the overdeterminedg.b.v.p.'s can as well be solved by condition equations. Constant radius approximation can be overcome in an iterative approach. For the Stokes problem this results in the solution of the “simple” Molodenskii problem. Finally defining an error covariance model with a Krarup-type kernel first results were obtained for a posteriori variance-covariance and reliability analysis.     

Second-order derivatives in a local frame developed in spheroidal and spherical coordinates

Second-order derivatives of a general scalar function of position (F) with respect to the length elements along a family of local Cartesian axes are developed in the spheroidal and spherical coordinate systems. A link between the two kinds of formulations is established when the results in spherical coordinates are confirmed also indirectly, through a transformation from spheroidal coordinates. IfF becomesW (earth's potential) the six distinct second-order derivatives—which include one vertical and two horizontal gradients of gravity—relate the symmetric Marussi tensor to the curvature parameters of the field. The general formulas for the second-order derivatives ofF are specialized to yield the second-order derivatives ofU (standard potential) and ofT (disturbing potential), which allows the latter to be modeled by a suitable set of parameters. The second-order derivatives ofT in which the property ΔT=0 is explicitly incorporated are also given. According to the required precision, the spherical approximation may or may not be desirable; both kinds of results are presented. The derived formulas can be used for modeling of the second-order derivatives ofW orT at the ground level as well as at higher altitudes. They can be further applied in a rotating or a nonrotating field. The development in this paper is based on the tensor approach to theoretical geodesy, introduced by Marussi [1951] and further elaborated by Hotine [1969], which can lead to significantly shorter demonstrations when compared to conventional approaches.     

A few basic principles and techniques of array algebra

This paper is intended to demonstrate the usefulness of array algebra techniques in certain multilinear least squares problems. A typical restriction of array algebra is the need for a gridded observational structure; however, the grid does not have to be uniform and in general is not limited to any particular coordinate system nor to two- or three-dimensional spaces. Another restriction comes to light when dealing with weighted multilinear least squares adjustments. The a—priori variance-covariance matrix cannot be completely arbitrary but must be expressible in terms of certain matrix products. There exist various practical ways (not discussed herein) to bridge these restrictions. The reward for using the array algebra technique when it is appropriate lies in the great computational savings. From the theoretical point of view, the backbone of most derivations are the “R-matrix multiplications” and a simple tool, demonstrated herein, called “fundamental transformation”. It follows that the least squares solution of “array observation equations” does not have to be sought by some new and complex mathematical means. The fundamental transformation allows such an adjustment problem to be rewritten in a conventional (monolinear) form; the familiar least squares solution is then written down and transformed back to the array form using the same tool. The statistical properties of the results (e.g. minimum variance) are known from the conventional approach and do not have to be rederived in the array case.     

New results in airborne vector gravimetry using strapdown INS/DGPS

A method for airborne vector gravimetry has been developed. The method is based on developing the error dynamics equations of the INS in the inertial frame where the INS system errors are estimated in a wave estimator using inertial GPS position as update. Then using the error-corrected INS acceleration and the GPS acceleration in the inertial frame, the gravity disturbance vector is extracted. In the paper, the focus is on the improvement of accuracy for the horizontal components of the airborne gravity vector. This is achieved by using a decoupled model in the wave estimator and decorrelating the gravity disturbance from the INS system errors through the estimation process. The results of this method on the real strapdown INS/DGPS data are promising. The internal accuracy of the horizontal components of the estimated gravity disturbance for repeated airborne lines is comparable with the accuracy of the down component and is about 4–8 mGal. Better accuracy (2–4 mGal) is achieved after applying a wave-number correlation filter (WCF) to the parallel lines of the estimated airborne gravity disturbances.     

The solution of the Korn–Lichtenstein equations of conformal mapping: the direct generation of ellipsoidal Gauß–Krüger conformal coordinates or the Transverse Mercator Projection

The differential equations which generate a general conformal mapping of a two-dimensional Riemann manifold found by Korn and Lichtenstein are reviewed. The Korn–Lichtenstein equations subject to the integrability conditions of type vectorial Laplace–Beltrami equations are solved for the geometry of an ellipsoid of revolution (International Reference Ellipsoid), specifically in the function space of bivariate polynomials in terms of surface normal ellipsoidal longitude and ellipsoidal latitude. The related coefficient constraints are collected in two corollaries. We present the constraints to the general solution of the Korn–Lichtenstein equations which directly generates Gau?–Krüger conformal coordinates as well as the Universal Transverse Mercator Projection (UTM) avoiding any intermediate isometric coordinate representation. Namely, the equidistant mapping of a meridian of reference generates the constraints in question. Finally, the detailed computation of the solution is given in terms of bivariate polynomials up to degree five with coefficients listed in closed form. Received: 3 June 1997 / Accepted: 17 November 1997     

Improvement of inertial surveys through post-mission network adjustment and self-calibration

Present day inertial surveys are limited to single traverse runs in which the number of unknown system parameters to be determined are few, depending on the number of control points available along the traverse. Further, conventional inertial surveys are generally restricted to the determination of coordinates with no possibility for a rigorous post-mission adjustment of the observations. The consequence is the continued presence of systematic trends in the residuals, even after the use of error models such as those proposed by Ball, Gregerson or Kouba. Future work aiming at higher accuracies obviously requires more comprehensive models and rigorous adjustment procedures. These can be accomplished by the development of such error models and by the use of “area surveys”, instead of the single traverses, together with rigorous adjustment procedures suitable for the network of criss-crossing lines inertially surveyed. In such a network the cross-over points serve as constraints for the geodetic parameters (latitude, longitude, height, gravity anomaly, deflection components) and allow the addition of hardware and software related error parameters. Thus an opportunity is provided to effectively self-calibrate the system—a concept successfully used, for example, in photogrammetry or in satellite tracking. The number and the strength of such parameters depend on the number of control and cross-over points. The adjustment, of course, also provides the necessary statistical information on the adjusted parameters, such as their precision and the correlation between them. The presentation will describe current work at OSU in this area. Presented at the Second International Symposium on Inertial Technology for Surveying and Geodesy, Banff, Canada, June 1–5, 1981.     

A framework for modelling kinematic measurements in gravity field applications

The determination of the local gravity field from sensors mounted in a fixed wing aircraft has long been a dream of geodesists and geophysicists. The progress in sensor technology during the last decade has brought its realization within reach and recent tests indicate that results at the level of a fewmGal are possible. To assess different sensor configurations and their effect on the resolution of the gravity field spectrum, a state model for motion in the gravity field of the earth is formulated. The resulting set of differential equations can accommodate first and second order gravity gradients, specific force, kinematic acceleration, vehicle velocity and position as input. It offers therefore a rather general framework for gravity field determination from a variety of kinematic sensors, such as gravity meters, gravity gradiometers, inertial systems, differentialGPS, laser altimeters and others. The derivation of the basic kinematic model and its linearization are given in detail, while sensor error models are discussed in a generic way. A few remarks on the modelling of gravity gradiometer measurements conclude the paper.     

Enhanced MEMS-IMU/odometer/GPS integration using mixture particle filter

Dead reckoning techniques such as inertial navigation and odometry are integrated with GPS to avoid interruption of navigation solutions due to lack of visible satellites. A common method to achieve a low-cost navigation solution for land vehicles is to use a MEMS-based inertial measurement unit (IMU) for integration with GPS. This integration is traditionally accomplished by means of a Kalman filter (KF). Due to the significant inherent errors of MEMS inertial sensors and their time-varying changes, which are difficult to model, severe position error growth happens during GPS outages. The positional accuracy provided by the KF is limited by its linearized models. A Particle filter (PF), being a nonlinear technique, can accommodate for arbitrary inertial sensor characteristics and motion dynamics. An enhanced version of the PF, called Mixture PF, is employed in this paper. It samples from both the prior importance density and the observation likelihood, leading to an improved performance. Furthermore, in order to enhance the performance of MEMS-based IMU/GPS integration during GPS outages, the use of pitch and roll calculated from the longitudinal and transversal accelerometers together with the odometer data as a measurement update is proposed in this paper. These updates aid the IMU and limit the positional error growth caused by two horizontal gyroscopes, which are a major source of error during GPS outages. The performance of the proposed method is examined on road trajectories, and results are compared to the three different KF-based solutions. The proposed Mixture PF with velocity, pitch, and roll updates outperformed all the other solutions and exhibited an average improvement of approximately 64% over KF with the same updates, about 85% over KF with velocity updates only, and around 95% over KF without any updates during GPS outages.     

On a self-consistent representation of earth models,with an application to the computing of internal flattening

Most realistic Earth models published as yet have been given in tabulated form, with the noticeable exception of three simple parametric Earth models derived by Dziewonski et al. (1975). Simple interpolation in these tables may lead to inconsistencies, when we consider certain effects which depend crucially on detailed density structure. We establish algorithmic formulae, which may be used to compute all the mechanical properties of a model in an entirely consistent way, once the density as well asP— andS— wave velocities are known. We then use this formulation to integrate Clairaut’s equation in a very efficient way, and thus obtain the hydrostatic flattening to the first order in smallness at any point inside the model. For most geodynamic purposes, we may suffice with this approximation. Finally, we show the results of some calculations of hydrostatic flattening to the first and second order, using an iterative technique of solving the integral figure equations, for an Earth model consistent with all geophysical data available at present. We find that the hydrostatic flattening at the surface should be about 1/298.8, instead of 1/296.961 as quoted by Nakiboglu (1979) for essentially the same model. Moreover, from our results, we estimate the actual flattening of the coremantle boundary to be about 1/390.3.     

Singularity free formulations of the geodetic boundary value problem in gravity-space

The idea of transforming the geodetic boundary value problem into a boundary value problem with a fixed boundary dates back to the 1970s of the last century. This transformation was found by F. Sanso and was named as gravity-space transformation. Unfortunately, the advantage of having a fixed boundary for the transformed problem was counterbalanced by the theoretical as well as practical disadvantage of a singularity at the origin. In the present paper two more versions of a gravity-space transformation are investigated, where none of them has a singularity. In both cases the transformed differential equations are nonlinear. Therefore, a special emphasis is laid on the linearized problems and their relationships to the simple Hotine-problem and to the symmetries between both formulations. Finally, in numerical simulation study the accuracy of the solutions of both linearized problems is studied and factors limiting this accuracy are identified.     

Geocentric indian geodetic datum by astrogeodetic — gravimetric method using smoothened geoidal heights

The astrogeodetic—gravimetric method based on the principle of least—squares solution has been used to determine the geocentric Indian geodetic datum making use of the available nongeocentric astrogeodetic data and the gravimetric geocentric geoidal heights in the form of smoothened values. Everett's method of interpolation has been used to obtain the smoothened geoidal heights at the astrogeodetic stations in India from the available generalized values at 1°×1° corners. The values of the geoidal height and deflections of the vertical at the geodetic datum Kalianpur H.S. so obtained have the negligible difference from the values computed earlier by the same method using directly computed gravimetric geoidal heights at the astrogeodetic stations, indicating that the use of the interpolated values in the astrogeodetic—gravimetric method employed would be an economical approach of absolute orientation of a nongeocentric system if the gravimetric geoidal heights are available at 1°×1° corners in the area of interest.     

The laplace condition in the adjustment of a chain of triangles

Summary The purpose of this treatise is to study in detail the influence of the approximations usually applied in the Laplace condition, on the adjusted elements of a chain of triangles. In § 1 the Laplace condition is derived once again from the well-known Laplace equation, because in our opinion the usual deriviations are not quite correct. In §§ 2–6 the Laplace condition is converted to a form—terms containing first powers of s: R inclusive—suitable for an application in the adjustment of a chain of triangles. The usual approximate form is derived from it by omitting the terms mentioned. In § 7 the influence of these approximations is examined by adjusting a chain of triangles twice over: applying the exact resp. the approximate Laplace condition and comparing the results. From this it is concluded that in all normal cases the influence of applying the approximations is negligible and that the validity of this conclusion is independent of a possible future increase of the accuracy of observation.     

Identification of Loriya Fault,Its Reactivation Due to 2001 Bhuj Earthquake, (Using Remote Sensing Data) and Its Bearing on the Kachchh Mainland Fault,Kachchh District,Gujarat

The 2001 Bhuj earthquake (M_w 7.7), one of the most severe earthquakes in the recent history of India, reactivated various existing active faults. It is manifested in the form of coseismic ground fissures/cracks and upheaval of land in the form of bumps. Identification and reactivation of Loriya Fault is established by 1—Geomorphic changes with the help of digital imagery (LISS III images). 2—Coseismic changes through ground checks and 3—Geophysical signatures through magnetic and gravity survey. A lineament cutting the north-western part of the Pur River alluvial fan has been revealed by satellite imagery. The streams flowing along the lineament add to the evidences of a weak plane, while the occurrence of coseismic ground fissures confirms the existence of an active fault. No dip slip movement is recorded in the trenches made across the Loriya active fault while the en-echelon pattern of ground fissures suggest strike slip movement along the fault due to 2001 earthquake.     

Contribution au probleme de la compensation des reseaux de triangulation

Résumé Dans cette étude on expose une adaptation de la méthode des directions —pour la compensation d’un réseau de triangulation—aux possibilités de l’ordinateur électronique. On présente les formules par lesquelles on calcule les coefficients des inconnues dans les équations de condition (équations aux angles et équations aux c?tés) et on dresse leur matrice. Ensuite on traite de la formation de la matrice des coefficients des équations normales et son inversion, qui fournit les quantités corrélatives de Lagrange. Enfin, après avoir déterminé les corrections à apporter aux directions observées, on calcule l’erreur moyenne quadratique d’une observation isolée et l’erreur moyenne de chacune des directions compensées.

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Summary In this study, an adaptation to the computers’ possibilities, regarding the method of directions, is disclosed in order to realise the adjustment of triangulation nets. The computing formulae of the coefficients of the unknown quantities in the condition equations and the creation of their matrix are given herebelow. The treatment leading towards the construction of the matrix of normal equations and its inversion, that furnishes Lagrange’s quantities, follows. After the computation of the corrections applied on the observed directions, the mean error of a single observation and the mean error of every adjusted direction are determined. The study is executed in such a way that it can be included in only one programme for automatic computation.