Showing papers in "Journal of Geodesy in 1998"

Estimating regional deformation from a combination of space and terrestrial geodetic data

Abstract: We discuss an approach for efficiently combining different types of geodetic data to estimate time-dependent motions of stations in a region of active deformation. The primary observations are analyzed separately to produce loosely constrained estimates of station positions and coordinate system parameters which are then combined with appropriate constraints to estimate velocities and coseismic displacements. We define noninteger degrees of freedom to handle the case of finite constraints and stochastic perturbation of parameters and develop statistical tests for determining compatibility between different data sets. With these developments, we show an example of combining space and terrestrial geodetic data to obtain the deformation field in southern California.

Success probability of integer GPS ambiguity rounding and bootstrapping

Abstract: Global Positioning System ambiguity resolution is usually based on the integer least-squares principle (Teunissen 1993). Solution of the integer least-squares problem requires both the execution of a search process and an ambiguity decorrelation step to enhance the efficiency of this search. Instead of opting for the integer least-squares principle, one might also want to consider less optimal integer solutions, such as those obtained through rounding or sequential rounding. Although these solutions are less optimal, they do have one advantage over the integer least-squares solution: they do not require a search and can therefore be computed directly. However, in order to be confident that these less optimal solutions are still good enough for the application at hand, one requires diagnostic measures to predict their rate of success. These measures of confidence are presented and it is shown how they can be computed and evaluated.

A discrimination test procedure for ambiguity resolution on-the-fly

Abstract: Ambiguity validation tests include the acceptance test and discrimination test, which are both important steps in the Global Positioning System ambiguity resolution process An ambiguity discrimination test procedure based on a test statistic which is constructed by the difference (not the ratio as used in current procedures) between the minimum and second minimum quadratic form of the residuals in ambiguity identification, and its standard deviation, is proposed The distribution function of the proposed test statistic is theoretically identified as a standard normal distribution when the known a priori variance factor is used, or as a Student's t distribution when the estimated variance factor is used With this procedure, the ambiguity discrimination test is based on a more rigorous test statistic whose critical value can be calculated with any chosen level of significance Test results indicate that the proposed ambiguity discrimination test procedure is reliable for use in ambiguity resolution on-the-fly

Improvement of the empirical thermospheric model DTM: DTM94 - a comparative review of various temporal variations and prospects in space geodesy applications

Abstract: The Drag Temperature Model (1978) has been improved to respond better to the actual requirements of space geodesy, especially under extreme solar and geomagnetic conditions. Extended data and an improved algorithm have been considered, leading to valuable improvements. Temporal variations of temperature of the thermopause, total density and major chemical constituent density are reviewed and compared to the DTM94, DTM78 and MSIS86 models. A comparison with data is performed, giving the mean ratios between observed and model values with their root mean squares for different physical and geometric conditions. This comparison is made for the three models and includes data used in the modelling as well as external data. The limits of the thermosphere modelling are discussed.

GPS error modeling and OTF ambiguity resolution for high-accuracy GPS/INS integrated system

Abstract: The Center for Mapping at The Ohio State University is currently developing a fully digital Airborne Integrated Mapping System (AIMS) for large-scale mapping and other precise positioning applications. AIMS, installed in an aerial platform, incorporates state-of-the-art positioning [differential Global Positioning System (GPS) integrated with an Inertial Navigation System (INS)] and imaging (Charge-Coupled Device) technologies. The project goal is to develop a low-cost hardware prototype that acquires position and orientation of an aerial platform with accuracy of 4–7 cm and below 10 arcsec, respectively, over long baselines, and performs essential processing of digital imagery in real-time and in post-processing mode. A tightly integrated GPS and high-accuracy INS provide orientation and position of the aerial platform. To the best of the authors' knowledge, AIMS is the first tightly integrated GPS/INS system whose existing prototype promises sub-decimeter accuracy in positioning, and below 10 arcsec in orientation, in the airborne scenario. However, the expected high performance still remains to be verified against the ground truth. The new GPS/INS tightly integrated system for high-precision airborne applications is described, with special emphasis on the GPS component of the system. The evaluation of the actual prototype and the airborne test results are discussed. A detailed description of the tight GPS/INS integration and Kalman filter implementation is presented elsewhere.

Inverse Vening Meinesz formula and deflection-geoid formula: applications to the predictions of gravity and geoid over the South China Sea

Abstract: Using the spherical harmonic representations of the earth's disturbing potential and its functionals, we derive the inverse Vening Meinesz formula, which converts deflection of the vertical to gravity anomaly using the gradient of the H function. The deflection-geoid formula is also derived that converts deflection to geoidal undulation using the gradient of the C function. The two formulae are implemented by the 1D FFT and the 2D FFT methods. The innermost zone effect is derived. The inverse Vening Meinesz formula is employed to compute gravity anomalies and geoidal undulations over the South China Sea using deflections from Seasat, Geosat, ERS-1 and TOPEX//POSEIDON satellite altimetry. The 1D FFT yields the best result of 9.9-mgal rms difference with the shipborne gravity anomalies. Using the simulated deflections from EGM96, the deflection-geoid formula yields a 4-cm rms difference with the EGM96-generated geoid. The predicted gravity anomalies and geoidal undulations can be used to study the tectonic structure and the ocean circulations of the South China Sea.

Robust Kalman filter for rank deficient observation models

Abstract: A robust Kalman filter is derived for rank deficient observation models. The datum for the Kalman filter is introduced at the zero epoch by the choice of a generalized inverse. The robust filter is obtained by Bayesian statistics and by applying a robust M-estimate. Outliers are not only looked for in the observations but also in the updated parameters. The ability of the robust Kalman filter to detect outliers is demonstrated by an example.

A Meissl-modified Vaníček and Kleusberg kernel to reduce the truncation error in gravimetric geoid computations

Abstract: A deterministic modification of Stokes's integration kernel is presented which reduces the truncation error when regional gravity data are used in conjunction with a global geopotential model to compute a gravimetric geoid. The modification makes use of a combination of two existing modifications from Vanicek and Kleusberg and Meissl. The former modification applies a root mean square minimisation to the upper bound of the truncation error, whilst the latter causes the Fourier series expansion of the truncation error to coverage to zero more rapidly by setting the kernel to zero at the truncation radius. Green's second identity is used to demonstrate that the truncation error converges to zero faster when a Meissl-type modification is made to the Vanicek and Kleusberg kernel. A special case of this modification is proposed by choosing the degree of modification and integration cap-size such that the Vanicek and Kleusberg kernel passes through zero at the truncation radius.

A Kalman-filter-based approach to combining independent Earth-orientation series

Abstract: An approach, based upon the use of a Kalman filter, that is currently employed at the Jet Propulsion Laboratory (JPL) for combining independent measurements of the Earth's orientation, is presented. Since changes in the Earth's orientation can be described as a randomly excited stochastic process, the uncertainty in our knowledge of the Earth's orientation grows rapidly in the absence of measurements. The Kalman-filter methodology allows for an objective accounting of this uncertainty growth, thereby facilitating the intercomparison of measurements taken at different epochs (not necessarily uniformly spaced in time) and with different precision. As an example of this approach to combining Earth-orientation series, a description is given of a combination, SPACE95, that has been generated recently at JPL.

Performance of three types of Stokes's kernel in the combined solution for the geoid

Abstract: When regional gravity data are used to compute a gravimetric geoid in conjunction with a geopotential model, it is sometimes implied that the terrestrial gravity data correct any erroneous wavelengths present in the geopotential model. This assertion is investigated. The propagation of errors from the low-frequency terrestrial gravity field into the geoid is derived for the spherical Stokes integral, the spheroidal Stokes integral and the Molodensky-modified spheroidal Stokes integral. It is shown that error-free terrestrial gravity data, if used in a spherical cap of limited extent, cannot completely correct the geopotential model. Using a standard norm, it is shown that the spheroidal and Molodensky-modified integration kernels offer a preferable approach. This is because they can filter out a large amount of the low-frequency errors expected to exist in terrestrial gravity anomalies and thus rely more on the low-frequency geopotential model, which currently offers the best source of this information.

Flight test results from a strapdown airborne gravity system

Abstract: In June 1995, a flight test was carried out over the Rocky Mountains to assess the accuracy of airborne gravity for geoid determination. The gravity system consisted of a strapdown inertial navigation system (INS), two GPS receivers with zero baseline on the airplane and multiple GPS master stations on the ground, and a data logging system. To the best of our knowledge, this was the first time that a strapdown INS has been used for airborne gravimetry. The test was designed to assess repeatability as well as accuracy of airborne gravimetry in a highly variable gravity field. An east-west profile of 250 km across the Rocky Mountains was chosen and four flights over the same ground track were made. The flying altitude was about 5.5km, i.e., between 2.5 and 5.0km above ground, and the average flying speed was about 430km/h. This corresponds to a spatial resolution (half wavelength of cutoff frequency) of 5.07.0km when using filter lengths between 90 and 120s. This resolution is sufficient for geoid determination, but may not satisfy other applications of airborne gravimetry. The evaluation of the internal and external accuracy is based on repeated flights and comparison with upward continued ground gravity using a detailed terrain model. Gravity results from repeated flight lines show that the standard deviation between flights is about 2mGal for a single profile and a filter length of 120s, and about 3mGal for a filter length of 90s. The standard deviation of the difference between airborne gravity upward continued ground gravity is about 3mGal for both filter lengths. A critical discussion of these results and how they relate to the different transfer functions applied, is given in the paper. Two different mathematical approaches to airborne scalar gravimetry are applied and compared, namely strapdown inertial scalar gravimetry (SISG) and rotation invariant scalar gravimetry (RISG). Results show a significantly better performance of the SISG approach for a strapdown INS of this accuracy class. Because of major differences in the error model of the two approaches, the RISG method can be used as an effective reliability check of the SISG method. A spectral analysis of the residual errors of the flight profiles indicates that a relative geoid accuracy of 23cm over distances of 200km (0.1 ppm) can be achieved by this method. Since these results present a first data analysis, it is expected that further improvements are possible as more refined modelling is applied.

Minimal detectable biases of GPS data

Abstract: In this contribution closed-form expressions are given for the minimal detectable biases of single- and dual-frequency pseudo-range and carrier-phase data. They are given for three different single-baseline models. These are the geometry-free model and two variants of the geometry-based model, namely the roving and stationary variants. The baselines are considered to be sufficiently short such that orbital uncertainties in the fixed orbits and residual ionospheric and tropospheric delays can be assumed absent. The stochastic model used is one that permits cross-correlation and the use of different variances for individual GPS observables, including the possibility to weigh the observables in dependence on which satellite is tracked.

A compendium of transformation formulas useful in GPS work

Abstract: In order properly to apply transformations when using data derived from different GPS solutions, the effects of plate motion on the coordinates should be accurately taken into consideration. Only then can a rigorous comparison be established between results observed at different epochs. Equations are given to relate GPS-derived Cartesian coordinates and velocities affected by changes of their reference frames and epochs.

Diurnal atmospheric forcing and temporal variations of the nutation amplitudes

Abstract: A 29-year time-series of four-times-daily atmospheric effective angular momentum (EAM) estimates is used to study the atmospheric influence on nutation. The most important atmospheric contributions are found for the prograde annual (77 μas), retrograde annual (53 as), prograde semiannual (45 as), and for the constant offset of the pole (δψsinɛ0=−86 as, δɛ=77 as). Among them only the prograde semiannual component is driven mostly by the wind term of the EAM function, while in all other cases the pressure term is dominant. These are nonnegligible quantities which should be taken into account in the new theory of nutation. Comparison with the VLBI corrections to the IAU 1980 nutation model taking into account the ocean tide contribution yields good agreement for the prograde annual and semiannual nutations. We also investigated time variability of the atmospheric contribution to the nutation amplitudes by performing the sliding-window least-squares analysis of both the atmospheric excitation and VLBI nutation data. Almost all detected variations of atmospheric origin can be attributed to the pressure term, the biggest being the in-phase annual prograde component (about 30 as) and the retrograde one (as much as 100200 as). These variations, if physical, limit the precision of classical modeling of nutation to the level of 0.1 mas. Comparison with the VLBI data shows significant correlation for the retrograde annual nutation after 1989, while for the prograde annual term there is a high correlation in shape but the size of the atmospherically driven variations is about three times less than deduced from the VLBI data. This discrepancy in size can be attributed either to inaccuracy of the theoretical transfer function or the frequency-dependent ocean response to the pressure variations. Our comparison also yields a considerably better agreement with the VLBI nutation data when using the EAM function without the IB correction for ocean response, which indicates that this correction is not adequate for nearly diurnal variations.

Possible improvement of Earth orientation forecast using autocovariance prediction procedures

Abstract: Autocovariance prediction has been applied to attempt to improve polar motion and UT1-UTC predictions. The predicted polar motion is the sum of the least-squares extrapolation model based on the Chandler circle, annual and semiannual ellipses, and a bias fit to the past 3 years of observations and the autocovariance prediction of these extrapolation residuals computed after subtraction of this model from pole coordinate data. This prediction method has been applied also to the UT1-UTC data, from which all known predictable effects were removed, but the prediction error has not been reduced with respect to the error of the current prediction model. However, the results show the possibility of decreasing polar motion prediction errors by about 50 for different prediction lengths from 50 to 200 days with respect to the errors of the current prediction model. Because of irregular variations in polar motion and UT1-UTC, the accuracy of the autocovariance prediction does depend on the epoch of the prediction. To explain irregular variations in x, y pole coordinate data, time-variable spectra of the equatorial components of the effective atmospheric angular momentum, determined by the National Center for Environmental Prediction, were computed. These time-variable spectra maxima for oscillations with periods of 100–140 days, which occurred in 1985, 1988, and 1990 could be responsible for excitation of the irregular short-period variations in pole coordinate data. Additionally, time-variable coherence between geodetic and atmospheric excitation function was computed, and the coherence maxima coincide also with the greatest irregular variations in polar motion extrapolation residuals.

Determination of post-glacial land uplift from the three precise levellings in Finland

Abstract: We have calculated vertical velocities and their change in time from three precise levellings in Finland, with central epochs 1902, 1946, and 1986. Models with and without simultaneous determination of heights give the same results for velocities. Evidence for velocity change in time appears at first sight strong but on closer examination turns out to be inconclusive. We offer an alternative explanation in terms of levelling errors and estimate them. A new uplift map closely reproduces main features of earlier maps from partial data, but does not support some details in them.

Robust biased estimation and its applications in geodetic adjustments

Abstract: In this paper, a number of robust biased estimators (e.g. ordinary robust ridge estimator, robust principal components estimator, robust combined principal components estimator, robust single-parametric principal components estimator, robust root-root estimator) are established by means of a unified expression of biased estimators and based on the principle of equivalent weight. The most attractive advantage of these new estimators is that they can not only overcome the ill-conditioning of the normal equation but also have the ability to resist outliers. A numerical example is used to illustrate that these new estimators are much better than the least-squares estimator and various biased estimators even when both ill-conditioning and outliers exist.

Calibration of satellite gradiometer data aided by ground gravity data

Abstract: Parametric least squares collocation was used in order to study the detection of systematic errors of satellite gradiometer data. For this purpose, simulated data sets with a priori known systematic errors were produced using ground gravity data in the very smooth gravity field of the Canadian plains. Experiments carried out at different satellite altitudes showed that the recovery of bias parameters from the gradiometer “measurements” is possible with high accuracy, especially in the case of crossing tracks. The mean value of the differences (original minus estimated bias parameters) was relatively large compared to the standard deviation of the corresponding second-order derivative component at the corresponding height. This mean value almost vanished when gravity data at ground level were combined with the second-order derivative data set at satellite altitude. In the case of simultaneous estimation of bias and tilt parameters from ∂2 T/∂z 2“measurements”, the recovery of both parameters agreed very well with the collocation error estimation.

An integrated wavelet concept of physical geodesy

Abstract: For the determination of the earth's gravity field many types of observations are nowadays available, including terrestrial gravimetry, airborne gravimetry, satellite-to-satellite tracking, satellite gradio-metry, etc. The mathematical connection between these observables on the one hand and gravity field and shape of the earth on the other is called the integrated concept of physical geodesy. In this paper harmonic wavelets are introduced by which the gravitational part of the gravity field can be approximated progressively better and better, reflecting an increasing flow of observations. An integrated concept of physical geodesy in terms of harmonic wavelets is presented. Essential tools for approximation are integration formulas relating an integral over an internal sphere to suitable linear combinations of observation functionals, i.e. linear functionals representing the geodetic observables. A scale discrete version of multiresolution is described for approximating the gravitational potential outside and on the earth's surface. Furthermore, an exact fully discrete wavelet approximation is developed for the case of band-limited wavelets. A method for combined global outer harmonic and local harmonic wavelet modelling is proposed corresponding to realistic earth's models. As examples, the role of wavelets is discussed for the classical Stokes problem, the oblique derivative problem, satellite-to-satellite tracking, satellite gravity gradiometry and combined satellite-to-satellite tracking and gradiometry.

The Ionosphere-weighted GPS baseline precision in canonical form

Abstract: In this contribution we consider the precision of the floated and the fixed GPS baseline for the case of weighted ionosphere. Canonical forms of the baseline variance matrices are developed for different measurement scenarios. These forms make the relation between the various variance matrices transparent and thus present a simple way of studying their relative merits. It is also shown how these canonical forms give an intrinsic description of the gain in baseline precision which is experienced when the carrier-phase double-differenced ambiguities are treated as integers instead of as reals. The dependence of this gain on the various contributing factors, such as the decorrelation of the ionosphere, is also included.

The exterior Airy/Heiskanen topographic-isostatic gravity potential, anomaly and the effect of analytical continuation in Stokes' formula

Abstract: This study deals with the external type of topographic–isostatic potential and gravity anomaly and its vertical derivatives, derived from the Airy/Heiskanen model for isostatic compensation. From the first and the second radial derivatives of the gravity anomaly the effect on the geoid is estimated for the downward continuation of gravity to sea level in the application of Stokes' formula. The major and regional effect is shown to be of order H 3 of the topography, and it is estimated to be negligible at sea level and modest for most mountains, but of the order of several metres for the highest and most extended mountain belts. Another, global, effect is of order H but much less significant

Unification of vertical datums by GPS and gravimetric geoid models with application to Fennoscandia

Abstract: The second Baltic Sea Level (BSL) GPS campaign was run for one week in June 1993. Data from 35 tide gauge sites and five fiducial stations were analysed, for three fiducial stations (Onsala, Metsahovi and Wettzell) fixed at the ITRF93 system. On a time-scale of 5 days, precision was several parts in 109 for the horizontal and vertical components. Accuracies were about 1 cm in comparison with the International GPS Geodynamical Service (IGS) coordinates in three directions. To connect the Swedish and the Finnish height systems, our numerical application utilises three approaches: a rigorous approach, a bias fit and a three-parameter fit. The results between the Swedish RH70 and the Finnish N 60 systems are estimated to −19.3 ± 6.5, −17 ± 6 and −15 ± 6 cm, respectively, by the three approaches. The results of the three indirect methods are in an agreement with those of a direct approach from levelling and gravity measurements.

Generalized inverses of nonlinear mappings and the nonlinear geodetic datum problem

Abstract: Motivated by the existing theory of the geometric characteristics of linear generalized inverses of linear mappings, an attempt is made to establish a corresponding mathematical theory for nonlinear gen- eralized inverses of nonlinear mappings in finite- dimensional spaces. The theory relies on the concept of fiberings consisting of disjoint manifolds (fibers) in which the domain and range spaces of the mappings are partitioned. Fiberings replace the quotient spaces gen- erated by some characteristic subspaces in the linear case. In addition to the simple generalized inverse, the minimum-distance and the x0-nearest generalized inverse are introduced and characterized, in analogy with the least-squares and the minimum-norm generalized in- verses of the linear case. The theory is specialized to the geodetic mapping from network coordinates to observ- ables and the nonlinear transformations (Baarda's S- transformations) between diAerent solutions are defined with the help of transformation parameters obtained from the solution of nonlinear equations. In particular, the transformations from any solution to an x0-nearest solution (corresponding to Meissl's inner solution) are given for two- and three-dimensional networks for both the similarity and the rigid transformation case. Finally the nonlinear theory is specialized to the linear case with the help of the singular-value decomposition and algebraic expressions with specific geometric meaning are given for all possible types of generalized inverses.

On some problems of the downward continuation of the 5 ' ×5 ' mean Helmert gravity disturbance

Abstract: This research deals with some theoretical and numerical problems of the downward continuation of mean Helmert gravity disturbances. We prove that the downward continuation of the disturbing potential is much smoother, as well as two orders of magnitude smaller than that of the gravity anomaly, and we give the expression in spectral form for calculating the disturbing potential term. Numerical results show that for calculating truncation errors the first 180∘ of a global potential model suffice. We also discuss the theoretical convergence problem of the iterative scheme. We prove that the 5′×5′ mean iterative scheme is convergent and the convergence speed depends on the topographic height; for Canada, to achieve an accuracy of 0.01 mGal, at most 80 iterations are needed. The comparison of the “mean” and “point” schemes shows that the mean scheme should give a more reasonable and reliable solution, while the point scheme brings a large error to the solution.

Spherical panel clustering and its numerical aspects

Abstract: In modern approximation methods, linear combinations in terms of (space localizing) radial basis functions play an essential role. Areas of application are numerical integration formulae on the unit sphere Ω corresponding to prescribed nodes, spherical spline interpolation and spherical wavelet approximation. The evaluation of such a linear combination is a time-consuming task, since a certain number of summations, multiplications and the calculation of scalar products are required. A generalization of the panel clustering method in a spherical setup is presented. The economy and efficiency of panel clustering are demonstrated for three fields of interest, namely upward continuation of the Earth's gravitational potential, geoid computation by spherical splines and wavelet reconstruction of the gravitational potential.

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

Abstract: 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 Gaus–Kruger 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.

Terrain corrections to power 3 in gravimetric geoid determination

Abstract: In precise geoid determination by Stokes formula, direct and primary and secondary indirect terrain effects are applied for removing and restoring the terrain masses. We use Helmert's second condensation method to derive the sum of these effects, together called the total terrain effect for geoid. We develop the total terrain effect to third power of elevation H in the original Stokes formula, Earth gravity model and modified Stokes formula. It is shown that the original Stokes formula, Earth gravity model and modified Stokes formula all theoretically experience different total terrain effects. Numerical results indicate that the total terrain effect is very significant for moderate topographies and mountainous regions. Absolute global mean values of 5–10 cm can be reached for harmonic expansions of the terrain to degree and order 360. In another experiment, we conclude that the most important part of the total terrain effect is the contribution from the second power of H, while the contribution from the third power term is within 9 cm.

Fast integration of low orbiter's trajectory perturbed by the earth's non-sphericity

Abstract: A fast algorithm is proposed to integrate the trajectory of a low obiter perturbed by the earth's non- sphericity. The algorithm uses a separation degree to define the low-degree and the high-degree acceleration components, the former computed rigorously, and the latter interpolated from gridded accelerations. An FFT method is used to grid the accelerations. An optimal grid type for the algorithm depends on the trajectory's permissible error, speed, and memory capacity. Using the non-spherical accelerations computed from EGM96 to harmonic degree 360, orbit integrations were per- formed for a low orbiter at an altitude of 170 km. For a separation degree of 50, the new algorithm, together with the predict-pseudo correct method, speeds up the integration by 145 times compared to the conventional algorithm while keeping the errors in position and velocity below 10 ˇ4 m and 10 ˇ7 m/s for a 3-day arc.

The optimal Mercator projection and the optimal polycylindric projection of conformal type – case-study Indonesia

Abstract: As a conformal mapping of the sphere S 2 R or of the ellipsoid of revolution E 2 A , B the Mercator projection maps the equator equidistantly while the transverse Mercator projection maps the transverse metaequator, the meridian of reference, with equidistance. Accordingly, the Mercator projection is very well suited to geographic regions which extend east-west along the equator; in contrast, the transverse Mercator projection is appropriate for those regions which have a south-north extension. Like the optimal transverse Mercator projection known as the Universal Transverse Mercator Projection (UTM), which maps the meridian of reference Λ0 with an optimal dilatation factor &ρcirc;=0.999 578 with respect to the World Geodetic Reference System WGS 84 and a strip [Λ0−Λ W ,Λ0 + Λ E ]×[Φ S ,Φ N ]= [−3.5∘,+3.5∘]×[−80∘,+84∘], we construct an optimal dilatation factor ρ for the optimal Mercator projection, summarized as the Universal Mercator Projection (UM), and an optimal dilatation factor ρ0 for the optimal polycylindric projection for various strip widths which maps parallel circles Φ0 equidistantly except for a dilatation factor ρ0, summarized as the Universal Polycylindric Projection (UPC). It turns out that the optimal dilatation factors are independent of the longitudinal extension of the strip and depend only on the latitude Φ0 of the parallel circle of reference and the southern and northern extension, namely the latitudes Φ S and Φ N , of the strip. For instance, for a strip [Φ S ,Φ N ]= [−1.5∘,+1.5∘] along the equator Φ0=0, the optimal Mercator projection with respect to WGS 84 is characterized by an optimal dilatation factor &ρcirc;=0.999 887 (strip width 3∘). For other strip widths and different choices of the parallel circle of reference Φ0, precise optimal dilatation factors are given. Finally the UPC for the geographic region of Indonesia is presented as an example.

Combining the geodetic models of vertical crustal deformation

Abstract: Different kinds of analytical models of vertical curstal deformation have both advantages and disadvantages, and are appropriate to different deformation patterns. It is difficult to identify which deformation model is most suitable for a particular deformation area. In order to obtain a more precise and reliable analytical result, the combined model based on forecast theory has been approached. As a result, the fitting ability of single models is significantly improved. The combined model still possesses the same important features as the single models, which enable transferal of the derived crustal deformation information, within or even outside the time span of the data coverage, from geodetic sites to any user-specified locations where geodetic data may not exist. Examples are presented of both numerical simulation and a real-life situation, and the results are analysed in detail to demonstrate the efficiency of the proposed methodology.