A new ellipsoidal gravimetric, satellite altimetry and astronomic boundary value problem, a case study: The geoid of Iran

A new gravimetric, satellite altimetry, astronomical ellipsoidal boundary value problem for geoid computations has been developed and successfully tested. This boundary value problem has been constructed for gravity observables of the type (i) gravity potential, (ii) gravity intensity (i.e. modulus of gravity acceleration), (iii) astronomical longitude, (iv) astronomical latitude and (v) satellite altimetry observations. The ellipsoidal coordinates of the observation points have been considered as known quantities in the set-up of the problem in the light of availability of GPS coordinates. The developed boundary value problem is ellipsoidal by nature and as such takes advantage of high precision GPS observations in the set-up. The algorithmic steps of the solution of the boundary value problem are as follows:

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and of the ellipsoidal centrifugal field for the removal of the effect of global gravity and the isostasy field from the gravity intensity and the astronomical observations at the surface of the Earth.

- Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the gravity intensity and the astronomical observations at the surface of the Earth the effect of the residual masses at the radius of up to 55 km from the computational point.

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and ellipsoidal centrifugal field for the removal from the geoidal undulations derived from satellite altimetry the effect of the global gravity and isostasy on the geoidal undulations.

- Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the geoidal undulations derived from satellite altimetry the effect of the water masses outside the reference ellipsoid within a radius of 55 km around the computational point.

- Least squares solution of the observation equations of the incremental quantities derived from aforementioned steps in order to obtain the incremental gravity potential at the surface of the reference ellipsoid.

- The removed effects at the application points are restored on the surface of reference ellipsoid.

- Application of the ellipsoidal Bruns’ formula for converting the potential values on the surface of the reference ellipsoid into the geoidal heights with respect to the reference ellipsoid.

- Computation of the geoid of Iran has successfully tested this new methodology.

A New Reference Equipotential Surface, and Reference Ellipsoid for the Planet Mars

Using the shape model of Mars GTM090AA in terms of spherical harmonics complete to degree and order 90 and gravitational field model of Mars GGM2BC80 in terms of spherical harmonics complete to degree and order 80, both from Mars Global Surveyor (MGS) mission, the geometry (shape) and gravity potential value of reference equipotential surface of Mars (Areoid) are computed based on a constrained optimization problem. In this paper, the Areoid is defined as a reference equipotential surface, which best fits to the shape of Mars in least squares sense. The estimated gravity potential value of the Areoid from this study, i.e. W ₀ = (12,654,875 ± 69) (m²/s²), is used as one of the four fundamental gravity parameters of Mars namely, {W ₀, GM, ω, J ₂₀}, i.e. {Areoid’s gravity potential, gravitational constant of Mars, angular velocity of Mars, second zonal spherical harmonic of gravitational field expansion of Mars}, to compute a bi-axial reference ellipsoid of Somigliana-Pizzetti type as the hydrostatic approximate figure of Mars. The estimated values of semi-major and semi-minor axis of the computed reference ellipsoid of Mars are (3,395,428 ± 19) (m), and (3,377,678 ± 19) (m), respectively. Finally the computed Areoid is presented with respect to the computed reference ellipsoid.     

The Galileo star scanner observations at Amalthea

In November of 2002, the Galileo spacecraft passed within 250 km of Jupiter's moon Amalthea. An onboard telescope, the star scanner, observed a series of bright flashes near the moon. It is believed that these flashes represent sunlight reflected from 7 to 9 small moonlets located within about 3000 km of Amalthea. From star scanner geometry considerations and other arguments, we can constrain the diameter of the observed bodies to be between 0.5 m to several tens of kilometers. In September of 2003, while crossing Amalthea's orbit just prior to Galileo's destruction in the jovian atmosphere, a single additional body seems to have been observed. It is suspected that these bodies are part of a discrete rocky ring embedded within Jupiter's Gossamer ring system.     

Multi-sensor approach to settlement analysis of earth dams

In this paper, a fusion method for the settlement study of the earth dams based on geodetic and geotechnical data is developed. The developed method can be algorithmically explained as follows: (a) interpolation of the geotechnical data to the epoch of geodetic observations by four degree polynomial fitting, which serves as a low-pass filter. (b) Conversion of the initial observations into time series of the dam heights at the geodetic and geotechnical stations. (c) Fusion of the data from the two sources at different fusion levels. (d) Final decision based on the deformation parameters derived from fused data. The significant innovation of the proposed method centered upon its ability to incorporate geodetic and geotechnical observation types into a one integrated solution through fusion. The method is numerically tested for the Karkhe earth dam by using geodetic and geotechnical data from 1997 till 2009. The numerical evaluation at 229 check points indicates more than 70% improvement in the settlementmodeling based on the fusion of the geodetic and geotechnical data, as compared to the settlement modeling based on geotechnical data alone.     

The spheroidal fixed–free two-boundary-value problem for geoid determination (the spheroidal Bruns' transform)

The target of the spheroidal Gauss–Listing geoid determination is presented as a solution of the spheroidal fixed–free two-boundary value problem based on a spheroidal Bruns' transformation (“spheroidal Bruns' formula”). The nonlinear spheroidal Bruns' transform (nonlinear spheroidal Bruns' formula), the spheroidal fixed part and the spheroidal free part of the two-boundary value problem are derived. Four different spheroidal gravity models are treated, in particular to determine whether they pass the test to fit to the postulate of a level ellipsoidal gravity field, namely of Somigliana–Pizzetti type. Received: 4 May 1999 / Accepted: 21 May 1999     

High-resolution regional geoid computation without applying Stokes’s formula: a case study of the Iranian geoid

A new theory for high-resolution regional geoid computation without applying Stokess formula is presented. Operationally, it uses various types of gravity functionals, namely data of type gravity potential (gravimetric leveling), vertical derivatives of the gravity potential (modulus of gravity intensity from gravimetric surveys), horizontal derivatives of the gravity potential (vertical deflections from astrogeodetic observations) or higher-order derivatives such as gravity gradients. Its algorithmic version can be described as follows: (1) Remove the effect of a very high degree/order potential reference field at the point of measurement (POM), in particular GPS positioned, either on the Earths surface or in its external space. (2) Remove the centrifugal potential and its higher-order derivatives at the POM. (3) Remove the gravitational field of topographic masses (terrain effect) in a zone of influence of radius r. A proper choice of such a radius of influence is 2r=4×10⁴ km/n, where n is the highest degree of the harmonic expansion. (cf. Nyquist frequency). This third remove step aims at generating a harmonic gravitational field outside a reference ellipsoid, which is an equipotential surface of a reference potential field. (4) The residual gravitational functionals are downward continued to the reference ellipsoid by means of the inverse solution of the ellipsoidal Dirichlet boundary-value problem based upon the ellipsoidal Abel–Poisson kernel. As a discretized integral equation of the first kind, downward continuation is Phillips–Tikhonov regularized by an optimal choice of the regularization factor. (5) Restore the effect of a very high degree/order potential reference field at the corresponding point to the POM on the reference ellipsoid. (6) Restore the centrifugal potential and its higher-order derivatives at the ellipsoidal corresponding point to the POM. (7) Restore the gravitational field of topographic masses ( terrain effect) at the ellipsoidal corresponding point to the POM. (8) Convert the gravitational potential on the reference ellipsoid to geoidal undulations by means of the ellipsoidal Bruns formula. A large-scale application of the new concept of geoid computation is made for the Iran geoid. According to the numerical investigations based on the applied methodology, a new geoid solution for Iran with an accuracy of a few centimeters is achieved.Acknowledgments. The project of high-resolution geoid computation of Iran has been support by National Cartographic Center (NCC) of Iran. The University of Tehran, via grant number 621/3/602, supported the computation of a global geoid solution for Iran. Their support is gratefully acknowledged. A. Ardalan would like to thank Mr. Y. Hatam, and Mr. K. Ghazavi from NCC and Mr. M. Sharifi, Mr. A. Safari, and Mr. M. Motagh from the University of Tehran for their support in data gathering and computations. The authors would like to thank the comments and corrections made by the four reviewers and the editor of the paper, Professor Will Featherstone. Their comments helped us to correct the mistakes and improve the paper.     

An alternative method for density variation modeling of the crust based on 3-D gravity inversion

In this paper we are proposing an alternative method for determination of density variations of the crust from constrained inversion of the terrestrial gravity data. The main features of the method can be summarized as follows: (i) Constructing a band-pass filter to remove the long and short wavelength signals from the terrestrial gravity data. (ii) Using an iterative method for stabilization and solution of the inverse problem. The mentioned regularization method is first validated by simulated gravity data and next the methodology is used for development of a new regional density variation model of the crust in three layers based on real gravity data in geographical area of Iran. Application of the band-pass filter to the latter data resulted the residual gravitation variations in the range of − 300 to 50 (mGal) which next based on the iterative method resulted following ranges for residual densities: −120 to 40 (kg/m³) in first layer, −40 to 40 (kg/m³) in second layer, and − 40 to 40 (kg/m³) in third layer.     

Improved vessel squat modeling for hydrographic and navigation applications using kinematic GNSS positioning

The squat phenomenon, that is, the sinkage of a vessel due to its motion can affect the safety of navigation and reduce the accuracy of hydrographic bathymetry. Therefore, it is necessary to model and predict the squat of vessels as a function of cruise speed. We present a Global Navigation Satellite Systems–based squat modeling method for both hydrographic and navigation applications. For implementation of the proposed method, onboard GPS antennae configurations are offered to model bow squat for full-form ships such as supertankers or ore–bulk–oil carriers as well as stern squat for fine-form vessels such as passenger liners or container ships. In the proposed methodology, the onboard GPS observations are used to determine cruise ground speed, heave, attitude, and controlling the quality of kinematic positioning via fixed baselines. The vessel squat is computed from ellipsoidal height differences of the onboard antennae with respect to a reference state, after removal of all disturbing effects due to roll, pitch, heave, tide, vessel load, and geoidal height variations. The final products of the proposed approach are the analytical squat models usable for hydrographic and navigation applications. As the case study, the method is applied to a survey vessel in the offshore waters of Kish harbor. Numerical results indicate that the experimental precision of the derived analytical squat models is in the range of 0.003–0.028 m. The computed navigation squat of the test vessel at a speed of 12.64 knots is 30 % of the vessel draft and about twice its hydrographic squat. Although the field test was performed on a survey vessel, the method can be applied to any ship at any waterway. The proposed method can address the inevitable demand of reliable squat models for delicate hydrographic projects and high-speed marine traffic.     

Piles in fully liquefied soils with lateral spread

The paper provides a new analysis procedure for the assessment of the lateral response of isolated piles/drilled shafts in saturated sands as liquefaction and lateral soil spread develop in response to dynamic loading such as that generated by the earthquake shaking. The presented method accounts for: (1) the development of full liquefaction in the free-field soil that could trigger the lateral spread of the overlying crust layer; (2) the driving force exerted by the crust layer based on the interaction between the pile and the upper non-liquefied soil (crust) layer; and (3) the variation of the excess pore water pressure (i.e. post-liquefaction soil strength) in the near-field soil with the progressive pile deflection under lateral soil spread driving force. A constitutive model for fully liquefied sands under monotonic loading and undrained conditions is developed in order to predict the zone of post-liquefaction zero-strength of liquefied sand before it rebounds with the increasing soil strain in the near-field. The analytical and empirical concepts employed in the Strain Wedge (SW) model allow the modeling of such a sophisticated phenomenon of lateral soil spread that could accompany or follow the occurrence of seismic events without using modifying parameters or shape corrections to account for soil liquefaction.