Robust Automatic Reduction of Multibeam Bathymetric Data Based on M-estimators

Multibeam echosounders have commonly been employed for a wide range of applications including offshore survey, navigation, hydrogeology, and oceanography. Because the tremendous volume of the bathymetric data is demanding for some purposes and requires significant storage space, the data reduction plays a prominent role in practice. Additionally, the multibeam soundings are inevitably contaminated with sporadic outliers, and as such, the data cleaning can be challenging especially in shallow waters. We present a speedily robust method for reliably reducing the volume of the bathymetric data within grid cells. In this respect, robust M-estimators are recursively applied to the data in a patch-wise manner to alleviate the undesirable effects of the outlying observations. Accordingly, the reduced bathymetry is automatically made unaffected by the possible outliers once their equivalent weights have been downweighted. The performance of the presented method has been demonstrated by synthetic datasets and an experimental dataset collected by an ATLAS FS 20/100 kHz shallow-water multibeam echosounder in the offshore waters of Kish wharf. The reliability, efficiency, and capability of the proposed method have been verified, which makes it quite possible to meet the IHO requirements for special-order seafloor mapping.     

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.     

Ellipsoidal Neumann geodetic boundary-value problem based on surface gravity disturbances: case study of Iran

An ellipsoidal Neumann type geodetic boundary-value problem (GBVP) for the computation of disturbing potential on the surface of the Earth based on the surface gravity disturbance as the boundary data is formulated. The solution methodology of the GBVP can be algorithmically summarized as follows: (i) using global navigation satellite systems (GNSS) coordinates of the gravity stations, the surface gravity disturbances are generated as the boundary data. (ii) Applying the deflection correction to the gravity disturbances to arrive at the derivative of the surface disturbing potential along the ellipsoidal normal. (iii) Removing the low frequencies part of the gravity field using harmonic expansion to degree and order 110. (iv) Using the short wavelength part of the corrected gravity disturbances derived in the previous section as the boundary data within the constructed GBVP to derive the short wavelength disturbing potential over the Earth surface. (v) The computed shortwave length signals of disturbing potentials are converted to disturbing potential values by restoring the removed effects.     

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.

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.     

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.     

World Geodetic Datum 2000

 Based on the current best estimates of fundamental geodetic parameters {W ₀,GM,J ₂,Ω} the form parameters of a Somigliana-Pizzetti level ellipsoid, namely the semi-major axis a and semi-minor axis b (or equivalently the linear eccentricity ) are computed and proposed as a new World Geodetic Datum 2000. There are six parameters namely the four fundamental geodetic parameters {W ₀,GM,J ₂,Ω} and the two form parameters {a,b} or {a,ɛ}, which determine the ellipsoidal reference gravity field of Somigliana-Pizzetti type constraint to two nonlinear condition equations. Their iterative solution leads to best estimates a=(6 378 136.572±0.053)m, b=(6 356 751.920 ± 0.052)m, ɛ=(521 853.580±0.013)m for the tide-free geoide of reference and a=(6 378 136.602±0.053)m, b=(6 356 751.860±0.052)m, ɛ=(521 854.674 ± 0.015)m for the zero-frequency tide geoid of reference. The best estimates of the form parameters of a Somigliana-Pizzetti level ellipsoid, {a,b}, differ significantly by −0.39 m, −0.454 m, respectively, from the data of the Geodetic Reference System 1980. Received: 1 February 1999 / Accepted: 31 August 1999     

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.     

An overdetermined geodetic boundary value problem approach to telluroid and quasi-geoid computations

In this paper an overdetermined Geodetic Boundary Value Problem (GBVP) approach for telluroid and quasi-geoid computations is presented. The presented GBVP approach can solve the problem of potential value computation on the surface of the Earth, which when applied to a mapping scheme, e.g., here minimum distance mapping, provides a point-wise approach to telluroid computation. Besides, we have succeeded in reducing the number of equations and unknowns of the minimum distance telluroid mapping by one. The sufficient condition of minimum distance telluroid mapping is also recapitulated. Since the introduced GBVP approach has the advantage of implementing various gravity observables simultaneously as input boundary data, it can be regarded as a data fusion technique that exploits all available gravity data. The developed GBVP is used for the computation of the quasi-geoid within a test area in Southwest Finland.