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.     

On the equivalence of spherical splines with least-squares collocation and Stokes’s 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.     

Short guide to direct gravitational field modelling with Hotine’s equations

This paper presents a unified approach to the least squares spherical harmonic analysis of the acceleration vector and Eötvös tensor (gravitational gradients) in an arbitrary orientation. The Jacobian matrices are based on Hotine’s equations that hold in the Earth-fixed Cartesian frame and do not need any derivatives of the associated Legendre functions. The implementation was confirmed through closed-loop tests in which the simulated input is inverted in the least square sense using the rotated Hotine’s equations. The precision achieved is at the level of rounding error with RMS about $10^{-12}{-}10^{-14}$  m in terms of the height anomaly. The second validation of the linear model is done with help from the standard ellipsoidal correction for the gravity disturbance that can be computed with an analytic expression as well as with the rotated equations. Although the analytic expression for this correction is only of a limited accuracy at the submillimeter level, it was used for an independent validation. Finally, the equivalent of the ellipsoidal correction, called the effect of the normal, has been numerically obtained also for other gravitational functionals and some of their combinations. Most of the numerical investigations are provided up to spherical harmonic degree 70, with degree 80 for the computation time comparison using real GRACE data. The relevant Matlab source codes for the design matrices are provided.     

Surface gravitational field and topography changes induced by the Earth’s fluid core motions

Using a Love number formalism, the elastic deformations of the mantle and the mass redistribution gravitational potential within the Earth induced by the fluid pressure acting at the core–mantle boundary (CMB) are computed. This pressure field changes at a decadal time scale and may be estimated from observations of the surface magnetic field and its secular variation. First, using a spherical harmonic expansion, the poloidal and toroidal part of the fluid velocity field at the CMB for the last 40 years is computed, under the hypothesis of tangential geostrophy. Then the associated geostrophic pressure, whose order of magnitude is about 1000 Pa, is computed. The surface topography induced by this pressure field is computed using Love numbers, and is a few millimetres. The mass redistribution gravitational potential induced by these deformations and, in particular, the zonal components of the related surface gravitational potential perturbation (J₂, J₃ and J₄ coefficients), are calculated. Overall perturbations for the J₂ coefficient of about 10^–10, for J₃ of about 10^–11 and for J₄ are found of about 0.3×10^–11. Finally, these theoretical results are compared with recent observations of the decadal variation of J₂ from satellite laser ranging. Results concerning J₂ can be described as follows: first, they are one order of magnitude too small to explain the observed decadal variation of J₂ and, second, they show a significant linear trend over the last 40 years, whose rate of decrease amounts to 7% of the observed value.     

Spatiotemporal filtering of regional GNSS network’s position time series with missing data using principle component analysis

The existing spatiotemporal analysis methods suppose that the involved time series are complete and have the same data interval. However missing data inevitably occur in the position time series of Global Navigation Satellite Systems networks for many reasons. In this paper, we develop a modified principal component analysis to extract the Common Mode Error (CME) from the incomplete position time series. The principle of the proposed method is that a time series can be reproduced from its principle components. The method is equivalent to the method of Dong et al. (J Geophys Res 111:3405–3421, 2006) in case of no missing data in the time series and to the extended ‘stacking’ approach under the assumption of a uniformly spatial response. The new method is first applied to extract the CME from the position time series of the Crustal Movement Observation Network of China (CMONOC) over the period of 1999–2009 where the missing data occur in all stations with the different gaps. The results show that the CMEs are significant in CMONOC. The size of the first principle components for the North, East and Up coordinates are as large as 40, 41 and 37 % of total principle components and their spatial responses are not uniform. The minimum amplitudes of the first eigenvectors are only 41, 15 and 29 % for the North, East and Up coordinate components, respectively. The extracted CMEs of our method are close to the data filling method, and the Root Mean Squared error (RMS) values computed from the differences of maximum CMEs between two methods are only 0.31, 0.52 and 1.55 mm for North, East and Up coordinates, respectively. The RMS of the position time series is greatly reduced after filtering out the CMEs. The accuracies of the reconstructed missing data using the two methods are also comparable. To further comprehensively test the efficiency of our method, the repeated experiments are then carried out by randomly deleting different percentages of data at some stations. The results show that the CMEs can be extracted with high accuracy at the non missing-data epochs. And at the missing-data epochs, the accuracy of extracted CMEs has a strong dependence on the number of stations with missing data.     

Corn Area Extraction by the Integration of MODIS-EVI Time Series Data and China’s Environment Satellite (HJ-1) Data

In this paper, we intent to use the remotely sensed MODerate resolution Imaging Spectroradiometer (MODIS) data and China’s Environment Satellite (HJ-1) data for extracting the corn cultivated area over a regional scale. The high resolution HJ-1 data was to extract corn distribution at a small scale class with Support Vector Machine (SVM). The mean Enhanced Vegetation Index (EVI) time series curve of corn from MODIS was derived for the reference area and validated in a larger area. The MODIS-EVI time series curve derived from the reference area instead of the MODIS-EVI time series curve derived from the study area after validation, which was taken as the standard MODIS-EVI time series curve in for generating a standard MODIS-EVI image of corn. The mean absolute distance (MAD) between the standard MODIS-EVI image of corn and the MODIS-EVI time series image was used to detect the maximum possible extent of corn distribution in the study area. The results showed that the overall accuracy of the method was 82.17 %, with commission and omission errors of 16.85 and 15.40 %, respectively; at the county level, the satellite-estimated corn area and statistical data were well correlated (R ²?=?0.85, N?=?50) for the whole Jilin Province. It indicated that the MODIS data integrated with higher spatial resolution of HJ-1 satellite data could be utilized to enhance the extraction accuracy of corn cultivated area at a larger scale.