The Iranian height datum offset from the GBVP solution and spirit-leveling/gravimetry data

Amir Ebadi; Alireza A. Ardalan; Roohollah Karimi

doi:10.1007/s00190-019-01237-x

Journal of Geodesy

pp 1–19 | Cite as

The Iranian height datum offset from the GBVP solution and spirit-leveling/gravimetry data

Authors
Authors and affiliations

Amir Ebadi
Alireza A. Ardalan
Roohollah Karimi

Original Article

First Online: 12 February 2019

Abstract

The gravity potential of the zero point of the Iranian height datum (IRHD) is determined as well as the IRHD offset from a global geoid. For this purpose, the geodetic boundary value problem (GBVP) solution based on the remove–compute–restore (RCR) technique is used. In the RCR technique, a global geopotential model (GGM) is required as a reference to remove and restore the long wavelengths of the gravity field. Since the GGMs do not have adequate accuracy over Iran, the IRHD offset is not precisely estimated by the GBVP solution. In this study, aiming to improve the latter, a combination solution based on the GBVP approach and spirit-leveling/gravimetry (LG) data, called the GBVP_LG solution, is proposed. To obtain the GBVP_LG solution, gravity potential obtained from the GBVP solution and the gravity potential differences derived from the LG data are used as two types of observations in a least-squares adjustment. The proper relative weight matrices are determined using the variance component estimation method. To evaluate the proposed method, the gravity potential differences between the start and end points of several check-lines in the leveling network derived from the GBVP and GBVP_LG solutions are compared with those of the LG data. The results show that the dependency of the GBVP_LG solution on the reference model used is much less than that of the GBVP solution. In addition, the results indicate that the GBVP_LG solution has a 42% improvement with respect to the GBVP solution in terms of root-mean-square error. As a result of the GBVP_LG solution, the gravity potential of the IRHD zero point is estimated equal to $ W_{0}^{\text{IRHD}} = 62,636,855.89 \pm 0.16\,{\text{m}}^{2} / {\text{s}}^{2} $. Therefore, the IRHD offset with respect to the geoid defined by $ W_{0} = 62,636,853.4\,{\text{m}}^{2} / {\text{s}}^{2} $ is obtained equal to $ - \,25.4 \pm 1.6\,{\text{cm}} $, which means that the IRHD is 25.4 cm below the geoid.

Keywords

Geodetic boundary value problem Spirit-leveling/gravimetry data Height system unification Iranian height datum offset Remove–compute–restore technique Variance component estimation

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Notes

Acknowledgements

We would like to thank the National Cartographic Center (NCC) of Iran for providing the data and supporting this study under contract No. 4142. We would also like to thank the editor-in-chief, Prof. J. Kusche, and the responsible editor, Prof. I.N. Tziavos, for taking the time and handling our manuscript. We are very grateful to three anonymous reviewers for very constructive and valuable comments, which helped us to improve the manuscript.

Appendix A: The uncertainties of $ W_{L} $, $ g_{L} $, $ V^{\text{RTM}} $ and $ g^{\text{RTM}} $

A1: The uncertainty of $ W_{L} $

The GGM-derived gravity potential of the Earth in terms of the spherical harmonics is represented as follows (Heiskanen and Moritz 1967):

$$ W_{L} (\phi ,\lambda ,r) = \frac{\text{GM}}{R}\mathop \sum \limits_{n = 0}^{L} \left( {\frac{R}{r}} \right)^{n + 1} \mathop \sum \limits_{m = 0}^{n} (\bar{C}_{nm} \cos (m\lambda ) + \bar{S}_{nm} \sin (m\lambda ))\bar{P}_{nm} (\sin (\phi )) + \frac{1}{2} r^{2} \omega^{2} \cos^{2} \phi $$

(22)

where $ {\text{GM}} $ is the geocentric gravitational constant, $ \omega $ is the angular velocity of the Earth, $ R $ is the mean radius of the Earth, $ \phi $ and $ \lambda $ are the spherical latitude and longitude, $ r $ is the radial distance, $ \bar{P}_{nm} $ are the fully normalized Legendre functions of the first kind, and $ \bar{C}_{nm} $ and $ \bar{S}_{nm} $ are the unit-less fully normalized spherical harmonic coefficients. By applying the error propagation law to Eq. (22) and neglecting the uncertainties of $ {\text{GM}} $, $ R $ and $ \omega $, the uncertainty of $ W_{L} $ is estimated from the following formula:

$$ \sigma_{{W_{L} }}^{2} = \left( {\frac{\text{GM}}{R}} \right)^{2} \mathop \sum \limits_{n = 0}^{L} \left( {\frac{R}{r}} \right)^{2(n + 1)} \mathop \sum \limits_{m = 0}^{n} \cos^{2} (m\lambda )\bar{P}_{nm}^{2} (\sin (\phi ))\sigma_{{\bar{C}_{nm} }}^{2} + \sin^{2} (m\lambda )\bar{P}_{nm}^{2} (\sin (\phi ))\sigma_{{\bar{S}_{nm} }}^{2} $$

(23)

where $ \sigma_{{\bar{C}_{nm} }} $ and $ \sigma_{{\bar{S}_{nm} }} $ are the uncertainties of the spherical harmonic coefficients. By considering the radial derivative of $ W_{L} $, the uncertainty of $ g_{L} $ can be approximated by:

$$ \sigma_{{g_{L} }}^{2} = \left( {\frac{\text{GM}}{{R^{2} }}} \right)^{2} \mathop \sum \limits_{n = 0}^{L} (n + 1)^{2} \left( {\frac{R}{r}} \right)^{2(n + 2)} \mathop \sum \limits_{m = 0}^{n} \cos^{2} (m\lambda )\bar{P}_{nm}^{2} (\sin (\phi ))\sigma_{{\bar{C}_{nm} }}^{2} + \sin^{2} (m\lambda )\bar{P}_{nm}^{2} (\sin (\phi ))\sigma_{{\bar{S}_{nm} }}^{2} $$

(24)

A2: The uncertainty of $ V^{\text{RTM}} $

The RTM-derived gravity potential using prism-integration forward-modeling is computed by (Nagy et al. 2000; Hirt et al. 2010):

$$ \begin{aligned} V^{\text{RTM}} & = G\rho |||xy\ln (z + r) + yz\ln (x + r) + zx\ln (y + r) \\ & \quad - \,\frac{{x^{2} }}{2}\tan^{ - 1} \frac{yz}{xr} - \frac{{y^{2} }}{2}\tan^{ - 1} \frac{zx}{yr} - \frac{{z^{2} }}{2}\tan^{ - 1} \frac{xy}{zr}|_{{x_{1} }}^{{x_{2} }} |_{{y_{1} }}^{{y_{2} }} |_{{z_{1} }}^{{z_{2} }} \\ \end{aligned} $$

(25)

where $ r $ is the distance between the point $ (x,y,z) $ and the origin of the coordinate system, and $ G $ is the gravitational constant, $ \rho $ is the topographic mass density, $ (x_{1} ,y_{1} ,z_{1} ) $ and $ (x_{2} ,y_{2} ,z_{2} ) $ are the corner coordinates of a single prism. To evaluate Eq. (25), we set $ z_{1} = 0 $ and $ z_{2} = z^{\text{SRTM}} - z^{{{\text{DTM}}\,2006.0}} $ such that the prism heights $ z_{2} - z_{1} $ represent the RTM elevations. To estimate the uncertainty of $ V^{\text{RTM}} $, we can approximate Eq. (25) with the following formula:

$$ V^{\text{RTM}} = \sum\limits_{i = 1}^{n} {V_{i}^{\text{prism}} } $$

(26)

where $ V_{i}^{\text{prism}} $ is the gravitational potential derived from ith prism which is computed from:

$$ V_{i}^{\text{prism}} = \frac{{G\rho_{i} \Delta x_{i} \Delta y_{i} \Delta z_{i} }}{{r_{i} }} $$

(27)

where $ \Delta x_{i} $, $ \Delta y_{i} $ and $ \Delta z_{i} $ are the sizes of ith prism, $ \rho_{i} $ is the mass density of ith prism and $ r_{i} $ is the distance between the center of ith prism and the origin of the coordinate system. Note that the origin of the coordinate system is located at the computational point. By applying error propagation law to Eq. (27), we have:

$$ \sigma_{{V^{\text{RTM}} }}^{2} = \sum\limits_{i = 1}^{n} {\sigma_{{V_{i}^{\text{prism}} }}^{2} } = \left( {\frac{G}{{r_{i} }}} \right)^{2} \sum\limits_{i = 1}^{n} {\rho_{i}^{2} (\Delta x_{i} \Delta y_{i} )^{2} \sigma_{{\Delta z_{i} }}^{2} + (\Delta x_{i} \Delta y_{i} \Delta z_{i} )^{2} \sigma_{{\rho_{i} }}^{2} } $$

(28)

where $ \sigma_{{V^{\text{RTM}} }} $ and $ \sigma_{{\rho_{i} }} $ are the uncertainties of $ V^{\text{RTM}} $ and $ \rho_{i} $, respectively. $ \sigma_{{\Delta z_{i} }}^{2} = \sigma_{{z_{i}^{\text{SRTM}} }}^{2} + \sigma_{{z_{i}^{{{\text{DTM}}\,2006.0}} }}^{2} $, where $ \sigma_{{z_{i}^{\text{SRTM}} }} $ and $ \sigma_{{z_{i}^{{{\text{DTM}}2006.0}} }} $ are the uncertainties of the SRTM and DTM2006.0 elevations, respectively. By considering the radial derivative of $ V_{i}^{\text{prism}} $, the uncertainty of $ g^{\text{RTM}} $ can be approximated by:

$$ \sigma_{{g^{\text{RTM}} }}^{2} = \left( {\frac{G}{{r_{i}^{2} }}} \right)^{2} \sum\limits_{i = 1}^{n} {\rho_{i}^{2} (\Delta x_{i} \Delta y_{i} )^{2} \sigma_{{\Delta z_{i} }}^{2} + (\Delta x_{i} \Delta y_{i} \Delta z_{i} )^{2} \sigma_{{\rho_{i} }}^{2} }. $$

(29)

Appendix B: Least-squares variance component estimation (LS-VCE) method

By taking initial values for $ \sigma_{1}^{2} $ and $ \sigma_{2}^{2} $, the unknowns in Eq. (16) are estimated by Eq. (17) as follows:

$$ {\hat{\mathbf{y}}} = \left( {\frac{1}{{\sigma_{1}^{2} }}{\mathbf{A}}_{1}^{\text{T}} {\mathbf{P}}_{1} {\mathbf{A}}_{1} + \frac{1}{{\sigma_{2}^{2} }}{\mathbf{A}}_{2}^{\text{T}} {\mathbf{P}}_{2} {\mathbf{A}}_{2} } \right)^{ - 1} \left( {\frac{1}{{\sigma_{1}^{2} }}{\mathbf{A}}_{1}^{\text{T}} {\mathbf{P}}_{1} {\mathbf{l}}_{1} + \frac{1}{{\sigma_{2}^{2} }}{\mathbf{A}}_{2}^{\text{T}} {\mathbf{P}}_{2} {\mathbf{l}}_{2} } \right) $$

(30)

In the LS-VCE method, the unknown variance factors $ \sigma_{1}^{2} $ and $ \sigma_{2}^{2} $ are estimated from:

$$ \hat{\sigma }^{2} = \left[ {\begin{array}{*{20}c} {\hat{\sigma }_{1}^{2} } \\ {\hat{\sigma }_{2}^{2} } \\ \end{array} } \right] = {\mathbf{N}}^{ - 1} {\mathbf{k}} $$

(31)

where $ {\mathbf{N}} $ is a $ 2 \times 2 $ matrix and $ {\mathbf{k}} $ is a $ 2 \times 1 $ vector. The entries of $ {\mathbf{N}} $ and $ {\mathbf{k}} $ matrices are derived from:

$$ \begin{aligned} n_{11} & = \frac{1}{2}{\text{trace}}({\mathbf{Q}}_{1} {\mathbf{WRQ}}_{1} {\mathbf{WR}}) \\ n_{12} & = \frac{1}{2}{\text{trace}}({\mathbf{Q}}_{1} {\mathbf{WRQ}}_{2} {\mathbf{WR}}) \\ n_{21} & = \frac{1}{2}{\text{trace}}({\mathbf{Q}}_{2} {\mathbf{WRQ}}_{1} {\mathbf{WR}}) \\ n_{22} & = \frac{1}{2}{\text{trace}}({\mathbf{Q}}_{2} {\mathbf{WRQ}}_{2} {\mathbf{WR}}) \\ \end{aligned} $$

(32)

$$ \begin{aligned} k_{1} & = \frac{1}{2}{\hat{\mathbf{v}}}^{\text{T}} {\mathbf{WQ}}_{1} {\mathbf{W\hat{v}}} \\ k_{2} & = \frac{1}{2}{\hat{\mathbf{v}}}^{\text{T}} {\mathbf{WQ}}_{2} {\mathbf{W\hat{v}}} \\ \end{aligned} $$

(33)

where

$$ {\mathbf{W}} = \left[ {\begin{array}{*{20}c} {\frac{{{\mathbf{P}}_{1} }}{{\sigma_{1}^{2} }}} & {\mathbf{0}} \\ {\mathbf{0}} & {\frac{{{\mathbf{P}}_{2} }}{{\sigma_{2}^{2} }}} \\ \end{array} } \right],\quad {\mathbf{Q}}_{1} = \left[ {\begin{array}{*{20}c} {{\mathbf{P}}_{1}^{ - 1} } & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right],\quad {\mathbf{Q}}_{2} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{P}}_{2}^{ - 1} } \\ \end{array} } \right] $$

(34)

and $ {\mathbf{R}} $ (the redundancy matrix) and $ {\hat{\mathbf{v}}} $ (the estimated residual vector) are computed from:

$$ {\mathbf{R}} = {\mathbf{I}} - \left[ \begin{aligned} {\mathbf{A}}_{1} \hfill \\ {\mathbf{A}}_{2} \hfill \\ \end{aligned} \right]\left( {\frac{1}{{\sigma_{1}^{2} }}{\mathbf{A}}_{1}^{\text{T}} {\mathbf{P}}_{1} {\mathbf{A}}_{1} + \frac{1}{{\sigma_{2}^{2} }}{\mathbf{A}}_{2}^{\text{T}} {\mathbf{P}}_{2} {\mathbf{A}}_{2} } \right)^{ - 1} \left[ {\begin{array}{*{20}c} {\frac{1}{{\sigma_{1}^{2} }}{\mathbf{A}}_{1}^{\text{T}} {\mathbf{P}}_{1} } & {\frac{1}{{\sigma_{2}^{2} }}{\mathbf{A}}_{2}^{\text{T}} {\mathbf{P}}_{2} } \\ \end{array} } \right] $$

(35)

$$ {\hat{\mathbf{v}}} = \left[ \begin{aligned} {\hat{\mathbf{v}}}_{1} \hfill \\ {\hat{\mathbf{v}}}_{2} \hfill \\ \end{aligned} \right] = \left[ \begin{aligned} {\mathbf{A}}_{1} \hfill \\ {\mathbf{A}}_{2} \hfill \\ \end{aligned} \right]{\hat{\mathbf{y}}} - \left[ \begin{aligned} {\mathbf{l}}_{1} \hfill \\ {\mathbf{l}}_{2} \hfill \\ \end{aligned} \right] $$

(36)

After estimating $ \hat{\sigma }_{1}^{2} $ and $ \hat{\sigma }_{2}^{2} $ from Eq. (31), $ \hat{\sigma }_{1}^{2} $ and $ \hat{\sigma }_{2}^{2} $ are substituted into Eqs. (30), (34), and (35), and the process will be repeated until $ \hat{\sigma }_{1}^{2} $ and $ \hat{\sigma }_{2}^{2} $ converge.

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Amir Ebadi
- 1
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Alireza A. Ardalan
- 1
Roohollah Karimi
- 2

1.School of Surveying and Geospatial Engineering, College of EngineeringUniversity of TehranTehranIran
2.Department of Geodesy and Surveying EngineeringTafresh UniversityTafreshIran

The Iranian height datum offset from the GBVP solution and spirit-leveling/gravimetry data

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