Deflections of the vertical from full-tensor and single-instrument gravity gradiometry

Abstract

Gravity gradiometry on a moving platform, whether ground or airborne, has the potential to offer an efficient and accurate determination of the deflection of the vertical by simple line integration. A significant error in this process is a trend error that results from the integration of systematic gradient errors. Using an airborne full-tensor gradiometry data set of regularly spaced and intersecting tracks over a 10 km square region and the USDOV2012 vertical deflection model to calibrate these long wavelength errors, it is shown that the gradient-derived deflections agree with the USDOV2012 model at the level of 0.6–0.9 arcsec. Moreover, it is shown by graphical inspection that these differences represent high-frequency signal rather than error. Another data processing technique is examined using only (simulated) single-gradiometer instrument data, i.e., the local differential curvature components, $(\varGamma _{{ yy}} - \varGamma _{{ xx}})/2$ and $\varGamma _{{ xy}}$, of the gravity field. While in theory these data can yield deflection components using two parallel data tracks, the results in the tested case are unsatisfactory due to implicit additional cross-track integration errors that accumulate systematically. The analysis thus demonstrates the importance of using the individual horizontal gradient components, $\varGamma _{{ xx}}$, $\varGamma _{{ yy}}$, to derive the deflection of the vertical.

Appendix

The planar approximation for gravity gradients means that one neglects the variation in the directional derivatives of the potential due to Earth’s curvature. The approximation assumes a constant direction for all derivatives regardless of location. However, the orientation of a typical gradiometer, or, more importantly, of its processed data, is maintained in a local-level, north-slaved coordinate frame (such as north-east-down). Thus, the difference between the actual and assumed data involves, in the first place, a rotation of the gradient tensor by the angles of Earth’s curvature. For local areas less than 60 km in dimension (as in the case studied here), that angle is less than 30 / R, where $R=6371\,\hbox { km}$, hence, less than $0.3^{\circ }$. Let these angles in the north and east directions be denoted, $\chi$ and $\zeta$, respectively. In addition, using the UTM projection, for example, for the Cartesian coordinates neglects the convergence of the meridians, which for the area under study is $\alpha \le 1.5^{\circ }$.

If $\mathbf{R}$ is a rotation matrix that describes these rotations of the local Cartesian coordinates from a curvilinear system then the error in the assumed gradient tensor is

$$\begin{aligned} {\varvec{\upvarepsilon }}\le \hat{{{\varvec{\Gamma }}}}^{\left( {{ xyz}} \right) }-\mathbf{R}{\varvec{\Gamma }}^{\left( {\mathrm{ned}} \right) }{} \mathbf{R}^{\mathrm{T}}, \end{aligned}$$

(A.1)

where ${\hat{{{\varvec{\Gamma }}}}}^{( {{ xyz}})}$ is the gradient tensor assumed in the local Cartesian system, but taken from the data in the curvilinear true-north, local-level system. The angles, $\chi$, $\zeta$, $\alpha$, are sufficiently small so that the error of approximation, itself, may be approximated with a “small-angle” rotation matrix,

$$\begin{aligned} \mathbf{R}= & {} \left( {{\begin{array}{ccc} 1&{}\quad \alpha &{}\quad {-\zeta } \\ \alpha &{}\quad 1&{}\quad \chi \\ \zeta &{}\quad {-\chi }&{}\quad 1 \\ \end{array} }} \right) =\mathbf{I}-\left( {{\begin{array}{ccc} 0&{}\quad {-\alpha }&{}\quad \zeta \\ \alpha &{}\quad 0&{}\quad {-\chi } \\ {-\zeta }&{}\quad \chi &{}\quad 0 \\ \end{array} }} \right) \nonumber \\= & {} \mathbf{I}-\delta \mathbf{R}. \end{aligned}$$

(A.2)

Setting ${\hat{{{\varvec{\Gamma }}}}}^{( {{ xyz}} }={{\varvec{\Gamma }}}^{( {\mathrm{ned}})}\equiv {{\varvec{\Gamma }}}$, and neglecting second-order terms, the error is

$$\begin{aligned} {\varvec{\upvarepsilon }}\le & {} {{\varvec{\Gamma }}}-\left( {\mathbf{I}-\delta \mathbf{R}} \right) {{\varvec{\Gamma }}}\left( {\mathbf{I}-\delta \mathbf{R}} \right) ^{\mathrm{T}}=\delta \mathbf{R}{\varvec{\Gamma }} +\left( {{{\varvec{\Gamma }} }\delta \mathbf{R}} \right) ^{\mathrm{T}} \nonumber \\= & {} \left( {{\begin{array}{ccc} {2\left( {\zeta \varGamma _{zx} -\alpha \varGamma _{yx} } \right) }&{}\quad {\zeta \varGamma _{zy} -\alpha \left( {\varGamma _{{ yy}} -\varGamma _{{ xx}} } \right) -\chi \varGamma _{zx} }&{}\quad {\chi \varGamma _{yx} +\zeta \left( {\varGamma _{zz} -\varGamma _{{ xx}} } \right) -\alpha \varGamma _{{ yz}} } \\ {\zeta \varGamma _{zy} -\alpha \left( {\varGamma _{{ yy}} -\varGamma _{{ xx}} } \right) -\chi \varGamma _{zx} }&{}\quad {2\left( {\alpha \varGamma _{{ xy}} -\chi \varGamma _{zy} } \right) }&{} \quad {\alpha \varGamma _{{ xz}} +\chi \left( {\varGamma _{{ yy}} -\varGamma _{zz} } \right) -\zeta \varGamma _{{ xy}} } \\ {\chi \varGamma _{yx} +\zeta \left( {\varGamma _{zz} -\varGamma _{{ xx}} } \right) -\alpha \varGamma _{{ yz}} }&{} \quad {\alpha \varGamma _{{ xz}} +\chi \left( {\varGamma _{{ yy}} -\varGamma _{zz} } \right) -\zeta \varGamma _{{ xy}} }&{} \quad {2\left( {\chi \varGamma _{{ yz}} -\zeta \varGamma _{{ xz}} } \right) } \\ \end{array} }} \right) . \end{aligned}$$

(A.3)

The gradients here are the disturbance gradients, on the order of 10–100 E (typically). Therefore, since $\chi ,\zeta <\alpha \le 3\times 10^{-2}\hbox { rad}$, the planar approximation error is at or below the level of the measurement error for present studies. Nevertheless, it is a systematic error that, when integrated, can cause trend errors in the DOV.