Downward continuation of gravitational field quantities to an irregular surface by spectral weighting

Abstract

In geophysical and geodetic studies, gravity inversion is typically performed such that observed gravity values are first continued downward onto a regular (planar, spherical or spheroidal) surface by solving an inverse integral transform, which originates from a classical solution to the first boundary-value problem in potential theory. A typical example is continuing gravity observed at the topographic surface down to the mean sea level (geoid). Nowadays, gravity-dedicated satellite missions and aerial gravimetry provide gravity data above the topographic surface in addition to classical terrestrial gravity observations. For specific purposes (such as gravity data combination and validation, or quasigeoid determination), satellite and aerial gravity observations have to be continued to the irregular topographic surface. In this study, we address this issue by formulating a functional model for a spectral downward continuation of selected gravitational field quantities to an irregular topographic surface. Moreover, we generalize this functional model to allow for transformation between different types of gravitational field quantities. In particular, we derive spectral weights for estimation of the disturbing potential or disturbing/anomalous gravity at the Earth’s surface by combining the first-, second- and third-order radial gradients of the disturbing potential (disturbing gradients). The correctness of the developed combined spectral estimator is verified in a closed-loop test based on synthetic satellite disturbing gradients. The combined spectral estimator is applied to simulated satellite disturbing gradients polluted by a realistic Gaussian noise. Results of the numerical experiments show that the combined spectral estimator puts the highest importance on the least polluted disturbing gradient, while the contribution of the least accurate disturbing gradient is negligible. An important advantage of this spectral combination method is that no matrix inversion with numerical instabilities requiring regularization is needed.

Appendices

Appendix A: Formulas for transforming the first-, second- and third-order disturbing gradients onto the disturbing potential, gravity disturbance and gravity anomaly

Integral transforms between the disturbing potential and its first-, second- and third-order disturbing gradients are defined as (Novák et al. 2017):

$$\begin{aligned} T_{{}}^{z} (r_{\text{Top}} ,\varOmega ) &= - \frac{{r_{\text{Top}} }}{4\pi }\\&\quad\times\int\limits_{{\varOmega^{\prime}}} { {\left[ {\sum\limits_{n}^{{N_{\text{max} } }} {\frac{{ {2n + 1} }}{n + 1}\frac{1}{{t^{{\left( {n + 2} \right)}} }}} P_{n} (\cos \psi )} \right] \, T_{z} (R_{\text{MOS}} ,\varOmega^{\prime})} } {\text{d}}\varOmega^{\prime}, \end{aligned}$$

(A.1)

$$\begin{aligned} T_{{}}^{zz} (r_{\text{Top}} ,\varOmega ) &= \frac{{r_{\text{Top}}^{2} }}{4\pi }\\ \quad&\times\int\limits_{{\varOmega^{\prime}}} { {\left[ {\sum\limits_{n}^{{N_{\text{max} } }} {\frac{{ {2n + 1} }}{(n + 1)(n + 2)}} \frac{1}{{t^{{\left( {n + 3} \right)}} }}P_{n} (\cos \psi )} \right] \, T_{zz} (R_{\text{MOS}} ,\varOmega^{\prime})} } {\text{d}}\varOmega^{\prime}, \end{aligned} $$

(A.2)

$$ \begin{aligned}T_{{}}^{zzz} (r_{\text{Top}} ,\varOmega ) &= - \frac{{r_{\text{Top}}^{3} }}{4\pi }\\&\quad\times\int\limits_{{\varOmega^{\prime}}} \left[ {\sum\limits_{n}^{{N_{\text{max} } }} {\frac{{ {2n + 1} }}{(n + 1)(n + 2)(n + 3)}} \frac{1}{{t^{{\left( {n + 4} \right)}} }}P_{n} (\cos \psi )} \right] \\&\quad \times T_{zzz} (R_{\text{MOS}} ,\varOmega^{\prime}) {\text{d}}\varOmega^{\prime}. \end{aligned}$$

(A.3)

Relations between the gravity disturbance and the first-, second- and third-order disturbing gradients read as follows (Novák et al. 2017):

$$\begin{aligned} \delta g_{{}}^{z} (r_{\text{Top}} ,\varOmega ) &= - \frac{1}{4\pi }\\&\quad\times\int\limits_{{\varOmega^{\prime}}} { {\left[ {\sum\limits_{n}^{{N_{\text{max} } }} {\left( {2n + 1} \right)\frac{1}{{t^{{\left( {n + 2} \right)}} }}} P_{n} (\cos \psi )} \right] \, T_{z} (R_{\text{MOS}} ,\varOmega^{\prime})} } {\text{d}}\varOmega^{\prime}, \end{aligned}$$

(A.4)

$$ \begin{aligned}\delta g_{{}}^{zz} (r_{\text{Top}} ,\varOmega ) &= \frac{{r_{\text{Top}}^{{}} }}{4\pi }\\&\quad\times\int\limits_{{\varOmega^{\prime}}} { {\left[ {\sum\limits_{n}^{{N_{\text{max} } }} {\frac{{ {2n + 1} }}{n + 2}} \frac{1}{{t^{{\left( {n + 3} \right)}} }}P_{n} (\cos \psi )} \right] \, T_{zz} (R_{\text{MOS}} ,\varOmega^{\prime})} } {\text{d}}\varOmega^{\prime}, \end{aligned}$$

(A.5)

$$\begin{aligned} \delta g_{{}}^{zzz} (r_{\text{Top}} ,\varOmega ) &= - \frac{{r_{\text{Top}}^{2} }}{4\pi }\\ &\quad\times\int\limits_{{\varOmega^{\prime}}} \left[ {\sum\limits_{n}^{{N_{\text{max} } }} {\frac{{ {2n + 1} }}{(n + 2)(n + 3)}} \frac{1}{{t^{{\left( {n + 4} \right)}} }}P_{n} (\cos \psi )} \right] \\ &\quad \times T_{zzz} (R_{\text{MOS}} ,\varOmega^{\prime}) {\text{d}}\varOmega^{\prime}. \end{aligned}$$

(A.6)

The gravity anomaly can be calculated from the first-, second- and third-order disturbing gradients via the following integral transformations (Novák et al. 2017):

$$\begin{aligned} \Delta g_{{}}^{z} (r_{\text{Top}} ,\varOmega ) &= - \frac{1}{4\pi }\\&\quad\times\int\limits_{{\varOmega^{\prime}}} { {\left[ {\sum\limits_{n}^{{N_{\text{max} } }} {\frac{{\left( {2n + 1} \right)\left( {n - 1} \right)}}{{ {n + 1} }}\frac{1}{{t^{{\left( {n + 2} \right)}} }}} P_{n} (\cos \psi )} \right] \, T_{z} (R_{\text{MOS}} ,\varOmega^{\prime})} } {\text{d}}\varOmega^{\prime}, \end{aligned}$$

(A.7)

$$\begin{aligned} \Delta g_{{}}^{zz} (r_{\text{Top}} ,\varOmega ) &= \frac{{r_{\text{Top}}^{{}} }}{4\pi }\\&\quad\times\int\limits_{{\varOmega^{\prime}}} { {\left[ {\sum\limits_{n}^{{N_{\text{max} } }} {\frac{{\left( {2n + 1} \right)\left( {n - 1} \right)}}{{\left( {n + 1} \right)(n + 2)}}} \frac{1}{{t^{{\left( {n + 3} \right)}} }}P_{n} (\cos \psi )} \right] \, T_{zz} (R_{\text{MOS}} ,\varOmega^{\prime})} } {\text{d}}\varOmega^{\prime}, \end{aligned}$$

(A.8)

$$\begin{aligned} \Delta g_{{}}^{zzz} (r_{\text{Top}} ,\varOmega ) &= - \frac{{r_{Top}^{2} }}{4\pi }\\&\quad\times\int\limits_{{\varOmega^{\prime}}} \left[ {\sum\limits_{n}^{{N_{\text{max} } }} {\frac{{\left( {2n + 1} \right)\left( {n - 1} \right)}}{{\left( {n + 1} \right)(n + 2)(n + 3)}}} \frac{1}{{t^{{\left( {n + 4} \right)}} }}P_{n} (\cos \psi )} \right] \\ &\quad\times T_{zzz} (R_{\text{MOS}} ,\varOmega^{\prime}) {\text{d}}\varOmega^{\prime}, \end{aligned}$$

(A.9)

Appendix B: Formulas for empirical signal and error degree-order variances of the disturbing gradients

The signal degree-order variances of the disturbing potential, gravity disturbance and gravity anomaly, respectively, are defined as follows:

$$ c_{n,P}^{{}} = \frac{{G^{2} M^{2} }}{{R^{2} }}\left( {\frac{R}{r}} \right)^{2n + 2} \sum\limits_{m} {\bar{C}_{n,m}^{2} } , $$

(B.1)

$$ c_{n,H}^{{}} = \frac{{G^{2} M^{2} }}{{R^{4} }}\left( {\frac{R}{r}} \right)^{2n + 4} (n + 1)^{2} \sum\limits_{m} {\bar{C}_{n,m}^{2} } , $$

(B.2)

$$ c_{n,S}^{{}} = \frac{{G^{2} M^{2} }}{{R^{4} }}\left( {\frac{R}{r}} \right)^{2n + 4} (n - 1)^{2} \sum\limits_{m} {\bar{C}_{n,m}^{2} } , $$

(B.3)

where $\bar{C}_{n,m}^{{}}$ are the fully normalized geopotential coefficients of the degree n and order m. The signal degree-order variances of the first-, second and third-order disturbing gradients are defined as follows:

$$ c_{{n,T_{z} }}^{{}} = \frac{{G^{2} M^{2} }}{{R^{4} }}\left( {\frac{R}{r}} \right)^{2n + 4} \left( {n + 1} \right)^{2} \sum\limits_{m} {\bar{C}_{n,m}^{2} } , $$

(B.4)

$$ c_{{n,T_{zz} }}^{{}} = \frac{{G^{2} M^{2} }}{{R^{6} }}\left( {\frac{R}{r}} \right)^{2n + 6} \left[ {(n + 1)(n + 2)} \right]^{2} \sum\limits_{m} {\bar{C}_{n,m}^{2} } , $$

(B.5)

$$ c_{{n,T_{zzz} }}^{{}} = \frac{{G^{2} M^{2} }}{{R^{8} }}\left( {\frac{R}{r}} \right)^{2n + 8} \left[ {(n + 1)(n + 2)(n + 3)} \right]^{2} \sum\limits_{m} {\bar{C}_{n,m}^{2} } . $$

(B.6)

The corresponding error degree-order variances of the disturbing gradients are given according to the following formulas:

$$ \sigma_{{n,T_{z} }}^{2} = \frac{{G^{2} M^{2} }}{{R^{4} }}\left( {\frac{R}{r}} \right)^{2n + 4} \left( {n + 1} \right)^{2} \sum\limits_{m} {\sigma_{{\bar{C}_{n,m} }}^{2} } , $$

(B.7)

$$ \sigma_{{n,T_{zz} }}^{2} = \frac{{G^{2} M^{2} }}{{R^{6} }}\left( {\frac{R}{r}} \right)^{2n + 6} [(n + 1)(n + 2)]^{2} \sum\limits_{m} {\sigma_{{\bar{C}_{n,m} }}^{2} } , $$

(B.8)

$$ \sigma_{{n,T_{zzz} }}^{2} = \frac{{G^{2} M^{2} }}{{R^{8} }}\left( {\frac{R}{r}} \right)^{2n + 8} \left[ {(n + 1)(n + 2)(n + 3)} \right]^{2} \sum\limits_{m} {\sigma_{{\bar{C}_{n,m} }}^{2} } . $$

(B.9)

In Eqs. (B.7)–(B.9), $\sigma_{{\bar{C}_{n,m} }}^{{}}$ represent errors of the fully normalized geopotential coefficients.

Appendix C: Estimation of noise levels

In this appendix, we provide an insight how the standard deviation of the Gaussian noise applied in numerical experiments with noisy gradient data is derived. We consider the same analytical error model for differential accelerometry as was presented by Šprlák et al. (2016). We provide parameters of a hypothetic satellite mission similar to the GOCE mission in Table 7. The accelerometers’ power spectral densities (PSD) for measuring the first-, second- and third-order radial derivatives of the gravitational potential are defined as follows (Šprlák et al. 2016):

Table 7 Parameters of a hypothetic satellite mission (similar to the GOCE mission) used for generation of the Gaussian noise

$$ Q_{{a_{z} }} = \frac{2D}{\pi }\frac{{G^{2} M^{2} }}{{R^{4} }}\left( {\frac{R}{R + h}} \right)^{2n + 4} \left( {n + 1} \right)^{2} \frac{{10^{ - 10} }}{{n^{4} }}, $$

(C.1)

$$ Q_{{a_{z} }} = \frac{{\Delta z^{2} D}}{\pi }\frac{{G^{2} M^{2} }}{{R^{6} }}\left( {\frac{R}{R + h}} \right)^{2n + 6} \left[ {\left( {n + 1} \right)\left( {n + 2} \right)} \right]^{2} \frac{{10^{ - 10} }}{{n^{4} }}, $$

(C.2)

$$ Q_{{a_{z} }} = \frac{{\Delta z^{4} D}}{3\pi }\frac{{G^{2} M^{2} }}{{R^{8} }}\left( {\frac{R}{R + h}} \right)^{2n + 8} \left[ {\left( {n + 1} \right)\left( {n + 2} \right)\left( {n + 3} \right)} \right]^{2} \frac{{10^{ - 10} }}{{n^{4} }}. $$

(C.3)

Equations (C.1), (C.2) and (C.3) yield the following values: $\sqrt {Q_{{a_{z} }} } = 2.5 \times 10^{ - 8}$ m s⁻² Hz^−0.5, $\sqrt {Q_{{a_{z} }} } = 4 \times 10^{ - 13}$ m s⁻² Hz^−0.5 and $\sqrt {Q_{{a_{z} }} } = 5.3 \times 10^{ - 18}$ m s⁻² Hz^−0.5, respectively. We use PSD calculated from Eq. (C.1) in further calculations. This noise level is achievable by current sensors while the noise from Eqs. (C.2) and (C.3) is beyond current technological limits. The relation between the accelerometers’ accuracy and accelerometers’ PSD is defined as (Sneeuw 2000, Sect. 5.4):

$$ \sigma_{{a_{z} }}^{2} = \frac{{Q_{{a_{z} }} }}{\Delta t}. $$

(C.4)

The noise levels of the first-, second- and third-order radial derivatives of the gravitational potential are defined as (Šprlák et al. 2016):

$$ \sigma_{{V_{z} }}^{2} = \sigma_{{a_{z} }}^{2} , $$

(C.5)

$$ \sigma_{{V_{zz} }}^{2} \approx \frac{2}{{\Delta z^{2} }}\sigma_{{a_{z} }}^{2} , $$

(C.6)

$$ \sigma_{{V_{zzz} }}^{2} \approx \frac{6}{{\Delta z^{4} }}\sigma_{{a_{z} }}^{2} . $$

(C.7)

According to the parameters of the satellite mission defined in Table 7 and to the previous equations, the error degree-order variances of the first-, second- and third-order radial derivatives are $\sigma_{{V_{z} }}^{2} = 6. 1 6\times 1 0^{ - 16} \;{\text{m}}^{2} \;{\text{s}}^{ - 2}$, $\sigma_{{V_{zz} }}^{2} = 4. 7 4\times 1 0^{ - 32} \;{\text{s}}^{ - 2}$ and $\sigma_{{V_{zzz} }}^{2} = 3.95 \times 1 0^{ - 33} \;{\text{m}}^{ - 2} \;{\text{s}}^{ - 2}$, respectively.