<p>Downward-continuing the Earth’s surface gravity observations to the reference ellipsoid is a key issue in gravity field modeling. In this paper, we present an iterative downward continuation approach completely in the oblate spheroidal coordinate system and ellipsoidal harmonics, based on the Taylor series with respect to the semi-minor axis of the confocal ellipsoid. The use of the oblate spheroidal coordinates simplifies the calculation of gradients. Through numerical tests, the accuracy of the Taylor series expansion was validated. In the closed-loop experiments, we applied the iterative technique to the global disturbing potential and gravity disturbance grids synthesized from truncated XGM2019e ellipsoidal harmonic coefficients. We found that 500 iterations are sufficient for practical applications of downward continuation up to degree 5399, to recover the gravity field with a global error of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sim 4.93 \times 10^{ - 4} {\text{ m}}^{{2}} {\text{s}}^{{ - 2}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>∼</mo> <mn>4.93</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>4</mn> </mrow> </msup> <msup> <mrow> <mspace width="0.333333em" /> <mtext>m</mtext> </mrow> <mn>2</mn> </msup> <msup> <mrow> <mtext>s</mtext> </mrow> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> in disturbing potential, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\sim 5.03 \times 10^{ - 5} {\text{ m}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>∼</mo> <mn>5.03</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>5</mn> </mrow> </msup> <mrow> <mspace width="0.333333em" /> <mtext>m</mtext> </mrow> </mrow> </math></EquationSource> </InlineEquation> in height anomaly at most. The iterative downward continuation approach was then applied to modeling a composite of gravity disturbance data set from SRTM2gravity and XGM2019e up to d/o 5399. The results show that short wavelength signals with a root mean square (RMS) of 0.60 mGal above the harmonic band are not modeled, so the model yields significant errors in some areas. At last, we augment the existing XGM2019e model on the continents up to d/o 10,799 via the iterative downward continuation approach; the evaluations by ground-truth data show that the augmented model improves XGM2019e by ~ 25% in Colorado and ~ 37% in Switzerland. In summary, the proposed iterative downward continuation method can provide a convenient and reliable solution to the modeling of the Earth’s surface gravity field. This may be beneficial to further enhance the resolution of global gravity field models.</p>

Iterative Downward Continuation and Modeling of the Earth’s Global Gravity Field in Ellipsoidal Harmonics

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发表日期：

2026年2月7日

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Cong Liu,
Zhengtao Wang,
Yu Gao,
Yang Xiao

摘要

Downward-continuing the Earth’s surface gravity observations to the reference ellipsoid is a key issue in gravity field modeling. In this paper, we present an iterative downward continuation approach completely in the oblate spheroidal coordinate system and ellipsoidal harmonics, based on the Taylor series with respect to the semi-minor axis of the confocal ellipsoid. The use of the oblate spheroidal coordinates simplifies the calculation of gradients. Through numerical tests, the accuracy of the Taylor series expansion was validated. In the closed-loop experiments, we applied the iterative technique to the global disturbing potential and gravity disturbance grids synthesized from truncated XGM2019e ellipsoidal harmonic coefficients. We found that 500 iterations are sufficient for practical applications of downward continuation up to degree 5399, to recover the gravity field with a global error of $\sim 4.93 \times 10^{ - 4} {\text{ m}}^{{2}} {\text{s}}^{{ - 2}}$ $\sim 4.93 \times 10^{- 4} m^{2} s^{- 2}$ in disturbing potential, $\sim 5.03 \times 10^{ - 5} {\text{ m}}$ $\sim 5.03 \times 10^{- 5} m$ in height anomaly at most. The iterative downward continuation approach was then applied to modeling a composite of gravity disturbance data set from SRTM2gravity and XGM2019e up to d/o 5399. The results show that short wavelength signals with a root mean square (RMS) of 0.60 mGal above the harmonic band are not modeled, so the model yields significant errors in some areas. At last, we augment the existing XGM2019e model on the continents up to d/o 10,799 via the iterative downward continuation approach; the evaluations by ground-truth data show that the augmented model improves XGM2019e by ~ 25% in Colorado and ~ 37% in Switzerland. In summary, the proposed iterative downward continuation method can provide a convenient and reliable solution to the modeling of the Earth’s surface gravity field. This may be beneficial to further enhance the resolution of global gravity field models.