# Two New Methods Based on Implicit Expressions and Corresponding Predictor-Correctors for Gravity Anomaly Downward Continuation and Their Comparison

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Two Explicit Expressions for Downward Continuation

#### 2.1.1. Numerical Solutions of the Mean-Value Theorem for Gravity Anomalies

#### 2.1.2. Explicit Adams–Bashforth and Explicit Milne Expressions for Downward Continuation

#### 2.2. Two Implicit Expressions and Their Predictor-Corrector Methods for Downward Continuation

#### 2.2.1. Two Implicit Expressions for Gravity Anomalies

#### 2.2.2. Predictor-Corrector Methods for Downward Continuation

## 3. Examples and Comparison

#### 3.1. Synthetic Models

^{3}. The green cuboid’s side lengths in the x, y, and z directions are 10 m, 20 m, and 20 m, respectively, with a center point at (65, 90, −22) m. Its top interface is 12 m beneath the ground observation surface, and its density contrast is 0.5 g/cm

^{3}. The x, y, and z direction side lengths of the blue cuboid are 10 m, 12 m, and 20 m, respectively, and the center point coordinate of this cuboid is (85, 90, −21) m. This cuboid’s buried depth of its top interface is 11 m, and its density contrast is 0.4 g/cm

^{3}. The observation surface has 150 measurement lines, 150 points per line, and a grid spacing of 1 m.

#### 3.1.1. Downward Continuation with Theoretical Gravity Anomalies and Their Vertical Derivatives at Different Heights from Forward Calculations

#### 3.1.2. Downward Continuation with the Theoretical Gravity Anomaly and Its Vertical Derivative at the Measurement Height of 0 m from Forward Calculations

#### 3.1.3. Downward Continuation with the Theoretical Gravity Anomaly at the Measurement Height of 0 m from the Forward Calculation

#### 3.1.4. Downward Continuation with the Theoretical Gravity Anomaly at the Measurement Height of 0 m from the Forward Calculation with Gaussian White Noise

#### 3.1.5. RMS Errors at Different Depths by Different Downward Continuation Methods

#### 3.2. Real Data

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Theoretical gravity anomalies and their vertical derivatives from forward calculations at different heights are used in downward continuations. All downward continuation depths are 8 m (eight times the depth interval) at the height of −8 m. (

**a**) The gravity anomaly to be downward continued, which is the theoretical gravity anomaly from forward calculation at the measurement height of 0 m regarded as the ground observation surface, (

**b**) The reference gravity anomaly for downward continuations, which is the theoretical gravity anomaly from forward calculation at the height of −8 m. (

**c**) The downward continuation of (

**a**) by the classical FFT method. (

**d**) The downward continuation of (

**a**) by the integral iteration method. (

**e**) The downward continuation of (

**a**) by the Milne method. (

**f**) The downward continuation of (

**a**) by the Milne–Simpson predictor-corrector method. (

**g**) The downward continuation of (

**a**) by the Adams–Bashforth method. (

**h**) The downward continuation of (

**a**) by the Adams–Bashforth–Moulton predictor-corrector method.

**Figure 3.**The theoretical gravity anomaly and its vertical derivative at the measurement height of 0 m from the forward calculation and corresponding gravity anomalies and their vertical derivatives at non-measurement heights above 0 m calculated by upward continuation are used in downward continuations. All downward continuation depths are 8 m (eight times the depth interval) which is at the measurement height of −8 m. (

**a**) The gravity anomaly to be downward continued, which is the theoretical gravity anomaly from forward calculation at the measurement height of 0 m regarded as the ground observation surface. (

**b**) The reference gravity anomaly for downward continuations, which is the theoretical gravity anomaly from forward calculation at the measurement height of −8 m. (

**c**) The downward continuation of (

**a**) by the classical FFT method. (

**d**) The downward continuation of (

**a**) by the integral iteration method. (

**e**) The downward continuation of (

**a**) by the Milne method. (

**f**) The downward continuation of (

**a**) by the Milne–Simpson predictor-corrector method. (

**g**) The downward continuation of (

**a**) by the Adams–Bashforth method. (

**h**) The downward continuation of (

**a**) by the Adams–Bashforth–Moulton predictor-corrector method.

**Figure 4.**The theoretical gravity anomaly from forward calculation at the measurement height of 0 m, corresponding calculated vertical derivatives of the gravity anomaly at the measurement height of 0 m by the ISVD method, and corresponding gravity anomalies and their vertical derivatives at non-measurement heights above 0 m calculated by upward continuation are used in downward continuations. All downward continuation depths are 8 m (eight times the depth interval) which is at the measurement height of −8 m. (

**a**) The gravity anomaly to be downward continued, which is the theoretical gravity anomaly from forward calculation at the measurement height of 0 m regarded as the ground observation surface. (

**b**) The reference gravity anomaly for downward continuations, which is the theoretical gravity anomaly from forward calculation at the measurement height of −8 m. (

**c**) The downward continuation of (

**a**) by the classical FFT method. (

**d**) The downward continuation of (

**a**) by the integral iteration method. (

**e**) The downward continuation of (

**a**) by the Milne method. (

**f**) The downward continuation of (

**a**) by the Milne–Simpson predictor-corrector method. (

**g**) The downward continuation of (

**a**) by the Adams–Bashforth method. (

**h**) The downward continuation of (

**a**) by the Adams–Bashforth–Moulton predictor-corrector method.

**Figure 5.**The differences between the downward continuations and the reference gravity anomaly from forward calculation at the height of −8 m. (

**a**) The difference between Figure 4b,c. (

**b**) The difference between Figure 4b,d. (

**c**) The difference between Figure 4b,e. (

**d**) The difference between Figure 4b,f. (

**e**) The difference between Figure 4b,g. (

**f**) The difference between Figure 4b,h.

**Figure 6.**The theoretical gravity anomaly from forward calculation with 2% Gaussian white noise at the measurement height of 0 m, corresponding calculated vertical derivatives of the gravity anomaly at the measurement height of 0 m by the ISVD method, and corresponding gravity anomalies and their vertical derivatives at non-measurement heights above 0 m calculated by upward continuation are used in downward continuations. All downward continuation depths are 8 m (eight times the depth interval) which is at the measurement height of −8 m. (

**a**) The gravity anomaly to be downward continued, which is the theoretical gravity anomaly from forward calculation with 2% Gaussian white noise at the measurement height of 0 m regarded as the ground observation surface. (

**b**) The reference gravity anomaly for downward continuations, which is the theoretical gravity anomaly from forward calculation without noise at the measurement height of −8 m. (

**c**) The downward continuation of (

**a**) by the classical FFT method. (

**d**) The downward continuation of (

**a**) by the integral iteration method. (

**e**) The downward continuation of (

**a**) by the Milne method. (

**f**) The downward continuation of (

**a**) by the Milne–Simpson predictor-corrector method. (

**g**) The downward continuation of (

**a**) by the Adams–Bashforth method. (

**h**) The downward continuation of (

**a**) by the Adams–Bashforth–Moulton predictor-corrector method.

**Figure 7.**Variations of RMS errors between reference gravity anomalies from forward calculations and downward continuation results by different methods from the height of −1 m to that of −10 m. Green lines represent the integral iteration method, blue lines represent the Milne method, magenta lines represent the Milne–Simpson predictor-corrector method, cyan lines represent the Adams–Bashforth method, and black lines represent the Adams–Bashforth–Moulton predictor-corrector method. (

**a**) Under the condition of Section 3.1.1 in general coordinates. (

**b**) Under the condition of Section 3.1.1 in logarithmic coordinates. (

**c**) Under the condition of Section 3.1.2 in general coordinates. (

**d**) Under the condition of Section 3.1.2 in logarithmic coordinates. (

**e**) Under the condition of Section 3.1.3 in general coordinates. (

**f**) Under the condition of Section 3.1.3 in logarithmic coordinates.

**Figure 8.**Both the observed airborne gravity anomaly and the observed vertical derivative are used to carry out downward continuation in the Nechako basin, with a downward continuation depth of 2000 m. (

**a**) The gravity anomaly to be downward continued, which is obtained from the measured airborne gravity anomaly by upward continuation to 2200 m. (

**b**) The observed gravity anomaly at the height of 200 m, which is taken as the reference gravity anomaly. (

**c**) The downward continuation of (

**a**) by the Milne method. (

**d**) The downward continuation of (

**a**) by the Milne–Simpson predictor-corrector method. (

**e**) The downward continuation of (

**a**) by the Adams–Bashforth method. (

**f**) The downward continuation of (

**a**) by the Adams–Bashforth–Moulton predictor-corrector method.

**Figure 9.**Only the observed airborne gravity anomaly is used to carry out downward continuation in the Nechako basin, with a downward continuation depth of 2000 m. (

**a**) The gravity anomaly to be downward continued, which is obtained from the measured airborne gravity anomaly by upward continuation to 2200 m. (

**b**) The observed gravity anomaly at the height of 200 m, which is taken as the reference gravity anomaly. (

**c**) The downward continuation of (

**a**) by the Milne method. (

**d**) The downward continuation of (

**a**) by the Milne–Simpson predictor-corrector method. (

**e**) The downward continuation of (

**a**) by the Adams–Bashforth method. (

**f**) The downward continuation of (

**a**) by the Adams–Bashforth–Moulton predictor-corrector method.

**Figure 10.**RMS errors between the observed airborne gravity anomaly, which is taken as the reference anomaly, and the downward continuations from the upward continuation gravity anomaly at the height of 200 m. Solid lines represent the real gravity anomaly and its vertical derivative (vertical derivative abbreviated as VD), which are obtained by upward continuation at the height of 2200 m as input. Dash lines represent only the real gravity anomaly obtained by upward continuation at the height of 2200 m as input. The blue lines are by the Milne method. The magenta lines are the Milne–Simpson predictor-corrector method. The cyan lines are the Adams–Bashforth method. The black lines are the Adams–Bashforth–Moulton predictor-corrector method.

**Table 1.**The RMS errors between the reference gravity anomaly and six downward continuation results at the height of −8 m under different conditions: Section 3.1.1 theoretical gravity anomalies and their vertical derivatives at different heights from forward calculations; Section 3.1.2 the theoretical gravity anomaly and its vertical derivative at the measurement height of 0 m from forward calculations and corresponding gravity anomalies and their vertical derivatives at non-measurement heights above 0 m calculated by upward continuation; Section 3.1.3 the theoretical gravity anomaly at the measurement height of 0 m from the forward calculation and corresponding calculated vertical derivatives of the gravity anomaly at the measurement height of 0 m by the ISVD method and corresponding gravity anomalies and their vertical derivatives at non-measurement heights above 0 m calculated by upward continuation; Section 3.1.4 adding noise to the third condition.

RMS Errors | Section 3.1.1 | Section 3.1.2 | Section 3.1.3 | Section 3.1.4 | |
---|---|---|---|---|---|

Methods | |||||

FFT | 0.42 × 10^{17} | 0.42 × 10^{17} | 0.42 × 10^{17} | 0.19 × 10^{20} | |

Integral iteration | 0.16 × 10^{−2} | 0.16 × 10^{−2} | 0.16 × 10^{−2} | 0.17 × 10^{−2} | |

Milne | 0.92 × 10^{−3} | 0.39 × 10^{−2} | 0.30 × 10^{−2} | 0.30 × 10^{−2} | |

Milne–Simpson predictor-corrector | 0.52 × 10^{−3} | 0.13 × 10^{−2} | 0.10 × 10^{−2} | 0.16 × 10^{−2} | |

Adams–Bashforth | 0.95 × 10^{−3} | 0.95 × 10^{−3} | 0.10 × 10^{−2} | 0.11 × 10^{−2} | |

Adams–Bashforth–Moulton predictor-corrector | 0.53 × 10^{−3} | 0.53 × 10^{−3} | 0.61 × 10^{−3} | 0.13 × 10^{−2} |

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**MDPI and ACS Style**

Zhang, C.; Qin, P.; Lü, Q.; Zhou, W.; Yan, J.
Two New Methods Based on Implicit Expressions and Corresponding Predictor-Correctors for Gravity Anomaly Downward Continuation and Their Comparison. *Remote Sens.* **2023**, *15*, 2698.
https://doi.org/10.3390/rs15102698

**AMA Style**

Zhang C, Qin P, Lü Q, Zhou W, Yan J.
Two New Methods Based on Implicit Expressions and Corresponding Predictor-Correctors for Gravity Anomaly Downward Continuation and Their Comparison. *Remote Sensing*. 2023; 15(10):2698.
https://doi.org/10.3390/rs15102698

**Chicago/Turabian Style**

Zhang, Chong, Pengbo Qin, Qingtian Lü, Wenna Zhou, and Jiayong Yan.
2023. "Two New Methods Based on Implicit Expressions and Corresponding Predictor-Correctors for Gravity Anomaly Downward Continuation and Their Comparison" *Remote Sensing* 15, no. 10: 2698.
https://doi.org/10.3390/rs15102698