^{1}

An inversion technique using a fast method is developed to estimate, successively, the depth, the shape factor, and the amplitude coefficient of a buried structure using residual gravity anomalies. By defining the anomaly value at the origin and the anomaly value at different points on the profile, the problem of depth estimation is transformed into a problem of solving a nonlinear equation of the form

Inversion of gravity data is nonunique in the sense that the observed gravity anomalies in the plane of observation can be explained by a variety of density distributions. One way to solve this ambiguity is to assign a suitable geometry to the anomalous body with a known density followed by inversion of gravity anomalies [

In this paper, an inversion technique based on nonlinear equation

The general vertical component of the gravity anomaly expression produced by a sphere (3D), an infinite long horizontal cylinder (2D), and a semiinfinite vertical cylinder (3D) is shown in Figure

Residual gravity anomalies and schematic diagrams for various simple geometrical structures: (a) vertical cylinder, (b) horizontal cylinder, and (c) sphere.

At the origin (

Using (

Let

Equation (

Once, the depth (

For each

We computed three different residual gravity anomalies, each consisting of the effect of local structure (semiinfinite vertical cylinder, horizontal cylinder, and sphere). The model equations representing the models are

The three gravity anomalies are shown in Figure

Numerical results of the present method applied to the semiinfinite vertical cylinder synthetic example (

Using synthetic data | Using data with 10% random errors | |||||

Computed depth | Computed shape factor | Computed amplitude factor | Computed depth | Computed shape factor | Computed amplitude factor | |

2.00 | 3.00 | 0.50 | 100.00 | 2.97 | 0.47 | 94.33 |

3.00 | 3.00 | 0.50 | 100.00 | 2.90 | 0.54 | 104.01 |

4.00 | 3.00 | 0.50 | 100.00 | 2.42 | 0.38 | 65.53 |

5.00 | 3.00 | 0.50 | 100.00 | 2.97 | 0.47 | 94.40 |

6.00 | 3.00 | 0.50 | 100.00 | 3.31 | 0.58 | 134.60 |

7.00 | 3.00 | 0.50 | 100.00 | 3.15 | 0.53 | 112.76 |

8.00 | 3.00 | 0.50 | 100.00 | 3.26 | 0.56 | 126.73 |

9.00 | 3.00 | 0.50 | 100.00 | 3.19 | 0.54 | 118.41 |

10.00 | 3.00 | 0.50 | 100.00 | 3.06 | 0.50 | 102.89 |

Average (km) | 3.00 | 0.50 | 100.00 | 3.03 | 0.51 | 105.96 |

Percent of error | 0.00 | 0.00 | 0.00 | 1.00 | 2.00 | 5.96 |

Numerical results of the present method applied to the horizontal cylinder synthetic example (

Using synthetic data | Using data with 10% random errors | |||||

Computed depth (km) | Computed shape factor | Computed amplitude factor (mGal) | Computed depth (km) | Computed shape factor | Computed amplitude factor (mGal) | |

2.00 | 4.00 | 1.00 | 300.00 | 4.66 | 1.19 | 350.25 |

3.00 | 4.00 | 1.00 | 300.00 | 4.18 | 1.12 | 349.38 |

4.00 | 4.00 | 1.00 | 300.00 | 4.15 | 1.11 | 327.14 |

5.00 | 4.00 | 1.00 | 300.00 | 4.00 | 1.03 | 331.93 |

6.00 | 4.00 | 1.00 | 300.00 | 4.05 | 1.06 | 362.68 |

7.00 | 4.00 | 1.00 | 300.00 | 3.78 | 0.93 | 336.27 |

8.00 | 4.00 | 1.00 | 300.00 | 4.00 | 1.03 | 329.87 |

9.00 | 4.00 | 1.00 | 300.00 | 4.28 | 1.01 | 318.85 |

10.00 | 4.00 | 1.00 | 300.00 | 4.49 | 1.11 | 350.93 |

Average (km) | 4.00 | 1.00 | 300.00 | 4.17 | 1.07 | 339.69 |

Percent of error | 0.00 | 0.00 | 0.00 | 4.25 | 7.00 | 13.23 |

Numerical results of the present method applied to the sphere synthetic example (^{2}; profile length = 20 km; sampling interval = 1 km) without and with 10% random noise.

Using synthetic data | Using data with 10% random errors | |||||

Computed depth (km) | Computed shape factor | Computed amplitude factor (mGal) | Computed depth (km) | Computed shape factor | Computed amplitude factor (mGal) | |

2.00 | 5.00 | 1.50 | 500.00 | 2.79 | 1.39 | 334.34 |

3.00 | 5.00 | 1.50 | 500.00 | 4.48 | 1.17 | 146.71 |

4.00 | 5.00 | 1.50 | 500.00 | 4.99 | 1.44 | 407.92 |

5.00 | 5.00 | 1.50 | 500.00 | 3.00 | 1.20 | 271.76 |

6.00 | 5.00 | 1.50 | 500.00 | 5.07 | 1.49 | 493.73 |

7.00 | 5.00 | 1.50 | 500.00 | 5.07 | 1.49 | 496.48 |

8.00 | 5.00 | 1.50 | 500.00 | 5.30 | 1.62 | 844.65 |

9.00 | 5.00 | 1.50 | 500.00 | 5.25 | 1.59 | 750.98 |

10.00 | 5.00 | 1.50 | 500.00 | 5.16 | 1.54 | 602.08 |

Average (km) | 5.00 | 1.50 | 500.00 | 4.56 | 1.44 | 483.18 |

Percent of error | 0.00 | 0.00 | 0.00 | 8.80 | 4.66 | 3.36 |

where

Table

In all cases examined, the exact values of the depth (

We compute a gravity anomaly due to a sphere model (profile length = 40 km, ^{2}; station separation interval = 1 km) buried at different depths. The computed gravity anomaly

where

Following the interpretation method, (

The effect of the depth of burial of a sphere model on the inverted parameters for synthetic data contaminated by 10% random noise.

The composite gravity anomaly in mGal, shown in Figure

Noisy composite gravity anomaly consisting of the combined effects of an intermediate structure (horizontal cylinder with

In Figure

Numerical results of the present method applied to the horizontal cylinder synthetic example (

Using synthetic data | Using data with 5% random errors | |||||

Computed depth (km) | Computed shape factor | Computed amplitude factor (mGal) | Computed depth (km) | Computed shape factor | Computed amplitude factor (mGal) | |

3.00 | 7.29 | 1.54 | 2252.78 | 5.38 | 0.76 | 393.01 |

4.00 | 5.87 | 1.01 | 1812.75 | 6.91 | 1.22 | 2686.23 |

5.00 | 5.30 | 0.82 | 590.16 | 5.93 | 0.91 | 706.47 |

6.00 | 4.94 | 0.72 | 435.48 | 5.73 | 0.85 | 561.92 |

7.00 | 4.67 | 0.64 | 359.25 | 5.46 | 0.78 | 423.99 |

8.00 | 4.43 | 0.58 | 211.87 | 5.02 | 0.67 | 280.06 |

9.00 | 4.20 | 0.52 | 179.18 | 4.50 | 0.54 | 187.71 |

10.00 | 4.00 | 0.47 | 156.33 | 4.18 | 0.47 | 152.74 |

Average (km) | 5.09 | 0.79 | 749.72 | 5.39 | 0.78 | 674.02 |

When interpreting real gravity data, inaccurate selection of the origin point of the gravity profile can lead to errors in estimating the gravity parameters. In order to examine this effect, we have introduced some successive errors (

The effect of the offset in the origin point of a semiinfinite vertical cylinder model: (a) on the inverted model parameters for noise free synthetic data and (b) on the inverted model parameters for synthetic data contaminated by 5% random noise.

The composite gravity anomaly (in mGal), shown in Figure ^{2}; profile length = 20 units) and a 2nd-order polynomial of regional structure. The model equation is

Composite gravity anomaly of a buried sphere model (^{2} and

Using a separation technique to remove the effect of the regional structure (graphical method; [

Numerical results of the present method applied to composite gravity anomaly (in mGal), which consists of a local structure (sphere with ^{2}; profile length = 20 units) and a 2nd-order polynomial of regional structure.

Composite synthetic gravity data inverted | Using graphical separation techniques, remaining residual data inverted | Using analytical method, the remaining residual data inverted | |||||||

^{2}) | ^{2}) | ^{2}) | |||||||

3.00 | 2.29 | 0.73 | 135.65 | 2.54 | 0.99 | 230.87 | 3.00 | 1.50 | 750.00 |

4.00 | 2.04 | 0.61 | 109.29 | 2.42 | 0.92 | 195.12 | 3.00 | 1.50 | 750.00 |

5.00 | 1.81 | 0.51 | 94.78 | 2.30 | 0.85 | 166.59 | 3.00 | 1.50 | 750.00 |

6.00 | 1.59 | 0.43 | 87.68 | 2.19 | 0.78 | 145.15 | 3.00 | 1.50 | 750.00 |

7.00 | 1.40 | 0.37 | 85.45 | 2.08 | 0.72 | 129.56 | 3.00 | 1.50 | 750.00 |

8.00 | 1.22 | 0.32 | 86.62 | 1.98 | 0.67 | 118.34 | 3.00 | 1.50 | 750.00 |

9.00 | 1.07 | 0.27 | 90.42 | 1.89 | 0.63 | 110.23 | 3.00 | 1.50 | 750.00 |

10.00 | 0.94 | 0.24 | 96.49 | 1.81 | 0.59 | 104.35 | 3.00 | 1.50 | 750.00 |

Average (unit) | 1.55 | 0.43 | 98.30 | 2.15 | 0.77 | 150.02 |

The fast algorithm has been adapted for interpreting residual gravity anomalies related to three different types of structures, for example, a sphere, a vertical cylinder, and a horizontal cylinder. The standard error (

where

The residual gravity anomaly profile (Figure

Numerical results of the present method applied to the Mobrun field example, Canada (best-fit in bold).

Depth | Shape factor | Amplitude coefficient | Standard error | ||
---|---|---|---|---|---|

33.50 | 67.00 | 25.68 | 0.61 | 55.75 | 0.49 |

33.50 | 100.50 | 26.54 | 0.64 | 55.48 | 0.34 |

33.50 | 134.00 | 28.66 | 0.71 | 55.51 | 0.13 |

67.00 | 100.50 | 28.64 | 0.68 | 56.07 | 0.16 |

100.50 | 134.00 | 42.36 | 0.93 | 72.38 | 0.21 |

−33.50 | −67.00 | 24.89 | 0.52 | 58.71 | 1.01 |

−33.50 | −100.50 | 29.53 | 0.64 | 57.14 | 0.16 |

−33.50 | −134.00 | 30.74 | 0.68 | 57.45 | 0.08 |

−67.00 | −100.50 | 41.30 | 0.84 | 68.43 | 0.14 |

−67.00 | −134.00 | 40.14 | 0.82 | 66.46 | 0.11 |

−100.50 | −134.00 | 38.15 | 0.79 | 63.86 | 0.06 |

Average (m) | 32.50 | 0.72 | 60.53 | 0.06 |

Comparative results of Mobrun field example, Canada.

UsingGrant and West [ | Using Roy et al. [ | Using present method | |
---|---|---|---|

30 | 29.44 | 33.3 | |

— | 0.77 | 0.78 | |

— | — | 59.1 |

The measured gravity response (black circles) over a Mobrun sulfide ore body in Noranda, QC, Canada, and the predicted response (white circles) computed from the present inversion method.

The problem of determining the appropriate depth, shape factor, and amplitude coefficient of a buried structure from the residual gravity data of a short or a long profile length can be solved using the present method. A simple and rapid inversion approach is formulated to use the anomaly values at the origin and two pairs of measured data points (

At the origin (

Using (

Let

by dividing (

From (

By taking an exponential to both sides, we get

Equation (

The author wishes to express his sincere thanks to Professor Steven Forman, the editor, Professor Michel Chouteau, and the reviewer for their excellent suggestions, keen interests and thorough review that improved the paper. Many thanks to Professor El-Sayed M. Abdelrahman, Geophysics Department, Faculty of Science, Cairo University, for his constant help and encouragement.