We 21 C08 Joint Inversion of Gravity Gradient Tensor ...

We 21 C08 Joint Inversion of Gravity Gradient Tensor at Vredefort Impact Crater V. Paoletti* (University Federico II), M. Fedi (University Federico II) & F. Italiano (Eni SpA Upstream and Technical Services)

SUMMARY In recent years, Gravity Gradient Tensor (GGT) has been successfully used in applied and environmental geophysics, also in light of the development of gradiometers. In this paper, we aim at analysing the inversion, either joint or separate, of different GGT components and of the sole gravity field vertical component. We perform our analysis by inspection of the Picard Plot, a well-known Singular Value Decomposition tool, and employ both synthetic data and gradiometer measurements carried out at the Vredefort structure, South Africa. We show that the main factors controlling the quality of the inversion are algebraic ambiguity and signal-to-noise ratio. Provided that algebraic ambiguity is kept low – by different combinations of GGT components and/or only gravity field data – the choice of components involved in the inversion is non-crucial to the quality of the reconstructions. Nonetheless, the use gradiometers allows a quicker and more effective way, with respect to the sole gravity field, to improve algebraic ambiguity.

Near Surface Geoscience Turin, Italy, 6-10 September 2015

Introduction In recent years, Gravity Gradient Tensor (GGT) has been successfully used in applied and environmental geophysics. In fact, many applications in geology, oil research, natural hazards, as well as hydrology, may benefit from detailed information derived from GGT data. Methods based on GGT permit a non-invasive investigation of the Earth's subsurface by interpretation of the spatial rate of change of the gravity field components. The possibility of measuring the GGT components represents an important technical improvement with respect to the conventional gravimeter surveys, in which only the vertical component Gz is measured. GGT maps bear better resolution and relative absence of regional effects and, thus, often exhibit interesting source features not apparent in gravity maps. This is because the gravity gradient field decays faster than the gravity field, meaning it is less sensitive to deeper structure effects and it provides greater sensitivity at short wavelengths. Gravity Gradient Tensor anomalies may affectively describe the density distribution within the Earth, delineating the geometrical parameters of geological bodies, source edges, faults, ore bodies, cavities, hydrocarbon deposits and salt intrusions (e.g., Zhdanov et al., 2004; Martinez et al., 2013). A recent and renewed interest in GGT applications is due to the development of gradiometers measuring all five independent components of the gradient tensor. In this paper, we aim at analyzing the inversion, either joint or separate, of different GGT components and of the sole gravity field vertical component. We perform our analysis by inspection of the Picard Plot, a well-known Singular Value Decomposition tool (e.g., Paoletti et al., 2014). We show, by synthetic and real data cases, that the main factors controlling the quality of the inversion are algebraic ambiguity and signal-to-noise ratio. Provided that algebraic ambiguity is kept low – by different combinations of GGT components and/or only gravity field data – the choice of components involved in the inversion is non-crucial to the quality of the reconstructions. Inversion of Gravity Gradient Tensor The gravity gradient tensor may be expressed in terms of second derivatives of the Earth’s gravitational potential in x, y, and z directions of the Cartesian coordinate system. The gravitational potential U from an excess density distribution, U, in volume V can be written as (e.g., Beiki, 2010): ఘሺ‫ܚ‬Ԣሻ ݀‫ݒ‬Ԣ ‫ܚ‬Ԣȁ

ܷሺ‫ܚ‬ሻ ൌ െߛ ‫׬‬௏ ȁ‫ିܚ‬

(1)

where r and r' denote observation and integration position vectors, respectively, and ߛ is the gravitational constant. Then the gravity gradient tensor T is: డమ ௎ డమ ௎ డమ ௎

  డ௫ మ డ௫௬ డ௫௭ ܶ‫ݖݔܶݕݔܶݔݔ‬ మ మ ተడ ௎ డ ௎ డ మ ௎ ተ ‫ ܂‬ൌ  อܶ‫ݖݕܶݕݕܶݔݕ‬อ  ൌ డ௬௫  డ௬ మ  డ௬௭ ተ ተ ܶ‫ݖݖܶݕݖܶݔݖ‬ డమ ௎ డమ ௎ డమ ௎   డ௭௫ డ௭௬ డ௭ మ

(2)

Outside of the source, U satisfies the Laplace equation‫׏‬ଶ ܷ (r) =0, and the trace of the tensor is equal to zero. Because T is symmetric, the independent components are reduced to five. Single tensor components, their combinations and/or the standard vertical gravity field component, Gz, can all be used in inversion. Different authors explored the performances of single and multiple components in inversion. They used all five independent tensor components Txx, Txy, Txz, Tyy, Tyz (e.g., Li, 2001; Martinez et al., 2013), a combination of horizontal components such as Txy and (TxxTyy)/2 (Zhdanov et al., 2004), and a combination of tensor components and gravity field Gz (Capriotti et al., 2014). The mentioned studies suggest that an improvement in the reconstructions is apparent when more components are included in the inversion. They also show that the differences in the performance of components may be model dependent and, thus, the choice of the GGT components to Near Surface Geoscience Turin, Italy, 6-10 September 2015

be used may be dictated by the purpose of surveys and the target characteristics (Pilkington, 2012; 2014). Synthetic Inversion In this section we illustrate that the choice of components involved in the inversion is non-crucial to the quality of the reconstructions, provided that algebraic ambiguity is kept low. We perform our study by the use of the Picard Plot, a known Singular Value Decomposition tool (Fedi et al., 2005; Paoletti et al., 2014), and we compare the inversion results obtained by the employment of: i) five different GGT components (Txx, Tyy, Tzx, Tzy, Txy); ii) the sole Tzz component; iii) the sole gravity field vertical component Gz. The source volume has dimensions 5 km × 5 km × 1 km, discretized with nx = ny = 40 and nz = 10 cells, i.e., the solution consists of n = 16 000 unknowns. Data grid covers an area of 5 km × 5 km and is located on the ground level. For the first case (i), data points are arranged in a 56 × 56 grid (m = 5 ˜ 3136 = 15 680 data). For the Tzz and Gz sole components, data points are arranged in a 126 × 126 grid (m = 15 876). Thus, for all three cases we have, roughly, an evendetermined problem. The source is a box with density 1 g/cm3 and dimensions 0.375 km × 0. 375 km × 0.2 km and its top is located at 0.2 km depth. We use white Gaussian noise added to all data (noise level 10-5). The optimal regularization parameters for this problem (the SVD truncation parameter) were chosen on the basis of the Picard Plots (a plot of the singular values, the SVD-coefficients of the right-hand side, and the SVD coefficients of the solution; Paoletti et al., 2014), cf. the left plots in Figure 1. These plots show that the signal-to-noise ratio allows the inclusion of up to 5000 SVD components. Right plots in Figure 1 show the reconstructions obtained with these optimal regularization parameters. We note that the three inversions retrieve the source position similarly, with some more information on the depth extent shown by the tensor joint inversion. We conclude that the lack of algebraic ambiguity, rather than the choice of tensor and gravity field components to be used, is the main factor controlling the quality of reconstructions.

Figure 1 Left: Picard Plots for the three data sets considered in this study: a) Combination of Txx, Tyy, Tzx, Tzy, Txy); b) Tzz component; c) Gz component. Right: inversion results obtained with 5000 SVD components for the three cases. Actual source position is shown by the white outline. Near Surface Geoscience Turin, Italy, 6-10 September 2015

Real Data Inversion In this section we compare the results of inversion of real data (gravity vertical component vs. gravity gradient tensor components) measured over the Vredefort structure (Witwatersrand basin, South Africa). The ca. 80-km wide Vredefort Dome is now widely accepted as the central uplift of a much larger impact structure (Gibson & Reimold, 2001). The Gravity Gradient Tensor (GGT) data used in this work were acquired using the FALCON system by Fugro Airborne surveys. The data, covering an area of 42 × 63 km, were acquired with a north-south line spacing of 1 km and a ground clearance of about 80 m (Dransfield, 2010). In the FALCON AGG system, the full GGT is derived from the measured horizontal curvature components Txy and Tuv =(Txx-Tyy)/2 (Dransfield & Lee, 2004). Figure 2 shows the gravity field Gz, the two measured components Txy and Tuv and the computed components (Tzz, Tzx, Tzy). The terrain effect was removed using a density of 2.67 g/cm3. For the inversion we used the sole gravity data Gz arranged in a 82 ×125 grid (m = 10 250, spacing 0.5 km) and four tensor components (Tzz, Txy, Tzx, Tzy) arranged in a 42 × 63 grid (m = 10 584, spacing 1 km). The source region (42 × 63 × 15 km along the x, y and z directions) was discretized with 25 x 25 x 15 cells (n = 9375), leading to a slightly over-determined problem for both cases.

Figure 2 Real data used in this study measured at Vredefort structure (South Africa) (Dransfield, 2010).

In Figure 3 we show the reconstructions obtained from the inversions of the sole gravity field (Gz) and the Tzz, Txy, Tzx, Tzy components. Both inversion models show the presence of a deeper central structure and a shallower outer collar. However, the joint gravity gradient tensor inversion highlights more details, especially if it is considered that they are obtained with a coarser data-sampling. Conclusions In this paper we aimed at gaining insights into the utility and effectiveness of Gravity Gradient Tensor (GGT) inversion in applied geophysics. The analysis was carried on synthetic data and on gradiometer measurements carried out at the Vredefort structure, South Africa. Our conclusion is that the main factor controlling the quality of the inversion reconstructions is algebraic ambiguity. Provided that algebraic ambiguity is kept low – by different combinations GGT components and/or only gravity field data – the choice of components involved in the inversion is non-crucial to the Near Surface Geoscience Turin, Italy, 6-10 September 2015

quality of the reconstructions. This result is in agreement with the outcome of the study carried out by Ialongo et al. (2014) on invariant models in the inversion of gravity and magnetic fields and their derivatives. Nonetheless, we note that the use gradiometers allows a quicker and more effective way, with respect to the sole gravity field, to improve algebraic ambiguity.

Figure 3 Reconstructions obtained from the SVD inversion of: (a) the sole gravity field Gz and (b) the tensor components Tzz, Txy, Tzx, Tzy at Vredefort structure (South Africa).

References Beiki, M. [2010] Analytic signals of gravity gradient tensor and their application to estimate source location. Geophysics, 75, I59-I74. Capriotti, J. and Li, Y. [2014] Gravity and gravity gradient data: Understanding their information content through joint inversions. 84th SEG Ann. Internat. Mtg., 1329-1333. Dransfield, M.H. [2010] Conforming Falcon gravity and the global gravity anomaly. Geophysical Prospecting, 58, 469-483. Gibson, R.L. and Reimold, W.U. [2001] The Vredefort impact structure, South Africa (the scientific evidence and a two-day excursion guide), Counc. Geosci., Pretoria, Mem, 92, 111 pp. Ialongo, S., Fedi, M. and Florio, G. [2014] Invariant models in the inversion of gravity and magnetic fields and their derivatives. Journal of Applied Geophysics, 110, 51-62. Li, Y. [2001] 3-D inversion of gravity gradiometer data. 71st SEG Ann. Internat. Mtg., 1470-1473. Martinez, C., Li, Y., Krahenbuhl, R. & Braga, M. [ 2013] 3D inversion of airborne gravity gradiometry data in mineral exploration: A case study in the Quadrilátero Ferrífero, Brazil. Geophysics, 78, B1-B11. Paoletti, V., Hansen, P.C., Hansen, M.F. and Fedi, M. [2014] A Computationally Efficient Tool for Assessing the Depth Resolution in Potential-Field Inversion. Geophysics, 79, A33-A38. Pilkington, M. [2012] Analysis of gravity gradiometer inverse problems using optimal design measures. Geophysics, 77, G25-G31. Pilkington, M. [2014] Evaluating the utility of gravity gradient tensor components. Geophysics, 79, G1-G14. Zhdanov, M.S., Ellis, R. and Mukherjee, S. [2004] Three-dimensional regularized focusing inversion of gravity gradient tensor component data, Geophysics, 69, 925-937. Near Surface Geoscience Turin, Italy, 6-10 September 2015