We present an inversion approach that estimates the basement relief of a fault-bounded sedimentary basin. We parameterize the Earth’s subsurface, containing the sedimentary pack, into prismatic cells, with known horizontal dimensions and known density contrast, and estimate their thicknesses, which represent the depths to the basement. To obtain depth-to-basement estimates we use the total-variation regularization as a stabilizing function that allows the estimation of a faulting basement relief. Tests on synthetic and on field data from Almada Basin, Brazil, and from Steptoe Valley, Nevada, confirmed the potential of our method in detecting and locating normal faults in the basement relief of a sedimentary basin.
The gravity inverse problem for depth-to-basement estimate is an ill-posed problem because its solution is neither unique nor stable. The standard Tikhonov regularization method (Tikhonov and Arsenin, 1977) is generally used to guarantee a stable solution. A common regularizing function used in geophysics, named the firstorder Tikhonov regularization, imposes a smooth character on the solution. Mathematically, the smoothing regularization function is the L2 norm of the first-order derivative of the parameters along the spatial directions. By using this smoothing regularization, the solution will be at most a smoothed model of the Earth subsurface. Then, if the parameters to be estimated from the gravity data are the depths to the basement relief, we recover an essentially smooth basement relief, even if the true relief is controlled by abrupt faults. Over the past 30 years, substantial effort has been directed to recover sharp parameters distribution from different geophysical data sets. Several nonsmoothing regularization methods have been applied to geophysical data to estimate nonsmooth physical-property distributions by assuming a piecewise constant function defined on a user-specified grid of cells. Most of these inversion methods may be grouped into two categories. The first one defines the nonsmoothing regularization as the L2 norm of a function of the parameters (e.g., Last and Kubik, 1983; Guillen and Menichetti, 1984, Barbosa and Silva, 1994; Portniaguine and Zhdanov, 1999; 2002; Silva and Barbosa, 2006; Barbosa and Silva, 2006; Ajo-Franklin et al., 2007; Silva Dias et al., 2009). The second category of inversion methods, aiming at retrieving a sharp image of the anomalous sources, defines a nonsmoothing regularization through a non-L2 measure of a function of the parameters (Silva et al., 2007; Pilkington, 2009). Still in the second category, the minimization of the L1 norm of the first-order derivative of the parameters along the horizontal and vertical directions, named Total-Variation (TV) regularization (Rudin et al. 1992; Vogel and Oman, 1998), allows recovering discontinuities in parameter distribution. In the geophysical literature, examples of inversions using the TV regularization are: Farquharson and Oldenburg (1998), Bertere-Aguirre et al. (2002), Loke et al. (2003), Farquharson (2008) and Burstedde and Ghattas (2009). All these methods retrieve a sharper image of geologic sources. Here, to estimate the basement relief of a fault-bounded sedimentary basin, we use the gravity data and the TV regularization as a stabilizing function that allows estimating normal faults in the basement relief.
