Generalized cross-validation and Regínska's methods for choosing the regularization parameter in 3D gravity inversion of basement relief - a hybrid MPI/OpenMP parallel algorithm

Abstract

This paper addresses the problem of gravity inversion to determine the 3D basement relief of sedimentary basins. It is a nonlinear optimization problem where the gravity anomalies attributable to basement interfaces above which the density contrast varies continuosly with the depth are analyzed. We use the Levenberg-Marquardt (LM) algorithm which needs a key input parameter called the regularization parameter λ to deal with the singularity of the normal equation matrix in the linearized problem at each iteration. The generalized cross-validation (GCV) and Reginska’s methods for estimating the optimal regularization parameter are tested through numerical experiments on a synthetic data set. Also, we implemented the LM algorithm with a hybrid message passing interface (MPI) and Open Multi-Processing (OpenMP) approach to avoid the high proccesing time. This lead to a fast and reliable algorithm, which produced satisfactory results in mapping the basement topography. Introduction One of the important applications of the gravimetric method is the estimation of the depth of the sediment-basement contact in a given sedimentary basin. The basement relief of a sedimentary basin generally controls the deposition of the sediments and overlying structures. Hence, the structural analysis of the basement plays a significant role in understanding the petroleum system. Generally, the gravity inversion methods for estimating the basement relief of a sedimentary basin may be grouped into two categories. The first group considers that the density contrast between the sediment and the basement is constant. The second group assumes a densisty contrast variation with the depth due to the compaction of the sediment. In the second category methods, Chakravarthi and Sundararajan (2007), for example, have assumed that the density contrast decays with the depth according to a parabolic law and have developed an inversion scheme based on the Levenberg-Marquardt algorithm to estimate the regional gravity anomaly and the depth of the 3D basement relief of the sedimentary basins. In this work, we followed the same approach proposed by Chakravarthi and Sundararajan (2007) and we introduced the generalized cross-validation and Reginska’s criteria for automatic selection of optimal regularization parameter. The performance of these criteria were compared by using a synthetic data set. The validation of the GCV method by synthetic data rather than real cases, has been completely demonstrated in a previous work (Mojica and Bassrei, 2014). It’s worthwhile to mention that the efficiency of the gravity inversion methods applied to the interpretation of sedimentary basins depends on the number of observations and parameters to be estimated (usuallly the number of observations and parameters are made equal), making it very poor when these are very large. Therefore, the development of efficient gravity inversion methods is of utmost importance. To adress this difficulty, recently Silva et al. (2014) proposed an improved Bott’s method, which overcomes the known limitations of the Bott’s method and allows a fast recovery of the basement relief. Alternatively, here we used a hybrid programming model combining MPI and OpenMP that tackles the most computationally expensive parts of the inversion procedure: The forward modeling, the Jacobian matrix computation and the search for the optimal regularization parameter through a regularization parameter choice method such as GCV or Reginska’s method. Methodology