Particle swarm optimization (PSO) is a global optimization technique that works similarly to swarms of birds searching for food. A MATLAB code in the PSO algorithm has been developed to estimate the depth to the bottom of a 2.5-D sedimentary basin and coefficients of regional background from observed gravity anomalies. The density contrast within the source is assumed to vary parabolically with depth. Initially, the PSO algorithm is applied on synthetic data with and without some Gaussian noise, and its validity is tested by calculating the depth of the Gediz Graben, western Anatolia, and the Godavari sub-basin, India. The Gediz Graben consists of Neogen sediments, and the metamorphic complex forms the basement of the graben. A thick uninterrupted sequence of Permian–Triassic and partly Jurassic and Cretaceous sediments forms the Godavari sub-basin. The PSO results are better correlated with results obtained by the Marquardt method and borehole information.

The gravity method is a natural source method which helps in locating masses of greater or lesser density than the surrounding formations. It is used as a reconnaissance survey in hydrocarbon exploration. Sedimentary basins, which are characterized by negative gravity anomalies, are the location of almost all of the world's hydrocarbon reserves. Interpretation of gravity data is a mathematical process of trying to optimize the parameters of the initial model in order to get a good match to the observed data. Interpretation of gravity data is always associated with the ambiguity problem. Ambiguity in gravity anomalies can be overcome by assigning a mathematical geometry to the anomaly-causing body with a known density contrast (Rama Rao and Murthy, 1978). Bott (1960) used stacked prism model to describe the cross-section of a sedimentary basin, whereas Talwani (1959) used the polygonal model to describe source geometry. The parabolic density function is used to remove the complications associated with exponential (Cordell, 1973), cubic (Garcia-Abdeslem, 2005) and quadratic (Gallardo-Delgado et al., 2003) density functions. The Marquardt inversion through residual gravity anomalies delineates the structure of a sedimentary basin by estimating regional background (Chakravarthi and Sundararajan, 2007; Marquardt, 1963). Several authors have developed 2-D/2.5-D local optimization techniques over the 2-D/2.5-D sedimentary basin (Annecchione et al., 2001; Barbosa et al., 1999; Bhattacharya and Navolio, 1975; Gadirov et. al, 2016; Litinsky, 1989; Morgan and Grant, 1963; Murthy et al., 1988; Murthyan and Rao, 1989; Rao, 1994; Won and Bavis, 1987) to interpret gravity anomalies with constant density function. In many publications over 3-D gravity field computation with an approximation of geological bodies by 3-D polygonal horizontal prism has been applied (Eppelbaum and Khesin, 2004; Khesin et al. 1996). Rao (1990) used a quadratic density function, which is comparatively reliable, to analyse gravity anomalies over basins having a limited thickness, whereas Chakravarthi and Rao (1993) carried out modelling and inversion of gravity anomalies with a parabolic density function.

Particle swarm optimization (PSO) is a robust stochastic optimization technique based on the movement and intelligence of swarms, which was developed by Kennedy and Eberhart (1995). PSO applies the concept of social optimization in problem solving in various fields. In this paper, a MATLAB code based on PSO is developed to interpret the gravity anomalies of 2.5-D sedimentary basins, where the density varies parabolically with depth. PSO-analysed results are consistent with and more accurate than other techniques and also agree significantly well with borehole information.

In Bott's approach, a sedimentary basin is approximated by a series of
vertical prisms. The gravity anomaly

The 2.5-D sedimentary basin and its approximation by juxtaposing prisms.

Finally, after integration of Chakravarthi and Sundararajan (2006),
Eq. (2) becomes

Since the profile

PSO uses a number of particles (solutions) that constitute a swarm moving
around in the search space looking for the best solution. Each particle
adjusts its “flying” according to its own flying experience as well as the
flying experience of other particles. Each particle keeps track of its
coordinates in the solution space which is associated with the best solution
(fitness) that has been achieved so far by that particle. This value is called
personal best,

The detail schematic diagram/flow chart of PSO techniques.

The MATLAB code based on PSO is applied to a synthetic model of a sedimentary basin and real field data sets from Gediz Graben, western Anatolia, and the Godavari sub-basin, India.

We have used a synthetic gravity anomaly of

We have used the same synthetic gravity anomaly for the PSO algorithm. The Fig. 3
shows the learning process of

Iteration verses RMSE of

Synthetic and calculated gravity anomalies with parabolic density function due to a synthetic model of a 2.5-D sedimentary basin, obtained from the PSO algorithm, and inferred depth structure obtained from the PSO and Marquardt algorithm.

Synthetic data with 5 % Gaussian noise and calculated gravity anomalies obtained from the PSO algorithm, and inferred depth structure obtained from the PSO and Marquardt algorithm.

Synthetic data with 10 % Gaussian noise and calculated gravity anomalies obtained from the PSO algorithm, and inferred depth structure obtained from the PSO and Marquardt algorithm.

The first field case study of the interpretation of gravity anomalies has
been taken from Gediz Graben, western Anatolia. The PSO technique has been
applied using 29 vertical prisms, each with equal width of

We have used a similar number of prisms in PSO to improve the results. So
with 65 iterations and 50 models, we achieve a good fit between observed and
PSO-analysed gravity anomalies with a RMSE of 0.0083. The maximum
thickness of the graben is inferred as

Observed and calculated gravity anomalies with parabolic density function, Gediz Graben, western Anatolia, obtained from the PSO algorithm, and inferred structure obtained from the PSO and Marquardt algorithm.

Residual Bouguer gravity anomaly map of the Godavari sub-basin (modified after Chakravarthi and Sundararajan, 2005) and gravity anomaly profile taken for study.

The Godavari sub-basin is one of the major basins of the Pranhita–Godavari
valleys (Rao, 1982), whose approximate strike length is

Observed and calculated residual Bouguer gravity anomalies of parabolic density function of the Godavari sub-basin obtained from the PSO algorithm, and inferred depth structure from the PSO and Marquardt algorithm.

Particle swarm optimization in the MATLAB environment has been developed to estimate the model parameters of a 2.5-D sedimentary basin where the density contrast varies parabolically with depth. We have implemented the PSO algorithm on synthetic data with and without Gaussian noise and two field data sets. An observation has made that PSO is affected by some levels of noise, but estimated depths are close to the true depths. The results obtained from PSO using synthetic and field gravity anomalies are well correlated with borehole samples and provide more geological viability. Despite its long computation time, PSO is very simple to implement and is controlled by only one operator.

Our paper presents the applicability and potentiality of PSO in gravity inverse problems. First, PSO is validated on synthetic gravity anomalies with and without noise, and the developed PSO-based algorithm is finally applied over two kinds of field gravity data taken from different geological terrains: (i) residual gravity anomaly over Gradiz Graben, western Anatolia (Sari and Salk, 2002), and (ii) residual gravity anomaly taken from Godavari sub-basin, India (Chakravarthi and Sundararajan, 2004).

The authors declare that they have no conflict of interest.

The authors express their sincere thanks to Lev Eppelbaum (associate editor), V. G. Gadirov (referee) and one more anonymous referee for their constructive criticism leading to significant improvement of our manuscript. First author Kunal K. Singh thanks IIT(ISM), Dhanbad (Jharkhand), India, for providing the financial support to develop the infrastructre. Edited by: L. Eppelbaum Reviewed by: Gadirov and one anonymous referee