Many kinds of particle swarm optimization (PSO) techniques are now available
and various efforts have been made to solve linear and non-linear problems
as well as one-dimensional and multi-dimensional problems of geophysical data.
Particle swarm optimization is a metaheuristic optimization method that
requires intelligent guesswork and a suitable selection of controlling
parameters (i.e. inertia weight and acceleration coefficient) for better
convergence at global minima. The proposed technique, tuned PSO, is an
improved technique of PSO, in which efforts have been made to choose the
controlling parameters, and these parameters have been selected after analysing
the responses of various possible exercises using synthetic gravity anomalies
over various geological sources. The applicability and efficacy of the
proposed method is tested and validated using synthetic gravity
anomalies over various source geometries. Finally, tuned PSO is applied over
field residual gravity anomalies of two different geological terrains to
find the model parameters, namely amplitude coefficient factor (

The gravity method is based on the measurement of gravity anomalies caused by the density variation due to source anomalies. The gravity method has been used in a wide range of applications as a reconnaissance method for oil exploration and as a secondary method for mineral exploration, to find out the approximate geometry of the source anomalies, bedrock depths and shapes of the earth. The interpretation of geophysical data involves solving an inverse problem; many techniques have been developed to invert the geophysical data to estimate the model parameters. These methods can be broadly categorized into two groups: (1) local search techniques (e.g. steepest descent method, conjugate gradient method, ridge regression, Levenberg–Marquardt method) and (2) global search techniques (e.g. simulated annealing, genetic algorithms, particle swarm optimization, ant colony optimization). Local search techniques are simple and require a very good initial presumption – one that is close enough to the true model for a successful convergence. On the other hand, global search methods may provide an acceptable solution but are computationally time intensive. Several local and global inversion techniques have been developed to interpret gravity anomalies (Thanassoulas et al., 1987; Shamsipour et al., 2012; Montesinos et al., 2005; Qiu, 2009; Toushmalani, 2013). However, PSO has been successfully applied to many fields, such as model construction, biomedical images, electromagnetic optimization, hydrological problems, etc. (Cedeno and Agrafiotis, 2003; Wachowiak et al., 2004; Boeringer and Werner, 2004; Kumar and Reddy, 2007; Eberhart and Shi, 2001; El-Kaliouby and Al-Garni, 2009) but in the geophysical field PSO has a limited number of applications (Alvarez et al., 2006; Shaw and Srivastava, 2007).

In this paper, improved particle swarm optimization, known as
tuned PSO, has been discussed. This PSO method is a global optimization
technique, has artificial intelligence, preserves its past experience to
avoid trapping in local minima and converges at global minima. This technique
has a very good exploration capability (searching capability) and global
convergence capability (ability of finding the optimal solution). These
capabilities are modulated by inertia weight (

A general expression of gravity anomaly caused by a sphere, an infinite long
horizontal cylinder and a semi-infinite vertical cylinder have been used for
generating the gravity anomalies in a forward problem that is given in
Eq. (1) (Abdelrahman et al., 1989) as follows:

Tuned particle swarm optimization (tuned PSO) is an improved particle swarm
optimization (PSO) method, named after the fine tuning of its learning parameters.
The PSO technique is an evolutionary computational technique inspired by the social behaviour of the particles (Eberhart and Kennedy,
1995). Each particle as a potential solution of the problem knows its best values
(

In this paper, a judicious selection of the parameters (i.e.

Inertia weight (

Maximum velocity (

It is quite a common practice in the PSO literature to limit the range of the number of particles. Van den Bergh and Engelbrecht (2001) have shown that, although there is a slight improvement of the optimal value with increasing swarm size, a larger swarm increases the number of function evaluations to converge to an error limit. However, Eberhart and Shi illustrated that the population size has hardly any effect on the performance of the PSO method. Therefore, in this paper, population size is set at 100.

Iteration versus rms error plot at different acceleration coefficients and inertia weights.

The acceleration coefficients are the learning coefficients which provide
stability for the exploration of the particle. There are two kinds of
acceleration coefficients: (i) the cognitive coefficient

Performance of the acceleration coefficients

Initially the set of the controlling parameters (

Two geometrical models, i.e. sphere and vertical cylinder, have been considered for testing the applicability and efficacy of tuned PSO. The synthetic gravity anomalies over the above-considered models are generated from Eq. (1). In addition, other data sets (noisy synthetic gravity anomalies) are also generated with 10 % Gaussian noise to perceive the efficacy of the proposed algorithm. In each case, the gravity profile length is 51 km and the data points are kept at equal intervals of 1 km. The proposed tuned PSO algorithm has been applied to the above-mentioned synthetic data sets. The optimized results obtained by tuned PSO for synthetic data without noise have been shown in Table 1a and Table 2a. Similarly, the results for synthetic data with noise have been shown in Table 1b and Table 2b.

Figure 1 shows the iteration versus error. It suggests that
the error is at a minimum and has a lower number of local minima at values of
controlling parameters

Tables 2a and 3a show the values of the rms error using the synthetic data
without noise. Also, Tables 2b and 3b show the rms error using the synthetic
data with noise. The analysis of the tables reflects that the rms error is
comparatively higher in the case of synthetic data with noise. However, the
horizontal location (

Iteration versus rms error of tuned PSO showing

The Mobrun polymetallic deposit near Rouyn-Noranda comprises two complexes of massive lenses within mainly felsic volcanic rocks of the Archean Blake River Group (Barrett et al., 1992). Host volcanic rocks of mainly sulfide ore bodies are mostly massive, breciated, and tuffaceous rhyolites. The Mobrun ore body is located at a shallow depth; the top of the body is at a depth of approximately 17 m and extends to 175 m (Aghajani et al., 2009).

Tuned PSO in MATLAB has been applied to field residual gravity anomalies. The
anomaly profile length of 268 m has been taken from the Mobrun sulfide body,
Noranda, Canada (Nettleton, 1976; Essa, 2012). It is seen from Fig. 5 that
both anomaly curves, i.e. analysed from tuned PSO and observed gravity
anomalies, are significantly well correlated with the optimal rms error of
0.0271 %. The results in terms of model parameters (amplitude coefficient
factor, shape factor and depth) over the Mobrun ore body, analysed using the
tuned PSO method, can seen in Table 4a. This table provides the optimum
results obtained from tuned PSO with a 0.0271 % error. This agrees well
with the results obtained from other methods. The calculated value of the
shape factor,

The case study area Louga, on the west coast of Senegal, is used for another interpretation of gravity data using tuned PSO. The Senegal basin is part of the north-western African coastal basin – a typical passive margin basin opening west to the Atlantic. The complexities of the rift tectonics of the Atlantic opening give rise to a series of sub-basins aligned north–south. The pre-rift (upper proterozoic to Palaeozoic), syn-rift (Permian to Lower Jurassic) and post-rift are divided into a number of sub-basins, controlled by east–west transform-related lineaments (Nettleton, 1962).

Observed field gravity anomaly versus tuned PSO calculated gravity anomaly over Mobrun sulfide ore body, Canada.

Observed field gravity anomaly versus tuned PSO calculated gravity anomaly over the western Senegal anomaly, Louga area, western Africa.

In this paper, tuned PSO in MATLAB environment has also been applied to
another field case study. The gravity anomaly of Louga area, west coast of
Senegal, western Africa (Essa, 2014) has been chosen for tuned PSO analysis
as shown in Fig. 6. It has a profile length of 32 km. The results in terms
of the model parameters (amplitude coefficient factor, shape factor and
depth) over the Louga anomaly analysed from tuned PSO method can be seen in
Table 5a. It is seen from Fig. 6 that both gravity anomalies curves analysed
with tuned PSO and observed gravity anomalies are extremely well correlated
with the optimal rms error of 0.0271 %. The optimum results of the model
parameters amplitude coefficient factor (

Synthetic source models, iterations and computation time (in seconds).

In this paper, various synthetic gravity anomalies and field gravity anomalies have been adopted to evaluate the applicability and efficacy of tuned PSO algorithms and also to determine the suitable range of settings for the learning parameters (i.e. inertia weight and acceleration coefficients). On the basis of the performance, a novel algorithm PSO with suitable learning parameters has been implemented for gravity anomalies of assuming models such as spheres and vertical cylinders. This technique has been tested and demonstrated on synthetic gravity anomalies with and without Gaussian noise and finally applied to field residual gravity anomalies over the Mobrun sulfide ore body, Noranda, QC, Canada and the Louga anomaly on the western coast of Senegal, western Africa. This technique provides robust and plausible results even in the presence of noise and is consistent with the results obtained from other classical methods. Thus, tuned PSO is a powerful tool that improves the results of classical PSO and other techniques significantly with less time and optimal error.

The main aim of our work deals with the applicability of the proposed algorithm tuned PSO in the geophysical potential field. Initially tuned PSO is demonstrated on synthetic data created by Eq. (1) in the text (Abdelrahman et al., 1989) and details about synthetic data were given in Sect. 4.1. Finally, proposed algorithms were applied over two kinds of field data, taken as follows: (i) residual gravity anomaly over Mobrun field area of profile length 268 m was digitized at interval of 8.4 m from Essa (2012) and (ii) the Louga anomaly, west coast of Senegal, western Africa, of profile length 32 km was digitized at interval of 1.0 km from Essa (2014).

The authors declare that they have no conflict of interest.

Authors would like to thank Jothiram Vivekanandan, chief executive editor for accepting our manuscript. Authors are especially grateful to the associate editor and reviewer Walter Schmidt and one of the reviewers, Sanjay Kumar Prajapati, for their constructive comments for improving our manuscript. The first author is also thankful to IIT(ISM), Dhanbad for funding the research work. Edited by: W. Schmidt Reviewed by: S. K. Prajapati and W. Schmidt

^{©}, IEEE Xplore, 2009.