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 Open Access
Comparison result of inversion of gravity data of a fault by particle swarm optimization and LevenbergMarquardt methods
 Reza Toushmalani^{1}Email author
 Received: 10 September 2013
 Accepted: 12 September 2013
 Published: 14 September 2013
Abstract
The purpose of this study was to compare the performance of two methods for gravity inversion of a fault. First method [Particle swarm optimization (PSO)] is a heuristic global optimization method and also an optimization algorithm, which is based on swarm intelligence. It comes from the research on the bird and fish flock movement behavior. Second method [The LevenbergMarquardt algorithm (LM)] is an approximation to the Newton method used also for training ANNs. In this paper first we discussed the gravity field of a fault, then describes the algorithms of PSO and LM And presents application of LevenbergMarquardt algorithm, and a particle swarm algorithm in solving inverse problem of a fault. Most importantly the parameters for the algorithms are given for the individual tests. Inverse solution reveals that fault model parameters are agree quite well with the known results. A more agreement has been found between the predicted model anomaly and the observed gravity anomaly in PSO method rather than LM method.
Keywords
 Particle swarm optimization
 LevenbergMarquardt method
 Inversion
 Gravity data
 Fault
Introduction
where D is the number of parameters to be optimized. Many population based algorithms were proposed for solving unconstrained optimization problems. Genetic algorithms (GA), particle swarm optimization (PSO), are most popular optimization algorithms which employ a population of individuals to solve the problem on hand. The success or failure of a population based algorithms depends on its ability to establish proper tradeoff between exploration and exploitation. A poor balance between exploration and exploitation may result in a weak optimization method which may suffer from premature convergence, trapping in a local optima and stagnation. PSO algorithm is another example of population based algorithms (Ardito et al. 2005). PSO is a stochastic optimization technique which is well adapted to the optimization of nonlinear functions in multidimensional space and it has been applied to several realworld problems (Boehner et al. 2007; Khan and Sahai 2012).
The gravity method was the first geophysical technique to be used in oil and gas exploration. Despite being eclipsed by seismology, it has continued to be an important and sometimes crucial constraint in a number of exploration areas. In oil exploration the gravity method is particularly applicable in salt provinces, over thrust and foothills belts, underexplored basins, and targets of interest that underlie highvelocity zones. The gravity method is used frequently in mining applications to map subsurface geology and to directly calculate ore reserves for some massive sulfide orebodies. There is also a modest increase in the use of gravity techniques in specialized investigations for shallow targets. Also it has application in agriculture and archeology. Data reduction, filtering, and visualization, together with lowcost, powerful personal computers and color graphics, have transformed the interpretation of gravity data. Also in gravity methods, Euler and Werner deconvolution depth and edge estimation techniques can help define the lateral location and depth of isolated faults and boundaries from gravity data. Complex geology with overlapping anomalies arising from different depths can, however, limit the effectiveness of deconvolution faultdetection results (Nabighian et al. 2005; Toushmalani 2010b; Toushmalani 2010c; Toushmalani 2010d; Toushmalani 2011).
The outline of this paper is as follows. In first section we discussed the gravity field of a fault, Section LevenbergMarquardt describes the algorithms of PSO and LM. Section Particle Swarm Optimization (PSO) presents application of LevenbergMarquardt backpropaction algorithm, and a particle swarm algorithm in solving inverse problem of a fault. Most importantly the parameters for the algorithms are given for the individual tests. Section Application of PSO and LM optimization in inverse problem solving presents conclusions and final comments.
Appplication to the gravity field of a fault
K: =6.672e3;
LevenbergMarquardt
Where J ^{ p } (w) is the Jacobian matrix of the error vector e ^{ p } (w) evaluated in w, and I is the identity matrix. The vector error e ^{ p } (w) is the error of the network for pattern p, that is, e ^{ p }(w) = t ^{ p } − o ^{ p }(w).
Particle swarm optimization (PSO)

Step 1.Initialize position and velocity of all the particles randomly in the N dimension space.

Step 2. Evaluate the fitness value of each particle, and update the global optimum position.

Step 3. According to changing of the gathering degree and the steady degree of particle swarm, determine whether all the particles are reinitialized or not.

Step 4. Determine the individual best fitness value. Compare the p _{ i } of every individual with its current fitness value. If the current fitness value is better, assign the current fitness value to p _{ i }.

Step 5. Determine the current best fitness value in the entire population. If the current best fitness value is better than the p _{ g }, assign the current best fitness value to p _{ g }.

Step 6. For each particle, update particle velocity,

Step 7. Update particle position.

Step 8. Repeat Step 2–7 until a stop criterion is satisfied or a predefined number of iterations are completed (Khan and Sahai 2012; Toushmalani 2013).
Application of PSO and LM optimization in inverse problem solving
Gravity anomaly for inversion
Gravity anomaly (mgal)  xcoordinate (m) 

−2.24  −15000 
−3.47  −10000 
−5.60  −5000 
0  0 
2.02  5000 
1.61  10000 
1.27  15000 
1.04  20000 
 a)
the thickness of the sheet,
 b)
the left distance to the middle of the sheet,
 c)
the right distance to the middle of the sheet, and
 d)
the angle of the fault.

Thickness of fault: 500 m

Fault angle (a): 60°

Depth to bottom of the fault (h1): 5780 m

Depth to top of the fault (h2): 1753 m

. Parameters of obtained solution with PSO:

Thickness of fault: 501.44 849 m;

Fault angle (a): 1.0 5*p  p =189180 = 9°,

Depth to bottom of the fault (h1): 6000 m;

Depth to top of the fault (h2): 2001.6431 m; (Toushmalani 2013). Table 2 shows Parameters of obtained solution.Table 2
Parameters of obtained solution
Calculated gravity with LM
Calculated gravity with PSO
Observed gravity
−2.23
−2.23
−2.24
−3.48
−3.47
−3.47
−5.84
−5.60
−5.60
0
0
0
2.15
2
2.02
1.67
1.63
1.61
1.30
1.29
1.27
1.06
1.05
1.04
2.5%
0.5%
RMS
The mean squared error function was used as the training error. The term root mean square error (RMSE) is the square root of mean squared error (MSE). RMSE measures the differences between values predicted by a hypothetical model and the observed values. In other words, it measures the quality of the fit between the actual data and the predicted model. RMSE is one of the most frequently used measures of the goodness of fit of generalized regression models.
Conclusion
The parameters which are optimized with these methods are: (a) the thickness of the sheet, (b) the left distance to the middle of the sheet, (c) the right distance to the middle of the sheet, and (d) the angle of the fault. Inverse solution reveals that fault model parameters are agree quite well with the known results. A more agreement has been found between the predicted model anomaly and the observed gravity anomaly in PSO Method rather than LM method.
Declarations
Authors’ Affiliations
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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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