- Open Access
Comparison result of inversion of gravity data of a fault by particle swarm optimization and Levenberg-Marquardt methods
© Toushmalani; licensee Springer. 2013
- Received: 10 September 2013
- Accepted: 12 September 2013
- Published: 14 September 2013
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 Levenberg-Marquardt 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 Levenberg-Marquardt 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.
- Particle swarm optimization
- Levenberg-Marquardt method
- Gravity data
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 trade-off 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 real-world 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 high-velocity zones. The gravity method is used frequently in mining applications to map subsurface geology and to directly calculate ore reserves for some massive sulfide ore-bodies. 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 low-cost, 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 fault-detection 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 Levenberg-Marquardt describes the algorithms of PSO and LM. Section Particle Swarm Optimization (PSO) presents application of Levenberg-Marquardt 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
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 re-initialized 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.
Application of PSO and LM optimization in inverse problem solving
Gravity anomaly for inversion
Gravity anomaly (mgal)
the thickness of the sheet,
the left distance to the middle of the sheet,
the right distance to the middle of the sheet, and
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 =189-180 = 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
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.
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.
- Ardito C, et al.: An approach to usability evaluation of eLearning applications. Univers Access Inform Soc 2005, 4(3):270-283.View ArticleGoogle Scholar
- Boehner K, DePaula R, Dourish P, Sengers P: How emotion is made and measured. Int J Hum Comput Stud 2007, 65: 275-291. 10.1016/j.ijhcs.2006.11.016View ArticleGoogle Scholar
- Chiu C, Hsu M, Sun S, Lin T, Sun P: Usability, quality, value and eLearning continuance decisions. Comput Educ 2005, 45: 4.View ArticleGoogle Scholar
- Council of Ministers of Education: UNESCO World Conference on Higher Education, Report. 1998.Google Scholar
- Hannula M: Motivation in mathematics: goals reflected in emotions. Educ Stud Math 2006., 63(2):Google Scholar
- ISO 9241: Software ergonomics requirements for office work with visual display terminal (VDT). Part 11: guidance on usability. International Organisation for Standardisation; 1998.Google Scholar
- Juutinen S, Saariluoma P: Usability and emotional obstacles in adopting e-learning-a case study. Vancouver, Canada: IRMA International Conference; 2007.Google Scholar
- Khan K, Sahai A: Comparison of BA, GA, PSO, BP and LM for training feed forward neural networks in e-Learning context. Int J Intell Syst Appl 2012, 7: 23-29.Google Scholar
- Nabighian MN, et al.: 75th Anniversary Historical development of the gravity method in exploration. Geophysics 2005, 70(6):63-89.View ArticleGoogle Scholar
- Telford WM, Geldart LP, Sheriff RE, Keys DA: Applied geophysics. Cambridge: Cambridge University Press; 1976:860.Google Scholar
- Thanassoulas C, Tselentis GA, Dimitriadis K: Gravity inversion of a fault by Marquardt’s method. Comput Geosci 1987, 13(4):399-404. 10.1016/0098-3004(87)90012-4View ArticleGoogle Scholar
- Toushmalani R: Application of gravity method in fault path detection. Aust J Basic Appl Sci 2010, 4(12):6450-6460.Google Scholar
- Toushmalani R: Application of Geophysical Methods in Agriculture. Aust J Basic Appl Sci 2010, 4(12):6433-6439.Google Scholar
- Toushmalani R: Application and Limitation of Geophysical Techniques in Archaeology. Aust J Basic Appl Sci 2010, 4(12):6440-6449.Google Scholar
- Toushmalani R: Using Gaussian Elimination for Determination of Structure Index in Euler Deconvolution. Aust J Basic Appl Sci 2010, 4(12):6390-6396.Google Scholar
- Toushmalani R: Application of Fuzzy c-means Algorithm in Euler deconvolution depth optimization. Int J Acad Res 2011, 3(2):44-48.Google Scholar
- Toushmalani R: Gravity inversion of a fault by particle swarm optimization (PSO). SpringerPlus 2013, 2: 315. 10.1186/2193-1801-2-315View ArticleGoogle Scholar
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