Proceedings of the 12th SEGJ International Symposium, 2015

3D inversion of gravity data using Cuckoo optimization algorithm Reza Toushmalani(1) and Hakim Saibi(2) (1)

Islamic Azad University (geoman110@gmail.com), (2) Kyushu University (saibi.hakim@mine.kyushu-u.ac.jp) ABSTRACT

This paper describes a new inversion algorithm for 3D gravity data based on a new evolutionary optimization algorithm, inspired by the life history of the avian family of cuckoos. The newly developed method was applied to synthetic data to demonstrate its suitability and then applied to real data. Two case studies are presented: (1) salt dome case study from Charak region (Iran) and (2) Obama geothermal field (SW Japan). The results of the 3D gravity inversion of two sets of real field data, yield geologically plausible models with the estimated depths and shape that fit well with previous geoscientific studies. KEY WORDS: gravity inversion, 3D, Cuckoo optimization. INTRODUCTION This study presents the application of a new optimization algorithm in gravity inversion. This technique is inspired by life history of the avian family of cuckoos, to address problems in geophysical sciences. Specifically, the reproductive strategy of cuckoos serves as the basis of the optimization algorithm. Adult cuckoos lay eggs in the nests of other birds, and if these eggs go unnoticed by host birds, they hatch, and the nestlings are raised by the host.

Having saved a great deal of energy by outsourcing reproductive responsibilities, adult cuckoos are able to expend more energy seeking out optimal habitats. This best environment is analogous to the global maximum of objective functions.

The main objective of this paper is to develop a new and fast 3D gravity inversion method that is capable of quickly reconstituting the 3D distribution of density underground. To achieve this, a new algorithm called Cuckoo Optimization Algorithm (COA) is applied.

In what follows, after a short introduction of the inverse procedure, the method is applied to synthetic data to show its suitability for 3D inversion. To this end, the new method was applied to two real data cases: (1) gravity data obtained from the Bandar Charak region located in south Iran and (2) gravity data obtained from Obama geothermal field (OGF) in southwest Japan. Mapped 3D density distributions in the Bandar Charak area show the known salt dome intrusion and the 3D inversion results from Obama geothermal field show.

METHODOLOGY Optimization is the process of adjusting the inputs to or characteristics of a device, mathematical process, or experiment to find the minimum or maximum output or result. The input consists of variables, while the process, or function, is known as the cost function, objective function, or fitness function, and the output is the cost or fitness (Haupt and Haupt, 2004; Bhandari et al., 2014).

There are different techniques for solving an optimization problem, some of which are inspired by natural processes. These methods usually start with an initial set of variables and then evolve to obtain the global minimum or maximum of the objective function. Historically, the Genetic Algorithm (GA) has been the most popular technique in evolutionary computation research (Rajabioun, 2011).

The COA approach requires the value of an array known as “habitat.” Given an issue of Nvar dimensions, the habitat is an array of 1x Nvar that indicates the position in which a cuckoo is currently lives. This can be summarized as follows:

Habitat = [x1, x2, …, xNvar] (1)

The profit of a habitat can be calculated through the profit function, fb, at a habitat of (x1, x2, …, xNvar).

Profit = fb (habitat) = fb (x1, x2, …, xNvar) (2)

By creating a candidate habitat matrix of size Npop x Nvar the optimization algorithm initiates. For each of these initial nests, a random number of eggs is generated. The range is known as “Egg Laying Radius” (ELR) because the cuckoos lay their eggs within the maximum distance from their habitat. This ELR can be defined as follows:

ELR = α [(Number of current Cuckoo’s eggs)/(Total number of eggs)] (varhi – varlow) (3)

Here, a is an integer quantifying the maximum

value of ELR, and varhi and varlow are the variables’ upper and lower limit, respectively. After laying, the P% of eggs (typically 10%) with the lowest profit values will be destroyed, having no chance to hatch. The remaining eggs potentially hatch and are raised by the host birds as their own nestlings. The matter is further complicated by groups of cuckoos that originate

Proceedings of the 12th SEGJ International Symposium, 2015

in certain areas but emigrate to other areas, making it exceedingly difficult to determine which cuckoo belongs to which group. To address this issue, we employed an algorithm using the K-means clustering method. When emigrating, cuckoos do not fly to the destination habitat; instead, they only fly part of the way back to the destination and have a deviation as well. Each cuckoo is considered to fly l % of the overall distance and has a deviation of f radians. By parameterizing l and f, we allow cuckoos to search many more places in various areas. These two parameters are defined for each cuckoo as follows: l ~ U (0, 1), j ~ U (-w, w) (4)

l~U (0,1), means that l is a random number uniformly distributed between 0 and 1. w is a parameter bounding the deviation from targeted habitat. We suggest an w of π/6 (radian) is appropriate. In summary, the cuckoo optimization algorithm (COA) is outlined as follows: 1. Starting cuckoo habitats by considering random

points on the profit function; 2. Apportioning a number of eggs to each cuckoo; 3. Calculating the ELR for each cuckoo; 4. Calculating the number of eggs inside the

corresponding ELR for each cuckoo; 5. Determining the number of eggs identified by host

birds; 6. Assuming eggs not recognized hatch and fledge; 7. Assessing the habitat of each newly fledged

cuckoo; 8. Destroying those cuckoos occupying lower quality

habitat according to the maximum number of cuckoo’s in the environment;

9. Clustering cuckoos and determining a best target habitat for each;

10. Allowing emigration to that target habitat; 11. If the condition is satisfied, stop. Otherwise return

to step 2. SYNTHETIC MODEL We assume a buried anomalous structure is made out of two bodies. That is portrayed by a density contrast of 0.4 g/cm3 concerning the non-anomalous subsurface matter and more or less aligned along the axis X. Fig. 1 gives perspective of this structure. Several depths define the limiting horizontal surfaces for the geometrical bodies (0,-50,-100,-200,-400,-500m). The anomalous masses and mass centers of these bodies are: (1) body 1 with a mass of 264x1011 kg, Z1 is -134 m and (2) body 2 with a mass of 360x1011 kg and Z2 is -360 m.

To begin with, we applied 3D Euler deconvolution to the model data. The majority of the Euler solutions happened in the vicinity of models (depth range between 0 to 500 m).

The inversion scheme built model spaces within a certain distance of the Euler solutions, and then searched systematically in those spaces. Fig. 1 shows representation of unique anomalous structure for first simulation test. Two geometrical bodies of positive density differentiate 400 kg/m3 show up at different depths, but the same density contrast.

Figure 1. Synthetic model with 2 bodies.

After several tests with increasing resolution, we

adopt the following final model: size of the smallest cells, number of cells in final step: desired_length_X=100; desired_length_Y=100; desired_length_Z = 50.

Figure 2. 3D view of the results (density contrast is 0.4 g/cm3).

REAL DATA CASE 1: BANDAR CHARAK AREA Badar charak case study is located in southern Iran. Based on recent density measurements from the study area, the mean density is approximately 2.3 g/cm3. The objective is to distinguish the salt dome in this area

Proceedings of the 12th SEGJ International Symposium, 2015

with a mean density of 2.1 g/cm3, which corresponds to a density contrast of −0.2 g/cm3.

For the real case study, the minimum and maximum depths were calculated somewhere around zero and 4,000 m using the Euler deconvolution technique. Subsequently, the study area was separated into 10000 prisms with dimensions of 500×500×500 m in the x, y, and z directions, respectively.

Using the formula of Nagy et al. (2000), a coefficients matrix was calculated, resulting in an 875×10000 matrix (875 gravity points and 10000 prisms). Using the developed 3D gravity inversion algorithm, all prisms with density contrasts of −0.2 g/cm3 were detected (Fig. 3). In fact, prisms with such a density contrast demonstrative of the figure of the salt dome positioned inside the study area, making these results compatible with the research findings from geological (Bosak et al., 1998) and geophysical investigations (Esmaeil Zadeh et al., 2010) in the study area.

Figure 3. 3D view of 3D gravity inversion results by COA algorithm (green cubes represent prisms with a

density contrast of -0.2 g/cm3). CASE 2: OBAMA GEOTHERMAL FIELD Obama area is located in southwestern Japan and known for its geothermal activity. The purpose of this case study are: (1) to know the structure of substratum beneath Obama area from 3D inversion of gravity data (Fig. 4) and (2) to compare the results of 3D gravity inversion deduced by Cordell and Henderson method (Fig. 5). For this, inverse interpretation method of gravity anomalies is used (COA) to approximate the structure of substratum.

Density contrast between substratum and sediment rock is -0.2 g/cm3 determined from the borehole well data; basement rock with average density of 2.4 g/cm3 composed by Pliocene (Neogene) formations and the sediment rock with average density of 2.2 g/cm3 composed essentially by Quaternary formations. All prism with density contrast: 0.2 g/cm3 have been detected (Fig. 6).

13.5013.4013.3013.2513.2013.1513.0513.0012.9512.9012.8512.8012.7812.7412.7212.6812.6412.6012.5512.5012.4812.4412.4012.3512.3012.2812.2412.2012.1512.1012.0512.0011.9011.8511.8011.7011.6011.5011.4011.20

mgal

Figure 4. Bouguer gravity map of OGF (saibi et al.,

2005).

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Depth in km

kmX

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Figure 5. 3D gravity inversion of OGF gravity data using Cordell & Henerson method (Saibi et al., 2005).

Figure 6. 2D plan view of 3D gravity inversion of OGF gravity data using COA algorithm (green area

represent a density contrast of 0.2 g/cm3).

Proceedings of the 12th SEGJ International Symposium, 2015

CONCLUSIONS The obtained 3D gravity inversion results for Obama geothermal field with COA method are in agreement with the 3D inversion results obtained from Cordell and Henderson method.

In conclusion, a new method for 3D inversion of gravity data that uses a Cuckoo optimization algorithm was presented. Synthetic and field data tests show that our newely developed is able to recover an anomaly source with different density contrasts. Moreover, our method is able to handle irregularity sampled data. REFERENCES

Bhandari, A.K., Singh, V.M., Kumar, A., and Singh,

G.K., 2014, Cuckoo search algorithm and wind driven optimization based study of satellite image segmentation for multilevel thresholding using Kapur’s entropy. Expert Systems with Applications, 41, 3538–3560.

Bosak, P., Jaros, J., Spudil, J., Sulovosky, P., Vaclavek, V., 1998, Salt plugs in the eastern Zagros, Iran: Results of regional geological reconnaissance. Ceolines 7, 178 pp.

Cordell, L., Henderson, R.G., 1968, Iterative three-dimensional solution of gravity anomaly data using a digital computer, Geophysics, 33, 596–601.

Esmaeil Zadeh, A., Doulati Ardejani, F., Ziaii, M., Mohammado Khorasani, M., 2010, Investigation of salt plugs intrusion into Dehnow anticline using image processing and geophysical magnetotelluric methods. Russ. J. Earth. Sci., 11, 1-9. doi:10.2205/2009ES000375.

Haupt, R.L. and Haupt, S.E., 2004, Practical Genetic Algorithms, 2nd edn. John Wiley & Sons, New Jersey, 253 p.

Rajabioun, R., 2011, Cuckoo Optimization Algorithm. Applied Soft Computing, 11, 5508–5518.

Saibi, H., 2005, Imaging of subsurface structure beneath Obama area, Japan using gravity data. Proceedings of the 2nd North African/Mediterranean Petroleum & Geosciences Conference and Exhibition, Algiers, April, Algeria.