Basics of Computational Geophysics

By Pijush Samui and Barnali Dixon

Basics of Computational Geophysics provides a one-stop, collective resource for practitioners on the different techniques and models in geoscience, their practical applications, and case studies. The reference provides the modeling theory in an easy-to-read format that is verified with onsite models for specific regions and scenarios, including the use of big data and artificial intelligence. This book offers a platform whereby readers will learn theory, practical applications, and the comparison of real-world problems surrounding geomechanics, modeling and optimizations.

• Covers various advanced computational techniques for solving different problems in geophysics, including the use of Big Data and artificial intelligence
• Includes case studies that provide examples surrounding practical applications
• Provides an assessment of the capabilities of commercial software

• Language: English
• Publisher: Elsevier Science
• Release date: Nov 27, 2020
• ISBN: 9780128209011

Book preview

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Preface

Computational geophysics is a new but rapidly growing field of study that uses any type of numerical computations to model and simulate complex earth and geophysical systems to improve the understanding and prediction capabilities, thus improving decision-making process. Computational approaches are used for both simulations and inverse problems.

In the context of recent explosion of interest in the field of computation, availability of large data sets, and enhanced modeling capabilities there is a growing interest in the applications of computational geophysics and geosciences including remote sensing applications. However, the field of computation is an emerging but complicated subject.

This book aims to complement technical journal articles that require advanced knowledge of the subject matters on computation theory and application from their readers and aims to bridge the knowledge gap by providing background information via case studies that recent graduates and new practitioners usually lack.

This book aims to provide examples of case studies to become a resource that is one-stop-shop for practitioners. This book provides collection of case studies and/or practical demonstration to illustrate applications of computation and different computational techniques and models in the field of geophysics and geosciences. Each case study included here illustrates the modeling theory in an easy-to-read format accompanied by specific applications in the context of a specific region and/or a scenario. The editors hope this book will offer a quick-start background on theory and applications of computation for geophysical and geosciences applications via case studies for recent graduates, early career practitioners or experts, who want to dabble into a new sub-field of computation and their diverse applications.

In addition, widespread adaption of computational approach necessitates the development of best practice guidelines for the application of computation in the field of geophysics and geoscience. There is a need for providing guidance to practitioners.

This book’s primary focus is the application of computation to geophysics; however, since this book is also about computation, we have included chapters to show computation application to other key areas of geosciences. Part I (Chapters 1–14) of the book discusses applications of computation for geophysics, where it discusses a wide range of topics related to applications of computation techniques to earthquakes, landslides, and liquefaction, while part II (Chapter 15–22) discusses applications of computation for geosciences and topics include prediction of petrophysical data, mineral explorations, tides, and cyclones as well as soil subsidence. This book covers computational techniques that include artificial neural network, stochastic model, particle swarm optimization, machine learning, adaptive neuro-fuzzy Inference System, etc. Goals of the case studies presented in this book using these computational techniques are to offer readers examples of supervised, unsupervised, and reinforcement learning strategies in the context of geophysics and geosciences applications.

Readers are expected to have the basic knowledge of geophysics (exploration seismic, earthquakes, global seismology, gravity, heat flow, and plate tectonics), and geosciences such as oil subsidence, hydrology, mineral explorations, and cyclone formation.

Part I

Computation & geophysics applications

Outline

Chapter 1 Synthetic ground motions of the October 8, 2005 Kashmir earthquake (Mw 7.6): An inference to the site response and seismic hazard of Kashmir basin, NW Himalaya

Chapter 2 Global particle swarm optimization technique in the interpretation of residual magnetic anomalies due to simple geo-bodies with idealized structure

Chapter 3 Emerging techniques to simulate strong ground motion

Chapter 4 Earthquakes: Basics of seismology and computational techniques

Chapter 5 Significance and limit of electrical resistivity survey for detection sub surface cavity: A case study from, Southern Western Ghats, India

Chapter 6 A review on geophysical parameters comparison in Garhwal and Kumaun Himalaya region, India

Chapter 7 Liquefaction susceptibility of high seismic region of Bihar considering fine content

Chapter 8 Evaluating the reliability of various geospatial prediction models in landslide risk zoning

Chapter 9 Fractals and complex networks applied to earthquakes

Chapter 10 Liquefaction as a seismic hazard: Scales, examples and analysis

Chapter 11 Landslide prediction and field monitoring for Darjeeling Himalayas: A case study from Kalimpong

Chapter 12 Improvement of shear strength of cohesive soils by additives: A review

Chapter 13 Static stress change from February 6, 2017 (M 5.8) earthquake Northwestern Himalaya, India

Chapter 14 Remote sensing for geology-geophysics

Chapter 1

Synthetic ground motions of the October 8, 2005 Kashmir earthquake (Mw 7.6): An inference to the site response and seismic hazard of Kashmir basin, NW Himalaya

Hamid Sana,    Department of Neotectonics and Thermochronology, Institute of Rock Structure and Mechanics, Czech Academy of Sciences, Prague, Czech Republic

Abstract

Synthetic ground motions of the October 8, 2005 Kashmir earthquake (Mw 7.6) generated at the bedrock and surface level in the Kashmir basin, NW Himalaya are presented here. This earthquake caused paramount amount of damage to life and property. The stochastic finite fault element method with dynamic corner frequency was used to simulate the ground motions of this earthquake throughout the Kashmir basin. The synthetic ground motions were generated at engineering bed rock and later transmitted to the surface using shear wave velocity (Vs30) of the alluvium in the basin. The results are in agreement with the damage trajectory of the earthquake and to the estimates of the ground motion from the field investigations. This study thus presents a deterministic scenario of the seismic hazard of the October 8, 2005 earthquake (Mw7.6) in the Kashmir basin. It is also evident from the value and pattern of synthetic ground motions at the surface that the site conditions played an important role in distribution of the ground motions of the October 8, 2005 Kashmir earthquake (Mw 7.6) at the surface. These results are therefore useful in carrying-out the detailed seismic hazard assessment of the Kashmir basin.

Keywords

Synthetic ground motions; Stochastic finite fault method; Site response; seismic hazard

1.1 Introduction

Kashmir valley is a northwest-southeast directed, Neogene-Quaternary tectonic basin in the NW Himalaya. Due to the active tectonic set-up in and around this basin, this region has been struck by various destructive earthquakes (Ambraseys and Douglas, 2004XXX). The October 8, 2005 Kashmir earthquake (Mw 7.6) was the most recent earthquake that shook the region. This earthquake caused paramount damage to life and property (Sana and Nath, 2017). Here the synthetic ground motions of this earthquake are presented at engineering bedrock and at the surface in the Kashmir basin. Due to the lack of the strong motion data, the stochastic finite fault element method with dynamic corner frequency was used to simulate the synthetic ground motions at the bedrock (Motezedin and Atkinson, 2005). The ground motions at bedrock were later transmitted through the geotechnical standard penetration test boreholes spread across the Kashmir basin to the surface (Sana et al., 2019).

The Kashmir Himalaya is one of the most tectonically active convergent boundary zones between Indian and Eurasian plate in the NW Himalayas. The prevailing seismotectonics of this terrain is discussed in Sana and Nath (2016b)XXX. The geodetic studies have revealed that the India-Eurasia convergence rates are up to 20 mm/year (Mohadjer et al., 2010XXX). The recent geodetic study shows that the convergence rates in the Kashmir Himalaya is of the order of 13 (±1) mm/year (Kundu et al., 2014XXX).

Due to the seismotectonic complexity and geological heterogeneity, this territory is divided into five seismotectonic zones based on seismicity clustering and prevailing tectonics, supported by characteristic b-values by Sana and Nath (2017). Theses seismogenic zones were used as sources in the probabilistic seismic hazard assessment (PSHA) of the Kashmir basin by Sana (2019)XXX. The PSHA map of the Kashmir basin and synthetic ground motions at the surface of the October 8, 2005 Kashmir earthquake of Mw 7.6 (Sana et al., 2019) were used as important hazard themes for the seismic microzonation of the Srinagar city in Kashmir valley (Sana, 2018XXX). The synthetic ground motions of the October 8, 2005 Kashmir earthquake (Mw 7.6) were also used to evaluate the liquefaction potential of the Kashmir valley alluvium (Sana and Nath, 2016aXXX). The seismotectonic map in and around the Kashmir basin is shown in Fig. 1.1.

Figure 1.1 Seismotectonic map in and around the Kashmir basin. ITSZ is Indus Tsangpa Suture Zone, MMT is Main Mantle Thrust, MBT is Main Boundary Thrust, HTS is Hazara Thrust System, and KT is Karakorum Thrust, M/J is Murree-Jhelum region, KW is Kishtwar Window, while NPS and HKS stand for Nanga Parbat Syntaxis and Hazara-Kashmir Syntaxis, respectively (Sana and Nath, 2017).

1.2 Input parameters

The source, path and site parameters are important for generating the synthetic ground motions of an earthquake.

1.2.1 Source and path parameters of the October 8, 2005 Kashmir earthquake (Mw 7.6)

The source and path parameters of the October 8, 2005 (Mw 7.6) Kashmir Earthquake from Singh et al. (2006)XXX, Mandal et al. (2007)XXX and Raghukanth (2008)XXX are provided in the Table 1.1 below.

Table 1.1

1.2.2 Shear wave velocity (Vs30)

Shear wave velocity (Vs30) is the most important parameter for the site response analysis during an earthquake. Geotechnically, the Vs30 is determined by the standard penetration test (SPT) blow counts referred to as N values using region or soil type specific relationships between SPT N values and Vs30 (Castelli et al., 2016XXX). Sana et al. (2019) have developed soil type specific relationships between SPT N values and Vs30 for the Kashmir basin soils, shown as Fig. 1.2. A total of 219 SPT boreholes spread across the Kashmir basin have been used in this study. The obtained Vs30 values of each borehole have been interpolated to generate a Vs30 map for the Kashmir basin. The Vs30 values have been classified according to the National Earthquake Hazard Reduction Program (NEHRP) scheme (Fig. 1.3).

Figure 1.2 Average shear wave velocity to a depth of 30 m (Vs30) versus SPT N value for different soil types in the Kashmir valley alluvium (Sana et al., 2019).

Figure 1.3 Shear wave velocity (Vs30) map of the Kashmir basin with NEHRP site classification (Sana et al., 2019).

1.3 Methodology

The strong ground motion prediction modeling is usually carried out either by the adopting the Frequency-Wave number integration (F-K) method or the widely used stochastic approach. In the F-K method the ground motion is calculated by an elastodynamic representation theorem in-which the fault rupture is represented by a slip function on a fault plane (Aki and Richards, 1980XXX). Although this approach seems to be useful especially in complex layered geological terrains (Bouchon and Aki, 1977XXX), however, stochastic method is preferred here due to ease in the compilation of the strong-motion simulation parameters. The stochastic point source model has been used widely for the observing the ground motions (Hanks and McGuire, 1981XXX; Boore, 1983, 2003XXX; Boore and Atkinson, 1987XXX; Toro et al., 1997XXX). This modeling technique was basically introduced by Boore (1983), since then it has been modified with very useful changes like the finite-fault modeling of Beresnev and Atkinson (1998) and the dynamic corner frequency approach of Motazedian and Atkinson (2005). In the finite-fault method a large fault is divided into several sub-faults, which are considered as point sources, and the contribution from each point source is summed-up with appropriate time delaying to get the combined effect. The basic framework of this method is to assume that the Fourier amplitude spectrum at the site of interest is produced as a result of the combined effect of the seismic source spectrum, the path attenuation and the site response. Stochastic finite-fault fault model as represented by Beresnev and Atkinson (1998) is:

(1.1)

where, ‘A(ω)’ is the Fourier amplitude site spectrum, ‘S(ω)’ is the seismic source spectrum, the term ‘exp (−πκf)’ represents the filtering function in terms of spectral decay parameter ‘κ’ (Anderson and Hough, 1984XXX), ‘R’ is the distance between site and source, ‘Q’ is the quality factor and ‘β’ is the share wave velocity (km/s).

Since, the Beresnev and Atkinson technique (1997) is not sensitive to higher frequencies, this problem was overcome by the introduction of dynamic corner frequency concept by Motazedian and Atkinson (2005).

By replacing ‘f'=ω/2π in Eq. (1.1) the acceleration spectrum (Aij) of a shear wave generated from the (ij th) sub-fault can be described as:

(1.2)

where ‘C’ is a constant, ‘M0ij’ is seismic moment, ‘f0ij’ is corner frequency, ‘Rij’ is distance from the observation point. The constant ‘C’ is a collective representation of radiation pattern (Rθφ), free surface amplification (F), density (ρ in g/cm³), and is equivalent to Rθφ FV/4πρβ³, ‘V’ represents the division of horizontal into two components. The corner frequency (f0ij) is given by:

(1.3)

where Δσ is stress drop (bars).

The stochastic finite fault element method with dynamic corner frequency was used to generate the ground motions at the engineering bedrock, corresponding to NEHRP site class B with Vs30 of 760 m/s, throughout the Kashmir basin.

The synthetic ground motion generated at the bedrock were later transmitted through the SPT N boreholes using equivalent linear approach on the DEEPSOIL platform, DEEPSOIL is a 1D site response analysis tool (Hashash et al., 2016).

1.4 Results

1.4.1 Synthetic seismograms at the bedrock level and at surface

The synthetic seismograms of the October 8, 2005 Kashmir earthquake (Mw 7.6) generated at the bedrock level using stochastic finite fault approach (Motazedian and Atkinson (2005)) of the four major towns of Anantnag, Baramulla, Kupwara and Srinagar (Sana et al., 2019) are shown as Fig. 1.4. While as the synthetic seismograms of this earthquake at the surface generated after transmitting the ground motions from the bedrock via 1D site response analysis (Hashash et al., 2016) of the previously mentioned four major towns of Kashmir basin (Sana et al., 2019) are shown as Fig. 1.5.

Figure 1.4 Synthetic ground motions of October 8, 2005 Kashmir earthquake (Mw 7.6) at Anantnag, Baramulla, Kupwara and Srinagar at the engineering bedrock level (Sana et al., 2019).

Figure 1.5 Synthetic ground motions of October 8, 2005 Kashmir earthquake (Mw 7.6) at Anantnag, Baramulla, Kupwara and Srinagar at the surface level (Sana et al., 2019).

1.4.2 Synthetic ground motions (PGA in g) at the bedrock level and at surface

The synthetic ground motions of the October 8, 2005 Kashmir (Mw 7.6) at the bedrock were generated at 650 grid points across the Kashmir basin using Motezedin and Atkinson, 2005. The synthetic ground motions at each grid point were later interpolated in the GIS environment to produce a synthetic ground motion map (Fig. 1.6) of the October 8, 2005 Kashmir earthquake (Mw 7.6) at the bedrock level for the Kashmir basin (Sana, 2017). The synthetic ground motions at the bedrock level were transmitted through 219 boreholes using 1D site response approach (Hashash et al., 2016) and were interpolated to generate the ground motion map (Fig. 1.7) of the October 8, 2005 Kashmir earthquake (Mw 7.6) at the surface level the Kashmir basin (Sana et al., 2019).

Figure 1.6 Synthetic ground motion map of the October 8, 2005 Kashmir earthquake (Mw 7.6) of the Kashmir basin at the bedrock level (Sana, 2017).

Figure 1.7 Synthetic ground motions map of the October 8, 2005 Kashmir earthquake (Mw 7.6) of the Kashmir basin at the surface level (Sana et al., 2019).

1.5 Conclusions

The synthetic ground motions of the October 8, 2005 Kashmir earthquake (Mw 7.6) at bedrock and at the surface are consistent with the reported ground motions and the damage pattern in the Kashmir basin. The synthetic ground motions maps at bedrock and surface of the Kashmir basin give an insight into the deterministic hazard scenario in the Kashmir basin vis-à-vis October 8, 2005 Kashmir earthquake (Mw 7.6). Also, the synthetic ground motion map of the basin depicts that the site conditions play an important role in the ground motion distribution in the Kashmir basin.

References

Aki and Richards, 1980 Aki K, Richards P. Quantitative Seismology: Theory and Methods. Vol. 1 and 2 San Francisco, CA: W. H. Freeman and Company; 1980;948.

Ambraseys and Douglas, 2004 Ambraseys NN, Douglas J. Magnitude calibration of north Indian earthquakes. Geophys J Int. 2004;159(1):165–206.

Anderson and Hough, 1984 Anderson J, Hough S. A model for the shape of the Fourier amplitude spectrum of acceleration at high frequencies. Bull Seismol Soc Am. 1984;74:1969–1993.

Beresnev and Atkinson, 1998 Beresnev I, Atkinson GM. FINSIM: a FORTRAN program for simulating stochastic acceleration time histories from finite faults. Seismol Res Lett. 1998;69:27–32.

Boore, 1983 Boore DM. Stochastic simulation of high-frequency ground motions based on seismological models of the radiated spectra. Bull Seismol Soc Am. 1983;73:1,865–1,894.

Boore, 2003 Boore DM. Simulation of ground motion using the stochastic method. Pure Appl Geophys. 2003;160(3–4):635–676 https://doi.org/10.1007/PL00012553.

Boore and Atkinson, 1987 Boore DM, Atkinson GM. Stochastic prediction of ground motion and spectral response parameters at hard-rock sites in eastern North America. Bull Seismol Soc Am. 1987;77:440–467.

Bouchon and Aki, 1977 Bouchon M, Aki K. Discrete wave number representation of seismic source wave fields. Bull Seismol Soc Am. 1977;67:259–277.

Castelli et al., 2016 Castelli F, Cavallaro A, Grasso S, Lentini V. Seismic microzoning from synthetic ground motion earthquake scenarios parameters: the case study of the City of Catania (Italy). Soil Dyn Earthq Eng. 2016;88:307–327.

Hanks and McGuire, 1981 Hanks TC, McGuire RK. Character of high frequency ground motion. Bull Seismol Soc Am. 1981;71:2,071–2,095.

Hashash et al., 2016 Hashash, Y.M.A., Musgrove, M.I., Harmon, J.A., Groholski, D.R., Phillips, C.A., Park, D., 2016. DEEPSOIL 6.1, user manual. <http://deepsoil.cee.illinois.edu/Files/DEEPSOIL_User_Manual_v6.pdf>.

Kundu et al., 2014 Kundu B, Yadav RK, Bali BS, Chowdhury S. Oblique convergence and slip partitioning in the NW Himalaya: implications from GPS measurements. Tectonics. 2014;33:2013–2024 https://doi.org/10.1002/2014TC003633.

Mandal et al., 2007 Mandal P, Chadha RK, Kumar N, Raju IP, Satyamurty C. Estimation of source parameters of the 8 October 2005 Kashmir earthquake of M w 7.6. Curr Sci. 2007;93(5):660–668.

Mohadjer et al., 2010 Mohadjer S, Bendick R, Ischuk A, et al. Partitioning of India-Eurasia convergence in the Pamir-Hindu Kush from GPS measurements. J Geophys Res. 2010;37:L04305.

Motezedin and Atkinson, 2005 Motezedin D, Atkinson G. Stochastic finite fault modeling based on a dynamic corner frequency. Bull Seismol Soc Am. 2005;95:995–1010.

Raghukanth, 2008 Raghukanth STG. Ground motion estimation during the Kashmir earthquake of 8th October 2005. Nat Hazards. 2008;46(1):1–13.

Sana, 2017 Sana, H., 2017. Synthetic ground motions of the 8 October 2005 Kashmir earthquake (Mw 7.6): a stochastic finite fault element approach. In: Southern California Earthquake Center Annual Meeting, 2017. Proceedings Volume XXVII, September 9–13, 2017. Poster Number 252.

Sana, 2018 Sana H. Seismic microzonation of Srinagar city, Jammu and Kashmir. Soil Dyn Earthq Eng. 2018;115:578–588. doi 10.1016/j.soildyn.2018.09.028.

Sana, 2019 Sana H. A probabilistic approach to the seismic hazard in Kashmir basin. NW Himalaya Geosci Lett. 2019;6:5 https://doi.org/10.1186/s40562-019-0136-0.

Sana and Nath, 2016a Sana H, Nath SK. Liquefaction potential analysis of the Kashmir valley alluvium, NW Himalaya. Soil Dyn Earthq Eng. 2016a;85:11–18.

Sana and Nath, 2016b Sana H, Nath SK. In and Around the Hazara-Kashmir Syntaxis: a seismotectonic and seismic hazard perspective. J Indian Geophys Union. 2016b;20(05):496–505.

Sana and Nath, 2017 Sana H, Nath SK. Seismic source zoning and maximum credible earthquake prognosis of the Greater Kashmir Territory, NW Himalaya. J Seismol. 2017;21(2):411–424.

Sana et al., 2019 Sana H, Nath SK, Gujral K. Site response analysis of the Kashmir valley during the 8 October 2005 Kashmir earthquake (Mw 7.6) using a geotechnical dataset. Bull Eng Geol Environ. 2019;78:2551–2563.

Singh et al., 2006 Singh SK, Iglesias A, Dattatrayam RS, et al. Muzaffarabad earthquake of 8 October 2005 (Mw 7.6): a preliminary report on source characteristics and recorded ground motions. Curr Sci. 2006;91(5):689–695.

Toro et al., 1997 Toro GR, Abrahamson NA, Schneider JF. Model of strong ground motions from earthquakes in central and eastern North America: best estimates and uncertainties. Seismol Res Lett. 1997;68:41–57.

Chapter 2

Global particle swarm optimization technique in the interpretation of residual magnetic anomalies due to simple geo-bodies with idealized structure

Anand Singh¹ and Arkoprovo Biswas²,    ¹1Department of Earth Science, Indian Institute of Technology Bombay, Mumbai, India,    ²2Department of Geology, Centre of Advanced Study, Institute of Science, Banaras Hindu University, Varanasi, India

Abstract

The Global Particle Swarm Optimization (GPSO) method using a MATLAB programme was developed for the interpretation of residual magnetic anomaly data produced by hidden subsurface bodies with simple geometrical bodies such as spheres, horizontal cylinders, and thin dyke and thin sheet-like structures. Inversion parameters such as the depth of the body, the location of the anomaly, amplitude coefficient, effective magnetization angle, and shape factor were inverted. The GPSO inversion method was tested on noise-free synthetic data, synthetic data with 10% Random noise and 20% Gaussian noise and four field examples were interpreted. The inversion results showed excellent fit with the earlier results got using different inversion and interpretation techniques. The present study also shows that the optimization technique is able to delineate all the model parameters correctly when shape factor is fixed. The computation period for GPSO optimization process is very short (less than 1 seconds) for a constrained number of swarm sizes. The GPSO method is enormously fast and does not require any suppositions about the nature of the source of the residual magnetic anomaly.

Keywords

Residual magnetic anomaly; Inversion; Particle swarm optimization

2.1 Introduction

The main purpose of the interpretation of residual magnetic anomalies is to determine the different geological structures which can be categorised into four categories viz. spheres, cylinders, and thin dyke and thin sheet-like structures. Various methods have been presented and are used for inversion and interpretation of magnetic field anomalies due to simple geometric bodies. These inversion methods were an attempt to elucidate the different magnetic parameters values, such as the depth of the buried body, the amplitude coefficient, the effective magnetization angle, and the location of the body/structure of magnetic anomalies produced by some geo-bodies with idealized shapes observed from magnetic data.

A lot of interpretation approaches have been established to interpret magnetic field data assuming either a priori assumed fixed source geometrical models either being a priori assumed, and/or by different interpretation methods without prior knowledge. The interpretation methods initiated with the standardized curve matching techniques (Gay, 1963, 1965XXX; McGrath, 1970). The characteristic points and distance approaches was proposed by Grant and West (1965)XXX and subsequently by Abdelrahman (1994)XXX. Other methods such as Fourier Transform, Hilbert Transforms and Nomograms were developed by Bhattacharyya (1965)XXX, Mohan et al. (1982)XXX and Prakasa Rao et al. (1986), respectively. Silva (1989) and McGrath and Hood (1973)XXX developed the least square minimization approaches. Salem et al. (2004)XXX developed the Linearized Least Squares and Salem and Smith (2005)XXX developed the Normalized Local Wave Number. Lately, interpretation of analytical signal derivatives was developed by Salem (2005)XXX. Other methods such as Euler Deconvolution, Second-Horizontal Derivatives, Simplex Algorithm, Fair Function Minimization and Deconvolution Technique was developed by Salem and Ravat (2003)XXX, Abdelrahman and Essa (2015), and Tlas and Asfahani (2015, 2011aXXX,b)XXX, respectively. Moreover, global optimization such as Simulated Annealing (Gokturkler and Balkaya, 2012XXX), Very Fast Simulated Annealing (Biswas, 2015XXX; Biswas and Sharma, 2014XXX, 2015XXX; Sharma and Biswas, 2011XXX, 2013XXX; Trivedi et al., 2020XXX), Regularized Inversion (Mehanee, 2014aXXX,bXXX), and Particle Swarm Optimization (Singh and Biswas, 2016) was developed to interpret various geophysical data and have been applied to solve nonlinear geophysical problems.

Global Particle Swarm Optimization (GPSO) was established over the past two decades (Eberhart and Kennedy, 1995XXX) but the method is used a few areas in geophysics (Alvarez et al., 2006XXX; Shaw and Srivastava, 2007XXX; Sweilam et al., 2008XXX; Juan et al., 2010XXX; Toushmalani, 2013aXXX,bXXX; Pekşen et al., 2014XXX; Singh and Biswas, 2016; Ekinci, 2016). The method was applied to Self-potential modeling (Monteiro Santos, 2010XXX; Pekşen et al., 2011XXX), Seismic wavelet inversion (Sanyi et al., 2009XXX), hydrological problems (Chau, 2008XXX), and interpretation of gravity anomaly (Singh and Biswas, 2016). GPSO methods are very simple to apply in the interpretation of any type of data because they can be understood very easily are, easy to write an algorithm, and computationally proficient (Eberhart and Shi, 2001XXX). The present study advocated the use of GPSO method and applied to evaluate all the parameters of simple geo-bodies (spheres, horizontal cylinders, and thin dyke and thin sheet-like structures) to produce a given magnetic anomaly by noise-free and noisy synthetic data. The method is used to interpret four field examples (the Pima Copper deposit, Arizona, United States, the Pishabo Lake anomaly, Ontario, Canada, the Bankura Anomaly, India, and the Parnaiba Anomaly, Brazil) and compared with other interpretations made using a number of interpretative techniques.

2.2 Forward formulation for magnetic anomaly

The simple mathematical equation for magnetic anomaly m(x) for different geo-bodies (Fig. 2.1) is given by the equations:

Figure 2.1 A diagram showing cross-sectional views, geometries and parameters of (A) sphere, (B) an infinitely long horizontal cylinder, (C) thin dyke, and (D) thin sheet.

For Spherical structure, the expression can be written as (Rao et al., 1977XXX; Prakasa Rao and Subrahmanyam, 1988):

(2.1)

For Horizontal Cylindrical structure, the expression can be written as (Prakasa Rao et al., 1986):

(2.2)

For Thin Dike like structure, the expression can be written as (Abdelrahman and Sharafeldin, 1996; Atchuta Rao et al., 1980XXX; Gay, 1963):

(2.3)

For Thin Sheet type structure, the expression can be written as (Gay, 1963):

(2.4)

where, k is the amplitude coefficient, z is the depth from the surface to the center of the body (Sphere, Cylinder) and depth from the surface to the top of the body (Dyke and Thin Sheet), x0 (i = 1,…,N) is the horizontal position coordinate, θ is the effective magnetization angle or the index parameter and q is the shape factor. The shape factor for sphere is 2.5, Horizontal Cylinder is 2, and for Thin Dyke and Thin Sheet is 1.

The forward expression of the Eq. (2.2)– (2.4) can be written in the discrete form as

(2.5)

where, f is a model vector, and dpre is a predicted data vector. Through measured data dobs in Eq. (2.5) is based on the minimization of data misfit function, which is written as:

(2.6)

where N is the total number of data point in dobs is the normalized root-mean square error, which generally expressed as a percentage. The subsequent constrained multi-variate problem is resolved using the particle swarm optimization algorithm:

Minimize

,

,

and

2.3 Global particle swarm optimization method

Particle Swarm Optimization (PSO) is swarm intelligence-based optimization algorithm (Kennedy and Eberhart, 1995XXX). It is stimulated from the behavior bird flocking jointly searching for food. In the present work, we adopted the optimization approach of Singh and Biswas (2016) which is based on Global Particle Swarm Optimization (GPSO). Numerically, the value of the ith model parameter for the jth particle in the swarm at iteration k + 1 is updated as:

(2.7)

is the corresponding updated velocity vector. The velocity vector of each value of model parameter is calculated as:

(2.8)

is the velocity vector of the ith model parameter for the jth particle in the swarm at iteration kmodel parameter for the jth particle in the swarm at iteration kis the value of the ith model parameter for the personal best misfit-function value recorded by the jth particle in the swarm, from initialization through iteration kis the ith model parameter of the global best misfit-function value recorded by the swarm, from initialization through iteration k; w each represents a random number in the interval [0, 1] at iteration k; c1 and c2 are the positive acceleration constants which are applied to weight the contribution of the cognitive parameter and social parameter, respectively (see Singh and Biswas, 2016).

Fig. 2.2 illustrates the updating the value of the ith model parameter and velocity for the jth and inertia weight w.

Figure 2.2 PSO model parameter and velocity update. Source: After Singh, A., Biswas, A., 2016. Application of global particle swarm optimization for inversion of residual gravity anomalies over geological bodies with idealized geometries. Nat. Resour. Res. 25 (3), 297–314.

A desktop PC with Intel Core2Duo was taken to perform the current work. The GPSO method took approximately 0.8 seconds in which five model parameters were inverted to obtain the model. The algorithm is developed in a Windows 10 environment using MATLAB (Version, 2015a).

2.4 Results and discussion

The current GPSO method developed for this study were used to interpret the various parameters for noise-free synthetic data, noisy data and field data. Sphere, horizontal cylinder and thin dyke and sheet type structures were taken for this study. Field examples from United States, Canada, India and Brazil were taken for this study. The testing was showed with 20 tests using the resulting parameters: population size = 40, the number of generations = 200, and c1 and c2 were 1.2 and 1.7, respectively. The maximum and minimum values of inertia weight were taken as 0.9 and 0.4, respectively.

2.4.1 Theoretical examples

The GPSO as defined in the above section, is used to infer the noise-free and noisy synthetic magnetic anomaly data. Forward models of the magnetic anomaly are interpreted for a sphere, cylinder, and thin sheet and dyke like structures, with noise free and with 10% Random noise (Multiplied by a random draw between 1.0 and 1.10) and 20% Gaussian noise added data (multiplied by a Gaussian random value with mean 1 and standard deviation 0.2). Earlier, GPSO was applied to interpret the synthetic gravity anomaly data (Singh and Biswas, 2016), where all model parameters were optimized for every data set. Consequently, the shape factor (q) was constrained to its actual value, 2.5 for sphere, 2.0 for cylinder and 1.0 for thin sheet and thin dyke like bodies. The same procedure using GPSO was repeated for magnetic anomaly considering different