research article 2.5d inversion algorithm of frequency ...research article 2.5d inversion algorithm...

Research Article2.5D Inversion Algorithm of Frequency-Domain AirborneElectromagnetics with Topography

Jianjun Xi1,2 and Wenben Li1

1College of Geo-Exploration Science and Technology, Jilin University, Changchun 130026, China2Heibei Electric Power Design & Research Institute, Shijiazhuang 050031, China

Correspondence should be addressed to Jianjun Xi; [email protected]

Received 10 July 2016; Revised 20 August 2016; Accepted 30 August 2016

Academic Editor: Yann Favennec

Copyright © 2016 J. Xi and W. Li.This is an open access article distributed under theCreative CommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We presented a 2.5D inversion algorithm with topography for frequency-domain airborne electromagnetic data. The forwardmodeling is based on edge finite element method and uses the irregular hexahedron to adapt the topography. The electric andmagnetic fields are split into primary (background) and secondary (scattered) field to eliminate the source singularity. For themultisources of frequency-domain airborne electromagnetic method, we use the large-scale sparse matrix parallel shared memorydirect solver PARDISO to solve the linear system of equations efficiently. The inversion algorithm is based on Gauss-Newtonmethod, which has the efficient convergence rate. The Jacobian matrix is calculated by “adjoint forward modelling” efficiently. Thesynthetic inversion examples indicated that our proposed method is correct and effective. Furthermore, ignoring the topographyeffect can lead to incorrect results and interpretations.

1. Introduction

As an important geophysical prospecting method, the freq-uency-domain airborne electromagnetics is widely used infields of mineral survey, geological mapping, groundwaterresource exploration, environmental monitoring, and soforth. Since the prospecting is often made in mountainousareas with large undulations which can cause severe impactupon the aeroelectromagnetic response, ignorance of thelandform will lead to errors in aeroelectromagnetic datainterpretation. By boundary element method, Liu andBecker simulated in 1992 the topography’s impact upon thefrequency-domain airborne electromagnetics [1]. Newmanand Alumbaugh made 3D finite difference calculation andsimulated the impact from simple topography upon theobserved data results in 1995 [2]. Sasaki and Nakazato sim-ulated the topography’s impact upon the observed result byfinite difference calculation andmade an attempt to find awayto eliminate this impact in 2003, concluding that no way butmaking inversionwith the topography taken into account caneliminate the impact from the topography [3]. In a word,

implementing inversion with the topography taken intoaccount is imperative in airborne electromagnetics.

The conductivity imaging technique and 1D layeredmedium inversions are still the main interpretation means infrequency-domain airborne electromagnetics [4–6]. Due tothe bad reconstruction capacity of 1D inversion for the high-dimensional models, lots of scholars made research on theinversion for high-dimensional models. Wilson and othersrealized 2.5D inversion in 2006 and applied it in theoreticaland measured data [7]. Cox and others made 3D inversionbased on footprint technique to inverse the measured datain 2010 [8]. Liu and Yin developed a finite difference 3Dinversion program in 2013, compared the NLCGwith LBFGSmethod, and drew the conclusion that the latter one is moreapplicable for the inversion in 3D frequency-domain airborneelectromagnetic [9]. Yi and Sasaki put forward a joint inver-sion program making use of aeroelectromagnetic data andground DC data in 2015 to restore the real model better andpromote the inversion resolution ratio [10]. Nevertheless, allthe above are conducted under the premise of a level surface,with no consideration of the topographical impact; thus in

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 1468514, 10 pageshttp://dx.doi.org/10.1155/2016/1468514

2 Mathematical Problems in Engineering

fact the research about inversion algorithm for the airborneelectromagnetics is still quite limited.

This paper, based on vector finite elementmethod, adoptsPARDISO to solve the linear system of equations, subdividesthe model in irregular hexahedral mesh, makes 3D for-ward modeling in topography, and employs Gauss-Newtonmethod to carry out 2D inversion of frequency-domain air-borne electromagnetics with topography, providing an effec-tive mean to calculate, process, and interpret the aeroelectro-magnetic data for mountainous areas.

2. Forward Modeling

To eliminate the source singularity, assume 𝑒𝑖𝜔𝑡 as the time-harmonic factor and divide the electromagnetic field intoprimary field (background) and secondary field (scattering).Thedouble curl electric field equation based on the secondaryfield can be as follows [11]:

∇ × ∇ × E𝑠 + 𝑖𝜔𝜇0𝜎E𝑠 = −𝑖𝜔𝜇0Δ𝜎E𝑝, (1)where E𝑠 is the secondary electric field, 𝜔 is the angularfrequency, 𝜇0 is themagnetic conductivity in vacuum, 𝜎 is theelectric conductivity,E𝑝 is the background field, andΔ𝜎 is thedifference between the total conductivity and the backgroundconductivity, expressed as Δ𝜎 = 𝜎 − 𝜎𝑝, with 𝜎𝑝 as thebackground electric conductivity.Theprimary field under theuniform whole space medium or under the layered mediumis calculated by analytic solution, while the secondary field(scattered field) is calculated by vector finite elementmethod.The double curl Maxwell equation satisfies the followingboundary condition:

∇ × ∇ × E𝑠 + 𝑖𝜔𝜇0𝜎E𝑠 = −𝑖𝜔𝜇0Δ𝜎E𝑝, E𝑠 ∈ ΩE𝑠 = 0, E𝑠 ∈ 𝜕Γ. (2)

Implement the Galerkin weighted residual method forformula (1); then

𝑅𝑘 = ∫ΩN𝑘 ⋅ [∇ × ∇ × E𝑠 + 𝑖𝜔𝜇𝜎E𝑠 + 𝑖𝜔𝜇Δ𝜎E𝑝] 𝑑𝑉

= 0.(3)

In the equation Ω = ∑𝑁𝑒𝑒=1Ω𝑒, Ω𝑒 represents a discreteelement and 𝑁𝑒 is the number of discrete elements; thusformula (3) can be expressed in discrete form as below:

𝑅𝑘 =𝑁𝑒∑𝑒=1

[K𝑒E𝑒𝑠 − 𝑖𝜔𝜇𝜎M𝑒E𝑒𝑠 − 𝑖𝜔𝜇Δ𝜎M𝑒E𝑒𝑝] = 0, (4)where K𝑒 andM𝑒 are element stiffness matrix, expressed as

K𝑒𝑘𝑙 = ∫Ω𝑒(∇ × N𝑒𝑘) ⋅ (∇ × N𝑒𝑙 ) 𝑑𝑉

M𝑒𝑘𝑙 = ∫Ω𝑒

N𝑒𝑘 ⋅ N𝑒𝑙𝑑𝑉.(5)

1 2

34

5 6

78

X

ZY 1 2

34

5 6

78

E12 E12

E10E10

E11E11

E9E9

E8 E8

E7E7

E5E5

E6E6

E2

E2

E4E4

E3E3

E1E1𝜁

𝜉

𝜂

Figure 1: Element global coordinates mapping on local coordinates.

N𝑒𝑘 is the vector basis function [12]. 27-point Gaussintegral is used for the calculations in formula (5).

The element global coordinates mapping on local coor-dinates is shown in Figure 1. Allocate the element stiffnessmatrix to the overall stiffness matrix according to certainstandard, and then the large-scale linear system of equationscan be drawn:

KF = b. (6)K is sparse complex symmetricmatrix, F is the field value,

and b is the source item.To guarantee a unique solution, the Dirichlet boundary

condition is employed, deeming that the secondary abnormalfield at the boundarywhich is far enough from the anomalousbody has attenuated nearly to zero. In other words, at theboundary,

n × E|𝜕Ω = 0. (7)Once the electric field is calculated, themagnetic field can

be solved by Faraday’s law:

H = (−𝑖𝜔𝜇0)−1 ∇ × E. (8)2.1. Precision Validation of Forward Modeling. To verify thecorrectness of this procedure, a 2D trapezoid mountainmodel the same as that used by Sasaki and Nakazato [3] isadopted with the medium electric conductivity of 100 ohm-m, as shown in Figure 2. The top and bottom width of thetrapezoid are, respectively, 20m and 220m, the height of thetrapezoid is 50m, and the gradient is 26.5∘. Subdivide themodeling area in 𝑥, 𝑦, and 𝑧 direction into 41 × 41 × 41 grids,with the one in the middle sized 10m × 10m × 10m and thelargest one at the boundary sized 1280m × 1280m × 1280m.HCP device is employed for calculation, with the transmitter-receiver coil pair keeping 30m above the ground in 10mtransmitter-receiver spacing in three frequencies of 1 kHz,4 kHz, and 16 kHz.The results in Figure 3 highly coincidewithSasaki’s finite difference method results, which illustratedthat the forward modeling program in this paper is highlyaccurate.

Mathematical Problems in Engineering 3

100 ohm·m50m

220 m

20 m

Figure 2: 2D trapezoid mountain model diagram.

This work 16k QSasaki 16k QThis work 16k ISasaki 16k IThis work 4k QSasaki 4k Q

This work 4k ISasaki 4k IThis work 1k QSasaki 1k QThis work 1k ISasaki 1k I

0

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1000

1500

2000

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Hz

(ppm

)

Figure 3: Comparison with Sasaki’s finite difference method result.

3. Inversion Algorithm

3.1. Objective Function Definition. Theobjective function canbe written as

Φ (m) = 12 d (m) − dobs2 +𝛽2 W (m −mref)2 , (9)

where ‖ ⋅ ‖ denotes the Euclidean norm, d is the data vectorobtained from forward modeling, dobs is the measured datavector,m is the model parameter vector,mref is the referencemodel vector or the prior information model vector, W isthe model smoothness matrix, and 𝛽 is the regularizationparameter that balances the effect of data misfit and modelregularization during minimization.

The objective function defined by formula (9) is dividedinto data fitting item and regularization item. The formerguarantees the inversed model to fit the measured data, whilethe latter controls the inversion result to draw close to theknown prior model. There are normally three cases for theregularization item: (1) smallest model, that is, smallest m −mref ; (2) flattest model, that is, smallest first-order differenceoperator form−mref; (3)most smoothmodel, that is, smallestsecond-order difference operator for m − mref. Due to theinversion algorithm in this paper, the third type is adopted,

that is, with W as the second-order difference operator, andhas the following form:

W =[[[[[[[[[[[[

−4 1 0 ⋅ ⋅ ⋅ 1 0 ⋅ ⋅ ⋅ 0 ⋅ ⋅ ⋅ 01 −4 1 0 ⋅ ⋅ ⋅ 1 0 ⋅ ⋅ ⋅ 0 ⋅ ⋅ ⋅... ...0 ⋅ ⋅ ⋅ 0 1 0 ⋅ ⋅ ⋅ 0 1 −4 10 ⋅ ⋅ ⋅ 0 ⋅ ⋅ ⋅ 0 1 ⋅ ⋅ ⋅ 0 1 −4

]]]]]]]]]]]]

. (10)

Assuming m𝑖 as the initial model, substitute it intoformula (9) and linearize it by Taylor expansion, ignoring thehigher order terms to reach

Φ (m) = 12(d +

𝜕d𝜕m𝑖 ⋅ Δm) − dobs

2

+ 𝛽2 W (m𝑖 −mref)2 ,(11)

whereΔm ism−m𝑖. Derive formula (11) and order it equal tozero so as to obtain the model updated iterative formula forGauss-Newton method:

(J𝑇J + 𝛽W𝑇W)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝐻

⋅ Δm

= − (J𝑇 (d − dobs) + 𝛽W𝑇W (m −mref))⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝑔

, (12)

where𝐻 is the approximate Hessianmatrix, 𝑔 is the objectivefunction gradient, and J is Jacobian matrix or sensitivitymatrix, with its elements expressed as

J =[[[[[[[[[

𝜕𝑑1𝜕𝑚1 ⋅ ⋅ ⋅𝜕𝑑1𝜕𝑚𝑛... d ...

𝜕𝑑𝑚𝜕𝑚1 ⋅ ⋅ ⋅𝜕𝑑𝑚𝜕𝑚𝑛

]]]]]]]]]

. (13)

3.2. Jacobian Matrix Calculation. From formula (12), it isknown that the key to the model update iteration is to solvethe linear system of equations, which in turn is decided byJacobian matrix or sensitivity matrix J; thus the calculationfor Jacobian matrix is crucial in the inversion algorithm, andits calculation efficiency is decisive to some extent for theinversion efficiency. The calculation methods for Jacobianmatrix include analytic method, difference method, andadjoint equation solution method [13–15]. Since the analyticmethod cannot solve the matrix in high-dimensional inver-sion and the difference method, though simple, is inefficient,here we choose to solve the adjoint equation for the solutionof Jacobian matrix. The calculation formula for the magneticfield is expressed as

H = (−𝑖𝜔𝜇0)−112∑𝑖=1

∇ × N𝑖𝐸𝑖. (14)


The magnetic field’s model parameter derivative isexpressed as

𝜕H𝜕𝑚 = (−𝑖𝜔𝜇0)−1

12∑𝑖=1

∇ × N𝑖 𝜕𝐸𝑖𝜕𝑚. (15)The magnetic field’s model parameter derivative can be

converted into the electric field’s model parameter derivative.The forward modeling can yield a large-scale complex num-ber linear system of equations:

Ax = b. (16)Derive𝑚 simultaneously from both sides of formula (15)

to work out

A 𝜕x𝜕𝑚 = −𝜕A𝜕𝑚x +

𝜕b𝜕𝑚. (17)

The electric field’s model parameter derivative can beworked out by solving formula (17). Formula (17) has nearlythe same coefficient matrix A as the forward modeling linearsystem of equations, with difference at the right-hand item.The solving process is similar to the forward modeling pro-cess, named as “accompanying forward modeling.” Accord-ing to interchange theorem, to obtain the model parametersensitivity of the magnetic field at the receiving point, allthat is needed is to put the unit source at the receiving pointand conduct “accompanying forward modeling” once. It isequivalent to the simple weighted summation with themodelparameter-correlated element field value when putting theunit source at the exact receiving point, which can be workedout from the right-hand item of formula (17). “Accompanyingforward modeling” is needed to be carried out just once,which refers to solving all themeasured points and frequencypoints, and then the Jacobian matrix can be obtained.

3.3. Solution for Model Updated Equations. As known fromformula (12), to obtain the model parameter update vectorquantity, the linear system of equations must be solved first,either solved directly or solved by iterative method. J𝑇J needsbe worked out for the solution of formula (12). If selectingdirect solution, Jacobianmatrix J is needed to be stored. Herethe preconditioned conjugate gradient iteration method isadopted, only with the multiplication of the matrix and thevector required to be worked out. As for whether to store theJacobian matrix in the iteration method, if storing, thoughunnecessary to solve J in each time of iteration, the memorydemand will be large, and Jacobian matrix must be storedentirely due to its denseness, while if not storing, thememoryinsufficiency can be avoided, but for each time of iteration,Jacobian matrix must be calculated once again, plus theforward modeling and “accompanying forward modeling”being conducted at each time.

3.3.1. Step Length Selection. After obtaining the model para-meter update vectorΔm byworking out formula (12), the nextiterative model is updated by

m𝑖+1 = m𝑖 + 𝛼Δm, (18)

where 𝛼 is the step length with the range as 0 < 𝛼 ≤ 1.The suitable step length is defined by the following steepestdescent formula [16]:

𝜙 (m𝑖+1) = 𝜙 (m𝑖 + 𝛼Δm)≤ 𝜙 (m𝑖) + 𝑐1𝛼∇𝜙 (m𝑖) Δm, (19)

where 𝑐1 is a constant that in practice takes a very smallvalue, for example, 10−4. Satisfying (19) guarantees that theour new model update adequately reduces the objectivefunction. When performing the model update in (18), theinitial value of 𝛼 is set to one and the newmodel is generated.The inequality in (19) was evaluated, and if the conditionis satisfied, the new model is acceptable., However, if thecondition is not satisfied, 𝛼 is reduced by one-half, and (19)was reevaluated. This procedure is repeated until we find anew model that meets the criterion of (19).

The iterative process is repeated until the objectivefunction and/or the data misfit have reduced to a specifiedlevel. Alternatively, the inversion can be stopped when themagnitude of the objective functional gradient falls belowa given tolerance, for example, 1.0𝑒 − 10, which indicatesthat the objective function has flattened out and that nofurther model improvements can be made. Moreover, theinversion can be stopped when the iteration number exceedsthemaximum iteration number.Our experience suggests thatit is usually not more than ten iterations. Otherwise, stoppingthe inversion once the data misfit reaches the noise level ofthe measure data usually provides reasonable results.

3.3.2. Regularization Factor Selection. The regularization fac-tor 𝛽 can balance the data misfit and model constraints.There are lots of schemes that have been recommended inliterature and used in practice for the selection of 𝛽 [17, 18].To deal with 𝛽, there are three most common approaches:(1) one fixed 𝛽 value in the whole inversion process; (2)decreasing 𝛽 value gradually in each time of iteration; (3)choosing the most appropriate 𝛽 value by L curve in eachtime of iteration. Normally, scheme (3) has the best effect butrequires additional calculation work to seek the appropriatevalue each time. In this paper, schemes (1) and (2) are adoptedin the inversion and are found capable of yielding satisfactoryresults as well. Furthermore, for a given problem one mustoften determine 𝛽 through trial and error. When using trialand error, one should seek the largest 𝛽 value which stillpermits that data to be fit down to the noise level.

3.4. Inversion Process. The inversion procedure is shown inFigure 4:

(1) Set the number of iterations as 𝑖 = 0, the accuracy offitting, and the largest number of iterations, and inputthe initial model and inversion data.

(2) Conduct the forwardmodeling and solve the forwardmodeling equationKF = b to work out the secondarymagnetic field𝐻𝑥 and𝐻𝑧.

(3) Calculate the fitting error, exit the program if theset accuracy or the largest number of iterations isachieved, and continue the program if not.


Yes

Start

ExitNo

Input inversion data and initial model

2.5D forward modeling

Is the misfit acceptable or iteration reaches the maximum

Update model 𝐦k+1 = 𝐦k + Δ𝐦

Figure 4: Inversion process.

10100

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(ohm

-m)

80604020

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th (m

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−100 0 100 200−200Distance (m)

Figure 5: Multiple anomalous bodies inversion model.

(4) Calculate Jacobian matrix by forward simulation toobtain the updated step length.

(5) Update the model parameter:m𝑘+1 = m𝑘 + Δm.4. Synthetic Inversion Examples

The synthetic data sets from two 2D models were used totest the inversion algorithm. The first is a flat-earth model.The second example represents a 2D feature with topography.The data sets are generated for frequency-domain airborneelectromagnetic with HCP or VCX using the vector finiteelement method.

4.1. Inversion in Flat-Earth Model. Figure 5 shows the modelused to test the inversion method for flat-earth model. Thereare four anomalies bodies with 10 ohm-m resistivity buried in1000 ohm-m half-space. HCP and VCX devices are employedunder the frequencies of 1000Hz, 2700Hz, 7400Hz, and20000Hz. The coil pair is keeping 20m above the groundwith the transmitter-receiver distance of 8m, and there aretotally 51 dots with spacing of 10m. The model domain issubdivided into 69×34×29 grids, with the one in the middlesized 10m× 10m× 10m and the largest one at the boundarysized 1280m×1280m×1280m.The forwardmodeling resultsfor HCP and VCX are shown in Figures 6 and 7, respectively,with the value normalized by the primary field in ppm.

The inversion model is subdivided into 52×1×10 pieces,with the 520 electric resistance unknown, while the initialmodel is even half-spatial with resistance of 300 ohm-m. By10 times of iterations, HCP and VCX are converged to 1.5%and 0.7% of the origin, respectively. Figure 8 shows theRMS variation curve along with the increasing iterations for

HCP and VCX device. Their inversion results are shown inFigures 9 and 10, respectively, from which it is observed thatboth HCP and VCX have satisfactorily reconstructed thereal model. Comparatively, HCP has better lateral resolutionwhile VCX acts better in vertical resolution.

4.2. Inversion in the Model with Topography. The followingtwo models are designed to analyze and check the inversioneffect in undulating surface condition.

4.2.1. Low Resistance Composite Pattern. The model is aneven half-spatial mountain type as shown in Figure 11,with two low resistance anomalous bodies buried therein.The electric resistance of the anomalous bodies and of thebackground is, respectively, 10 ohm-m and 300 ohm-m. Themodel top is 20m wide and 50m from the bottom, and thegradient is 26.6∘. Here we change the coordinate of 𝑧-axisin the grids to simulate the undulating surface. HCP deviceis employed under the frequencies of 1000Hz, 2700Hz,7400Hz, and 20000Hz. The coil is 20m above the groundwith transmit-receive distance of 8m, and there are totally51 dots in spacing of 10m. The model domain is subdividedinto 69 × 34 × 29 grids, with the one in the middle sized10m× 10m× 10m and the largest one at the boundary sized1280m× 1280m× 1280m.The forward modeling results forHCP are shown in Figure 12, with the value normalized by theprimary field in ppm.

In the inversion, the forward modeling uses a grid of69 × 34 × 29 cells. The inversion model region is subdividedinto 52 × 1 × 10 pieces, with the 520 unknown electricresistivity, and the startingmodel is half-space with resistanceof 300 ohm-m. After 10 iterations, the HCP RMS is reducedto 0.5% of the first iteration. The RMS variation with theincreasing of iteration is shown in Figure 13. The inversionresult is shown in Figure 14, from which it is observed thatHCP has satisfactorily reconstructed the real model.

To illustrate the effectiveness of the inversion algorithmwith the landform being taken into account, the inversionresults for different landforms are shown in Figure 15. It isobserved that ignorance of the landform will cause undesira-ble inversion result; for example, the low resistance anoma-lous body, though being reflected, is far different from the real


HCP

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20000 Hz7400 Hz

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(b) Imaginary component

Figure 6: HCP forward modeling result.

VCX

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20000 Hz7400 Hz

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Figure 7: VCX forward modeling result.

model, especially for the lateral stretch. Besides, much false-ness and abnormities arise in the inversion profile, and parti-cularly a high resistance anomalous body arises right belowthe mountain due to the topographic impact.

4.2.2. High-Low Resistance Composite Pattern. The high-lowresistance composite pattern is shown in Figure 16, with theanomalous bodies in 10 ohm-m low and 3000 ohm-m highresistance. The other parameters are the same as those in lowresistance composite pattern. The forward modeling result isshown in Figure 17.

The initial model for inversion is even half-spatial withresistance of 300 ohm-m. By 10 times of iterations, the fitting

difference reduces to 2.2% of the first iteration misfit, refer-ring to Figure 18 for its variation along with the iterations.From Figure 19 showing inversion result for high-low resis-tance composite pattern (with the landform being takeninto account), it is observed that the inversion result hassatisfactorily reconstructed the real geoelectricmodelwith noredundant abnormity, whereas the high resistance anomalousbody, though being located correctly, is observed with itsresistance value as just 600 ohm-m which is far differentfrom the real value of 3000 ohm-m. Oldenburg (2012) andothers have asserted that this phenomenon is normal [19],because the pure secondary abnormal field collected byfrequency-domain airborne electromagnetics is insensitive


0

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rcen

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(%)

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0

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f firs

t ite

ratio

n RM

S (%

)

(b) VCX device

Figure 8: RMS variation along with the times of iterations.

101001000

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80604020

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Figure 9: HCP inversion result.

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101001000

(ohm

-m)

Figure 10: VCX inversion result.

−200 −100 0 100 200

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th (m

)

10

67

448

3000

Figure 11: Mountain low resistance composite pattern.

to high resistance body, and, consequently, the inductivesourcemagnetic field data is essentially difficult to inverse thehigh resistance body. In Figure 20 showing inversion resultfor high-low resistance composite pattern (without takinginto account the landform), it is observed that the inversionresult, though capable of reflecting the outline of high orlow resistance anomalous bodies, is far different from thereal model, particularly with large lateral stretch arising forthe low resistance anomalous body. Moreover, the same as

the low resistance composite pattern, much falseness andabnormities arise, and particularly a high resistance anoma-lous body arises right below the mountain.

5. Conclusion

We have developed the 2.5D inversion algorithm for freq-uency-domain airborne electromagnetic data and applied itto synthetic data, such as flat-earthmodel and themodel withtopography. Commencing from analysis for the inversioneffect of HCP and VCX device at horizontal landform, thepaper makes research for multiple low resistance compositepattern of anomalous bodies and indicates that HCP hasbetter lateral resolution while VCX acts better in vertical res-olution. Therefore, data for various measuring devices shallbe made use of to promote the inversion efficiency. After-wards, through research for the mountain low resistancecomposite pattern and high-low resistance composite pat-tern, it is concluded that inversion with the topography being


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10

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Figure 14: Inversion result for mountain low resistance compositepattern (with the landform being taken into account).

taken into account can better reconstruct the real model;otherwise much falseness and abnormities will arise, andparticularly a high resistance anomalous body will arise rightbelow the mountain. Thus to obtain an accurate geoelectricmodel, the impact from the topography must be taken intoaccount in the measured data processing.

10674483000

Dep

th (m

)

−100 0 100 200−200Distance (m)

80604020

0

Figure 15: Inversion result for mountain low resistance compositepattern (without taken into account the landform).

−200 −100 0 100 200

−50

0

50

Distance (m)

Dep

th (m

)

10

67

448

3000

Figure 16: Mountain high-low resistance composite pattern.


HCP

0

200

400

600

800

1000

In-p

hase

(ppm

)

−200 −100 0 100 200Distance (m)

20000 Hz7400 Hz

2700 Hz1000 Hz

(a) Real component

HCP

0

200

400

600

800

1000

1200

1400

Qua

drat

ure (

ppm

)

20000 Hz7400 Hz

2700 Hz1000 Hz

−100 0 100 200−200Distance (m)



0

20

40

60

80

100

Perc

ent o

f firs

t ite

ratio

n RM

S (%

)



−50

0

50

−200 −100 0 100 200Distance (m)

Dep

th (m

)

10

67

448

3000

Figure 19: Inversion result for high-low resistance compositepattern (with the landform being taken into account).

Competing Interests

The authors declare that they have no competing interests.

10674483000

Dep

th (m

)

−100 0 100 200−200Distance (m)

80604020

0

Figure 20: Inversion result for high-low resistance composite pat-tern (without taking into account the landform).

Acknowledgments

The authors thank the financial support of the Doctoral FundProject of the Ministry of Education (no. 20130061110060,class tutors), the National Natural Science Foundation ofChina (no. 41504083), and National Basic Research Programof China (973 Program) (no. 2013CB429805).

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[16] L. Armijo, “Minimization of functions having Lipschitz con-tinuous first partial derivatives,” Pacific Journal of Mathematics,vol. 16, pp. 1–3, 1966.

[17] S. C. Constable, R. L. Parker, and C. G. Constable, “Occam’sinversion: a practical algorithm for generating smooth modelsfrom electromagnetic sounding data,” Geophysics, vol. 52, no. 3,pp. 289–300, 1987.

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