Research ArticleSeismic Waveform Inversion Using the Finite-DifferenceContrast Source Inversion Method

Bo Han1 Qinglong He1 Yong Chen1 and Yixin Dou2

1 Department of Mathematics Harbin Institute of Technology Harbin 150001 China2 School of Finance Harbin University of Commerce Harbin 150028 China

Correspondence should be addressed to Bo Han bohanhiteducn

Received 23 May 2014 Revised 9 August 2014 Accepted 11 August 2014 Published 26 August 2014

Academic Editor Filomena Cianciaruso

Copyright copy 2014 Bo Han et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper extends the finite-difference contrast source inversion method to reconstruct the mass density for two-dimensionalelastic wave inversion in the framework of the full-waveform inversionThe contrast source inversionmethod is a nonlinear iterativemethod that alternatively reconstructs contrast sources and contrast function One of the most outstanding advantages of thisinversionmethod is the highly computational efficiency since it does not need to simulate a full forward problem for each inversioniteration Another attractive feature of the inversion method is that it is of strong capability in dealing with nonlinear inverseproblems in an inhomogeneous background medium because a finite-difference operator is used to represent the differentialoperator governing the two-dimensional elastic wave propagation Additionally the techniques of a multiplicative regularizationand a sequential multifrequency inversion are employed to enhance the quality of reconstructions for this inversion methodNumerical reconstruction results show that the inversion method has an excellent performance for reconstructing the objectsembedded inside a homogeneous or an inhomogeneous background medium

1 Introduction

Full-waveform inversion (FWI) is a powerful tool to gethigh-resolution parameter models in the context of seismictomography This feature of FWI makes it very suitablefor estimating subsurface parameters which is the ultimategoal in exploration seismology In recent years with thedevelopment of computer technology and the requirementof higher quality of seismic imaging FWI becomes a verypromising technique for exploration seismology as acqui-sition improves in size and density The process of FWIis to find an optimal parameter model by minimizing anobjective functional that measures the discrepancy betweenthe observed seismograms and the synthetic seismogramsIn other words the specific implementation of FWI is aniterative procedure that is usually based on gradient-basedoptimization either in time domain or frequency domain(see [1ndash9]) In this paper we only focus on frequencydomain case Simulation in frequency domain has its inherentadvantages it can obtain a relatively good reconstructionresult with only a few frequencies which means that itcan significantly reduce the data redundancy in addition

a parallel computation programming can be easily imple-mented for different frequency inversions which results fromthe independence of each frequency Most methods in theframework of FWI are based on gradient or Newton methodA common feature of these gradient-based optimizationmethods is the highly expensive forwarding calculation forestimating the gradient of the object functional with respectto the interested parameter One special method to estimatethe gradient for a minimization problem based on gradientmethod is the adjointmethod (see [10]) Additionally seismicinverse problems usually are nonlinear and illposed whichmake the seismic inversion difficult to handle In short FWIincorporated both traveltime and amplitude information is ahighly computational process no matter it is implemented intime domain or frequency domain and this feature greatlyprevents from its widespread applications in explorationseismology

An alternative method is proposed by van den Bergand Kleinman (see [11]) extended by van den Berg andAbubakar to effectively reconstruct the interested param-eters in the field of the microwave imaging (see [12])

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 532159 11 pageshttpdxdoiorg1011552014532159

2 Journal of Applied Mathematics

The method is referred to as the so-called contrast sourceinversion (CSI) method which has two versions IE-CSIbased on integral equation formulation (see [11 12]) andFD-CSI based on the finite-difference operator (see [13])The IE-CSI method is very efficient for the homogeneousor simple layered background medium inverse problemssince the IE-CSI method needs to calculate Green functionof the background medium while the FD-CSI method hasa good performance for solving high complex problems inan inhomogeneous background medium Another highlyefficient scattering inversion method called subspace-basedoptimization method (SOM) shares this good propertyof CSI method which also has two variants adopted inhomogeneous and inhomogeneous background mediumrespectively (see [14 15]) The comparisons in detail betweenSOM and CSI are covered in [14] and this reference alsoconcludes that the CSI is close in spirit to a special case ofSOM and the SOM could be convergent for a few specialproblems The interesting advantage of CSI method is thatit does not require solving any full forward problem in eachinversion iteration therefore it can significantly reduce thecomputational cost and it may be very efficient for large scaleor high dimension inverse problems (see [16]) Both the IE-CSI and FD-CSI methods incorporating with multiplicativeregularization constraint have been successfully applied inelectromagnetic wave imaging and acoustic wave imaging(see [17ndash21])

In this paper we extend the FD-CSI method to recon-struct the mass density based on a two-dimensional elasticwave equation (in fact all theories in this paper are suitablefor three-dimensional problems without any changes) in anisotropic medium either homogeneous or inhomogeneousbackground medium and its IE-version is presented in [22]which is successfully employed in elastodynamics when thebackgroundmedium is homogeneous For our forwardmod-eling a perfectly matched layer (PML) absorbing boundarycondition is employed to reduce the boundary reflections(see [23 24]) and considering the fact that the stiffnessmatrix usually is large and highly sparse the techniquesof matrix compression store and LU decomposition of asparse matrix are employed to decrease the storage spaceand increase the computation efficiency (see [25]) For theinversion processing since all the finite-difference operatorsinvolving the FD-CSI algorithm are only dependent on thebackground medium and angular frequency that both ofthem are invariant throughout the inversion process we usea direct solver such as an LU decomposition method to solvethese finite-difference responses to improve the efficiencyof the forward modeling We invert these finite-differenceoperators only once at the start of the inversion processingand the results can be reused for multiple source positionsand successive iterations of the inversion

In order to get a better reconstruction profile and somegeometrical information of an abnormality a multiplica-tive regularization factor called FD-MRCSI is incorporatedwhich is first introduced by van den Berg et al (see [26])Themultiplicative regularization factor has the same function asthe total variation (TV) regularization The most promisingadvantage of the multiplicative regularization is that the

regularization parameter is automatically calculated duringeach inversion Furthermore in order to mitigate the nonlin-earity and enhance the reconstruction results the sequentialmultifrequency inversion approach is employed that is theseismic inversion starts from low frequency and movesupward to high frequency To illustrate the performanceof the finite-difference contrast source inversion method inseismic inversion problems we present several numericalexperiments either in a homogeneous or an inhomogeneousbackground medium

2 Forward Modeling

In this section we present the two-dimensional forwardmodeling using finite-difference method which is of greatimportance for the inversion methodThe forward modelingsimulates the propagation process of the elastic wave in anisotropic inhomogeneous medium and drives the inversionalgorithm In time domain the governing PDE system calledelastodynamic system is written as the following differentialequation system

1205881205972

119905u = nabla sdot 120590 + f (1)

where u is the displacement vector 120588 is the mass density 120590denotes the stress tensor of the elastic medium nablasdot representsthe divergence operator with respect to spatial variablesand the external force is denoted by the term f In anisotropicmedium the stress tensor120590 is given by the followingformulas

120590119894119895= 120582120575119894119895120576119896119896

+ 2120583120576119894119895 (2)

120576119894119895=

1

2(120597119906119894

120597119909119895

+120597119906119895

120597119909119894

) (3)

where the indices 119894 and 119895 can be 1 or 2 120575119894119895is the well-known

Kronecker delta symbol 120576119894119895denotes the strain tensor and 120582

and 120583 are the Lame parameters of the isotropic mediumFor simplicity all physical variables in frequency domain

are also denoted by the same symbols as in time domainunless it makes an ambiguity For example u also representsthe displacement vector in frequency domain By substituting(2) and (3) into (1) and employing the Fourier transformationon both sides of (1) we get the Navier equations as

1205962120588 (r) u (r 120596 r119904) + nabla [120582 (r) nabla sdot u (r 120596 r119904)]

+ nabla sdot [120583 (r) nablau (r 120596 r119904) + [nablau (r 120596 r119904)]119879]

= f (r 120596 r119904)

(4)

where r denotes the spatial variable r119904 represents the sourceposition 120596 is the angular frequency the source term f is aRicker wavelet represented in frequency domain throughoutthis paper 119879 is a matrix transpose and nabla is a gradientoperator with respect to the spatial variable r

System (1) and (4) defines the wave propagation in aninfinitemedium in time and frequency domains respectively

Journal of Applied Mathematics 3

However the wave field is numerically modeled in a finitedomain Therefore waves will reflect from the computationedges of the domain and the artificial waves will be recordedIn time domain these reflections can be eliminated by thetime windowing however such reflections cannot be elimi-nated in frequency domain Therefore we should attenuateor suppress the unwanted waves to simulate a modelingin a finite domain and one type of absorbing boundaryconditions should be applied For our forward modelinga stabilized unsplit convolutional perfectly matched layer(PML) is employed (see [24]) and the source term f we addedis in term of p-wave incident excitation

The parameters and wave fields are uniformly discretizedon a regular two-dimensional domain at intervals of Δ119909 andΔ119911 by using the second-order finite-differences described in[27 28] Note that the spatial sampling intervals Δ119909 and Δ119911

must be small enough so that the numerical solutions of thediscretization linear equation system of (4) are sufficientlyaccurate Generally this five-point stencil requires about 10grid points per shear wave wavelength in the lowest velocityzone for an accurate modeling (see Pratt [28]) After theunknowns at each grid point are ordered by row-major rulewe obtain a linear equation system as

A (120596)U (120596) = F (120596) (5)

where A is the finite-difference matrix or stiffness matrixwhich is asymmetric and extremely sparse U is theunknowns vector and F denotes the discrete vector of thesource term To solve the linear equation system (5) wechoose a direct solver method based on the sparse matrixLU decomposition rather than an iterative solver The mainreason we employ a direct solver method is that the stiffnessmatrix A is only dependent on the background mediumand an angular frequency 120596 Hence we can get all solutionsfor all source excitations simultaneously by factoring thefinite-difference matrix A only once However a new matrixdecomposition is required for another frequency

3 Finite-Difference Contrast SourceInversion Algorithm

31 Problem Formulation We consider the two-dimensionalelastic wave inversion problem in a homogeneous or aninhomogeneous backgroundmediumThroughout the paperthe inversion domain or object domain inwhich the scatterersare embedded is denoted by symbol 119863 and the data domain119878 is the surface where the sources and receivers are locatedWe assume that both of the domains are contained withinthe total domain 119863

1called finite-difference computation

domain By this configuration of this inverse problem we canslightly reduce the inversion domain Assume that there are119873119904 sources successively illuminating the inversion domain

and the index for the 119895th source is denoted by subscript 119895For each source 119895 the total field vector u

119895(r 120596) and incident

field vector uinc119895

(r 120596) satisfy the following Navier equationsrespectively

N [u119895(r 120596)]

equiv 1205962120588 (r) u

119895(r 120596 r119904) + nabla [120582 (r) nabla sdot u

119895(r 120596 r119904)]

+ nabla sdot [120583 (r) nablau119895(r 120596 r119904) + [nablau

119895(r 120596 r119904)]

119879

]

= f (r 120596 r119904) r isin 1198631

(6)

N119887[uinc119895

(r 120596)]

equiv 1205962120588119887(r) uinc119895

(r 120596 r119904) + nabla [120582 (r) nabla sdot uinc119895

(r 120596 r119904)]

+ nabla sdot [120583 (r) nablauinc119895

(r 120596 r119904) + [nablauinc119895

(r 120596 r119904)]119879

]

= f (r 120596 r119904) r isin 1198631

(7)

In (7) the parameter involving the subscript 119887 denotesthe relative parameters when the equation is employed todescribe the seismic wave propagation in a known back-ground medium Subtracting (7) from (6) and using therelationship between the total filed u

119895 incident filed uinc

119895 and

the scattered filed usct119895 the scattered fields satisfy

N119887[usct119895

(r 120596)] = minus1205962120588119887(r)w119895(r 120596) r isin 119863

1 (8)

where w119895(r 120596) are the contrast source vectors defined as

w119895(r 120596) = 120594 (r) u

119895(r 120596) r isin 119863 (9)

in which the contrast function 120594(r) is given by

120594 (r) =120588 (r) minus 120588

119887(r)

120588119887(r)

=120588 (r)120588119887(r)

minus 1 r isin 119863 (10)

According to (7) the differential operator N119887with the

known Lame parameters 120582 and 120583 only depends on thebackground mass density 120588

119887(r) The inverse of the operator

N119887is formally given by the linear operator L

119887[sdot] =

Nminus1119887[minus1205962120588119887(r)(sdot)] Therefore the solution vectors of (8) can

be symbolically denoted by the following equation

usct119895

(r 120596) = Nminus1

119887[minus1205962120588119887(r)w119895(r 120596)]

= L119887[w119895(r 120596)] r isin 119863

1

(11)

By using (11) the measured scattered field data vectors atthe receiver r

119877location can be formally presented by the

following equation

d119895(r119877 120596) = M

119878L119887[w119895(r 120596)] r

119877isin 119878 r isin 119863

1 (12)

and the object equation or state equation satisfies

w119895(r 120596) = 120594 (r) uinc

119895(r 120596)

+ 120594 (r)M119863 L119887[w119895(r1015840 120596)] r isin 119863 r1015840 isin 119863

1

(13)

4 Journal of Applied Mathematics

whereM119878 is an operator selecting the scattered fields on themeasurement surface 119878 among the scattered fields in the totalcomputation domain and the operatorM119863 selects the fieldsinside the inversion domain119863

Note that (12) and (13) are the fundamental equationsfor the CSI method which are used to establish the costfunctional From (12) and (13) the data error r119878

119895(r 120596) and the

object (or state) error r119863119895(r 120596) are denoted by the following

equations respectively

r119878119895(r 120596) = d

119895(r119877 120596)

minusM119878L119887[w119895(r 120596)] r

119877isin 119878 r isin 119863

1

(14)

r119863119895(r 120596) = 120594 (r) uinc

119895(r 120596) + 120594 (r)M119863 L

119887[w119895(r1015840 120596)]

minus w119895(r 120596) r isin 119863 r1015840 isin 119863

1

(15)

To establish the objective functional we should firstly givethe specific expressions of the inner product and 119871

2-norm as

⟨u k⟩Ω = intΩ

u (r) sdot k (r)119889r (16)

r2Ω= intΩ

u (r) sdot u (r)119889r (17)

where the overbar denotes the complex conjugate for eachcomponent of a complex vector and symbol sdot represents theproduct of two vectors Ω is the integral domain which canbe either 119878 or119863

For simplicity the explicit dependence of the field quan-tities on r and r1015840 is dropped in the remainder of this paper

32 Finite-Difference Contrast Source Inversion Method Inthe framework of the contrast source inversion method(see [11]) it does not eliminate the contrast source quan-tities w

119895in (12) and (13) On the contrary it alternatively

reconstructs contrast sources w119895and contrast function 120594 by

minimizing a cost functional In order to avoid minimizinga nonquadratic objective function caused by the presence ofcontrast function 120594 in both numerators and denominatorswe borrow the idea from [14] and slightly modify the costfunctional of CSI method Therefore the modified costfunctional to be minimized at the 119899th step is defined as thefollowing form

119865119899(120594w119895) =

sum119895

10038171003817100381710038171003817r119878119895

10038171003817100381710038171003817

2

119878

sum119895

10038171003817100381710038171003817d119878119895

10038171003817100381710038171003817

2

119878

+

sum119895

10038171003817100381710038171003817r119863119895

10038171003817100381710038171003817

2

119863

sum119895

10038171003817100381710038171003817120594119899minus1

uinc119895

10038171003817100381710038171003817

2

119863

= 119865119878(w119895) + 119865119863

119899(120594w119895)

(18)

where 119865119878(w119895) represents the data equation error and

119865119863

119899(120594w119895) represents the object equation error The scattered

field data is obtained by subtracting the simulated back-ground field from the simulated model field From (14)(15) and (18) we note that the background medium does

not change throughout the inversion process therefore thecomputation cost of the inversion of the finite-differenceoperator N

119887is relatively cheap for each frequency which

results from the LU decomposition required only onceThe main idea of the CSI method is to reconstruct

two interlaced sequences w119895119899

and 120594119899 by minimizing the

objective functional (18)Thefirst step of FD-CSImethod is toupdate contrast source vectorsw

119895119899by the conjugate-gradient

methodwith Polak-Ribiere search directionsWhile updatingthe contrast source the contrast function 120594 is assumed to be120594119899minus1

At the 119899th step the contrast sources w119895119899

are updated bythe following formula

w119895119899

= w119895119899minus1

+ 120572120596

119895119899k119895119899 (19)

where 120572120596119895119899

is an update-step size and k119895119899

is the Polak-Ribieresearch directions (see [29]) given by

k1198950

= 0

k119895119899

= 119892120596

119895119899+

sum119896⟨119892120596

119895119899 119892120596

119895119899minus 119892120596

119895119899minus1⟩

sum119896

10038171003817100381710038171003817119892120596

119895119899minus1

10038171003817100381710038171003817

2

119863

k119895119899minus1

(20)

in which 119892120596

119895119899is the gradient of the cost function with respect

to w119895119899

evaluated at the (119899 minus 1)th iteration The details ofthis type of the conjugate-gradient can be found in [30]After updating the gradient 119892120596

119895119899 the update-steps 120572120596

119895119899can be

obtained by minimizing the cost functional 119865119899 that is

120572120596

119895119899= argmin

120572119865119899(120594119899minus1

w119895119899minus1

+ 120572k119895119899) (21)

and the explicit expression of the minimizer 120572120596119895119899

of (21) canbe founded by setting the derivative with respect to 120572 beingequal to zero that is

120572120596

119895119899= (minus⟨119892

120596

119895119899 k119895119899⟩119863)

times (12057811987810038171003817100381710038171003817M119878L119887[k119895119899]10038171003817100381710038171003817

2

119878

+120578119863

119899

10038171003817100381710038171003817k119895119899

minus 120594119899minus1

M119863L119887[k119895119899]10038171003817100381710038171003817

2

119863)

minus1

(22)

where the normalization factors 120578119878 and 120578119863

119899are

120578119878= (sum

119895

10038171003817100381710038171003817d119895

10038171003817100381710038171003817

2

119878)

minus1

120578119863

119899= (sum

119895

10038171003817100381710038171003817120594119899minus1

uinc119895

10038171003817100381710038171003817

2

119863)

minus1

(23)

The second step of the CSI method is to update contrastfunction 120594 Once we update the contrast source vectors w

119895

with conjugate gradient method the contrast function 120594 isupdated by finding theminimizer of the functional119865119863

119899(120594w119895)

We assume the contrast source vectorsw119895are constant during

Journal of Applied Mathematics 5

this process It can be shown that the updating formula hasthe closed form as follows

120594119899=

sum119895Re (w

119895119899sdot u119895119899)

sum119895

10038161003816100381610038161003816u119895119899

10038161003816100381610038161003816

2 (24)

where

u119895119899

= uinc119895119899

+M119863L119887[w119895119899] (25)

In (24) the symbol Re indicates getting the real part of acomplex number The details of finding the minimizer withrespect to 120594 of the objective functional (18) can be found in[11]

To start the FD-CSI algorithm Considering the non-linearity and illposedness of the inverse problem and thelocal convergence of the FD-CSI method a good initial guessis very important to get a better reconstruction result Weinitialize the algorithm by the following way

w1198950

=

10038171003817100381710038171003817Llowast119887M119878lowast [d

119895]10038171003817100381710038171003817

2

119863

10038171003817100381710038171003817M119878 (L

119887[Llowast119887M119878lowast [d

119895]])

10038171003817100381710038171003817

2

119878

Llowast

119887M119878lowast

[d119895]

(26)

and the contrast function is calculated by using (24) and (25)In (26)Llowast

119887is the adjoint operator of the operatorL

119887 which

is defined as

Llowast

119887[sdot] = (N

lowast

119887)minus1

[minus1205962120588119887(r) (sdot)] (27)

where

Nlowast

119887[sdot] equiv 1205962120588 (r) (sdot) + nabla [120582 (r)nabla sdot (sdot)]

+ nabla sdot [120583 (r) nabla (sdot) + [nabla (sdot)]119879]

(28)

and the symbol M119878lowast has the same meaning with Llowast119887 which

maps the field values on the measurement surface intothe total computation domain This initialization method iscalled back-propagation method and it has been proved thatit can provide a good initial guess (see [8])

To improve the quality of reconstruction results themultiplicative regularization technique introduced by vanden Berg et al (see [26]) is applied to this inverse problemwhich has the same effect as the total variational regularizedmethod After incorporating themultiplicative regularizationfactor the cost functional is rewritten as

119862119899(120594w119895) = [119865

119878(w119895) + 119865119863

119899(120594w119895)] 119865119877

119899(120594) (29)

where the regularization cost function119865119877

119899(120594) for the weighted

1198712-norm regularization is defined as

119865119877

119899(120594) = int

119863

1198872

119899(1003816100381610038161003816nabla120594 (r)1003816100381610038161003816

2

+ 1205752

119899) 119889r (30)

where

1198872

119899(r) = 1

119860 (1003816100381610038161003816nabla120594119899minus1 (r)

10038161003816100381610038162

+ 1205752119899)

1205752

119899=

119865119863

119899(120594119899minus1

w119895119899minus1

)

Δ119909Δ119911

(31)

where 119860 is the area of the inversion domain and Δ119909Δ119911 is thecell areaMore details on finite-differencemultiplicative regu-larization contrast source inversion (FD-MRCSI)method canbe found in Appendix A of [8]

It is very similar to minimizing functional (29) usingconjugate-gradient method There is no change while updat-ing the contrast source vectors w

119895 since the regularization

factor 119865119877

119899(120594) is only dependent on contrast function 120594 and

119865119877

119899(120594119899minus1

) equiv 1 Nevertheless the contrast function 120594 isupdated by using the conjugate-gradient method with Polak-Ribiere search directions

Compared with an additive regularization method it isnot necessary to determine the artificial weighting regular-ization parameter because it is automatically calculated ateach iteration As we know that selecting the regularizationparameter is a time-consuming and difficult work especiallythere is no prior information about the measurement datamultiplicative regularizationmethod is very suitable to invertexperimental data as shown in [20]

33 Computational Complexity Analysis The highest com-putational cost part of FD-MRCSI is to calculate theseoperators L

119887and Llowast

119887to estimate the conjugate 119892

119895and

update-steps 120572120596

119895 Hence an efficient method to compute

the action of these operators is important to increase theefficiency of the inversion algorithm Considering the factthat these operators only depend on the backgroundmediumwhich does not change throughout the inversion process for agiven frequency 120596 we choose an efficient direct method thatis the LU decomposition and the decomposition for each ofthe two discretization operatorsL

119887andLlowast

119887is done only once

at the beginning of the inversion process and then reusedat each subsequent step The decomposition process takes119874(11989915) operations where 119899 is the number of the unknowns

of the discrete system (119899 = 2119873 119873 is the number of thegrid cells) (see [25]) The back-substitution needs 119874(119899 log 119899)operations for each source position In the framework ofFD-MRCSI back-substitution is required twice for each ofthe two operators L

119887and Llowast

119887per iteration Hence the

computational complexity of FD-MRCSI for each frequencyis approximately given by

119879 (119899119873iter

119873119904) asymp 119874 (2119899

15) + 119873

iter119874 (2119873

119904119899 log 119899) (32)

where119873iter and119873119904 are the number of inversion iteration and

the number of sources respectivelyFor usual inversion methods (UIM) the objective func-

tional involving the scattered observed data d119895and scattered

synthetic data usct119895

is given in the following formula

119865 (120594) = sum

119895

10038171003817100381710038171003817d119895minus usct119895

(120594)10038171003817100381710038171003817

2

119878+ 120572119877 (120594) (33)

The first term of (33) ensures that the minimizer 120594120572of 119865

will be an optimal parameter model while the second isthe regularized term which guarantees the stabilization ofthe problem and forces the minimizer 120594

120572of (33) to satisfy

certain regularity properties and120572 denotes the regularization

6 Journal of Applied Mathematics

Table 1 Computational complexity for FD-MRCSI and UIM

Method LU decomposition Back-substitution Total costUIM 119873

iter119874(211989915) 119873

iter119874(2119873

S119899 log 119899) 119873

iter119874(2119899

15) + 119874(2119873

S119899 log 119899)

FD-MRCSI 119874(211989915) 119873

iter119874(2119873

S119899 log 119899) 119874(2119899

15) + 119873

iter119874(2119873

S119899 log 119899)

119873iter is the iteration number119873S presents the number of source and 119899 denotes the number of the grid cells

parameter that balances the data error and regularized termIf functional (33) is also minimized by using a nonlinearconjugate gradient method then the computational com-plexity of this algorithm for this inverse problem can beapproximately estimated in analogy to FD-MRCSImethod asfollows

1198791(119899119873

iter 119873119904) asymp 119873

iter119874 (2119899

15) + 119874 [(2119873

119878) 119899 log 119899]

(34)

in which 119874(11989915) represents the LU decomposition compu-

tational cost for forward modeling and 119874(119899 log 119899) denotesthe back-substitution computation for each source duringeach iteration process Table 1 indicates the details of com-putational complexity for FD-MRCSI and usual inversionmethods

From Table 1 we observe that the FD-MRCSI methoddoes not need to do an LU decomposition for each iterationwhile usual methods require it and this is the highestcomputational cost part Therefore the FD-MRCSI methodis a greatly efficient method

4 Numerical Results

In this section we present some numerical examples basedon the FD-MRCSI algorithm All inversion domains of thesetests are inside a square with side length 119871 In order tomeasure the discrepancy between the exact profile and recon-structed profile the relative error with respect to contrastquantity 120594 is introduced which is given by the followingformula

Err =1003817100381710038171003817120594119899 (r) minus 120594exact (r)

10038171003817100381710038171198631003817100381710038171003817120594exact (r)

1003817100381710038171003817119863

times 100 (35)

where 120594119899is the reconstructed result after 119899 iterations and

120594exact is the exact solution in which we are interestedSince the ldquoinverse crimerdquo phenomenon may happen

the synthetic data are generated by using a finite-differencetime-domain forward method A Ricker wavelet f with adominant frequency 30Hz is employed for generating thesynthetic data However the inversion algorithm is carriedout in frequency domain and hence the datasets generatedby using time-domain forward method are transformed intofrequency domain using a fast Fourier transform (FFT) (see[31]) After that we select some frequencies for the FD-MRCSI algorithm Once we get the synthetic data a randomnoise is added to each frequency synthetic data using thisformula as follows

dnoise119895

= d119895+max119895

(10038161003816100381610038161003816d119895

10038161003816100381610038161003816)120578

2(120585

ReI + 119894120585ImI) (36)

where dnoise119895

and d119895are the field data vectors with noise and

the noiseless data respectively 120585Re and 120585Im are uniformly

distributed random numbers on the interval (minus1 1) I is atwo- dimensional vector whose components are equal to oneand 120578 is the desired fraction of the noise

41 Reconstruction in a Homogeneous Background MediumThe first example is concerned with recovering a squarescatterer from noise observed data dnoise

119895in a homogeneous

background medium and the exact model is given byFigure 1(a) The inversion domain 119863 is set as a square withside length 6 km which does not include PML thickness Forthis square scatterer the second Lame parameter 120583 is equal to20GPa and a Poissonrsquos ratio is 025The square with the sidelength 15 km is located in the inversion domain and its centreis at (33) km the mass density of the scatterer is 18 gcm315 gcm3 outside the scatterer Hence the contrast 120594 for thescatterer is 02 During the whole inversion procedure theangular frequency 120596 is 8Hz

The inversion domain 119863 is divided into 119873 times 119873 uniformgrids using the second-order scheme adding the 119899

1grids of

PML so the size of the total computation domain is (119873 +

21198991) times (119873 + 2119899

1) For this experiment 119873 = 61 and 119899

1=

15 therefore the grid dimension of the total computationdomain is 91times91 Both the sources and receivers are assignedto 16 which are equally distributed on the four sides of theinversion domain and all sources are added in the form of 119875-wave incident excitation To show the denoising performanceof the FD-MRCSI method the numerical inversion resultsare presented when the measurement datasets are added 5random noise Figures 1(b) and 1(c) are the reconstructionprofile of the experiment and discrepancy between the truemodel and recoveringmodel after 1000 iterations Figure 1(d)shows the slices along the depth at horizontal distances of3 km for the true model and inversion model The relativeerror is 1226

In this type of numerical tests of a homogeneous back-ground medium we will test that the FD-MRCSI methodpossesses the ability of reconstructing multiple scattererswith different geometrical information and different phys-ical property parameters For this purpose we employ amodel consisting of two rectangular scatterers as shown inFigure 2(a) The configuration consists of two rectangularobjects which have different dimensions One is 12 km times

12 km and another is 08 km times 1 kmWe also assume that theinversion domain is a square domain with the dimension of6 km times 6 km and has a second Lame coefficient 120583 of 52GPaa Poissonrsquos ratio of 025 and a mass density of 15 gcm3 Thelarge obstacle has a mass density of 27 gcm3 and the small

Journal of Applied Mathematics 7

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1500

1550

1600

1650

1700

1750

1800

1850

1900

(a) True model

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1500

1550

1600

1650

1700

1750

1800

1850

1900

(b) Reconstruction result

3000

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 4000 5000 6000

minus150

minus100

minus50

0

50

100

(c) Discrepancy

x (m)

1500

1550

1600

1650

1700

1750

1800

1850

1900

0 1000 2000 3000 4000 5000 6000

InversionTrue

Den

sity

(kg

m3)

(d) Model slice

Figure 1 Applying the FD-MRCSI method to one abnormity inversion (a) True model (b) Reconstruction result after 1000 inversioniterations (c) Discrepancy between true model and reconstruction result (d) Slices at horizontal distances of 3 km for the true model andreconstruction result

one has a mass density of 21 gcm3 Hence the contrasts forthe two abnormities are 08 and 04 respectively

In the inversion process we discretize the total compu-tation domain into 151 times 151 uniform grids including 15PML grids and 24 sources and 24 receivers are used for thisexampleThese sources and receivers are distributed along theinversion domain six sources or receivers for each side Afterthe measurement data is generated using the same way as theprevious example we select them at frequency 8Hz to drivethe inversion algorithm and 5 noise is added After 1000iterations the reconstruction profile is given in Figure 2(b)Figure 2(c) denotes the discrepancy between the true modeland recovering model after 1000 iterations and Figure 2(d)shows the variation of the data equation term error withrespect to iterations The relative error is 1605

At first glance we should solve the forward modeling formany times for these 1000 inversion iterations Since the con-trast source w and contrast function 120594 are incorporated intothis algorithm whichmakes the forward stiffnessmatrix only

involve the background medium for each given frequencythe LU decomposition is required only once throughout thewhole inversion procedure therefore the forward modelingcomputation cost is very cheap However the price that wehave to pay is the storing of the LU decomposition arrays ofthe stiffness matrix of the background medium

From these numerical results we observe that the FD-MRCSI method can effectively reconstruct the scatterersincluding their locations and shapes using the measurementscattered data and the method is of strong denoising abilityAdditionally Figure 2(d) demonstrates that the FD-MRCSImethod is convergent

42 Reconstruction in an Inhomogeneous BackgroundMedium In this section we examine the performance of theFD-CSI using an inhomogeneous medium as a backgroundmedium The configuration of the background consists of acompressional wave speed of 2500ms a shear wave speed of1443ms and a four-layer distribution of amass density of 18

8 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1400

1600

1800

2000

2200

2400

2600

2800

(a) True model

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1400

1600

1800

2000

2200

2400

2600

2800

(b) Reconstruction result

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

minus800

minus600

minus400

minus200

0

200

400

600

(c) Discrepancy

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

0 200 400 600 800 1000

log(x

)

Iteration

Error of data term

(d) Variation of the data equation error

Figure 2 Applying the FD-CSImethod to two abnormities inversion (a) Truemodel (b) Reconstruction result after 1000 inversion iterations(c) Discrepancy between true model and reconstruction result (d) Variation of the data term error with respect to iterations

16 15 and 18 gcm3 from top to bottom Figure 3(a) showsthe mass density distribution of the background mediumThe inversion model consisting of a circle and an ellipse ispresented as shown in Figure 3(b) The circular abnormitywith a contrast of 08 has a radius of 0550 km which iscentered at (34 36) km For the elliptical scatterer it has asemimajor axis of 07 km and a semiminor axis of 036 kmand its centre is (23 2) kmWe also assume that the inversiondomain is a square with dimension of 119871times119871 In our numericalexperiments 119871 is assigned to 6 km

In the inversion the contrast function 120594 contrast sourcesw and state variables u uinc and usct in the inversion domainare uniformly discretized on a regular two-dimensional gridat intervals of Δ119909 and Δ119911 Both Δ119909 and Δ119911 are equal to005 km in other words the grid size is 121 times 121 Themeasurement domain surface 119878 where 24 receivers and 24sources are distributed is a square around the inversiondomain After the synthetic datasets are generated by usingFFT on the finite-difference time-domain (FD-DT) data atfrequency 10Hz 5 noise is added by using (35) Afterrunning 1000 iterations the reconstruction results are givenin Figure 3(c) The relative errors are 1883

As shown in Figure 3 the FD-MRCSI method can suc-cessfully reconstruct the objects including their locationsand shapes as those present in a homogeneous backgroundmediumThe feature of the inversionmethod results from theflexibility of the finite-difference operator of the governingdifferential equations

43 Reconstruction Improvement Using Multiple FrequenciesIn these examples we use the previous lower frequencyinversion reconstruction results as an initial guess for anotherfrequency inversion which is the so-called sequential multi-frequency inversion approach that one usually employs thistechnique in theGauss-Newton or nonlinear conjugate gradi-ent method This technique applied to improve the inversionprofiles is guaranteed by the following two observations (1)each frequency is independent of FD-MRCSI algorithm (2)the inversion algorithm is locally convergent and hence agood initial guess will result in a better inversion resultsAdditionally multiple frequencies inversion technique canmigrate the nonlinearity of the elastic inverse problem

To drive the inversion algorithm we should use anotherinitialization method which was introduced by Abubakar et

Journal of Applied Mathematics 9

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(a) Background medium

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(b) True model

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(c) Reconstruction result

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

0 200 400 600 800 1000

log(x

)

Iteration

Error of data term

(d) Variation of the data equation error

Figure 3 Applying the FD-CSI method to two abnormities inversion in an inhomogeneous background medium (a) Background medium(b) True model (c) Reconstruction result after 1000 inversion iterations (d) Variation of the data term with respect to iterations

al (see [8]) Firstly use the previous inversion result as 1205940

then solve the forward problemwith 120588 = (1+1205940)120588119887to get the

total field vectors u1198950 finally the initial contrast sources w

1198950

are obtained with the equation w1198950

= 1205940u1198950

We employ the strategy to rebuild the two test examplesgiven in Section 41 and all relative parameters are thesame as those present in Section 41 For the one scatterermodel case we successively select three frequencies (4 6and 8Hz) For the two scatterers model case the sequentialfrequency applied during the multiple frequencies inversionis 5Hz 7Hz and 10Hz Figures 4(a) and 4(b) show thecorresponding reconstruction results The relative errors are813 and 1260 respectively which are smaller than thosein Section 41 Table 2 indicates the reconstruction relativeerrors of the two models using FD-MRCSI and FD-MRCSIwith multiple frequencies technique

From Figure 4 and Table 2 we can observe that theuse of the sequential multifrequency approach can improvereconstructed model to some extent This can be explainedby the fact that a good initial guess can result in a bettercomputation result for a locally convergent method

Table 2 Relative errors using FD-CSI and FD-CSI with multiplefrequencies technique

Model FD-CSI FD-CSI with multifrequencyOne scatterer model 1226 813Two scatterers model 1605 1260

5 Conclusions

In this paper the finite-difference contrast source inversionmethod incorporating a multiplicative regularization factoris successfully employed in seismic full-waveform inversionto reconstruct the mass density The method is very efficientbecause it does not require solving a full forward problem ineach inversion iteration and the highly efficient calculationof the forward modeling makes it have a potential for dealingwith large scale computational inverse problems

From the results of numerical experiments the algorithmcan well recognize the shapes of the scatterers and preciselyidentify their locations no matter the background mediums

10 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1500

1550

1600

1650

1700

1750

1800

1850

1900z

(m)

x (m)

(a) Multifrequency inversion result

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

1400

1600

1800

2000

2200

2400

2600

2800

x (m)

(b) Multifrequency inversion result

Figure 4 Applying sequential multifrequency technique to improve reconstruction results (a) Reconstruction result of one square scatterermodel (b) Reconstruction result of two rectangle scatterers model

are homogeneous or inhomogeneousmediumThe algorithmis very robust when the measurement data is polluted byrandom noise Furthermore the FD-MRCSI using sequen-tial multifrequency technique can successfully improve thereconstruction results which is very hopeful for gettinghigher resolution in exploration seismology Estimating othersubsurface parameters using this method such as Lamecoefficients and velocity is our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Professor van den Berg who givesus many helpful suggestions on both theory and numericalimplementation of the contrast source inversion methodThis work is supported by the National Natural ScienceFoundation of China underGrants no 91230119 no 41474102and no 11301119

References

[1] A Tarantola ldquoA strategy for nonlinear elastic inversion ofseismic reflection datardquo Geophysics vol 51 no 10 pp 1893ndash1903 1986

[2] PMora ldquoNonlinear two-dimensional elastic inversion ofmulti-offset seismic datardquoGeophysics vol 52 no 9 pp 1211ndash1228 1987

[3] R G Pratt and M H Worthington ldquoInverse theory appliedto multi-source cross-hole tomography Part 1 acoustic wave-equation methodrdquo Geophysical Prospecting vol 38 no 3 pp287ndash310 1990

[4] R G Pratt ldquoInverse theory applied to multi-source cross-holetomography Part 2 elastic wave-equationmethodrdquoGeophysicalProspecting vol 38 no 3 pp 311ndash329 1990

[5] B Han H S Fu and Z Li ldquoA homotopy method for theinversion of a two-dimensional acoustic wave equationrdquo InverseProblems in Science and Engineering vol 13 no 4 pp 411ndash4312005

[6] H Ben-Hadj-Ali S Operto and J Virieux ldquoVelocity model-building by 3D frequency-domain full-waveform inversionof wide-aperture seismic datardquo Geophysics vol 73 no 5 ppVE101ndashVE117 2008

[7] W W Symes ldquoMigration velocity analysis and waveforminversionrdquo Geophysical Prospecting vol 56 no 6 pp 765ndash7902008

[8] A Abubakar W Hu T M Habashy and P M van den BergldquoApplication of the finite-difference contrast-source inversionalgorithm to seismic full-waveform datardquo Geophysics vol 74no 6 pp WCC47ndashWCC58 2009

[9] J J Cao Y F Wang J T Zhao and C Yang ldquoA reviewon restoration of seismic wavefields based on regularizationand compressive sensingrdquo Inverse Problems in Science andEngineering vol 19 no 5 pp 679ndash704 2011

[10] A Fichtner P H Bunge and H Igel ldquoThe adjoint methodin seismology I Theoryrdquo Physics of the Earth and PlanetaryInteriors vol 157 no 1-2 pp 86ndash104 2006

[11] P M van den Berg and R E Kleinman ldquoA contrast sourceinversion methodrdquo Inverse Problems vol 13 no 6 pp 1607ndash1620 1997

[12] P M van den Berg A L van Broekhoven and A AbubakarldquoExtended contrast source inversionrdquo Inverse Problems vol 15no 5 pp 1325ndash1344 1999

[13] A Abubakar W Hu P M van den Berg and T M HabashyldquoA finite-difference contrast source inversion methodrdquo InverseProblems vol 24 no 6 Article ID 065004 pp 1ndash17 2008

[14] X Chen ldquoSubspace-based optimization method for solvinginverse-scattering problemsrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 48 no 1 pp 42ndash49 2010

[15] X Chen ldquoSubspace-based optimization method for inversescattering problems with an inhomogeneous backgroundmediumrdquo Inverse Problems vol 26 Article ID 074007 2010

[16] A Abubakar G Pan M Li L Zhang T M Habashy and PM van den Berg ldquoThree-dimensional seismic full-waveform

Journal of Applied Mathematics 11

inversion using the finite-difference contrast source inversionmethodrdquo Geophysical Prospecting vol 59 no 5 pp 874ndash8882011

[17] R F Bloemenkamp A Abubakar and P M van den BergldquoInversion of experimental multi-frequency data using thecontrast source inversion methodrdquo Inverse Problems vol 17 no6 pp 1611ndash1622 2001

[18] A Abubakar and P M van den Berg ldquoThe contrast sourceinversion method for location and shape reconstructionsrdquoInverse Problems vol 18 no 2 pp 495ndash510 2002

[19] L P Song C Yu and Q H Liu ldquoThrough-wall imaging(TWI) by radar 2-D tomographic results and analysesrdquo IEEETransactions on Geoscience and Remote Sensing vol 43 no 12pp 2793ndash2798 2005

[20] M Li A Abubakar and P M van den Berg ldquoApplication of themultiplicative regularized contrast source inversion method on3D experimental Fresnel datardquo Inverse Problems vol 25 no 2Article ID 024006 23 pages 2009

[21] A Zakaria C Gilmore and J LoVetri ldquoFinite-element contrastsource inversionmethod for microwave imagingrdquo Inverse Prob-lems vol 26 no 11 Article ID 115010 21 pages 2010

[22] G Pelekanos A Abubakar and P M van den Berg ldquoContrastsource inversion methods in elastodynamicsrdquo Journal of theAcoustical Society of America vol 114 no 5 pp 2825ndash28342003

[23] D Komatitsch and R Martin ldquoAn unsplit convolutional per-fectly matched layer improved at grazing incidence for theseismic wave equationrdquo Geophysics vol 72 no 5 pp SM155ndashSM167 2007

[24] R Martin D Komatitsch and S D Gedney ldquoA variational for-mulation of a stabilized unsplit convolutional perfectlymatchedlayer for the isotropic or anisotropic seismic wave equationrdquoComputer Modeling in Engineering amp Sciences vol 37 no 3 pp274ndash304 2008

[25] T A Davis Direct Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2006

[26] P M van den Berg A Abubakar and J T Fokkema ldquoMul-tiplicative regularization for contrast profile inversionrdquo RadioScience vol 38 no 2 2003

[27] K R Kelly R WWard S Treitel and R M Alford ldquoSyntheticseismograms a finite difference approachrdquo Geophysics vol 41no 1 pp 2ndash27 1976

[28] R G Pratt ldquoFrequency-domain elastic wave modeling by finitedifferences a tool for crosshole seismic imagingrdquo Geophysicsvol 55 no 5 pp 626ndash632 1990

[29] K W Brodlie and D A H Jacobs ldquoUnconstrained minimiza-tionrdquo in State of the Art in Numerical Analysis pp 229ndash268Academic Press 1997

[30] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 2006

[31] P Duhamel and M Vetterli ldquoFast Fourier transforms a tutorialreview and a state of the artrdquo Signal Processing vol 19 no 4 pp259ndash299 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

The method is referred to as the so-called contrast sourceinversion (CSI) method which has two versions IE-CSIbased on integral equation formulation (see [11 12]) andFD-CSI based on the finite-difference operator (see [13])The IE-CSI method is very efficient for the homogeneousor simple layered background medium inverse problemssince the IE-CSI method needs to calculate Green functionof the background medium while the FD-CSI method hasa good performance for solving high complex problems inan inhomogeneous background medium Another highlyefficient scattering inversion method called subspace-basedoptimization method (SOM) shares this good propertyof CSI method which also has two variants adopted inhomogeneous and inhomogeneous background mediumrespectively (see [14 15]) The comparisons in detail betweenSOM and CSI are covered in [14] and this reference alsoconcludes that the CSI is close in spirit to a special case ofSOM and the SOM could be convergent for a few specialproblems The interesting advantage of CSI method is thatit does not require solving any full forward problem in eachinversion iteration therefore it can significantly reduce thecomputational cost and it may be very efficient for large scaleor high dimension inverse problems (see [16]) Both the IE-CSI and FD-CSI methods incorporating with multiplicativeregularization constraint have been successfully applied inelectromagnetic wave imaging and acoustic wave imaging(see [17ndash21])

In this paper we extend the FD-CSI method to recon-struct the mass density based on a two-dimensional elasticwave equation (in fact all theories in this paper are suitablefor three-dimensional problems without any changes) in anisotropic medium either homogeneous or inhomogeneousbackground medium and its IE-version is presented in [22]which is successfully employed in elastodynamics when thebackgroundmedium is homogeneous For our forwardmod-eling a perfectly matched layer (PML) absorbing boundarycondition is employed to reduce the boundary reflections(see [23 24]) and considering the fact that the stiffnessmatrix usually is large and highly sparse the techniquesof matrix compression store and LU decomposition of asparse matrix are employed to decrease the storage spaceand increase the computation efficiency (see [25]) For theinversion processing since all the finite-difference operatorsinvolving the FD-CSI algorithm are only dependent on thebackground medium and angular frequency that both ofthem are invariant throughout the inversion process we usea direct solver such as an LU decomposition method to solvethese finite-difference responses to improve the efficiencyof the forward modeling We invert these finite-differenceoperators only once at the start of the inversion processingand the results can be reused for multiple source positionsand successive iterations of the inversion

In order to get a better reconstruction profile and somegeometrical information of an abnormality a multiplica-tive regularization factor called FD-MRCSI is incorporatedwhich is first introduced by van den Berg et al (see [26])Themultiplicative regularization factor has the same function asthe total variation (TV) regularization The most promisingadvantage of the multiplicative regularization is that the

regularization parameter is automatically calculated duringeach inversion Furthermore in order to mitigate the nonlin-earity and enhance the reconstruction results the sequentialmultifrequency inversion approach is employed that is theseismic inversion starts from low frequency and movesupward to high frequency To illustrate the performanceof the finite-difference contrast source inversion method inseismic inversion problems we present several numericalexperiments either in a homogeneous or an inhomogeneousbackground medium

2 Forward Modeling

In this section we present the two-dimensional forwardmodeling using finite-difference method which is of greatimportance for the inversion methodThe forward modelingsimulates the propagation process of the elastic wave in anisotropic inhomogeneous medium and drives the inversionalgorithm In time domain the governing PDE system calledelastodynamic system is written as the following differentialequation system

1205881205972

119905u = nabla sdot 120590 + f (1)

where u is the displacement vector 120588 is the mass density 120590denotes the stress tensor of the elastic medium nablasdot representsthe divergence operator with respect to spatial variablesand the external force is denoted by the term f In anisotropicmedium the stress tensor120590 is given by the followingformulas

120590119894119895= 120582120575119894119895120576119896119896

+ 2120583120576119894119895 (2)

120576119894119895=

1

2(120597119906119894

120597119909119895

+120597119906119895

120597119909119894

) (3)

where the indices 119894 and 119895 can be 1 or 2 120575119894119895is the well-known

Kronecker delta symbol 120576119894119895denotes the strain tensor and 120582

and 120583 are the Lame parameters of the isotropic mediumFor simplicity all physical variables in frequency domain

are also denoted by the same symbols as in time domainunless it makes an ambiguity For example u also representsthe displacement vector in frequency domain By substituting(2) and (3) into (1) and employing the Fourier transformationon both sides of (1) we get the Navier equations as

1205962120588 (r) u (r 120596 r119904) + nabla [120582 (r) nabla sdot u (r 120596 r119904)]

+ nabla sdot [120583 (r) nablau (r 120596 r119904) + [nablau (r 120596 r119904)]119879]

= f (r 120596 r119904)

(4)

where r denotes the spatial variable r119904 represents the sourceposition 120596 is the angular frequency the source term f is aRicker wavelet represented in frequency domain throughoutthis paper 119879 is a matrix transpose and nabla is a gradientoperator with respect to the spatial variable r

System (1) and (4) defines the wave propagation in aninfinitemedium in time and frequency domains respectively

Journal of Applied Mathematics 3

However the wave field is numerically modeled in a finitedomain Therefore waves will reflect from the computationedges of the domain and the artificial waves will be recordedIn time domain these reflections can be eliminated by thetime windowing however such reflections cannot be elimi-nated in frequency domain Therefore we should attenuateor suppress the unwanted waves to simulate a modelingin a finite domain and one type of absorbing boundaryconditions should be applied For our forward modelinga stabilized unsplit convolutional perfectly matched layer(PML) is employed (see [24]) and the source term f we addedis in term of p-wave incident excitation

The parameters and wave fields are uniformly discretizedon a regular two-dimensional domain at intervals of Δ119909 andΔ119911 by using the second-order finite-differences described in[27 28] Note that the spatial sampling intervals Δ119909 and Δ119911

must be small enough so that the numerical solutions of thediscretization linear equation system of (4) are sufficientlyaccurate Generally this five-point stencil requires about 10grid points per shear wave wavelength in the lowest velocityzone for an accurate modeling (see Pratt [28]) After theunknowns at each grid point are ordered by row-major rulewe obtain a linear equation system as

A (120596)U (120596) = F (120596) (5)

where A is the finite-difference matrix or stiffness matrixwhich is asymmetric and extremely sparse U is theunknowns vector and F denotes the discrete vector of thesource term To solve the linear equation system (5) wechoose a direct solver method based on the sparse matrixLU decomposition rather than an iterative solver The mainreason we employ a direct solver method is that the stiffnessmatrix A is only dependent on the background mediumand an angular frequency 120596 Hence we can get all solutionsfor all source excitations simultaneously by factoring thefinite-difference matrix A only once However a new matrixdecomposition is required for another frequency

3 Finite-Difference Contrast SourceInversion Algorithm

31 Problem Formulation We consider the two-dimensionalelastic wave inversion problem in a homogeneous or aninhomogeneous backgroundmediumThroughout the paperthe inversion domain or object domain inwhich the scatterersare embedded is denoted by symbol 119863 and the data domain119878 is the surface where the sources and receivers are locatedWe assume that both of the domains are contained withinthe total domain 119863

1called finite-difference computation

domain By this configuration of this inverse problem we canslightly reduce the inversion domain Assume that there are119873119904 sources successively illuminating the inversion domain

and the index for the 119895th source is denoted by subscript 119895For each source 119895 the total field vector u

119895(r 120596) and incident

field vector uinc119895

(r 120596) satisfy the following Navier equationsrespectively

N [u119895(r 120596)]

equiv 1205962120588 (r) u

119895(r 120596 r119904) + nabla [120582 (r) nabla sdot u

119895(r 120596 r119904)]

+ nabla sdot [120583 (r) nablau119895(r 120596 r119904) + [nablau

119895(r 120596 r119904)]

119879

]

= f (r 120596 r119904) r isin 1198631

(6)

N119887[uinc119895

(r 120596)]

equiv 1205962120588119887(r) uinc119895

(r 120596 r119904) + nabla [120582 (r) nabla sdot uinc119895

(r 120596 r119904)]

+ nabla sdot [120583 (r) nablauinc119895

(r 120596 r119904) + [nablauinc119895

(r 120596 r119904)]119879

]

= f (r 120596 r119904) r isin 1198631

(7)

In (7) the parameter involving the subscript 119887 denotesthe relative parameters when the equation is employed todescribe the seismic wave propagation in a known back-ground medium Subtracting (7) from (6) and using therelationship between the total filed u

119895 incident filed uinc

119895 and

the scattered filed usct119895 the scattered fields satisfy

N119887[usct119895

(r 120596)] = minus1205962120588119887(r)w119895(r 120596) r isin 119863

1 (8)

where w119895(r 120596) are the contrast source vectors defined as

w119895(r 120596) = 120594 (r) u

119895(r 120596) r isin 119863 (9)

in which the contrast function 120594(r) is given by

120594 (r) =120588 (r) minus 120588

119887(r)

120588119887(r)

=120588 (r)120588119887(r)

minus 1 r isin 119863 (10)

According to (7) the differential operator N119887with the

known Lame parameters 120582 and 120583 only depends on thebackground mass density 120588

119887(r) The inverse of the operator

N119887is formally given by the linear operator L

119887[sdot] =

Nminus1119887[minus1205962120588119887(r)(sdot)] Therefore the solution vectors of (8) can

be symbolically denoted by the following equation

usct119895

(r 120596) = Nminus1

119887[minus1205962120588119887(r)w119895(r 120596)]

= L119887[w119895(r 120596)] r isin 119863

1

(11)

By using (11) the measured scattered field data vectors atthe receiver r

119877location can be formally presented by the

following equation

d119895(r119877 120596) = M

119878L119887[w119895(r 120596)] r

119877isin 119878 r isin 119863

1 (12)

and the object equation or state equation satisfies

w119895(r 120596) = 120594 (r) uinc

119895(r 120596)

+ 120594 (r)M119863 L119887[w119895(r1015840 120596)] r isin 119863 r1015840 isin 119863

1

(13)

4 Journal of Applied Mathematics

whereM119878 is an operator selecting the scattered fields on themeasurement surface 119878 among the scattered fields in the totalcomputation domain and the operatorM119863 selects the fieldsinside the inversion domain119863

Note that (12) and (13) are the fundamental equationsfor the CSI method which are used to establish the costfunctional From (12) and (13) the data error r119878

119895(r 120596) and the

object (or state) error r119863119895(r 120596) are denoted by the following

equations respectively

r119878119895(r 120596) = d

119895(r119877 120596)

minusM119878L119887[w119895(r 120596)] r

119877isin 119878 r isin 119863

1

(14)

r119863119895(r 120596) = 120594 (r) uinc

119895(r 120596) + 120594 (r)M119863 L

119887[w119895(r1015840 120596)]

minus w119895(r 120596) r isin 119863 r1015840 isin 119863

1

(15)

To establish the objective functional we should firstly givethe specific expressions of the inner product and 119871

2-norm as

⟨u k⟩Ω = intΩ

u (r) sdot k (r)119889r (16)

r2Ω= intΩ

u (r) sdot u (r)119889r (17)

where the overbar denotes the complex conjugate for eachcomponent of a complex vector and symbol sdot represents theproduct of two vectors Ω is the integral domain which canbe either 119878 or119863

For simplicity the explicit dependence of the field quan-tities on r and r1015840 is dropped in the remainder of this paper

32 Finite-Difference Contrast Source Inversion Method Inthe framework of the contrast source inversion method(see [11]) it does not eliminate the contrast source quan-tities w

119895in (12) and (13) On the contrary it alternatively

reconstructs contrast sources w119895and contrast function 120594 by

minimizing a cost functional In order to avoid minimizinga nonquadratic objective function caused by the presence ofcontrast function 120594 in both numerators and denominatorswe borrow the idea from [14] and slightly modify the costfunctional of CSI method Therefore the modified costfunctional to be minimized at the 119899th step is defined as thefollowing form

119865119899(120594w119895) =

sum119895

10038171003817100381710038171003817r119878119895

10038171003817100381710038171003817

2

119878

sum119895

10038171003817100381710038171003817d119878119895

10038171003817100381710038171003817

2

119878

+

sum119895

10038171003817100381710038171003817r119863119895

10038171003817100381710038171003817

2

119863

sum119895

10038171003817100381710038171003817120594119899minus1

uinc119895

10038171003817100381710038171003817

2

119863

= 119865119878(w119895) + 119865119863

119899(120594w119895)

(18)

where 119865119878(w119895) represents the data equation error and

119865119863

119899(120594w119895) represents the object equation error The scattered

field data is obtained by subtracting the simulated back-ground field from the simulated model field From (14)(15) and (18) we note that the background medium does

not change throughout the inversion process therefore thecomputation cost of the inversion of the finite-differenceoperator N

119887is relatively cheap for each frequency which

results from the LU decomposition required only onceThe main idea of the CSI method is to reconstruct

two interlaced sequences w119895119899

and 120594119899 by minimizing the

objective functional (18)Thefirst step of FD-CSImethod is toupdate contrast source vectorsw

119895119899by the conjugate-gradient

methodwith Polak-Ribiere search directionsWhile updatingthe contrast source the contrast function 120594 is assumed to be120594119899minus1

At the 119899th step the contrast sources w119895119899

are updated bythe following formula

w119895119899

= w119895119899minus1

+ 120572120596

119895119899k119895119899 (19)

where 120572120596119895119899

is an update-step size and k119895119899

is the Polak-Ribieresearch directions (see [29]) given by

k1198950

= 0

k119895119899

= 119892120596

119895119899+

sum119896⟨119892120596

119895119899 119892120596

119895119899minus 119892120596

119895119899minus1⟩

sum119896

10038171003817100381710038171003817119892120596

119895119899minus1

10038171003817100381710038171003817

2

119863

k119895119899minus1

(20)

in which 119892120596

119895119899is the gradient of the cost function with respect

to w119895119899

evaluated at the (119899 minus 1)th iteration The details ofthis type of the conjugate-gradient can be found in [30]After updating the gradient 119892120596

119895119899 the update-steps 120572120596

119895119899can be

obtained by minimizing the cost functional 119865119899 that is

120572120596

119895119899= argmin

120572119865119899(120594119899minus1

w119895119899minus1

+ 120572k119895119899) (21)

and the explicit expression of the minimizer 120572120596119895119899

of (21) canbe founded by setting the derivative with respect to 120572 beingequal to zero that is

120572120596

119895119899= (minus⟨119892

120596

119895119899 k119895119899⟩119863)

times (12057811987810038171003817100381710038171003817M119878L119887[k119895119899]10038171003817100381710038171003817

2

119878

+120578119863

119899

10038171003817100381710038171003817k119895119899

minus 120594119899minus1

M119863L119887[k119895119899]10038171003817100381710038171003817

2

119863)

minus1

(22)

where the normalization factors 120578119878 and 120578119863

119899are

120578119878= (sum

119895

10038171003817100381710038171003817d119895

10038171003817100381710038171003817

2

119878)

minus1

120578119863

119899= (sum

119895

10038171003817100381710038171003817120594119899minus1

uinc119895

10038171003817100381710038171003817

2

119863)

minus1

(23)

The second step of the CSI method is to update contrastfunction 120594 Once we update the contrast source vectors w

119895

with conjugate gradient method the contrast function 120594 isupdated by finding theminimizer of the functional119865119863

119899(120594w119895)

We assume the contrast source vectorsw119895are constant during

Journal of Applied Mathematics 5

this process It can be shown that the updating formula hasthe closed form as follows

120594119899=

sum119895Re (w

119895119899sdot u119895119899)

sum119895

10038161003816100381610038161003816u119895119899

10038161003816100381610038161003816

2 (24)

where

u119895119899

= uinc119895119899

+M119863L119887[w119895119899] (25)

In (24) the symbol Re indicates getting the real part of acomplex number The details of finding the minimizer withrespect to 120594 of the objective functional (18) can be found in[11]

To start the FD-CSI algorithm Considering the non-linearity and illposedness of the inverse problem and thelocal convergence of the FD-CSI method a good initial guessis very important to get a better reconstruction result Weinitialize the algorithm by the following way

w1198950

=

10038171003817100381710038171003817Llowast119887M119878lowast [d

119895]10038171003817100381710038171003817

2

119863

10038171003817100381710038171003817M119878 (L

119887[Llowast119887M119878lowast [d

119895]])

10038171003817100381710038171003817

2

119878

Llowast

119887M119878lowast

[d119895]

(26)

and the contrast function is calculated by using (24) and (25)In (26)Llowast

119887is the adjoint operator of the operatorL

119887 which

is defined as

Llowast

119887[sdot] = (N

lowast

119887)minus1

[minus1205962120588119887(r) (sdot)] (27)

where

Nlowast

119887[sdot] equiv 1205962120588 (r) (sdot) + nabla [120582 (r)nabla sdot (sdot)]

+ nabla sdot [120583 (r) nabla (sdot) + [nabla (sdot)]119879]

(28)

and the symbol M119878lowast has the same meaning with Llowast119887 which

maps the field values on the measurement surface intothe total computation domain This initialization method iscalled back-propagation method and it has been proved thatit can provide a good initial guess (see [8])

To improve the quality of reconstruction results themultiplicative regularization technique introduced by vanden Berg et al (see [26]) is applied to this inverse problemwhich has the same effect as the total variational regularizedmethod After incorporating themultiplicative regularizationfactor the cost functional is rewritten as

119862119899(120594w119895) = [119865

119878(w119895) + 119865119863

119899(120594w119895)] 119865119877

119899(120594) (29)

where the regularization cost function119865119877

119899(120594) for the weighted

1198712-norm regularization is defined as

119865119877

119899(120594) = int

119863

1198872

119899(1003816100381610038161003816nabla120594 (r)1003816100381610038161003816

2

+ 1205752

119899) 119889r (30)

where

1198872

119899(r) = 1

119860 (1003816100381610038161003816nabla120594119899minus1 (r)

10038161003816100381610038162

+ 1205752119899)

1205752

119899=

119865119863

119899(120594119899minus1

w119895119899minus1

)

Δ119909Δ119911

(31)

where 119860 is the area of the inversion domain and Δ119909Δ119911 is thecell areaMore details on finite-differencemultiplicative regu-larization contrast source inversion (FD-MRCSI)method canbe found in Appendix A of [8]

It is very similar to minimizing functional (29) usingconjugate-gradient method There is no change while updat-ing the contrast source vectors w

119895 since the regularization

factor 119865119877

119899(120594) is only dependent on contrast function 120594 and

119865119877

119899(120594119899minus1

) equiv 1 Nevertheless the contrast function 120594 isupdated by using the conjugate-gradient method with Polak-Ribiere search directions

Compared with an additive regularization method it isnot necessary to determine the artificial weighting regular-ization parameter because it is automatically calculated ateach iteration As we know that selecting the regularizationparameter is a time-consuming and difficult work especiallythere is no prior information about the measurement datamultiplicative regularizationmethod is very suitable to invertexperimental data as shown in [20]

33 Computational Complexity Analysis The highest com-putational cost part of FD-MRCSI is to calculate theseoperators L

119887and Llowast

119887to estimate the conjugate 119892

119895and

update-steps 120572120596

119895 Hence an efficient method to compute

the action of these operators is important to increase theefficiency of the inversion algorithm Considering the factthat these operators only depend on the backgroundmediumwhich does not change throughout the inversion process for agiven frequency 120596 we choose an efficient direct method thatis the LU decomposition and the decomposition for each ofthe two discretization operatorsL

119887andLlowast

119887is done only once

at the beginning of the inversion process and then reusedat each subsequent step The decomposition process takes119874(11989915) operations where 119899 is the number of the unknowns

of the discrete system (119899 = 2119873 119873 is the number of thegrid cells) (see [25]) The back-substitution needs 119874(119899 log 119899)operations for each source position In the framework ofFD-MRCSI back-substitution is required twice for each ofthe two operators L

119887and Llowast

119887per iteration Hence the

computational complexity of FD-MRCSI for each frequencyis approximately given by

119879 (119899119873iter

119873119904) asymp 119874 (2119899

15) + 119873

iter119874 (2119873

119904119899 log 119899) (32)

where119873iter and119873119904 are the number of inversion iteration and

the number of sources respectivelyFor usual inversion methods (UIM) the objective func-

tional involving the scattered observed data d119895and scattered

synthetic data usct119895

is given in the following formula

119865 (120594) = sum

119895

10038171003817100381710038171003817d119895minus usct119895

(120594)10038171003817100381710038171003817

2

119878+ 120572119877 (120594) (33)

The first term of (33) ensures that the minimizer 120594120572of 119865

will be an optimal parameter model while the second isthe regularized term which guarantees the stabilization ofthe problem and forces the minimizer 120594

120572of (33) to satisfy

certain regularity properties and120572 denotes the regularization

6 Journal of Applied Mathematics

Table 1 Computational complexity for FD-MRCSI and UIM

Method LU decomposition Back-substitution Total costUIM 119873

iter119874(211989915) 119873

iter119874(2119873

S119899 log 119899) 119873

iter119874(2119899

15) + 119874(2119873

S119899 log 119899)

FD-MRCSI 119874(211989915) 119873

iter119874(2119873

S119899 log 119899) 119874(2119899

15) + 119873

iter119874(2119873

S119899 log 119899)

119873iter is the iteration number119873S presents the number of source and 119899 denotes the number of the grid cells

parameter that balances the data error and regularized termIf functional (33) is also minimized by using a nonlinearconjugate gradient method then the computational com-plexity of this algorithm for this inverse problem can beapproximately estimated in analogy to FD-MRCSImethod asfollows

1198791(119899119873

iter 119873119904) asymp 119873

iter119874 (2119899

15) + 119874 [(2119873

119878) 119899 log 119899]

(34)

in which 119874(11989915) represents the LU decomposition compu-

tational cost for forward modeling and 119874(119899 log 119899) denotesthe back-substitution computation for each source duringeach iteration process Table 1 indicates the details of com-putational complexity for FD-MRCSI and usual inversionmethods

From Table 1 we observe that the FD-MRCSI methoddoes not need to do an LU decomposition for each iterationwhile usual methods require it and this is the highestcomputational cost part Therefore the FD-MRCSI methodis a greatly efficient method

4 Numerical Results

In this section we present some numerical examples basedon the FD-MRCSI algorithm All inversion domains of thesetests are inside a square with side length 119871 In order tomeasure the discrepancy between the exact profile and recon-structed profile the relative error with respect to contrastquantity 120594 is introduced which is given by the followingformula

Err =1003817100381710038171003817120594119899 (r) minus 120594exact (r)

10038171003817100381710038171198631003817100381710038171003817120594exact (r)

1003817100381710038171003817119863

times 100 (35)

where 120594119899is the reconstructed result after 119899 iterations and

120594exact is the exact solution in which we are interestedSince the ldquoinverse crimerdquo phenomenon may happen

the synthetic data are generated by using a finite-differencetime-domain forward method A Ricker wavelet f with adominant frequency 30Hz is employed for generating thesynthetic data However the inversion algorithm is carriedout in frequency domain and hence the datasets generatedby using time-domain forward method are transformed intofrequency domain using a fast Fourier transform (FFT) (see[31]) After that we select some frequencies for the FD-MRCSI algorithm Once we get the synthetic data a randomnoise is added to each frequency synthetic data using thisformula as follows

dnoise119895

= d119895+max119895

(10038161003816100381610038161003816d119895

10038161003816100381610038161003816)120578

2(120585

ReI + 119894120585ImI) (36)

where dnoise119895

and d119895are the field data vectors with noise and

the noiseless data respectively 120585Re and 120585Im are uniformly

distributed random numbers on the interval (minus1 1) I is atwo- dimensional vector whose components are equal to oneand 120578 is the desired fraction of the noise

41 Reconstruction in a Homogeneous Background MediumThe first example is concerned with recovering a squarescatterer from noise observed data dnoise

119895in a homogeneous

background medium and the exact model is given byFigure 1(a) The inversion domain 119863 is set as a square withside length 6 km which does not include PML thickness Forthis square scatterer the second Lame parameter 120583 is equal to20GPa and a Poissonrsquos ratio is 025The square with the sidelength 15 km is located in the inversion domain and its centreis at (33) km the mass density of the scatterer is 18 gcm315 gcm3 outside the scatterer Hence the contrast 120594 for thescatterer is 02 During the whole inversion procedure theangular frequency 120596 is 8Hz

The inversion domain 119863 is divided into 119873 times 119873 uniformgrids using the second-order scheme adding the 119899

1grids of

PML so the size of the total computation domain is (119873 +

21198991) times (119873 + 2119899

1) For this experiment 119873 = 61 and 119899

1=

15 therefore the grid dimension of the total computationdomain is 91times91 Both the sources and receivers are assignedto 16 which are equally distributed on the four sides of theinversion domain and all sources are added in the form of 119875-wave incident excitation To show the denoising performanceof the FD-MRCSI method the numerical inversion resultsare presented when the measurement datasets are added 5random noise Figures 1(b) and 1(c) are the reconstructionprofile of the experiment and discrepancy between the truemodel and recoveringmodel after 1000 iterations Figure 1(d)shows the slices along the depth at horizontal distances of3 km for the true model and inversion model The relativeerror is 1226

In this type of numerical tests of a homogeneous back-ground medium we will test that the FD-MRCSI methodpossesses the ability of reconstructing multiple scattererswith different geometrical information and different phys-ical property parameters For this purpose we employ amodel consisting of two rectangular scatterers as shown inFigure 2(a) The configuration consists of two rectangularobjects which have different dimensions One is 12 km times

12 km and another is 08 km times 1 kmWe also assume that theinversion domain is a square domain with the dimension of6 km times 6 km and has a second Lame coefficient 120583 of 52GPaa Poissonrsquos ratio of 025 and a mass density of 15 gcm3 Thelarge obstacle has a mass density of 27 gcm3 and the small

Journal of Applied Mathematics 7

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1500

1550

1600

1650

1700

1750

1800

1850

1900

(a) True model

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1500

1550

1600

1650

1700

1750

1800

1850

1900

(b) Reconstruction result

3000

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 4000 5000 6000

minus150

minus100

minus50

0

50

100

(c) Discrepancy

x (m)

1500

1550

1600

1650

1700

1750

1800

1850

1900

0 1000 2000 3000 4000 5000 6000

InversionTrue

Den

sity

(kg

m3)

(d) Model slice

Figure 1 Applying the FD-MRCSI method to one abnormity inversion (a) True model (b) Reconstruction result after 1000 inversioniterations (c) Discrepancy between true model and reconstruction result (d) Slices at horizontal distances of 3 km for the true model andreconstruction result

one has a mass density of 21 gcm3 Hence the contrasts forthe two abnormities are 08 and 04 respectively

In the inversion process we discretize the total compu-tation domain into 151 times 151 uniform grids including 15PML grids and 24 sources and 24 receivers are used for thisexampleThese sources and receivers are distributed along theinversion domain six sources or receivers for each side Afterthe measurement data is generated using the same way as theprevious example we select them at frequency 8Hz to drivethe inversion algorithm and 5 noise is added After 1000iterations the reconstruction profile is given in Figure 2(b)Figure 2(c) denotes the discrepancy between the true modeland recovering model after 1000 iterations and Figure 2(d)shows the variation of the data equation term error withrespect to iterations The relative error is 1605

At first glance we should solve the forward modeling formany times for these 1000 inversion iterations Since the con-trast source w and contrast function 120594 are incorporated intothis algorithm whichmakes the forward stiffnessmatrix only

involve the background medium for each given frequencythe LU decomposition is required only once throughout thewhole inversion procedure therefore the forward modelingcomputation cost is very cheap However the price that wehave to pay is the storing of the LU decomposition arrays ofthe stiffness matrix of the background medium

From these numerical results we observe that the FD-MRCSI method can effectively reconstruct the scatterersincluding their locations and shapes using the measurementscattered data and the method is of strong denoising abilityAdditionally Figure 2(d) demonstrates that the FD-MRCSImethod is convergent

42 Reconstruction in an Inhomogeneous BackgroundMedium In this section we examine the performance of theFD-CSI using an inhomogeneous medium as a backgroundmedium The configuration of the background consists of acompressional wave speed of 2500ms a shear wave speed of1443ms and a four-layer distribution of amass density of 18

8 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1400

1600

1800

2000

2200

2400

2600

2800

(a) True model

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1400

1600

1800

2000

2200

2400

2600

2800

(b) Reconstruction result

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

minus800

minus600

minus400

minus200

0

200

400

600

(c) Discrepancy

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

0 200 400 600 800 1000

log(x

)

Iteration

Error of data term

(d) Variation of the data equation error

Figure 2 Applying the FD-CSImethod to two abnormities inversion (a) Truemodel (b) Reconstruction result after 1000 inversion iterations(c) Discrepancy between true model and reconstruction result (d) Variation of the data term error with respect to iterations

16 15 and 18 gcm3 from top to bottom Figure 3(a) showsthe mass density distribution of the background mediumThe inversion model consisting of a circle and an ellipse ispresented as shown in Figure 3(b) The circular abnormitywith a contrast of 08 has a radius of 0550 km which iscentered at (34 36) km For the elliptical scatterer it has asemimajor axis of 07 km and a semiminor axis of 036 kmand its centre is (23 2) kmWe also assume that the inversiondomain is a square with dimension of 119871times119871 In our numericalexperiments 119871 is assigned to 6 km

In the inversion the contrast function 120594 contrast sourcesw and state variables u uinc and usct in the inversion domainare uniformly discretized on a regular two-dimensional gridat intervals of Δ119909 and Δ119911 Both Δ119909 and Δ119911 are equal to005 km in other words the grid size is 121 times 121 Themeasurement domain surface 119878 where 24 receivers and 24sources are distributed is a square around the inversiondomain After the synthetic datasets are generated by usingFFT on the finite-difference time-domain (FD-DT) data atfrequency 10Hz 5 noise is added by using (35) Afterrunning 1000 iterations the reconstruction results are givenin Figure 3(c) The relative errors are 1883

As shown in Figure 3 the FD-MRCSI method can suc-cessfully reconstruct the objects including their locationsand shapes as those present in a homogeneous backgroundmediumThe feature of the inversionmethod results from theflexibility of the finite-difference operator of the governingdifferential equations

43 Reconstruction Improvement Using Multiple FrequenciesIn these examples we use the previous lower frequencyinversion reconstruction results as an initial guess for anotherfrequency inversion which is the so-called sequential multi-frequency inversion approach that one usually employs thistechnique in theGauss-Newton or nonlinear conjugate gradi-ent method This technique applied to improve the inversionprofiles is guaranteed by the following two observations (1)each frequency is independent of FD-MRCSI algorithm (2)the inversion algorithm is locally convergent and hence agood initial guess will result in a better inversion resultsAdditionally multiple frequencies inversion technique canmigrate the nonlinearity of the elastic inverse problem

To drive the inversion algorithm we should use anotherinitialization method which was introduced by Abubakar et

Journal of Applied Mathematics 9

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(a) Background medium

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(b) True model

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(c) Reconstruction result

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

0 200 400 600 800 1000

log(x

)

Iteration

Error of data term

(d) Variation of the data equation error

Figure 3 Applying the FD-CSI method to two abnormities inversion in an inhomogeneous background medium (a) Background medium(b) True model (c) Reconstruction result after 1000 inversion iterations (d) Variation of the data term with respect to iterations

al (see [8]) Firstly use the previous inversion result as 1205940

then solve the forward problemwith 120588 = (1+1205940)120588119887to get the

total field vectors u1198950 finally the initial contrast sources w

1198950

are obtained with the equation w1198950

= 1205940u1198950

We employ the strategy to rebuild the two test examplesgiven in Section 41 and all relative parameters are thesame as those present in Section 41 For the one scatterermodel case we successively select three frequencies (4 6and 8Hz) For the two scatterers model case the sequentialfrequency applied during the multiple frequencies inversionis 5Hz 7Hz and 10Hz Figures 4(a) and 4(b) show thecorresponding reconstruction results The relative errors are813 and 1260 respectively which are smaller than thosein Section 41 Table 2 indicates the reconstruction relativeerrors of the two models using FD-MRCSI and FD-MRCSIwith multiple frequencies technique

From Figure 4 and Table 2 we can observe that theuse of the sequential multifrequency approach can improvereconstructed model to some extent This can be explainedby the fact that a good initial guess can result in a bettercomputation result for a locally convergent method

Table 2 Relative errors using FD-CSI and FD-CSI with multiplefrequencies technique

Model FD-CSI FD-CSI with multifrequencyOne scatterer model 1226 813Two scatterers model 1605 1260

5 Conclusions

In this paper the finite-difference contrast source inversionmethod incorporating a multiplicative regularization factoris successfully employed in seismic full-waveform inversionto reconstruct the mass density The method is very efficientbecause it does not require solving a full forward problem ineach inversion iteration and the highly efficient calculationof the forward modeling makes it have a potential for dealingwith large scale computational inverse problems

From the results of numerical experiments the algorithmcan well recognize the shapes of the scatterers and preciselyidentify their locations no matter the background mediums

10 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1500

1550

1600

1650

1700

1750

1800

1850

1900z

(m)

x (m)

(a) Multifrequency inversion result

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

1400

1600

1800

2000

2200

2400

2600

2800

x (m)

(b) Multifrequency inversion result

Figure 4 Applying sequential multifrequency technique to improve reconstruction results (a) Reconstruction result of one square scatterermodel (b) Reconstruction result of two rectangle scatterers model

are homogeneous or inhomogeneousmediumThe algorithmis very robust when the measurement data is polluted byrandom noise Furthermore the FD-MRCSI using sequen-tial multifrequency technique can successfully improve thereconstruction results which is very hopeful for gettinghigher resolution in exploration seismology Estimating othersubsurface parameters using this method such as Lamecoefficients and velocity is our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Professor van den Berg who givesus many helpful suggestions on both theory and numericalimplementation of the contrast source inversion methodThis work is supported by the National Natural ScienceFoundation of China underGrants no 91230119 no 41474102and no 11301119

References

[1] A Tarantola ldquoA strategy for nonlinear elastic inversion ofseismic reflection datardquo Geophysics vol 51 no 10 pp 1893ndash1903 1986

[2] PMora ldquoNonlinear two-dimensional elastic inversion ofmulti-offset seismic datardquoGeophysics vol 52 no 9 pp 1211ndash1228 1987

[3] R G Pratt and M H Worthington ldquoInverse theory appliedto multi-source cross-hole tomography Part 1 acoustic wave-equation methodrdquo Geophysical Prospecting vol 38 no 3 pp287ndash310 1990

[4] R G Pratt ldquoInverse theory applied to multi-source cross-holetomography Part 2 elastic wave-equationmethodrdquoGeophysicalProspecting vol 38 no 3 pp 311ndash329 1990

[5] B Han H S Fu and Z Li ldquoA homotopy method for theinversion of a two-dimensional acoustic wave equationrdquo InverseProblems in Science and Engineering vol 13 no 4 pp 411ndash4312005

[6] H Ben-Hadj-Ali S Operto and J Virieux ldquoVelocity model-building by 3D frequency-domain full-waveform inversionof wide-aperture seismic datardquo Geophysics vol 73 no 5 ppVE101ndashVE117 2008

[7] W W Symes ldquoMigration velocity analysis and waveforminversionrdquo Geophysical Prospecting vol 56 no 6 pp 765ndash7902008

[8] A Abubakar W Hu T M Habashy and P M van den BergldquoApplication of the finite-difference contrast-source inversionalgorithm to seismic full-waveform datardquo Geophysics vol 74no 6 pp WCC47ndashWCC58 2009

[9] J J Cao Y F Wang J T Zhao and C Yang ldquoA reviewon restoration of seismic wavefields based on regularizationand compressive sensingrdquo Inverse Problems in Science andEngineering vol 19 no 5 pp 679ndash704 2011

[10] A Fichtner P H Bunge and H Igel ldquoThe adjoint methodin seismology I Theoryrdquo Physics of the Earth and PlanetaryInteriors vol 157 no 1-2 pp 86ndash104 2006

[11] P M van den Berg and R E Kleinman ldquoA contrast sourceinversion methodrdquo Inverse Problems vol 13 no 6 pp 1607ndash1620 1997

[12] P M van den Berg A L van Broekhoven and A AbubakarldquoExtended contrast source inversionrdquo Inverse Problems vol 15no 5 pp 1325ndash1344 1999

[13] A Abubakar W Hu P M van den Berg and T M HabashyldquoA finite-difference contrast source inversion methodrdquo InverseProblems vol 24 no 6 Article ID 065004 pp 1ndash17 2008

[14] X Chen ldquoSubspace-based optimization method for solvinginverse-scattering problemsrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 48 no 1 pp 42ndash49 2010

[15] X Chen ldquoSubspace-based optimization method for inversescattering problems with an inhomogeneous backgroundmediumrdquo Inverse Problems vol 26 Article ID 074007 2010

[16] A Abubakar G Pan M Li L Zhang T M Habashy and PM van den Berg ldquoThree-dimensional seismic full-waveform

Journal of Applied Mathematics 11

inversion using the finite-difference contrast source inversionmethodrdquo Geophysical Prospecting vol 59 no 5 pp 874ndash8882011

[17] R F Bloemenkamp A Abubakar and P M van den BergldquoInversion of experimental multi-frequency data using thecontrast source inversion methodrdquo Inverse Problems vol 17 no6 pp 1611ndash1622 2001

[18] A Abubakar and P M van den Berg ldquoThe contrast sourceinversion method for location and shape reconstructionsrdquoInverse Problems vol 18 no 2 pp 495ndash510 2002

[19] L P Song C Yu and Q H Liu ldquoThrough-wall imaging(TWI) by radar 2-D tomographic results and analysesrdquo IEEETransactions on Geoscience and Remote Sensing vol 43 no 12pp 2793ndash2798 2005

[20] M Li A Abubakar and P M van den Berg ldquoApplication of themultiplicative regularized contrast source inversion method on3D experimental Fresnel datardquo Inverse Problems vol 25 no 2Article ID 024006 23 pages 2009

[21] A Zakaria C Gilmore and J LoVetri ldquoFinite-element contrastsource inversionmethod for microwave imagingrdquo Inverse Prob-lems vol 26 no 11 Article ID 115010 21 pages 2010

[22] G Pelekanos A Abubakar and P M van den Berg ldquoContrastsource inversion methods in elastodynamicsrdquo Journal of theAcoustical Society of America vol 114 no 5 pp 2825ndash28342003

[23] D Komatitsch and R Martin ldquoAn unsplit convolutional per-fectly matched layer improved at grazing incidence for theseismic wave equationrdquo Geophysics vol 72 no 5 pp SM155ndashSM167 2007

[24] R Martin D Komatitsch and S D Gedney ldquoA variational for-mulation of a stabilized unsplit convolutional perfectlymatchedlayer for the isotropic or anisotropic seismic wave equationrdquoComputer Modeling in Engineering amp Sciences vol 37 no 3 pp274ndash304 2008

[25] T A Davis Direct Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2006

[26] P M van den Berg A Abubakar and J T Fokkema ldquoMul-tiplicative regularization for contrast profile inversionrdquo RadioScience vol 38 no 2 2003

[27] K R Kelly R WWard S Treitel and R M Alford ldquoSyntheticseismograms a finite difference approachrdquo Geophysics vol 41no 1 pp 2ndash27 1976

[28] R G Pratt ldquoFrequency-domain elastic wave modeling by finitedifferences a tool for crosshole seismic imagingrdquo Geophysicsvol 55 no 5 pp 626ndash632 1990

[29] K W Brodlie and D A H Jacobs ldquoUnconstrained minimiza-tionrdquo in State of the Art in Numerical Analysis pp 229ndash268Academic Press 1997

[30] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 2006

[31] P Duhamel and M Vetterli ldquoFast Fourier transforms a tutorialreview and a state of the artrdquo Signal Processing vol 19 no 4 pp259ndash299 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

However the wave field is numerically modeled in a finitedomain Therefore waves will reflect from the computationedges of the domain and the artificial waves will be recordedIn time domain these reflections can be eliminated by thetime windowing however such reflections cannot be elimi-nated in frequency domain Therefore we should attenuateor suppress the unwanted waves to simulate a modelingin a finite domain and one type of absorbing boundaryconditions should be applied For our forward modelinga stabilized unsplit convolutional perfectly matched layer(PML) is employed (see [24]) and the source term f we addedis in term of p-wave incident excitation

The parameters and wave fields are uniformly discretizedon a regular two-dimensional domain at intervals of Δ119909 andΔ119911 by using the second-order finite-differences described in[27 28] Note that the spatial sampling intervals Δ119909 and Δ119911

must be small enough so that the numerical solutions of thediscretization linear equation system of (4) are sufficientlyaccurate Generally this five-point stencil requires about 10grid points per shear wave wavelength in the lowest velocityzone for an accurate modeling (see Pratt [28]) After theunknowns at each grid point are ordered by row-major rulewe obtain a linear equation system as

A (120596)U (120596) = F (120596) (5)

where A is the finite-difference matrix or stiffness matrixwhich is asymmetric and extremely sparse U is theunknowns vector and F denotes the discrete vector of thesource term To solve the linear equation system (5) wechoose a direct solver method based on the sparse matrixLU decomposition rather than an iterative solver The mainreason we employ a direct solver method is that the stiffnessmatrix A is only dependent on the background mediumand an angular frequency 120596 Hence we can get all solutionsfor all source excitations simultaneously by factoring thefinite-difference matrix A only once However a new matrixdecomposition is required for another frequency

3 Finite-Difference Contrast SourceInversion Algorithm

31 Problem Formulation We consider the two-dimensionalelastic wave inversion problem in a homogeneous or aninhomogeneous backgroundmediumThroughout the paperthe inversion domain or object domain inwhich the scatterersare embedded is denoted by symbol 119863 and the data domain119878 is the surface where the sources and receivers are locatedWe assume that both of the domains are contained withinthe total domain 119863

1called finite-difference computation

domain By this configuration of this inverse problem we canslightly reduce the inversion domain Assume that there are119873119904 sources successively illuminating the inversion domain

and the index for the 119895th source is denoted by subscript 119895For each source 119895 the total field vector u

119895(r 120596) and incident

field vector uinc119895

(r 120596) satisfy the following Navier equationsrespectively

N [u119895(r 120596)]

equiv 1205962120588 (r) u

119895(r 120596 r119904) + nabla [120582 (r) nabla sdot u

119895(r 120596 r119904)]

+ nabla sdot [120583 (r) nablau119895(r 120596 r119904) + [nablau

119895(r 120596 r119904)]

119879

]

= f (r 120596 r119904) r isin 1198631

(6)

N119887[uinc119895

(r 120596)]

equiv 1205962120588119887(r) uinc119895

(r 120596 r119904) + nabla [120582 (r) nabla sdot uinc119895

(r 120596 r119904)]

+ nabla sdot [120583 (r) nablauinc119895

(r 120596 r119904) + [nablauinc119895

(r 120596 r119904)]119879

]

= f (r 120596 r119904) r isin 1198631

(7)

In (7) the parameter involving the subscript 119887 denotesthe relative parameters when the equation is employed todescribe the seismic wave propagation in a known back-ground medium Subtracting (7) from (6) and using therelationship between the total filed u

119895 incident filed uinc

119895 and

the scattered filed usct119895 the scattered fields satisfy

N119887[usct119895

(r 120596)] = minus1205962120588119887(r)w119895(r 120596) r isin 119863

1 (8)

where w119895(r 120596) are the contrast source vectors defined as

w119895(r 120596) = 120594 (r) u

119895(r 120596) r isin 119863 (9)

in which the contrast function 120594(r) is given by

120594 (r) =120588 (r) minus 120588

119887(r)

120588119887(r)

=120588 (r)120588119887(r)

minus 1 r isin 119863 (10)

According to (7) the differential operator N119887with the

known Lame parameters 120582 and 120583 only depends on thebackground mass density 120588

119887(r) The inverse of the operator

N119887is formally given by the linear operator L

119887[sdot] =

Nminus1119887[minus1205962120588119887(r)(sdot)] Therefore the solution vectors of (8) can

be symbolically denoted by the following equation

usct119895

(r 120596) = Nminus1

119887[minus1205962120588119887(r)w119895(r 120596)]

= L119887[w119895(r 120596)] r isin 119863

1

(11)

By using (11) the measured scattered field data vectors atthe receiver r

119877location can be formally presented by the

following equation

d119895(r119877 120596) = M

119878L119887[w119895(r 120596)] r

119877isin 119878 r isin 119863

1 (12)

and the object equation or state equation satisfies

w119895(r 120596) = 120594 (r) uinc

119895(r 120596)

+ 120594 (r)M119863 L119887[w119895(r1015840 120596)] r isin 119863 r1015840 isin 119863

1

(13)

4 Journal of Applied Mathematics

whereM119878 is an operator selecting the scattered fields on themeasurement surface 119878 among the scattered fields in the totalcomputation domain and the operatorM119863 selects the fieldsinside the inversion domain119863

Note that (12) and (13) are the fundamental equationsfor the CSI method which are used to establish the costfunctional From (12) and (13) the data error r119878

119895(r 120596) and the

object (or state) error r119863119895(r 120596) are denoted by the following

equations respectively

r119878119895(r 120596) = d

119895(r119877 120596)

minusM119878L119887[w119895(r 120596)] r

119877isin 119878 r isin 119863

1

(14)

r119863119895(r 120596) = 120594 (r) uinc

119895(r 120596) + 120594 (r)M119863 L

119887[w119895(r1015840 120596)]

minus w119895(r 120596) r isin 119863 r1015840 isin 119863

1

(15)

To establish the objective functional we should firstly givethe specific expressions of the inner product and 119871

2-norm as

⟨u k⟩Ω = intΩ

u (r) sdot k (r)119889r (16)

r2Ω= intΩ

u (r) sdot u (r)119889r (17)

where the overbar denotes the complex conjugate for eachcomponent of a complex vector and symbol sdot represents theproduct of two vectors Ω is the integral domain which canbe either 119878 or119863

For simplicity the explicit dependence of the field quan-tities on r and r1015840 is dropped in the remainder of this paper

32 Finite-Difference Contrast Source Inversion Method Inthe framework of the contrast source inversion method(see [11]) it does not eliminate the contrast source quan-tities w

119895in (12) and (13) On the contrary it alternatively

reconstructs contrast sources w119895and contrast function 120594 by

minimizing a cost functional In order to avoid minimizinga nonquadratic objective function caused by the presence ofcontrast function 120594 in both numerators and denominatorswe borrow the idea from [14] and slightly modify the costfunctional of CSI method Therefore the modified costfunctional to be minimized at the 119899th step is defined as thefollowing form

119865119899(120594w119895) =

sum119895

10038171003817100381710038171003817r119878119895

10038171003817100381710038171003817

2

119878

sum119895

10038171003817100381710038171003817d119878119895

10038171003817100381710038171003817

2

119878

+

sum119895

10038171003817100381710038171003817r119863119895

10038171003817100381710038171003817

2

119863

sum119895

10038171003817100381710038171003817120594119899minus1

uinc119895

10038171003817100381710038171003817

2

119863

= 119865119878(w119895) + 119865119863

119899(120594w119895)

(18)

where 119865119878(w119895) represents the data equation error and

119865119863

119899(120594w119895) represents the object equation error The scattered

field data is obtained by subtracting the simulated back-ground field from the simulated model field From (14)(15) and (18) we note that the background medium does

not change throughout the inversion process therefore thecomputation cost of the inversion of the finite-differenceoperator N

119887is relatively cheap for each frequency which

results from the LU decomposition required only onceThe main idea of the CSI method is to reconstruct

two interlaced sequences w119895119899

and 120594119899 by minimizing the

objective functional (18)Thefirst step of FD-CSImethod is toupdate contrast source vectorsw

119895119899by the conjugate-gradient

methodwith Polak-Ribiere search directionsWhile updatingthe contrast source the contrast function 120594 is assumed to be120594119899minus1

At the 119899th step the contrast sources w119895119899

are updated bythe following formula

w119895119899

= w119895119899minus1

+ 120572120596

119895119899k119895119899 (19)

where 120572120596119895119899

is an update-step size and k119895119899

is the Polak-Ribieresearch directions (see [29]) given by

k1198950

= 0

k119895119899

= 119892120596

119895119899+

sum119896⟨119892120596

119895119899 119892120596

119895119899minus 119892120596

119895119899minus1⟩

sum119896

10038171003817100381710038171003817119892120596

119895119899minus1

10038171003817100381710038171003817

2

119863

k119895119899minus1

(20)

in which 119892120596

119895119899is the gradient of the cost function with respect

to w119895119899

evaluated at the (119899 minus 1)th iteration The details ofthis type of the conjugate-gradient can be found in [30]After updating the gradient 119892120596

119895119899 the update-steps 120572120596

119895119899can be

obtained by minimizing the cost functional 119865119899 that is

120572120596

119895119899= argmin

120572119865119899(120594119899minus1

w119895119899minus1

+ 120572k119895119899) (21)

and the explicit expression of the minimizer 120572120596119895119899

of (21) canbe founded by setting the derivative with respect to 120572 beingequal to zero that is

120572120596

119895119899= (minus⟨119892

120596

119895119899 k119895119899⟩119863)

times (12057811987810038171003817100381710038171003817M119878L119887[k119895119899]10038171003817100381710038171003817

2

119878

+120578119863

119899

10038171003817100381710038171003817k119895119899

minus 120594119899minus1

M119863L119887[k119895119899]10038171003817100381710038171003817

2

119863)

minus1

(22)

where the normalization factors 120578119878 and 120578119863

119899are

120578119878= (sum

119895

10038171003817100381710038171003817d119895

10038171003817100381710038171003817

2

119878)

minus1

120578119863

119899= (sum

119895

10038171003817100381710038171003817120594119899minus1

uinc119895

10038171003817100381710038171003817

2

119863)

minus1

(23)

The second step of the CSI method is to update contrastfunction 120594 Once we update the contrast source vectors w

119895

with conjugate gradient method the contrast function 120594 isupdated by finding theminimizer of the functional119865119863

119899(120594w119895)

We assume the contrast source vectorsw119895are constant during

Journal of Applied Mathematics 5

this process It can be shown that the updating formula hasthe closed form as follows

120594119899=

sum119895Re (w

119895119899sdot u119895119899)

sum119895

10038161003816100381610038161003816u119895119899

10038161003816100381610038161003816

2 (24)

where

u119895119899

= uinc119895119899

+M119863L119887[w119895119899] (25)

In (24) the symbol Re indicates getting the real part of acomplex number The details of finding the minimizer withrespect to 120594 of the objective functional (18) can be found in[11]

To start the FD-CSI algorithm Considering the non-linearity and illposedness of the inverse problem and thelocal convergence of the FD-CSI method a good initial guessis very important to get a better reconstruction result Weinitialize the algorithm by the following way

w1198950

=

10038171003817100381710038171003817Llowast119887M119878lowast [d

119895]10038171003817100381710038171003817

2

119863

10038171003817100381710038171003817M119878 (L

119887[Llowast119887M119878lowast [d

119895]])

10038171003817100381710038171003817

2

119878

Llowast

119887M119878lowast

[d119895]

(26)

and the contrast function is calculated by using (24) and (25)In (26)Llowast

119887is the adjoint operator of the operatorL

119887 which

is defined as

Llowast

119887[sdot] = (N

lowast

119887)minus1

[minus1205962120588119887(r) (sdot)] (27)

where

Nlowast

119887[sdot] equiv 1205962120588 (r) (sdot) + nabla [120582 (r)nabla sdot (sdot)]

+ nabla sdot [120583 (r) nabla (sdot) + [nabla (sdot)]119879]

(28)

and the symbol M119878lowast has the same meaning with Llowast119887 which

maps the field values on the measurement surface intothe total computation domain This initialization method iscalled back-propagation method and it has been proved thatit can provide a good initial guess (see [8])

To improve the quality of reconstruction results themultiplicative regularization technique introduced by vanden Berg et al (see [26]) is applied to this inverse problemwhich has the same effect as the total variational regularizedmethod After incorporating themultiplicative regularizationfactor the cost functional is rewritten as

119862119899(120594w119895) = [119865

119878(w119895) + 119865119863

119899(120594w119895)] 119865119877

119899(120594) (29)

where the regularization cost function119865119877

119899(120594) for the weighted

1198712-norm regularization is defined as

119865119877

119899(120594) = int

119863

1198872

119899(1003816100381610038161003816nabla120594 (r)1003816100381610038161003816

2

+ 1205752

119899) 119889r (30)

where

1198872

119899(r) = 1

119860 (1003816100381610038161003816nabla120594119899minus1 (r)

10038161003816100381610038162

+ 1205752119899)

1205752

119899=

119865119863

119899(120594119899minus1

w119895119899minus1

)

Δ119909Δ119911

(31)

where 119860 is the area of the inversion domain and Δ119909Δ119911 is thecell areaMore details on finite-differencemultiplicative regu-larization contrast source inversion (FD-MRCSI)method canbe found in Appendix A of [8]

It is very similar to minimizing functional (29) usingconjugate-gradient method There is no change while updat-ing the contrast source vectors w

119895 since the regularization

factor 119865119877

119899(120594) is only dependent on contrast function 120594 and

119865119877

119899(120594119899minus1

) equiv 1 Nevertheless the contrast function 120594 isupdated by using the conjugate-gradient method with Polak-Ribiere search directions

Compared with an additive regularization method it isnot necessary to determine the artificial weighting regular-ization parameter because it is automatically calculated ateach iteration As we know that selecting the regularizationparameter is a time-consuming and difficult work especiallythere is no prior information about the measurement datamultiplicative regularizationmethod is very suitable to invertexperimental data as shown in [20]

33 Computational Complexity Analysis The highest com-putational cost part of FD-MRCSI is to calculate theseoperators L

119887and Llowast

119887to estimate the conjugate 119892

119895and

update-steps 120572120596

119895 Hence an efficient method to compute

the action of these operators is important to increase theefficiency of the inversion algorithm Considering the factthat these operators only depend on the backgroundmediumwhich does not change throughout the inversion process for agiven frequency 120596 we choose an efficient direct method thatis the LU decomposition and the decomposition for each ofthe two discretization operatorsL

119887andLlowast

119887is done only once

at the beginning of the inversion process and then reusedat each subsequent step The decomposition process takes119874(11989915) operations where 119899 is the number of the unknowns

of the discrete system (119899 = 2119873 119873 is the number of thegrid cells) (see [25]) The back-substitution needs 119874(119899 log 119899)operations for each source position In the framework ofFD-MRCSI back-substitution is required twice for each ofthe two operators L

119887and Llowast

119887per iteration Hence the

computational complexity of FD-MRCSI for each frequencyis approximately given by

119879 (119899119873iter

119873119904) asymp 119874 (2119899

15) + 119873

iter119874 (2119873

119904119899 log 119899) (32)

where119873iter and119873119904 are the number of inversion iteration and

the number of sources respectivelyFor usual inversion methods (UIM) the objective func-

tional involving the scattered observed data d119895and scattered

synthetic data usct119895

is given in the following formula

119865 (120594) = sum

119895

10038171003817100381710038171003817d119895minus usct119895

(120594)10038171003817100381710038171003817

2

119878+ 120572119877 (120594) (33)

The first term of (33) ensures that the minimizer 120594120572of 119865

will be an optimal parameter model while the second isthe regularized term which guarantees the stabilization ofthe problem and forces the minimizer 120594

120572of (33) to satisfy

certain regularity properties and120572 denotes the regularization

6 Journal of Applied Mathematics

Table 1 Computational complexity for FD-MRCSI and UIM

Method LU decomposition Back-substitution Total costUIM 119873

iter119874(211989915) 119873

iter119874(2119873

S119899 log 119899) 119873

iter119874(2119899

15) + 119874(2119873

S119899 log 119899)

FD-MRCSI 119874(211989915) 119873

iter119874(2119873

S119899 log 119899) 119874(2119899

15) + 119873

iter119874(2119873

S119899 log 119899)

119873iter is the iteration number119873S presents the number of source and 119899 denotes the number of the grid cells

parameter that balances the data error and regularized termIf functional (33) is also minimized by using a nonlinearconjugate gradient method then the computational com-plexity of this algorithm for this inverse problem can beapproximately estimated in analogy to FD-MRCSImethod asfollows

1198791(119899119873

iter 119873119904) asymp 119873

iter119874 (2119899

15) + 119874 [(2119873

119878) 119899 log 119899]

(34)

in which 119874(11989915) represents the LU decomposition compu-

tational cost for forward modeling and 119874(119899 log 119899) denotesthe back-substitution computation for each source duringeach iteration process Table 1 indicates the details of com-putational complexity for FD-MRCSI and usual inversionmethods

From Table 1 we observe that the FD-MRCSI methoddoes not need to do an LU decomposition for each iterationwhile usual methods require it and this is the highestcomputational cost part Therefore the FD-MRCSI methodis a greatly efficient method

4 Numerical Results

In this section we present some numerical examples basedon the FD-MRCSI algorithm All inversion domains of thesetests are inside a square with side length 119871 In order tomeasure the discrepancy between the exact profile and recon-structed profile the relative error with respect to contrastquantity 120594 is introduced which is given by the followingformula

Err =1003817100381710038171003817120594119899 (r) minus 120594exact (r)

10038171003817100381710038171198631003817100381710038171003817120594exact (r)

1003817100381710038171003817119863

times 100 (35)

where 120594119899is the reconstructed result after 119899 iterations and

120594exact is the exact solution in which we are interestedSince the ldquoinverse crimerdquo phenomenon may happen

the synthetic data are generated by using a finite-differencetime-domain forward method A Ricker wavelet f with adominant frequency 30Hz is employed for generating thesynthetic data However the inversion algorithm is carriedout in frequency domain and hence the datasets generatedby using time-domain forward method are transformed intofrequency domain using a fast Fourier transform (FFT) (see[31]) After that we select some frequencies for the FD-MRCSI algorithm Once we get the synthetic data a randomnoise is added to each frequency synthetic data using thisformula as follows

dnoise119895

= d119895+max119895

(10038161003816100381610038161003816d119895

10038161003816100381610038161003816)120578

2(120585

ReI + 119894120585ImI) (36)

where dnoise119895

and d119895are the field data vectors with noise and

the noiseless data respectively 120585Re and 120585Im are uniformly

distributed random numbers on the interval (minus1 1) I is atwo- dimensional vector whose components are equal to oneand 120578 is the desired fraction of the noise

41 Reconstruction in a Homogeneous Background MediumThe first example is concerned with recovering a squarescatterer from noise observed data dnoise

119895in a homogeneous

background medium and the exact model is given byFigure 1(a) The inversion domain 119863 is set as a square withside length 6 km which does not include PML thickness Forthis square scatterer the second Lame parameter 120583 is equal to20GPa and a Poissonrsquos ratio is 025The square with the sidelength 15 km is located in the inversion domain and its centreis at (33) km the mass density of the scatterer is 18 gcm315 gcm3 outside the scatterer Hence the contrast 120594 for thescatterer is 02 During the whole inversion procedure theangular frequency 120596 is 8Hz

The inversion domain 119863 is divided into 119873 times 119873 uniformgrids using the second-order scheme adding the 119899

1grids of

PML so the size of the total computation domain is (119873 +

21198991) times (119873 + 2119899

1) For this experiment 119873 = 61 and 119899

1=

15 therefore the grid dimension of the total computationdomain is 91times91 Both the sources and receivers are assignedto 16 which are equally distributed on the four sides of theinversion domain and all sources are added in the form of 119875-wave incident excitation To show the denoising performanceof the FD-MRCSI method the numerical inversion resultsare presented when the measurement datasets are added 5random noise Figures 1(b) and 1(c) are the reconstructionprofile of the experiment and discrepancy between the truemodel and recoveringmodel after 1000 iterations Figure 1(d)shows the slices along the depth at horizontal distances of3 km for the true model and inversion model The relativeerror is 1226

In this type of numerical tests of a homogeneous back-ground medium we will test that the FD-MRCSI methodpossesses the ability of reconstructing multiple scattererswith different geometrical information and different phys-ical property parameters For this purpose we employ amodel consisting of two rectangular scatterers as shown inFigure 2(a) The configuration consists of two rectangularobjects which have different dimensions One is 12 km times

12 km and another is 08 km times 1 kmWe also assume that theinversion domain is a square domain with the dimension of6 km times 6 km and has a second Lame coefficient 120583 of 52GPaa Poissonrsquos ratio of 025 and a mass density of 15 gcm3 Thelarge obstacle has a mass density of 27 gcm3 and the small

Journal of Applied Mathematics 7

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1500

1550

1600

1650

1700

1750

1800

1850

1900

(a) True model

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1500

1550

1600

1650

1700

1750

1800

1850

1900

(b) Reconstruction result

3000

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 4000 5000 6000

minus150

minus100

minus50

0

50

100

(c) Discrepancy

x (m)

1500

1550

1600

1650

1700

1750

1800

1850

1900

0 1000 2000 3000 4000 5000 6000

InversionTrue

Den

sity

(kg

m3)

(d) Model slice

Figure 1 Applying the FD-MRCSI method to one abnormity inversion (a) True model (b) Reconstruction result after 1000 inversioniterations (c) Discrepancy between true model and reconstruction result (d) Slices at horizontal distances of 3 km for the true model andreconstruction result

one has a mass density of 21 gcm3 Hence the contrasts forthe two abnormities are 08 and 04 respectively

In the inversion process we discretize the total compu-tation domain into 151 times 151 uniform grids including 15PML grids and 24 sources and 24 receivers are used for thisexampleThese sources and receivers are distributed along theinversion domain six sources or receivers for each side Afterthe measurement data is generated using the same way as theprevious example we select them at frequency 8Hz to drivethe inversion algorithm and 5 noise is added After 1000iterations the reconstruction profile is given in Figure 2(b)Figure 2(c) denotes the discrepancy between the true modeland recovering model after 1000 iterations and Figure 2(d)shows the variation of the data equation term error withrespect to iterations The relative error is 1605

At first glance we should solve the forward modeling formany times for these 1000 inversion iterations Since the con-trast source w and contrast function 120594 are incorporated intothis algorithm whichmakes the forward stiffnessmatrix only

involve the background medium for each given frequencythe LU decomposition is required only once throughout thewhole inversion procedure therefore the forward modelingcomputation cost is very cheap However the price that wehave to pay is the storing of the LU decomposition arrays ofthe stiffness matrix of the background medium

From these numerical results we observe that the FD-MRCSI method can effectively reconstruct the scatterersincluding their locations and shapes using the measurementscattered data and the method is of strong denoising abilityAdditionally Figure 2(d) demonstrates that the FD-MRCSImethod is convergent

42 Reconstruction in an Inhomogeneous BackgroundMedium In this section we examine the performance of theFD-CSI using an inhomogeneous medium as a backgroundmedium The configuration of the background consists of acompressional wave speed of 2500ms a shear wave speed of1443ms and a four-layer distribution of amass density of 18

8 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1400

1600

1800

2000

2200

2400

2600

2800

(a) True model

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1400

1600

1800

2000

2200

2400

2600

2800

(b) Reconstruction result

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

minus800

minus600

minus400

minus200

0

200

400

600

(c) Discrepancy

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

0 200 400 600 800 1000

log(x

)

Iteration

Error of data term

(d) Variation of the data equation error

Figure 2 Applying the FD-CSImethod to two abnormities inversion (a) Truemodel (b) Reconstruction result after 1000 inversion iterations(c) Discrepancy between true model and reconstruction result (d) Variation of the data term error with respect to iterations

16 15 and 18 gcm3 from top to bottom Figure 3(a) showsthe mass density distribution of the background mediumThe inversion model consisting of a circle and an ellipse ispresented as shown in Figure 3(b) The circular abnormitywith a contrast of 08 has a radius of 0550 km which iscentered at (34 36) km For the elliptical scatterer it has asemimajor axis of 07 km and a semiminor axis of 036 kmand its centre is (23 2) kmWe also assume that the inversiondomain is a square with dimension of 119871times119871 In our numericalexperiments 119871 is assigned to 6 km

In the inversion the contrast function 120594 contrast sourcesw and state variables u uinc and usct in the inversion domainare uniformly discretized on a regular two-dimensional gridat intervals of Δ119909 and Δ119911 Both Δ119909 and Δ119911 are equal to005 km in other words the grid size is 121 times 121 Themeasurement domain surface 119878 where 24 receivers and 24sources are distributed is a square around the inversiondomain After the synthetic datasets are generated by usingFFT on the finite-difference time-domain (FD-DT) data atfrequency 10Hz 5 noise is added by using (35) Afterrunning 1000 iterations the reconstruction results are givenin Figure 3(c) The relative errors are 1883

As shown in Figure 3 the FD-MRCSI method can suc-cessfully reconstruct the objects including their locationsand shapes as those present in a homogeneous backgroundmediumThe feature of the inversionmethod results from theflexibility of the finite-difference operator of the governingdifferential equations

43 Reconstruction Improvement Using Multiple FrequenciesIn these examples we use the previous lower frequencyinversion reconstruction results as an initial guess for anotherfrequency inversion which is the so-called sequential multi-frequency inversion approach that one usually employs thistechnique in theGauss-Newton or nonlinear conjugate gradi-ent method This technique applied to improve the inversionprofiles is guaranteed by the following two observations (1)each frequency is independent of FD-MRCSI algorithm (2)the inversion algorithm is locally convergent and hence agood initial guess will result in a better inversion resultsAdditionally multiple frequencies inversion technique canmigrate the nonlinearity of the elastic inverse problem

To drive the inversion algorithm we should use anotherinitialization method which was introduced by Abubakar et

Journal of Applied Mathematics 9

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(a) Background medium

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(b) True model

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(c) Reconstruction result

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

0 200 400 600 800 1000

log(x

)

Iteration

Error of data term

(d) Variation of the data equation error

Figure 3 Applying the FD-CSI method to two abnormities inversion in an inhomogeneous background medium (a) Background medium(b) True model (c) Reconstruction result after 1000 inversion iterations (d) Variation of the data term with respect to iterations

al (see [8]) Firstly use the previous inversion result as 1205940

then solve the forward problemwith 120588 = (1+1205940)120588119887to get the

total field vectors u1198950 finally the initial contrast sources w

1198950

are obtained with the equation w1198950

= 1205940u1198950

We employ the strategy to rebuild the two test examplesgiven in Section 41 and all relative parameters are thesame as those present in Section 41 For the one scatterermodel case we successively select three frequencies (4 6and 8Hz) For the two scatterers model case the sequentialfrequency applied during the multiple frequencies inversionis 5Hz 7Hz and 10Hz Figures 4(a) and 4(b) show thecorresponding reconstruction results The relative errors are813 and 1260 respectively which are smaller than thosein Section 41 Table 2 indicates the reconstruction relativeerrors of the two models using FD-MRCSI and FD-MRCSIwith multiple frequencies technique

From Figure 4 and Table 2 we can observe that theuse of the sequential multifrequency approach can improvereconstructed model to some extent This can be explainedby the fact that a good initial guess can result in a bettercomputation result for a locally convergent method

Table 2 Relative errors using FD-CSI and FD-CSI with multiplefrequencies technique

Model FD-CSI FD-CSI with multifrequencyOne scatterer model 1226 813Two scatterers model 1605 1260

5 Conclusions

In this paper the finite-difference contrast source inversionmethod incorporating a multiplicative regularization factoris successfully employed in seismic full-waveform inversionto reconstruct the mass density The method is very efficientbecause it does not require solving a full forward problem ineach inversion iteration and the highly efficient calculationof the forward modeling makes it have a potential for dealingwith large scale computational inverse problems

From the results of numerical experiments the algorithmcan well recognize the shapes of the scatterers and preciselyidentify their locations no matter the background mediums

10 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1500

1550

1600

1650

1700

1750

1800

1850

1900z

(m)

x (m)

(a) Multifrequency inversion result

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

1400

1600

1800

2000

2200

2400

2600

2800

x (m)

(b) Multifrequency inversion result

Figure 4 Applying sequential multifrequency technique to improve reconstruction results (a) Reconstruction result of one square scatterermodel (b) Reconstruction result of two rectangle scatterers model

are homogeneous or inhomogeneousmediumThe algorithmis very robust when the measurement data is polluted byrandom noise Furthermore the FD-MRCSI using sequen-tial multifrequency technique can successfully improve thereconstruction results which is very hopeful for gettinghigher resolution in exploration seismology Estimating othersubsurface parameters using this method such as Lamecoefficients and velocity is our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Professor van den Berg who givesus many helpful suggestions on both theory and numericalimplementation of the contrast source inversion methodThis work is supported by the National Natural ScienceFoundation of China underGrants no 91230119 no 41474102and no 11301119

References

[1] A Tarantola ldquoA strategy for nonlinear elastic inversion ofseismic reflection datardquo Geophysics vol 51 no 10 pp 1893ndash1903 1986

[2] PMora ldquoNonlinear two-dimensional elastic inversion ofmulti-offset seismic datardquoGeophysics vol 52 no 9 pp 1211ndash1228 1987

[3] R G Pratt and M H Worthington ldquoInverse theory appliedto multi-source cross-hole tomography Part 1 acoustic wave-equation methodrdquo Geophysical Prospecting vol 38 no 3 pp287ndash310 1990

[4] R G Pratt ldquoInverse theory applied to multi-source cross-holetomography Part 2 elastic wave-equationmethodrdquoGeophysicalProspecting vol 38 no 3 pp 311ndash329 1990

[5] B Han H S Fu and Z Li ldquoA homotopy method for theinversion of a two-dimensional acoustic wave equationrdquo InverseProblems in Science and Engineering vol 13 no 4 pp 411ndash4312005

[6] H Ben-Hadj-Ali S Operto and J Virieux ldquoVelocity model-building by 3D frequency-domain full-waveform inversionof wide-aperture seismic datardquo Geophysics vol 73 no 5 ppVE101ndashVE117 2008

[7] W W Symes ldquoMigration velocity analysis and waveforminversionrdquo Geophysical Prospecting vol 56 no 6 pp 765ndash7902008

[8] A Abubakar W Hu T M Habashy and P M van den BergldquoApplication of the finite-difference contrast-source inversionalgorithm to seismic full-waveform datardquo Geophysics vol 74no 6 pp WCC47ndashWCC58 2009

[9] J J Cao Y F Wang J T Zhao and C Yang ldquoA reviewon restoration of seismic wavefields based on regularizationand compressive sensingrdquo Inverse Problems in Science andEngineering vol 19 no 5 pp 679ndash704 2011

[10] A Fichtner P H Bunge and H Igel ldquoThe adjoint methodin seismology I Theoryrdquo Physics of the Earth and PlanetaryInteriors vol 157 no 1-2 pp 86ndash104 2006

[11] P M van den Berg and R E Kleinman ldquoA contrast sourceinversion methodrdquo Inverse Problems vol 13 no 6 pp 1607ndash1620 1997

[12] P M van den Berg A L van Broekhoven and A AbubakarldquoExtended contrast source inversionrdquo Inverse Problems vol 15no 5 pp 1325ndash1344 1999

[13] A Abubakar W Hu P M van den Berg and T M HabashyldquoA finite-difference contrast source inversion methodrdquo InverseProblems vol 24 no 6 Article ID 065004 pp 1ndash17 2008

[14] X Chen ldquoSubspace-based optimization method for solvinginverse-scattering problemsrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 48 no 1 pp 42ndash49 2010

[15] X Chen ldquoSubspace-based optimization method for inversescattering problems with an inhomogeneous backgroundmediumrdquo Inverse Problems vol 26 Article ID 074007 2010

[16] A Abubakar G Pan M Li L Zhang T M Habashy and PM van den Berg ldquoThree-dimensional seismic full-waveform

Journal of Applied Mathematics 11

inversion using the finite-difference contrast source inversionmethodrdquo Geophysical Prospecting vol 59 no 5 pp 874ndash8882011

[17] R F Bloemenkamp A Abubakar and P M van den BergldquoInversion of experimental multi-frequency data using thecontrast source inversion methodrdquo Inverse Problems vol 17 no6 pp 1611ndash1622 2001

[18] A Abubakar and P M van den Berg ldquoThe contrast sourceinversion method for location and shape reconstructionsrdquoInverse Problems vol 18 no 2 pp 495ndash510 2002

[19] L P Song C Yu and Q H Liu ldquoThrough-wall imaging(TWI) by radar 2-D tomographic results and analysesrdquo IEEETransactions on Geoscience and Remote Sensing vol 43 no 12pp 2793ndash2798 2005

[20] M Li A Abubakar and P M van den Berg ldquoApplication of themultiplicative regularized contrast source inversion method on3D experimental Fresnel datardquo Inverse Problems vol 25 no 2Article ID 024006 23 pages 2009

[21] A Zakaria C Gilmore and J LoVetri ldquoFinite-element contrastsource inversionmethod for microwave imagingrdquo Inverse Prob-lems vol 26 no 11 Article ID 115010 21 pages 2010

[22] G Pelekanos A Abubakar and P M van den Berg ldquoContrastsource inversion methods in elastodynamicsrdquo Journal of theAcoustical Society of America vol 114 no 5 pp 2825ndash28342003

[23] D Komatitsch and R Martin ldquoAn unsplit convolutional per-fectly matched layer improved at grazing incidence for theseismic wave equationrdquo Geophysics vol 72 no 5 pp SM155ndashSM167 2007

[24] R Martin D Komatitsch and S D Gedney ldquoA variational for-mulation of a stabilized unsplit convolutional perfectlymatchedlayer for the isotropic or anisotropic seismic wave equationrdquoComputer Modeling in Engineering amp Sciences vol 37 no 3 pp274ndash304 2008

[25] T A Davis Direct Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2006

[26] P M van den Berg A Abubakar and J T Fokkema ldquoMul-tiplicative regularization for contrast profile inversionrdquo RadioScience vol 38 no 2 2003

[27] K R Kelly R WWard S Treitel and R M Alford ldquoSyntheticseismograms a finite difference approachrdquo Geophysics vol 41no 1 pp 2ndash27 1976

[28] R G Pratt ldquoFrequency-domain elastic wave modeling by finitedifferences a tool for crosshole seismic imagingrdquo Geophysicsvol 55 no 5 pp 626ndash632 1990

[29] K W Brodlie and D A H Jacobs ldquoUnconstrained minimiza-tionrdquo in State of the Art in Numerical Analysis pp 229ndash268Academic Press 1997

[30] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 2006

[31] P Duhamel and M Vetterli ldquoFast Fourier transforms a tutorialreview and a state of the artrdquo Signal Processing vol 19 no 4 pp259ndash299 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

whereM119878 is an operator selecting the scattered fields on themeasurement surface 119878 among the scattered fields in the totalcomputation domain and the operatorM119863 selects the fieldsinside the inversion domain119863

Note that (12) and (13) are the fundamental equationsfor the CSI method which are used to establish the costfunctional From (12) and (13) the data error r119878

119895(r 120596) and the

object (or state) error r119863119895(r 120596) are denoted by the following

equations respectively

r119878119895(r 120596) = d

119895(r119877 120596)

minusM119878L119887[w119895(r 120596)] r

119877isin 119878 r isin 119863

1

(14)

r119863119895(r 120596) = 120594 (r) uinc

119895(r 120596) + 120594 (r)M119863 L

119887[w119895(r1015840 120596)]

minus w119895(r 120596) r isin 119863 r1015840 isin 119863

1

(15)

To establish the objective functional we should firstly givethe specific expressions of the inner product and 119871

2-norm as

⟨u k⟩Ω = intΩ

u (r) sdot k (r)119889r (16)

r2Ω= intΩ

u (r) sdot u (r)119889r (17)

where the overbar denotes the complex conjugate for eachcomponent of a complex vector and symbol sdot represents theproduct of two vectors Ω is the integral domain which canbe either 119878 or119863

For simplicity the explicit dependence of the field quan-tities on r and r1015840 is dropped in the remainder of this paper

32 Finite-Difference Contrast Source Inversion Method Inthe framework of the contrast source inversion method(see [11]) it does not eliminate the contrast source quan-tities w

119895in (12) and (13) On the contrary it alternatively

reconstructs contrast sources w119895and contrast function 120594 by

minimizing a cost functional In order to avoid minimizinga nonquadratic objective function caused by the presence ofcontrast function 120594 in both numerators and denominatorswe borrow the idea from [14] and slightly modify the costfunctional of CSI method Therefore the modified costfunctional to be minimized at the 119899th step is defined as thefollowing form

119865119899(120594w119895) =

sum119895

10038171003817100381710038171003817r119878119895

10038171003817100381710038171003817

2

119878

sum119895

10038171003817100381710038171003817d119878119895

10038171003817100381710038171003817

2

119878

+

sum119895

10038171003817100381710038171003817r119863119895

10038171003817100381710038171003817

2

119863

sum119895

10038171003817100381710038171003817120594119899minus1

uinc119895

10038171003817100381710038171003817

2

119863

= 119865119878(w119895) + 119865119863

119899(120594w119895)

(18)

where 119865119878(w119895) represents the data equation error and

119865119863

119899(120594w119895) represents the object equation error The scattered

field data is obtained by subtracting the simulated back-ground field from the simulated model field From (14)(15) and (18) we note that the background medium does

not change throughout the inversion process therefore thecomputation cost of the inversion of the finite-differenceoperator N

119887is relatively cheap for each frequency which

results from the LU decomposition required only onceThe main idea of the CSI method is to reconstruct

two interlaced sequences w119895119899

and 120594119899 by minimizing the

objective functional (18)Thefirst step of FD-CSImethod is toupdate contrast source vectorsw

119895119899by the conjugate-gradient

methodwith Polak-Ribiere search directionsWhile updatingthe contrast source the contrast function 120594 is assumed to be120594119899minus1

At the 119899th step the contrast sources w119895119899

are updated bythe following formula

w119895119899

= w119895119899minus1

+ 120572120596

119895119899k119895119899 (19)

where 120572120596119895119899

is an update-step size and k119895119899

is the Polak-Ribieresearch directions (see [29]) given by

k1198950

= 0

k119895119899

= 119892120596

119895119899+

sum119896⟨119892120596

119895119899 119892120596

119895119899minus 119892120596

119895119899minus1⟩

sum119896

10038171003817100381710038171003817119892120596

119895119899minus1

10038171003817100381710038171003817

2

119863

k119895119899minus1

(20)

in which 119892120596

119895119899is the gradient of the cost function with respect

to w119895119899

evaluated at the (119899 minus 1)th iteration The details ofthis type of the conjugate-gradient can be found in [30]After updating the gradient 119892120596

119895119899 the update-steps 120572120596

119895119899can be

obtained by minimizing the cost functional 119865119899 that is

120572120596

119895119899= argmin

120572119865119899(120594119899minus1

w119895119899minus1

+ 120572k119895119899) (21)

and the explicit expression of the minimizer 120572120596119895119899

of (21) canbe founded by setting the derivative with respect to 120572 beingequal to zero that is

120572120596

119895119899= (minus⟨119892

120596

119895119899 k119895119899⟩119863)

times (12057811987810038171003817100381710038171003817M119878L119887[k119895119899]10038171003817100381710038171003817

2

119878

+120578119863

119899

10038171003817100381710038171003817k119895119899

minus 120594119899minus1

M119863L119887[k119895119899]10038171003817100381710038171003817

2

119863)

minus1

(22)

where the normalization factors 120578119878 and 120578119863

119899are

120578119878= (sum

119895

10038171003817100381710038171003817d119895

10038171003817100381710038171003817

2

119878)

minus1

120578119863

119899= (sum

119895

10038171003817100381710038171003817120594119899minus1

uinc119895

10038171003817100381710038171003817

2

119863)

minus1

(23)

The second step of the CSI method is to update contrastfunction 120594 Once we update the contrast source vectors w

119895

with conjugate gradient method the contrast function 120594 isupdated by finding theminimizer of the functional119865119863

119899(120594w119895)

We assume the contrast source vectorsw119895are constant during

Journal of Applied Mathematics 5

this process It can be shown that the updating formula hasthe closed form as follows

120594119899=

sum119895Re (w

119895119899sdot u119895119899)

sum119895

10038161003816100381610038161003816u119895119899

10038161003816100381610038161003816

2 (24)

where

u119895119899

= uinc119895119899

+M119863L119887[w119895119899] (25)

In (24) the symbol Re indicates getting the real part of acomplex number The details of finding the minimizer withrespect to 120594 of the objective functional (18) can be found in[11]

To start the FD-CSI algorithm Considering the non-linearity and illposedness of the inverse problem and thelocal convergence of the FD-CSI method a good initial guessis very important to get a better reconstruction result Weinitialize the algorithm by the following way

w1198950

=

10038171003817100381710038171003817Llowast119887M119878lowast [d

119895]10038171003817100381710038171003817

2

119863

10038171003817100381710038171003817M119878 (L

119887[Llowast119887M119878lowast [d

119895]])

10038171003817100381710038171003817

2

119878

Llowast

119887M119878lowast

[d119895]

(26)

and the contrast function is calculated by using (24) and (25)In (26)Llowast

119887is the adjoint operator of the operatorL

119887 which

is defined as

Llowast

119887[sdot] = (N

lowast

119887)minus1

[minus1205962120588119887(r) (sdot)] (27)

where

Nlowast

119887[sdot] equiv 1205962120588 (r) (sdot) + nabla [120582 (r)nabla sdot (sdot)]

+ nabla sdot [120583 (r) nabla (sdot) + [nabla (sdot)]119879]

(28)

and the symbol M119878lowast has the same meaning with Llowast119887 which

maps the field values on the measurement surface intothe total computation domain This initialization method iscalled back-propagation method and it has been proved thatit can provide a good initial guess (see [8])

To improve the quality of reconstruction results themultiplicative regularization technique introduced by vanden Berg et al (see [26]) is applied to this inverse problemwhich has the same effect as the total variational regularizedmethod After incorporating themultiplicative regularizationfactor the cost functional is rewritten as

119862119899(120594w119895) = [119865

119878(w119895) + 119865119863

119899(120594w119895)] 119865119877

119899(120594) (29)

where the regularization cost function119865119877

119899(120594) for the weighted

1198712-norm regularization is defined as

119865119877

119899(120594) = int

119863

1198872

119899(1003816100381610038161003816nabla120594 (r)1003816100381610038161003816

2

+ 1205752

119899) 119889r (30)

where

1198872

119899(r) = 1

119860 (1003816100381610038161003816nabla120594119899minus1 (r)

10038161003816100381610038162

+ 1205752119899)

1205752

119899=

119865119863

119899(120594119899minus1

w119895119899minus1

)

Δ119909Δ119911

(31)

where 119860 is the area of the inversion domain and Δ119909Δ119911 is thecell areaMore details on finite-differencemultiplicative regu-larization contrast source inversion (FD-MRCSI)method canbe found in Appendix A of [8]

It is very similar to minimizing functional (29) usingconjugate-gradient method There is no change while updat-ing the contrast source vectors w

119895 since the regularization

factor 119865119877

119899(120594) is only dependent on contrast function 120594 and

119865119877

119899(120594119899minus1

) equiv 1 Nevertheless the contrast function 120594 isupdated by using the conjugate-gradient method with Polak-Ribiere search directions

Compared with an additive regularization method it isnot necessary to determine the artificial weighting regular-ization parameter because it is automatically calculated ateach iteration As we know that selecting the regularizationparameter is a time-consuming and difficult work especiallythere is no prior information about the measurement datamultiplicative regularizationmethod is very suitable to invertexperimental data as shown in [20]

33 Computational Complexity Analysis The highest com-putational cost part of FD-MRCSI is to calculate theseoperators L

119887and Llowast

119887to estimate the conjugate 119892

119895and

update-steps 120572120596

119895 Hence an efficient method to compute

the action of these operators is important to increase theefficiency of the inversion algorithm Considering the factthat these operators only depend on the backgroundmediumwhich does not change throughout the inversion process for agiven frequency 120596 we choose an efficient direct method thatis the LU decomposition and the decomposition for each ofthe two discretization operatorsL

119887andLlowast

119887is done only once

at the beginning of the inversion process and then reusedat each subsequent step The decomposition process takes119874(11989915) operations where 119899 is the number of the unknowns

of the discrete system (119899 = 2119873 119873 is the number of thegrid cells) (see [25]) The back-substitution needs 119874(119899 log 119899)operations for each source position In the framework ofFD-MRCSI back-substitution is required twice for each ofthe two operators L

119887and Llowast

119887per iteration Hence the

computational complexity of FD-MRCSI for each frequencyis approximately given by

119879 (119899119873iter

119873119904) asymp 119874 (2119899

15) + 119873

iter119874 (2119873

119904119899 log 119899) (32)

where119873iter and119873119904 are the number of inversion iteration and

the number of sources respectivelyFor usual inversion methods (UIM) the objective func-

tional involving the scattered observed data d119895and scattered

synthetic data usct119895

is given in the following formula

119865 (120594) = sum

119895

10038171003817100381710038171003817d119895minus usct119895

(120594)10038171003817100381710038171003817

2

119878+ 120572119877 (120594) (33)

The first term of (33) ensures that the minimizer 120594120572of 119865

will be an optimal parameter model while the second isthe regularized term which guarantees the stabilization ofthe problem and forces the minimizer 120594

120572of (33) to satisfy

certain regularity properties and120572 denotes the regularization

6 Journal of Applied Mathematics

Table 1 Computational complexity for FD-MRCSI and UIM

Method LU decomposition Back-substitution Total costUIM 119873

iter119874(211989915) 119873

iter119874(2119873

S119899 log 119899) 119873

iter119874(2119899

15) + 119874(2119873

S119899 log 119899)

FD-MRCSI 119874(211989915) 119873

iter119874(2119873

S119899 log 119899) 119874(2119899

15) + 119873

iter119874(2119873

S119899 log 119899)

119873iter is the iteration number119873S presents the number of source and 119899 denotes the number of the grid cells

parameter that balances the data error and regularized termIf functional (33) is also minimized by using a nonlinearconjugate gradient method then the computational com-plexity of this algorithm for this inverse problem can beapproximately estimated in analogy to FD-MRCSImethod asfollows

1198791(119899119873

iter 119873119904) asymp 119873

iter119874 (2119899

15) + 119874 [(2119873

119878) 119899 log 119899]

(34)

in which 119874(11989915) represents the LU decomposition compu-

tational cost for forward modeling and 119874(119899 log 119899) denotesthe back-substitution computation for each source duringeach iteration process Table 1 indicates the details of com-putational complexity for FD-MRCSI and usual inversionmethods

From Table 1 we observe that the FD-MRCSI methoddoes not need to do an LU decomposition for each iterationwhile usual methods require it and this is the highestcomputational cost part Therefore the FD-MRCSI methodis a greatly efficient method

4 Numerical Results

In this section we present some numerical examples basedon the FD-MRCSI algorithm All inversion domains of thesetests are inside a square with side length 119871 In order tomeasure the discrepancy between the exact profile and recon-structed profile the relative error with respect to contrastquantity 120594 is introduced which is given by the followingformula

Err =1003817100381710038171003817120594119899 (r) minus 120594exact (r)

10038171003817100381710038171198631003817100381710038171003817120594exact (r)

1003817100381710038171003817119863

times 100 (35)

where 120594119899is the reconstructed result after 119899 iterations and

120594exact is the exact solution in which we are interestedSince the ldquoinverse crimerdquo phenomenon may happen

the synthetic data are generated by using a finite-differencetime-domain forward method A Ricker wavelet f with adominant frequency 30Hz is employed for generating thesynthetic data However the inversion algorithm is carriedout in frequency domain and hence the datasets generatedby using time-domain forward method are transformed intofrequency domain using a fast Fourier transform (FFT) (see[31]) After that we select some frequencies for the FD-MRCSI algorithm Once we get the synthetic data a randomnoise is added to each frequency synthetic data using thisformula as follows

dnoise119895

= d119895+max119895

(10038161003816100381610038161003816d119895

10038161003816100381610038161003816)120578

2(120585

ReI + 119894120585ImI) (36)

where dnoise119895

and d119895are the field data vectors with noise and

the noiseless data respectively 120585Re and 120585Im are uniformly

distributed random numbers on the interval (minus1 1) I is atwo- dimensional vector whose components are equal to oneand 120578 is the desired fraction of the noise

41 Reconstruction in a Homogeneous Background MediumThe first example is concerned with recovering a squarescatterer from noise observed data dnoise

119895in a homogeneous

background medium and the exact model is given byFigure 1(a) The inversion domain 119863 is set as a square withside length 6 km which does not include PML thickness Forthis square scatterer the second Lame parameter 120583 is equal to20GPa and a Poissonrsquos ratio is 025The square with the sidelength 15 km is located in the inversion domain and its centreis at (33) km the mass density of the scatterer is 18 gcm315 gcm3 outside the scatterer Hence the contrast 120594 for thescatterer is 02 During the whole inversion procedure theangular frequency 120596 is 8Hz

The inversion domain 119863 is divided into 119873 times 119873 uniformgrids using the second-order scheme adding the 119899

1grids of

PML so the size of the total computation domain is (119873 +

21198991) times (119873 + 2119899

1) For this experiment 119873 = 61 and 119899

1=

15 therefore the grid dimension of the total computationdomain is 91times91 Both the sources and receivers are assignedto 16 which are equally distributed on the four sides of theinversion domain and all sources are added in the form of 119875-wave incident excitation To show the denoising performanceof the FD-MRCSI method the numerical inversion resultsare presented when the measurement datasets are added 5random noise Figures 1(b) and 1(c) are the reconstructionprofile of the experiment and discrepancy between the truemodel and recoveringmodel after 1000 iterations Figure 1(d)shows the slices along the depth at horizontal distances of3 km for the true model and inversion model The relativeerror is 1226

In this type of numerical tests of a homogeneous back-ground medium we will test that the FD-MRCSI methodpossesses the ability of reconstructing multiple scattererswith different geometrical information and different phys-ical property parameters For this purpose we employ amodel consisting of two rectangular scatterers as shown inFigure 2(a) The configuration consists of two rectangularobjects which have different dimensions One is 12 km times

12 km and another is 08 km times 1 kmWe also assume that theinversion domain is a square domain with the dimension of6 km times 6 km and has a second Lame coefficient 120583 of 52GPaa Poissonrsquos ratio of 025 and a mass density of 15 gcm3 Thelarge obstacle has a mass density of 27 gcm3 and the small

Journal of Applied Mathematics 7

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1500

1550

1600

1650

1700

1750

1800

1850

1900

(a) True model

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1500

1550

1600

1650

1700

1750

1800

1850

1900

(b) Reconstruction result

3000

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 4000 5000 6000

minus150

minus100

minus50

0

50

100

(c) Discrepancy

x (m)

1500

1550

1600

1650

1700

1750

1800

1850

1900

0 1000 2000 3000 4000 5000 6000

InversionTrue

Den

sity

(kg

m3)

(d) Model slice

Figure 1 Applying the FD-MRCSI method to one abnormity inversion (a) True model (b) Reconstruction result after 1000 inversioniterations (c) Discrepancy between true model and reconstruction result (d) Slices at horizontal distances of 3 km for the true model andreconstruction result

one has a mass density of 21 gcm3 Hence the contrasts forthe two abnormities are 08 and 04 respectively

In the inversion process we discretize the total compu-tation domain into 151 times 151 uniform grids including 15PML grids and 24 sources and 24 receivers are used for thisexampleThese sources and receivers are distributed along theinversion domain six sources or receivers for each side Afterthe measurement data is generated using the same way as theprevious example we select them at frequency 8Hz to drivethe inversion algorithm and 5 noise is added After 1000iterations the reconstruction profile is given in Figure 2(b)Figure 2(c) denotes the discrepancy between the true modeland recovering model after 1000 iterations and Figure 2(d)shows the variation of the data equation term error withrespect to iterations The relative error is 1605

At first glance we should solve the forward modeling formany times for these 1000 inversion iterations Since the con-trast source w and contrast function 120594 are incorporated intothis algorithm whichmakes the forward stiffnessmatrix only

involve the background medium for each given frequencythe LU decomposition is required only once throughout thewhole inversion procedure therefore the forward modelingcomputation cost is very cheap However the price that wehave to pay is the storing of the LU decomposition arrays ofthe stiffness matrix of the background medium

From these numerical results we observe that the FD-MRCSI method can effectively reconstruct the scatterersincluding their locations and shapes using the measurementscattered data and the method is of strong denoising abilityAdditionally Figure 2(d) demonstrates that the FD-MRCSImethod is convergent

42 Reconstruction in an Inhomogeneous BackgroundMedium In this section we examine the performance of theFD-CSI using an inhomogeneous medium as a backgroundmedium The configuration of the background consists of acompressional wave speed of 2500ms a shear wave speed of1443ms and a four-layer distribution of amass density of 18

8 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1400

1600

1800

2000

2200

2400

2600

2800

(a) True model

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1400

1600

1800

2000

2200

2400

2600

2800

(b) Reconstruction result

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

minus800

minus600

minus400

minus200

0

200

400

600

(c) Discrepancy

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

0 200 400 600 800 1000

log(x

)

Iteration

Error of data term

(d) Variation of the data equation error

Figure 2 Applying the FD-CSImethod to two abnormities inversion (a) Truemodel (b) Reconstruction result after 1000 inversion iterations(c) Discrepancy between true model and reconstruction result (d) Variation of the data term error with respect to iterations

16 15 and 18 gcm3 from top to bottom Figure 3(a) showsthe mass density distribution of the background mediumThe inversion model consisting of a circle and an ellipse ispresented as shown in Figure 3(b) The circular abnormitywith a contrast of 08 has a radius of 0550 km which iscentered at (34 36) km For the elliptical scatterer it has asemimajor axis of 07 km and a semiminor axis of 036 kmand its centre is (23 2) kmWe also assume that the inversiondomain is a square with dimension of 119871times119871 In our numericalexperiments 119871 is assigned to 6 km

In the inversion the contrast function 120594 contrast sourcesw and state variables u uinc and usct in the inversion domainare uniformly discretized on a regular two-dimensional gridat intervals of Δ119909 and Δ119911 Both Δ119909 and Δ119911 are equal to005 km in other words the grid size is 121 times 121 Themeasurement domain surface 119878 where 24 receivers and 24sources are distributed is a square around the inversiondomain After the synthetic datasets are generated by usingFFT on the finite-difference time-domain (FD-DT) data atfrequency 10Hz 5 noise is added by using (35) Afterrunning 1000 iterations the reconstruction results are givenin Figure 3(c) The relative errors are 1883

As shown in Figure 3 the FD-MRCSI method can suc-cessfully reconstruct the objects including their locationsand shapes as those present in a homogeneous backgroundmediumThe feature of the inversionmethod results from theflexibility of the finite-difference operator of the governingdifferential equations

43 Reconstruction Improvement Using Multiple FrequenciesIn these examples we use the previous lower frequencyinversion reconstruction results as an initial guess for anotherfrequency inversion which is the so-called sequential multi-frequency inversion approach that one usually employs thistechnique in theGauss-Newton or nonlinear conjugate gradi-ent method This technique applied to improve the inversionprofiles is guaranteed by the following two observations (1)each frequency is independent of FD-MRCSI algorithm (2)the inversion algorithm is locally convergent and hence agood initial guess will result in a better inversion resultsAdditionally multiple frequencies inversion technique canmigrate the nonlinearity of the elastic inverse problem

To drive the inversion algorithm we should use anotherinitialization method which was introduced by Abubakar et

Journal of Applied Mathematics 9

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(a) Background medium

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(b) True model

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(c) Reconstruction result

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

0 200 400 600 800 1000

log(x

)

Iteration

Error of data term

(d) Variation of the data equation error

Figure 3 Applying the FD-CSI method to two abnormities inversion in an inhomogeneous background medium (a) Background medium(b) True model (c) Reconstruction result after 1000 inversion iterations (d) Variation of the data term with respect to iterations

al (see [8]) Firstly use the previous inversion result as 1205940

then solve the forward problemwith 120588 = (1+1205940)120588119887to get the

total field vectors u1198950 finally the initial contrast sources w

1198950

are obtained with the equation w1198950

= 1205940u1198950

We employ the strategy to rebuild the two test examplesgiven in Section 41 and all relative parameters are thesame as those present in Section 41 For the one scatterermodel case we successively select three frequencies (4 6and 8Hz) For the two scatterers model case the sequentialfrequency applied during the multiple frequencies inversionis 5Hz 7Hz and 10Hz Figures 4(a) and 4(b) show thecorresponding reconstruction results The relative errors are813 and 1260 respectively which are smaller than thosein Section 41 Table 2 indicates the reconstruction relativeerrors of the two models using FD-MRCSI and FD-MRCSIwith multiple frequencies technique

From Figure 4 and Table 2 we can observe that theuse of the sequential multifrequency approach can improvereconstructed model to some extent This can be explainedby the fact that a good initial guess can result in a bettercomputation result for a locally convergent method

Table 2 Relative errors using FD-CSI and FD-CSI with multiplefrequencies technique

Model FD-CSI FD-CSI with multifrequencyOne scatterer model 1226 813Two scatterers model 1605 1260

5 Conclusions

In this paper the finite-difference contrast source inversionmethod incorporating a multiplicative regularization factoris successfully employed in seismic full-waveform inversionto reconstruct the mass density The method is very efficientbecause it does not require solving a full forward problem ineach inversion iteration and the highly efficient calculationof the forward modeling makes it have a potential for dealingwith large scale computational inverse problems

From the results of numerical experiments the algorithmcan well recognize the shapes of the scatterers and preciselyidentify their locations no matter the background mediums

10 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1500

1550

1600

1650

1700

1750

1800

1850

1900z

(m)

x (m)

(a) Multifrequency inversion result

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

1400

1600

1800

2000

2200

2400

2600

2800

x (m)

(b) Multifrequency inversion result

Figure 4 Applying sequential multifrequency technique to improve reconstruction results (a) Reconstruction result of one square scatterermodel (b) Reconstruction result of two rectangle scatterers model

are homogeneous or inhomogeneousmediumThe algorithmis very robust when the measurement data is polluted byrandom noise Furthermore the FD-MRCSI using sequen-tial multifrequency technique can successfully improve thereconstruction results which is very hopeful for gettinghigher resolution in exploration seismology Estimating othersubsurface parameters using this method such as Lamecoefficients and velocity is our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Professor van den Berg who givesus many helpful suggestions on both theory and numericalimplementation of the contrast source inversion methodThis work is supported by the National Natural ScienceFoundation of China underGrants no 91230119 no 41474102and no 11301119

References

[1] A Tarantola ldquoA strategy for nonlinear elastic inversion ofseismic reflection datardquo Geophysics vol 51 no 10 pp 1893ndash1903 1986

[2] PMora ldquoNonlinear two-dimensional elastic inversion ofmulti-offset seismic datardquoGeophysics vol 52 no 9 pp 1211ndash1228 1987

[3] R G Pratt and M H Worthington ldquoInverse theory appliedto multi-source cross-hole tomography Part 1 acoustic wave-equation methodrdquo Geophysical Prospecting vol 38 no 3 pp287ndash310 1990

[4] R G Pratt ldquoInverse theory applied to multi-source cross-holetomography Part 2 elastic wave-equationmethodrdquoGeophysicalProspecting vol 38 no 3 pp 311ndash329 1990

[5] B Han H S Fu and Z Li ldquoA homotopy method for theinversion of a two-dimensional acoustic wave equationrdquo InverseProblems in Science and Engineering vol 13 no 4 pp 411ndash4312005

[6] H Ben-Hadj-Ali S Operto and J Virieux ldquoVelocity model-building by 3D frequency-domain full-waveform inversionof wide-aperture seismic datardquo Geophysics vol 73 no 5 ppVE101ndashVE117 2008

[7] W W Symes ldquoMigration velocity analysis and waveforminversionrdquo Geophysical Prospecting vol 56 no 6 pp 765ndash7902008

[8] A Abubakar W Hu T M Habashy and P M van den BergldquoApplication of the finite-difference contrast-source inversionalgorithm to seismic full-waveform datardquo Geophysics vol 74no 6 pp WCC47ndashWCC58 2009

[9] J J Cao Y F Wang J T Zhao and C Yang ldquoA reviewon restoration of seismic wavefields based on regularizationand compressive sensingrdquo Inverse Problems in Science andEngineering vol 19 no 5 pp 679ndash704 2011

[10] A Fichtner P H Bunge and H Igel ldquoThe adjoint methodin seismology I Theoryrdquo Physics of the Earth and PlanetaryInteriors vol 157 no 1-2 pp 86ndash104 2006

[11] P M van den Berg and R E Kleinman ldquoA contrast sourceinversion methodrdquo Inverse Problems vol 13 no 6 pp 1607ndash1620 1997

[12] P M van den Berg A L van Broekhoven and A AbubakarldquoExtended contrast source inversionrdquo Inverse Problems vol 15no 5 pp 1325ndash1344 1999

[13] A Abubakar W Hu P M van den Berg and T M HabashyldquoA finite-difference contrast source inversion methodrdquo InverseProblems vol 24 no 6 Article ID 065004 pp 1ndash17 2008

[14] X Chen ldquoSubspace-based optimization method for solvinginverse-scattering problemsrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 48 no 1 pp 42ndash49 2010

[15] X Chen ldquoSubspace-based optimization method for inversescattering problems with an inhomogeneous backgroundmediumrdquo Inverse Problems vol 26 Article ID 074007 2010

[16] A Abubakar G Pan M Li L Zhang T M Habashy and PM van den Berg ldquoThree-dimensional seismic full-waveform

Journal of Applied Mathematics 11

inversion using the finite-difference contrast source inversionmethodrdquo Geophysical Prospecting vol 59 no 5 pp 874ndash8882011

[17] R F Bloemenkamp A Abubakar and P M van den BergldquoInversion of experimental multi-frequency data using thecontrast source inversion methodrdquo Inverse Problems vol 17 no6 pp 1611ndash1622 2001

[18] A Abubakar and P M van den Berg ldquoThe contrast sourceinversion method for location and shape reconstructionsrdquoInverse Problems vol 18 no 2 pp 495ndash510 2002

[19] L P Song C Yu and Q H Liu ldquoThrough-wall imaging(TWI) by radar 2-D tomographic results and analysesrdquo IEEETransactions on Geoscience and Remote Sensing vol 43 no 12pp 2793ndash2798 2005

[20] M Li A Abubakar and P M van den Berg ldquoApplication of themultiplicative regularized contrast source inversion method on3D experimental Fresnel datardquo Inverse Problems vol 25 no 2Article ID 024006 23 pages 2009

[21] A Zakaria C Gilmore and J LoVetri ldquoFinite-element contrastsource inversionmethod for microwave imagingrdquo Inverse Prob-lems vol 26 no 11 Article ID 115010 21 pages 2010

[22] G Pelekanos A Abubakar and P M van den Berg ldquoContrastsource inversion methods in elastodynamicsrdquo Journal of theAcoustical Society of America vol 114 no 5 pp 2825ndash28342003

[23] D Komatitsch and R Martin ldquoAn unsplit convolutional per-fectly matched layer improved at grazing incidence for theseismic wave equationrdquo Geophysics vol 72 no 5 pp SM155ndashSM167 2007

[24] R Martin D Komatitsch and S D Gedney ldquoA variational for-mulation of a stabilized unsplit convolutional perfectlymatchedlayer for the isotropic or anisotropic seismic wave equationrdquoComputer Modeling in Engineering amp Sciences vol 37 no 3 pp274ndash304 2008

[25] T A Davis Direct Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2006

[26] P M van den Berg A Abubakar and J T Fokkema ldquoMul-tiplicative regularization for contrast profile inversionrdquo RadioScience vol 38 no 2 2003

[27] K R Kelly R WWard S Treitel and R M Alford ldquoSyntheticseismograms a finite difference approachrdquo Geophysics vol 41no 1 pp 2ndash27 1976

[28] R G Pratt ldquoFrequency-domain elastic wave modeling by finitedifferences a tool for crosshole seismic imagingrdquo Geophysicsvol 55 no 5 pp 626ndash632 1990

[29] K W Brodlie and D A H Jacobs ldquoUnconstrained minimiza-tionrdquo in State of the Art in Numerical Analysis pp 229ndash268Academic Press 1997

[30] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 2006

[31] P Duhamel and M Vetterli ldquoFast Fourier transforms a tutorialreview and a state of the artrdquo Signal Processing vol 19 no 4 pp259ndash299 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

this process It can be shown that the updating formula hasthe closed form as follows

120594119899=

sum119895Re (w

119895119899sdot u119895119899)

sum119895

10038161003816100381610038161003816u119895119899

10038161003816100381610038161003816

2 (24)

where

u119895119899

= uinc119895119899

+M119863L119887[w119895119899] (25)

In (24) the symbol Re indicates getting the real part of acomplex number The details of finding the minimizer withrespect to 120594 of the objective functional (18) can be found in[11]

To start the FD-CSI algorithm Considering the non-linearity and illposedness of the inverse problem and thelocal convergence of the FD-CSI method a good initial guessis very important to get a better reconstruction result Weinitialize the algorithm by the following way

w1198950

=

10038171003817100381710038171003817Llowast119887M119878lowast [d

119895]10038171003817100381710038171003817

2

119863

10038171003817100381710038171003817M119878 (L

119887[Llowast119887M119878lowast [d

119895]])

10038171003817100381710038171003817

2

119878

Llowast

119887M119878lowast

[d119895]

(26)

and the contrast function is calculated by using (24) and (25)In (26)Llowast

119887is the adjoint operator of the operatorL

119887 which

is defined as

Llowast

119887[sdot] = (N

lowast

119887)minus1

[minus1205962120588119887(r) (sdot)] (27)

where

Nlowast

119887[sdot] equiv 1205962120588 (r) (sdot) + nabla [120582 (r)nabla sdot (sdot)]

+ nabla sdot [120583 (r) nabla (sdot) + [nabla (sdot)]119879]

(28)

and the symbol M119878lowast has the same meaning with Llowast119887 which

maps the field values on the measurement surface intothe total computation domain This initialization method iscalled back-propagation method and it has been proved thatit can provide a good initial guess (see [8])

To improve the quality of reconstruction results themultiplicative regularization technique introduced by vanden Berg et al (see [26]) is applied to this inverse problemwhich has the same effect as the total variational regularizedmethod After incorporating themultiplicative regularizationfactor the cost functional is rewritten as

119862119899(120594w119895) = [119865

119878(w119895) + 119865119863

119899(120594w119895)] 119865119877

119899(120594) (29)

where the regularization cost function119865119877

119899(120594) for the weighted

1198712-norm regularization is defined as

119865119877

119899(120594) = int

119863

1198872

119899(1003816100381610038161003816nabla120594 (r)1003816100381610038161003816

2

+ 1205752

119899) 119889r (30)

where

1198872

119899(r) = 1

119860 (1003816100381610038161003816nabla120594119899minus1 (r)

10038161003816100381610038162

+ 1205752119899)

1205752

119899=

119865119863

119899(120594119899minus1

w119895119899minus1

)

Δ119909Δ119911

(31)

where 119860 is the area of the inversion domain and Δ119909Δ119911 is thecell areaMore details on finite-differencemultiplicative regu-larization contrast source inversion (FD-MRCSI)method canbe found in Appendix A of [8]

It is very similar to minimizing functional (29) usingconjugate-gradient method There is no change while updat-ing the contrast source vectors w

119895 since the regularization

factor 119865119877

119899(120594) is only dependent on contrast function 120594 and

119865119877

119899(120594119899minus1

) equiv 1 Nevertheless the contrast function 120594 isupdated by using the conjugate-gradient method with Polak-Ribiere search directions

Compared with an additive regularization method it isnot necessary to determine the artificial weighting regular-ization parameter because it is automatically calculated ateach iteration As we know that selecting the regularizationparameter is a time-consuming and difficult work especiallythere is no prior information about the measurement datamultiplicative regularizationmethod is very suitable to invertexperimental data as shown in [20]

33 Computational Complexity Analysis The highest com-putational cost part of FD-MRCSI is to calculate theseoperators L

119887and Llowast

119887to estimate the conjugate 119892

119895and

update-steps 120572120596

119895 Hence an efficient method to compute

the action of these operators is important to increase theefficiency of the inversion algorithm Considering the factthat these operators only depend on the backgroundmediumwhich does not change throughout the inversion process for agiven frequency 120596 we choose an efficient direct method thatis the LU decomposition and the decomposition for each ofthe two discretization operatorsL

119887andLlowast

119887is done only once

at the beginning of the inversion process and then reusedat each subsequent step The decomposition process takes119874(11989915) operations where 119899 is the number of the unknowns

of the discrete system (119899 = 2119873 119873 is the number of thegrid cells) (see [25]) The back-substitution needs 119874(119899 log 119899)operations for each source position In the framework ofFD-MRCSI back-substitution is required twice for each ofthe two operators L

119887and Llowast

119887per iteration Hence the

computational complexity of FD-MRCSI for each frequencyis approximately given by

119879 (119899119873iter

119873119904) asymp 119874 (2119899

15) + 119873

iter119874 (2119873

119904119899 log 119899) (32)

where119873iter and119873119904 are the number of inversion iteration and

the number of sources respectivelyFor usual inversion methods (UIM) the objective func-

tional involving the scattered observed data d119895and scattered

synthetic data usct119895

is given in the following formula

119865 (120594) = sum

119895

10038171003817100381710038171003817d119895minus usct119895

(120594)10038171003817100381710038171003817

2

119878+ 120572119877 (120594) (33)

The first term of (33) ensures that the minimizer 120594120572of 119865

will be an optimal parameter model while the second isthe regularized term which guarantees the stabilization ofthe problem and forces the minimizer 120594

120572of (33) to satisfy

certain regularity properties and120572 denotes the regularization

6 Journal of Applied Mathematics

Table 1 Computational complexity for FD-MRCSI and UIM

Method LU decomposition Back-substitution Total costUIM 119873

iter119874(211989915) 119873

iter119874(2119873

S119899 log 119899) 119873

iter119874(2119899

15) + 119874(2119873

S119899 log 119899)

FD-MRCSI 119874(211989915) 119873

iter119874(2119873

S119899 log 119899) 119874(2119899

15) + 119873

iter119874(2119873

S119899 log 119899)

119873iter is the iteration number119873S presents the number of source and 119899 denotes the number of the grid cells

parameter that balances the data error and regularized termIf functional (33) is also minimized by using a nonlinearconjugate gradient method then the computational com-plexity of this algorithm for this inverse problem can beapproximately estimated in analogy to FD-MRCSImethod asfollows

1198791(119899119873

iter 119873119904) asymp 119873

iter119874 (2119899

15) + 119874 [(2119873

119878) 119899 log 119899]

(34)

in which 119874(11989915) represents the LU decomposition compu-

tational cost for forward modeling and 119874(119899 log 119899) denotesthe back-substitution computation for each source duringeach iteration process Table 1 indicates the details of com-putational complexity for FD-MRCSI and usual inversionmethods

From Table 1 we observe that the FD-MRCSI methoddoes not need to do an LU decomposition for each iterationwhile usual methods require it and this is the highestcomputational cost part Therefore the FD-MRCSI methodis a greatly efficient method

4 Numerical Results

In this section we present some numerical examples basedon the FD-MRCSI algorithm All inversion domains of thesetests are inside a square with side length 119871 In order tomeasure the discrepancy between the exact profile and recon-structed profile the relative error with respect to contrastquantity 120594 is introduced which is given by the followingformula

Err =1003817100381710038171003817120594119899 (r) minus 120594exact (r)

10038171003817100381710038171198631003817100381710038171003817120594exact (r)

1003817100381710038171003817119863

times 100 (35)

where 120594119899is the reconstructed result after 119899 iterations and

120594exact is the exact solution in which we are interestedSince the ldquoinverse crimerdquo phenomenon may happen

the synthetic data are generated by using a finite-differencetime-domain forward method A Ricker wavelet f with adominant frequency 30Hz is employed for generating thesynthetic data However the inversion algorithm is carriedout in frequency domain and hence the datasets generatedby using time-domain forward method are transformed intofrequency domain using a fast Fourier transform (FFT) (see[31]) After that we select some frequencies for the FD-MRCSI algorithm Once we get the synthetic data a randomnoise is added to each frequency synthetic data using thisformula as follows

dnoise119895

= d119895+max119895

(10038161003816100381610038161003816d119895

10038161003816100381610038161003816)120578

2(120585

ReI + 119894120585ImI) (36)

where dnoise119895

and d119895are the field data vectors with noise and

the noiseless data respectively 120585Re and 120585Im are uniformly

distributed random numbers on the interval (minus1 1) I is atwo- dimensional vector whose components are equal to oneand 120578 is the desired fraction of the noise

41 Reconstruction in a Homogeneous Background MediumThe first example is concerned with recovering a squarescatterer from noise observed data dnoise

119895in a homogeneous

background medium and the exact model is given byFigure 1(a) The inversion domain 119863 is set as a square withside length 6 km which does not include PML thickness Forthis square scatterer the second Lame parameter 120583 is equal to20GPa and a Poissonrsquos ratio is 025The square with the sidelength 15 km is located in the inversion domain and its centreis at (33) km the mass density of the scatterer is 18 gcm315 gcm3 outside the scatterer Hence the contrast 120594 for thescatterer is 02 During the whole inversion procedure theangular frequency 120596 is 8Hz

The inversion domain 119863 is divided into 119873 times 119873 uniformgrids using the second-order scheme adding the 119899

1grids of

PML so the size of the total computation domain is (119873 +

21198991) times (119873 + 2119899

1) For this experiment 119873 = 61 and 119899

1=

15 therefore the grid dimension of the total computationdomain is 91times91 Both the sources and receivers are assignedto 16 which are equally distributed on the four sides of theinversion domain and all sources are added in the form of 119875-wave incident excitation To show the denoising performanceof the FD-MRCSI method the numerical inversion resultsare presented when the measurement datasets are added 5random noise Figures 1(b) and 1(c) are the reconstructionprofile of the experiment and discrepancy between the truemodel and recoveringmodel after 1000 iterations Figure 1(d)shows the slices along the depth at horizontal distances of3 km for the true model and inversion model The relativeerror is 1226

In this type of numerical tests of a homogeneous back-ground medium we will test that the FD-MRCSI methodpossesses the ability of reconstructing multiple scattererswith different geometrical information and different phys-ical property parameters For this purpose we employ amodel consisting of two rectangular scatterers as shown inFigure 2(a) The configuration consists of two rectangularobjects which have different dimensions One is 12 km times

12 km and another is 08 km times 1 kmWe also assume that theinversion domain is a square domain with the dimension of6 km times 6 km and has a second Lame coefficient 120583 of 52GPaa Poissonrsquos ratio of 025 and a mass density of 15 gcm3 Thelarge obstacle has a mass density of 27 gcm3 and the small

Journal of Applied Mathematics 7

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1500

1550

1600

1650

1700

1750

1800

1850

1900

(a) True model

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1500

1550

1600

1650

1700

1750

1800

1850

1900

(b) Reconstruction result

3000

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 4000 5000 6000

minus150

minus100

minus50

0

50

100

(c) Discrepancy

x (m)

1500

1550

1600

1650

1700

1750

1800

1850

1900

0 1000 2000 3000 4000 5000 6000

InversionTrue

Den

sity

(kg

m3)

(d) Model slice

Figure 1 Applying the FD-MRCSI method to one abnormity inversion (a) True model (b) Reconstruction result after 1000 inversioniterations (c) Discrepancy between true model and reconstruction result (d) Slices at horizontal distances of 3 km for the true model andreconstruction result

one has a mass density of 21 gcm3 Hence the contrasts forthe two abnormities are 08 and 04 respectively

In the inversion process we discretize the total compu-tation domain into 151 times 151 uniform grids including 15PML grids and 24 sources and 24 receivers are used for thisexampleThese sources and receivers are distributed along theinversion domain six sources or receivers for each side Afterthe measurement data is generated using the same way as theprevious example we select them at frequency 8Hz to drivethe inversion algorithm and 5 noise is added After 1000iterations the reconstruction profile is given in Figure 2(b)Figure 2(c) denotes the discrepancy between the true modeland recovering model after 1000 iterations and Figure 2(d)shows the variation of the data equation term error withrespect to iterations The relative error is 1605

At first glance we should solve the forward modeling formany times for these 1000 inversion iterations Since the con-trast source w and contrast function 120594 are incorporated intothis algorithm whichmakes the forward stiffnessmatrix only

involve the background medium for each given frequencythe LU decomposition is required only once throughout thewhole inversion procedure therefore the forward modelingcomputation cost is very cheap However the price that wehave to pay is the storing of the LU decomposition arrays ofthe stiffness matrix of the background medium

From these numerical results we observe that the FD-MRCSI method can effectively reconstruct the scatterersincluding their locations and shapes using the measurementscattered data and the method is of strong denoising abilityAdditionally Figure 2(d) demonstrates that the FD-MRCSImethod is convergent

42 Reconstruction in an Inhomogeneous BackgroundMedium In this section we examine the performance of theFD-CSI using an inhomogeneous medium as a backgroundmedium The configuration of the background consists of acompressional wave speed of 2500ms a shear wave speed of1443ms and a four-layer distribution of amass density of 18

8 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1400

1600

1800

2000

2200

2400

2600

2800

(a) True model

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1400

1600

1800

2000

2200

2400

2600

2800

(b) Reconstruction result

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

minus800

minus600

minus400

minus200

0

200

400

600

(c) Discrepancy

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

0 200 400 600 800 1000

log(x

)

Iteration

Error of data term

(d) Variation of the data equation error

Figure 2 Applying the FD-CSImethod to two abnormities inversion (a) Truemodel (b) Reconstruction result after 1000 inversion iterations(c) Discrepancy between true model and reconstruction result (d) Variation of the data term error with respect to iterations

16 15 and 18 gcm3 from top to bottom Figure 3(a) showsthe mass density distribution of the background mediumThe inversion model consisting of a circle and an ellipse ispresented as shown in Figure 3(b) The circular abnormitywith a contrast of 08 has a radius of 0550 km which iscentered at (34 36) km For the elliptical scatterer it has asemimajor axis of 07 km and a semiminor axis of 036 kmand its centre is (23 2) kmWe also assume that the inversiondomain is a square with dimension of 119871times119871 In our numericalexperiments 119871 is assigned to 6 km

In the inversion the contrast function 120594 contrast sourcesw and state variables u uinc and usct in the inversion domainare uniformly discretized on a regular two-dimensional gridat intervals of Δ119909 and Δ119911 Both Δ119909 and Δ119911 are equal to005 km in other words the grid size is 121 times 121 Themeasurement domain surface 119878 where 24 receivers and 24sources are distributed is a square around the inversiondomain After the synthetic datasets are generated by usingFFT on the finite-difference time-domain (FD-DT) data atfrequency 10Hz 5 noise is added by using (35) Afterrunning 1000 iterations the reconstruction results are givenin Figure 3(c) The relative errors are 1883

As shown in Figure 3 the FD-MRCSI method can suc-cessfully reconstruct the objects including their locationsand shapes as those present in a homogeneous backgroundmediumThe feature of the inversionmethod results from theflexibility of the finite-difference operator of the governingdifferential equations

43 Reconstruction Improvement Using Multiple FrequenciesIn these examples we use the previous lower frequencyinversion reconstruction results as an initial guess for anotherfrequency inversion which is the so-called sequential multi-frequency inversion approach that one usually employs thistechnique in theGauss-Newton or nonlinear conjugate gradi-ent method This technique applied to improve the inversionprofiles is guaranteed by the following two observations (1)each frequency is independent of FD-MRCSI algorithm (2)the inversion algorithm is locally convergent and hence agood initial guess will result in a better inversion resultsAdditionally multiple frequencies inversion technique canmigrate the nonlinearity of the elastic inverse problem

To drive the inversion algorithm we should use anotherinitialization method which was introduced by Abubakar et

Journal of Applied Mathematics 9

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(a) Background medium

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(b) True model

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(c) Reconstruction result

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

0 200 400 600 800 1000

log(x

)

Iteration

Error of data term

(d) Variation of the data equation error

Figure 3 Applying the FD-CSI method to two abnormities inversion in an inhomogeneous background medium (a) Background medium(b) True model (c) Reconstruction result after 1000 inversion iterations (d) Variation of the data term with respect to iterations

al (see [8]) Firstly use the previous inversion result as 1205940

then solve the forward problemwith 120588 = (1+1205940)120588119887to get the

total field vectors u1198950 finally the initial contrast sources w

1198950

are obtained with the equation w1198950

= 1205940u1198950

We employ the strategy to rebuild the two test examplesgiven in Section 41 and all relative parameters are thesame as those present in Section 41 For the one scatterermodel case we successively select three frequencies (4 6and 8Hz) For the two scatterers model case the sequentialfrequency applied during the multiple frequencies inversionis 5Hz 7Hz and 10Hz Figures 4(a) and 4(b) show thecorresponding reconstruction results The relative errors are813 and 1260 respectively which are smaller than thosein Section 41 Table 2 indicates the reconstruction relativeerrors of the two models using FD-MRCSI and FD-MRCSIwith multiple frequencies technique

From Figure 4 and Table 2 we can observe that theuse of the sequential multifrequency approach can improvereconstructed model to some extent This can be explainedby the fact that a good initial guess can result in a bettercomputation result for a locally convergent method

Table 2 Relative errors using FD-CSI and FD-CSI with multiplefrequencies technique

Model FD-CSI FD-CSI with multifrequencyOne scatterer model 1226 813Two scatterers model 1605 1260

5 Conclusions

In this paper the finite-difference contrast source inversionmethod incorporating a multiplicative regularization factoris successfully employed in seismic full-waveform inversionto reconstruct the mass density The method is very efficientbecause it does not require solving a full forward problem ineach inversion iteration and the highly efficient calculationof the forward modeling makes it have a potential for dealingwith large scale computational inverse problems

From the results of numerical experiments the algorithmcan well recognize the shapes of the scatterers and preciselyidentify their locations no matter the background mediums

10 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1500

1550

1600

1650

1700

1750

1800

1850

1900z

(m)

x (m)

(a) Multifrequency inversion result

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

1400

1600

1800

2000

2200

2400

2600

2800

x (m)

(b) Multifrequency inversion result

Figure 4 Applying sequential multifrequency technique to improve reconstruction results (a) Reconstruction result of one square scatterermodel (b) Reconstruction result of two rectangle scatterers model

are homogeneous or inhomogeneousmediumThe algorithmis very robust when the measurement data is polluted byrandom noise Furthermore the FD-MRCSI using sequen-tial multifrequency technique can successfully improve thereconstruction results which is very hopeful for gettinghigher resolution in exploration seismology Estimating othersubsurface parameters using this method such as Lamecoefficients and velocity is our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Professor van den Berg who givesus many helpful suggestions on both theory and numericalimplementation of the contrast source inversion methodThis work is supported by the National Natural ScienceFoundation of China underGrants no 91230119 no 41474102and no 11301119

References

[1] A Tarantola ldquoA strategy for nonlinear elastic inversion ofseismic reflection datardquo Geophysics vol 51 no 10 pp 1893ndash1903 1986

[2] PMora ldquoNonlinear two-dimensional elastic inversion ofmulti-offset seismic datardquoGeophysics vol 52 no 9 pp 1211ndash1228 1987

[3] R G Pratt and M H Worthington ldquoInverse theory appliedto multi-source cross-hole tomography Part 1 acoustic wave-equation methodrdquo Geophysical Prospecting vol 38 no 3 pp287ndash310 1990

[4] R G Pratt ldquoInverse theory applied to multi-source cross-holetomography Part 2 elastic wave-equationmethodrdquoGeophysicalProspecting vol 38 no 3 pp 311ndash329 1990

[5] B Han H S Fu and Z Li ldquoA homotopy method for theinversion of a two-dimensional acoustic wave equationrdquo InverseProblems in Science and Engineering vol 13 no 4 pp 411ndash4312005

[6] H Ben-Hadj-Ali S Operto and J Virieux ldquoVelocity model-building by 3D frequency-domain full-waveform inversionof wide-aperture seismic datardquo Geophysics vol 73 no 5 ppVE101ndashVE117 2008

[7] W W Symes ldquoMigration velocity analysis and waveforminversionrdquo Geophysical Prospecting vol 56 no 6 pp 765ndash7902008

[8] A Abubakar W Hu T M Habashy and P M van den BergldquoApplication of the finite-difference contrast-source inversionalgorithm to seismic full-waveform datardquo Geophysics vol 74no 6 pp WCC47ndashWCC58 2009

[9] J J Cao Y F Wang J T Zhao and C Yang ldquoA reviewon restoration of seismic wavefields based on regularizationand compressive sensingrdquo Inverse Problems in Science andEngineering vol 19 no 5 pp 679ndash704 2011

[10] A Fichtner P H Bunge and H Igel ldquoThe adjoint methodin seismology I Theoryrdquo Physics of the Earth and PlanetaryInteriors vol 157 no 1-2 pp 86ndash104 2006

[11] P M van den Berg and R E Kleinman ldquoA contrast sourceinversion methodrdquo Inverse Problems vol 13 no 6 pp 1607ndash1620 1997

[12] P M van den Berg A L van Broekhoven and A AbubakarldquoExtended contrast source inversionrdquo Inverse Problems vol 15no 5 pp 1325ndash1344 1999

[13] A Abubakar W Hu P M van den Berg and T M HabashyldquoA finite-difference contrast source inversion methodrdquo InverseProblems vol 24 no 6 Article ID 065004 pp 1ndash17 2008

[14] X Chen ldquoSubspace-based optimization method for solvinginverse-scattering problemsrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 48 no 1 pp 42ndash49 2010

[15] X Chen ldquoSubspace-based optimization method for inversescattering problems with an inhomogeneous backgroundmediumrdquo Inverse Problems vol 26 Article ID 074007 2010

[16] A Abubakar G Pan M Li L Zhang T M Habashy and PM van den Berg ldquoThree-dimensional seismic full-waveform

Journal of Applied Mathematics 11

inversion using the finite-difference contrast source inversionmethodrdquo Geophysical Prospecting vol 59 no 5 pp 874ndash8882011

[17] R F Bloemenkamp A Abubakar and P M van den BergldquoInversion of experimental multi-frequency data using thecontrast source inversion methodrdquo Inverse Problems vol 17 no6 pp 1611ndash1622 2001

[18] A Abubakar and P M van den Berg ldquoThe contrast sourceinversion method for location and shape reconstructionsrdquoInverse Problems vol 18 no 2 pp 495ndash510 2002

[19] L P Song C Yu and Q H Liu ldquoThrough-wall imaging(TWI) by radar 2-D tomographic results and analysesrdquo IEEETransactions on Geoscience and Remote Sensing vol 43 no 12pp 2793ndash2798 2005

[20] M Li A Abubakar and P M van den Berg ldquoApplication of themultiplicative regularized contrast source inversion method on3D experimental Fresnel datardquo Inverse Problems vol 25 no 2Article ID 024006 23 pages 2009

[21] A Zakaria C Gilmore and J LoVetri ldquoFinite-element contrastsource inversionmethod for microwave imagingrdquo Inverse Prob-lems vol 26 no 11 Article ID 115010 21 pages 2010

[22] G Pelekanos A Abubakar and P M van den Berg ldquoContrastsource inversion methods in elastodynamicsrdquo Journal of theAcoustical Society of America vol 114 no 5 pp 2825ndash28342003

[23] D Komatitsch and R Martin ldquoAn unsplit convolutional per-fectly matched layer improved at grazing incidence for theseismic wave equationrdquo Geophysics vol 72 no 5 pp SM155ndashSM167 2007

[24] R Martin D Komatitsch and S D Gedney ldquoA variational for-mulation of a stabilized unsplit convolutional perfectlymatchedlayer for the isotropic or anisotropic seismic wave equationrdquoComputer Modeling in Engineering amp Sciences vol 37 no 3 pp274ndash304 2008

[25] T A Davis Direct Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2006

[26] P M van den Berg A Abubakar and J T Fokkema ldquoMul-tiplicative regularization for contrast profile inversionrdquo RadioScience vol 38 no 2 2003

[27] K R Kelly R WWard S Treitel and R M Alford ldquoSyntheticseismograms a finite difference approachrdquo Geophysics vol 41no 1 pp 2ndash27 1976

[28] R G Pratt ldquoFrequency-domain elastic wave modeling by finitedifferences a tool for crosshole seismic imagingrdquo Geophysicsvol 55 no 5 pp 626ndash632 1990

[29] K W Brodlie and D A H Jacobs ldquoUnconstrained minimiza-tionrdquo in State of the Art in Numerical Analysis pp 229ndash268Academic Press 1997

[30] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 2006

[31] P Duhamel and M Vetterli ldquoFast Fourier transforms a tutorialreview and a state of the artrdquo Signal Processing vol 19 no 4 pp259ndash299 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

Table 1 Computational complexity for FD-MRCSI and UIM

Method LU decomposition Back-substitution Total costUIM 119873

iter119874(211989915) 119873

iter119874(2119873

S119899 log 119899) 119873

iter119874(2119899

15) + 119874(2119873

S119899 log 119899)

FD-MRCSI 119874(211989915) 119873

iter119874(2119873

S119899 log 119899) 119874(2119899

15) + 119873

iter119874(2119873

S119899 log 119899)

119873iter is the iteration number119873S presents the number of source and 119899 denotes the number of the grid cells

parameter that balances the data error and regularized termIf functional (33) is also minimized by using a nonlinearconjugate gradient method then the computational com-plexity of this algorithm for this inverse problem can beapproximately estimated in analogy to FD-MRCSImethod asfollows

1198791(119899119873

iter 119873119904) asymp 119873

iter119874 (2119899

15) + 119874 [(2119873

119878) 119899 log 119899]

(34)

in which 119874(11989915) represents the LU decomposition compu-

tational cost for forward modeling and 119874(119899 log 119899) denotesthe back-substitution computation for each source duringeach iteration process Table 1 indicates the details of com-putational complexity for FD-MRCSI and usual inversionmethods

From Table 1 we observe that the FD-MRCSI methoddoes not need to do an LU decomposition for each iterationwhile usual methods require it and this is the highestcomputational cost part Therefore the FD-MRCSI methodis a greatly efficient method

4 Numerical Results

In this section we present some numerical examples basedon the FD-MRCSI algorithm All inversion domains of thesetests are inside a square with side length 119871 In order tomeasure the discrepancy between the exact profile and recon-structed profile the relative error with respect to contrastquantity 120594 is introduced which is given by the followingformula

Err =1003817100381710038171003817120594119899 (r) minus 120594exact (r)

10038171003817100381710038171198631003817100381710038171003817120594exact (r)

1003817100381710038171003817119863

times 100 (35)

where 120594119899is the reconstructed result after 119899 iterations and

120594exact is the exact solution in which we are interestedSince the ldquoinverse crimerdquo phenomenon may happen

the synthetic data are generated by using a finite-differencetime-domain forward method A Ricker wavelet f with adominant frequency 30Hz is employed for generating thesynthetic data However the inversion algorithm is carriedout in frequency domain and hence the datasets generatedby using time-domain forward method are transformed intofrequency domain using a fast Fourier transform (FFT) (see[31]) After that we select some frequencies for the FD-MRCSI algorithm Once we get the synthetic data a randomnoise is added to each frequency synthetic data using thisformula as follows

dnoise119895

= d119895+max119895

(10038161003816100381610038161003816d119895

10038161003816100381610038161003816)120578

2(120585

ReI + 119894120585ImI) (36)

where dnoise119895

and d119895are the field data vectors with noise and

the noiseless data respectively 120585Re and 120585Im are uniformly

distributed random numbers on the interval (minus1 1) I is atwo- dimensional vector whose components are equal to oneand 120578 is the desired fraction of the noise

41 Reconstruction in a Homogeneous Background MediumThe first example is concerned with recovering a squarescatterer from noise observed data dnoise

119895in a homogeneous

background medium and the exact model is given byFigure 1(a) The inversion domain 119863 is set as a square withside length 6 km which does not include PML thickness Forthis square scatterer the second Lame parameter 120583 is equal to20GPa and a Poissonrsquos ratio is 025The square with the sidelength 15 km is located in the inversion domain and its centreis at (33) km the mass density of the scatterer is 18 gcm315 gcm3 outside the scatterer Hence the contrast 120594 for thescatterer is 02 During the whole inversion procedure theangular frequency 120596 is 8Hz

The inversion domain 119863 is divided into 119873 times 119873 uniformgrids using the second-order scheme adding the 119899

1grids of

PML so the size of the total computation domain is (119873 +

21198991) times (119873 + 2119899

1) For this experiment 119873 = 61 and 119899

1=

15 therefore the grid dimension of the total computationdomain is 91times91 Both the sources and receivers are assignedto 16 which are equally distributed on the four sides of theinversion domain and all sources are added in the form of 119875-wave incident excitation To show the denoising performanceof the FD-MRCSI method the numerical inversion resultsare presented when the measurement datasets are added 5random noise Figures 1(b) and 1(c) are the reconstructionprofile of the experiment and discrepancy between the truemodel and recoveringmodel after 1000 iterations Figure 1(d)shows the slices along the depth at horizontal distances of3 km for the true model and inversion model The relativeerror is 1226

In this type of numerical tests of a homogeneous back-ground medium we will test that the FD-MRCSI methodpossesses the ability of reconstructing multiple scattererswith different geometrical information and different phys-ical property parameters For this purpose we employ amodel consisting of two rectangular scatterers as shown inFigure 2(a) The configuration consists of two rectangularobjects which have different dimensions One is 12 km times

12 km and another is 08 km times 1 kmWe also assume that theinversion domain is a square domain with the dimension of6 km times 6 km and has a second Lame coefficient 120583 of 52GPaa Poissonrsquos ratio of 025 and a mass density of 15 gcm3 Thelarge obstacle has a mass density of 27 gcm3 and the small

Journal of Applied Mathematics 7

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1500

1550

1600

1650

1700

1750

1800

1850

1900

(a) True model

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1500

1550

1600

1650

1700

1750

1800

1850

1900

(b) Reconstruction result

3000

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 4000 5000 6000

minus150

minus100

minus50

0

50

100

(c) Discrepancy

x (m)

1500

1550

1600

1650

1700

1750

1800

1850

1900

0 1000 2000 3000 4000 5000 6000

InversionTrue

Den

sity

(kg

m3)

(d) Model slice

Figure 1 Applying the FD-MRCSI method to one abnormity inversion (a) True model (b) Reconstruction result after 1000 inversioniterations (c) Discrepancy between true model and reconstruction result (d) Slices at horizontal distances of 3 km for the true model andreconstruction result

one has a mass density of 21 gcm3 Hence the contrasts forthe two abnormities are 08 and 04 respectively

In the inversion process we discretize the total compu-tation domain into 151 times 151 uniform grids including 15PML grids and 24 sources and 24 receivers are used for thisexampleThese sources and receivers are distributed along theinversion domain six sources or receivers for each side Afterthe measurement data is generated using the same way as theprevious example we select them at frequency 8Hz to drivethe inversion algorithm and 5 noise is added After 1000iterations the reconstruction profile is given in Figure 2(b)Figure 2(c) denotes the discrepancy between the true modeland recovering model after 1000 iterations and Figure 2(d)shows the variation of the data equation term error withrespect to iterations The relative error is 1605

At first glance we should solve the forward modeling formany times for these 1000 inversion iterations Since the con-trast source w and contrast function 120594 are incorporated intothis algorithm whichmakes the forward stiffnessmatrix only

involve the background medium for each given frequencythe LU decomposition is required only once throughout thewhole inversion procedure therefore the forward modelingcomputation cost is very cheap However the price that wehave to pay is the storing of the LU decomposition arrays ofthe stiffness matrix of the background medium

From these numerical results we observe that the FD-MRCSI method can effectively reconstruct the scatterersincluding their locations and shapes using the measurementscattered data and the method is of strong denoising abilityAdditionally Figure 2(d) demonstrates that the FD-MRCSImethod is convergent

42 Reconstruction in an Inhomogeneous BackgroundMedium In this section we examine the performance of theFD-CSI using an inhomogeneous medium as a backgroundmedium The configuration of the background consists of acompressional wave speed of 2500ms a shear wave speed of1443ms and a four-layer distribution of amass density of 18

8 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1400

1600

1800

2000

2200

2400

2600

2800

(a) True model

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1400

1600

1800

2000

2200

2400

2600

2800

(b) Reconstruction result

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

minus800

minus600

minus400

minus200

0

200

400

600

(c) Discrepancy

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

0 200 400 600 800 1000

log(x

)

Iteration

Error of data term

(d) Variation of the data equation error

Figure 2 Applying the FD-CSImethod to two abnormities inversion (a) Truemodel (b) Reconstruction result after 1000 inversion iterations(c) Discrepancy between true model and reconstruction result (d) Variation of the data term error with respect to iterations

16 15 and 18 gcm3 from top to bottom Figure 3(a) showsthe mass density distribution of the background mediumThe inversion model consisting of a circle and an ellipse ispresented as shown in Figure 3(b) The circular abnormitywith a contrast of 08 has a radius of 0550 km which iscentered at (34 36) km For the elliptical scatterer it has asemimajor axis of 07 km and a semiminor axis of 036 kmand its centre is (23 2) kmWe also assume that the inversiondomain is a square with dimension of 119871times119871 In our numericalexperiments 119871 is assigned to 6 km

In the inversion the contrast function 120594 contrast sourcesw and state variables u uinc and usct in the inversion domainare uniformly discretized on a regular two-dimensional gridat intervals of Δ119909 and Δ119911 Both Δ119909 and Δ119911 are equal to005 km in other words the grid size is 121 times 121 Themeasurement domain surface 119878 where 24 receivers and 24sources are distributed is a square around the inversiondomain After the synthetic datasets are generated by usingFFT on the finite-difference time-domain (FD-DT) data atfrequency 10Hz 5 noise is added by using (35) Afterrunning 1000 iterations the reconstruction results are givenin Figure 3(c) The relative errors are 1883

As shown in Figure 3 the FD-MRCSI method can suc-cessfully reconstruct the objects including their locationsand shapes as those present in a homogeneous backgroundmediumThe feature of the inversionmethod results from theflexibility of the finite-difference operator of the governingdifferential equations

43 Reconstruction Improvement Using Multiple FrequenciesIn these examples we use the previous lower frequencyinversion reconstruction results as an initial guess for anotherfrequency inversion which is the so-called sequential multi-frequency inversion approach that one usually employs thistechnique in theGauss-Newton or nonlinear conjugate gradi-ent method This technique applied to improve the inversionprofiles is guaranteed by the following two observations (1)each frequency is independent of FD-MRCSI algorithm (2)the inversion algorithm is locally convergent and hence agood initial guess will result in a better inversion resultsAdditionally multiple frequencies inversion technique canmigrate the nonlinearity of the elastic inverse problem

To drive the inversion algorithm we should use anotherinitialization method which was introduced by Abubakar et

Journal of Applied Mathematics 9

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(a) Background medium

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(b) True model

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(c) Reconstruction result

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

0 200 400 600 800 1000

log(x

)

Iteration

Error of data term

(d) Variation of the data equation error

Figure 3 Applying the FD-CSI method to two abnormities inversion in an inhomogeneous background medium (a) Background medium(b) True model (c) Reconstruction result after 1000 inversion iterations (d) Variation of the data term with respect to iterations

al (see [8]) Firstly use the previous inversion result as 1205940

then solve the forward problemwith 120588 = (1+1205940)120588119887to get the

total field vectors u1198950 finally the initial contrast sources w

1198950

are obtained with the equation w1198950

= 1205940u1198950

We employ the strategy to rebuild the two test examplesgiven in Section 41 and all relative parameters are thesame as those present in Section 41 For the one scatterermodel case we successively select three frequencies (4 6and 8Hz) For the two scatterers model case the sequentialfrequency applied during the multiple frequencies inversionis 5Hz 7Hz and 10Hz Figures 4(a) and 4(b) show thecorresponding reconstruction results The relative errors are813 and 1260 respectively which are smaller than thosein Section 41 Table 2 indicates the reconstruction relativeerrors of the two models using FD-MRCSI and FD-MRCSIwith multiple frequencies technique

From Figure 4 and Table 2 we can observe that theuse of the sequential multifrequency approach can improvereconstructed model to some extent This can be explainedby the fact that a good initial guess can result in a bettercomputation result for a locally convergent method

Table 2 Relative errors using FD-CSI and FD-CSI with multiplefrequencies technique

Model FD-CSI FD-CSI with multifrequencyOne scatterer model 1226 813Two scatterers model 1605 1260

5 Conclusions

In this paper the finite-difference contrast source inversionmethod incorporating a multiplicative regularization factoris successfully employed in seismic full-waveform inversionto reconstruct the mass density The method is very efficientbecause it does not require solving a full forward problem ineach inversion iteration and the highly efficient calculationof the forward modeling makes it have a potential for dealingwith large scale computational inverse problems

From the results of numerical experiments the algorithmcan well recognize the shapes of the scatterers and preciselyidentify their locations no matter the background mediums

10 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1500

1550

1600

1650

1700

1750

1800

1850

1900z

(m)

x (m)

(a) Multifrequency inversion result

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

1400

1600

1800

2000

2200

2400

2600

2800

x (m)

(b) Multifrequency inversion result

Figure 4 Applying sequential multifrequency technique to improve reconstruction results (a) Reconstruction result of one square scatterermodel (b) Reconstruction result of two rectangle scatterers model

are homogeneous or inhomogeneousmediumThe algorithmis very robust when the measurement data is polluted byrandom noise Furthermore the FD-MRCSI using sequen-tial multifrequency technique can successfully improve thereconstruction results which is very hopeful for gettinghigher resolution in exploration seismology Estimating othersubsurface parameters using this method such as Lamecoefficients and velocity is our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Professor van den Berg who givesus many helpful suggestions on both theory and numericalimplementation of the contrast source inversion methodThis work is supported by the National Natural ScienceFoundation of China underGrants no 91230119 no 41474102and no 11301119

References

[1] A Tarantola ldquoA strategy for nonlinear elastic inversion ofseismic reflection datardquo Geophysics vol 51 no 10 pp 1893ndash1903 1986

[2] PMora ldquoNonlinear two-dimensional elastic inversion ofmulti-offset seismic datardquoGeophysics vol 52 no 9 pp 1211ndash1228 1987

[3] R G Pratt and M H Worthington ldquoInverse theory appliedto multi-source cross-hole tomography Part 1 acoustic wave-equation methodrdquo Geophysical Prospecting vol 38 no 3 pp287ndash310 1990

[4] R G Pratt ldquoInverse theory applied to multi-source cross-holetomography Part 2 elastic wave-equationmethodrdquoGeophysicalProspecting vol 38 no 3 pp 311ndash329 1990

[5] B Han H S Fu and Z Li ldquoA homotopy method for theinversion of a two-dimensional acoustic wave equationrdquo InverseProblems in Science and Engineering vol 13 no 4 pp 411ndash4312005

[6] H Ben-Hadj-Ali S Operto and J Virieux ldquoVelocity model-building by 3D frequency-domain full-waveform inversionof wide-aperture seismic datardquo Geophysics vol 73 no 5 ppVE101ndashVE117 2008

[7] W W Symes ldquoMigration velocity analysis and waveforminversionrdquo Geophysical Prospecting vol 56 no 6 pp 765ndash7902008

[8] A Abubakar W Hu T M Habashy and P M van den BergldquoApplication of the finite-difference contrast-source inversionalgorithm to seismic full-waveform datardquo Geophysics vol 74no 6 pp WCC47ndashWCC58 2009

[9] J J Cao Y F Wang J T Zhao and C Yang ldquoA reviewon restoration of seismic wavefields based on regularizationand compressive sensingrdquo Inverse Problems in Science andEngineering vol 19 no 5 pp 679ndash704 2011

[10] A Fichtner P H Bunge and H Igel ldquoThe adjoint methodin seismology I Theoryrdquo Physics of the Earth and PlanetaryInteriors vol 157 no 1-2 pp 86ndash104 2006

[11] P M van den Berg and R E Kleinman ldquoA contrast sourceinversion methodrdquo Inverse Problems vol 13 no 6 pp 1607ndash1620 1997

[12] P M van den Berg A L van Broekhoven and A AbubakarldquoExtended contrast source inversionrdquo Inverse Problems vol 15no 5 pp 1325ndash1344 1999

[13] A Abubakar W Hu P M van den Berg and T M HabashyldquoA finite-difference contrast source inversion methodrdquo InverseProblems vol 24 no 6 Article ID 065004 pp 1ndash17 2008

[14] X Chen ldquoSubspace-based optimization method for solvinginverse-scattering problemsrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 48 no 1 pp 42ndash49 2010

[15] X Chen ldquoSubspace-based optimization method for inversescattering problems with an inhomogeneous backgroundmediumrdquo Inverse Problems vol 26 Article ID 074007 2010

[16] A Abubakar G Pan M Li L Zhang T M Habashy and PM van den Berg ldquoThree-dimensional seismic full-waveform

Journal of Applied Mathematics 11

inversion using the finite-difference contrast source inversionmethodrdquo Geophysical Prospecting vol 59 no 5 pp 874ndash8882011

[17] R F Bloemenkamp A Abubakar and P M van den BergldquoInversion of experimental multi-frequency data using thecontrast source inversion methodrdquo Inverse Problems vol 17 no6 pp 1611ndash1622 2001

[18] A Abubakar and P M van den Berg ldquoThe contrast sourceinversion method for location and shape reconstructionsrdquoInverse Problems vol 18 no 2 pp 495ndash510 2002

[19] L P Song C Yu and Q H Liu ldquoThrough-wall imaging(TWI) by radar 2-D tomographic results and analysesrdquo IEEETransactions on Geoscience and Remote Sensing vol 43 no 12pp 2793ndash2798 2005

[20] M Li A Abubakar and P M van den Berg ldquoApplication of themultiplicative regularized contrast source inversion method on3D experimental Fresnel datardquo Inverse Problems vol 25 no 2Article ID 024006 23 pages 2009

[21] A Zakaria C Gilmore and J LoVetri ldquoFinite-element contrastsource inversionmethod for microwave imagingrdquo Inverse Prob-lems vol 26 no 11 Article ID 115010 21 pages 2010

[22] G Pelekanos A Abubakar and P M van den Berg ldquoContrastsource inversion methods in elastodynamicsrdquo Journal of theAcoustical Society of America vol 114 no 5 pp 2825ndash28342003

[23] D Komatitsch and R Martin ldquoAn unsplit convolutional per-fectly matched layer improved at grazing incidence for theseismic wave equationrdquo Geophysics vol 72 no 5 pp SM155ndashSM167 2007

[24] R Martin D Komatitsch and S D Gedney ldquoA variational for-mulation of a stabilized unsplit convolutional perfectlymatchedlayer for the isotropic or anisotropic seismic wave equationrdquoComputer Modeling in Engineering amp Sciences vol 37 no 3 pp274ndash304 2008

[25] T A Davis Direct Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2006

[26] P M van den Berg A Abubakar and J T Fokkema ldquoMul-tiplicative regularization for contrast profile inversionrdquo RadioScience vol 38 no 2 2003

[27] K R Kelly R WWard S Treitel and R M Alford ldquoSyntheticseismograms a finite difference approachrdquo Geophysics vol 41no 1 pp 2ndash27 1976

[28] R G Pratt ldquoFrequency-domain elastic wave modeling by finitedifferences a tool for crosshole seismic imagingrdquo Geophysicsvol 55 no 5 pp 626ndash632 1990

[29] K W Brodlie and D A H Jacobs ldquoUnconstrained minimiza-tionrdquo in State of the Art in Numerical Analysis pp 229ndash268Academic Press 1997

[30] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 2006

[31] P Duhamel and M Vetterli ldquoFast Fourier transforms a tutorialreview and a state of the artrdquo Signal Processing vol 19 no 4 pp259ndash299 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1500

1550

1600

1650

1700

1750

1800

1850

1900

(a) True model

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1500

1550

1600

1650

1700

1750

1800

1850

1900

(b) Reconstruction result

3000

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 4000 5000 6000

minus150

minus100

minus50

0

50

100

(c) Discrepancy

x (m)

1500

1550

1600

1650

1700

1750

1800

1850

1900

0 1000 2000 3000 4000 5000 6000

InversionTrue

Den

sity

(kg

m3)

(d) Model slice

Figure 1 Applying the FD-MRCSI method to one abnormity inversion (a) True model (b) Reconstruction result after 1000 inversioniterations (c) Discrepancy between true model and reconstruction result (d) Slices at horizontal distances of 3 km for the true model andreconstruction result

one has a mass density of 21 gcm3 Hence the contrasts forthe two abnormities are 08 and 04 respectively

In the inversion process we discretize the total compu-tation domain into 151 times 151 uniform grids including 15PML grids and 24 sources and 24 receivers are used for thisexampleThese sources and receivers are distributed along theinversion domain six sources or receivers for each side Afterthe measurement data is generated using the same way as theprevious example we select them at frequency 8Hz to drivethe inversion algorithm and 5 noise is added After 1000iterations the reconstruction profile is given in Figure 2(b)Figure 2(c) denotes the discrepancy between the true modeland recovering model after 1000 iterations and Figure 2(d)shows the variation of the data equation term error withrespect to iterations The relative error is 1605

At first glance we should solve the forward modeling formany times for these 1000 inversion iterations Since the con-trast source w and contrast function 120594 are incorporated intothis algorithm whichmakes the forward stiffnessmatrix only

involve the background medium for each given frequencythe LU decomposition is required only once throughout thewhole inversion procedure therefore the forward modelingcomputation cost is very cheap However the price that wehave to pay is the storing of the LU decomposition arrays ofthe stiffness matrix of the background medium

From these numerical results we observe that the FD-MRCSI method can effectively reconstruct the scatterersincluding their locations and shapes using the measurementscattered data and the method is of strong denoising abilityAdditionally Figure 2(d) demonstrates that the FD-MRCSImethod is convergent

42 Reconstruction in an Inhomogeneous BackgroundMedium In this section we examine the performance of theFD-CSI using an inhomogeneous medium as a backgroundmedium The configuration of the background consists of acompressional wave speed of 2500ms a shear wave speed of1443ms and a four-layer distribution of amass density of 18

8 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1400

1600

1800

2000

2200

2400

2600

2800

(a) True model

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1400

1600

1800

2000

2200

2400

2600

2800

(b) Reconstruction result

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

minus800

minus600

minus400

minus200

0

200

400

600

(c) Discrepancy

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

0 200 400 600 800 1000

log(x

)

Iteration

Error of data term

(d) Variation of the data equation error

Figure 2 Applying the FD-CSImethod to two abnormities inversion (a) Truemodel (b) Reconstruction result after 1000 inversion iterations(c) Discrepancy between true model and reconstruction result (d) Variation of the data term error with respect to iterations

16 15 and 18 gcm3 from top to bottom Figure 3(a) showsthe mass density distribution of the background mediumThe inversion model consisting of a circle and an ellipse ispresented as shown in Figure 3(b) The circular abnormitywith a contrast of 08 has a radius of 0550 km which iscentered at (34 36) km For the elliptical scatterer it has asemimajor axis of 07 km and a semiminor axis of 036 kmand its centre is (23 2) kmWe also assume that the inversiondomain is a square with dimension of 119871times119871 In our numericalexperiments 119871 is assigned to 6 km

In the inversion the contrast function 120594 contrast sourcesw and state variables u uinc and usct in the inversion domainare uniformly discretized on a regular two-dimensional gridat intervals of Δ119909 and Δ119911 Both Δ119909 and Δ119911 are equal to005 km in other words the grid size is 121 times 121 Themeasurement domain surface 119878 where 24 receivers and 24sources are distributed is a square around the inversiondomain After the synthetic datasets are generated by usingFFT on the finite-difference time-domain (FD-DT) data atfrequency 10Hz 5 noise is added by using (35) Afterrunning 1000 iterations the reconstruction results are givenin Figure 3(c) The relative errors are 1883

As shown in Figure 3 the FD-MRCSI method can suc-cessfully reconstruct the objects including their locationsand shapes as those present in a homogeneous backgroundmediumThe feature of the inversionmethod results from theflexibility of the finite-difference operator of the governingdifferential equations

43 Reconstruction Improvement Using Multiple FrequenciesIn these examples we use the previous lower frequencyinversion reconstruction results as an initial guess for anotherfrequency inversion which is the so-called sequential multi-frequency inversion approach that one usually employs thistechnique in theGauss-Newton or nonlinear conjugate gradi-ent method This technique applied to improve the inversionprofiles is guaranteed by the following two observations (1)each frequency is independent of FD-MRCSI algorithm (2)the inversion algorithm is locally convergent and hence agood initial guess will result in a better inversion resultsAdditionally multiple frequencies inversion technique canmigrate the nonlinearity of the elastic inverse problem

To drive the inversion algorithm we should use anotherinitialization method which was introduced by Abubakar et

Journal of Applied Mathematics 9

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(a) Background medium

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(b) True model

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(c) Reconstruction result

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

0 200 400 600 800 1000

log(x

)

Iteration

Error of data term

(d) Variation of the data equation error

Figure 3 Applying the FD-CSI method to two abnormities inversion in an inhomogeneous background medium (a) Background medium(b) True model (c) Reconstruction result after 1000 inversion iterations (d) Variation of the data term with respect to iterations

al (see [8]) Firstly use the previous inversion result as 1205940

then solve the forward problemwith 120588 = (1+1205940)120588119887to get the

total field vectors u1198950 finally the initial contrast sources w

1198950

are obtained with the equation w1198950

= 1205940u1198950

We employ the strategy to rebuild the two test examplesgiven in Section 41 and all relative parameters are thesame as those present in Section 41 For the one scatterermodel case we successively select three frequencies (4 6and 8Hz) For the two scatterers model case the sequentialfrequency applied during the multiple frequencies inversionis 5Hz 7Hz and 10Hz Figures 4(a) and 4(b) show thecorresponding reconstruction results The relative errors are813 and 1260 respectively which are smaller than thosein Section 41 Table 2 indicates the reconstruction relativeerrors of the two models using FD-MRCSI and FD-MRCSIwith multiple frequencies technique

From Figure 4 and Table 2 we can observe that theuse of the sequential multifrequency approach can improvereconstructed model to some extent This can be explainedby the fact that a good initial guess can result in a bettercomputation result for a locally convergent method

Table 2 Relative errors using FD-CSI and FD-CSI with multiplefrequencies technique

Model FD-CSI FD-CSI with multifrequencyOne scatterer model 1226 813Two scatterers model 1605 1260

5 Conclusions

In this paper the finite-difference contrast source inversionmethod incorporating a multiplicative regularization factoris successfully employed in seismic full-waveform inversionto reconstruct the mass density The method is very efficientbecause it does not require solving a full forward problem ineach inversion iteration and the highly efficient calculationof the forward modeling makes it have a potential for dealingwith large scale computational inverse problems

From the results of numerical experiments the algorithmcan well recognize the shapes of the scatterers and preciselyidentify their locations no matter the background mediums

10 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1500

1550

1600

1650

1700

1750

1800

1850

1900z

(m)

x (m)

(a) Multifrequency inversion result

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

1400

1600

1800

2000

2200

2400

2600

2800

x (m)

(b) Multifrequency inversion result

Figure 4 Applying sequential multifrequency technique to improve reconstruction results (a) Reconstruction result of one square scatterermodel (b) Reconstruction result of two rectangle scatterers model

are homogeneous or inhomogeneousmediumThe algorithmis very robust when the measurement data is polluted byrandom noise Furthermore the FD-MRCSI using sequen-tial multifrequency technique can successfully improve thereconstruction results which is very hopeful for gettinghigher resolution in exploration seismology Estimating othersubsurface parameters using this method such as Lamecoefficients and velocity is our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Professor van den Berg who givesus many helpful suggestions on both theory and numericalimplementation of the contrast source inversion methodThis work is supported by the National Natural ScienceFoundation of China underGrants no 91230119 no 41474102and no 11301119

References

[1] A Tarantola ldquoA strategy for nonlinear elastic inversion ofseismic reflection datardquo Geophysics vol 51 no 10 pp 1893ndash1903 1986

[2] PMora ldquoNonlinear two-dimensional elastic inversion ofmulti-offset seismic datardquoGeophysics vol 52 no 9 pp 1211ndash1228 1987

[3] R G Pratt and M H Worthington ldquoInverse theory appliedto multi-source cross-hole tomography Part 1 acoustic wave-equation methodrdquo Geophysical Prospecting vol 38 no 3 pp287ndash310 1990

[4] R G Pratt ldquoInverse theory applied to multi-source cross-holetomography Part 2 elastic wave-equationmethodrdquoGeophysicalProspecting vol 38 no 3 pp 311ndash329 1990

[5] B Han H S Fu and Z Li ldquoA homotopy method for theinversion of a two-dimensional acoustic wave equationrdquo InverseProblems in Science and Engineering vol 13 no 4 pp 411ndash4312005

[6] H Ben-Hadj-Ali S Operto and J Virieux ldquoVelocity model-building by 3D frequency-domain full-waveform inversionof wide-aperture seismic datardquo Geophysics vol 73 no 5 ppVE101ndashVE117 2008

[7] W W Symes ldquoMigration velocity analysis and waveforminversionrdquo Geophysical Prospecting vol 56 no 6 pp 765ndash7902008

[8] A Abubakar W Hu T M Habashy and P M van den BergldquoApplication of the finite-difference contrast-source inversionalgorithm to seismic full-waveform datardquo Geophysics vol 74no 6 pp WCC47ndashWCC58 2009

[9] J J Cao Y F Wang J T Zhao and C Yang ldquoA reviewon restoration of seismic wavefields based on regularizationand compressive sensingrdquo Inverse Problems in Science andEngineering vol 19 no 5 pp 679ndash704 2011

[10] A Fichtner P H Bunge and H Igel ldquoThe adjoint methodin seismology I Theoryrdquo Physics of the Earth and PlanetaryInteriors vol 157 no 1-2 pp 86ndash104 2006

[11] P M van den Berg and R E Kleinman ldquoA contrast sourceinversion methodrdquo Inverse Problems vol 13 no 6 pp 1607ndash1620 1997

[12] P M van den Berg A L van Broekhoven and A AbubakarldquoExtended contrast source inversionrdquo Inverse Problems vol 15no 5 pp 1325ndash1344 1999

[13] A Abubakar W Hu P M van den Berg and T M HabashyldquoA finite-difference contrast source inversion methodrdquo InverseProblems vol 24 no 6 Article ID 065004 pp 1ndash17 2008

[14] X Chen ldquoSubspace-based optimization method for solvinginverse-scattering problemsrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 48 no 1 pp 42ndash49 2010

[15] X Chen ldquoSubspace-based optimization method for inversescattering problems with an inhomogeneous backgroundmediumrdquo Inverse Problems vol 26 Article ID 074007 2010

[16] A Abubakar G Pan M Li L Zhang T M Habashy and PM van den Berg ldquoThree-dimensional seismic full-waveform

Journal of Applied Mathematics 11

inversion using the finite-difference contrast source inversionmethodrdquo Geophysical Prospecting vol 59 no 5 pp 874ndash8882011

[17] R F Bloemenkamp A Abubakar and P M van den BergldquoInversion of experimental multi-frequency data using thecontrast source inversion methodrdquo Inverse Problems vol 17 no6 pp 1611ndash1622 2001

[18] A Abubakar and P M van den Berg ldquoThe contrast sourceinversion method for location and shape reconstructionsrdquoInverse Problems vol 18 no 2 pp 495ndash510 2002

[19] L P Song C Yu and Q H Liu ldquoThrough-wall imaging(TWI) by radar 2-D tomographic results and analysesrdquo IEEETransactions on Geoscience and Remote Sensing vol 43 no 12pp 2793ndash2798 2005

[20] M Li A Abubakar and P M van den Berg ldquoApplication of themultiplicative regularized contrast source inversion method on3D experimental Fresnel datardquo Inverse Problems vol 25 no 2Article ID 024006 23 pages 2009

[21] A Zakaria C Gilmore and J LoVetri ldquoFinite-element contrastsource inversionmethod for microwave imagingrdquo Inverse Prob-lems vol 26 no 11 Article ID 115010 21 pages 2010

[22] G Pelekanos A Abubakar and P M van den Berg ldquoContrastsource inversion methods in elastodynamicsrdquo Journal of theAcoustical Society of America vol 114 no 5 pp 2825ndash28342003

[23] D Komatitsch and R Martin ldquoAn unsplit convolutional per-fectly matched layer improved at grazing incidence for theseismic wave equationrdquo Geophysics vol 72 no 5 pp SM155ndashSM167 2007

[24] R Martin D Komatitsch and S D Gedney ldquoA variational for-mulation of a stabilized unsplit convolutional perfectlymatchedlayer for the isotropic or anisotropic seismic wave equationrdquoComputer Modeling in Engineering amp Sciences vol 37 no 3 pp274ndash304 2008

[25] T A Davis Direct Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2006

[26] P M van den Berg A Abubakar and J T Fokkema ldquoMul-tiplicative regularization for contrast profile inversionrdquo RadioScience vol 38 no 2 2003

[27] K R Kelly R WWard S Treitel and R M Alford ldquoSyntheticseismograms a finite difference approachrdquo Geophysics vol 41no 1 pp 2ndash27 1976

[28] R G Pratt ldquoFrequency-domain elastic wave modeling by finitedifferences a tool for crosshole seismic imagingrdquo Geophysicsvol 55 no 5 pp 626ndash632 1990

[29] K W Brodlie and D A H Jacobs ldquoUnconstrained minimiza-tionrdquo in State of the Art in Numerical Analysis pp 229ndash268Academic Press 1997

[30] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 2006

[31] P Duhamel and M Vetterli ldquoFast Fourier transforms a tutorialreview and a state of the artrdquo Signal Processing vol 19 no 4 pp259ndash299 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1400

1600

1800

2000

2200

2400

2600

2800

(a) True model

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1400

1600

1800

2000

2200

2400

2600

2800

(b) Reconstruction result

z (m

)

x (m)

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

minus800

minus600

minus400

minus200

0

200

400

600

(c) Discrepancy

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

0 200 400 600 800 1000

log(x

)

Iteration

Error of data term

(d) Variation of the data equation error

Figure 2 Applying the FD-CSImethod to two abnormities inversion (a) Truemodel (b) Reconstruction result after 1000 inversion iterations(c) Discrepancy between true model and reconstruction result (d) Variation of the data term error with respect to iterations

16 15 and 18 gcm3 from top to bottom Figure 3(a) showsthe mass density distribution of the background mediumThe inversion model consisting of a circle and an ellipse ispresented as shown in Figure 3(b) The circular abnormitywith a contrast of 08 has a radius of 0550 km which iscentered at (34 36) km For the elliptical scatterer it has asemimajor axis of 07 km and a semiminor axis of 036 kmand its centre is (23 2) kmWe also assume that the inversiondomain is a square with dimension of 119871times119871 In our numericalexperiments 119871 is assigned to 6 km

In the inversion the contrast function 120594 contrast sourcesw and state variables u uinc and usct in the inversion domainare uniformly discretized on a regular two-dimensional gridat intervals of Δ119909 and Δ119911 Both Δ119909 and Δ119911 are equal to005 km in other words the grid size is 121 times 121 Themeasurement domain surface 119878 where 24 receivers and 24sources are distributed is a square around the inversiondomain After the synthetic datasets are generated by usingFFT on the finite-difference time-domain (FD-DT) data atfrequency 10Hz 5 noise is added by using (35) Afterrunning 1000 iterations the reconstruction results are givenin Figure 3(c) The relative errors are 1883

As shown in Figure 3 the FD-MRCSI method can suc-cessfully reconstruct the objects including their locationsand shapes as those present in a homogeneous backgroundmediumThe feature of the inversionmethod results from theflexibility of the finite-difference operator of the governingdifferential equations

43 Reconstruction Improvement Using Multiple FrequenciesIn these examples we use the previous lower frequencyinversion reconstruction results as an initial guess for anotherfrequency inversion which is the so-called sequential multi-frequency inversion approach that one usually employs thistechnique in theGauss-Newton or nonlinear conjugate gradi-ent method This technique applied to improve the inversionprofiles is guaranteed by the following two observations (1)each frequency is independent of FD-MRCSI algorithm (2)the inversion algorithm is locally convergent and hence agood initial guess will result in a better inversion resultsAdditionally multiple frequencies inversion technique canmigrate the nonlinearity of the elastic inverse problem

To drive the inversion algorithm we should use anotherinitialization method which was introduced by Abubakar et

Journal of Applied Mathematics 9

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(a) Background medium

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(b) True model

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(c) Reconstruction result

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

0 200 400 600 800 1000

log(x

)

Iteration

Error of data term

(d) Variation of the data equation error

Figure 3 Applying the FD-CSI method to two abnormities inversion in an inhomogeneous background medium (a) Background medium(b) True model (c) Reconstruction result after 1000 inversion iterations (d) Variation of the data term with respect to iterations

al (see [8]) Firstly use the previous inversion result as 1205940

then solve the forward problemwith 120588 = (1+1205940)120588119887to get the

total field vectors u1198950 finally the initial contrast sources w

1198950

are obtained with the equation w1198950

= 1205940u1198950

We employ the strategy to rebuild the two test examplesgiven in Section 41 and all relative parameters are thesame as those present in Section 41 For the one scatterermodel case we successively select three frequencies (4 6and 8Hz) For the two scatterers model case the sequentialfrequency applied during the multiple frequencies inversionis 5Hz 7Hz and 10Hz Figures 4(a) and 4(b) show thecorresponding reconstruction results The relative errors are813 and 1260 respectively which are smaller than thosein Section 41 Table 2 indicates the reconstruction relativeerrors of the two models using FD-MRCSI and FD-MRCSIwith multiple frequencies technique

From Figure 4 and Table 2 we can observe that theuse of the sequential multifrequency approach can improvereconstructed model to some extent This can be explainedby the fact that a good initial guess can result in a bettercomputation result for a locally convergent method

Table 2 Relative errors using FD-CSI and FD-CSI with multiplefrequencies technique

Model FD-CSI FD-CSI with multifrequencyOne scatterer model 1226 813Two scatterers model 1605 1260

5 Conclusions

In this paper the finite-difference contrast source inversionmethod incorporating a multiplicative regularization factoris successfully employed in seismic full-waveform inversionto reconstruct the mass density The method is very efficientbecause it does not require solving a full forward problem ineach inversion iteration and the highly efficient calculationof the forward modeling makes it have a potential for dealingwith large scale computational inverse problems

From the results of numerical experiments the algorithmcan well recognize the shapes of the scatterers and preciselyidentify their locations no matter the background mediums

10 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1500

1550

1600

1650

1700

1750

1800

1850

1900z

(m)

x (m)

(a) Multifrequency inversion result

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

1400

1600

1800

2000

2200

2400

2600

2800

x (m)

(b) Multifrequency inversion result

Figure 4 Applying sequential multifrequency technique to improve reconstruction results (a) Reconstruction result of one square scatterermodel (b) Reconstruction result of two rectangle scatterers model

are homogeneous or inhomogeneousmediumThe algorithmis very robust when the measurement data is polluted byrandom noise Furthermore the FD-MRCSI using sequen-tial multifrequency technique can successfully improve thereconstruction results which is very hopeful for gettinghigher resolution in exploration seismology Estimating othersubsurface parameters using this method such as Lamecoefficients and velocity is our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Professor van den Berg who givesus many helpful suggestions on both theory and numericalimplementation of the contrast source inversion methodThis work is supported by the National Natural ScienceFoundation of China underGrants no 91230119 no 41474102and no 11301119

References

[1] A Tarantola ldquoA strategy for nonlinear elastic inversion ofseismic reflection datardquo Geophysics vol 51 no 10 pp 1893ndash1903 1986

[2] PMora ldquoNonlinear two-dimensional elastic inversion ofmulti-offset seismic datardquoGeophysics vol 52 no 9 pp 1211ndash1228 1987

[3] R G Pratt and M H Worthington ldquoInverse theory appliedto multi-source cross-hole tomography Part 1 acoustic wave-equation methodrdquo Geophysical Prospecting vol 38 no 3 pp287ndash310 1990

[4] R G Pratt ldquoInverse theory applied to multi-source cross-holetomography Part 2 elastic wave-equationmethodrdquoGeophysicalProspecting vol 38 no 3 pp 311ndash329 1990

[5] B Han H S Fu and Z Li ldquoA homotopy method for theinversion of a two-dimensional acoustic wave equationrdquo InverseProblems in Science and Engineering vol 13 no 4 pp 411ndash4312005

[6] H Ben-Hadj-Ali S Operto and J Virieux ldquoVelocity model-building by 3D frequency-domain full-waveform inversionof wide-aperture seismic datardquo Geophysics vol 73 no 5 ppVE101ndashVE117 2008

[7] W W Symes ldquoMigration velocity analysis and waveforminversionrdquo Geophysical Prospecting vol 56 no 6 pp 765ndash7902008

[8] A Abubakar W Hu T M Habashy and P M van den BergldquoApplication of the finite-difference contrast-source inversionalgorithm to seismic full-waveform datardquo Geophysics vol 74no 6 pp WCC47ndashWCC58 2009

[9] J J Cao Y F Wang J T Zhao and C Yang ldquoA reviewon restoration of seismic wavefields based on regularizationand compressive sensingrdquo Inverse Problems in Science andEngineering vol 19 no 5 pp 679ndash704 2011

[10] A Fichtner P H Bunge and H Igel ldquoThe adjoint methodin seismology I Theoryrdquo Physics of the Earth and PlanetaryInteriors vol 157 no 1-2 pp 86ndash104 2006

[11] P M van den Berg and R E Kleinman ldquoA contrast sourceinversion methodrdquo Inverse Problems vol 13 no 6 pp 1607ndash1620 1997

[12] P M van den Berg A L van Broekhoven and A AbubakarldquoExtended contrast source inversionrdquo Inverse Problems vol 15no 5 pp 1325ndash1344 1999

[13] A Abubakar W Hu P M van den Berg and T M HabashyldquoA finite-difference contrast source inversion methodrdquo InverseProblems vol 24 no 6 Article ID 065004 pp 1ndash17 2008

[14] X Chen ldquoSubspace-based optimization method for solvinginverse-scattering problemsrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 48 no 1 pp 42ndash49 2010

[15] X Chen ldquoSubspace-based optimization method for inversescattering problems with an inhomogeneous backgroundmediumrdquo Inverse Problems vol 26 Article ID 074007 2010

[16] A Abubakar G Pan M Li L Zhang T M Habashy and PM van den Berg ldquoThree-dimensional seismic full-waveform

Journal of Applied Mathematics 11

inversion using the finite-difference contrast source inversionmethodrdquo Geophysical Prospecting vol 59 no 5 pp 874ndash8882011

[17] R F Bloemenkamp A Abubakar and P M van den BergldquoInversion of experimental multi-frequency data using thecontrast source inversion methodrdquo Inverse Problems vol 17 no6 pp 1611ndash1622 2001

[18] A Abubakar and P M van den Berg ldquoThe contrast sourceinversion method for location and shape reconstructionsrdquoInverse Problems vol 18 no 2 pp 495ndash510 2002

[19] L P Song C Yu and Q H Liu ldquoThrough-wall imaging(TWI) by radar 2-D tomographic results and analysesrdquo IEEETransactions on Geoscience and Remote Sensing vol 43 no 12pp 2793ndash2798 2005

[20] M Li A Abubakar and P M van den Berg ldquoApplication of themultiplicative regularized contrast source inversion method on3D experimental Fresnel datardquo Inverse Problems vol 25 no 2Article ID 024006 23 pages 2009

[21] A Zakaria C Gilmore and J LoVetri ldquoFinite-element contrastsource inversionmethod for microwave imagingrdquo Inverse Prob-lems vol 26 no 11 Article ID 115010 21 pages 2010

[22] G Pelekanos A Abubakar and P M van den Berg ldquoContrastsource inversion methods in elastodynamicsrdquo Journal of theAcoustical Society of America vol 114 no 5 pp 2825ndash28342003

[23] D Komatitsch and R Martin ldquoAn unsplit convolutional per-fectly matched layer improved at grazing incidence for theseismic wave equationrdquo Geophysics vol 72 no 5 pp SM155ndashSM167 2007

[24] R Martin D Komatitsch and S D Gedney ldquoA variational for-mulation of a stabilized unsplit convolutional perfectlymatchedlayer for the isotropic or anisotropic seismic wave equationrdquoComputer Modeling in Engineering amp Sciences vol 37 no 3 pp274ndash304 2008

[25] T A Davis Direct Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2006

[26] P M van den Berg A Abubakar and J T Fokkema ldquoMul-tiplicative regularization for contrast profile inversionrdquo RadioScience vol 38 no 2 2003

[27] K R Kelly R WWard S Treitel and R M Alford ldquoSyntheticseismograms a finite difference approachrdquo Geophysics vol 41no 1 pp 2ndash27 1976

[28] R G Pratt ldquoFrequency-domain elastic wave modeling by finitedifferences a tool for crosshole seismic imagingrdquo Geophysicsvol 55 no 5 pp 626ndash632 1990

[29] K W Brodlie and D A H Jacobs ldquoUnconstrained minimiza-tionrdquo in State of the Art in Numerical Analysis pp 229ndash268Academic Press 1997

[30] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 2006

[31] P Duhamel and M Vetterli ldquoFast Fourier transforms a tutorialreview and a state of the artrdquo Signal Processing vol 19 no 4 pp259ndash299 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(a) Background medium

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(b) True model

0

1000

2000

3000

4000

5000

6000 0 1000 2000 3000 4000 5000 6000

z (m

)

x (m)

1600

1800

2000

2200

2400

(c) Reconstruction result

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

0 200 400 600 800 1000

log(x

)

Iteration

Error of data term

(d) Variation of the data equation error

Figure 3 Applying the FD-CSI method to two abnormities inversion in an inhomogeneous background medium (a) Background medium(b) True model (c) Reconstruction result after 1000 inversion iterations (d) Variation of the data term with respect to iterations

al (see [8]) Firstly use the previous inversion result as 1205940

then solve the forward problemwith 120588 = (1+1205940)120588119887to get the

total field vectors u1198950 finally the initial contrast sources w

1198950

are obtained with the equation w1198950

= 1205940u1198950

We employ the strategy to rebuild the two test examplesgiven in Section 41 and all relative parameters are thesame as those present in Section 41 For the one scatterermodel case we successively select three frequencies (4 6and 8Hz) For the two scatterers model case the sequentialfrequency applied during the multiple frequencies inversionis 5Hz 7Hz and 10Hz Figures 4(a) and 4(b) show thecorresponding reconstruction results The relative errors are813 and 1260 respectively which are smaller than thosein Section 41 Table 2 indicates the reconstruction relativeerrors of the two models using FD-MRCSI and FD-MRCSIwith multiple frequencies technique

From Figure 4 and Table 2 we can observe that theuse of the sequential multifrequency approach can improvereconstructed model to some extent This can be explainedby the fact that a good initial guess can result in a bettercomputation result for a locally convergent method

Table 2 Relative errors using FD-CSI and FD-CSI with multiplefrequencies technique

Model FD-CSI FD-CSI with multifrequencyOne scatterer model 1226 813Two scatterers model 1605 1260

5 Conclusions

In this paper the finite-difference contrast source inversionmethod incorporating a multiplicative regularization factoris successfully employed in seismic full-waveform inversionto reconstruct the mass density The method is very efficientbecause it does not require solving a full forward problem ineach inversion iteration and the highly efficient calculationof the forward modeling makes it have a potential for dealingwith large scale computational inverse problems

From the results of numerical experiments the algorithmcan well recognize the shapes of the scatterers and preciselyidentify their locations no matter the background mediums

10 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1500

1550

1600

1650

1700

1750

1800

1850

1900z

(m)

x (m)

(a) Multifrequency inversion result

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

1400

1600

1800

2000

2200

2400

2600

2800

x (m)

(b) Multifrequency inversion result

Figure 4 Applying sequential multifrequency technique to improve reconstruction results (a) Reconstruction result of one square scatterermodel (b) Reconstruction result of two rectangle scatterers model

are homogeneous or inhomogeneousmediumThe algorithmis very robust when the measurement data is polluted byrandom noise Furthermore the FD-MRCSI using sequen-tial multifrequency technique can successfully improve thereconstruction results which is very hopeful for gettinghigher resolution in exploration seismology Estimating othersubsurface parameters using this method such as Lamecoefficients and velocity is our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Professor van den Berg who givesus many helpful suggestions on both theory and numericalimplementation of the contrast source inversion methodThis work is supported by the National Natural ScienceFoundation of China underGrants no 91230119 no 41474102and no 11301119

References

[1] A Tarantola ldquoA strategy for nonlinear elastic inversion ofseismic reflection datardquo Geophysics vol 51 no 10 pp 1893ndash1903 1986

[2] PMora ldquoNonlinear two-dimensional elastic inversion ofmulti-offset seismic datardquoGeophysics vol 52 no 9 pp 1211ndash1228 1987

[3] R G Pratt and M H Worthington ldquoInverse theory appliedto multi-source cross-hole tomography Part 1 acoustic wave-equation methodrdquo Geophysical Prospecting vol 38 no 3 pp287ndash310 1990

[4] R G Pratt ldquoInverse theory applied to multi-source cross-holetomography Part 2 elastic wave-equationmethodrdquoGeophysicalProspecting vol 38 no 3 pp 311ndash329 1990

[5] B Han H S Fu and Z Li ldquoA homotopy method for theinversion of a two-dimensional acoustic wave equationrdquo InverseProblems in Science and Engineering vol 13 no 4 pp 411ndash4312005

[6] H Ben-Hadj-Ali S Operto and J Virieux ldquoVelocity model-building by 3D frequency-domain full-waveform inversionof wide-aperture seismic datardquo Geophysics vol 73 no 5 ppVE101ndashVE117 2008

[7] W W Symes ldquoMigration velocity analysis and waveforminversionrdquo Geophysical Prospecting vol 56 no 6 pp 765ndash7902008

[8] A Abubakar W Hu T M Habashy and P M van den BergldquoApplication of the finite-difference contrast-source inversionalgorithm to seismic full-waveform datardquo Geophysics vol 74no 6 pp WCC47ndashWCC58 2009

[9] J J Cao Y F Wang J T Zhao and C Yang ldquoA reviewon restoration of seismic wavefields based on regularizationand compressive sensingrdquo Inverse Problems in Science andEngineering vol 19 no 5 pp 679ndash704 2011

[10] A Fichtner P H Bunge and H Igel ldquoThe adjoint methodin seismology I Theoryrdquo Physics of the Earth and PlanetaryInteriors vol 157 no 1-2 pp 86ndash104 2006

[11] P M van den Berg and R E Kleinman ldquoA contrast sourceinversion methodrdquo Inverse Problems vol 13 no 6 pp 1607ndash1620 1997

[12] P M van den Berg A L van Broekhoven and A AbubakarldquoExtended contrast source inversionrdquo Inverse Problems vol 15no 5 pp 1325ndash1344 1999

[13] A Abubakar W Hu P M van den Berg and T M HabashyldquoA finite-difference contrast source inversion methodrdquo InverseProblems vol 24 no 6 Article ID 065004 pp 1ndash17 2008

[14] X Chen ldquoSubspace-based optimization method for solvinginverse-scattering problemsrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 48 no 1 pp 42ndash49 2010

[15] X Chen ldquoSubspace-based optimization method for inversescattering problems with an inhomogeneous backgroundmediumrdquo Inverse Problems vol 26 Article ID 074007 2010

[16] A Abubakar G Pan M Li L Zhang T M Habashy and PM van den Berg ldquoThree-dimensional seismic full-waveform

Journal of Applied Mathematics 11

inversion using the finite-difference contrast source inversionmethodrdquo Geophysical Prospecting vol 59 no 5 pp 874ndash8882011

[17] R F Bloemenkamp A Abubakar and P M van den BergldquoInversion of experimental multi-frequency data using thecontrast source inversion methodrdquo Inverse Problems vol 17 no6 pp 1611ndash1622 2001

[18] A Abubakar and P M van den Berg ldquoThe contrast sourceinversion method for location and shape reconstructionsrdquoInverse Problems vol 18 no 2 pp 495ndash510 2002

[19] L P Song C Yu and Q H Liu ldquoThrough-wall imaging(TWI) by radar 2-D tomographic results and analysesrdquo IEEETransactions on Geoscience and Remote Sensing vol 43 no 12pp 2793ndash2798 2005

[20] M Li A Abubakar and P M van den Berg ldquoApplication of themultiplicative regularized contrast source inversion method on3D experimental Fresnel datardquo Inverse Problems vol 25 no 2Article ID 024006 23 pages 2009

[21] A Zakaria C Gilmore and J LoVetri ldquoFinite-element contrastsource inversionmethod for microwave imagingrdquo Inverse Prob-lems vol 26 no 11 Article ID 115010 21 pages 2010

[22] G Pelekanos A Abubakar and P M van den Berg ldquoContrastsource inversion methods in elastodynamicsrdquo Journal of theAcoustical Society of America vol 114 no 5 pp 2825ndash28342003

[23] D Komatitsch and R Martin ldquoAn unsplit convolutional per-fectly matched layer improved at grazing incidence for theseismic wave equationrdquo Geophysics vol 72 no 5 pp SM155ndashSM167 2007

[24] R Martin D Komatitsch and S D Gedney ldquoA variational for-mulation of a stabilized unsplit convolutional perfectlymatchedlayer for the isotropic or anisotropic seismic wave equationrdquoComputer Modeling in Engineering amp Sciences vol 37 no 3 pp274ndash304 2008

[25] T A Davis Direct Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2006

[26] P M van den Berg A Abubakar and J T Fokkema ldquoMul-tiplicative regularization for contrast profile inversionrdquo RadioScience vol 38 no 2 2003

[27] K R Kelly R WWard S Treitel and R M Alford ldquoSyntheticseismograms a finite difference approachrdquo Geophysics vol 41no 1 pp 2ndash27 1976

[28] R G Pratt ldquoFrequency-domain elastic wave modeling by finitedifferences a tool for crosshole seismic imagingrdquo Geophysicsvol 55 no 5 pp 626ndash632 1990

[29] K W Brodlie and D A H Jacobs ldquoUnconstrained minimiza-tionrdquo in State of the Art in Numerical Analysis pp 229ndash268Academic Press 1997

[30] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 2006

[31] P Duhamel and M Vetterli ldquoFast Fourier transforms a tutorialreview and a state of the artrdquo Signal Processing vol 19 no 4 pp259ndash299 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

1500

1550

1600

1650

1700

1750

1800

1850

1900z

(m)

x (m)

(a) Multifrequency inversion result

0

1000

2000

3000

4000

5000

60000 1000 2000 3000 4000 5000 6000

z (m

)

1400

1600

1800

2000

2200

2400

2600

2800

x (m)

(b) Multifrequency inversion result

Figure 4 Applying sequential multifrequency technique to improve reconstruction results (a) Reconstruction result of one square scatterermodel (b) Reconstruction result of two rectangle scatterers model

are homogeneous or inhomogeneousmediumThe algorithmis very robust when the measurement data is polluted byrandom noise Furthermore the FD-MRCSI using sequen-tial multifrequency technique can successfully improve thereconstruction results which is very hopeful for gettinghigher resolution in exploration seismology Estimating othersubsurface parameters using this method such as Lamecoefficients and velocity is our future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Professor van den Berg who givesus many helpful suggestions on both theory and numericalimplementation of the contrast source inversion methodThis work is supported by the National Natural ScienceFoundation of China underGrants no 91230119 no 41474102and no 11301119

References

[1] A Tarantola ldquoA strategy for nonlinear elastic inversion ofseismic reflection datardquo Geophysics vol 51 no 10 pp 1893ndash1903 1986

[2] PMora ldquoNonlinear two-dimensional elastic inversion ofmulti-offset seismic datardquoGeophysics vol 52 no 9 pp 1211ndash1228 1987

[3] R G Pratt and M H Worthington ldquoInverse theory appliedto multi-source cross-hole tomography Part 1 acoustic wave-equation methodrdquo Geophysical Prospecting vol 38 no 3 pp287ndash310 1990

[4] R G Pratt ldquoInverse theory applied to multi-source cross-holetomography Part 2 elastic wave-equationmethodrdquoGeophysicalProspecting vol 38 no 3 pp 311ndash329 1990

[5] B Han H S Fu and Z Li ldquoA homotopy method for theinversion of a two-dimensional acoustic wave equationrdquo InverseProblems in Science and Engineering vol 13 no 4 pp 411ndash4312005

[6] H Ben-Hadj-Ali S Operto and J Virieux ldquoVelocity model-building by 3D frequency-domain full-waveform inversionof wide-aperture seismic datardquo Geophysics vol 73 no 5 ppVE101ndashVE117 2008

[7] W W Symes ldquoMigration velocity analysis and waveforminversionrdquo Geophysical Prospecting vol 56 no 6 pp 765ndash7902008

[8] A Abubakar W Hu T M Habashy and P M van den BergldquoApplication of the finite-difference contrast-source inversionalgorithm to seismic full-waveform datardquo Geophysics vol 74no 6 pp WCC47ndashWCC58 2009

[9] J J Cao Y F Wang J T Zhao and C Yang ldquoA reviewon restoration of seismic wavefields based on regularizationand compressive sensingrdquo Inverse Problems in Science andEngineering vol 19 no 5 pp 679ndash704 2011

[10] A Fichtner P H Bunge and H Igel ldquoThe adjoint methodin seismology I Theoryrdquo Physics of the Earth and PlanetaryInteriors vol 157 no 1-2 pp 86ndash104 2006

[11] P M van den Berg and R E Kleinman ldquoA contrast sourceinversion methodrdquo Inverse Problems vol 13 no 6 pp 1607ndash1620 1997

[12] P M van den Berg A L van Broekhoven and A AbubakarldquoExtended contrast source inversionrdquo Inverse Problems vol 15no 5 pp 1325ndash1344 1999

[13] A Abubakar W Hu P M van den Berg and T M HabashyldquoA finite-difference contrast source inversion methodrdquo InverseProblems vol 24 no 6 Article ID 065004 pp 1ndash17 2008

[14] X Chen ldquoSubspace-based optimization method for solvinginverse-scattering problemsrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 48 no 1 pp 42ndash49 2010

[15] X Chen ldquoSubspace-based optimization method for inversescattering problems with an inhomogeneous backgroundmediumrdquo Inverse Problems vol 26 Article ID 074007 2010

[16] A Abubakar G Pan M Li L Zhang T M Habashy and PM van den Berg ldquoThree-dimensional seismic full-waveform

Journal of Applied Mathematics 11

inversion using the finite-difference contrast source inversionmethodrdquo Geophysical Prospecting vol 59 no 5 pp 874ndash8882011

[17] R F Bloemenkamp A Abubakar and P M van den BergldquoInversion of experimental multi-frequency data using thecontrast source inversion methodrdquo Inverse Problems vol 17 no6 pp 1611ndash1622 2001

[18] A Abubakar and P M van den Berg ldquoThe contrast sourceinversion method for location and shape reconstructionsrdquoInverse Problems vol 18 no 2 pp 495ndash510 2002

[19] L P Song C Yu and Q H Liu ldquoThrough-wall imaging(TWI) by radar 2-D tomographic results and analysesrdquo IEEETransactions on Geoscience and Remote Sensing vol 43 no 12pp 2793ndash2798 2005

[20] M Li A Abubakar and P M van den Berg ldquoApplication of themultiplicative regularized contrast source inversion method on3D experimental Fresnel datardquo Inverse Problems vol 25 no 2Article ID 024006 23 pages 2009

[21] A Zakaria C Gilmore and J LoVetri ldquoFinite-element contrastsource inversionmethod for microwave imagingrdquo Inverse Prob-lems vol 26 no 11 Article ID 115010 21 pages 2010

[22] G Pelekanos A Abubakar and P M van den Berg ldquoContrastsource inversion methods in elastodynamicsrdquo Journal of theAcoustical Society of America vol 114 no 5 pp 2825ndash28342003

[23] D Komatitsch and R Martin ldquoAn unsplit convolutional per-fectly matched layer improved at grazing incidence for theseismic wave equationrdquo Geophysics vol 72 no 5 pp SM155ndashSM167 2007

[24] R Martin D Komatitsch and S D Gedney ldquoA variational for-mulation of a stabilized unsplit convolutional perfectlymatchedlayer for the isotropic or anisotropic seismic wave equationrdquoComputer Modeling in Engineering amp Sciences vol 37 no 3 pp274ndash304 2008

[25] T A Davis Direct Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2006

[26] P M van den Berg A Abubakar and J T Fokkema ldquoMul-tiplicative regularization for contrast profile inversionrdquo RadioScience vol 38 no 2 2003

[27] K R Kelly R WWard S Treitel and R M Alford ldquoSyntheticseismograms a finite difference approachrdquo Geophysics vol 41no 1 pp 2ndash27 1976

[28] R G Pratt ldquoFrequency-domain elastic wave modeling by finitedifferences a tool for crosshole seismic imagingrdquo Geophysicsvol 55 no 5 pp 626ndash632 1990

[29] K W Brodlie and D A H Jacobs ldquoUnconstrained minimiza-tionrdquo in State of the Art in Numerical Analysis pp 229ndash268Academic Press 1997

[30] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 2006

[31] P Duhamel and M Vetterli ldquoFast Fourier transforms a tutorialreview and a state of the artrdquo Signal Processing vol 19 no 4 pp259ndash299 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

inversion using the finite-difference contrast source inversionmethodrdquo Geophysical Prospecting vol 59 no 5 pp 874ndash8882011

[17] R F Bloemenkamp A Abubakar and P M van den BergldquoInversion of experimental multi-frequency data using thecontrast source inversion methodrdquo Inverse Problems vol 17 no6 pp 1611ndash1622 2001

[18] A Abubakar and P M van den Berg ldquoThe contrast sourceinversion method for location and shape reconstructionsrdquoInverse Problems vol 18 no 2 pp 495ndash510 2002

[19] L P Song C Yu and Q H Liu ldquoThrough-wall imaging(TWI) by radar 2-D tomographic results and analysesrdquo IEEETransactions on Geoscience and Remote Sensing vol 43 no 12pp 2793ndash2798 2005

[20] M Li A Abubakar and P M van den Berg ldquoApplication of themultiplicative regularized contrast source inversion method on3D experimental Fresnel datardquo Inverse Problems vol 25 no 2Article ID 024006 23 pages 2009

[21] A Zakaria C Gilmore and J LoVetri ldquoFinite-element contrastsource inversionmethod for microwave imagingrdquo Inverse Prob-lems vol 26 no 11 Article ID 115010 21 pages 2010

[22] G Pelekanos A Abubakar and P M van den Berg ldquoContrastsource inversion methods in elastodynamicsrdquo Journal of theAcoustical Society of America vol 114 no 5 pp 2825ndash28342003

[23] D Komatitsch and R Martin ldquoAn unsplit convolutional per-fectly matched layer improved at grazing incidence for theseismic wave equationrdquo Geophysics vol 72 no 5 pp SM155ndashSM167 2007

[24] R Martin D Komatitsch and S D Gedney ldquoA variational for-mulation of a stabilized unsplit convolutional perfectlymatchedlayer for the isotropic or anisotropic seismic wave equationrdquoComputer Modeling in Engineering amp Sciences vol 37 no 3 pp274ndash304 2008

[25] T A Davis Direct Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2006

[26] P M van den Berg A Abubakar and J T Fokkema ldquoMul-tiplicative regularization for contrast profile inversionrdquo RadioScience vol 38 no 2 2003

[27] K R Kelly R WWard S Treitel and R M Alford ldquoSyntheticseismograms a finite difference approachrdquo Geophysics vol 41no 1 pp 2ndash27 1976

[28] R G Pratt ldquoFrequency-domain elastic wave modeling by finitedifferences a tool for crosshole seismic imagingrdquo Geophysicsvol 55 no 5 pp 626ndash632 1990

[29] K W Brodlie and D A H Jacobs ldquoUnconstrained minimiza-tionrdquo in State of the Art in Numerical Analysis pp 229ndash268Academic Press 1997

[30] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 2006

[31] P Duhamel and M Vetterli ldquoFast Fourier transforms a tutorialreview and a state of the artrdquo Signal Processing vol 19 no 4 pp259ndash299 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of