Applied Mathematics and Computation 208 (2009) 189–196

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

A wavelet adaptive-homotopy method for inverse problemin the fluid-saturated porous media q

Ying He *, Bo HanDepartment of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China

a r t i c l e i n f o

Keywords:WaveletAdaptive-homotopy methodInversionFluid-saturated porous media

0096-3003/$ - see front matter � 2008 Elsevier Incdoi:10.1016/j.amc.2008.11.033

q Project supported by the National Natural Scienc* Corresponding author.

E-mail address: happybirdzhp@sohu.com (Y. He)

a b s t r a c t

In this paper, we consider a parameter identification problem for the elastic wave equa-tions in the fluid-saturated porous media. A wavelet adaptive-homotopy method is pro-posed for the recovery of porosity. This algorithm combines the wavelet multiscaleinversion idea with the adaptive-homotopy method. And the adaptive-homotopy methodprovides a simple way to adapt computational refinement to the choice of the homotopyparameter. Numerical results illustrate that the wavelet adaptive-homotopy method isglobally convergent and highly effective for solving the inverse problem.

� 2008 Elsevier Inc. All rights reserved.

1. Introduction

The elastic wave propagation in porous media has received considerable attention in resent years because of increasingapplications in science and engineering. Applications of porous media include Mechanics, Engineering, Geosciences, Materialscience. In many engineering problems, such as those found in petroleum and geotechnical engineering, rocks have to bemodeled as porous materials saturated with fluid. In 1956, Biot [1,2] firstly considered the propagation of elastic wave inthe fluid-saturated porous media. He summarized a general and systematic theory to describe a porous solid containing aviscous fluid. Since then, various numerical techniques have appeared in the literatures of the inverse problem in porousmedia. The inverse problem in the fluid-saturated porous media can be viewed as a parametric data-fitting problem. It ispossible to formalize such a problem in the framework of optimization where a functional defined in terms of discrepancybetween observed and computed data is minimized over a model space. In general, such problem is very difficult to solve,since it is nonlinear and ill-posed. The nonlinear property causes the presence of numerous local minimum, while the ill-posedness often affects the efficiency of the numerical scheme significantly. For the nonlinear and ill-posed problem, tradi-tional linearized inversion algorithms, such as conjugate gradient method, time convolution iteration method [3] and per-turbation method [4], are locally convergent. The widely popular recent approaches such as simulated annealingalgorithm, genetic algorithm, trust region method, and neural networks [5], are globally convergent. However, along withthe searching space decreasing, the efficiency of these are worse than before. Therefore, the shortages of the above inversionmethods motive us to design a highly efficient, numerically stable and globally convergent algorithm.

Wavelet multiscale method is a newly developmental inversion strategy which accelerate convergent rate, enhance sta-bility, and overcome disturbance of local minimum. Because of these features, the applications of the wavelet multiscaleinversion approach are giving rise to much attention. Liu [6] proposed wavelet multiresolution method for distributedparameter estimation of the elliptic problem. Ebrahimi and Sahimi [7] has described scale up of geological models offield-scale porous media using a method based on the wavelet transformation. Zhang et al. [8] firstly recovered the porosity

. All rights reserved.

e Foundation of China (No. 40774056).

.

190 Y. He, B. Han / Applied Mathematics and Computation 208 (2009) 189–196

in the fluid-saturated porous media by using wavelet method associated with the Gauss–Newton iterative method, and theresults showed that the Gauss–Newton method performed much better when employed with wavelet multiscale method.

Homotopy method is a powerful tool for solving inverse problems due to its globally convergent property. It is a veryinteresting research to apply homotopy method to solve the inverse problem. Various formulations for the homotopy meth-od have been used in the literature [9–13]. Although the homotopy method is applied in many fields, the literatures aboutthe availability of the homotopy method in the fluid-saturated porous media have rarely been found except [14], where Hanet al. recovered the parameter of 1D wave equation. All these works testified that the homotopy method is effective for theinverse problem.

As mentioned above, in this paper, we will design an inversion strategy called wavelet adaptive-homotopy method, forparameter identification problem in the fluid-saturated porous media. This new algorithm combines the property of goodstability and fast convergent rate of wavelet multiscale method with the advantage of global minimum of homotopy method.In order to search the global minimum efficiently, we provide a simple way to adapt computational refinements to the choiceof the homotopy parameter.

The structure of the paper is organized as follows: we start with presenting the elastic wave equations and correlativeparameters in the fluid-saturated porous media based on Biot theory in Section 2. In Section 3, an adaptive-homotopy meth-od is proposed and the adaptive property of this algorithm provides a simple way to adapt a computational refinements tothe choice of the homotopy parameter. In Section 4, a wavelet adaptive-homotopy is proposed for solving the inverse prob-lem in the fluid-saturated porous media. Then we describe how this algorithm can be carried out by combining the waveletmultiscale method with the adaptive-homotopy method. Finally, in order to illustrate the performance of this algorithm, wecarry out some numerical simulations for the inversion of porosity in fluid-saturated porous media. The results obtainedshow that the wavelet adaptive-homotopy method is effective and feasible to the inverse problem.

2. Governing equations

The propagation of elastic wave in the fluid-saturated porous media is different from that in the single-phase media.Based on continuum mechanics and macroscopic constitutive relationship, Biot developed a theory of wave motion in anelastic solid saturated with a viscous compressible fluid.

We ignore viscous effects and consider the 2D elastic wave propagation equations in fluid-saturated porous media asfollows:

2o

oxl oux

ox

� �þ o

ozl oux

ozþ l ouz

ox

� �þ o

oxk

oux

oxþ k

ouz

oz

� �þ o

oxaM

owx

oxþ aM

owz

oz

� �¼ q

o2ux

ot2 þ qfo2wx

ot2 � f1; ð2:1Þ

o

oxl oux

ozþ l ouz

ox

� �þ 2

o

ozl ouz

oz

� �þ o

ozkoux

oxþ k

ouz

oz

� �þ o

ozaM

owx

oxþ aM

owz

oz

� �¼ q

o2uz

ot2 þ qfo2wz

ot2 � f2; ð2:2Þ

o

oxaM

oux

oxþ aM

ouz

ozþM

owx

oxþM

owz

oz

� �¼ qf

o2ux

ot2 þmo2wx

ot2 � f1; ð2:3Þ

o

ozaM

oux

oxþ aM

ouz

ozþM

owx

partialxþM

owz

oz

� �¼ qf

o2uz

ot2 þmo2wz

ot2 � f2: ð2:4Þ

where u ¼ ðux; uzÞT denotes the solid-frame displacement, w ¼ ðwx;wzÞT is the fluid displacement relative to solid-frame, k isthe Lam�e coefficient, the quantity j is the Darcy permeability coefficient, g is the viscosity of the pore fluid, qf is the densityof the pore fluid, qs is the density of the solid grain, q is the bulk density defined by q ¼ bqf þ ð1� bÞqs, and b is the porosity.The relationship of the tortuosity a, added mass density m, and coupling constant M can be given by

a ¼ 1� Ks

Kr; M ¼ K2

r

Dr � Ks; Dr ¼ Kr 1þ b

Kr

Kf� 1

� �� �; m ¼

qf

b;

where Kr is the bulk modulus of the grain, Kf is the bulk modulus of the pore fluid, Ks is the bulk modulus of the skeletalframe.

The boundary conditions for the problem considered here can be given by

ouxðx; z; tÞox

����x¼0¼ ouxðx; z; tÞ

ox

����x¼L

¼ 0; ð2:5Þ

ouxðx; z; tÞoz

����z¼H

¼ 0; ð2:6Þ

ouzðx; z; tÞox

jx¼0 ¼ouzðx; z; tÞ

ox

����x¼L

¼ 0; ð2:7Þ

Y. He, B. Han / Applied Mathematics and Computation 208 (2009) 189–196 191

ouzðx; z; tÞoz

����z¼H

¼ 0; ð2:8Þ

owxðx; z; tÞox

����x¼0¼ owxðx; z; tÞ

ox

����x¼L

¼ 0; ð2:9Þ

owxðx; z; tÞoz

����z¼H

¼ 0; ð2:10Þ

owzðx; z; tÞox

����x¼0¼ owzðx; z; tÞ

ox

����x¼L

¼ 0; ð2:11Þ

owzðx; z; tÞoz

����z¼H

¼ 0; ð2:12Þ

where ðx; zÞ 2 ½0; L� � ½0;H�; t 2 ½0; T�.The initial conditions are given as follows:

uxðx; z; 0Þ ¼ 0;ouxðx; z; 0Þ

ot¼ 0; ð2:13Þ

uzðx; z; 0Þ ¼ 0;ouzðx; z; 0Þ

ot¼ 0; ð2:14Þ

wxðx; z;0Þ ¼ 0;owxðx; z;0Þ

ot¼ 0; ð2:15Þ

wzðx; z;0Þ ¼ 0;owzðx; z;0Þ

ot¼ 0; ð2:16Þ

and the additional condition is

uxðx;0; tÞ ¼ u�xðx; tÞ; ð2:17Þ

where u�x is the observed data.If given a value of the porosity b, Eqs. (2.1)–(2.6), (2.8), (2.7), (2.9)–(2.16) enable to form the direct problem for wave fields

simulations, i.e. we can solve the wave field functions ux;uz;wx;wz using a numerical method. Consequently, the additionalcondition uxðx;0; tÞ can be obtained. Note that since the porosity b is unknown in practice, governing equations (2.1)–(2.4)with the boundary conditions (2.5), (2.6), (2.8), (2.7), (2.9)–(2.12), the initial conditions (2.13)–(2.16) and the additional con-dition (2.17) completely define the inverse problem for the elastic wave equations in the fluid-saturated porous media.

3. An adaptive-homotopy method

In general, the inverse problem in the fluid-saturated porous media can be viewed as a parametric data-fitting problem. Itis possible to formalize such a problem in the framework of optimization where a function defined in terms of discrepancybetween observed and computed data is minimized over a model space. Now, we discretize Eqs. (2.1)–(2.6), (2.8), (2.7),(2.9)–(2.17) by the second order finite difference method as follows:

uxði; j; kþ 1Þ ¼mU1 � qf U2

mqði; jÞ � q2f

; uzði; j; kþ 1Þ ¼mU3 � qf U4

mqði; jÞ � q2f

;

wxði; j; kþ 1Þ ¼qði; jÞU2 � qf U1

mqði; jÞ � q2f

; wzði; j; kþ 1Þ ¼qði; jÞU4 � qf U3

mqði; jÞ � q2f

;

uxð0; j; kÞ ¼ uxð1; j; kÞ; uxðp� 1; j; kÞ ¼ uxðp; j; kÞ; uxði; n� 1; kÞ ¼ uxði;n; kÞ;uzð0; j; kÞ ¼ uzð1; j; kÞ; uzðp� 1; j; kÞ ¼ uzðp; j; kÞ; uzði; n� 1; kÞ ¼ uzði;n; kÞ;wxð0; j; kÞ ¼ wxð1; j; kÞ; wxðp� 1; j; kÞ ¼ wxðp; j; kÞ; wxði;n� 1; kÞ ¼ wxði;n; kÞ;wzð0; j; kÞ ¼ wzð1; j; kÞ; wzðp� 1; j; kÞ ¼ wzðp; j; kÞ; wzði;n� 1; kÞ ¼ wzði;n; kÞ;uxði; j; 0Þ ¼ uxði; j;1Þ ¼ 0; uzði; j;0Þ ¼ uzði; j;1Þ ¼ 0;

wxði; j;0Þ ¼ wxði; j;1Þ ¼ 0; wzði; j;0Þ ¼ wzði; j;1Þ ¼ 0;

uxði;0; kÞ ¼ u�xði; kÞ;

where b is the model to be recovered, u�x is the observed data, uxði; j; kÞ ¼ uxði� hx; j� hz; k� sÞ, hx;hz are the step sizes of therectangle grid in the x- and z-direction, respectively, s is the time step size, p ¼ L

hx;n ¼ H

hz; l ¼ T

s.The above difference equations define an operator equation

F : b! U; i:e: FðbÞ ¼ U;

192 Y. He, B. Han / Applied Mathematics and Computation 208 (2009) 189–196

where b and U denote vectors, which are composed of bði; jÞ and u�xði; kÞ in the following sequence:

Table 1The dec

Scale sp

V0

V1

V2

bT ¼ ðbð1;1Þ; . . . ;bð1;nÞ;bð2;1Þ; . . . ;bð2;nÞ; . . . ; bðp;nÞÞ;UT ¼ ðu�xð1;1Þ; . . . ;u�xðp;1Þ;u�xð2;1Þ; . . . ;u�xðp;2Þ; . . . ;u�xðp; lÞÞ;

where p;n; l denote the number of mesh nodes for the x; z; t components, respectively.In order to identify the porosity b from the observed data U containing noise, we define a nonlinear operator equation

TðbÞ, FðbÞ � U ¼ 0: ð3:1Þ

Eq. (3.1) is usually highly nonlinear and ill-posed, therefore we consider a globally convergent algorithm, namely an adap-tive-homotopy method, to solve the nonlinear operator equation (3.1).

Let us construct a homotopy regularization method

Hðb;aÞ ¼ ð1� aÞkFðbÞ � Uk2 þ akb� b0k2; ð3:2Þ

where b0 is the initial guess, the parameter a 2 ½0;1� is not only the homotopy parameter bur also the regularization param-eter. When a ¼ 1, the solution of Hðb;1Þ ¼ 0 is b0, which is known. When a ¼ 0, the solution of Hðb;0Þ ¼ 0 is the desired solu-tion of the nonlinear operator equation (3.1).

Above all, we divide the interval [0,1] into 1 ¼ a0 > a1 > � � � > aN ¼ 0. For some ak, we use some iterative method to solveHðb;akÞ ¼ 0 sequently. As the solution b0 of Hðb;1Þ ¼ 0 is known, it can be used as the initial approximation of the next equa-tion. Assume that the approximation solution bk of the Hðb;akÞ ¼ 0 has already been found. For computing the solution bkþ1

of Hðb;akþ1Þ ¼ 0, we use the damped Gauss–Newton method. In practice, we need not compute the accurate solution of eachequation. So we only iterate one step when using the damped Gauss–Newton method to solve the equation Hðb;akÞ ¼ 0. Theiterative formulation is given as follows:

bkþ1 ¼ bk � ½ð1� akÞF 0ðbkÞ�F 0ðbkÞ þ akI��1 � ½ð1� akÞF 0ðbkÞ�ðFðbkÞ � UÞ þ akbk�; k ¼ 0;1; . . . ;N � 1: ð3:3Þ

From formulation (3.3), the approximation solution bN of the Hðb;1Þ ¼ 0 can be solved, which can be taken as the initialguess to the problem (3.1). Then iterate repeatedly by using the damped Gauss–Newton algorithm

bkþ1 ¼ bk � ½F 0ðbkÞ�F 0ðbkÞ��1 � ½F 0ðbkÞ�ðFðbkÞ � UÞ�; k ¼ N;N þ 1; . . . ð3:4Þ

The formulations (3.3) and (3.4) constitute a globally convergent method for the inverse problem.As we all know, the homotopy parameters akðk ¼ 0;1; . . .Þ plays an important role. But the homotopy parameters always

can be given by ak ¼ kN in the existing literatures. Such choice may loss some convergent approximation solutions of the Eq.

(3.1). Therefore, in this paper, we propose a modified homotopy method named by the adaptive-homotopy method and ap-ply the new approach to find the global minimum of the inverse problem. This algorithm can adaptively choose the homot-opy parameters ak.

Algorithm 1. Choose an initial model b0, small threshold e and initial homotopy step size h. Set a1 ¼ a0 � h, a2 ¼ a1 � h.Calculate the solution b1 of Hðb;a1Þ ¼ 0 (using damped Gauss–Newton method). For k ¼ 2; . . ., do:

� Calculate the solution bk of Hðb;akÞ ¼ 0 (using damped Gauss–Newton method).� Calculate j ¼ kbk�bk�1k

Vertbk�1�bk�2k.� Check if j < e, then set h ¼ 2h, else if j P e, then set h ¼ h

2.� Check the convergence, if not converged: set k ¼ kþ 1 ak ¼ ak�1 � h: and go to step 1.

4. Wavelet adaptive-homotopy method

The wavelet multiscale method is a technique that improves the performance of inversion method by decomposing theproblem into subproblem on different scales. At coarser scale there are fewer local minima and those that remain are furtherapart from each other, and then inversion method can get the solution closer to the neighborhood of the global minimum. Asthe scale increases, the complexity of the original problem is reduced and the size of the corresponding domain of conver-gence is decreasing. In this section, we try to use wavelet multiscale method to decompose the original operator equation(3.1). Once the problem has been decomposed into different subproblems, it is natural to solve first the subproblem on

omposition of the original problem by wavelet.

ace Source Wave field Solved problem Solution of the problem Initial guess

f u�x FðbÞ ¼ U b b1

f 1 u1�x F1ðbÞ ¼ U1 b1 b2

f 2 u2�x F2ðbÞ ¼ U2 b2 b0

0 20 40 60 80 100 120 140 160 180 200

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time axis (0.001s)

Fig. 1. The source function.

Table 2The results of inversion by wavelet adaptive-homotopy method.

Number Initial value Real value Iterative value Iterative times Relative error (%)

1 0.045 0.266000 0.26472 27 0.4832 0.237 0.266000 0.26579 14 0.0793 0.384 0.266000 0.26678 17 0.2944 0.659 0.266000 0.26713 30 0.425

Table 3The results of inversion by homotopy method.

Number Initial value Real value Iterative value Iterative times Relative error (%)

1 0.045 0.266000 0.26464 34 0.5122 0.237 0.266000 0.26554 21 0.1743 0.384 0.266000 0.26714 26 0.4274 0.659 0.266000 0.26758 39 0.593

0 5 10 15 20 25 300

5

10

15

20

25

30

Surface axis (×10m)

Dep

th a

xis

(x10

m)

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

Fig. 2. The true value of porosity model.

Y. He, B. Han / Applied Mathematics and Computation 208 (2009) 189–196 193

194 Y. He, B. Han / Applied Mathematics and Computation 208 (2009) 189–196

the coarsest scale by using adaptive-homotopy method, and to use the resulting solution as initial guess for the subproblemon the finer scale.

Without loss of generality, we describe the wavelet adaptive-homotopy method by decomposing the original probleminto two scales. Assume that the original problem is on the scale space V0 and decomposed on the coarser scale space V1

as well as the coarsest scale space V2. We utilize the compactly supported orthogonal wavelet ‘DB4’ to decompose the wavefield functions and source function. The algorithm can be described as follows:

Step 1. Given an initial model b0, the true porosity model b�, the source function f. Calculate the forward problem to obtainthe model data U.

Step 2. Decompose f and U at each scale, such as V1 and V2 by the compactly supported orthogonal wavelet ‘DB4’ (usingMallat algorithm [15]), i.e. the original problem (3.1) can be decomposed as F1ðbÞ ¼ U1 and F2ðbÞ ¼ U2, which isshown in the following Table 1.

Step 3. At the coarsest scale space V2, calculate the solution b2 of F2ðbÞ ¼ U2 with the initial guess b0 (using the adaptive-homotopy method).

Step 4. At the finer scale space V1, we regard the solution b2 as the initial guess, and then calculate the solution b1 ofF1ðbÞ ¼ U1 (using the adaptive-homotopy method).

0 5 10 15 20 25 300

5

10

15

20

25

30

Surface axis (x10m)

Dep

th a

xis

(×10

m)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Fig. 3. The inversion result at scale 3.

0 5 10 15 20 25 300

5

10

15

20

25

30

Surface axis (×10m)

Dep

th a

xis

(×10

m)

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Fig. 4. The inversion result at scale 2.

Y. He, B. Han / Applied Mathematics and Computation 208 (2009) 189–196 195

Step 5. At the initial space V0, we regard the solution b1 as the initial guess, and then calculate the solution b of FðbÞ ¼ U(using the adaptive-homotopy method).The obtained solution b is the solution of original problem (3.1).

5. Numerical simulations

In this section, we deal with numerical experiments using the wavelet adaptive-homotopy method. We choose to recoverthe porosity for the elastic wave equations in the fluid-saturated porous media.

In the first example, let us consider a homogenous model. The source function we selected is Ricker function with theamplitude 0.8 m and the frequency 40 Hz, which is shown in Fig. 1. The parameters describing the physical properties ofthe medium are given by

kb ¼ 3:3568� 106; l ¼ 2:32� 106; kf ¼ 1:25� 106 Pa;

ks ¼ 6:296106 Pa;qf ¼ 1:000 kg=m3; qs ¼ 2:400 kg=m3:

In this numerical test, we choose the spatial step size hx ¼ hz ¼ 10 m, the temporal step size s ¼ 0:001 s and 32 grid points inthe horizonal and vertical directions, respectively. We decompose the original problem into five scales using the wavelet

0 5 10 15 20 25 300

5

10

15

20

25

30

Surface axis (×10m)

Dep

th a

xis

(×10

m)

0.2

0.25

0.3

0.35

0.4

Fig. 5. The inversion result at scale 1.

0 5 10 15 20 25 300

5

10

15

20

25

30

Surface axis (×10m)

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

Dep

th a

xis

(×10

m)

Fig. 6. The inversion result at the original scale.

196 Y. He, B. Han / Applied Mathematics and Computation 208 (2009) 189–196

‘DB4’, and then apply the wavelet adaptive-homotopy method proposed in Section 4 to recover the porosity in the fluid-sat-urated porous media. The inversion results are shown in Table 2. In order to perform the advantages of the wavelet adaptive-homotopy method, we also apply the homotopy method for recovering the porosity and the results are shown in Table 3.Comparison of Table 2 with Table 3 on the aspects of relative errors and iterative times, we find that the wavelet adap-tive-homotopy method is faster than the homotopy method when searching the global minimum.

In the second example, a complicatedly exact porosity model is considered in Fig. 2. The parameters are the same as thefirst example. Note that the digitization process would have added 5% random noisy to the observed data, and then we recov-ery the porosity from noise data. We decompose the original problem into three scales and the inversion results at differentscales are shown in Figs. 3–6. These results illustrate the fact that the method has the ability of noise suppression.

The above numerical results show that the wavelet adaptive-homotopy method is globally convergent and it is feasible touse this method to recover the porosity in the fluid-saturated porous media.

6. Conclusions

In order to accelerate convergence and overcome the problem of local minimum, we have designed the wavelet adaptive-homotopy method consisting of the wavelet multiscale technique and the adaptive-homotopy method. And then we carryout numerical simulations by using the proposed method. The present results indicate that the method has clear advantagesin comparison with other well established numerical algorithms. Numerical results demonstrate the ability of the method torecovery the parameter for the elastic wave equation in the fluid-saturated porous media.

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