We develop a new full waveform inversion (FWI) method for slowness with the crosshole data based on the acoustic wave equation in the time domain. The method combines the total variation (TV) regularization with the constrained optimization together which can inverse the slowness effectively. One advantage of slowness inversion is that there is no further approximation in the gradient derivation. Moreover, a new algorithm named the skip method for solving the constrained optimization problem is proposed. The TV regularization has good ability to inverse slowness at its discontinuities while the constrained optimization can keep the inversion converging in the right direction. Numerical computations both for noise free data and noisy data show the robustness and effectiveness of our method and good inversion results are yielded.
Seismic exploration is one of the ways of identifying media properties and structures by propagation of waves. The wave is ignited by sources on the surface and propagates into underground. Due to different properties of media, various waves such as reflection wave and refraction wave are produced. The nature of wave reflection and refraction gives the physical information of media. So the inverse of seismic waves can give geological properties of underground materials. Unlike the observation on the surface, crosshole configuration arranges sources in one well or hole and receivers in another well. Most approaches to the inversion of crosshole data involve traveltime tomography. Traveltime tomography is efficient and robust. However, the resolution of traveltime tomography is limited by its foundation on ray theory and its typical use of first arrivals rather than the full waveform.
The full waveform inversion (FWI) is a high resolution method to inverse the media parameter by using the whole wavefield information such as amplitude, phase and arrival time. The FWI can be divided into two categories, the time- domain method and the frequency-domain method. The FWI was originally developed in the time domain [
The FWI is a typical ill-posed problem for its nonlinearity and cycle-skipping problem. To overcome the ill-posedness, various methods have been developed, for example the multiscale method [
In this paper, we develop an effective full waveform slowness inversion method for crosshole data by combing the total variation regularization with the constrained optimization together. We also propose the new skip algorithm to implement the constrained optimization problem. The rest of this paper is arranged as follows. The theoretical method is described in Section 2 in detail. It includes the forward method and the inverse method. In Section 3, numerical computations both for the noise free data and noisy date are implemented. Finally, the conclusion is drawn in Section 4.
There are three subsections in this section. In Section 2.1, the forward problem and the staggered-grid scheme with the perfectly matched layer are described. In Section 2.2, the inversion method and the corresponding algorithm by combing TV regularization with bound constraints are described. Section 2.3 involves the computations for the gradient of objective function and the step length, which is important to the success of inversion.
We consider the acoustic wave equation excited by the source f ( T ) located at the point ( x s , z s ) , which the mathematical form of the pressure u ( x , z , t ) satisfies
σ ∂ 2 u ∂ t 2 − ∇ ⋅ ( ∇ u ) = f ( t ) δ ( x − x s ) δ ( z − z s ) , in Ω × ( 0 , T ) (1)
u ( x , z , t ) | t = 0 = 0 , ∂ u ( x , z , t ) ∂ t | t = 0 = 0 , on Ω (2)
where Ω ⊂ R 2 is the area of computational domain, T is the final observation time, σ = 1 / v 2 is the square of slowness parameter, v is the wave velocity. The system (1)-(2) is the forward problem. In this paper, we solve the pressure u numerically with the finite difference method for its high computational efficiency. Among various of finite difference schemes, the staggered-grid scheme is a typical one [
∂ w x ∂ t = ∂ u ∂ x , (3)
∂ w z ∂ t = ∂ u ∂ z , (4)
σ ∂ u ∂ t = ∂ w x ∂ x + ∂ w z ∂ z + ∫ T f ( t ) ⋅ δ ( x − x s ) δ ( z − z s ) d t . (5)
Now we construct the difference scheme of (3)-(5) on the staggered grids. The schematic map of grid points for different variables is shown in
( w x ) i + 1 2 , j n + 1 2 − ( w x ) i + 1 2 , j n − 1 2 = Δ t Δ x ( u i + 1 , j n − u i , j n ) , (6)
( w z ) i , j + 1 2 n + 1 2 − ( w z ) i , j + 1 2 n − 1 2 = Δ t Δ z ( u i , j + 1 n − u i , j n ) , (7)
u i , j n + 1 − u i , j n = Δ t σ i , j Δ x ( ( w x ) i + 1 2 , j n + 1 2 − ( w x ) i − 1 2 , j n + 1 2 ) + Δ t σ i , j Δ z ( ( w z ) i , j + 1 2 n + 1 2 − ( w z ) i , j − 1 2 n + 1 2 ) + σ i , j Δ t 2 ∑ s = 0 n f i , j s , (8)
with initial conditions
u i , j − 1 = 0 , u i , j 0 = 0 , (9)
( w x ) i + 1 2 , j − 1 2 = 0 , ( w z ) i , j + 1 2 − 1 2 = 0 . (10)
The sufficient and necessary stability for the scheme (6)-(8) is (see Appendix A)
Δ t ≤ σ ( 1 Δ x 2 + 1 Δ z 2 ) . (11)
Since the computational domain is limited, we need to use absorbing boundary conditions to eliminate the boundary reflections. Several methods can be applied, for example, the paraxial approximation [
∂ u x ∂ t + d ( x ) u x = 1 σ ∂ w x ∂ x , (12)
∂ u z ∂ t + d ( z ) u z = 1 σ ∂ w z ∂ z , (13)
∂ w x ∂ t + d ( x ) w x = ∂ u ∂ x , (14)
∂ w z ∂ t + d ( z ) w z = ∂ u ∂ z , (15)
where d ( x ) and d ( z ) are designated to attenuate the refractions in absorbing zone.
We use the following model for the damping parameter d ( p ) [
d ( p ) = d 0 ( p L ) 2 , d 0 = − 3 v 2 L ln ( R ) , p = x , z , (16)
where L is the thickness of PML absorbing layer, v is the media velocity, x or z is the distance from current position to PML inner boundary, and R is a parameter chosen as 10 − 3 ~ 10 − 6 . Note that the coefficient d ( p ) is zero in the original computational domain. The staggered-grid scheme for system (12)-(15) is
( w x ) i + 1 2 , j n + 1 2 = 1 − Δ t d i / 2 1 + Δ t d i / 2 ( w x ) i + 1 2 , j n − 1 2 + 1 1 + Δ t d i / 2 Δ t Δ x ( u i + 1 , j n − u i , j n ) , (17)
( w z ) i , j + 1 2 n + 1 2 = 1 − Δ t d j / 2 1 + Δ t d j / 2 ( w z ) i , j + 1 2 n − 1 2 + 1 1 + Δ t d j / 2 Δ t Δ z ( u i , j + 1 n − u i , j n ) , (18)
( u x ) i , j n + 1 = 1 − Δ t d i / 2 1 + Δ t d i / 2 ( u x ) i , j n + 1 1 + Δ t d i / 2 Δ t Δ x σ i , j ( ( w x ) i + 1 2 , j n + 1 2 − ( w x ) i − 1 2 , j n + 1 2 ) , (19)
( u z ) i , j n + 1 = 1 − Δ t d j / 2 1 + Δ t d j / 2 ( u z ) i , j n + 1 1 + Δ t d j / 2 Δ t Δ z σ i , j ( ( w z ) i , j + 1 2 n + 1 2 − ( w z ) i , j − 1 2 n + 1 2 ) , (20)
u i , j n + 1 = ( u x ) i , j n + 1 + ( u z ) i , j n + 1 . (21)
The forward computations can be extrapolated explicitly in the time direction based on (17)-(21). In
Consider the inversion of slowness fields in the vector σ = ( σ 1 , σ 2 , ⋯ , σ s ) T .
Define the objective function J ( σ ) as
J ( σ ) = 1 2 ∑ s h o t ∫ 0 T ∫ Ω ( u ( x , z , t ) − u o b s ( x , z , t ) ) 2 δ ( x − x r ) d x d z d t + η ∫ Ω ε 2 + | ∇ σ | 2 d x d z , (22)
where the summation is for all the shots or sources in the well. In Equation (22), the first term comes from the misfit between the forward data u and the observed data u o b s , the second term is the TV regularization with positive parameter ε 2 , and η > 0 is the regularization parameter. The TV regularization is applied because it can yield better inversion result at jump discontinuities theoretically [
min σ J ( σ ) . (23)
It is a high dimensional complex optimization problem. The dimensions are N x N z , where N x and N z are the discretization points along x and z directions respectively. In this paper, the iterative decent method named the global Barzilai-Borwein (GBB) method is applied because it is suitable for solving high-dimensional complex minimization problem [
σ m + 1 = σ m − α m ∇ f ( σ m ) , (24)
where the superscript m denotes the iteration number, α m is the step length and ∇ f is the gradient of f at the point σ m .
Now we propose to solve the minimization problem (23) as a constrained optimization problem subject to the constraints a ≤ σ ≤ b . Here a and b are the lower bound and upper bound for each component of the vector σ respectively. The solution of constraint problem for nonlinear optimization is challenging. Usually, the penalty function technique is used to change the constrainted optimization to unconstrained optimization problem [
For minimizing the problem (23), the gradient and the step length are required. The gradient of the objection functional (22) with respect to σ is
∂ J ( σ ) ∂ σ = − ∑ s h o t ∫ 0 T ∫ Ω ϕ ∂ 2 u ∂ 2 t d x d z d t − η ∫ Ω ∇ ⋅ ∇ σ ε 2 + | ∇ σ | 2 d x d z , (25)
where ϕ is the backward wavefield satisfying
σ ∂ 2 ϕ ∂ t 2 − ∇ ⋅ ∇ ϕ = − ( u − u o b s ) ⋅ δ ( x − x r ) , (26)
ϕ | t = T = 0 , ∂ ϕ ∂ t | t = T = 0 , (27)
which can be solved effectively like the forward problem in Section 2.1. The backward problem (26)-(27) is introduced in the gradient derivation naturally in Appendix B. There are two terms on the right hand side of Equation (25). The first is the gradient of the misfit functional between the forward data u and the observed data u o b s , which is derived in Appendix B. We remark that there is no Born approximation in the gradient derivation in Appendix B. The second is the gradient of TV regularization term and its discrete form can be obtained by the approximation of the gradient operator, i.e.,
| ∇ σ | 2 ≈ ( D x σ i , j ) 2 + ( D z σ i , j ) 2 , (28)
where
D x σ i , j = σ i + 1 , j − σ i − 1 , j 2 Δ x , D z σ i , j = σ i , j + 1 − σ i , j − 1 2 Δ z . (29)
The parameter ε in Equation (25) is a small number to prevent the singularity of the denominator.
Now we consider the step length which is important to the success of inversion. For a general objective function ϕ :
ϕ ( α k ) = f ( x k + α k p k ) , (30)
the one dimensional line search is applied to find the step length, where p k is the search direction and α k is the step length. Line search condition gives the step length α k which guarantees the sufficient decrease in the objective function f by inequality
f ( x k + α k p k ) ≤ f ( x k ) + c 1 α k ∇ f k T p k
for some constant c 1 ∈ ( 0 , 1 ) . The reduction of the objective function is proportional to both the search direction and the step length. The sufficient condition states that α k is the step length only if
ϕ ( α k ) ≤ f ( x k ) + c 1 α k ∇ f k T p k .
The decrease is not enough if it only satisfies the sufficient condition because it can satisfy for the very small value of α . For this reason it requires another essential condition called the curvature condition. If α k satisfies
∇ f ( x k + α k p k ) T p k ≥ c 2 ∇ f k T p k
for some constant c 2 ∈ ( c , 1 1 ) , it is called to satisfy the curvature condition. The sufficient decrease condition and the curvature condition together are called Wolfe condition [
f ( x k + α k p k ) ≤ f ( x k ) + c 1 α k ∇ f k T p k , (31)
| ∇ f ( x k + α k p k ) T p k | ≤ c 2 | ∇ f k T p k | , (32)
where 0 < c 1 < c 2 < 1 . For the importance of line search in inversion, the algorithm for line search is given by Algorithm 1 and Algorithm 2. The FWI algorithm with both TV regularization and bound constraints is given by Algorithm 3.
We implement numerical computations with C-language code. The computational domain is Ω = X × Z , where X = [ 0 , 0.25 km ] and Z = [ 0 , 0.25 km ] . The spatial increment is Δ x = Δ z = 8.33 m and time step Δ t = 0.001 s . In numerical computations, the regularization parameter η is around 1.0 × 10 − 9 . The crosshole configuration is shown in
f ( t ) = A ( 2 ( π f 0 ( t − t 0 ) ) 2 − 1 ) e − ( π f 0 ( t − t 0 ) ) 2 , (33)
where f 0 = 25 Hz is the peak frequency, A is the maximum amplitude and
t 0 = 0.01 s is the time when maximum amplitude occurs. The squared slowness of the model exact is
σ = { 0.111 , ( x , z ) ∈ C 0.250 , ( x , z ) ∈ Ω \ C (34)
where
C = { ( x , z ) : ( x − 0.125 ) 2 + ( z − 0.125 ) 2 < 0.000851 } .
As shown in
First we consider the inverse for the noise free data. The inverse result after 54 iterations without TV regularization and bound constraints in Algorithm 3 is shown in
of the objective function for the first 200 iterations. In
decreases fastest. The inverse results can be improved further if more iterations are carried out.
The recorded data are usually affected by noise due to error in measurements or the influence of natural environment. So it is necessary to consider noise analysis. The Gaussian white noise is one of the effective way to add noise. It has the normal distribution and standard deviation of the data. Let the discrete signal be { S i } i = 1 N t and the noisy signal { S ˜ i } i = 1 N t . To obtain { S ˜ i } i = 1 N t , we simply add { S i } i = 1 N t to a white Gaussian noise which is random number with the mean zero and standard deviation under different signal-to-noise ratio (SNR)
S ˜ i = S i + N i , N i ~ Ν ( 0 , ‖ S ‖ 2 / N t SNR ) , (35)
where Ν ( m , s 2 ) denotes a normal distribution with mean m and standard derivation s 2 , and ‖ ⋅ ‖ is the Euclidean mean. We first present some noisy data
at different receivers with different SNR levels. Figures 8-10 are the noisy data received from a source with SNR = 100, SNR = 10, SNR = 1, respectively. In each figure, the data received at three different receivers z r , i.e., z r = N z / 2 ,
Full waveform inversion is an effective method for recovering the media parameter. It is an optimization iterative process by minimizing the misfit between the recorded and synthesized data. In this paper, we have developed a new numerical method for crosshole waveform slowness inversion. This method combines TV regularization with constrained optimization together. The TV regularization method can yield better images of slowness at its discontinuities while the bound constraints from logging data can give more reliable constraints to keep the inversion converging to the right result. The combination is necessary to improve the robustness and inversion precision. The novel skip method is proposed to implement the constrained optimization problem effectively. Numerical computations both for noise free data and noisy data with lower SNR show the effectiveness and robustness of the proposed method. In the future, we will design preconditioners to improve the computational efficiency further.
This work is supported by China National Natural Science Foundation under grand number 11471328. It is also supported by the Chinese Academy of Sciences and World Academy of Sciences for CAS-TWAS fellowship and the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences. The computations are completed in the State Key Laboratory of Scientific and Engineering, ICMSEC.
Zhang, W.S. and Joardar, A.K. (2018) Acoustic Based Crosshole Full Waveform Slowness Inversion in the Time Domain. Journal of Applied Mathematics and Physics, 6, 1086-1110. https://doi.org/10.4236/jamp.2018.65094
We apply the Fourier method to analyze the stability. Since the source term in Equation (8) has no influence on stability, we consider the source-free formulation of the scheme (6)-(8) for brevity, i.e.,
where
Subtracting (39) from (38) and applying (36)-(37), we obtain
Let
where
and
The sufficient and necessary condition for stability is that the eigenvalues of matrix A is within or on the unit circle. The characteristic equation of matrix A is
The stability requires
which gives
or
which is just the stability (11).
For convenience, we denote the first term in Equation (22) by
and the forward operator A by
where A is given by
The operator A is a self-conjugate operator. See the following Lemma 1.
Lemma 1. For any
where
Proof. Based on the definition of the operator A, we have
Applying the Green formula in time and using the conditions (52), the first term in Equation (54) becomes
Similarly, applying the Green formula in space and noting boundary conditions of
Inserting Equations (55) and (56) into Equation (54), we have
The proof is complete. Making variation for the objective function (49), we obtain
where
Note that
we obtain
Theorem 2 If
then the gradient of the objective function (49) is
Proof Based on Equation (57) and the definition of
Using boundary conditions and Lemma 1, we obtain
From the definition we know
Subtracting Equation (66) from equation (65), we have
Inserting Equation (67) into Equation (64), we have
Let
The proof is completed. The result (69) is just the first term on the right hand side of Equation (25). We can see that there is no Born approximation in the derivation above.