An Application of Elastic-Net Regularized Linear Inverse Problem in Seismic Data Inversion

Abstract

In exploration geophysics, seismic impedance is a physical characteristic parameter of underground formations. It can mark rock characteristics and help stratigraphic analysis. Hence, seismic data inversion for impedance is a key technology in oil and gas reservoir prediction. To invert impedance from seismic data, one can perform reflectivity series inversion first. Then, under a simple exponential integration transformation, the inverted reflectivity series can give the final inverted impedance. The quality of the inverted reflectivity series directly affects the quality of impedance. Sparse-spike inversion is the most common method to obtain reflectivity series with high resolution. It adopts a sparse regularization to impose sparsity on the inverted reflectivity series. However, the high resolution of sparse-spike-like reflectivity series is obtained at the cost of sacrificing small reflectivity. This is the inherent problem of sparse regularization. In fact, the reflectivity series from the actual impedance well log is not strictly sparse. It contains not only the sparse major large reflectivity, but also small reflectivity between major reflectivity. That is to say, the large reflectivity is sparse, but the small reflectivity is dense. To combat this issue, we adopt elastic-net regularization to replace sparse regularization in seismic impedance inversion. The elastic net is a hybrid regularization that combines sparse regularization and dense regularization. The proposed inversion method was performed on a synthetic seismic trace, which is created from an actual well log. Then, a real seismic data profile was used to test the practice application. The inversion results showed that it provides an effective new alternative method to invert impedance.

1. Introduction

Seismic inversion is a way to estimate the impedance of underground formations from post-stack seismic data. The inverted impedance can be used in rock characterization, stratigraphic correlation, lithology identification and further reservoir prediction, and many other areas in the oil and gas industry [1,2,3].

Usually, the post-stack seismic data are first inverted to obtain the reflectivity series of the underground formation. Then, under a simple exponential integration transformation, the inverted reflectivity series can give the final impedance [4,5]. Hence, the quality of the inverted reflectivity series directly affects the quality of impedance.

In seismology, we think the observed seismic data are the convolution result between reflectivity series and a seismic source wavelet. A seismic pulse incident at an interface is partly reflected and partly transmitted into the following layer. In a multi-layer medium, this is repeated at each interface present in the way of the advancing wave. A detector placed on the surface receives the reflected wave sequentially, at time intervals depending on the depths of the reflectors. The source pulse reaches the detector after being affected by various types of modifications, including the process of reflection, in which amplitude and polarity are governed by the reflectivity of the reflector. From each reflector, a wavelet representing the seismic source wavelet, scaled by the reflectivity, will arrive at the detection point. Thus, a series of such wavelets, each of which is shifted in time with respect to the preceding wavelet, are superimposed on each other to form a record that is a seismic data trace [6]. This process, which involves multiplication of the source wavelet by the reflectivity, time shifting, and superposition, is similar to the mathematical convolution process taking place between the source wavelet, in this case, and the reflectivity series. This is the construction mechanism model which is accepted by geophysicists for the formation of the seismic trace. The concept that the seismic data can be represented as the convolution of the seismic source wavelet with the earth reflectivity series was developed by Robinson [7]. Hence, we can say that a seismic wavelet is a waveform with the bulk of its energy confined to a finite interval on the time scale. It can be defined as a source energy signal that is concentrated in a certain time interval [8]. A seismic wavelet, or simply a wavelet, is one of the basic building blocks that is used for the construction of seismic signal models. These models are used as a basis for various seismic data inversions or processing algorithms. Hence, the reflectivity series inversion is, in fact, seismic data deconvolution, which is a geophysical inverse problem.

Zhadnov gave the definition and conditions of the well-posedness for the geophysical inverse problem [9]. The well-posed inverse problem possesses all the properties of the “good” solution: the solution exists, is unique, and is stable [9]. However, the majority of geophysical inverse problems are ill posed, because at least one of the well-posedness conditions fails. Like many other geophysical inverse problems, the seismic data deconvolution is also ill posed [10]. To solve the ill-posed inverse problem, the most effective way is regularization constraint [10].

The first used regularization method in geophysical inverse problems is Tikhonov regularization, which is the L2-norm of the model parameter or its attributes [9,11,12]. The regularized inverse problem theory can also be interpreted from Bayesianism. The misfit between observed data and synthetic data corresponds to the likelihood function, and the regularization corresponds to a priori distribution. The solution of the regularized inverse problem corresponds to maximum posterior estimation [9,12]. From this point of view, the Tikhonov regularization corresponds to a priori Gaussian distribution of model parameters. Hence, Tikhonov regularization is an additional term after misfit in the objective function of inverse problems.

In the early days, Tikhonov regularization was used to constrain seismic inverse problems [13,14]. For example, pre-whitening least-square deconvolution is frequently used in seismic data processing [15]. The pre-whitening is, in fact, zero-order Tikhonov regularization. In addition, due to the band-limited feature of seismic data, i.e., lack of effective low-frequency and high-frequency information, the important low-frequency geological trend and high-frequency information characterizing thin layers will be missing in seismic inversion results. To solve this issue, a priori model constraint regularization was developed based on conventional Tikhonov regularization [16].

However, Tikhonov regularization yields a non-sparse solution. A dense reflectivity series will be obtained through Tikhonov regularization seismic inversion. The estimated impedance model from the dense reflectivity series is not blocky, i.e., the formation boundary is not sharp and precise. To invert a sparse reflectivity series with high resolution and estimate the blocky impedance model, sparse regularization is introduced in the seismic reflectivity inversion. The basic assumption behind sparse regularization in seismic inversion is that the reflectivity series of the underground formation is sparsely distributed. Hence, the final impedance from the sparse reflectivity series is blocky through the exponential integration transformation [2].

As early as the 1970s, sparse regularization has been used in seismic data inverse problems [17]. However, the impact is confined to the geophysical community. The extensive research of sparse regularization originates from the signal sparse representation and recovery and compressed sensing theory [18,19,20,21]. There are many kinds of sparse regularization in the existing literature, such as Huber regularization [22], Cauchy regularization [23,24,25], modified Cauchy regularization [24], minimum entropy regularization [9,26], t regularization [27], L1-norm regularization [4,28], weighted L1-norm regularization [29], Lp-norm (0 < p < 1) regularization [30], L0-norm regularization [31,32,33], and so forth.

Similarly, sparse regularization can also be interpreted from Bayesianism. In fact, the sparse regularization corresponds to a priori long-tail distribution of model parameters [12]. For example, L1-norm regularization corresponds to the bilateral exponential distribution. Other sparse regularizations include Lp-norm (0 < p < 1), Cauchy norm, and Huber norm, which, respectively, correspond to the generalized Gaussian distribution, Cauchy distribution, Huber distribution, and so forth.

However, due to the inherent problems of sparse regularization, i.e., its tendency to protect large coefficients and suppress small coefficients [29,34], the sparse regularization seismic inversion method cannot estimate some weak reflectors [1]. The high resolution of sparse-spike inversion is obtained at the cost of sacrificing small reflectivity.

In fact, the reflectivity series from the actual impedance well log is not strictly sparse. It contains not only the sparse major large reflectivity, which corresponds to the main stratigraphic sequence interface of the underground formation, but also small reflectivity, which represents weak reflectors of a thin interbed between major reflectivity.

To combat this issue, we use elastic-net regularization as an alternative to sparse regularization in seismic inversion. The elastic net is a hybrid regularization that combines sparse regularization and dense regularization. Next, the proposed inversion method is performed on a synthetic seismic trace, which is created from an actual well log. Then, a real seismic data profile is used to test the practice application.

2. Methods

2.1. Sesimic Inversion for Impedance

Usually, in seismic inversion process, the seismic data are first inverted to obtain the reflectivity series of underground formation. In seismology, we think the observed seismic data are the convolution result from reflectivity series of underground formation and a source wavelet [6]. For calculation, the convolution relationship is discretized as

$$d=Wr+n$$

(1)

where d is the observed seismic data vector, W is the wavelet convolution matrix, and r is the reflectivity series vector; n is the noise in observed seismic data.

Hence, the reflectivity series inversion is, in fact, a linear inverse problem. Based on least square inverse problem theory, the objective function of seismic inversion is

$$\min f(r)=\left|\right|Wr-d|{|}_2^2$$

(2)

where,$\left|\right|\cdot |{|}_2$ is L2-norm of a vector.

However, the solution of seismic inversion is ill posed. To solve the ill-posed inverse problem, the most effective way is regularization constraint [6]. In the early days, Tikhonov regularization was used to stabilize seismic inversion [13,14]. Hence, the objective function of seismic inversion can be updated as

$$\min f(r)=\left|\right|Wr-d|{|}_2^2+{\lambda }_2|\left|r\right|{|}_2^2$$

(3)

where ${\lambda }_2$ is the regularization parameter of Tikhonov regularization. In fact, it is pre-whitening least-square deconvolution [15].

Next, the final inverted impedance can be obtained from inverted reflectivity series through a simple exponential integration transformation [35], i.e.,

$$z(i)=z(0)\exp [2{\displaystyle \sum _{j=0}^{i}r(j)}],i=1,2,\dots ,n$$

(4)

where, z(i) and r(j) are the impedance and reflectivity at samples i and j, n is the number of reflectivity series.

2.2. Sesimic Inversion with Sparse Regularization

A big problem of Tikhonov regularization seismic inversion is that it yields non-sparse solution, i.e., dense reflectivity series. The final inverted impedance from dense reflectivity series is not blocky, i.e., the formation boundary is not sharp and precise. The resolution of inversion results is poor [22,23,24].

To invert sparse reflectivity series with high resolution and estimate blocky impedance model, sparse regularization is introduced in seismic reflectivity inversion as an alternative to Tikhonov regularization. The commonly used sparse regularization is L1-norm. Hence, the objective function of seismic inversion with L1-nom sparse regularization is

$$\min f(r)=\left|\right|Wr-d|{|}_2^2+{\lambda }_1|\left|r\right|{|}_1$$

(5)

where ${\lambda }_1$ is the regularization parameter of L1-nom sparse regularization. This is the so-called sparse-spike seismic inversion [2].

2.3. Sesimic Inversion with Elastic-Net Regularization

The basic assumption behind sparse regularization in seismic inversion is that the reflectivity series of underground formation is sparsely distributed. Hence, the final impedance from sparse reflectivity series is blocky through the exponential integration transformation [2].

In fact, the reflectivity series of underground formation is not strictly sparse from the observation of actual well log. For example, Figure 1a shows the reflectivity series calculated from an actual impedance well log shown in Figure 1b. The well log has been transformed into time domain and scaled into the frequency band of seismic data. We can see that the major large reflectivity is sparse, which corresponds to main stratigraphic sequence interface of underground formation, such as stratigraphic unconformity, lithology boundary, and so forth. However, there are also many small reflectivity series, which represent weak reflectors of thin interbed between major reflectivity. The large reflectivity is sparse, but small reflectivity is dense. In addition, small reflectivity is highly correlated to the major reflectivity right above it.

To combat this issue, we adopt elastic-net regularization as an alternative to sparse regularization. The objective function with elastic-net regularization is

$$\min f(r)=\left|\right|Wr-d|{|}_2^2+{\lambda }_1|\left|r\right|{|}_1+{\lambda }_2\left|\right|r|{|}_2^2$$

(6)

The elastic net was first proposed by Zou and Hastie to solve LASSO (Least Angle Selection and Shrinkage Operator) problem in statistics [36]. In elastic-net regularization, the L1-norm and L2-norm have opposing effects. L1-norm imposes a sparse solution, whereas L2-norm promotes density of solution. Hence, it is called elastic net. The combined effect of elastic-net regularization in seismic inversion is that it yields sparse large reflectivity, but the dense effect of L2-norm yields only correlated small reflectivity [36,37].

Usually, seismic inversion can greatly improve the identification capability of formations compared with the original seismic data by compensating for the lacking low- and high-frequency components in the original seismic data [16,32]. However, it may be unreasonable just through L1-norm or L2-norm regularization. This is because the low-frequency components compensated by these regularizations may not coincide with the true geological background [38].

The a priori model with effective low-frequency components can be used to serve as a regularization term to compensate for the lacked low-frequency components in the original seismic data [4]. This a priori model can be obtained from some credible prior information, such as well-log data, seismic velocity analysis, and geological knowledge. The a priori model regularization can make the low-frequency information compensated by the inversion results more consistent with the actual work area [32]. Hence, objective function (6) is updated as

$$\min f(r)=\left|\right|Wr-d|{|}_2^2+{\lambda }_1|\left|r\right|{|}_1+{\lambda }_2\left|\right|r|{|}_2^2+\mu ||Cr-\mathsf{\eta }|{|}_2^2$$

(7)

where μ is the regularization parameter of a priori model, η is the logarithmic impedance from a priori impedance model, C is the integral operator matrix. The a priori model can be obtained from some credible prior information, such as well-log data, seismic velocity analysis information, and prior geological knowledge.

To solve objective function (7), we can rewrite it as

$$\min f(r)=\left|\right|Ar-b|{|}_2^2+{\lambda }_1|\left|r\right|{|}_1$$

(8)

where $A={[W^{\mathrm{T}},\sqrt{\mu }{C}^{\mathrm{T}},\sqrt{{\lambda }_2}I]}^{\mathrm{T}}$, $b={[d^{\mathrm{T}},{\mathsf{\eta }}^{\mathrm{T}},0^{\mathrm{T}}]}^{\mathrm{T}}$, I is identity matrix, 0 is a zero-value vector.

Equation (8) takes the standard form of L1-norm regularized inverse problem. However, its physical meaning is completely different from Equation (5). In this paper, we use iterative soft-thresholding algorithm (ISTA) to solve it. The specific procedure of ISTA can be found in the existing literature [18,19,20,21].

3. Synthetic Data Tests

We first use a synthetic seismic trace to test the effects of inversion with elastic-net regularization and its differences compared to conventional inversion methods. The synthetic trace is shown in Figure 2a, which is obtained from the convolution between a 40 Hz Ricker wavelet and the reflectivity series shown in Figure 1a.

Then, three different regularized inversion methods are performed on this synthetic trace. The first one is an inversion with elastic-net regularization (ENRI), the second one is an inversion with L1-norm regularization (L1RI), and the last one is an inversion with L2-norm regularization (L2RI). The initial solution of inversion is obtained by smoothing the actual impedance well log through a high-cut filter with a threshold value of 15 Hz. In addition, this low-frequency trend log is served as the a priori model. The inverted reflectivity series traces and impedances from these three inversion methods are plotted in Figure 3, Figure 4 and Figure 5, respectively. In Figure 3, Figure 4 and Figure 5, the black curves are the actual reflectivity series or impedances well log, and the red curves are the inverted reflectivity series or impedances.

From Figure 3, Figure 4 and Figure 5, we can see that, generally speaking, all of the inverted reflectivity series can match with the actual reflectivity series, and all of the inverted impedances follow the relative trend of the actual well log very nicely. However, the inverted reflectivity series and impedances by the three inversion methods are very different in detail. First, the L2RI swings some major large reflectivity series to make the inverted reflectivity series over oscillate. Hence, the inverted impedance by L2RI is over oscillating, too. The formation boundary of the main stratigraphic sequence interface is not precise. Second, compared to the L2RI, the L1RI can obtain more blocky impedance. Additionally, the inverted reflectivity series is sparse with high resolution. However, many small reflectivity series are over suppressed. Hence, the inverted impedance corresponding to weak reflectors is underestimated. Furthermore, some interbeds are completely suppressed. Last, in the inverted reflectivity series by ENRI, both large and small reflectivity series are all recovered well. Hence, the inverted impedance by ENRI is the best match compared with the actual impedance well log.

We calculate the relative errors (RE) of the inverted reflectivity series by different inversion methods through the following formula

$$RE=\frac{\left|\right|r-r_{\mathrm{inverted}}|{|}_2^2}{\left|\right|r|{|}_2^2}$$

(9)

where $r_{\mathrm{inverted}}$ is the inverted reflectivity series, and r is the actual reflectivity series. The three calculated REs of the inverted reflectivity series shown in Figure 3, Figure 4 and Figure 5 are listed in

Table 1. The REs of inverted reflectivity series by different methods.

Inversion Method	RE
L2RI	0.1327
L1RI	0.1189
ENRI	0.0511

. The RE for ENRI is much smaller than the other two inversion methods.

Next, the synthetic seismic trace is added to different level zero-mean Gaussian random noise to test the anti-noise performance of ENRI. Figure 2b,c show the synthetic seismic trace with SNR = 4, and the synthetic seismic trace with SNR = 1, respectively. Then, ENRI is performed on these two noise-contaminated synthetic traces. The inverted reflectivity series and impedances from synthetic traces with different levels of noise are shown in Figure 6 and Figure 7, respectively. In Figure 6 and Figure 7, the black curves are the actual reflectivity series or impedances well log, and the red curves are the inverted reflectivity series or impedances by ENRI. We can see that, even though noise existed in seismic data, the inversion procedure is still relatively stable.

We calculate the REs of the inverted reflectivity series by ENRI performed on synthetic traces with different levels of noise. The calculated REs are listed in

Table 2. The REs of inverted reflectivity series by ENRI performed on noise-contaminated synthetic traces.

Noise Level	RE
SNR = 4	0.0657
SNR = 1	0.1073

. The RE for ENRI performed on the noise-contaminated synthetic traces is still relatively low.

4. Field Data Applications

After performing the synthetic seismic traces test of ENRI, a real seismic data profile from East China is used to test its practice application. Figure 8 shows this original seismic data profile including CDP 595–655 (CDP means common depth point). In Figure 8, the actual impedance well log curve near this profile is overlaid.

Then, ENRI is performed on this seismic data profile. In the process of inversion, the initial and a priori model is obtained through the following procedures. First, we use a local kriging estimate with actual impedance log curves from this work area under the constraint of seismic interpretation horizons to obtain a well-log interpolating impedance model. Next, the interpolating impedance model is smoothed through a high-cut filter with a 15 Hz threshold value. The result is an impedance model with a low-frequency trend, which is served as the initial and a priori model.

Figure 9 and Figure 10 show the inverted reflectivity series profile and the inverted impedance profile, respectively. In Figure 10, the actual impedance well log curve near this inline is overlaid, too.

We can see that the inverted reflectivity series is well matched with original seismic data with good structural configurations and stratigraphic lateral distributions. Between the sparse major large reflectivity series, there are small reflectivity series, which match with the actual reflectivity series. In inverted impedance, the major formation boundary is precise. In addition, the impedances corresponding to weak reflectors are also clearly estimated. The clear vertical and spatial variation features of the inverted reflectivity series profile and impedance profile can be used well in further reservoir prediction and description. Hence, ENRI is an effective new alternative method to invert impedance.

5. Discussions

As in most inversion methods, the set of good regularization parameters is very important to obtain a satisfying solution in seismic inversion. In this paper, we dealt with elastic-net regularization. It is, in fact, the efficient combination of sparse regularization and dense regularization. Hence, the effect of elastic-net regularization is determined by two parameters: ${\lambda }_1$ and ${\lambda }_2$. As usual, the method most widely used to pick suitable regularization parameters is the L-curve criterion [9,14,39]. The best regularization parameter is closest to the corner of the L-curve. However, it is not suitable for ENRI. During the inversion procedure, the regularization parameters of elastic-net act as a tradeoff between the sparsity and density of the reflectivity series. Hence, the choice of the regularization parameters depends on the actual geological setting of underground formations and the aim of seismic inversion.

If the underground formations are main thick, or the aim of inversion is to invert the main stratigraphic sequence interface of the underground formation, the value of ${\lambda }_1$ needs to be large. In this case, it is, in fact, equivalent to sparse-spike inversion. On the other hand, if the underground formations contain a thin interbed, which is important for reservoir characterization, we need to relatively increase the value of ${\lambda }_2$. One can argue that one is able to tune these regularization parameters, which is necessarily the case since there is now more flexibility in the elastic-net mathematical model. However, the flexibility increase provided by the regularization parameters of elastic net should be effectively taken advantage of. It is also one of the aims of the above discussions. Of course, the flexibility must match the actual geological setting. The point we want to express is that one has the flexibility to choose the appropriate regularization parameters according to the actual geological setting of underground formations and the aim of seismic inversion when they perform ENRI.

To achieve this goal, we use quality control to determine these regularization parameters in this paper. It needs available actual well logs. In quality control, we think the actual well logs are the “answer” to the inversion of near-well seismic traces. The actual well logs represent the actual geological setting of underground formations. The best value of regularization parameters is determined by doing quality control at well locations. That is, adjust the value of regularization parameters, obtain the inversion result from the near well seismic traces for each set of regularization parameters, and choose the one whose inversion result has the best match with the well log. Then, the chosen regularization parameters are adopted when we perform inversion for other seismic traces.

6. Conclusions

The quality of the inverted reflectivity series directly affects the quality of impedance. Sparse-spike inversion adopts a sparse regularization to impose sparsity on the inverted reflectivity series. The high resolution of the sparse-spike-like reflectivity series is obtained at the cost of sacrificing small reflectivity. However, the reflectivity series of underground formations from the actual impedance well log is not strictly sparse. It contains not only the sparse major large reflectivity, which corresponds to the main stratigraphic sequence interface of underground formation, but also small reflectivity, which represents weak reflectors of a thin interbed between major reflectivity. The large reflectivity is sparse, but the small reflectivity is dense.

In this paper, we adopt elastic-net regularization to replace sparse regularization. The elastic net is, in fact, the efficient combination of sparse regularization and dense regularization. Hence, ENRI combines the advantages of sparse regularization and dense regularization. From the synthetic seismic trace tests, we can see that the L2RI swings some major large reflectivity series to make the inverted reflectivity series and impedance over oscillate; the L1RI can obtain more blocky impedance, but will suppress many small reflectivity series; and in the inverted reflectivity series by ENRI, both large and small reflectivity series are all recovered well. From the field seismic data application, we can conclude that ENRI provides an effective new alternative method to invert impedance.