1 Introduction

Seismic impedance is an important parameter in reservoir characterization. Conventional impedance inversion from post-stack zero-offset seismic data is based on the convolution model. Wave-equation based inversion is rare, although it is capable to precisely describe seismic wave propagation and invert impedance model with higher resolution. One of the factors that make the wave-equation based inversion difficult are the multiple physical parameters involved in conventional wave equations. To obtain the impedance model from these wave equations, one can perform multi-parameter inversion (Tarantola, 1986) or assume one parameter known to reduce the unknown parameters or use the empirical formulas for velocity and density. All these methods are possible to cause errors in the inverted impedance model due to the cross-talks in multi-parameter inversion (Köhn et al., 2012; Pan et al., 2016), the inaccurate assumptions and the empirical formulae in single parameter inversion.

An alternative to avoid the multiple parameter problem in impedance inversion is to use a wave equation that contains only the impedance parameter. It is converted from the conventional one-dimensional (1D) wave equation according to a depth to time transformation (Santosa & Schwetlick, 1982), which only works with the 1D wave equation. With this equation, 1D impedance inversion can be performed using waveform tomography, an efficient method for reconstructing subsurface parameters (Tarantola, 1984; Virieux & Operto, 2009; Wang & Rao, 2009). For three-dimensional (3D) models, this 1D impedance inversion procedure will invert the 3D impedance model trace by trace, which may lead to lateral discontinuities, especially for models with salt structure.

To obtain spatially continuous inversion results, the truncated Fourier series parameterization scheme may be applied to the 3D impedance inversion. In this scheme, the impedance parameter is parameterized to the Fourier coefficients and then the inverted parameter will be changed from impedance to the Fourier coefficients. After finishing the inversion of Fourier coefficients, the final 3D impedance model will be reconstructed using these coefficients. This parameterization scheme was first applied to seismic traveltime and amplitude inversion for interface geometry and velocity variation (Wang, 1999; Wang & Houseman, 1995; Wang & Pratt, 1997, 2000). It has also been implemented in two-dimensional (2D) impedance inversion (Wang, 2016) and elastic parameter inversion from amplitude versus offset (AVO) data (Liu & Wang, 2020). With this scheme, the large- and small-scale components of the model can be reconstructed stepwise by gradually increasing the number of Fourier coefficients in a model-driven mode.

Combination of the 1D waveform tomography and the model parameterization scheme can be used for the inversion of a 3D impedance model without salt structure using the full-band seismic data (Gao et al., 2023). However, in the inversion of impedance model with salt structure, the salt and subsalt structures cannot be recovered when the full-band data are used for the inversion (Chen et al., 2020). This is because the inversion of the salt structures depends on the low-frequency seismic data. One can apply the data-driven strategy, where the seismic data is inverted from low to high frequency bands, for the Fourier coefficients inversion. In each frequency band, small- to large-valued numbers of Fourier coefficients will be updated. This strategy will increase the computation time. Therefore, the question is how to develop efficient inversion strategies to effectively invert the Fourier coefficients for a 3D impedance model with salt structure.

In this study, we first present the wave equation containing only the impedance parameter, the model parameterization and the theory of seismic waveform tomography. Secondly, we clarify the properties of the model parameterization and propose the joint data-driven and model-driven strategy for efficient inversion of the Fourier coefficients. Finally, we test the inversion method and strategy on a 3D impedance model with salt structure.

2 Theories for Impedance Inversion

2.1 Wave Equation Defined by Impedance

The wave equation defined in terms of impedance is as follows (Wang, 2016).

$$\frac{\partial }{\partial \tau }\left( {Z(\tau )\frac{\partial u(\tau ,t)}{{\partial \tau }}} \right) - \frac{Z(\tau )}{4}\frac{{\partial^{2} u(\tau ,t)}}{{\partial t^{2} }} = 0,$$
(1)

where \(u\) is the displacement, \(\tau\) is the vertical two-way traveltime, and \(Z\) is the impedance, the product of density and velocity. In the subsurface model, we refer to vertical two-way traveltime \(\tau\) as the depth-time, treated equally as two spatial variables. Because of the use of the depth-time \(\tau\), the wave equation contains only a single elastic parameter, the impedance parameter. We solve this equation numerically using a finite difference method (Hall & Wang, 2009).

2.2 Impedance Parameterization

We parameterize a 3D impedance model using a truncated Fourier series:

$$\begin{aligned} & \hat{Z} \left( {p,q,r} \right) = \\ &\sum\limits_{(l,m,n) = (0,0,0)}^{{\left( {L - 1,M - 1,N - 1} \right)}} {[a_{lmn} \cos (l\Delta k_{x} p)} \cos (m\Delta k_{y} q)\cos (n\Delta k_{\tau } r) \\ & \quad + b_{lmn} \sin (l\Delta k_{x} p)\cos (m\Delta k_{y} q)\cos (n\Delta k_{\tau } r) \\ & \quad + c_{lmn} \cos (l\Delta k_{x} p)\sin (m\Delta k_{y} q)\cos (n\Delta k_{\tau } r) \\ & \quad + d_{lmn} \sin (l\Delta k_{x} p)\sin (m\Delta k_{y} q)\cos (n\Delta k_{\tau } r) \\ & \quad + e_{lmn} \cos (l\Delta k_{x} p)\cos (m\Delta k_{y} q)\sin (n\Delta k_{\tau } r) \\ & \quad + f_{lmn} \sin (l\Delta k_{x} p)\cos (m\Delta k_{y} q)\sin (n\Delta k_{\tau } r) \\& \quad + g_{lmn} \cos (l\Delta k_{x} p)\sin (m\Delta k_{y} q)\sin (n\Delta k_{\tau } r) \\ & \quad + h_{lmn} \sin (l\Delta k_{x} p)\sin (m\Delta k_{y} q)\sin (n\Delta k_{\tau } r)], \end{aligned}$$
(2)

where \(\hat{Z}\left( {p,q,r} \right) \equiv \hat{Z}\left( {p\Delta x,q\Delta y,r\Delta \tau } \right)\) is an approximation to the true impedance model, \((\Delta x, \, \Delta y)\) are the space intervals, \(\Delta \tau\) is the time interval, \(\{ a, \, b, \, \cdots , \, h\}\) with subscripts \((l, \, m, \, n)\) are the Fourier coefficients, (L, M, N) are the numbers of Fourier coefficients in each dimension, \((\Delta k_{x}\),\(\Delta k_{y}\),\(\Delta k_{\tau } )\) are the fundamental wavenumbers, set as \(\Delta k_{x} = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } {N_{x} }}} \right. \kern-0pt} {N_{x} }}\), \(\Delta k_{y} = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } {N_{y} }}} \right. \kern-0pt} {N_{y} }}\) and \(\Delta k_{\tau } = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } {N_{\tau } }}} \right. \kern-0pt} {N_{\tau } }}\), where \((N_{x} ,N_{y} ,N_{\tau } )\) are the number of samples for the model in \((x,y,\tau )\) direction, respectively.

After the model parameterization, the parameters inverted will be changed from impedance to Fourier coefficients. When the inverted Fourier coefficients are obtained, the impedance model will be reconstructed by Eq. 2.

2.3 The Inversion Method for Fourier Coefficients

In the waveform tomography for the Fourier coefficients, we define the objective function in terms of data misfit,

$$\phi \left( {\mathbf{m}} \right) = \frac{1}{2}\sum\limits_{p}^{{}} {\sum\limits_{q}^{{}} {\left\| {{\mathbf{u}}_{obs}^{{\left( {pq} \right)}} - {\mathbf{u}}^{{\left( {pq} \right)}} ({\mathbf{m}})} \right\|^{2} } } ,$$
(3)

where \({\mathbf{u}}_{obs}^{{\left( {pq} \right)}} \equiv {\mathbf{u}}_{obs} (p\Delta x,q\Delta y)\) represents a single observed seismic trace at \((p\Delta x,q\Delta y)\), \({\mathbf{m}}\) is the “model” consisting of the Fourier coefficients, and \({\mathbf{u}}^{{\left( {pq} \right)}} ({\mathbf{m}})\) is the synthetic seismic trace simulated by Eq. 1.

The gradient vector, which is the first-order derivative of the objective function \(\phi \left( {\mathbf{m}} \right)\) with respect to the model parameters, taking coefficients \(a_{lmn}\) as an example, is

$$\frac{{\partial \phi \left( {\mathbf{m}} \right)}}{{\partial a_{lmn} }} = - \sum\limits_{p}^{{}} {\sum\limits_{q}^{{}} {({\mathbf{u}}_{obs}^{{\left( {pq} \right)}} - {\mathbf{u}}^{{\left( {pq} \right)}} ({\mathbf{m}}))\frac{{\partial {\mathbf{u}}^{{\left( {pq} \right)}} ({\mathbf{m}})}}{{\partial a_{lmn} }}} } .$$
(4)

The derivative \({{\partial {\mathbf{u}}^{{\left( {pq} \right)}} ({\mathbf{m}})} \mathord{\left/ {\vphantom {{\partial {\mathbf{u}}^{{\left( {pq} \right)}} ({\mathbf{m}})} {\partial a_{lmn} }}} \right. \kern-0pt} {\partial a_{lmn} }}\), according to the chain rule, can be explicitly expressed as

$$\frac{{\partial {\mathbf{u}}^{{\left( {pq} \right)}} ({\mathbf{m}})}}{{\partial a_{lmn} }}{ = }\sum\limits_{r} {\frac{{\partial {\mathbf{u}}^{{\left( {pq} \right)}} ({\mathbf{m}})}}{{\partial Z_{r}^{(pq)} }}\frac{{\partial Z_{r}^{(pq)} }}{{\partial a_{lmn} }}} ,$$
(5)

where \(Z_{r}^{(pq)} { = }Z{(}p\Delta x{,}q\Delta y,r\Delta \tau {)}\) is the 1D impedance variable associated with the specific pth in x and qth in y trace. The first-order derivative \({{\partial {\mathbf{u}}^{{\left( {pq} \right)}} ({\mathbf{m}})} \mathord{\left/ {\vphantom {{\partial {\mathbf{u}}^{{\left( {pq} \right)}} ({\mathbf{m}})} {\partial Z_{r}^{(pq)} }}} \right. \kern-0pt} {\partial Z_{r}^{(pq)} }}\) is the derivative of the (p, q)th seismic trace with respect to the rth impedance in the same trace, and the first-order derivative \({{\partial Z_{r}^{(pq)} } \mathord{\left/ {\vphantom {{\partial Z_{r}^{(pq)} } {\partial a_{lmn} }}} \right. \kern-0pt} {\partial a_{lmn} }}\) can be explicitly written as

$$\frac{{\partial Z_{r}^{(pq)} }}{{\partial a_{lmn} }}{ = }\cos (l\Delta k_{x} p)\cos (m\Delta k_{y} q)\cos (n\Delta k_{\tau } r).$$
(6)

Combining Eqs. 4–6, the first order derivative of the objective function with respect to \(a_{lmn}\) can be expressed as

$$\begin{aligned} & \frac{{\partial \phi \left( {\mathbf{m}} \right)}}{{\partial a_{lmn} }} = \\ & - \sum\limits_{p}^{{}} {\sum\limits_{q}^{{}} {\sum\limits_{r} {\left[ {\frac{{\partial {\mathbf{u}}^{{\left( {pq} \right)}} ({\mathbf{m}})}}{{\partial Z_{r}^{(pq)} }}({\mathbf{u}}_{obs}^{{\left( {pq} \right)}} - {\mathbf{u}}^{{\left( {pq} \right)}} ({\mathbf{m}}))} \right]\frac{{\partial Z_{r}^{(pq)} }}{{\partial a_{lmn} }}} } } .\end{aligned}$$
(7)

The derivatives to other Fourier coefficients are derived in the same way.

The “model” m (i.e., Fourier coefficients) are updated along the negative gradient direction, and the step-length is determined by a linear search method (Vigh et al., 2009).

3 The Inversion Strategy

In seismic waveform tomography, a commonly used hierarchical strategy is to invert the seismic data from low-frequency band to high-frequency band (Bunks et al., 1995; Wang & Rao, 2009, 2020) to get the large-scale and small-scale components of the model gradually. This strategy is referred to as the data-driven mode.

In this study, the model is approximated by the truncated Fourier series in Eq. 2. The values of (L, M, N) in Eq. 2 determine the closeness of the reconstructed model to the true model. Figure 1 presents the reconstructed impedance profiles at x = 200 with different (L, M, N)s. It can be seen that with small-valued (L, M, N), the reconstructed model represents the background trend of the true model (Fig. 1a). With large-valued (L, M, N), the reconstructed model is further refined (Fig. 1b). In the right panel of Fig. 1, we also show their differences between the true and the reconstructed models. It can be seen that when the values of (L, M, N) are increased, the differences are smaller. When (L, M, N) = (201, 201, 211), there is no difference between the true and the reconstructed model (Fig. 1c), meaning the reconstructed model is exactly the same to the true impedance model. Note that the true profile in Fig. 1c is extracted from a 3D model at x = 200, which will be shown in Fig. 2a.

Fig. 1
figure 1

Impedance profiles at x = 200 reconstructed by the Fourier coefficients (left panel) and the differences between the reconstructed and true models (right panel) when a (L, M, N) = (15, 15, 15), b (L, M, N) = (40, 40, 40), c (L, M, N) = (201, 201, 211)

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Fig. 2
figure 2

a The 3D impedance model. b The synthetic seismic cube

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This characterization of the model parameterization scheme is suitable for designing a model-driven strategy in the inversion. This strategy updates the Fourier coefficients with small valued (L, M, N) first, which represents the large-scale components of the model, and then updates the Fourier coefficients with large-valued (L, M, N), which indicates a further refined model.

In this study, we propose to combine the data-driven and model-driven strategies together. Specifically, in this joint strategy, Fourier coefficients with small-valued (L, M, N) will be updated by the low-frequency seismic data, while Fourier coefficients with large-valued (L, M, N) will be inverted using the high-frequency seismic data.

4 Application in Seismic Impedance Inversion

We test the proposed inversion scheme on the inversion for a 3D impedance model with salt structure.

Figure 2a is a 3D impedance model. It has the features of (a) complex structure; (b) high impedance contrast and steep interfaces between the salt and surrounding sedimentary layers; (c) thin layers inside/beneath the salt body; (d) thick salt structure and thin sediment layers. In the model, there are 400 samples in both the x and y directions with space interval of 25 m, and 420 samples in time direction with interval of 6 ms. Figure 2b is the corresponding seismic cube generated by the 1D wave equation simulation. It is assumed to have full frequency band components that is it has the low, medium and high frequency components. There are 1300 samples in each trace and the time interval is 2 ms. Figure 2b shows only the time range of 0.18–2.7 s. The wavelet used for the simulation is Ricker wavelet with the peak frequency of 25 Hz (Wang, 2015a). For field data test, the wavelet may be estimated by the generalized wavelet estimation method, which is calculated from the observed seismic data and well-independent (Wang, 2015b). With this estimated wavelet, the proposed method is able to invert the impedance model reliably.

In the waveform tomography, the starting model is a smoothed version of the true impedance model, calculated by a 3D smooth function. Figure 3a and b are two profiles at x = 200, extracted from the true model and the starting model, respectively. We first invert the impedance directly without using the model parameterization scheme by the Fourier coefficients, which is implemented trace by trace. This method is referred to as trace-by-trace method. In this test, each impedance trace is inverted from its corresponding seismic trace by using the data-driven strategy, which means that the impedance in each trace is inverted by the seismic trace from the low-frequency band to high-frequency band gradually. To recover the high-valued salt body/layers, the inversion starts from an ultra-low frequency band, [0, 1] Hz. In the test, five frequency bands are used for the impedance inversion, which are [0, 1], [0, 3], [0, 5], [0, 15], [0, 25] Hz. In each frequency band, the inversion will terminate when the iteration reaches 10. Figure 3c is the inverted impedance profile at x = 200 by the trace-by-trace method. It can be seen that impedance values beneath the salt body in positions around y = 150 and y = 250 are not well recovered. Besides, there are also some lateral discontinuities between 2.0 and 2.7 s.

Fig. 3
figure 3

Impedance profiles at x = 200 for the a true model, b starting model, c inverted model by the trace-by-trace method, d reconstructed model using the inverted Fourier coefficients from the full frequency band seismic data

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We then try to use the model-driven strategy to invert the Fourier coefficients by using the full frequency band seismic data. In the model-driven inversion procedure, considering the computation time and the convenience for sub-groups setting in the inversion, we select approximately twentieth to tenth of \(\left( {{{N_{x} } \mathord{\left/ {\vphantom {{N_{x} } {2}}} \right. \kern-0pt} {2}}{ + 1,}{{N_{y} } \mathord{\left/ {\vphantom {{N_{y} } {2}}} \right. \kern-0pt} {2}}{ + 1,}{{N_{\tau } } \mathord{\left/ {\vphantom {{N_{\tau } } {2}}} \right. \kern-0pt} {2}}{ + 1}} \right)\) as the starting values of (L, M, N), where \({N_{x}}, {N_{y}}, {N_{\tau }}\) are the number of samples of the model in three directions, and the increment of (L, M, N) is usually set as 1–2 times of the starting values of (L, M, N) (Gao et al., 2023). Therefore, in this test, the values of (L, M, N) of the Fourier coefficients start from (15, 15, 15) and terminate at (90, 90, 90). The increments of (L, M, N) is (25, 25, 25). Figure 3d is a profile at x = 200 from the reconstructed impedance model using the inverted Fourier coefficients by the model-driven strategy. It is obvious that thick salt body/layers with high impedance values are not recovered. The reason is that the seismic data is not divided into low to high frequency bands, instead the full frequency band seismic data is directly used to invert for the Fourier coefficients. The full frequency band data is equivalent to the high-frequency bands data in data-driven strategy, which mainly works for the recovery of the small-scale components of the model. Since the starting model in Fig. 3b is far away from the true model, only using the high frequency band seismic data cannot reconstruct the large-scale components of the model, as shown in Fig. 3d.

When the proposed strategy is applied in impedance inversion, the low-frequency seismic data and Fourier coefficients with small-valued (L, M, N) will be applied for the reconstruction of large-scale components of the salt body and boundaries between the salt body and sedimentary layers, while the high-frequency seismic data and Fourier coefficients with large-valued (L, M, N) will be used for the refinement of the boundaries and recovery of the thin layers. In this test, the frequency bands used are the same to that in the trace-by-trace method. For each frequency band, different numbers of Fourier coefficients are set. When the seismic data are filtered with [0, 1] Hz and [0, 3] Hz, the (L, M, N)s start with (15, 15, 15) and terminate at (40, 40, 40). When the seismic data are within [0, 5] Hz, the starting and final (L, M, N)s are (15, 15, 15) and (65, 65, 65), respectively. When the seismic data are filtered by high frequency bands, i.e. [0, 15] and [0, 25] Hz, the (L, M, N)s begin at (40, 40, 40) and (65, 65, 65), and stop at (65, 65, 65) and (90, 90, 90), respectively.

Figure 4 compares the impedance profiles at x = 200 between the true and reconstructed models at different stages. From Fig. 4, it can be seen that even when the starting impedance profile (Fig. 3b) is far away from the true model (Fig. 3a), with the ultra-low frequency seismic data, the large-scale components of the impedance model can be recovered, as shown in Fig. 4a and b. Based on the large-scale components, with large valued numbers of the Fourier coefficients and high frequency seismic data, the small-scale components of the impedance model are reconstructed and finally a gradually refined model is obtained in Fig. 4c and d. Compared to Figs. 3c and d, Fig. 4d gives a continuous impedance profile with well recovered salt and subsalt structure. However, if the seismic data lacks the low frequency components, which is useful for inverting the large-scale components of the model, the starting model is better to include these large-scale components.

Fig. 4
figure 4

Impedance profiles in different stages at x = 200. a The reconstructed impedance profile when (L, M, N) = (40, 40, 40) and the seismic data used for inversion is within [0, 1] Hz. b The reconstructed impedance profile when (L, M, N) = (40, 40, 40) and the seismic data is within [0, 3] Hz. c The reconstructed impedance profile when (L, M, N) = (65, 65, 65) and the seismic data is within [0, 5] Hz. d The reconstructed impedance profile when (L, M, N) = (90, 90, 90) and the seismic data is within [0, 25] Hz

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Figure 5 compares 8 pairs of true and reconstructed impedance profiles, which are at x = 20, 140, 260, 380 in Fig. 5a and b and at y = 50, 150, 250, 350 in Fig. 5c and d, respectively. The reconstructed impedance profiles by the proposed strategy at different positions are close to the true impedance profiles.

Fig. 5
figure 5

Comparisons between the true and reconstructed impedance profiles. The left panel is the impedance profiles at x = 20, 140, 260, and 380. The right panel is impedance profiles at y = 50, 150, 250, and 350. a and c are the true ones, while b and d are the reconstructed counterparts. The colour bar is the same to that in Fig. 3

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Figure 6 shows 4 groups of impedance traces at (x, y) = (20, 50), (140, 150), (260, 250), and (380, 350). In each group, the true, starting and reconstructed impedance traces are presented. The true and reconstructed impedance traces are nearly the same except some small fluctuations around the salt body.

Fig. 6
figure 6

Impedance trace comparisons at a (x, y) = (20, 50), b (x, y) = (140, 150), c (x, y) = (260, 250), and d (x, y) = (380, 350). The true, starting and reconstructed impedance traces are presented by the red, black and blue curves, respectively

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Figure 7 presents the true and reconstructed impedance slices at time 0.9 s, 1.8 s and 2.4 s, respectively. The true and reconstructed slices are also nearly the same, which prove the validity of the proposed strategy in 3D impedance inversion.

Fig. 7
figure 7

Time slices comparison between the true and reconstructed models. The left panel is for the true impedance model and the right panel is for the reconstructed model. The time slices are at 0.9 s (a), 1.8 s (b) and 2.4 s (c)

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5 Conclusion

In this study, the wave equation containing only the impedance parameter is used in seismic waveform tomography for the impedance model. The subsurface impedance model is parameterized by a truncated Fourier series and the inverted parameters are changed to Fourier coefficients. Waveform tomography and model parameterization are combined for the inversion of the salt model. We have proposed a joint data- and model-driven strategy to invert the Fourier coefficients with small-valued numbers by the low-frequency seismic data and to update these coefficients with large-valued numbers by the high-frequency seismic data. Model tests show that with this inversion strategy the salt and subsalt structures can be well recovered with high resolution and good spatial continuity.