Seismic Multi-Parameter Full-Waveform Inversion Based on Rock Physical Constraints

Abstract

Seismic multi-parameter full-waveform inversion (FWI) integrating velocity and density parameters can fully use the kinematic and dynamic information of observed data to reconstruct underground models. However, seismic multi-parameter FWI is a highly ill-posed problem due to the strong dependence on the initial model. An inaccurate initial model often leads to cycle skipping and convergence to local minima, resulting in poor inversion results. The introduction of prior information can regularize the inversion problem, not only improving the crosstalk phenomenon between parameters, but also effectively constraining the inversion parameters, enhancing the inversion efficiency. Multi-parameter FWI based on rock physical constraints can introduce prior information of underground media into the objective function of FWI. Taking a simple layered model as an example, the results show that the inversion strategy based on rock physical constraints can enhance the stability of inversion and obtain high-precision inversion results. Application to the international standard 1994BP model further confirms that the proposed inversion strategy has good applicability to complex geological models.

1. Introduction

Full-waveform inversion (FWI) for multi-parameter seismic wavefield reconstruction presents a strongly nonlinear and ill-posed optimization challenge, wherein conventional gradient-based algorithms frequently converge to local minima [1,2,3,4]. Furthermore, multi-parameter FWI is significantly influenced by crosstalk between parameters, meaning it is easy to map the error of one parameter to the update of other parameters [5,6,7,8], thus undermining its reliability as a basis for lithological interpretation. The regularization methodology enhances the stability of ill-posed inverse problems by integrating prior information about the model during the inversion process and takes the estimation of model parameters as constraint conditions [9,10].

The Tikhonov regularization method constitutes one of the most frequently utilized approaches in addressing ill-posed inversion problems. By integrating a regularization term with prior model constraints, it enhances stability to the inversion procedure. However, its isotropic smoothing nature inevitably introduces edge smear artifacts across geological interfaces. The Total Variation (TV) model constraint methodology can effectively mitigate the over-smoothing of boundary values that transpires during Tikhonov regularization inversion [11,12]. However, the improvement effect is highly contingent upon the selection of the regularization factor. When the regularization factor is properly chosen, more accurate inversion results can be obtained. In contrast, an improper selection of the regularization factor undermines the equilibrium between model parameters and residuals, consequently impairing the inversion accuracy. In practical engineering applications, seismic data volumes are customarily of considerable magnitude. The selection experiment of regularization factors not only demands copious amounts of time but also depends on the experience of professionals, which restricts the practical deployment of FWI in practical engineering to a certain extent. Domestic and foreign scholars have developed a variety of methods for the selection of regularization parameters. Among them, the Generalized Cross-Validation (GCV) method [13] and the L-curve method [14] are the most commonly adopted. Nevertheless, the application of adaptive regularization parameter selection in practical applications still confronts several predicaments. For example, the minimization of the GCV function is typically extremely smooth, rendering it difficult to locate mathematically [15]. Additionally, when errors are highly correlated, it becomes exceedingly difficult to obtain accurate regularization parameters [11]. With regard to the L-curve method, it requires a large number of experiments to determine the appropriate selection of regularization parameters [16]. Hence, it is necessary to study the optimization theory and method for the multi-parameter FWI of seismic waves.

Based on geological information, this paper considers regularized FWI based on geological prior information. In practical engineering, the prior information of underground strata, such as lithological information and petrophysical relationship information, can be obtained in various ways, such as drilling, laboratory analysis, and acoustic logging in research areas [17]. Such information, sourced directly from underground media, holds substantial referential significance and serves as dependable prior knowledge for inversion. The approach of introducing the above prior information as constraints into the inversion can be termed “rock physical constraint inversion”. The multi-parameter FWI expounded in this paper is intended to realize simultaneous high-precision imaging of wave velocity and formation density. To this end, the focus of this paper is primarily focused on prior information relevant to wave velocity and formation density and the conversion of such information into constrained information to achieve constrained inversion.

2. Materials and Methods

2.1. Construction of Rock Physical Prior Information

Subsequently, the commonly employed prior information regarding wave velocity and formation density in seismic exploration, along with their acquisition channels, are expounded.

2.1.1. Wave Velocity and Lithology

Seismic exploration serves as a pivotal geophysical method for characterizing subsurface geological architectures through the analysis of seismic wave propagation dynamics in underground strata. The development of underground media is inherently regulated by sedimentary environments, including the natural geographical conditions, climatic conditions, structural conditions, physical conditions of sedimentary media, and geochemical conditions of media [18]. As a result, in a given sedimentary setting, the strata have certain similarities, which provides a theoretical basis for the petrophysical constraint based on the prior information of lithology [19]. Prior lithological information can be acquired through multiple phases in seismic exploration, such as drilling, coring, and laboratory determination.

2.1.2. Wave Velocity and Formation Density

For the strata situated in an identical sedimentary environment, this study constructs the mathematical expression of prior information with the help of the petrophysical relationship between physical parameters. In practical engineering, petrophysical relationships are derived from well logging data, which are usually obtained in the ultrasonic frequency band (kHz level). Due to the petrophysical dispersion effect [20,21,22,23], the logging data obtained in the ultrasonic frequency band (kHz level) are different from the data obtained in the seismic frequency band (0–100 Hz). Consequently, when transferring petrophysical relationships from well logging data to seismic FWI workflows, two strategies can be adopted: (1) implementing frequency-dependent dispersion corrections to the logging data [24] or (2) utilizing frequency-independent empirical relationships.

Gardner’s equation [25] represents an empirical model for estimating bulk density from P-wave velocity. Derived through the statistical fitting of a large dataset of experimental measurements, this relationship assumes a stable power-law correlation between density and velocity under specific lithological conditions. The robustness of Gardner’s equation in seismic applications stems from three key attributes: (1) the statistical consistency of its empirical foundation; (2) the inherently low frequency sensitivity of bulk density; and (3) parameter insensitivity to viscoelastic dispersion mechanisms. These characteristics collectively ensure its applicability across the seismic frequency spectrum. Gardner’s equation is expressed as follows:

$$\ln (\rho )=a\ln (v_p)+b$$

(1)

where v_p represents P-wave velocity, ρ is rock density, and a and b are constants.

Parameters a and b in Equation (1) exhibit regional variations. With reference to the measured data (presented in

Table 1. Actual measurement data from outcrop in New Zealand. Formation abbreviations: UDCL (Unaltered Dense Coherent Lava), ADCL (Altered Dense Coherent Lava), UBLM (Unaltered Brecciated Lava Margin), ABLM (Altered Brecciated Lava Margin), UI (Unaltered Intrusions), and AI (Altered Intrusions) [26].

Abbreviation of Formation	Number of Samples	Velocity of P-Wave (m∙s−1)	Density of Rock (kg∙m−3)
UDCL	16	4472	2630
ADCL	59	3701	2330
UBLM	6	2390	1911
ABLM	33	3112	2161
UI	41	4767	2645
AI	28	4118	2523

) collected in [26] from a particular outcrop in New Zealand, the measured data were plotted within a logarithmic coordinate system (presented in Figure 1) to validate the empirical equation proposed in [25]. Red dots represent the measured data, and the blue solid line represents the fitted line of the velocity and density of the measured data.

The following parameters were selected for a quantitative assessment of the goodness of fit:

$$\begin{array}{c}Sum Squard Error (SSE):S={\displaystyle \sum _{i=1}^m{\left(y_i-{\stackrel{⌢}{y}}_i\right)}^2}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}R\text{-}square:R^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^m{\left(y_i-{\stackrel{⌢}{y}}_i\right)}^2}}{\frac{1}{m}{\displaystyle \sum _{i=1}^m{\left(y_i-{\overline{y}}_i\right)}^2}}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}Adjusted R\text{-}square:{R^2}_{adjusted}= 1-{\displaystyle \frac{(1-R^2)(m-1)}{m-p-1}}\\ Root Mean Squard Error (RMSE):R=\sqrt{{\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m{\left(y_i-{\stackrel{⌢}{y}}_i\right)}^2}}\end{array}$$

(2)

where y represents the real value, $\stackrel{⌢}{y}$ represents the predicted value, m is the number of samples, and p is the number of features. Through calculation, the evaluation results were as follows: the SSE value is 0.0007242, R-square is 0.9911, adjusted R-square is 0.9889, and RMSE is 0.01346. The results show that the above relationship between velocity and density is applicable to the outcrop data and has a high degree of fitting. This provides a theoretical basis for subsequent rock physical constraints based on the relationships among physical parameters.

In summary, the rock physical constraints discussed in this paper can be divided into two levels. At the primary level, presupposing that extant data validate the presence of several rock types in the study area, and the physical property parameters are known, an inversion strategy circumscribed by lithological information can be adopted. At the secondary level, it is assumed that the existing data show that there is a certain correlation among the physical property parameters of formation in the study area, and the constrained inversion can be realized through the petrophysical relationship.

2.2. The Objective Function of Inversion Based on Rock Physical Constraints

FWI is a data-processing technique used to reconstruct the quantitative characteristics of subsurface media from seismic wave propagation data. Numerically, the estimation procedure of FWI typically depends on minimizing the misfit function between the model data and the observed data. The inversion iteration starts with an initial conjecture of the parameter model. Subsequently, FWI records the propagation data of a wave and computes the residual between it and the observed model. The update direction is determined by calculating the gradient of the objective function with respect to the model parameters. Eventually, the step size of the search is determined based on a line search algorithm, and the model parameters are iteratively updated to minimize the objective function. By repeating the above steps, the closest estimate to the true model can be obtained. In this paper, we initially introduce the construction approach of the objective function and the equation for calculating the gradient in the multi-parameter FWI of seismic waves. According to the core principle of FWI, the objective function Q(m) for the multi-parameter FWI of seismic waves can be defined in the following form:

$$Q(\mathbf{m})={\displaystyle \frac{1}{2}}{\displaystyle \sum _{x_s}{{\displaystyle \sum _{x_r}{\displaystyle \sum _t\left(\mathbf{R}\mathbf{w}\left(t,\mathbf{x},{\mathbf{x}}_s;\mathbf{m}\right)-{\mathbf{d}}_{obs}\left(t,{\mathbf{x}}_r,{\mathbf{x}}_s\right)\right)}}}^2}$$

(3)

where m denotes the parameter vector of the model medium, x_s represents the spatial coordinates of the excitation point, x_r stands for the spatial coordinates of the receiver point, t is the observation time, R signifies the constraint on the receiver point position, w(t,x,x_s;m) is the simulated full wavefield information, and d_obs(t,x_r,x_s) is the observed wavefield.

In this study, the variable-density acoustic wave equation is adopted as an example for multi-parameter FWI testing. In this equation, m = [v(p), ρ(r)]^T, where v is the velocity of the acoustic wave, ρ denotes the medium density, p is the acoustic pressure, r is the spatial coordinate vector of the receiver point, and ^T is the transpose operator.

To mitigate the ill-posed problem in the inversion problem, the traditional TV regularization strategy is used to formulate the regularization term based on the L1 norm. The research in [11] indicates that TV regularization can effectively mitigate the excessive smoothing of boundaries encountered in Tikhonov regularization, resulting in more pronounced distinctions between features and background regions, as well as sharper and more well-defined edges in the reconstructed target structures. As a regularization method based on penalty functions, TV regularization introduces an additional TV regularization constraint term to the conventional FWI objective function, thereby constructing a new objective function:

$$Q_{TV}(\mathbf{m})=Q(\mathbf{m})+{\lambda }_{TV}{\mathrm{\Phi }}_m(\mathbf{m})$$

(4)

where λ_TV is the TV regularization factor, Φm(m) is the model parameter objective function, and Q_TV(m) is the objective function under TV regularization.

The objective function for the multi-parameter FWI of seismic waves, incorporating rock physical constraints as proposed in this study, is formulated as follows.

2.2.1. The Inversion Objective Function of Lithological Constraints

When incorporating prior information that encompasses lithological information, the inversion object function is structured as follows:

$$Q_L(\mathbf{m})=Q(\mathbf{m})+{\lambda }_L{\displaystyle \sum _{i=1}^N{\mathrm{\Phi }}_L(\mathbf{m})}$$

(5)

where Q_L(m) is the objective function under lithological constraints, λ_L is the lithological regularization factor, Φ_L(m) is the parameter penalty term, and N is the total number of spatial positions of model parameters. If the prior geological information indicates that there are n potential rock types in the study area, and the physical property parameters of each type of lithology are normally distributed, the average value m* and standard deviation σ of the model parameters of each type of lithology are derived from the outcomes of laboratory analyses.

In order to integrate the clustering characteristics of physical parameters offered by the prior information, model parameter Φ_L(m) is defined as

$${\mathrm{\Phi }}_L(\mathbf{m})={\left({\displaystyle \frac{{\mathbf{m}}_i-{\mathbf{m}}_{n(i)}^{*}}{{\mathbf{\sigma }}_{n(i)}}}\right)}^2$$

(6)

where i is the index of the model element, and n(i) represents the category of the cluster with the smallest penalty term for the ith element. Logging data and laboratory analysis generally facilitate the retrieval of parameters such as velocity and density. In this regard, the corresponding model penalty term can be formulated as follows:

$${\mathrm{\Phi }}_L(\mathbf{m})={\left({\displaystyle \frac{v_i-v_{n(i)}^{*}}{{\sigma }_{vn(i)}}}\right)}^2+{\left({\displaystyle \frac{{\rho }_i-{\rho }_{n(i)}^{*}}{{\sigma }_{\rho n(i)}}}\right)}^2$$

(7)

2.2.2. The Inversion Objective Function of Petrophysical Relationship Constraints

The incorporation of novel constraint terms mandates a redefinition of the inversion objective function:

$$Q_R(\mathbf{m})=Q(\mathbf{m})+{\lambda }_R{\mathrm{\Phi }}_R(\mathbf{m})$$

(8)

where Φ_R(m) is the model term based on the model penalty function norm, and λ_R is the petrophysical relationship regularization factor, which controls the trade-off parameter between the weight of the data misfit and the model term in the objective function under petrophysical relationship constraints Q_R(m). In order to integrate the petrophysical relationships (commonly in explicit forms) among different parameters, this study formulates the model constraints as follows:

$${\mathrm{\Phi }}_R(\mathbf{m})={\displaystyle \frac{1}{2}}{\displaystyle \sum _x{\left[m_1-f\left(m_2\right)\right]}^2}$$

(9)

where x is the model space coordinate, m₁ = m₁(x) and m₂ = m₂(x) are different physical parameters, and f = f(x) is a predefined function that maps m₂ to m₁. Insights from geological exploration indicate that, governed by the uniform sedimentary environment and sedimentary facies, a robust correlation exists between the seismic wave velocity of strata and rock density. Moreover, drilling data can provide a reliable petrophysical parameter relationship. Therefore, m₂ is the density, m₁ is the velocity, and Equation (9) becomes

$${\mathrm{\Phi }}_R(\mathbf{m})={\displaystyle \frac{1}{2}}{\displaystyle \sum _x{\left[\rho -f\left(v\right)\right]}^2}$$

(10)

where Φ_R(m) plays a pivotal role in constraining the updated model to remain within the trend delineated by ρ − f(v).

The gradient of Φ_R(m) relative to m is:

$${\nabla }_{\mathbf{m}}{\mathrm{\Phi }}_R(\mathbf{m})=\left[\rho -f\left(v\right)\right]{\displaystyle \frac{\partial \left[\rho -f\left(v\right)\right]}{\partial \mathbf{m}}}$$

(11)

where ∂ is the partial derivation operator, and $\nabla$ represents the gradient.

3. Results

3.1. Data

3.1.1. The Layered Model

A layered model was established to verify the applicability of the inversion framework proposed in this paper. In assuming that the model parameter information was obtained from drilling data and laboratory experiments, it is confirmed that there are four potential rock types in the target area. The average values and standard deviations of their physical property parameters are as follows:

$$\begin{array}{l}{v}_{n(1)}^{*}=1500 \mathrm{m}/\mathrm{s}, {\rho }_{n(1)}^{*}=1929.2 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3, {\sigma }_{n(1)}=200;\\ v_{n(2)}^{*}=2000 \mathrm{m}/\mathrm{s}, {\rho }_{n(2)}^{*}=2073.1 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3, {\sigma }_{n(2)}=200;\\ v_{n(3)}^{*}=2500 \mathrm{m}/\mathrm{s}, {\rho }_{n(3)}^{*}=2192 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3, {\sigma }_{n(3)}=200;\\ v_{n(4)}^{*}=3000 \mathrm{m}/\mathrm{s}, {\rho }_{n(4)}^{*}=2294.3 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3, {\sigma }_{n(4)}=200.\end{array}$$

(12)

The prior information represented by Equation (12) is graphically illustrated in Figure 2. White solid circles represent the average values of model parameters corresponding to different lithologies, the circular regions of different colors represent the range of the standard deviation of the corresponding lithology, and the blue solid line represents the fitted line of the velocity and density. It can be seen from Figure 2 that the lithological information based on the average value and standard deviation of physical parameters, as well as the petrophysical relationship information between velocity and density, can aptly describe the relationship between seismic wave velocity and medium density. Consequently, the prior information represented by Equation (12) can be employed as constraint terms in the multi-parameter FWI of seismic waves.

The velocity model and density model constructed from the above data are shown in Figure 3a,b. The layered model comprises five strata with increasing velocity and density trends at greater depths as shown in Figure 3a,b. Two basal strata exhibit normal faults, inducing displacement and interface undulations between adjacent layers. When conducting multi-parameter inversion, homogeneous geological models were constructed as the initial model to validate the reliability of the proposed algorithm. Referencing the top-layer parameters of the layered model, the initial velocity and density were set to 2250 m/s and 2135 kg/m³, respectively, as shown in Figure 3c,d.

3.1.2. International Standard Model

In this study, the 1994 BP statics benchmark model (1994BP model) [27] was adopted as the international standard model to verify the applicability of the proposed inversion strategies. The 1994BP velocity model is shown in Figure 4a. It comprises multiple geological units with distinct velocity characteristics: a seawater layer at 0 m depth, layered sedimentary layers developed in the shallow subsurface zones, fault structures developed in the middle and deep formations, and high-speed dense rock masses developed in the base. If the relationship between velocity and density satisfies lnρ = 0.25 lnv_p + 5.7366, derived from the data of the layered model, the density model is obtained, as shown in Figure 4b. When conducting multi-parameter inversion, the Gaussian smoothing model of the 1994BP model was selected as the initial model, as shown in Figure 4c,d.

3.2. Numerical Experiments of the Layered Model

3.2.1. Numerical Experiments Based on Total Variation Regularization

To validate the influence of TV regularization on FWI, the model shown in Figure 3 was selected to carry out TV regularization inversion experiments. In Figure 3a,b, the black crosses denote the positions of the source points, while the red circles signify the positions of the receiver points. Both the source and receiver points were situated at a depth of z = 50 m. Spanning from x = 0 m to 600 m, a source point was established every 20 m, amounting to a total of 30 shot points. A receiver point was set every 10 m, and a total of 60 channels of data were collected. The forward simulation was carried out with a 12.5 Hz Ricker wavelet, a time interval of 0.5 ms, and a total simulation duration of 0.6 s. The initial model was a homogeneous medium model with a velocity of 2250 m/s and a density of 2135 kg/m³, as illustrated in Figure 3c,d. There were three termination criteria for the inversion iteration: first, the total number of iterations is 300; second, the ratio of the relative objective function Q_k/Q₀ is less than 10⁻³; third, the step size is smaller than 10⁻⁵. Equation (5) was used as the objective function to conduct the FWI experiment based on L-BFGS (details are provided in Appendix A).

When conducting the TV regularization experiments, regularization factor λ_TV was determined through manual adjustment. To identify an optimal regularization factor, this study carried out model experiments under different TV regularization factors while maintaining other inversion parameters constant. The model experiments were the following: without loading TV regularization (λ_TV = 0), with λ_TV = 500, and with λ_TV = 1000. The inversion results obtained are shown in Figure 5.

From the velocity inversion results without regularization, the interfaces of middle and deep formations were indistinct, particularly the top boundary of the bottom high-velocity formation, which is difficult to identify in the inversion result. In the density inversion result, the inversion outcomes on both sides of the middle low-density formation exhibit significant deviation from the true model, and the top boundary of the deep high-density formation is also obscure. After the implementation of TV regularization, the interface of the low-velocity formation was clear under λ_TV = 500, and the model parameters showed a closer resemblance to those of the true model. The density inversion results also yield analogous conclusions. In the density inversion result, formation boundaries can be distinguished more distinctly, and the density value is closer to that of the true model. When λ_TV was set to 1000, the contour of the high-velocity body and the inversion accuracy of parameters were relatively unsatisfactory in the velocity inversion results. The reconstructed parameter values of the high-density formation in the middle and deep parts were remarkably close to those of the true model, though the shallow formation density inversion result still shows deviations from the true model.

A cross section (the black dashed line AA′ in Figure 3a) was selected for the comparative analysis of model parameters. Figure 6a,b shows the velocity and density curves under section AA′, respectively. Black solid lines represent the true model value, blue dashed lines represent the inversion results with λ_TV = 0, green dotted lines represent the inversion results with λ_TV = 1000, and red dashed lines represent the inversion results with λ_TV = 500. From the comparison curves of the velocity inversion results in Figure 6a, it can be seen that the addition of TV regularization can improve the accuracy of FWI, especially for the low-velocity formation with a depth ranging from 100 m to 180 m. It can be observed from the enlarged view in Figure 6a that the inversion results under λ_TV = 500 were closest to the true model; the velocity parameter curve does not have as strong oscillations as under unconstrained conditions. However, when the regularization factor was not appropriately selected, the inversion model deviated significantly from the true values. The same conclusion can be drawn from the comparison curves of the density inversion results shown in Figure 6b. The addition of TV regularization can effectively improve the inversion accuracy, especially for the middle and shallow formation, the oscillation of the density parameter curve was the least when λ_TV = 500, and the model density value was closer to the true model.

Figure 7a shows the objective function curves of the multi-parameter FWI based on TV regularization under different selections of TV regularization factors. Black solid lines represent the inversion results with λ_TV = 0, blue dashed lines represent the inversion results with λ_TV = 1000, and red dotted lines represent the inversion results with λ_TV = 500. Compared with the inversion results without TV regularization, the number of inversion iterations was larger, the objective function was smaller, and the convergence during the inversion process was relatively stable when λ_TV = 500. The value of the inversion objective function was relatively large when λ_TV = 1000. Figure 7b shows the velocity and density reconstruction error curves corresponding to the selection of TV regularization factors. The number of inversion iterations was larger, the reconstruction error of the model parameters was smaller, and the inversion was more accurate when λ_TV = 500.

Table 2. Comparison of number of iterations, objective function, and parameter reconstruction errors of the layered model based on TV regularization with different TV regularization factors.

Inversion Parameters	λTV = 0	λTV = 500	λTV = 1000
Number of iterations	39	48	38
Objective function	3.7335 × 108	3.4017 × 108	6.8019 × 108
Reconstruction error of velocity	0.4339	0.4208	0.4356
Reconstruction error of density	0.4066	0.3950	0.4084

lists the number of inversion iterations, the objective function values, and the parameter reconstruction errors of the layered model under different selections of TV regularization factors. In the three sets of numerical experiments conducted, the number of inversion iterations was larger, the inversion objective function was smaller, and the reconstruction errors of velocity and density were smaller when λ_TV = 500.

3.2.2. Numerical Experiments Based on Lithological Constraints

In order to validate the influence of lithology information constraints on FWI, the model depicted in Figure 3 was chosen to perform inversion experiments. FWI experiments were conducted with Equation (5) as the objective function. It was assumed that the average values and standard deviations of the physical parameters of potential rock types within the target area were acquired, presented in Equation (12). The model penalty term based on lithology information can be obtained by substituting the above prior information into Equation (7). Another critical challenge in FWI is cycle skipping [28]. To reduce the dependence of inversion on the initial model, the inversion framework proposed in this paper employs the multiscale strategy with stage-wise data injection from low to high frequencies. Multiscale inversion was carried out using a 3 Hz starting frequency, 7.5 Hz band data, and 12.5 Hz original data following Wiener low-pass filtering.

The assignment of regularization factors needs to balance the contributions of the data misfit term Q(m) and the constraint term Φ_L(m). Considering the orders of magnitude of Q(m) and Φ_L(m), lithological regularization factors λ_L were set to 0 and 10⁶, respectively, representing the unconstrained and lithological constrained inversion strategies. The obtained inversion results are illustrated in Figure 8. It can be seen from Figure 8a,c that the interface information of the middle formation can be effectively recovered in the velocity inversion model with λ_L = 0 and λ_L = 10⁶. However, in terms of the formation velocity parameter values, the inversion results with constraints show closer agreement with the true model. Specifically, compared to the unconstrained inversion results, the inverted values of the deep high-velocity formation after adding constraints are closer to 3000 m/s. As is shown in Figure 8b,d, for the density inversion in the deep strata, the inversion result obtained under lithological constrained inversion was closer to the true density model.

Section AA′ was selected for the purpose of comparing the inversion parameters under lithological constraint with different lithological regularization factors. Figure 9a,b presents the velocity and density curve diagrams of section AA′. Black solid lines represent the true model, blue dotted lines show the inversion results under constrained strategy, and red dashed lines represent the inversion results under the lithological constrained strategy. According to the comparison curves of the velocity inversion results, adding lithological constraints can make the inversion parameter values closer to the true model, especially in the deep zones of the layered model. Meanwhile, higher-precision density inversion results can be obtained by introducing lithological constraints, especially in the deep zones. In this numerical experiment, the introduction of lithological constraints could enhance the inversion accuracy, especially in the deep zones of the model, and produce inversion models that are more in line with the prior information.

To further analyze the influence of lithological constraints on the inversion results, a scatter plot of the velocity and density is shown in Figure 10. White circles represent the true model’s results, blue circles represent the inversion results under the unconstrained strategy, and red circles represent the inversion results under the lithological constrained strategy. When there was no constraint, the inversion results of the model were rather scattered around the true model dataset, and it is impossible to clearly distinguish the lithology corresponding to the model parameters. However, after adding lithological constraints, the inversion results can move towards the clustering direction contained in the prior information. Most of the results fall within the standard deviation range provided by the prior information, and the number of main lithology types developed in the underground media of the research area can be more intuitively distinguished.

3.2.3. Numerical Experiment Based on Petrophysical Relationship Constraints

To verify the effect of petrophysical relationship constraints on multi-parameter FWI, experiments were conducted using the model shown in Figure 3. The corresponding model penalty term can be formulated as follows:

$${\mathrm{\Phi }}_R(\mathbf{m})={\displaystyle \frac{1}{2}}{\displaystyle \sum _x{\left(\rho -310v^{\frac{1}{4}}\right)}^2}$$

(13)

The constrained FWI objective function listed in Equation (8) can be expressed as:

$$Q_R(\mathbf{m})=Q(\mathbf{m})+{\lambda }_R{\displaystyle \frac{1}{2}}{\displaystyle \sum _x{\left(\rho -310v^{\frac{1}{4}}\right)}^2}$$

(14)

The gradient of Φ_R(m) with respect to m(v, ρ) is

$$\begin{array}{l}{\nabla }_v{\mathrm{\Phi }}_R(\mathbf{m})=\left(\rho -310v^{{\displaystyle \frac{1}{4}}}\right)\cdot \left(-{\displaystyle \frac{165}{2}}{v}^{-{\displaystyle \frac{3}{4}}}\right)\\ {\nabla }_{\rho }{\mathrm{\Phi }}_R(\mathbf{m})=\rho -310v^{{\displaystyle \frac{1}{4}}}\end{array}$$

(15)

The multi-parameter FWI experiments based on the L-BFGS algorithm were carried out by setting the petrophysical relationship regularization factor λ_R = 0 and λ_R = 3000, respectively. Other inversion parameters were identical to those in the experiment of 3.2.2. The resultant inversion outcomes are graphically presented in Figure 11. Upon analyzing the velocity inversion results, it becomes evident that with the incorporation of constraints, the inversion image of the middle—layer formation interface attains a higher degree of clarity. Moreover, the parameter values of the deep high-velocity layer demonstrate a closer approximation to those of the true model. When examining the density inversion results, it is observable that after the addition of petrophysical relationship constraints, the formation interfaces become more distinct. The artifact issues within the middle low-density formation are notably ameliorated. The inversion parameters of the low-density formation with depths of 110 m to 170 m and deep-seated formation show a closer proximity to the true model.

Section AA′ was selected for the comparison of inversion parameters. Figure 12a,b displays the velocity and density curves, respectively, under section AA′. Black solid lines represent the true model results, blue dotted lines show the inversion results under the unconstrained strategy, and red dashed lines represent the results under the petrophysical relationship constrained strategy. From the velocity inversion results, it can be inferred that the addition of petrophysical relationship constraints can reduce the inversion oscillations, bringing the inversion parameter values closer to those of the true model and resulting in higher inversion accuracy. The same conclusion can be drawn from the density inversion results.

To further analyze the influence of petrophysical relationship constraints on the model inversion results, the scatter plot of the velocity and density is shown in Figure 13. White circles represent the true model results, blue circles represent the inversion results under unconstrained strategy, and red circles represent the inversion results under the petrophysical relationship constrained strategy. The inversion results of the model are rather dispersedly distributed around the fitted line without constraints. After adding petrophysical relationship constraints, the inversion results fit in the direction approaching the fitted line, thus obtaining inversion results that were more consistent with the geological prior information.

The inversion results of the above three constraint strategies (corresponding to the numerical experimental results in Section 3.2.1, Section 3.2.2 and Section 3.2.3, respectively) are compared with each other (presented in

Table 3. The number of iterations, objective function, and reconstruction error of the layered model under different inversion strategies.

Inversion Parameters	TV Regularization Strategy	Lithological Constrained Strategy	Petrophysical Relationship Constrained Strategy
Number of iterations	48	61	87
Objective function	3.4017 × 108	1.5627 × 108	1.0651 × 108
Reconstruction error of velocity	0.4208	0.3947	0.4026
Reconstruction error of density	0.3950	0.3923	0.3476

). Compared with TV regularization, the inversion strategy constrained by rock physical information features a simpler process, eliminating the need for parameter selection, and the inversion results are closer to the true model.

Next, the applicability of the multi-parameter FWI strategy based on rock physical constraints proposed in this paper was verified in a complex geological model.

3.2.4. Numerical Experiment of Standard Geological Model

In the development of energy resources, the acquisition of logging information is more convenient than that of laboratory analysis results, and the amount of logging data is larger. Therefore, prior information containing petrophysical relationships is of greater practical significance than lithological information in inversion. In this study, numerical experiments were carried out with the 1994BP model to verify the applicability of the petrophysical relationship constrained inversion strategy proposed in this paper. Multiscale inversion was carried out using a 10 Hz starting frequency, 15 Hz band data, and 40 Hz original data following Wiener low-pass filtering. The inversion results obtained by setting the regularization factor λ_R to 0 and 500, respectively, are shown in Figure 14.

The velocity inversion results demonstrate that the constrained inversion strategy proposed in this paper improved the continuity of middle formations and achieved closer alignment with the true-velocity model in the middle and deep zones. In the density inversion results, under unconstrained inversion, there are many inversion artifacts in the middle zone of the model, especially in the low-density formations below the relatively high-density formation. After adding constraints, the artifact phenomenon of the middle formation improved, and the interfaces and parameter values of the deep high-density formation were closer to those of the true model.

A horizontal cross section of the 1994BP model (referring to the z = 250 m section indicated by the white dashed line AA′ in Figure 4a) and a vertical cross section (referring to the x = 800 m section indicated by the white dashed line BB’ in Figure 4a) were selected for the comparative analysis of velocity and density, and the inversion results are shown in Figure 15a,c and Figure 15b,d, respectively. Black solid lines represent the true model results, blue dotted lines show the inversion results under the unconstrained strategy, and red dashed lines represent the results under petrophysical relationship constraints. The accuracy of velocity inversion under these two inversion strategies was comparable. In the regions at both lateral sides and in the middle–deep part of the model, the results obtained after incorporating petrophysical relationship constraints were more congruent with the true model. After the addition of petrophysical relationship constraints, the density inversion accuracy significantly improved, with local oscillations mitigated, and the model values closer to those of the true model.

To better analyze the impact of prior information on the inversion results, a velocity–density scatter plot was drawn, as shown in Figure 16. It can be seen that after adding the prior information of the petrophysical relationship, the inversion objective function incorporates a penalty term based on the petrophysical relationship. As a result, the inversion results converge towards the direction of the prior petrophysical relationship. In the scatter plot, this is manifested as the inversion data points being closer to the fitted line than the data points without constraints.

Table 4. Number of iterations, and the objective function and reconstruction errors of the 1994BP model under the unconstrained strategy and petrophysical relationship constraints.

Inversion Parameters	Unconstrained Strategy	Petrophysical Relationship Constrained Strategy
Number of iterations	152	302
Objective function	7.0867 × 109	1.2560 × 109
Reconstruction error of velocity	0.4269	0.3575
Reconstruction error of density	0.6454	0.4869

shows the comparison of the objective functions and model parameter reconstruction errors for the 1994BP model between unconstrained inversion and petrophysical relationship constrained inversion. White circles represent the true model results, blue circles represent the inversion results under the unconstrained strategy, red circles represent the inversion results under petrophysical relationship constraints, and the green line is the fitted line of the true model. From the data in the table, it can be seen that the value of the inversion objective function, velocity reconstruction error, and density reconstruction error decrease by approximately 82%, 12%, and 25%, respectively. The value of the inversion objective function, and the inversion accuracies of the velocity and density, were all improved.

In summary, introducing petrophysical relationships among different physical property parameters into the inversion objective function can reduce the degrees of freedom of the inversion problem, mitigate the ill-posed nature of the inversion, enhance the stability and accuracy of the inversion, and obtain a reconstructed model that is more consistent with geological prior information. This constrained inversion method achieved good application results in both simple models and standard geological models.

4. Conclusions

To mitigate the ill-posed nature of the multi-parameter FWI of seismic waves, this paper proposes a rock physical constraint inversion framework. The proposed framework synergistically combines the following: lithological constraints and petrophysical relationship constraints embedded into the FWI objective function through regularization terms, the multiscale inversion strategy to reduce the cycle skipping problem by progressively incorporating higher-frequency data components, and the L-BFGS optimization algorithm to enhance convergence efficiency.

The inversion results of the simple layered model show that the proposed rock physical constraint inversion strategies based on lithological information and the relational expressions of physical property parameters can enhance the stability of inversion, reduce the parameter oscillation, and obtain multi-parameter inversion results with higher accuracy. Compared with unconstrained FWI, the constrained inversion framework proposed in this study can improve the density inversion accuracy without reducing the velocity inversion accuracy, thereby both the reconstruction error of velocity and density were reduced. The inversion results of the standard geological model 1994BP indicate that the multi-parameter inversion strategy of seismic waves based on rock physical constraints has good applicability to complex models.

While the proposed FWI framework demonstrates compatibility with diverse prior constraints, its current implementation is restricted to lithological and petrophysical relationship constraints. Future work will focus on other types of constraints to improve the effect of inversion. Meanwhile, measured data will be used to verify the applicability of the proposed FWI framework.