# A Case Study Comparing Methods for Coal Thickness Identification in Complex Geological Conditions
## 3 Methods  
### Seismic Wave Impedance Inversion Based on Well Log Constraints  
Seismic wave impedance inversion is a technique used to infer the wave impedance characteristics of strata (or rock layers) by utilizing seismic data [25]. This method links seismic reflection profiles with lithology and physical property profiles, enabling a comparison between seismic data and drilling data [26; 27]. The technical process primarily includes the following steps: **Initial Model Setup:** Set an initial wave impedance model of the strata and perform seismic forward modeling to generate synthetic seismic records. **Comparison and Adjustment:** Compare the synthetic records with actual seismic data to adjust the main parameters of the geological model. Perform forward modeling again to obtain new synthetic records. **Iteration:** Continuously compare the new synthetic records with the actual seismic records, iteratively modifying the geological model until the synthetic seismic data best match the actual seismic records. The geological model obtained when the error is minimized is considered the result of the wave impedance model inversion, as shown in Figure 5.  
Given a seismic record, as follows:  
\[S(t)=R(t)\times W(t)+N(t) \tag{1}\]  
\(S(t)\) is the seismic record, \(R(t)\) is the reflection coefficient of the interfaces, \(W(t)\) is the seismic wavelet, and \(N(t)\) is noise. The goal of seismic inversion is to obtain the reflection coefficient series \(R(t)\), which only reflects the changes in subsurface interfaces. Through the reflection coefficient series, the relationship between the velocity and density  
Figure 5: Workflow diagram for well log-based impedance inversion. of each reflection layer is established, allowing for the prediction of the distribution of subsurface media. Assume the initial seismic wavelet is \(W_{1}(t)\), and the initial reflectivity series is \(R_{1}(t)\). The formula for the synthetically generated record \(y_{1}(t)\) is as follows:  
\[y_{1}(t)=R_{1}(t)\times W_{1}(t)+N_{1}(t) \tag{2}\]  
The cross-correlation between \(x(t)\) and \(y_{1}(t)\) is given by the following correlation coefficient \(R_{xy}\):  
\[R_{xy}(s)=\frac{1}{n}{\sum_{t=1}^{n}{x(t)y_{1}(t+s)}} \tag{3}\]  
By adjusting \(y_{1}(t)\) to maximize Rxy, the initial seismic wavelet \(W_{1}(t)\) and the initial reflectivity series \(R_{1}(t)\) are modified. Once \(R_{xy}\) reaches a satisfactory value, an initial model is established using density, velocity, and reflectivity through Kriging interpolation to generate the initial impedance data volume. This process is a forward modeling procedure that involves adjusting the well logs, initial reflectivity, and initial wavelet. Based on the initial model, all interpolated impedances are subjected to a limited number of adjustments within a certain range to find an optimal stratigraphic model. This model minimizes the error energy between the synthetic seismic data generated from it and the actual observed data, achieving the minimum value of the objective function (Equation (4)). The result is the final inversion profile, where \(e_{1}(t)\) represents the degree of match between the model and the seismic record. This process is an inversion procedure that modifies the impedance of the initial model [28], as follows:  
\[e_{1}(t)=x(t)-y_{1}(t) \tag{4}\]  
**Processing Well Log Data**: Well log data, particularly acoustic and density logs, are crucial for constructing the initial model and geological interpretation [29; 30]. However, acoustic logs often face interference from borehole conditions, such as borehole collapse and mud invasion, potentially leading to errors that require correction. Additionally, the time-domain thickness of converted acoustic log curves may exhibit inaccuracies. To mitigate such velocity errors, a common approach is to accurately determine the correspondence between the main wave group in the synthetic record and the nearby seismic trace.