Enhanced Seismic Imaging of Complex Geological Structures Using Model-Constrained Kirchhoff Pre-Stack Depth Migration: Numerical Validation and Field Application

Abstract

Seismic imaging in areas with complex geological structures, such as steeply dipping strata and lateral velocity variations, remains a significant challenge in geophysical exploration. In this paper, a Kirchhoff pre-stack depth and pre-stack time migration imaging method under the constraint of an initial model is proposed. By establishing the initial velocity model, the method is iteratively optimized under the horizon constraint, and the travel time difference is used to update the model. Finally, Kirchhoff pre-stack imaging is realized. Numerical simulations using a synthetic five-layer velocity model demonstrate that removing direct wave interference and incorporating horizon constraints significantly improve the signal-to-noise ratio and structural accuracy of the migration results. A field case study in a coalfield with monoclinic structures and high-angle faults further validates the method’s effectiveness. Comparative analysis with pre-stack time migration reveals that Kirchhoff pre-stack depth migration achieves superior fault delineation, diffraction wave homing, and event continuity, particularly in steeply dipping formations. The results highlight the method’s potential for improving seismic interpretation accuracy in complex structural settings, offering practical value for coal mine safety and resource exploration.

1. Introduction

Seismic wave propagation follows Snell’s law [1]; if the velocities of adjacent layers are equal, transmitted waves pass through without refraction or reflection. Assuming that the medium is uniform or horizontally layered, the vertex of the diffraction time–distance curve (imaging point) and the diffraction point (subsurface reflection point) have the same position in the horizontal direction, which is a prerequisite for the application of time misalignment. However, when the velocity changes laterally, the transmitted wave produces refraction time difference, and there is a deviation in the real imaging position of the underground reflection point. Pre-stack depth misalignment imaging can ensure the amplitude fidelity of seismic waves while realizing misalignment imaging and no longer needs to consider the influence of velocity on the time misalignment profile, so it is a research hotspot [2,3].

The development of seismic migration dates back to the 1920s, when foundational concepts such as Snell’s law and wavefront geometry were applied to manual reflection mapping [4]. A pivotal shift occurred in the 1970s with Claerbout’s wave equation-based migration methods, which replaced ray-tracing approaches and enabled efficient wavefield reconstruction via finite difference solutions [5]. Schneider further advanced wave equation migration by incorporating amplitude preservation and boundary conditions [6], while Gazdag and Stolt introduced frequency–wavenumber domain solutions [7,8]. Despite theoretical progress in full-wave and reverse-time migration during the 1980s, computational limitations hindered practical adoption. The 1990s marked a turning point with the application of Kirchhoff pre-stack depth migration in complex exploration settings (e.g., Gulf of Mexico subsalt imaging) [9], alongside advancements in amplitude fidelity and velocity model optimization [10,11]. Recent innovations, such as frequency division migration and Q-compensation techniques, have enhanced resolution in challenging reservoirs [12,13], solidifying migration as a cornerstone of modern seismic imaging.

At present, there are two kinds of pre-stack depth migration methods commonly used in China: Kirchhoff integral pre-stack depth migration based on a wave equation integral solution and pre-stack depth migration based on wave equation micro-decomposition [14,15]. Gray used a space-varying frequency-selective integral operator to suppress the spurious frequency of the migration operator [16]. Zhang et al. discussed the theoretical expression of the amplitude weighting of the integral method, but it is not applicable to variable-speed media [17]. Zhang Xiaodan et al. applied the fast-marching method (FMM) to the travel time calculation of Kirchhoff integral pre-stack depth migration and found that this method improves the migration speed, and the migration result is good [18]. Zhang et al. proposed a new seismic travel time algorithm based on quasi-differential operator theory and the Lie algebra integral method, which is suitable for Kirchhoff pre-stack depth migration in laterally varying media. The effectiveness and accuracy of the algorithm were verified by numerical experiments [19]. Yuan et al. studied the Kirchhoff integral vector migration formula based on the homogeneous isotropic elastic wave equation, realized wavefield separation, improved the migration imaging accuracy of multi-component seismic data, and effectively eliminated the wave mode leakage phenomenon [20]. Shan et al. studied the Kirchhoff pre-stack depth migration method based on model constraints and compared it with pre-stack time migration. The results show that pre-stack depth migration imaging has a higher signal-to-noise ratio and resolution [21]. Wang Huazhong et al. studied the parallel implementation scheme of Kirchhoff pre-stack depth migration for large-scale 3D seismic data based on dynamic programming travel time calculation and MPI + OpenMp hybrid programming mode and realized efficient and high-precision migration imaging [22]. Gray et al. used the difference method to solve the eikonal equation for Kirchhof migration and pointed out that the method of solving the eikonal equation to calculate the travel time (without interpolation) is more advantageous than the traditional method of ray tracing [23]. Genovese and Palmeri (2025) proposed a stochastic wavelet method with exponential auto-correlation to generate artificial accelerograms, optimizing time–frequency trade-offs for seismic engineering [24]. The Kirchhoff integral method is the most widely used pre-stack depth migration method in actual production. Its main advantage is that it has the ability to process complex velocity fields relatively efficiently, and its computational efficiency is high. However, its reliance on Sensitivity to velocity model errors, especially laterally; high computational cost for wide-aperture and 3D surveys; and its neglect of finite-frequency and elastic effects are disadvantages. In contrast, pre-stack depth migration based on wave equation micro-decomposition leverages wave equation solutions to improve the imaging of steep dips and multi-scale structures but requires careful frequency sampling to avoid aliasing, a larger memory footprint for storing wavefields, and more complex implementation and parameter tuning, which restricts its real-time applicability [25,26,27].

At present, in areas with simple geological structures, seismic data imaging is mainly based on post-stack time migration technology. For the areas where underground structures are developed, the above methods often cause the fault position to swing, the back and syncline to return poorly, and so on. The structure of seismic interpretation has a displacement swing with the structure exposed by mining, and the size of the structure is very different, which seriously affects the fine interpretation of seismic data and is not conducive to guiding the safe mining of coal mines. Therefore, in this paper, the Kirchhoff integral pre-stack depth migration imaging method based on a model constraint is proposed. Firstly, the synthetic model is used to verify the accuracy of the method. Secondly, the model-constrained pre-stack migration technology is used to process and interpret the original seismic data in the complex structural area. The position and direction of the fault are accurately divided; the problem of the low imaging resolution of faults, steeply dipping strata, and other structures is solved; and the problem of multiple solutions of data interpretation is focused on. This technology not only has certain theoretical significance but also has great practical value.

2. Kirchhoff Pre-Stack Depth Migration Imaging Principle

2.1. Method Principle

The theoretical basis of Kirchhoff integral pre-stack depth migration is the solution of the Green’s function of the wave equation. Green’s function represents the impulse response between a secondary source and a calculation point. By establishing the integral relationship between the underground wavefield and the reflected waves using the Green’s function, seismic wave migration and imaging can be realized based on imaging constraints. The implementation process can be divided into two steps: model building and migration imaging. The formula for Kirchhoff integral pre-stack depth migration is given in [28,29,30,31].

$$R\left(x,x_S\right)={\int }_{\sum }n{\nabla }_{\tau R}\left(x_R,x\right)\zeta \left(x_R,x,x_S\right){\displaystyle \frac{\partial u\left[x_R,{\tau }_S\left(x,x_S\right)+{\tau }_R\left(x_R,x\right),x_S\right]}{\partial t}}\mathrm{d}{x}_R$$

(1)

In Equation (1), $R\left(x,x_S\right)$ is the reflection coefficient; n is the outward normal vector of the reflector $\sum$; $x_S$, $x_R$, and x represent the source point, receiver point, and imaging point, respectively; ${\tau }_S$ and ${\tau }_R$ are the travel times from the source point to the imaging point and from the imaging point to the receiver point, respectively; and ${\nabla }_{\tau R}\left(x_R,x\right)$ describes the local propagation direction of the wave from the imaging point $x$ to the receiving point $x_R$. Its direction is tangent to the propagation path of the wave, and its size is related to the reciprocal of the propagation velocity. $\zeta \left(x_R,x,x_S\right)$ is the wavefield spreading factor, and $\partial u\left[x_R,{\tau }_S\left(x,x_S\right)+{\tau }_R\left(x_R,x\right),x_S\right]$ is the displacement of the seismic wave propagating in the medium.

2.2. Pre-Stack Depth Migration Imaging Process

In the process of pre-stack depth migration, we should pay attention to the following points: Firstly, we must obtain an accurate time migration velocity field, which lays a good and solid foundation for the establishment of the initial layer velocity model. Secondly, in order to obtain a high-quality time migration velocity field, the velocity point can be encrypted to improve the accuracy of velocity picking. Thirdly, the pre-stack time migration data layer can be picked up, paying attention to how the required layer can be reasonably selected and ensuring that the established layer model can conform to the geological law. This layer can also act as a reasonable and effective constraint in the process of velocity modeling. Fourthly, in the process of optimizing the layer velocity model, it should be taken into account that the inaccuracy of the interpretation model will directly affect the effects of migration imaging. Therefore, in this data processing method, the first choice is to use the horizon constraint to iterate 2~3 times to optimize the velocity model and then use the grid tomography to iterate 1~2 times until the ideal depth domain layer velocity body is obtained, and finally, the data are offset as a whole. Fifth, the appropriate migration parameters are selected, including migration aperture and anti-aliasing parameters, to ensure that migration imaging obtains good results. The specific process is shown in Figure 1.

2.3. Applications of Model-Constrained Kirchhoff Pre-Stack Depth Migration

The proposed model-constrained Kirchhoff pre-stack depth migration method is versatile and can be applied to a wide range of subsurface investigations—including groundwater flow modeling, hydrocarbon reservoir characterization, contaminant plume delineation, geothermal exploration, carbon capture and storage monitoring, engineering site assessment, and archeological prospection—where its ability to preserve true-amplitude responses and deliver high-resolution images of stratigraphic interfaces, fault networks, and small-scale heterogeneities leads to more reliable predictive modeling, resource quantification, and risk evaluation.

3. Methods

3.1. Model Constraint Method

3.1.1. Model Building

The quality of the depth migration model directly determines the results of pre-stack depth migration imaging. Currently, the ray tracing-based tomographic velocity inversion method is one of the key modeling techniques for pre-stack depth migration imaging. It mainly uses migration and tomographic iteration for velocity inversion to recover high- and low-frequency information in the velocity field and is widely used in industry for velocity model building.

The tomographic velocity inversion process includes modeling and model updating. First, the model is discretized into a grid, and ray tracing is performed for each reflection point in the grid. Each ray path can be represented by an equation, and the entire model’s ray paths [32] can be expressed as

$$t_u=\sum _{p=1}^P\sum _{j=1}^J\sum _{m=1}^M{d}_{pjm}\cdot s_{pjm}$$

(2)

where P, J, and M are the number of grid points; t_u is the seismic wave travel time; dpjm is the ray path length at the pjm-th grid point in the model; and s_pjm is the slowness at the pjm-th grid point. The travel time difference is given by

$${∆t}_u=\sum _{p=1}^P\sum _{j=1}^J\sum _{m=1}^M{d}_{pjm}\cdot {∆s}_{pjm}$$

(3)

where ${∆t}_u$ is the seismic wave travel time difference, and ${∆s}_{pjm}$ is the slowness error at the pjm-th grid point. By updating the velocity model using the travel time difference ${∆t}_u$, iterative tomographic velocity modeling can be achieved.

3.1.2. Model Constraints

The Kirchhoff integral pre-stack depth migration method with model constraints is an extension of conventional depth migration modeling, incorporating constraints from target geological body horizons. Horizon constraints can effectively improve the precision of model updates, thereby enhancing the accuracy and resolution of Kirchhoff integral pre-stack depth migration imaging. The primary issue addressed by constrained Kirchhoff integral pre-stack depth migration is the reduction in travel time differences during tomographic velocity iteration. Tomographic velocity modeling inversion is a multi-solution problem, and to reduce this ambiguity, known geological horizon constraints, such as accurate grid velocities of horizons, can be incorporated during the iterative inversion process.

Equation (3) can be rewritten as

$${∆t}_u=d\cdot ∆s$$

(4)

where $d$ is the coefficient matrix represented by $d_{pjm}$, and $\mathsf{\Delta }s$ is the vector of the variable $\Delta s_{pjm}$.

For known grid velocities, regularization constraints can be applied by incorporating the accurate geological horizon grid velocities. Equation (4) can then be expressed as

$$\left(\begin{array}{c}\mathsf{\Delta }{t}_u\\ \lambda W\left(s^{\prime }-s_k\right)\end{array}\right)=\left(\begin{array}{c}d\\ \lambda W\end{array}\right)\mathsf{\Delta }s$$

(5)

where s′ is the slowness vector of the accurate geological horizon grid velocities; $s_k$ is the solution at the k-th iteration; $W$ is an L × N matrix, $W_i$ = [0…1…0], i ∈ [1, L], indicating that the constrained parameter is 1; and λ is the regularization factor.

3.1.3. Simulation Parameters

To demonstrate the effectiveness of this method, a numerical test was conducted using a model dataset for Kirchhoff integral pre-stack depth migration imaging. The designed two-dimensional model is shown in Figure 2. The model dimensions are 1005 m × 615 m, consisting of five layers of media and four reflection interfaces from top to bottom. The parameters of each layer are listed in

Table 1. Model medium parameters.

Layer Number	Medium Velocity (m/s)	Density (g/cm3)	Computing Time (s)
1	1000	2.250	206.75
2	1200	2.250
3	1500	2.250
4	2000	2.250
5	3000	2.250

. It is noteworthy to focus exclusively on the influence of velocity heterogeneity, so a uniform density of 2.25 g/cm³—representative of average sedimentary formations—is assumed, thereby minimizing the confounding effects of density variations on migration imaging.

The elastic wave finite difference forward modeling method was used, and the observation system was designed as follows: The trace spacing is 10 m, with one shot, and each shot has 100 traces. The shot point is located at x = 500 m. The sampling interval is 0.5 ms, and the number of sampling points is 2001. The Kirchhoff integral pre-stack depth migration imaging calculations of the subsequent synthesis models in this paper were completed on an Intel (R) Core (TM) i7–8700 K laptop with a CPU main frequency of 3.70 GHz with 48 GB RAM, and the calculation time of the model is only 206.75 s. The excitation method adopts P28M28 controllable seismic source excitation, using single-point excitation. The wireless node seismic acquisition system adopts SmartSolo IGU-BD3C-5 broadband integrated 3 component seismograph reception. The detailed parameters are shown in

Table 2. SmartSolo IGU-BD3C-5 parameter table.

SmartSolo IGU-BD3C-5
Physical Specs	Channel Performance
Seismic data channel(s): 3 ADC resolution: 32 bits Sample intervals: 0.25, 0.5, 1, 2, 4, 8, 10, 20 ms Preamplifier gain: 0 dB to 36 dB in 6 dB steps Instrument noise floor: Whole frequency band lower than the NHNM curve, 5 s~1 Hz lower than the NLNM curve Operating temperature: −40 °C~+70 °C Waterproof: IP68 Data harvesting: USB 3.0	Maximum Input Signal: ±2.5 Vpeak @ Gain 0 dB Common Mode Rejection: ≥100 dB Gain Accuracy: <1% GPS Time Standard: 1 ppm Timing Accuracy: ±10 μs, GPS Disciplined Cross-Feed: <−110 dB Inter-channel Phase Offset: <0.1 ms Transverse Vibration Rejection: Better than 0.1% Inter-channel Amplitude Coherence: 5% System Dynamic Range: 140 dB Frequency Response: 0~1652 Hz

.

3.2. Field Test

3.2.1. General Geology

As shown in Figure 3, the strata in the study area include the Cambrian Gushan Formation and Changshan Formation; Upper Carboniferous Benxi Formation and Taiyuan Formation; Lower Permian Shanxi Formation and Lower Shihezi Formation; Upper Permian Shihezi Formation; and Paleogene, Neogene, and Quaternary strata. The strata are distributed from old to new. The upper Cambrian strata include the Gushan Formation and Changshan Formation, which are mainly gray and light gray limestone, dolomite, and mudstone, with thicknesses of 189 m and 64 m, respectively. The Upper Carboniferous Benxi Formation and Taiyuan Formation are interbedded with limestone, sandstone, and mudstone, with a thickness of about 70 m, showing parallel unconformity contact with Cambrian strata. The Shanxi Formation, Lower Shihezi Formation, and Upper Shihezi Formation of the Permian are mainly composed of sandstone and mudstone, containing important coal seams. The Shanxi Formation is the main exploration object, with an average thickness of 72 m. The Paleogene and Neogene are alternately composed of conglomerate, sandstone, and mudstone, and the thickness is between 330 and 1230 m. The Quaternary is a yellow-brown, light red sandy clay and gravel layer, which is gradually thickened and covers the Neogene strata.

In terms of coal seam, the T₁ coal seam is the main mineable coal seam, which is located in the lower part of the Shanxi Formation. The thickness of the coal seam is 3.94–8.59 m, with an average thickness of 6.01 m. The T₃ coal seam is located in the middle and lower parts of the Shanxi Formation, with a thickness of 0–2.51 m and local recoverability. The T6 coal seam is located in the middle of the fourth coal section of the lower Shihezi Formation, with a thickness of 0–1.97 m and local recoverability.

In terms of stratigraphic structure, the mine field is characterized by a monoclinic structure trending northwest and dipping southeast. The stratigraphic trend is 300–330°, and the dip angle is 30–35°. It is mainly composed of three groups of high-angle normal faults, which are NE, NWW, and near-EW faults.

3.2.2. Seismic Geologic Condition

The terrain in the study area is flat, the ground elevation is 94 m–118 m, and the transportation is convenient, which is suitable for the field construction of seismic exploration. However, the villages in the area are relatively dense, which may bring some challenges to the construction. The shallow seismic geological conditions are more complicated. The surface layer in the area is covered by Quaternary loess, and the groundwater level is 2–3 m. The excitation conditions are good. However, it is difficult to form holes in the clay stone below 2 m, which affects exploration efficiency.

Deep seismic exploration mainly tracks the Cenozoic bottom interface, T₁ coal seam and T₃ coal seam, and the Cambrian limestone top interface. The Cenozoic bottom interface is composed of clay, sand, and gravel layers, which has a large difference in wave impedance, which is conducive to the formation of reflected wave T_Q. The coal seam has obvious wave impedance differences. The reflected wave energy of the T₁ coal seam is strong, and the waveform is stable, which is suitable for continuous tracking in the whole area. The top interface of Cambrian limestone is affected by weathering and denudation, and the quality of the T_∈ reflection wave is poor, but it can still be effectively tracked through stable inter-layer spacing. In general, seismic exploration in the area has good feasibility.

3.2.3. Seismic Observational System

The main parameters of the three-dimensional observation system are as follows: a total of 4 shots were exciting, the midpoint excitation was adopted, the longitudinal shot row spacing was 100 m, and the transverse shot point spacing was 20 m. The number of receiving channels was 60/line × 12 = 720, the channel spacing was 20 m, the receiving line spacing was 20 m, and the CDP grid size was 10 m × 10 m. The full coverage was 24 times, and the maximum number of coverages was 52.

4. Results and Analysis

4.1. Numerical Simulation Results and Analysis

4.1.1. Seismic Source

In the numerical simulation study, the Ricker wavelet is used as the source signal. Its zero-phase characteristics and narrowband spectrum distribution are conducive to generating high-quality reflection data. As shown in Figure 4, the main frequency of the Ricker wavelet is set to 25 Hz, the time sampling interval is 0.0005 s, the total time length is 0.1 s, and its mathematical expression is as follows:

$$w\left(t\right)=\left(\begin{array}{c}1-2{\pi }^2{f}_0^2{t}^2\end{array}\right)e^{-{\pi }^2{f}_0^2{t}^2}$$

(6)

where $f_0$ represents the dominant frequency.

4.1.2. Synthetic Seismic Profile

As shown in Figure 2, a five-layer velocity model is constructed for synthesizing seismic profiles. The model’s velocities are 1000 m/s, 1200 m/s, 1500 m/s, 2000 m/s, and 3000 m/s. These values correspond to 150 m, 150 m, 105 m, 105 m, and 105 m formation thicknesses. With a grid sampling interval of 5 m, the model’s extent measured 1005 m (lateral) by 615 m (vertical).

As shown in Figure 5, the seismic wave propagation is simulated using the finite difference method, and the synthetic seismic profile contains a variety of wavefield components such as direct, reflected, and multiple waves. Preliminary analysis shows that the reflection events at different stratigraphic interfaces have obvious time characteristics and energy distributions, with stronger energy in shallow reflected waves and slightly lower signal-to-noise ratios in deeper reflected waves. In addition, the strong energy of the direct wave is obvious in the shallow time interval, which interferes with the resolution of the deep reflection events. Therefore, it is necessary to remove the direct wave.

4.1.3. Kirchhoff Integral Pre-Stack Depth Migration Imaging of Synthetic Model

Figure 6 and Figure 7 show the imaging results of the pre-stack depth migration of the data obtained from the forward modeling of the model in Figure 2 by using the Kirchhoff integral pre-stack depth migration imaging method. Figure 6 shows the pre-stack depth migration imaging result of the Kirchhoff integral method without removing the direct wave, and Figure 7 shows the migration imaging result after removing the direct wave. It can be seen from Figure 6 and Figure 7 that the actual velocity model is used as the migration velocity field in the imaging process, and the migration results can effectively restore the geological structure of the target layer boundary. The imaging section shows that the shape of the five-layer geological interface is basically consistent with the input model. The continuity of the reflected wave event is good, the energy distribution of each interface is uniform, and the resolution is high. By comparing the offset seismic profiles, the offset results significantly eliminate the interference of direct and multiple waves, and the imaging clarity and accuracy are effectively improved, which verifies the accuracy of the method.

As shown in Figure 6, for the Kirchhoff pre-stack depth migration imaging results affected by unremoved direct waves, the interference of direct waves significantly affects the imaging accuracy of shallow and deep reflection events. There is discontinuity in shallow reflection events, and the signal-to-noise ratio of the deep reflection interface is low, resulting in a decrease in the resolution of the formation interface.

As shown in Figure 7, the results of Kirchhoff pre-stack offset imaging with the effects of the excised direct wave show that after the interference of the direct wave is excised by the wavefield separation method, Kirchhoff pre-stack depth offset processing is performed again, and the results show that the reflection events of the shallow interfaces are clear and continuous, and the signal-to-noise ratio of the deep interfaces is significantly improved, and the imaging accuracy is significantly improved. The results of migration imaging restore the stratigraphic morphology and interface characteristics of the input model well, and the waveform continuity and energy distribution of each interface are enhanced.

The final results show that the existence of the direct wave is one of the main factors affecting the accuracy of Kirchhoff pre-stack migration imaging. After removing the influence of direct waves, the interface characteristics of the imaging section in the shallow and deep parts are significantly improved, which provides method support and a theoretical basis for improving imaging accuracy under complex geological conditions in actual exploration.

Figure 8 shows a comparison of the Kirchhoff pre-stack depth offset imaging results with the addition of model constraints (Figure 8a) and without (Figure 8b). As can be seen in Figure 8, the offset imaging results with model constraints added are clearer, the results match the modeled structural morphology, and the resolution of the laminar interface is higher than that of the imaging results without model constraints, which verifies the accuracy of the method.

4.2. Field Measurement Results and Analysis

4.2.1. Pre-Stack Combination Denoising

The noise suppression of seismic data is a key link in data processing. Linear interference waves, such as surface waves and multiple refraction waves, and random interference waves are mainly developed in the study area, which are the main noise interference affecting pre-stack migration imaging. According to the characteristics of the interference wave in the area, the combined denoising method is used to suppress the interference wave and improve the signal-to-noise ratio of the reflected wave. For linear interference such as surface waves and multiple refraction, the t-x domain subtraction method is used to denoise step by step in the common shot point domain, common detection point domain, and common offset domain. For random noise, adaptive high-frequency noise attenuation technology is used to effectively suppress high-frequency environmental noise and burst noise. The combined denoising results are shown in Figure 9. The seismic interference wave is effectively suppressed by the combined denoising method, the reflected wave signal-to-noise ratio is significantly improved, and a better amplitude-preserving denoising effect is achieved.

4.2.2. Actual Data Velocity Modeling

The focus of seismic data pre-stack depth migration processing is to accurately obtain a migration velocity model. Without an accurate velocity model, accurate migration imaging is impossible. Velocity–depth modeling contributes to the difficulty and core of pre-stack depth offset imaging, and the acquisition of the velocity model can be divided into two methods: the first is the velocity model correction by meshing; the second is the modeling of velocities along layers. The former does not require layer constraints and is greatly affected by the information signal-to-noise ratio, which makes it difficult for the speed iteration to converge for information with a poor signal-to-noise ratio, while the latter is constrained by the accuracy of the information interpretation. The solution idea of velocity modeling in this paper is based on 3D seismic high-quality channel set data and adopts the scheme of using the above two methods alternately. Firstly, the velocity model is established under the constraints of the geological tectonic model, and the model constraints are used to adjust the velocity model iteratively two to three times. Since the accuracy of the velocity model in the upper layer directly affects the imaging results in the lower layer, the velocity should be adjusted from shallow to deep, layer by layer. When the velocity model reaches a certain accuracy, the grid lamination is used for one or two iterations for global velocity optimization to accurately find the velocity in the domain of the pre-stack time offset and ultimately realize the fine adjustment of the inter-layer velocity and finally obtain the results of the pre-stack time offset (as shown in Figure 10(a-1,b-1)). In this way, this method not only solves the influence of the accuracy of seismic data interpretation but also solves the problem of the slow convergence of iteration speed using only grid tomography. A time domain velocity model is built based on the pre-stack time offset results (Figure 10(a-2)), and then the time domain velocity body is converted into a depth domain layer velocity body by the constrained velocity inversion (CVI) technique (Figure 10(b-2)). Finally, the time domain model is offset to the depth domain to model the depth domain velocity.

4.2.3. Stretching Correction in Pre-Stack Depth Migration

When the velocity model is iterated in the depth domain, the CRP gather should be flattened as the basic principle so that the imaging effect will be improved. When the model constraint velocity iteration reaches a certain accuracy, the global velocity model is optimized and adjusted by using the three-dimensional grid tomography method. Based on the tomographic optimization velocity model, the data attribute body in the depth domain (the continuity body of the seismic data event, the dip angle body, and the azimuth body; see Figure 11(a-1–a-3)) is extracted. According to the continuity of the formation, the internal reflection layer of the seismic data is automatically extracted to form multiple reflection internal layers in different regions. According to the common imaging point gathered and obtained by pre-stack depth migration, the depth residual velocity of the target line is picked up to form the depth residual velocity body. The above three seismic attribute bodies, depth residual velocity bodies, and previous depth migration layer velocity bodies are considered. Several data volumes such as the internal reflection layer are merged to create a pencil database (see Figure 11b). Each seismic record contains the above information, which lays the foundation for travel time calculation. A global grid tomography matrix containing multiple layers is established. The least squares method is used to solve the grid tomography matrix under the constraints of the above information. Finally, the optimized depth domain layer velocity body is obtained, and migration imaging is performed. Figure 11(c-1–c-3) show the flattening of CRP gathers at the same position during the velocity iteration. When the residual delay of each layer is basically zero, the CRP gathers are considered to have been fully flattened. At this time, the layer velocity is accurate, and the iteration of the velocity model is completed.

4.2.4. Migration Parameter Selection

a. 

Offset aperture

The selection of migration parameters directly affects the migration running time and the resolution and signal-to-noise ratio of migration imaging. In migration, the migration aperture can be said to be the parameter that has the greatest impact on pre-stack depth migration imaging in addition to the depth domain velocity model. If the offset aperture is too small, the steep dip phase axis will be destroyed, the amplitude will change dramatically, and a large amount of random noise will be generated, which may lead to a false horizontal phase axis. When the migration aperture is too large, it will not only waste a lot of computing time and reduce the processing efficiency but also reduce the signal-to-noise ratio of the data. Deep noise affects shallow data, resulting in a decrease in the quality of migration imaging.

For the same work area, it is best to use a unified migration aperture for the migration imaging of all survey lines so that the migration profile can maintain a unified amplitude characteristic. In practical applications, considering factors such as noise and target horizon, the selection of the migration aperture needs to be determined by experiments. The test results are shown in Figure 12. Taking the Inline80 line as an example, in the Inline direction, when the migration aperture is increased to 3000 m, the imaging quality of the steeply dipping formation is obviously improved, and the signal-to-noise ratio is relatively low. When it is increased to 3500 m, it has no effect on the imaging of inclined strata, but it brings more migration noise to shallow migration imaging. Therefore, it is determined that 3000 m is the best migration aperture in the Inline direction. In the Crossline direction, when the aperture increases to 2000 m, the imaging quality of the inclined formation is better, and the signal-to-noise ratio is relatively low. Therefore, 2000 m is the best migration aperture in the Crossline direction (see Figure 12).

b. 

Anti-aliasing operator

The Kirchoff integral method involves summing the discrete input data in time and space, without considering the frequency component of the data. Therefore, when the operator inclination angle of the offset summation trajectory is too steep or contains high frequency energy or the data space sampling is sparse, it is very easy for aliasing to appear, so it is necessary to carry out anti-aliasing filtering processing. The Inline80 line is still selected as an example, and the triangle accurate anti-aliasing method [33] is used. The anti-aliasing operators are 0.25, 1, and 2. The test results are shown in Figure 13. The imaging profiles of the three anti-aliasing factors show that the imaging results of the inclined strata are basically the same, and the signal-to-noise ratio of the shallow layer is relatively low. Because this processing uses a square grid of 10 m × 10 m, and the sampling rate is high, the possibility of generating spatial aliasing is small, and a smaller anti-aliasing operator should be used. Therefore, the anti-aliasing operator is 0.25.

4.2.5. Kirchhoff Pre-Stack DEPTH Migration Imaging Results

By using the comprehensive velocity modeling method, the Kirchhoff pre-stack depth migration imaging processing of seismic data in the study area is completed by using the methods of velocity update and grid tomography. In order to compare the effects of migration imaging, the imaging results of the pre-stack depth migration profile and pre-stack time migration profile are compared and analyzed. It can be seen from Figure 14 that the imaging quality of the pre-stack depth migration section is greatly improved compared with that of pre-stack time migration, which is manifested in the higher signal-to-noise ratio of the target layer, clearer reflection wave characteristics, and better event continuity. On the depth migration profile, the fault distribution, breakpoint location, and contact relationship near the fault near the target layer are obviously clearer, and the diffraction wave homing is more reasonable. In general, Kirchhoff pre-stack depth migration significantly improved the signal-to-noise ratio, resolution, and imaging effects compared with pre-stack time migration, which can be used to further explore the geological structure of the study area.

5. Conclusions and Recommendations

Kirchhoff pre-stack depth migration is the most flexible and effective method for pre-stack seismic data imaging and has high computational efficiency. Our principal innovation lies in the integrated use of model constraints within the Kirchhoff pre-stack depth migration framework. Unlike traditional approaches that apply true-amplitude migration or velocity inversion in isolation, our methodology synergistically unites iterative velocity inversion, denoising, and model constraint mechanisms. This novel combination not only elevates imaging fidelity under severe lateral velocity contrasts but also streamlines parameter tuning and enhances reproducibility, representing a significant advancement in pre-stack seismic imaging technology. The following conclusions are drawn:

(1)

The Kirchhoff pre-stack depth migration method based on model constraints is used to study the pre-stack depth migration imaging of theoretical model data. The results show that after removing the direct wave, the clarity and continuity of the shallow and deep reflection events are significantly improved, and the energy distribution of each interface is uniform, which significantly improves the imaging accuracy and signal-to-noise ratio. The addition of model constraints further optimizes the imaging results, making the layered interface features clearer and in good agreement with the model structure. The numerical model verifies the effectiveness of the proposed method in improving the imaging accuracy under complex geological conditions.

(2)

The signal-to-noise ratio of seismic data in complex structural areas is often very low. The initial velocity and input gather of depth migration are highly dependent on the signal-to-noise ratio of the data. The combined denoising technology, comprehensive velocity modeling method, and constrained velocity inversion (CVI) method are used to maximize the protection of effective signals and provide high-fidelity data for depth migration. The Kirchoff integral migration technique improves the continuity of the phase axis of the pre-stack depth migration imaging profile, the imaging is clearer, and the resolution is significantly improved.

(3)

Comparing the imaging results of pre-stack time migration and pre-stack depth migration, the Kirchoff integral pre-stack depth migration method based on model constraints can solve the technical difficulties of inaccurate migration imaging caused by drastic changes in lateral velocity, effectively improving the signal-to-noise ratio and resolution of seismic wave imaging and better solve the problem of complex structural imaging. The fault breakpoint position is clearer, and the wave group characteristics are obvious, which provides an effective technical method for complex geological structure detection imaging.

(4)

Through the application of pre-stack depth migration imaging technology in complex geological structures, the boundary characterization of geological structures can be enhanced, the imaging accuracy can be greatly improved, the exploration accuracy can be effectively improved, and the problem of poor post-stack migration imaging can be solved. This provides a solid data foundation for the geological structure of the study area.

Based on the numerical simulation and field application results, we recommend conducting the following for the effective implementation of model-constrained Kirchhoff pre-stack depth migration in complex geological settings: (1) Ensure accurate initial velocity modeling through a combination of conventional analysis and constrained velocity inversion to provide a reliable foundation for depth migration; (2) apply robust noise attenuation techniques to enhance the signal-to-noise ratio, particularly in areas with structurally complex and low-quality data; (3) optimize migration parameters such as aperture, anti-aliasing filters, and weighting functions to balance computational efficiency and imaging resolution; and (4) validate the migration results using well and geological data to confirm structural consistency. These strategies help maximize the method’s potential in improving imaging accuracy and resolving complex subsurface structures.