A Variational Data Assimilation Framework for Mining Subsidence Reconstruction from Heterogeneous D-InSAR and TLS Observations

Highlights

What are the main findings?

• A multi-source subsidence data fusion method based on a Variational Data Assimilation Framework is proposed. By constructing an objective function with five constraint terms, the D-InSAR-derived boundary observations and TLS-measured central subsidence are integrated into a unified optimization framework. The fusion results achieve an overall RMSE of 0.12 m and an RRMSE of 2.4%.
• Parameter sensitivity analysis indicates that the smoothness parameter ${\lambda }_s$ has the most significant influence on fusion accuracy, whereas the background constraint weight has negligible effect over four orders of magnitude. The joint coefficient of variation for all parameter pairs is below 1%, demonstrating the overall robustness of the proposed method.

What are the implications of the main findings?

• The VDAF method effectively resolves the complementary fusion problem between D-InSAR decorrelation in high-gradient deformation areas and the limited spatial coverage of TLS, providing a theoretically rigorous and accuracy-controllable solution for the complete three-dimensional reconstruction of mining-induced subsidence basins. The proposed framework can also be extended to multi-source deformation monitoring in other geohazard scenarios.
• Comparative analysis shows that D-InSAR completely fails in high-gradient subsidence areas (RMSE up to 3.18 m), while TLS exhibits large errors in certain segments or individual points. By fully exploiting the spatial complementarity of the two data sources, the VDAF method achieves significantly higher accuracy than any single sensor along both observation lines, demonstrating the practical effectiveness of the multi-source fusion strategy in engineering applications.

Abstract

Accurate characterization of mining-induced surface subsidence is essential for safety assessment in mining areas; however, single monitoring techniques have inherent limitations. Spaceborne interferometric synthetic aperture radar (InSAR) provides large-area coverage but suffers from low signal-to-noise ratio in the subsidence center, whereas terrestrial laser scanning offers high accuracy but limited spatial coverage. To achieve physically consistent quantitative fusion, a multi-source subsidence fusion framework based on variational data assimilation is proposed. By constructing an objective function that incorporates a background prior, D-InSAR-derived boundary constraints, TLS observations, spatial smoothness constraints, and gradient penalty terms, multi-source data are integrated into a unified optimization framework. The results show that, compared with RTK observations, the fused subsidence field achieves an RMSE of 0.12 m and an RRMSE of 2.4% approximately. Parameter sensitivity analysis indicates that the smoothing strength has the greatest influence on fusion accuracy, whereas the observation weight and gradient penalty coefficient exhibit relatively wide stable intervals, and the background constraint has a minor effect on the results. Parameter interaction analysis further demonstrates that the coupling between smoothing strength and observation weight is the most significant. The proposed method provides a physically consistent and parameter-controllable framework for multi-source deformation data fusion in mining subsidence monitoring.

1. Introduction

Large-scale industrial coal extraction induces irreversible mechanical disturbances in the overburden, which ultimately manifest as extensive surface subsidence, posing severe threats to surface infrastructure, disrupting hydrological balance, and accelerating environmental degradation in the ecologically fragile regions of western China [1,2]. As coal resources in eastern China have gradually been depleted, mining activities have increasingly shifted toward the Loess Plateau and semi-arid coalfields in western China, where both the magnitude and morphological complexity of surface subsidence have increased significantly [3,4]. Western mining areas are typically characterized by shallow burial depth, high extraction intensity, and rapid face advance. Under repeated mining conditions, stress redistribution from successive extractions accumulates, and the presence of reserved coal pillars locally constrains overburden movement, resulting in complex subsidence patterns. The maximum surface subsidence of a single working face commonly reaches 2–6 m, with daily subsidence rates up to several tens of centimeters [2,5]. In such scenarios, the constraint effect of protective coal pillars often produces a pronounced multi-bowl or “W-shaped” subsidence morphology, which deviates substantially from the idealized bell-shaped basin predicted by traditional single-seam subsidence models [6]. The resulting surface cracking, groundwater drainage, and ecological degradation generate cascading impacts in the already sensitive western environment [4,7]. Therefore, a spatially complete and consistent characterization (i.e., maintaining uniform measurement accuracy and physical continuity from boundary to center) of the entire subsidence basin—from centimeter-level deformation at the basin boundary to large-gradient subsidence in the basin center—is a fundamental prerequisite for mine safety evaluation, geohazard prevention, and the study of overburden movement mechanisms [8].

Conventional surveying methods, including precise leveling, GNSS geodetic measurements, and total-station traverse observations, have long served as the fundamental means of monitoring mining-induced subsidence, and their point-wise accuracy remains the gold standard for parameter calibration in subsidence models [8,9]. However, under the conditions of rapid and large deformation, complex terrain, and ecological protection constraints in western mining regions, ground markers are frequently destroyed, discrete observations provide insufficient spatial resolution for basin boundary characterization, and the cost of high-frequency repeated surveys becomes prohibitive. The emergence of spaceborne interferometric synthetic aperture radar (InSAR) has fundamentally changed this situation. By extracting interferometric phase differences between multi-temporal SAR images, deformation fields covering hundreds of square kilometers can be obtained without ground contact [10,11]. Time-series techniques, including PS-InSAR, SBAS-InSAR, and intermittent SBAS [12,13,14,15], have significantly improved deformation monitoring continuity over vegetated and dynamically changing surfaces, and have been successfully applied in many coal mining areas in Europe and China [16,17,18].

Meanwhile, low-altitude UAV photogrammetry and TLS provide a complementary observation pathway. By constructing multi-temporal digital elevation models (DEMs) and performing DEM differencing, subsidence fields with decimeter- to centimeter-level accuracy can be directly measured, thereby avoiding the coherence limitation inherent to radar observations [19,20,21,22]. However, each of these techniques has intrinsic structural deficiencies, and these deficiencies are often complementary. InSAR loses coherence in the basin center where rapid deformation exceeds the half-wavelength phase ambiguity threshold [23,24], and the measured maximum subsidence may be less than 10% of leveling results [25]. Conversely, although UAV and TLS measurements maintain decimeter-level elevation accuracy, their errors exceed the signal itself in millimeter-level deformation zones near basin boundaries, leading to relative errors of 15–25% in angle parameter inversion when relying solely on UAV-derived data [26]. This complementary blind-zone characteristic makes any single-sensor solution structurally incapable of reconstructing a complete subsidence basin, which directly motivates the development of multi-source data fusion methods.

Fusion studies for such complementary datasets have evolved through four progressive stages: (1) multi-mode SAR fusion; (2) physics-constrained fusion; (3) cross-sensor deformation-domain fusion; and (4) data-driven and optimal-estimation approaches. A summary of these four categories of methods is presented in

Table 1. Summary of four categories.

Category	Description	Representative Methods	Limitations
Multi-mode SAR fusion	Extending the coherent monitoring coverage toward the subsidence basin center by incorporating additional SAR processing modes	Pixel Offset Tracking (POT); Sub-band InSAR; GIS-based spatial decision fusion [27,28,29,30,31,32,33,34,35]	Limited by SAR spatial resolution, resulting in insufficient accuracy for angular parameter inversion; threshold-based zoning may introduce structural discontinuities in transition areas.
Physics-constrained fusion	Employing the Probability Integral Method (PIM) as a physically constrained interpolation operator to bridge the gap between coherent InSAR observations at the basin margins and decorrelated areas in the basin center.	PIM-guided InSAR infill; Coherence-adaptive PIM; Multi-source data Kriging–PIM joint workflow [36,37,38,39]	Assumes homogeneous overburden—inadequate for fault/topo modulation; PIM initialization requires leveling data; fails under repeated/asymmetric mining
Cross-sensor deformation fusion	Directly integrating InSAR deformation maps with UAV/LiDAR-derived differential DEMs through spatial mask segmentation, feature-level boundary extraction, or accuracy-weighted fusion strategies.	InSAR boundary + UAV/LiDAR center masking; Accuracy-weighted local polynomial; Prior-weighted (PW) GNSS calibration DS-InSAR + UAV [26,40,41,42,43,44,45,46]	Fixed/empirical deformation thresholds → structural discontinuities in transition zone; cannot handle spatially heterogeneous accuracy within one observation; fails when observations are incomplete
Data-driven & optimal estimation	Incorporating machine-learning enhancement, formal statistical estimation, and data assimilation principles into multi-source data integration.	GAN phase recovery; Physics-informed regularized fusion; VCE + neural network GNSS/InSAR 3D [47,48,49,50]	Fusion still treated as spatial mosaicking / weighted averaging; no variational framework enforcing physical continuity; no globally consistent cost function balancing prior + obs + regularization

.

Nevertheless, in all existing studies, fusion is essentially treated as a spatial mosaicking or weighted averaging problem. The variational data assimilation paradigm—long established in meteorology [51] and physical oceanography [52] as a rigorous framework for combining heterogeneous observations with background physical knowledge—has, to the best of our knowledge based on the literature surveyed here, not been systematically applied to mining subsidence basin reconstruction. Variational assimilation minimizes a cost function that simultaneously constrains the analysis field to background priors, weighted observations, and regularization terms, enabling globally consistent solutions without threshold stitching or strict parametric assumptions. Under repeated mining, strong local gradients, and complex geological conditions where no single parametric model is sufficient, the variational framework provides a mathematically rigorous pathway for smooth and complete basin reconstruction.

Motivated by these gaps, this study addresses two unresolved problems. First, existing cross-sensor fusion methods treat all datasets as equivalent contributors divided by static thresholds, ignoring the inverse accuracy distribution in which InSAR is most accurate near basin boundaries while UAV/LiDAR is least accurate in these regions—by elevating D-InSAR observations to dominant boundary constraints rather than weighted inputs [26,40]. Second, constructing a geo-physically consistent transition zone between incoherent SAR gaps and LiDAR-measured central deformation remain an open challenge within a unified optimization framework [6,38].

To address these issues, VDAF is proposed for high-precision reconstruction of mining-induced subsidence basins using dual constraints from D-InSAR boundary observations and TLS large-deformation measurements. The method first constructs a background field jointly constrained by the basin boundary identified from D-InSAR and the central deformation measured by TLS, and then minimizes a cost function consisting of background constraint, observation constraint, Laplacian smoothing, and gradient regularization terms, solved efficiently using the L-BFGS quasi-Newton algorithm. D-InSAR observations are assigned high weights as high-precision boundary constraints, directly anchoring the spatial structure of the fused field without relying on PIM parameterization. Combined with large-gradient central deformation measurements, the proposed framework produces a physically smooth transition zone that cannot be reproduced by threshold-based fusion methods, representing a methodological advancement for high-precision monitoring of mining-induced subsidence under complex geological and mining conditions.

2. Study Area and Data

2.1. Study Area

The study area is located at the working face 208 in the second panel of the Huojitu Mine, Daliuta mining district, Shendong Coalfield, Yulin City, Shaanxi Province, China (Figure 1). The Huojitu Mine lies on the Shaanxi side of the boundary between Shaanxi Province and the Inner Mongolia Autonomous Region and is administratively under the jurisdiction of Daliuta Town, Shenmu City. The mine field covers an area of approximately 63.8 km², extending about 9.76 km in the north–south direction and 10.6 km in the east–west direction, the coal seam structure ranges from simple to moderately complex. The parting rocks are mainly composed of mudstone and siltstone. The geographic coordinates range from 110°7′50″ to 110°16′28″E and from 39°11′27″ to 39°16′49″N. The working face 208 has a strike length of approximately 2700 m and a dip length of about 330 m. Mining began in 1 April 2023, with an average advance rate of about 14 m/d. The extracted seam belongs to the No. 2-2 coal seam, with an average mining thickness of 4.2 m and average mining depth of approximately 140 m. The seam is nearly horizontal and is mined using fully mechanized retreat longwall mining with full caving roof management. The 1-2 upper seam and the 1-2 seam above the working face have already been mined, and most of the area has experienced extraction of two overlying seams; therefore, the present operation represents the third repeated mining in this region. Repeated extraction operations lead to cumulative stress redistribution in the overburden, giving rise to superimposed subsidence effects and consequently enlarging the extent of the subsidence basin. In contrast to single-panel mining, the alternating distribution of multiple working faces and coal pillars creates a “working face–coal pillar–working face” structural configuration. As a result, subsidence troughs over mined-out areas and deformation highs maintained by coal-pillar support emerge alternately along the ground-subsidence profile, generating a characteristic undulating deformation pattern with a distinct W-shaped geometry.

2.2. Data Acquisition

2.2.1. RTK Measurements at the Working Face 208

To obtain high-precision surface elevation and ground-truth deformation measurements in the study area, real-time kinematic positioning (Real-Time Kinematic, RTK) surveying was carried out. The RTK measurements were performed using a cooperative operation of a base station and a rover station. Based on carrier-phase differential positioning, the three-dimensional coordinates of the rover station were solved in real time, enabling centimeter-level accuracy in both horizontal position and elevation. During the field survey, the rover station collected data point by point along predefined observation lines and measurement points. The observations were designed to cover the subsidence center, boundary transition zones, and representative geomorphic areas to ensure spatial representativeness and measurement stability. In this study, two observation lines were arranged along the strike (H) and dip (C) directions of the working face 208 (Figure 1c), with a total of 77 monitoring points. The observation period lasted from 31 March to 15 June 2023, with a measurement interval of approximately 1–3 days, A total of 44 monitoring epochs were obtained from the surface observation stations.

2.2.2. Acquisition of TLS Point Cloud Data

To obtain high-density and high-resolution three-dimensional surface deformation data, a RIEGL VZ-1000 TLS system was used to collect point cloud data in the study area (Figure 2). TLS acquires three-dimensional coordinates of target surfaces through synchronized laser pulse ranging and angular encoding, enabling rapid generation of high-density and high-resolution surface point clouds. This technique is particularly suitable for detailed representation of micro-topographic features within subsidence basins under complex terrain conditions. Prior to field acquisition, eight scanning stations were designed based on the local topography, occlusion distribution, and the extent of the monitoring area. The overlapping regions between adjacent stations were approximately 30–40% to ensure accurate point cloud registration and complete spatial coverage. The resulting point cloud density was approximately 150 pts/m², and additional instrument parameters are summarized in

Table 2. Detailed parameters of TLS.

Performance Indicators	Parameters
Maximum Range	1400 m
Minimum Range	2.5 m
Accuracy	8 mm
Repeatability	5 mm
Maximum Number of Returns	Infinite Returns
Scan Angle Range	360°
Angular Resolution	0.0005°

. After data collection, the point clouds were registered and transformed using RiscanPro software (version 2.1.2). During the fine registration stage, a maximum planar error of 0.01 m was set, and an iterative least-squares approach was applied to minimize residual errors, yielding a complete point cloud dataset covering the study area. Subsequently, noise filtering was performed using Lidar 360 software (version 5.2), and a 1-m resolution DEM was generated via inverse distance weighting (IDW) interpolation. A total of three TLS surveys were conducted, capturing the surface morphology of the working face at different time points. The acquisition dates were 18 March, 14 April, and 15 June 2023.

2.2.3. Sentinel-1 Data Acquisition

Sentinel-1A, operated by the European Space Agency (ESA), is equipped with a C-band synthetic aperture radar (SAR) sensor with a central frequency of 5.405 GHz, corresponding to a wavelength of approximately 5.6 cm. In this study, three Sentinel-1A single look complex (SLC) images were collected. The data were acquired in the Interferometric Wide Swath (IW) mode, which employs the Terrain Observation with Progressive Scans SAR (TOPSAR) technique. The IW mode consists of three sub-swaths with a total swath width of approximately 250 km. The nominal spatial resolution is about 5 m in the range direction and 20 m in the azimuth direction. All images were acquired in ascending orbit with VV polarization. To improve the geometric accuracy of the SAR data, the precise orbit ephemerides (POEORB) provided by ESA were applied during data processing. The detailed information is shown in

Table 3. Data information of Sentinel-1A for study area.

Interference Paris	Acquisition Date		Temporal Baseline (d)	Spatial Baseline (m)	Datatype	Polarization Mode
	Master Image	Slave Image
1	5 March 2023	22 April 2023	48	97	IW(SLC)	VV
2	22 April 2023	21 June 2023	60	49

. The summary of acquisition time for difference sensors is shown in

Table 4. Summary of the acquisition times for different sensors.

Data Type	Period (2023)	Frequency
RTK	31 March~15 June	≈2 days
TLS	19 March, 14 April, 15 June	3 times
SAR	5 March, 22 April, 21 June	3 phases

.

3. Methodology

The workflow of this study is illustrated in Figure 2. (1) Multi-temporal D-InSAR processing is first applied to delineate the subsidence basin boundary using the −10 mm deformation contour, while the TLS point cloud data are used to generate digital elevation models (DEMs) to obtain high-gradient deformation in the basin center. (2) Taking advantage of the complementary characteristics of SAR observations and TLS measurements, the proposed variational data assimilation fusion method is employed to integrate the two datasets, thereby reconstructing the three-dimensional subsidence basin. (3) Sensitivity analysis of the key model parameters is then performed to evaluate the influence of parameter variations on the fusion accuracy. (4) Finally, the deformation results derived from RTK, TLS, and D-InSAR observations are compared with the VDAF fusion results for error assessment, providing validation and support for practical engineering applications.

3.1. Processing of Multi-Temporal Stacked D-InSAR

In this study, a pairwise differential interferometry stacking strategy is adopted to obtain the cumulative surface deformation from 5 March 2023 to 21 June 2023. Specifically, the processing workflow was implemented using the SAR-scape module in ENVI (version 5.6). For each Sentinel-1 SLC image, precise orbit ephemerides (POEORB) provided by the European Space Agency (ESA) were applied to reduce orbital phase errors. The Generic Atmospheric Correction Online Service (GACOS) was employed to correct tropospheric path delays. Topographic phase removal was performed using the 30-m-resolution Digital Elevation Model (DEM) from the Shuttle Radar Topography Mission (SRTM), which was resampled to the SAR geometry using bilinear interpolation. During interferometric processing, a multi-looking factor of 4 × 1 (range × azimuth) was adopted to suppress phase noise and improve the signal-to-noise ratio of the interferograms. Subsequently, the interferograms were filtered using the Goldstein adaptive spectral filtering algorithm. Prior to phase unwrapping, pixels with coherence values lower than 0.3 were masked out. Phase unwrapping was then carried out using the Minimum Cost Flow (MCF) algorithm with an unwrapping threshold of 0.3. Finally, based on the relationship between deformation phase and the line-of-sight (LOS) displacement (Equation (1)), the LOS displacements for each observation interval, denoted as $d_1$ and $d_2$, were derived separately. The total cumulative displacement during the study period, $D$, was subsequently obtained through linear superposition of the two displacement fields (Equation (2)):

$$d_{Los}=-{\displaystyle \frac{\lambda }{4\pi }}\cdot {\varphi }_{def}$$

(1)

$$D=d_1+d_2=-{\displaystyle \frac{\lambda }{4\pi }}({\varphi }_{def}^{(1)}+{\varphi }_{def}^{(2)})$$

(2)

This strategy effectively avoids phase decorrelation caused by long temporal baselines, while improving the reliability and accuracy of deformation monitoring through the combination of short-baseline interferometric pairs.

3.2. Principle of the VDAF

In this study, a multi-source data fusion method based on a VDAF is proposed to fully exploit the advantages of heterogeneous observations and obtain a more accurate estimation of the target deformation field. The proposed method integrates prior information from a background field with multi-source observations into a unified objective function, and the optimal fused result is obtained by minimizing this function. Specifically, a background field representing the large-scale deformation trend is first constructed as the initial state. Observation operators are then defined to introduce discrete measurements into the assimilation framework, allowing heterogeneous datasets to be incorporated in a consistent mathematical form. Based on these components, a variational objective function is formulated, which includes a background constraint term, observation constraint terms, and smoothness regularization terms. The optimal analysis field is obtained by solving the minimization problem using a gradient-based optimization algorithm. Through this procedure, multi-source datasets are assimilated within a unified variational framework, enabling physically consistent fusion and significantly improving the accuracy and spatial continuity of the reconstructed deformation field.

3.2.1. Construction of the Background Field

The background field represents the prior estimate of the target field and generally reflects the large-scale deformation trend. In this study, the background field is constructed using both the D-InSAR-derived boundary observations and the TLS measurements. In practice, scattered data interpolation is employed to fit the observed values onto the analysis grid, and the Thin Plate Spline (TPS) interpolation method is adopted for this purpose. TPS interpolation is a powerful two-dimensional interpolation technique based on the principle of minimizing the bending energy of a surface while forcing the fitted surface to pass smoothly through all observation points. Owing to its theoretical optimality in minimizing surface curvature, TPS is particularly suitable for constructing background fields over irregularly distributed terrain observations. The TPS solution is obtained by minimizing the following functional (Equation (3)):

$${\min }_f {\displaystyle \sum _{i=1}^n{[f\left(x_i,y_i\right)-z_i]}^2+\lambda {\displaystyle \iint }\left[{\left(\frac{{\partial }^{2}f}{\partial x^2}\right)}^2+2{\left(\frac{{\partial }^{2}f}{\partial x\partial y}\right)}^2+{\left(\frac{{\partial }^{2}f}{\partial y^2}\right)}^2\right]dxdy}$$

(3)

where $f\left(x_i,y_i\right)$ is the fitted value of the spline function at the $i$-th observation point,$z_i$ the measured subsidence at the $i$-th observation point, and $\lambda$ is the bending-energy weight controlling the trade-off between fitting accuracy and smoothness. The interpolated surface is used as the initial state $z_0$ of the variational optimization. This initialization incorporates the spatial trends contained in both dataset A (D-InSAR observations) and dataset B (TLS observations), while avoiding iterative optimization starting from a null field, thereby significantly improving the convergence efficiency of the assimilation process.

3.2.2. Formulation of the Variational Assimilation Model

The core of the fusion analysis lies in the construction of a variational assimilation objective function J that incorporates multiple constraint terms. This objective function penalizes the deviation in the analysis field from both the background field and the observations, while additional regularization terms, such as smoothness constraints, are introduced to ensure that the optimized solution remains physically reasonable while fitting the measurements. The objective function is formulated as follows (Equation (4)):

$$J\left(z\right)=J_b+J_A+J_B+J_{\mathrm{smooth}}+J_{\mathrm{grad}}$$

(4)

In the above equation, the terms $J_b$, $J_A$ and $J_B$, $J_{\mathrm{smooth}}$, and $J_{\mathrm{grad}}$ denote the background constraint term, observation constraint terms, smoothness constraint term, and gradient penalty term, respectively. The individual terms are defined as follows:

(1)

Background constraint term ($J_b$):

$$J_b={\displaystyle \frac{{\omega }_b}{2}}|H_{B}z-y_B{|}^2={\displaystyle \frac{{\omega }_b}{2}}{\displaystyle \sum _i{(z\left(x_{B,i}\right)-y_{B,i})}^2, i = 1,2,3,\dots n}$$

(5)

This term (Equation (5)) is constructed from the background field generated by joint interpolation of A-type and B-type observations and represents the prior estimate of the subsidence trend. It measures the deviation in the analysis field from the background field. By introducing the background field as a reference, this term prevents the solution from drifting in regions where observations are sparse or absent. A larger background weight leads to a solution that follows the background field more closely.

(2)

Observation constraint term ($J_A$, $J_B$):

$$J_A={\displaystyle \frac{{\omega }_A}{2}}|H_{A}z-y_A{|}^2={\displaystyle \frac{{\omega }_A}{2}}{\displaystyle \sum _i{(z\left(x_{A,i}\right)-y_{A,i})}^2}, i=1,2,3,\dots n$$

(6)

$$J_B={\displaystyle \frac{{\omega }_B}{2}}|H_{B}z-y_B{|}^2={\displaystyle \frac{{\omega }_B}{2}}{\displaystyle \sum _i{(z\left(x_{B,i}\right)-y_{B,i})}^2, i=1,2,3,\dots n}$$

(7)

This term (Equations (6) and (7)) enforces consistency between the analysis field and the observations at measurement locations. A-type and B-type observations are incorporated with weights ${\omega }_A$ and ${\omega }_B$, respectively. Larger weights indicate higher confidence in the corresponding observations, forcing the analysis field to better fit that dataset. By setting ${\omega }_A>>{\omega }_B$, higher-accuracy A-type observations are emphasized, while deviations from the lower-accuracy B-type observations are penalized less strictly.

(3)

Smoothness constraint term ($J_{\mathrm{smooth}}$):

$$J_{\mathrm{smooth}}={\displaystyle \frac{{\lambda }_s}{2}}|{\nabla }^{2}z{|}^2={\displaystyle \frac{{\lambda }_s}{2}}{\displaystyle \sum _{i,j}{({\nabla }^2{z}_{i,j})}^2,} i = 1,2,3,\dots n$$

(8)

Surface subsidence in mining areas is transmitted progressively through overburden strata and is controlled by the continuity and elastic–plastic deformation characteristics of the rock mass. Therefore, the subsidence field is physically expected to be spatially continuous and smooth. The Laplacian smoothing term provides a mathematical representation of this physical continuity by penalizing the curvature of the analysis field. Minimizing the second-order derivative suppresses unrealistic local oscillations, allows reasonable interpolation in regions lacking observations, and reduces the influence of noise (Equation (8)).

(4)

Gradient penalty term ($J_{\mathrm{grad}}$):

$$J_{\mathrm{grad}}={\displaystyle \frac{{\lambda }_g}{2}}|{\displaystyle \frac{\partial z}{\partial x}}{|}^2+|{\displaystyle \frac{\partial z}{\partial y}}{|}^2={\displaystyle \frac{{\lambda }_g}{2}}{\displaystyle \sum _{i,j}\left[{\left(\frac{\partial z}{\partial x}\right)}_{i,j}^2+{\left(\frac{\partial z}{\partial y}\right)}_{i,j}^2\right]}, i,j = 1,2,3,\dots n$$

(9)

In mining subsidence, the deformation gradient is physically limited by the allowable surface tilt. The gradient penalty term (Equation (9)) reflects this constraint by imposing a mild penalty on the magnitude of the first-order gradient of the analysis field. This term restricts excessive spatial variation and prevents unrealistic steep gradients. As a regularization term, its weight is usually set to a relatively small value so that it mainly suppresses high-frequency noise without over-smoothing the true deformation gradients.

The sum of the above four terms forms the total objective function $J$. By appropriately selecting the weighting parameters, a balance can be achieved between physical prior constraints and observational fitting. When $J$ reaches its minimum, the corresponding analysis field ($z$) represents the optimal fused deformation field that simultaneously satisfies the background prior, the observations, and the required smoothness conditions.

3.2.3. Gradient Formulation

In this study, analytical gradients are used to accelerate the L-BFGS iteration and to avoid truncation errors introduced by finite-difference approximations. The gradient of the cost function with respect to the state vector $z$ can be expressed as (Equation (10)):

$$\nabla J\left(z\right)=\underset{\mathrm{background}}{\underbrace{{\alpha }_b\left(z-z_b\right)}}+\underset{\mathrm{obs}. \mathrm{term} \mathrm{A}}{\underbrace{H_A^{\mathrm{T}}[{\alpha }_A\left(H_{A}z-y_A\right)]}}+\underset{\mathrm{obs}. \mathrm{term} \mathrm{B}}{\underbrace{H_B^{\mathrm{T}}[{\alpha }_B\left(H_{B}z-y_B\right)]}}-\underset{\mathrm{smooth}}{\underbrace{{\lambda }_s{\nabla }^{4}z}}-\underset{\mathrm{gradient}}{\underbrace{{\lambda }_g{\nabla }^{2}z}}$$

(10)

where $H_A^{\mathrm{T}}$ and $H_B^{\mathrm{T}}$ denote the adjoint operators of the observation operators, which map the observation residuals back to the analysis grid. The operator ${\nabla }^4={\nabla }^2\cdot {\nabla }^2$ represents the biharmonic operator, corresponding to the gradient of the smoothness term. Numerically, it is implemented by applying the Laplacian operator to the Laplacian of the analysis field.

3.2.4. Design of the Observation Operator

The observation operator $H$ maps the continuous grid-based state to discrete observation locations. To reduce computational complexity, a nearest-neighbor mapping strategy is adopted (Equation (11)):

$${\left(H_A\right)}_{ij}=\delta \left(j-\arg {\min }_k\sqrt{{\left(x_i-x_k\right)}^2+{\left(y_i-y_k\right)}^2}\right) i,j=1,2,3,\dots$$

(11)

Each observation point is mapped to the grid node with the minimum Euclidean distance, and the observation operator is constructed as a sparse 0–1 matrix. This design reduces the computational complexity of the observation-term gradient evaluation to O(m), completely avoiding the memory bottleneck associated with storing an n × m dense Jacobian matrix, which is essential for the efficient solution of large-scale problems.

3.2.5. Optimization Algorithm: L-BFGS

After constructing the objective function and its analytical gradient, the minimization problem is solved using the L-BFGS algorithm (Limited-Memory Broyden–Fletcher–Goldfarb–Shanno). L-BFGS is a quasi-Newton iterative method characterized by low memory consumption and fast convergence. Its fundamental idea is to iteratively approximate the inverse Hessian matrix using gradient information, without explicitly computing or storing the full Hessian matrix. In the classical BFGS method, the update formula for the inverse Hessian approximation is given by (Equation (12)):

$$H_{k+1}=(I-{\rho }_k{s}_k{y}_k^T)H_k(I-{\rho }_k{y}_k{s}_k^T)+{\rho }_k{s}_k{s}_k^T,\hspace{1em}{\rho }_k={\displaystyle \frac{1}{y_k^T{s}_k}}$$

(12)

This update formula provides curvature correction for the quasi-Newton method, allowing the search direction ${\rho }_k=-H_k\nabla f_k$ to incorporate second-order derivative information, thereby accelerating convergence toward the optimum compared with first-order descent methods. The L-BFGS algorithm is developed based on this formulation by retaining $m$ pairs of vectors $(s_i,y_i)$ to approximate the inverse Hessian matrix $H_k^{-1}$. Compared with the standard Newton method, which requires explicit computation and storage of the full Hessian matrix, L-BFGS stores only a limited sequence of vectors to implicitly construct the inverse Hessian approximation. Therefore, it is particularly suitable for high-dimensional optimization problems such as the one in this study, where the Hessian matrix is difficult to compute or store explicitly.

3.2.6. Accuracy Assessment

To assess the reliability of the fused results, the Root Mean Square Error (RMSE) was used to quantify the discrepancies between the predicted and observed values. Furthermore, the Relative Root Mean Square Error (RRMSE) was adopted to evaluate whether the reconstruction meets the required accuracy level. The corresponding metrics are defined as follows (Equations (13) and (14)):

$$RMSE=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^n(D_{RTK}-D_{VDAF})}}{n}}}$$

(13)

$$RRMSE={\displaystyle \frac{RMSE}{V_m}}\times 100\%$$

(14)

In these equations, $D_{RTK}$ represents the subsidence values obtained from RTK surface observations, whereas $D_{VDAF}$ denotes the corresponding subsidence values extracted from the VDAF fusion results at the RTK observation coordinates. $n$ is the total number of samples, and $V_m$ is the maximum observed subsidence value.

4. Results and Analysis

4.1. Results of Stacked D-InSAR

It should be noted that the surface deformation obtained from D-InSAR measurements is primarily in the radar line-of-sight (LOS) direction. According to the imaging geometry of SAR satellites, LOS deformation can be converted to vertical deformation. Since mining subsidence basins are dominated by vertical displacement, and the horizontal component is negligible compared with the vertical component, horizontal displacement is not considered in this study [29,39,40,53]. According to the “Specifications for Coal Pillar Retention and Coal Mining under Buildings, Water Bodies, Railways and Main Roadways” [54], the boundary of a subsidence basin is defined by the −10 mm deformation contour. To determine the effective subsidence boundary up to 21 June, multi-temporal stacked D-InSAR processing was applied. The cumulative deformation between 5 March and 21 June was then obtained through linear superposition of the two differential results, yielding the subsidence distribution of the working face 208 as shown in Figure 3.

The deformation pattern obtained from D-InSAR indicates that the subsidence basin exhibits an approximately elliptical shape, and the maximum subsidence derived from D-InSAR is only about −77 mm. Massonnet and Feigl (1998) [11] demonstrated that, under constant pixel spacing, the theoretical maximum detectable deformation gradient by InSAR is approximately λ/2. Following multi-looking of the Sentinel-1 data, the effective measurable deformation rate is approximately 4 mm/m. In contrast, RTK observations show that the maximum cumulative deformation over the same period reaches about 5 m, corresponding to a deformation rate of approximately 300 mm/m, which clearly demonstrates the limitation of D-InSAR in areas with large deformation gradients. Nevertheless, for small-magnitude deformation, D-InSAR provides high measurement accuracy.

To obtain a more reliable boundary of the subsidence basin, the coherence-dependent limit of detectable deformation must be considered. According to the analysis of Baran et al. (2005) [23], the maximum measurable deformation gradient increases with increasing interferometric coherence. For Sentinel imagery, when the coherence coefficient is lower than 0.3, reliable deformation retrieval is generally not possible. Therefore, considering the balance between coherence quality and effective pixel coverage, a coherence threshold of 0.3 was adopted in this study. Based on this threshold, the −10 mm deformation contour was extracted to delineate the boundary of the subsidence basin.

4.2. Construction of the TLS-Derived Subsidence Basin

During the acquisition of TLS point cloud data, factors such as instrument measurement errors, scanning angle, scanning direction, atmospheric interference affecting laser pulses, and interpolation algorithms can significantly influence the density and completeness of the point cloud, thereby affecting the accuracy of the generated DEM [55]. Therefore, the DEMs derived from two TLS surveys were compared with synchronous RTK elevation measurements to evaluate their accuracy. Partial results are listed in

Table 5. Data Partial elevation deviation between RTK and TLS in two phases.

First Phase of DEM				Second Phase of DEM
Point No.	TLS (m)	RTK (m)	Error (m)	Point No.	TLS (m)	RTK (m)	Error (m)
1	1198.921	1198.968	0.047	1	1198.58	1198.744	0.164
2	1198.604	1198.791	0.187	2	1198.598	1198.504	−0.094
3	1197.854	1197.939	0.085	3	1197.454	1197.48	0.026
4	1197.576	1197.404	−0.172	4	1196.833	1196.844	0.011
5	1197.867	1197.861	−0.006	5	1196.789	1196.718	−0.071
6	1197.335	1197.36	0.025	6	1195.421	1195.543	0.122
7	1197.409	1197.208	−0.201	7	1194.546	1194.528	−0.018
8	1197.395	1197.104	−0.291	8	1193.816	1193.879	0.063
9	1197.427	1197.269	−0.158	9	1193.994	1193.797	−0.197
10	1197.132	1196.971	−0.161	10	1193.649	1193.449	−0.2
…				…
RMSE/MAE	0.18/0.12			0.21/0.14

. As shown in the table, the RMSE and AAE of the DEM from the first survey are 0.18 m and 0.12 m, respectively, while the RMSE and AAE of the DEM from the second survey are 0.21 m and 0.14 m, respectively.

By performing differential analysis on the DEMs obtained from the two TLS surveys, the vertical surface deformation during this period can be derived, as shown in Figure 4.

It can be observed that the maximum subsidence within the surface subsidence basin of the working face 208 reaches approximately 5.2 m. Previous studies have shown that TLS is capable of capturing high-density point clouds in areas with rapid and large-magnitude deformation, allowing detailed characterization of surface subsidence. In contrast, at the basin boundary, where deformation develops slowly and the magnitude is very small, subtle changes are difficult to distinguish from point cloud noise, resulting in reduced observation accuracy [56]. In this study, RTK measurements located far from the basin center were compared with elevations extracted from the DEM. The results indicate that the farther a point is from the subsidence center, the larger the discrepancy between the RTK measurements and the DEM-derived elevations, with the average error approaching 0.11 m. Conversely, in the high-gradient deformation zone near the basin center, the TLS-derived elevations show smaller differences relative to RTK measurements and are closer to the true surface deformation. To ensure the accuracy of subsidence basin reconstruction and to eliminate the influence of unreliable data, the mean absolute error (MAE) between synchronous RTK observations and TLS-derived elevations was calculated (−0.22 m). In this study, TLS-derived deformation values smaller than −0.22 m were selected as valid data to be included in the fusion process.

4.3. Data Fusion Results

By combining the subsidence basin boundary derived from D-InSAR with the high-gradient subsidence center observed by TLS as dual physical constraints, and applying the proposed VDAF data fusion method, the transition zone between the two datasets can be effectively reconstructed. This approach enables a more realistic representation of the surface subsidence field, as shown in Figure 5.

As shown in Figure 5a, the VDAF method successfully integrates the D-InSAR-derived basin boundary and the TLS-observed high-gradient subsidence center into a unified deformation field, while the transition zone between the two datasets exhibits both trend consistency and spatial smoothness. Due to repeated mining and the constraint effect of reserved coal pillars (Figure A1), the overall subsidence basin presents a distinct “W-shaped” morphology rather than the idealized “bell-shaped” basin typical of single-seam extraction. The two troughs correspond to the locations of maximum subsidence, both approaching approximately −5.2 m (Figure 5c,d). As illustrated in Figure 5b, from a geometric perspective, the high-gradient subsidence center represented by TLS observations is not located at the geometric center enclosed by the D-InSAR-derived boundary. Instead, the distance from the −10 mm subsidence boundary on the southwest side to the open-off cut is significantly larger than the distance from the opposite side to the TLS-derived center (R₁ > R₂). According to mining subsidence theory, this asymmetry is mainly caused by high-intensity mining and rapid face advance in the study area. Near the open-off cut, mining had been completed for nearly three months, and surface subsidence had gradually stabilized and approached its final state. In contrast, at locations farther from the open-off cut, underground extraction was still ongoing, and the deformation induced by mining had not yet fully propagated to the surface. As a result, the surface subsidence and the extent of the influence zone had not reached their peak values, producing a noticeable lag effect in the basin geometry.

RTK monitoring records (Figure A2) indicate that measurable surface subsidence (≥10 mm) generally occurs 3–7 days after the underground working face passes beneath a given surface location, whereas peak subsidence is typically reached after 20–45 days. This lag effect is primarily controlled by the thickness and mechanical properties of the immediate roof strata and the advance rate of the longwall face. As a result, the subsidence basin observed during the monitoring period represents a spatial assemblage of different deformation stages, with areas near the open-off cut approaching final subsidence conditions and areas near the active working face remaining under active deformation.

4.4. Spatial Error Analysis

To investigate the spatial distribution of errors, the subsidence values at the corresponding coordinates of 55 points along line H and 22 points along line C in the RTK dataset were extracted from the fused results. Errors were then calculated by comparison with the RTK measurements.

As shown in Figure 6a,b, the RTK-observed and predicted values along both lines exhibit an approximately linear relationship, with R² values exceeding 0.98, indicating a good fit. The RMSE along line H (−0.09 m) is slightly lower than that along line C (−0.14 m), accounting for 2.1% and 2.6% of the maximum subsidence (−5.2 m), respectively. The MAE indicates that the deviations in predicted values from measured values are larger along line C than along line H. Figure 6c shows the spatial error distribution along line H. Errors between points H02 and H10 (C₁) are relatively large, with an average of approximately 0.18 m, and the maximum error occurring at H09 (0.37 m). The remaining points exhibit smaller errors with a more uniform distribution. Figure 6d illustrates the error distribution along line C, where larger errors are concentrated between points C05 and C11 (C₂), with an average of approximately 0.26 m and a maximum of 0.35 m at C09. Based on their spatial locations, both C₁ and C₂ are located in the outer boundary zones of the subsidence basin. As noted in Section 4.2, point cloud measurements are less sensitive to subtle deformations in these regions, resulting in reduced monitoring accuracy. As a result, a clear trend can be observed: areas with larger subsidence gradients generally exhibit smaller monitoring errors, whereas areas with smaller gradients tend to show larger errors.

4.5. Comparative Analysis of D-InSAR, TLS, and RTK Data

To validate the effectiveness of the proposed VDAF method, deformation values from D-InSAR, TLS-derived DEM, and VDAF fusion results were extracted at a total of 77 RTK observation points distributed along the H and C directions. A comparative analysis was then conducted based on these datasets. The spatial distribution of the RTK observation points is shown in Figure 7.

Due to the large spatial extent of the study area, it was not possible to ensure full coverage of RTK observation points at the basin boundaries during TLS. As indicated by the final DEM, points H01–H05 were not captured by the TLS observations. Therefore, for consistency among the three datasets, only points H06–H55 along the strike direction and all points along the dip (C) direction were selected for extraction and analysis. The comparison results are shown in Figure 8 and Figure 9.

As shown in Figure 8, for the strike (H) observation points, the D-InSAR results exhibit relatively small differences from the RTK measurements at the basin boundary, whereas they fail almost completely in the high-gradient subsidence region. This further confirms that D-InSAR has advantages in detecting small-magnitude deformation but is limited in areas of large deformation. The TLS-derived results show a strong overall correlation with the RTK observations in terms of trend; however, some measurement points exhibit overestimation or underestimation (apparent uplift or excessive subsidence), resulting in relatively large local errors. In contrast, the VDAF results demonstrate high fitting accuracy for most observation points, except for a few larger deviations at the basin edges. Overall, the predicted deformation field shows a high degree of agreement with the RTK measurements.

As shown in Figure 9, for the dip (C) observation points, the D-InSAR results exhibit a pattern similar to that along the H line, characterized by higher accuracy at the basin boundaries and decreasing accuracy toward the basin center. The TLS-derived subsidence values show an overall underestimation, with no clear pattern in the error distribution and relatively large errors occurring in certain segments or individual points. Compared with the H line, the VDAF fusion results along the C line show slightly reduced accuracy, with larger discrepancies mainly occurring near the basin boundaries. Nevertheless, the overall predicted deformation still exhibits strong agreement with the RTK measurements. The quantitative accuracy assessment results for both observation lines are presented in

Table 6. Accuracy statistics table.

Line	Method	MAE (m)	RMSE (m)	Max-Error (m)	R2
H	TLS	0.1261	0.1782	0.5789	0.9731
	D-InSAR	3.3768	3.5487	4.9588	−9.6527
	VDAF	0.0639	0.0948	0.3746	0.9947
C	TLS	0.1462	0.2189	0.7084	0.9806
	D-InSAR	1.4744	2.1259	3.5647	−0.828
	VDAF	0.1425	0.174	0.3502	0.9878
Combined	TLS	0.1322	0.1916	0.7084	0.9846
	D-InSAR	2.7955	3.1822	4.9588	−3.2527
	VDAF	0.0807	0.1168	0.3746	0.9943

.

Considering all observation points along both the H and C lines, the RMSE of TLS is approximately 0.19 m, while D-InSAR exhibits the poorest performance with an RMSE of up to 3.18 m. In contrast, the proposed method achieves an overall RMSE of approximately 0.12 m, with the RMSE along the H line being lower than that along the C line. Using the maximum subsidence value obtained from RTK observations (approximately −4.98 m), RRMSE is calculated to be 2.4%, further demonstrating the effectiveness of the proposed method.

5. Discussion

5.1. Parameter Sensitivity Analysis

To further investigate the influence of parameter selection on the fusion results, a sensitivity analysis was conducted by adjusting the key parameters in the VDAF method, including the background constraint weight (${\omega }_b$), the weight ratio between D-InSAR and TLS observations (${\omega }_A/{\omega }_B$), the smoothness regularization parameter (${\lambda }_s$), and the gradient penalty parameter (${\lambda }_g$). The variation in the resulting RMSE under different parameter settings was evaluated to provide a systematic justification for the rational selection of these parameters. The complete derivation process and the response curves of all parameters are provided in Appendix A and Appendix B. The main results are summarized as follows.

Parameter weights were derived based on the statistical properties of sensor noise within the optimal estimation framework. The D-InSAR boundary precision is approximately ${\sigma }_A$ ≈ 5 mm [57,58,59], and the residual standard deviation in TLS is ${\sigma }_B$ ≈ 22 mm, yielding a theoretical weight ratio of ${({\sigma }_B/{\sigma }_A)}^2$ ≈ 19.4. A conservative weight ratio of ${\omega }_A/{\omega }_B$ = 10 was adopted in this study, as sensitivity tests indicated that varying this ratio within the range [10, 50] caused an RMSE change of less than 0.5 mm, confirming the robustness of the solution within this interval. The ranges of all parameters were set based on physical considerations: the weight ratio was allowed to vary within [1, 5 × theoretical value] to avoid numerical ill-conditioning of the Hessian matrix in the L-BFGS algorithm; the background weight ${\omega }_b$ was set within [10⁻⁴, 1] to capture the transition from negligible to dominant prior influence; the smoothing parameter ${\lambda }_s$ range [10⁻⁴, 10] was determined via L-curve analysis; and the gradient penalty ${\lambda }_g$ was varied within [0, 1] to encompass physically meaningful slope constraints.

Single-parameter sensitivity analyses (Figure A3a–e and

Table 7. Summary table of parameter sensitivity analysis.

Parameters	Sensitivity	Robustness	Current Setting Evaluation
${\omega }_A/{\omega }_B$	Medium	Strong	Slightly conservative, can be moderately increased
${\lambda }_{\mathrm{s}}$	High	Weak	Supported by the L-curve and physically reasonable
${\omega }_b$	low	Extremely Strong	Within the stable region, reasonable
${\lambda }_g$	Medium–low	Relatively Strong	Reasonable, slightly above optimal but acceptable

) indicate that the smoothing regularization parameter ${\lambda }_s$ is the dominant factor: its RMSE varies by up to 10.1 mm across the entire range, and the L-curve analysis identifies a clear optimal value. Both the observation weight ratio ${\omega }_A/{\omega }_B$ and the gradient penalty ${\lambda }_g$ exhibit broad stability plateaus, with RMSE variations of less than 5 mm and 2.2 mm, respectively, demonstrating strong robustness. The background constraint weight ω_b has a negligible impact over four orders of magnitude (RMSE < 10⁻⁵ m), confirming that the final solution is primarily driven by observational data rather than prior information.

Joint parameter interaction analyses (Figure A4 and Figure A5) reveal that ${\lambda }_s$ participates in all of the top-ranked parameter pairs. Specifically, the coupling effects between ${\lambda }_s$ and ${\lambda }_g$, as well as between ${\omega }_b$ and ${\lambda }_s$, rank first and second, respectively, whereas the combination of ${\omega }_A/{\omega }_B$ and ${\omega }_b$ ranks last. The combined coefficients of variation for all parameter pairs remain below 1%, reflecting the overall robustness of the system. The selected parameter combination lies within the stable-accuracy plateaus of each parameter; in practical applications for specific sites, only ${\lambda }_s$ may warrant further fine-tuning.

5.2. Limitations of the VDAF Method

(1)

Manual specification of weighting parameters without adaptive calibration. In this study, the five weighting parameters in the cost function (${\omega }_b$, ${\omega }_A$, ${\omega }_B$, ${\lambda }_{\mathrm{s}}$, and ${\lambda }_{\mathrm{g}}$) are manually assigned based on empirical judgment, without systematic calibration. Although the current parameter settings achieve satisfactory accuracy on the validation dataset, the optimal parameter combination may vary significantly under different geological conditions, mining depths, or sensor configurations. The subjectivity in parameter selection limits the generality and reproducibility of the method. Future work should consider incorporating automatic calibration strategies, such as the L-curve criterion, generalized cross-validation (GCV), or Bayesian hyperparameter optimization.

(2)

Static framework incapable of modeling spatiotemporal evolution of subsidence. The proposed method adopts a three-dimensional variational (3D-Var) framework, which performs snapshot-based reconstruction for a single observation epoch and does not account for the temporal evolution of the subsidence field during mining. For multi-temporal monitoring data, each epoch is processed independently, and the temporal correlation between successive observations is not exploited, resulting in a loss of information along the time dimension. Extending the current framework to a four-dimensional variational (4D-Var) scheme or an Ensemble Kalman Filter framework would enable joint spatiotemporal estimation of subsidence evolution.

(3)

Limitations in physical model applicability. Classical prediction models in mining subsidence, such as the Probability Integral Method (PIM) and the Knothe influence function, are supported by well-established mechanical theories. However, in this study area, a significant time lag exists between underground mining and the manifestation of surface subsidence. As a result, the observed subsidence basin exhibits strong asymmetry, abrupt local slope variations, and irregular boundary morphology, deviating from the idealized “bell-shaped” profile described by traditional physical models. Imposing such idealized prior constraints on the background field would introduce theoretical bias into the fusion results and distort the transition zone. Therefore, this study adopts a data-driven approach by constructing the background field using thin plate spline interpolation, avoiding strong assumptions about basin morphology and instead relying on multi-source observations to constrain the deformation field. This strategy provides greater adaptability under dynamic mining conditions where physical prior models are inadequate. However, it also implies that the spatial reliability of the fusion results depends entirely on the coverage and accuracy of the observational data. In regions with sparse observations, the lack of physical constraints may reduce the reliability of the reconstructed deformation field, representing an inherent limitation of the method in terms of physical interpretability.

(4)

Limitations due to data availability. Owing to practical constraints such as the observation schedule of surface monitoring stations, cost considerations, and the uncertainty in the revisit times of SAR satellites, only three Sentinel-1 scenes actually covered the study area during the three-month observation period. This limitation is inherent to the data acquisition process. It should be emphasized that the primary focus of this study is the fusion of heterogeneous observations within VDAF, rather than performing multi-temporal time-series deformation analysis. The objective is to achieve a spatially complete reconstruction of the subsidence basin at a specific time epoch. Two interferometric pairs adequately capture both the active mining period and the subsequent deformation stabilization phase, providing the boundary deformation gradients required by the VDAF cost function; hence, constructing a dense temporal sequence is not necessary. Nevertheless, access to a more densely sampled SAR time series would facilitate improved accuracy in delineating basin boundaries. Accordingly, in future work, we plan to incorporate multi-temporal SAR stack techniques to extend the framework toward four-dimensional spatiotemporal deformation reconstruction.

6. Conclusions

This study addresses the core challenge of multi-source heterogeneous data fusion in mining-induced surface subsidence monitoring by proposing a data-driven VDAF for subsidence field reconstruction. The method constructs a five-component objective function, including a background constraint term, two observation-fitting terms, a spatial smoothness term, and a gradient penalty term. Within this unified optimization framework, boundary constraints provided by spaceborne InSAR and central subsidence observations from TLS are jointly assimilated to reconstruct a complete three-dimensional subsidence basin over the mined-out area. Furthermore, a systematic parameter sensitivity analysis is conducted to comprehensively evaluate the reliability and robustness of the proposed method. The main conclusions are summarized as follows:

(1)

Validation against independent RTK measurements shows that the proposed method achieves an RMSE better than 0.12 m and an RRMSE of 2.4%, demonstrating that the method effectively exploits the spatial complementarity of the two data sources and significantly improves subsidence reconstruction accuracy.

(2)

Comparative analysis of D-InSAR, TLS, and RTK data indicates that D-InSAR performs well in detecting small-magnitude deformation but is substantially constrained in areas with steep deformation gradients. TLS observations show an overall strong correlation with RTK measurements but exhibit relatively large errors in certain segments. In contrast, the VDAF method achieves high overall accuracy, with only minor deviations at a few boundary points.

(3)

Parameter sensitivity analysis reveals that the smoothness parameter ${\lambda }_{\mathrm{s}}$ is the dominant factor controlling fusion accuracy. Its influence is significantly stronger than that of other parameters, with parameter combinations involving ${\lambda }_{\mathrm{s}}$ consistently ranking among the top three in interaction influence, and the maximum RMSE variation reaching 10.1 mm. The observation weight ratio and gradient penalty parameter exhibit strong robustness, while the background constraint weight has minimal influence on the fusion results. In this framework, the background field mainly serves as an initialization and stabilization constraint, and the final solution is primarily driven by observational data, indicating a low dependence on prior assumptions.

(4)

Joint stability analysis shows that the coefficient of variation (CV) for all parameter combinations is below 1%, indicating that the VDAF method maintains high overall stability within the defined physical parameter ranges. This demonstrates that the parameter selection is both reproducible and generalizable.

In summary, the proposed method demonstrates strong performance in terms of theoretical completeness, objective parameter selection, and robustness of results, providing a systematically validated framework for the quantitative fusion of multi-source subsidence monitoring data in mining areas. Future work will focus on extending the method to time-series subsidence monitoring scenarios and exploring the integration of additional heterogeneous data sources into the fusion framework.