Symmetry

Regional-scale gravity imaging is limited by the size of the sensitivity matrix: a direct three-dimensional formulation of the gravity operator requires on the order of ${10}^{13}$ coefficients, making explicit storage infeasible on standard hardware. We exploit two symmetry properties of the gravitational sensitivity kernel: translational invariance, giving the operator a Block Toeplitz with Toeplitz Blocks (BTTB) structure, and central symmetry, halving the number of unique kernel evaluations, to reformulate the Linear Back-Projection (LBP) adjoint operator as depth-wise two-dimensional convolutions. With finite-support truncation at cutoff radius $R_{\max }$, memory requirements drop from terabyte-scale dense arrays to compact per-layer kernels. We implement and benchmark two equivalent backends: an OpenMP spatial-domain stencil and an FFT convolution via BTTB circulant embedding. For a production-scale model ( cells, $R_{\max }=150$), the stencil completed the LBP computation in 7187 s on one thread and 354 s on 32 threads ($S_{32}=20.29$), while the FFT backend required 13.8 s; both agree to within an invariant checksum ($3.3464\times {10}^{10}$). Applied to Bouguer gravity data over the Chicxulub impact crater, northern Yucatán Peninsula, the raw LBP amplitudes were calibrated to physical density contrasts through a two-parameter rescaling solved in closed form as a box-constrained convex quadratic program and cross-validated against an independent simulated-annealing search, reducing the RMS misfit from $27.3$ to $5.3$ mGal (an $81\%$ reduction). The resulting model shows density patterns consistent with the crater’s known structural organization. As a single-pass adjoint method, LBP does not resolve the non-uniqueness of the gravity inverse problem, and the model is interpreted for regional structure rather than for pointwise density values; the formulation extends directly to iterative regularized inversion.

To investigate the progressive asymmetry damage characteristics of intersecting fracture rock under the coupled effect of freeze–thaw cycles and loading, nuclear magnetic resonance tests were conducted on intact, single fracture, and intersecting fracture sandstone under freeze–thaw cycles. Uniaxial and triaxial compression tests were conducted to investigate the mechanical behavior and failure characteristics of fractured rock, while the effects of freeze–thaw cycles on pore structure evolution, microscopic damage, and mechanical degradation were further evaluated. The results showed that the intersecting fracture rock was more sensitive to microscopic damage caused by freeze–thaw cycles. As external loading was applied, the initiation stress and peak stress of the intersecting fracture rock were the lowest of the three rock types, and the failure mode changed from shear failure to tensile failure. Based on damage localization theory and fracture mechanics, the expansion of macro-fissures in the rock were found to be equivalent to the growth process of crack clusters at the tip, and a progressive damage constitutive model considering the coupled effect of freeze–thaw cycles and loading was developed. This model was able to reproduce the main characteristics of the mechanical response and damage evolution of intersecting fracture rock. It provides a potential reference for understanding the mechanical behavior engineering rock in cold regions.

Although the Kalman filter is optimal for linear Gaussian systems, its performance degrades in practical scenarios when noise is colored rather than white. Additionally, the Kalman filter requires substantial domain knowledge to derive parametric forms. To overcome these limitations, this paper proposes a data-driven filtering method that operates in the presence of colored noise and with partial information without requiring exact prior knowledge. We propose CKFNet, a novel real-time filtering framework that integrates a linear state-space model under colored noise with a gated recurrent unit. The network replaces the traditional Kalman gain computation, enabling the model to implicitly learn noise statistics from data and adapt to colored noise dynamics. Experiments across various colored noise conditions and model mismatch scenarios show that CKFNet outperforms baseline methods in most tested scenarios, and remains robust where model-based filters degrade. The results indicate that CKFNet effectively combines the interpretability and efficiency of state-space models with the adaptive learning capability of neural networks, offering a reliable filtering solution under realistic noise assumptions.