Multi-medium geological data cross-manifold topological consistency expression method

Abstract

The application discloses a kind of multi-medium geologic data cross-manifold topological consistency expression methods, method includes: obtaining borehole core, well logging curve, remote sensing image and three-dimensional seismic data, establishes each medium original Riemann manifold and feature point set;Cross-manifold alignment mapping and parallel transmission operator are constructed, and geometric consistency quantity is determined according to closed loop transmission deviation and the aggregation weight is adjusted;Based on alignment feature, construct Vitoris-Rips filter flow and persistent graph, and locate topological conflict with bottleneck distance;According to physical field operator output, physical equation residual and topological deviation, construct joint drift constraint, solve Schrodinger bridge random transport path, update conflict data and encode with remaining data as consistent expression data.The application realizes the integration consistency processing of multi-medium geologic data from original geometry preservation, topological conflict identification to physical constraint rectification and continuous expression.

Description

Technical Field

[0001] This application relates to the field of geological information processing technology, and in particular to a method for expressing cross-manifold topological consistency of multi-media geological data. Background Technology

[0002] Borehole cores, well logging curves, remote sensing images, and 3D seismic volumes are crucial data for oil and gas exploration, mineral surveys, and geological modeling. Due to significant differences in acquisition methods, spatial resolution, physical response mechanisms, and data structures, existing technologies often use coordinate registration, scale normalization, feature encoding, or common spatial mapping to transform multi-source heterogeneous geological data into a unified parameter domain before fusing them. While these methods improve data processability, they typically assume that correct pairwise mappings between different media represent overall consistency, easily overlooking the issue that multi-media features cannot be restored to their initial state after transmission along a closed mapping path. For example, even with small matching errors between cores and well logging, or between well logging and seismic data, the closed-loop transmission along the "core—well logging—seismic—core" path may still result in directional rotation, scale scaling, or feature shifts, causing the fusion results to change with the processing order. Existing distance or statistical distribution indicators are insufficient to identify such global geometric inconsistencies.

[0003] Furthermore, existing geological data cleaning, interpolation, and generative correction methods often evaluate correction effectiveness based on numerical errors, physical residuals, or model quality before and after processing, easily overlooking transient changes in the topology of geological bodies during the correction process. When two originally isolated reservoir units are incorrectly connected, continuous seepage channels are severed, or trap cavities disappear during intermediate iterations, the data before and after correction may still meet numerical and physical thresholds, but reservoir connectivity, fault sealing, and trap integrity have already changed. Therefore, current technologies lack a digital consistency representation method that can both maintain the original geometry of each medium, verify the consistency of cross-medium closed-loop transport, and identify multi-scale topological conflicts and constrain the correction process. Summary of the Invention

[0004] To address the aforementioned problems, embodiments of the present invention provide a method for expressing the topological consistency of multi-media geological data across manifolds, the method comprising:

[0005] Acquire borehole cores, logging curves, remote sensing images, and 3D seismic data; establish the native Riemannian manifold for each type of medium and form a set of medium feature points;

[0006] Establish cross-manifold alignment mapping and parallel transport operator for the corresponding nodes. Combine the parallel transport operator along the closed path that passes through four types of native Riemannian manifolds in sequence and returns to the starting manifold to obtain the closed-loop transport operator. Determine the geometric consistency quantity based on the deviation of the closed-loop transport operator from the identity transformation.

[0007] The cross-medium aggregation weights are adjusted according to the geometric consistency quantity to obtain the aligned feature point set;

[0008] Based on the Riemann geodesic distance of the aligned feature point set, a Vitoris-Lipps filter is constructed, a media persistence map is extracted, and the topological conflict media pairs, conflict coherence dimensions, and conflict scale intervals are determined according to the bottleneck distance between the media persistence maps.

[0009] The nodes to be corrected are determined based on the topological conflict media pairs, conflict coherence dimensions, and conflict scale intervals. A reference distribution is constructed using the output of the multi-media physical field operator, and a reference persistence map is extracted. A joint drift constraint is constructed based on the geophysical equation residuals and the topological deviation between the transport state persistence map and the reference persistence map.

[0010] Using the node data to be corrected and the reference distribution as boundary distributions, the Schrödinger bridge random transport path is solved according to the joint drift constraint. The node data to be corrected is updated along the Schrödinger bridge random transport path, and the updated data and the remaining observation data are encoded into consistent representation data.

[0011] Furthermore, the primary Riemannian manifold includes: a borehole core manifold using mineral composition probability vectors as manifold elements and employing the Fisher-Rao information metric; a logging curve manifold induced by the Archie rock electrical relation in the space of porosity, water saturation, and cementation parameters; a remote sensing image manifold using spectral reflectance functions as manifold elements and employing the function space inner product metric; and a three-dimensional seismic manifold using elastic wave impedance distribution as manifold elements and employing the Wasserstein optimal transmission metric.

[0012] Furthermore, the cross-manifold alignment mapping establishes corresponding node relationships based on spatial coordinates, stratigraphic horizons, and sampling depth. Nodes of the preceding primary Riemannian manifold in the closed path are determined as source nodes, and nodes of the following primary Riemannian manifold are determined as target nodes. Nodes in the neighborhood of the source node are transformed to the source tangent space via a logarithmic mapping with the source node as the base point, then transformed to the target tangent space via a tangent space mapping, and finally formed into aligned nodes via an exponential mapping with the target node as the base point. The parallel transport operator is established according to the source node, the target node, and the alignment path between them.

[0013] Furthermore, the closed path connects corresponding nodes in the order of borehole core manifold, well logging curve manifold, remote sensing image manifold, 3D seismic manifold, and borehole core manifold; each parallel transmission operator is compounded according to the transmission order of the closed path, and the resulting closed-loop transmission operator is determined as a Hele group element; the geometric consistency quantity is determined based on the normalized Frobenius norm difference between the Hele group element and the initial tangent space identity transformation, and the monotonically decaying transformation of the geometric consistency quantity is used to modulate the cross-medium aggregation weight.

[0014] Furthermore, the Vitoris-Lipps filter is constructed according to the Riemann geodesic distance and increasing adjacency scale between nodes in the aligned feature point set; the media persistence map records the birth and death scales of the corresponding homology features of connected components, loops, and cavities; the bottleneck distance is determined by partial bijection between a persistence point in one media persistence map and a persistence point in another media persistence map, as well as the diagonal projection point; based on the media to which the persistence points participating in the partial bijection belong, the homology dimension, the birth scale, and the death scale, the topological conflict media pair, the conflict homology dimension, and the conflict scale interval are determined.

[0015] Further, the topological generators corresponding to the persistent conflict points involved in determining the bottleneck distance are obtained, and the simplex constituting the topological generators is mapped to the nodes in the alignment feature point set. Then, the nodes to be corrected are determined based on the spatial correspondence between the nodes and the sampling positions in borehole cores, well logging curves, remote sensing images, and 3D seismic data. The reference persistent map corresponding to the conflict coherence dimension and conflict scale interval is extracted from the reference distribution, and the transport state persistent map is extracted from the current transport state of the Schrödinger bridge random transport path. The topological deviation is determined based on the bottleneck distance between the two.

[0016] Furthermore, the multi-medium physical field operator includes a feature lifting layer, a Fourier operator layer, and a physical property field output layer; the Fourier operator layer performs a Fourier transform on the input function corresponding to the aligned feature point set, performs a learnable spectral transform on the preset frequency domain mode, performs an inverse Fourier transform on the spectral transform result, and combines the inverse Fourier transform result with the point-by-point linear transform result; the physical property field output layer generates a unified geological physical property field, and determines the geological physical equation residuals based on the Darcy seepage equation residuals, Maxwell's quasi-static electromagnetic equation residuals, and Archie rock electrical relation residuals.

[0017] Further, a physical drift term is constructed based on the residuals of the geophysical equations, and a topological drift correction term is constructed based on the topological deviation. The physical drift term and the topological drift correction term form the joint drift constraint. The forward Kolmogorov equation and the inverse Kolmogorov equation, with the empirical distribution formed by the data of the node to be corrected and the reference distribution as boundary distributions, are solved to obtain the drift field of the Schrödinger bridge random transport path. After each discrete update, the persistent transport state map is re-extracted, and the topological drift correction term is updated based on the newly determined topological deviation.

[0018] Furthermore, the updated data and the remaining observation data are used as training data for the implicit neural representation network, which takes spatial coordinates as input and geological attributes as output. After each network parameter update, a discrete update of the Schrödinger bridge random transport path is invoked according to the joint drift constraint to perform a projection update on the geological attributes output by the implicit neural representation network. Multi-resolution spatial sampling is performed on the trained implicit neural representation network, and isosurfaces are extracted based on the sampled geological attributes to obtain the consistent representation data.

[0019] Furthermore, a method for expressing the topological consistency of multi-media geological data across manifolds further includes: continuously matching persistent conflict points and their topological generators in adjacent transport states based on the conflict coherence dimension, the conflict scale interval, and the bottleneck distance; determining the node topological responsibility weight based on the number of consecutive occurrences of the node, the distance from the node to the spatial center of the topological generator, and the contribution of the node to the bottleneck distance, and allocating the topological drift correction term accordingly.

[0020] The technical solution provided by this invention has at least the following technical effects or advantages:

[0021] This invention achieves integrated consistency processing of multi-media geological data, from preserving the original geometry and identifying topological conflicts to correcting physical constraints and continuous representation. It preserves the original geometric structure of each medium and evaluates overall self-consistency through closed-loop transmission bias, identifying hidden problems with normal pairwise matching but global transmission distortion, reducing the dependence of fusion results on the processing order. By utilizing persistent topological features to locate conflicting media, homology types, and scale ranges, it can discover differences in connectivity, loops, and cavities, compensating for the inadequacy of numerical errors and statistical indicators in reflecting topological changes. The residuals of geophysical equations and topological biases jointly constrain the random transport correction process, ensuring that data evolution balances physical rationality and topological stability, reducing erroneous connectivity, channel breaks, and the disappearance of closed structures. Unifying the correction data with the remaining observation data and performing projection updates results in a continuous, multi-resolution consistent geological representation, improving the reliability of subsequent geological interpretation, modeling, and analysis. Attached Figure Description

[0022] Figure 1 This is a flowchart of a method for expressing the topological consistency of multi-media geological data across manifolds, as shown in Example 1.

[0023] Figure 2 This is a flowchart of the cross-manifold closed-loop transmission and geometrically consistent modulation method in Example 1;

[0024] Figure 3 This is a flowchart of the topology generator continuous tracking and topology drift correction method in Example 2. Detailed Implementation

[0025] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0026] Example 1:

[0027] Please see Figure 1 As shown, embodiments of the present invention provide a method for expressing the topological consistency of multi-media geological data across manifolds, the method comprising:

[0028] S1. Acquire borehole cores, logging curves, remote sensing images, and 3D seismic data; establish the native Riemannian manifold for each type of medium and form a set of medium feature points.

[0029] S2. Establish cross-manifold alignment mapping and parallel transport operator for the corresponding nodes. Combine the parallel transport operator along the closed path that passes through four types of native Riemannian manifolds in sequence and returns to the starting manifold to obtain the closed-loop transport operator. Determine the geometric consistency quantity based on the deviation of the closed-loop transport operator from the identity transformation.

[0030] S3. Adjust the cross-medium aggregation weight according to the geometric consistency quantity to obtain the aligned feature point set;

[0031] S4. Construct the Vitoris-Lipps filter based on the Riemann geodesic distance of the aligned feature point set, extract the media persistence map, and determine the topological conflict media pairs, conflict coherence dimension, and conflict scale interval according to the bottleneck distance between the media persistence maps.

[0032] S5. Determine the nodes to be corrected based on the topological conflict media pairs, conflict coherence dimensions, and conflict scale intervals. Construct a reference distribution and extract a reference persistence map using the output of the multi-media physical field operator. Construct joint drift constraints based on the geophysical equation residuals and the topological deviation between the transport state persistence map and the reference persistence map.

[0033] S6. Using the node data to be corrected and the reference distribution as the boundary distribution, solve the Schrödinger bridge random transport path according to the joint drift constraint, update the node data to be corrected along the Schrödinger bridge random transport path, and encode the updated data and the remaining observation data into consistent representation data.

[0034] In step S1, borehole cores, logging curves, remote sensing images, and 3D seismic data are uniformly converted to the same spatial coordinate system, and geological reference units are established based on spatial location, stratigraphic horizon, and sampling depth. Each geological reference unit is associated with the core sampling section, logging sampling point, remote sensing pixel, and seismic voxel neighborhood. Each medium node records spatial location, stratigraphic horizon, original data index, manifold element, and metric information, and forms a medium feature point set according to medium type.

[0035] For borehole core data, the content of preset mineral categories within the core sampling section is statistically analyzed and normalized according to the total mineral content to obtain mineral component probability vectors. Each mineral component probability vector is used as a manifold element in the borehole core manifold. The distance between core nodes is determined using the Fisher-Lao information metric. Specifically, the mineral component probability vectors are subjected to square root transformation, and the distance between core nodes is calculated based on the spherical geodesic length between the transformed vectors.

[0036] For well logging data, porosity, water saturation, and cementation parameters are extracted from the same sampling depth and then dimensionlessly converted to form a well logging parameter vector. Based on the Argyle rock electrical relationship, the formation resistivity response changes caused by the changes in each parameter are determined, and a positive definite local metric matrix is ​​established in combination with the parameter measurement uncertainty. The well logging parameter vector and its local metric matrix together constitute the manifold elements in the well logging curve manifold. The distance between two well logging nodes is determined by the cumulative length of the local metrics on the connection path.

[0037] For remote sensing image data, radiometric calibration, atmospheric correction, geometric registration, and cloud shadow removal are performed sequentially. The band reflectance of the same pixel is arranged in wavelength order and interpolated to form a spectral reflectance function. Each spectral reflectance function is used as a manifold element in the remote sensing image manifold. The inner product of the function space is used as a metric, and the distance between remote sensing nodes is calculated based on the difference in reflectance functions within a unified wavelength range.

[0038] For 3D seismic data, the voxel neighborhood corresponding to the geological reference unit is extracted from the elastic wave impedance volume. The impedance values ​​are nonnegated and normalized. The voxel coordinates and normalized impedance values ​​are combined to form an elastic wave impedance distribution. Each elastic wave impedance distribution is used as a manifold element in the 3D seismic manifold. The Wasserstein optimal transmission metric is used to determine the distance between seismic nodes by minimizing the cumulative transmission cost to satisfy the mass conservation condition.

[0039] After step S1 is completed, the data of the four types of media are retained in their respective native Riemannian manifolds without direct vector splicing. Geological reference units are used to establish the correspondence between the nodes of the four types of media. The media feature point set and its metric information are used as inputs for establishing cross-manifold alignment mapping and parallel transport operators in step S2.

[0040] In step S2, the set of medium feature points formed in step S1 is read, and the corresponding node relationship is established based on the geological reference unit. Specifically, the four types of medium nodes are converted to a unified spatial coordinate system, and their vertical positions are corrected according to the stratigraphic position and sampling depth of the nodes. Among the nodes with consistent stratigraphic positions and corresponding sampling ranges, the correspondence between borehole core nodes, well logging curve nodes, remote sensing image nodes and three-dimensional seismic nodes is determined according to their spatial positions.

[0041] Following the transmission order of the closed path, the node in the current primary Riemannian manifold is determined as the source node, and the corresponding node in the next primary Riemannian manifold is determined as the target node. The source node is used as the local base point of the source manifold. Neighboring nodes belonging to the same stratigraphic level as the source node are selected. Through the logarithmic mapping of the source manifold, the neighboring nodes are transformed into source tangent vectors in the source tangent space. The source tangent vectors represent the local change direction and magnitude of the neighboring nodes relative to the source node.

[0042] Using the target node as the local base point of the target manifold, a logarithmic mapping is performed on the target neighbor nodes that have a geological reference unit correspondence with the source neighbor nodes to obtain the target tangent vector in the target tangent space. The source tangent vector set and the target tangent vector set are formed according to the arrangement relationship of the corresponding nodes. The tangent space mapping is obtained according to the correspondence between the two sets of tangent vectors. In the process of obtaining the mapping, the two sets of tangent vectors are first scaled according to the metric information of the source manifold and the target manifold. Then, a linear transformation that reduces the deviation between the source tangent vector transformation result and the corresponding target tangent vector is determined. This linear transformation is used as the tangent space mapping. When the dimensions of the source tangent space and the target tangent space are inconsistent, a corresponding subspace is established according to the common change components of the two sets of tangent vectors, and the tangent space mapping is determined in the corresponding subspace.

[0043] The source node is transformed into the source tangent space representation by the logarithmic mapping of the source manifold. Then, it is transformed into the target tangent space representation by the tangent space mapping. Finally, the target tangent space representation is restored to the node on the target manifold by the exponential mapping of the target manifold, resulting in the aligned node. The aligned node maintains the same manifold data structure as the target node, so that the alignment result can continue to participate in the geodesic distance calculation in the target manifold.

[0044] The alignment path is formed by sequentially connecting the source geodesic path in the source manifold, the tangent space mapping, and the target geodesic path in the target manifold. The parallel transmission operator is established in a segmented composite manner: first, the source tangent vector is transmitted along the source geodesic path to the mapping position corresponding to the source node, then transformed to the target tangent space through the tangent space mapping, and then transmitted along the target geodesic path to the target node. Parallel transmission operators between adjacent medium nodes in the closed path are established in the above manner, and the obtained parallel transmission operators are input into the subsequent closed-loop transmission processing.

[0045] For example: After establishing the correspondence between core nodes and logging nodes in the same reservoir segment, the changes in mineral composition in the neighborhood of the core node are converted into the source tangent vector of the borehole core manifold, and the changes in rock electrical parameters in the neighborhood of the corresponding logging node are converted into the target tangent vector of the logging curve manifold. Then, a tangent space mapping is established based on the correspondence between the two sets of local changes.

[0046] like Figure 2 As shown, in steps S2 and S3, for the borehole core node, well logging curve node, remote sensing image node and 3D seismic node corresponding to the same geological reference unit, a closed path is established in the order of borehole core manifold, well logging curve manifold, remote sensing image manifold, 3D seismic manifold and borehole core manifold. Adjacent nodes in the closed path are connected by the parallel transmission operator established in step S2.

[0047] To enable adjacent parallel transport operators to be sequentially composited, the tangent vector directions of the four parallel transport operators with effective mapping relationships are read to form a closed path corresponding subspace. The tangent vector directions are orthogonalized according to the metric information of the corresponding native Riemannian manifold. Each parallel transport operator is represented in the closed path corresponding subspace, and the output basis of the previous parallel transport operator corresponds to the input basis of the next parallel transport operator.

[0048] Using the tangent space where the borehole core node is located as the initial tangent space, the basis vectors in the initial tangent space are sequentially input into the parallel transport operators from the borehole core manifold to the logging curve manifold, from the logging curve manifold to the remote sensing image manifold, from the remote sensing image manifold to the 3D seismic manifold, and from the 3D seismic manifold to the borehole core manifold. The four parallel transport operators are combined according to the transport order to obtain a closed-loop transport operator whose input and output ends are both located in the initial tangent space. The closed-loop transport operator is then determined as a Hele group element.

[0049] To avoid misjudging accidentally formed approximate identity transformations as geometrically consistent when node correspondences are unstable, this embodiment considers both closed-loop transmission bias and cross-manifold alignment reliability. It performs degree-preserving orthogonalization on each segment of the tangent space mapping and restricts the four parallel transmission operators to the subspace corresponding to the commonly valid closed paths, specifically targeting the geological reference unit. By combining four parallel transmission operators in the transmission order, a closed-loop transmission operator is obtained. And determine the geometric consistency quantity according to the following formula:

[0050] ;

[0051] In the formula, Geological reference unit Geometric consistency quantity; The closed-loop transport operator is obtained by combining the manifold of borehole core, the manifold of well logging curve, the manifold of remote sensing image, the manifold of 3D seismic data, and the manifold of borehole core in that order. This is the identity transformation in the subspace corresponding to the closed path; Let be the dimension of the subspace corresponding to the closed path; It is the Frobenius norm.

[0052] To determine the closed-loop alignment reliability, the geometric mean of the reliability of the four tangent space mappings is taken. The reliability of each tangent space mapping is determined based on the normalized residual between the source tangent vector after tangent space mapping and the corresponding target tangent vector. The value range is greater than zero and not greater than one. When the alignment residual decreases, the closed-loop alignment reliability increases; when the alignment residual increases, the closed-loop alignment reliability decreases.

[0053] Since each segment of the tangent space mapping has undergone degree-preserving orthogonalization, the closed-loop transport operator is represented by an orthogonal matrix in the corresponding subspace of the closed path, and the denominator... To eliminate the influence of subspace dimension on the Frobenius norm, the geometric consistency quantity approaches zero when the closed-loop transmission operator is close to the identity transformation and the four-segment mapping has high reliability; when the closed-loop transmission deviation increases or the closed-loop alignment reliability decreases, the geometric consistency quantity increases accordingly.

[0054] The identity transformation represents the baseline state where the tangent vector in the initial tangent space retains its original direction and scale after being transmitted through a closed path. Using the same initial tangent space basis to represent the elements of the Hele group and the identity transformation respectively, the difference between the matrix elements of the two is calculated term by term. The square root of the sum of the squared differences is obtained to obtain the Frobenius norm difference. Then, it is normalized according to the dimension of the subspace corresponding to the closed path to obtain the geometric consistency quantity. The geometric consistency quantity is a non-negative scalar, and its value increases with the degree to which the closed-loop transmission operator deviates from the identity transformation.

[0055] When determining cross-media aggregation weights, the media The target medium is identified, and the medium is... The source medium is identified; the target medium is... Nodes in The source medium is identified as the target node. Nodes that have spatial and stratigraphic correspondence with the target node The source node was identified; the geological reference unit was adopted. serial number, Indicates the source node The geological reference unit to which it belongs.

[0056] Source node The characteristics are achieved through the source medium Pointing to target medium The cross-manifold alignment mapping is transformed to the native Riemannian manifold of the target medium, and the cross-medium aggregation weight is determined according to the following formula based on the Riemann geodesic distance between the transformed features and the target node features, and the geometric consistency of the geological reference unit to which the source node belongs:

[0057] ;

[0058] In the formula, For the source node To the target node Cross-media aggregation weights when transferring features; For the target node In the target medium Features in the original Riemannian manifold; For the source node In the source medium Features in the original Riemannian manifold; For the source medium Pointing to target medium Cross-manifold alignment mapping; For target medium Riemann geodesic distance on the primary Riemannian manifold; For the source node Geological reference unit Geometric consistency quantity; Source medium In and target node Establish a set of neighboring nodes with spatial and stratigraphic correspondences; The summation index in the set of neighboring nodes; Neighboring nodes The geological reference unit to which it belongs; and These are the scale adjustment coefficients for the geodesic distance term and the geometric consistency term, respectively.

[0059] Before calculating the cross-medium aggregation weight, the Riemann geodesic distance and geometric consistency quantity are dimensionless. When the Riemann geodesic distance between the mapped features of the source node and the features of the target node increases, the basic correlation of the source node decreases. When the geometric consistency quantity of the geological reference unit to which the source node belongs increases, the geometric self-consistency of the corresponding closed path of the geological reference unit decreases, and the cross-medium aggregation weight of the source node decreases accordingly.

[0060] The denominator represents the set of neighboring nodes of the source medium. All source nodes in the process are normalized so that the sum of cross-medium aggregation weights corresponding to the same target node is one. Based on the cross-medium aggregation weights, the source medium features mapped to the tangent space of the target node are weighted and aggregated, and then the corresponding alignment nodes are formed through the exponential mapping of the original Riemannian manifold of the target medium.

[0061] When the source node With the target node When belonging to the same geological reference unit, The target node's geological reference unit is the same as the target node's geological reference unit. When a source medium neighbor node crosses an adjacent geological reference unit, the geometric consistency value of each source node's geological reference unit is read separately to avoid directly replacing the geological reference unit index with the target node index.

[0062] The medium features within the target tangent space are weighted and aggregated according to the cross-medium aggregation weight. Then, alignment nodes are formed by the exponential mapping of the original Riemannian manifold where the target node is located. The above processing is completed by taking the borehole core node, well logging curve node, remote sensing image node and 3D seismic node as target nodes respectively, forming the alignment feature point set in step S3. The alignment feature point set is then input into the Vitoris-Ripple filter construction process in step S4.

[0063] In step S4, the alignment feature point set formed in step S3 is divided into borehole core feature point set, well logging curve feature point set, remote sensing image feature point set and 3D seismic feature point set according to the medium type. The Riemann geodesic distance between nodes of the same type is calculated respectively, and dimensionless is performed according to the distance reference scale of the corresponding original Riemann manifold, so that the four types of media adopt a unified incremental adjacency scale.

[0064] For a set of characteristic points of a medium, nodes are used as vertices. When the Riemann geodesic distance between two vertices is not greater than the current adjacent scale, an edge is established between the two vertices. When a set of vertices satisfies the distance condition pairwise, a simplex is constructed with the set of vertices. The adjacent scale is increased step by step, and the simplex formed at the previous adjacent scale is retained to the next adjacent scale, thus forming the Vitoris-Ripse filter.

[0065] The boundary matrix of the Vitoris-Lipps filter is reduced, and the generation and disappearance of zero-dimensional connected components, one-dimensional loops, and two-dimensional cavities are tracked sequentially. The adjacency scale when a homology feature first appears is used as the birth scale, and the adjacency scale when the homology feature is merged or filled by a higher-dimensional simplex is used as the extinction scale. The birth and extinction scales of the homology feature are combined to form persistent points. The set of persistent points of the same medium and the same homology dimension constitutes the medium persistence graph. The Vitoris-Lipps filter can be constructed based on the distance matrix of the finite metric space, and the persistence graph records the continuation process of topological features with birth and extinction scales.

[0066] Comparing media persistence maps of the same homology dimension in two types of media, a partial bijection is established between persistence points in the two media persistence maps. If a persistence point cannot form a corresponding persistence point in the other media persistence map, it is matched to its projection point on the diagonal of the persistence map. The partial bijection is adjusted to minimize the maximum scale difference among all matching relationships, and this maximum scale difference is determined as the bottleneck distance. The distance from the persistence point to the diagonal projection point reflects the persistence of the homology feature represented by the persistence point. The bottleneck distance is used to characterize the maximum topological difference between the two media persistence maps.

[0067] The two types of media containing the persistent point that forms the bottleneck distance are identified as the topological conflict media pair, and the coherence dimension to which the persistent point belongs is identified as the conflict coherence dimension. When two persistent points match each other, the conflict scale interval is the earlier birth scale to the later extinction scale of the two. When a persistent point matches to a diagonal projection point, the conflict scale interval is the birth scale to the extinction scale of that persistent point. The topological conflict media pair, conflict coherence dimension, conflict scale interval and corresponding persistent point index are written into the topological conflict record as the input for step S5 to determine the node to be corrected.

[0068] In steps S5 and S6, the topological conflict record formed in step S4 is read, the persistent conflict point involved in determining the bottleneck distance and its corresponding birth simplex, extinction simplex and homology dimension are obtained, and the topological generator corresponding to the persistent conflict point is restored based on the boundary matrix reduction record retained in the persistent homology calculation process.

[0069] Topological generators are used to represent the constitutive range of conflicting and homogeneous features in the alignment feature point set. The topological generator corresponding to a connected component is represented by the set of vertices within the connected component, the topological generator corresponding to a loop is represented by the set of edges that are closed at both ends, and the topological generator corresponding to a cavity is represented by the set of triangular faces surrounding the cavity. The vertex indices of the simplex constituting the topological generator are read and mapped to the nodes in the alignment feature point set. Then, based on the geological reference unit identifier carried by the node, the corresponding sampling positions in borehole cores, well logging curves, remote sensing images, and 3D seismic data are found. The found sampling positions and their associated nodes are determined as nodes to be corrected.

[0070] Data corresponding to the nodes to be corrected are extracted from the unified geological property field output by the multi-media physical field operator in step S5 to form a reference distribution. The same distance normalization method and adjacent scale sequence as in step S4 are used to construct a filter flow on the reference distribution, and the reference persistent map corresponding to the conflict coherence dimension and conflict scale interval is extracted.

[0071] During the random transport path update process of the Schrödinger Bridge, the data of the nodes to be corrected under the current transport state are read. The transport state filter is constructed according to the same node order, distance calculation rules and adjacency scale sequence. The transport state persistent graph is extracted. The persistent graph of the transport state is matched with the persistent points in the reference persistent graph that are within the conflict scale interval. The bottleneck distance obtained by the matching is determined as the topology deviation. The topology deviation is recalculated as the state of the random transport path is updated and used as the input for the subsequent construction of the topology drift correction term.

[0072] The boundary matrix reduction result of persistent homology can simultaneously preserve persistent point pairings and representative topological generators. Based on this, conflicting persistent points in the persistent graph can be traced back to the simplex that constitutes the corresponding connected components, loops and cavities. This section uses the same node order, distance rules and adjacency scale to process the reference distribution and the current transport state respectively, so as to keep the basis for comparison of the two persistent graphs consistent.

[0073] In step S5, the aligned feature point set obtained in step S3 is converted to a unified geological spatial grid. During the conversion, the corresponding grid position is determined according to the spatial coordinates of the node and the stratigraphic position. The node features, spatial coordinates and data validity markers are arranged according to a fixed channel. The grid positions not covered by the observed data are interpolated according to the neighboring nodes and distinguished from the measured nodes by the data validity markers, thereby forming the input function of the multi-media physical field operator.

[0074] The feature enhancement layer performs a linear combination of the input channels at each grid location, converting the alignment features of different media into internal features of the same dimension. The Fourier operator layer performs a Fourier transform on the internal features along the spatial direction, reads the preset frequency domain pattern from the transform result, and establishes the correlation between features at different locations and in different media through a learnable spectral transform. After performing an inverse Fourier transform on the spectral transform result, it is added to the pointwise linear transform result at the same grid location and then input into the next Fourier operator layer. The basic implementation of the Fourier neural operator is to parameterize the operator kernel in the frequency domain and combine the frequency domain transform branch with the pointwise transform branch.

[0075] The physical property field output layer converts the internal features of the last layer into a unified geological physical property field, which includes permeability field, pressure field, electrical conductivity field, electric potential field, porosity field, water saturation field, and formation resistivity field. The aforementioned physical property fields use the same spatial grid and formation horizon index, so that different physical relationships can be calculated at the same grid location.

[0076] The underground fluid flux is determined based on the permeability and pressure fields, and the differences between inflow, outflow, and source-sink rates are calculated to form the Darcy flow equation residuals. The electric field and current density are determined based on the conductivity and potential fields, and the deviation of the current continuity is calculated to form the Maxwell quasi-static electromagnetic equation residuals. The formation resistivity corresponding to the rock-electric relationship is calculated based on the porosity field, water saturation field, and cementation parameters. The calculation results are compared with the formation resistivity field generated by the physical property field output layer to form the Archie rock-electric relationship residuals. The Archie rock-electric relationship is used to describe the relationship between porosity, water saturation, and formation resistivity. The quasi-static Maxwell equations can describe the low-frequency electromagnetic response with the medium conductivity as a parameter.

[0077] When training the multi-media physical field operator, the difference between the predicted and measured values ​​of physical properties is calculated at the grid position corresponding to the measured data, and the aforementioned three types of residuals are calculated within the geological space grid. The difference in physical property prediction and the three types of residuals are used together to update the feature enhancement layer, the Fourier operator layer, and the physical property output layer. The unified geological physical property field after training is sampled at the position corresponding to the node to be corrected. The obtained physical property samples are used to construct the reference distribution in step S5. The observation data constraints and partial differential equation constraints are used together to train the neural operator, which belongs to the existing implementation basis of the physical constraint neural operator.

[0078] In steps S5 and S6, an initial boundary empirical distribution is constructed based on the node data to be corrected, and the reference distribution is used as the termination boundary distribution. The two sets of samples are normalized using the same node arrangement order and attribute dimensions. The current transport state in the random transport path is the set of intermediate node attributes formed after the node data to be corrected has undergone one discrete update.

[0079] For the current transport status, the node attributes are input into the multi-medium physical field operator in step S5 to calculate the Darcy seepage equation residuals, Maxwell's quasi-static electromagnetic equation residuals, and Archie rock electrical relation residuals. The attribute components of the node to be corrected are subjected to equal-amplitude positive and negative perturbations, respectively. The geophysical equation residuals before and after the perturbation are compared. The physical drift term is formed according to the attribute change direction that reduces the residual. The physical drift term includes the node index, attribute component, update direction, and update amplitude.

[0080] The topology drift correction term is formed based on the aforementioned topology deviation. The persistent point matching relationship that generates the bottleneck distance and the corresponding topology generator are read. The nodes to be corrected that participate in the connected components, loops and cavities are determined. Equal amplitude positive and negative perturbations are applied to the attribute components of the nodes to be corrected. The transport state persistent graph is reconstructed and the bottleneck distance between it and the reference persistent graph is calculated. The topology drift correction term is formed according to the attribute change direction that reduces the bottleneck distance.

[0081] The physical drift term and the topological drift correction term are uniformly represented according to the normalized scale of the node attributes, and then superimposed according to the preset combined weights to form a joint drift constraint. The physical drift term acts on the node attributes corresponding to the residuals of the geophysical equations, and the topological drift correction term acts on the topological generator nodes corresponding to the conflict coherence dimension and conflict scale interval.

[0082] To enable the physical drift term and topological drift correction term to directly participate in the state update of the Schrödinger bridge random transport path, this embodiment generates adjacent discrete states according to the following formula:

[0083] ;

[0084] In the formula, For the first Geological attribute vectors of nodes to be corrected in discrete states; To perform a discrete update of the geological attribute vector; For the first The transport time corresponding to each discrete state; The discrete interval between adjacent transport states; Boundary matching drift determined by the forward and inverse Kolmogorov equations; This indicates the direction of decrease in the residuals of the geophysical equations; This indicates the direction of decreasing topological deviation; and These are the scaling factors for the physical drift term and the topology drift correction term, respectively; The intensity of random diffusion; It is a standard normal random vector with the same dimension as the geological attribute vector.

[0085] The geophysical equation residuals are formed by combining the Darcy seepage equation residuals, Maxwell's quasi-static electromagnetic equation residuals, and Archie rock electrical relation residuals after dimensionless transformation. The topological bias is the bottleneck distance between the current transport state persistence map and the reference persistence map in the conflict coherence dimension and conflict scale interval.

[0086] Sure At that time, equal-amplitude positive and negative perturbations are applied sequentially to the effective attribute components of the node to be corrected, and the geophysical equation residuals are recalculated respectively. The update direction and relative amplitude of the corresponding attribute components are determined based on the difference between the two residual calculation results. At the same time, the transport state persistence graph is re-extracted using the same perturbation method, and the update direction and relative amplitude of the corresponding attribute components are determined based on the changes in topological deviation before and after the perturbation.

[0087] The aforementioned symmetric perturbation method does not require the bottleneck distance to be globally differentiable, and can transform topology deviations into executable node attribute updates, completing each iteration. After the discrete update, the geophysical equation residuals and topological biases are recalculated, and the physical drift term and topological drift correction term used in the next discrete state are updated accordingly.

[0088] Using joint drift constraints as the drift part of state evolution, the forward Kolmogorov equation is solved in the direction from the data of the node to be corrected to the reference distribution, and the inverse Kolmogorov equation is solved in the direction from the reference distribution to the data of the node to be corrected. During the solution, the transport period is divided into sequentially connected discrete states. The forward state update is performed first, followed by the inverse boundary correction. A random diffusion increment is added to the state update, thereby obtaining the Schrödinger bridge random transport path connecting the two boundary distributions.

[0089] After each discrete update, the transport state persistent map is re-extracted according to the node distance, adjacency scale and homology dimension used in step S4. The topological deviation between the transport state persistent map and the reference persistent map is calculated, and the topological drift correction term is updated. At the same time, the geophysical equation residuals are recalculated and the physical drift term is updated. After completing the forward update and reverse correction corresponding to the preset termination condition, the final transport state is determined as the updated node data to be corrected.

[0090] In step S6, the updated node data to be corrected is merged with the remaining observation data, and training samples are established according to unified geological spatial coordinates. Each training sample includes spatial coordinates, stratigraphic position, geological attributes, and data source identifier. After scale unification, the spatial coordinates are input into the implicit neural representation network, and the network outputs the geological attributes of the corresponding location. The implicit neural representation network consists of sequentially connected fully connected layers and is used to establish the mapping between continuous spatial coordinates and geological attributes. It does not rely on fixed voxels to store geological attributes. The coordinate-type implicit neural representation can represent spatial signals and their spatial changes in the form of continuous functions.

[0091] During training, the geological properties at the coordinates of the training samples are first calculated based on the current network parameters. The calculation results are compared with the updated data and the remaining observation data, and the network parameters are updated accordingly. For the nodes to be corrected corresponding to the topological conflict medium pairs, after each network parameter update, the current output of the network is read as the current transport state of the random transport path of the Schrödinger bridge. The drift field obtained in step S6 is called, and a discrete update is performed according to the update rules of adjacent discrete states.

[0092] A discrete update employs the aforementioned joint drift constraint, where the physical drift term corrects attribute changes that do not satisfy the geophysical equation, and the topological drift correction term corrects attribute changes that deviate from the reference persistent map. The node attributes obtained from the discrete update are used as the projection target, and the difference between the network output and the projection target is calculated again. Based on this, the network parameters are corrected. The projection update referred to in this paper means adjusting the network output to the state defined by the joint drift constraint through the above discrete update, and then making the implicit neural representation network fit the adjusted node attributes.

[0093] After training is completed, a sampling grid from coarse to fine is established in the target geological space. First, the geological attributes of the coarse grid nodes are calculated. For grid regions where the attributes of adjacent nodes cross a preset attribute threshold, the sampling density is increased, and the geological attributes of the refined nodes are recalculated. For each grid cell that crosses the attribute threshold, the position where the attribute value reaches the attribute threshold is determined on its edge, and adjacent positions are connected to form a triangular patch. The isosurface of the corresponding geological attribute is extracted in this way. Constructing triangular patches based on the threshold intersections in the three-dimensional regular grid is an existing implementation method for isosurface extraction.

[0094] Example 2:

[0095] like Figure 3 As shown, this embodiment further improves upon the design of Embodiment 1. The difference lies in the fact that, in the actual operation of Embodiment 1, it was found that the topology generators corresponding to the same persistent conflict point have multiple homological equivalent representations. When the random transport state undergoes local changes, the topology generators obtained from boundary matrix reduction may switch between different spatial locations, causing the nodes to be corrected determined by adjacent discrete states to jump. The topology drift correction term alternates between different geological reference units, failing to maintain the continuous positioning of topology conflicts corresponding to the same geological structure during random transport. Based on this, a method for expressing the topology consistency of multi-media geological data across manifolds further includes: establishing a continuous tracking record of the topology generators for persistent conflict points; determining the node topology responsibility weight based on the continuous tracking record of the topology generators; and allocating the topology drift correction term according to the node topology responsibility weight.

[0096] Specifically, in step S5, for each persistent conflict point in the topological conflict record, its homology dimension, birth scale, extinction scale, and boundary matrix reduction record are read. For connected components, candidate topological generators are determined from the set of connected nodes corresponding to the persistent conflict point. For loops and cavities, the representative chain obtained from the boundary matrix reduction is used as the initial representative chain. By superimposing the same-dimensional boundary chain onto the initial representative chain, candidate topological generators belonging to the same homology class as the persistent conflict point are formed.

[0097] The simplex contained in each candidate topological generator is mapped to the nodes in the aligned feature point set, and further mapped to the corresponding geological reference unit. Based on the number of nodes contained in the candidate topological generator, the Riemann geodesic distance between nodes, the spatial distribution range of nodes, and the geophysical equation residuals at the corresponding nodes, the support cost of the candidate topological generator is determined. The candidate topological generator with the minimum support cost is determined as the baseline topological generator in the initial transport state, and its node set, simplex set, spatial range, and corresponding persistent point index are recorded.

[0098] After the Schrödinger Bridge random transport path completes a discrete update, the transport state persistence graph is re-extracted according to the distance calculation rules and adjacency scale in step S4. Based on the order of consistent homology dimension, intersection of birth and death scale intervals, and minimum persistent point matching distance, a continuous correspondence is established between the persistent points in the current transport state and the conflicting persistent points in the previous transport state.

[0099] For persistent points that establish continuous correspondence, candidate topological generators are obtained again. The degree of node overlap, simplex overlap, and spatial geodesic displacement between the candidate topological generators and the topological generators selected in the previous transport state are calculated respectively. Candidate topological generators with high degree of node and simplex overlap, small spatial geodesic displacement, and no increase in the residual of the corresponding geophysical equation are selected as continuous topological generators in the current transport state. The continuous topological generators of the previous and previous transport states are written into the continuous topological generator tracking record in sequence.

[0100] The node topological responsibility weight is determined based on the number of times a node appears in a continuous topological generator, the Riemann geodesic distance between the node and the spatial center of the topological generator, and the node's contribution to the bottleneck distance. Nodes located in the core region of a continuous topological generator and continuously participating in the conflict and homology features have a larger node topological responsibility weight; edge nodes that appear only in a single transport state have a smaller node topological responsibility weight.

[0101] When forming the topology drift correction term, the correction magnitude of each node to be corrected is allocated according to the node topology responsibility weight, so that the topology drift correction is concentrated on the nodes that continuously constitute conflict and coherence characteristics. The physical drift term is still determined according to the geophysical equation residual at the corresponding node. Then, the topology drift correction term allocated by the node topology responsibility weight is combined with the physical drift term to form the joint drift constraint of the current transport state.

[0102] If the current transport state fails to establish a persistent point continuous correspondence with the previous transport state, or if the spatial displacement of the current topological generator relative to the previous topological generator exceeds the continuous coverage range of the corresponding geological reference unit, the current discrete update is cancelled, the state change amplitude of the discrete update is reduced, and the transport state persistent map is re-extracted until the continuous correspondence between the conflict persistent points and their topological generators is restored. Thus, when updating the node data to be corrected along the Schrödinger Bridge random transport path in step S6, the topological drift correction term is always updated based on the continuous spatial support range of the same topological conflict.

[0103] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention also intends to include these modifications and variations.

[0104] The above description is merely a preferred embodiment of the present application, but the scope of protection of the present application is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present application, based on the technical solution and concept of the present application, should be covered within the scope of protection of the present application.

Claims

1. A method for expressing cross-manifold topological consistency in multi-media geological data, characterized in that the method... include: Acquire borehole cores, logging curves, remote sensing images, and 3D seismic data; establish the native Riemannian manifold for each type of medium and form a set of medium feature points; Establish cross-manifold alignment mapping and parallel transport operator for the corresponding nodes. Combine the parallel transport operator along the closed path that passes through four types of native Riemannian manifolds in sequence and returns to the starting manifold to obtain the closed-loop transport operator. Determine the geometric consistency quantity based on the deviation of the closed-loop transport operator from the identity transformation. The cross-medium aggregation weights are adjusted according to the geometric consistency quantity to obtain the aligned feature point set; Based on the Riemann geodesic distance of the aligned feature point set, a Vitoris-Lipps filter is constructed, a media persistence map is extracted, and the topological conflict media pairs, conflict coherence dimensions, and conflict scale intervals are determined according to the bottleneck distance between the media persistence maps. The nodes to be corrected are determined based on the topological conflict media pairs, conflict coherence dimensions, and conflict scale intervals. A reference distribution is constructed using the output of the multi-media physical field operator, and a reference persistence map is extracted. A joint drift constraint is constructed based on the geophysical equation residuals and the topological deviation between the transport state persistence map and the reference persistence map. Using the node data to be corrected and the reference distribution as boundary distributions, the Schrödinger bridge random transport path is solved according to the joint drift constraint. The node data to be corrected is updated along the Schrödinger bridge random transport path, and the updated data and the remaining observation data are encoded into consistent representation data. The cross-manifold alignment mapping establishes corresponding node relationships based on spatial coordinates, stratigraphic horizons, and sampling depth. Nodes of the preceding primary Riemannian manifold in the closed path are determined as source nodes, and nodes of the following primary Riemannian manifold are determined as target nodes. Nodes in the neighborhood of the source node are transformed to the source tangent space via a logarithmic mapping with the source node as the base point, then transformed to the target tangent space via a tangent space mapping, and finally aligned nodes are formed via an exponential mapping with the target node as the base point. The parallel transport operator is established according to the source node, the target node, and the alignment path between them. The closed path connects corresponding nodes in the order of borehole core manifold, well logging curve manifold, remote sensing image manifold, 3D seismic manifold, and borehole core manifold; each parallel transmission operator is compounded according to the transmission order of the closed path, and the resulting closed-loop transmission operator is determined as a Hele group element; the geometric consistency quantity is determined according to the normalized Frobenius norm difference between the Hele group element and the initial tangent space identity transformation, and the monotonically decaying transformation of the geometric consistency quantity is used to modulate the cross-medium aggregation weight.

2. The method for expressing the topological consistency of multi-media geological data across manifolds according to claim 1, characterized in that, The native Riemannian manifolds include: borehole core manifolds with mineral composition probability vectors as manifold elements and Fisher-Lao information metric; well logging curve manifolds induced by Archie rock electrical relations in the space of porosity, water saturation, and cementation parameters; remote sensing image manifolds with spectral reflectance functions as manifold elements and function space inner product metric; and three-dimensional seismic manifolds with elastic wave impedance distribution as manifold elements and Wasserstein optimal transmission metric.

3. The method for expressing the topological consistency of multi-media geological data across manifolds according to claim 1, characterized in that, The Vitoris-Lipps filter is constructed according to the Riemann geodesic distance and increasing adjacency scale between nodes in the aligned feature point set; the media persistence graph records the birth and death scales of the homology features corresponding to connected components, loops, and cavities; the bottleneck distance is determined by partial bijection between a persistence point in one media persistence graph and a persistence point in another media persistence graph, as well as the diagonal projection point; the topological conflict media pairs, conflict homology dimensions, and conflict scale intervals are determined based on the media to which the persistence points participating in the partial bijection belong, the homology dimension, the birth scale, and the death scale.

4. The method for expressing the topological consistency of multi-media geological data across manifolds according to claim 3, characterized in that, The topological generators corresponding to the persistent conflict points involved in determining the bottleneck distance are obtained. The simplex constituting the topological generators is mapped to the nodes in the alignment feature point set. The nodes to be corrected are then determined based on the spatial correspondence between the nodes and the sampling positions in borehole cores, well logging curves, remote sensing images, and 3D seismic data. The reference persistent map corresponding to the conflict coherence dimension and conflict scale interval is extracted from the reference distribution. The transport state persistent map is extracted from the current transport state of the Schrödinger bridge random transport path. The topological deviation is determined based on the bottleneck distance between the two.

5. The method for expressing the topological consistency of multi-media geological data across manifolds according to claim 4, characterized in that, The multi-medium physical field operator includes a feature enhancement layer, a Fourier operator layer, and a physical property field output layer. The Fourier operator layer performs a Fourier transform on the input function corresponding to the aligned feature point set, performs a learnable spectral transform on the preset frequency domain mode, performs an inverse Fourier transform on the spectral transform result, and combines the inverse Fourier transform result with the point-by-point linear transform result. The physical property field output layer generates a unified geological physical property field and determines the geological physical equation residuals based on the Darcy seepage equation residuals, Maxwell's quasi-static electromagnetic equation residuals, and Archie rock electrical relation residuals.

6. The method for expressing the topological consistency of multi-media geological data across manifolds according to claim 5, characterized in that, A physical drift term is constructed based on the residuals of the geophysical equations, and a topological drift correction term is constructed based on the topological deviation. The physical drift term and the topological drift correction term form the joint drift constraint. The forward Kolmogorov equation and the inverse Kolmogorov equation, with the empirical distribution formed by the data of the node to be corrected and the reference distribution as boundary distributions, are solved to obtain the drift field of the Schrödinger bridge random transport path. After each discrete update, the persistent transport state map is re-extracted, and the topological drift correction term is updated based on the newly determined topological deviation.

7. A method for expressing the topological consistency of multi-media geological data across manifolds according to claim 6, characterized in that, The updated data and the remaining observation data are used as training data for an implicit neural representation network, which takes spatial coordinates as input and geological attributes as output. After each network parameter update, a discrete update of the Schrödinger bridge random transport path is invoked according to the joint drift constraint to perform a projection update on the geological attributes output by the implicit neural representation network. Multi-resolution spatial sampling is performed on the trained implicit neural representation network, and isosurfaces are extracted based on the sampled geological attributes to obtain the consistent representation data.

8. A method for expressing the topological consistency of multi-media geological data across manifolds according to claim 7, characterized in that, It also includes: continuously matching persistent conflict points and their topology generators in adjacent transport states based on the conflict coherence dimension, the conflict scale interval, and the bottleneck distance; determining the node topology responsibility weight based on the number of consecutive occurrences of the node, the distance from the node to the center of the topology generator space, and the node's contribution to the bottleneck distance, and allocating the topology drift correction term accordingly.

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